Vol.:(0123456789)

https://doi.org/10.1007/s11340-024-01047-z

RESEARCH PAPER

Stress Evaluation Through the Layers of a Fibre‑Metal Hybrid 
Composite by IHD: An Experimental Study

J. P. Nobre1,2  · T. C. Smit2 · R. Reid2 · Q. Qhola2 · T. Wu3 · T. Niendorf4

Received: 23 October 2023 / Accepted: 19 February 2024 
© The Author(s) 2024

Abstract
Background Incremental hole-drilling (IHD) has shown its importance in the measurement of the residual stress distribu-
tion within the layers of composite laminates. However, validation of these results is still an open issue, especially near the 
interfaces between plies. 
Objectives In this context, this study is focused on experimentally verifying its applicability to fibre metal laminates.
Methods Tensile loads are applied to cross-ply GFRP-steel [0/90/steel]s samples. Due to the difference in the mechanical properties 
of each ply, Classical Lamination Theory (CLT) predicts a distribution of the uniform stress within each layer, with pulse gradients 
between them. The interfaces act as discontinuous regions between the plies. The experimental determination of such stress vari-
ation is challenging and is the focus of this research. A horizontal tensile test device was designed and built for this purpose. A 
differential method is used to eliminate the effect of the existing residual stresses in the samples, providing a procedure to evaluate 
the ability of the IHD technique to determine the distribution of stress due to the applied tensile loads only. The experimentally 
measured strain-depth relaxation curves are compared with those determined numerically using the finite element method (FEM) 
to simulate the hole-drilling. Both are used as input for the IHD stress calculation method (unit pulse integral method). The distri-
bution of stress through the composite laminate, determined by classical lamination theory (CLT), is used as a reference.
Results Unit pulse integral method results, using the experimental and numerical strain-depth relaxation curves, compare 
reasonably well with those predicted by CLT, provided that there is no material damage due to high applied loads.
Conclusions IHD seems to be an important measurement technique to determine the distribution of residual stresses in fibre 
metal laminates and should be further developed for a better assessment of the residual stresses at the interfaces between plies.

Keyword Fibre-metal laminate · Composite laminate · Residual stress · Hole-drilling method

Introduction

Advances in composite technology have allowed significant 
weight reduction in structural design. Fibre reinforced poly-
mers (FRP) usually present greater strength and stiffness to 

weight ratios compared to metallic alloys, also providing 
excellent fatigue behaviour and corrosion resistance [1]. 
Fibre metal laminates (FML), primarily developed to be 
used in aerospace structures, are hybrid composite materi-
als consisting of layers of metallic alloys bonded to FRP 
plies during the autoclave curing process [2]. The use of 
these two types of materials to form a hybrid composite 
material enhances its mechanical behaviour by combining 
the best properties of each constituent [3, 4]. The curing 
process of FRPs and FMLs always induces residual thermal 
stresses due to the mismatch in the coefficients of thermal 
expansion and mechanical properties between the constitu-
ents, the polymer cure shrinkage, and other parameters that 
affect their magnitude and distribution [5]. Curing of the 
FML is usually followed by a post-stretching process to 
mitigate the induced residual stresses [1, 2]. The presence 
of residual stress alters the midrange stress (or mean stress) 

 * J. P. Nobre 
 joao.nobre@dem.uc.pt

1 Univ Coimbra, CFisUC, Department of Physics, Coimbra, 
Portugal

2 School of Mechanical, Industrial and Aeronautical 
Engineering, University of the Witwatersrand, Johannesburg, 
South Africa

3 Fraunhofer Institute for Ceramic Technologies and Systems, 
Dresden, Germany

4 Institute of Materials Engineering, University of Kassel, 
Moenchebergstr. 3, 34125 Kassel, Germany

/ Published online: 5 March 2024

Experimental Mechanics (2024) 64:487–500

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http://orcid.org/0000-0002-9616-9518


within materials and, therefore, the stress ratio during cyclic 
loading. Fatigue strength can either be enhanced or com-
promised depending on the contribution of residual stress 
to the stress ratio [6]. The overall material strength may 
be adversely affected by the introduction of fibre waviness 
resulting from the presence of residual stresses in FMLs [7]. 
Moreover, tensile residual stress may initiate microcracks 
at pre-existing defects, such as voids within FRP plies, irre-
spective of whether there is any external loading imposed 
on the laminate [8, 9]. In addition, the use of high perfor-
mance thermoplastics requires greater processing tempera-
tures compared to thermoset epoxy matrices, for example 
in GLARE (GLAss REinforced Aluminium [2, 3]) that was 
successfully used in large parts of the Airbus A-380 fuselage 
[10]. Therefore, it is a crucial design consideration for resid-
ual stresses to be thoroughly understood and quantified in 
FML components that are used in applications where struc-
tural integrity and safety are of high importance. Moreover, 
as recently stated by Vu et al. [11], at the minimum, reliable 
experimental methods are needed for the evaluation and 
verification of the residual stresses predicted using micro-
mechanical models in FRPs. In this context, the incremental 
hole-drilling (IHD) technique has been demonstrated to be 
a viable option to experimentally evaluate residual stresses 
in composite laminates [12–19]. 

However, beyond the difficulty in calculating residual 
stresses from the measured strain relaxation during IHD in 
composite laminates, other issues related to the technique 
itself remain. Firstly, the thermomechanical effects of fric-
tion and plastic deformation in the drilling process generate 
heat around the hole, which can modify the residual stresses 
before they are measured. This is particularly true in the 
case of polymer materials [20]. Secondly, the IHD tech-
nique assumes linear elastic material behaviour, but local 
damage such as delamination can arise due to high residual 
stresses induced by the stress concentration around the hole. 
In these materials, the geometric stress concentration factor 
also depends on the material and ply stacking sequence, and 
can be greater than that observed in isotropic materials [21].

In this work, all these issues will be simultaneously  
analysed in a FML. FML samples are subjected to tensile 
loading, where the applied loads can be finely controlled. In  
composite laminates subjected to pure tensile loading the 
stress through the thickness is not uniform as in isotropic 
materials, but uniformly distributed in each ply with step/
pulse gradients at interfaces. Theoretically, the interfaces are 
discontinuous stress regions where stresses change abruptly, 
as predicted by Classical Lamination Theory (CLT) [22, 23]. 
The experimental determination of such stress distributions 
is attempted in this work, which is a difficult experimental 
case for IHD. The main goal is to experimentally determine 
the strain-depth relaxation curves during IHD in “stress free” 
FML samples subjected to a tensile load. This is a difficult 

issue, since the samples are not stress free, but are subjected 
to residual stresses arising from the manufacturing process.  
However, based on a hybrid experimental-numerical method 
(HENM) [24], the effect of these initial stresses can be elimi- 
nated, allowing the experimental determination of the IHD 
strain-depth relaxation curves arising only from the applied 
tensile load and possible cutting effects. Simulation of the 
hole-drilling process is performed using the finite element 
method (FEM), considering the same data as in the experi-
mental tests. During the numerical simulation, incremental 
hole-drilling is simulated by removing the elements from the 
mesh corresponding to a given depth increment. Numerical 
strain-depth relaxation curves, arising only from the applied 
loads, can thereby be determined. The experimental and 
numerical curves differ only in the possible thermomechani-
cal effects that develop during hole-drilling of the samples. 
This allows the through-thickness variation in stress distri-
bution in the FML to be compared between the IHD unit 
pulse integral method and CLT.

The Incremental Hole‑Drilling (IHD) Technique 
and its Stress Calculation Methods

IHD involves stepwise drilling of a small hole in a sample  
and measuring the surface strain relief around the hole after 
each incremental depth increase. However, there is no direct 
relationship between the relieved strain measured at the sur-
face and the residual stress existing at each depth increment. 
The well-accepted integral method can be used to relate the 
relieved strain to the residual stresses. This method, pro-
posed by Schajer in its final format in 1988 [25], considers  
that the strain relief at the surface is the accumulated result 
of the residual stresses originally existing in the region of 
each successive increment, along the total hole depth. The 
residual stresses are considered constant within each depth 
increment. To completely define the plane stress state at each  
depth increment, a typical standard three-gauge rosette is  
commonly used [26]. For isotropic materials [26], the inte-
gral method relates strain relaxation measurements and 
residual stresses through an integral equation that can be 
solved considering a set of standard calibration coefficients,  
which are determined by the finite element method, depend-
ing only on the increment size, geometry of the hole and 
rosette used. These coefficients are nearly material independ- 
ent, and therefore, a standard procedure could be proposed 
(ASTM E837) [26]. However, since composite laminates are 
orthotropic and layered in nature there is a full dependency 
on the material and ply stacking sequence. Additionally, the 
trigonometric relation that can be used to relate strains and 
stresses around the hole in the isotropic case is not valid and 
the stresses cannot be decoupled [15]. In 2014, Akbari et al. 
[12] extended the pioneering method of Schajer and Yang 

488 Experimental Mechanics (2024) 64:487–500



[15] to the incremental hole-drilling measurement of non-
uniform residual stresses in composite laminates, proposing 
the fully coupled unit pulse integral method, which can be 
written mathematically as:

where εk(i) is the relaxed strain measured by strain gauge 
number k when the hole depth is i increments deep, σjl cor-
responds to the residual stress at depth j in the direction of 
the strain gauge l and  Cijkl is the compliance matrix.

Equation (1) considers all three stress and strain direc-
tions simultaneously and is valid provided that linear elas-
tic material behaviour exists. In composite laminates, each 
calibration coefficient in the calibration matrix is replaced 
by a 3 × 3 matrix to consider the elastic constants of the 
orthotropic plies. Each element of this matrix contains the 
measured strain of gauge k for a hole of a depth i, when the 
l component of the residual stress in the range j-1 ≤ i ≤ j is 
equal to a unit stress. Other researchers have followed this 
approach for residual stress measurement in composite lami-
nates [13, 14, 16, 19, 27].

In this work the strain gauges are fixed at three known 
positions. For example, strain gauge 1 is parallel to the fibre 
direction x, strain gauge 3 is in the transverse direction y and 
strain gauge 2 is positioned at 45° (between strain gauges 1 
and 3 as in the ASTM type B rosette [26]) or at -135° from 
strain gauge 1 (for the ASTM type A rosette [26]). Coeffi-
cients Ckl are calculated for a specific alignment between the 
strain gauges and the fibres, which needs to be considered 
during measurement. These calibration coefficients reflect 
the strains measured at the surface due to the relief of the 
different residual stresses at each incremental depth of the 
hole. For example, the presence of a unit stress value σx 
at increment j, contributes to strain values ε1(i), ε2(i) and 
ε3(i) and the contribution of this unit stress value to the 
measured strains is given by the coefficients Cij11, Cij21 and 
Cij31, respectively. The same applies for unit stresses σy and 
τxy. Each matrix Ckl in the main compliance matrix Cijkl can 

(1)�k(i) =

3∑
l=1

i∑
j=1

�jl ⋅ Cijklk, l = 1, 2, 3 1 ≤ i ≤ n,

be obtained by FEM. Figure 1 summarises the process of 
obtaining these influence matrices for the case of the ASTM 
type A rosette and the final compliance matrix Cijkl.

In 2018, Smit and Reid [17] proposed a new approach, 
extending the power series method initially proposed by 
Schajer [28] in the 80’s. In the new approach, power series 
are applied separately to each ply orientation instead of being 
applied to the total hole depth as in Schajer’s work. In the 
original method, due to effects of numerical ill-conditioning, 
only first order polynomials could be used which reduced its 
spatial resolution since only linear residual stress distribu-
tions could be determined through the total hole depth. The 
advantage of this method, despite its greater complexity and 
more difficult practical application, is that errors decrease as 
the number of the depth increments increases. The integral 
method behaves inversely due to the error propagation in the 
inverse problem involved. For this reason, Tikhonov regu-
larization [29] was proposed to approximate the ill-posed 
problem of the integral method with a well-posed problem to 
reduce scattering and variance in the final results, while also 
allowing and even benefiting from the use of an increased 
number of depth increments [30]. This regularization pro-
cedure was adopted in the 2013 revision of the ASTM E837 
standard [26] for the case of isotropic materials. As pointed 
out by Akbari et al. [12], based on the work of Zuccarello 
[31], the number of the calculation steps should be less than 
10 to achieve stable results when the integral method is used 
without Tikhonov regularization in composite laminates. 
Attempts have been made to apply Tikhonov regularization 
to the unit pulse integral method in composite laminates 
[16, 18]. In such laminates, the released strain at a depth 
increment is influenced by all components of residual stress 
in a form that cannot be described analytically at present 
and the stresses cannot be decoupled, since the relationship 
between residual stress and released strain does not have 
a trigonometric form. These facts prevent direct applica-
tion of Tikhonov regularization to IHD in FML composites, 
unless the effect of secondary contributions to each strain 
component is neglected [16, 18]. If the configuration of a 
composite laminate, as in the case of the samples used in the 

Fig. 1  (a) Construction of 
the compliance matrix  Cijkl. 
(b) Strategy to determine the 
compliance matrix elements Ckl 
based on the unit stresses σx = 
1 MPa and σy = τxy = 0; σy = 1 
MPa and σx = τxy = 0; τxy = 1 
MPa and σx = σy = 0 

489Experimental Mechanics (2024) 64:487–500



present work, results in diagonal coefficients of each 3 × 3 
matrix in equation (1) that are dominant due to Poisson’s 
effects, strain and stress in the x direction can be related by:

Similar equations can be found for strain and stress 
components in y and xy directions. Regularization must be 
adapted to account for the discontinuities at the interfaces 
between material types. The rows of the regularization oper-
ator, L, on both sides of an interface must be set to zero. This 
ensures that no regularization is applied across the interface 
between material types [32]. Tikhonov second-derivative 
regularization can then be implemented separately for each 
stress component. For example, for the x component:

where α is the regularization parameter that controls the 
degree of regularization applied. Similar equations can be 
written for the other components.

The quality of the calculated stress distribution can be 
significantly affected by the degree of regularization, which 
can be optimised through the Morozov Discrepancy Princi-
ple [33] to maximally remove noise effects without distort-
ing the residual stress distribution. The required value of α 
can be found iteratively by making the chi-squared statistic, 
x
2, equal to the number of depth increments. Each ply must 

be treated separately because the estimated standard error, e, 
varies between plies. The average local misfit norm [34] is 
used to estimate the standard strain error separately in each 
ply. This ensures that smooth stress variations exist within 
each ply and that deviations from the smooth variation are a 
result of the measurement noise [35]. The local misfit norm 
must exclude the misfit across interfaces between plies of 
different material properties because the released strain at 
these interfaces can be discontinuous in slope. This ensures 
that the standard error due to slope discontinuities is not 
over-estimated, thereby avoiding distortion of the stress 
results through excessive regularization. The chi-squared 
statistics when there are n plies with x increments per ply 
can be determined by [16, 34]:

where �meas
i

 is the measured strain and �calc
i

 is the regularized 
fit to the strain data that can be calculated using equation (2). 

(2)�
x
= C

x
⋅ �

x
⇔ �x(i) =

i∑
j=1

Cijxx ⋅�jx.

(3)
(
C
T

x
C
x
+ �

x
L
T
L
)
�
x
= C

T

x
�
x
,

(4)

�2 =

x∑
i=1

(
�calc
i

− �meas
i

eply1

)2

+

2x∑
i=x+1

(
�calc
i

− �meas
i

eply2

)2

+⋯ +

nx∑
i=x(n−1)+1

(
�calc
i

− �meas
i

eplyn

)2

,

Considering that each ply has x ≥ 4 increments per layer, the 
standard error in each ply, eplyi, can be estimated by [34]:

After applying Tikhonov regularization to each stress 
component, the calculated regularized strain fit in each 
experimental measurement direction can be combined into 
a full strain vector, εcalc, in the form of equation (1). Equa-
tion (1) with the fully coupled calibration matrix, C, and the 
regularized strain vector, εcalc, is then used to calculate the 
regularized residual stress distribution. The regularization 
parameters may require adjustment if the stress solution 
shows signs of unstable behaviour or distortion.

Smit et al. [16] performed a comparison between the 
calculation methods referred to above in cross-ply GFRP-
steel samples, as used in this work, using neutron diffrac-
tion (ND) to evaluate the residual stresses in the metallic 
layers. Their results showed a good agreement between 
IHD and ND in the metallic layers. However, the residual 
stresses determined in the fibre reinforced plies were not 
experimentally validated. The present work was developed 
to achieve this goal.

Materials and Methods

Fibre‑Metal Laminate Samples

Symmetric laminates with a GFRP/Steel/GFRP lay-up 
were used in this work. The glass fibres used in the sam-
ples were G U300-0/NF-E506/26% type (with 26% resin 
content), and HC340LA micro-alloyed sheets of steel. The 
samples were manufactured using an advanced intrinsic 
manufacturing technology, in which the fibre compound 
and core metal layer were bonded during the formation 
of the fibre reinforced polymer. More specifically, a pre-
pared glass fibre prepreg comprising two plies (with a 
stacking sequence of [0°/90°]) were placed in a heated 
die of dimensions 25 mm × 250 mm × 100 mm. A metal 
sheet, with sand blasted surfaces (to enhance bonding 
behaviour) was placed on the prepreg, and GFRP with 
a [90°/0°] stacking completed the symmetric stacking as 
shown in Fig. 2. A punch heated to 160 °C was used to 
apply a pressure of 0.3 MPa to the laminate. The curing 
process occurred, at atmospheric conditions, for a period 
of 18 min. The adhesive bond was simultaneously created 
within the GFRP itself (between the matrix and fibres) and 
between the GFRP layers and the metal sheet layer.

The mechanical properties of the samples were obtained 
according to DIN EN ISO 527, DIN 256 EN ISO 14129 and 

(5)e2
plyi

≈

x−3∑
i=1

(
�i − 3�i+1 + 3�i+2 − �i+3

20(x − 3)

)2

.

490 Experimental Mechanics (2024) 64:487–500



DIN EN ISO 6892-1 as already detailed in previous work 
[36]. Table 1 presents the mechanical properties determined 
for the GFRP plies and steel core of the hybrid composite 
used in the present work.

Experimental Procedure and the HENM Method

The first studies on induced drilling stresses during the 
application of the hole-drilling technique were performed 
in metallic materials, where suitable thermal treatments can 
be applied to relieve residual stresses to achieve an initial 
“stress-free” state. Beaney and Procter [37] introduced the 
air abrasion technique for “stress-free” hole-drilling. This 
“stress-free” state was roughly estimated by Flaman and 
Herring [38] for different metals and alloys. Previously, Fla-
man had proposed the use of a pressurized air turbine for 
ultra-high-speed milling (up to ~ 400,000 rpm), for a “stress-
free” application of the hole-drilling technique to metals and 
their alloys [39]. Nowadays, all commercial equipment for 
the hole-drilling technique uses Flaman’s drilling proce-
dure. Approaches to study induced drilling stresses during 
hole-drilling were investigated further by Weng et al. [40] 
using samples cut by electric discharging machining (EDM). 
However, all these approaches are essentially qualitative and 
can only give a rough estimate of induced drilling stresses 

during IHD. In addition, these approaches cannot be applied 
in fibre reinforced polymers where possible thermal damage  
to the matrix and the mismatch in the thermal expansion 
coefficients of the constituents prohibits their application. 
A hybrid experimental-numerical method (HENM) was  
proposed to overcome these difficulties in carbon fibre rein-
forced polymers [41]. Further studies include the successful 
application of the method to quantify the effect of the drill-
ing operation in glass fibre reinforced polymers [24].

The HENM method is based on the comparison between 
the strain relaxation field, measured during incremental 
hole-drilling on a specimen subjected to a well-known 
applied tensile load (calibration load), and the strain relaxa-
tion field calculated by hole simulation on a semi-infinite 
plate subjected to the same loading using FEM. More pre-
cisely, during incremental hole-drilling a set of curves of 
strain relaxation as a function of depth, corresponding to a 
given stress state and possible cutting effects, is obtained. 
This set of curves can also be obtained considering only 
the applied tensile load without cutting effects through 
hole-drilling simulation using FEM. The direct comparison 
between the experimental and numerical curves allows esti-
mation of the effect of the cutting procedure itself. However, 
the resulting experimental strain relaxation is the superpo-
sition of the applied loading, cutting effects and existing 

Fig. 2  Fibre-metal laminate 
sample. (a) Stacking ply con-
figuration; (b) sample dimen-
sions; (c) ASTM E837 type A 
strain-gauge rosette [26] applied 
on the surface of the samples

Table 1  Mechanical properties 
of the GFRP-steel samples 
constituents [36]

Yield Stress
[MPa]

Young’s Modulus
[GPa]

Poisson’s Ratio Shear Modulus [GPa]

Material Ex Ey Ez vxy vxz vyz Gxy Gxz Gyz

GFRP - 34 8.8 8.8 0.33 0.33 0.37 5.23 5.23 3.21
Steel 380 210 0.29

491Experimental Mechanics (2024) 64:487–500



residual stresses in the material, i.e., those existing prior 
to hole-drilling. Therefore, it is necessary to eliminate the 
effect of the existing residual stresses. To accomplish this, 
a differential load is applied instead of an absolute one, as 
presented in Fig. 3.

In previous works [24, 41], hole-drilling is performed for 
the minimum applied load  F1. After each depth increment, the 
strains are recorded, and the sample is loaded up to the maxi-
mum load  F2. The strains are recorded again, and the sample 
unloaded up to the minimum load  F1. This process is repeated 
for all incremental depths until the total hole depth. In the cur-
rent work, to minimize possible errors due to the recentring 
process of the hole and to mitigate the possible influence of 
cutting effects, hole-drilling was performed twice, i.e., for 
the minimum and maximum applied loads. Using this differ-
ential method, the effect of the existing residual stresses can 
be cancelled and only the applied load affects the measured 
strain relaxation. Thus, the strain relaxation vs. depth obtained 
experimentally for the calibration load ΔF (related to Δσ) dur-
ing incremental hole-drilling can be used as input for the unit 
pulse integral method, which returns the stresses in the com-
posite laminate due to the calibration load. Considering all 
experimental data, the finite element method is further used 
to simulate the hole-drilling in a sample subjected to the same 
external load ΔF (Δσ). The strain relaxation vs. depth obtained 
numerically can also be used as input for the unit pulse integral 
method. When an external tensile load is applied to a compos-
ite laminate, uniform stress distributions exist in each ply, with 
discontinuities at interfaces. Classical lamination theory can be 
used to analytically determine the stress distribution through 
the laminate’s plies [22, 23], as explained in next section.

To conduct the experimental evaluation of IHD in sam-
ples subjected to well-known tensile loads, a horizontal 

tensile-test machine (HTTM), depicted in Fig.  4, was 
designed and built with rigidity as a main design criterion. It 
was important, amongst many design factors, for the HTTM 
to be able to accommodate simultaneous use of the SINT 
Technology MTS3000-Restan IHD hole-drilling machine. 
The hand driven HTTM can be operated at tensile/compres-
sion speeds of approximately 0.1 mm/min, allowing quasi-
static tensile conditions to be attained on samples.

Before mounting the samples in the machine, a set of type 
A standard strain gauge rosettes (062UL, Vishay Precision 
Group, Inc.) [26] were bonded on their surfaces. A dummy 
rosette was always used for temperature compensation. All 
strain gauges were connected in a 1/4. Wheatstone bridge 
circuit to a data acquisition system (HBM Quantum X) with 
sixteen channels. Three channels were used for IHD meas-
urements, three for temperature compensation and an addi-
tional one to connect the precision load cell (HBM S9M). 
In the present study, two differential loads equal to 700 N 
and 3500 N were selected. The first calibration load is based 
on a minimum and maximum load of 500 N and 1200 N, 
respectively, and the second based on 500 N and 4000 N, 
respectively. It should be noted that initial residual stresses 
in the samples were determined in a previous study by IHD 
and neutron diffraction [16]. The greatest residual stresses 
(approximately 150 MPa and 200 MPa in the longitudinal 
and transverse directions of the samples, respectively) were 
compressive and were determined inside the steel layer near 
the interface with the 90° ply. In the steel layer, a maximum 
tensile residual stress of slightly greater than 50 MPa was 
determined at a depth greater than 0.8 mm, where the uncer-
tainty from IHD is already high. Additionally, CLT predicts 
a maximum tensile stress of 146 MPa in the steel layer when 
the tensile load of 4000 N is applied. Therefore, there was no 

Fig. 3  Principle of superpo-
sition used in the proposed 
method. σRS is the existing 
residual stress, σ1cal and σ2cal 
are stresses resulting from the 
applied load  F1 and  F2, ΔF and 
Δσ are the calibration load and 
stress

492 Experimental Mechanics (2024) 64:487–500



plastic deformation in the steel layer during the tensile tests. 
The preliminary tests also showed full recovery of strain when 
the maximum tensile load was removed. Even when consider-
ing the stress concentration induced by the hole during IHD, 
the maximum stress is below 60% of the steel’s yield stress 
and no plasticity effects on IHD results would occur in the 
steel layer [42] (the ASTM E837 standard suggests a limit of 
80% for uniform stresses [26]). However, the stress induced 
by the greatest load is likely to produce transverse cracking in 
the 90° plies of the GFRP laminate, since the residual stress 
transverse to the fibers in the GFRP plies already exceeds 
50 MPa in tension. IHD was performed in the tensile samples 
using depth increments of 10 μm, to reduce drilling induced 
heating effects in the GFRP plies and to improve accuracy of 
polynomial interpolation, up to 1 mm depth. Polynomial inter-
polation was used to fit and smooth strain relaxation data, and 
the stress evaluation was performed considering four incre-
ments per GFRP ply, which correspond to 55 μm incremental 
depths. In all tests, a feed rate lower than 0.2 mm/min and a 
time delay greater than 25 s were used. At least three hole 
combinations for F1 and F2 were performed in each sample 
for a total of six cases for F1 and F2, respectively.

Classical Lamination Theory (CLT)

CLT was used to predict the theoretical stress variation 
across the thickness of each layer of the FML sample while 
subjected to traction, considering the orthotropy of its plies. 
Stress [σ]θ remains uniform across each layer θ and relates 
to the mid-plane strain [ε]0 through the lamina stiffness 
matrix [Q] as shown in equation (6) [22]. The lamina stiff-
ness matrix Qij components of each ply (and θ fibre direc-
tion) are evaluated from the reduced stiffness matrix Qij 
components through a series of equation (7) [22]. Note that 
in the position (3,3), the subscripts (6,6) are related to the 
constitutive equation for an orthotropic material obtained 

from an anisotropic one. In addition, Q16 and Q26 are zero 
for the current samples, since extension-shear coupling is 
not present in laminae loaded at angles of 0° and 90° relative 
to the fibre direction.

where:

The reduced stiffness matrix components Qij at each ply 
are directly dependent on the material properties: elastic 
moduli (E1, E2 and G12) and Poisson's ratios (v12 and v21), 
as can be determined from equation (8) [22, 23].

In the case of symmetric laminates, the mid-plane strain 
matrix (in equation (6)) relates to the calibration stress 
resultant Nx and the extensional stiffness matrix [A], for 
each respective layer thickness tk, through equations (9–10) 
[22, 23].

and.

(6)
⎡⎢⎢⎣

�x
�y
�xy

⎤⎥⎥⎦
�

=

⎡⎢⎢⎣

Q11 Q12 0

Q21 Q22 0

0 0 Q66

⎤⎥⎥⎦
⋅

⎡⎢⎢⎣

�x
�y
�xy

⎤⎥⎥⎦
0

(7)

⎧⎪⎪⎨⎪⎪⎩

Q11 = Q11 cos
4 � + 2

�
Q12 + 2Q66

�
cos2 � sin2 � + Q22 sin

4 �

Q12 =
�
Q11 + Q22 − 4Q66

�
cos2 � sin2 � + Q12

�
cos4 � + sin

4
�

�

Q22 = Q11 sin
4 � + 2

�
Q12 + 2Q66

�
cos2 � sin2 � + Q22cos

4 �

Q66 =
�
Q11 + Q22 − 2Q12 − 2Q66

�
cos2 � sin2 � + Q66

�
cos4 � + sin

4
�

�

(8)

⎧⎪⎪⎨⎪⎪⎩

Q11 =
E1

1−�12�21

Q22 =
E2

1−�12�21

Q12 = Q21 =
�12E2

1−�12�21
=

�21E1

1−�12�21

Q66 = G12

(9)Aij =

n∑
k=1

[
Qij

]
k
⋅ tk

Fig. 4  Horizontal tensile-test machine used in the experimental calibration tests. The samples had a span length of approximately 160 mm

493Experimental Mechanics (2024) 64:487–500



Inverting equation (10), the mid-plane strains �0
ij
 can be 

determined and stresses through the plies can be evaluated 
from equation (6). If necessary, stress–strain transformation 
equations can be used to determine the stresses according to 
the principal material coordinate system.

Numerical Simulation

Numerical simulation was performed using two finite ele-
ment models. For both cases, APDL scripts were developed 
for ANSYS Mechanical finite element analysis software. 
Higher order 3-D 20-node solid elements (SOLID186) 
[43] that exhibit quadratic displacement behaviour, were 
used to model the different material layers. These elements 
are defined by 20 nodes having three degrees of freedom 
per node: translations in the nodal x, y, and z directions. 
These elements were used with two available key options 
in ANSYS; the layered structural solid option to model the 
layered thick shells, which allow modelling of the differ-
ent glass fibre plies, and the homogeneous structural solid 
option to model the metallic (steel) core.

The first model was developed to determine the calibration 
coefficients that relate the relieved strains to the stresses pre-
sent in the material and can be easily modified for a given com-
posite laminate. Nine elastic constants (Ex, Ey, Ez, νxy, νxz, νyz, 
Gxy, Gxz, Gyz) are required to relate the stresses and strains in an 

(10)
⎡
⎢⎢⎣

Nx

Ny

Nxy

⎤
⎥⎥⎦
=

⎡
⎢⎢⎣

A11 A12 0

A21 A22 0

0 0 A66

⎤
⎥⎥⎦
⋅

⎡
⎢⎢⎣

�x
�y
�xy

⎤
⎥⎥⎦
0

orthotropic and layered material – see Table 1. To determine 
the whole compliance  Cijkl matrix, according to the description 
provided previously (see Fig. 1), a full 3D cylindrical model 
with an external diameter equal to 5 times the average radius 
of the strain gauge rosette (to avoid edge effects) was used. 
Figure 5(a) shows the FEM mesh used in this case with the 
constraints applied only on the outer diameter of the model and 
Fig. 5(b) shows an enlarged view of the mesh near the hole, 
showing the position of the strain gauge grids of the ASTM 
type A rosette used (Fig. 5(c)). Only the nodes at outer diam-
eter, as shown in Fig. 5(a), were constrained to avoid free body 
motion. Since, in this case, the thickness can be considered a 
sensitive parameter in FE analysis, as previously observed [44, 
45], the model should only be constrained far from the hole 
position to avoid boundary effects.

For obtaining the whole compliance  Cijkl matrix, the 
effects of σx, σy and τxy on the strain, εk, measured by each 
gauge of the rosette, are considered separately. All matrix 
values can be determined considering three different unit 
stress states, as shown in Fig. 1(b). As stated before, the 
presence of a unit stress value σx  = 1 and σy  = τxy  = 0, at 
increment j, contributes to strain values ε1(i), ε2(i) and ε3(i) 
and the contribution of this unit stress value to the meas-
ured strains is given by the coefficients Cij11, Cij21 and Cij31, 
respectively, for the first column. The second column, con-
sidering σy  = 1, σx = τxy = 0, and the last column considering 
a pure shear stress state τxy  = 1, σx = σy= 0. The unit loads are 
converted into the cylindrical coordinate system by equation 
(11) [14], and σr and τrθ are applied at the hole boundary, 
within each depth increment, using the SURF154 3-D struc-
tural surface effect element [43].

Fig. 5  (a) 3D-FEM mesh for the determination of calibration coefficients matrix  Cijkl. (b) Enlarged view near the hole (1 mm depth), showing 
the strain gauge grids (1.57 mm length) of the (c) 3-element ASTM type A rosette used (062UL, Vishay Precision Group, Inc.)

494 Experimental Mechanics (2024) 64:487–500



A second model was developed for the hole-drilling 
simulation in the samples subjected to a given applied uni-
axial tensile load, as per the HENM method – see Fig. 3. 
To determine the strain-depth relaxation curves when the 
samples are subjected to a given tensile calibration load (in 
the present study 700 N and 3500 N, respectively), consider-
ing the existing symmetries, a ¼ 3D plate model was used 
as shown in Fig. 6(a). Strain gauge grids are also shown in  
Fig. 6(b). The superposition of the grids here is apparent, 
because strain gauge 2 is artificially placed in the first quad-
rant. As for the other model, it was only constrained at the 
basal nodes of the external boundary of the plate to prevent 
rigid body motion. Both models have the same mesh con-
figuration and refinement within a radius that extends to the 
outer edges of the gauges.

A total sample thickness of 1.88 mm, corresponding to 
0.22 mm thickness for each GFRP ply and 1 mm thickness 
for the internal steel core was considered (as per Fig. 2). 
A mean hole diameter of 1.8 mm, as observed during the 
experimental tests was used. The incremental hole depth 
simulation was performed using the “birth and death of 
elements” ANSYS code features, considering four depth 
increments per GFRP ply (55 μm each) and a further ten 
increments in the steel core, for a total of eighteen depth 
increments up to a depth of roughly 1 mm. After each depth 
increment, the nodal displacement values were integrated 
over an area equivalent to the strain gauge grids, following 
a procedure proposed by Schajer [46].

(11)

�
σr
τrθ

�
=

�
cos2 θ sin2 θ sin 2 θ

−0.5 ⋅ sin 2 θ 0.5 ⋅ sin 2 θ cos 2 θ

�
⋅

⎡⎢⎢⎣

σx
σy
τxy

⎤⎥⎥⎦

Results and Discussion

First, it was necessary to verify the linear elastic hypoth-
esis between the applied load and the measured strains dur-
ing the tensile tests. Figure 7(a) shows the evolution of the 
measured strain in the longitudinal direction of the sample 
with the applied tensile load, during loading and unload-
ing. A linear relationship between the applied load and the 
strain measured is observed. A coefficient of determination 
R squared near 1 was determined and, therefore, the hypoth-
esis is satisfied.

Thus, considering the CLT theory for an applied tensile 
load of 700 N and a 25 mm wide sample, an Nx = 28 N/mm 
force resultant is obtained and used to evaluate the stresses in 
each ply through equation (6). For this loading (Ny = Nxy = 0), 
the distribution of the longitudinal stress through the plies 
of the laminate can be determined. Longitudinal uniform 
stresses in each layer (σx) of 4.2, 1.0 and 25.7 MPa, respec-
tively, for the GFRP fibre orientation at 0°, 90° and the 
steel layer are determined for the FML sample shown in 
Fig. 2. Considering the cross-sectional area of the samples 
(1.88 mm × 25 mm), a uniform mean stress of 14.9 MPa 
through the thickness would be expected for the case of an 
isotropic material. However, since each ply of the laminate 
presents a different mechanical behaviour, there is a distribu-
tion of stress through the plies of the laminate, as predicted 
by both CLT theory and numerical simulation using FEM, as 
shown in Fig. 7(b). There is an almost exact match between 
CLT and FEM results. Note the discontinuity in stress when 
passing from one ply to another. Similar results are obtained 
when a tensile load of 3500 N is applied. In this case, lon-
gitudinal uniform stresses in each layer (σx) of 20.8, 5.0 and 

Fig. 6  (a) 3D-FEM mesh for hole-drilling simulation of the GFRP-steel plate subjected to a given tensile load. (b) Enlarged view near the hole 
(1 mm depth), showing the strain gauge grids (1.57 mm length) (ASTM type A - 062UL, Vishay Precision Group, Inc.)

495Experimental Mechanics (2024) 64:487–500



128.6 MPa, respectively, for the GFRP fibre orientation at 
0°, 90° and the steel layer are determined. However, for the 
3500 N load, transverse cracking is likely to occur at lay-
ers with the fibres oriented at 90° to the direction of the 
tensile load due to the stress concentration induced by the 
hole and the initial residual stresses within the FML. The 
main research question is whether the IHD technique can 
accurately determine such stress distributions.

As explained previously, the experimental IHD is per-
formed when the samples are subjected to a minimum and 
maximum load. These minima and maxima led to a calibration 
load of 700 N and 3500 N, respectively. The resultant strain 
relaxation curve versus depth for each tensile load can be seen 
in Fig. 8. Uncertainty analysis was performed considering the 
total number of tests carried out and the standard deviation 

is included for the average strain relaxation observed in each 
incremental depth. As expected, maximum values are observed 
at strain gauge 1 which is oriented with the longitudinal direc-
tion of the samples and coincides with the direction of the 
applied load. In the transverse direction, strain measurement 
in strain gauge 3 is only due to Poisson’s effects.

In Fig. 8(a), a variation in slope is observed in the strain 
relaxation curves when passing from one layer to another. 
For the 3500 N load in Fig. 8(b), this variation is not evident. 
In both Figs., dotted lines correspond to the best polynomial 
fit to the relieved strain in each ply. This interpolation is 
necessary to determine the strain relaxation when different 
numbers of depth increments per ply are considered. These 
curves ( �exp

i
(z)) will be further used as input for the integral 

method for residual stress calculation by IHD.

Fig. 7  (a) Applied axial load vs. measured strain in the longitudinal direction of the samples (in x direction as per Fig. 2). (b) Longitudinal stress 
(x) arising in each laminate’s ply for a tensile load of 700 N, according to CLT and numerical simulation using FEM

Fig. 8  Experimental strain relaxation vs. hole depth. (a) Tensile load of 700 N and (b) 3500 N

496 Experimental Mechanics (2024) 64:487–500



Considering the FEM simulation of the incremental hole-
drilling, based on the data obtained during the experimental 
tests, a set of numerical curves ( �num

i
(z)) can also be deter-

mined, which can be superimposed on the experimental 
ones. Figure 9 shows the obtained results. Some observations 
should be pointed out. Hole-drilling simulation by FEM, 
assuming the material behaves linearly and elastically, pre-
sents strain-depth relaxation curves with the same trend for 
both applied loads, regardless of the difference in the mag-
nitude of the measured strain relief. As seen in Fig. 9(a), for 
the lowest applied load of 700 N, there is a good agreement 
between the experimental and numerical results in all glass 
fibre layers, but a slight deviation in the steel layer. The most 
important results are those in the direction of the strain gauge 
1 which is oriented with the direction of the tensile load. 
The stresses calculated from these curves for IHD are more 
related with their slopes than with their absolute values. For 
the 3500 N load, as shown in Fig. 9(b), there is only a good 
agreement in the first ply, corresponding to the layer with the 
fibres oriented at 0° (GFRP 0°). The divergence between the 
experimental and numerical results observed in the second 
ply (GFRP 90°) is probably a result of the appearance of 
transverse cracking or delamination occurring in this layer. 
To attain the calibration load of 3500 N, a maximum load of 
4000 N was applied to the samples. This high applied load, 
together with the stress concentration induced by the hole 
and the existing tensile residual stresses, as determined in 
previous work [16], has a high probability of having caused 
transverse cracking or delamination and consequently having 
increased the measured strain relaxation. Similar concerns 
have been expressed in previous work [16]. This effect can 
explain the divergence observed in the experimental strain 
relaxation values compared to the numerical ones at this load.

The experimental, �exp
i

(z) , and numerical, �num
i

(z), strain-
depth relaxation curves can be used together with the  
calibration coefficients matrix,  Cijkl, determined by FEM, 
to evaluate the stress distribution in the FML samples via 
IHD. Equation (1) or equation (3), for using the unit pulse 
integral method with or without regularization, respectively, 
can then be used to compute the stress distribution. For the 
numerical strain-depth relaxation curves no regularization 
is needed since there is no noise or outliers. For the residual 
stress calculation by the integral method, strong numeri-
cal ill-conditioning effects begin when the total hole depth 
becomes greater than half of the hole diameter [25]. This 
behaviour has also been confirmed in GFRP laminates [47]. 
Therefore, stress calculation is performed up to a hole depth 
equal to the half of the hole diameter (~ 0,9 mm). Consid-
ering four increments per FRP ply and a total of sixteen 
depth increments, the unit pulse integral method without 
regularization (equation (1)) was used with experimental and 
numerical strain-depth relaxation curves. Tikhonov regulari-
zation (equation (3)) was only used with the experimental 
strain-depth relaxation curves. Figure 10 shows the results 
obtained, compared with those determined by CLT.

For both applied loads, IHD numerical results agree very 
well with the results predicted by CLT in all glass fibre lay-
ers. The difference observed in the steel layer (around 3% 
avg.) could be related to the small numerical differences 
induced by the fact that two FEM models are used (a full 
3D cylindrical model for the determination of C matrix and 
a ¼ 3D plate for the hole-drilling simulation in the tensile 
samples). The full 3D cylindrical model developed can be 
used for any composite laminate subjected to IHD, while the 
¼ 3D plate models the samples subjected to tensile loading 
during the performed experiments only.

Fig. 9  �exp
i

(z) vs. �num
i

(z) , where z corresponds to the hole depth. (a) Tensile load of 700 N and (b) 3500 N

497Experimental Mechanics (2024) 64:487–500



For the lowest applied load of 700 N, a good agreement 
is observed in all GFRP plies, as shown in Fig. 10(a), which 
corroborates previous observations made by Nobre et al. 
[24]. The average stress value is near the stress predicted 
by the CLT theory and IHD numerical results. This is also 
valid for the first GFRP layer, for the case of the 3500 N 
applied load. However, for this load, the experimental stress 
determined by IHD in the 90° layer is not uniform, as pre-
dicted by the CLT theory and numerical IHD, but presents 
an increasing trend through the layer, which is believed to 
be due to the appearance of transverse cracking or delamina-
tion, as previously mentioned. The high strain values in this 
layer also affect the stress results observed in the steel layer, 
which are much greater than that predicted by the CLT the-
ory and numerical IHD. For the steel layer and for the 700 N 
applied load, experimental IHD stress results are much more 
reliable than those obtained for the 3500 N load, but not 
uniform as expected and as theoretically and numerically 
predicted. There is a parabolic stress distribution, instead 
of a perfectly uniform one. Tikhonov regularization slightly 
improves the results. The small experimental strain errors 
in the GFRP layers lead to higher errors in stress at deeper 
increments corresponding to the steel layer. This relates with 
the intrinsically high error sensitivity and error propagation 
of the integral method itself (ill-posed inverse problem). 
The polynomial interpolation conducted in each layer is 
less constrained at its ends. Consequently, larger errors are 
to be expected near the interfaces. At the GFRP-steel inter-
face, this effect, coupled with the high modulus of the steel, 
results in a large error in the first stress measured within the 
steel. The stress near the interface in the GFRP is, however, 
accurately determined by IHD. Additional work should be 
carried out regarding this issue to improve the accuracy of 
IHD in the tensile samples, such as using cubic splines for 

the interpolation of the experimental strain relaxation val-
ues and improving the regularization procedures for IHD in 
composite laminates.

Conclusions

Based on the results of this study, incremental hole-drilling 
can evaluate the stress distribution in fibre-metal hybrid 
composites subjected to tensile loading, using commercially 
available equipment for IHD, provided that the tensile load 
is low enough to avoid transverse cracking or delamination 
in the fibre reinforced plies.

Stresses near interfaces, where the stress is discontinu-
ous, were determined with acceptable accuracy in the GFRP 
layers, whilst at the deeper GFRP-steel interface a discrep-
ancy was observed. The discrepancy near this interface 
seems to be related with the large stress variation observed 
at this interface, the uncertainty of the strain relaxation 
measured during IHD and the polynomial fit needed to 
describe the experimental strain-depth relaxation curves 
through the thickness of the composite laminate. Stresses 
near singularity regions at FRP-metal interface at deeper 
layers are challenging to be accurately determined, due to 
the greater uncertainty associated with measured strain 
relaxation at deeper increments. Nonetheless, IHD seems 
to be an appropriate measurement technique to provide the 
necessary information about the residual stresses arising in 
these regions.

Additional research is needed to extend regularization 
procedures to these materials for a better assessment of non-
uniform residual stresses through the plies of the laminate, 
particularly near the interfaces at deeper layers. Improving 
accuracy in measuring strain relief, the use of cubic splines 

Fig. 10  Stress �exp

xi
(z) vs. �num

xi
(z) , vs. �theor

xi
(z) . (a) Applied load of 700 N and (b) 3500 N

498 Experimental Mechanics (2024) 64:487–500



for fitting the measured strain relaxation through the depth 
(especially at interface regions) or the use of the separate 
series expansion method could enable reducing the size of 
depth increments and, therefore, enhance the determination 
of residual stresses in these regions.

Acknowledgements Professor T. Tröster and his group at University 
of Paderborn (Automotive Lightweight Design, (LiA)) are thanked for 
providing the samples used in this research.

Author Contributions Conceptualization, J.P. Nobre; methodology, 
J.P. Nobre and T.C. Smit; software, J.P. Nobre, T.C. Smit and T. Wu; 
validation, J.P. Nobre and T.C. Smit; formal analysis, J.P. Nobre and 
T.C. Smit; investigation, Q. Qhola, J.P. Nobre and T.C. Smit; resources, 
T. Niendorf, J.P. Nobre and R. Reid; writing—original draft prepara-
tion, J.P Nobre; writing—review and editing, T.C. Smit, R. Reid, T. 
Wu and T. Niendorf; supervision, T. Niendorf, R. Reid, J.P. Nobre and 
T.C. Smit; project administration, T. Niendorf, J.P. Nobre and T. Wu; 
funding acquisition, T. Niendorf and J.P. Nobre. All authors have read 
and agreed to the published version of the manuscript.

Funding Open access funding provided by FCT|FCCN (b-on).  The 
research leading to these results received funding from German 
Research Foundation (DFG – Deutsche Forschungsgemeinschaft) 
under Grant Agreement No 399304816 and from the National Research 
Foundation of South Africa (NRF) under Grant Agreement No 106036. 
This work was also supported by funds from FCT – Fundação para a 
Ciência e a Tecnologia, I.P., within the projects UIDB/04564/2020 and 
UIDP/04564/2020.

Data Availability  All data presented in this manuscript will be avail-
able on request.

Declarations 

Conflict of Interest The authors declare that they have no conflict 
of interest.

Statements on Human and Animal Rights Not applicable.

Informed Consent Not applicable.

Open Access This article is licensed under a Creative Commons Attri-
bution 4.0 International License, which permits use, sharing, adapta-
tion, distribution and reproduction in any medium or format, as long 
as you give appropriate credit to the original author(s) and the source, 
provide a link to the Creative Commons licence, and indicate if changes 
were made. The images or other third party material in this article are 
included in the article’s Creative Commons licence, unless indicated 
otherwise in a credit line to the material. If material is not included in 
the article’s Creative Commons licence and your intended use is not 
permitted by statutory regulation or exceeds the permitted use, you will 
need to obtain permission directly from the copyright holder. To view a 
copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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	Stress Evaluation Through the Layers of a Fibre-Metal Hybrid Composite by IHD: An Experimental Study
	Abstract
	Background 
	Objectives 
	Methods 
	Results 
	Conclusions 

	Introduction
	The Incremental Hole-Drilling (IHD) Technique and its Stress Calculation Methods
	Materials and Methods
	Fibre-Metal Laminate Samples
	Experimental Procedure and the HENM Method
	Classical Lamination Theory (CLT)
	Numerical Simulation

	Results and Discussion
	Conclusions
	Acknowledgements 
	References