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 .