Issue 48

A. Fesenko et alii, Frattura ed Integrità Strutturale, 48 (2019) 768-792; DOI: 10.3221/IGF-ESIS.48.70 788 here         , 1 1 ( ) ( cos ) sin( cos )( sin ) sin( sin ) A B t S tA tA tB tB As it may be seen function  , ( ) A B t S is continuously differentiated function with regard of variable  . This function is even also, so after changing the variable    sin one can obtain the correspondence         1 , , , 2 2 1 2 ( , ) ( , ) 1 A B A B t t AB d J x y F x y , (A6) where                  2 , 2 , 2 sin 1 sin ( , ) cos 1 cos( ) 1 A B t tA tB F x y tx ty tB tA . Based on the quadrature formula of the highest degree of accuracy, one can obtain for (A6)      ( ) , , , 1 2 1 ( , ) ( , ) N k N A B A B t t k AB J x y F x y N ,     ( ) 2 1 cos , 2 N k k N  1, k N  ( ) N k are zeros of the Chebyshov polynomials of the first kind. So, after substitution we get                     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 A PPENDIX 2. D ERIVING THE FORMULA (13) For the analysis of the reliability of the calculations, we perform a test based on the obtained formula (12) for the fulfillment of the boundary condition of the problem. To this end, we consider the temperature (12) in the corner point of the layer    0, 0, x y z h                        2 , , 2 1 1 0 0 sin 1 sin 2 2 (0, 0, ) (0, 0) 1 N N k k A B t k k k k tA tB C C T h t F dt dt N N tA B (A7) Using the formula (3.741(1), [41])               2 0 sin sin 1 ln , 0, 0, 4 mx nx m n dx m n m n x m n (A8) gives                        2 2 2 2 1 1 ( / ) 1 (0, 0, ) ln 2 1 1 ( / ) N k k k k k k k B A C T h N B A . (A9)