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Journal of Thermal Stresses, 34: 68 74, 2011 Copyright Taylor & Francis Group, LLC ISSN: 0149-5739 print/1521-074x online DOI: 10.1080/01495739.2010.511934 DISSIPATION FUNCTION IN HYPERBOLIC THERMOELASTICITY Keywords: MOTIVATION Martin Ostoja-Starzewski Department of Mechanical Science and Engineering, Institute for Condensed Matter Theory and Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA The equations governing two hyperbolic thermoelasticity theories (with one or with two relaxation times are examined from the standpoint of thermodynamics with internal variables, i.e., employing the free energy and the dissipation function explicitly. In the case of thermoelasticity with one relaxation time (the Lord Shulman model, the derivation relies on an extended state space and a representation theory of Edelen. In the case of thermoelasticity with two relaxation times (the Green Lindsay model, the derivation process by simply using the thermodynamic orthogonality of Ziegler. Dissipation; Heat conduction; Relaxation times; Thermoelasticity with finite wave speeds The Fourier heat conduction has always been recognized in physics as a dissipative process. Thus, it is described by a diffusion equation in a rigid conductor. Therefore, it should be possible to derive the constitutive equations of the thermoelasticity with two relaxation times which may well be viewed as a generalization of the Fourier heat conduction to elastic solids from the free energy and the dissipation function rather then only from the free energy as done in the original paper by Green and Lindsay [1]. In particular, their contains t 1 k 2 0 i j, which is a dissipative (! term. The Maxwell Cattaneo heat conduction is a dissipative process inso far as its diffusive nature is concerned, and at the same time a conservative process relative to its hyperbolic nature. Appropriately then, it is described by a telegraph equation in a rigid conductor. Therefore, it should be possible to derive the constitutive equations of the thermoelasticity with one relaxation time which may well be viewed as a generalization of the Maxwell Cattaneo heat conduction to elastic solids from another pair of and rather than only from the free energy as done in the original paper by Lord and Shulman [2]. More specifically, their contains t 0 2 0 q i q j, which again is a dissipative (! term. Received 21 February 2010; accepted 27 April 2010. Address correspondence to Martin Ostoja-Starzewski, Department of Mechanical Science and Engineering, Institute for Condensed Matter Theory and Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA; E-mail: martinos@illinois.edu 68
DISSIPATION FUNCTION IN HYPERBOLIC THERMOELASTICITY 69 In this paper we resolve these two issues from the standpoint of thermodynamics with internal variables (TIV based on two appropriately chosen pairs. The subject matter is important because both theories form the basis of generalized thermoelasticity models [3]. THEORY WITH ONE RELAXATION TIME To derive the constitutive equations of the theory with one relaxation time [2] begin with the (specific, per unit mass internal energy u as a function of the infinitesimal (elastic strain E, the entropy, and the heat flux q i The specific power of deformation is defined by u = u q i E (1 l = 1 S Ė (2 where S is the Cauchy stress. Also, the classical relation holds = u whereby, in view of (1 and by a partial Legendre transformation switching the argument to the absolute temperature, we recognize that = q i E and = q i E The first fundamental law takes the form u = S Ė q i i (3 while the second fundamental law is written in terms of the reversible r and irreversible i parts of entropy production rate ( = r + i with r qi = i and i 0 (4 Proceeding just like in TIV [4, 5], from (1 (4 we obtain or u = S Ė q i i ( u S Ė = E E ( u + q i q i ( u Ė + q i + q i i i (5 + i (6 If we consider the case of a rigid heat conductor, from (6 at zero net flow of heat, we get u = (7
70 M. OSTOJA-STARZEWSKI and since both sides are state functions (and hence independent of the particular process, this equation must be valid in general. Thus, noting = u, S Ė = Ė E + q q i + q i i + i (8 i Here we set up the the quasi-conservative stress as well as the entropy as S q = and = E This leads to the Clausius Duhem inequality in the form where S d q i i q q i + S d Ė = i 0 (10 i is the dissipative stress, the full stress being S = S q (9 + S d (11 The free energy is now taken in the form (with = 0 and 0 being the reference temperature q i E = 1 2 E C kl E kl + M E C E 2 0 2 + t 0 2 0 q i q j (12 while the (specific, per unit mass dissipation is a function of the heat flux, its rate q i, and the strain rate Ė : i q i q i Ė (13 Clearly, the inequality (10 may be written as a scalar product where Y v 0 (14 ( Y = i S d q i (15 is the vector of dissipative thermodynamic forces, and v = ( q i q i Ė v1 v 2 v 3 (16 is the vector of conjugate thermodynamic velocities. In (16 we define three subvectors v i. A general procedure based on the representation theory due to Edelen [6, 7] allows a derivation of the most general form of the constitutive relation either for
DISSIPATION FUNCTION IN HYPERBOLIC THERMOELASTICITY 71 v as a function of Y or for Y as a function of v, subject to (14. Pursuing the second alternative, the following steps are involved: Assume Yv, and determine it as Y = v + U or Y i = v i + U i (17 where the vector U = u 1 u 2 u 3 does not contribute to the entropy production (and hence may be called a powerless dissipative force according to U v = 0 (18 while the dissipation function is = 1 and U is uniquely determined, for a given Y, by U i = Also, the symmetry relations 1 0 0 v j [ Yi v v j Y i v U i v j v Yvd (19 Y ] jv d (20 v i = [ ] Yj v U j v i must hold, and these reduce to the classical Onsager reciprocity conditions Y i v v j Now, we can satisfy (14 and (18 by adopting with the vector U made of three sub-vectors: = Y jv v i iff U = 0 (21 v q i q i Ė = 1 q i q j (22 u 1i = t 0 q j u 2i = t 0 q j u 3 = 0 (23 On account of (17, we obtain i Y 1i = 1 + u 2 q 1i = 1 i q j + t 0 q j t 0 q j = Y 0 q 2i = + u i q 2i = t 0 i q j (24 S d Y 3 = Ė + u 3 = 0
72 M. OSTOJA-STARZEWSKI Observe: (i The first of (24 immediately yields the Maxwell Cattaneo model q i + t 0 q i = k j (25 where k = 1 is the Fourier-type thermal conductivity and the overdot on q i as a tensor density needs to be interpreted as the Lie (i.e., the Oldroyd upperconvected derivative [5, 8]. In the case of a rigid conductor the overdot is the material time derivative as shown by the requirement of Galilean invariance [9] or, independently, by continuum thermodynamics [10]. (ii The second of (24 is satisfied for 0, which is the same approximation as in the original model [2]; see also equations (R.1.1 (R.1.11 in [3]. (iii The third relation in (24 yields, as expected, a null dissipative mechanical stress, so that, on account of (9 (12 S = S q + S d = C kl E kl + M and = M E + C E 0 (26 Clearly, (24 and (25 coincide with the constitutive relations of the Lord Shulman theory, but their derivation has now followed a more logical route treating the heat conduction as a partially dissipative process, and thus as a process also described by the dissipation function rather than by the free energy function alone. (iv One can form a dissipation function in the Y-space, and derive the constitutive equations directly by thermodynamic orthogonality in that space. THEORY WITH TWO RELAXATION TIMES To obtain the constitutive equations of the theory with two relaxation times [1] from TIV, adopt the (specific, per unit mass internal energy u as a function of the infinitesimal (elastic strain E, the temperature, and its rate u = ue (27 along with the (specific, per unit mass dissipation functional as a function of the strain rate Ė, the heat flux, and its rate = Ė q i q i (28 An overdot denotes a material derivative /t for absolute tensors (like the temperature gradient and deformation rate and an Oldroyd derivative for tensor densities (like the heat flux and stress tensor. Agina proceeding just like in TIV [4], we obtain u = S Ė q i i i (29
DISSIPATION FUNCTION IN HYPERBOLIC THERMOELASTICITY 73 or ( u S Ė = E E ( u + q i q i ( u Ė + q i + q i i + i (30 Relation (7 holds as before, and hence (30 becomes S Ė = Ė E + q q i + q i i + i (31 i Since (9 and (11 carry through as well, the Clausius Duhem inequality becomes so that S q Adopt q i i q q i + S d Ė = i 0 (32 i = E = 1 2 E C kl E kl + M E C E 2 0 2 C E 0 t 0 (33 = E = C kl E kl + M and = = M E + C E 0 C E 0 t 0 (34 Adopt so that, with U = 0, i Y 1i = 1 2 i Ė q i = t 1 M Ė + q iq j (35 = q i q j and S d Combining the above relations, we obtain S = S q + S d Y 2 = Ė = t 1 M (36 = C kl E kl + M E + t 1 0 = 0 M E + C E + t 0 (37 q i = k j where, again, k = 1 is the Fourier thermal conductivity. Observe: (i These constitutive relations are the same as those of the Green Lindsay theory, but their derivation has followed a more logical route treating the Fourier-type heat conduction as a purely dissipative process, and thus as a process described by the dissipation function rather than by the free energy function.
74 M. OSTOJA-STARZEWSKI (ii In contradistinction to the L-S theory, the derivation of the G-L theory neither involves any approximation nor does it require the general theory of Edelen but only its special case, the thermodynamic orthogonality of Ziegler [4]. An interesting discussion of various aspects of both theories can also be found in [11]. REFERENCES 1. H. W. Lord and Y. Shulman, A Generalized Dynamical Theory of Thermoelasticity, J. Mech. Phys. Solids, vol. 15, pp. 299 309, 1967. 2. A. E. Green and K. A. Lindsay, Thermoelasticity, J. Elast., vol. 2, no. 1, pp. 1 7, 1972. 3. J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds, Oxford University Press, 2009. 4. H. Ziegler, An Introduction to Thermomechanics, North Holland, Amsterdam, 1983. 5. G. A. Maugin, The Thermomechanics of Nonlinear Irreversible Behaviors An Introduction, World Scientific, Singapore, 1999. 6. D. G. B. Edelen, On the Existence of Symmetry Relations and Dissipative Potentials, Arch. Rat. Mech. Anal., vol. 51, pp. 218 227, 1973. 7. D. G. B. Edelen, Primitive Thermodynamics: A New Look at the Clausius Duhem Inequality, Int. J. Eng. Sci., vol. 12, pp. 121 141, 1974. 8. C. I. Christov, On Frame Indifferent Formulation of the Maxwell Cattaneo Model of Finite-Speed Heat Conduction, Mech. Res. Comm., vol. 36, pp. 481 486, 2008. 9. C. I. Christov and P. M. Jordan, Heat Conduction Paradox Involving Second-Sound Propagation in Moving Media, Phys. Rev. Lett., vol. 94, pp. 154301-1 4, 2005. 10. M. Ostoja-Starzewski, A Derivation of the Maxwell Cattaneo Equation from the Free Energy and Dissipation Potentials, Intl. J. Eng. Sci., vol. 47, pp. 807 810, 2009. 11. R. B. Hetnarski and M. R. Eslami, Thermal Stresses Advanced Theory and Applications, Springer, 2009.