Traditionally, it is assumed that a deformable body yields under load at the moment when the stress-state or the stored energy parameters at the most critical points in the body reach the maximum allowable values and that the moment of failure is uniquely determined by the strength constants of the material. However, there is experimental evidence that-the rigidity of the loading system also affects the resistance to failure. Here the loading system includes both the loading device (the testing machine, the structural elements which transmit the load, the working fluid or gas, etc.) and area of the deformable body around the failure zone [1, 2].If the load is "soft," i.e., if the loading forces are independent of the resistance of a body in a homogeneous stress state, then failure really does correspond to the maximum stresses. In the other limiting case (a "rigid" load), the boundary points are displaced by a given amount, and damage can accumulate in an equilibrium process, which is reflected by a descending section in the stress-strain diagram [3][4][5][6][7][8][9]. If the loading system has finite rigidity, the moment that load-carrying capability is lost can correspond to one point or another on the descending part of the stress-strain diagram. The material state corresponding to the highest point on the stress-strain diagram is the critical point, but -to be more precise -failure is the final rapid non-equilibrium stage of this process, and can be viewed as the result of the loss of stability of accumulated damage at a supercritical strain stage. Also, the concept of a supercritical strain stage allows the reserves of load-carrying capacity to be used in optimizing structure design.A more precise calculation that uses the total strain diagram requires the formulation and solution of boundary-value problems that consider material yield [10][11][12][13], and also possible stability losses in the weakened zones [4, 13, 14]. Here we present new boundary conditions that consider the rigidity of the loading system, formulate the defining equations, introduce supercritical strain conditions, obtain stability criteria for damage accumulation at the supercritical strain stage for an elementary material particle, and give a formulation of boundary-value problems that considers these effects within the framework of the mechanics of deformable solids.1. Equation of State. For a material with microdamage, the stress tensor a is related to the strain tensor e in terms of a fourth-order damage-vulnerability tensor fl by a defining equation in the form [15] ( 1.1) where C is the elastic modulus tensor; the Iklra n = (1/2)(~/on~tn + ~/m) are the components of the unit tensor, and ~ is the Kronecker delta.In this model, all processes that change the material state are described by the damage-vulnerability tensor operator 0, whose components are uniquely defined by the strain (loading) process. If the stresses can be defined by knowing the strains only at the current moment of time, then fl is a function. When experimental...