Issue 48

A. Fesenko et alii, Frattura ed Integrità Strutturale, 48 (2019) 768-792; DOI: 10.3221/IGF-ESIS.48.70 787 A PPENDIX 1. D ERIVING THE FORMULA (12) FOR THE TEMPERATURE FUNCTION The corresponding formula for  f ,           0 ( , )cos i y f f x y x e dydх is substituted at (11) and the order of integration operators is changed at (11). To calculate the integrals exactly, the formula (1.314(3), [41]) is used                                 2 1 0 0 ( , , ) ( , )cos cos i y i chNz T x y z f e d d x e d d chNh (A1) Let’s consider the internal integrals at formula (A1). One must use the fact that function   ( , ) f is the even one and to use Euler’s formula. After some additional transforms the solution of the boundary value problem (7-9) is written as                                        2 ( ) ( ) ( ) 1 4 0 ( , , ) ( , ) i x i x i y chNz T x y z f e e e d d d d chNh (A2) One should note that value N at the solution (A2) is defined as     2 2 2 N . It gives opportunity to simplify the expression (A2) with the formula [40]                         2 2 2 2 1 0 2 0 ( ) i x i y F e d d t F t J t x y dt (A3) where 0 ( ) J t is a Bessel’s function of the first kind. With regard of formula (A3), the solution (A2) will be transformed                       * 1 0 2 0 0 ( , , ) ( , ) ( , , , , ) chtz T x y z f t J t x y dt d d chth (A4) where             * 2 2 2 2 0 0 0 ( , , , , ) ( ) ( ) ( ) ( ) J t x y a b J t x a y b J t x a y b . If at the boundary condition (9) one takes   , f C C is constant, then exact solution of boundary value problem (7-9) will be obtained                   * 0 2 0 0 ( , , ) ( , , , , ) A B C B chtz T x y z t J t x y dt d d chth . (A5) Let’s simplify the internal integral at the equality (A5), using the known integral representation of Bessel function [42]                       2 2 2 0 0 2 ( ) ( ) cos cos ( ) cos sin ( ) J t x y t x t y d After some elementary transformations one obtains           2 , 2 0 0 4 ( , , ) ( )cos( cos )cos( sin ) A B t chtz AB T x y z C t S tx ty d dt chth