System of Equations of the Dynamic Problem of Thermoelasticity in Stresses for an Elliptic Cylinder
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SYSTEM OF EQUATIONS OF THE DYNAMIC PROBLEM OF THERMOELASTICITY IN STRESSES FOR AN ELLIPTIC CYLINDER H. B. Stasyuk
UDC 539.3
On the basis of the equations of motion, Cauchy formulas, generalized Hooke’s law, and compatibility conditions for the Saint-Venant strains, a system of determining equations of the dynamic problem of thermoelasticity in stresses is deduced for a homogeneous isotropic cylinder in an elliptic cylindrical coordinate system. This system is reduced to a system of consecutively correlated wave equations in which the equation for the first invariant of the stress tensor is independent. The initial conditions for the resolving functions are presented.
In the direct problems of the theory of elasticity and thermoelasticity, it is customary to specify bulk forces F and surface loads P [1 – 5]. Therefore, it is reasonable to solve static [1, 5] and dynamic [3, 4, 6 – 9] problems of this sort in stresses. In finding the solutions of boundary-value problems for the systems of equations in stresses by using polynomial approximations of the required components of the stress tensor in spatial variables, it is possible to get higher orders of accuracy of approximations than in the case of the analysis of the problem in displacements with the help of polynomials of the same order [9]. In [10 – 13], one can find the systems of determining equations in stresses for the solution of one-, two-, and three-dimensional dynamic problems of thermoelasticity in Cartesian, cylindrical, and spherical coordinates. These systems of six coupled equations for the six components of the stress tensor are reduced, without introducing additional functions, to systems of consecutively correlated wave equations for the key functions, namely, for the first invariant of the stress tensor, linear combinations of the normal components of stresses, and tangential components of stresses. In what follows, the equations of the dynamic problem of thermoelasticity in stresses deduced for an elliptic cylinder related to an elliptic cylindrical coordinate system are reduced to a system of wave equations in which the role of the key function is also played by the first invariant of the stress tensor. Statement of the Problem Consider a homogeneous isotropic body related to an elliptic cylindrical coordinate system ( u, w, z, a ). Note that the Cartesian coordinates ( x, y, z ) are expressed via the indicated cylindrical coordinates as follows: x = a cos u cosh w,
y = a sin u sinh w,
z = z,
where a is a focus of the ellipse. The body is subjected to the action of nonstationary bulk forces F(u, w, z, t ) = { Fu, Fw , Fz } and a temperature field T ( u, w, z, t ), where t is time. The source system of relations specifying the stress–strain state of the body includes the equations of motion 2 ⎛S ∂ + S ⎞ σ − S σ ⎛ S ∂ + 2S ⎞ σ + ∂ σ + F = ρ ∂ U , + 1 2 uu 2 ww 1 3 uw uz u u ⎝ ∂u ⎠ ⎝ ∂w ⎠ ∂z ∂t 2
(1a)
“L’vivs’ka Politekhnika” National University, Lviv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 40, No. 5, pp. 57– 62, S
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