Issue 48

A. Fesenko et alii, Frattura ed Integrità Strutturale, 48 (2019) 768-792; DOI: 10.3221/IGF-ESIS.48.70 773 T HE DERIVING OF NORMAL STRESS WITH REGARD TO ITS PROPER WEIGHT o solve this problem, the displacements    u , v , w will be searched as the function depending on the variable z       0 1 2 u ( ), v ( ), w ( ) f z f z f z (14) Because of problem’s symmetry one can take     u 0, v 0 . Hence, the tangent stress   0 xy . That’s why the displacements were taken in the form (14), the boundary condition of the ideal contact (3) is satisfied automatically at the edge of the layer. The function  w satisfies the Lame's equation              0 w w 0 , (15) where   / , z z q G q is the weight of an elastic material,      1 0 (1 2 ) . The face  z h is supposed free of stress           0, 0, 0 z zx zy z h z h z h The lower face  0 z is in ideal contact conditions          0 0 0 w 0, 0, 0 zx zy z z z Finally, one gets the boundary problem         0 2 (1 ) ( ) 0, 0 f z z h    2 2 (0) 0, ( ) 0 f f h The solution has the form          1 2 0 w (1 ) 2 hz z Hence, the stress   x for the layer with regard to its proper weight is constructed        1 (1 ) ( ) x z q h z (16) The solution of the initial stated problem (2-5) will be searched in the form                * * * * u u u , v v v , w w w , x x x (17) Here  u( , , ), v( , , ), w( , , ), ( , , ) x x y z x y z x y z x y z are displacements and stress appearing in a body without regard to its proper weight. T