With reference to plane strain, incompressible elasticity, it is shown that bifurcation of elastic materials deformed incrementally can be interpreted as the occurrence of waves propagating at null speed. After the plane wave propagation is solved for an infinite medium as a perturbation superimposed on a finitely and homogeneously strained elastic material, a wave propagation analysis in elastoplasticity elucidates the meaning of divergence instability (occurrence of negative eigenvalues of the acoustic tensor) and the difficulties (related to the fact that the constitutive tangent operator is piece-wise linear) connected with the interpretation of flutter instability. Finally, the treatment of acceleration waves reveals that the condition of localisation of deformation in elastoplasticity can be understood as the condition of vanishing speed of acceleration waves.
Wave propagation in solids is a topic strictly connected with stability and bifurcation. It will be shown in this chapter that the condition for incremental bifurcation analysed in chapter 12 for elastic solids is equivalent to the condition of vanishing propagation speed for an incremental wave mode, whereas instability corresponds to a blow up of the wave mode amplitude during propagation.
The simple example of small-amplitude vibrations of a beam superimposed on a given axial stress (‘pre-stress’) is sufficient to clarify the above-mentioned issues. To this purpose, we reconsider the beam illustrated in Section 10.2.3, subjected to axial load F corresponding to a longitudinal Cauchy stress - σ parallel to the beam axis x1.