Couple-Stresses in the Theory of Thermoelasticity
The aim of the present paper is to generalize some theorems on the coupled thermoelasticity of a medium characterized by two vectors independent from each other: the displacement vector u and the rotation vector ω.
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W. Nowacki Warsaw (Poland) Summary The aim of the present paper is to generalize some theorems on the coupled thermoelasticity of a medium characterized by two vectors independent from each other: the displacement vector u and the rotation vector w. Basing on the thermodynamics of irreversible processes the constitutive equations and the expanded equation of heat conductivity for an isotropic medium are derived. The author succeeded in obtaining a basic system of differential equations of coupled thermoelasticity. The propagation of thermoelastic waves in an unbounded medium is discussed. Moreover, a generalization of the virtual work principle to dynamic problem of coupled thermoelasticity is advanced. Finally, the reciprocity theorem is derived and some conclusions resulting from this theorem are discussed.
1. Introduction The asymmetric theory of elasticity was first advanced in 1887 by VOIGT [1], and then developed in 1909 by the brothers E. and F. Co SSERAT [2]. It is assumed in this theory that the entire action upon a material volume bounded by a surface is described completely in terms of the field of stress vectors and "couple-stress" vectors. Recently we witness a further development of this theory, particularly since it proved useful in explaining some regularities of propagation of short acoustic waves in crystals, in polycrystalline structures as well as in high polymers. Let us mention here the papers by TRUESDELL and TOUPIN [3], and by AERO and KUVSHINSKII [4]. GRIOLI [5] and TOUPIN [6] succeeded in generalizing the theory of the Cosserat medium to finite deformations. Finally, MINDLIN and TIERSTEN [7] presented an exhaustive discussion of the linear theory of a homogeneous, isotropic and centrosymmetric medium. 17*
H. Parkus et al. (eds.), Irreversible Aspects of Continuum Mechanics and Transfer of Physical Characteristics in Moving Fluids © Springer-Verlag/Wien 1968
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NOWACKI
In subsequent years KUVSHINSKII and AERO (1963, [8]), PALMOV [9] and ERINGEN and SURUBI [10] developed the theory of asymmetric elasticity describing the deformation of a body by the vectors of displacement u and rotation w, mutually independent thus departing from the kinematic assumption w
= ~
curl u, which is the very basis of
the theory of the Cosserat medium. The purpose of the present work is to consider, within the framework of the theory of asymmetric elasticity, the interaction between the fields of displacements u, rotation wand temperature 0. Confining our considerations to the elastic, homogeneous, isotropic and centrosymmetric medium we derive constitutive equations based on the thermodynamics of irreversible processes. A complete set of differential equations of asymmetric thermoelasticity is given as well as equations of motion and an generalized equation of heat conductivity. Finally, a variational theorem and a theorem on reciprocity are given, and some conclusions are derived from these theorems.
2. Equations of Motion. Energy Equation and Entropy Balance The system of equations of motion cons
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