Thermodynamic Approach to Generalized Continuaberez/texts/71.Van-Berezovski-Papenfuss... · 2020-01-01 · Continuum Mechanics and Thermodynamics manuscript No. (will be inserted

Continuum Mechanics and Thermodynamics manuscript No.(will be inserted by the editor)

Thermodynamic Approach to Generalized Continua

Peter Van · Arkadi Berezovski · ChristinaPapenfuss

Received: date / Accepted: date

Abstract Governing equations of dissipative generalized solid mechanics are de-rived by thermodynamic methods in the Piola-Kirchhoff framework using the Liuprocedure. The isotropic small strain case is investigated in more detail. The connec-tion to the Ginzburg-Landau type evolution, dual internal variables, and a thermody-namic generalization of the standard linear solid model of rheology is demonstrated.Specific examples are chosen to emphasize experimental confirmations and predic-tions beyond less general approaches.

Keywords Generalized solid mechanics · Liu procedure · Dual internal variables

1 Introduction

Conventional theories of continua do not provide the description of a microstructuralinfluence because material elements are considered as indistinct pieces of matter.Generalized continuum theories (higher-order or higher-grade) are first examples of

P. VanDept. of Theoretical Physics, KFKI Research Institute for Particle and Nuclear Physics, H-1525 Budapest,P.O.Box 49, HungaryTel.: +36-1 392 2222/3803Fax: +36-1 ..........E-mail: [email protected] Dept. of Energy Engineering, Budapest University of Technology and Economics, H-1111, Budapest,Bertalan Lajos u. 4-6, Hungaryand Montavid Thermodynamic Research Group

A. BerezovskiCentre for Nonlinear Studies, Institute of Cybernetics at Tallinn University of Technology, Akadeemia tee21, 12618 Tallinn, Estonia

C. PapenfussTechnical University of Berlin, Strasse des 17. Juni 135, 10623 Berlin, Germany

2 Peter Van et al.

what has been proposed to describe the macroscopic behavior of materials with in-ner material structure. Though their constitutive structure is restricted by the secondlaw of thermodynamics, the contribution of small-scale events to entropy fluxes andsources is still not completely investigated.

Governing equations of generalized continuum mechanics can be obtained bydifferent ways. Three most widely accepted approaches are the following: the varia-tional approach by Mindlin [1], the microhomogenization procedure by Eringen anSuhubi [2,3], and the virtual power method by Germain [4].

Mindlin [1] derived governing equations in the small strain approximation withthe help of a variational principle analogous to that in ideal elasticity. In his the-ory of microdeformations, Mindlin introduced kinetic and potential energy for bothmicro- an macro-displacements as well as a tensor characterizing micro-inertia. Inthe Mindlin theory, the potential energy density is a quadratic function of the macro-strain, the relative strain, and the micro-deformation gradient. With this variationalfoundation, the Mindlin theory is an idealized one, which does not include any dissi-pation.

In their approach to generalized continua, Eringen and Suhubi [2,3] did not startfrom a variational principle. They obtained an evolution equation of the micro-strainextending mechanical concepts of inertia, stress, strain, and energy onto the mi-crolevel and calculating velocity moments of the microevolution of the momentum.The zeroth moment of the mesoscopic momentum balance results in the macroscopicCauchy equation and the first moment gives the evolution equation of the micro-momentum. In their constitutive theory, Eringen and Suhubi consider the microde-formation and the gradient of the microdeformation as general internal variables andcalculate the entropy production accordingly. They obtained the following expressionfor the entropy production T ∆ ([2] (5.13)):

T ∆ =qi∂i logT +(τi j − τi j)∂iv j +(skl − τkl − σkl)νkl+

+(µklm − µklm)∂mνkl −ρζaξa ≥ 0.(1)

Here qi is the heat flux, i.e. flux of the internal energy density, T is the temperature,τi j is the stress, ρ is the matter density, si j is the relative stress, νi j is the microveloc-ity gradient, µi jk is the double stress, the derivative of the free energy with respect tothe microdeformation gradient. Tilde denotes the reversible, nondissipative parts ofthe corresponding quantities. Further, σi j = ti j − si j is the reversible microstress, thederivative of the free energy with respect to the microdeformation. Equation (1) de-fines si j, the reversible relative stress, too. The last term gives the entropy productiondue to additional internal variables ξa, a = 1, ..n, and ζa denotes the related intensivevariable, the derivative of the entropy density with respect to ξa. Here the notation ofderivatives and indices follow Mindlin as far as it was possible. Remarkable is thatthe Eringen-Suhubi theory defines the entropy flux in its classical form Ji = qi/T .

As we can see, the descriptions of generalized continua are weakly nonlocal fromthe beginning because the gradient of the microdeformation is introduced as a statevariable. This is true also for the third approach that uses the principle of virtual powerin order to derive the evolution equation of the microdeformation. The virtual powermethod is essentially dealing with statics. Dynamics in this method is introduced by

Thermodynamic Approach to Generalized Continua 3

an assumption that inertia is connected to virtual displacements directly [4,5] or withthe help of dissipation potentials. In the first case, the dissipation requires separateconsiderations. In the second case, the dissipation is introduced together with inertialterms.

Let us underline some common properties of above mentioned approaches:

1. The connection of new microstructural variables to mechanical effects has a kine-matic background. However, the microdeformation can be originated in differentstructural changes (e.g. microcracking), which is not necessarily connected toenergy alterations due to the change of a Riemann geometry of the material man-ifold.

2. The evolution equations of microstructural variables are originated in mechan-ics. Variational principles, moment series developement as well as virtual powerwith dissipation potentials are mechanical concepts. The dissipative effects areconsidered as secondary and frequently disregarded.

3. The entropy flux has the same form as that in the simplest case of Cauchy con-tinua.

4. In all these approaches, it is customary to introduce internal variables indepen-dently of the microdeformation, representing already identified structural changesof continua, e.g. damage or cracking [6,7].

Intuitively, it is natural to expect that the microstructure affects also dissipative phe-nomena. It is needed, therefore, to investigate in detail how dissipation effects aredescribed by generalized continuum theories.

The response of materials to external loads can be expressed explicitly as a func-tional [8]. On the other hand, additional internal variables can be introduced to definethis functional in an implicit manner by means of their evolution equations.

It is well known, that variational principles may exist for a dissipative evolution,but they are not of the usual Hamiltonian kind. At best, they need to be modified be-cause they do not work without any further ado [9–11]. Regarding the homogeniza-tion technics of Eringen and Suhubi for internal variables [2], it should be noted thatthere is no primary microscopic candidate for an evolution equation, therefore theirmethod cannot be applied beyond the kinematic determinations. Direct statistical orkinetic theory related calculations (as for example [12]) would require a particularmicrostructure and an interpretation of the internal variable. Finally, regarding thethird treatment, the principle of virtual power is a mechanical concept, and internalvariables (if any) are not related to any kind of spacial changes of continua.

It is known, that there are two alternate approaches to obtain the evolution equa-tions of internal variables directly. The thermodynamic approach is related to internalvariables of state and the variational one is regarded to dynamic degrees of free-dom, respectively [13,14]. In the thermodynamic approach, the entropy productionis calculated considering additional constraints (e.g. balances), and the evolution ofinternal variables is determined as a part of the constitutive theory by means of thedissipation inequality. Evolution equations obtained in such a way contain typicallyonly first order time derivatives [15]. The most important issues here are the stabilityof the weakly nonlocal extension (see critical remarks in [16]), the thermodynamicconsistency, and seemingly missing boundary conditions. In the case of dynamic de-

4 Peter Van et al.

grees of freedom, a Hamiltonian variational principle is applied to the nondissipativepart of the evolution, and the dissipative contribution is calculated by dissipation po-tentials.

Recently, it was shown that one can get a unified description of the two meth-ods introducing weakly nonlocal dual internal variables [17]. Then the exact andconstructive exploitation of the entropy inequality, e.g. the Liu procedure, combinedwith an Onsagerian linear approximation of constitutive functions leads to completelysolvable constraints. An essential ingredient of the approach of dual internal vari-ables is the observation that in the case of higher-order gradient theories, gradientsof constraints of the entropy inequality are constraints on the constitutive state space[18–20].

It has been observed recently [21] that the structure of the dual internal variablesystem of evolution equations in the nondissipative case corresponds exactly to theevolution equation in the Mindlin theory. In this case, one of the internal variablescan be interpreted as the microdeformation and another as the conjugated momen-tum. It is worth to extend the dual internal variable approach to a broader class ofmaterials taking into account dissipative effects. This suggests a more general pro-cedure to construct evolution equations than in [17]. The corresponding procedureis presented in the paper. It consists in the extension of the state space, the formula-tion of constraints, the application of the Liu procedure to the entropy inequality, thesolution of obtained Liu equations, and the specification of a general form of evo-lution equations for internal variables following from linear relationships betweenthermodynamic fluxes and forces. The relation to linear viscoelasticity, to the patternformation equations of the Ginzburg-Landau-type, and to the standard linear solidmodel is demonstrated in the small-strain approximation. The same approximation isused to point out the extension of generalized continua descriptions onto dissipativematerials and microstructural thermal effects.

2 Construction of evolution equations

2.1 Balance laws

We start with the formulation of thermodynamic constraints for continuum mechanicswith dual internal variables in the Piola-Kirchhoff framework (PK frame). This meansthat our treatment is only partially objective.

The balances of momentum and energy can be represented as follows:

ρ0vi −∂ jti j = 0, (2)

ρ0e+∂iqi = ti j∂ jvi, (3)

where ρ0 is the matter density, vi is the velocity field, ti j is the first Piola-Kirchhoffstress, e is the specific internal energy and qi is the flux of internal energy densityin the PK frame. The dot denotes the material time derivative, which is a partialtime derivative on the material manifold. ∂ j denotes the (material) space derivative.The Einstein summation rule is applied for repeated indices. Our index notation is ab-stract, does not refer to any particular system of coordinates, and denotes the tensorial


degree and contractions in accordance with the traditional coordinate free treatmentin continuum mechanics [22]. In the notation we use uniformly lowercase indices,i.e. do not distinguish between material and spatial indices and vectors and covec-tors. This way it is easier to follow calculations, and examples are considered in thesmall strain approximation where the differences are negligible.

The balances are introduced without source terms, the momentum and the totalenergy are conserved, because source terms are irrelevant in a constitutive theory.The relation between the deformation gradient Fi j and the velocity field is consideredas a constraint:

Fi j −∂ jvi = 0. (4)

Evolution equations of the internal variables ψi j and βi j are formally represented as

ψi j = fi j, βi j = gi j, (5)

where the constitutive functions fi j and gi j depend on the whole set of state variables.Here the notation of Mindlin was applied for the first internal variable, ψi j, which isthe microdeformation there [1].

The entropy inequality is given as follows:

ρ0s+∂iJi ≥ 0, (6)

where s is the specific entropy and Ji is the flux of the entropy density.It is assumed that constitutive functions

ti j,qi, fi j,gi j,s,Ji, (7)

are defined on the weakly nonlocal state space spanned by the following variables:

∂ jvi,Fi j,∂kFi j,e,∂ie,ψi j,∂kψi j,∂klψi j,βi j,∂kβi j,∂klβi j. (8)

For the sake of simplicity, we consider a weakly nonlocal constitutive state space ofthe first order in the deformation gradient and in the internal energy, but of the secondorder in the internal variables, i.e., their second gradients are included. The velocityfield is distinguished, because only its derivative is present in the constitutive statespace. This assumption allows us to avoid velocity related problematic aspects of thematerial frame indifference (see e.g. [23]).

2.2 Liu procedure

Balance of linear momentum (2), balance of internal energy (3), kinematic relation(4), and evolution equations of internal variables (5) are constraints of the entropyinequality. Taking into account that the constitutive state space is weakly nonlocal ofthe second order in the internal variables, we should introduce additional constraintsfor derivatives of their evolution equations. We consider a derivative of a constraintas a new constraint in case of higher order weakly nonlocal constitutive state space.

6 Peter Van et al.

It is an important aspect for the development of correct thermodynamic conditions inweakly nonlocal thermodynamic theories [24,25]:

∂kψi j = ∂k fi j, ∂kβi j = ∂kgi j. (9)

Then we introduce Lagrange-Farkas multipliers λi,κ,Λi,Ai j,Ai jk,Bi j,Bi jk for con-straints (2), (3), (4), (5)1, (9)1, (5)2, (9)2, respectively. The constrained entropy im-balance is, therefore,

ρ0s+∂ jJ j −Λ ji(Fi j −∂ jvi

)−λi (ρ0vi −∂ jti j)−κ (ρ0e+∂iqi − ti j∂ jvi)+

+A ji (ψi j − fi j)+Ak ji (∂kψi j +∂k fi j)+

+B ji

(βi j −gi j

)+Bk ji

(∂kβi j −∂kgi j

)≥ 0.

(10)

Liu equations are obtained as coefficients of higher derivatives after a straightforwardcalculation:

vi : 0 = λi (11)∂ j vi : ∂∂ jvi s = 0 (12)

Fi j : ρ0 ∂Fi j s = Λi j (13)

∂kFi j : ∂∂kFi j s = 0 (14)

e : ∂es = κ (15)∂ie : ∂∂ies = 0 (16)ψi j : ρ0 ∂ψi j s =−A ji (17)

∂kψi j : ρ0 ∂∂kψi j s =−Ak ji (18)

∂klψi j : ρ0 ∂∂klψi j s = 0 (19)

βi j : ρ0 ∂βi j s =−B ji (20)

∂kβi j : ρ0 ∂∂kβi j s =−Bk ji (21)

∂kl βi j : ρ0 ∂∂klβi j s = 0 (22)

∂k jvi : ∂∂kviJ j −∂es ∂∂kviq j −ρ0∂∂kψlms ∂∂ jvi flm −

−ρ0∂∂kβlms ∂∂ jvi glm = 0 (23)

∂klFi j : ∂∂kFi j Jl −∂es ∂∂kFi j ql −ρ0 ∂∂kψnms∂∂lFi j fnm −−ρ0 ∂∂kβnms ∂∂lFi j gnm = 0 (24)

∂i je : ∂∂ieJ j −∂es ∂∂ie q j −ρ0 ∂∂ jψlms ∂∂ie flm −

−ρ0 ∂∂ jβlms ∂∂ie glm = 0 (25)

∂klmψi j : ∂∂lmψi j Jk −∂es ∂∂lmψi j qk −ρ0 ∂∂kψop s ∂∂lmψi j fop −−ρ0 ∂∂kβop s ∂∂lmψi j gop = 0 (26)

∂klmβi j : ∂∂lmβi j Jk −∂es ∂∂lmβi j qk −ρ0 ∂∂kβop s ∂∂lmβi j fop −−ρ0 ∂∂kβop s ∂∂lmβi j gop = 0 (27)


Liu equations (11), (13), (15), (17), (18), (20), and (21) determine Lagrange-Farkasmultipliers by corresponding entropy derivatives. The solution of Liu equations (12),(14), (16), (19), and (22) reduces the constitutive form of the entropy to the followingone: s = s(Fi j,e,ψi j,∂kψi j,βi j,∂kβi j). The dissipation inequality then follows consid-ering Liu equations (23)-(27) for the entropy flux. Together with the mentioned formof the entropy function, these equations can be solved and we obtain the entropy fluxin the form:

Ji = ∂esqi +ρ0∂∂iψlms flm +ρ0∂∂iβlm

s glm + J0i . (28)

Here the dependence of J0i is reduced similarly to that of the specific entropy J0

i =J0

i (Fi j,e,ψi j,∂kψi j,βi j,∂kβi j). It should be noted that Eq. (28) is not the most generalsolution of corresponding Liu equations, because we did not consider symmetriesof functions in Eqs. (19), (22) and (23)-(27). For example, Eq. (23) is obtained asmultiplier of ∂k jvi, therefore only the symmetric part of Eq. (23) must be zero, butwe consider, however, solutions that are more restrictive.

The dissipation inequality then follows as

∂kFi j(∂Fi j Jk −∂es ∂Fi j qk −ρ0 ∂∂kψlm

s ∂Fi j flm −ρ0 ∂∂kβlms ∂Fi j glm

)+

∂ie(∂eJi −∂es ∂e qi −ρ0∂∂iψlm

s ∂e flm −ρ0 ∂∂iβlms ∂eglm

)+

∂k ψi j(∂ψi j Jk −∂es ∂ψi j qk −ρ0∂∂kψlm

s ∂ψi j flm −ρ0 ∂∂kβlms ∂ψi j glm

)+

∂lkψi j

(∂∂lψi j Jk −∂es ∂∂lψi j qk −ρ0 ∂∂kψoms ∂∂lψi j fom −ρ0 ∂∂kβoms ∂∂lψi j gom

)+

∂kβi j

(∂βi j Jk −∂es ∂βi j qk −ρ0 ∂∂kψlm

s ∂βi j flm −ρ0 ∂∂kβlms ∂βi j glm

)+

∂lk βi j

(∂∂lβi j Jk −∂es ∂∂lβi j qk −ρ0 ∂∂kψoms ∂∂lβi j fom −ρ0 ∂∂kβoms ∂∂lβi j gom

)+

ρ0 ∂Fi j s Fi j +∂es ti j∂iv j −ρ0 ∂ψi j s fi j −ρ0 ∂βi j s gi j ≥ 0.(29)

By substituting Eq. (28) into the dissipation inequality we arrive at the followingexpression:

∂k (∂es)qk +∂es ti j∂iv j +∂ jviρ0∂Fi j s−

−(

∂ψi j s−∂k∂∂kψi j s)

ρ0 fi j −(

∂βi j s−∂k∂∂kβi j s)

ρ0gi j +∂iJ0i ≥ 0.

(30)

Here ∂es = 1/T , and we may identify thermodynamic fluxes and forces as follows:

∂i

(1T

)qi +

1T

(ti j +ρ0T ∂Fi j s

)︸︷︷︸tvi j

∂iv j −(


︸︷︷︸−Xi j

ρ0 fi j−

−(


︸︷︷︸−Yi j

ρ0gi j +∂iJ0i ≥ 0,

(31)

where tvi j is the viscous stress. The entropy is a distinguished constitutive function,

that fixes the static information of the system. All other constitutive functions aredetermined by means of entropy derivatives.

8 Peter Van et al.

Table 1 Thermodynamic fluxes and forces.

Thermal Mechanical Internal 1 Internal 2

Fluxes qi

(ti j +T ρ0∂Fi j s

)/T fi j gi j

Forces ∂i( 1

T

)∂iv j ρ0

(∂ψi j s−∂k∂∂kψi j s

)ρ0

(∂βi j s−∂k∂∂kβi j s

)

2.3 Evolution equations

The entropy production in the dissipation inequality is represented as a sum of prod-ucts, and there is an undetermined constitutive function in each term multiplied by agiven function of the constitutive state space. Therefore, it is straightforward to pointout the simplest solution of the dissipation inequality assuming linear relationshipsbetween thermodynamic fluxes (terms with undetermined constitutive functions) andtheir multipliers, thermodynamic forces (see Table 1).

The classical thermal interaction is vectorial, while other terms are tensorial. Themechanical term is responsible for viscoelasticity if no other terms are present. Thelast two terms constitutively determine evolution equations of second-order tenso-rial internal variables ψi j and βi j. In isotropic materials, tensorial mechanical andinternal variables may be coupled independently of the vectorial thermal constitutivefunction:

qi = λ∂i1T, (32)

1T

tvi j =

1T

(ti j +ρ0T ∂Fi j s

)= L11

i jkl∂kvl +L12i jklXkl +L13

i jklYkl , (33)

ψi j = fi j = L21i jkl∂kvl +L22

i jklXkl +L23i jklYkl , (34)

βi j = gi j = L31i jkl∂kvl +L32

i jklXkl +L33i jklYkl . (35)

So we have the three times three matrix of fourth order tensors LIJi jkl called conduc-

tivity tensors. In isotropic materials they are characterized by three scalar materialparameters each and can be represented for all I,J = 1,2,3 as follows:

LIJi jkl = lIJ

1 δikδ jl + lIJ2 δilδ jk + lIJ

3 δi jδkl . (36)

where δi j is the Kronecker delta. Therefore, there are 1+3x9 = 28 material conductiv-ity coefficients in isotropic media with dual internal variables. Representation (36)can be decomposed into traceless symmetric, antisymmetric (deviatoric) and spheri-cal parts, i.e., is equivalent to

LIJi jkl = sIJδi⟨kδ jl⟩+aIJδi[kδ jl]+ lIJδi jδkl , (37)

where braces ⟨⟩ denote the traceless symmetric part of the corresponding tensor inrelated indices δi⟨kδ jl⟩ = (δikδ jl +δ jkδil)/2−δi jδkl/3 and the rectangular parenthesis[ ] denotes the antisymmetric part as δi[kδ jl] = (δikδ jl − δ jkδil)/2. Therefore, sIJ =

lIJ1 + lIJ

2 , aIJ = lIJ1 − lIJ

2 , and lIJ = (3lIJ3 + lIJ

1 + lIJ2 )/3. This kind of decomposition is

instructive because symmetric, antisymmetric, and spherical second-order tensors are


mutually orthogonal in the “double dot” product, i.e. taking the trace of their product.Therefore, constitutive equations (33)-(35) can be decomposed into three parts: fivecomponent traceless symmetric, three component antisymmetric, and one componentspherical parts are independent. The spherical part is determined as

tvkk = l11∂kvk + l12Xkk + l13Ykk, (38)

ψkk = l21∂kvk + l22Xkk + l23Ykk, (39)βkk = l31∂kvk + l32Xkk + l33Ykk. (40)

In its turn, for the symmetric traceless, deviatoric part we have

tv⟨i j⟩ = s11∂⟨ivl⟩+ s12X⟨i j⟩+ s13Y⟨i j⟩, (41)

ψ⟨i j⟩ = s21∂⟨ivl⟩+ s22X⟨i j⟩+ s23Y⟨i j⟩, (42)

β⟨i j⟩ = s31∂⟨ivl⟩+ s32X⟨i j⟩+ s33Y⟨i j⟩, (43)

and the antisymmetric part is given in a tensorial form instead of the vectorial invari-ants:

tv[i j] = a11∂[ivl]+a12X[i j]+a13Y[i j], (44)

ψ[i j] = a21∂[ivl]+a22X[i j]+a23Y[i j], (45)

β[i j] = a31∂[ivl]+a32X[i j]+a33Y[i j]. (46)

Therefore, conductivity matrices of spherical, traceless symmetric, and antisymmet-ric components are

lIJ =

l11 l12 l13

l21 l22 l23

l31 l32 l33

, sIJ =

s11 s12 s13

s21 s22 s23

s31 s32 s33

, aIJ =

a11 a12 a13

a21 a22 a23

a31 a32 a33

. (47)

The second law requires that symmetric parts of 3x3 conductivity matrices lIJ ,sIJ , and aIJ are positive definite.

With this requirement the thermodynamically consistent construction of contin-uum mechanical theory with second order tensorial dual internal variables is com-plete. Balances of momentum and energy (2), (3) and evolution equations of the inter-nal variables (5) together with isotropic constitutive functions (32)-(35) form a com-plete system, if a particular form of the entropy function s(Fi j,e,ψi j,∂kψi j,βi j,∂kβi j)is given.

2.4 Boundary conditions

It is remarkable that one can get natural boundary conditions for internal variableswith the requirement that the intrinsic part of entropy flux (28) is zero at the boundary.There are three basic possibilities:

1. Combined condition. In this case ∂∂iψlms and ∂∂iβlm

s are orthogonal to flm and glm.

10 Peter Van et al.

2. No change condition. Internal variables do not change at the boundary, ψi j andβi j are zero.

3. Gradient condition. ti∂∂iψlms = 0lm and ti∂∂iβlm

s = 0lm for any vector ti that isparallel to the boundary. In the case of a quadratic dependence on gradients, thecondition is that gradients of internal variables are orthogonal to the boundary.

2.5 Remarks on reciprocity relations

Classical irreversible thermodynamics requires special conductivity matrices obtainedby reciprocity relations. In the case of symmetric matrices, the reciprocity relationsare of Onsager type [26,27], if the matrices are antisymmetric, then they are ofCasimir type [28]. These restrictions are justified by arguments from statistical physicsrequiring a microscopic interpretation of thermodynamic variables. If these variablesare even or odd functions of the microscopic velocities, then the conductivity matrixis symmetric or antisymmetric, respectively. However, in our case one cannot spec-ify the conductivity matrices, the conditions of Onsager or Casimir type reciprocityrelations are not fulfilled. In the following we will show some possible interpreta-tions of the dual internal variables, and we will see that the most straightforwardinterpretation, namely, the micromorphic generalized mechanics when the internalvariables are of deformation and deformation rate type, requires general forms of theconductivity matrices. Moreover, the background of internal variables in continuummechanics is originated in structural changes in a material, but not on the micro-scopic, atomic, or mesoscopic level [17]. Therefore we simply do not see any reasonsto further specialize the theory and we keep our treatment universal, independent ofmicroscopic or mesoscopic interpretations.

Idealized theories are characterized by a nondissipative behavior. In the case ofheat conduction, the heat conduction coefficient λ is zero and, therefore, the heat flux,the flux of the internal energy density, is zero, too. In the case of pure mechanicalinteraction without internal variables, the idealized theory is characterized by zeroviscosities, the bulk viscosity l11 = ηv = 0 and the shear viscosity s11 = 2η = 0are zero, together with the condition a11 = 0. Therefore, the viscous stress is zero,the stress in the momentum balance is the static stress t i j = −ρT ∂Fi j s. However,antisymmetric terms of conductivity matrices do not produce entropy, therefore inour case the coupling between different terms may result in nondissipative transport,too.

3 Spatial representation

In this section, we represent governing equations in spatial framework and at thesame time introduce the small strain approximation. Technical details are given inthe Appendix.

We can transform balances of internal energy and momentum (2), (3) into spatialform due to the Nanson theorem to obtain

ρ e+∂iqi = ti j∂ jvi, (48)


ρ vi −∂ jti j = 0. (49)

Here ρ ≈ ρ0 is the density in the actual configuration, ti j is the Cauchy stress tensor,and qi is the spatial flux of the internal energy, the upper dot denotes the substantialtime derivative, e = ∂te + vi∂ ie. Spatial and material forms of these balances areidentical in the small strain approximation.

The spatial form of evolution equations of internal variables is the following:

⋄ψ i j= fi j,

⋄β i j= gi j. (50)

The symbol ⋄ denotes the spatial form of the material time derivative of a second-order tensor. The material time derivative is an upper convected one, because internalvariables are defined on the material manifold, supposedly independent of the motionof the continuum (see Appendix):

⋄ψ i j= ψi j −∂kviψk j −∂kv jψ jk, (51)

i.e., spatial and material forms of constitutive functions for internal variables are thesame in the small strain approximation if velocity gradients are negligible. The spatialform of the entropy flux is represented as

Ji = ∂esqi +ρ∂∂iψlms flm +ρ∂∂iβlm

s glm + J0i . (52)

We do not deal here with any interpretation of the extra entropy flux J0i and, therefore,

we assume that it is equal to zero. This requirement is not necessary, and there arephenomena which can be modelled only with the help of nonzero extra entropy flux[30–32].

Finally, the spatial form of the entropy production in the small strain approxima-tion is the following:

σ = ∂i

(1T

)qi +

1T

(ti j +ρT ∂εi j s

)εi j−

−(


ρ fi j −(


ρgi j ≥ 0.(53)

Here εi j is the Cauchy strain.The original dependency of the entropy function on the deformation gradient Fi j

turns to a dependency on εi j because of the small strain approximation. Moreover, itis convenient to introduce the specific free energy function instead of the specific en-tropy as a thermodynamic potential. In our case, the expression of the correspondingpartial Legendre transformation is:

w(T,εi j,ψi j,∂kψi j,βi j,∂kβi j) = e−T s(e,εi j,ψi j,∂kψi j,βi j,∂kβi j). (54)

Therefore,∂ s∂e

=1T

and∂w∂T

=−s, (55)

12 Peter Van et al.

Table 2 Thermodynamic fluxes and forces in the small strain approximation.

Thermal Mechanical Internal 1 Internal 2

Fluxes qk = qk+ tvi j = ρ fi j ρ gi j

ρ fi j∂∂kψi j w+ρgi j∂∂kβi j w ti j −ρ∂εi j wForces ∂k lnT εi j Xi j = Yi j =

∂ψi j w−∂k∂∂kψi j w ∂βi j w−∂k∂∂kβi j w

keeping fixed all other variables. Other partial derivatives of w and s are related asfollows:

∂ s∂εi j

∣∣∣∣e= − 1

T∂w∂εi j

∣∣∣∣T,

∂ s∂ψi j

∣∣∣∣e= − 1

T∂w

∂ψi j

∣∣∣∣T,

∂ s∂βi j

∣∣∣∣e= − 1

T∂w∂βi j

∣∣∣∣T,

∂ s∂∂kψi j

∣∣∣∣e= − 1

T∂w

∂∂kψi j

∣∣∣∣T,

∂ s∂∂kβi j

∣∣∣∣e= − 1

T∂w

∂∂kβi j

∣∣∣∣T.

(56)

All not indicated variables are kept fixed in the partial derivatives. In terms of the freeenergy function, we can transform entropy production (53) to the form

T σ = ∂i lnT(

qi +ρ fi j∂∂kψi j w+ρgi j∂∂kβi j w)+(ti j −ρ∂εi j w

)εi j+

+(

∂ψi j w−∂k(∂∂kψi j w))

ρ fi j +(

∂βi j w−∂k(∂∂kβi j w))

ρgi j ≥ 0.(57)

It is worth to introduce thermodynamic forces and fluxes in the small strain approx-imation according to Eq. (57) (Table.2). It is easy to see that thermodynamic forcesand fluxes in the small strain approximation are very similar to those in the PK frame.The most important difference is the regrouping of terms which are proportional tothe temperature gradient. This representation of forces and fluxes is more conve-nient for the separation (or coupling) of thermodynamic and mechanical parts of theentropy production, especially in the case of thermal stresses. The solution of dissi-pation inequality (57) is provided by conductivity equations:

qk = qk +ρ fi j∂∂kψi j w+ρgi j∂∂kβi j w = λ∂k lnT, (58)

tvi j = ti j −ρ∂εi j w = L11

i jkl εkl + L12i jklXkl + L13

i jklYkl , (59)

ρ⋄ψ i j = ρ fi j = L21


i jklYkl , (60)

ρ⋄β i j = ρ gi j = L31


i jklYkl . (61)

In the case of isotropic materials, a decomposition of conductivity matrices can beperformed introducing symmetric, antisymmetric and spherical parts of correspond-ing tensorial forces and fluxes similarly to previous case:

LIJi jkl = sIJδi⟨kδ jl⟩+aIJδi[kδ jl]+ lIJδi jδkl . (62)

In the following, we treat different special cases, all of them with constant conduc-tivity matrices.


4 Examples

4.1 Linear viscoelasticity, relaxation, and Ginzburg-Landau equation

We start with the simplest case when there is not any coupling between evolutionequations (58)–(61). In this case, the balance equation of the linear momentum andevolution equations for internal variables are independent.

This means that the conductivity hypermatrix is diagonal L12 = L21 = L13 =L31 = L32 = L23 = 0, and we assume that the heat conduction coefficient λ is zero.The free energy is additively decomposed into parts which are dependent on ψi j, βi j,and εi j separately. Dissipation inequality (57) reduces to

T σ =(ti j −ρ∂εi j w

)εi j+

+(


ρ fi j +(

∂βi j w−∂k(∂∂kβi j w))

ρgi j ≥ 0,(63)

and we see that terms related to internal variables are completely similar. This meansthat in the absence of couplings it is enough to analyze only one of them.

For the viscous part of the stress we have, therefore,

ti j − telai j = tv

i j = l11εkkδi j + s11ε(i j). (64)

where the elastic stress is introduced as usually

ρ∂w∂εi j

= telai j = ρλεkkδi j +ρ2µεi j, (65)

l11 =ηv corresponds to the bulk viscosity and s11 =η is the shear viscosity. All othercoefficients in conductivity matrices are zero due to the absence of any coupling. Theevolution equation for the internal variable ψi j results in:

ρ⋄ψ i j= L22

i jlm(∂ψlm w−∂k∂∂kψlm

w). (66)

Assuming the isotropy of the conductivity tensor L22, we can decompose Eq. (66) intosix independent evolution equations for the spherical, symmetric traceless and anti-symmetric parts of the internal variable tensor. These evolution equations for the in-ternal variable ψi j give the generalization of the Ginzburg-Landau-Khalatnikov equa-tion, introduced first as a scalar equation in the case of superconductors [34,35]. Foreach free energy functional, the structure of such equations is universal and widelyused with different thermodynamic arguments [36–39]. The Ginzburg-Landau equa-tion was derived by pure thermodynamic arguments as the evolution equation for asecond-order weakly nonlocal internal variable in [24] (see also [21]). That deriva-tion shows a universal character of the Ginzburg-Landau equation: the second lawrequires an evolution equation of this form for an internal variable without any otherconstraints independently of the microscopic background.

The symmetric traceless form part of Eq. (66) gives the de Gennes-Landau theoryof liquid crystals, if a suitable quadratic dependence of the free energy on ψ⟨i j⟩ isintroduced [40]. This very particular example of the proposed approach shows therichness of the mathematical structure and its physical interpretation.

14 Peter Van et al.

4.2 Generalized standard linear solid

In this section we consider a coupling between a single tensorial internal variable anddeformation, which results in the thermodynamic theory of rheology [41,42]. There-fore, we still assume zero heat conduction coefficient and L13 = L31 = L32 = L23 = 0.However, L12 = 0 and L21 = 0 anymore. Let us further reduce the treatment by in-troducing a local theory for the single internal variable with the simplest quadraticdependence of the free energy on the internal variable without its gradients. In thiscase, the free energy can be written as

w(εi j,ψi j) =λ2

ε2ii +µεi jεi j+

+b1

2ψ2

ii +b2

2ψi jψi j +

b3

2ψi jψ ji +g1ψiiε j j +g2(ψi j +ψ ji)εi j.

(67)

Thermodynamic stability requires the convexity of the free energy, hence the inequal-ities follow:

3µ +2λ ≥ 0, µ ≥ 0, b2 ≥ 0, b2 +b3 ≥ 0, 3b1 +2b2 ≥ 0,

µ(b2 +b3)−g21 ≥ 0, (3λ +2µ)(3b1 +b2 +b3)− (3g1 +2g2)

2 ≥ 0.(68)

The entropy production contains two tensorial terms that may be coupled:


)εi j −∂ψi j wρ f i j ≥ 0. (69)

In isotropic materials, spherical, deviatoric and antisymmetric parts of the tensorsare independent. For the deviatoric, symmetric traceless part we obtain the followingconductivity equations from Eqs. (67) and (69):

t⟨i j⟩−2ρ(µε⟨i j⟩+g2ψ⟨i j⟩) = s1ε⟨i j⟩− s12 ((b2 +b3)ψ⟨i j⟩+2g2ε⟨i j⟩)

(70)

ρ⋄ψ⟨i j⟩ = s21ε⟨i j⟩− s2 ((b2 +b3)ψ⟨i j⟩+2g2ε⟨i j⟩

). (71)

Equation )70) is a constitutive equation for the deviatoric part of the stress t⟨i j⟩, andEq. (71) is the evolution equation of ψ⟨i j⟩. Moreover, the second law requires that thesymmetric part of the conductivity matrix is positive definite, therefore,

s1 ≥ 0, s2 ≥ 0, s1s2 − s12s21 −(

s12 − s21

2

)2

≥ 0. (72)

The role of the internal variable may be better understood if we eliminate it from

Eqs. (70)–(71). Taking the material time derivative of Eq. (70) and substituting⋄ψ⟨i j⟩

from Eq. (71) and ψ⟨i j⟩ from Eq. (70) into the obtained form, the following relationfollows:

⋄t ⟨i j⟩ +s2ρ(b2 +b3)t⟨i j⟩ = s1ε⟨i j⟩+

+ρ[(b2 +b3)(2µ +(s1s2 − s12s21)+(s21 − s12)2g2)

]ε⟨i j⟩+

+ s2ρ2((b2 +b3)µ −2g2)ε⟨i j⟩

(73)


The positive definiteness of the free energy w requires that the coefficient (b2 +b3)µ −2g2 is non-negative and Eq. (73) can be transformed to

τ⋄t ⟨i j⟩ +t⟨i j⟩ = τd ε⟨i j⟩+2νε⟨i j⟩+Eε⟨i j⟩, (74)

where

τ =1

ρs2(b2 +b3), τd = s1τ, E = s2ρ2((b2 +b3)µ −2g2)τ,

ν = ρ[(b2 +b3)(2µ +(s1s2 − s12s21)+(s21 − s12)2g2)

]τ.

(75)

Constitutive relation (74) reduces to the Zener body of rheology or standard linearsolid [43,44], if τd = 0 and the nonlinear part of the time derivative is neglected. Thisrheological model is widely used in different fields from biology [45] to engineering[46]. The complete form of this constitutive relation (with τd = 0) is called the in-ertial Poynting-Thomson body. The proposed thermodynamic model has remarkableproperties which are distinctive in comparison with more intuitive approaches:

1. Neither the inertial term beyond the standard model, nor the coupled volumetric-deviatoric effect is neglectable in general, both are important, e.g. in experimentalrock rheology [47–49], where an inertial Poynting-Thomson body is used for bothdeviatoric and spherical parts of the deformation in order to get a good agreementwith experimental data.

2. Reciprocity has not been required in the proposed approach. The same is truein the simplest case of a standard linear solid body, otherwise the second law ofthermodynamics contradicts to observations [41].

3. The complete form of constitutive equation (70)-(71) is preferable instead of (74)if dynamical problems with the coupled balance of momentum (2) are needed tobe solved. This is an important advantage of the thermodynamic approach.

4.3 Dual internal variables

Now we consider dual (coupled) tensorial internal variables which are independentof mechanical and thermal interactions (λ = 0, L11 = L12 = L13 = L21 = L31 = 0).The evolution equations of the coupled tensorial internal variables again follow fromthe dissipation inequality. In the small strain approximation and with small velocitygradients, Eqs. (60)-(61) are simplified to

ψi j = L22i jklXkl + L23

i jklYkl , (76)

βi j = L32i jklXkl + L33

i jklYkl . (77)

Here Xi j = ∂ψi j w−∂k

(∂∂kψi j w

), and Yi j = ∂βi j w−∂k

(∂∂kβi j w

). Equations (76)–(77)

are independent of balances of linear momentum and energy, if the free energy canbe decomposed into a sum of functions depending on two sets of variables, e,εi jand ψi j,βi j separately, and if the objective time derivatives can be substituted by the

16 Peter Van et al.

substantial time derivative, that is the nonlinear terms are neglectable in the upperconvected derivatives.

In the case of isotropic materials, tensorial equations (76)–(77) are decomposedinto a spherical, deviatoric and antisymmetric parts with scalar coefficients. The re-markable difference between the evolution described by a Ginzburg-Landau-typeequation based on a single internal variable and this dual structure becomes appar-ent after the separation of the symmetric and antisymmetric parts of the particulardecompositions.

As an example, we consider completely decoupled deviatoric evolution equa-tions. Then conductivity matrices are two-dimensional and evolution equations arethe following:

ψ⟨i j⟩ = s1X⟨i j⟩+(s+a)Y⟨i j⟩, (78)

β⟨i j⟩ = (s−a)X⟨i j⟩+ s2Y⟨i j⟩. (79)

The above evolution equations are decomposed into a symmetric part, which repre-sents a dissipative evolution and therefore produces entropy, and the antisymmetricpart that does not produce entropy and represents a non-dissipative part of the evo-lution. The role of the non-dissipative part can be better understood with the help ofthe following free energy function, where we assume a local theory for β⟨i j⟩:

w(β⟨i j⟩,ψ⟨i j⟩,∂kψ⟨i j⟩) =c2

β⟨i j⟩β⟨i j⟩+wh(ψ⟨i j⟩)+wg(∂kψ⟨i j⟩) =

=c2

β⟨i j⟩β⟨i j⟩+b2

ψ⟨i j⟩ψ⟨i j⟩+f1

2∂kψ⟨i j⟩∂kψ⟨i j⟩+

+f2

2∂kψ⟨i j⟩∂iψ⟨ jk⟩+

f3

2∂kψ⟨ik⟩∂ jψ⟨i j⟩.

(80)

Here wh is the homogeneous, local part and wg is the gradient dependent, weaklynonlocal part of the free energy related to the variable ψ⟨i j⟩. The second and thirdlines of Eq. (80) shows a particular quadratic form of corresponding functions. Iff2 = f3 = 0, then we obtain the usual second gradient theory. The non-dissipativepart of evolution equations (78)-(79) has the form

ψ⟨i j⟩ = acβ⟨i j⟩, (81)

β⟨i j⟩ =−a(

∂ψ⟨i j⟩ws −∂k(∂∂kψ⟨i j⟩wg)). (82)

It is easy to eliminate β⟨ik⟩ and obtain an evolution equation for ψ⟨ik⟩, which is secondorder in time:

1a2c

ψ⟨ik⟩+∂ψ⟨ik⟩ws −∂k

(∂∂kψ⟨ik⟩wg

)= 0. (83)

The last equation can be considered as the Euler-Lagrange equation of the Lagrangian

L(ψ⟨ik⟩, ψ⟨ik⟩) =1

2a2cψ⟨ik⟩ψ⟨ik⟩−w(ψ⟨ik⟩,∂kψ⟨ik⟩). (84)

It is remarkable that natural thermodynamic boundary conditions of the zero entropyflux requirement correspond exactly to natural boundary conditions of the variationalprinciple.


4.4 Dissipative generalized continua

It was already observed by Berezovski, Engelbrecht and Maugin [21] that generalizedthermomechanics of solids is a particular case of the dual internal variables theory.They observed that evolution equations of the non-dissipative theory of Mindlin cor-respond exactly to non-dissipative evolution equations of the dual internal variablestheory. However, they did not perform a complete thermodynamic analysis and, there-fore, their observation is restricted to the idealized non-dissipative case.

The corresponding thermodynamic analysis was performed in the second sec-tion of the present paper exploiting the dissipation inequality. The coupled evolutionequations are represented in the form of linear conductivity equations. If one of theinternal variables is interpreted as a microdeformation, then our calculations are tobe considered as a pure thermodynamic derivation of a generalized dissipative con-tinuum theory. The particular example of the theory of Mindlin arises then under thefollowing conditions:

1. No thermal and viscous dissipation.2. Pure antisymmetric coupling between internal variables.3. Quadratic free energy function.

Another generalized continuum theory has been introduced by Eringen and Suhubi[2]. To compare the entropy production, let us consider for simplicity a continuumwithout additional internal variables in the small strain approximation and uniformtemperature field. Then the entropy production in the Eringen-Suhubi theory (1) lackthe first and the last terms and can be written with our notation as


)εi j+

(si j − τi j −ρ∂ψi j w

)ψi j+

(µi jk −∂∂kψi j w

)∂kψi j ≥ 0. (85)

Here the internal variable ψi j is identified with the microdeformation gradient χ ′i j of

the Eringen-Suhubi theory and therefore the material time derivative is the substantialtime derivative due to the deformation interpretation. For the comparison, let us repeathere our entropy production (63):


)εi j+

+(


ρ⋄ψ i j +

(∂βi j w−∂k(∂∂kβi j w)

)ρ

⋄β i j≥ 0.

(86)

The difference in the entropy flux in the two theories and the different concept ofconstitutive quantities determine the diversity in the entropy production. We haveintroduced the evolution equations of internal variables as constitutive relations to bedetermined from the entropy inequality. Eringen and Suhubi [2] simply indicated theform of the dissipation inequality following from their definition of stresses.

Regarding the exploitation of the second law one should observe the following:

– The micromomentum balance and the evolution equation of ψi j are not con-structed from the dissipation inequality.

– The micromomentum balance is not used as a constraint for the entropy inequalityin the Eringen-Suhubi derivation. However, it is implicitly considered during theapplication of the Coleman-Noll procedure assuming that the multipliers of thetime derivatives should be zero.

18 Peter Van et al.

– The entropy flux is not an arbitrary constitutive function, but it is restricted to theclassical Ji = qi/T in the case of Eringen and Suhubi.

– The constitutive state space is not weakly nonlocal and not fixed in the Eringen-Suhubi derivation.

As we have shown above, not only the basic assumptions, but also the final equa-tions of the Eringen-Suhubi theory are particular and can be obtained from our gener-alized approach if several dissipative terms are neglected. Virtual power approachesalso introduce an entropy production that have a similar form and similar limitationsas the Eringen-Suhubi approach has (see e.g. [50,51]).

4.5 Heat conduction and weakly nonlocal internal variables: Microtemperature

Finally, let us consider the case with a non-zero heat conduction coefficient λ . Ne-glecting the viscosity influence (L11 = L12 = L13 = L21 = L31 = 0), we chose the freeenergy dependence on internal variables in the form of Eq. (80), but with the reducedconductivity matrix for deviatoric evolution equations

ψ⟨i j⟩ = aY⟨i j⟩, (87)

β⟨i j⟩ =−aX⟨i j⟩+ s2Y⟨i j⟩. (88)

which corresponds to the choice s = s1 = 0 in Eqs. (78)-(79). The free energy depen-dence (80) allows to represent the evolution equations in the form

ψ⟨i j⟩ = acβ⟨i j⟩, (89)

β⟨i j⟩ =−a(

∂ψ⟨i j⟩ws −∂k(∂∂kψ⟨i j⟩wg))+ s2cβ⟨i j⟩, (90)

which can be reduced to the single second-order evolution equation for the primaryinternal variable ψ⟨i j⟩

1a2c

ψ⟨ik⟩−s2

aψ⟨ik⟩+∂ψ⟨ik⟩ws −∂k

(∂∂kψ⟨ik⟩wg

)= 0, (91)

which is similar to the Jeffreys type modification of the Maxwell-Cattaneo-Vernotteequation [52].

The thermal part of the dissipation inequality is satisfied by the modified Fourierlaw that follows from Eq. (58)

qk +ρ fi j∂∂kψi j w+ρgi j∂∂kβi j w = λ∂k lnT. (92)

As it was shown [21] on the example of one-dimensional thermoelasticity, the pri-mary internal variable ψ⟨i j⟩ can be interpreted in this case as a microtemperature.In this context, it is understood as a fluctuation of the macrotemperature due to theinfluence of the existing microstructure. The solution of the equations shows thatinfluence of microtemperature may result in a wavelike propagation of temperatureif the corresponding damping effects are small [21].


5 Summary and discussion

The paper is devoted to the answer of the following question: How could we obtainevolution equations of physical quantities, about which we do not know anything,i.e., only general principles can be considered? There are essentially two basic ap-proaches. The first one postulates a variational principle of Hamiltonian type comingfrom mechanics. In this case dissipation is something additional to the non-dissipativebasic mechanical evolution. The second approach is coming from thermodynamics:one can assume that the evolution of new variables is not exception from the secondlaw and generate their governing differential equations accordingly. This is the situa-tion in the case of internal variables in general as it was summarized by Maugin andMuschik [13,14]. The two approaches can be generalized. Thermodynamic princi-ples and dual internal variables in the framework of a second order weakly nonlocaltheory give a straightforward and simple way of the generalization [17].

As a result, the thermodynamic consistency of continuum mechanics with dualtensorial internal variables was analyzed in the present paper by the Liu approach tothe exploitation of the second law in the Piola-Kirchhoff framework. Then local evo-lution was considered in isotropic materials in the small strain approximation. Theentropy production was calculated and thermodynamic forces and fluxes were identi-fied. Then a quadratic free energy and linear conductivity relations closed the systemof equations. The final evolution equations in a non-dissipative case are equivalentof those for micromorphic continua, therefore, the thermodynamic method gave adissipative extension of the original Mindlin theory.

We have given several particular examples that arise as special cases of the gen-eral theory. Our goal was only partially a justification, but also the identification ofthe most important differences from other theories and the interpretation of somequalitative predictions of our approach. We have seen that phenomena of microtem-perature, sophisticated couplings in generalized rheology, and special properties ofthe dissipative extension without assuming reciprocity relations, all are open for ex-perimental testing. We think that this approach is essential in the case of generalizedcontinua, where additional coefficients are considered hardly measurable.In this re-spect we have analyzed the limitations of the dissipation in other classical approaches,in particular in the Eringen-Suhubi theory, which is one of the most developed fromthis point of view. We have seen, that due to the restrictive starting assumptions (me-chanical interpretation of the internal variable, locality, special entropy flux, etc.)the considered dissipation is extremely limited. For example, under the traditionalapproaches one cannot recover neither the Ginzburg-Landau equation, nor simpleviscoelasticity.

It is remarkable, that the finite deformation part of our approach shares shortcom-ings of the Piola-Kirchhoff framework. There are indications that the requirementof objectivity and material frame indifference are not treated properly in this case[53,54]. Beyond the reservations of using material manifolds in general, it is alsoremarkable that the exclusion of velocity field vi from the constitutive state space isnot necessary [23,55]. We expect several interesting phenomena by the analysis ofhigher-order nonlocality at the mechanical and thermal side. Here the comparison tophase field approaches looks like a promising direction (see e.g. [56–58]).

20 Peter Van et al.

6 Appendix

In this section we shortly derive the material time derivatives, introduce the smallstrain approximation and describe the transformation of the balances between a Piola-Kirchhoff and local frameworks. Here we distinguish between contra and covariantas well as between material manifold and space-time vectors and tensor components.The covariant and contravariant vectors are denoted by lower and upper indices, thespace vector and tensor components are denoted by minuscules, and the vector andtensor components at the material manifold by capital letters. We assume here that thereference configuration is relaxed, stress free, therefore the transformation betweenthe material and spatial descriptions is standard (see e.g. [33]). For a more detailedkinematics, considering general bodies, see [53].

The material vectors are denoted by X i, the spatial ones by xi. Therefore, thedeformation gradient, the material manifold derivative of the motion, χ i(t,XJ), isgiven as:

F iJ = ∂J χ i. (93)

The transformation between material and spatial vectors and covectors is the follow-ing:

aJ = (F−1)Jia

i, ai = F iJaJ , (94)

bJ = F iJbi, bi = (F−1)J

ibJ . (95)

In particular, the transformation of space derivatives follows the lower indexed cov-ector rule:

∂J = F iJ∂i, ∂i = (F−1)J

i∂J . (96)

The summation over repeated indices is still assumed.

6.1 Material time derivatives

The transformation of time derivatives is different for quantities with different tenso-rial character. For scalars, the partial time derivative on the material manifold, ∂t , isthe substantial derivative for local quantities. We use the convenient dot notation forlocal quantities that corresponds to the partial time derivative on the material mani-fold for scalars, for the velocity and for acceleration fields:

∂ta(t,X i) = a = ∂ta(t,xi)+ vi∂ia(t,xi), (97)

∂t χ i = vi, ∂tt χ i = ∂tvi = vi. (98)

The material time derivative of internal variables with various tensorial characterdiffer from each other. Here we give the calculation for the second-order tensorialvariable, ψ i j. According to the definition of material time derivatives, the partial time


derivative on the material manifold expressed by spatial fields gives the material timederivative:

∂tψ IJ = ∂t((F−1)I

i(F−1)J

jψ i j)== (F−1)I

i(F−1)J

jψ i j − (F−1)Ik∂tFk

L(F−1)L

i(F−1)J

jψ i j−

− (F−1)Ii(F

−1)Jk∂tFk

L(F−1)L

jψ i j =

= (F−1)Ii(F

−1)Jj

(ψ i j −∂kviψk j −∂kv jψ ik

).

(99)

Therefore, the spatial form of the abovementioned formula, the material time deriva-tive of the tensor, is given as

⋄ψ

i j= F i

IFjJ ∂tψ IJ = ψ i j −∂kviψk j −∂kv jψ ik. (100)

Here we used the kinematic relation for the spatial velocity gradient and the timederivative of the deformation gradient ∂ jvi = ∂tF i

J(F−1)J

j. In case of cotensors ormixed tensors, the spatial form of the material time derivative is different.

6.2 Small strains

The small spatial strains are defined with the left Cauchy-Green deformation Ai j =F i

JF jJ , as

ε i j :=12(Ai j −δ i j). (101)

This choice is the best considering the requirement of objectivity [53]. The materialtime derivative of the strain in the small strain approximation is the symmetric partof the velocity gradient

ε i j =12

Ai j ≈ 12(∂ iv j +∂ jvi). (102)

It should be noted that in the small strain approximation one may obtain identicalresults starting from different deformation concepts.

The spatial form of the second material time derivative of the strain is particular

ε i j =12

Ai j ≈ ∂ (l vi)+∂kvi∂kv j. (103)

Here we have used that ∂l viAl j = F iJ(F

−1)JlF

lKF j

K .

22 Peter Van et al.

6.3 Spatial balances

The relation between the local density ρ and the material density ρ0 is

ρ =ρ0

detF,

where detF is the determinant of F iK .

The local and Piola-Kirchhoff forms of the heat flux and the stress are, respec-tively,

qi = (detF)−1F iJqJ , qI = detF(F−1) I

j q j, (104)

t i j = (detF)−1F iKtK j, tI j = detF(F−1)I

ktk j. (105)

For the transformation of basic balances, the Nanson theorem is essential. It can bewritten as:

∂J(detF(F−1)J

i)= 0i. (106)

The proof is straightforward, when considering that the derivative of the determi-nant is ∂J(detF) = detF(F−1)K

l ∂JF lK and the derivative of the inverse deformation

gradient is ∂J(F−1)Ij =−(F−1)I

l∂JF lK(F

−1)Kj.

Then the transformation of the balance of internal energy follows by substitutingthe definitions:

ρ0e+∂KqK = detFρ e+detF(F−1)Ij∂Iq j =

= detF(ρ e+∂ jq j) = tI j∂Iv j = detF tk j∂kv j.(107)

Similarly, the balance of linear momentum can be easily obtained:

ρ0vi +∂KtKi = ...= detF(ρ vi +∂ktki) = 0i. (108)

Therefore, we obtain usual local balances of internal energy and momentum (48) and(49) without approximations.

Acknowledgements The work was supported by the grants Otka K81161, K104260 and TT 10-1-2011-0061/ZA-15-2009. The authors thank Tamas Fulop and Csaba Asszonyi for valuable discussions.

References

1. Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Rat. Mech. Anal. 16, 51–78 (1964)2. Eringen, A.C., Suhubi, E.S.: Nonlinear theory of simple micro-elastic solids I. Int. J. Engng. Sci. 2,

189–203 (1964)3. Eringen, A.C.: Microcontinuum Field Theories. vol.I. Springer, New York (1999)4. Germain, P.: The method of virtual power in continuum mechanics. Part 2: Microstructure. SIAM J.

Appl. Math. 25, 556–575 (1973)5. Forest, S., Amestoy, M.: Hypertemperature in thermoelastic solids. C. R. Mecanique 336, 347–353

(2008)6. Aslan, O., Forest, S.: The micromorphic versus phase field approach to gradient plasticity and damage

with application to cracking in metal single crystals. In: de Borst, R., Ramm, E., (eds.) MultiscaleMethods in Computational Mechanics, pp. 135–154, Springer, New York (2011)

7. Papenfuss, C., Forest, S.: Thermodynamical frameworks for higher grade material theories with internalvariables or additional degrees of freedom. J. Non-Equilib. Thermodyn. 31(4), 319–353, (2006)


8. Haupt, P.: Non-Equilibrium Thermodynamics with Applications to Solids, CISM-Course, pp. 65 - 138,Springer, Wien (1993)

9. Gyarmati, I.: Non-equilibrium Thermodynamics /Field Theory and Variational Principles, Springer,Berlin (1970)

10. Van, P., Muschik, W,: The structure of variational principles in nonequilibrium thermodynamics. Pe-riodica Polytechnica, Physics and Nuclear Sciences, 2/1-2, 111–122 (1994)

11. Van, P., Nyıri, B.: Hamilton formalism and variational principle construction. Ann. Phys. (Leipzig),8, 331–354 (1999)

12. Irwing, J. H., Kirkwood, J. G.: The statistical mechanical theory of transport processes. IV. the equa-tions of hydrodynamics. J. Chem. Phys. 18, 817–830 (1950)

13. Maugin, G. A., Muschik, W,: Thermodynamics with internal variables. Part I. General concepts. J.Non-Equilib. Thermodyn. 19, 217–249 (1994)

14. Maugin, G. A., Muschik, W,: Thermodynamics with internal variables. Part II. Applications. J. Non-Equilib. Thermodyn. 19, 250–289 (1994)

15. Coleman, B. D., Gurtin, M. E.: Thermodynamics with internal state variables. J. Chem. Phys. 47(2),597–613 (1967)

16. Muller, I., Weiss, W.: Thermodynamics of irreversible processes - past and present. Eur. Phys. J. H,37, 139–236 (2012)

17. Van, P., Berezovski, A., Engelbrecht, J.: Internal variables and dynamic degrees of freedom. J. Non-Equilib. Thermodyn. 33, 235–254 (2008)

18. Van, P.: Weakly nonlocal continuum theories of granular media: restrictions from the Second Law.Int. J. Solids Struct. 41(21), 5921–5927 (2004)

19. Van, P., Fulop, T.: Weakly nonlocal fluid mechanics - the Schrodinger equation. Proc. Roy. Soc.London A, 462(2066), 541–557 (2006)

20. Cimmelli, V. A.: An extension of Liu procedure in weakly nonlocal thermodynamics. J. Math. Phys.48, 113510 (2007)

21. Berezovski, A., Engelbrecht, J., Maugin, G.A.: Thermoelasticity with dual internal variables. J. Ther-mal Stresses 34, 413–430 (2011)

22. Wald, R. M.: General Relativity, The University of Chicago Press, Chicago (1984)23. Matolcsi, T., Van, P.: Can material time derivative be objective? Phys. Lett. A, 353, 109–112 (2006)

math-ph/0510037.24. Van, P.: Exploiting the Second Law in weakly nonlocal continuum physics. Periodica Polytechnica,

Ser. Mech. Engng. 49(1), 79–94 (2005)25. Van, P.: Weakly nonlocal non-equilibrium thermodynamics - variational principles and Second Law.

In: Quak. E., Soomere, T., (eds.) Applied Wave Mathematics, pp. 153–186. Springer, Berlin (2009)26. Onsager, L.: Reciprocal relations of irreversible processes I. Phys. Rev. 37, 405–426 (1931)27. Onsager, L.: Reciprocal relations of irreversible processes II. Phys. Rev. 38, 2265–2279 (1931)28. Casimir, H. G. B.: On Onsager’s principle of microscopic reversibility. Rev. Mod. Phys. 17, 343–350

(1945)29. Morro, A.: Evolution equations and thermodynamic restrictions for dissipative solids. Math. Comput.

Model. 52, 1869–1876 (2010)30. Cimmelli, V. A., Van, P.: The effects of nonlocality on the evolution of higher order fluxes in non-

equilibrium thermodynamics. J. Math. Phys. 46(11), 112901–15 (2005)31. Ciancio, V., Cimmelli, V. A., Van, P.: On the evolution of higher order fluxes in non-equilibrium

thermodynamics. Math. Comput. Model. 45, 126–136 (2007)32. Van, P., Fulop, T.: Universality in heat conduction theory: weakly nonlocal thermodynamics. Ann.

Phys. 524(8), 470–478 (2012)33. Gurtin, M. E., Fried, E., Anand, L.: The mechanics and thermodynamics of continua, Cambridge

University Press (2010)34. Landau, L. D., Ginzburg, V. L.: K teorii sverkhrovodimosti. Zh. Eksper. Teor. Fiz. 20, 1064–1082

(1950)35. Landau, L. D., Khalatnikov, I. M.: Ob anomal’nom pogloshehenii zvuka vblizi tochek fazovo

perekhoda vtorovo roda. Doklady Akademii Nauk SSSR, 96, 469–472 (1954)36. Grmela, M.: Weakly nonlocal hydrodynamics. Phys. Rev. E, 47(1), 351–602 (1993)37. Grmela, M., Ottinger H. C.: Dynamics and thermodynamics of complex fluids. I. Development of a

general formalism. Phys. Rev. E, 56(6), 6620–6632 (1997)38. Fabrizio, M.: An evolution model for the Ginzburg-Landau equations. Riv. Mat. Univ. Parma, 2(6),

155–169 (1999)

24 Peter Van et al.

39. Fabrizio, M. Lazzari, B., Morro, A.: Thermodynamics of nonlocal electromagnetism and supercon-ductivity. Math. Models Methods Appl. Sci. 13(7), 945–969 (2003)

40. de Gennes, P. G.: The physics of liquid crystals, Oxford University Press, New York, 2 edition (1993)41. Fulop, T.: Thermodynamics of rheology: the standard model. private communication42. Verhas, J.: Thermodynamics and Rheology, Akademiai Kiado and Kluwer Academic Publisher, Bu-

dapest (1997)43. Ferry, J. D.: Viscoelastic properties of polymers, Wiley, New York, 3d edition (1980)44. Tschoegl, N. W.: The phenomenological theory of linear viscoelastic behavior: An introduction,

Springer, Berlin (1989)45. Klatt, D., Hamhaber, U., Asbach, P., Braun, J., Sack1, I.: Noninvasive assessment of the rheological

behavior of human organs using multifrequency mr elastography: a study of brain and liver viscoelastic-ity. Phys. Med. Biol. 52, 7281–7294 (2007)

46. Haan, Y. M., Sluimer, G. M.: Standard linear solid model for dynamic and time dependent behaviourof building materials. Heron, 46(1), 49–76 (2001)

47. Matsuki, K., Takeuchi, K.: Three-dimensional in situ stress determination by anelastic strain recoveryof a rock core. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 30, 1019–1022, (1993)

48. Matolcsi, T.: Hengerszimmetrikus deformacio, laboratoriumi probatestek meretfuggese. In: Fulop T.,(ed.) Izotrop kontinuumok anyagtorvenye, pp. 121–136, Muegyetemi Kiado, Budapest (2008)

49. Lin, W., Kuwahara, Y., Satoh, T., Shigematsu, N., Kitagawa, Y., Kiguchi, T., ... & Koizumi, N.: A casestudy of 3d stress orientation determination in Shikoku island and Kii peninsula, Japan. In: Vrkljan, I.,(ed.), Rock Engineering in Difficult Ground Conditions (Soft Rock and Karst), pp. 277–282, London,Balkema (2010)

50. Forest, S., Sievert, R.: Nonlinear microstrain theories. Int. J. Solids Struct. 43, 7224–7245 (2006)51. Lee, J. D., Wang, X.: Generalized micromorphic solids and fluids. Int. J. Engng. Sci. 49, 1378–1387

(2010)52. Joseph, D.D., Preziosi, L.: Heat waves. Rev. Mod. Phys. 61, 41–73 (1989)53. Fulop T., Van, P.: Kinematic quantities of finite elastic and plastic deformations. Math. Methods Appl.

Sci. 35, 1825–1841 (2012)54. Noll, W., Seguin, B.: Basic concepts of thermomechanics. J. Elasticity, 101, 121–151 (2010)55. Van, P., Papenfuss, C.: Thermodynamic consistency of third grade finite strain elasticity. Proc. Esto-

nian Acad. Sci. 59(2), 126–132 (2010)56. Fabrizio, M., Morro, A.: Thermodynamics and second sound in a two-fluid model of helium II; Re-

visited. J. Non-Equilib. Thermodyn. 28, 69–84 (2003)57. McFadden, G. B., Anderson, D. M., Wheeler, A. A.: Diffuse-interface methods in fluid mechanics.

Ann. Rev. Fluid Mech. 30, 139–165 (1998)58. Goddard, J. D.: A note on Eringen’s moment balances. Int. J. Engng. Sci. 49, 1486–1493 (2011)

Documents

Thermodynamic Approach to Generalized Continuaberez/texts/71.Van-Berezovski-Papenfuss... · 2020-01-01 · Continuum Mechanics and Thermodynamics manuscript No. (will be inserted