Linear Stability of a Two-Phase Process Involving a Steadily Propagating Planar Phase Boundary in a Solid. Part 1. Purely Mechanical Case.

摘要

This work investigates the linear stability of an antiplane shear motion which involves a planar phase boundary in an arbitrary element of a wide class of nonelliptic generalized neo-Hookean materials which have two distinct elliptic phases. It is shown, via a normal mode analysis, that, in the absence of inertial effects, such a process is linearly unstable with respect to a large class of disturbances if and only if the kinetic response function-a constitutively supplied entity which gives the normal velocity of a phase boundary in terms of the driving traction which acts on it or vice versa-is locally decreasing as a function of the appropriate argument. An alternate analysis, in which the linear stability problem is recast as a functional equation for the interface position, allows the interface to be tracked subsequent to perturbation. A particular choice of the initial disturbance is used to show that, in the case of an unstable response, the morphological character of the phase boundary evolves to qualitatively resemble the plate-like structures which are found in displacive solid-solid phase transformations. In the presence of inertial effects a combination of normal mode and energy analyses are used to show that the condition which is necessary and sufficient for instability with respect to the relevant class of perturbations in the absence of inertia remains necessary for the entire class of perturbations and sufficient for all but a very special, and physically unrealistic, subclass of these perturbations. The linear stability of the relevant process depends, therefore, entirely upon the transformation kinetics intrinsic to the kinetic response function.