Stability of rectangular bar with distributed dislocations

The problem of the influence of distributed dislocations on the equilibrium stability of a rectangular bar loaded with a longitudinal force is investigated. In the subcritical (unperturbed) state, the body experiences a finite plane inhomogeneous deformation. The dislocation density tensor depends on the coordinate measured along the bar thickness and has only one nonzero component corresponding to the distribution of edge dislocations. Within the framework of the compressible semilinear material model, the unperturbed state is defined as an exact solution to the equations of the nonlinear continuum theory of dislocations. Stability analysis is performed using the Euler bifurcation method, which consists in finding nontrivial solutions to the linearized homogeneous boundary value problem of the equilibrium of a prestressed elastic body. The influence of different types of dislocation distribution on the critical values of the longitudinal force and the form of stability loss of the bar is studied. It is established, in particular, that dislocations significantly affect the number of waves along the length of the bar, characterizing the form of stability loss.