Концентрації напружень у пружному тілі з тонким в’язкопружним включенням

Abstract

Отримано розв’язок задачі про напружено-деформований стан пружного тіла з тонким в’язкопружним включенням. Для включення еліпсоїдальної форми і матеріалу з реологічними властивостями тіла Кельвіна розв’язок отримано в замкненому аналітичному вигляді. Знайдено напруження у включенні та їх концентрацію в матриці в околі включення.

The known in literature solutions of problems of stress concentration in the vicinity of inclusions, in thin ones, in particular concern the elastic materials of matrix and heterogeneity. In engineering practice the problems arise, for solution of which it is not enough to use the models of elastic bodies only. Such is, for example, the problem of cracks “healing” with injection liquid materials that are able to harden in certain time. As a result the material damaged by cracks hardens and is able to bear certain loads. Since the injection materials are usually high molecular compounds, which are characterized by creep and stress relaxation, the problem of these phenomena consideration within the framework of complex rheological models of materials becomes very urgent. The elastic space subjected to the effect of constant uniaxial tension and in which a defect is as an ellipsoidal inclusion V, one characteristic size of which is much smaller than the other two, is considered. The inclusion material is considered to be a visco-elastic one. Taking into account the small thickness of the inclusion the boundary conditions from its surface are moved to the middle region. As result the boundary value problem for a space with the cross section along the middle region, on which the appropriate boundary conditions are set, has been obtained. The solution of the problem for each moment of time is obtained within the static theory of elasticity and is expressed in terms of a harmonic function, which is presented as a Fourier integral expansion. After satisfying boundary conditions a two-dimensional singular integral equation for displacements determination, in which time is presented as a parameter, has been obtained. After some transformations, the solution of the integral equation is reduced to the Volterra equation f(t). As an example the material the creep of which is described by the generalized Kelvin model is considered. After corresponding calculations the relationships, which characterize the stress relaxation in the inclusion, are obtained. Plots of the stress relaxation in the inclusion and changes of stress concentration with time in its vicinity due to the inclusion material creep are constructed. Calculations testified that the stress concentration increases with time and finally stabilizes at a certain level. It is found that the stress concentration obtained in the elastic and viscoelastic models can vary considerably and it must be taken into account when calculating the long-term strength of the heterogeneous materials containing visco-elastic inclusions.