Static thermoelasticity problems for layered thermosensitive plates with cubic dependence of the coefficients of heat conductivity on temperature

Abstract

We describe an analytic-numerical method of solution of one-dimensional static thermoelasticity problems for layered plates, heated in different ways. We take into account the cubic dependence of the coefficients of heat conductivity and arbitrary nature of the dependence of other physicomechanical parameters on temperature. Here, using the constructed exact solution of an auxiliary problem, we have reduced the heat conduction problems, irrespective of the number of layers, to the solution of one or a system of two nonlinear algebraic equations. We have also studied the temperature fields and stresses in four-layer plates under conditions of complex heat exchange.The determination of the thermoelastic state of both homogeneous and inhomogeneous bodies, in particular, those with plane-parallel boundaries, with regard for their thermal sensitivity (the dependence of physicomechanical characteristics on temperature) has attracted the attention of numerous researchers [4,[15][16][17][18][19][20]. Even in the case of uncoupled thermoelasticity problems, the corresponding heat conduction problems remain nonlinear. For their solution, it is customary to apply the Kirchhoff substitution [1, 2, 5, 6, 9-12, 14, 16]. This technique enables one to linearize the problems under study partially or even completely and, correspondingly, to simplify their solution or, in some cases, to obtain an exact solution. However, for layered bodies, one does not succeed in completely linearizing heat conduction problems, including the simplest, one-dimensional, because nonlinearity is preserved in one of the conditions of thermal contact. Obviously, the solution of heat conduction problems and, hence, of the corresponding thermoelasticity problems becomes more difficult both in the case of convective or convective-radiative heat exchange with the environment and taking into account more complex temperature dependence of the coefficients of heat conductivity than linear. Therefore, such problems are solved, as a rule, by numerical and approximate analytic methods or by their combination.In the present work, we describe an analytic-numerical procedure, based on using the approach [7], of the solution of one-dimensional static thermoelasticity problems for layered plates, heated in different ways, with regard for the cubic dependence of the coefficients of heat conductivity and arbitrary character of the dependence of other physicomechanical characteristics on temperature. Using the exact solutions of the corresponding equations under special boundary conditions, constructed with the help of the Kirchhoff substitution and generalized functions, we reduce the heat conduction problems under study, irrespective of the quantity of layers, to the solution of one or a system of two nonlinear algebraic equations. The thermoelasticity problem is solved in terms of stresses. The boundary conditions on the cylindrical surface are satisfied integrally. We also analyze numerically the temperature fields and thermostressed state in four-layer plates under ...

Numerical-Analytic Determination of the Static Thermoelastic State of Plane Multilayer Thermosensitive Structures

“…In the general case, the integration constants C 1 and C 2 are determined from the system of two nonlinear algebraic equations obtained as a result of the substitution of expression (10), with regard for (13), in the boundary conditions (3). Note that, in the case where the conditions of heat exchange of the first or second kind are imposed on one of the boundary surfaces z = z i , i = 0, n, the procedure of determination of C 1 and C 2 is reduced to the solution of a single nonlinear algebraic equation.…”

“…The thermal stressed state of the analyzed structure under a given thermal and force load is described by the relations [4,10] σ .. = σ yy = σ 0 (z),…”

“…The methods aimed at finding one-dimensional stationary temperature fields and stresses caused by these fields in layered bodies for linear, quadratic, and cubic temperature dependences of the thermal conductivities of the materials of layers were proposed in [3,6,10,13]. Depending on the conditions of heat exchange, for any number of layers and any given form of temperature dependences of their thermal conductivities, these methods reduce the analyzed problem by means of the Kirchhoff transformation (change) [4] to the solution either of a single equation or of a system of two nonlinear algebraic equations.…”

“…In this case, the choice of the initial approximation and the analysis of the problem of existence and uniqueness of the solution require additional investigations. An integral representation of the solution of a one-dimensional stationary problem of heat conduction that does not impose restrictions on the character of temperature dependences of the thermal conductivities was proposed in the appendix of [10] for a special case of heating of multilayer bodies of simple geometric shapes whose inner surface is subjected to the action of a heat flux q 0 = const and the outer surface is kept at a temperature t 0 = const .…”

Thermal stress analysis of a thermosensitive functionally graded rectangular plate due to thermally induced resultant moments

“…They developed a numerical method and derived the solution of plane thermoelasticity problem for thermosensitive strip and temperature fields in thermosensitive structural elements. Protsyuk (2012) described an analytic-numerical method of solution of one-dimensional static thermoelasticity problems for layered plates, heated in different ways. Reddy and Chin (1998) studied the dynamic thermoelastic response of FG cylinders and plates by including thermo-mechanical coupling.…”