Issue 49

A. Baryakh et alii, Frattura ed Integrità Strutturale, 49 (2019) 257-266; DOI: 10.3221/IGF-ESIS.49.25 258 F INITE - DIMENSIONAL ANALOG CONSTRUCTION METHOD onsider a system of N plane-parallel elastic layers of arbitrary thickness 2 n h , which is under plane strain conditions (Fig. 1). There are no mass forces. Due to the fact that these forces are usually constant, they, if necessary, can be easily taken into account based on the usual superposition. Let us assume that homogeneous conditions, for example, zero displacements, are preset at the distant side boundaries. In addition, for the sake of simplicity we will assume that cohesion between the contact layers is perfect. The displacement boundary conditions are as follows:     U U  for S. Here and in all following examples there is a line of symmetry at the left boundary. X Y 0 U H h n -h n Y n X n 0 Figure 1 : General computational scheme Let us write down the principle of additional virtual work [8]:         T T V S dV T U dS       (1) where     ,   - strain and stress vectors       T ij T n    respectively, n - cosine of the outer normal to S . As is known [8], this variational principle is true for arbitrary infinitely small stress variations that satisfy the equilibrium equations and the specified stress boundary conditions. For each layer in the local Cartesian coordinate system х0у (the x axis runs across the middle line of a layer) stress at any point can be represented as a Fourier series expansion:       1 1 ( , ) ( ) ( ) ( ) ( ) c c a a k k k k k k x y t x y t x y                    (2) where ( , ) ( , ) ( , ) ( , ) xx yy xy x y x y x y x y                , ( ) ( ) ( ) ( ) ( ) ( ) ( ) c a xxk c a c a k yyk c a xyk y y y                  cos 0 0 ( ) 0 cos 0 0 0 sin k c k k k x t x x x                   , sin 0 0 ( ) 0 sin 0 0 0 cos k a k k k x t x x x                   C