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The problem of uniaxial compression of an incompressible 2D thin rectangle is studied in this paper. We consider the case where the two ends of the rectangle are welded to two rigid bodies. The focus is on the bifurcation analysis and post-buckling solutions. The combined series–asymptotic expansion method is used to derive two coupled non-linear ordinary differential equations (ODEs), which govern the leading-order axial strain and shear strain. Through an analysis on the linearized equations, an algebraic equation for determining the critical stress values of buckling is obtained. For the non-linear coupled ODEs, the numerical solutions of the non-trivial solutions are obtained by providing proper initial guesses. Energy analysis shows that for the first mode of buckling material failure may first happen at the middle point of the bottom surface. [ABSTRACT FROM PUBLISHER]

This paper concerns the global structure stability of impact-generated tensile waves in a 1D bar made of a rubber-like material. Because the stress–strain curve changes from concave to convex as the strain increases, the governing quasi-linear system of partial differential equations, though hyperbolic, fails to be ‘genuinely non-linear’ so that the standard form of the initial-boundary value problem corresponding to impact is not well-posed at all levels of loading. However, Knowles (2002, SIAM J. Appl. Math., 62, 1153–1175) constructed the solutions of the initial-boundary value problem corresponding to impact. Based on this, in this paper we prove the global structure stability of the impact-generated tensile waves constructed by Knowles. The method of the proof is constructive. [ABSTRACT FROM PUBLISHER]

This paper studies the propagation of disturbances superimposed on a pre-stressed incompressible hyperelastic thin rod. Starting from the incremental equations given by Haughton and Ogden (1979, J. Mech. Phys. Solids 27, 179-212 and 489-512), we derive a three parameter-dependent one-dimensional rod equation as the governing equation. In particular, it is found that one parameter plays a crucial role. Depending on whether it is larger or smaller than or equal to a critical value, the shear-wave velocity is larger or smaller than or equal to the bar-wave velocity. In the case that these two velocities are equal, there exist travelling-wave solutions of arbitrary form. This implies that for this particular case the initial disturbance would propagate along the rod without distortion. To see the influence of the pre-stress in detail, we further consider an initial-value problem with an initial singularity in the shear strain. The solutions are expressed in terms of integrals through the method of Fourier transform. We then conduct an asymptotic analysis for the solutions. For a material point in a neighbourhood behind the shear-wave front, the phase function of these integrals has a stationary point at infinity. Here, we use a technique of uniform asymptotic expansion to handle this case. An asymptotic expansion, correct up to order O(t-l), for the shear strain, which is uniformly valid in a neighbourhood behind the shear-wave front, is derived. For material points in other spatial domains, the method of stationary point is applicable, and asymptotic expansions (correct up to order O(t-1)) are obtained. A novelty is that we are able to deduce precise qualitative information about the waves in the far field from our analytic results. Wave profiles for two concrete examples are also provided. [ABSTRACT FROM AUTHOR]