Issue 48

A. Fesenko et alii, Frattura ed Integrità Strutturale, 48 (2019) 768-792; DOI: 10.3221/IGF-ESIS.48.70 781                      3 2 0 2 2 1 0 0 ( , , ) 1 1 w ( , , )= ( , )cos 1 cos( ) 2 1 h N k k k k k k C F z t x y z F t tx ty dtd N t                            2 3 2 0 1 3 2 1 0 0 ( , , / ) 1 1 u ( , , )= ( , )cos 1 cos( ) . 2 1 h N y x k k k h h k k k h C F zh h t h x y z F t t t dtd x N t The expression for displacement 3 v ( , , ) x y z can be constructed by analogy. The expressions (38) correspond to the solution of the problem of a semi-infinite layer loading. The normal stress is constructed by the formula [20] in the form                        2 1 0 1 ( , ) 2 ( , , ) ( 1) (1 ) (1 ) (1 ) N k AB k x k k t F t x y z z cht z z cht z dt N D                         1 2 0 2 1 0 1 ( , ) (1 ) (1 ) 1 N k k k t k k F t sht z sht z dt D t The final solution of the initially stated uncoupled thermoelasticity problem for a semi-infinite layer with its proper weight was derived in the form         * ( , , ) ( , , ) 2 (1 ) (1 ), T AB x x z x y z x y z q h z         ( , , ) ( , , ) (1 ) T AB AB x x x y z x y z G C                                        1 1 2 1 2 0 0 1 2 2 2 1 1 0 0 0 0 2 1 ( , ) ( , ) 1 , 1 1 N N k k k k k k k k k G C F t F t F d dt F d dt N cht cht where the functions 1 2 , F F are defined in Аppendix 5,    * ,  is a coefficient of linear expansion,   2 2 t D sh t t ,               2 2 ( , ) sin 1 sin cos 1 cos( ) k h k h k h k h k F t tA tB tx ty . D ISCUSSION AND NUMERICAL RESULTS uring the calculations, two types of materials were selected: Copper − µ = 1/3, G = 44.7 GPa , α = 16.5 · 10 -6 1/С 0 , q z = 0.00896 kg/м 3 ; Glass − µ = 1/4, G = 26.2 GPa , α = 6 ·10 -6 1/С 0 , q z = 0.00119 kg/м 3 ; the layer’s thickness h = 1 м ; In all diagrams and tables, the units of measurement for stress are Pa ; The investigation of the influence of the load area shape on the stress was carried out for the case of unit temperature. For glass, in Figures 2, 3 the distributions of normal stress  , AB x   AB T x along the lateral wall of the layer for   0 z h depending on the shapes of the distributed load section and the temperature are represented. Here  AB x indicates the normal stress caused by mechanical loading, distributed over the site    [0, ], [ , ] x A y B B ,   AB T x − normal stress under the action of distributed load and temperature influence. The case  2 B A corresponds to the distribution of compression loading on the layer’s face  z h along the rectangle, elongated along the axis y ;  / 4 B A − along a rectangle elongated along the axis x ;  / 2 B A − quadratic spread. These graphs correspond to the case for glass. The stress graphs for copper, where   1/ 3 , are similar, but with larger absolute values. So, they are not shown here. As it can be seen from the graphs, in the case of a shape  / 2 B A , maximal compressing stress is observed. In the case D