5. Generalized solution of equations for displacements and its regularization.
Taking into account (1.2) Eq (3.2) can be written in the form:
(5.1)
where is Lamé operator of motion equations of an elastic medium:
To determine the displacements, we use the Green tensor of these equations:
(5.2)
and radiation conditions:
by and by . (5.3)
At plane deformation it is expressed by the formula [10,14]:
(5.4)
It has the following symmetry properties:
Investigation of the asymptotic properties of this tensor showed that it has a weak singularity at wave fronts , does not have singularities for fixed t> 0, r→0.
Using the properties of the Green tensor and the differentiation properties of the convolution, we represent the generalized solution (5.1) in the form of a tensor-functional convolution:
(5.5)
Further, it is not possible to write down relation (5.5) in integral form, because the derivatives in the second term are hypersingular at the wave fronts .
To regularize formula (5.5), we introduce the Green tensor antiderivative on t :
(5.6)
The singularities of the tensor were studied in [14]. It is continuous at wave fronts and has a logarithmic singularity at .
Using and the rules for differentiating the convolution, the second term in relation (5.5) can be represented in a regularized form:
.
As a result, the generalized solution (5.5) takes the regularized form:
(5.7)
which can be written in integral form. To do this, let consider the stress tensor and related tensors.
6. Fundamental stress tensors. Regularization. We introduce the following fundamental stress tensors generated :
(6.1)
Tensor describes stresses on an area with a normal n generated by impulse concen-trated forces at the origin. The tensor is a solution to the Lamé equations at and describes the displacement of the medium under the action of a pulsed concentrated source of the multipole type [14,15]. It is antisymmetric on x and n :
.
For integral representation of (5.7), we introduce the antiderivative ont tensor of fundamental stresses W(x,t,n):
(6.2)
We integrate this tensor at wave fronts and represent it in the form:
(6.3)
The dynamic tensor has a weak integrable singularity at wave fronts.
The tensor is the fundamental Green stress tensor of static Lamé equations. It satisfies the homogeneous Lamé equations at and has only one singularity at x = 0: