Issue 49

A. Baryakh et alii, Frattura ed Integrità Strutturale, 49 (2019) 257-266; DOI: 10.3221/IGF-ESIS.49.25 259 ; 2 k k l l    - interval of expansion into the Fourier series, upper indexes c and a denote, respectively, the values related to the symmetric and antisymmetric state of a layer in relation to the y axis. The Fourier coefficients in expansion (2) can be expressed as constants within the layer of vectors   c k C and   a k C [9]:       ( ) ( ) c c k k k y r y C   ;       ( ) ( ) a a k k k y r y C   , (3) where   ( ) k r y - the known matrix [6],   ( ) ( ) ( ) 2 ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k sh y ch y y sh y ch y y ch y sh y r y sh y ch y y sh y y ch y ch y sh y y ch y sh y y sh y ch y                                                               . The representation of stresses in the form of expansion (2), with (3) taken into account, satisfies the equilibrium equations, which, as already noted, is an integral condition for the validity of the principle of virtual work. Let us express constants ( ) c a k C in (3) in terms of the relevant coefficients of decomposition of force vectors ( ) c a k p at the upper ( ) c a kt p and lower ( ) c a kb p boundaries of a layer:       1 ( ) ( ) 0 c a c a k k k C R p   , (4) where     ( ) ( ) ( ) , c a c a c a k kt kb p p p  ;   ( ) ( ) ( ) ( ) ( ) c a yyk c a kt c a xyk h p h              ;   ( ) ( ) ( ) ( ) ( ) c a yyk c a kb c a xyk h p h               ;   * 0 * ( ) ( ) k k k r h R r h           ; * ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) k k k k k k k k k k k k k k k k k k k k sh y ch y y sh y y ch y r y ch y sh y y ch y sh y y sh y ch y                                             . Then, the expression for stresses (2) can be represented as:        1 0 1 ( , ) ( ) ( ) c c k k k k k x y t x r y R p                1 0 1 ( ) ( ) a a k k k k k t x r y R p          , (5) and the strains will be determined by the following relation:          1 0 1 ( , ) ( ) ( ) c c k k k k k x y C t x r y R p                  1 0 1 ( ) ( ) a a k k k k k C t x r y R p          , (6) where   C - compliance matrix that is for an isotropic layer with Young's modulus E and Poisson's ratio  is written as: