,
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 โโof 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).