On the influence of microstructure on heat conduction in solids

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1 On the influence of microstructure on heat conduction in solids Arkadi Berezovski Institute of Cybernetics at Tallinn University of Technology, Akadeemia tee 21, Tallinn, Estonia Abstract The description of heat conduction in microstructured solids is presented in the framework of the dual internal variable approach. One of the internal variables is identified with microtemperature, i.e. the fluctuation of macroscopic temperature due to the inhomogeneity of the body. It is shown that the microstructure influence may result in a hyperbolic heat propagation for the microtemperature. The macroscale heat conduction is described by a parabolic equation which is coupled with the hyperbolic equation for the microtemperature. Keywords: Heat conduction, Single and dual internal variables, Microstructured solids 2010 MSC: 80A20, 74A60, 74F05 1. Introduction 5 10 In the one-dimension setting, heat conduction in homogeneous solids without internal heat sources is governed by the energy conservation equation (Ericksen, 1998) E t +Q x = 0, (1) where E is the internal energy density, Q is the heat flux, indices denote time and space derivatives. Considering a rigid body we suppose that the internal energy depends only on absolute temperature θ, i.e., E = E(θ). As every physical process, heat conduction satisfies the second law of thermodynamics. The second law is expressed in the form of the Clausius-Duhem inequality (Ericksen, 1998) S t +(θ 1 Q) x 0, (2) where S is the entropy per unit volume. The Helmholtz free energy density W(θ) = E θs connects the internal energy and entropy and the following relation is fulfilled: S = W θ. (3) Preprint submitted to International Communications in Heat and Mass TransferJuly 26, 2016

2 15 The energy conservation equation can be represented in the canonical form (Maugin, 2006) in terms of the entropy and the free energy (Sθ) t +Q x = h th, h th = W t = Sθ t. (4) It follows from the latter equation that θs t +Q x = 0, (5) and the Clausius-Duhem inequality can be represented as θs t +θ(θ 1 Q) x 0, (6) or, even simpler, accounting Eq. (5), θ 1 Qθ x 0. (7) 20 Since absolute temperature is non-negative by definition, the second law will be fulfilled automatically under the choice Q = λ 2 θ x. (8) 25 This is nothing else but the Fourier law of heat conduction (Narasimhan, 1999). It should be noted that Eq. (7) contains the product of the thermodynamic flux Q and the thermodynamic force θ x. However, the Fourier law is not the unique choice for the relation between the heat flux and the temperature gradient. Well known other possibilities can be classified as follows (Ván and Fülöp, 2012): Q = λ 2 θ x Fourier (1822) τq t +Q = λ 2 θ x Cattaneo (1948), Vernotte (1958) τ GK Q t +Q = λ 2 θ x +aq xx Guyer and Krumhansl (1966) τq t +Q = λ 2 θ x +bτθ xt Jeffreys type(joseph and Preziosi, 1989) Q t = λ 2 θ x +aq xx Green and Naghdi (1991) 30 where τ and τ GK are relaxation times (specific for each model), a and b are appropriate coefficients. These models are examined in detail in reviews by Joseph and Preziosi (1989); Tamma and Zhou (1998); Cimmelli (2009) and in books by Wang et al. (2007); Straughan (2011). The considered relations between the heat flux and the temperature gradient can be unified and extended as follows (Ván and Fülöp, 2012): τq t +Q = λ 2 θ x +aq xx +bτθ xt +cq xxt, 35 where an additional coefficient c is introduced and the relaxation time τ varies according to a model. It should be noted that all the heat transfer models mentioned above are elaborated for homogeneous bodies. The same is true for the more recently developed dual-phase-lagging model (Tzou, 1995; Zhang et al., 2013) τq t +Q = λ 2 (θ x +τ θ θ xt ), 2

3 and for the thermomass model (Cao and Guo, 2007; Zhang et al., 2013) τ TM Q t +Q = λ 2 Q θ x +τ TM θ θ Q t τ TM ρc v θ Q Q Q x +τ TM ρc v θ θ θ x, where τ θ and τ TM are specific relaxation times for dual-phase lagging and thermomass models, ρ is the density of a material, and C v is the heat capacity at constant volume. In spite of the controversy of recent experimental results concerning the non-fourier heat conduction (Tang and Araki, 2000; Herwig and Beckert, 2000; Roetzel et al.,2003;scott et al.,2009;bright and Zhang,2009;Both et al.,2016), only heat conduction models for homogeneous materials are used for planing and interpretation of experiments. This means that the heat transport mechanisms in materials with nonhomogeneous inner structures is clearly not understood to date, as pointed out by Tamma and Zhou (1998). At the same time, it has been demonstrated how the influence of a microstructure can be taken into account in generalized continua by means of the internal variables (Maugin, 2006; Berezovski et al., 2011a, 2014). Moreover, this technique reveals its descriptive capability also in the thermoelastic case as shown recently in (Berezovski et al., 2011b; Berezovski and Berezovski, 2013; Berezovski and Engelbrecht, 2013; Berezovski et al., 2014). It is worst, therefore, to apply the internal variables method to the examination of the microstructure impact in heat conduction problems. Inner inhomogeneities in solids induce temperature fluctuations due to the variation in material properties. Though such fluctuations are, as a rule, small in magnitude, their gradients may be not necessarily small. The aim of the paper is to describe how internal variables can be used for the accounting of the microstructural influence on heat conduction in solids. We start with the well established single internal variable theory to explain main features of the internal variables formalism. Then this technique is extended by the introduction of an additional dual internal variable. The similarity and the difference between these two approaches are demonstrated explicitly. For the simplicity, all the considerations are presented in the one-dimensional setting. 2. Heat conduction in microstructured solids. Single internal variable explication 70 It is supposed that the aggregate effect of a microstructure is characterized be a certain additional field ϕ (Rice, 1971). Therefore this variable and its gradient are included into the set of state variables: W = W(θ,ϕ,ϕ x ). (9) Introducing internal variables we extend the thermodynamic state space. To be able to use the thermodynamic formalism, we accept the concept of local equilibrium state (Maugin and Muschik, 1994)]. This assumes that there always exists a local accompanying equilibrium state, onto which the local non-equilibrium 3

4 75 80 state can be projected (Muschik, 1990; Kestin, 1993). Although this mapping or projection may not be unique and one-to-one, the concepts of thermostatics are assumed to be applicable to the local accompanying equilibrium state, and then to the corresponding local constrained non-equilibrium state. Then partial derivatives of the free energy W with respect to the state variables define the entropy, S, the ϕ-affinity, υ, and the force conjugated to the gradient of the internal variable, η, in the standard way (Maugin, 1999; Lebon et al., 2008) S := W θ, υ := W, η := W. (10) ϕ ϕ x The canonical energy conservation equation keeps its form (Sθ) t +Q x = h int, h int := W t, (11) 85 where the right-hand side of Eq. (11) 1 is formally an internal heat source (Maugin, 2006). The energy conservation equation is accompanied by the second law of thermodynamics, represented in the form of the Clausius-Duhem inequality S t +(θ 1 Q+K) x 0, (12) 90 where, in contrast to the homogeneous case, the extra entropy flux K is appended to the classical entropy flux (Maugin, 2006). Multiplying the Clausius- Duhem inequality (12) by θ and taking into account Eq. (11), we obtain θs t +θ(θ 1 Q+K) x 0, (13) ( W t +Sθ t ) +(θk)x (θ 1 Q+K)θ x 0. (14) The last equation can be represented in the form Sθ t +(θ 1 Q+K)θ x h int +(θk) x. (15) The internal heat sourceh int is calculated following the constitutive assumption (9) h int = W t = W θ θ t W ϕ ϕ t W ϕ x ϕ xt = Sθ t +υϕ t +ηϕ xt. (16) 95 Accounting for Eq. (16), dissipation inequality (15) can be rewritten as υϕ t +ηϕ xt (θ 1 Q+K)θ x +(θk) x 0. (17) Torearrangethe dissipationinequality, we add and subtract the sameterm η x ϕ t υϕ t +η ϕ x η x ϕ t +η x ϕ t (θ 1 Q+K)θ x +(θk) x 0, (18) 4

5 which leads to (υ η x )ϕ t (θ 1 Q+K)θ x +(ηϕ t +θk) x 0. (19) 100 As one can see, the first two terms in Eq. (19) represent products of thermodynamical forces and fluxes, but the third one is related to the divergence of a certain combination depending on the internal variable and the extra entropy flux. It is clear that the elimination of this divergence term leads to the pure thermodynamical flux-force relation. This idea has been formulated explicitly by Maugin(1990). Utilizing this idea, we define the extra entropy flux as follows: Then the dissipation inequality reduces to K = θ 1 ηϕ t. (20) θ(υ η x )ϕ t (Q ηϕ t )θ x 0. (21) This is the basis for the derivation of the evolution equation for the internal variable Evolution equation for the single internal variable Following de Groot and Mazur (1962), we represent thermodynamic fluxes ϕ t and (Q ηϕ t ) as linear functions of conjugated thermodynamic forces which delivers the solution of dissipation inequality (21) ( ) ϕ t = M (Q ηϕ t ) ( ) θ(υ ηx ), where M = θ x ( ) M11 M 12, (22) M 21 M 22 where components M ij of the matrix M are considered as constants for simplicity. The non-negativity of the entropy production (21) results in the positive semidefiniteness of the symmetric part of the matrix M, which requires M 11 0, M 22 0, M 11 M 22 (M 12 +M 21 ) 2 Thus, the evolution equation for the internal variable ϕ 2 0. (23) ϕ t = M 11 θ(υ η x ) M 12 θ x, (24) 115 depends on temperature and its gradient. The same is valid for the generalized heat flux Q ηϕ t = M 21 θ(υ η x ) M 22 θ x. (25) Returning to energy conservation equation (11) we represent the internal heat source h int in the form (Sθ) t +Q x = h int, (26) h int = Sθ t +υϕ t +ηϕ xt = Sθ t +(υ η x )ϕ t +(ηϕ t ) x. (27) 5

6 It follows then that the energy conservation equation can be rewritten as (Sθ) t +(Q ηϕ t ) x = Sθ t +(υ η x )ϕ t = h th +h intr. (28) 120 Accounting for Eq. (25), we can eliminate the heat flux from the energy conservation equation which results in θs t M 22 θ xx = (υ η x )ϕ t (M 21 θ(υ η x )) x. (29) This is the most general form of the energy conservation equation in the case of linear relation between thermodynamic forces and fluxes within the single internal variable approach Quadratic free energy To be more specific, we will use a quadratic free energy density W = ρc p 2θ 0 (θ θ 0 ) Bϕ Cϕ2 x, (30) where c p is the heat capacity, θ 0 is the reference temperature, B and C are material parameters. It follows from equations of state that S = W θ = ρc p (θ θ 0 ), υ := W θ 0 ϕ and evolution equation (24) is reduced to = Bϕ, η := W ϕ x = Cϕ x, (31) ϕ t = M 11 θ(cϕ xx Bϕ) M 12 θ x. (32) 130 Correspondingly, energy conservation equation (29) has the form θs t M 22 θ xx = (Cϕ xx Bϕ)ϕ t (M 21 θ(cϕ xx Bϕ)) x. (33) For small deviations of temperature from the reference value θ 0, we obtain then the heat conduction equation ρc p θ t M 22 θ xx = (Cϕ xx Bϕ)ϕ t M 21 θ x (Cϕ xx Bϕ) M 21 θ 0 (Cϕ xx Bϕ) x. (34) Evolution equation for the internal variable (32) and heat conduction equation (34) are coupled parabolic equations. Together they describe the transient temperature distribution in a body with microstructure. It is natural to consider the internal variable ϕ as a microtemperature, i.e. the fluctuation of temperature relative to the mean macroscopic value. While the microtemperature can be small in magnitude, its gradient may be not necessarily small. Neglecting the explicit dependence of the free energy of the microtemperature, i.e. W = W(θ,ϕ x ), we obtain simplified governing equations ϕ t = M 11 θ 0 Cϕ xx M 12 θ x, (35) 6

7 ρc p θ t M 22 θ xx = Cϕ xx ϕ t M 21 θ x Cϕ xx M 21 θ 0 Cϕ xxx. (36) It may be instructive to point out that in the case of M 11 = 0 the internal variable ϕ can be interpreted as the thermal displacement gradient in the spirit by Green and Naghdi (1991). The introduction of an internal variable for the description of heat conduction in solids with microstructure allows us to identify this internal variable with the microtemperature, i.e. with fluctuations of the macroscopic temperature due to the inhomogeneity of the body. However, this description does not change the mathematical structure of heat conduction equations: they remain parabolic both for macroscopic and microscopic temperatures. The possible extensions of the Fourier law mentioned in Introduction lead, as a rule, to a hyperbolic heat conduction equation. We examine, therefore, a more general approach with two dual internal variables (Ván et al., 2008) Heat conduction in microstructured solids with dual internal variables Now we extend the internal variable techique described in previous Section onto the case of two internal variables. Let us suppose that the free energy density depends on the internal variables ϕ, ψ and their gradients W = W(θ,ϕ,ϕ x,ψ,ψ x ). (37) The equations of state in the case of two internal variables read 160 S = W θ, υ := W, η := W ϕ, ξ := W ϕ x ψ The canonical energy conservation equation is not changed, ζ := W ψ x. (38) as well as the Clausius-Duhem inequality (Sθ) t +Q x = h int, h int := W t, (39) S t +(θ 1 Q+K) x 0. (40) 165 As previously, the internal heat source can be calculated in the considered case as follows h int = Sθ t +υϕ t +ηϕ xt +ξψ t +ζψ xt. (41) The non-zero extra entropy flux is set again to eliminate the divergence term in the Clausius-Duhem inequality K = θ 1 ηϕ t θ 1 ζξ t. (42) The latter means that the dissipation inequality reads θ(υ η x )ϕ t +θ(ξ ζ x )ψ t (Q ηϕ t ζξ t )θ x 0. (43) 7

8 The solution of the dissipation inequality is again determined by the thermodynamic flux-force relations ϕ t ψ t = L θ(υ η x) θ(ξ ζ x ), where L = L 11 L 13 L 21 L 22 L 23. (44) (Q ηϕ t ζξ t ) θ x L 31 L 32 L Nonnegativity of the entropy production (43) results in the positive semidefiniteness of the symmetric part of the conductivity matrix L, which requires L 11 0, L 22 0, L 33 0, L 11 L 22 ( +L 21 ) 2 0, 2 L 22 L 33 (L 32 +L 23 ) 2 0, det 1 (45) 2 2 (L+LT ) 0. Components of the matrix L are considered as constants. Evolution equations for internal variables have the form ϕ t = L 11 θ(υ η x )+ θ(ξ ζ x ) L 13 θ x, (46) 175 ψ t = L 21 θ(υ η x )+L 22 θ(ξ ζ x ) L 23 θ x, (47) and the generalized heat flux has the similar structure (Q ηϕ t ζξ t ) = L 31 θ(υ η x )+L 32 θ(ξ ζ x ) L 33 θ x. (48) The energy conservation equation keeps its canonical form (Sθ) t +(Q ηϕ t ζψ t ) x = Sθ t +(υ η x )ϕ t +(ξ ζ x )ψ t, (49) Eliminating heat flux by means of Eq. (48), we arrive at the most general heat conduction equation for microstructured solids in the framework of the dual internal variables approach θs t L 33 θ xx = (L 31 θ(υ η x )+L 32 θ(ξ ζ x )) x +(υ η x )ϕ t +(ξ ζ x )ψ t. (50) Quadratic free energy As previously, we specify the free energy density to a quadratic one W = ρc p 2θ 0 (θ θ 0 ) Bϕ Cϕ2 x Dψ Fψ2 x. (51) Calculating the quantities defined in Eq. (38) S = W θ = ρc p (θ θ 0 ), υ := W θ 0 ϕ = Bϕ, η := W ϕ x = Cϕ x, (52) ξ := W ψ = Dψ, ζ := W ψ x = Fψ x, (53) 8

9 we can represent system of Eqs. (46)- (48) in the form ϕ t = L 11 θ( Bϕ+Cϕ xx )+ θ( Dψ +Fψ xx ) L 13 θ x, (54) ψ t = L 21 θ( Bϕ+Cϕ xx )+L 22 θ( Dψ +Fψ xx ) L 23 θ x. (55) (Q ηϕ t ζξ t ) = L 31 θ( Bϕ+Cϕ xx )+L 32 θ( Dψ +Fψ xx ) L 33 θ x. (56) Accordingly, the heat conduction equation reads θs t L 33 θ xx = (L 31 θ( Bϕ+Cϕ xx )+L 32 θ( Dψ +Fψ xx )) x + +( Bϕ+Cϕ xx )ϕ t +( Dψ +Fψ xx )ψ t. (57) Up to now, the formal structure of evolution equations for internal variables and the expression for the generalized heat flux looks very similar to the case of the single internal variable. However, the introduction of the dual internal variable leads to non-trivial results as we will demonstrate below Hyperbolicity of evolution equations for internal variables To demonstrate the qualitative difference between the evolution of internal variables in this case and in the case of the single internal variable, we will derive a single evolution equation for the internal variable ϕ. For this purpose we differentiate evolution equation (54) with respect to time ϕ tt = L 11 θ t ( Bϕ+Cϕ xx )+L 11 θ( Bϕ+Cϕ xx ) t + + θ t ( Dψ +Fψ xx )+ θ( Dψ +Fψ xx ) t L 13 θ xt. (58) What we need is to eliminate the terms with the internal variable ψ. First, we use Eq. (54) expressing ( Dψ +Fψ xx ) in terms of ϕ θ( Dψ +Fψ xx ) = ϕ t L 11 θ( Bϕ+Cϕ xx )+L 13 θ x. (59) As the result, we have 200 ϕ tt = L 11 θ( Bϕ t +Cϕ xxt )+ θ t θ ϕ t + L 13θ t θ x + θ + θ( Dψ t +Fψ xxt ) L 13 θ xt. (60) Substituting relation (59) into evolution equation (55) we obtain its expression in terms of the internal variable ϕ ψ t = L 21 θ( Bϕ+Cϕ xx )+ L 22 ϕ t + L 23 θ x, (61) where L 21 = L 21 L 11 L 22 and L 23 = L 13 L 22 L 23 are introduced for convenience. Differentiation of the latter relation with respect to space coordinate represents ψ xxt ψ xxt = L 21 θ xx ( Bϕ+Cϕ xx )+2 L 21 θ x ( Bϕ+Cϕ xx ) x + + L 21 θ( Bϕ+Cϕ xx ) xx + L 22 ϕ xxt + L 23 θ xxx. (62) 9

10 Collecting all the obtained relations, we have finally for the internal variable ϕ ϕ tt (L 11 L 22 L 21 )(θ 2 (BF +CD) CFθθ x )ϕ xx = = (L 11 L 22 L 21 )(BFθθ xx BDθ 2 )ϕ+ +( θ t θ L 11θB L 22 θd)ϕ t +2(L 11 L 22 L 21 )FBθθ x ϕ xx + +θ(l 11 C +L 22 F)ϕ xxt 2(L 11 L 22 L 21 )FCθθ x ϕ xxx (63) (L 11 L 22 L 21 )FCθ 2 ϕ xxxx + L 13θ t θ x + θ D(L 13 L 22 L 23 )θθ x +F(L 13 L 22 L 23 )θθ xxx L 13 θ xt To simplify the consideration, we suppose again that the free energy depends only on the gradient of the internal variable ϕ x but not on the internal variable itself. Additionally, we assume that the gradient of the second internal variable is negligible. This results in the choice of the values of material parameters B = 0 and F = 0. Then the evolution equation for the internal variable ϕ is reduced to ϕ tt (L 11 L 22 L 21 )θ 2 CDϕ xx = ( θ t θ L 22θD)ϕ t + +θl 11 Cϕ xxt + L 13θ t θ x D(L 13 L 22 L 23 )θθ x L 13 θ xt. θ (64) Since the free energy density W is non-negative by default, material parameters C and D are also non-negative. This means that Eq. (64) is a hyperbolic wave equation with dissipation. The corresponding evolution equation for the second internal variable can be derived similarly Parabolicity of heat conduction equation Returning to the heat conduction equation for the temperature at the macroscale, θs t L 33 θ xx = (L 31 θ( Bϕ+Cϕ xx )+L 32 θ( Dψ +Fψ xx )) x + +( Bϕ+Cϕ xx )ϕ t +( Dψ +Fψ xx )ψ t, (65) we will also eliminate one internal variable. Applying the previous results, we have for B = 0 and F = θs t L 33 L 32 L 13 θ xx = (L 31 θcϕ xx ) x L 32 (ϕ t L 11 θcϕ xx ) x + +Cϕ xx ϕ t + 1 (ϕ t L 11 θcϕ xx +L 13 θ x ) (66) θl ( 12 L21 L 11 L 22 θcϕ xx + L 22 ϕ t + L ) 13L 22 L 23 θ x. It follows that the heat conduction equation for the macroscopic temperature remains parabolic. Its complicated right hand side depends on internal variables and the temperature gradient. 10

11 4. Summary and discussion The Fourier law for heat conduction in solids is sufficient for many practical applications (Nellis and Klein, 2009; Jiji, 2009). Attempts for its generalization remain restricted by the case of homogeneous bodies (Straughan, 2011; Ván and Fülöp, 2012; Carlomagno et al., 2016). The influence of microstructure persists negligible in this case. At the same time, generalized continuum theories take the effects of microstructure into account (Mindlin, 1964; Eringen and Suhubi, 1964). However, these theories are, as a rule, nondissipative and do not include heat conduction (Eringen, 1999). It was shown recently that thermal effects can be incorporated in the framework of generalized continua theories by means of the dual internal variables approach (Berezovski et al., 2011b; Berezovski and Berezovski, 2013; Berezovski and Engelbrecht, 2013; Berezovski et al., 2014). Nevertheless, this approach was never applied to the pure heat conduction. As it is demonstrated in the paper, the dual internal variable approach is able to predict a hyperbolic character of heat conduction at the microscale. One of the internal variables is identified with microtemperature, i.e. the fluctuation of macroscopic temperature due to the inhomogeneity of the body. The macroscopic heat conduction equation remains parabolic, but coupled with the hyperbolic evolution equation for the microtemperature. The effect of microstructure may be small or even neglected for sufficiently high temperatures and slow or lengthy processes. For a fast heating or low temperatures this influence may not be disregarded. Acknowledgments The work was supported by the EU through the European Regional Development Fund and by the Estonian Research Council grant PUT434. References 250 Berezovski, A., Berezovski, M., Influence of microstructure on thermoelastic wave propagation. Acta Mechanica 224, Berezovski, A., Engelbrecht, J., Thermoelastic waves in microstructured solids: dual internal variables approach. Journal of Coupled Systems and Multiscale Dynamics 1, Berezovski, A., Engelbrecht, J., Maugin, G.A., 2011a. Generalized thermomechanics with dual internal variables. Archive of Applied Mechanics 81, Berezovski, A., Engelbrecht, J., Maugin, G.A., 2011b. Thermoelasticity with dual internal variables. Journal of Thermal Stresses 34,

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