PSI - Issue 33

Anna Fesenko et al. / Procedia Structural Integrity 33 (2021) 509–527 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

511

3

2. Statement of the problem. An elastic layer of thickness h ( G is a shear modulus,  is a Poisson’s ratio,  is density), describing in the cylindrical coordinate system by the correspondences: a r    ,     , 0 z h   is weakened by a cylindrical cavity   0 r a ,     , 0 z h   (Fig. 1).

Fig. 1. Geometry of the problem

The layer’s upper face z h  is in the conditions of ideal contact with a rigid base (the layer is supported by a smooth foundation without a friction) and the bottom face 0 z  is rigidly fixed ( ,0, ) 0 r u r t  , ( ,0, ) 0 z u r t  , ( , , ) 0 z u r h t  , ( , , ) 0 zr r h t   (1) The cylindrical cavity’s surface r a  is under the influence of the normal dynamic tensile force ( , ) P P z t  , applied at the initial moment 0 t  , the tangential loading is absent ( , , ) ( , ) r a z t P z t   , ( , , ) 0 rz a z t   (2) Thus, the problem was reduced to solving axisymmetric equations of motion with respect to the functions ( , , ) ( , , ) r u r z t u r z t  , ( , , ) ( , , ) z u r z t w r z t  in a cylindrical coordinate system (Novazkiy, 1975)

2

2

r r       





 

( , , )

2

2 u r z t t 

 

1

2 r u r z t 



( , , ) r u r z t

( , , )

( , , )

( , , )

r

u r z t

w r z t

1 1

2

1 1







 

 

 

1

G



2

r z

 

z



2

r r       



r z       

  

  

( , , )

2

2 w r z t t 

 

1

1





( , , ) r w r z t

( , , )

( , , ) r u r z t

r

w r z t

r

1 1

2

(3)







 

1

G



2

z



where 3 4     and subjected to the mixed boundary conditions (1), (2). Here 2 1 1 1 / c G   

 − squared velocity

/ c G   − squared velocity of shear wave propagation. So, 2 c

of longitudinal wave propagation, 2

1 2 c 1

.

  

1

The following change of the variables was done

1 ca t    ,

1 a r    ,

1 h z    ,

1 ( , , ) ( , , ) u a h c a U         ,

1 ( , , ) ( , , ) w a h c a W         (4)

Consequently, the motion equations (3) can be written in the form