MODELLING AND PREDICTION OF THE
ENVIRONMENTAL DEGRADATION OF FIBRE
REINFORCED PLASTICS
Etienne Kolomoni Ngoy
A thesis submitted to the Faculty of Engineering and the Built Environment,
University of the Witwaterstrand, Johannesburg, in fulfillment of the requirements
for the degree of Doctor of Philosophy.
Johannesburg, 2010.
i
DECLARATION
I declare that this thesis is my own unaided work. It is being submitted for the Degree
of Doctor of Philosophy to the University of the Witwatersrand, Johannesburg. It has
not been submitted before for any degree or examination in any other university.
______________________________________
__________day of _______________year____
ii
ABSTRACT
In their service life, fibre reinforced plastics (FRP) face a variety of environmental
conditions resulting from natural or artificial factors. These include variable
temperature and humidity conditions, energetic radiations such as ultraviolet rays
from the sun, and diverse chemical reactants such as liquid in storage tanks and pipes.
These factors are always combined and negatively affect the material properties over
the time.
Therefore, optimized utilization of FRP material requires reliable methods for
quantifying, controlling, and predicting environmental effects. This allows for
optimal handling of issues related to component design, economic assessment and
safety considerations, as well as the technical problems relating to equipment
maintenance.
Efforts worldwide are devoted to the modelling of FRP environmental degradation.
However, modelling efforts have been hindered by the complexity of the process.
This analysis presents a comprehensive model of the environmental degradation of
FRP and a prediction method. The modelling method consists of a theoretical
demonstration based on material science theories. An analytical approach is
proposed. It resolves the complexity of the process into only three components: the
chemical degradation, the physical degradation, and the stress state modification. A
method to represent the real service environment as a constant environment in
laboratory is also introduced.
iii
Then, the comprehensive model is expressed as a dynamic constitutive equation
resulting from the combination of the historical variation in chemical link density and
cohesive forces and the stress history of the material. It is shown that:
? The average of the chemical and physical degradation as well as its upper and
lower limits can be determined in a laboratory, in a constant environment, as
exponential functions of the degradation time.
? The environmental degradation can be comprehensively measured as a stress
relaxation.
? Acceleration of the predictive test can be obtained from a modified time
temperature shift principle.
iv
RELATED PUBLICATION
Ngoy E.K., Campbell I.M.D., R. D. Reid, Paskaramoorthy R. ?Modelling and
prediction of the chemical and physical degradation of fibre reinforced plastics?
Journal of Materials Science Vol. 44 (9), 2009, p. 2393, doi: 10.1007/s10853-009-
3303-4.
v
ACKNOWLEDGEMENTS
The author wishes to acknowledge the valuable support received from the University
of the Witwatersrand, THRIP, and DENEL.
The support of the DST/NRF Centre of Excellence in Strong Material (CoE-SM)
towards this research is hereby acknowledged.
To my Supervisor and to all the staffs and colleagues from the school of Mechanical,
Industrial, and Aeronautical Engineering, I wish to express my sincere gratefulness
for all the support received.
vi
TABLE OF CONTENTS
DECLARATION .................................................................................................... I
ABSTRACT .......................................................................................................... II
RELATED PUBLICATION .............................................................................. IV
ACKNOWLEDGEMENTS .................................................................................. V
TABLE OF CONTENTS .................................................................................... VI
LIST OF TABLES .............................................................................................. XI
LIST OF SYMBOLS ........................................................................................ XII
LIST OF ABBREVIATIONS ......................................................................... XIII
1 INTRODUCTION .............................................................................................. 1
2 ENVIRONMENTAL DEGRADATION MECHANISMS AND
EFFECTS, A LITERATURE SURVEY .......................................................... 7
2.1 MOISTURE DIFFUSION ............................................................................................................. 7
2.1.1 Kinetics of moisture diffusion through FRP laminates ................................................... 8
2.1.2 Factors affecting the diffusion process in FRP laminates ............................................ 12
Fibre volume fraction and orientation effect .................................................................................... 12
Effect of solvent molecules .............................................................................................................. 13
Temperature effects ......................................................................................................................... 13
Applied load and hydrostatic pressure effects .................................................................................. 14
2.1.3 Humidity effects on FRP laminates .............................................................................. 14
2.2 TEMPERATURE EFFECTS ........................................................................................................ 16
2.2.1 Thermolysis and Thermoxidation ................................................................................. 17
2.2.2 Residual stresses due the Temperature ......................................................................... 18
2.2.3 The Arrhenius Law ....................................................................................................... 19
2.3 ULTRAVIOLET RADIATIONS EFFECTS AND MECHANISM. ....................................................... 19
2.3.1 Photoxidation ................................................................................................................ 20
2.3.2 Photo-degradation mechanism ..................................................................................... 21
2.4 CHEMICAL AGENTS EFFECTS ................................................................................................. 22
2.5 STRESS CORROSION ............................................................................................................... 25
2.6 CONCLUSION ......................................................................................................................... 27
3 MODELLING AND PREDICTION METHODS OF
ENVIRONMENTAL DEGRADATION, A LITERATURE SURVEY ...... 29
3.1 MODELLING AND LIMITATIONS ............................................................................................. 29
3.2 ACCELERATED TEST METHODS ............................................................................................. 33
3.2.1 Acceleration using higher temperature based on Arrhenius law .................................. 34
3.2.2 General method using extrapolation based on the temperature ................................... 34
3.2.3 Accelerated or artificial humidification ........................................................................ 35
3.2.4 Limitation of accelerated methods ................................................................................ 36
3.3 STANDARD TEST METHODS ................................................................................................... 37
3.4 ANALYTICAL METHODS ........................................................................................................ 37
3.5 CONCLUSION ......................................................................................................................... 38
vii
4 METHODS TO REPRESENT THE REAL SERVICE ENVIRONMENT
IN LABORATORY .......................................................................................... 40
4.1 CONSTANT ENVIRONMENT MODELS ..................................................................................... 40
4.1.1 The model based on statistical control charts .............................................................. 40
4.1.2 Model based on the mean value .................................................................................... 42
4.2 EXPERIMENT ......................................................................................................................... 44
4.2.1 Experimental procedure and results ............................................................................. 44
Experiment 1: heat flow across a reference point ............................................................................ 44
Experiment 2: Variation in temperature at the reference point ........................................................ 48
4.2.2 Discussion ..................................................................................................................... 52
4.3 CONCLUSION ......................................................................................................................... 54
5 THE CHEMICAL AND PHYSICAL DEGRADATION MODEL ............. 56
5.1 THE MODELLING APPROACH .................................................................................................. 56
5.2 DEFINITIONS .......................................................................................................................... 58
5.2.1 Degradation index of the chemical link density: Ld ..................................................... 58
5.2.2 Degradation index of the cohesive forces: Cf ............................................................... 59
5.2.3 The degradation index of the material stiffness: Ed ..................................................... 60
5.3 MATERIAL RHEOLOGY AS A FUNCTION OF CHEMICAL LINK DENSITY .................................. 60
5.4 MATERIAL RHEOLOGY AS A FUNCTION OF MOISTURE CONTENT .......................................... 61
5.5 MATERIAL RHEOLOGY AS A FUNCTION OF TEMPERATURE .................................................... 62
5.6 CHEMICAL CONCENTRATION AS A FUNCTION OF MATERIAL RHEOLOGY AND DIFFUSION
EFFECT .................................................................................................................................. 63
5.7 RHEOLOGY DEPENDANT FUNCTION OF DEGRADATION RATES .............................................. 66
5.7.1 Chemical degradation rate as a function of material rheology .................................... 66
5.7.2 Physical degradation rate as a function of the material rheology ................................ 68
5.8 DEGRADATION AS A FUNCTION OF ENVIRONMENTAL FACTORS ............................................ 68
5.8.1 The chemical degradation rate as a function of environmental factors ....................... 69
5.8.2 Physical degradation rate as a function of environmental factors ............................... 69
5.9 THE MODEL OF CHEMICAL AND PHYSICAL DEGRADATION ................................................... 70
5.9.1 The mathematical model ............................................................................................... 70
5.9.2 The Physical and Chemical degradation in a constant environment ............................ 70
Exponential equation of the chemical and physical degradation ...................................................... 70
Equivalence between indices Ld and Ed .......................................................................................... 71
Statistical control limits of the degradation rate ............................................................................... 72
5.10 EXPERIMENTATION .............................................................................................................. 72
5.10.1 Material ...................................................................................................................... 73
5.10.2 Lamination method ..................................................................................................... 73
5.10.3 Experimental procedure ............................................................................................. 75
Experiment 1 .................................................................................................................................... 75
Experiment 2 .................................................................................................................................... 80
Experiment 3 .................................................................................................................................... 80
5.10.4 Experimental results and discussion ........................................................................... 81
Tensile strength ................................................................................................................................ 81
Moisture curve ................................................................................................................................. 82
Micrograph ...................................................................................................................................... 83
Raman spectra .................................................................................................................................. 84
Correlation between the model and experimental results ................................................................. 89
Comparing calculated and experimental material lifetime based on tensile strength evolution ....... 91
Variation of shear strength under physical and chemical degradation ............................................. 92
Variation of storage modulus under physical and chemical degradation ......................................... 93
5.11 CONCLUSION ....................................................................................................................... 93
viii
6 COMPREHENSIVE MODELLING AND PREDICTION OF FRP
ENVIRONMENTAL DEGRADATION ........................................................ 96
6.1 THE CONSTITUTIVE EQUATION OF ENVIRONMENTAL DEGRADATION IN FRP ....................... 96
6.2 DETERMINATION OF THE LABORATORY TEST CONDITIONS ................................................... 99
6.3 EXPERIMENTAL METHOD .................................................................................................... 100
6.3.1 Relaxation test procedure ........................................................................................... 100
6.3.2 Results and discussion ................................................................................................ 101
Amplification factor ( )tKenv ...................................................................................................... 101
Time Temperature correspondence ................................................................................................ 104
6.4 CONCLUSION ....................................................................................................................... 111
7 FINAL CONCLUSION ................................................................................. 112
7.1 DETERMINATION OF THE LONG-TERM DEGRADATION AS AN AVERAGE VALUE WITH UPPER
AND LOWER LIMITS ............................................................................................................. 112
7.2 THE CONSTITUTIVE EQUATION OF ENVIRONMENTAL DEGRADATION IN FRP. .................... 115
7.3 SHORT-TERM TEST METHOD ............................................................................................... 116
7.3.1 Determination of the laboratory test conditions ......................................................... 116
7.3.2 Prediction of long-term stiffness coefficient based on ( )tKenv ................................ 116
7.3.3 Prediction of relaxation times based on time temperature shift principle .................. 117
7.4 RECOMMENDATIONS FOR FUTURE WORKS .......................................................................... 118
REFERENCES .................................................................................................. 119
APPENDIX A .................................................................................................... 127
APPENDIX B .................................................................................................... 129
APPENDIX C .................................................................................................... 131
APPENDIX D .................................................................................................... 133
ix
LIST OF FIGURES
Figure 2.1 Moisture absorption trends in FRP .................................................... 11
Figure 2.2 Degradation due to UV rays .............................................................. 20
Figure 2.3 Chemical attack on industrial pipes ................................................... 23
Figure 4.1 Test specimen for heat flow ............................................................... 45
Figure 4.2 Temperature variations under variable source ................................... 45
Figure 4.3 Determination of the equivalent constant temperature ...................... 46
Figure 4.4 Temperature variations under constant source .................................. 46
Figure 4.5 Comparing average variations in temperature at the reference point 47
Figure 4.6 Comparing integrated heat flows at the reference point .................... 47
Figure 4.7 Test specimen for the change in temperature at the reference point
in the material ................................................................................... 48
Figure 4.8 Temperature variations under the variable source ............................. 49
Figure 4.9 Temperature variations under the equivalent constant source ........... 49
Figure 4.10 Comparing the change in temperature under variable and constant
temperature source ........................................................................... 50
Figure 4.11 Variation of temperature during moisturization .............................. 50
Figure 4.12 Scattering of moisture content after exposure under variable and
constant temperature sources ........................................................... 51
Figure 4.13 Comparing moisture content after exposure .................................... 51
Figure 4.14 Temperature difference between the material and the environment
in the course of exposure .................................................................. 52
Figure 4.15 Transmission power in the course of exposure ............................... 53
Figure 4.16 Schematic representation of a transformation from a continuous
variable environment (a) to a succession of constant environments
(b) ..................................................................................................... 54
Figure 5.1 Environmental degradation process ................................................... 57
Figure 5.2 Portion of pipe wall 4 ......................................................................... 64
Figure 5.3 Mould compressed under vacuum bag for even resin spread ............ 74
Figure 5.4 External view of the expository chamber .......................................... 76
Figure5.5 Internal view of the expository chamber ............................................ 76
Figure 5.6 Sample cut in circular shape, the fibre direction marked, (1) and
ready to be fitted to the expository cell (2). ..................................... 78
Figure 5.7 Sample from the expository chamber (1) and cut for tensile strength
test (2) ............................................................................................... 78
Figure 5.8 Tensile test on dog bon samples of 3mm width, 15mm long ............ 79
Figure 5.9 Raman spectrometer .......................................................................... 79
Figure 5.10 hydrolysis apparatus ........................................................................ 80
Figure 5.11 UV and temperature post-curing effect on tensile strength. ............ 82
Figure 5.12 Variation of tensile strength during degradation. ............................ 82
Figure 5.13 Moisture variation during degradation ............................................ 83
Figure 5.14 Micrograph ...................................................................................... 83
Figure 5.15 Spectrum of non-degraded samples from 370 cm-1 to 2000 cm-1 .... 84
x
Figure 5.16 Spectrum of non-degraded sample from 2000 cm-1 to 3600 cm-1 ... 84
Figure 5.17 Variation of Monosubstituted aromatic ring peaks (1001cm-1). ...... 86
Figure 5.18 Decreasing of ortho substituted aromatic ring peaks (1040 cm-1). .. 87
Figure 5.19 Decreasing peaks at 1040 cm-1 ........................................................ 87
Figure 5.20 Peaks shifting from 1040 cm-1 (days1, 6, 7) to 1032 cm-1 ............... 88
Figure 5.21 Variation of 1600 cm-1 (Carbonyl) Raman peaks during
degradation. ...................................................................................... 88
Figure 5.23 Index of chemical links degradation deduced from ester groups
reduction ........................................................................................... 90
Figure 5.24 Correlation between the model and experimental values ................ 90
Figure 5.25 Predicted lifetimes compared to experimental lifetimes. ................ 92
Figure 5.26 Shear strength variation of samples subjected to chemical
degradation ....................................................................................... 92
Figure 5.27 Variation of storage modulus during the course of degradation ..... 93
Figure 6.1 Testing apparatus for stress relaxation ............................................ 101
Figure 6.2 Relaxation curves. a) under mechanical stress only, b) under
mechanical stress and chemical and physical degradation. ........... 102
Figure 6.3 Amplification of the relaxation due to environmental degradation . 103
Figure 6.4 Accumulated environmental degradation measured as 1 / Kenv ...... 104
Figure 6.5 Relaxation under chemical and physical degradation at various
temperatures ................................................................................... 104
Figure 6.6 Experimental determination of the activation energy for different
relaxation amplitudes ..................................................................... 106
Figure 6.7 Activation energy as a function of the relaxation amplitude ........... 107
Figure 6.8 Experimental determination of the shift factor as a function of the
relaxation amplitude ....................................................................... 107
Figure 6.9 Experimental measurement of the relaxation under chemical and
physical degradation ....................................................................... 109
Figure 6.10 Experimental relaxation times compared to the relaxation times
predicted from times measured at 80?C ......................................... 110
xi
LIST OF TABLES
Table 4.1: Materials ............................................................................................ 44
Table 4.2: Integrated heat flow .......................................................................... 47
Table 5.1: Materials ............................................................................................ 73
Table 5.2: Exposure conditions ........................................................................... 75
Table 5.3 Tensile Strength of exposed samples .................................................. 81
Table 5.4: Calculated value of index Ed and experimental tensile strength ....... 90
Table 5.5: Predicted and experimental lifetime .................................................. 91
Table 6.1: Materials .......................................................................................... 100
Table 6.2: Test conditions ................................................................................. 100
xii
LIST OF SYMBOLS
Coefficient of chemical activity
Coefficient of linear correlation
Constants
Density
Diffusion coefficient
Diffusivity
Enthalpy
Ideal gas constant
Kinetic constant
Mathematical divergence
Natural logarithm
Power
Pressure
Relative humidity
Sodium hydroxide
Temperature
Thermal flux
Time
Variation
Index of material rheology
Volume
Work and energy
? (dimensionless number)
R2
?
(dimensionless number)
i, ?i, ai, bi,
? (kg/m
c, const
3
D (m
)
2
J (l/m
/s)
2.s, moles/m2
H (J, cal)
.s)
R (J/ ?K.mole, l.atm/?K.mole)
K
?
Ln
P (W)
Pr
? (%)
(Pa)
NaOH
T (?C, ?K)
?T (W/m
2
t (s)
)
?
?
V (m3
W (J, cal)
, l)
xiii
LIST OF ABBREVIATIONS
Chop strand mat
Fibre Reinforced Plastics
Glass Reinforced Plastics
Woven roving
CSM
FRP
GRP
WR
1
1 INTRODUCTION
Fibre reinforced plastics (FRP) offer a number of advantages including light
weight, better corrosion resistance, easy shaping, aesthetics, excellent mechanical
properties and have found a wide variety of applications in modern industry.
However, one limitation that is likely to slow down the use of FRP concerns the
method for environmental degradation evaluation. The major unresolved issues
involved concern the availability of a comprehensive model of the degradation
process, capable of providing a quantitative basis for an acceptable predictive
method of assessing the material performance in service, using short-term test
results [1-3].
In their service life, FRP materials face a variety of environmental conditions
resulting from natural or artificial factors. These include variable temperature and
humidity conditions, energetic radiation such as ultraviolet (UV) rays from the
sun or other artificial sources, and diverse chemical reactants such as liquid in
storage tanks and pipes, or atmospheric oxygen and ozone.
FRP materials interact with environmental factors. This results in a change of the
chemical and physical structure of the material and its composition. Effects of
such changes are irreversible degeneration of the mechanical properties of the
material for the most part. Aesthetic characteristics, such as colour and glow,
undergo changes. The material may become brittle and cracks appear. In practice,
any change affecting the material properties relative to the initial desirable
properties is called degradation.
Environmental degradation factors are always combined and negatively affect the
material properties over time. The top of a boat for instance is subjected to
ultraviolet rays in combination with corrosive humidity and temperature cycles.
Chapter One: Introduction
2
The inner surface of a pipe or a storage tank faces wet and corrosive conditions in
combination with temperature cycles. The resulting degradation mechanism is
complex. The degradability in service conditions varies from one polymer to
another depending on the chemical structure and the presence of impurities that
considerably contribute to the initiation of many degradation processes. For most
of the cases, the environmental degradation is a slow process lasting up to several
decades before effects be manifested. However, there are cases where effects are
seen at short-term scale.
Therefore, optimized utilization of a fibre reinforced plastic material requires the
availability of a reliable method for quantifying environmental effects and for
predicting material lifetime. This allows for optimal handling of issues related to
component design, economic assessment and safety considerations, as well as the
technical problems relating to equipment maintenance. In this regard, efforts
worldwide are devoted to the modelling of FRP environmental degradation.
However, the high complexity of the process explains why no general model or
viable corrosion resistance test method has been available so far [1, 2].
A great number of analyses have been published on the environmental
degradation of fibre-reinforced plastics. In this regard, four predominant trends
have been surveyed in the literature.
The first trend is supported by abundant literature and focuses on the
characterization of effects and/or on the description of mechanisms [1-42]. The
second trend in the literature deals with modelling and is also the focus of many
published works [1, 16, 39, 43-50]. Modelling efforts, for the most part, are
limited to partial models based on a single mechanism dominating the whole
process. Such models are mostly based on moisture and/or on temperature effects
and are empirical. Another modelling approach is that based on chemical reaction
mechanisms especially in the case of hydrolysis and oxidation or photo-oxidation
[23, 47-49]. The third trend in the literature goes beyond the characterization of
effects and mechanisms and suggests prediction methods. These methods are
Chapter One: Introduction
3
based on the assumption that the dominating mechanism is thermally activated
and follows the Arrhenius law. The prediction relies on linear extrapolation based
on temperature variation.
Due to the lack of predictive models, some researchers resort to exposure of the
FRP material in typical service environments in order to assess the environmental
resistance of the material. In this fourth trend, the method requires many years of
exposure and tests must be conducted for each climatic area [1, 36]. Similarly, in
industry, standards specify the material lifetime based on statistics resulting from
many years of practice in the field. This implies that many years of
experimentation are required prior to setting the lifetime standard for each new
specification.
The lack of a reliable method of environmental effect prediction has been a
hindrance for extended use of FRP material in fields such as construction [2, 17,
19] and is a cause of concern in the chemical industry where cases of catastrophic
failure due to environmental degradation were reported [51].
Experimental methods show limitation with regards to the comprehensive
description of FRP environmental degradation process. The work presented here
suggests an analytical approach based on well-established material science laws. It
is felt that, there is a need for a theoretical analysis of the environmental
degradation of FRP. Such analysis should lead to a theoretical basis capable of
providing a common frame where all the processes involved in the environmental
degradation of FRP may be treated comprehensively. Without such a frame, direct
experimental observation can only lead to partial and limited treatment of the
problem.
This analysis is aimed at the development of a short-term test method for the
prediction of the environmental degradation of mechanical strength of FRP
composites. The method is based on a comprehensive mathematical model,
involving all the factors that affect the degradation process. These factors include
Chapter One: Introduction
4
the chemical degradation, the ultraviolet rays attack, the temperature and humidity
effects, and the stress corrosion. In addition, this method must be practical, and
relatively fast.
The suggested model is a mathematical function logically derived from material
science theories and expresses a qualitative relation between the material
degradation and environmental factors. The development of the model is based on
a theoretical demonstration. The demonstration is based on a deductive argument
using material science laws as premises. This means that, well established theories
are applied to FRP environmental degradation process through a deductive
reasoning in such a way that, results arrived at are logical and rigorous
consequence of pre-existent material science theories. Therefore, the validation of
the model is provided by the validity of the material science laws because the
deductive reasoning shows that it would be self-contradictory to assert these laws
and deny the model. Or, the negation of the model would be contradictory to the
veracity of the material science laws. However, the deductive reasoning is further
supported by experimental results obtained from a laboratory.
The suggested analytical approach resolves the complexity of FRP environmental
degradation into only three processes: the chemical degradation, the physical
degradation, and the modification of the stress state. The first step of the
demonstration deals with the chemical and physical degradation. The method
consists of deriving a qualitative mathematical relation between the degradation
rate and the environmental factors including the chemical concentration, the
moisture, the diffusion coefficient, UV rays, and the temperature.
In the second step, effects of mechanical stresses are introduced through a
dynamic constitutive equation that takes account of the evolution of the material
stiffness. In that equation, the material stiffness is expressed as a function of time
arising from two contributions: the mechanical reaction due to the material
viscoelasticity and the chemical and physical degradation. The resulting dynamic
Chapter One: Introduction
5
constitutive equation provides the comprehensive model of FRP environmental
degradation.
The analysis introduces also a method of reliably relating laboratory test results to
the real service conditions. This method is based on the use of constant
environment. From all the above, a short-term test method of FRP environmental
degradation is deduced and is based on the test of stress relaxation.
Additionally, the modelling method suggested in this analysis is based on the
environmental resistance of the matrix. This is because in industrial applications,
the environmental resistance of a laminate is determined by that of its barrier coat,
made of a resin rich layer. It is also observed, in general, that the degradation
mechanism of FRP initiates in the matrix and primarily affects the matrix
dominated properties [42, 46, 50]. Though the method can be applied to any
thermoset matrix, experimental results are limited to those obtained from an
orthophtalic polyester matrix. The laminate under investigation was designed to be
comparable to a standard barrier coat and exposure of the material corresponds to
that of a pipe or storage tank environment.
From the general introduction to the general conclusions that constitute
respectively the first and the last chapter, the analysis goes over five other
chapters to present successively a literature survey, a method for the
determination of the laboratory test conditions, the chemical and the physical
degradation, and the constitutive equation of environmental degradation in FRP.
The first chapter, this introduction, presents a description of the problem dealt
with and the motivation of the research as resulting from unfulfilled need for a
reliable method to predict FRP environmental degradation and the subsequent
need for optimization of the design, economic assessment of the investment, safe
use of GRP equipments, and optimization of maintenance operations.
Chapter One: Introduction
6
The following two chapters are devoted to the literature survey. Focus is on two
main topics as arising from the number of works published on these topics. These
include environmental degradation mechanism and effects in chapter 2, and
modelling efforts in chapter 3.
Prior to mathematical modelling of the environmental degradation process, the
chapter 4 deals with a method to represent the real service environment into
laboratory and a model of the environment that allow taking into account its
complex variability when modelling the degradation process.
The chapter 5 constitutes the first step of the mathematical modelling of the
environmental degradation process. It presents details of the modelling approach,
and the chemical and physical degradation model. This part deals with case where
no mechanical stress is involved in the degradation process. But, it provides the
basis for the introduction of the chemical and physical degradation in the general
case where mechanical stresses are combined with the remaining environmental
factors. This general case is presented in chapter 6, which deals with the
constitutive equation of environmental degradation in FRP and the resulting
prediction methods. The constitutive equation provides the general model
allowing a comprehensive test of environmental degradation of FRP.
The last chapter is the general conclusion where the analysis is summarized, and
the main results presented. Of great importance, the short-term test method
resulting from this work is presented. Some links to possible future works are also
defined.
Appendix A is introduced to deal with the demonstration of some energy and
mass transfer equations relating to the modelling process. Specials issues relating
to the mathematical resolution of equations are dealt with in appendix B and C.
Appendix D provides an illustration of the computation method relative to the
prediction of the physical and chemical degradation of FRP.
7
2 ENVIRONMENTAL DEGRADATION MECHANISMS AND EFFECTS,
A LITERATURE SURVEY
The literature describes the environmental degradation as a complex mechanism
resulting from several factors. Moisture and temperature are viewed as the most
common and deleterious environmental factors and many analyses focus on this.
Ultraviolet irradiation and attack by chemical agents constitute also two of the
most important aspects of the environmental degradation mechanism studied in
the literature. This chapter presents four sections devoted to each of these
environmental factors. Attention is also given to the stress corrosion aspect.
2.1 Moisture Diffusion
Moisture results from natural humidity, rainfall, industrial vapours or any liquid in
contact with the material such as liquid circulating in pipes or stored in tanks. The
relative humidity in the material service environment, can take values from
moderate to fully moisture-saturated environment 100%. The humidity effects on
FRP are chemical, physical, and mechanical.
Once humidity encounters the FRP laminate surface, it dissolves on the substrate
surface and then migrates through the bulk of the material. The diffusion process
is facilitated when the permeant molecules present compatibility with the
polymeric structure in terms of polarity and cohesive energy. Plasticization occurs
and this, subsequently, increases the diffusion rate. In the other side, when there is
incompatibility, clustering occurs resulting in retardation of the penetration [24].
Many other factors, both internal and external, affect the diffusion process.
However, depending on the environmental conditions and material characteristics,
the diffusion results in a steady state concentration of the diffused humidity. The
presence of moisture inside the FRP laminate structure as a foreign material,
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
8
negatively affects the cohesive characteristics of the laminate. The resulting
effects are swelling, softening, and reduction of the glass transition temperature.
Moisture diffusion through FRP laminate has been extensively investigated
throughout the last century. Main issues dealt with were firstly, the determination
of the diffusion kinetic in terms of humidity distribution inside the laminate as a
function of time. Secondly, the determination of humidity effects on the
mechanical properties of the laminate. Thus, this section deals firstly with the
kinetic of diffusion in the FRP laminate and secondly, with the effects of moisture
diffusion on the mechanical properties of the laminate.
2.1.1 Kinetics of moisture diffusion through FRP laminates
The general kinetic law controlling the diffusion of liquid through a solid material
is known as Fick?s law [52]. According to this law, the mass of liquid entering a
unit volume of a solid material per unit time is given by:
CD
t
C 2?=
?
?
(2.1)
where C and t represent the liquid concentration and the time respectively. D is
the diffusion coefficient that accounts for the material characteristics. Barrer [52]
presents solutions of this equation for variable cases.
According to Springer [16], reporting on several researches conducted on
environmental effects on composites materials, though for most of FRP materials
encountered Fick?s law constitutes a reasonable approximation of the diffusion
kinetic, there are still many cases where this law remains inapplicable.
In the reported researches [16], the case of E-glass polyester composite, exposed
to several kinds of liquid and humid air at variable temperature, shows that, under
most conditions, Fick?s law may be used to assess effectively the kinetic of
diffusion. The same experiment shows also that the maximum value of moisture
uptake is a function of the ambient relative humidity and does not depend on the
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
9
temperature for most of the times [6]. Shen and Springer [10] have studied the
moisture absorption and desorption of composite material. They suggested the
following relation between the ambient relative humidity ? and the maximum
moisture content Mm
b
m aM ?=
of laminate exposed to that environment:
(2.2)
In this relation a and b are constants depending on the material.
Based on Fick?s law, the same authors suggested for the moisture content M
(percentage weight gain) the following equation:
iim MMMGM +?= )( (2.3)
In equation (2.3), Mi is the initial moisture content of the material, Mm
?
?
= +
+?
?=
0
2
22
2 )12(
)]()12(exp[
8
1
j
x
j
s
tD
j
G
?
?
is the
maximum moisture content that can be attained under the given environmental
conditions, and G is a time dependent parameter given by:
(2.4)
G can be approximated by the following expression:
75.0
2
)(3.7exp[1
s
tD
G x??= ] (2.5)
where Dx
is the material diffusivity in the direction normal to the surface and s is
a parameter relative to the material thickness. This relation was developed for
one-dimensional problems where the ambient moisture and temperature were
assumed constants. The initial moisture and temperature distribution inside the
material also must be uniform. The application of the proposed model to the case
of a unidirectional graphite T-300 fiberite 1034 composite, fully immersed in
water or exposed to humid air, has shown good correlation between experimental
data and the theory.
Some investigators reported that moisture absorption level is history-dependent.
Due to clustering, the diffusion proved to be a non-fickian two stage process. The
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
10
diffusion coefficient decreases with increasing permeation [18] and consequently,
sorption behaviour under temperature cycles is not the same as under constant
humidity and temperature level [19].
In material science, the diffusion mechanism through solid material is explained
by the theory of vacant sites. According to this theory, solid polymers contain a
steady-state distribution of molecular-size voids [24]. The diffused particles move
through the solid network by interchange with the voids or vacant sites [21].
Though contested by some theorists who estimated, for the case of epoxies for
instance, that the only mechanism governing the diffusion into solid material is
the molecular interactions between the permeant and the polymeric solid [22], this
theory allowed understanding successfully the kinetic of the diffusion for many
cases. However, it is known that the diffused humidity interacts with the
polymeric material and this leads to the transformation of the polymer. At least
three types of transformation that mostly affect the permeation process have been
surveyed by Gesner [23]: cross-linking, crystallization, and micro porosity. The
cross linking reduces the material permeability likewise the crystallization. The
micro porosity allows the convective diffusion to take place. This, subsequently,
increases the total diffusion rate. All of the above explains the multiple behaviour
of the diffusion coefficient that has been observed.
Srivastava [7] has reported a case where moisture affects the material rheology by
increasing the viscosity. This was shown by the increased shear strength of a
quasi-isotropic glass-fiber reinforced epoxy vinyl ester composite that was
immersed in water with variable immersion time. This implies that the coefficient
of diffusion decreases during the course of moisturization. Tang et al [8] mention
a different behaviour. The plot of shear strength versus moisture content in a
Fiberite 976 composite showed a decreasing trend of the shear strength
corresponding to the increase in moisture content.
In fact, moisture diffusion through FRP appears to follow variable kinetic curves
depending on the physical transformation occurring in the material or chemical
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
11
reactions between the permeant and the material [19]. This is illustrated in Figure
2.1:
Figure 2.1 Moisture absorption trends in FRP [19]
a. Curve LF represents linear Fickian behaviour where the moisture weight gain
gradually attains equilibrium after a rapid initial take-off.
b. Curve A represents the so-called pseudo-Fickian behaviour where the
moisture weight gain never reaches equilibrium after the initial take-off. This
depicts a transformation of the material structure accompanying the
absorption.
c. Curve B displays a two-stage diffusion denoting a change of environment
such as temperature, applied load, relative humidity, or physical
transformation occurred in the material.
d. Curve C represents the case where large deformation or damage occurs in the
material. Such large deformation may be fiber/matrix debonding or matrix
cracking.
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
12
e. Curve D shows the case where the gain in moisture weight presents a
decreasing trend after the initial take-off. The process is irreversible as a result
of leaching out of the material from the bulk following chemical or physical
break-down. This, in fact, means that the weight variation measured does not
represent the real humidity uptake but the loss of laminate mass due to
leaching.
2.1.2 Factors affecting the diffusion process in FRP laminates
Many factors both internal and external to the material affect the kinetic of
moisture diffusion in FRP.
Fibre volume fraction and orientation effect
Rao et al [25] showed that the equilibrium moisture content in a composite is a
function of the fibre volume fraction and the fibre orientation to the diffusion
path. These authors have studied the case of three-dimensional diffusion in a jute-
epoxy laminate and in a glass-epoxy laminate. The moisture diffusion is as higher
as the fibre orientation to the diffusion path ? is increased. The fibre orientation
effect is inversely proportional to the fibre permeability. The resulting diffusion
coefficient Dc may be assessed as function of fibre and resin diffusion coefficient
respectively denoted Df and Dr,
??
?
?
??
?
?
+= ?? 2sin2cos
11
22
D
D
DVD ffc
as follows:
if Df >> Dr
( ) ??
?
?
??
?
?
+?= ?? 2sin2cos1
11
22
D
D
VDD frc
(2.6)
if Df << Dr
In equations (2.6) and (2.7),
(2.7)
? , 11D and 22D respectively represent the fibre
orientation angle, and the diffusivity in the directions parallel and normal to the
fibre (longitudinal and transverse).
The equilibrium moisture content Mm is a linear function of the fibre volume
fraction:
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
13
baVM fm += (2.8)
In equation (2.8), a and b are constants.
Effect of solvent molecules
The molecule size of the permeant liquid, likewise the molecule polarity,
significantly affects the permeation. Larger molecules diffuse more slowly and
polar molecules diffuse more easily in polar thermosetting structure. The analysis
conducted by Sonawala and Spontak [27] showed, for example, that the
isophtalicpolyester offers a high solubility to aqueous solutions due to the high
concentration of ester-linkages available for hydrogen bonding with permeated
water. The analysis of Bellenger et al quoted by Sonawala [27] showed that the
equilibrium moisture content does not depend on the temperature and the packing
density. Instead, it is correlated to the concentration of hydroxyl, ether, and ester
groups. It was shown that the solubility of water into the vinyl ester laminate
increases with the number of carbonyl groups. Verleg and Van der Wal [53]
studied the absorption of several types of solvents in thermosetting unsaturated
polyester and vinyl ester. Their results showed that the penetration time as well as
the penetration flux, increases from the lowest solvent to the heaviest one.
Temperature effects
Throughout the literature, it is generally observed that temperature affects the
diffusion coefficient and effects on the equilibrium moisture content depend on
the material type. Loos et al [5-6], reporting on a case of fiberglass polyester
composites, showed that the diffusion followed Fick?s law and the equilibrium
moisture content does not depend on the temperature. However, in the case of
epoxy composites reported by Marsh et al [18], the solubilization followed
Henry?s Law given by the equation (2.9). According to this law, the equilibrium
moisture content is a function of the humidity pressure rP and depends on the
temperature T as follows:
RT
H
rm ePM
?
= ? (2.9)
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
14
In equation (2.9),? is the coefficient of chemical activity. H , and R respectively
represent the solubilization enthalpy, and the ideal gas constant.
The temperature dependence function of the diffusion coefficient for permeable
and impermeable material can be represented by an Arrhenius type relationship.
RT
WD
eDD
?
= 0 (2.10)
Where DW is the activation energy of diffusion and the subscript 0 refers to the
initial temperature conditions. This relation has been successfully used by several
researchers [8, 21, 54]. However, it is reported that this holds only over a limited
range of temperature below the temperature of glassy transition [2].
Applied load and hydrostatic pressure effects
The load applied on the material also affects the moisture uptake. Maron et al [29]
observed the moisture penetration into both glass fibre and carbon fibre epoxies
composite under stressed and unstressed conditions at 95?C. From their
observation, they concluded that the diffusion rate as well as the equilibrium
moisture content is increased under external load. The same behaviour was
observed in the course of studies conducted in the composite facility of the school
of mechanical engineering at the University of the Witwatersrand on the
polyester, vinyl ester resin, and a corrosion barrier coat material, all of them
subjected to tensile load [55-57].
It is reported [2] that the hydrostatic pressure does not influence the diffusion
coefficient but affects equilibrium percentage moisture in the material. Avena et
al [4] observed that high pressure is susceptible to reduce moisture uptake as it
tends to compress the matrix and close the micro voids or defects.
2.1.3 Humidity effects on FRP laminates
Major concerns about moisture uptake in FRP laminate are related to the
deterioration of the material performance. As said earlier, the moisture diffusion
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
15
into FRP laminate induces swelling, plasticization, clustering, and reduction of
glass transition temperature. Some liquids are susceptible to react chemically with
the polymeric matrix as well as with the reinforcing fibres. For instance, water,
likewise the remaining polar solvent such as alcohols, hydrazine, and ammonia,
attacks the polymeric material by a solvolytic reaction whereby the hydroxyl
reacts with the polymeric chain. This leads to a chain scission and the formation
of alcohol and carboxylic acid. The solvolytic reaction is catalyzed by acidic or
basic environment. Moisture attacks the fibre material as well as the interface
fibre-matrix by hydrolysis.
Another known effect of moisture on FRP is related to the cycling humidification.
The material exposed to the natural environment is often subjected to cycle effects
of moisturization during rainy time followed by drying during sunny time. The
cyclic moisturization induces residual stresses inside the material leading to
cracks apparition.
Hygroscopic expansion resulting from the absorption of moisture by a FRP
composite is also a well-known effect. The material is subjected to a strain
assumed to be proportional to moisture content expressed as weight percentage.
The constant of proportionality is called the coefficient of the hygroscopic
expansion. The hygroscopic expansion effect is generally associated with the
thermal strain resulting from the thermal expansion. When the material is
constrained, as are the individual layers of fibres and matrix in a laminate, the
thermal or hygroscopic strain cannot develop freely. This leads to the
hygrothermal stresses [58] of which effects on the FRP material has been
recognized as one of the most deleterious. For example, in construction, Sachin et
al [59] have observed that the main cause of environmental damage into the FRP
wrapped concrete cylinders is the combined effects of moisture and elevated
temperature on the tensile strength of E-glass fiber.
Thus, the humidity negatively affects the internal structure of FRP material and
accelerates the failure of the laminate under load.
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
16
Many reported works in the literature show that moisture affects both the tensile
strength and the elastics modulus of the composite material. For example, the
degradation of material properties can reach 20 to 25% decrease in modulus, for 3
to 4% moisture content, depending on the material characteristics, the temperature
and the degree of moisturization. This is illustrated by the case of brittle epoxies
studied by Tang and al [8]. In a survey into durability of fluid containment vessels
marine structures and aircrafts with up to nineteen years of service, Lieblein [60]
suggested that, over a period of twenty years, in addition to the reduction in
strength as result of exposure to moisture, the reduction in modulus is of the order
of 10 %.
Moisture effects are often associated with temperature. Shen and Springer [10]
have assessed the change in the ultimate tensile strength of composite materials
exposed to air in which humidity varied from 0 to 100% and temperature ranges
from 200?K to 450?K. The material used was the Thornel 300/fiberite 1034
graphite epoxy composite with 0?, 45?, and 90? lay-ups. For 90? lay-up, they
observed 60 to 90% reduction in tensile strength for about 1% to 1.5% moisture
content combined with temperature. For 0? and 45? lay-up, in the same
conditions, effects were negligible. The same authors, Shen and Springer [11],
investigated also the change in elastic modulus for the same material in the same
conditions than that used for the ultimate tensile strength. The results were similar
where the deterioration was about 60 to 90% for 90? lay-up while no more than
20% reduction was noted for the 0? and 45?
lay-up.
2.2 Temperature Effects
The temperature is part of all environments the FRP materials usually encounter.
In the natural environment, FRP materials may be subjected to temperature raging
from ?30?C to 60?C depending on the geographical region, the climatic season,
and the moment of the day. More elevated temperatures may be encountered in
industrial environments. They result from industrial vapours and hot media
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
17
transported in pipes or stored in tanks. High temperatures over the range of 150?C
are associated only with particular circumstances such as fire. Long-term exposure
to moderate temperature accounts for the failure of many FRP materials. The
absorption of thermal energy leads to a set of deleterious effects on the matrix.
The thermal energy attacks also the bonding region between the matrix and the
fibre as well as the overall composite laminate. These effects are chemical as well
as physical and irreversibly affect the material mechanical properties for most of
the time. These effects are well known. In the literature, authors dealing with the
temperature effects on the fibre reinforced plastics focus mainly on the
mechanism whereby the temperature affects the mechanical properties of the
material.
2.2.1 Thermolysis and Thermoxidation
In an inert atmosphere, the temperature induces the thermolysis of polymeric
materials. The thermolysis is an energetic action where chemical bonds are broken
by imparting to electrons sufficient energy to pull them out of the bond. This leads
to the scission of the molecular chain. Three mechanisms at least have been
recognized for the thermolysis process [61]:
? The depolymerization in which reduction of macromolecular size occurs
without change in chemical composition or alteration of the monomer unit
structure. The cleavage occurs by random chain scission to yield either the
monomer or the lower molecular weight product
? The elimination whereby small molecules are eliminated without man chain
scission, giving an alternated double bond chain responsible for the coloration
of the material.
? The cyclization whereby a macromolecular chain presents potential reactant
groups located in close proximity. At elevated temperature, these groups react
and the intramolecular cyclization occurs with or no elimination of small
molecules.
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
18
Thermolysis is restricted to some particular cases where high temperatures are
encountered in the absence of oxygen. The thermoxidation, however, is a more
common degradation mode polymeric materials face in normal service conditions.
Reid Shelton [30] comments that even relatively stable polymers undergo
significant deterioration on long-term exposure at ordinary temperature in air. The
thermoxidation is an autocatalytic process in which the major product is a
hydroperoxide that decomposes under appropriate conditions to give free radicals
capable of initiating a free-radical chain reaction. The decomposition of hydro
peroxide is accelerated by heat [29]. Peroxide molecules in many plastics material
are originated from the manufacturing process and play the role of catalysis that
contribute in large part to the initiation of the degradation process. It is also
mentioned that the kinetics of this process is oxygen diffusion dependant [12].
2.2.2 Residual stresses due the Temperature
The temperature is also responsible for residual stresses in the FRP laminate. One
of the well-known deleterious effects results from the cycling temperature. A long
exposure to the sun increases the material temperature. Following such exposure,
a sudden cooling resulting for example from the rain or due to night conditions,
yields transient stresses on the material surface. Such stresses are known to be the
cause of cracks initiation [12] and cause the material to become brittle.
The residual stresses due to the temperature in the composite material result also
from the mismatch of the coefficient of thermal expansion between the laminate
components. This causes the laminate deformation and leads also to cracks
apparition [13]. Residual stresses resulting from post-curing can approach or
exceed the design load and acceptable material limits. The magnitude can be
sufficient to promote environmentally assisted cracking [31]. A simple experiment
was conducted in the RP/Composite facility at the University of the
Witwaterstrand. During this experiment, a post-cured specimen along with a non
post-cured specimen were allowed to stand for five days with a content of 75% of
sulphuric acid. It was observed that the post-cured sample experienced
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
19
environmental cracking while the non post-cured sample remained undamaged
[62].
2.2.3 The Arrhenius Law
The most important temperature effects are related to the temperature dependence
functions of the remaining environmental agents. It is known that the temperature
affects both the moisture diffusion and the chemical reaction of degradation
according to an Arrhenius type dependence function. According to Arrhenius law,
each temperature increase of about 10?C results in doubling the reaction rate. This
shows the high impact of temperature variation on chemical reactions such as the
oxidation of the polymeric material due to UV rays and the hydrolysis in humid
environment.
2.3 Ultraviolet Radiations Effects and Mechanism.
UV rays are the part of the solar radiation spectrum covering the wavelength
range from 100 nm to 400 nm and constitute about 5 to 7% of the total energy
emitted by the sun. Only a fraction of this radiation reaches the earth?s surface as
most of the radiation is absorbed by the atmosphere.
The UV-B range (280 nm ? 315 nm) is known to be the most destructive part of
the ultraviolet light spectrum [20]. Its effects on polymeric materials have been
discussed in the literature for many years [12, 20, 32, 33, 47]. UV rays interact
with the matrix material of the FRP laminate by a mechanism that includes
photolysis and photo-oxidation.
Photolysis induces a molecular chain breakage process. The chain scission results
from an energetic action initiated by light energy that the electrons of the chemical
bond absorb. Photolysis produces free radicals that react with oxygen to initiate
photo-oxidation. In the natural environment, atmospheric oxygen is always
present and so photo-oxidation proceeds from photolysis.
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
20
2.3.1 Photoxidation
In the course of photo-oxidation, photons attack the organic chain and produce
free peroxide-radicals. These radicals subsequently react with the organic
molecular chain to produce hydro-peroxides that can be decomposed by UV rays
of wavelength below 360 nm. Photo-oxidation is thus a complex free-radical
chain reaction involving several steps in its mechanism [12, 20, 32, 33, 47].
The reaction is catalyzed by the hydro-peroxide and carbonyl molecules
commonly found in the polymer as products of thermal oxidation during
polymerization or processing. Likewise, transition metal ions are important
catalysts for the photo-oxidation process. Traces of these ions are present in the
polymer as residue of Ziegler-Natta catalysts used during the polymerization
reaction [33].
Effects of photo-oxidation on FRP include chemical as well as physical changes.
The chemical changes include cross-linking and chain scission [20, 35] associated
with the formation of double bonds and oxygen-containing structures such as
carboxylic acids, alcohols, ketones, and peroxides. The changes are physically
manifested by the discoloration of the polymer (Figure 2.2), which changes from
white to yellow, red, or reddish black as degradation occurs.
Figure 2.2 Degradation due to UV rays identified as discoloration or flaking of
the surface on the topside of a pipe (with the permission of SASOL)
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
21
Surface cracking and embrittlement often accompany photochemical discoloration
[33]. Observations of FRP exposed to UV rays [34, 35] show evidence of
morphological changes on the degraded surface including formation of crazes and
cracks due to chain scission, formation of holes and voids by the venting of
volatile degradation products, and gelation processes associated with cross-
linking.
2.3.2 Photo-degradation mechanism
Efforts worldwide are devoted to the elucidation of the physical and chemical
mechanisms that lead to the deterioration of the mechanical properties of the
material [42, 46, 50, 53, 63, 64]. Most of the analyses of the photo-degradation
reported in the literature are based on the description of complex chemical
kinetics [1, 32, 47].
It is generally observed that the photo-degradation process affects primarily the
composite matrix and subsequently the matrix dominated properties [42, 46, 50].
It has been shown that effects of UV irradiation are strongly dependant on
mechanical stresses [46, 65-67] and other environmental factors. These factors
include moisture [36, 42, 63, 64, 67] and temperature [36, 68]. These effects can
be described by a thickness profile showing the moisture and oxygen diffusion
dependency [66, 69].
Startsev et al [36] observed that the mechanical properties of polymer composite
materials used in aviation vary in a layered way after exposure to a natural
environment. A gradient is generated through the laminate thickness. Because of
such a gradient, effects on the mechanical strength depend strongly on the
laminate thickness. This is consistent with the results reported by Larsson [9] who
exposed Kevlar 49?epoxy composites with various thicknesses (0.13 mm, 0.25
mm, and 0.50 mm) to UV rays from a xenon source over 1000 and 2000 hours.
He noted that no perceptible effect of exposure was observed on the 0.25 mm and
0.50 mm thickness laminate, while the loss in strength on the 0.13 mm thickness
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
22
laminate was about 40%. The latter observation provided the basis for a model
proposed by Sevostianov et al [50] whereby the degradation process is considered
to be a progressing damaged front that moves through the laminate.
The literature report on many other approaches based on the characterization of
the mechanical, physical, and chemical degradation mechanism. Hartly and
Guillet [70] investigated the chemical mechanism of the ketone polymers
photochemistry. They showed that the degradation occurs according to Norrish
type II and I, and that the type II is predominant at ordinary temperature.
2.4 Chemical Agents Effects
Apart from water and atmospheric gasses such as oxygen and ozone, which are
the most important chemical reactants in the natural environment, FRP material
face also a variety of chemical attacks (Figure 2.3). These result from air
pollutants, industrial smog, and liquid circulating in pipes or stored in tanks made
of FRP.
Industrial smog content includes dyes, solvents, detergents, metals, and many
others, while air pollutants resulting also from industrial smog includes acidic and
basic gasses such as dioxide sulphur and oxide of nitrogen.
Chemical reactions of degradation are facilitated by the thermal or light energy
encountered in the usual FRP environment. The degradation mechanism is
particular to each kind of chemical agent. However, the common characteristics
are the diffusion dependence and the Arrhenius type temperature dependence of
chemical attack processes.
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
23
a)
b)
Figure 2.3 Chemical attacks on industrial pipes (with the permission of SASOL)
a) Inner of a pipe attacked by chemical: The glass surface tissue hanging from the walls where the
resin has been removed by chemicals.
b) Advanced corrosion on the surface of a pipe: The structural laminate becomes exposed, which
looks like dry glass, with no resin bonding it together.
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
24
Authors dealing with the chemical degradation of FRP materials focus mainly on
water caused hydrolysis. Very little information has been presented on the
remaining known chemical agents. Polymers containing ester or amide links are
susceptible to hydrolytic attack. Fibreglass also can be hydrolyzed at the siloxy
bond. Bonds formed by silane groups and constituting the interface matrix?fibre
are also subjected to hydrolysis by water as already mentioned in section 2.1.3.
Effects of hydrolysis include molecular chain length reduction resulting into the
embrittlement of the material and the deterioration of tensile strength, shear
strength, and module.
Springer et al [46] have surveyed results of researches conducted on glass-
reinforced polyester and vinyl ester, at two temperatures, in a variety of
environments relevant to civil applications including humid air, saturated solution
of NaCl in water, diesel fuel, lubricant oil, antifreeze, and gasoline. It is shown the
significant reduction in tensile strength and in shear strength while the modulus
was only slightly affected. It is also emphasized the fact that high effects are
related to the presence of water humidity and that high or low pH humid
environment is highly detrimental to fibres.
Comparing the degradation of two GRP laminates namely pultruded isopthalic
polyester and a hand moulded vinyl ester, in 5% brine solution and 10% NaOH
solution, Sonawala and Spontak [27] demonstrated a case where the degradation
kinetic is determined by the diffusion or by the solubility of the chemical solution
into the laminate. They noticed that polyesters undergo dramatic tensile strength
deterioration while vinyl esters present higher properties retention. This was
explained by the fact that polyesters offer higher solubility to aqueous solutions
due to a higher concentration of esters linkage available for hydrogen bonding
with permeated water, while vinyl esters present higher resistance to permeation
[27, 28].
Prian and Barkatt [40] investigated the degradation mechanism in the case of
fibreglass plastic composites exposed to aqueous media. They showed that the
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
25
leaching of alkaline components out of the fibre and the resulting increase in pH
at the interface matrix-fibre determine the fundamental mechanism that explains
the drastic increase in degradation rate during long-term exposure. The increase in
pH leads to the acceleration of the hydrolytic attack at the interface. The
degradation and opening up of the interface allows further penetration of
moisture. This subsequently accelerates the hydrolytic attack. The degradation
occurring mainly at the fibre?matrix interface was also noticed earlier by Harper
and Naeem [71] who studied the moisture absorption of glass reinforced
vinylester and polyester composites.
2.5 Stress Corrosion
The stress corrosion constitutes an important aspect of environmental degradation
the literature deals with. It has been observed that environmental factors affect
consistently the stress-rupture time of stressed material exposed also to
environmental factors [45] or inversely the severity of the environmental
degradation increases when the material is also subjected to mechanical stresses
[1].
Pritchard and Speake [45] carried out an experience on a glass-polyester
composite laminate immersed in water under load at three different temperatures.
The experiment allowed them to observe that the combination of water and
temperature effects could plasticize the material or increase the fracture toughness
of the resin and the stress-rupture time was affected accordingly. Many works
have been devoted to stress corrosion throughout the literature and effects vary
according to particular conditions of experimentations.
Barkatt [2] reports a relation that was found between the crack velocity and the
stress intensity in an environmentally assisted cracking:
( )
n
bKXA
V
n )exp(0= (2.11)
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
26
In this equation, V is the velocity of crack propagation, X0
n
XCD
V w
?
0=
is the partial pressure
of water, n is the order of the chemical reaction, and K is the stress intensity
factor. A and b are constants. This formula describes the process kinetic in the
initial and most important stage controlled by the stress corrosion rate. The
mechanism includes a second stage controlled by the moisture diffusion rate
where the crack propagation velocity is given by:
(2.12)
In the above equation, Dw
is the moisture diffusivity, ? is the boundary layer
thickness, and C is a constant.
A quite different approach has been suggested by White and Turnbull [1] whereby
the problem is presented as a stress-aided chemical reaction. It is stated that the
most highly stressed bonds will be the most likely to react. Experimentation
carried out on a polypropylene specimen showed that the rate of oxidation
increases with the load at high loads. It was explained that the stress alters the
activation energy of oxidation by reducing it and according to Arrhenius law, the
rate of oxidation r as suggested by Zhurkov, quoted by White and Turnbull [1]
can be given as follows:
)
)(
exp(
RT
BG
Ar
????
= (2.13)
In equation (2.13), A is the frequency factor. G? and ? respectively represent the
energy barrier and the stress magnitude. B is a constant that has the dimension of a
volume.
White and Turnbull [1] reported some other results found by Popov et al (1991)
showing that the tensile stresses accelerate the degradation while compressive
stresses retard it.
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
27
2.6 Conclusion
Mechanisms and effects studied in the literature include moisture diffusion,
temperature, ultraviolet irradiation, chemical attack and the stress corrosion.
Attack by moisture involves a diffusion step, followed by hydrolysis or hydration.
Though, most of diffusion occurring in FRP laminate follow Fick?s law, variable
kinetics have been observed depending on the type of transformation occurring in
the material and the type of physical and chemical interaction between the
permeant and the material. Factors influencing the moisture diffusion include fibre
volume fraction and orientation, size and polarity of solvent molecules,
temperature, the applied load and the hydrostatic pressure. Humidity affects the
material properties and accelerates failure. Cycling humidity, as well as cycling
temperature results in hygrothermal residual stresses that can approach or exceed
the design load. The synergic effect resulting from the combined action of
temperature and humidity is known to be very deleterious. The temperature
influences considerably ultraviolet irradiation and chemical action through
Arrhenius law. Temperature effects include also thermolysis and thermo-
oxidation.
Ultraviolet irradiation is one of the most important natural degradation factors for
FRP and one of the most studied in the literature. The UV-B range (280 ? 315
nm) is known to be the most destructive for FRP. The chemical mechanism is
complex, and the process includes photolysis and photo-oxidation. In natural
atmosphere, due to oxygen, the photo-oxidation is the most common. The process
is strongly dependent on mechanical stresses, humidity and temperature.
Ultraviolet irradiation causes the material discoloration, the surface cracking,
embrittlement, and reduction in mechanical properties.
FRP encounter several kinds of chemicals in their service environment. The
literature provides only limited information on chemical attacks. The most known
effects are the hydrolytic or solvolytic attack that results in mechanical strength
deterioration. The process is boosted in acidic or alkaline medium. The combined
Chapter Two: Environmental Degradation Mechanism and Effects, a Literature Survey
28
action of chemical and mechanical stresses produces a synergic action known as
stress corrosion that result in the acceleration of the material failure.
In general, environmental degradation primarily affects the composite matrix and
subsequently the matrix dominated properties. The effects can be described by a
thickness profile showing the moisture and oxygen diffusion dependency.
29
3 MODELLING AND PREDICTION METHODS OF ENVIRONMENTAL
DEGRADATION, A LITERATURE SURVEY
Optimal utilization of FRP material requires a good understanding of the
environmental degradation effects and the availability of reliable method for
quantifying and predicting these effects. The previous chapter presented an overview
of the degradation mechanism and effects reported in the literature. Those
investigations were aimed at the development of models allowing to set assessment
and prediction methods of environmental effects. The present chapter presents an
overview of modelling issues including limitations to global modelling, and
predictive methods based on partial models. Attention will be given also to analytical
methods used.
3.1 Modelling and Limitations
White and Turnbull [1] presented a comprehensive review of the modelling and
prediction issues relative to the environmental degradation of polymers. They
underlined the necessity to develop models that allow short-term laboratory data to be
used to give an accurate forecast of the lifetime of the component, to assist in material
selection and to permit economic planned replacement.
Their review, however, noted that no general or accurate model was available at the
time. They pointed out that the main difficulty was the great number of chemical and
physical processes involved in the environmental degradation, and their interactions.
White and Turnbull [1] explained that the problem was as complex as identifying the
various reaction?s pathways, then measuring the rate constants, including the effects
of intermediate reactions products, determining the diffusion coefficient for main
Chapter Three: Modelling and Prediction Methods of Environmental Degradation, a Literature Survey
30
chemical agents as well for intermediate reactions, and this should take account of
effects resulting from the variation of the matrix morphology during the degradation.
These observations were consistent with the conclusions from a review by Schutte [3]
who reported that the major unresolved issues are firstly, the difficulty to take into
account all the processes involved and their interactions in a comprehensive model
and secondly, the uncertainty related to the translatability of laboratory tests
conditions to the real service environment [3].
A more recent review of the subject by Barkatt [2] in 2001 restated the same
conclusion. Barkatt recognized that considerable efforts had been made to model the
degradation of FRP, but unfortunately, no comprehensive models were yet available
to provide a quantitative basis for evaluating the performance of FRP.
A great number of published works focus on the characterization of effects and/or on
the description of mechanisms [1- 43]. A large range of material properties have been
observed for a large range of materials in a variety of environments. It has been
observed that environmental factors negatively affect the material properties over
time. Concerns have arisen regarding the degradation mechanism since this is
understood as a prerequisite to any modelling effort. For most of the cases, the
mechanisms deduced were applicable only to some particular processes.
In this regard, considerable progress was made in understanding the diffusion
mechanism as well as some specific processes like oxidation, and hydrolysis as
already presented in the previous chapter. Such analysis allowed mainly the
development of stabilizing systems applied in polymeric materials against
environmental agents attack. The use of stabilizer allows reducing the acuity of
problems related to the environmental degradation. Stabilizers have sometimes
produced spectacular improvement in the lifetime of polymeric material [1].
However, in some cases, they may not be effective, or their effect may wear off over
Chapter Three: Modelling and Prediction Methods of Environmental Degradation, a Literature Survey
31
the time. Predictive models would then be useful at least for the assessment of the
stabilizer effectiveness.
Modelling efforts, for the most part, are limited to partial models based on a single
mechanism dominating the whole process [1, 16, 39, 43-50]. Such models are mostly
based on moisture and/or on temperature effects and are empirical.
An example is the model based on hygrothermal stress distributions reported by
Springer [44]. This model is based on a three-step method for effects evaluation.
Firstly, analyses are conducted to determine the temperature and moisture distribution
inside the material. Secondly, hygrothermal stresses and strains are calculated based
on the distribution of temperature and moisture. In the third step, the change in
material performance is evaluated based on calculated hygrothermal stresses.
However, this method is subject to limiting hypotheses, such as that the diffusion
should obey Fick?s law.
A second example is the empirical model based on moisture effects proposed by
Prichart et al. [45]. They reported on a case where the kinetic equation was deduced
empirically by mathematical regression of experimental data. The changes in tensile
strength and modulus of a fibreglass?polyester resin composite were plotted versus
the moisture content in the course of an exposure. Two temperatures were used with
several fibreglass orientations. It was reported that the predicted behaviour based on
that model was good for up to three years. Unfortunately, no information on longer
time scales is provided.
Nakamura et al. [46] suggested an empirical model that allowed for quantification of
the combined effect of UV rays, humidity, and cyclic load on the flexural strength of
cross-ply laminates of carbon fibre reinforced epoxy.
Chapter Three: Modelling and Prediction Methods of Environmental Degradation, a Literature Survey
32
Another modelling approach is based on chemical reaction mechanisms especially in
the case of hydrolysis and oxidation or photo-oxidation [23, 47-49]. This approach
leads to complex mathematical formulae and parameters in these formulae cannot
always be measured.
Some models of photo-oxidation have been surveyed from the literature. Generally,
these models relate the degradation time to UV rays absorption and the time is only
indirectly related to the material properties. Parameters in these models are not easily
measurable from a practical point of view and the applicability of any model
proposed is limited to specific polymers. Illustration of the above is given by the
following cases:
Reich and Stivala [47] have described a kinetic model based on the reaction
mechanism proposed by Ershov et al [47]. This model was successfully applied to the
photo-degradation of polystyrene. In that model the degradation rate is measured by
the rate of oxygen absorption as follows:
( ) 02654
3212 ]][[
][
ISO
kkk
kkk
dt
Od
+
=? (3.1)
In equation (3.1), [O2], [S], I0 are respectively the oxygen concentration, the polymer
concentration and the UV radiation intensity; while k1, k2, k3, k4, k5, and k6
are
respective kinetic constants associated with chemical steps in the mechanism.
A review by White and Turnbull [1] has reported some alternative mechanisms that
have been described in the literature. For example, polyethylene oxidation under
ultraviolet radiation was determined by Karphukhin to follow the following rate
equation:
2
1
bIaI
dt
dc
+= (3.2)
Chapter Three: Modelling and Prediction Methods of Environmental Degradation, a Literature Survey
33
In the above equation c is the carbonyl concentration, I is the ultraviolet intensity and
a, and b are constants. A kinetic model proposed by Minsker et al. applicable to
Polyvinyl Chloride was also reported in the review. In the model, the humidity and
UV rays effects are correlated to determine the degradation time ? as follows:
1
0
??= HWe T
U
??? (3.3)
In equation (3.3), T, W and H represent the temperature, the relative percentage
humidity and the UV rays dose respectively. U is a thermodynamic parameter
proportional to the activation energy. In the report, no description of the parameter ?
was given.
Beak-Su Lee et al [38] presented an approach in which they correlated the chemical
transformation on a UV-treated surface of an epoxy/glass composite with electrostatic
changes and suggested a control model based on the electrostatic behaviour.
Based on the observation that the degradation process can be considered as a
progressing damage front that moves through the laminate, Sevostianov et al [50]
proposed a mathematical model. They suggested a mathematical relation that
calculates the overall modulus as a function of time arising from the contributions of
the modulus of both damaged and undamaged layers. However, this model is affected
by several limiting hypotheses and by the omission of the effects of moisture
diffusion.
3.2 Accelerated Test Methods
One important trend in the literature goes beyond the characterization of effects and
mechanisms and suggests prediction methods. These methods are based on partial
models where a single mechanism dominates the whole degradation process.
Chapter Three: Modelling and Prediction Methods of Environmental Degradation, a Literature Survey
34
Assumption is made that the dominating mechanism is thermally activated and
follows Arrhenius law. The prediction relies on linear extrapolation based on
temperature variation. The material is subjected to an accelerated degradation in a
laboratory and results are extrapolated to real services conditions using acceleration
factors based on the partial model. Acceleration factors are based not only on
temperature increase, but also on media that are more corrosive or the application of
mechanical loads [2, 16]. Examples of these are given below.
3.2.1 Acceleration using higher temperature based on Arrhenius law
This method is reported by Mill [12]. It is assumed that there is a single process that
is thermally activated. Then according to Arrhenius law the degradation rate is
proportional to ( )RTWA /exp ? which implies that the degradation time is
proportional to ( )RTWA /exp . Then the plot of ( )timelog versus T/1 that should be
a straight line allows extrapolating the lifetime from one temperature to a different
temperature. Mill [12] reported that the application of this model to the hydrolysis of
polycarbonate showed good results. But caution is given concerning the applicable
range of temperature. Rana et al (1961) and Holland (1996) quoted by Barkatt [2]
reported as well the case of silicate glasses hydrolysis for which the method was
applied successfully. For these materials, the Arrhenius behavior was observed over a
broad range of temperature up to 90?C.
3.2.2 General method using extrapolation based on the temperature
The procedure in this method consists of performing tests in accelerated conditions as
well as in real service condition for a short time. The extrapolation factor F is
experimentally deduced by comparing the rates in accelerated and real service
conditions respectively represented by Ka and Kr.
Chapter Three: Modelling and Prediction Methods of Environmental Degradation, a Literature Survey
35
r
a
K
K
F =
The lifetime in real service conditions represented by T
(3.4)
r is now calculated from the
lifetime in accelerated conditions represented by Ta
??
?
?
??
?
?
==
r
a
aar K
K
TFTT
:
(3.5)
However, precaution should be taken to assure that the acceleration does not cause
the change of the controlling degradation mechanism. Collin et al [72] reported that
this method was used to predict moisture and temperature effect, within the range of
expected service environment for FRP used for structural applications in the aircraft
industry. It was also demonstrated that elevated temperature could be used to
reproduce the same reduction in Tg obtained by the moisture and consequently the
same effects on the mechanical properties of the material.
3.2.3 Accelerated or artificial humidification
The accelerated humidification allows the moisture content in the material to reach,
within a shortened time, the level it would normally reach after a long-term exposure.
This is achieved by exposing the material to higher percentage humidity. The
reduction of the exposure time is approximately 50% [43].
Artificial humidification consists of using physical models whereby the real exposure
conditions are physically simulated in a laboratory so that to obtain the same moisture
content as in real service conditions. This method was also used to predict the
moisture distribution over time. Springer [73] reported on the simulation of the
environmental humidity and temperature cycle allowing to calculate the moisture
content and distribution of a FRP for twenty years service life of an aircraft.
Chapter Three: Modelling and Prediction Methods of Environmental Degradation, a Literature Survey
36
Physical model are extensively used nowadays through environmental chambers.
This is applied not only for humidity but also for all the remaining environmental
agents.
3.2.4 Limitation of accelerated methods
According to Rana et al (1961) and Holland (1966) quoted by Barkatt [2],
extrapolation based on Arrhenius law holds only over a limited range of temperatures.
A recent review by Celina [74] shows the considerable limitation of this law in many
cases due to two competing processes. Similarly, studies reported by Prian and Barkat
[40] show the non applicability of this method in many cases due to supra or
sublinear kinetics arising during the degradation. The most important limitation faced
is the incontrollable deviation of the kinetic curve after a certain period of exposure
due to accelerated conditions. This occurs when the material reaches its glass
transition temperature Tg
and at the swelling point during moisture diffusion.
Moderate temperature increase causes also the material post-curing. The extent of
cross-linking caused by the post-curing affects the diffusion process and consequently
the degradation kinetic also. Another complication noted is related to the chemical
reaction products in course of degradation. These products modify the environment
composition affecting variables such as pH and chemical media reactivity.
An important aspect is that related to the translatability of laboratory test conditions
to real services environment [3]. White and Turnbull [1] underlined the difficulty to
correlate accelerated conditions used in the laboratory with the real service
environment conditions. From their survey of the literature, they stated that there was
a general agreement that no correlation exists between natural and artificial
weathering, and these views were even expressed in some standards. However, an
interesting view is this one expressed by David and Sims quoted by White and
Turnbull [1]. They said: ? Accelerated tests should only be considered therefore as
Chapter Three: Modelling and Prediction Methods of Environmental Degradation, a Literature Survey
37
giving a rough indication of the relationship between natural and artificial weathering
?[Artificial weathering] is neither as perfect as its disciples claim nor as useless as
its detractors state?The degree of correlation between natural and artificial ageing
seems to be inversely related to the degree of acceleration?.
Obviously as one can conclude, there was controversy denoting that the subject was
not yet correctly elucidated and that much investigation still was needed.
3.3 Standard Test Methods
In industry, some practical methods have been standardized and allow providing
qualitative indications to serve only as practical guide for the assessment of material
corrosion resistance. The ASTM C581-74 provides the standard method of assessing
the chemical resistance of thermosetting resin used in glass-reinforced structures. The
test procedure includes immersion of samples into the corrosive liquid, visual
inspection of the sample surface for colour change, pits, cracking, loss of gloss,
etching, and mechanical test for hardness, flexural strength, and modulus. The
inspection is conducted in regular intervals during the exposure that may last up to
180 days or a year. The obtained results may be plotted to check for kinetic trends.
Such evaluation, as said previously, provides a rough qualitative indication of the
material chemical resistance as ?no attempt is made to incorporate into the method all
the various factors which may enter into the serviceability of a glass fiber reinforced
resin structure when subjected to chemical environment.? (ASTM C581-74).
3.4 Analytical Methods
A major aspect of the problem concerns analytical methods for environmental
degradation effects quantification. The ASTM Standard Test Methods for Chemical
resistance of Thermosetting Resin used in glass fiber reinforced structures, ASTM
C581-74 for instance, resorts to mechanicals properties measurement for effects
quantification. Maier et al [75] has demonstrated that utilization of mechanical
Chapter Three: Modelling and Prediction Methods of Environmental Degradation, a Literature Survey
38
properties for effect quantification, requires a drying step during which the material
undergoes supplemental transformations in such a way that the results are subject to
considerable uncertainty.
In fact, a number of methods are found in the literature. Apart from methods using the
direct measurement of the mechanical material properties, the alternate way consists
of monitoring chemical or physical transformation occurring in the material because
of the degradation. Some example are provided by vibrational methods such as
Raman spectrometry, Infrared spectrometry [76, 77] and Nuclear Magnetic
Resonance (NMR). Likewise, these analytical methods like electron-scanning
microscopy (ESM), thermo gravimetric analysis (TGA) or differential scanning
calorimetry (DSC) have been abundantly used for monitoring of chemical structure
changes. But the use of these methods is validated only by the existence of known
good correlation between the monitored change and the material mechanical
properties.
3.5 Conclusion
The design, selection and economic evaluation of FRP material require the
availability of a reliable method for long-term environmental effects quantification.
Modelling of environmental degradation have been hindered by the complexity of the
process and the resulting difficulty to take into account all the processes involved and
their interactions in a comprehensive model. Modelling efforts, for the most part, are
limited to partial models based on a single mechanism dominating the whole process.
Such models are mostly based on moisture and/or on temperature effects and are
empirical. No comprehensive models or viable prediction method is yet available.
The prediction methods used are based on the assumption that dominating
mechanism is thermally activated. However, application of this method has been
hindered by irregular kinetics caused by transformations occurring in the material in
the course of acceleration. Irregular kinetics were observed because no method was
Chapter Three: Modelling and Prediction Methods of Environmental Degradation, a Literature Survey
39
yet defined to take comprehensively into account all the processes involved in the
degradation. The literature shows also that another unresolved issue is related to the
translation of laboratory test results into the real service conditions.
.
40
4 METHODS TO REPRESENT THE REAL SERVICE ENVIRONMENT
IN LABORATORY
One major unresolved issue in predicting FRP environmental degradation is that
related to the translation of real service conditions into laboratory. The main
problem results from the difficulty to reproduce the complex variability of the real
service environment in laboratory. It also results from the difficulty to correlate
accelerated test conditions to the real conditions. In addition, an effective
modelling of the degradation process requires a reliable method to take into
account the complex variability of the environment that causes the degradation.
This chapter demonstrates a simplified method to represent the variability of the
real service conditions into laboratory to allow for an effective modelling of the
degradation process and accelerated predictive tests.
4.1 Constant Environment Models
4.1.1 The model based on statistical control charts
The concept suggested in this subsection is based on the fact that the variability of
a parameter is generally represented by an average value with a variation range
determined by an absolute error. Alternatively, the variability of a process is
expressed in terms of a statistical control chart determined by constant control
limits [78].
Therefore, the variability of any process or variable is expressed in terms of
constant limits that fix the variation range.
Chapter Four: Methods to Represent the Real Service Environment in Laboratory
41
This implies that, if a defined relation exists between the variable and its effects,
then the variation limits or the statistical control limits of the effects can also be
exactly determined.
Therefore, effects can be determined in a range that corresponds exactly to the
variability of the variable.
Let fj jbe an environmental factor. is an integer such that nj ??1 and n
represents the number of environmental factors in the service environment. The
service environment may comprise various environmental factors such as
temperature, moisture, chemicals (oxygen, ozone, acids, etc...), energetic
radiations (UV rays, nuclear radiations or other artificial radiations), and different
types of environmental stresses. Environmental stresses are of several kinds: the
hygrothermal stresses from humidity and temperature cycles, erosive stresses
from friction with circulating liquid in pipes or tanks or friction with wind and
dust, stresses from rainfall impact, and vibration stresses in the vicinity of
industrial engines. However, in this analysis all of these environmental variables
will be represented into only five categories including: environmental chemical
(C0), temperature (T), ultraviolet rays (IUV
nenv
?
), moisture (?m) and hygrothermal
stresses ( ).
The environmental factor jf is a continuous function of time (except for special
cases such as explosion or fire). Its variation over the time can be represented by a
statistical control chart determined by the upper control limits
UCLj
f and the lower
control limits
;LCLj
f .
Let ( )jd fE be the environmental degradation resulting from . If ( )jd fE is a
defined function of , then the control limits of the degradation correspond
respectively to ( )
UCLjd
fE and ( )
LCLjd
fE . Therefore, the control limits of the
degradation can be determined in laboratory in a constant environment defined by
jf
jf
Chapter Four: Methods to Represent the Real Service Environment in Laboratory
42
and .
The above means that the real service environment can be reliably reproduced in
laboratory in terms of constant environment that express exactly the variability of
environmental factors.
However, when substituting a variable environment by a constant one, care must
be taken to compensate the fatigue effect caused by temperature and humidity
cycles. To this end, the material exposed to a constant environment must
additionally and in the same time, be subjected to cyclic stresses equivalent to
those caused by the temperature and humidity cycles. The amplitude of the cyclic
stress in laboratory is also determined by the upper and lower control limits of the
cyclic stress in the real service environment.
4.1.2 Model based on the mean value
The present concept stems from the need to manage the complex variability of the
service environment when modelling the degradation process. The idea consists of
replacing a variable environment by a constant one that produces the same effect
on the material for the same exposure duration. To this end, environmental effect
is treated in terms of mass and energy transfer. The constant environment is
required to produce identical amount of work implying the same amount of
transferred mass and energy, at a comparable rate (power) during the same
exposure time.
The total work an environmental factor achieves in transforming the material
during degradation depends on the amount of transferred mass and energy. The
latter is a function of the environmental factor magnitude and the exposure
duration. Effect of the environmental factor magnitude over the exposure time is
expressed as an integrated value of over the time. So, if ( )jenv fW represents
the transferred mass and energy under the environmental factor , then:
UCLj
f
;LCLj
f
jf
jf
Chapter Four: Methods to Represent the Real Service Environment in Laboratory
43
( )dttfkfW
t
t
jjenv ?=
2
1
0)( (4.1)
where 0k represents a constant and t is the time.
Equation (4.1) is actually a common expression of well-established laws that
govern energy and mass transfer in materials [79, 80]. It corresponds respectively
to the Newton?s law for calorific flux, Fick?s law for moisture or chemical
solution flux and the definition of UV ray intensity (see Appendix A).
If is a continuous function over the time interval 12 ttt ?=? , then it can be
represented by a constant value denoted jcf and determined by the theorem of
mean values for integration [81]. Resulting from that theorem, the mass or energy
transferred by over the time interval is equivalent to that
transferred by the constant over the same time interval. This is shown here
below.
The theorem of mean value for integration is expressed as [81]:
( )cjjc tff =? = constant, such that 21 ttt c << and ( ) tfdttf jc
t
t
j ?=?
2
1
Taking account of equation (4.1), this implies that
( ) ( ) ( )jenv
t
t
jjc
t
t
jcjcenv fWdttfktfkdtfkfW ==?== ??
2
1
2
1
000 (4.2)
An experimental illustration of equation (4.2) is provided in section 4.2 for the
case of storage tanks or pipes exposure conditions. This equation implies that all
environmental histories can be modelled as a continuous sequence of constant
environmental states comprising constant environmental factors. A useful
application of this conclusion is shown in the next chapter (section 5.9.2)
jf
jf 12 ttt ?=?
jcf
Chapter Four: Methods to Represent the Real Service Environment in Laboratory
44
4.2 Experiment
Samples of polyester composite laminate were exposed to an environment
comparable to a pipe or storage tank service conditions. The experiment was
based on thermal energy and mass transfer. The thermal energy and mass
transferred under a variable source of temperature was compared to that
transferred under an equivalent constant temperature source. It was considered
that, the humid environment in a storage tank or pipe, is constant (laminate
entirely wet in a liquid). Thus, only the temperature variation was considered. The
material used is shown in Table 4.1.
Table 4.1: Materials
LAMINATE RESIN FORMULATION
Composition Resin/fibre/resin Resin
Orthophtalic
polyester Crystic
196
Glass fibre WR 400 g / m Resin (phr) 2 100
Vf
AVG
(%)
39 Catalyst (phr) 2
StD 0.6 Accelerator (phr) 0.6
Thickness (mm) 3.0 ? 0.1 Curing cycle
24h/ 25o
3h/ 80
C,
o
Void (visual)
C
No voids
4.2.1 Experimental procedure and results
Experiment 1: heat flow across a reference point
1. The laminate sample was exposed to a variable temperature source. The
variation in temperature at a reference point inside the laminate was measured
by mean of a thermocouple. In the first experiment, the thermal energy was
allowed to flow across the reference point (Figure 4.1) and the transferred heat
was measured as an integrated heat flow across the reference point over the
exposure time.
Chapter Four: Methods to Represent the Real Service Environment in Laboratory
45
Figure 4.1 Test specimen for heat flow
2. The source temperature and the temperature inside the laminate were recorded
respectively as TSV and TA
V where V stands for variable conditions. Results
are presented in Figure 4.2.
Figure 4.2 Temperature variations under variable source
3. An interval of time was chosen from zero to 20 minutes and the equivalent
constant temperature was determined for that interval according to the
theorem of mean value (Figure 4.3).
0
10
20
30
40
50
60
0 5 10 15 20 25 30
Te
m
pe
ra
tu
re
(?
C
)
Time (minutes)
TVA1 TVA2 TVA3 TVA4 TVS1 TVS2 TVS3 TVS4A1V A2V A3V A4V TS1V TS2V S3V TS4V
S
TA
S: Temperature source
A: reference point inside the laminate
TS: source temperature
TA: temperature inside the laminate
: Laminate sample
: Thermo conductive material
: Thermal insulator
--- : thermocouple wires
A
TS
Chapter Four: Methods to Represent the Real Service Environment in Laboratory
46
Figure 4.3 Determination of the equivalent constant temperature
4. The laminate was conditioned in such a way as to obtain the same initial
temperature as under the variable source and then exposed to the constant
temperature source. The temperature inside the laminate was recorded for the
same time interval and the very same reference point as for the variable
temperature source. Source and reference point temperatures are respectively
denoted TSC and TA
C where C refers to constant conditions (Figure 4.4).
Figure 4.4 Temperature variations under constant source
5. The integrated heat flow under variable temperature source was compared to
that obtained under the equivalent constant temperature source (Figures 4.5
and 4.6, Table 4.3). It is shown that, comparatively to the variable source, the
constant source creates exactly the same integrated heat flow at the reference
point in the material, over the same time interval.
0
10
20
30
40
50
60
0 5 10 15 20 25
Te
m
pe
ra
tu
re
(?
C
)
Time (seconds)
Variable source from zero to 20 minutes
Equivalent constant temperature
fjc=??t fj(t)dt /?t
0
5
10
15
20
25
30
35
40
0 2 4 6 8 10 12 14 16 18 20 22
Te
m
pe
ra
tu
re
(?
C
)
Time (minutes)
TCA1 TCA2 TCA3 TCA4 TCS1 TCS2 TCS3 TCS4TA1C TA2C A3C A4C TS1C TS2C S3C S4C
Chapter Four: Methods to Represent the Real Service Environment in Laboratory
47
Figure 4.5 Comparing average variations in temperature at the reference point
Table 4.2: Integrated heat flow 1
Test
1 2 3 4 AVG StD
Variable source (?C.S) 26.3 28.1 24.0 24.3 25.7 1.9
Constant source (?C.S) 25.5 26.6 25.7 26.3 26.1 0.5
Figure 4.6 Comparing integrated heat flows at the reference point
1Integrated heat flow was measured as Q/k0 = ? ?
20
0
Tdt according to Fourrier law for heat conduction.
k0=Ar.h/l where Ar, h, and l respectively stand for the transmission surface area, the coefficient of thermal
conductivity, and the laminate thickness.
0
5
10
15
20
25
30
35
40
0 2 4 6 8 10 12 14 16 18 20 22
Te
m
pe
ra
tu
re
(
?C
)
Time (minutes)
TCA
TVA
TAC
TAV
0
5
10
15
20
25
30
1 2 3 4 AVG
Q
/k
0
(?
C
.s
ec
on
ds
)
Test number
Heat flow under variable source Heat flow under equivalent constant source
Chapter Four: Methods to Represent the Real Service Environment in Laboratory
48
Experiment 2: Variation in temperature at the reference point
1. The laminate sample was exposed to a variable temperature source. The
variation in temperature at a reference point inside the laminate was
measured by mean of a thermocouple as in the first experiment. In this
experiment, the heat flow was stopped at the reference point by mean of a
thermal insulator (Figure 4.7). The transferred thermal energy was
measured by the accumulated heat at the reference point. Following the
calorimetric equation TsMQ ?=? .. , where Q, M, and s respectively
stand for the heat, mass heated, and specific heat of the sample [82], the
accumulated heat is expressed as a variation in temperature denoted ?T.
Figure 4.7 Test specimen for the change in temperature at the reference point in
the material
2. The same steps 2 to 4 as in the first experiment were repeated. The
working time interval was chosen from 5 to 15 minutes. The Figures 4.8
and 4.9 show respectively the variation in temperature under variable
source and under equivalent constant source.
TS
TA
S
S: Temperature source
A: reference point inside the laminate
TS: source temperature
TA: temperature inside the laminate
: Laminate sample
: Thermal insulator
--- : thermocouple wires
A
Chapter Four: Methods to Represent the Real Service Environment in Laboratory
49
Figure 4.8 Temperature variations under the variable source
Figure 4.9 Temperature variations under the equivalent constant source
3. The change in temperature at the reference point under variable
temperature source was compared to that obtained under equivalent
constant temperature source, for the same time interval (Figure 4.10). It is
shown that, under the equivalent constant source, the change in
temperature at the reference point is the same as under the variable source.
This denotes that the amount of transmitted energy is preserved.
0
10
20
30
40
50
60
0 2 4 6 8 10 12 14 16 18
T
em
pe
ra
tu
re
(?
C
)
Time (minutes)
TVA1 TVA2 TVA3 TVA4 TVS1 TVS2 TVS3TA1V A2V TA3V A4V S1V TS2V TS3V
0
5
10
15
20
25
30
35
40
45
0 1 2 3 4 5 6 7 8 9 10 11
Te
m
pe
ra
tu
re
(?
C
)
Time (minutes)
TCA1 TCA2 TCA3 TCA4 TCA5 TCA6 TCSTSCA1C T 2C TA3C TA4C TA5C TA6C
Chapter Four: Methods to Represent the Real Service Environment in Laboratory
50
Figure 4.10 Comparing the change in temperature under variable and constant
temperature source
Experiment 3
1. A dozen of polyester laminate samples were moisturized in distilled water
under variable temperature source and one other dozen of the very same
polyester laminate were moisturized under an equivalent constant
temperature source (Figure 4.11).
Figure 4.11 Variation of temperature during moisturization
0
10
20
30
40
50
60
0 5 10 15 20
Te
m
pe
ra
tu
re
(
?C
)
Time (minutes)
TVA TVS TCA TCSTACTAV SCTSV
0
10
20
30
40
50
60
0 20 40 60 80 100
T
em
pe
ra
tu
re
(?
C
)
Time (minutes)
Variable source temperature
Constant source temperature
?T=12?C
Chapter Four: Methods to Represent the Real Service Environment in Laboratory
51
2. Moisture diffusion under the two different temperature sources was
compared (Figure 4.12 and 4.13)
Figure 4.12 Scattering of moisture content after exposure under variable and
constant temperature sources
Figure 4.13 Comparing moisture content after exposure
The above results (Figures 4.12 and 4.13) indicate that the change from the
variable source to the equivalent constant one does not affect moisture diffusion.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 1 2 3 4 5 6 7 8 9 10 11 12 13
M
oi
st
ur
e
co
nt
en
t (
%
)
Sample number
Moisture % under variable temperature
Moisture % under equivalent constant temperature
1.42 1.40
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Under variable source Under constant source
M
oi
st
ur
e
(%
)
Chapter Four: Methods to Represent the Real Service Environment in Laboratory
52
4.2.2 Discussion
Experimental results show that it is possible, for suitably chosen time interval, to
replace a variable source of temperature by an equivalent constant source that
transmits exactly the same amount of energy. It is also shown that such
transformation from the variable source to the constant one does not affect the
diffusion results. Therefore, the amount of transmitted mass and energy is
conserved.
However, in order to insure comparable transformation or degradation of the
material, the transmission rate or power under the constant source is also required
to be in the same range than that generated under the variable source. This means,
their difference should be negligible.
In the above experiment, the transformation from the variable source to the
constant one modifies the distribution of the difference in temperature between the
material and the environment (Figure 4.14).
Figure 4.14 Temperature difference between the material and the environment in
the course of exposure
0
2
4
6
8
10
12
14
16
18
0 5 10 15 20
Te
m
pe
ra
tu
re
d
iff
er
en
ce
(?
C
)
Exposure time (minutes)
under variable source
under constant source
Chapter Four: Methods to Represent the Real Service Environment in Laboratory
53
Subsequently, the distribution of transmission rate or transmission power is also
modified as the latter depends proportionally on the temperature difference
according to Fourrier law for heat conduction [80] (Figure 4.15).
Figure 4.15 Transmission power in the course of exposure2
In Figure 4.15, the maximal difference in transmission power is represented by
maxP? . The figure shows that maxP? is proportional to the length of the time
interval. Then, it is trivial that maxP? can get negligible if the time interval is
sufficiently reduced.
Therefore, a variable temperature source over a determined time interval can
theoretically be represented as constant temperature that produces comparable
degradation over the same time interval.
2 Expression of transmission power is obtained from the calorimetric law: ?Q = M.s.?T by
differentiation, as the value of k0 in the Fourrier equation is not determined. s and M respectively
stand for the specific heat and the mass of the heated material
0.4
0.9
1.4
1.9
2.4
4 5 6 7 8 9 10 11 12 13 14 15 16
M
-1
.s
-1
.d
Q
/d
t (
?C
.m
in
ut
e-
1 )
Time (minutes)
Under constant source
Under variable source
?Pmax
Chapter Four: Methods to Represent the Real Service Environment in Laboratory
54
The above, directly results from equation (4.1) and therefore, can be extended to
the remaining environmental factors. This means that the real service environment
history can be determined as a theoretical succession of constant environmental
states based on the theorem of the mean value. The time intervals can sufficiently
be minimized in such a way as to obtain a discrete pattern as the one represented
in Figure 4.16b. In such a pattern, the difference in transmission power is
minimized and effect of cycling temperature or humidity over the overall time
interval is preserved. A useful application of this concept is presented in the next
chapter.
Figure 4.16 Schematic representation of a transformation from a continuous
variable environment (a) to a succession of constant environments
(b)
Transformation time intervals are sufficiently reduced to minimize the difference in transmission
power and preserve the cycling effect over the overall exposure time
4.3 Conclusion
A simplified method to represent the variability of the real service environment
into laboratory has been demonstrated. The method is based on the statistical
control chart and the use of a mean value. It is shown that, the real service
environment can be reliably reproduced in laboratory in terms of constant
environment that expresses exactly its variability. The constant environment is
provided by the statistical control limits of the real service conditions.
0 5 10
Time
a)
0 5 10
Time
b)
Chapter Four: Methods to Represent the Real Service Environment in Laboratory
55
It is also shown that, the environment history can be modelled as a sequence of
constant environments by replacing variable factors by constant values over a
determined time interval. Experimental observation based on the storage tank
model indicates that the amount of transmitted mass and energy is preserved. The
transmission power also is preserved if the time interval is sufficiently reduced.
The above models provide a method to manage the complexity relating to the
variability of real service conditions. This constitutes a solution to issues relating
to the translatability of laboratory test results to real service conditions.
56
5 THE CHEMICAL AND PHYSICAL DEGRADATION MODEL
Modelling and prediction of the environmental degradation of fibre reinforced
plastics has been hindered by the complexity of the process. Published works are
limited to effects and mechanism characterization or partial models, most of the
time empirical.
In this chapter an analytical approach is presented which resolves the degradation
process into only three components: the chemical link density variation, the
cohesion force variation, and the stress state modification. The first two are
referred to as chemical and physical degradation.
The present chapter demonstrates that in a constant environment an exponential
function correlates the chemical and physical degradation of the material to the
environmental factors. It is also shown that the chemical and physical degradation
rates in a real service environment can be determined in a laboratory in a constant
environment based only on the variation of the chemical link density. The
suggested model is a mathematical function logically derived from material
science theories and expresses a qualitative relation between the material
degradation and environmental factors.
5.1 The modelling approach
In order to resolve the complexity of the environmental degradation process, the
analysis relies on the fact that, irrespective of its cause, all environmental
degradation results only from one of the following three sub-processes:
? Chemical degradation, corresponding to the modification of the density of
chemical links caused either by chemical attack, thermal attack or ultra violet
rays.
Chapter Five: The Chemical and Physical Degradation
57
? Physical degradation, corresponding to the deterioration of cohesive forces or
plasticization caused by either moisture absorption or temperature increase.
? Mechanical degradation, corresponding to the modification of the stress state
caused by temperature cycles or humidity.
The effects of these three sub-processes are only of two kinds. Firstly, the material
stiffness is altered. This results from the modification of chemical link density and
from the variation of cohesive forces. These effects will be referred to as
?chemical and physical degradation?. Cases where environmental factors cause
crosslinking (post-cure by UV rays or temperature) or stiffen the material are not
seen as degradation. This is because the mechanical strength of the material is not
negatively affected. These cases are consequently not considered in the following
development. Secondly, the stress state is modified as a result of hygrothermal
stresses. The ultimate effect of these two processes is the transformation of the
material rheology. Therefore, material rheology provides the common parameter
allowing the integration of the two processes into a common mathematical
relation which provides the mathematical model. The analysis focuses therefore
on the rheological changes in the material. This view is schematically presented in
Figure 5.1.
Figure 5.1 Environmental degradation process
Temperature T
Moisture, ?m
Chemicals, C0
UV Rays, IUV
Variation in
chemical link
density
Hygrothermal
stresses
Physical degradation Change in stress state
Effects
Environmental
factors
Degradation
Change in stiffness
Rheology
Chemical degradation
Variation in
cohesive force
Stress corrosion
Chapter Five: The Chemical and Physical Degradation
58
For the sake of methodology, this chapter deals only with the chemical and
physical degradation. The method consists of deriving a qualitative mathematical
relation between the degradation rate and the environmental factors including the
chemical concentration, the moisture, the diffusion coefficient, the temperature,
and the UV rays. The qualitative mathematical relation contains theoretical
parameters that can be determined experimentally. At first, based on material
science theories, the environmental factors are mathematically correlated to the
material rheology. Secondly, the chemical and physical degradation is expressed
as a function of the material rheology. Then, based on the mathematical relation
between the material rheology and the environmental factors, the chemical and
physical degradation is expressed as a mathematical function of the environmental
factors to derive the mathematical model. The validity of the mathematical model
is measured by the degree of correlation between the variation of the degradation
index calculated from the model and the variation of the mechanical strength of
the material measured experimentally over the course of the degradation.
5.2 Definitions
Following the above description, the alteration of the material stiffness during the
chemical and physical degradation is caused by the variation in chemical link
density and the variation in cohesive forces. Indices Ld, Cf, Ed
, are now
introduced and are respectively the chemical link degradation index, the cohesion
force deterioration index, and the stiffness degradation index.
5.2.1 Degradation index of the chemical link density: L
d
It is assumed that for a given material, if the cohesive forces are held constant,
only the modification of chemical link density determines the degradation. The
variation in mechanical properties is then directly related to the variation in
chemical link density and thereby the variation in diffusivity. Theoretically, the
mechanical resistance of a given material may therefore be expressed as a given
critical chemical link density that should assure the material structure will hold
Chapter Five: The Chemical and Physical Degradation
59
against breaking stresses. The index of the degradation of the chemical link
density Ld
defines the material degradation due only to chemical link breakage
and is an increasing value over the course of the degradation.
The chemical link density index Ld is modified only by attack from chemicals,
UV rays and temperature, respectively, represented by symbols ch, UV and Th.
Mathematically the total variation of Ld
is the sum of its variation due to the
effects of chemical agents, UV rays and thermal attack, expressed as follows:
ThdUVdchdd dLdLdLdL ++= (5.1)
Variation in the rate of Ld
is obtained by derivation of equation (5.1) as follows:
dt
Thd
Th
L
dt
UVd
UV
L
dt
chd
ch
L
dt
dL dddd )(
)(
)(
)(
)(
)( ?
?
+
?
?
+
?
?
= (5.2)
5.2.2 Degradation index of the cohesive forces: C
f
For a given material, assuming that the chemical link density is held constant, the
degradation is directly related to the decrease of cohesive forces and consequently
the reduction in the mechanical strength. So, theoretically, the mechanical
resistance of a given material may be expressed as a given critical cohesive forces
level that should assure the material structure will hold against breaking stresses.
The cohesive force degradation index defines the material environmental
degradation resulting only from the variation of cohesive forces. The index Cf
increases when the cohesive forces decrease.
The variation of the cohesive force results only from diffused moisture and from
temperature variation. On this basis, the degradation rate of the cohesive forces
can be mathematically expressed as the sum of only two contributions arising
from the variation in the rates of the diffused moisture and temperature:
( )
( )
( )
( )
dt
Thd
Th
C
dt
md
m
C
dt
dC fff
?
?
+
?
??
?
= (5.3)
Chapter Five: The Chemical and Physical Degradation
60
where ?m represents the diffused moisture.
5.2.3 The degradation index of the material stiffness: E
d
The index Ed
represents the degradation of any material property such as tensile
strength, modulus, etc. The chemical and physical degradation of the material
stiffness is the sum of two contributions including chemical link degradation and
cohesive force degradation. This can be mathematically expressed as follows:
fdd CpLpE 21 +=
(5.4)
The factors p1 and p2
dt
dC
p
dt
dL
p
dt
dE fdd
21 +=
are weighting factors relating the contribution of the
chemical link density and cohesive forces to the degree of degradation. The
stiffness degradation rate can then be deduced from equation (5.4) by
differentiation as follows:
(5.5)
The next step in this analysis is aimed at the determination of each of the terms in
the second part of the equation (5.5). It is intended to express these terms as a
function of environmental factors. To this end, the following section introduces
theoretical assumptions based on material science theories.
5.3 Material Rheology as a Function of Chemical Link Density
It is assumed that the material rheological state is linearly correlated to a power of
the chemical link density. This is mathematically expressed as follows:
( )ad
Ld
Lk1
1
=?
?
?
?
?
?
?
(5.6)
In equation (5.6), ? represents a material rheology index such as the viscosity or
stiffness; in which case its inverse expresses the material compliance and a is an
empirical positive constant. In the rest of this text, the symbol ki represents a
positive constant and subscripts following parentheses mean ?due to?. Thus,
Chapter Five: The Chemical and Physical Degradation
61
equation (5.6) gives the material rheological state due to the degradation state of
the chemical link density.
The equation (5.6) is a logical assumption asserting that the stiffness of a solid
polymer is proportional to its chemical link density or is inversely proportional to
the index of chemical link degradation. This means that a specific reduction of the
chemical link density results in a corresponding specific reduction of the material
stiffness and that the reduction is affected by factors such as molecular chain
length and molecular spatial configuration. This kind of correlation provides the
basis of rheometric measures. For instance, in rubber vulcanization, the crosslink
level or molecular weight distribution is linearly correlated to the shear torque
resistance [83]. A similar principle was also used in the equation of Mark-
Houwink [84] for the determination of polymer molecular weight in dilute
solution. The study of the melt viscosity of polymers has also established the same
kind of correlation between the molecular weight and the polymer melt viscosity.
In this case, the exponent of the molecular weight varies from 1.5 to 3.5 [85]. The
experimental verification of this relation for the polyester resin shows that the
value of a in equation (5.6) is 1.0 (see subsection 5.4.3).
5.4 Material Rheology as a Function of Moisture Content
It is assumed that the material stiffness is inversely proportional to a power of the
moisture content and this assumption may be mathematically expressed as
follows:
b
moist
m
k ??
?
?
??
?
? ?
=?
?
?
?
?
?
?? ?2
1
(5.7)
In this equation ?m is the mass of the diffused moisture measured by the variation
in the sample weight over time and ? is the specific mass of the diffused liquid.
This assumption results from the consideration that the diffused moisture creates
additional separation space between the polymer molecules. The intermolecular
distance r as well as the dielectric parameter and consequently the intermolecular
Chapter Five: The Chemical and Physical Degradation
62
attraction forces, Fcf,
?
m
V
?
=?
are thus modified. The additional separation space can be
expressed as a volume determined by the moisture content as follows:
(5.8)
Considering Van der Waal?s law for cohesive forces, one can deduce the
following equation and hence equation (5.7?) where b is a constant depending on
the material:
( ) ( )
3/
43/
43
d
ddcf
m
k
V
k
r
k
F
?
??
?
?
??
?
? ?
=
?
=
?
=?
?
(5.9)
b
moistcfmoist
m
k
F
k
??
?
?
??
?
? ?
=?
?
?
?
?
?
?
?
?
=?
?
?
?
?
?
?? ?6
51 (5.7?)
The equation (5.7) assumes that the laminate has been plasticized by the diffused
liquid but it can also be applied to the case where the penetrating moisture forms
clusters as noted by Marsh et al [18]. In this case, the resistance to the diffusion
process is proportional to the interaction area between the diffused mass and the
laminate. This area is equal to the external surface area of the diffused volume and
it can also be expressed as a power of the volume.
5.5 Material Rheology as a Function of Temperature
It is assumed that the material stiffness is inversely proportional to a power of the
temperature variation and this assumption may be mathematically expressed by
the following equation where ?T is the temperature variation and c is a constant.
( )c
T
Tk ?=?
?
?
?
?
?
?? 7
1
(5.10)
The temperature variation affects the material rheology by modifying the thermal
kinetic energy of molecules and consequently the level of segmental motions
along with the activation energy for flow. This effect is manifested by variation of
the material free volume. Common experimental measurement of this effect
shows a linear correlation between the temperature and the free volume with a
Chapter Five: The Chemical and Physical Degradation
63
slope change at the glass transition temperature [85, 86]. This can be expressed as
follows:
Tkv ?=? 8 (5.11)
In this equation ?v is the free volume that can also be expressed in terms of
intermolecular radius, r, as shown in equation (5.9). Then Van der Waal law of
cohesive forces can be written in terms of free volume and in terms of temperature
as follows:
( ) ( ) ( ) 3/
10
3/
93
dddcf T
k
v
k
r
k
F
?
=
?
=
?
=? (5.12)
Equation (5.12) can also be written in terms of material rheology as follows:
( )c
TTcf
Tk
F
k
?=?
?
?
?
?
?
??
=?
?
?
?
?
?
?
?
? 11
5 1 (5.13)
where c is a positive constant depending on the material. This equation accounts
only for the physical degradation. Chemical degradation including thermolysis
and post-curing is already considered in equation (5.6).
5.6 Chemical Concentration as a Function of Material Rheology and
Diffusion Effect
A portion of a pipe or storage tank wall is shown in Figure 5.2. The wall laminate
is exposed to a chemical denoted ch, and to moisture at a temperature T. l is the
laminate thickness. C0 and Cl
are respectively the chemical concentrations at the
internal and external surfaces of the laminate.
According to Fick?s law, the concentration Cch
l
C
DJ Ch
?
?
?=
is a function of the diffusive rate J
which can be expressed as
(5.14)
Integration of equation (5.14) (see Appendix B) leads to
Chapter Five: The Chemical and Physical Degradation
64
l
CC
DkC lch
?
= 012 (5.15)
Figure 5.2 Portion of pipe wall
Since C0 >> Cl, one can assume that C0 - Cl ? C0
l
C
DkCch
0
12=
. Equation (5.15) can therefore
be written as
(5.16)
Analogically to the Stokes-Einstein equation [87, 88], it is assumed that the
diffusion coefficient and the material rheology are correlated in the following
manner:
?
=
T
kD 130 (5.17)
Equation (5.17) represents the effect of material rheology on diffusion and
expresses the obvious fact that in a solid material moisture diffusion increases
with temperature and decreases with an increase in viscosity. However, the
diffusion coefficient is also related to the variation in the thermal kinetic energy of
the diffusing molecules according to the Arrhenius law [8, 21, 26] as follows:
RT
W
RT
W DD
e
T
keDD
??
?
== 130 (5.18)
Barrier coat
l
C0 Cch Cl
?
UV rays
Iabs
?m, T
Inner surface Outer surface
Chapter Five: The Chemical and Physical Degradation
65
In equation (5.18) DW is the activation energy of diffusion. The relation between
the concentration chC and the material rheology is obtained by combining
equations (5.16) and (5.18):
RT
W
ch
D
e
l
CT
kC
?
?
= 014 (5.19)
This relation shows that the concentration of the diffused material inside the
laminate depends on its concentration at the laminate surface, on the temperature,
on the material viscosity at the given temperature and on the laminate thickness.
However, the material rheology is modified over the course of degradation. The
variation of material rheology arises from the variation of chemical link density,
dL , and from the variation of cohesive forces due to moisture and temperature. In
order to account for this effect, let the total differential of the material rheology be
given as a function of these three variables as follows:
( ) ( ) dTTmdmddLLd dd ?
?
?
?
?
?
?
?
?
+?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
=?
?
?
?
?
?
?
111
1
(5.20)
Integration of equation (5.20) gives the expression of the material rheology as
follows:
TmoistLd
?
?
?
?
?
?
?
+?
?
?
?
?
?
?
+?
?
?
?
?
?
?
=
?
1111
(5.21)
Equation (5.21) is now substituted into equation (5.19). The chemical
concentration is then given as a function of the material rheology as follows:
RT
W
TmoistLd
ch
D
e
TTT
l
C
kC
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+?
?
?
?
?
?
?
+?
?
?
?
?
?
?
= 014 (5.22)
Equations (5.6), (5.7), (5.10), and (5.22) provide qualitative relations between the
material rheology and the environmental factors. These relations provide the basis
for combining effects of these environmental factors into a single mathematical
relation. To this end, in the next section, the material rheology is related to the
Chapter Five: The Chemical and Physical Degradation
66
degradation rate. The degradation rate is subsequently related to environmental
factors, based on the material rheology.
5.7 Rheology Dependant Function of Degradation Rates
5.7.1 Chemical degradation rate as a function of material rheology
Considering again a portion of the pipe wall as represented in Figure 5.2, the
chemical reaction occurs between the polymeric matrix and the chemical reagent.
As the polymeric material constitutes the reaction medium, only the chemical
reagent concentration Cch
chch
ch
d Ck
t
L
=?
?
?
?
?
?
?
?
will determine the chemical reaction rate and the law of
chemical reaction rates can be expressed as
(5.23)
In equation (5.23) kch
RT
W
ch
ch
eAk
?
= 0
is the kinetic constant given by the Arrhenius law
(5.24)
In equation (5.24), Wch is the activation energy of the chemical reaction, R is the
ideal gas constant and A0
0
0
Ck
t
L
ch
ch
d =?
?
?
?
?
?
?
?
the frequency factor. There are two cases to be
considered. Firstly, the reaction at the material surface which is not influenced by
diffusion and the chemical degradation rate does not depend on the material
rheology. According to equation (5.23), the reaction kinetic can be expressed as
(5.25)
where the superscript 0 refers to the laminate surface. Secondly, the reaction
inside the laminate where the chemical concentration, and consequently the
chemical degradation rate, is a function of the material rheology, which can be
obtained by substituting equation (5.22) into equation (5.23):
RT
W
TmoistLd
ch
ch
d
D
e
TTT
l
C
kk
t
L ?
?
?
?
?
?
?
?
?
?
?
?
?
?
+?
?
?
?
?
?
?
+?
?
?
?
?
?
?
=?
?
?
?
?
?
?
? 0
14 (5.26)
Chapter Five: The Chemical and Physical Degradation
67
This equation covers in general all types of chemical attack where the process
comprises a diffusion step. These also include thermo-oxidation and photo-
oxidation. However, the thermal attack and the UV irradiation comprise also
thermolysis and photolysis respectively. These are energetic actions where bonds
are broken by imparting sufficient energy to electrons to pull them out of the
bond.
The thermolysis reaction can be schematized as follows:
CD + ?Hr
The rate of reaction is proportional to the thermal flux in the material [47]:
? C + D
r
T
Th
d
t
L
??
?
=?
?
?
?
?
?
?
?
3? (5.27)
In equation (5.27), ?T is the thermal flux, ?Hr is the thermal energy yield per
chain scission, and ?3 is a constant. For a given material at constant temperature,
the expression ?3?T/?Hr
is a constant. The case where temperature is variable is
irrelevant in this analysis as shown later in subsection 5.9.2.
Taking nh? as the photon energy yield per chain scission, the photolysis reaction
can similarly be schematized as
CD + nh?
The reaction rate is proportional to absorbed UV intensity I
? C + D
abs
?
?
nh
I
t
L abs
UV
d
4=?
?
?
?
?
?
?
?
:
(5.28)
Equations (5.27) and (5.28) express the chemical degradation rate as function of
environmental factors. No further transformation will be operated on these
equations because the goal is to express the degradation rate as function of
environmental factors. Now equations (5.25), (5.26), (5.27), and (5.28) are
substituted into equation (5.2) and the chemical degradation rate is expressed as a
function of materiel rheology as follows:
Chapter Five: The Chemical and Physical Degradation
68
RT
W
TmoistLd
chch
d
D
e
TTT
l
C
kkCk
t
L ?
?
?
?
?
?
?
?
?
?
?
?
?
?
+?
?
?
?
?
?
?
+?
?
?
?
?
?
?
+=
?
? 0
140
?
??
nh
I abs
r
T
43 +??
?
+ (5.29)
5.7.2 Physical degradation rate as a function of the material rheology
The determination of the physical degradation rate as a function of the rheology is
based on the fact that variation in the material rheology resulting from the change
in temperature and moisture, is proportional to the cohesive force:
( ) Tmoistcf kF ,15 ?= (5.30)
According to the definition given in subsection 5.2.2, Cf increases when the
cohesive forces are reduced. This means that a positive variation in Cf
( ) ( )??=?= dkFddC cff 15
corresponds to a negative variation in the cohesive forces as expressed in the
following equation:
(5.31)
The physical degradation rate as function of the material rheology is obtained by
derivation of equation (5.31) as follows:
( ) ( )
( )
( ) ( )
??
?
?
??
?
?
?
???
+
?
??
???
=
??
=
dt
dT
Tdt
md
m
k
dt
d
k
dt
dC f
1515 (5.32)
5.8 Degradation as a Function of Environmental Factors
The material rheology has been correlated to environmental factors (subsection
5.1.3). Equations (5.6), (5.7) and (5.10) are qualitative relations resulting from
materials science laws. Based on materials science laws, it has also been shown
that the variation of material rheology is a function of the chemical and physical
degradation rates (subsection 5.1.4). In the following subsection, the chemical and
Chapter Five: The Chemical and Physical Degradation
69
physical degradation rates are related to environmental factors, based on the
material rheology.
5.8.1 The chemical degradation rate as a function of environmental factors
The chemical degradation rate can be expressed as a function of environmental
factors by substituting equations (5.6), (5.7) and (5.10) into equation (5.29).This
leads to
( )[ ]
( )
RT
WW
Cb
dch
d
Dch
TeCTmLCk
dt
dL +?
+?++= 02100 ???
?
??
nh
I abs
r
T
43 +??
?
+ (5.33)
In equation (5.33) ?0 = k1k14 / l; ?1 = k2k14/ l?
b; and ?2 = k7 / l. These three
parameters ?0, ?1, ?2 and generally the symbol ?i
in the rest of the description, are
positive constants depending on the material type and environmental conditions.
All terms of equation (5.33) are positive and increasing functions of
environmental factors. This shows that the chemical degradation rate is an
increasing monotonic function of the environment.
5.8.2 Physical degradation rate as a function of environmental factors
In order to determine the physical degradation rate as a function of environmental
factors, equations (5.7) and (5.10) are substituted in equation (5.32) which gives:
( ) ( )
??
?
?
??
?
? ?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
=?
?
?
?
?
? ??=
?
dt
Td
k
dt
m
d
k
dt
d
k
dt
dC c
b
f
151515
?
(5.34)
By applying the chain rule of differentiation to the terms on the right side of
equation (5.34), one obtains
( )
( )
dt
dT
Tdt
md
mdt
dC
cb
b
f
1615
1
++
+
?
?
= ??? (5.35)
Chapter Five: The Chemical and Physical Degradation
70
5.9 The Model of Chemical and Physical Degradation
5.9.1 The mathematical model
In this section, the rate of chemical and physical degradation is determined as a
function of environmental factors to derive the mathematical model. To this end,
equations (5.33) and equations (5.35) are substituted into equation (5.5) as
follows:
( )[ ]
( )
r
TRT
WW
Cb
dch
d
Dch
TeTCmCLCCk
dt
dE
??
?
++?++=
+?
30201000 ????
( )
( )
dt
dT
Tdt
md
mnh
I
Cb
b
abs
15144
1
++
+
?
?
++ ???
?
? (5.36)
In equation (5.36), the environmental factors C0
, T, and ?m vary with time in a
way that is not always controllable. The exact solution of equation (5.36) requires
the determination of the time dependence function of each of the environmental
factors. It is not obvious that such a task may be achieved successfully and the
solution of the equation would be quite complex. Nevertheless, the problem can
be resolved by treating the environmental history as a sequence of constant
environments as explained in the next section.
5.9.2 The Physical and Chemical degradation in a constant environment
In chapter 4, it was shown that, the real service environment could be modelled as
a continuous succession of constant environmental states (section 4.2). Now
considering that concept, the above equation (5.36) is applied in one of the
constant environments and this leads to the following:
Exponential equation of the chemical and physical degradation
In a constant environment, equation (5.36) can be rewritten in a simple form as
?? += d
d EkTC
dt
dE
00 (5.37)
Chapter Five: The Chemical and Physical Degradation
71
where k and ? are constants defined by the following equations:
( )
RT
WW chD
e
p
k
+?
=
1
1
(5.38)
( )[ ]
( )
( )
( )
dt
md
mnh
I
TeTCmC
b
b
abs
r
TRT
WW
Cb
Dch ?
?
++
??
?
++?=
+
+?
15430201
??
?
?????
?
?
?
?
?
?
?
?
+?
?
?
?
?
?
?
?+ + c
b
C Tm
pkCT
dt
dT
T
11
8720016
????? (5.39)
Equation (5.37) was obtained by rewriting equation (5.4) as:
)(
1
2
1
fdd CpEp
L ?= (5.40)
and then by substituting equation (5.40) in equation (5.36), taking into account
that the environment is constant.
The rate of change of Ed
tkTCd e
dt
dE
00??=
can be obtained from the solution of equation (5.37). The
former is given by the exponential function below, correlating the chemical and
physical degradation to time t (see Appendix C):
(5.41)
Equivalence between indices Ld and E
In a constant environment, according to equation (5.5) and equation (5.35), the
degradation rate of the material stiffness is linearly correlated to the degradation
of the chemical link density as shown here below:
d
dt
dL
p
dt
dE dd
1= (5.42)
This implies that index Ld is equivalent to index Ed
. Practically, this means that in
a constant environment, the degradation rate of the material stiffness is
determined by the degradation rate of the chemical link density.
Chapter Five: The Chemical and Physical Degradation
72
Statistical control limits of the degradation rate
Since the degradation rate of the chemical link density is a monotonic ascending
function of the environment (see subsection 5.8.1), equation (5.42) implies that
the degradation rate of the material stiffness is also a monotonic ascending
function of the environment:
( ) ( ) ( )??
?
??
? ??? 1212 j
d
j
d
jj fdt
dE
f
dt
dE
ff (5.43)
Subsequently, as explained in chapter four, degradation rates measured at the
control limits of the environment correspond to the control limits of the
degradation rate. In the same way, the degradation rate obtained at the average
value of , corresponds to the average degradation rate.
This means that the control limits of the degradation rate and its average value can
be determined as simple exponential functions in constant environmental
conditions respectively determined by the control limits of the real service
environment and its average conditions.
5.10 Experimentation
The above analysis has provided a theoretical demonstration of the assertion that,
in a constant environment, the chemical and physical degradation of a polymeric
material follows an exponential law. Experimental evidence given in the present
section provides further validation. Experiments were designed in such a way as
to simulate storage tank or pipe exposure conditions and a range of measurement
methods were used. These include Raman spectrometry for chemical structure
changes, rheometry for storage modulus, and mechanical testing of tensile and
shear strength evolution during degradation. The material used was a polyester
laminate formulated in such a way as to be comparable to a standards barrier coat.
jf
Chapter Five: The Chemical and Physical Degradation
73
5.10.1 Material
The materials are selected in such a way as to represent a significant choice for the
local composite industry. The laminate formulation is presented in Table 5.1.
Table 5.1: Materials
RESIN
Experiment 1 Experiment 2 Experiment 3
Material Orthophtalic
polyester Crystic
196
Orthophtalic
polyester
NCS 901 PA
Orthophtalic
polyester NCS 901
PA
Resin phr 100 100 100
Catalyst(Butanox
M50) phr
2 2 2
Accelerator
(Crystic E) phr
0.6 - -
Curing cycle 24h/ 25oC, 3h/ 80o 24h/ 25C oC, 3h/ 80o 24h/ 25C oC, 3h/ 80o
LAMINATE
C
Composition Resin/fiber/resin Res/fib/res/fib/res/fib/res Res/fib/res/fib/ res
Glass fiber WR 27.4 g /m CSM 300 g /m2 CSM 300 g /m2
V
2
f
AVG
(%)
14 41 14
StD 1.16 1.52 3.1
Thickness
(mm)
AVG 0.21 2.89 2.00
StD 0.02 0.06 -
Void(visual) No voids No voids No voids
5.10.2 Lamination method
The method for making samples was chosen so as to reduce optimally all kinds of
variation between samples: minimal variation in thickness and weight, minimal
void content, and constant volume fraction. To this end, three methods were
experimented: Vacuum bag, hand lay up followed by short vacuum bag, and hand
lay up followed by compression in the mould.
In the first case, the procedure was as follows:
? Cut the fibre in such a way as it fits exactly into the mould and according to
the number of layers.
? Clean the mould carefully and apply the releasing wax.
? Lay down the fibre material in the mould and close the mould firmly.
Chapter Five: The Chemical and Physical Degradation
74
? Then the resin is sucked into the mould by means of a vacuum bag.
This required preparing the resin in such way as to get a long enough gelling time
to allow for a good spread of the resin. However, results were not good due to air
bulbs. The resin used was so thick that air bulbs retained during the mixing could
not be released before the gelling. The laminate contained too much voids. A
different procedure was then applied so that to evacuate air bulbs. Hand lay up
followed by a strong rolling, then by the laminate compression under a mould or
in the vacuum bag (Figure 5.3). This should allow an even resin flow throughout
the laminate and thus, a uniform thickness. Obtained laminates presented good
properties as shown in Table 5.1
Figure 5.3 Mould compressed under vacuum bag for even resin spread
Chapter Five: The Chemical and Physical Degradation
75
5.10.3 Experimental procedure
Experiment 1
Ten sets of five material samples were exposed to a corrosive environment. The
expository chamber (Figures 5.4 and 5.5) was set in such a way as to simulate the
service environment of a pipe or storage tank where samples were simultaneously
exposed to the chemical and physical degradation. The chamber comprised two
compartments thermally and optically isolated. Each compartment comprised the
following:
a. A set of UV-B lamps. The light spectrum emitted by the fluorescent lamp
comprised peaks around 313 nm at about 500 microWatt/cm2/30?. The
emitted energy was very low for higher wavelength (less than 100
microWatt/cm2/30? between 350 nm and 400nm), while the sun spectrum put
out more than 1000 microWatt/cm2
b. Heating device 900 Watts
/30? in the same region.
c. Digital temperature controller
d. 24 expository cells
e. A grill with a capacity of 25 expository cells. The laminate was exposed to
ultraviolet rays at one face and entirely wet with a chemical solution to the
second face while the temperature inside the expository chamber is maintained
constant.
Exposure conditions were determined from a table of chemical resistance (Crystic
resin) in such a way as to obtain notable degradation in relatively short time
(Table 5.2)
Table 5.2: Exposure conditions
Temperature Constant at 40?C
UV rays from outside cell 450 ?Watt /cm2
Humidity from inside cell 100% sample entirely wet
Chemical reagent from inside 5% sodium hydroxide
Maximal duration 10 days
Chapter Five: The Chemical and Physical Degradation
76
Figure 5.4 External view of the expository chamber
Figure 5.5 Internal view of the expository chamber
Chapter Five: The Chemical and Physical Degradation
77
The experimental procedures are as follows:
a. Ten samples of material 1 (Table 5.1) were cut in such a way as they fit in the
expository cells and the fibre directions were marked (Figure 5.6).
b. The samples were initially post cured for three hours at 80o
c. A fixed volume of 5% sodium hydroxide solution was injected into each cell.
C according to the
manufacturer?s recommendations for optimal mechanical strength. The
samples were then carefully checked for voids or cracks prior to being
mounted in the cells of exposure chamber.
d. All the cells were then immediately mounted on the tray in the exposure
chamber where the temperature was already set at 40o
e. Five samples were simultaneously exposed to ultraviolet rays, in order to
assess the effects of post curing due to ultra violet rays only. Five others were
put into an entirely closed aluminium container which stopped ultraviolet rays
and the container was then placed in the exposure chamber at the same time in
order to determine the post curing effects due to temperature only.
C.
f. Samples were unloaded one by one from the chamber at 24 hour intervals.
Unloaded samples were immediately tested for tensile strength and moisture
content (Figures 5.7 and 5.8).
g. All the samples were then scanned using a Raman spectrometer (Figure 5.9)
for changes in chemical structure. Three parameters were used to monitor the
chemical change: the ester group reduction, the carbonyl variation, and the
orthopthalicbenzene ring variation peaks
Chapter Five: The Chemical and Physical Degradation
78
Figure 5.6 Sample cut in circular shape, the fibre direction marked, (1) and ready
to be fitted to the expository cell (2)
Figure 5.7 Sample from the expository chamber (1) and cut for tensile strength
test (2)
2
1
1 2
Chapter Five: The Chemical and Physical Degradation
79
Figure 5.8 Tensile test on dog bone samples of 3mm width, 15mm long
Figure 5.9 Raman spectrometer
Raman spectra were acquired using the micro-Raman attachment of a Jobin-Yvon T64000 Raman
spectrometer operated in single spectrograph mode with a 600 grooves/mm grating and the
647.1nm line of a krypton ion laser as excitation source. Laser plasma lines were removed using a
narrow band interference filter, and the dispersed signal was detected using a cooled CCD
detector. So that to not modify the analysis spot through localized heating the laser power at the
sample was kept low. The diameter of the laser spot with the 20x objective was ~2micron.
Chapter Five: The Chemical and Physical Degradation
80
Experiment 2
Ten sets of 5 samples from material 2 (Table 5.1) were exposed to 10% sodium
hydroxide aqueous solution for a maximum of eight hours at 80o
C (Figure 5.10).
Sets were unloaded consecutively at 60 minute intervals. Unloaded samples were
immediately rinsed in abundant distilled water in order to stop further reaction.
Samples were then dried in desiccators for twenty four hours and then tested in
shear strength.
Figure 5.10 Hydrolysis apparatus
The hydrolysis apparatus was mounted and allowed for an accelerated chemical degradation at
constant temperature (80?C). It comprised two identical compartments; each made of a
temperature controlled heating device, a 500 ml glass flask, and a condensing tube.
Experiment 3
The same procedure as for experiment 2 was followed with ten sets of 3 samples
of polyester composite laminates (see table 5.1) over nine hours. The samples
were then tested for shear modulus on an Anton Paar Physica MCR rheometer.
Chapter Five: The Chemical and Physical Degradation
81
5.10.4 Experimental results and discussion
Tensile strength
The tensile strength was measured for samples subjected only to the temperature,
for samples exposed to UV rays and temperature only, and for samples exposed to
UV rays, temperature and chemical attack. Results are presented in Table 5.3. The
plot in Figure 5.11 shows that the laminate strength is not affected by the post
cure and that the degradation of the tensile strength occurs according an
exponential law (Figure 5.12).
Table 5.3 Tensile Strength of exposed samples
Duration
(Days)
Tensile strength
( MPa)
Degraded
samples
Post-curing effect
UV rays Temperature
AVG StD AVG StD AVG StD
1 89 0.019 95 0.016 95 0.016
2 68 0.001
3 - 91 0.009 86 0.004
4 36 0.008
5 39 0.000 89 0.015 138 0.011
6 19 0.002
7 17 0.003 121 0.031 90 0.008
8 19 0.002 99 0.042 90 0.008
Chapter Five: The Chemical and Physical Degradation
82
Figure 5.11 UV and temperature post-curing effect on tensile strength
Figure 5.12 Variation of tensile strength during degradation
Moisture curve
Figure 5.13 shows that the moisture percentage increases during degradation,
denoting that the more the material structure is degraded the more the moisture
penetrates inside the laminate. This explains the dramatic increase in absorbed
moisture from the sixth day where the laminate has reached a high disintegration
0
20
40
60
80
100
0 2 4 6 8
T
en
si
le
s
tre
ng
th
(M
P
a)
Time (days)
Chemical and physical degradation
UV rays and temperature effect
Temperature effect
y = 84.71e-0.25x
R? = 0.92
0
10
20
30
40
50
60
70
80
90
100
0 2 4 6 8
Te
ns
ile
s
tre
ng
th
(M
P
a)
Time (days)
Chapter Five: The Chemical and Physical Degradation
83
level corresponding to the lowest value of tensile strength recorded in Figure 5.12.
This shows that the moisture recorded inside the laminate is related to material
disintegration from chemical attack and not due to Fickian diffusion. The curve
shows an exponential trend corresponding to the suggested theoretical degradation
model.
Figure 5.13 Moisture variation during degradation
Micrograph
SEM pictures, showed broken fibres following resin depletion (figure 5.14).
Figure 5.14 Micrograph
(1) Non attacked area, (2) progressive depletion of resin and fibre denudation, (3) attacked area
showing pits and resin depletion, (4) inside of pits showing broken fibres
0
2
4
6
8
10
12
14
16
18
20
0 2 4 6 8 10
P
er
ce
nt
ag
e
m
oi
st
ur
e(
%
)
Time(Days)
1 2
3 4
Chapter Five: The Chemical and Physical Degradation
84
Raman spectra
Raman spectra were acquired both for non-degraded and degraded samples. The
spectra were compared to show the change in chemical structures as below.
Spectrum of non-degraded samples
Figure 5.15 Spectrum of non-degraded samples from 370 cm-1 to 2000 cm
-1
Figure 5.16 Spectrum of non-degraded sample from 2000 cm-1 to 3600 cm
-1
The spectrum shows peaks at the following bands (Figures 5.15 & 5.16)
- 3060 cm
- 2939 cm
-1
- 1729 cm
-1
-1
- 1600 cm
- 1450 cm
-1
- 1280-1300 cm
-1
- 1200 cm
-1
- 1040 cm
-1
-1
- 1001 cm
- 850 cm
-1
- 650 cm
-1
- 620 ?625 cm
-1
-1
15
16
17
18
19
20
21
22
0 500 1000 1500 2000
R
am
an
In
te
ns
ity
x
1
0-
3
Wave number Cm-1
6
7
8
9
10
11
12
13
2000 2500 3000 3500 4000
R
am
an
In
te
ns
ity
x
1
0-
3
Wave Number cm-1
Chapter Five: The Chemical and Physical Degradation
85
Figures 5.15 and 5.16 show typical spectra of ortho-para?phthalic polyesters.
Peaks at 1729 cm-1 band denote the stretching of an ester carbonyl conjugated
with a C=C bond or with an aromatic ring. The peak at 1280 cm-1 indicates that
such aromatic ring may be a phthalate or benzoate. The ester groups conjugated
with an aromatic ring are normally characterized by a strong absorption band due
to the carbonyl C=O stretching between 1740 and 1715 cm-1 and involving the
stretch of C-O near 1200 cm-1 [89]. The presence of aromatic rings is confirmed
by peaks at 1001 cm-1 for the mono-substituted rings and 1040 cm-1 for the ortho
substituted aromatic rings. These peaks are usually strong in Raman and normally
expected to arise respectively at 1000 -/+ 5 cm-1 and 1033-/+ 11 cm-1 bands. The
aromatics rings are possibly also responsible for the absorption at 650 cm-1 for the
para substituted and 625 cm-1
for the mono substituted. These peaks are usually
medium strong in Raman.
The resin structure comprises carbonyl group in 1,3 Diketone. The enol form of
the 1,3 Diketone is shown by the absorption at 1600 cm-1. The 1500 cm-1, 1450
cm-1, and 1260 cm-1 peaks confirm this. The absorption around 1450cm-1
may
possibly also represent the C-H bending in alkane branches of the polymeric
chain.
The spectra show another absorption band around 3060 cm-1 and 2930 cm-1. This
may be related to the bonded OH groups in the enol form of 1,3 Diketone
structures which usually absorbs around 3000 to 2700 cm-1. It may also be related
to the methylene group -CH2- in a 6 membered ring that may absorb at 2930 cm-1
or also to the diene bonds C=C which peaks arises above 3000 cm-1
.
Change in chemical structure showed by Raman spectra
Raman spectrometry shows the change in chemical structure resulting from the
hydrolysis of ester groups and photo-oxidation. The spectra show increasing peaks
at 1001 cm-1 after an earlier decreasing period (Figure 5.17), a decrease of peaks
at 1040 cm-1 (Figures 5.18 and 5.19), and a shift of peaks from 1040 cm-1 to 1032
cm-1 over the course of the degradation (Figures 5.20).
Chapter Five: The Chemical and Physical Degradation
86
The peaks at 1001 cm-1 and at 1040 cm-1 represent the mono-substituted and the
ortho-substituted aromatic rings respectively. These peaks are usually strong in
Raman and are normally expected to arise at 1000 -/+ 5 cm-1 and 1033-/+ 11 cm-1
bands respectively [89].
Therefore, the observed decrease and shift may denote the modification of the
initial structure (substituants sensitive bands) resulting from environmental attack
followed by the production of alcohol structures that normally absorb between
1075 cm-1 and 1000 cm-1 for aromatic secondary alcohol, and between 1036 and
970 cm-1 for axial cyclic secondary alcohol [89]. This occurs as well around 1600
cm-1
(figure 5.21) denoting the production of carbonyl structure due to ultraviolet
rays attack and around 3060 denoting the arise of diene bond also due to
ultraviolet rays attacks.
Figure 5.22 shows the gradual hydrolysis of ester groups that absorb at 1729 cm-1
.
The peaks decrease according to an exponential law.
Figure 5.17 Variation of monosubstituted aromatic ring peaks (1001cm-1
)
0
5
10
15
20
25
30
35
40
45
50
0 2 4 6 8 10 12
R
am
an
In
te
ns
ity
x
1
0-
2
Time (Days)
Chapter Five: The Chemical and Physical Degradation
87
Figure 5.18 Decreasing of ortho substituted aromatic ring peaks (1040 cm-1
)
Figure 5.19 Decreasing peaks at 1040 cm
-1
90
95
100
105
110
115
120
125
1015 1020 1025 1030 1035 1040 1045 1050
R
am
an
In
te
ns
ity
x
1
0-
2
Wave Number cm-1
day1
day2
day3 day4
day10
y = 2170.e -0.333x
R 2 = 0.803
0
5
10
15
20
25
0 2 4 6 8 10
Time (days)
R
am
an
In
te
ns
ity
x
10
-2
Chapter Five: The Chemical and Physical Degradation
88
Figure 5.20 Peaks shifting from 1040 cm-1 (days1, 6, 7) to 1032 cm
-1
Figure 5.21 Variation of 1600 cm-1
(Carbonyl) Raman peaks during degradation
95
100
105
110
115
120
125
1015 1020 1025 1030 1035 1040 1045 1050
R
am
an
In
te
ns
ity
x
1
0-
2
Wave Number cm-1
Day 1 (peak at 1040 cm-1)
Day 8 (peak at 1033 cm-1)
Day 7 (peak at 1031 cm-1)
Day 6 (peak at 1040 cm-1)
138
140
142
144
146
148
150
152
154
156
158
0 2 4 6 8 10
R
am
an
In
te
ns
ity
x
1
0-
2
Time (Days)
Chapter Five: The Chemical and Physical Degradation
89
Figure 5.22 Decrease of ester peaks (1729 cm-1
) during degradation
Correlation between the model and experimental results
According to the model suggested in this analysis, in a constant environment the
degradation of the material stiffness is linearly correlated to the variation in
chemical link density and this variation rate is an exponential function of time. In
order to experimentally assess the validity of this theoretical assertion, the degree
of correlation between Ld and the material strength was numerically measured.
The numerical regression of Raman peaks provided the degradation model
(Figures 5.22) corresponding to the variation in chemical structure. Values of Ld
were obtained from the Raman peaks of ester groups. The index Ld corresponds to
the degree of degradation. As Raman peaks decreases, the degradation increases.
Thus, the variation rate of Ld is taken by the opposite of variation rate of Raman
peaks. The degradation model corresponding to Ld was subsequently determined
by integration of Ld variation rate (see Appendix D), and represented in figure
5.23. The material strength was taken by the tensile strength (Table 5.4 and Figure
5.12). The correlation coefficient R2
is 0.97 (Figure 5.24) indicating very good
correlation between the model and experimental values.
y = 1993.0e-0.260x
R2 = 0.968
0
2
4
6
8
10
12
14
16
18
0 2 4 6 8 10
R
am
an
In
te
ns
ity
x
10
-2
Time (days)
Chapter Five: The Chemical and Physical Degradation
90
Table 5.4: Calculated value of index Ld
Days
and experimental tensile strength
0 1 3 4 5 6 7
Calculated L 0 d 451.5 800.6 1070.7 1279.6 1441.1 1566.1
Experimental
Tensile Strength
(MPa)
89 68 36 39 19 17 19
Figure 5.23 Index of chemical links degradation deduced from ester groups
reduction
Figure 5.24 Correlation between the model and experimental values
Ld(t) = 1992.7(1-e-0.260t)
R2 = 0.999
0
2
4
6
8
10
12
14
16
18
0 2 4 6 8
L d
x
10
-2
Time (Days)
y = -0.04x + 88.39
R? = 0.97
0
10
20
30
40
50
60
70
80
90
100
0 5 10 15 20
T
en
si
le
S
tre
ng
th
(
M
P
a)
Ld x10-2
Chapter Five: The Chemical and Physical Degradation
91
This good linear correlation also provides the experimental verification of
equation (5.6) showing a linear correlation between the material stiffness and the
degree of chemical links degradation.
Comparing calculated and experimental material lifetime based on tensile
strength evolution
In order to perform the experimental comparison, the problem was set as follows:
?what is the material lifetime if the minimal admissible value for its tensile
strength is determined as a given percentage of the initial value??
From the degradation model of chemical structure in Figure 5.22, the theoretical
degradation rate for the material strength was determined by derivation (see
Appendix D):
td e
dt
dL 260.09.511 ?= (5.44)
Based on equation (5.44), the lifetimes were predicted for a range of property
retention values (see Appendix D). The predicted values are compared to
experimental values determined from table 5.4. The comparison is presented in
Figure 5.25.
Table 5.5: Predicted and experimental lifetime
Percentage
reduction
(%)
Tensile
strength
value
( MPa )
Predicted Life-time
(Days)
Experimental
Life-time
(Days) Calculated Interpretation
80 17.9 6.3 6 minimal 5 to 6
70 26.8 4.7 4.5 minimal 4 to 5
60 35.7 3.6 3.5 minimal 3
50 44.7 2.7 2.5 minimal 2
40 53.6 2.0 2 minimal 1 to 3
30 62.5 1.4 1 minimal 1
20 71.5 0.9 Less than 1 Less than 1
Chapter Five: The Chemical and Physical Degradation
92
Figure 5.25 Predicted lifetimes compared to experimental lifetimes
Variation of shear strength under physical and chemical degradation
Test results are presented in Figure 5.26. The figure shows that the variation of
shear strength resulting from the degradation follows an exponential trend.
Figure 5.26 Shear strength variation of samples subjected to chemical degradation
1
2
2.5
3.5
4.5
6
1
2 2
3
4.5
5.5
0
1
2
3
4
5
6
7
70 60 50 40 30 20
lif
et
im
e
(d
ay
s)
property retention (%)
predicted
experimental
y = 20.28e-0.29x
R? = 0.95
0
2
4
6
8
10
12
14
16
0 2 4 6 8 10
S
he
ar
s
S
tre
ng
th
(
M
P
a)
Time (hours)
Chapter Five: The Chemical and Physical Degradation
93
Variation of storage modulus under physical and chemical degradation
Figure 5.27 presents the storage modulus variation for attacked samples at several
temperatures below TG
. The variation follows an exponential trend.
Figure 5.27 Variation of storage modulus during the course of degradation
5.11 Conclusion
Throughout the literature, the environmental degradation of FRP is described as a
complex process. Modelling efforts are limited to partial models, most of the time
empirical. No viable prediction method is yet available for the environmental
degradation of the material mechanical strength. The analytical framework
presented in this chapter relies on resolving the degradation process into only
three components consisting of chemical degradation, physical degradation, and
mechanical degradation. Based on material science theories, the analysis has
demonstrated that, in a constant environment, the chemical and physical
degradation of a polymeric matrix follows an exponential law. This results from
the effect of the material rheology transformation on the diffusion process. The
resulting mathematical model correlates the degradation rate directly to the
material lifetime.
y = 2.29E+9e-0.15x
R2 = 0.96
0.00E+00
5.00E+08
1.00E+09
1.50E+09
2.00E+09
2.50E+09
0 2 4 6 8 10
S
to
ra
ge
M
od
ul
us
(
P
a)
Reaction time (hours)
100*C"
90*C
80*C
70*C
60*C
35*C
50*C
Chapter Five: The Chemical and Physical Degradation
94
Experiments have been conducted using a range of measurement methods. These
include Raman spectrometry for chemical structure changes, rheometry for
storage modulus, and mechanical testing of tensile and shear strength evolution
during degradation. All experiments confirm the validity of the suggested model.
The degree of correlation between the model and the experimental process is very
good.
The analysis also demonstrates that the chemical and physical degradation rate of
a polymeric matrix is an ascending monotonic function of the environment.
Consequently, the evaluation of environmental degradation in a laboratory can be
conducted in a constant environment and laboratory test results can be directly
translated to the real service environment. This is an important conclusion because
many authors [1-3] have difficulty relating laboratory tests to actual service
conditions.
Based on this mathematical model, a simple and practical prediction method has
been suggested. The environmental degradation rate of a material in a real service
environment can be determined in a laboratory based on tests conducted in a
constant environment. This method requires monitoring only of the chemical
structure change or any other material property linearly correlated to the chemical
structure. For the environmental conditions to be used in a laboratory, the method
requires only the availability of statistical information such as the control chart of
environmental variables including moisture, temperature, chemical concentration
and ultraviolet ray intensity. Experiments conducted in a laboratory show that
predicted degradation of the tensile strength of polyester fibreglass composite,
based on this prediction method, is in good agreement with experimentally
measured degradation.
The theoretical analysis presented in this chapter assumes that the material
undergoing the physical and chemical degradation is not subject to any kind of
mechanical load. Results obtained in this chapter apply only to the case where
mechanical stresses can be neglected. However, the suggested model provides a
Chapter Five: The Chemical and Physical Degradation
95
useful tool for assessing the chemical and physical degradation factors in cases
involving both mechanical stress effects and physical and chemical degradation.
This will be dealt with in the next chapter.
96
6 COMPREHENSIVE MODELLING AND PREDICTION OF FRP
ENVIRONMENTAL DEGRADATION
A comprehensive model of the environmental degradation of fibre reinforced
plastics is presented. This model includes all the processes involved in the
environmental degradation of these materials. The model is expressed as a
dynamic constitutive equation resulting from the combination of the historical
variation in chemical link density and cohesive forces and the stress history of the
material. Prediction of long-term behaviour can be obtained from short-term tests
based on the time-temperature correspondence and the Boltzmann superposition
principle.
In the literature, it has been observed that the evolution of the material rheological
state constitutes one of the major consequences of environmental degradation.
Experimental attempts to use this effect to assess the environmental degradation
of polymeric materials have also been reported [1, 90-91]. This analysis presents
an analytical demonstration of the fact that the ultimate and comprehensive result
of environmental degradation caused by any factor is the change in the material
rheology. This effect is expressed through the dynamic constitutive equation.
Detailed modelling of the variation in chemical link density and cohesive forces
has been presented in the previous chapter. The present chapter is concerned
mainly with presenting a dynamic constitutive equation resulting from the
combination of the chemical and physical degradation history with the stress
degradation history.
6.1 The Constitutive Equation of Environmental Degradation in FRP
Due to resin viscoelasticity, stress effects are time dependant as the stiffness
coefficient changes with time. The Boltzmann superposition principle [86, 92]
Chapter Six: Comprehensive Modelling and Prediction of FRP Environmental Degradation
97
takes account of the material stress or strain history and expresses the material
stiffness evolution as a dynamic constitutive equation. Let 't represent the instant
when a strain ?( 't ) is applied to the material. The stiffness coefficient at time t can
be written as ( )'ttC ? representing its residual value after relaxation over a period
equal to 'tt ? . The Boltzmann superposition principle is then written as:
( ) ( ) ( ) '
'
'
' dt
dt
td
ttCt
t ?? ?
??
?= (6.1)
Now at the same time, chemical and physical degradation also modifies the
material stiffness over time. Therefore, the stiffness coefficient is also affected by
the chemical and physical degradation history that can be expressed as a factor
( )tKenv representing the accumulated effect of degradation up to time t. Thus the
actual value of the stiffness coefficient at time t is ( ) ( )'ttCtKenv ? . The Boltzmann
superposition principle can therefore be modified as:
( ) ( ) ( ) ( ) '
'
'
' dt
dt
td
ttCtKt
t
env
?? ?
??
?= (6.2)
The above equation expresses the evolution of the material stiffness resulting from
stress effects combined with chemical and physical degradation. It provides the
constitutive equation of the material under environmental degradation and shows
that the ultimate and comprehensive result of environmental degradation caused
by any factor, is the change in the material stiffness. This equation can, therefore,
be used to measure the comprehensive environmental degradation effect caused
by all environmental factors.
When the degradation function is known, the chemical and physical degradation
history can be determined through the retention of specific material properties
over time. So, if ( )0tEd is the index that represents the initial value of the material
property and ( )tEd? represents the reduction in the material property at instant t ,
( )tKenv is defined as:
Chapter Six: Comprehensive Modelling and Prediction of FRP Environmental Degradation
98
( )
( )0
1)(
tE
tE
tK
d
d
env
?
?= (6.3)
This means that ( )tKenv can be determined from equation (5.37) that represents the
physical and chemical degradation function. Subsequently, if ( )tKenv can be
calculated, then at any time, the stiffness of a viscoelastic material undergoing
environmental degradation can also be calculated by multiplying by ( )tKenv , its
stiffness when not subjected to environmental degradation. This provides an
analytical method to predict the environmental degradation of the stiffness
coefficient.
However, it has been shown that stresses affect chemical reactions by modifying
the activation energy of the reaction as well as that of diffusion [1, 65-67]. The
exact value of ( )tKenv would thus be calculated by introducing a factor accounting
for stress into the equation used to determine the kinetic constants as is done in the
equation of Zurkov [1, 65]. The latter equation has been successfully used in quite
a number of works where there has been a need to evaluate the effects of stresses
on chemical reactions [1, 65]. Such calculations are not pursued in this analysis.
An alternative method based on experimental measurements is suggested in the
next paragraph.
It is apparent from equations (6.1) and (6.2) that stress relaxation is amplified due
to environmental degradation and this amplification is a determined function of
the degradation time and results from thermally activated processes, namely
modification of the chemical link density and modification of the cohesive forces.
These observations are sufficient to provide a method to measure and to predict
exactly the environmental degradation. They lead to logical consequences that:
? The environmental degradation can be measured through stress relaxation or
creep.
? The amplification of stress relaxation by the environmental degradation is
thermally activated and obeys, therefore, the time temperature superposition
principle.
Chapter Six: Comprehensive Modelling and Prediction of FRP Environmental Degradation
99
6.2 Determination of the Laboratory Test Conditions
The above theoretical result show that, the prediction of long-term environmental
degradation of FRP matrix can be done by measuring long-term stress relaxation
of the material as commonly done through dynamic mechanical analysis (DMA)
or rheometry. In this case, the test conditions in the DMA equipment or rheometer
must be set in such a way as to represent the degrading environment of interest.
As shown in Chapter four, the upper and lower statistical control limits of the real
service conditions provide the constant environmental conditions to be used in
laboratory:
It has been shown that the stress relaxation under environmental factors is caused
by the physical and chemical degradation as well as the viscoelasticity of the
material. Given that the physical and chemical degradation is a monotonic
increasing function of environmental factors (subsection 5.9.2), and that the
viscoelastic relaxation is also a monotonic increasing function of the temperature,
it follows that the stress relaxation during environmental degradation is also a
monotonic increasing function of environment factors. Experimental results of
Figures 6.5 and 6.9 provide also an illustration of the above. Therefore, the upper
and lower control limits of the long-term relaxation can be determined in
laboratory from the constant conditions of the control limits of the environment
control chart. Alternatively the average long-term relaxation can be determined
from the average of the environment control chart.
However, care must be taken to compensate the fatigue effect caused by
temperature and humidity cycles as shown in Chapter four.
From results of section 6.1, it follows also that, accelerated test conditions can be
determined based on the time temperature shift principle.
Chapter Six: Comprehensive Modelling and Prediction of FRP Environmental Degradation
100
6.3 Experimental Method
Two series of stress relaxation tests were conducted in a laboratory. In the first
experiment, the material was subjected to mechanical stress only. The second
experiment combines the application of stresses, the degradation of chemical links
by the sodium hydroxide, and the deterioration of cohesive forces by moisture and
temperature. The materials used and the test conditions are presented in Tables 6.1
and 6.2.
Table 6.1: Materials
LAMINATE RESIN FORMULATION
Composition Resin/fibre/resin Resin
Orthophtalic
polyester NCS 901
PA
Glass fibre WR 27.4 g / m Resin (phr) 2 100
Vf
AVG
(%)
2.7 Catalyst-ButanoxM50
(phr)
2
StD 0.2
Curing cycle
24h/ 25o
3h/ 80
C,
oThickness (mm) C 1.0
? 0.1
Void (visual) No voids
Table 6.2: Test conditions
6.3.1 Relaxation test procedure
A device was designed to be used on an ordinary tensile machine as shown in
Figure 6.1. Samples were wet in a solution of sodium hydroxide and subjected to
a tensile stress as follows:
1. The test device was mounted in the weathering chamber of a tensile machine
and preheated to the test temperature.
Temperatures 50o C, 60o C, 70o C, 80o
Initial tensile stress
C
11.4 ? 0.2 MPa
Humidity 100 % (sample entirely wet)
Chemical reagent 5 % sodium hydroxide
Chapter Six: Comprehensive Modelling and Prediction of FRP Environmental Degradation
101
2. The sodium hydroxide solution was also preheated to the test temperature
separately.
3. A dog bone laminate sample was mounted in the test device. The
preconditioned sodium hydroxide solution was then poured into the device so
that it entirely wet the laminate sample. The weathering chamber was then
closed and the sample was left to condition for 30 minutes.
4. The data logger was started and the sample was tensioned to an initial stress of
11.4 ? 0.2 MPa. This value was selected to ensure that the resulting strains
are in the linear viscoelastic region.
5. The sample was left to relax until the relaxation curve became fairly flat.
Figure 6.1 Testing apparatus for stress relaxation
6.3.2 Results and discussion
Amplification factor ( )tKenv
Figure 6.2a presents the results of stress relaxation tests under mechanical stresses
only while Figure 6.2b presents the results for the combined mechanical stress and
Weathering chamber
Chemical container
FRP sample entirely wet in
the chemical solution
Tensile machine
clamps
Chapter Six: Comprehensive Modelling and Prediction of FRP Environmental Degradation
102
environmental degradation.
a)
b)
Figure 6.2 Relaxation curves. a) under mechanical stress only, b) under
mechanical stress and chemical and physical degradation
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1000 2000 3000 4000 5000 6000 7000 8000
N
or
m
al
iz
ed
s
tre
ss
Relaxation time (seconds)
Series1
Series2
Series3
Series4
Series5
Test temperature 80 C
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1000 2000 3000 4000 5000 6000 7000 8000
N
or
m
al
iz
ed
s
tre
ss
Relaxation time (seconds)
Series1
Series2
Series3
Series4
Series5
Test temperature 80 C
Chapter Six: Comprehensive Modelling and Prediction of FRP Environmental Degradation
103
A comparison of average curves from each of the two relaxations is presented in
Figure 6.3. It is apparent that the humid environment slows down the rate of
relaxation during the first 1000 seconds. This behaviour is typical of more than
ten tests conducted in identical conditions. At longer time scales, however, the
comparison shows that the relaxation is amplified by the environmental
degradation.
Figure 6.3 Amplification of the relaxation due to environmental degradation
Figure 6.4 presents the inverse of the environmental degradation factor ( )tKenv
which was calculated from the experimental data of Figure 6.3 by using the
following expression:
( )
( ) ( )tKt
t
env
env =
?
?
(6.4)
The factor ( )tKenv provides a measure of the increasing effect of environmental
degradation including stress effects on the chemical and physical degradation. It is
shown that ( )tKenv is a function of the environmental degradation time. It can be
experimentally determined from short-term test and subsequently used to calculate
long-term stiffness coefficient.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1000 2000 3000 4000 5000 6000
N
or
m
al
iz
ed
s
tre
ss
Time (seconds)
relaxation under mechanical stress only
relaxation under environmental degradation
and mechanical stress
Chapter Six: Comprehensive Modelling and Prediction of FRP Environmental Degradation
104
Figure 6.4 Accumulated environmental degradation measured as 1 / K
env
Time Temperature correspondence
The stress relaxation under environmental degradation was also measured at
various temperatures. Figure 6.5 shows the relaxation amplitude expressed as a
percentage reduction of the initial stress value for 50o C, 60o C, 70?C and 80o
C.
Figure 6.5 Relaxation under chemical and physical degradation at various
temperatures
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 1000 2000 3000 4000 5000 6000
E
nv
iro
nm
en
ta
l f
ac
to
r 1
/ K
ev
Time (seconds)
0
10
20
30
40
50
60
0 200 400 600 800 1000 1200
R
el
ax
at
io
n
am
pl
itu
de
(%
)
Time (seconds)
50?C 60?C 70?C 80?C
Chapter Six: Comprehensive Modelling and Prediction of FRP Environmental Degradation
105
As expected, the elevated temperature environment causes substantial stress
relaxation. The role of temperature in stress relaxation (Figure 6.5) is not
surprising since the environmental degradation results from the modification of
chemical link density and cohesive forces which are thermally activated
processes. Therefore, prediction of long-term environmental degradation can be
obtained trough the time temperature shift principle based on Arrenhius law. The
experimental data of Figure 6.5 are now used to make a prediction of long-term
relaxation based on the time temperature shift principle. The method uses a shift
factor
iT
? to determine long-term relaxation time [86, 92]. The time
iT
t required to
achieve a given stress relaxation at temperature iT is determined from the time
refT
t required to achieve the same stress relaxation measured at a reference
temperature refT . This requires the measurements made at the reference
temperature to be shifted by
iT
?ln on the time logarithm scale. The shift factor is
expressed as [86]:
refT
iT
T t
t
i
=? (6.5)
Or alternatively as:
( ) ( )
refii TTT
tt lnlnln += ? (6.6)
In this experiment, the reference temperature is fixed at 80?C and the shift factor
iT
? is calculated from the Arrhenius equation as follows [86]:
?
?
?
?
?
?
?
?
?=
refi
T TTR
W
i
11
ln? (6.7)
where W is the activation energy of the process.
Since
refT
iT
T t
t
i
=? , the factor
R
W
can be determined by plotting
refT
iT
t
t
ln against
iT
1
for a fixed value of the relaxation amplitude. Equation (6.7) indicates that the plot
is a straight line and that
R
W
corresponds to the slope of this line. This procedure
Chapter Six: Comprehensive Modelling and Prediction of FRP Environmental Degradation
106
is applied to the curves fitted to the experimental data of Figure 6.5 for several
values of the relaxation amplitude. The straight lines resulting from such plots are
usually parallel since the magnitude of the activation energy is normally constant.
In the present study, the measured slopes are instead an increasing linear function
of the relaxation amplitude (Figures 6.6 and 6.7), which implies that the activation
energy also is an increasing linear function of the relaxation amplitude. This
results from the interaction between the mechanical stress and the chemical
reaction. The above experimental observation is not surprising since it is known
that tensile stress reduces the activation energy of chemical reactions [1, 41, 65].
Due to the relaxation, stress decreases over the course of the degradation and so
its effect on the activation energy reduces also. As a consequence, the activation
energy increases with increasing degradation.
Figure 6.6 Experimental determination of the activation energy for different
relaxation amplitudes
0
0.5
1
1.5
2
2.5
3
290 295 300 305 310 315
Ln
(t
i
/ t
re
f)
1x10-5 /T (?K-1)
5%
7.50%
10%
12.50%
15%
17.50%
20%
22.50%
25%
27.50%
30%
32.50%
Chapter Six: Comprehensive Modelling and Prediction of FRP Environmental Degradation
107
Figure 6.7 Activation energy as a function of the relaxation amplitude
Resulting from the above, an alternative method to determine the shift factor is
suggested. Since the activation energy is not constant but is a linear function of
the relaxation amplitude, equation (6.6) requires that the shift factor is an
exponential function of the relaxation amplitude and can be directly determined
from experimental data by plotting
refT
iT
t
t
against relaxation amplitude (Figure
6.8).
Figure 6.8 Experimental determination of the shift factor as a function of the
relaxation amplitude
y = 332.8x + 1807
R2 = 0.999
0
2000
4000
6000
8000
10000
12000
14000
0 5 10 15 20 25 30 35
W
/
R
(?
K
-1
)
Relaxation amplitude (%)
0
1
2
3
4
5
6
7
0 5 10 15 20 25 30 35 40
t i
/ t
re
f
Relaxation amplitude (%)
T = 50?C
T = 60?C
T = 70?C
?70 C=0.913e0.016x R2 = 0.999
?60 C = 1.353e0.036x R2 = 0.999
?50 C = 1.253e0.081x R2 = 0.999
Chapter Six: Comprehensive Modelling and Prediction of FRP Environmental Degradation
108
The above results establish that the process follows the Arrhenius law but, due to
the activation energy not being constant, the shift factor depends on the relaxation
amplitude and consequently can be theoretically calculated as below.
From the results of Figure 6.8, the shift factor can be expressed as a function of
the relaxation amplitude, represented by x :
xk
T eki
6
5=? (6.8)
The above equation can be rewritten as:
xkk
iT 65
lnln +=? (6.9)
Furthermore, since the activation energy is a linear function of the relaxation
amplitude, equation (6.7) can be written as:
?
?
?
?
?
?
?
?
?
+
=
refi
T TTR
xWW
i
11
ln 10? (6.10)
or
?
?
?
?
?
?
?
?
?+?
?
?
?
?
?
?
?
?=
refirefi
T TTR
xW
TTR
W
i
1111
ln 10? (6.11)
Comparing equations (6.9) and (6.11), it becomes apparent that:
?
?
?
?
?
?
?
?
?=
refi TTR
W
k
11
ln 05 (6.12)
and
?
?
?
?
?
?
?
?
?=
refi TTR
W
k
111
6 (6.13)
In the above, 0W is the value of the activation energy prior to relaxation. It
represents the maximum effect of the applied stress on the chemical reaction. 1W
provides a measure of the rate at which the stress effect on the chemical reaction
decreases.
Chapter Six: Comprehensive Modelling and Prediction of FRP Environmental Degradation
109
The shift factor is thus determined from Figure 6.8. A shift as per equation (6.6) is
then applied to the measurements taken at 80?C. Long-term results up to 6000
seconds are predicted for 50?C, 60?C, and 70?C. The relaxation was also
measured experimentally up to 6000 seconds at the three temperatures (Figure
6.9). Experimental relaxation times obtained from Figure 6.9, compare excellently
to the predicted results, as shown in Figure 6.10.
Figure 6.9 Experimental measurement of the relaxation under chemical and
physical degradation
It is worth mentioning that, in the above experiment, the limit of relaxation goes
up to 60 % (at 70?C) and the temperature up to 80?C. This covers the range of
relaxation and temperatures experienced in the real service environment (storage
tank and pipe).
It can therefore be concluded that long-term environmental degradation can be
predicted using the time-temperature shift principle. The demonstrated method of
applying this shift takes account of the variation of the activation energy resulting
from the interaction between the stress and the chemical reaction over the course
of the degradation.
0
10
20
30
40
50
60
70
80
0 1000 2000 3000 4000 5000 6000
R
el
ax
at
io
n
am
pl
itu
de
(%
)
Time (seconds)
80 C
70 C
60 C
50 C
Chapter Six: Comprehensive Modelling and Prediction of FRP Environmental Degradation
110
a)
b)
c)
Figure 6.10 Experimental relaxation times compared to the relaxation times
predicted from times measured at 80?C. a) T= 70?C, B) T= 60?C,
c) T = 50?C
0
1000
2000
3000
4000
5000
6000
35 40 45 50 55 60
T
im
e
(s
ec
on
ds
)
Relaxation amplitude (%)
Predicted
Experimental
0
1000
2000
3000
4000
5000
6000
7000
30 35 40 45 50
Ti
m
e
(s
ec
on
ds
)
Relaxation amplitude (%)
Predicted
Experimental
0
1000
2000
3000
4000
5000
6000
7000
20 25 30 35
T
im
e
(s
ec
on
ds
)
Relaxation amplitude (%)
Predicted
Experimental
Chapter Six: Comprehensive Modelling and Prediction of FRP Environmental Degradation
111
Relaxation measurements taken at one temperature can consequently be used to
predict the degradation rate at any other temperature once short-term
measurements are taken at both the reference temperature and the temperature of
interest.
6.4 Conclusion
The complexity of the environmental degradation of FRP is resolved into only
three components including chemical degradation, physical degradation, and
modification of the stress state. The common effect of these three sub processes is
to change the material rheology over time. This has permitted the development of
a qualitative model based on the evolution of the material stiffness over time. This
model is expressed as a dynamic constitutive equation arising from the
combination of the historical variation in chemical link density and cohesive
forces with the stress or strain history of the material. This equation shows that the
environmental degradation causes amplification of stress relaxation and the
amplification factor is a determined function of the degradation time. Therefore,
the amplification factor provides a method to predict comprehensively FRP
environmental degradation. It can be determined experimentally from accelerated
stress relaxation test. The dynamic constitutive equation indicates also that the
process is thermally activated and prediction of long-term degradation can be
obtained from time-temperature superposition principle as experimentally
demonstrated. Application of this principle, in the case of environmental
degradation, takes account of the variation of the activation energy resulting from
the interaction between the stress and the chemical reaction over the course of the
degradation.
112
7 FINAL CONCLUSION
Efficient use of FRP materials requires reliable method to quantify, predict, and
control the long-term environmental effects on the material, hence the need for
modelling the degradation process. Throughout the literature, it was shown that
modelling and prediction efforts have been hindered by the complexity of the
process. Most of the published works are limited to characterization of effects and
mechanism or partial models, most of the time empirical. No viable prediction
method was yet available for the environmental degradation of the material
mechanical strength. The literature showed that major unresolved issues were
firstly, the difficulty to take into accounts all the processes involved and their
interactions in a comprehensive model and secondly, the uncertainty related to the
translation of laboratory tests conditions to the real service environment
conditions [1 - 3].
This analysis has suggested a theoretical approach resolving the complexity of the
degradation process into only three components including a chemical degradation,
a physical degradation, and an alteration of the stress state. Methods to manage
the variability of the real service environment were also introduced. The main
outcomes are described in the next three subsections.
7.1 Determination of the Long-term Degradation as an Average Value with
Upper and Lower Limits
This analysis has demonstrated that the average of the degradation rate as well as
its upper and lower limits can be determined in laboratory in a constant
environment. This was shown as below.
Based on material science theories, the analysis has demonstrated that, in a
constant environment, the chemical and physical degradation of a polymeric
Chapter Seven: Final Conclusion
113
matrix follows an exponential law. This results from interactions between the
transformation of material rheology and the diffusion process. The transformation
of the material rheology results from changes in the internal structure of the
material caused by temperature, moisture, chemical attack, or irradiation by UV
rays.
It was shown that the material rheology is mathematically related to the chemical
link density and environmental factors including moisture, temperature, and
concentration of the chemical in the material. The material rheology is
proportional to the chemical link density, inversely proportional to a power of the
diffused moisture, and proportional to a power of the material temperature.
Then, the material rheology was related to the diffusion coefficient via the Stokes-
Einstein equation [84, 85]. This has permitted effects of the change in material
rheology caused by environmental factors to be introduced in the diffusion
coefficient equation besides the temperature effects on the activation energy
defined by the Arrhenius law.
Subsequently, the material rheology was related to the chemical degradation
through the law of chemical rates and Fick?s law.
All the above has led to the integration of the chemical and physical degradation
into a mathematical relation too complex to solve, due to the variability of
environmental factors.
To be able to manage the variability of the environment in the modelling process,
a method to transform a variable environment into a constant environment was
suggested. It was shown that environmental condition history could always be
modelled as a sequence of constant environmental states.
Therefore, the complex mathematical relation of the chemical and physical
degradation was applied in a constant environment. The resulting mathematical
Chapter Seven: Final Conclusion
114
model is a simple exponential equation that relates the degradation rate directly to
the material lifetime.
Experiments have been conducted using variable measurement methods. These
include Raman spectrometry for chemical structure changes, rheometry for
storage modulus, and mechanical testing of tensile and shear strength evolution in
the course of degradation. All experiments confirm the suggested model. The
degree of correlation between the model and the experimental process is very
good as shown by the coefficient of linear correlation being very close to one.
The analysis has also demonstrated that the chemical and physical degradation
rate of a polymeric matrix is an ascending monotonic function of the environment.
This has provided the basis for the determination of laboratory test condition.
Since the degradation rate is a determined function of the service environment, it
was concluded that the statistical control limits of the degradation correspond to
degradations measured at the statistical control limits of the service environment.
Thus, laboratory tests can be conducted in constant environments that express
exactly the variability of the real service environment. This constitutes a solution
to problems related to the translation of laboratory tests conditions to the real
service environment conditions [1 - 3].
The use of a constant environment in laboratory also implies that, the
determination of the physical and chemical degradation requires monitoring only
of the chemical structure change or any other material property linearly correlated
to the chemical structure.
The ultimate outcome from all the above is that the average degradation rate, the
upper and lower limits of the degradation rate, hence the average long-term
degradation and its limits can be determined in a laboratory in a constant
environment, as exponential functions of the degradation time.
Chapter Seven: Final Conclusion
115
7.2 The Constitutive Equation of Environmental Degradation in FRP.
Following, the analysis has also shown that, the common effect of the three sub
processes constituting the environmental degradation is to change the material
rheology over time. This has permitted to develop a qualitative comprehensive
model based on the evolution of the material stiffness over the time. This model is
a dynamic constitutive equation arising from the combination of the historical
variation in chemical link density and cohesive forces with the stress or strain
history of the material. This equation shows that, the environmental degradation
causes the amplification of the stress relaxation and that the process obeys the
time temperature superposition principle. This has permitted to conclude that:
? The environmental degradation can comprehensively be quantified as stress
relaxation amplitude.
? The prediction of long-term effects is possible using the time temperature
superposition principle.
Experimental evidence of the above was provided. A polyester composite
laminate was used and exposure mode corresponded to that of a storage tank or
pipe.
The amplification function ( )tKenv was experimentally determined by mean of a
stress relaxation test and was shown to be a determined function of the
environmental degradation time.
Experimental data obtained up to 1000 seconds on the degradation time scale,
were used to predict longer time scale degradation up to 6000 seconds. Agreement
between predicted and experimental result was excellent showing that the time
temperature shift principle can be used to accelerated prediction test of the
environmental degradation of FRP. However, due to the interaction between
stresses and chemical degradation, a modified method to determine the shift factor
was introduced. The method requires the amplification factor to be directly
Chapter Seven: Final Conclusion
116
determined from short-term experimental data by plotting
refT
iT
t
t
against
relaxation amplitude.
Therefore, the average long-term degradation and its upper and lower limits can
be comprehensively measured in a constant environment through a stress
relaxation test in laboratory. The resulting short-term test method is as described
in the next section.
7.3 Short-term Test Method
Resulting from all the above a short term test method is suggested as follows:
7.3.1 Determination of the laboratory test conditions
Laboratory tests are conducted in a constant environment determined by the upper
and lower control limits of the environment control chart or its average.
Care must be taken to compensate the fatigue effect caused by temperature and
humidity cycles. To this end, the material exposed to a constant environment must
additionally and in the same time, be subjected to cyclic stresses equivalent to
those caused by the temperature and humidity cycles. The amplitude of the cyclic
stress in laboratory is also determined by the upper and lower control limits of the
cyclic stress in the real service environment.
7.3.2 Prediction of long-term stiffness coefficient based on ( )tKenv
Tests are to be conducted on specimen representing a barrier coat according the
following steps:
a. Short-term stress relaxation under mechanical stress only
b. Short-term stress relaxation under physical and chemical degradation.
Chapter Seven: Final Conclusion
117
c. From experimental results of stress relaxations, the amplification factor
( )tKenv is determined according to equation (6.4).
d. Determination of the long-term value of the stiffness coefficient ( )tC by
applying the time temperature shift principle.
e. Determination of the long-term stiffness coefficient under environmental
degradation by multiplying ( )tC by ( )tKenv .
7.3.3 Prediction of relaxation times based on time temperature shift principle
The time temperature shift principle can be applied to determine long-term
relaxation time as follow:
a. Determination of the reference temperature.
b. Short-term stress relaxation both at the reference temperature and the
temperature of interest.
c. Experimental determination of the shift functions by plotting
refT
iT
t
t
against
relaxation amplitude.
d. Determination of relaxation time by multiplying the time measured at the
reference temperature by the corresponding shift factor as per equation
(6.5)
It should be noticed that, in cases involving liquid environment, the reference
temperature is limited by the boiling point of the liquid and so acceleration of the
predictive test is limited. Thus, for prediction to longer time scales, the
amplification factor ( )tKenv offers better capability. The time temperature shift
principle will only be helpful, accelerating the determination of the amplification
factor.
Chapter Seven: Final Conclusion
118
7.4 Recommendations for Future Works
The implementation of the methods suggested in this analysis requires the
availability of reliable method for environmental stresses determination. As
mentioned in section 4.2, these stresses are of many kinds including hygrothermal
stresses from humidity and temperature cycles, erosive stresses from friction with
circulating liquid in pipes or tanks or friction with wind and dust, stresses from
rainfall impact, and vibration stresses in the vicinity of industrial engines. As part
of the real service environment, all these stresses need to be known and taken into
account when determining the laboratory test conditions. Therefore, the design of
methods to determine these stress constitute a compulsory next step in this
analysis.
Theoretical principles demonstrated in this analysis lay also useful basis for the
design of practical instruments for experimental evaluation of FRP environmental
degradation. The need for practical instruments to allow for rheological test in a
multifactor environment has been demonstrated. A typical basic idea for the
design of instruments to be used for the test of FRP environmental degradation is
that represented by the device used in this research (Figure 6.1). This offers
further opportunity for research in the field of environmental degradation.
The contribution of the present analysis is mostly to demonstrate theoretical
principles that will provide the basis for the development of useful analytical
techniques for the measurement of environmental effects on FRP. The
development of the mathematical model was mainly based on a theoretical
demonstration. However, experimental evidences provided further support to
theoretical conclusions. The resulting prediction method is a rigorous and logical
consequence of the mathematical model.
119
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APPENDIX A
It is shown that equation (4.1) is a common expression of well-established laws
that govern energy and mass transfer in materials. It corresponds respectively to
the Newton?s law for calorific flux, Fick?s law for moisture or chemical solution
flux and the definition of UV ray intensity. Equation (4.1) can be rewritten as
( )tfk
dt
dW
j
env
0= (A.1)
? If =envW heat energy, ( ) Ttf j ?= the temperature difference and L
hA
k ar=0 ,
where arA is the exchange surface area, L the material thickness, and h the
heat conduction or heat convection coefficient (in which case L =1), then
equation (A.1) corresponds to the Fourrier law or the Newton?s law for heat
transfer.
? If =envW photon energy from UV rays, and 10 =k , then equation (A.1)
corresponds to the definition of the UV rays intensity and ( ) UVj Itf = .
? Equation (A.1) corresponds also to the first law of Fick for mass diffusion. It
is understood that the diffusing mass conveys the mechanical or chemical
energy that causes the physical degradation (hygroscopic expansion) or the
chemical degradation. The conveyed energy is proportional to the diffusing
mass or volume. The first law of Fick is expressed as
x
C
DJ x
?
?
?= (A.2)
where J , D , and XC are respectively the diffusivity, the coefficient of diffusion,
and the concentration of the diffusing material at a distance x from the exposed
surface. If 18k being the coefficient of proportionality between the diffusing mass
128
and the energy, A the diffusion area, then the diffusing energy through an
elementary thickness dx is expressed from equation (A.2) as
( )
x
C
Ddx
txAk
txW Xenv
?
?
?=
??
?
18
2 ,
(A.3)
The diffusing energy through a laminate of thickness L is then obtained by
integration of equation (A.3) as
dtdx
x
C
DAkW
t
t
l
x
arenv ? ? ?
?
?=
2
1 0
18 (A.4)
Taking as reference the unexposed side of the laminate, the integration of equation
(A.4) shows that the first law of Fick corresponds exactly to the equation (A.1)
where ( ) 0Ctf j =
dtCDAkW
t
t
arenv ?=
2
1
018 (A.5)
129
APPENDIX B
DISCRETE INTEGRATION OF EQUATION (5.14)
Let the figure i1 represents a laminate. According to pipe or tank model the
diffusion occurs in the direction perpendicular to the laminate. It is admitted that
in the direction perpendicular to the diffusion, the material is homogeneous and
therefore the diffusion coefficient is constant throughout a surface perpendicular
to the diffusion direction.
A: total laminate area
?s: unit area A
?xi
C: chemical concentration
: unit thickness
l: total thickness ?s
l
Figure B.1: Laminate portion
The flux of the diffused material through ?s is given as follows:
i
S x
C
DJ
?
?
?=? [B.1]
Assuming that the total area is made of n unit areas, the total flux trough the area
A will be as follows:
S
n
S i
n
S
S x
C
DJ ??
==
??
?
?
??
?
?
?
?
?=?
11
[B.2]
?x1?x2 ?xL
130
As the diffusion coefficient is constant throughout A the equation [B.2] may be
written as follows:
i
n
S
S x
C
nDJ
?
?
?=??
=1
[B.3]
The total flux through the laminate thickness l is given by the sum of the
individual flux crossing each single surface layer as expressed below.
???
== = ?
?
?=?
l
i i
l
i
n
S
iS x
C
nDJ
00 1
, [B.4]
Now the sum of all the partial concentration gradients through the unit thickness
?x i
is equal to the total concentration gradient throughout the total thickness l
?
=
?
=
?
?l
i
l
i l
CC
x
C
0
0 [B.5]
The total diffused material content in the laminate, at any instant, is exactly given
by the total flux crossing the laminate at this instant. And this flux is given by the
equation [B.4]. Then taking into account equation [B.5]
l
CC
nDC lch
?
= 0 [B.6]
In practice as C0 >> Cl, one can approximate the difference C0 ? Cl to C0
.
l
C
nDCch
0= [B.7]
131
APPENDIX C
SOLUTION OF THE DIFFERENTIAL EQUATION OF THE
DEGRADATION RATE
The degradation rate equation was found to be as follow:
(See subsection 5.9.2)
?? += d
d ETC
dt
dE
00 (C.1)
To simplify the notation, let be:
Ed= y, ?0TC0
Then equation (C.1) may be noted as follows:
= constant = u, ? = constant= v, and t = x.
y? = uy + v (C.2)
The derivation of equation (C.2) gives
y?? = uy? (C.3)
Thereby,
u
y
y
=
'
''
(C.4)
Equation (C.4) can be integrated as follows:
?? =
tt
udxdx
y
y
00 '
''
(C.5)
[ ] [ ]tt uxy 00'ln = (C.6)
132
( )
( ) uty
ty
=
0'
'
ln (C.7)
( ) ( ) uteyty 0'' = (C.8)
From equation (C.2), we take:
y?(0) = uy(0) + v (C.9)
Now y(0) = Ed(t =0)
= 0 as at the initial instant there is no degradation. Thus
y?(0) = v = ? (C.10)
Equation (C.8) then becomes:
tTCd e
dt
dE
00??= (C.11)
133
APPENDIX D
DETERMINATION OF THE MATERIAL LIFETIME BASED ON THE
VARIATION IN CHEMICAL LINK DENSITY
The rate of variation in chemical link density was obtained from Raman peaks by
numerical regression followed by derivation. Then, the variation rate of index Ld
( ) ( )
dt
tdR
dt
tdL pd ?=
was determined by the opposite of the variation rate of Raman peaks:
(D.1)
where ( )tRp represents the Raman peak at time t. This has led to equation (5.44)
(page 91).
Integrating equation (D.1):
( ) ( )tRtL pd ??=? (D.2)
( ) ( ) ( ) ( )tRRLtL ppdd ?=? 00 (D.3)
( ) 00 =dL because there is no degradation at the initial time. The value of ( )0pR
is obtained from equation (5.44) and is equal to 1992.7. Then from equation (D.3),
the degradation index of chemical link density is expressed as
( ) )1(7.1992 260.0 td etL ??= (D.4)
Values of Ld
were calculated from equation (D.4) for various value of time and
represented in Table 5.4
Resolving equation (D.4) with respect to time, the material lifetime for a
maximum allowable degradation d? is expressed as
( )dt ???= 1ln
260.0
1
(D.5)
134
In equation (D.5), the degradation d? is expressed as a fraction of the initial value
of the material tensile strength.
According to the suggested model, the degradation of Ld
corresponds to the
degradation of the mechanical strength. Equation (D.5) was then used to predict
material lifetime for various degradation in tensile strength shown in Table 5.5.
For illustration, the maximum allowable reduction in tensile strength is given as
80 % of the initial value. Then, the material lifetime is determined as a maximum
allowable exposure time as
( ) 2.68.01ln
260.0
1
=?
?
=t (D.6)
Now, taking account of the scale used for the measurement of time, this value is
reduced to 6 days.