Stability and possible bifurcations for a Gent-Thomas elastic parallelepiped subject to dead-load surface tractions

We study the equilibrium and the local stability for an incompressible elastic solid under arbitrary dead-load surface tractions on the boundary. We particularize the analysis to homogeneous deformations of a homogeneous, isotropic parallelepiped, finding necessary and sufficient algebraic local stability conditions consistent with the further requirement of the zero moment condition. We specialize the study for a uniform distribution of dead-load surface tractions s > 0 on two pairs of faces and -s on the remaining two faces, and we find that two classes of equilibrium solutions may occur: symmetric and asymmetric solutions, respectively. For the symmetric solutions we also determine local stability inequalities. For the special case of a Gent-Thomas material we show that both equilibrium symmetric and asymmetric solutions may occur if the material parameters satisfy certain inequalities. Then, we completely describe the response of the parallelepiped in a loading process starting from the unloaded state for five ranges of the values of the material parameters. In particular, for one of these ranges we show that symmetric solutions are the unique locally stable homogenous equilibrium deformations until a critical value scr of the load; at scr, symmetric solutions lose their uniqueness, and a bifurcation into asymmetric solutions may occur.