Thermodynamic consistency of objective rate equations

Accepted Manuscript Title: Thermodynamic consistency of objective rate equations Author: A. Morro PII: DOI: Reference: S0093-6413(16)30348-2 http://dx.doi.org/doi:10.1016/j.mechrescom.2017.06.008 MRC 3176 To appear in: Received date: Revised date: Accepted date: 15-12-2016 6-6-2017 9-6-2017 Please cite this article as: A. Morro, Thermodynamic consistency of objective rate equations, <![CDATA[Mechanics Research Communications]]> (2017), http://dx.doi.org/10.1016/j.mechrescom.2017.06.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Thermodynamic consistency of objective rate equations A. Morro DIBRIS, University of Genoa, Via Opera Pia 13, 16145 Genoa, Italy t Highlights Ac ce pt ed M an us cr ip Deformable and heat-conducting solids are modelled by objective rate equations for stress and heat flux. Objective time derivatives are those of Jaumann, Green-Naghdi, Oldroyd, and Truesdell. Thermodynamic consistency is established for the Jaumann, Green-Naghdi, Oldroyd derivatives in the spatial description. Thermodynamic consistency of equations involving the Truesdell rate is shown in the material description. 1 Page 1 of 8 Thermodynamic consistency of objective rate equations us cr ip t A. Morro∗ Abstract M an The paper addresses the modelling of solids via rate equations for the heat flux and the stress. As with any constitutive property, the rate equations are required to be both objective and consistent with the second law of thermodynamics. Upon a review of a connection between objective time derivatives, attention is restricted to rate equations where the time derivatives are those named after Jaumann, Green and Naghdi, Oldroyd, and Truesdell. Equations involving the Truesdell rate are investigated within the material description thanks to the identity between the time derivative of the material fluxes and the Truesdell rate in the current configuration. The remaining equations are examined within the spatial description. The occurrence of a skew tensor in the objective derivative results in a further restriction on the constitutive properties. The thermodynamic requirements are found to be satisfied and the corresponding free energy is determined by direct integrations. Ac ce pt ed Keywords: Rate equations; Objectivity; Compatibility with thermodynamics; Internal variables ∗ Corresponding author at: DIBRIS, University of Genoa, Via Opera Pia 13, 16145 Genoa, Italy. Email address: angelo.morro@unige.it Preprint submitted to Mech. Res. Commun. June 6, 2017 Page 2 of 8 4 5 6 7 8 9 10 11 12 of the material field (of vectors and tensors) is the material field of the corresponding Truesdell rate. The modelling of solid behaviour is often performed by means of internal variables, namely scalar, vector, or tensor fields governed by rate equations. Internal variables trace back to Duhem [1], §I.8, and Bridgman [2]. Thermodynamic analyses within continuum approaches were given by Eckart [3], and next by Green and Naghdi [4], and Coleman and Gurtin [5]. It is customary to describe the constitutive properties related to the internal variables via rate equations providing the evolution by means of a function of the state of the material. There are many examples of models based on rate equations of internal variables. The so-called Maxwell-Cattaneo equation relates the heat flux q to the temperature gradient g, in the form 40 2. Connection between rate equations q + τ q̇ = −κg, τ, κ > 0. (1) t 3 39 38 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 M We can then view q as an internal variable, such that q̇ is a (linear) function of g and q itself. The Cauchy stress T is often modelled by a rate equation, mainly on the basis of rheological models. Let E be the infinitesimal strain tensor. The rate equation 1 Ṫ = k Ė − T, k, τ > 0, τ characterizes the Maxwell model. Similar rate equations describe the standard linear solid, the Wiechert model, and the Jeffreys model. Two questions arise in connection with the rate equations. The time derivative, of vectors and tensors, is required to be objective namely it has to transform like vectors and tensors under a change of frames of reference. The well-known derivatives named after Jaumann, Green and Naghdi, Oldroyd and Truesdell are objective and moreover are connected to each other in a sense that is shown in §2. The second question is related to compatibility of rate equations, as constitutive equations, with thermodynamics. This is the main topic of the paper. The subject is of interest because objective time derivatives involve appropriate spin tensors. The analysis of Jaumann, Green-Naghdi, and Oldroyd time derivatives is developed in the spatial description and the occurrence of a spin tensor is shown to provide an additional restriction relative to the corresponding equations in non-objective form. Instead, the Truesdell rate is more naturally framed and investigated within the material description. This procedure is allowed since the time derivative We let a body occupy a three-dimensional region. The points of the body are labelled by their position vector X in a reference configuration R while x is the position in the current configuration. We denote by F the deformation gradient, relative to R, FiK = ∂XK xi , and by C = FT F the CauchyGreen tensor. Moreover, v is the velocity, L is the velocity gradient, Lij = ∂xj vi , D is the stretching tensor and W is the spin tensor so that L = D+W. ∇ is the gradient in the current configuration while ∇X is the gradient relative to the reference position X. The superscript T means transpose. We let θ be the absolute temperature and hence g = ∇θ. Also, sym denotes the symmetric part while Sym is the set of symmetric tensors. For any vector p, p2 stands for p · p. Let F , F ∗ be two frames of reference. The vector positions of a point x and x∗ , relative to F and F ∗ , are related by ([6], §17) us cr ip 2 1. Introduction an 1 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 ed pt 15 58 59 60 ce 14 57 x∗ = c(t) + Q(t)x, (2) where c(t) is an arbitrary vector-valued function, of time t, while Q(t) is a proper orthogonal tensor function, det Q = 1. Moreover we denote by Ω = Q̇QT = −ΩT the spin tensor associated with Q. Let K be any objective tensor so that the values in F ∗ and F are related by K∗ = QKQT . The material time derivative gives K˙ ∗ = ΩK∗ + K∗ ΩT + QK̇QT . Ac 13 56 (3) Let A be a tensor which transforms to A∗ such that A∗ = QAQT + Ω. (4) Substitution of Ω from (4) into (3) results in K˙ ∗ − A∗ K∗ − K∗ A∗ T = Q(K̇ − AK − KAT )QT . This means that, for any tensor A satisfying (4) and any objective tensor K, the derivative ◦ K= K̇ − AK − KAT 61 (5) is objective. If p is an objective vector, p∗ = Qp, then p˙∗ = ΩQp + Qṗ. 2 Page 3 of 8 82 3. Entropy inequalities 75 p˙∗ − A∗ p∗ = Q(ṗ − Ap). 76 77 This means that, subject to (4), 78 ◦ (6) p= ṗ − Ap 79 80 is an objective derivative. The velocity gradient L and the spin W obey the transformation law (4) (see [7], §20.3) L∗ = QLQT + Ω, 63 W∗ = QWQT + Ω Let ε be the energy density (per unit mass) and ρ the mass density. The balance of energy is written in the form whereas D is objective and ∇ · v is invariant. Let R be the rotation provided by the polar decomposition theorem, so that F = RU, U2 = FT F. Since F∗ = QF and R∗ = QR then ρε̇ = T · D − ∇ · q + ρr, an 62 67 the objective time derivatives (5) and (6) become the Jaumann, Green-Naghdi, and Oldroyd time derivatives of tensors and vectors [7, 8]. Let A△ and A▽ be two tensors which transform to A∗△ and A∗▽ according to (4). Hence the two derivatives ed 66 Hence ṘRT too satisfies (4). Letting A = W, ṘRT , L, △ ▽ K= K̇ − A▽ K − KAT▽ ce K= K̇ − A△ K − KAT△ , pt 65 83 84 are objective and the two rate equations △ K △ = K ▽ − (A△ − A▽ )K − K(AT△ − AT▽ ) 69 70 71 72 73 (11) where ψ = ε − θη is the Helmholtz free energy density. Let J = det F and ρR = J ρ. Moreover let TRR = JF−1 TF−T . (12) Hence ρR and qR are the mass density and the heat flux in the reference configuration. Also, TRR is the second Piola-Kirchhoff stress tensor. In the material description the Clausius-Duhem inequality can be written in the form [12, 13] (7) holds. The analogous statement is true for vectors. It then follows that, subject to the connection (7), the Jaumann, Green-Naghdi, and Oldroyd derivatives describe the same material response. Of course, the two material responses are different if K △ and K ▽ do not satisfy (7). Look at the Truesdell rate given by [9] 1 −ρR (ψ̇ + η θ̇) + 21 TRR · Ċ − qR · ∇X θ ≥ 0. (13) θ 85 86 87 88  p := ṗ − Lp + (∇ · v)p, (8) K := K̇ − LK − KLT + (∇ · v)K. (9)  1 −ρ(ψ̇ + η θ̇) + T · D − q · g ≥ 0, θ qR = JqF−T , ▽ are equivalent if only 68 Substitution of ρr −∇·q from the balance of energy allows inequality (10) to be written in the form K= K ▽ , Ac K= K △ , r being the heat supply. Let η be the entropy density. For the present purposes there is no loss in generality by letting the entropy flux be equal to q/θ so that the entropy inequality is assumed in the Clausius-Duhem form q ρr ρη̇ + ∇ · − ≥ 0. (10) θ θ M R˙ ∗ R∗ T = Ω + QṘRT QT . 64 t 81 Equations (8)-(9) take the forms (6)-(5) by letting A = L − (∇ · v)1 and A = L − 21 (∇ · v)1. In [10] an objective re-formulation of the Maxwell-Cattaneo equation is derived in a form  that, in the present notation, reads τ q +q = −kg.  The Truesdell rate p is referred to as Oldroyd’s upper convected derivative in [10] and a particular Lie-Oldroyd derivative in [11]. 74 us cr ip Substitution of Ω from (4) provides Hence we can state the second law of thermodynamics by saying that inequality (11), or (13), holds for any thermodynamic process compatible with the balance equations. Sometimes the second law of thermodynamics is assumed to be expressed by, or to comprise, q · g ≤ 0, (14) 3 Page 4 of 8 which is true for Fourier’s law and ensures that heat flows from hot to cold regions at any time t (see [14, 15, 16]). Now, by (10) we have The linearity and the arbitrariness of Ḋ, ġ, θ̇ imply 1 1 1 −η̇ + (ε̇ − T · D) + 2 q · g ≤ 0. θ ρ ρθ whence ψ = ψ(C, θ, q, T ). The remaining inequality suggests that we make the identification T = T − 2ρ F ∂C ψ FT . 105 93 94 The linearity and the arbitrariness of g, D, and A require that an 92 1 ρβ∂q ψ − q = 0, θ which reduces to (14) if q · g has a constant sign. That the approximation ψ = ψ(θ) provides (14) is consistent with the derivation of (14) in [16] where ρ is constant and ε̇ = θη̇ is regarded to hold (locally). 4. Thermodynamic consistency of spatial descriptions ed Denote by T a dissipative stress to be identified by the thermodynamic analysis. To fix ideas we let A = −AT and consider the rate equations T + T A = −γT T + µD, T˙ − AT 97 98 99 100 101 102 103 104 T − ρµ∂T ψ = 0, (19) T − T A) + ρ∂q ψ · Aq = 0, ρ∂T ψ · (AT (20) ργ∂T ψ · T + ρν∂qψ · q ≥ 0. (21) Substitution for ∂T ψ and ∂q ψ from (19) into (21) gives γ ν 2 T ·T + q ≥ 0. µ βθ Since γ, ν > 0 it follows that µ > 0, β > 0. (22) Moreover, in view of (19), eq. (20) becomes in the spatial description, where ν, β, γ, µ are allowed to depend on temperature. The left-hand sides of (15) and (16) become the Jaumann or the Green-Naghdi time derivative according as A = W or A = ṘRT . Equations (15) and (16) reduce to the Maxwell-Cattaneo equation and the Maxwell model when A = 0 and D is replaced by Ė. We assume ν > ν0 > 0 and γ > γ0 > 0 so that q and T are bounded in time. Let Γ = (C, θ, g, D, q, T ) be the set of independent variables so that ψ and η are functions of Γ T are given by (15) and (16). Moreover while q̇ and Ṫ let ψ(Γ) be differentiable. Since 1 1 T − T A) + q · Aq = 0. T · (AT µ β ce 96 (16) (23) The skew symmetry of A makes each term vanish, T = 0, T · AT Ac 95 (15) pt q̇ − Aq = −νq − βg, (18) Hence we have M 91 (17) T − T A − γT T ) + (T T − ρµ∂T ψ) · D −ρ∂T ψ · (AT 1 (ρβ∂q ψ − q) · g − ρ∂q ψ · (Aq − νq) ≥ 0. θ If no work is done, D = 0, and ψ = ψ(θ), so that ε̇ = θη̇, then the inequality simplifies to Z t2  1  q · g dt ≤ 0, 2 ρθ t1 90 η = −∂θ ψ. us cr ip If the system undergoes a cyclic process as t ∈ [t1 , t2 ], and hence η(t2 ) = η(t1 ), then Z t2 1  1 1 (ε̇ − T · D) + 2 q · g dt ≤ 0. θ ρ ρθ t1 89 ∂g ψ = 0, t ∂D ψ = 0, T · T A = 0, q · Aq = 0. Finally, by (19) two direct integrations give ψ = Ψ(C, θ) + 1 1 q2 + T ·T . 2ρβθ 2ρµ ∂C ψ · Ċ = 2(F ∂C ψ FT ) · D Equations (17)-(21) are then necessary and sufficient for the thermodynamic consistency of (15) and (16), subject to the identification of T . Now, then, upon evaluation of ψ̇ and substitution in (11), T = 2ρ F ∂C Ψ FT + T , −ρ(∂θ ψ + η)θ̇ + (T − 2ρF ∂C ψ FT ) · D 1 − q · g − ρ∂g ψ · ġ − ρ∂D ψ · Ḋ θ −ρ∂q ψ · (Aq − νq − βg) T − T A − γT T + µD) ≥ 0. −ρ∂T ψ · (AT 106 107 108 109 110 where T is given by (16), may be viewed as the generalized stress of the standard linear solid, with Ė replaced by D. Accordingly, while the Maxwell model is said to describe viscoelastic fluids, the whole stress T describes a viscoelastic solid. 4 Page 5 of 8 T = 0, A = 0) In stationary conditions (q̇ = 0, Ṫ 116 Since ν, γ > 0, the thermodynamic requirements (22) result in the positive value of the heat conductivity β/ν and of the shear viscosity µ/γ. The thermodynamic requirement (23) holds for every skew tensor A. Hence the conclusions hold for the Green-Naghdi and Jaumann derivatives. 117 4.1. A model with the Oldroyd derivative 111 112 113 114 115 t T = (µ/γ)D. τ (θ)q̇ = −κ(θ)g − q thus showing a relaxation time τ (θ). This may indicate that we look for models where the appropriate internal variable, p say, is just τ (θ)q [18]. To account for deformation and hence for objectivity of the time derivative we may replace (15) by The occurrence of D in A = L leads to different thermodynamic results. For definiteness we investigate the rate equation ṗ − Ap = ν̂p − β̂g, ν̂ and β̂ being functions of θ. We then let Γ = (C, θ, g, D, p, T ), with T subject to (16), and repeat the previous procedure to obtain again (17) and 1 ρβ̂∂p ψ − p = 0, T − ρµ∂T ψ = 0, (27) θτ T − T A) + ρ∂p ψ · Ap = 0, ρ∂T ψ · (AT (28) (24) ⋄ 119 120 121 ργ∂T ψ · T + ρν̂∂p ψ · p ≥ 0. ed 122 for q. So we replace q̇−Aq in (15) with q := q̇−Lq whereas no rate equation for the stress is considered. Accordingly we let Γ = (C, θ, g, D, q) be the set of independent variables. Upon substitution of ψ̇, the entropy inequality (11) becomes M 118 1 −ρ(∂θ ψ + η)θ̇ − ρ∂g ψ · ġ − ρ∂D ψ · Ḋ − q · g θ T · D − ρ∂q ψ · (Wq + Dq − νq − βg) ≥ 0, +T ce pt where T is given by (18). The linearity and the arbitrariness of ġ, Ḋ, and θ̇ imply that ∂g ψ = 0, ∂D ψ = 0, and η = −∂θ ψ. The remaining inequality holds if and only if T̂ · D ≥ 0, T̂ := T − ρ∂q ψ ⊗ q, Ac 1 βρ∂q ψ − q = 0, θ 128 ∂q ψ ⊗ q ∈ Sym ∂q ψ · q ≥ 0. (25) 1 ψ = Ψ(C, θ) + q2 , 2βρθ 129 β > 0. 130 131 132 Hence ∂q ψ ⊗ q ∈ Sym holds identically. As a consequence T = 2ρ F ∂C ψ FT + 124 125 126 (29) Hence we find that ψ = Ψ(C, θ) + 1 2ρβ̂θτ p2 + 1 T ·T , 2ρµ where µ > 0, β̂ > 0. Remark. It is sometimes claimed that the Maxwell-Cattaneo equation is not compatible with the second law of thermodynamics. This objection stems from the heat conduction inequality (14) as the content of the second law for undeformable bodies [14, 15, 16]. It is true in fact that, by (1), 1 (q · q + τ q · q̇) κ and the right-hand side is not positive definite because of the time derivative q̇. Even worse would be the case with objective rate equations. No inconsistency though holds because, for time-dependent fields, the second law is expressed by (11) or (13). −q · g = By (25) it follows that 123 (26) an q̇ − Lq = −νq − βg 4.2. A model for second sound in solids The appropriate internal variables need not be the physical variables q and T . As an example, in [17] the heat flux equation is found to fit data on wave speed at cryogenic temperatures by taking the rate equation in the form us cr ip q = −(β/ν)g, 127 1 q ⊗ q + T̂, βθ 133 134 135 5. Thermodynamic consistency of material descriptions Rate equations involving the Truesdell derivative are naturally framed within a material description. This is so since the definition (12) and use of where T̂ is the dissipative part of T, possibly T̂ = µD, µ > 0. Hence the rate equation for q via the Oldroyd derivative implies that the stress embodies the q-dependent term q ⊗ q/βθ. J˙ = J∇ · v, ˙ F−1 = −F−1 Ḟ F−1 5 Page 6 of 8 make the material heat flux qR and the second Piola-Kirchhoff stress TRR satisfy −1 ṪRR = JF 144 145 146 147 148 149 150 151 152 153 154 155 156 157 ẇ + λw = −α∇X θ, (32) Ẏ + ΛY = ΦĊ, (33) for the invariant vector w and the invariant tensor Y within the material description. We assume that λ, Λ, α, and Φ are invariant scalars so that the rate equations (32) and (33) are invariant. We assume that λ ≥ λ0 > 0, Λ ≥ Λ0 > 0. The field w may be taken as a scalar, ξ say, times qR . The tensor Y will be identified in a moment by thermodynamic arguments. Let ψ, η, qR , TRR be continuous functions of the set of independent variables Γ = (C, θ, w, Y, ∇Xθ, Ċ). Moreover we let ψ be continuously differentiable while ξ, ..., Φ are allowed to depend on θ. Evaluation of ψ̇ and substitution in (13) provides +( 12 TRR − ρR ∂C ψ − ρR Φ∂Y ψ) · Ċ 1 −ρR (∂θ ψ + η)θ̇ + (ρR α ∂w ψ − qR ) · ∇X θ θ +ρR λ∂w ψ · w + ρR Λ∂Y ψ · Y ≥ 0. 1 Y − ρR Φ∂Y ψ = 0, 2Ξ 158 ∂Ċ ψ = 0, η = −∂θ ψ. ρR α∂w ψ − (36) 1 w = 0, ξθ ρR λ∂w ψ · w + ρR Λ∂Y ψ · Y ≥ 0. (37) The direct integrations with respect to w and Y provide ψ = Ψ(θ, C) + 1 1 Y·Y+ w2 . (38) 4ρR ΦΞ 2ρR αξθ Substitution in (37) shows that the inequality holds if and only if 159 160 161 162 α > 0, 164 165 166 167 (34) The remaining inequality, Φ > 0. (39) The requirements (34), (38), and (39) are necessary and sufficient for the thermodynamic consistency of the rate equations (32) and (33) and the assumptions (35), (36). Equations (32) and (33) are now integrated. Subject to λ ≥ λ0 > 0 and Λ ≥ Λ0 > 0 we can integrate on (−∞, t] to obtain Z t t Y(t) = Φ(τ ) exp(− ∫ Λ(s)ds)Ċ(τ )dτ, (40) −∞ The linearity and the arbitrariness of the values ∇X θ̇, C̈, θ̇ imply that ∂∇X θ ψ = 0, ξ = ξ(θ) > 0. The linearity of Ċ, ∇Xθ and the arbitrariness of Ċ, ∇X θ, w, Y require that 163 −ρR ∂∇X θ ψ · ∇X θ̇ − ρR ∂Ċ ψ · C̈ (35) us cr ip 142 143 1 Y. Ξ an 141 M 140 1 w, ξ qR = ed 139 Moreover we let Hence the effect of the Truesdell derivative in the spatial description can be determined by considering the time derivative in the material description. The material fields qR and TRR are invariant under a change of frame. As a consequence, the time derivatives are also invariant under a change of frame. Indeed, if λ, ξ, and Λ, Ξ are invariant then λqR , (ξqR )˙, ΛTRR , and (ΞTRR )˙ are invariant too. The temperature gradient ∇X θ, in the reference configuration, and the Cauchy-Green tensor C, and hence Ċ, are invariant. Borrowing again from the Maxwell-Cattaneo equation for the heat flux and the equation for a solid modelled by the Maxwell element we consider the rate equations pt 138 TRR = 2ρR ∂C ψ + . (31) ce 137 (Ṫ−LT−TL +(∇ · v)T)F Ac 136 (30) −T T t q̇R = J(q̇ − Lq + (∇ · v)q)F−T , indicates that we identify Y with a positive scalar function 2Ξ(θ) times 21 TRR − ρR ∂C ψ so that τ and the like for w. Hence by (35) the second PiolaKirchhoff stress TRR is the sum of an elastic part, 2ρR ∂C Ψ, and a memory functional, Y/Ξ, of the history of Ċ. By (36) the reference heat flux qR , is a memory functional, w/ξ, of the history of ∇Xθ. We now look for the rate equations, in the spatial description, associated with (32) and (33). First we observe that, by viewing ψ as a function of F through C, the chain rule gives ∂F ψ = 2 F ∂C ψ. As a consequence ρ ∂F ψ FT = 2 ρ F ∂C ψ FT and hence 2 ρR ∂C ψ = ρ J F−1 ∂F ψ = J F−1 (ρ ∂F ψ FT )F−T . ( 21 TRR − ρR ∂C ψ − ρR Φ∂Y ψ) · Ċ 1 +(ρR α∂w ψ − qR ) · ∇Xθ θ +ρR λ∂w ψ · w + ρR Λ∂Y ψ · Y ≥ 0, This in turn implies that TdRR := Y/Ξ = J F−1T F−T , T := T−ρ ∂F ψ FT , (41) 6 Page 7 of 8 where T is the (Cauchy) stress T, deprived of the elastic part ρ ∂F ψ FT , in the spatial description. As with (31), since TdRR = J F−1T F−T we find that is thermodynamically consistent if and only if so is ◦ p = fO (p, g) + Dp. d Let w = ξqR and Y = ΞTRR . From (32) and (33) we have (ξqR )˙+λξqR = −α∇X θ, 182 References 178 179 180 (ΞTdRR )˙+ΛΞTdRR = ΦĊ. Hence by (30) and (42) we obtain α ˙ q̇ − Lq + ∇ · v q + (λ + ξ/ξ)q = − F ∇Xθ, Jξ 183 184 185 Φ T −T T LT +∇·v T +(Λ+ Ξ̇/Ξ)T T = T˙ −LT F Ċ FT . JΞ To determine T we go back to (41) and find that 186 187 188 189 190 170 171 1 (t)F(t)Y(t)F−T (t) JΞ 191 192 193 where Y(t) is given by (40). The analogue holds for q with obvious changes. 194 195 196 ed 197 172 198 6. Conclusions 199 pt The rate equations involving objective time derivatives (for a vector p) are relations of the form ṗ − Ap = fA(p, g), A = −AT , (43) ce for the Jaumann and Green-Naghdi time derivatives, and  p = fT (p, g), (44) Ac for the Truesdell derivative. The thermodynamic consistency is shown in the spatial description for (43), with fA (q, g) = −νq − βg, and in the material description for (44) with fT (p, g) = −λp − (α/Jξ)Bg. The crucial role of the Truesdell rate is related to the identity  ˙ JF−1 KF−T = JF−1 K F−T , 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 173 174 175 176 [1] C. Truesdell, Rational Thermodynamics, Springer, New York (1984), ch. 5. [2] P.W. Bridgman, The Nature of Thermodynamics, Harvard University Press, Cambridge, MA, 1961. [3] C. Eckart, The thermodynamics of irreversible processes. IV. The theory of elasticity and anelasticity, Phys. Rev. 73 (1948) 373-382. [4] A.E. Green, P.M. Naghdi, A general theory of an elastic-plastic continuum. Arch. Rational Mech. Anal. 1 (1965) 251-281. [5] B.D. Coleman, M.E. Gurtin, Thermodynamics with internal state variables, J. Chem. Phys. 47 (1967) 597613. [6] C.A. Truesdell, W. Noll, The non-linear field theories of mechanics. In Encyclopedia of Physics, ed. S. Flügge, vol. III/3, Springer, Berlin, 1965. [7] M.E. Gurtin, E. Fried, L. Anand, The Mechanics and Thermodynamics of Continua, Cambridge University Press, 2011. [8] J.G. Oldroyd, On the formulation of rheological equations of state, Proc. Roy. Soc. A 200 (1950) 523-541. [9] C.A. Truesdell, The simplest rate theory of pure elasticity, Comm. Pure Appl. Math. 8 (1955) 123-132. [10] C.I. Christov, On frame indifferent formulation of the Maxwell-Cattaneo model of finite-speed heat conduction, Mech. Res. Comm. 36 (2009) 481-486. [11] I.C. Christov, P.M. Jordan, On an instability exhibited by the ballistic-diffusive heat conduction model of Xu and Hu, Proc. R. Soc. A 470 (2014) issue 2161. [12] A. Morro, Evolution equations and thermodynamic restrictions for dissipative solids, Math. Comp. Modelling 52 (2010) 1869-1876. [13] A. Morro, Evolution equations for non-simple viscoelastic solids, J. Elasticity 105 (2011) 93-105. [14] F. Franchi, B. Straughan, Continuous dependence on the relaxation time and modelling, and unbounded growth, in theories of heat conduction with finite propagation speeds, J. Math. Anal. Appl. 185 (1994) 726-746. [15] M.B. Rubin, Hyperbolic heat conduction and the second law, Int. J. Engng Sci. 30 (1992) 1665-1676. [16] A. Barletta, E. Zanchini, Hyperbolic heat conduction and local equilibrium: a second law analysis, Int. J. Heat Mass Transfer 40 (1997) 1007-1016. [17] D.D. Coleman, D.C. Newman, Implications of a nonlinearity in the theory of second sound in solids, Phys. Rev. B37 (1988) 1492-1498. [18] A. Morro, T. Ruggeri, Second sound and internal energy in solids, Int. J. Non-Linear Mech. 22 (1987) 27-36. M T (t) = t 181 Yet the occurrence of Dp makes the analysis of ⋄ the thermodynamic consistency of p = fO (p, g) quite different from, or inequivalent to, that for ◦ p = fO (p, g) + Dp. For definiteness this is shown via the example in §4. 177 T −T T LT )F−T . (42) (TdRR )˙ = JF−1 (T˙ +∇·v T −LT us cr ip 169 an 168 for any tensor K and the analogue for vectors. The thermodynamic restrictions so established are all satisfied and the corresponding free energy is determined. ⋄ Since the Oldroyd derivative p is related to the ◦ ⋄ ◦ Jaumann derivative p = ṗ − Wp by p = p −Dp then the rate equation 222 223 224 225 226 227 228 229 230 ⋄ p = fO (p, g) 7 Page 8 of 8