Пружно-пластична задача для обмеженої пластинки з м’яким еліптичним включенням

Abstract

Розглянуто пружно-пластичну задачу для обмеженої пластинки із м’яким еліптичним включенням. Визначення напруженого стану проведено за допомогою методів граничних елементів та змінних параметрів. Досліджено напружено-деформований стан пластинок залежно від розміщення включення та прикладеного навантаження з урахуванням зміцнення матеріалу.

Experimentally it is found that most of the materials are more or less macroscopically inhomogeneous. In particular, structural elements contain many inclusions, which can induce crack initiation. When the stress is sufficient to deform permanently the material of a solid or an inclusion, elasto-plastic deformations should be also accounted for, because plastic deformation involves the breaking of a limited number of atomic bonds by the movement of dislocations. Thus, determination of the elasto-plastic state in the vicinity of inclusion tips is essential in the study of the plate-like structural elements strength. Similar studies were provided mainly for the truly elastic case, or for the infinite plates. However, in practice the size of inclusions are often comparable with the size of the structural element. This significantly influences stress field and stress concentration inside the inclusion and near it. This type of problems can be solved numerically; however, for complex geometry it will require high computational performance and large amount of time. On the other hand, the solution of elasto-plastic problem for a soft inclusion in the infinite plate is quite simple and is easy-to-use in the engineering applications. Therefore, it is of considerable interest to test whether the latter can be applied to the calculation of finite plates with inclusions. This paper provides the analysis of elasto-plastic state of a soft inclusion embed into a finite square plate and the stress/strain field of inclusion and a plate using the boundary element method combined with the method of variable elastic parameters. In this approach, the problem of elastic-plastic deformation of the inclusion is reduced to the solution of the sequence of linear algebraic equations for Mises stress inside the inclusion, which is obtained based on the boundary element method, in which the kernels are related to that Mises stress by constitutive relations of the method of variable elastic parameters. Based on the obtained numerical solution the elastic-plastic state of plates with inclusions of different size and placement is analyzed. Stress/strain state of the plate is studied depending on the position of inclusion and the applied load. Material hardening is accounted for. Obtained results are compared with the solutions for infinite plates under the same input data. It is found that for the inclusion, which dimensions are two or more times smaller than that of a plate, the divergence of the numerical results and the closed-form solution for infinite plate does not exceed 1%, which is quite acceptable for engineering calculations. Critical load and orientation of the inclusion are studied as well. Thus, it is found that the calculation of Mises stress inside the inclusion under relatively low loads can be provided based on the solution for infinite plates. It is also found that the nature of the stress distribution in the inclusion and the plate is different. In particular, the maximal Mises stress inside the inclusion are the greatest for tension of the plate at an angle of relatively to the bigger semi-axis of the inclusion, and in the matrix for the transverse tension.