Nonlinear elastic waves in materials described by a subclass of implicit constitutive equations

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.