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Nonlinear and Linearized Models in Thermoviscoelasticity

We consider a quasistatic nonlinear model in thermoviscoelasticity at a finite-strain setting in the Kelvin-Voigt rheology where both the elastic and viscous stress tensors comply with the principle of frame indifference under rotations. The force balance is formulated in the reference configuration by resorting to the concept of nonsimple materials whereas the heat transfer equation is governed by the Fourier law in the deformed configurations. Weak solutions are obtained by means of a staggered in-time discretization where the deformation and the temperature are updated alternatingly. Our result refines a recent work by Mielke&Roub\'{\i}\v{c}ek [arXiv:1903.11094] since our approximation does not require any regularization of the viscosity term. Afterwards, we focus on the case of deformations near the identity and small temperatures, and we show by a rigorous linearization procedure that weak solutions of the nonlinear system converge in a suitable sense to solutions of a system in linearized thermoviscoelasticity. The same property holds for time-discrete approximations and we provide a corresponding commutativity result.


Introduction
Nonlinear and large strain continuum mechanics has become a thriving field of research over the last decades which is still subject of important advancements and, at the same time, offers many challenging open questions. For instance, rigorous studies on large strain viscoelastic materials [19,24,26,31] or nonlinear models in thermoviscoelasticity [33] have been initiated only recently. Besides analytical intricacies, the usage of large strain models in engineering practice is often impeded due to nonconvex behavior that complicates numerical implementations. At many occasions, however, linearized models are still sufficient to describe observed phenomena and are significantly easier to treat, both analytically and numerically. Roughly speaking, heuristic calculations suggest that, if the deformation of the body is "close" to the identity, nonlinear models can be replaced by linear ones with a negligible error. Clearly, the reliability of such predictions depends on the rigorous derivation of simplified linearized models, e.g., via Γ-convergence [7,12]. This is an intensive research program that has been initiated in the context of linearized elastostatics in [13]. Subsequently, this work was extended in various directions, among others, models for incompressible materials [23,27,28], atomistic models [8,40], or problems without Dirichlet boundary conditions [29] have been considered. For multiwell energies allowing for phase transitions we refer to [1,14,39], and we mention also settings beyond elasticity such as plasticity [34] or fracture [18,20]. As to evolutionary models, we refer to [19] where viscoelasticity in the Kelvin-Voigt rheology and its linearized version are treated.
The goal of this contribution is to couple the nonlinear equations of viscoelasticity with a heat transfer equation. We first analyze a corresponding frame-indifferent and thermodynamically-consistent model of thermoviscoelasticity at large strains, and refine the results obtained recently by Mielke & Roubíček [33]. Afterwards, in the spirit of the isothermal result [19], we pass to a linearized limit in terms of rescaled displacement fields and different regimes of rescaled temperatures.
We start by introducing the large strain model. Neglecting inertial effects, a nonlinear viscoelastic material in Kelvin-Voigt rheology obeys the following system of equations − div ∂ F W (∇y, θ) + ∂Ḟ R(∇y, ∇ẏ, θ) = f in [0, T ] × Ω. (1.1) Here, [0, T ] is a process time interval with T > 0, Ω ⊂ R d is a bounded domain representing the reference configuration, y : [0, T ] × Ω → R d is a deformation mapping, ∇y is the deformation gradient, θ denotes the temperature, W : R d×d × [0, ∞) → R ∪ {+∞} is a stored energy density, which represents a potential of the first Piola-Kirchhoff stress tensor ∂ F W , and F ∈ R d×d is the placeholder of ∇y. Finally, R : R d×d × R d×d × [0, ∞) → R denotes a (pseudo)potential of dissipative forces, whereḞ is the time derivative of F , and f : [0, T ] × Ω → R d is a volume density of external forces acting on Ω. The density W respects frame indifference under rotations and positivity of the determinant of the deformation gradient, i.e., local non-self-penetration is realized. (In contrast to [24], we do not consider conditions implying global non-self-penetration.) At the same time, we focus on physically correct viscous stresses, i.e., as observed by Antman [3], R must comply with a time-continuous frame indifference principle meaning that for all F it holds that R(F,Ḟ , θ) =R(C,Ċ, θ), for some nonnegative functionR, where C := F F andĊ :=Ḟ T F + F TḞ .
In contrast to the rapidly developed static theory at large strains, already in the isothermal case existence of solutions to (1.1) remains a challenging problem and results for models respecting the physically relevant frame indifference for both W and R are scarce. We refer, e.g., to [26] for local in-time existence or to [15] for the existence of measure-valued solutions. To date, weak solutions in finite strain isothermal viscoelasticity [19,24,33] can only be guaranteed by using the concept of second-grade nonsimple materials where the stored energy density (and consequently the first Piola-Kirchhoff stress tensor) additionally depends on the second gradient of the deformation. This idea was first introduced by Toupin [41,42] and proved to be useful in mathematical continuum mechanics, see e.g. [4,5,32,35]. In this spirit, we consider a version of (1.1) for nonsimple materials where the stored energy density depends also on the second gradient of y, and (1.1) is replaced by − div ∂ F W (∇y, θ) − div(∂ G H(∇ 2 y)) + ∂Ḟ R(∇y, ∇ẏ, θ) = f in [0, T ] × Ω, (1.2) which corresponds to an additional convex term Ω H(∇ 2 y) dx in the stored energy. Let us mention that a main justification of this model lies in the observation that, in the small strain limit and under suitable scaling, the problem leads to the standard system of linear viscoelasticity without second gradient [19].
In the present contribution, we focus on a nonlinear coupling of the system (1.2) with a heat transfer equation of the form c V (∇y, θ)θ = div(K(∇y, θ)∇θ) + ∂Ḟ R(∇y, ∇ẏ, θ) : ∇ẏ + θ∂ F θ W cpl (∇y, θ) : ∇ẏ in [0, T ] × Ω, (1.3) where W cpl denotes a thermo-mechanical coupling potential, c V (F, θ) = −θ∂ 2 θ W cpl (F, θ) is the heat capacity, K denotes the matrix of the heat-conductivity coefficients, and the last term plays the role of an adiabatic heat source. This corresponds to heat transfer modeled by the Fourier law in the deformed configuration which is however pulled back to the reference configurations, whence K depends on the deformation gradient. Here, following [33], we assume a rather weak thermal coupling by using the splitting of the free energy W via the explicit ansatz implying ∂ θ W = ∂ θ W cpl . The coupled system (1.2)-(1.3) is equipped with suitable initial and boundary conditions, see (2.17)-(2.18) below. Thermoviscoelasticity is a notoriously difficult problem already at small strains, e.g., there is no obvious variational structure of the thermal part due to the low regularity of data. New developments in the the L 1 -theory for the nonlinear heat equation [9,10] paved the way to advancements in small strain thermoviscoelasticity (for example, see [6,11,37]) which eventually culminated in the analysis of a physically sound large-strain model by Mielke & Roubíček [33]. We refer to [33,Introduction; items (α)-(ε)] for the main properties and challenges for this model which coincides with ours up to minor points, see Remark 2.1. Their existence result is based on a time-incremental approach for a regularized system which does not comply with the above mentioned frame indifferent principles, e.g., in (1.2) a term λ∇ẏ is added for λ > 0. Then, they first pass to the time-continuous limit in the regularized problem and eventually recover the original system in the limit of the vanishing regularization parameter λ.
The first result of our work (Theorem 2.3) revisits their study by proposing a slightly different semidiscretization in time which directly approximates the PDE system in the limit for vanishing time steps and comes along without any regularization. Although establishing the same existence result on weak solutions, our approach sheds new light on the issue as we propose a time-discrete approximation scheme complying with frame indifference. This combined with a spatial discretization, see e.g. [25,Section 9.3], could be the basis for a numerical implementation. As in [33], our scheme is staggered, i.e., first the deformation is updated at fixed temperature from the previous time step and then the temperature is updated. Our scheme differs in the usage of explicit or implicit steps, i.e., whether in certain terms the 'old' or the 'new' temperature is used, see Remark 2.4. By means of delicate estimates on the coupling potential, we are hereby able to establish the necessary a priori bounds without any regularization. At this point, we derive a priori estimates for different scalings of the elastic strains and the temperature which is at the basis of our subsequent analysis on small-strain limits.
In the second part of our work, we are interested in the case of small strains and temperatures, i.e., when ∇u := ∇y − Id is of order ε for some small ε > 0 and θ is of order ε α for any exponent α > 0. Here, u := y − id is the displacement corresponding to y with id and Id standing for the identity map and identity matrix, respectively. Such properties are certainly reasonable if initial values and boundary values for the deformation and the temperature are close to the identity or zero, respectively. Therefore, it is convenient to introduce the rescaled displacement u ε = ε −1 (y − id) and rescaled temperature µ ε = ε −α θ, and to replace f by εf . We write (1.2)-(1.3) in terms of the rescaled quantities and multiply (1.2) with ε −1 and (1.3) with ε −α . Then, formally, we can pass to the limit and obtain the system D e(u) : e(u), (1.5) where C W := ∂ 2 F W el (Id) is the tensor of elastic constants (W el is defined in (1.4)), C D := ∂Ḟ 2 R(Id,Ḟ , 0) is the tensor of viscosity coefficients, B (α) represents a thermal expansion matrix,c V is the heat capacity at zero temperature and the stress free material state, and K 0 := K(Id, 0). Finally, e(u) := (∇u + (∇u) )/2 denotes the linear strain tensor and e(u) the strain rate. By different scaling properties of the two equations, it turns out that the limit is α-dependent and, as we point out later, only meaningful in the regime α ∈ [1,2]. The matrix B (α) is only active for α = 1 and in this case it is related to the coupling potential, namely B (α) = ∂ F θ W cpl (Id, 0). On the other hand, C (α) D is nonzero only for α = 2 and then it coincides with C D . Interestingly, although the nonlinear thermoviscoelasticity system is written for a nonsimple material, in the limit we obtain linear equations without spatial gradients of e(u).
Formal derivations of such PDE systems is not new and can be found, e.g., in [21,Section 59]. The second main contribution of our work (Theorem 2.7) is to make this limit passage rigorous, i.e., we show that solutions to the nonlinear system (1.2)-(1.3) converge in a suitable sense to the unique solution of the linear system (1.5) as ε → 0. Besides this convergence result, we also get analogous convergences for time-discretized problems, and we confirm that convergences for vanishing time step and ε → 0 commute, see Theorem 2.8.
To our best knowledge, it is the first linearization result of a mechanical model coupled with heat transfer in the material. We perform linearization near the natural (i.e., stress free) state and zero temperature. Without further details, let us however mention that by a shifting argument our techniques would allow to linearize about a fixed, positive temperature θ c , whenever the initial and boundary data lie above θ c and the coupling potential W cpl (F, θ) vanishes for θ ≤ θ c .
We now give an outline of the paper and present some fundamental ingredients of the proof. After some basic notation, we introduce the nonlinear setting in Subsection 2.1. In Subsection 2.2, we formulate our semi-discrete approximation result in the nonlinear setting and briefly highlight the differences to the scheme in [33], see Remark 2.4. In Subsection 2.3, we introduce the linearized setting and present our results on convergence of solutions in the nonlinear-to-linear passage.
In Subsections 3.1-3.2, we address the well-definedness of the staggered time-incremental scheme. The core of our approach is an inductive bound on the total energy, see Lemma 3.11: this is achieved by suitably testing the momentum balance and the heat-transfer equation, adding the two equations, and exploiting cancellation of the dissipation. In contrast to [33], see particularly [33,Remark 6.1], this cancellation is already possible in the time-discrete setting as we use a simpler, explicit, thermo-mechanical coupling term in the scheme allowing us to proceed without the necessity of regularizing terms. This, however, comes at the expense of the fact that the argument to guarantee nonnegativity of the temperature in the thermal step is more sophisticated. For this, we need a delicate estimate for the coupling potential, see Proposition 3.8.
As a preparation for the passage to the linearized system, we need an adaption of the bound on the total energy, see Subsection 3.3. In fact, due to the different scaling ε and ε α of the mechanical and the heat-transfer equation, the above mentioned cancellation cannot be used in general for small ε. Thus, novel techniques are required to tame the contribution of the dissipation including higher integrability of the temperature variable, see Lemma 3.15 for details. Section 3 is closed with a priori bounds derived from the energy bound, see Subsection 3.4. As in [33], the main ingredients here are Gagliardo-Nirenberg interpolation inequalities and special test functions developed by Boccardo and Gallouët [10] for parabolic equations with measure-valued right-hand side. For convenience of the reader, almost complete proofs are provided since in addition to [33] we need scaling invariant estimates in terms of the small parameter ε.
In Section 4 we then address the passage to vanishing time steps in the nonlinear model. At this point, having settled the a priori estimates, we can essentially follow [33]. Since we work without regularization terms, however, we need to combine and adapt the techniques from Sections 5-6 of [33], and therefore we elaborate the proofs to some extent. Eventually, Section 5 is devoted to the linearization. In Subsection 5.1 we first deal with the passage to the time-continuous problem. The strategy in the proof is similar to the one in the nonlinear setting in Section 4, with the additional challenge that in each term we need to ensure that higher order terms in Taylor expansions are asymptotically negligible. In particular, we show that contributions of the second gradient vanish in the limit. As in the nonlinear setting, strong convergence of the strains and the strain rates is necessary to pass to the limit, see Lemma 4.5 and Lemma 5.4. Due to rescaling of the equations, however, this is more demanding in the passage to the linerized setting as higher integrability of the temperature is needed to control the coupling term, cf. Remark 4.3. Eventually, Subsection 5.2 is devoted to time-discrete problems which particularly involves a Γ-convergence result for the mechanical part, see Proposition 5.7.

The model and main results
2.1. The setting and modeling assumptions. In what follows, we use standard notation for Lebesgue and Sobolev spaces. The lower index + means nonnegative elements, i.e., L 2 + (Ω) denotes the convex cone of nonnegative functions belonging to L 2 (Ω) and a similar notation is used for H 1 + (Ω). We also set R + := [0, +∞). Let a ∧ b := min{a, b} for a, b ∈ R. Denoting by d ≥ 2 the dimension, we let Id ∈ R d×d be the identity matrix, and id(x) := x stands for the identity map on R d . We define the subsets SO(d) := {A ∈ R d×d : A T A = Id, det A = 1}, GL + (d) := {F ∈ R d×d : det(F ) > 0}, and R d×d sym := {A ∈ R d×d : A T = A}. Furthermore, F −T := (F −1 ) T = (F T ) −1 , and given a tensor (of arbitrary dimension), |F | will denote its Frobenius norm. We denote the scalar product between vectors, matrices, or 3rd-order tensors by ·, :, and . . . , respectively. As usual, in the proofs generic constants C may vary from line to line. If not stated otherwise, constants depend only on d, p > d, Ω, α > 0, and the potentials introduced in the sequel. We frequently use a scaled version of Young's inequality with constant λ ∈ (0, 1) by which we mean ab ≤ λa p + Cb q /λ for a, b ≥ 0, exponents p, q ≥ 1 with 1/p + 1/q = 1, and C > 0 large enough.
Consider an open set Ω ⊂ R d with Lipschitz boundary Γ := ∂Ω. Let Γ D , Γ N be disjoint Borel subsets of Γ such that H d−1 (Γ D ) > 0, H d−1 (Γ N ) > 0, and Γ = Γ D ∪ Γ N , representing Dirichlet and Neumann parts of the boundary, respectively. For p > d, we introduce the set of admissible deformations by and we say that the absolute temperature θ is admissible if θ ∈ L 1 + (Ω). We also introduce the space Next, we discuss our variational setting. In this regard, let c 0 , C 0 with 0 < c 0 < C 0 < ∞ be some fixed constants.
Mechanical energy and coupling energy: The elastic energy W el : Y id → R + is given by where W el : GL + (d) → R + is a frame indifferent elastic energy potential with the usual assumptions in nonlinear elasticity. More precisely, we require that Adopting the concept of 2nd-grade nonsimple materials, see [41,42], we also consider a strain gradient energy term H : Y id → R + , defined as where its potential H : The mechanical energy M : Y id → R + is then defined as the sum M(y) := W el (y) + H(y). (2.5) Besides the mechanical energy, we introduce a coupling energy W cpl : Y id × L 1 + (Ω) → R given by where W cpl : GL + (d) × R + → R describes mutual interactions of mechanical and thermal effects (see e.g. [21]), and satisfies (C.1) W cpl is continuous and , and θ ≥ 0; (C.5) For all F ∈ GL + (d) and θ > 0 it holds that Notice that, by (C.3) and the second bound in (C.5), ∂ F W cpl can be continuously extended to zero temperatures with ∂ F W cpl (F, 0) = 0. For F ∈ GL + (d) and θ ≥ 0, we define the total free energy potential sym , and θ ≥ 0. Notice that the fact that R can be written as a function depending on the right Cauchy-Green tensor C = F T F and its time derivativeĊ is equivalent to dynamic frame indifference (see also [2]). Condition (D.1) also implies that the viscous stress ∂Ḟ R(F,Ḟ , θ) is linear in the time derivativeĊ as indeed a simple calculation shows ∂Ḟ R(F,Ḟ , θ) = 2F (D(C, θ)Ċ).
(2.8) The choice of a linear material viscosity is crucial in our approach and is a relevant modeling assumption for non-activated dissipative processes with rather moderate rates. We emphasize, however, that the geometrical nonlinearity of finite elasticity is still present due toĊ in (2.8), and that ∂Ḟ R necessarily also depends on F , even for constant functions D. We also define the associated dissipation rate ξ : where the second identity follows from (2.8),Ċ =Ḟ T F + F TḞ , and the symmetries in (D.1).
Heat conductivity and internal energy: The map K : Ω×R + → R d×d sym will denote the heat conductivity tensor of the material in the deformed configuration. We require that K is continuous, symmetric, uniformly positive definite, and bounded. More precisely, for all x ∈ Ω and θ ≥ 0 it holds that where the inequalities are meant in the eigenvalue sense. We define the pull-back K : Ω × GL + (d) × R + → R d×d sym of K into the reference configuration by (see [33, (2.24) From this point on, we will usually omit stating the x-dependence of K and K explicitly, shortly writing K(θ) for K(x, θ) and K(F, θ) for K(x, F, θ).
Internal and total energy: The internal energy W in : Using (C.3) and the third bound in (C.5), we can easily see that W in can be continuously extended to zero temperatures by setting W in (F, 0) = 0 for all F ∈ GL + (d). Also by the third bound in (C.5), the internal energy is controlled by the temperature in the following sense: (2.14) Eventually, the total energy functional E : Y id × L 1 + (Ω) → R + is then given by Remark 2.1 (Comparison to [33]). We close this part on modeling assumptions by highlighting the differences to the assumptions in [33]: Our condition in (W.3) is slightly more general than the corresponding one in [33, (2.30a)], where the term |F | 2 is replaced by |F | s for s > 2. We do not assume that W cpl is bounded from below. Condition (C.3) as well as bounds similar to (C.4)-(C.5) are also required in [33], see [33, (2.15), (2.30)]. There, the bound on ∂ F θ W cpl is slightly more general for θ near zero, and only an upper bound on the eigenvalues of ∂ 2 F W cpl (F, θ) is required, see [33, (2.30c)]. This similarity of the assumptions will in particular allow us to employ several intermediate steps proven in [33]. For models complying with the above assumptions we refer to [33,Examples 2.4,2.5].
Equations of nonlinear thermoviscoelasticity: Fixing a finite time horizon T > 0, let us from now on shortly write I := [0, T ]. We fix a constant ε ∈ (0, 1] which represents the magnitude of the elastic strain. In the first part of the paper, we are mainly interested in the large strain setting, where ε = 1. However, later we perform the passage to the small strain limit ε → 0. To allow for a consistent notation, we include the parameter ε throughout the entire paper. Let εf with f ∈ W 1,1 (I; L 2 (Ω; R d )) be a time-dependent dead force, εg with g ∈ W 1,1 (I; L 2 (Γ N ; R d )) be a boundary traction, and let ε α θ with θ ∈ W 1,1 (I; L 2 + (Γ)) and α > 0 be an external temperature. We study the coupled system which, as in [33], is complemented with the boundary conditions Here, ν denotes the outward pointing unit normal on Γ and κ ≥ 0 is a phenomenological heat-transfer coefficient on Γ. Moreover, div S represents the surface divergence, defined by div S (·) = tr(∇ S (·)), where tr denotes the trace and ∇ S := (Id − ν ⊗ ν)∇ denotes the surface gradient. We refer to [33, (2.28)-(2.29)] for an explanation and derivation of the boundary conditions. Note that by (2.9) the system (2.16) indeed coincides with (1.2)-(1.3). The mechanical evolution (2.16a) is the quasistatic version of the Kelvin-Voigt rheological model (neglecting inertia), corresponding to the sum of the conservative and the dissipative forces. The equation (2.16b) follows from the entropy equation θṡ = ξ − div q, where the entropy s is expressed in terms of the free energy by s = −∂ θ W = −∂ θ W cpl . Furthermore, the dissipation rate ξ is defined in (2.9) and the heat flux q is modeled by the Fourier law in the deformed configuration, pulled back to the reference configuration, i.e., q = −K(F, θ)∇θ. The term −θ∂ 2 θ W cpl (∇y, θ) corresponds to the heat capacity at constant volume and the last term in (2.16b) is an adiabatic heat source. We again refer to [33] or to [25,Section 8.1] for details. Notice that the the purely mechanical stored energy W el , see (2.3), does not influence the heat production and transfer in (2.16b).

2.2.
Approximation of solutions in the nonlinear setting. In this subsection, we study the nonlinear system and therefore we fix ε = 1. (In the notation, ε is still included, as before.) The existence of energyconserving weak solutions to (2.16) in the sense of Definition 2.2 has been proven in [33,Theorem 2.2]. In contrast to this work, we show here that the solutions can be obtained directly as limits of a staggered time-incremental scheme without using any additional regularization. We fix a discrete time step size τ ∈ (0, 1]. For the sake of notational clarity, we assume without a further mention that any τ we encounter evenly divides the time interval [0, T ]. Given any sequence (a l ) l≥0 , it will be useful to introduce the following notation for discrete differences Our time-discrete staggered scheme is initialized by setting where y 0,ε and θ 0,ε are as in (2.18). We then alternate between a mechanical step, deforming the material while keeping the temperature fixed, and a thermal step, adjusting the temperature distribution inside the material without changing the deformation, see Figure 1. More precisely, suppose that we have already as a solution of the minimization problem where h τ plays the role of a heat source given by and θ (k) ,τ := 1 τ kτ (k−1)τ θ (t) dt. The underlying idea is that the Euler-Lagrange equations associated to (2.22) and (2.24) lead to time-discretized variants of the equations (2.16), see (3.7) and (3.11) below. Supposing that the steps y τ , . . . , θ (T /τ ) τ as described above exist, we define interpolations as follows: for k ∈ {0, . . . , T /τ }, we let y τ (kτ ) = y τ (kτ ) =ŷ τ (kτ ) := y (k) τ and for t ∈ ((k − 1)τ, kτ ) (2.26) A similar notation is employed for y τ , y τ , andŷ τ . We now formulate our first main result concerning the convergence of solutions to the staggered scheme towards a weak solution of (2.16)-(2.18). (ii) (Convergence to solutions) There exist y ∈ L ∞ (I; Y id ) ∩ H 1 (I; H 1 (Ω; R d )) and θ ∈ L 1 (I; W 1,1 (Ω)) such that the couple (y, θ) is a weak solution to (2.16)- (2.18) in the sense of Definition 2.2, and up to selecting a subsequence, it holds that andẏ τ →ẏ ε strongly in L 2 (I; H 1 (Ω; R d )), (2.27) θ τ → θ in L s (I × Ω) andθ τ θ weakly in L r (I; W 1,r (Ω)) (2.28) as τ → 0 for any r ∈ [1, d+2 d+1 ) and s ∈ [1, d+2 d ). The same holds true if we replaceŷ τ by y τ or y τ in the first part of (2.27), andθ τ by θ τ or θ τ in (2.28).
Let us mention that the proof shows that weak solutions satisfy a total energy balance of the form i.e., the total energy is conserved up to the work of the external loadings and the heat flux through Γ.
Remark 2.4 (Difference to scheme in [33]). The scheme has several differences to the one considered in [33, (4.5)-(4.7)]. On the one hand, both steps in [33] are suitably regularized. More precisely, in (2.22) an additional dissipative term λ 2τ ∇y−∇y is considered, where λ > 0 is a regularization parameter (called ε there), and in (2.24) the dissipation rate ξ is replaced by a smoothly truncated version ξ 1+λξ . On the other hand, the term ∂ F W cpl (∇y τ , s) ds. One of the main novelties in the present work is that the same result on existence and time-discrete approximations is achieved for the simpler, explicit, thermo-mechanical coupling term ) and without regularizing terms.

2.3.
Passage to linearized thermoviscoelasticity. We are now interested in the passage to a small strain regime ε → 0. This is induced by small external loading, boundary traction, and external temperature as ε → 0, see (2.16a) and the boundary conditions in (2.17). In a similar fashion, we suppose that the initial values are small when ε is small, see (2.18). At this point, we additionally need to require (W.4) W el (F ) ≥ c 0 dist 2 (F, SO(d)) for all F ∈ GL + (d), and W el (F ) = 0 if F ∈ SO(d); (H.4) H(0) = 0; (C.6) The heat capacity c V (F, θ) := −θ∂ 2 θ W cpl (F, θ) for F ∈ GL + (d) and θ > 0 as well as ∂ F θ W cpl can be continuously extended to GL + (d) × R + ; (C.7) For all F ∈ GL + (d) and θ > 0 it holds that ∂ F F θ W cpl (F, θ) ≤ C0 max{θ,1} . In order to ensure the compatibility of (W.4) with (W.3), we assume C 0 ≥ c 0 (d+1) from now on. We write the equations (2.16) and the boundary conditions (2.17) equivalently in terms of the rescaled displacement field u = ε −1 (y − id) and the rescaled temperature µ = ε −α θ. Then, for α ∈ [1,2], rescaling the equations by ε −1 and ε −α , respectively, and letting ε → 0 we obtain, at least formally, the system and initial conditions Here, e(u) := 1 2 (∇u + (∇u) T ) denotes the linear strain tensor, and the tensors of elasticity and viscosity coefficients are defined by Moreover, the heat conductivity tensor and the heat capacity (see also (C.6)) at zero temperature and the natural material state are given by Eventually, we have the α-dependent quantities where B (α) plays the role of a thermal expansion matrix. Notice that in the formal analysis above the elasticity tensor does not depend on the coupling potential. This is due to the fact that ∂ 2 F W cpl (Id, 0) = 0, see (C.3).
Although the nonlinear system is given for a nonsimple material, in the limit we obtain equations without spatial gradients of e(u). This is a consequence of the growth conditions in (H.3). Moreover, there is an interesting decoupling effect due to the different scaling of coupling terms in the mechanical and the heat-transfer equation, expressed in terms of the α-dependent quantities in (2.34). This computation also shows why we restrict to the range α ∈ [1,2]. Indeed, formally, we would have B (α) = +∞ for α < 1 while C (α) The second main goal of this article is to show that the above formal linearization can be made rigorous. In the case α ∈ [1, 2), our analysis requires a regularization of the thermal evolution. More precisely, we define the k-th thermal step through Due to the different scaling of the mechanical and the heat-transfer equation, the existence of a solution to the scheme is more delicate for ε small and α = 2. More specifically, we need higher integrability of W in defined in (2.12) in L 2/α which can be guaranteed by the choice in (2.35). We refer to Subsection 3.3 below for details. We also emphasize that for α = 2 no regularization is applied as ξ reg α = ξ. A similar result as Theorem 2.3 holds true in the regularized setting. Proposition 2.5 (Vanishing time-discretization in the regularized nonlinear setting). Given any T > 0 there exists ε 0 , τ 0 ∈ (0, 1] such that for any τ ∈ (0, τ 0 ) and ε ∈ (0, ε 0 ) the following holds: (i) (Existence of the scheme) The sequences y (ii) (Convergence to solutions) The convergences (2.27)-(2.28) towards a limit (y ε , θ ε ) hold true for the interpolations of the steps from (i). Here (y ε , θ ε ) is a weak solution to the system in a sense similar to Definition 2.2, namely (2.19) is satisfied and (2.20) holds with ξ replaced by ξ reg α . We will prove that (2.29) admits a unique weak solution and that solutions of the above described regularization guaranteed by Proposition 2.5(ii) converge to the solution of (2.29) in a suitable sense. Setting we have the following definition of weak solutions for the linearized system. Definition 2.6 (Weak solution of the linearized system). A couple (u, µ) : I ×Ω → R d ×R is called a weak solution to the initial-boundary-value problem (2.29)-(2.31) if u ∈ H 1 (I; H 1 Γ D (Ω; R d )) with u(0, ·) = u 0 , µ ∈ L 1 (I; W 1,1 (Ω)) with µ ≥ 0 a.e., and if it satisfies the identities Indeed, it is a standard matter to check that sufficiently smooth weak solutions lead to the classical formulation (2.29). Next, we state the relation between time-continuous or time-discrete solutions of the nonlinear system and solutions to (2.29)-(2.31).
Theorem 2.7 (Passage to linearized thermoviscoelasticity). Under the above assumptions we have: (i) There exists a unique weak solution (u, µ) to (2.29)-(2.31) in the sense of Definition 2.6.
Note particularly that we obtain strong convergence of strains and strain rates. Finally, we study the relation between the time-discrete solutions in the nonlinear and the linear setting, as well as the convergence of time-discrete solutions in the linearized setting under vanishing time-discretization. Theorem 2.8 (Passage to linearized thermoviscoelasticity, time-discrete solutions). The following properties hold: is the unique weak solution of (2.29)-(2.31) in the sense of Definition 2.6. Apart from the time derivative, the convergences in (2.42) also hold for the other interpolations.
is the unique solution of the minimization problem (2.43) From the a priori bounds in the nonlinear setting, we will be only able to prove that C D e(δ τ u Consequently, the functional in (2.43) might not be well-defined on H 1 (Ω) for α = 2. Nevertheless, for sufficiently smooth Γ, smooth functions f and θ , and Γ D = Γ, it follows by elliptic regularity theory that C D e(δ τ u is a minimizer of (2.43) also for α = 2. Section 3 is devoted to existence of the staggered time-incremental scheme leading to Theorem 2.3(i) and Proposition 2.5(i). Then, in Section 4 we pass to the limit τ → 0 and show Theorem 2.3(ii) and Proposition 2.5(ii). Eventually, in Section 5 we address the limit ε → 0 and prove Theorems 2.7 and 2.8.

Staggered time-incremental scheme
This section is devoted to the analysis of the staggered time-incremental scheme described in the previous section. Let us start with some fundamental auxiliary results.
If W additionally satisfies (W.4), there exists a universal constant C and a constant C * Proof. For a proof of the first part we refer to [33, Theorem 3.1] relying on a result in [22]. The second part can be found in [19,Lemma 4.2], where S M δ therein simply corresponds to M(y) ≤ M δ 2 .
Proof. The statement can be found in [33,Corollary 3.4], relying on the result in [36].

Lemma 3.3 (Heat conductivity). For any
Proof. By Lemma 3.1 we see that (∇y(x)) −1 exists for every x ∈ Ω which shows the well-definedness of K(∇y, θ), see (2.11). The bound in (3.4) is a direct consequence of the latter three estimates in (3.1) combined with (2.10).
Lemma 3.4 (Estimate on coupling potential). For all F ∈ GL + (d) and θ ≥ 0 it holds that Proof. We start by proving (3.5) for θ ≤ 1. To this end, we use that ∂ F W cpl (F, 0) = 0 (see (C.3) and comments thereafter), (C.5), and apply the Fundamental Theorem of Calculus to get On the other hand, for θ ≥ 1, we use ( 3.1. Existence of solutions to time-discretized schemes. In this subsection, we show that for sufficiently small τ ∈ (0, 1], depending on a bound of the mechanical energy of the previous deformation, a single time step of the staggered time-discretization scheme introduced in (2.22)-(2.24) is well-defined.
be as in (2.23). Suppose that we have already constructed y (We always add an index ε for clarification.) We first investigate the existence of the k-th mechanical step.
attains a solution. Furthermore, such a minimizer y Proof. We provide the proof for the coercivity in W 2,p (Ω; R d ). The remaining argument coincides with the one in [33, Proposition 4.1], and we only include a brief sketch for convenience of the reader. Let us shortly writeỹ := y . Let (y n ) n ⊂ Y id be a minimizing sequence for the problem in (3.6). Usingỹ as a competitor we can, without loss of generality, assume that for all n ∈ N M(y n ) + W cpl (y n ,θ) and therefore By (3.5), the Fundamental Theorem of Calculus, Young's inequality with constant c M /(2τ ), and (W.3) we derive for C M sufficiently large depending on M and c 0 . By using Poincaré's inequality, the trace estimate on the bulk and surface term, respectively, and Young's inequality with constant c M /(4Cτ ε) we derive that Combining the aforementioned estimates with (3.8), and using M ≥ W el we get For τ 0 sufficiently small such that C M τ 0 ≤ 1/2, Lemma 3.1 then shows the desired coercivity in W 2,p (Ω; R d ).
The functional is weakly lower semicontinuous on W 2,p (Ω; R d ) by the convexity of H, see (H.1), the com- From the previous proof, we directly deduce the following. (3.10) Proof. Let C M as in (3.9). For τ 0 sufficiently small with respect to C M we derive ε,τ in place of y n ) by 1 − C M τ we get the desired estimate, up to changing the constants C M and c M .
Remark 3.7. By 1 ∧ s ≤ √ s for s ≥ 0, (2.14), (2.15), by the definition below (2.23), and by a standard application of Hölder's inequality, we deduce from (3.10) that In fact, we have f (Ω) dt and a similar computation holds for g.
In the next lemma we discuss the well-definedness of the thermal step.
Proposition 3.8 (Thermal step). For any M > 0 there exists τ 0 ∈ (0, 1] such that if the minimizer given by Proposition 3.5 exists, τ ∈ (0, τ 0 ), and M(y is finite on H 1 (Ω) and attains a unique minimizer θ ε,τ ). Remarkably, the nonnegativity constraint in the minimization problem (2.24) does not influence the stationarity condition (3.11). We also emphasize that in contrast to [33] we can ensure uniqueness of the minimizer. This is due to the fact that we use a simpler (explicit) thermo-mechanical coupling term in the scheme, see Remark 2.4 for details. Proof.
Step 1 (Finiteness): We start by showing that all terms of T are well-defined and integrable. First, by (2.14) we find that a.e. on Ω which both lie in L 1 (Ω) by Hölder's inequality. By Lemma 3.3, ) is well-defined in Ω, and the corresponding term in T is integrable. Finally, by (3.5), (D.2), (2.9), and the second estimate in (3.1) we get that the term h τ defined in (2.25) satisfies h τ (y ) ∈ L ∞ (Ω), i.e., the third term is also well-defined. This completes the proof of the well-definedness of T .
Step 2 (Existence): The functional is coercive on H 1 ε,τ , s) ds ≥ c0 2 θ 2 by (3.12), the estimate ∇θ · K(∇y (3.4), and the fact that all other terms are either nonnegative or linear in θ. Moreover, the functional is weakly lower semicontinuous on H 1 + (Ω). To see this, we again use (3.4), the weak continuity of the trace operator in H 1 (Ω), and the fact that all other bulk terms are continuous in L 2 (Ω) by the reasoning in Step 1. This shows that a minimizer θ (k) ε,τ exists.
Step 3 (Euler-Lagrange equation): In order to prove (3.11) for test functions ϕ ∈ H 1 (Ω) which are not constrained to be nonnegative, we extend the minimization problem (2.24) to possibly negative functions θ ∈ H 1 (Ω) and we show that θ (k) ε,τ minimizes T on H 1 (Ω). To this end, recalling that W in (F, 0) = 0 for F ∈ GL + (d) (see below (2.12)), we continuously extend W in to negative temperatures by setting W in (F, θ) = 0 for θ < 0. It now suffices to check that there exists a constant c M > 0 such that for all where θ − := max{−θ, 0} and θ + := max{θ, 0}, i.e., θ = θ + − θ − . This guarantees that minimizers of T are nonnegative, and because T is strictly convex (to see this, use (2.13)), θ (k) ε,τ is its unique minimizer on H 1 (Ω). Once this is achieved, in view of (2.25) and (3.12), by taking first variations it is a standard matter to check that (3.11) holds true. Hence, it remains to prove (3.13). First, as θ (3.14) Next, by using (3.4) we see that where for brevity we have set K . Eventually, we consider the terms involving h τ and w . At this point, our argument for proving nonnegativity of the temperature is more delicate compared to [33] as we use the backward approximation θ cpl is symmetric which implies with the chain rule that , we derive by the second and third bound in (3.1), (3.5), (3.17), and the fact that t ∧ 1 ≤ √ t for all t ≥ 0 that for C M > 0 sufficiently large. Let us further defineĊ ε,τ . By the symmetry of ∂ CŴ cpl we have for all F ∈ GL + (d), G ∈ R d×d , and θ ≥ 0 where, again, C := F T F . We now use this identity with F = ∇y ε,τ . By (2.14), (3.17), (3.18), and Young's inequality with constant τ it follows that Choosing τ 0 sufficiently small such that C 2 M τ 0 ≤ c 0 , we derive by (D.1), (D.2), and (2.9) for all τ ∈ (0, τ 0 ) that Combining the estimates (3.14)-(3.20) leads to (3.13) which concludes the proof.
Remark 3.9 (Nonnegativity of temperature without dissipation rate). To derive estimate (3.19), it was essential that ξ(F,Ḟ , θ) ≥ c|Ḟ T F + F TḞ | q for some q > 1 . The pointwise nonnegativity can still be established only under the assumption that ξ ≥ 0, at the expense of assuming that M(y ε,τ ) ≤ η for some η sufficiently small, and that W additionally satisfies (W.4). Indeed, in this case we can show a.e. in Ω which along with ξ ≥ 0 implies (3.20). To see this, by (2.14), (3.3), and (3.5), we can estimate Since C * η → 0 as η → 0, (3.21) indeed follows for η small enough. This property will be exploited in the adaptions to the case α < 2 in Subsection 3.3 below.
For any y (k) ε,τ and θ (k) ε,τ as given in this subsection, we define from now on

3.2.
Well-definedness of the scheme. For fixed time horizon T > 0 and time step τ ∈ (0, 1] small enough, we will now prove the well-definedness of the staggered time-discretization scheme described in the previous subsection. In this part, we are interested in the large-strain setting, and treat the case ε = 1 and α = 2, where ξ is not regularized. For later purposes, we again include ε in the estimates. (The reader only interested in large strains, can readily set ε = 1.) As before, we assume for the sake of simplicity that T /τ is an integer. Although not being necessary, for convenience we suppose that (W.4) holds. At the end of the subsection, we briefly indicate the changes if (W.4) is not assumed. We start with a bound on the total energy E defined in (2.15). We also need to take the work of the external forces into account. To this end, similar to the notation in (2.23), we consider for each t ∈ I the functionals (t) on H 1 (Ω; R d ) defined by and, hence, by the Fundamental Theorem of Calculus in W 1,1 (I; L 2 (Ω)) and W 1,1 (I; L 2 (Γ N )) we get for a constant C T only depending on T . Given the sequences y ε,τ for some k ∈ {1, . . . , T /τ }, as described in Subsection 3.1, we define for l ∈ {0, . . . , k} and observe the following relation between F (l) and the total energy E(y Lemma 3.10. There exists a constant C T > 0 only depending on T such that for all l ∈ {0, . . . , k} with k ∈ {1, . . . , T /τ } it holds that with C f,g as defined in (3.23).
Proof. By y Hence, by (3.24) and Young's inequality with constant λ/ε (to be chosen below) it follows that Now, take λ small enough such that Cλ ≤ 1 2 . Then, by the definition of F (l) we discover and the statement follows.
We now proceed with the bound on the total energy. For definiteness, we set (t) = 0 for t / ∈ I. ε,τ , as described in Subsection 3.1, for some k ∈ {1, . . . , T /τ } exist satisfying F (l) ≤ M for all l = 0, . . . , k − 1 with F (l) defined in (3.25), it holds that where C is a universal constant, C T a constant only depending on T , and Proof.
Step 2: We continue by bounding the first two sums on the right-hand side of (3.30) from below. By the convexity of H (see (H.1)) it follows for l ∈ {1, . . . , k} that ε,τ − ∇ 2 y (l) ε,τ ). Integrating the above inequality over Ω and summing over l = 1, . . . , k leads to where we recall the notation in (2.4). By using the piecewise affine functionŷ ε,τ introduced in (2.26), and that W el is Gateaux differentiable (see [33,Proposition 3.2]) we get that Together with Lemma 3.1 we get that there exists a compact convex set K, only depending on M , T , f , and g, such that ∇y (l) ε,τ ∈ K a.e. on Ω for all l ∈ {0, . . . , k}. Then, by the regularity of W el , setting C M := sup F ∈K |∂ F F W el (F )|, we can estimate for any t ∈ [(l − 1)τ, lτ ] with l ∈ {1, . . . , k} that Combined with (3.32) this leads to In a similar fashion, using the first bound in (C.5), we can estimate Now, employing (3.31), (3.33), and (3.34) in (3.30), and using the definition of the total energy E we conclude that Step 3: It remains to estimate the last two terms on the right-hand side of (3.35). By the nonnegativity of θ Note that for any l ∈ {1, . . . , k} and t ∈ ((l − 1)τ, lτ ) we have that δ τ y (l) ε,τ =ẏ ε,τ (t). Consequently, integration by parts yields By Poincaré's inequality, (3.2), and (W.4), for t ∈ [(l − 1)τ, lτ ], we have . Therefore, by Lemma 3.10, (3.24), and √ s ≤ s/ε for all s ≥ ε 2 we get Then, using an index shift and C f,g ≤ 2 for a possibly larger C T > 0. We plug this into (3.37) and use (3.36) to estimate the terms on the right-hand side of (3.35), which by (3.25) concludes the proof.
We proceed with a bound on the (discrete) strain rates V k defined in (3.27).
We are ready to prove the well-definedness of our time-discretization scheme, i.e., Theorem 2.3(i). At the same time, we will also derive two useful a priori bounds, namely on the total energy and on the (discrete) strain rate, respectively. Theorem 3.13 (Well-definedness of the scheme). For any T > 0 there exist a constantC T > 0, corresponding constants as well as constants C M > 0 and τ 0 ∈ (0, 1] depending on M such that the following holds true: For each τ ∈ (0, τ 0 ) such that T /τ ∈ N the sequences y Proof.
Step 1: Let C T be the maximum of the constants C T from Lemmas 3.10, 3.11, 3.12, and equation (3.24), and let C be the universal constant of Lemma 3.11. DefineC T = max{2CT C T , 2C T , 2}, and let M and M be as in (3.42). Then, let C M > 0 be the maximum of the constants C M from Lemmas 3.11 and 3.12. Moreover, let τ 0 ∈ (0, 1] be chosen sufficiently small so that Lemma 3.5, Lemma 3.6, Proposition 3.8, Lemma 3.11, and Lemma 3.12 hold true (all applied for M from (3.42)). In place of (3.43), we focus on showing ε,τ = θ 0,ε . Given K ∈ {1, . . . , T /τ }, let us assume that the statement as well as (3.45) hold true for K − 1. We now show that the statement holds true for K. Applying first Proposition 3.5 and then Proposition 3.8 we see that y (k) ε,τ and θ (k) ε,τ exist, where for both propositions we use the induction hypothesis (3.43) for K − 1.
Step 2: In this step, we prove that for τ 0 small enough we have that where V K is defined in (3.27). By ε 2 M ≤ M , Remark 3.7, and (3.23) there exists a constantC M only depending on M such that 1 where we again used the hypothesis (3.43) for K − 1. Hence, by possibly further decreasing τ 0 (depending only on M , f , g, T , and the initial values) we can ensure that Furthermore, by possibly decreasing τ 0 (depending only on M , u 0 , µ 0 , f , g, and T ) and using the hypothesis (3.44) for K − 1 in place of K we get C M τ V K−1 ≤ ε 2 /2. Consequently, combining the previous estimates and using τ V K = τ V K−1 + ∇y , the desired bound (3.46) follows.
Step 3: By hypothesis the energy bound in (3.45) is satisfied for k ∈ {0, . . . , K − 1}. Consequently, Lemma 3.11 applies for any k ∈ {0, . . . , K}. By (3.46) we have that We now use the following discrete version of Gronwall's Lemma: if β > 0, (a l ) l is a nonnegative sequence, (b l ) l ⊂ (0, 1/2), and Indeed, as b l ≤ 1/2, we get a k ≤ 2β + k−1 l=0 2b l a l , and then the statement follows from the elementary discrete Gronwall inequality. We apply this result for where we note that b l ≤ 1/2 for all l, provided that τ 0 is chosen small enough depending on f and g. In view of (3.24), we then see that (3.45) for K is true. Finally, (3.44) directly follows from the application of Lemma 3.12 and the fact that the last term in (3.41) can be controlled by E, see e.g. Remark 3.7.
Eventually, if (W.4) is not assumed, we get additional additive constants in Lemma 3.10 and in the derivation of (3.40), leading to an additional constant in (3.26) which however does not scale as ε 2 . This does influence the proof of the well-definedness, only the scaling of the energy in terms of ε.

3.3.
Adaptions for exponents α < 2. In this subsection, we prove Proposition 2.5(i). This part can be skipped by a reader only interested in the proof of Theorem 2.3. In the previous subsection, we have already established the well-definedness of the scheme in the large-strain setting, as well as the energy bound (3.43). The latter will be essential to obtain a priori bounds for the limit passage τ → 0 in Section 4. In the case α < 2, for the passage to the linearized setting ε → 0, however, the bound (3.43) and the induced a priori bounds are not expedient. This is due to the different scaling of the internal and mechanical energy, being of order ε α and ε 2 , respectively. To this end, it is necessary to establish energy bounds for rescaled versions of the energy functionals from Subsection 2.1, namely M ε := 1 where both 'types of energy' are of the same order. Controlling this energy is more delicate compared to Proposition 3.11, as the mechanical and thermal equation (3.28)-(3.29) scale with different powers of ε and cannot simply be added up. Therefore, novel ideas are required to control the contributions of W cpl and ξ. To achieve this, higher integrability of W in in L 2/α is needed which can be guaranteed by using the regularization of ξ reg α introduced in (2.35). This in turn induces new challenges for the analysis of the time-discrete scheme since showing the nonnegativity of the temperature in the thermal step, see Proposition 3.8, is more delicate. For this, it will be essential to assume that strains are small, i.e., we suppose that the parameter ε ∈ (0, 1] is sufficiently small. Note that for the entire subsection we can assume that α ∈ [1, 2) since in the case α = 2 there is no regularization of the dissipation rate, the existence of the scheme is already guaranteed by Theorem 2.3(i), and also an energy bound for E ε follows already from (3.43). The mechanical step is not affected by the regularization, but Proposition 3.8 needs to be adapted. ε,τ satisfying (3.11) for all ϕ ∈ H 1 (Ω) with ξ replaced by ξ reg α .
Proof. As ξ ≥ ξ reg α , the existence and uniqueness of θ (k) ε,τ follows by the same reasoning as in Steps 1-2 of the proof of Proposition 3.8. Since ξ reg α ≥ 0, the nonnegativity of the temperature follows by Remark 3.9 for ε 0 sufficiently small, where we use M(y Our next goal is to adapt Proposition 3.11 to the present setting. As a preparation, supposing that for k ∈ {0, . . . , T /τ } the steps y (k) ε,τ and θ (k) ε,τ exist, we define where is defined in (3.22). By repeating the proof of Lemma 3.10 we find Proof. As a preliminary step, we show that the assumption F Consequently, we can choose ε 0 sufficiently small such that M(y (l) ε,τ ) ≤ 1 for l ∈ {0, . . . , k − 1}. Then, we apply (3.10) for M = 1 to get τ 0 such that for τ ∈ (0, τ 0 ] it holds that where C 1 is a universal constant. By (2.14) and the fact that 1 ∧ t ≤ t 1/α for t ≥ 0 we find , (3.52) such that, dividing the above estimate by ε 2 and recalling (3.48) as well as Remark 3.7 we get This along with (3.51) shows that, possibly decreasing ε 0 , we have M ε (y (l) ε,τ ) ≤ 1 for all l ∈ {0, . . . , k}. This induces that in the following proof the constants coming from Lemmas 3.1, 3.6, and 3.12 are universal and denoted by C 1 .
As in the proof of Proposition 3.11, the strategy relies on a suitable test of the mechanical and the thermal equation, see also (3.28)-(3.29). In contrast, however, the resulting equations cannot be summed up, but have to be treated separately. This will allow us to show the estimates where C is a universal constant and C T possibly depends on T . Then, in view of (3.48), (3.49), and (3.51) for l = 0, the result follows by summing up the two estimates. We now treat (3.53) and (3.54) separately.
Step 1 (Inductive bound on the mechanical energy): The first part is achieved by bounds similar to the ones obtained in the proof of Proposition 3.11, and we therefore refer to estimates therein. Testing (3.7) for l in place of k with z = δ τ y (l) ε,τ we get (3.28). Then, multiplying both sides by τ ε 2 , summing over l = 1, . . . , k, and using W = W el + W cpl , (3.31), as well as (3.33), by possibly increasing C 1 we derive that Here, we also used the definition of V k in (3.27), and the fact that the initial value is given by y 0,ε . By (3.5), (3.52), and Young's inequality it follows that τ Using the definition of the total energy in (3.48), the definition of W el ε , and (W.4), we insert this in (3.56) to obtain Next, by repeating the argument in (3.37)-(3.40) we find Employing (3.57) and (3.59) in (3.55), and using again (3.50) we arrive at (3.53).
as well as constants ε 0 , τ 0 ∈ (0, 1] depending also on M such that the following holds true: for each ε ∈ (0, ε 0 ) and τ ∈ (0, τ 0 ) such that T /τ ∈ N the sequences y Proof. The theorem is a consequence of Lemma 3.15 and Lemma 3.12. The argument is similar to the one of Theorem 3.13 and we therefore omit the details. Let us just mention that the energy bound follows in the same way by induction, up to using different values β, a l , and b l in (3.47), and by employing (3.50) in place of Lemma 3.10. Based on the uniform energy bound, Proposition 3.14 indeed shows that the scheme is well-defined, provided that ε 0 is chosen sufficiently small. Eventually, the bound on the strain rates follows from Lemma 3.12, see particularly (3.57) in the previous proof.
Remark 3.17. Due to our regularization of the dissipation rate, in the case α ∈ [1, 2) we obtain the additional control for a constant C α depending on α, but independent of ε and τ , where we shortly wrote µ ε,τ := ε −α θ ε,τ (see also (2.26) for the definition of θ ε,τ ). This follows by using the positive term on the right-hand side of (3.66).
Proof. Let us first suppose that (W.4) holds. The energy bound on E ε for α = 2 and α ∈ [1, 2) follows directly from Theorem 3.13 and Theorem 3.16, respectively. The first two estimates can be shown from the uniform bound on the energy, (W.4), (3.2), (5.21), and Poincaré's inequality. In a similar way, the bound on w ε,τ in (3.70d) follows from the bound on the total rescaled energy, (3.48), and Hölder's inequality. Then, the proof of (3.70d) is concluded by (2.14). Finally, (3.70c) is a direct consequence of (3.44) and (3.68), respectively. Eventually, for α = 2 and ε near 1, the result also holds without assuming (W.4) as (W.3) allows us to derive (3.70a)-(3.70b) with C in place of Cε 2/p and Cε on the right-hand side.
In order to pass to the limit τ → 0 in the next section, we need additional a priori bounds for the temperature. Testing the equation (3.11) turns out to be delicate since for α = 2 the viscous dissipation ξ(∇y ) is only bounded in L 1 (I ×Ω). Thus, to obtain improved estimates that work in this case, we employ special test functions developed by Boccardo and Gallouët [10] for parabolic equations with a measure-valued right-hand side, see also [16]. We follow here the approach in [33]. However, almost complete proofs are provided since compared to their setting we perform the estimates in the time discrete setting and we derive fine estimates in terms of the small parameter ε. Lemma 3.19 (Weighted L 2 -bound). For any η ∈ (0, 1) there exists a constant C independent of ε, τ , and α such that Actually, this statement is needed only for α = 2 since for α ∈ [1, 2) we have a better estimate by Remark 3.17. Still, we state and prove the result for any α since the following argument does not depend on α. Proof.
Step 2: We are now ready to show (3.71). In this regard, first notice the following relation between ∇w (k) ε,τ and ∇θ By (3.5), (C.5), and (3.70a), we find that the abbreviationsW ∈ [c 0 , C 0 ], respectively. Then, using (3.77), Lemma 3.3, and the energy bound from Lemma 3.18 we see that there exists a constant c > 0 such that We now control the second term above. By t ∧ 1 ≤ t p−1 p for all t ≥ 0 and Young's inequality with constant λ ∈ (0, 1) (to be chosen later), we estimate by (2.14) Using the elementary fact Theorem 3.20 (Further a priori bounds on the temperature). For any q ∈ [1, d+2 d ) and r ∈ [1, d+2 d+1 ) there exist constants C q and C r , respectively, both independent of ε and τ such that Moreover, we can find a constant C independent of ε and τ such that Proof. Let q, r be as in the statement. As w (k) ε,τ ∈ H 1 (Ω) (see (2.24)), it follows that w ε,τ L ∞ (I;H 1 (Ω)) < ∞. Therefore, by using the a priori estimate 1 + ε −α w ε,τ L ∞ (I;L 1 (Ω)) ≤ C + L d (Ω) (see (3.70d)) as well as Lemma 3.19, we can repeat the argument from the proof of [33, Proposition 6.3, equation (6.6)] for ε −α w ε,τ in place of w ε , cf. also Remark 3.21 below. This gives the existence of constants C q , C r such that By (2.14) we then directly see that (for a possibly larger C q ) To conclude the proof of (3.81)-(3.82), it remains to control the gradient of the temperature. Employing the relation between ∇w Consequently, using t ∧ 1 ≤ t p−1 p for all t ≥ 0 and Young's inequality with powers p/(p − r) and p/r we derive that As r was chosen strictly smaller than d+2 d+1 , we see by p ≥ 2 that Consequently, multiplying (3.86) with τ , summing over k = 1, . . . , T /τ , and using (3.70a), (3.84), and (3.85) we conclude the proof of (3.82). Here, we again used α ≤ 2. Lastly, we show (3.83). Testing (3.11) for the k-th step with arbitrary ϕ ∈ W 1,∞ (Ω), and using the shorthand notation for h (k) ε,τ and K (k−1) ε,τ from (3.73), we see by (3.4) and the continuity of the trace operator By the arbitrariness of ϕ this shows that We have already seen in the proof of Lemma 3.19 (see in particular (3.76)) that Consequently, by (3.81)-(3.82) for q = r = 1 and (3.87) the desired bound (3.83) follows.

Existence of solutions in the nonlinear setting
In this section we pass from time-discrete to time-continuous solutions by letting τ → 0 and establish Proposition 2.5(ii). Notice that for the special case α = 2 and ε = 1 this will lead to Theorem 2.3(ii). For the deformation and the momentum balance we can closely follow [33,Section 5], and therefore proofs are omitted or sketched only. For the limit passage in the heat equation, however, our arguments are different as we work without regularization terms, cf. Remark 2.4. We first use the a priori estimates on the interpolants in order to extract convergent subsequences. Afterwards, we pass to the limit in the discretized weak forms of the momentum balance and the heat equation. Here, the most delicate term is the dissipation rate ξ which is quadratic inḞ . Therefore, strong convergence in L 2 (I; H 1 (Ω)) for the strain rates is required.
We start with the convergence of the deformations under vanishing time-discretization.
Proof. First, (4.1a) follows from the a priori estimates (3.70a), (3.70c) and by Banach's selection principle. For (4.1b), one uses the embedding W 2,p (Ω; R d ) ⊂ C 1,1− d p (Ω; R d ) to obtain a Hölder estimate in space and (3.70c) for a Hölder estimate in time. Then, by an interpolation estimate one can show that the sequence is bounded in C γ (I; C 1,γ (Ω; R d )) for some γ > 0, and the uniform convergence of the gradients follows then from the Arzelà-Ascoli theorem. We refer to [33, Proof of Proposition 5.1, Step 1] for more details. To conclude that the first convergences in (4.1a) and (4.1b) also hold for y ε,τ or y ε,τ one again uses (3.70b)-(3.70c) to see ∇ŷ ε,τ − ∇y ε,τ L ∞ (I;L 2 (Ω)) ≤ Cτ We proceed with the convergence of the temperatures.
We are ready to pass to the limit in the time-discrete mechanical evolution. Proof. The statement is proved in [33, Proof of Proposition 5.1, Step 2] and we include a sketch for the reader's convenience. For y ∈ Y id we define a functional on X := W 2,p (Ω; R d ) by Note that H is a hemicontinuous and monotone operator as H is convex. We further choose b ετ , b ε ∈ X * such that (3.7) can be written as H(y ε,τ ), z = b ετ , z (4.8) for all z ∈ Y 0 and (2.19) can be written as for all z ∈ Y 0 . Note that (4.8) holds by Proposition 3.5, and that our goal is to confirm (4.9). First, b ετ * b ε weakly* in X * for τ → 0 as in each of the three terms of b ετ (i.e., ∂ F W , ∂Ḟ R, and (k) τ , respectively, see (3.7)) one can pass to the limit by using weak convergence of (∇ẏ ε,τ ) τ in L 2 (I; H 1 (Ω; R d )) (see (4.1a)), uniform convergence of (∇y ε,τ ) τ , (∇y ε,τ ) τ on I ×Ω (see (4.1b)), and pointwise a.e. convergence of (θ ε,τ ) τ on I × Ω (up to a subsequence, see (4.2b)). At this point, we use in particular that ∂Ḟ R is linear in ∇ẏ ε,τ and that ∂ F W (y ε,τ , θ ε,τ ) is bounded due to (W.1), (3.5), and (3.70a). Moreover, due to uniform convergence of the gradients we also have b ετ , y ε,τ → b ε , y ε . We now use Minty's trick for the monotone operator H: identity (4.8) and the convergences y ε,τ y ε weakly in X, b ετ * b ε weakly* in X * , and b ετ , y ε,τ → b ε , y ε imply H(y ε ) = b ε as elements of X * , i.e., (4.9) holds.
Consequently, we get that ξ reg α (∇y ε,τ , ∇ẏ ε,τ , θ ε,τ ) τ is equi-integrable. Using the pointwise convergence of (∇y ε,τ ) τ and (θ ε,τ ) τ as well as the continuity of ξ reg α , we can also pass to the limit in the ξ reg α -term by an application of Vitali's convergence theorem. As we passed to the limit in each term, the proof is concluded.

Passage to the linearized system
This section is devoted to the proofs of Theorems 2.7-2.8. In the following, let (ε k ) k and (τ k ) k be sequences with ε k → 0 and either τ k = τ constant or τ k → 0. Suppose that initial data (y 0,ε k , θ 0,ε k ) as in (2.18) are given. For brevity, we denote the corresponding time-discrete interpolations by y k := y ε k ,τ k , y k := y ε k ,τ k , andŷ k :=ŷ ε k ,τ k , see (2.26). A similar shorthand notation is also used for the interpolation of the temperatures as well as the internal energies. Recall that the objects exist by Proposition 2.5(i). In a similar way, we denote the time-continuous solutions obtained in Proposition 2.5(ii) by (y ε k , θ ε k ). It will be useful to use a similar notation for the rescaled quantities: for time-discrete solutions we define and for time-continuous solutions we let For any v ∈ L 2 (I; H 1 (Ω; R d )) we denote the symmetrized gradient by e(v) := 1 2 (∇v + ∇v T ). Finally, all constants we encounter in this section are implicitly assumed to be independent of k.
We start with compactness results for the rescaled quantities which directly follow from the a priori estimates for the nonlinear system. Recall the definition of H 1 Γ D in (2.36). Lemma 5.1 (Compactness for the rescaled displacements). There exist u,ũ ∈ H 1 (I; H 1 Γ D (Ω; R d )) with u(0) =ũ(0) = u 0 such that, up to possibly taking a subsequence, it holds that Moreover, if τ k → 0, we also have Later, by uniqueness of the solution to the linear system, we will see that actually u =ũ.
Now, (5.2) and the other properties ofũ again follow by the Aubin-Lions' theorem.

(5.7)
Moreover, if τ k → 0, we also have µ k ,μ k → µ in L s (I × Ω). (5.8) Later, by uniqueness of the solution to the linear system, we will see that actually µ =μ.
Proof. Let r and s be as in the statement. The proof of (5.6) relies on the a priori bounds on the internal energy in Theorem 3.20, i.e., θ k L r (I;W 1,r (Ω)) + w k L s (I×Ω) + w k L r (I;W 1,r (Ω)) + ẇ k L 1 (I;W 1,∞ (Ω) * ) ≤ Cε α k . (5.9) In fact, we can follow closely the lines of the proof of Lemma 4.2, see Remark 4.3(ii). In particular, one first shows the convergence of the internal energies and then by (4.7) the convergence of the temperatures.
We now proceed with the proofs of Theorems 2.7 and 2.8 which we split into two subsections. 5.1. Proof of Theorem 2.7. We will only prove Theorem 2.7(iii) as item (ii) of the statement can be obtained along similar lines by performing the linearization directly in the weak formulation (2.19)- (2.20) in place of the Euler-Lagrange equations (3.7) and (3.11). Note that the proof of Theorem 2.7(iii) will also imply the existence statement in Theorem 2.7(i). In this subsection, we also address the uniqueness of the solutions to the linearized system. Proof. Let z be as in the statement. As z ∈ Y 0 , we can multiply (3.7) with τ k /ε k and sum over all steps 1, . . . , T /τ to get where τ k (t) := (l) τ k for t ∈ ((l − 1)τ, lτ ] and l ∈ {1, . . . , T /τ }. Our goal now is to show that (2.37) arises as the limit of the above equation as k → ∞. First, recalling (2.23) we can easily check that as k → ∞. By (H.3) for ∂ G H, (3.70a), and Hölder's inequality with powers p p−1 and p we derive that as p > d ≥ 2. We now address the coupling term. In view of ∂ F W cpl (Id, 0) = 0, (3.70a), and (3.5), a Taylor expansion implies 14) pointwise a.e. in I × Ω. Thus, by (5.1) and (5.8), along with t 2 ∧ 1 ≤ t s for t ≥ 0 for some fixed s ∈ (1, d+2 d ) it follows that Recalling By a Taylor expansion, (3.70a), and the fact that W el is C 3 we have Integrating the above inequality over I × Ω and using (3.70b) we get Note that the second identity is obtained by an elementary computation using the symmetries of D stated in (D.1). By (3.70a) and (5.1) we then see thaṫ C k 2e(u) weakly in L 2 (I × Ω; R d×d sym ). (5.19) Using (D.2) we also have that |D(C k , θ k )(∇z T ∇y k + ∇y T k ∇z)| ≤ 2C 0 ∇z L ∞ (Ω) ∇y k L ∞ (Ω) . Up to taking a subsequence (not relabeled), we can suppose that ∇y k → Id and θ k → 0 a.e. in I × Ω. Thus, Dominated Convergence implies D(C k , θ k )(∇z T ∇y k + ∇y T k ∇z) → D(Id, 0)(∇z + ∇z T ) = 2D(Id, 0)∇z strongly in L 2 (I × Ω; R d×d ). This along with (5.17) and (5.19) leads to Similarly as in Section 4, for the limit passage in the heat-transfer equation, we will need the strong convergence of the strain rates (∇u k ) k in L 2 (I; L 2 (Ω; R d×d )) since the dissipation rate is quadratic in ∇u k . We now improve the compactness in Lemma 5.1 as follows. At this state, we need the additional assumption (H.4) which combined with the bound on ∂ G H(G) from (H.3) leads to Lemma 5.4 (Strong convergence of the rescaled strains and strain rates). With u as in Lemma 5.1, up to possibly taking a subsequence, we havê u k (t) → u(t) strongly in H 1 (Ω; R d ) for all t ∈ I, ∇u k → ∇u strongly in L 2 (I; L 2 (Ω; R d×d )). (5.22) The first convergence also holds with u k or u k in place ofû k .

]) show
Step 2 (Convergence of elastic energies and dissipations): Our next goal is to show the reverse inequalities for the lim sup. To this end, we draw ideas from the proof of Lemma 4.5 and compare an energy balance on the nonlinear time-discrete level with a time-continuous energy balance in the linearized setting. First, recall from (4.13) that for K ∈ N with Kτ k ∈ [t, t + τ k ) it holds that where Λ > 0 does not depend on k. Here, we also used (2.9) to replace R by ξ. Now, in a similar fashion, testing (2.37) with z =u we see that For α = 1, by arguing similarly as in (5.14)-(5.15), and using (3.70a) as well as µ k → µ strongly in L 2 (I × Ω) by Remark 4.3(iii) we find where we also used the definition of B (α) in (2.34). For α ∈ (1, 2], (5.29) also holds (with B (α) = 0), since by Remark 3.21 we find that µ k is bounded in L q (I; L q (Ω)) for some q ∈ (2/α, 2], and therefore using t ∧ 1 ≤ t q/2 for t ≥ 0 and Young's inequality with constant ε αq/2 k we get Eventually, we get lim In fact, for the convergence of the elastic energy we repeat the Taylor expansion in (5.23) (with equality), and for the second-gradient term we get by (5.21), u 0 ∈ W 2,p (Ω; R d ), and p > 2 that Combining ( where we also used the definition of C D in (2.32).
Step 3 (Strong convergence): Strong convergence for u k in H 1 (Ω; R d ), i.e., the first part of (5.22), follows directly from (5.32), Korn's and Poincaré's inequality, and the fact that C W is positive definite on R d×d sym . In the same way we obtain convergence of u k by employing y k (t) in place of y k (t) in (5.23). Hence, the statement also holds forû k .
For the second part of (5.22), we will first show strong convergence of (Ċ k ) k defined in (5.18): by (D.2) we estimate By (5.33) for t = T , the pointwise convergence of (∇y k ) k and (θ k ) k to Id and 0, respectively (see (5.1)-(5.3) and (5.6)), and the already shown weak convergence ofĊ k towards 2e(u) (cf. (5.19)) we see that the above derived upper bound converges to 0 as k → ∞. Then, the desired strong convergence of (∇u k ) k is derived as follows: by using Poincaré's and Korn's inequality, (5.1), and (3.70a) we get This concludes the proof. Lemma 5.6 (Uniqueness of the linearized system). There exists at most one solution in the sense of Definition 2.6.
Proof. We start with α ∈ (1,2]. In this case, (2.37) is independent of the variable µ. We show uniqueness of u. To this end, we suppose that there exist two solutions u 1 , u 2 , and set u := u 1 − u 2 . Then u = 0 on I × Γ D and u(0) = 0. Subtracting the weak formulations (2.37) for both u 1 and u 2 , we see that for any z ∈ C ∞ (I × Ω; R d ) with z = 0 on I × Γ D it holds that By the arbitrariness ofφ it then follows for almost all t ∈ I that Ω C D e(u(t)) + C W e(u(t)) : ∇u(t) dx = 0.
As a(0) = 0, it follows that a = 0, and therefore u = 0. Now, given a unique u ∈ H 1 (I; H 1 (Ω)), we see that (2.38) is an equation in the variable µ only. More precisely, it corresponds to the weak formulation of a heat equation with L 1 -data. Uniqueness has been provided in [37,Proposition 1]. This finishes the proof in the case α ∈ (1, 2]. We now briefly give the argument for α = 1. In this case, (2.38) does not depend on u and uniqueness follows again from [37,Proposition 1]. Then, the term T 0 Ω B α µ : ∇z dx dt in (2.37) is only a datum, and uniqueness of u follows by repeating the argument starting with (5.37).
We are now ready to prove Theorem 2.7.
Proof of Theorem 2.7. We start with the proof of Theorem 2.7(iii). First, by Lemmas 5.1-5.2, we obtain limits u ∈ H 1 (I; H 1 Γ D (Ω; R d )) and µ ∈ L 1 (I; W 1,1 (Ω)). In view of (5.1), (5.6), and Lemma 5.4, the convergences stated in the statement hold, up to selecting a subsequence. In particular, (5.3) and (5.8) show that the convergence holds for all three different interpolations. By Propositions 5.3 and 5.5 we see that (u, µ) is a weak solution in the sense of Definition 2.6. As the weak solution is unique by Lemma 5.6, Urysohn's subsequence principle implies that the convergence holds for the whole sequence. This concludes the proof of Theorem 2.7(i),(iii).
We briefly describe the adaptions for Theorem 2.7(ii). First, in the compactness result we replace (5.1) and (5.6) by (5.2) and (5.7), respectively. The linearization of the mechanical equation and the heat-transfer equation in Propositions 5.3 and 5.5, respectively, can be derived along similar lines, by replacing the time discrete equations (5.11) and (5.34) with their time-continuous analogs in (2.19) and (2.20), respectively. In a similar fashion, for the proof Lemma 5.4, we use the time-continuous energy balance (4.11) in place of (5.26). The rest of the argument remains unchanged. 5.2. Proof of Theorem 2.8. We start with a Γ-convergence result. With the notation from Subsections 2.1-2.2 we define for k ∈ {1, . . . , T /τ } the functional E (k) ε (u) = +∞ if u / ∈ W 2,p (Ω; R d ) and if u ∈ W 2,p (Ω; R d ). Although the last term in (5.38) does not influence the minimizers of E (k) ε for fixed k, it is needed to ensure the boundedness of (|E (k) ε |) ε as ε → 0 along sequences of minimizers. Recall also E ε from (3.48).
Proposition 5.7. Suppose that sup ε>0 E ε (y Proof. All constants we encounter in this proof are implicitly assumed to be independent of ε. We will work with the equivalent representation E (k) ε (u) = ) and using the definition of W in in (2.12).
Step 3 (Γ-lim sup): Let u ∈ H 1 (Ω; R d ) with u = 0 on Γ D . By a standard approximation argument in Sobolev spaces we can assume without loss of generality that u ∈ C ∞ (Ω; R d ). Choose u ε = u for all ε. We only need to check the convergence of the energy. First, notice that by (5.21) and p > 2 1 where y ε := id + εu. By a Taylor expansion we also see that 1 Furthermore, using (D.1) we can write 1 ε 2 R(y (k−1) where C ε := (∇y The convergence of the terms (5.43) and (5.44) follows as in the previous step. This concludes the proof.
We close with the proof of Theorem 2.8.
Proof of Theorem 2.8. We prove the result by induction on k. For the base case k = 0, we only need to check the convergences and the energy convergence. In fact, setting u τ := µ 0 , this directly follows from (2.18) and repeating the argument in the Γ-lim sup above.