UNCOUPLED THERMOELASTODYNAMICS
Abstract: Nonstationary boundary value problems of uncoupled
thermoelasticity are considered. A method of boundary integral equations
in the initial space-time has been developed for solving boundary value
problems of thermoelasticity by plane deformation.
According to generalized functions method the generalized solutions of
boundary value problems are constructed and their regular integral
representations are obtained. These solutions allow, using known
boundary values and initial conditions (displacements, temperature,
stresses and heat flux), to determine the thermally stressed state of
the medium under the influence of various forces and thermal loads.
Resolving singular boundary integral equations are constructed to
determine the unknown boundary functions.
Key words: uncoupled thermoelasticity, fundamental solutions,
displacements, temperature, stresses, heat flux.
Introduction. The development of thermoelasticity research is
associated with the need to develop new mechanical structures, the
elements of which operate under conditions of uneven and unsteady
heating (in aviation and rocket technology, in nuclear reactor
protection systems, in a number of machine-building complexes, industry,
etc.). This leads to the appearance of temperature gradients in the
medium and deterioration of the strength properties of materials.
Thermal shock causes some materials to become brittle and degrade.
In 1956, the work of M. Biot [1] was published, in which a complete
substantiation of basic relations and equations of coupled
thermoelasticity, based on the laws of thermodynamics of irreversible
processes, was given for the first time. This author also formulated the
basic variational principles and developed some methods for solving the
thermoelasticity equations. In the ensuing publications of V. Novatskiy
[2,3], various methods of solving the differential equations of
thermoelasticity are proposed, and the models of coupled and uncoupled
thermoelasticity are substantiated. According to methods of complete and
incomplete separation of variables, he constructed and investigated a
number of solutions of these equations and considered a whole class of
quasi-static and dynamic problems of thermoelasticity.
In works devoted to dynamic problems of thermoelasticity, the thermal
shock problems stand out separately. In formulating such a problem, it
is assumed that at the initial moment the object is at rest, and in the
subsequent moment there is a sharp change in the thermoelastic state due
to the action of heat and power sources, both external and in the medium
itself.
Thus, the problem of the propagation of a thermoelastic wave in a
half-space due to the instantaneous heating of its boundary for the case
of small times was first considered by V.I. Danilovskaya [4] and
solved by the small parameters method. A large review of works on
thermoelasticity is given by Hetnarski R. in the encyclopedia [5].
In [6,7], the method boundary integral equations (BIE) was developed
to solve boundary value problems of coupled and uncoupled thermoelastic
elastodynamics. When solving these problems, BIE were constructed in the
space of Laplace transforms in time. One of the main problems of the
method BIE in the Laplace transform space, which is well known, is the
instability of the numerical procedures for inverting transformants of
solutions with increasing time, which does not allow constructing
solutions in calculations at even small times.
In order to avoid these problems, the method BIE in the initial
space-time is being developed here to solve boundary value problems
(BVP) of thermoelasticity under plane deformation.
1. Mane relations of thermoelasiticity. An isotropic
thermoelastic medium is characterized by a finite number of
thermodynamic parameters: mass density ρ, elastic Lamé constants λ, µ,
and thermoelastic constants γ, η and κ. In Cartesian coordinate system,
such medium is described by the system of equations [2,3,8]:
(1.1)
Here are the components of the displacement vector , , is the relative
temperature, are the components of the mass force, is the power of the
heat source, The stress tensor (x,t) is associated with displacements by
Duhamel-Neumann law:
. (1.2)
Everywhere there is the summation throughout the repeated indices within
the specified limits of variation. Substituting (1.2) into (1.1), we
obtain a closed system of equations for which we write in the form:
(1.3)
where the following differential operators are introduced:
This system has mixed hyperbolic-parabolic type. Waves, propagating in a
thermoelastic medium, can be shock waves. The equation of the wave frontF has the form:
=0 , (1.4)
where is the main part of the operator , containing only the highest
derivatives of the second order, and is a differential operator of
motion equations of corresponding elastic medium with parameters
(λ, µ, ρ).
We denote by the normal vector to in . It follows from (1.4) that either
, (1.5)
or either
. (1.6)
Equation (1.5) describes the characteristic surface of the classical
parabolic equation, which has the form and does not determine the wave
front in space . Equation (1.6) describes wave fronts moving in with the
speed:
, , (1.7)
where , is the velocity of dilatation waves, is the velocity of shear
waves. Consequently, wave fronts (thermoelastic shock waves) in the
medium move with the speeds of elastic waves.
In order to continuity conditions of a medium to be preserved and to be
a solution (1.1), the following conditions for jumps on characteristic
surfaces must be satisfied [7]:
, , (1.8)
, , .
Let us introduce the wave vector in , , directed towards the propagation
of the shock wave. These equalities imply the conditions on jumps at the
shock fronts in :
, , (1.9)
, (1.10)
. (1.11)
Here n is the wave vector perpendicular to and has the direction
of wave propagation. Equality (1.9) is the condition for conservation
the continuity of the medium, (1.10) coincides with the well-known law
of conservation of momentum at the fronts of shock waves in elastic
media [9].
It follows from (1.9) and (1.11) that the temperature is continuous at
the wave fronts, but its gradient suffers a jump proportional to the
jump of the normal component to the front of the velocity of the medium
displacements.
We will call a solution of equations (1.1), satisfying conditions (1.9)
- (1.11) on wave fronts, classical solution.
2. Statement of boundary value problems of uncoupled
thermoelasticity. If to set loads at the boundary of a body or mass
forces in the body itself, this leads to deformation. At low strain
rates in the equation for the temperature field (1.1), the rate of
volumetric deformation of the medium (η = 0) can be neglected. Then the
boundary value problem is divided into two tasks: determination of the
temperature field, after which it becomes possible to determine the
field of displacements and stresses in the medium. This model is calleduncoupled thermoelasticity (thermal stress theory).
Consider the following boundary value problems for this model. Let a
thermoelastic medium occupies a region bounded by a closed Lyapunov
surface S with an external normal n (x ). The
equations of medium motion in this model are as follows:
(2.1)
Here , is volume force in medium.
Initial conditions (Cauchy conditions ) are given:
, , , , , (2.2)
.
BVP 1. At the boundary (), the acting load and heat flow are
known:
, , (2.3)
where - functions enterable on S.
BVP 2. For given loads and temperature:
, , , (2.4)
.
BVP 3. For the given displacement and temperature:
, , , , (2.5)
.
BVP 4. For given displacements and heat flux:
, , , (2.6)
.
Here C (…) is the class of continuous functions on the indicated
set, C ’(…) are piecewise continuous bounded functions,
C1(…) are continuously differentiable
functions.
At the fronts, the solutions satisfy the jump conditions (1.9)-(1.11).
It is required to find displacements, temperature, stresses in the
medium.
To solve problems, we use the method of generalized functions (GFM), the
main ideas of which are presented in [10-13].
3. Statement of BVP in the space . Let consider
equations (2.1) in the space-time of generalized vector functions:
=,
where is generalized function, [13].
We introduce the characteristic function of a domain :
(3.1)
We consider the following regular generalized functions:
where H (t ) is Heaviside function, are classical solutions
of BVP.
Using generalized derivatives of functions:
where is the singular generalized function - a simple layer on S
[13], we get the their partial derivatives:
,
,
Then Eqs (2.1) in take the form:
(3.2)
(3.3)
Here is the tensor of elastic constants, which for an isotropic medium
has the form:
. (3.4)
It is seen from formulas (3.2) and (3.3) that the boundary conditions
entered in the form of densities of simple and double layers on S, as
surface forces and heat sources, and the initial conditions as impulse
conditions acting at the moment t = 0. Note that, when differentiating,
we took into account the conditions at the fronts (1.9)-(1.11), which
nullify simple and double layers at the fronts of shock waves.
Next, we construct generalized solutions to these boundary value
problems for plane deformation ().
4. Determination of the temperature field at plane deformation.To determine the temperature of the medium, we use the Green function of
the heat equation (2.1)2, corresponding to a pulsed
concentrated heat source [12]:
. (4.1)
We get the generalized solution (3.3) in the form of a convolution of
the Green function with acting sources in (3.3):
Here we use differentiation properties of a convolution [12]. A
variable under the convolution (*) means that it’s incomplete
convolution, which is taken only for this variable. If there is no such
symbol, then this is a complete convolution over (x, t).
This formula for can be represented in the following integral form:
(4.2)
where is surface differential.
Formula (4.2) allows for the given values of temperature and heat flux
at the boundary and the initial temperature to determine the temperature
field inside the region under the action of various heat sources.
For x∈S, t>0 , the same formula gives a singular BIE
for determining the unknown temperature or heat flux at the boundary:
(4.3)
Here , .
The proof of this formula for the boundary points can be carried out in
the original space-time. However, it directly follows from the BIE for
the Laplace transform of temperature [6].
After solving this equation, the unknown boundary functions of the BVP
are determined on the boundary. After that, using formula (4.2), one can
determine the temperature in the entire domain .