The article gives an overview and analysis of the models of the damage accumulation in monolithic materials when exposed to prolonged and repeatedly applied load. The application of the principle of equivalence stress in the continuous and damage body allowed introducing into the strength criterion of Pisarenko-Lebedev and the three-parameter plasticity condition of Coulomb-Mohr the measures of the theory to accumulate damage in capacity of which the damage of Y.N. Rabotnova and the continuity of L.M. Kachanov are used. It was found that when exposed to repeated load, the process of the reduction of the continuity and the increase in the damage is hereditary. Therefore, to predict changes in these measures under the action of cyclic loads the integral equations of the theory of heredity are applied. This simulation was carried out in two ways. The essence of the first method is that the change of the damage and the continuity in the effects of a repeated stress is determined by the integral equations where the integrand functions are given by means of the degree of dependence. The integration of these functions allowed receiving generalizing logarithmic and exponential models to calculate the reduction of the continuity and the increase in the damage when subjected to cyclic loading. The second way is to apply to the calculation of the damage the principle of equivalence of the deformation or energy where a decrease in the elasticity modulus of the damaged body is described by an integral equation. It was demonstrated that the first method of the calculation of the measures of the theory of the damage accumulation is similar to the principle of equivalence of deformations. There is some brief information about the experimental technique to determine the measures of the theory of the damage accumulation by means of which the parameters of the generalized models for some asphalt concretes are set. The difference of the proposed methods to predict the changes of the damage from analogue is shown.