Keywords: nonlinear thermoelasticity, thermo-mechanical coupling, first-order shear deformation theory, finite elements, nonlinear shell theory.

Due to thermo-elastic coupling solids and structures exhibit non-classical physical effects, like e.g. strain rate dependent cooling or heating due to the action of tensile or compressive stresses, respectively, or damping of vibrations due to heat loss. These effects cannot be modelled by merely accounting for the effect of temperature changes on the strains in the constitutive equations. The truly thermo-mechanically coupled analysis of solids and structures is based on coupling of the thermal and mechanical quantities in the field equations.

Thermodynamically consistent theories of three-dimensional thermoelasticity that account for the thermo-mechanical coupling have been developed in the early works of Biot [1], Boley [2], Nowacki [3,4,5]. Tauchert [6] presents the coupled equations of motion and heat transfer governing small displacements and temperature changes in thin isotropic plates.

In the present paper, a thermodynamically consistent continuum mechanics based framework of thermoelasticity in convective coordinates is used as described in [7]. It includes the conservation laws of mass, linear and angular momentum, and energy in a geometrically and thermally non-linear approach. The second principle of thermodynamics is used to derive the restrictions for the constitutive equations. On this basis weak formulations of the conditions of equilibrium and conservation of energy are derived that serve as a starting point for the non-linear finite element analysis of structures. For the transition to two-dimensional plate and shell theory, the first-order shear deformation (Reissner-Mindlin) hypothesis is used along with the hypothesis of a cubic through-thickness distribution of the temperature field. Two examples are computed, where in the first example heating of a rod due to compression is analysed and the numerical results are compared with experimental results available in literature. The second example deals with damping effects on thermally induced vibrations of a hinged square plate subject to a sudden change of temperature on one bounding surface.

1

M.A. Biot, "Thermoelasticity and Irreversible Thermodynamics", J. Appl. Phys., 27, 240-253, 1956. doi:10.1063/1.1722351

2

B.A. Boley, J.H. Weiner, "Theory of Thermal Stresses", John Wiley & Sons, Inc., New York London Sydney, 1960.

3

W. Nowacki, "Thermoelasticity", Addison-Wesley, Reading, MA, 1962; Pergamon, Oxford, 1986.

4

W. Nowacki, "Baudynamik", Translation of the 2nd Polish ed., Springer-Verlag, Wien New York, 1974.

5

W. Nowacki, "Dynamic Problems of Thermoelasticity", Noordhoff, Leyden, 1975.

6

T.R. Tauchert, "Thermal Stresses in Plates - Dynamical Problems", Thermal Stresses II, ed. R.B. Hetnarski, Elsevier Science Publishers B.V., Amsterdam, 1987.

7

R. Schmidt, "A Thermodynamically Consistent Theory of Elastic Plates and Shells at Finite Displacements and Rotations of the Surface Director" (in German), Habilitation Thesis, University of Wuppertal, 1993.

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