4 nonlinear version of the phenomenological theory of long-term strength of polymer materials (viscoelastic bodies) is proposed. It is based on the introduction of a function accounting for the damage accumulation connected with changes in the load intensity. The form of this function may be determined from the results of testingThe phenomenological approach is widely and efficiently used for estimating the long-term strength (durability) of materials and structures. It is based on the notion that a material subjected to a long-term load fails duo to damage accumulation, quantitatively characterized by a scalar or tonsorial quantity (which, at the moment of failure, reaches a certain limiting value). For viscoelastic materials (including polymeric ones), this approach is presented, in the most general form, in the well-known paper of A. A. II'yushin [1]. Some studies in this feld (see, for example, [2][3][4]) are mainly dedicated to the application of the general II'yushin's criterion in estimating the durability of some particular materials under different modes of loading and in selecting the respective effective functions (kernels) and nonlinear stress functions, as well as in formulating the limiting conditions of failure under combined stress states using different strength theories. The results of these studies provide a necessary basis for reliable estimation of the durability of polymer materials under constant loads and, in general, under all loads whose intensity varies only slightly with time.At significantly variable and particularly periodic (cyclic) loads, the durability of materials calculated by the known procedures, as a rule, proves to be overestimated as compared to its experimental values. This is apparently explained by the fact that the constitutive relations of the phenomenological theory of long-term strength of viscoelastic bodies, constructed by a formal analogy with the hereditary theory of viscoelasticity, do not reflect quite adequately some specific features of the process of damage accumulation in the material (fast of all, its essential irreversibility) and allow us to take them into account only indirectly ~ in the form of additional corrections. In particular, the accelerated failure of the material in cyclic loading is usually associated with its "softening" due to self-heating [2]. However, the same regularity is also observed for thermostable materials (for example, reinforced plastics based on thermosetting resins) at rather low loading frequencies when the effect of self-heating is practically absent. This allows us to assume the existence of a specific mechanism of damage accumulation in the materials which is associated with the changes in the load intensity with time. In what follows, we propose a variant of the phcnomenological theory of long-term strength taking into account the above-mentioned fe, atures. The damage of the material is represented by a symmetric second-order tensor H(d# ), the material is quasi-isotropic, and the moment (gradient) effects are neglected....