On a stability of polar symmetrical deformation of bodies from softening materials

Special case of continuum mechanical systems is considered. It is believed that deforming is carried out under conditions of polar symmetry of stresses and strains. Also it is assumed that material properties are described by Hencky model with softening under nonpositivity of volume deformation. Then union curve has region decreasing to zero. Aforementioned conditions are realized in such problems as expansion of spherical cavity in softening space and deforming of thick-walled spherical vessel by equable external pressure (it maybe bathyscaphe which is gradually submerged to the deep). Based on the Lagrange formalism integral quadratic functional is investigated. This functional is increment of total potential energy in the form of Lagrangian for mentioned problems. This study allows to formulate conditions of buckling for active loading which changes quasistatically. For considered problems sets of possible deformations are obtained. These possible deformations perturb the equilibrium position and do not break kinematic constraints. Obtained sets of possible deformations allow to write criterion of buckling of deformation process in explicit form for mentioned problems. It is established that only with sufficiently developed softening zone buckling of deformation process is possible.