Existence results for a coupled viscoplasticdamage model in thermoviscoelasticity
Abstract.
In this paper we address a model coupling viscoplasticity with damage in thermoviscoelasticity. The associated PDE system consists of the momentum balance with viscosity and inertia for the displacement variable, at small strains, of the plastic and damage flow rules, and of the heat equation. It has a strongly nonlinear character and in particular features quadratic terms on the righthand side of the heat equation and of the damage flow rule, which have to be handled carefully. We propose two weak solution concepts for the related initialboundary value problem, namely ‘entropic’ and ‘weak energy’ solutions. Accordingly, we prove two existence results by passing to the limit in a carefully devised time discretization scheme. Finally, in the case of a prescribed temperature profile, and under a strongly simplifying condition, we provide a continuous dependence result, yielding uniqueness of weak energy solutions.
2010 Mathematics Subject Classification: 35Q74, 74H20, 74C05, 74C10, 74F05. 74R05.
Key words and phrases: Thermoviscoelastoplasticity, Damage, Entropic Solutions, Time Discretization.
Dedicated to Tomáš Roubíček on the occasion of his 60th birthday
1. Introduction
This paper focuses on two phenomena related to inelastic behavior in materials, namely damage and plasticity. Damage can be interpreted as a degradation of the elastic properties of a material due to the failure of its microscopic structure. Such macroscopic mechanical effects take their origin from the formation of cracks and cavities at the microscopic scale. They may be described in terms of an internal variable, the damage parameter, on which the elastic modulus depends, in such a way that stiffness decreases with ongoing damage. Plasticity produces residual deformations that remain after complete unloading.
Recently, models combining plasticity with damage have been proposed in the context of geophysical modeling [RSV13, RV16] and, more in general, within the study of the thermomechanics of damageable materials under diffusion [RT15]. Perfect plasticity is featured in [RSV13, RV16], where the evolution of the damage variable is governed by viscosity, i.e. it is ratedependent. Conversely, in [RT15] damage evolves rateindependently, while the evolution of plasticity is ratedependent. In a different spirit, a fully rateindependent system for the evolution of the damage parameter, coupled with a tensorial variable which stands for the transformation strain arising during damage evolution, is analyzed in [BRRT16]. Finally, let us mention the coupled elastoplastic damage model from [AMV14, AMV15], analyzed in [Cri16a, Cri16b, CL16]. While the first two papers deal with the fully rateindependent case in [Cri16a], in [CL16], the model is regularized by adding viscosity to the damage flow rule, while keeping the evolution of the plastic tensor rateindependent. A vanishingviscosity analysis is then carried out, leading to an alternative solution concept (‘rescaled quasistatic viscosity evolution’) for the rateindependent elastoplastic damage system. A common feature in [RSV13, RV16, Cri16a, CL16] is that the plastic yield surface, and thus the plastic dissipation potential, depends on the damage variable.
In this paper we aim to bring temperature into the picture. Thermoplasticity models, both in the case of rateindependent evolution of the plastic variable and of ratedependent one, have been the object of several studies, cf. e.g. [KS97, KSS02, KSS03, BR08, BR11, Rou13b, HMS17, Ros16]. In recent years, there has also been a growing literature on the analysis of (rateindependent or ratedependent) damage models with thermal effects: We quote among others [BB08, Rou10, RR14, HR15, RR15, LRTT14]. As for models coupling plasticity and damage with temperature, one of the examples illustrating the general theory developed in [RT15] concerns geophysical models of lithospheres in short time scales, which couple the (smallstrain) momentum balance, damage, ratedependendent plasticity, the heat equation, as well as the porosity and the water concentration variables. Here we shall neglect the latter two variables and tackle the weak solvability and the existence of solutions, for a (fully ratedependent) viscoplastic (gradient) damage model, with viscosity and inertia in the momentum balance (the first according to KelvinVoigt rheology), and with thermal effects encompassed through the heat equation, whereas in [RT15] the enthalpy equation was analyzed, after a transformation of variables. We plan to address the vanishingviscosity and inertia analysis for our model, and discuss the weak solution concept thus obtained, in a future contribution.
In what follows, we shortly comment on the model. Then, we illustrate the mathematical challenges posed by its analysis, motivate a suitable regularization, and introduce the two solution concepts, for the original system and its regularized version, for which we will prove two existence results.
1.1. The thermoviscoelastoplastic damage system
The PDE system, posed in , where the reference configuration is a bounded, open, Lipschitz domain in , , and is a given time interval, consists of

the kinematic admissibility condition
(1.1a) which provides a decomposition of the linearized strain tensor into the sum of the elastic and plastic strains and . In fact, (the space of symmetric matrices), while (the space of symmetric matrices with null trace).  
The momentum balance
(1.1b) according to KelvinVoigt rheology for materials subject to thermal expansion (with the matrix of the thermal expansion coefficients). Here, is a given body force. Observe that both the elasticity and viscosity tensors and depend on the damage parameter , but we restrict to incomplete damage. Namely, the tensors and are definite positive uniformly w.r.t. , meaning that the system retains its elastic properties even when damage is maximal.  The damage flow rule for (1.1c) where the dissipation potential (density) encompasses the unidirectionality in the evolution of damage. We denote by its subdifferential in the sense of convex analysis. As in several other damage models, we confine ourselves to a gradient theory. However, along the lines of [KRZ13] and [CL16], we adopt a special choice of the gradient regularization, i.e. through the Laplacian operator , with . This technical condition ensures the compact embedding for the associated SobolevSlobodeckij space . Furthermore, a key role in the analysis, especially for the regularized system with the flow rule (1.3c) ahead, will be played by the linearity of the operator . As in [CL16], the term will have a singularity at , which will enable us to prove the positivity of the damage variable. Combining this with the unidirectionality constraint a.e. in , we will ultimately infer that all emanating from an initial datum take values in the physically admissible interval .  The flow rule for the plastic tensor reads (1.1d) with the deviatoric part of the stress tensor . Here, the plastic dissipation potential (density) depends on the plastic strain rate but also (on the space variable ), on the temperature, and on the damage variable; the symbol denotes its convex subdifferential w.r.t. the plastic rate. Typically, one may assume that the plastic yield surface decreases as damage increases, although this monotonicity property will not be needed for the analysis developed in this paper. Observe that (1.1d) is in fact a viscous regularization of the flow rule of perfect plasticity.  The heat equation (1.1e) with the heat conductivity coefficient and a given, positive, heat source.
 
We will supplement system (1.1) with the boundary conditions
(1.2a)  
(1.2b) 
where is the outward unit normal to , e , the Dirichlet/Neumann parts of the boundary, respectively, and with initial conditions.
In fact, system (1.1) can be seen as the extension of the model considered in [CL16], featuring the static momentum balance and a mixed ratedependent/rateindependent character in the evolution laws for the damage/plastic variables, respectively, to the case where viscosity is included in the plastic flow rule and in the momentum balance, the latter also with inertia, and the evolution of temperature is also encompassed.
1.2. Analytical challenges and weak solution concepts
Despite the fact that the ‘viscous’ plastic flow rule (1.1d) does not bring all the technical difficulties attached to perfect plasticity (cf. [DMDM06], see also [Rou13b] for the coupling with temperature), the analysis of system (1.1, 1.2) still poses some mathematical difficulties. Namely,

The overall nonlinear character of (1.1) and, in particular, the quadratic terms on the righthand side of the damage flow rule (1.1c), and on the righthand side of the heat equation (1.1e). Both sides are, thus, only estimated in as soon as , , and are estimated in , , and , respectively, as guaranteed by the dissipative estimates associated with (1.1).
Observe that the particular character of the momentum balance, where the elasticity and the viscosity contributions only involve the elastic part of the strain and its rate , instead of the full strain and strain rate , does not allow for elliptic regularity arguments which could at least enhance the spatial regularity/summability of the righthand side of the damage flow rule.

Another obstacle is given by the presence of the unbounded maximal monotone operator in the damage flow rule. Because of this, no comparison estimates can be performed. In particular, a pointwise formulation of (1.1c) would require a separate estimate of the terms and of (a selection in) . This cannot be obtained by standard monotonicity arguments due to the nonlocal character of the operator .
All of these issues shall be reflected in the weak solution concept for system (1.1, 1.2) proposed in the forthcoming Definition 2.3 and referred to as ‘entropic solution’. This solvability notion consists of the socalled entropic formulation of the heat equation, and of a weak formulation of the damage flow rule, in the spirit of the KarushKuhnTucker conditions. The entropic formulation originates from the work by E. Feireisl in fluid mechanics [Fei07] and has been first adapted to the context of phase transition systems in [FPR09], and later extended to damage models in [RR15]. It is given by an entropy inequality, formally obtained by dividing the heat equation by and testing the resulting relation by a sufficiently regular, positive test function (cf. the calculations at the beginning of Section 2.3), combined with a total energy inequality. The weak formulation of the damage flow rule has been first proposed in the context of damage modeling in [HK11, HK13]: the subdifferential inclusion for damage is replaced by a onesided variational inequality, with test functions reflecting the sign constraint imposed by the dissipation potential , joint with a (mechanical) energydissipation inequality also incorporating contributions from the momentum balance and the plastic flow rule.
Clearly, one of the analytical advantages of the entropy inequality for the heat equation, and of the onesided inequality for the damage flow rule, is that the troublesome quadratic terms on the righthand sides of (1.1c) and of (1.1e) feature as multiplied by a negative test function, cf. (2.34a) and (2.37) ahead, respectively. This allows for upper semicontinuity arguments in the limit passage in suitable approximations of such inequalities. Instead, the total and mechanical energy inequalities can be obtained by lower semicontinuity techniques.
We will also consider a regularized version of system (1.1, 1.2), where the damage flow rule features the additional term , modulated by a positive constant , and, accordingly, the term (with the bilinear form associated with ) occurs on the righthand side of the heat equation. This leads to the regularized thermoviscoelastoplastic damage system
(1.3a)  
(1.3b)  
(1.3c)  
(1.3d)  
(1.3e) 
with , supplemented with the boundary conditions
(1.4a)  
(1.4b) 
For this regularized system we will be able to show the existence of an enhanced type of solution. It features a conventional weak formulation of the heat equation with suitable test functions, and the formulation of the damage flow rule as a subdifferential inclusion in . To obtain the latter, a key role is played by the regularizing term , ensuring that a.e. in . Thanks to this feature, it is admissible to test the subdifferential inclusion rendering (1.3c) by itself. This is at the core of the validity of the total energy balance. That is why, also in accordance with the nomenclature from [RR15, Ros16], we shall refer to these enhanced solutions as weak energy solutions.
1.3. Our results
We will prove the existence of entropic solutions, see Theorem 2.5, and of weak energy solutions, see Theorem 2.6, to (the Cauchy problems for) systems (1.1, 1.2) and (1.3, 1.4), respectively, by passing to the limit in a timediscretization scheme carefully devised in such a way as to ensure the validity of discrete versions of the entropy and energy inequalities along suitable interpolants of the discrete solutions. One of our standing assumptions will be a suitable growth of the heat conductivity coefficient , namely
(1.5) 
Mimicking the calculations from [FPR09, RR15], we will exploit (1.5) in a key way to derive an estimate for in by testing the (discrete) heat equation by a suitable negative power of .
Under a more restrictive condition on , in fact depending on the space dimension , (i.e., if , if ), we will also be able to obtain a intime estimate for , with values in a suitable dual space, which will be at the core of the proof of the enhanced formulation of the heat equation for the weak energy solutions to system (1.3, 1.4). Concerning the physical interpretation of our growth conditions, we refer to [Kle12] for a discussion of experimental findings suggesting that a class of polymers exhibit a subquadratic growth for .
Finally, with Proposition 2.7 we will provide a continuous dependence estimate, yielding uniqueness, for the weak energy solutions to (1.3, 1.4) in the case of a prescribed temperature profile, and with a plastic dissipation potential independent of the state variables and .
Plan of the paper.
In Section 2 we fix all our assumptions, motivate and state our two weak solvability notions for systems (1.1, 1.2) & (1.3, 1.4), and finally give the existence Theorems 2.5 & 2.6, and the continuous dependence result Proposition 2.7. In Section 3 we set up a common time discretization scheme for systems (1.1, 1.2) and (1.3, 1.4), and prove the existence of discrete solutions, while Section 4 is devoted to the derivation of all the a priori estimates on the approximate solutions, obtained by interpolation of the discrete ones. In Section 5 we conclude the proofs of Thms. 2.5 & 2.6 by passing to the timecontinuous limit, while in Section 6 we perform the proof of Prop. 2.7.
We conclude by fixing some notation that shall be used in the paper.
Notation 1.1 (General notation).
Throughout the paper, shall stand for . For a given , we will use the notation for its positive part . We will denote by () the space of (, respectively) matrices. We will consider endowed with the Frobenius inner product for two matrices and , which induces the matrix norm . Therefore, we will often write in place of . The symbol stands for the subspace of symmetric matrices, and for the subspace of symmetric matrices with null trace. We recall that ( denoting the identity matrix), since every can be written as with the orthogonal projection of into . We will refer to as the deviatoric part of .
For a given Banach space , the symbol will stand for the duality pairing between and ; if is a Hilbert space, will denote its inner product. For simpler notation, we shall often write both for the norm on , and on the product space . With the symbol we will denote the closed unitary ball in . We shall use the symbols
for the spaces of functions from with values in that are defined at every and (i) are measurable; (ii) are weakly continuous on ; (iii) have bounded variation on .
Finally, throughout the paper we will denote various positive constants depending only on known quantities by the symbols , whose meaning may vary even within the same line. Furthermore, the symbols , , will be used as placeholders for several integral terms (or sums of integral terms) occurring in the various estimates: we warn the reader that we will not be selfconsistent with the numbering, so that, for instance, the symbol will have different meanings.
2. Setup and main results for the thermoviscoelastoplastic damage system
After fixing the setup for our analysis in Section 2.1, in Sec. 2.2 we motivate the notion of ‘weak energy’ solution to system (1.3, 1.4) by unveiling its underlying energetics. This concept is then precisely fixed in Definition 2.1. Sec. 2.3 is devoted to the introduction of the considerably weaker concept of ‘entropic’ solutions. Our existence theorems are stated in Sec. 2.4, while in Sec. 2.5 we confine the discussion to the case of a given temperature profile, and give a continuous dependence result for weak energy solutions.
2.1. Setup
The reference configuration.
Let be a bounded domain, with Lipschitz boundary; we set . The boundary is given by
(2.$\Omega$) 
We will denote by the Lebesgue measure of . On the Dirichlet part , assumed with we shall prescribe the displacement, while on we will impose a Neumann condition on the displacement. The trace of a function on or shall be still denoted by the symbol .
Sobolev spaces, the Laplacian, Korn’s inequality.
In what follows, we will use the notation The symbol , shall denote the analogous space. Further, we will use the notation
(2.1) 
Throughout the paper, we shall extensively resort to Korn’s inequality (cf. [GS86]): for every there exists a constant such that there holds
(2.2) 
We will denote by
We will also use the notation and the analogously defined notation . We recall that is a Hilbert space, with inner product , where the bilinear form is defined by
(2.3) 
We denote by the associated operator
(2.4) 
Kinematic admissibility and stress.
Given a function , we say that a triple is kinematically admissible with boundary datum , and write , if
(2.5a)  
(2.5b)  
(2.5c) 
As for the elasticity and viscosity tensors, we will suppose that
(2.${\small(\mathbb{C},\mathbb{D})_{1}}$)  
where denotes the space of linear operators from to . Furthermore, we will suppose that for every the map is continuously differentiable on and fulfills  
(2.${\small(\mathbb{C},\mathbb{D})_{2}}$)  
Finally, for technical reasons (cf. Remark 3.2 later on) it will be convenient to require that the map is convex, i.e. for every there holds  
(2.${\small(\mathbb{C},\mathbb{D})_{3}}$)  
It follows from the convexity ((2.${\small(\mathbb{C},\mathbb{D})_{3}}$)) that  
(2.6a)  
whence . In particular, due to the first of ((2.${\small(\mathbb{C},\mathbb{D})_{3}}$)), we find that  
(2.6b)  
Finally, we also suppose that the thermal expansion tensor fulfills  
(2.${\small\mathbb{E}}$) 
Observe that with and ((2.${\small\mathbb{E}}$)) we encompass in our analysis the case of an anisotropic and inhomogeneous material.
External heat sources.
For the volume and boundary heat sources and we require
(2.$\mathrm{G}_{1}$)  
(2.$\mathrm{G}_{2}$) 
Indeed, the positivity of and is crucial for obtaining the strict positivity of the temperature .
Body force and traction.
Our basic conditions on the volume force and the assigned traction are
(2.$\mathrm{L}_{1}$) 
where is the space of functions such that there exists with in .
For technical reasons, in order to allow for a nonzero traction , we will need to additionally require a uniform safe load type condition. Observe that this kind of assumption usually occurs in the analysis of perfectly plastic systems. In the present context, it will play a pivotal role in the derivation of the First a priori estimate for the approximate solutions constructed by time discretization, cf. the proof of Proposition 4.3 later on as well as [Ros16, Rmk. 4.4] for more detailed comments. Namely, we impose that there exists a function , with and , solving for almost all the following elliptic problem
(2.$\mathrm{L}_{2}$) 
When not explicitly using ((2.$\mathrm{L}_{2}$)), to shorten notation we will incorporate the volume force and the traction into the induced total load, namely the function given at by
(2.7) 
which fulfills in view of ((2.$\mathrm{L}_{1}$)).
Dirichlet loading.
We will suppose that the hard device to which the body is subject on is the trace on of a function, denoted by the same symbol, fulfilling
(2.$\mathrm{w}$) 
Also condition ((2.$\mathrm{w}$)) will be used in the proof of Prop. 4.3 ahead; we again refer to [Ros16, Rmk. 4.4] for more comments.
The weak formulation of the momentum balance.
The plastic dissipation potential.
Our assumptions on the multifunction involve the notions of measurability, lower semicontinuity, and upper semicontinuity for general multifunctions. For such concepts and the related results, we refer, e.g., to [CV77]. Hence, we suppose that
is measurable w.r.t. the variables ,  (2.$\mathrm{K}_{1}$)  
Furthermore, we require that
(2.$\mathrm{K}_{2}$)  
Therefore, the support function associated with the multifunction , i.e.
(2.10) 
is positive, with convex and positively homogeneous for almost all and for all . By the first of ((2.$\mathrm{K}_{1}$)), the function is measurable. Moreover, by the second of ((2.$\mathrm{K}_{1}$)), in view of [CV77, Thms. II.20, II.21] (cf. also [Sol09, Prop. 2.4]) the function
(2.11a)  
for almost all , i.e. is a normal integrand, and  
(2.11b) 
Finally, it follows from the second of ((2.$\mathrm{K}_{2}$)) that for almost all and for all there holds
(2.12a)  
(2.12b) 
Finally, we also introduce the plastic dissipation potential given by
(2.13) 
From now on, throughout the paper we will most often omit the dependence of the tensors , and of the dissipation density .
Nonlinearities in the damage flow rule.
Along the footsteps of [CL16], we will suppose that
(2.$\mathrm{W}_{1}$) 
The latter coercivity condition will play a key role in the proof that the damage variable takes values in the feasible interval . In this way, we will not have to include the indicator term in the potential energy. This will greatly simplify the analysis of the damage flow rule.
Furthermore, we shall require that
(2.$\mathrm{W}_{2}$) 
Observe that ((2.$\mathrm{W}_{2}$)) is equivalent to imposing that the function is convex. Therefore, we have the convex/concave decomposition
(2.14) 
Let us point out that (2.14) will be expedient in devising the time discretization scheme for the (regularized) thermoviscoelastoplastic damage system, in such a way that its solutions comply with the discrete version of the total energy inequality. We refer to Remark 3.2 ahead for more comments.
Cauchy data.
We will supplement the thermoviscoelastoplastic damage system with initial data
(2.15a)  
(2.15b)  
(2.15c)  
(2.15d) 
In the remainder of this section, we shall suppose that the functions , the data , and the initial data fulfill the conditions stated in Section 2.1. We will first address the weak solvability of the regularized system in Sec. 2.2, and then turn to examining the nonregularized one in Sec. 2.3. We will then state our existence results for both in Sec. 2.4.
2.2. Energetics and weak solvability for the (regularized) thermoviscoelastoplastic damage system
Prior to stating the precise notion of weak solution for the regularized thermoviscoelastoplastic damage system in Definition 2.1 ahead, we formally derive the mechanical & total energy balances associated with systems (1.1, 1.2) and (1.3, 1.4) (in the ensuing discussion, we shall take the parameter ).
The mechanical and total energy balances.
The free energy of the system is given by
(2.16) 
The total energy balance can be (formally) obtained by testing the momentum balance (1.3b) by , the damage flow rule (1.3c) by , the plastic flow rule (1.3d) by , the heat equation (1.3e) by , adding the resulting relations and integrating in space and over a generic interval .