Про контактну задачу для циліндричного штампа і шару з початковими (залишковими) напруженнями

Abstract

У рамках лінеаризованної теорїї пружності представлена осесиметрична змішана задача про тиск пружного циліндричного штампу на шар з початковими (залишковими) напруженнями. Дослідження представлені в загальному вигляді для теорії великих початкових деформацій та різних варіантів теорії малих початкових деформацій при довільній структурі пружного потенціалу.

The article deals with the coaxial mixed type task of measuring pressure of an elastic cylinder punch upon a layer with initial (residual) stresses within the framework of linear elasticity theory. Two cases will be viewed in the article 1) the layer is placed on an elastic surface without frictioning; 2) the layer is fastened to an elastic surface. In general, the research was carried out for the theory of great initial (ultimate) deformations and different variants of the theory of small initial deformations with arbitrary structure of elastic potential. It is assumed that elastic potentials are two continuously differentiated functions of algebraic invariants of the Green tensor deformation (the initial states of the layer and of the cylindrical punch remains uniform and equal). The research is carried out within the coordinates of the initial deformed state, which are interrelated with the Lagrangian coordinates (natural state). Besides, it is also assumed that the influence of the die causes small disorders of the main elastic deformed state. It is assumed that the elastic die as well as the layer are made of different isotropous, transversely isotropic or composite materials and they are interacting on one of the die’s surfaces. The mode of deformation in elastic layer with initial (residual) stress will be defined with the help of harmonic functions as the Henkel integrals. We should note, although the Henkel-method does not provide exact solutions, but it lets us reduce the task to the Fredholm equations, which let us use the method of consecutive approximations. Consequently we got a components of potential vector and tenzor of deformations in the case of unequal roots of axis-symetrical type task. So, the received solutions are defined by lines with the help of very many constants. These constants are defined with regular and linear algebraic systems. The research was carried out on the problem of the influence of initial stresses on the law of distribution of contact disorders in the elastic layer with initial (residual) stresses. The article deals with the coaxial mixed type task of measuring pressure of an elastic cylinder die upon a layer with initial stresses within the framework of linear elasticity theory. In general, the research was carried out for the theory of great initial (ultimate) deformations and different variants of the theory of small initial deformations with arbitrary structure of elastic potential.