Issue 48

A. Fesenko et alii, Frattura ed Integrità Strutturale, 48 (2019) 768-792; DOI: 10.3221/IGF-ESIS.48.70 772     0 ( , , ) 0 x T x y z x ,     0 ( , , ) 0 z T x y z z (8) Here function ( , , ) T x y z is the unknown temperature of the layer. At the upper face  z h of the layer, the temperature is given along the segment    [0, ], [ , ] x A y B B  ( , , ) ( , ) T x y h f x y ,    [0, ], [ , ] x A y B B (9) The integral transform method is used to derive the one-dimensional boundary problem. The Fourier cosine transform with regard to the variable x and full Fourier’s transform with regard to the variable y are applied consecutively to the Laplace equation (7) and boundary conditions (8, 9). It leads to the one-dimensional problem at the transforms' domain      2 ( ) ( ) 0 T z N T z ,  (0, ) z h (10)    (0) 0 T ,    ( ) T h f where           0 ( , )cos i y f f x y x e dydх ,     2 2 2 N ,   , are Fourier’s transform parameters. Usage of the general solution of the equation (10)    1 2 ( ) T z C shNz C chNz ,    1 2 0, / C C f chNh leads to the final solution in the transform domain, where the inverse integral transform was applied                  2 1 2 0 ( , , ) cos i y chNz T x y z f x e d d chNh (11) The final solution of the stated conductivity problem (7-9) is presented in the form                     2 2 2 1 0 sin 1 sin ( ) 2 ( , , ) cos 1 cos( ) ( ) 1 N k k k k k k k tA tB ch tz C T x y z tx ty dt N ch ht tA B (12) where  ( ) N k are zeros of Chebyshov polynomials of the first kind, C is the constant value of temperature, distributed along the segment    [0, ], [ , ] x A y B B . A more detailed derivation of the formula (12) is given in the Appendix 1. An important property                        2 2 2 2 1 1 1 ln 2 1 1 B N k kA B k k k k kA C C N (13) was obtained based on solution (12) and will be used for calculating stress at the corner point of a layer. A more detailed derivation of the formula (13) is given in Appendix 2. The next step is to find the solution of the problem for the elastic layer            , , 0 x y z h under its proper weight.