Issue 48

A. Fesenko et alii, Frattura ed Integrità Strutturale, 48 (2019) 768-792; DOI: 10.3221/IGF-ESIS.48.70 780 The formulas   2 βα βα u ( ) α ( ) z N Z z ,   2 βα βα v ( ) β ( ) z i N Z z were used to find the originals of the displacements 2 2 u ( , , ), v ( , , ) x y z x y z So,          βα βα 2 2 βα 2 βα 2 2 2 ( ) ( ) β u ( ) ( , ), v ( ) ( , ) 2 2 N N T h T h i z F N z z F N z D N D N . By analogy, it was found                                          2 2 2 , 2 2 0 0 2 , 2 2 0 0 ( , ) 2 u ( , , ) ( )cos( cos )cos( sin ) , ( , ) 2 v ( , , ) ( )cos( cos )cos( sin ) . A B t t A B t t F t z C x y z S tx ty d dt x t D F t z C x y z S tx ty d dt y t D Let’s consider the terms containing the integrals in (37). Using the solution of the stationary thermal conductivity problem (10), the temperature transformation and its derivative are written in the form    βα βα ( ) , chN T f chNh     βα βα ( ) . shN T Nf chNh After substitution of this transform into (40), the formula for 3 βα w ( ) z was constructed        3 0 βα βα 0 w ( ) ( , , ) 4 h z f F z N d where                                 (11) (12) 1 * 1 ( , , ) ( ) . N z N shN N chN F z N e N z shN z N chN chNh D In order to find final expression for displacement (40)                   3 0 βα 2 0 0 w ( , , ) ( , , ) cos 4 h i y x y z f F z N e xd d d (42) the representation of the function  f should be substituted into (42)                              3 0 2 0 0 0 w ( , , ) ( , ) ( , , ) cos cos . 4 h i y i x y z f F z N e e x d d d d d Using formula (A3) in the Appendix 1 and (401.06., [41]), it was derived                     3 * 0 0 0 0 0 w ( , , ) ( , , ) ( , , , , ) 8 B A h B C x y z t F z t J t x y d d dtd , where the temperature is defined on a finite interval and supposed to be constant C . Further, after changing the order of integration, and calculating the double integral of the Bessel function, the integral is represented in the form