Issue 49

A. Baryakh et alii, Frattura ed Integrità Strutturale, 49 (2019) 257-266; DOI: 10.3221/IGF-ESIS.49.25 260   1 0 1 1 0 0 0 2 C E                      . Based on the constructed relations for stresses (5) and strains (6), the left part of Eqn. (1) determining the virtual work of strains can be represented as follows:     ( , ) ( , T V x y x y dV                      1 1 0 0 1 1 ( ) ( ) ( ) ( ) T T T T c c c c k k k k m m m m k m V p R r y t x C t x r y dV R p                                   1 1 0 0 ( ) ( ) ( ) ( ) T T T T c c a a k k k k m m m m V p R r y t x C t x r y dV R p                               1 1 0 0 ( ) ( ) ( ) ( ) T T T T a a a a k k k k m m m m V p R r y t x C t x r y dV R p                               1 1 0 0 ( ) ( ) ( ) ( ) T T T T a a c c k k k k m m m m V p R r y t x C t x r y dV R p                  (7) If in Eqn. (1):       * * 1 1 ( ) ( ) c c a a k k k k k k T t x t x                    where * cos 0 0 0 0 sin 0 0 0 0 cos 0 0 0 0 sin k k c k k k x x t x x                      , * sin 0 0 0 0 cos 0 0 0 0 sin 0 0 0 0 cos k k a k k k x x t x x                      then its right-hand part that determines the external virtual work is given as:              * * 1 ( ) ( ) ( ) ( ) T T T T T c c a a k k k k k S S S T U dS p t x U x dS p t x U x dS                        (8) Substituting of (7), (8) into variational Eqn. (1), makes it possible to use the standard procedures of the finite element method: integrating for the relevant areas and boundaries of a layer; constructing the local compliance matrices; proceeding to global coordinates; combining the generated matrices and, finally, forming the system of independent algebraic equations for the coefficients of the force vector decomposition at the boundary of each layer:       k k k S p F  (9) where   k S is the global compliance matrix, which has a band structure,   k F is the k -th harmonic of a given displacement vector along the boundaries of the layers.