THE SOLUTION OF A CONTACT PROBLEM FOR A PRE-STRESSED CYLINDRICAL STAMP AND A LAYER WHICH IS LYING ON THE BASE WITHOUT FRICTION AND WITHOUT INITIAL STRESSES - YaretskaN. О. (Computer Science and Applied Mathematics, 2021)
The article is devoted to the study of the contact problem for a pre-stressed cylindrical stamp and a layer with initial stresses. The layer is lying without friction on the base without initial stresses. The initial stresses are absent at the base. The problem is solved in the case of unequal roots of the defining equation. The study is presented in general for the theory of large initial deformations and two variants of the theory of small initial deformations within the framework of the linearized theory of elasticity with an arbitrary structure of elastic potential. It is assumed that the initial states of the elastic cylindrical punch and the layer and the base are homogeneous and equal. The study is carried out in the coordinates of the initial deformed state, which are associated with Lagrangian coordinates (natural state). In addition, the influence of the cylindrical punch causes small perturbations of the corresponding values of the basic stress-strain state. It is also assumed that the elastic cylindrical punch and the elastic layer and base are made of different isotropic, transversely isotropic or composite materials. The general solutions of the basic differential equations of the linearized theory of elasticity in the case of axisymmetric deformation for a finite cylindrical domain are given. As a result, the problem is presented in the form of infinite series. The coefficients of this series are determined from an infinite quasi-regular system of algebraic equations. The influence of initial stresses in the layer and the cylinder punch and the base on the distribution of contact stresses in the contact region is studied. In the case of unequal roots for the chemically active rubber of UWC-6 and the Treloar potential (Neogukov type body) the results of numerical analysis are given. This results are presented in the form of graphs. It is illustrate a fairly significant effect of the initial stresses. So, influence of initial stresses on the stress-strain state of an elastic cylinder punch, is that the initial stress in the layer in the case of compression reduce the stress in the elastic punch, and in the case of stretching increase it.