,
where Kij are bounded.
An analogue of the Gauss formula has been proved for it [14,15]:
. (6.4)
The integral is singular only for boundary points and taken in the sense of the value principal sence.
7. Regular integral representation of displacements. Using the antiderivative fundamental stress tensor (6.2) and their symmetry properties, the solution (5.7) can be written in integral form:
(7.1)
Here, for , all integrals are regular with weak singularities of kernels at wave fronts. With known field temperature and boundary values of displacements and stresses, these formulas make it possible to determine displacements at any point of an elastic medium for any mass forces, including singular ones, which describe the effect of concentrated and impulse power sources of various types. In this case, the component wise convolution should be taken according to the convolution rules in the space of generalized functions.
To solve BVPs, it is necessary to determine the displacements or stresses at S.
8. Resolving singular integral equations. The following theorem is true.
Theorem. Solutions of BVPs 1-4 satisfy the boundary integral equations on S :
(8.1)
for .
Proof. Let , Let consider the second term in the first integral in (7.1), which, taking into account (6.3), can be transformed to the form:
In the integral
(8.2)
we consider the first term on the right for :
The limit of the second integral in (8.2) in the sum with the second integral in (7.1) is reduced to the form:
Taking into account these relations from formula (7.1), passing to the limit to the boundary of the domain, we obtain the formula of the theorem.
For the solution of the first or second boundary value problem, the formulas of the theorem are resolving singular BIEs for determining the displacements. For the third and fourth BVP, the stresses on the boundary are unknown in the formulas of the theorem, i.e. we have a BIEs with a weakly singular kernel:
(8.2)
where all are known, are calculated by the formulas of the theorem.
After determining the unknown boundary functions, using formulas (7.1), you can determine the displacement at any point in the region. After determining the displacements and temperatures using the Duhamel-Neumann formulas (1.2), the stresses in the medium are calculated. That solves the BVP.
Conclusion. The constructed boundary integral equations are no classical type. They are very different from BIEs of BVPs problems for elliptic and parabolic equations for which various mathematical methods are well developed. In particular, the use of the method of successive approximations is difficult here due to the presence of an unknown velocity of displacements (for the 1st and 2nd BVP). However, the use of numerical methods based on the boundary element method makes it possible to effectively solve this type of equations.
The resulting formulas (4.2) and (7.1) have an important engineering application. They make it possible to determine the thermally stressed state of the medium by the boundary values โ€‹โ€‹of stresses, displacements, temperature and heat flux, without solving singular BIEs. Because for real engineering problems these process characteristics can be experimentally measured at the boundary. Moreover, the formulas allow to calculate the influence of each of these characteristics of the process on its stress-strain state. The last one is very important in designing structures made of thermoelastic materials.
This work was supported by grant from the Ministry of Education and Science of the Republic of Kazakhstan (โ„– AP09258948).