ABSTRACT
To solve the direct problems of the elasticity theory, that is, to determine 15 unknown functions ui, σij, εij (i, j = 1, 2, 3), we use the equilibrium equations, Cauchy relations and Hooke’s law:
, 0ij j iFσ ρ+ = , (3.1) 12 ( , , )ij i j j iu uε = + , (3.2) 2ij ij ijσ με λθδ= + . (3.3)
Consequently, we have 15 linear equations in partial derivatives. In the case the displacements are not included into the number of unknowns, equations (3.2) are substituted for the deformation compatibility conditions (without summation over the repeated indices)
Here and further on we assume the following boundary conditions as postulated. We shall take that the surface of a body is S = S
us accept that the following conditions are given in each point of the surface:
0 ( )i iu u x= on uS , i ij jR lν σ= on σS . (3.5) This simplest case does not exhaust all potentialities. For example, when a rigid
punch is pressed into an elastic body, friction is often assumed to be absent. So, if to direct axis х3 normally to the body surface, the boundary conditions under the punch will be:
0 13 23, 0.i iu u σ σ= = = However, it makes no sense in the general theory to set the task like this on one
surface for both displacements and stresses. The discourse presented below can be easily extended on above cases.
