Nonlinear elastic waves in materials described by a subclass of implicit constitutive equations
No Thumbnail Available
Date
2018
Authors
Magan, Avnish Bhowan
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
The propagation of displacement and stress waves for a subclass of implicit
constitutive equations in a rectangular slab and a circular cylinder is investigated.
The general class of implicit constitutive equations contain Cauchy
elastic materials and hyperelastic materials as subclasses. We consider a special
subclass of implicit constitutive equations where the strain is prescribed
in terms of a non-invertible function of the stress. Two constitutive equations
are studied. The rst constitutive equation is called the power-law constitutive
equation due to its analogy with the constitutive equation for a power-law
uid
with exponent n in the expression for stress. This constitutive equation can
describe elastic responses where the stress and linearised strain are nonlinearly
related. Classical Cauchy elasticity and hyperelasticity cannot capture such
a phenomenon. The second constitutive equation is called the strain-limiting
constitutive equation. A feature of this constitutive equation is that it can
describe materials that exhibit limiting stretch. To derive the mathematical
models we assume a special semi-inverse solution where a speci c form for both
the displacement and stress are sought. This assumption leads to a system
of nonlinear partial di erential equations. The system of partial di erential
equations can be reduced to a single nonlinear hyperbolic partial di erential
equation which describes the propagation of solitary stress waves. The perturbation
solutions for the system of partial di erential equations describes
either travelling waves or standing waves. To nd travelling wave solutions
for the displacement and stress in a rectangular slab we reduce the perturbation
equations at each order to canonical form and solve the resulting wave
equations. For the circular cylinder we could not obtain travelling wave solutions
by reduction to canonical form. We nd standing wave solutions for
the displacement and stress in both the rectangular slab and circular cylinder.
For the rectangular slab the solutions to the perturbation equations contain
a secular term. However, the straightforward perturbation expansion breaks
down outside the range of interest. The standing wave solution in the circular
cylinder can only be solved at the zero and rst order since the equations at
the second order could not be solved analytically. The solutions at this order
are however, su cient to describe the physical properties of the wave. In the
standing wave solutions for the displacement and the stress at each end, at the
centre and surface of the cylinder, either the displacement or the stress vanish
or the spatial gradients of the displacement or stress vanish. We nd expressions
for the speed of the solitary stress wave for both constitutive equations
in both the rectangular slab and circular cylinder. The speed of propagation
decreases in parts of the wave for large stress magnitudes for the power-law
constitutive equation and increases in parts of the wave for large stress magnitudes
for the strain-limiting constitutive equation. The solitary stress wave
develops a shock front at the front of the wave for the strain-limiting constitutive
equation and at the back of the wave for the power-law constitutive
equation. The shock develops at the front for the strain-limiting constitutive
equation at a much earlier time than at the back for the power-law constitutive
equation. For the travelling wave solutions in the rectangular slab the wave
front is determined from the condition that the displacement at the wave front
is zero. The stress is non-zero at the wave front and propagates as a shock
wave with a strong discontinuity. We nd that the speed of propagation of the
displacement wave and the stress wave is slower for the power-law exponent
n > 0. Further the amplitude of the displacement waves are approximately
the same for n = 0 and n > 0 while the amplitude of the stress wave is less
for n > 0. The standing waves for both constitutive equations in both the
rectangular slab and circular cylinder showed that for the power-law constitutive
equation the period of oscillation remained approximately the same for
n = 0 and n > 0 while it increased for n > 0 for the strain-limiting constitutive
equation. In general the elastic response is enhanced for materials described
by the power-law constitutive equation and inhibited when described by the
strain-limiting constitutive equation.
Description
A thesis submitted to the Faculty of Science, University of the
Witwatersrand, Johannesburg, in ful llment of the requirements for the
Doctor of Philosophy. Johannesburg, December 2017
Keywords
Citation
Magan, Avnish Bhowan (2018) Nonlinear elastic waves in materials described by a subclass of implicit constitutive equations, University of the Witwatersrand, Johannesburg, <http://hdl.handle.net/10539/25870>