A Variational Framework for the Thermomechanics of Gradient-Extended Dissipative Solids -- with Applications to Diffusion, Damage and Plasticity
aa r X i v : . [ m a t h . NA ] S e p A Variational Framework for the Thermomechanics ofGradient-Extended Dissipative Solids – with Applicationsto Diffusion, Damage and Plasticity
S. Teichtmeister , M.-A. KeipInstitute of Applied Mechanics, Chair IUniversity of Stuttgart, 70569 Stuttgart, Pfaffenwaldring 7, Germany Abstract
The paper presents a versatile framework for solids which undergo nonisothermal processeswith irreversibly changing microstructure at large strains. It outlines rate-type and incremen-tal variational principles for the full thermomechanical coupling in gradient-extended dissipativematerials. It is shown that these principles yield as Euler equations essentially the macro- andmicro-balances as well as the energy equation. Starting point is the incorporation of the entropyand entropy rate as additional arguments into constitutive energy and dissipation functionswhich depend on the gradient-extended mechanical state and its rate, respectively. By meansof (generalized) Legendre transformations, extended variational principles with thermal as wellas mechanical driving forces can be constructed. On the thermal side, a rigorous distinctionbetween the quantity conjugate to the entropy and the quantity conjugate to the entropy rateis essential here. Formulations with mechanical driving forces are especially suitable when con-sidering possibly temperature-dependent threshold mechanisms. With regard to variationallyconsistent incrementations, we suggest an update scheme which renders the exact form of theintrinsic dissipation and is highly suitable when considering adiabatic processes. To underlinethe broad applicability of the proposed framework, we exemplarily set up three model problems,namely Cahn-Hilliard diffusion coupled with temperature evolution as well as thermomechanicsof gradient damage and gradient plasticity. In a numerical example we study the formation ofa cross shear band.
Keywords: variational principles, gradient theories, multifield problems, nonisothermal pro-cesses
1. Introduction
In order to model dissipative size effects in solid materials that are, for example re-lated to the width of shear bands or the grain size in polycrystals, nonstandard continuumtheories have to be elaborated that are based on characteristic length scale parameters.The idea is to incorporate at least first-order spatial gradients of microstructural variablesthat describe (possibly in a homogenized sense) the irreversibly evolving microstructure.It is of great importance here to account for thermomechanical coupling effects such asthermal softening and temperature increase due to intrinsic dissipation. Practical appli-cations include technological processes like sheet-metal forming or extrusion of metals.Within this work we present a unified framework for the fully coupled thermomechanicsof gradient-extended dissipative solids that is applicable to a wide range of model prob-lems such as diffusion, (crystal) plasticity and damage. The formulation is embedded intothe concept of standard dissipative solids that is characterized by energy and dissipationfunctions, see
Biot [6],
Ziegler [82] and
Halphen & Nguyen [31]. In particular,the strongly coupled multifield problem will exhibit an incremental variational structurewhich is an extension of the framework of gradient-extended dissipative solids presentedin
Miehe [46, 47] towards nonisothermal processes. Corresponding author. Tel.: +4971168566378, fax: +4971168566347. E-mail address:[email protected] (S. Teichtmeister) ariational Thermomechanics of Gradient-Extended Continua
Cosserat & Cosserat [12]. They investigated amicrostructure with rigid particles by introducing an additional microrotation governedby three additional degrees of freedom. Since then, a lot of research has been done onso called micropolar, micromorphic and microstrain continua, i.e. by
Eringen [16], seealso
Leismann & Mahnken [35] for a comparative study. The book of
Capriz [10]provides a general setting for materials with microstructure where order parameters areconsidered as components of elements of an abstract manifold. Here, we also refer to thework of
Mariano [41]. In
Maugin [42] the standard thermodynamic theory of localinternal variables is extended by an additional dependency of the energy function of thefirst-order spatial gradient of a (not necessarily scalar) internal variable, see also
Maugin& Muschik [43] and
Fr´emond [23]. An important ingredient of such theories is aproper energetic treatment of the microstructural processes yielding additional balance-type equations that are coupled with the standard macroscopic balance equations (mass,linear/angular momentum and energy). For an embedding into the theory of microforceswe refer to
Gurtin [25].Typical applications of above mentioned nonlocal theories including additional balanceequations are theories of phase transformation, gradient plasticity and gradient damage.Theories of gradient plasticity are dealt with on a phenomenological level in
Aifantis [1],
Fleck & Hutchinson [17],
Gurtin [25],
Forest & Sievert [21],
Gurtin & Anand [26] just to name a few. Gradient damage theories that are basically in the same spiritare considered in
Peerlings et al. [64],
Dimitrijevic & Hackl [15],
Pham et al. [67],
Fr´emond [23] among many others.An important role in the modeling of dissipative processes in solids is played by in-cremental variational principles . They offer an elegant way to couple different physicalfields, possess inherent symmetry and require only Jacobian and Hessian matrices of theunderlying potential density for an implementation into typical finite element codes. Inaddition, if a minimization structure is at hand, structural and material stability analysisis possible, i.e. the formation of complex microstructures can be described. For contribu-tions on the variational theory of local plasticity we refer to
Hill [32],
Simo & Honein [72],
Hackl [27],
Ortiz & Stainier [63],
Ortiz & Repetto [62],
Carstensen et al. [11],
Miehe et al. [50] and
Mosler & Bruhns [56]. The works of
Miehe [45],
Petryk [66] and
Hackl & Fischer [28] deal with general inelastic behavior. Extensions towardsincorporation of (at least) the first gradient of the internal variable into constitutive func-tions can be found in
M¨uhlhaus & Aifantis [57],
Fleck & Willis [18],
Mainik &Mielke [40],
Nguyen [59],
Miehe [47] and
Lancioni et al. [34] for gradient plastic-ity and in
Lorentz & Andrieux [37],
Mielke & Roub´ıˇcek [54],
Dimitrijevic &Hackl [15] and
Pham et al. [67] for gradient damage. In addition, we mention the gen-eral formulations of gradient-extended dissipative solids by
Svendsen [76],
Francfort& Mielke [22] and
Miehe [46]. A gradient-plasticity theory that incorporates a frac-tional derivative of the plastic strain is presented in
Dahlberg & Ortiz [13]. Note, thatall works citet so far in this paragraph deal with dissipative processes under isothermalconditions except
Hackl [27] in the last section. In the latter work, an original idea of
Simo & Miehe [73] is taken up, namely to introduce a plastic configurational entropy asan additional internal variable that as part of the total entropy leaves the internal energyunchanged. From the viewpoint of variational principles however, the main difficulty offully coupled thermomechanics is to account for the internal dissipation in the energy . Teichtmeister, M.-A. Keip single unified potential from which all coupled thermome-chanical field equations follow is outlined in the seminal work of
Yang et al. [81]. Theyintroduce a specific integrating factor that makes the linearized integral expressions show-ing up in the fully coupled weak form symmetric. This factor is based on distinguishingthe equilibrium temperature from a so-called external temperature. Both temperaturescoincide only at equilibrium. With this variational principle at hand, the fully coupledthermomechanical boundary value problem can be solved monolithically by a sequence ofsymmetric algebraic systems. Alternatively, staggered algorithms were mainly used beforethat lead to symmetric finite element stiffness matrices for each subproblem, see
Simo &Miehe [73] and
Simo & Armero [71]. In the recent years the work of
Yang et al. [81] has been the basis for further model developments in thermoplasticity, see
Stainier& Ortiz [74],
Canadija & Mosler [9],
Su et al. [75],
Bartels et al. [5],
Mielke [53] and
Fohrmeister et al. [19]. In the latter, a specific phenomenological model ofgradient plasticity together with a micromorphic extension in the sense of
Forest [20]is considered. It should be noted that the micromorphic approach has successfully beenapplied to a wide range of formulations incorporating length scales such as e.g. gradientcrystal plasticity, see
Scherer et al. [69]. Beside incorporating the first gradient ofa generic internal variable into the energy function,
Mielke [53] also discusses in detailthe gradient structure of thermoplasticity. In
Bartels et al. [5] special focus is puton the thermodynamic (in)consistency of the Taylor-Quinney factor. This factor, usuallyintroduced in an ad-hoc way, takes into account that only a certain amount of the plasticpower is transformed into heat as observed in the experiments of
Taylor & Quinney [77], see also
Rosakis et al. [68] and
Oliferuk & Raniecki [61].According to the authors’ knowledge, a universal variational framework for the ther-momechanics of solids with energetic as well as dissipative gradient extensions is stillmissing in literature. The present work outlines a generalization of the versatile frame-work of
Miehe [46, 47] for gradient-extended dissipative solids towards nonisothermalprocesses . Point of departure is the definition of two constitutive functions, namely (i)the internal energy function that depends on the (gradient-extended) mechanical stateand the entropy and (ii) a dissipation potential function that depends on the rates of the(gradient extended) mechanical state and the entropy. Then, at current time the ratesof the macro- and micro-motion and the entropy rate are governed by a canonical saddlepoint principle in case of an adiabatic process and a canonical minimization principle incase of a process including heat conduction. However, since specific forms of both con-stitutive functions are difficult to access, (generalized) Legendre transformations of thosefunctions are considered. Here, the concept of dual variables is rigorously pursued, i.e. thequantity conjugate to the entropy (the temperature) is distinguished from the quantityconjugate to the entropy rate (the thermal driving force). Then, the necessary conditionof the (generalized) Legendre transformation of the dissipation potential function mustrender the structure of the energy equation in form of an entropy evolution equation.The obtained rate-type variational principle is in line with
Yang et al. [81] but de-rived in an alternative way and extended by an additional long-range micro-motion thataccounts for effects related to length scales. In addition, extended rate-type variationalprinciples are constructed that incorporate dissipative mechanical driving forces and atemperature-dependent scalar threshold function via a Lagrange multiplier. By construc-tion of consistent time integration algorithms, variational principles valid for the timeincrement under consideration are obtained. Here, we especially point out an alternative ariational Thermomechanics of Gradient-Extended Continua algorithm that in contrast to a fully implicit scheme evaluates the incremental integrationfactor of Yang et al. [81] to one and hence gives the algorithmically correct form of theintrinsic dissipation.The paper is organized as follows: In Section 2 we consider an adiabatic thermome-chanical material element and derive rate-type as well as incremental variational prin-ciples. Latter govern the time-discrete evolution of the thermomechanical state. InSection 3 we set up general forms of governing equations for the thermomechanics ofgradient-extended dissipative solids in a three-dimensional large-strain continuum set-ting. Section 4 shows that this fully coupled system is related to a variational statement.In addition, we construct extended rate-type and incremental variational principles thatincorporate dissipative mechanical driving forces and threshold mechanisms that dependon the thermal state. The formulations outlined in Section 4 are general and can beapplied to a wide spectrum of problems. Exemplarily, we specify in Section 5 the devel-oped setting to Cahn-Hilliard diffusion, gradient damage and (additive) gradient plasticitywith strong coupling to temperature evolution. For latter model problem we also show anumerical example that is concerned with shear band localization in softening plasticity.
2. Variational Principles for Nonisothermal Rheology
We consider a rheological material element de-picted in Figure 1 describing a linear thermo-visco-elastic behavior. The material elementis understood to be embedded in a thermal environment characterized by the absolutetemperature θ >
0. The spring describes thermoelastic behavior by Hooke’s law coupledwith thermal expansion. The dashpot device characterizes viscous response via Newton’slaw. In addition, the material element is able to store heat. We have the fundamentalconstitutive relationships • Hooke’s law σ e = E [ ε e − α T ( θ − θ ) ] • Newton’s law σ d = H ˙ ε d • Heat storage e θ = C ( θ − θ )Here, σ denotes as usual a stress, ε a strain and e an internal energy. The four involvedmaterial parameters are Young’s modulus E , the thermal expansion coefficient α T , theviscosity H and the heat capacity C at constant internal and external deformation. With-out loss of generality these material parameters are assumed to be constant, i.e. they donot dependent on the temperature. Finally, θ stands for a reference temperature. Sucha simple model is able to describe classical thermomechanical coupling effects, such asthe Gough-Joule effect or thermal expansion, as well as heating due to dissipation in thedashpot. The mechanical state of the rheological device is described by the total strain ε and the internal variable q characterizing the strain of the inner dashpot. We summarizethese two quantities in the mechanical state array c = ( ε, q ) . (1) . Teichtmeister, M.-A. Keip q ε σ ext σ ext H H E, α T C θ
Figure 1:
Rheological material element with internal thermoelastic and dissipative (vis-cous) mechanisms, loaded by an external stress σ ext ( t ). The time-discrete evolution of thematerial’s state ( ε, q , η ) is described by a series of incremental variational principles. Identifying the strain in the spring device by the simple kinematic relationship ε e = ε − q ,we define the well known thermoelastic free energy function b ψ ( c , θ ) = 12 E ( ε − q ) | {z } =: b ψ e ( c ) + Eα T ( q − ε )( θ − θ ) | {z } =: b ψ e − θ ( c ,θ ) + C [( θ − θ ) − θ ln θθ ] | {z } =: b ψ θ ( θ ) (2)that is concave in θ and penalizes nonpositive temperatures by b ψ θ → + ∞ . In addition,we define the dissipation potential function associated with the two dashpot devices b φ ( ˙ c ) = 12 H ˙ ε + 12 H ˙ q . (3) The thermomechanical response of thematerial element within a time interval T = (0 , t e ) is fully governed by the two constitutivefunctions (2) and (3) and characterized by four equations (4)–(7). First, on the mechanicalside, the equilibrium equation ∂ ε b ψ + ∂ ˙ ε b φ = σ ext in T (4)is a relationship between internal and external stresses. Second, the evolving internalstrain state is governed by Biot’s equation ∂ q b ψ + ∂ ˙ q b φ = 0 in T (5)and characterizes an internal dissipative mechanism. Third, on the thermal side, the stateequation η = − ∂ θ b ψ in T (6)determines the current entropy η or in an inverse manner the current temperature θ .Finally, the evolution of the entropy is governed by a nonnegative dissipation˙ η = 1 θ D in T with D = ∂ ˙ c b φ · ˙ c ≥ . (7)Equation (7) is a form of the first law of thermodynamics, i.e. the balance of energyfor an adiabatic process under consideration. The inequality (7) is the second law of ariational Thermomechanics of Gradient-Extended Continua a priori by a dissipation potential function b φ ( · ) that is(i) nonnegative, (ii) zero in the origin b φ ( ) = 0 and (iii) convex. In case of a dissipationpotential function that is positively homogeneous of degree ℓ , i.e. b φ ( γ ˙ c ) = γ ℓ b φ ( ˙ c ) for all γ >
0, the dissipation can be written alternatively as D = ℓ b φ ( ˙ c ) . (8)Aim of the subsequent treatment is the construction of variational principles that accountfor the governing equations (4)–(7) of the thermomechanical device. Due to the dissipativenature of the process, we consider principles that determine the evolution at time t ∈ T .They are based on potentials that are defined for given state at the same time t . Consistentincrementations of these potentials lead to a series of variational principles determiningthe time-discrete evolution of the state. The central difficulty in the formulation of rate-type variational principles in thermo-mechanics is to account for the full update of the thermal state via the energy equation(7) . For the nonisothermal case, we start by modeling the internal energy e , the cor-responding function of which naturally depends on the entropy η . Hence, we start ourinvestigation of thermomechanics in generalized standard materials by assuming the con-stitutive functions e ( t ) = b e ( c , η ) and v ( t ) = b v ( ˙ c , ˙ η ; c , η ) . (9)The internal energy function b e determines the current internal energy stored in the mate-rial element. The dissipation potential function b v is assumed to be a function of the rates( ˙ c , ˙ η ) and might additionally depend on the current thermomechanical state ( c , η ). Alter-natively, instead of the current entropy η the dissipation potential function b v can dependon the current temperature θ by making use of the constitutive relationship (6). Com-paring b v with the dissipation potential function b φ defined in (3), we observe an additionalthermal slot with the dependence on ˙ η . This assumed dependency is the key ingredientof the subsequent construction of variational principles in dissipative thermomechanics.Based on the two constitutive functions (9), we construct the rate-type potential b p ( ˙ c , ˙ η ; c , η ) = ddt b e ( c , η ) + b v ( ˙ c , ˙ η ; c , η ) (10)at given thermomechanical state ( c , η ), that accounts for energetic as well as dissipativemechanisms. Next, we define an external load function b p ext ( ˙ ε ; t ) = σ ext ( t ) ˙ ε (11)depending on the given external stress σ ext , that loads the material element, see Figure 1.Then, the rate of the thermomechanical state at current time t under adiabatic conditionsis determined by the canonical saddle point principle { ˙ c , ˙ η } = Arg { inf ˙ c sup ˙ η [ b p ( ˙ c , ˙ η ; c , η ) − b p ext ( ˙ ε ; t ) ] } . (12) In case of viscous dissipation we have ℓ = 2, see (3). . Teichtmeister, M.-A. Keip as-sumed concavity of the dissipation potential function b v with respect to ˙ η , see Section 2.3.2.Taking the first derivative of the potential (10) yields as necessary conditions of the vari-ational principle (12) three Euler equations for the rate of the thermomechanical state ofthe material element at current time t , namely1 . Evolving external state ∂ ˙ ε b p ≡ ∂ ε b e + ∂ ˙ ε b v = σ ext . Evolving internal state ∂ ˙ q b p ≡ ∂ q b e + ∂ ˙ q b v = 03 . Evolving thermal state ∂ ˙ η b p ≡ ∂ η b e + ∂ ˙ η b v = 0 . (13)The first equation determines the rate ˙ ε of the external strain state and is a form ofthe stress equilibrium (4) without inertia terms. The second equation governs the rate˙ q of the internal variable and is identical to (5). Finally, the third equation is a form ofthe balance of energy, i.e. the first law of thermodynamics (7) combined with an inverserepresentation of (6), which determines the rate ˙ η of the entropy. However, this is a quiteunusual representation. The main difficulty in this canonical setting is the formulationof the two constitutive functions b e and b v in terms of the entropy η and entropy rate ˙ η ,respectively . Hence, a formulation in terms of the temperature is developed subsequently. First, we define the internal energy functionoccuring in the rate-type potential (10) by a partial Legendre transformation b e ( c , η ) = sup θ [ b ψ ( c , θ ) + θη ] , (15)where b ψ is the free energy function depending on the thermal variable θ . The necessaryoptimality condition of this Legendre transformation defines the entropy η = − ∂ θ b ψ ( c , θ ) (16)in terms of the thermal variable θ . Latter is the dual quantity to the entropy η , i.e. the temperature of the material element. In a second step, we define the dissipa-tion potential function b v occuring in the rate-type potential (10) by a partial Legendretransformation b v ( ˙ c , ˙ η ; c , η ) = inf T [ e φ ( ˙ c , T ; c , θ ) − T ˙ η ] (17)in terms of the dissipation potential function e φ , that depends on the thermal variable T and, by making use of (16), is defined at current state ( c , θ ). The necessary condition ofthis Legendre transformation defines the evolution of the entropy˙ η = ∂ T e φ ( ˙ c , T ; c , θ ) (18) For a single linear viscous dashpot with current temperature θ and entropy η , respectively, and rates( ˙ ε, ˙ η ) the internal energy function b e and the dissipation potential function b v take the nonintuitive forms b e ( η ) = Cθ ( e η/C −
1) and b v ( ˙ ε, ˙ η ; θ ) = − θ H (cid:18) ˙ η ˙ ε (cid:19) . (14) ariational Thermomechanics of Gradient-Extended Continua T . We identify T as the quantity dual to the entropy rate ˙ η and call it the thermal driving force that, up to this point, is rigorously distinguished fromthe temperature θ of the material element. For the adiabatic case under consideration,(18) must have the form of the balance of energy (7) and we identify ∂ T e φ ! = 1 θ ∂ ˙ c b φ · ˙ c . (19)This relationship restricts the form of the functional dependence of the dissipation poten-tial function e φ and is fulfilled if a multiplicative dependence on the thermal driving force T is assumed e φ ( ˙ c , T ; c , θ ) = b φ ( Tθ ˙ c ; c , θ ) . (20)To arrive at (19), we use the result that at equilibrium the thermal driving force T can beidentified with the temperature θ , which is an outcome of the mixed variational principle(23) below. The scaling factor T /θ first arised as integrating factor in the seminal workof
Yang et al. [81]. Note, that b φ occuring in (20) is just the standard mechanicaldissipation potential function as for example given in (3) for the model problem underconsideration. Starting from the rate-type potential (10),the two partial Legendre transformations (15) and (17) motivate the definition of a mixedrate-type potential b π ( ˙ c , ˙ η, ˙ θ, T ) := ddt b ψ ( c , θ ) + ( θ − T ) ˙ η + η ˙ θ + b φ ( Tθ ˙ c ) (21)at given thermomechanical state ( c , η, θ ) in terms of the two constitutive functions b ψ and b φ . Inserting the necessary condition (16) on the given thermomechanical state yields thereduced mixed rate-type potential b π red ( ˙ c , ˙ η, T ) = ∂ c b ψ · ˙ c + ( θ − T ) ˙ η + b φ ( Tθ ˙ c ) (22)at given thermomechanical state ( c , η, θ ). Based on this definition, we obtain the mixedsaddle point principle { ˙ c , ˙ η, T } = Arg { inf ˙ c sup ˙ η inf T [ b π red ( ˙ c , ˙ η, T ) − b p ext ( ˙ ε ; t ) ] } (23)that determines at current time t the rates of the external strain, internal variable andentropy as well as the thermal driving force. The Euler equations of this saddle pointprinciple are 1 . Evolving external state ∂ ˙ ε b π red ≡ ∂ ε b ψ + ∂ ˙ ε b φ = σ ext . Evolving internal state ∂ ˙ q b π red ≡ ∂ q b ψ + ∂ ˙ q b φ = 03 . Thermal driving force ∂ ˙ η b π red ≡ T − θ = 04 . Evolving thermal state ∂ T b π red ≡ − ˙ η + ∂ T b φ = 0 . (24) To keep notation short, we subsequently do not write explicitly the dependence of functions on givenstates. . Teichtmeister, M.-A. Keip
Yang et al. [81]. The last equation represents the first law ofthermodynamics in the form of an evolution equation for the entropy. Clearly, withinthis mixed setting the entropy rate ˙ η plays the role of a Lagrange multiplier . With knownrates ( ˙ c , ˙ η ) of mechanical state and entropy, the temperature rate at current time t canbe computed by taking the time derivative of (6) yielding˙ θ = − [ ∂ θθ b ψ ] − ( ˙ η + ∂ θ c b ψ · ˙ c ) . (25)A constrained minimization formulation dual to the variational principle (12) is obtainedby commuting the order of the infimum and the supremum in (23) (which we assume tobe possible) such thatinf ˙ c sup ˙ η inf T [ b π red ( ˙ c , ˙ η, T ) − b p ext ( ˙ ε ; t ) ] = inf ˙ c inf T ∈ (24) [ ∂ c b ψ · ˙ c + b φ ( Tθ ˙ c ) − b p ext ( ˙ ε ; t ) ] . (26) For the model problem depicted in Figure 1 with the con-stitutive functions defined in (2) and (3), the Euler equations (24) of the mixed variationalprinciple (23) read1 . Evolving external state E [ ε − q − α ( θ − θ )] + ( T /θ ) H ˙ ε = σ ext . Evolving internal state − E [ ε − q − α ( θ − θ )] + ( T /θ ) H ˙ q = 03 . Thermal driving force T − θ = 04 . Evolving thermal state − ˙ η + T /θ ( H ˙ ε + H ˙ q ) = 0 . (27)Paying attention to (27) we observe that specific forms of equations (4), (5) and (7) withthe (viscous) dissipation D = H ˙ ε + H ˙ q are recovered. Note, that (6) is a necessarycondition on the given thermomechanical state and was inserted into the mixed rate-typepotential b π defined in (21). Finally, the temperature rate reads according to (25)˙ θ = θC [ ˙ η − Eα T ( ˙ ε − ˙ q ) ] (28)which by (27) − can be recast into the classical form of the temperature evolution equa-tion C ˙ θ = D − H in terms of the dissipation D and the latent heat H = Eα T θ ( ˙ ε − ˙ q ). We consider a finite time interval [ t n , t n +1 ] ⊂ T with step length τ := t n +1 − t n > t n to be known. The goal isthen to determine all state quantities at time t n +1 based on variational principles validfor the time increment under consideration. Subsequently all variables without subscriptare understood to be evaluated at time t n +1 . A mixed variational principle associated withthe finite time interval [ t n , t n +1 ] is based on the incremental potential b π τ ( c , η, θ, T ) = Algo { Z t n +1 t n b π ( ˙ c , ˙ η, ˙ θ, T ) dt } (29) ariational Thermomechanics of Gradient-Extended Continua b π defined in (21). Here, Algo stands for anintegration algorithm in time within the interval [ t n , t n +1 ]. The incremental potential b π τ is understood to be defined at given thermomechanical state ( c n , η n , θ n ) at time t n . Then,the finite-step sized incremental mixed variational principle { c , η, θ, T } = Arg { inf c sup η,θ inf T [ b π τ ( c , η, θ, T ) − σ ext ( t n +1 ) ε ] } (30)determines the mechanical state, the entropy, the temperature and the thermal drivingforce at current time t n +1 . The Algo operator is constructed such that the incrementalvariational principle (30) yields as Euler equations consistent algorithmic counterparts ofthe Euler equations (27) stemming from the rate-type variational principle (23). In whatfollows two different forms of the
Algo operator are discussed.
As a first approach, we construct an algo-rithm that performs an exact incrementation of the internal energy Z t n +1 t n ddt [ b ψ ( c , θ ) + θη ] dt = b ψ ( c , θ ) + θη − b ψ ( c n , θ n ) − θ n η n (31)and an incrementation of the dissipative term according to a backward Euler scheme Z t n +1 t n [ − T ˙ η + b φ ( Tθ ˙ c ) ] dt ≈ − T ( η − η n ) + τ b φ ( Tθ n ˙ c τ ) . (32)Here, ˙ c τ = ( c − c n ) /τ denotes an algorithmic expression of the rate of the mechanicalstate. The incremental potential (29) takes the form b π τ ( c , η, θ, T ) = b ψ ( c , θ ) − b ψ ( c n , θ n ) + θη − θ n η n − T ( η − η n ) + τ b φ ( Tθ n ˙ c τ ) . (33)Then, the incremental saddle point principle (30) gives the Euler equations1 . Update external state ∂ ε b π τ ≡ ∂ ε b ψ + τ ∂ ε b φ = σ ext . Update internal state ∂ q b π τ ≡ ∂ q b ψ + τ ∂ q b φ = 03 . Thermal driving force ∂ η b π τ ≡ T − θ = 04 . Current thermal state ∂ θ b π τ ≡ ∂ θ b ψ + η = 05 . Update entropy ∂ T b π τ ≡ − ( η − η n ) + τ ∂ T b φ = 0 . (34)The first equation governs the update of the strain ε and is a time-discrete form ofthe stress equilibrium. The second equation is a time-discrete form of Biot’s equationand determines the update of the internal variable q . The third equation identifies thecurrent thermal driving force T with the current temperature θ . The fourth equationconstitutively defines the current entropy η in terms of the current mechanical state c and the current temperature θ . Finally, the fifth equation represents a time-discrete formof the energy equation and governs the update of the entropy η . On the thermal side, thekey equations are (34) and (34) η = − ∂ θ b ψ ( c , θ ) , η = η n + τθ n ℓ b φ ( θθ n ˙ c τ ) (35) . Teichtmeister, M.-A. Keip T has been eliminated by (34) and a positively homogeneous dissipation potentialfunction b φ of degree ℓ is assumed. By inserting (35) into (35) an update equation forthe temperature θ is obtained that includes contributions from thermoelastic heating andheating due to dissipation. The last term in (35) contains the algorithmic counterpartof the dissipation (7) and (8), respectively. However, the argument of the dissipationpotential function b φ in (35) is scaled by a factor θ/θ n which is a consequence of theimplicit variational update, see Yang et al. [81]. The same scaling factor arises in thedissipative terms of the Euler equations (34) and (34) . Finally, note that the incrementalupdates of the mechanical and thermal variables are fully coupled . For the construction of a variationalupdate alternative to the implicit approach, we consider an approximate incrementation of the internal energy Z t n +1 t n ddt [ b ψ ( c , θ ) + θη ] dt ≈ b ψ ( c , θ ) − b ψ ( c n , θ n ) + ( θ − θ n ) η n + ( η − η n ) θ n (36)based on an explicit integration of the second thermal term in the integrand. By (36) andthe incrementation (32) of the dissipative term, the incremental potential (29) takes theform b π τ ( c , η, θ, T ) = b ψ ( c , θ ) − b ψ ( c n , θ n ) + ( θ − θ n ) η n + ( θ n − T )( η − η n ) + τ b φ ( Tθ n ˙ c τ ) . (37)Then, the incremental saddle point principle (30) gives the Euler equations1 . Update external state ∂ ε b π τ ≡ ∂ ε b ψ + τ ∂ ε b φ = σ ext . Update internal state ∂ q b π τ ≡ ∂ q b ψ + τ ∂ q b φ = 03 . Thermal driving force ∂ η b π τ ≡ T − θ n = 04 . Current thermal state ∂ θ b π τ ≡ ∂ θ b ψ + η n = 05 . Update entropy ∂ T b π τ ≡ − ( η − η n ) + τ ∂ T b φ = 0 . (38)Comparing this set of equations with (34), we observe essential modifications in (38) and (38) . These equations identify the current thermal driving force T with the given temperature θ n at time t n and define the current temperature θ in terms of the currentmechanical state c and the given entropy η n at time t n . The key equations (38) and (38) on the thermal side can be recast into ∂ θ b ψ ( c , θ ) = − η n , η = η n + τθ n ℓ b φ ( ˙ c τ ) (39)where T has been eliminated by (38) and again a positively homogeneous dissipationpotential function b φ of degree ℓ is assumed. One observes, that in sharp contrast tothe governing equation (35) stemming from the implicit variational approach, equation(38) does not contain the scaling factor. Hence, by the explicit variational update the algorithmically correct form ℓ b φ ( ˙ c τ ) of the dissipation (7) and (8), respectively, is obtained.Furthermore, also the dissipative terms in the stress equilibrium (38) and Biot’s equation(38) do not have the scaling factor. Finally, note that as a consequence of the temperatureupdate (39) for given entropy η n a decoupled incremental formulation is at hand. ariational Thermomechanics of Gradient-Extended Continua The two update equations (39) characterize a variational-based staggered algorithmic treatment of dissipative thermomechanics based on the po-tential b π τ defined in (37): (i) update the mechanical state c as well as the temperature θ at frozen entropy η n and given (predicted) thermal driving force T = θ n , and (ii) updatethe entropy η as well as the thermal driving force T which turns out to be identical to thepredicted one. Hence, the explicit variational update in the time interval [ t n , t n +1 ] can beseen as a composition of two fractional steps Algo = Algo η,T ◦ Algo c ,θ (40)which are both variational. This algorithm falls under the category of incrementallyisentropic operator splits , specific forms of which are considered in Armero & Simo [3] and
Miehe [44]. Within the isentropic predictor step we first solve the variationalsub-problem(
Algo c ,θ ) : { c ∗ , θ ∗ } = Arg { inf c sup θ [ b π τ ( c , η n , θ, θ n ) − σ ext ( t n +1 ) ε ] } (41)that gives the mechanical state and the temperature at time t n +1 . The resulting opti-mality conditions are the equations (38) − and (38) . Within a second step, the entropycorrector , the entropy as well as the thermal driving force at time t n +1 are obtained viathe variational sub-problem( Algo η,T ) : { η ∗ , T ∗ } = Arg { sup η inf T b π τ ( c ∗ , η, θ ∗ , T ) } (42)for given mechanical state c ∗ and temperature θ ∗ stemming from the isentropic predictor(41). The resulting optimality conditions are the equations (38) and (38) .
3. Thermomechanics of Gradient-Extended Dissipative Solids
We generalize the above outlined variational framework towards the thermomechani-cal coupling in gradient-extended dissipative continua. Let
B ⊂ R d with d ∈ { , } be thereference placement ( d -manifold) of a material body B into Euclidean space with smoothboundary ∂ B . We study the thermomechanical behavior of the body under mechanicalas well as thermal loadings in a time interval T = (0 , t e ) ⊂ R . To this end, we focus ona multiscale viewpoint in the sense that we relate dissipative effects at ( X , t ) ∈ B × T tochanges in the microstructure (besides the dissipative effect stemming from heat conduc-tion). In the phenomenological context, we account for these microstructural mechanismsby micro-motion fields that generalize the classical concept of internal variables governedby ODEs to global fields governed by PDEs. Without loss of generality we assume withinour treatment homogeneous solids only, i.e. material properties do not depend on the po-sition X ∈ B . In what follows we denote by ˙( · ) = ∂∂t ( · )( X , t ) the material time derivativeand by ∇ ( · )( X , t ) the gradient on the reference manifold B . In addition, to keep notationshort, we understand the operation a · b either as duality product between a vector anda one-form a · b = a c b c or as inner product between two vectors a · b = a c b d δ cd andtwo one-forms a · b = a c b d δ cd , respectively. The Kronecker deltas δ ab and δ ab representthe coefficients of the metric and inverse metric tensor in a Cartesian coordinate system,respectively. . Teichtmeister, M.-A. Keip Within the geometrically nonlinear theory, themacroscopic motion of the body is described by the macro-motion field ϕ : (cid:26) B × T → R d ( X , t ) x = ϕ ( X , t ) , (43)which maps at time t ∈ T points X ∈ B of the reference configuration B onto points x ∈ ϕ t ( B ) of the current configuration ϕ t ( B ). Let G = δ AB E A ⊗ E B and g = δ ab e a ⊗ e b denotethe Euclidean metric tensors associated with the reference and current configuration,where the Kronecker symbols refer to Cartesian coordinate systems on both manifolds.The deformation gradient F = D ϕ ( X , t ) , F aA = ∂ϕ a ∂X A (44)is defined as the tangent corresponding to the macro-motion (43). Next, the convectedcurrent metric C = F T gF , C AB = F aA F bB δ ab (45)is introduced, that is often denoted as the right Cauchy-Green tensor. The macro-motion(43) is constrained by a positive Jacobian J = √ det C > , J = p det[ δ AB C BD ] > ∂ B ϕ and ∂ B t such that ∂ B = ∂ B ϕ ∪ ∂ B t . We prescribe the macro-motion (Dirichlet boundarycondition) ϕ ( X , t ) = ¯ ϕ ( X , t ) on ∂ B ϕ × T (47)and the macro-tractions (Neumann boundary condition) on ∂ B t × T as specified in Sec-tion 3.2 below. Microstructural dissipative changes of the bodyare described by additional fields related to the concept of internal variables. Thesevariables are assembled in the micro-motion field of the solid q : (cid:26) B × T → R δ ( X , t ) q ( X , t ) . (48)The array q with in total δ scalar valued entries may contain internal variables of anytensorial rank that describe in a homogenized sense the micro-motion of the material dueto structural changes on lower scales. Classical examples for members of q are damagevariables, plastic strains or phase fractions. In addition, we assume all tensorial elementsof q to be Lagrangian objects , i.e. they are not affected by rigid body macro-motionssuperimposed onto the current configuration ϕ t ( B ). We distinguish between long-rangevariables q l that are governed by PDEs in form of additional balance equations and ariational Thermomechanics of Gradient-Extended Continua PSfrag replacements e P n =¯ t q · n = ¯ q H l n = n n n X ∈ B X ∈ B X ∈ B ϕϕ = ¯ ϕ q T q l = T = ¯ T macro-motion field micro-motion field thermal driving force Figure 2:
Primary fields in thermomechanics of gradient-extended dissipative solids. Attime t , the macro-motion field ϕ ( X , t ) is constrained by Dirichlet and Neumann boundaryconditions ϕ = ¯ ϕ on ∂ B ϕ and e P n = ¯ t on ∂ B t . The long-range micro-motion field q l ( X , t )is at time t restricted by the homogeneous conditions q l = on ∂ B q and H l n = on ∂ B H . Finally, for a heat conducting solid the thermal driving force field T ( X , t ) is at time t constrained by the conditions T = ¯ T on ∂ B T and q · n = ¯ q on ∂ B q . connected to given length scale parameters, and short-range variables q s that are deter-mined by ODEs and represent the standard concept of local internal variables . Note, thatlong-range variables degenerate to short-range variables if associated gradient terms aredropped in the constitutive energy and dissipation functions. We summarize both, long-as well as short-range variables in the array q = ( q s , q l ) . (49)For the treatment of each long-range micro-motion field, we decompose the boundaryof the reference configuration into nonoverlapping parts ∂ B q and ∂ B H such that ∂ B = ∂ B q ∪ ∂ B H . We prescribe the (clamped) micro-motion (Dirichlet boundary condition) q l ( X , t ) = on ∂ B q × T (50)and the micro-tractions (Neumann boundary condition) on ∂ B H × T as specified in Sec-tion 3.3.3 below. As a third primary field in the thermomechanicsof dissipative materials, we introduce the (macroscopic) thermal driving force field T : (cid:26) B × T → R + ( X , t ) T ( X , t ) . (51)As outlined in the motivating Section 2.3.2, the thermal driving force appears as thedual quantity to the entropy rate and is identified as the temperature. It is governed bythe generalized Legendre transformation (84) introduced below. With respect to thermalloading of a heat conducting solid, we decompose the boundary of the reference configu-ration into nonoverlapping parts ∂ B T and ∂ B q such that ∂ B = ∂ B T ∪ ∂ B q . We prescribethe thermal driving force (Dirichlet boundary condition) T ( X , t ) = ¯ T ( X , t ) on ∂ B T × T (52)and the heat flux (Neumann boundary condition) on ∂ B q × T as specified in Section 3.2below. The primary fields, namely the macro-motion, the long-range micro-motion andin case of heat conduction the thermal driving force are depicted in Figure 2. . Teichtmeister, M.-A. Keip PSfrag replacements c f η θ X ∈ B X ∈ B X ∈ B X ∈ B dualdualmechanical state mechanical forceentropy temperature Figure 3:
State variables in thermomechanics of gradient-extended dissipative spolids. Themechanical constitutive state c = ( C , q , ∇ q ) containing Lagrangian variables at ( X , t ) ∈ B×T is determined by the macro- and micro-motion fields ϕ ( X , t ) and q ( X , t ). The mechanicalforces f = ( m , d , g ) are dual to the state c and the intrinsic dissipation is D int = f · ˙ c . Thethermal state at ( X , t ) ∈ B × T is described by the entropy η . Its dual quantity is thetemperature θ = T that can be identified with the thermal driving force field T ( X , t ). The key global equations in thermomechanics are the local forms of (i) the balance oflinear momentum − Div e P = ¯ γ in B × T with e P n = ¯ t on ∂ B t × T (53)in terms of the (contra-variant) first Piola-Kirchhoff stress tensor e P , (ii) the balance ofangular momentum e P F T = F e P T in B × T (54)and (iii) the first law of thermodynamics , e.g. in the form of a local balance of internalenergy ˙ e = P : ˙ F − Div q + ¯ r in B × T with q · n = ¯ q on ∂ B q × T . (55)Here, P = g e P is the (mixed-variant) first Piola-Kirchhoff stress tensor and q the materialheat flux vector. The coupled thermomechanical process is driven by given externalloadings. On the mechanical side, we prescribe a body force field ¯ γ ( X , t ) and nominalsurface tractions ¯ t ( X , t ). On the thermal side, we prescribe a heat source field ¯ r ( X , t )and material surface heat fluxes ¯ q ( X , t ). Note, that the first Piola-Kirchhoff stress tensor P as well as the material heat flux vector q are determined by constitutive functions. The constitutive modeling must be consistentwith the second law of thermodynamics that demands nonnegative entropy production ariational Thermomechanics of Gradient-Extended Continua γ ≥ γ arises in the balance of entropy, which reads in itslocal form ˙ η = − Div[ q /θ ] + ¯ r/θ + γ in B × T . (56)Inserting the balance of energy (55) and resolving for the dissipation D = θγ gives theclassical Clausius-Duhem inequality D = P : ˙ F + θ ˙ η − ˙ e − q · ∇ θ/θ ≥ . (57)Demanding the intrinsic part and the heat conduction part of the dissipation to be nonneg-ative separately, we end up with the Clausius-Planck inequality and the
Fourier inequality D int = P : ˙ F − η ˙ θ − ˙ ψ ≥ D con = − q · ∇ θ/θ ≥ , (58)where the Legendre transformation e = ψ + θη was inserted. The free energy storage in continua is governed bya free energy function. We specify it by focusing on simple materials of grade one, i.e. weinclude as arguments the first gradients of the micro- and macro-motions ψ ( X , t ) = b ψ ( c , θ ) with c = ( C , q , ∇ q ) (59)defining the mechanical constitutive state. This function is invariant under any rigid bodymacro-motion ϕ + ( x , t ) = Q ( t ) x + c ( t ) superimposed on the current configuration ϕ t ( B )with proper orthogonal (rotation) tensors Q ∈ SO ( d ) and (translation) vectors c , i.e. c + = c . With a free energy function (59) at hand, we define the state variables P e = ∂ F b ψ and η = − ∂ θ b ψ (60)and the Clausius-Planck inequality reduces to the form D int = P d : ˙ F − ∂ q b ψ · ˙ q − ∂ ∇ q b ψ · ∇ ˙ q ≥ . (61)Here, P d = P − P e denotes the dissipative contribution to the first Piola-Kirchhoff stresstensor. Note, that for a given stress response function b P e ( F , q , ∇ q ) a correspondingpotential b ψ exists only if the integrability condition ∂ ( b P e ) aA ∂F bB = ∂ ( b P e ) bB ∂F aA (62)pointing out major symmetry is fulfilled. Integrating the intrinsic dissipation (61) overthe reference domain and doing integration by parts yields Z B D int dV = Z B { P d : ˙ F − δ q b ψ · ˙ q } dV − Z ∂ B H [ ∂ ∇ q b ψ · n ] · ˙ q dA (64)where we made use of the boundary condition ˙ q = on ∂ B q × T in line with (50). For the seek of a compact notation, we assume from now on the presence of long-range variables q = q l only. In case of short-range variables q s , the corresponding gradients vanish and no micro-mechanicalboundary conditions are prescribed. Variational Derivative.
Throughout this text, we use the notation of variational derivatives δ q b ψ ( q , ∇ q ) = ∂ q b ψ ( q , ∇ q ) − Div[ ∂ ∇ q b ψ ( q , ∇ q ) ] . (63) . Teichtmeister, M.-A. Keip Dissipative mechanisms are described bydissipation potential functions. First, we take mechanical effects into account via anobjective intrinsic dissipation potential function φ int ( X , t ) = b φ int ( ˙ c ; c , θ ) (65)at given thermomechanical state ( c , θ ). With such a function at hand, we define thedissipative stress P d = ∂ ˙ F b φ int (66)that arises in the reduced Clausius-Planck inequality (61). In analogy to (62), a potential b φ int corresponding to a given stress response function b P d ( ˙ F , ˙ q , ∇ ˙ q ) exists only if theintegrability condition ∂ ( b P d ) aA ∂ ˙ F bB = ∂ ( b P d ) bB ∂ ˙ F aA (67)pointing out major symmetry holds. Assuming evolution equations for the micro-motionfield together with Neumann boundary conditions δ q b ψ + δ ˙ q b φ int ∋ in B × T with [ ∂ ∇ q b ψ + ∂ ∇ ˙ q b φ int ] n ∋ on ∂ B H × T (68)as suggested in Miehe [46, 47], we can rewrite (64) in the form Z B D int dV = Z B { P d : ˙ F + δ ˙ q b φ int · ˙ q } dV + Z ∂ B H [ ∂ ∇ ˙ q b φ int · n ] · ˙ q dA . (69)Note, that the micro-force balance (68) is an outcome of the variational principle (102)set up below. From (69) we see that the intrinsic dissipation D int can alternatively berepresented in terms of the intrinsic dissipation potential and we conclude for the Clausius-Planck inequality D int = ∂ ˙ c b φ int · ˙ c ≥ . (70)Here, we made use of the identity for the dissipative stress power per unit undeformedvolume P d : ˙ F = S d : ˙ C with S d = 2 ∂ ˙ C b φ int (71)denoting the dissipative part of the second Piola-Kirchhoff stress tensor. As a secondcontribution to entropy production, we take into account heat conduction via an objectivedissipation potential φ con ( X , t ) = b φ con ( g ; c , θ ) with g = −∇ θ/θ (72)at given thermomechanical state ( c , θ ). With such a function at hand, we define thematerial heat flux vector q = ∂ g b φ con (73) ariational Thermomechanics of Gradient-Extended Continua . Latter can now be recast into the form D con = ∂ g b φ con · g ≥ . (74)The inequalities (70) and (74) serve as fundamental physically-based constraints on thedissipation potential functions b φ int and b φ con . These two conditions are satisfied a priori for dissipation potential functions that are (i) nonnegative b φ ( · ; c , θ ) ≥
0, (ii) zero in theorigin b φ ( ; c , θ ) = 0 and (iii) convex b φ ( α ˙ c + (1 − α ) ˙ c ; c , θ ) ≤ α b φ ( ˙ c ; c , θ ) + (1 − α ) b φ ( ˙ c ; c , θ ) (75)for any admissible ˙ c , ˙ c and α ∈ [0 , Fr´emond [23].
Aim is now the combination of the material-independent balance equations set up inSection 3.2 with the constitutive framework introduced in Section 3.3. Inserting (60) and (66) into (53) gives the balance of linear momentum in the form δ ϕ b ψ + δ ˙ ϕ b φ int ∋ g ¯ γ in B × T with [ ∂ F b ψ + ∂ ˙ F b φ int ] n ∋ g ¯ t on ∂ B t × T (76)which governs the evolving macro-motion. Note, that this PDE generalizes the ODE (4)for the rheological device in Figure 1 to the large-strain continuum setting. The balance ofangular momentum (54) is a priori satisfied by the objectivities of the free energy function(59) and the dissipation potential function (65). The evolving micro-motion is governedby the micro-force balance equation (68) δ q b ψ + δ ˙ q b φ int ∋ in B × T with [ ∂ ∇ q b ψ + ∂ ∇ ˙ q b φ int ] n ∋ on ∂ B H × T (77)that generalizes the ODE (5) for the rheological device to the large-strain continuumsetting. We write (76) and (77) in forms of differential inclusions in order to account fora possibly nonsmoothness of the intrinsic dissipation potential function with respect to ˙ c .On the thermal side, the entropy is defined by the local state equation (60) η = − ∂ θ b ψ in B × T (78)in terms of mechanical variables and the temperature. This local equation is identical tothat for the rheological device (6). It can also be understood as an inverse definition ofthe temperature, if the entropy as well as the mechanical variables are known. Finally, theentropy balance equation (56) governs the evolving thermal state and can be rewritten as − ˙ η + 1 θ ∂ ˙ c b φ int · ˙ c − δ θ b φ con = − ¯ rθ in B × T with 1 θ ∂ g b φ con · n = ¯ qθ on ∂ B q × T (79)when inserting the constitutive law (73) for the material heat flux vector and substitutingthe representations (70) and (74) into the entropy production γ = ( D int + D con ) /θ . Note,that this PDE extends the ODE (7) for the rheological device to the large-strain continuumsetting and includes intrinsic gradient-type dissipative effects as well as heat conduction.The above four equations provide the modeling framework for the thermomechanics ofgradient-extended dissipative solids and are related to a variational statement as shownsubsequently. . Teichtmeister, M.-A. Keip
4. Variational Principles for Thermomechanics of Gradient-Extended Solids
The variational framework for thermomechanics of gradient-extended dissipative solidsis ultimately based on the definition of energy and dissipation potential functionals interms of the constitutive functions introduced in Section 3. The evolutions of the me-chanical and thermal states are determined by rate-type variational principles.
We generalize the variational framework for the rheological model presented in Sec-tion 2 to the large-strain continuum setting of gradient-extended dissipative solids. Tothis end, based on the internal energy and dissipation potential functions e ( X , t ) = b e ( c , η ) and v ( X , t ) = b v ( ˙ c , ˙ η ; c , η, X , t ) , (80)we introduce the energy and dissipation potential functionals E ( ϕ , q , η ) = Z B b e ( c , η ) dV and V ( ˙ ϕ , ˙ q , ˙ η ; ϕ , q , η, t ) = Z B b v ( ˙ c , ˙ η ; c , η, X , t ) dV . (81) E represents the internal energy stored in the entire body B due to coupled micro-macrodeformations and thermal effects. V is related to intrinsic dissipative mechanisms andentropy production due to heat conduction. In analogy to (9), the entropy η as well asthe entropy rate ˙ η are used as canonical thermal variables. On the mechanical side, theconstitutive functions (80) are assumed to depend on the constitutive state array c definedin (59) that makes those functions a priori objective. Note, that the dissipation potentialfunction b v in general depends on the current state ( c , η ) as well as explicitly on positionand time ( X , t ) ∈ B × T stemming from a possible inhomogeneous and time-dependentthermal loading, see below. For practical modeling, we transform the above energy and dissipation potential func-tionals into functionals that depend additionally on the temperature.
First, the definition of the internal energy func-tion by the Legendre transformation b e ( c , η ) = sup θ [ b ψ ( c , θ ) + θη ] (82)in terms of the free energy function b ψ motivates the introduction of a mixed internalenergy functional E + ( ϕ , q , η, θ ) = Z B b e + ( c , η, θ ) dV with b e + ( c , η, θ ) = b ψ ( c , θ ) + θη . (83)The necessary optimality condition of (82) is the statement (78) which identifies thethermal state variable θ as the dual quantity to the entropy η , i.e. as the temperature. ariational Thermomechanics of Gradient-Extended Continua In a second step, we define at time t thedissipation potential functional by a generalized Legendre transformation V ( ˙ ϕ , ˙ q , ˙ η ; ϕ , q , θ, t ) = sup T [ V ∗ ( ˙ ϕ , ˙ q , T ; ϕ , q , θ ) − Z ( T, ˙ η ; θ, t ) ] (84)in terms of the “dual” dissipation potential functional V ∗ ( ˙ ϕ , ˙ q , T ; ϕ , q , θ ) = Z B e φ ( ˙ c , T, ∇ T ; c , θ ) dV (85)governed by a dissipation potential function e φ and the mixed functional Z ( T, ˙ η ; θ, t ) = Z B T ˙ η dV + Z B b b T ( T ; θ, X , t ) dV + Z ∂ B q b s T ( T ; θ, X , t ) dA , (86)where the second and third integrals represent thermal body and surface loadings, respec-tively. Note, that by use of the local state equation (78) the dissipation potential function e φ depends on the current state ( c , θ ). The maximization in (84) at time t is performedunder the constraint T = ¯ T on ∂ B T and defines as necessary condition the evolution ofthe entropy along with a thermal boundary condition˙ η = δ T e φ − ∂ T b b T in B and ∂ ∇ T e φ · n = ∂ T b s T on ∂ B q . (87)These equations generalize the local statement (18) for the rheological device to a large-strain continuum setting including intrinsic gradient-type dissipative effects as well asheat conduction. As in Section 2.3.2, we call T the thermal driving force that is dual tothe entropy rate ˙ η . The entropy evolution (87) must have the form (79) and we identify δ T e φ ! = 1 θ ∂ ˙ c b φ int · ˙ c − δ θ b φ con and ∂ T b b T ! = − ¯ rθ . (88)The first of these conditions is fulfilled for T = θ in B if the dissipation potential functionis specified to e φ ( ˙ c , T, ∇ T ; c , θ ) = b φ int ( Tθ ˙ c ; c , θ ) − b φ con ( − T ∇ T ; c , θ ) , (89)which is in line with Yang et al. [81]. The second condition (88) is satisfied for avolumetric thermal loading function b b T ( T ; θ, X , t ) = − Tθ ¯ r ( X , t ) . (90)It remains to find an expression for the thermal surface load function b s T . Note, that − ∂ ∇ T b φ con = 1 /θ ∂ g b φ con for T = θ in B and we identify from (79) ∂ T b s T ! = ¯ qθ , (91)which is fulfilled for a thermal surface load function b s T ( T ; θ, X , t ) = Tθ ¯ q ( X , t ) . (92) . Teichtmeister, M.-A. Keip V via the generalized Legendre trans-formation (84) now motivates the introduction of a mixed dissipation potential functional V + ( ˙ ϕ , ˙ q , ˙ η, T ; ϕ , q , θ ) = Z B v + ( ˙ c , ˙ η, T, ∇ T ; c , θ ) dV (93)in terms of the mixed dissipation potential function v + ( ˙ c , ˙ η, T, ∇ T ; c , θ ) = e φ ( ˙ c , T, ∇ T ; c , θ ) − T ˙ η (94)governed by the specific form (89). Beside the mixed energy and dissipation potential func-tionals (83) and (93) that govern the internal mechanisms of energy storage and entropyproduction, respectively, we have an external thermomechanical load functional P ext ( ˙ ϕ , T ; θ, t ) = P ϕ ext ( ˙ ϕ ; t ) + P Text ( T ; θ, t ) (95)with decoupled mechanical and thermal contributions. On the mechanical side, we definethe load functional P ϕ ext ( ˙ ϕ ; t ) = Z B ¯ γ ( X , t ) · ˙ ϕ dV + Z ∂ B t ¯ t ( X , t ) · ˙ ϕ dA (96)in terms of a given body force field ¯ γ and nominal surface traction field ¯ t . The thermalload functional as a part of (86) attains with the identifications (90) and (92) the form P Text ( T ; θ, t ) = − Z B Tθ ¯ r ( X , t ) dV + Z ∂ B q Tθ ¯ q ( X , t ) dA (97)in terms of a given heat source field ¯ r and material surface heat flux ¯ q . Based on the internal energy and dissipation po-tential functionals E + and D + and the thermomechanical load functional P ext , we defineat current time t the rate-type potential Π + ( ˙ ϕ , ˙ q , ˙ η, ˙ θ, T ) = ddt E + ( ϕ , q , η, θ ) + V + ( ˙ ϕ , ˙ q , ˙ η, T ) − P ext ( ˙ ϕ , T ) (98)with given state ( ϕ , q , η, θ ). We write this potential with its internal and external contri-butions Π + ( ˙ ϕ , ˙ q , ˙ η, ˙ θ, T ) = Z B b π + ( ˙ c , ˙ η, ˙ θ, T, ∇ T ) dV − P ext ( ˙ ϕ , T ) (99)in terms of the internal potential density b π + ( ˙ c , ˙ η, ˙ θ, T, ∇ T ) = ddt b ψ ( c , θ ) + ( θ − T ) ˙ η + η ˙ θ + b φ int ( Tθ ˙ c ) − b φ con ( − T ∇ T ) . (100) To keep notation short, we subsequently do not write explicitly the dependence of functions andcorresponding functionals on given quantities. ariational Thermomechanics of Gradient-Extended Continua b π + red ( ˙ c , ˙ η, T, ∇ T ) = ∂ c b ψ · ˙ c + ( θ − T ) ˙ η + b φ int ( Tθ ˙ c ) − b φ con ( − T ∇ T ) . (101)Then, the rates of the macro- and the micro-motion as well as the rate of the entropy andthe thermal driving force at current time t are governed by the variational principle { ˙ ϕ , ˙ q , ˙ η, T } = Arg { inf ˙ ϕ , ˙ q , ˙ η sup T [ Z B b π + red ( ˙ c , ˙ η, T, ∇ T ) dV − P ext ( ˙ ϕ , T ) ] } . (102)Here, one has to account for the rate forms of the Dirichlet boundary conditions (47) and(50) for the macro- and micro-motions, i.e.˙ ϕ = ˙¯ ϕ on ∂ B ϕ and ˙ q = on ∂ B q (103)as well as for the Dirichlet boundary condition (52) for the thermal driving force. By thefirst variation of the functional (99) we have the necessary optimality conditions δ ˙ ϕ Π + + δ ˙ q Π + + δ ˙ η Π + ≥ , δ T Π + ≤ δ ˙ η and ( δ ˙ ϕ , δ ˙ q , δT ) fulfilling homogeneous forms of theDirichlet boundary conditions. We get the Euler equations1 . Evolving macro-motion δ ˙ ϕ b π + red ≡ δ ϕ b ψ + δ ˙ ϕ b φ int ∋ g ¯ γ . Evolving micro-motion δ ˙ q b π + red ≡ δ q b ψ + δ ˙ q b φ int ∋ . Thermal driving force ∂ ˙ η b π + red ≡ T − θ = 04 . Evolving thermal state δ T b π + red ≡ − ˙ η + ∂ T b φ int − δ T b φ con = − ¯ r/θ (105)in B along with the Neumann boundary conditions[ ∂ F b ψ + Tθ ∂ Tθ ˙ F b φ int ] n ∋ g ¯ t , [ ∂ ∇ q b ψ + Tθ ∂ Tθ ∇ ˙ q b φ int ] n ∋ , − ∂ ∇ T b φ con · n = ¯ q/θ (106)on ∂ B t , ∂ B H and ∂ B q , respectively. In contrast to (24), the equations are now exclusivelygoverned by variational derivatives of the reduced potential density b π + red defined in (101).The central three field equations are the quasi-static mechanical equilibrium δ ˙ ϕ b π + red (cid:12)(cid:12) T = θ ≡ − Div[ ∂ F b ψ + ∂ ˙ F b φ int ( ˙ c )] ∋ g ¯ γ (107)that governs the rate ˙ ϕ of the macro-motion, the micro-force balance δ ˙ q b π + red (cid:12)(cid:12) T = θ ≡ [ ∂ q b ψ + ∂ ˙ q b φ int ( ˙ c ) ] − Div[ ∂ ∇ q b ψ + ∂ ∇ ˙ q b φ int ( ˙ c )] ∋ (108)determining the rate ˙ q of the micro-motion and the energy equation δ T b π + red (cid:12)(cid:12) T = θ ≡ − ˙ η + 1 θ ∂ ˙ c b φ int ( ˙ c ) · ˙ c − θ Div[ ∂ g b φ con ( g )] = − ¯ rθ (109)for the evolution ˙ η of the entropy. Note, that the standard concept of local internalvariables is recovered for vanishing divergence terms in (108), i.e. when the constitutivefunctions (80) do not depend on the gradients of q and ˙ q , respectively. . Teichtmeister, M.-A. Keip Consider a finite time interval [ t n , t n +1 ] ⊂ T withstep length τ = t n +1 − t n > t n to be known. The goal is then to determine all fields at time t n +1 based on variationalprinciples valid for the time increment under consideration. Subsequently all variableswithout subscript are understood to be evaluated at time t n +1 . We may formulate theincremental potentialΠ + τ ( ϕ , q , η, θ, T ) = E + τ ( ϕ , q , η, θ ) + D + τ ( ϕ , q , η, T ) − P τext ( ϕ , T ) (110)where E + τ , D + τ and P τext are incremental energy, dissipation and load functionals as-sociated with the time interval [ t n , t n +1 ]. These functionals are defined at given state( ϕ n , q n , η n , θ n ) at time t n . In analogy to the rate-type formulation (99), we rewrite theincremental potentialΠ + τ ( ϕ , q , η, θ, T ) = Z B b π + τ ( c , η, θ, T, ∇ T ) dV − P τext ( ϕ , T ) (111)in terms of an incremental internal potential density b π + τ which is defined at given state( c n , η n , θ n ). Such a function is obtained by an integration algorithm b π + τ ( c , η, θ, T, ∇ T ) = Algo { Z t n +1 t n b π + ( ˙ c , ˙ η, ˙ θ, T, ∇ T ) dt } (112)that has to be constructed in such a way that the subsequent incremental variationalprinciple gives consistent algorithmic counterparts of the Euler equations (105). In whatfollows, we construct an implicit as well as a semi-explicit integration algorithm, compareSection 2.4. As a typical example we consider an integration using the approximations ofthe rates of state quantities˙ c τ = ( c − c n ) /τ , ˙ η τ = ( η − η n ) /τ , ˙ θ τ = ( θ − θ n ) /τ (113)and the incremental internal potential density b π + τ = b ψ ( c , θ ) + τ [ ( θ k − T ) ˙ η τ + η n ˙ θ τ + b φ int ( Tθ n ˙ c τ ; c n , θ n ) − b φ con ( − T ∇ T ; c n , θ n ) ] (114)with k = n + 1 for an implicit integration algorithm according to (31) and k = n for asemi-explicit integration algorithm according to (36). In (114) we dropped terms that areassociated with previous time t n . Then, defining the incremental load functional P τext ( ϕ , T ; ϕ n , θ n , t n +1 ) = P ϕ ext ( ϕ − ϕ n ; t n +1 ) + τ P Text ( T ; θ n , t n +1 ) (115)the incremental variational principle { ϕ , q , η, θ, T } = Arg { inf ϕ , q ,η sup θ,T Π + τ ( ϕ , q , η, θ, T ) } (116)determines all thermomechanical fields at time t n +1 . Note, that the optimization has tobe done considering the Dirichlet boundary conditions (47), (50) and (52) at time t n +1 . ariational Thermomechanics of Gradient-Extended Continua . Update macro-motion δ ϕ b π + τ ≡ δ ϕ b ψ + τ δ ϕ b φ int ∋ g ¯ γ . Update micro-motion δ q b π + τ ≡ δ q b ψ + τ δ q b φ int ∋ . Thermal driving force ∂ η b π + τ ≡ T − θ k = 04 . Current temperature ∂ θ b π + τ ≡ ∂ θ b ψ + η k = 05 . Update entropy δ T b π + τ ≡ − ( η − η n ) + τ ∂ T b φ int − τ δ T b φ con = − τ ¯ r/θ n (117)in B along with the Neumann boundary conditions[ ∂ F b ψ + Tθ n ∂ Tθn ˙ F τ b φ int ] n ∋ g ¯ t , [ ∂ ∇ q b ψ + Tθ n ∂ Tθn ∇ ˙ q τ b φ int ] n ∋ , − ∂ ∇ T b φ con · n = ¯ q/θ n (118)on ∂ B t , ∂ B H and ∂ B q , respectively. These equations are the time-discrete forms of (78)and (105)–(106). As a fundamental difference to the fully implicit algorithm, a semi-explicit update identifies the thermal driving force with the given temperature at time t n .Hence, the scaling factor results in T /θ n = 1 in B such that the algorithmically correctform of the intrinsic dissipation defined in (61) and reformulated in (70) is obtained.Especially, the incremental energy equation reads η = η n + τθ n ∂ ˙ c τ b φ int ( ˙ c τ ) · ˙ c τ + τθ n (¯ r − Div[ ∂ − T ∇ T b φ con ] (cid:12)(cid:12) T = θ n ) . (119)Additionally, also the dissipative terms in the quasi-static equilibrium (117) , micro-forcebalance (117) and the boundary conditions (118) − do not contain the scaling factor.However, there are two issues that arise when using the semi-explicit update for a heatconduction process : (i) on the thermal side it might be restricted to homogeneous Neu-mann boundary conditions (118) on the whole boundary, i.e. ¯ q = 0 on ∂ B and (ii) thetime step size τ is restricted by a CFL condition. Note, that the semi-explicit update canbe seen as an incrementally isentropic operator split that consists of two fractional steps Algo = Algo η,T ◦ Algo ϕ , q ,θ . (120)First, in the isentropic predictor step we optimize the incremental potential (111) withrespect to the macro- and micro-motions ϕ and q as well as the temperature θ , i.e.( Algo ϕ , q ,θ ) : { ϕ ∗ , q ∗ , θ ∗ } = Arg { stat ϕ , q ,θ Π + τ ( ϕ , q , η n , θ, θ n ) } , (121)where the entropy is frozen. The resulting optimality conditions in B are the equations(117) − and (117) with k = n . Then, the entropy η and the thermal driving force T areupdated via the entropy corrector step ( Algo η,T ) : { η ∗ , T ∗ } = Arg { stat η,T Π + τ ( ϕ ∗ , q ∗ , η, θ ∗ , T ) } . (122)The resulting optimality conditions in B are the equations (117) and (117) with k = n . . Teichtmeister, M.-A. Keip We consider the equivalent representation of the intrinsic dissipation (70) D int = ∂ ˙ c b φ int ( ˙ c ; c , θ ) · ˙ c ⇐⇒ D int = f · ∂ f b φ ∗ int ( f ; c , θ ) (123)by the dual mechanical dissipation potential function b φ ∗ int depending on the mechanicaldriving forces f = ( m , d , g ) conjugate to c = ( C , q , ∇ q ) . (124)Taking into account the representation (89), the Legendre transformation b φ int ( ˙ c ; c , θ ) = sup f [ f · ˙ c − b φ ∗ int ( f ; c , θ ) ] (125)motivates the definition of an extended dissipation potential functional D ∗ ( ˙ ϕ , ˙ q , ˙ η, T, f ; ϕ , q , θ ) = Z B d ∗ ( ˙ c , ˙ η, T, ∇ T, f ; c , θ ) dV (126)in terms of the extended dissipation potential function d ∗ ( ˙ c , ˙ η, T, ∇ T, f ; c , θ ) = Tθ f · ˙ c − T ˙ η − b φ ∗ int ( f ; c , θ ) − b φ con ( − T ∇ T ; c , θ ) (127)which governs the subsequent extended mixed variational principle. The necessary con-dition of (125) ˙ c ∈ ∂ f b φ ∗ int ( f ; c , θ ) (128)can be understood as an inverse definition of the driving forces f in terms of the rate ˙ c of the constitutive state. Note, that for generality a nonsmooth dual intrinsic dissipationpotential function was assumed. Perzyna-type dual dissipation potential function.
An important example of a smooth intrinsicdual dissipation potential function is b φ ∗ int ( f ; c , θ ) = 12 η f h b f ( f ; c , θ ) i (129)in terms of a function b f that differs from a gauge just by a constant and serves as a threshold functionin the (adiabatically) rate-independent setting considered in Section 4.5 below. η f > h·i + : R → R + , x ( | x | + x ) the ramp function. b φ ∗ int ( · ; c , θ ) defined in (129) is ahomogeneous function of degree two and accordingly the dissipation potential function b φ int ( · ; c , θ ) aswell. Hence, (129) is related to (rate-dependent) viscous behavior, see Perzyna [65] on the treatment ofvisco-plasticity. For the particular choice (129), the evolution equations (128) take the specific form˙ c = λ ∂ f b f with λ = 1 η f h b f i + (130)which regularizes the rate-independent structure (150)–(151) to be discussed later in Section 4.5. ariational Thermomechanics of Gradient-Extended Continua Based on the internal energy and dissipation po-tential functionals E + in (83) and D ∗ in (127) we are in the position to formulate a mixedrate-type variational principle that accounts for the mechanical driving forces f . We defineat current time t the rate-type potential Π ∗ ( ˙ ϕ , ˙ q , ˙ η, ˙ θ, T, f ) = ddt E + ( ϕ , q , η, θ ) + D ∗ ( ˙ ϕ , ˙ q , ˙ η, T, f ) − P ext ( ˙ ϕ , T ) (131)with given state ( ϕ , q , η, θ ). We write this potential with its internal and external contri-butions Π ∗ ( ˙ ϕ , ˙ q , ˙ η, ˙ θ, T, f ) = Z B b π ∗ ( ˙ c , ˙ η, ˙ θ, T, ∇ T, f ) dV − P ext ( ˙ ϕ , T ) (132)in terms of the extended internal potential density b π ∗ ( ˙ c , ˙ η, ˙ θ, T, ∇ T, f ) = ddt b ψ ( c , θ ) + ( θ − T ) ˙ η + η ˙ θ + Tθ f · ˙ c − b φ ∗ int ( f ) − b φ con ( − T ∇ T ) . (133)Inserting the necessary condition (78) on the given thermomechanical state yields thereduced extended internal potential density b π ∗ red ( ˙ c , ˙ η, T, ∇ T, f ) = ∂ c b ψ · ˙ c + ( θ − T ) ˙ η + Tθ f · ˙ c − b φ ∗ int ( f ) − b φ con ( − T ∇ T ) . (134)Then, the rates of the macro- and micro-motion as well as the rate of the entropy and thethermal and mechanical driving forces at current time t are governed by the variationalprinciple { ˙ ϕ , ˙ q , ˙ η, T, f } = Arg { inf ˙ ϕ , ˙ q , ˙ η sup T, f [ Z B b π ∗ red ( ˙ c , ˙ η, T, ∇ T, f ) dV − P ext ( ˙ ϕ , T ) ] } . (135)Like in (101) one has to account for the Dirichlet boundary conditions (103) and (52).By the first variation of the functional (132) we have the necessary optimality conditions δ ˙ ϕ Π ∗ + δ ˙ q Π ∗ + δ ˙ η Π ∗ ≥ , δ T Π ∗ + δ f Π ∗ ≤ δ ˙ η, δ f ) and ( δ ˙ ϕ , δ ˙ q , δT ) fulfilling homogeneous forms ofthe Dirichlet boundary conditions. We obtain the Euler equations1 . Evolving macro-motion δ ˙ ϕ b π ∗ red ≡ δ ϕ b ψ + δ ˙ ϕ ( Tθ f · ˙ c ) = g ¯ γ . Evolving micro-motion δ ˙ q b π ∗ red ≡ δ q b ψ + δ ˙ q ( Tθ f · ˙ c ) = . Thermal driving force ∂ ˙ η b π ∗ red ≡ T − θ = 04 . Evolving thermal state δ T b π ∗ red ≡ − ˙ η + ∂ T ( Tθ f · ˙ c ) − δ T b φ con = − ¯ r/θ . Mechanical driving forces ∂ f b π ∗ red ≡ Tθ ˙ c − ∂ f b φ ∗ int ∋ (137) To keep notation short, we subsequently do not write explicitly the dependence of functions andcorresponding functionals on given states. . Teichtmeister, M.-A. Keip B along with the Neumann boundary conditions[ ∂ F b ψ + 2 gF Tθ m ] n = g ¯ t , [ ∂ ∇ q b ψ + Tθ g ] n = , − ∂ ∇ T b φ con · n = ¯ q/θ (138)on ∂ B t , ∂ B H and ∂ B q , respectively. The central three field equations are the quasi-staticequilibrium δ ˙ ϕ b π ∗ red (cid:12)(cid:12) T = θ ≡ − Div[ ∂ F b ψ + 2 gF m ] = g ¯ γ , (139)the micro-force balance δ ˙ q b π ∗ red (cid:12)(cid:12) T = θ ≡ [ ∂ q b ψ + d ] − Div[ ∂ ∇ q b ψ + g ] = (140)and the energy equation δ T b π ∗ red (cid:12)(cid:12) T = θ ≡ − ˙ η + 1 θ f · ˙ c − θ Div[ ∂ g b φ con ( g ) ] = − ¯ r/θ . (141)They are complemented by the inverse definitions (137) of the mechanical driving forceswhich split into three evolution equations˙ C ∈ ∂ m b φ ∗ int , ˙ q ∈ ∂ d b φ ∗ int , ∇ ˙ q ∈ ∂ g b φ ∗ int , (142)where the identity T = θ in B from (137) has been used. Within a time interval [ t n , t n +1 ] ⊂ T a varia-tional principle can be constructed by the same avenue as outlined in Section 4.3.2. Itis based on a variationally consistent time integration algorithm of the extended internalpotential density (134) b π ∗ τ ( c , η, θ, T, ∇ T, f ) = Algo { Z t n +1 t n b π ∗ ( ˙ c , ˙ η, ˙ θ, T, ∇ T, f ) dt } (143)in analogy to (112). Using the algorithmic approximations (113) of rates of state quanti-ties, we get the incremental internal potential density b π ∗ τ = b ψ ( c , θ ) + τ [ ( θ k − T ) ˙ η τ + η n ˙ θ τ + Tθ n f · ˙ c τ − b φ ∗ int ( f ; c n , θ n ) − b φ con ( − T ∇ T ; c n , θ n ) ](144)where again k = n + 1 corresponds to a fully implicit and k = n to a semi-explicit update.In addition, we dropped terms that are associated with previous time t n . Then, with theuse of the incremental load functional (115) the incremental variational principle { ϕ , q , η, θ, T, f } = Arg { inf ϕ , q ,η sup θ,T, f [ Z B b π ∗ τ ( c , η, θ, T, ∇ T, f ) dV − P τext ( ϕ , T ) ] } (145)determines all thermomechanical fields at time t n +1 . The corresponding Euler equationsin B are time-discrete forms of (137) together with (117) stemming from the variationwith respect to the temperature θ . As before, the semi-explicit integration of the rate of ariational Thermomechanics of Gradient-Extended Continua T /θ n = 1 in B and one obtains the algo-rithmically correct form of the intrinsic dissipation, i.e. the incremental energy equationreads η = η n + τθ n f · ˙ c τ + τθ n (¯ r − Div[ ∂ − T ∇ T b φ con ] | T = θ n ) . (146)In addition, the dissipative terms in the time-discrete forms of the quasi-static equilibrium(137) , the micro-force balance (137) and the boundary conditions (138) − as well asthe dissipative terms in the time discrete forms of the evolution equations (137) do notcontain the scaling factor. The incrementally isentropic operator split is modified by anadditional optimization in the isentropic predictor step with respect to the mechanicaldriving forces f . Intrinsic dissipation potential functions are often modelled by the principle of maxi-mum dissipation . For the classical local theories of plasticity, this principle can be tracedback among others to the work of
Hill [32], see also
Moreau [55],
Simo [70] and
Lubliner [39]. For a general discussion of this principle and its connection to evolutionlaws governed by dissipation potentials we refer to
Hackl & Fischer [28] and
Hacklet al. [29, 30]. The intrinsic dissipation potential function is defined by the constrainedmaximum principle b φ int ( ˙ c ; c , θ ) = sup f ∈ E ( c ,θ ) f · ˙ c (147)that includes at given thermomechanical state ( c , θ ) a set of admissible mechanical drivingforces E ( c , θ ) = { f | b f ( f ; c , θ ) ≤ } . (148)Clearly, the function b φ int ( · ; c , θ ) defined in (147) is positively homogeneous of degree one.The set (148) is governed by a threshold function b f ( f ; c , θ ) = b w ( f ; c , θ ) − b c ( c , θ ) where b c ( c , θ ) > b w a level set function that is a gauge , i.e. (i) nonnegative b w ( · ; c , θ ) ≥ b w ( ; c , θ ) = 0, (iii) convex in f and (iv) positively homogeneous ofdegree one in f . By the use of the Lagrange multiplier method, we can put the constrainedoptimization problem (147) into the form b φ int ( ˙ c ; c , θ ) = sup f inf λ ≥ [ f · ˙ c − λ b f ( f ; c , θ ) ] , (149)the necessary optimality condition of which defines the evolution of the constitutive state ˙ c = λ ∂ f b f , (150)where the Lagrange multiplier λ satisfies classical loading-unloading conditions in Kuhn-Tucker form λ ≥ , b f ( f ; c , θ ) ≤ , λ b f ( f ; c , θ ) = 0 . (151) Throughout the text we assume the threshold function b f to be smooth. . Teichtmeister, M.-A. Keip λ cannot bedetermined by a usual local consistency condition for given state of stress, deformationand temperature rates at ( X , t ) ∈ B × T . The optimization problem (149) now motivatesthe definition of a modified dissipation potential functional D ∗ λ ( ˙ ϕ , ˙ q , ˙ η, T, f , λ ; ϕ , q , θ ) = Z B d ∗ λ ( ˙ c , ˙ η, T, ∇ T, f , λ ; c , θ ) dV (152)in terms of the modified dissipation potential function d ∗ λ ( ˙ c , ˙ η, T, ∇ T, f , λ ; c , θ ) = Tθ f · ˙ c − T ˙ η − λ b f ( f ; c , θ ) − b φ con ( − T ∇ T ; c , θ ) (153)that governs the subsequent modified mixed variational principle. Based on the internal energy and dissipation po-tential functionals E + in (83) and D ∗ λ in (152) we are in the position to formulate a mixedrate-type variational principle that accounts for dissipative threshold mechanisms. Wedefine at current time t the rate-type potentialΠ ∗ λ ( ˙ ϕ , ˙ q , ˙ η, ˙ θ, T, f , λ ) = ddt E + ( ϕ , q , η, θ ) + D ∗ λ ( ˙ ϕ , ˙ q , ˙ η, T, f , λ ) − P ext ( ˙ ϕ , T ) (154)with given state ( ϕ , q , η, θ ). We write this potential with its internal and external contri-butions Π ∗ λ ( ˙ ϕ , ˙ q , ˙ η, ˙ θ, T, f , λ ) = Z B b π ∗ λ ( ˙ c , ˙ η, ˙ θ, T, ∇ T, f , λ ) dV − P ext ( ˙ ϕ , T ) (155)in terms of the extended internal potential density b π ∗ λ ( ˙ c , ˙ η, ˙ θ, T, ∇ T, f , λ ) = ddt b ψ ( c , θ ) + ( θ − T ) ˙ η + η ˙ θ + Tθ f · ˙ c − λ b f ( f ) − b φ con ( − T ∇ T ) . (156)Again, inserting the necessary condition (78) on the given thermomechanical state yieldsthe reduced extended internal potential density b π ∗ λ,red ( ˙ c , ˙ η, T, ∇ T, f , λ ) = ∂ c b ψ · ˙ c + ( θ − T ) ˙ η + Tθ f · ˙ c − λ b f ( f ) − b φ con ( − T ∇ T ) . (157)Then, the rates of the macro- and micro-motion as well as the rate of the entropy, thethermal and mechanical driving forces and the Lagrange multiplier at current time t aregoverned by the variational principle { ˙ ϕ , ˙ q , ˙ η, T, f , λ } = Arg { inf ˙ ϕ , ˙ q , ˙ η sup T, f inf λ ≥ [ Z B b π ∗ λ,red ( ˙ c , ˙ η, T, ∇ T, f , λ ) dV − P ext ( ˙ ϕ , T ) ] } . (158)Like in (102) one has to account for the Dirichlet boundary conditions (103) and (52).By the first variation of the functional (155) we have the necessary optimality conditions δ ˙ ϕ Π ∗ λ + δ ˙ q Π ∗ λ + δ ˙ η Π ∗ λ ≥ , δ T Π ∗ λ + δ f Π ∗ λ ≤ , δ λ Π ∗ λ ≥ ariational Thermomechanics of Gradient-Extended Continua δ ˙ η, δ f , δλ ) with λ + δλ ≥ B and ( δ ˙ ϕ , δ ˙ q , δT ) fulfillinghomogeneous forms of Dirichlet boundary conditions. We obtain the Euler equations1 . Evolving macro-motion δ ˙ ϕ b π ∗ λ,red ≡ δ ϕ b ψ + δ ˙ ϕ ( Tθ f · ˙ c ) = g ¯ γ . Evolving micro-motion δ ˙ q b π ∗ λ,red ≡ δ q b ψ + δ ˙ q ( Tθ f · ˙ c ) = . Thermal driving force ∂ ˙ η b π ∗ λ,red ≡ T − θ = 04 . Evolving thermal state δ T b π ∗ λ,red ≡ − ˙ η + ∂ T ( Tθ f · ˙ c ) − δ T b φ con = − ¯ r/θ . Mechanical driving forces ∂ f b π ∗ λ,red ≡ Tθ ˙ c − λ ∂ f b f = . Loading Conditions ∂ λ b π ∗ λ,red ≡ − b f ≥ , λ ≥ , λ b f = 0 (160)in B along with the Neumann boundary conditions (138). Note, that the condition b f ≤ follows from (159) if we set λ = 0, which necessarily demands δλ ≥
0. On theother hand, by choosing λ > δλ can have any sign and we obtain from(159) the equality b f = 0, or in summary λ b f = 0 as given in (160) . The central threefield equations are identical to (139)–(141) and are complemented by the set of evolutionequations (160) which read˙ C = λ ∂ m b f , ˙ q = λ ∂ d b f , ∇ ˙ q = λ ∂ g b f , (161)where the identity T = θ in B from (160) has been used. In addition, we have the nonlocal consistency conditions λ ≥ , ddt b f ( f ; c , θ ) ≤ , λ ddt b f ( f ; c , θ ) = 0 (162)in B which at current time t are supplemented by rate forms of the equations (78) and(160) − and the rate forms of the Neumann boundary conditions (138) related to themacro- and micro-motion, respectively. To see conditions (162), consider at current time t a nonzero evolution of the mechanical constitutive state, i.e. λ t >
0. Then, the first vari-ation of the dissipation potential functional (152) with respect to the Lagrange mulitpliervanishes δ λ D ∗ λ | t = Z B − δλ b f | t dV = 0 (163)for all δλ . Next, at time t + τ , τ ≥ λ t + τ ≥ δ λ D ∗ λ | t + τ = Z B − δλ b f | t + τ dV ≥ λ t + τ + δλ ≥
0. Subtracting (163) from (164) and dividing by τ yields, see Foot-note 10, 1 τ [ δ λ D ∗ λ | t + τ − δ λ D ∗ λ | t ] = Z B [ − δλ ( ddt b f + O ( τ ) τ ) ] dV ≥ λ t + τ + δλ ≥
0. For τ → λ t + τ → λ t > O ( τ ) /τ → ddt b f = 0since δλ can have any sign. For τ small enough, we assume λ t + τ = 0 and δλ must benonnegative yielding ddt b f ≤
0. When summarizing, we arrive at the nonlocal consistencyconditions (162). . Teichtmeister, M.-A. Keip Within a time interval [ t n , t n +1 ] ⊂ T a varia-tional principle can be constructed by the same avenue as outlined in Section 4.3.2. Itis based on a variationally consistent time integration algorithm of the extended internalpotential density (157) b π ∗ τλ ( c , η, θ, T, ∇ T, f , λ ) = Algo { Z t n +1 t n b π ∗ λ ( ˙ c , ˙ η, ˙ θ, T, ∇ T, λ ) dt } (166)in analogy to (112). Using the algorithmic approximations (113) of rates of state quanti-ties, we get the incremental internal potential density b π ∗ τλ = b ψ ( c , θ ) + τ [ ( θ k − T ) ˙ η τ + η n ˙ θ τ + Tθ n f · ˙ c τ − λ b f ( f ; c n , θ n ) − b φ con ( − T ∇ T ; c n , θ n ) ](167)where like before k = n + 1 corresponds to a fully implicit and k = n to a semi-explicitupdate. Again, terms that are associated with previous time t n are dropped. Then, withthe use of the incremental load functional (115) the incremental variational principle { ϕ , q , η, θ, T, f , λ } = Arg { inf ϕ , q ,η sup θ,T, f inf λ ≥ [ Z B b π ∗ τλ ( c , η, θ, T, ∇ T, f , λ ) dV − P τext ( ϕ , T ) ] } (168)determines all fields at time t n +1 . The corresponding Euler equations in B are time-discrete forms of (160) together with (117) stemming from the variation with respect tothe temperature θ . Considering the semi-explicit integration, the incrementally isentropicoperator split is modified by an additional optimization in the isentropic predictor stepwith respect to the Lagrange multiplier λ ≥
5. Representative Model Problems
We consider as a first model problem the Cahn-Hilliard theory of diffusive phase separa-tion in a rigid solid coupled with temperature evolution. In the following c : B ×T → [0 , c = − Div H (169)where H : B × T → R d is the species flux vector field. Note, that we neglect the phe-nomenon of thermal diffusion (Soret effect) that is species flow caused by temperaturegradient, see De Groot & Mazur [14]. We specify the constitutive state related todiffusion as c = ( c, ∇ c ) (170)that contains beside the concentration also its first gradient. The free energy functiondecomposes into a local, nonlocal and purely thermal contribution b ψ ( c , θ ) = b ψ l ( c ) + b ψ ∇ ( ∇ c ) + b ψ θ ( θ ) , (171) ariational Thermomechanics of Gradient-Extended Continua Cahn & Hilliard [8] we choose b ψ l ( c ) = A [ c ln c + (1 − c ) ln(1 − c )] + Bc (1 − c ) and b ψ ∇ ( ∇ c ) = D |∇ c | (172)in terms of the threshold and mixing energy parameters A and B and the diffusive interfaceparameter D . Note the nonconvexity of b ψ l for B > A which is related to phase seperation.The evolution of the concentration is driven by the chemical potential µ given by µ = δ c b ψ = A ln c − c + B (1 − c ) − D ∆ c . (173)It can be understood as a constitutive representation of a micro-force balance in the senseof Gurtin [24].
Point of depar-ture is the definition of the energy functional E ( c ) = Z B b ψ ( c ) dV (174)that characterizes the energy stored in the entire body. Its rate at current concentrationfield c can alternatively be considered as a functional of the species flux H if the rate ˙ c of the concentration field is expressed by the balance equation (169) yielding ddt E ( c ) = E ( H ; c ) = − Z B δ c b ψ Div H dV − Z ∂ B ( ∂ ∇ c b ψ · n ) Div H dA (175)in line with Miehe et al. [52]. Next, we define at current concentration field c thecanonical dissipation potential functional that depends on the species flux H as D ( H ; c ) = Z B b φ ( H ; c ) dV (176)in terms of a dissipation potential function b φ accounting for dissipative diffusion mecha-nisms. It has the simple quadratic form b φ ( H ; c ) = 12 1 M c (1 − c ) H · H (177)where M > b φ ( · ; c ) is a positively ho-mogeneous function of degree two and its image coincides with half the dissipation in amaterial element. Finally, we have the external load functional P ext ( H ; t ) = − Z ∂ B µ ¯ µ ( X , t ) H · n dA , (178)where ¯ µ is a prescribed fluid potential on the Neumann part of the boundary ∂ B = ∂ B h ∪ ∂ B µ with ∂ B h ∩ ∂ B µ = ∅ .With the functionals (175), (176) and (178) at hand, we define at current time t thepotential Π( H ; c, t ) = E ( H ; c ) + D ( H ; c ) − P ext ( H ; t ) (179) . Teichtmeister, M.-A. Keip c . Its minimization with regard to Dirichlet-type boundaryconditions H · n = ¯ h ( X , t ) and − Div H = ˙¯ c ( X , t ) on ∂ B h determines at current time t the species flux field. We obtain the Euler equations1 . Species flux ∇ δ c b ψ + ∂ H b φ = in B . Prescribed fluid potential − δ c b ψ + ¯ µ = 0 on ∂ B µ . Vanishing micro-force ∂ ∇ c b ψ · n = 0 on ∂ B µ . (180)They contain necessary compatibility conditions for the given concentration field. Note,that the rate ˙ c of the concentration at current time t is determined via (169) by the speciesflux obtained from the minimization principle. In addition to(177) we consider the conductive dissipation potential function b φ con ( g ; θ ) = kθ ( g · g ) / g = −∇ θ/θ and where k > c = c n − τ Div H (182)and specify the incremental internal potential density (114) to b π + τ = b ψ ( c n − τ Div H , ∇ c n − τ ∇ [ Div H ] , θ )+ τ [( θ k − T ) ˙ η τ + η n ˙ θ τ + b φ dif ( Tθ n H ; c n , θ n ) − b φ con ( − T ∇ T ; θ n )] . (183)Here, b φ dif is the dissipation potential function (177) related to diffusion mechanisms wherean additional temperature dependence M = c M ( θ ) of the mobility parameter is taken intoaccount. We obtain the Euler equations δ H b π + τ ≡ ∇ µ + [ c M ( θ n ) c n (1 − c n )] − ( Tθ n ) H = ∂ η b π + τ ≡ θ k − T = 0 ∂ θ b π + τ ≡ − C ln θθ + η k = 0 δ T b π + τ ≡ − ( η − η n ) + τθ n [ Tθ n c M ( θ n ) c n (1 − c n ) | H | ] + τT Div[ k θ n T ∇ T ] = − τ ¯ r/θ n (184)in B , where we recall the definition of the chemical potential (173) together with (182). We consider as a second application the thermomechanics of a gradient damage modelwith an elastic stage . The scalar micro-motion field d : B×T → [0 ,
1] referred to as damagevariable measures at a macroscopic point X ∈ B the ratio between an arbitrary orientedarea of microcracks and a representative reference surface in which the mentioned cracksurfaces are embedded, see e.g. Lemaitre [36]. In this sense, a value d = 0 characterizesan unbroken state, whereas d = 1 represents a fully broken state. The irreversibility ariational Thermomechanics of Gradient-Extended Continua d ( X , t ) ≥ c = ( C , d, ∇ d ) (185)and contains the right Cauchy-Green tensor C , the damage variable as well as its firstgradient. In addition, we introduce the elastic right Cauchy-Green tensor C e = F eT gF e that is based on the definition of an elastic, stress producing part F e = J − / θ F of thedeformation gradient in terms of a volumetric thermal expansion J θ = exp[3 α T ( θ − θ )],see Lu & Pister [38]. One can then write C e by means of C as C e = J − / θ C . Asimple model of thermo-gradient-damage at large deformations may then be based on theobjective free energy function b ψ ( c , θ ) = b g ( d ) b ψ e ( C e ) + b ψ θ ( θ ) with b ψ e = µ C e −
3) + µδ [ (det C e ) − δ − b g ( d ) = (1 − d ) is a degradation function and b ψ θ the purely thermal contribution asgiven in (2). Note, that the gradient of damage does not arise in this constitutive functionbut will exclusively enter the dissipation potential function, see below. µ > δ > P e = ∂ F b ψ = b g ( d ) J − θ gF [ µ G − − µ (det C e ) − δ C e − ] β e = ∂ d b ψ = − − d )[ µ (tr C e −
3) + µδ ((det C e ) − δ −
1) ]˜ η = ∂ θ b ψ = − b g ( d ) α T [ µ tr C e − µ (det C e ) − δ C e : C e − ] − C ln θθ (187)that represent constitutive relationships for driving forces. We consider theintrinsic dissipation potential function b φ int ( ˙ d, ∇ ˙ d ; ∇ d, θ ) = b φ l ( ˙ d ; θ ) + b φ ∇ ( ∇ ˙ d ; ∇ d ) (188)as the sum of local and nonlocal parts b φ l ( ˙ d ; θ ) = b c ( θ ) ˙ d + I + ( ˙ d ) and b φ ∇ ( ∇ ˙ d ; ∇ d ) = µl ∇ d · ∇ ˙ d . (189)Here, irreversibility of damage is ensured by the indicator function I + ( ˙ d ) of the set ofpositive real numbers defined as I + ( ˙ d ) = (cid:26) d ≥ ∞ otherwise and ∂I + ( ˙ d ) = d > R − for ˙ d = 0 ∅ otherwise (190)with ∂ denoting the subdifferential. The parameter b c ( θ ) > ddθ b c < l a length scaleparameter. Note, that (188) is a positively homogeneous function of degree one in ( ˙ d, ∇ ˙ d )and hence models for an adiabatic process a rate-independent evolution of damage. Inaddition, we have the conductive dissipation potential function b φ con ( g ; C , d, θ ) = θ b k ( d ) C − : ( g ⊗ g ) / , (191) . Teichtmeister, M.-A. Keip b k ( d ) = b g ( d ) k b in terms of the heat conduction coefficient k b > b π + red = ∂ C b ψ : ˙ C + β e ˙ d + ( θ − T ) ˙ η + Tθ [ b c ( θ ) ˙ d + µl ∇ d · ∇ ˙ d + I + ( ˙ d ) ] − b φ con ( − T ∇ T ) . (192)Then, the variational principle (102) determines at current time t the rates of deformation,damage and entropy as well as the thermal driving force and gives the Euler equations δ ˙ ϕ b π + red ≡ − Div P e = g ¯ γ δ ˙ d b π + red ≡ β e + Tθ b c ( θ ) − µl ∆( Tθ d ) + ∂I + ( ˙ d ) ∋ ∂ ˙ η b π + red ≡ θ − T = 0 δ T b π + red ≡ − ˙ η + θ b φ int ( ˙ d, ∇ ˙ d ) + T Div[ θ b k ( d ) C − T ∇ T ] = − ¯ r/θ (193)in B . Observe, that the evolution of the entropy is driven by the rates of damage andgradient of damage. Note, that for the determination of the rates of deformation anddamage we conclude from the differential inclusion (193) and the relation (193) thenonlocal consistency condition˙ d ≥ , − ˙ β e − ddθ b c ( θ ) ˙ θ + µl ∆ ˙ d ≤ , ˙ d [ − ˙ β e − ddθ b c ( θ ) ˙ θ + µl ∆ ˙ d ] = 0 (194)in B , where ˙ β e = ∂ d C b ψ : ˙ C + ∂ dd b ψ ˙ d + ∂ dθ b ψ ˙ θ and ˙ θ follows from taking the time derivativeof the state equation (78). In addition, the rate formDiv[ ∂ F P e : ˙ F + ∂ d P e ˙ d + ∂ θ P e ˙ θ ] = g ˙¯ γ (195)of mechanical equilibrium (193) has to be considered together with the boundary condi-tions [ ∂ F P e : ˙ F + ∂ d P e ˙ d + ∂ θ P e ˙ θ ] n = ˙¯ t and ∇ ˙ d · n = 0 (196)on ∂ B t and ∂ B H , respectively. (196) are rate forms of the Neumann boundary conditions P e n = ¯ t on ∂ B t and ∇ d · n = 0 on ∂ B H which are outcomes of the variational principle.Finally, following the steps in Section 3.3 we obtain for the intrinsic dissipation Z B D int dV = − Z B β e ˙ d dV ≥ We specify thearray (124) of dissipative driving forces to f = ( , β, ) (198)where β is the quantity conjugate to d . The Legendre transform of the local part (189) of the intrinsic dissipation potential function reads b φ ∗ l ( β ; θ ) = sup ˙ d [ ( β − b c ) ˙ d − I + ( ˙ d ) ] = sup ˙ d ≥ [ ( β − b c ) ˙ d ] = (cid:26) β − b c ≤ ∞ for β − b c > ariational Thermomechanics of Gradient-Extended Continua b φ ∗ l is the indicatorfunction of the set (148) of admissible driving forces governed by the threshold function b f ( β ; θ ) = β − b c ( θ ) . (202)Latter defines the local intrinsic dissipation potential function by the constrained opti-mization problem b φ l ( ˙ d ; θ ) = sup β inf λ ≥ [ β ˙ d − λ b f ( β ; θ ) ] . (203)Then, with the exact time integration of the nonlocal term (189) of the intrinsic dissipa-tion potential function b φ τ ∇ ( ∇ d ; ∇ d n ) = Z t n +1 t n µl ∇ d · ∇ ˙ d dt = 12 µl ( |∇ d | − |∇ d n | ) (204)the incremental internal potential density (167) takes the form b π ∗ τλ = b ψ ( c , θ ) + τ [ ( θ k − T ) ˙ η τ + η n ˙ θ τ + Tθ n ( β ˙ d τ + 1 τ b φ τ ∇ ) − λ b f ( β ; θ n ) − b φ con ( g ; C n , d n , θ n ) ] . (205)As a result, the incremental variational principle (168) gives the Euler equations δ ϕ b π + τλ ≡ − Div P e = g ¯ γ δ d b π + τλ ≡ β e + Tθ n β − µl ∆( Tθ n d ) = 0 ∂ η b π + τλ ≡ θ k − T = 0 ∂ θ b π + τλ ≡ ˜ η + η k = 0 δ T b π + τλ ≡ − ( η − η n ) + θ n [ β ( d − d n ) + b φ τ ∇ ] + τT Div[ θ n b k ( d n ) C − n T ∇ T ] = − τ ¯ r/θ n ∂ β b π + τλ ≡ Tθ n ( d − d n ) − τ λ = 0 ∂ λ b π + τλ ≡ − τ [ β − b c ( θ n ) ] ≥ , λ ≥ , τ λ [ β − b c ( θ n ) ] = 0 (206)in B , which represent time-discrete forms of the general equations (78) and (160). Wecan reduce this set of equations by expressing for k = n the dissipative driving force β = − β e + µl ∆ d via (206) and the Lagrange multiplier τ λ = d − d n via (206) yieldingthe explicit nonlocal form of the Karush-Kuhn-Tucker conditions d ≥ d n , µl ∆ d − β e − b c ( θ n ) ≤ , ( d − d n )[ µl ∆ d − β e − b c ( θ n ) ] = 0 , (207) An alternative intrinsic local dissipation potential function may be given by b φ l ( ˙ d ; d, θ ) = b c ( θ ) d ˙ d + I + ( ˙ d ) (200)that in contrast to (189) depends explicitly on the given damage state d . We obtain the thresholdfunction b f ( β ; d, θ ) = β − b c ( θ ) d (201)which due to occurence of the damage variable in the resistance term corresponds to a model without anelastic stage , i.e. the damage starts to evolve from d = 0 at the instant of loading. . Teichtmeister, M.-A. Keip .As an alternative to this setting, which is fully rate-independent in the adiabatic case,we may consider a (regularized) viscous over-force formulation based on the smooth duallocal intrinsic dissipation potential function, see Footnote 8, b φ η ∗ l ( β ; θ ) = 12 η f h b f ( β ; θ ) i (208)that approaches (199) for the vanishing viscosity limit η f →
0. Then, with (204) theincremental internal potential density (143) takes the form b π ∗ τ = b ψ + τ [ ( θ k − T ) ˙ η τ + η n ˙ θ τ + Tθ n ( β ˙ d τ + 1 τ b φ τ ∇ ) − η f h b f ( β ; θ n ) i − b φ con ( − T ∇ T ; C n , d n , θ n ) ] (209)and the incremental variational principle (145) yields (206) –(206) together with ∂ β b π ∗ τ ≡ Tθ n ( d − d n ) − τη f h β − b c ( θ n ) i + = 0 (210)as Euler equations in B . We can reduce this set of equations by expressing the dissipativedriving force β from (206) yielding for k = n the nonlocal (regularized) update equation d = d n + τη f h µl ∆ d − β e − b c ( θ n ) i + (211)for the damage variable. As third model problem, we consider a thermomechanically coupled formulation ofadditive gradient plasticity. Beside the standard metrics G and g we introduce on thereference configuration the (covariant) plastic metric tensor G p ∈ Sym + (3) that evolvesin time starting from the initial state G p ( X , t ) = G . Following Miehe et al. [49] andas visualized in Figure 4, a Lagrangian elastic strain variable may be based on an explicitdependence on the right Cauchy-Green tensor C , that is the current metric pulled backto the reference configuration, and the plastic metric G p in an additive format ε e = ε − ε p (212)where the total and plastic Hencky strain tensors ε = 12 ln C and ε p = 12 ln G p (213)are introduced. Hence, instead of G p we consider in what follows the logarithmic plasticstrain ε p as the local internal variable the evolution of which from ε p ( X , t ) = isgoverned by a standard flow rule. Note, that within this framework the condition ofplastic incompressibility det G p = 1 is equivalent to a standard statement of vanishingtrace tr ε p = 0. We specify the mechanical constitutive state ariational Thermomechanics of Gradient-Extended Continua T X B T X B T x S T x S T ∗ X B T ∗ X B T ∗ x S T ∗ x S C g FF F − T F − T G p g p a) b) Figure 4:
Geometry of additive plasticity. a) Definition of the total Hencky strain tensor ε = ln C in terms of the pull-back C of the standard current metric g . b) Definition of theplastic metric tensor G p on the reference configuration governing the plastic Hencky strain ε p = ln G p , the evolution of which is given by a local flow rule. Then, ε e = ε − ε p is theelastic strain measure entering the free energy function. c = ( ε , ε p , α, ∇ α ) (214)which contains a scalar hardening variable α as well as its first gradient. In the followingwe focus on metal plasticity characterized by small elastic but large plastic deformationsand consider the free energy function b ψ ( c , θ ) = b ψ e ( ε , ε p ) + b ψ p ( α, θ ) + 12 µl |∇ α | + b ψ e − θ ( ε , θ ) + b ψ θ ( θ ) . (215)Here, b ψ e is the purely elastic contribution that is assumed to have the quadratic form b ψ e ( ε , ε p ) = κ ε ) + µ | Dev ε e | (216)where κ > µ > b ψ p is an isotropichardening function that also takes into account thermally induced softening. The gradientextension related to a length scale parameter l is assumed to affect the scalar hardeningvariable only. The coupled thermoelastic response is modelled by the constitutive function b ψ e − θ ( ε , θ ) = − κα T (tr ε )( θ − θ ) (217)in line with (2), where also the pure thermal contribution b ψ θ is specified. Note, that thefunction (216) known as Hencky energy is not polyconvex with respect to the deformationgradient F = D ϕ in the sense of Ball [4], but rank-one convex for a moderately highelastic deformation range, see
Bruhns et al. [7]. Hence, it is applicable to the typicalscenario of metal plasticity. With the free energy function (215) at hand we can derivethe tensor functions σ e = ∂ ε b ψ = κ [ tr ε − α T ( θ − θ ) ] G − + 2 µ Dev[ G − ε e G − ] β e = ∂ ε p b ψ = − µ Dev[ G − ε e G − ] β e = δ α b ψ = ∂ α b ψ p − µl ∆ α ˜ η = ∂ θ b ψ = ∂ θ b ψ p − κα T tr ε − C ln θθ (218)that represent constitutive relationships for driving forces. Neff et al. [58] introduced an exponentiated Hencky energy which is polyconvex in the two-dimensional case. . Teichtmeister, M.-A. Keip We specify thearray (124) of dissipative driving forces to f = ( , s , β, ) (219)where ( s , β ) are the quantities conjugate to ( ε p , α ). For von-Mises-type gradient plasticitywe define the yield function b f ( s , β ; θ ) = | s | − r
23 [ b y ( θ ) − β ] (220)that restricts the set of admissible driving forces according to (148). b y ( θ ) is a temperaturedependent yield stress function with ddθ b y <
0. The corresponding intrinsic dissipationpotential function is defined by the constrained optimization problem b φ int ( ˙ ε p , ˙ α ; θ ) = sup s ,β inf λ ≥ [ s : ˙ ε p + β ˙ α − λ b f ( s , β ; θ ) ] . (221)The intrinsic dissipation follows as D int = − ( β e : ˙ ε p + β e ˙ α ) = λ ( s : ∂ s b f + β∂ β b f ) = r λ b y ( θ ) ≥ s = − β e and β = − β e which are outcomes of the rate-typevariational principle (158). Note, that the last equality in (222) follows from b w ( s , β ) = | s | + p / β being a gauge, i.e. b w is a positively homogeneous function of degree onewith the resulting property s : ∂ s b w + β ∂ β b w = b w . With (221) the incremental internalpotential density (167) is specified to b π ∗ τλ = b ψ ( c , θ ) + τ [ ( θ k − T ) ˙ η τ + η n ˙ θ τ + Tθ n ( s : ˙ ε pτ + β ˙ α τ ) − λ b f ( s , β ; θ n ) − b φ con ( − T ∇ T ; C n , θ n ) ] (223)where b φ con is the conductive dissipation potential function defined in (191) however witha constant heat conduction coefficient. Then, the incremental variational principle (168)gives the Euler equations δ ϕ b π + τλ ≡ − Div[ σ e : ∂ F ε ] = g ¯ γ ∂ ε p b π + τλ ≡ β e + Tθ n s = δ α b π + τλ ≡ β e + Tθ n β = 0 ∂ η b π + τλ ≡ θ k − T = 0 ∂ θ b π + τλ ≡ ˜ η + η k = 0 δ T b π + τλ ≡ − ( η − η n ) + τθ n [ s : ˙ ε pτ + β ˙ α τ ] + τT Div[ θ n k C − n T ∇ T ] = − τ ¯ r/θ n ∂ s b π + τλ ≡ Tθ n ( ε p − ε pn ) − τ λ s / | s | = ∂ β b π + τλ ≡ Tθ n ( α − α n ) − τ λ p / ∂ λ b π + τλ ≡ − τ b f ( s , β ; θ n ) ≥ , λ ≥ , τ λ b f ( s , β ; θ n ) = 0 (224) ariational Thermomechanics of Gradient-Extended Continua B which represent time-discrete forms of the general equations (78) and (160). Algo-rithms for the computation of the derivative ∂ F ε can be found in Miehe & Lambrecht [48]. We can reduce the set of equations (224) by expressing for k = n the dissipa-tive driving forces s = − β e and β = − β e via (224) and (224) , respectively, and theLagrange multiplier τ λ = p / α − α n ) via (224) yielding the nonlocal form of theKarush-Kuhn-Tucker conditions in strain space α ≥ α n , | β e | − p / b y ( θ n ) + β e ) ≤ , ( α − α n )[ | β e | − p / b y ( θ n ) + β e ) ] = 0(225)where we recall the definitions of the driving forces (218) − . Note the occurance of theLaplacian term µl ∆ α in the yield resistance that is in line with the approach of Aifantis,see e.g. Aifantis [1].As an alternative to this setting, which is fully rate-independent in the adiabatic case,we may consider a (regularized) viscous over-force formulation based on the smooth dualintrinsic dissipation potential function b φ η ∗ int ( s , β ; θ ) = 12 η f h b f ( s , β ; θ ) i (226)in terms of the threshold function defined in (220). Then, the incremental internal po-tential density (143) takes the form b π ∗ τ = b ψ + τ [ ( θ k − T ) ˙ η τ + η n ˙ θ τ + Tθ n ( s : ˙ ε pτ + β ˙ α τ ) − η f h b f ( s , β ; θ n ) i − b φ con ( − T ∇ T ; C n , θ n ) ] (227)and the incremental variational principle (145) yields (224) − together with ∂ s b π ∗ τ ≡ Tθ n ( ε p − ε pn ) − ( τ /η f ) h b f ( s , β ; θ n ) i + s / | s | = ∂ β b π ∗ τ ≡ Tθ n ( α − α n ) − ( τ /η f ) h b f ( s , β ; θ n ) i + p / B . This set of equations can again be reduced by expressing thedissipative driving forces s and β from (224) and (224) yielding for k = n the nonlocal(regularized) update equations ε p = ε pn − ( τ /η f ) h | β e | − p / b y ( θ n ) + β e ) i + β e / | β e | α = α n + ( τ /η f ) h | β e | − p / b y ( θ n ) + β e ) i + p / Simo & Miehe [73] is proposed by
Ulz [78]. For a comparison of rate-independent and rate-dependent formulations in isothermalgradient-plasticity of Fleck-Willis-type we refer to
Nielsen & Niordson [60].
For softening visco-plasticity of von-Mises-type, we analyze the development of shear bands in a rectangularplate B = (0 , L ) × (0 , L ) with L = 50 mm subject to tensile loading under the condition . Teichtmeister, M.-A. Keip Table 1:
Parameters of representative numerical example.No. Parameter Name Value Unit1 κ bulk modulus 164 . µ shear modulus 80 . l plastic length scale { . , . , . } mm4 α T thermal expansion coefficient 10 − C heat capacity 3 . · − kN/(mm K)6 k heat conduction coefficient 0 . θ reference temperature 293 . h initial hardening parameter − .
13 kN/mm y initial yield stress 0 .
45 kN/mm w h thermal softening coefficient 0 . w thermal softening coefficient 0 .
002 1/K12 η f viscosity parameter 10 − kNs/mm PSfrag replacements 2
L L ¯ u ¯ u XY B Figure 5:
Cross Shear Localization. Geometry and mechanical loading. The process ofheat conduction is neglected. Due to the symmetry of the boundary value problem, onlythe top right quarter of the domain is discretized by finite elements. To trigger plasticity inthe center, the initial yield stress y is reduced by 3% in the dark grey element. of plane strain. The geometric setup is depicted in Figure 5. We use the viscous over-forceformulation of the mixed large deformation setting from Section 5.3.1 above and specifythe isotropic hardening function in (216) to b ψ p ( α, θ ) = 12 b h ( θ ) α . (230)Here, b h is a temperature dependent hardening function which together with the temper-ature dependent yield stress function is specified to b h ( θ ) = h [ 1 − w h ( θ − θ ) ] and b y ( θ ) = y [ 1 − w ( θ − θ ) ] , (231) ariational Thermomechanics of Gradient-Extended Continua a) b) c)d) e) f) 2.30.0472293 αθ [K] Figure 6:
Cross Shear Localization. Contour plots of equivalent plastic strain α andtemperature θ at final displacement ¯ u = 5 mm for a discretization of one quarter of thespecimen by 20 ×
40 finite elements. The chosen plastic length scale parameters are a), d) l = 0 .
05 mm; b), e) l = 0 . l = 0 . see Simo & Miehe [73]. The used material parameters are summarized in Table 1. Totrigger plasticity in the center, the initial yield stress is reduced by 3% in the elementshaded in dark grey in Figure 5. For simplicity, we neglect the effect of heat conductionwhich is a reasonable assumption in case of a fast formation of the shear band gener-ated by a high loading rate. We stretch the specimen with a constant displacement rate˙¯ u = 5 mm/s within the time interval T = (0 ,
1) s that is divided into 1000 equidis-tant increments . We use the semi-explicit variational update with index k = n inthe incremental internal potential density (227). Due to the variational structure, theresulting stiffness matrix within a typical Newton-Raphson iteration step is symmetric.As (global) primary fields we take the macroscopic deformation ϕ , the scalar harden-ing/softening variable α and its dual driving force β . The temperature θ is calculatedvia the implicit local equation (224) . Due to symmetry, only one quarter of the domainis discretized by 15 ×
30, 20 ×
40 and 25 ×
50 quadrilateral finite elements. We use aQ E -Q -Q element pairing which bases on a (local) enhancement of the macroscopicdisplacement gradient according to Simo & Armero [71]. Figure 6 shows contour plots Since the process of heat conduction is neglected and the viscosity is chosen very low in order to havea formulation that is close to the nonsmooth setting, the overall thermomechanical material behavior is defacto rate-independent. Hence, the specific loading rate applied on the specimen is practically irrelevant. . Teichtmeister, M.-A. Keip isothermal15 ×
30 el.15 ×
30 el. 20 ×
40 el.20 ×
40 el. 25 ×
50 el.25 ×
50 el. 0000 0.5 11 1.5 22 2.5 3 4 555 1010 1515 2020 2525 3030 displacement ¯ u [mm]displacement ¯ u [mm] l oa d F [ k N ]l oa d F [ k N ] a) b) Figure 7:
Cross Shear Localization. Load-deflection curves. a) Mesh dependent structuralresponse for local theory with l = 0 mm and b) mesh objective response for gradient theorywith l = 0 . . Note, that in b) the final displacement is ¯ u = 5 mm, whereas in a) it is ¯ u = 2 . of the equivalent plastic strain α and the temperature θ at final displacement ¯ u = 5 mmfor three different plastic length scale parameters l ∈ { . , . , . } mm. Clearly, thespecimen experiences a rise in the temperature in the region of plastic dissipation. Whenincreasing the plastic length scale, the equivalent plastic strain α as well as the tempera-ture θ spread over more elements. At the same time, one observes decreasing maximumvalues of α and θ , see also Aldakheel & Miehe [2] and the references citet therein,i.e.
Voyiadjis & Faghihi [79]. As widely known, in case of a local theory l = 0 mmthe plastic deformation localizes within one element width. This mesh dependency alsomanifests itself in the load-displacement curve of the structure as shown in Figure 7a).In contrast, the regularization provided by the gradient theory yields mesh independentresults and the structural response is objective, see Figure 7b). There, one also observesthe additional softening effect in the nonisothermal case due to locally decreasing yieldstresses according to (231) . At this point, we want to note that Wcis lo & Pamin [80]incorporate a gradient-enhanced relative temperature field in their formulation in orderto regularize adiabatic thermal softening behavior.For the used mixed setting of gradient plasticity, alternative finite element formulationsare presented in
Miehe et al. [51]. Note, that in this context the construction of inf-sup stable finite element pairings is a difficult task which in detail has recently beeninvestigated by
Krischok & Linder [33]. Especially, our chosen element pairing resultsin a nonphysical oscillatory behavior of the driving force field β . However, this instabilityseems to have no visible influence on the macroscopic deformation field ϕ , the scalarplastic strain field α and the temperature field θ .
6. Conclusion
We presented a unified framework for the fully coupled thermomechanics of gradient-extended dissipative solids which undergo large deformations. The key of the formulationis the consideration of the entropy and the entropy rate as canonical variables which enterbeside the gradient-extended mechanical state and the rate of this state, respectively, theinternal energy and dissipation functions. Here, the rate-type formulation of local ther-moplasticity outlined in
Yang et al. [81] is recovered by the concept of dual variables. ariational Thermomechanics of Gradient-Extended Continua
Acknowledgement.
Support for this research was provided by the German ResearchFoundation (DFG) under Grant MI 295/20–1.
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