Змішана статична контактна задача для пружного шару і пружного циліндричного штампа з початковими (залишковими) напруженнями

Abstract

У рамках лінеаризованної теорїї пружності представлено осесиметричну змішану задачу про тиск пружного циліндричного штампа на шар з початковими (залишковими) напруженнями. Дослідження представлено 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 mixed type problem of measuring pressure of an elastic cylinder punch upon a layer with initial stresses within the linearized elasticity theory. Two cases are analyzed: 1) the layer is placed on an elastic surface without friction; 2) the layer is fastened to an elastic surface. In general the research was carried out for the theory of finite initial (large) deformations and different variants of the theory of small initial deformations with the 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 influence of the punch causes slight disorders of the main stress-strain state. It is assumed that the elastic cylindrical punch as well as the layer is made of different isotropous, transversely isotropic or composite materials and they are interacting on one of the punch’s surfaces. The stress-strain state in the elastic layer with initial stresses will be defined with the help of harmonic functions as the Henkel integrals. It should be noted, that although the Henkel-method does not provide exact solutions, bat they make it possible to reduce the task to the Fredholm equations, which result in the effective use of the method of consecutive approximations. Thus, the relation for components of potential vector and tensor of deformations in the case of equal roots of axis-symmetrical type problem has been obtained. As the result the solution is presented as series with the help of infinitive system of constants. These constants are defined from the regular linear algebraic systems. The problem of the influence of initial stresses on the law of distribution of contact disorders in the elastic layer with initial stresses has been investigated.