Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion

DOI:10.1051/m2an/2010063 www.esaim-m2an.org

THERMO-VISCO-ELASTICITY WITH RATE-INDEPENDENT PLASTICITY IN ISOTROPIC MATERIALS UNDERGOING THERMAL EXPANSION

S¨ oren Bartels

and Tom´ aˇ s Roub´ ıˇ cek

Abstract. We consider a viscoelastic solid in Kelvin-Voigt rheology exhibiting also plasticity with hardening and coupled with heat-transfer through dissipative heat production by viscoplastic effects and through thermal expansion and corresponding adiabatic effects. Numerical discretization of the thermodynamically consistent model is proposed by implicit time discretization, suitable regularization, and finite elements in space. Finea-priori estimates are derived, and convergence is proved by careful successive limit passage. Computational 3D simulations illustrate an implementation of the method as well as physical effects of residual stresses substantially depending on rate of heat treatment.

Mathematics Subject Classification. 35K85, 49S05, 65M60, 74C05, 80A17.

Received November 30, 2009. Revised June 17, 2010.

Published online October 11, 2010.

1. Introduction

Thermal expansion in metallic bodies may create enormous elastic stresses if the temperature profile varies considerably. It occurs both within manufacturing processes (especially heat treatment of large bulks) and sometimes in working regimes, too. These “thermo-elastic” stresses may trigger activated inelastic processes, typically slip plasticity or even damage. Here we focus on plasticity and consider also hardening effects. Me- chanicalenergy dissipatedduring the plastic deformation is converted to changes of an internal structure of the material due to hardening but also to heat, which ultimately couples the mechanical and heat parts. Moreover, thermal expansion leads to heat production/consumption due toadiabatic effects.

Keywords and phrases.Thermodynamics of plasticity, Kelvin-Voigt rheology, hardening, thermal expansion, adiabatic effects, finite element method, implicit time discretization, convergence.

∗The authors warmly thank Professor Alexander Mielke and Dr. Jiˇr´ı Pleˇsek for many fruitful discussions, and an anonymous referee for many helpful comments on an earlier version of the manuscript. S.B. acknowledges support by “Neˇcas center for mathematical modeling” LC 06052 (MˇSMT ˇCR). T.R. acknowledges the hospitality of SFB 611 “Singular phenomena and scaling in mathematical models” of the University of Bonn, as well as partial support from the grants A 100750802 (GA AV ˇCR), 201/09/0917, 106/09/1573, and 201/10/0357 (GA ˇCR), and MSM 21620839 (MˇSMT ˇCR), and from the research plan AV0Z20760514 ( ˇCR).

1Institut f¨ur Numerische Simulation, Rheinische Friedrich-Wilhelms-Universit¨at Bonn, Wegelerstraße 6, 53115 Bonn, Germany.

bartels@ins.uni.bonn.de

2 Mathematical Institute, Charles University, Sokolovsk´a 83, 186 75 Praha 8, Czech Republic.

3 Institute of Thermomechanics of the ASCR, Dolejˇskova 5, 182 00 Praha 8, Czech Republic.

Article published by EDP Sciences c EDP Sciences, SMAI 2010

There is an extensive engineering literature addressing thermoplasticity in thermally expanding materials, employing computationally sophisticated models, e.g. [1,7,9,19,26,27], sometimes even at large strains [10,16, 20,32,33] but lacking a rigorous mathematical justification.

Mathematically supported theories seem, however, nearly missing for a long time. This was mainly because the relevantL¹-theory for the heat equation was developed only in the 1990s [4–6], and the mathematical theory for rate-independent processes is even more recent, cf. [13,15,21–24], as well as the interpolation technique of the adiabatic-heat term in three-dimensional case [29], and the coupling with rate-independent processes with viscous/inertial effects [30] and thermal effects [31].

The main mathematical difficulties are related to finite strain and multiplicative plasticity at finite strains, in evolution driven even by mere elastic response if kinetic effects are counted, and coupling of rate-independent processes with rate-dependent ones. In fact, each of the above mentioned three difficulties represents itself a hard open problem, especially in a three-dimensional setting and if no regularization (e.g. by capillarity or higher viscosity is involved). This is why we adopt the following simplifications: small strainsand strain-driven linearized additive plasticity, and alinear viscoelastic response. On the other hand, we allow for a fully rate- independent plastic flow rule although, of course, the whole system is necessarily rate dependent due to the heat transfer, and here also due to considered kinetic and viscous effects. As mentioned above, we also consider hardening to avoid spatial concentration of plastic strains as has been studied in the isothermal case in [13]

which, in general, would lead to awkward interactions of concentrating plastic-strain rate with thermal effects.

Recently, rigorous mathematical studies for thermoviscoplasticity at small strains had been performed in [3]

(considering, however, a rate-dependent plastic flow rule and no thermal expansion effects) and, as mentioned above, in [31] (considering general analytical scheme for a slightly different class of generalized standard materials with gradient theories for internal parameters but without numerical analysis and the modification for the linearized non-gradient plasticity only outlined in [31], Rem. 4.5 with Ex. 5.1).

The model will be formulated in Section 2 where also its thermodynamics will be exposed. We confine ourselves to trace-free plastic strain and to isotropic materials as far as both elastic response and thermal expansion are concerned. This implies a mathematically important cancellation effect owing to the fact that the thermal-expansion strain is diagonal and thus orthogonal to the “plastic stress”,cf.(2.24) below. Although this restricts generality and excludes e.g. single-crystal plasticity which is remarkably anisotropic, important applications in engineering which standardly treat polycrystalline (and thus isotropic) metals are allowed by the presented theory.

The main purpose of this paper, performed in Sections 3 and 4, is to develop an implementable numerical scheme for this model and prove its stability,i.e. a-priori estimates, and also convergence to a suitably defined weak solution of the model. In Section5, the efficiency of the proposed numerical strategy is demonstrated on a 3D example. We illustrate it on a physically motivated example, exhibiting rate-dependent effects of thermal coupling producing residual elastic stresses and plastic deformation depending on rate of heat treatment of a steel workpiece.

2. The model within thermodynamics

We consider a bounded Lipschitz domain Ω ⊂ R^d, d ≤ 3. The state variables will be the displacement u : Ω → R^d, the plastic strain π : Ω → R^d×d_dev, possibly a scalar isotropic hardening parameter η, and the temperatureθ: Ω→R, where

R^d×d_dev :=

A∈R^d×d_sym; tr(A) = 0

and R^d×d_sym:=

A∈R^d×d; A=A}. (2.1) The variables (π, η) play the role of internal parameters. We consider plastic response determined by a convex closed neighbourhood of the origin, say K ⊂ R^d×d_sym×R, defining an elasticity domain, while its boundary is called theyield surfaceand has the meaning of the stress that triggers the evolution of plastic strains; we refer to Section5for a specific example. Letδ_K denote its indicator function andδ_K^∗ the Fenchel-Legendre conjugate functional to δ_K with respect to the inner product σ : e =_d

i,j=1σ_ije_ij. Note that the physical dimension

of σ : e is P a = J·m⁻³ so that K determining the degree-1 positively homogeneous “plastic” dissipation potentialδ_K^∗, acting on the dimensionless tensorπand on the dimensionless internal hardening variableη, has indeed the dimension J·m⁻³. For a simpler notation, we write

ζ₁( ˙π,η) :=˙ δ^∗_K( ˙π,η).˙ (2.2) We remark that the condition 0 ∈ int(K) implies that ζ₁ = δ_K^∗ is coercive. The set K must be unbounded.

More specifically, we assume that

K=K₀⊕K₁, whereK₀⊂R^d×d_dev ×Ris convex, K₁is the orthogonal complement ofR^d×d_dev ×R. (2.3) This implies thatζ₁is finite only onR^d×d_dev ×Rand thatπ(t,·)∈R^d×d_dev a.e. in Ω provided thatπ₀(·)∈R^d×d_dev a.e.

in Ω. Therefore, no deviator operator occurs in (2.5) below. We remark that, alternatively, one may consider a (bounded) set K₀ ⊂R^d×d_dev and involve the deviator operator in (2.5) and (2.10b) below; then ζ₁ would be real-valued.

Considering a Kelvin-Voigt-type viscous material, our model will consist of theequilibrium equationbalancing inertial, viscous, and elastic mechanical forces,

∂²u

∂t² −div

D∂e(u)

∂t

−div

C(e(u)−π−Eθ)

= 0, (2.4)

whereis the mass density,Dthe tensor determining the viscous-type response,Cthe tensor determining the elastic response, and E the thermal expansion tensor, while the evolution of the internal parametersπ and η are governed by the inclusion

∂ζ₁

∂π

∂t,∂η

∂t

+ Cπ+Hπ bη

Ce(u) 0

(2.5) whereb >0 is anisotropic hardening coefficient andHis a symmetric positive semidefinite fourth-order tensor determining thekinematic hardening, and theheat transfer/productionis governed by the equation

c_v(θ)∂θ

∂t −div

K(θ)∇θ

=ζ₁

∂π

∂t,∂η

∂t

+ 2ζ₂

∂e(u)

∂t

−θE:C∂e(u)

∂t (2.6)

wherec_v=c_v(θ) is theheat capacity andK=K(θ) is thethermal conductivitytensor, ζ₂( ˙e) =1

2De˙: ˙e (2.7)

is the pseudopotential of viscous-dissipative forces, “ : ” denotes the product of two (d×d)-tensors, ande(u) is the small-strain tensor defined as

e_ij(u) =1 2

∂u_i

∂x_j +∂u_j

∂x_i

· (2.8)

Throughout this paper, we assumeisotropic thermal expansion,i.e.

E=αI (2.9)

withαa single thermal-expansion coefficient.

Using the identity [∂ζ₁]⁻¹ =∂ζ₁^∗ and taking into account (2.9), the equation and inclusion (2.4)–(2.5) can equivalently be written in a form which is more standard in engineering literature, namely

∂²u

∂t² = divσ−α∇θ with σ=σ_visc+σ_elast, σ_visc=D∂e(u)

∂t , σ_elast=C(e(u)−π), (2.10a)

∂π/∂t

∂η/∂t

∈∂ζ₁^∗ −σplast

−bη

with σ_plast=−σelast+Hπ, (2.10b)

which reveals the total stressσ to be composed from the viscous and the elastic partsσ_visc andσ_elast (which expresses just the Kelvin-Voigt rheology), and the plastic stressσ_plast involvingHπis in the position of theback stress to the elastic stressσ_elast. This is also known as Ziegler’s type model [34]. Alternatively, σ_plast would have involved only the deviatoric part ofσ_elast and Hwould have had to be assumed as R^d×d_dev →R^d×d_dev if the mentioned variant of ζ₁ valued inRwere considered.

The above equations/inclusion (2.4)–(2.8) are to hold on the space/time domainQ:= (0, T)×Ω withT >0 a fixed time horizon.

As we focus on processes in the bulk, we consider only the simplest boundary conditions, namely a prescribed normal stress and heat flux on Γ :=∂Ω:

D∂e(u)

∂t +C

e(u)−π−Eθ

ν=g on Γ, (2.11a)

K(θ)∇θ

·ν =f on Γ, (2.11b)

where “·” denotes the scalar product of two vectors andν is the outward normal to Γ.

The energetics of the model is based on themechanical part of the internal energy Φ(u, π, η) :=1

2

Ω

C(e(u)−π) : (e(u)−π) +Hπ:π+bη²dx, (2.12)

thekinetic energy

T_kin( ˙u) :=1 2

Ω

|u|˙ ²dx, (2.13)

thedissipation energy rate

Ξ( ˙u,π,˙ η) :=˙

Ω

ζ₁

˙ π,η˙

+ 2ζ₂

e( ˙u)) dx (2.14)

withζ₁from (2.2) andζ₂ from (2.7), the thermal part of the internal energy E(θ) :=

Ω

h(θ) dx, where h(θ) :=

_θ

0

c_v(w)dw, (2.15)

and thepower of external mechanical forces and heating F(t,u) =˙

Γ

g(t, x)·u(x) +˙ f(t, x) dS. (2.16) The energetics of the model (2.4)–(2.6) can be obtained by testing (2.4)–(2.6) respectively by the velocity ^∂u_∂t, by the plastic strain rate ^∂π_∂t, and by 1, which gives after using Green’s formula for both (2.4) and (2.6) together

with the boundary conditions (2.11) and eventually by summation theenergy balance d

dt T_kin ∂u

∂t

+ Φ(u, π, η) +E(θ)

=F

t,∂u

∂t

; (2.17)

cf.(3.7d) below.

We consider an initial-boundary-value problem for the system (2.4)–(2.8). Hence, we take the initial conditions

u(0,·) =u₀, ∂u

∂t(0,·) = ˙u₀, π(0,·) =π₀, η(0,·) =η₀, θ(0,·) =θ₀. (2.18) Also, thenon-negativity of temperatureis ensured providedθ₀≥0 and provided the boundary heat fluxf is non-negative,cf.also the proof of Proposition4.4below.

Throughout the article, we will rely on the following data qualification. The positive definite fourth order tensorsC= [Cijkl],D= [Dijkl] are assumed to be symmetric and isotropic so that in fact

C_ijkl=λ_eδ_ijδ_kl+μ_e

δ_ikδ_jl+δ_ilδ_jk

, D_ijkl =λ_vδ_ijδ_kl+μ_v

δ_ikδ_jl+δ_ilδ_jk , withμ_e, μ_v>0, λ_e>−2

dμ_e, λ_v>−2

dμ_v, (2.19)

withδdenoting here the Kronecker symbol,λ’s andμ’s are the Lam´e constants. Thus the elastic stress isCe= λ_etr(e)I+2μeewithI= [δ_ij] denoting the unit matrix, and corresponding energy is¹₂Ce:e= ¹₂λ_e|tr(e)|²+μ_e|e|² and, as a quadratic form ofe, it is positive definite, and similarly also the quadratic forme→ ¹₂De:eis positive definite. In fact, for the analysis presented below, we need the isotropy only for Cbut it would be physically inconsistent to haveDanisotropic.

One can derivethermodynamics of the above model by postulating the Helmholtzfree energyas ψ(e, π, η, θ) = 1

2C

e−π−Eθ :

e−π−Eθ +1

2Hπ:π+b

2η²−θ²

2CE:E−φ₀(θ). (2.20) Thenentropy is given by

s=s(e, θ) :=− ∂

∂θψ=φ₀(θ)−E:Ce. (2.21)

Note that the thermo-mechanically coupling terms in (2.20) are linear in terms ofθ, so that the mechanical variables separate from temperature in (2.21) and thus c_v = c_v(e, θ) = θ_∂θ^∂ s(e, θ) =θφ₀(θ) in (2.6) does not depend on these mechanical variables, which facilitates the analysis of the heat equation considerably.

The equation (2.6) itself can be derived from theentropy equation θ∂s

∂t −div(K∇θ) =ξ with the dissipation rate ξ:=ζ₁

∂π

∂t,∂η

∂t

+ 2ζ₂

∂e(u)

∂t

, (2.22)

which takes, in general, in the form θψ_θθ(θ)∂θ

∂t −div

K(θ)∇θ

=ξ−θψ_θe(e, π, η, θ) : ∂e(u)

∂t −θψ_θπ (e, π, η, θ) : ∂π

∂t −θψ_θη(e, π, η, θ)∂η

∂t· (2.23) Note that

Ωξdx= Ξ(^∂u_∂t,^∂π_∂t,^∂η_∂t) with Ξ from (2.14). Note also that, as b is independent ofθ, the last term in (2.23) vanishes. Moreover, the important facts are that alsoHis independent of θand that, owing to (2.3) together with (2.9) and (2.19) we have theorthogonality

Cπ:E=α

λ_etr(π)I+ 2μ_eπ

:I=α

dλ_e+ 2μ_e

tr(π) = 0 (2.24)

provided also tr(π₀) = 0. This guarantees that the term θψ_θπ(e, π, η, θ) : ^∂π_∂t = θE : C^∂π_∂t in (2.23) vanishes.

This facilitates the analysis in our rate-independent plasticity substantially, because otherwise anL¹-type term

∂π

∂t would be in product with temperatureθwhich hardly can be “compact” in anL^∞-space.

At least formally, assuming positivity of temperature andf ≥0, and realizing that alwaysξ≥0, from (2.22) we can see the Clausius-Duhem inequality

d dt

Ω

sdx=

Ω

div

K∇θ θ

+K∇θ·∇θ θ² +ξ

θdx=

Ω

K∇θ·∇θ θ² +ξ

θdx+

Γ

f

θdS≥0; (2.25) obviously, K∇θ·∇θ/θ²+ξ/θ is the entropy-production rate. Note also that a combination of (2.9), (2.19), and (2.24) allows us to writeψfrom (2.20) in the more specific form

ψ(e, π, η, θ) =λ_e

2 tr(e−π)²+μ_ee−π²−α

dλ_e+ 2μ_e

θtr(e) +1

2Hπ:π+b

2η²−φ₀(θ) (2.26) where (2.24) was used, too. In fact, (2.25) in the form _dt^d

Ωsdx≥

ΩK∇θ·∇θ

θ² +_θ^ξdx+

Γ f

θdS can conversely serve as the origin of the constitutional relation for the entropy s = −_∂θ^∂ ψ, cf. (2.21), and the elastic stress σ= _∂e^∂ ψ,cf.(2.10b), as well as the driving force for the flow rule _∂(π,η)^∂ ψ= (Hπ−σ, bζ).

3. Enthalpy transformation and weak formulation

It is desirable to allow for a certain growth of c_v(·) if we have the viscosity in the formDe(^∂u_∂t) in order to be able to treat the adiabatic term, cf.[29]. On the other hand, the technique from [29] specifically relies on Galerkin’s method and does not seem directly transferable if also the time discretization is involved, which is in turn needed both for designing a fully discrete scheme and for efficient treatment of the rate-independent flow rule. The particular difficulty is in establishing the limit of a time-discretization of the nonlinear term c_v(θ)^∂θ_∂t. Therefore, we first write the original system (2.4)–(2.6) in terms of enthalpy instead of temperature, using so-called enthalpy transformation

w=h₀(θ) :=

_θ

0

c_v(r) dr; (3.1)

thush₀is a primitive function to c_v normalized such that h₀(0) = 0. Further, we define Θ(w) :=

h⁻¹₀ (w) ifw≥0,

0 ifw <0, K(w) := K(Θ(w))

c_v(Θ(w)), (3.2)

whereh⁻¹₀ here denotes the inverse function to h₀. This transforms the system (2.4)–(2.6) into the form

∂²u

∂t² −div

De

∂u

∂t

+C

e(u)−π−Θ(w)E

= 0, (3.3a)

∂ζ₁

∂π

∂t,∂η

∂t

+ Cπ+Hπ bη

Ce(u) 0

, (3.3b)

∂w

∂t −div

K(w)∇w

=ζ₁

∂π

∂t,∂η

∂t

+ 2ζ₂

∂e(u)

∂t

+ Θ(w)E:Ce

∂u

∂t

· (3.3c)

We will call (3.3c) shortly theenthalpy equationrather than the heat-transfer equation in the enthalpy formu- lation. The boundary conditions (2.11) transforms to

De

∂u

∂t

+C

e(u)−π−EΘ(w)

ν=g on Γ, (3.4a)

K(w)∇w

·ν =f on Γ, (3.4b)

while the initial conditions (2.18) transform into u(0,·) =u₀, ∂u

∂t(0,·) = ˙u₀, π(0,·) =π₀, η(0,·) =η₀, w(0,·) =h₀(θ₀). (3.5) The following definition of a certain sort of a weak solution has been devised in [31], based on the concept of so- called energetic solution invented by Mielkeet al.[15,21,23,24] for the theory of rate independent processes and adapted also for coupling with viscous/inertial effects in [30]. We refer to [31], Proposition 3.2, for justification (and not entirely obvious fact) that this definition is indeed selective in the sense that, under an additional absolute continuity of ^∂π_∂t and ^∂η_∂t, it gives indeed a conventional notion of a weak solution. For an isothermal situation,cf.also [30], Proposition 5.2. It should be however emphasized that this additional regularity of ^∂π_∂t and ^∂η_∂t hardly can be expected due to the fully rate-independent flow rule, which just makes the devised concept properly fitted with this problem.

We consider an evolution in the time interval I := (0, T) with a fixed time horizon T > 0 and denote Q := (0, T)×Ω, Σ := (0, T)×∂Ω, and ¯I := [0, T]. We will use a standard notation for function spaces, namely the space of the continuousR^k-valued functionsC( ¯Ω;R^k), its dualM( ¯Ω;R^k) (i.e., up to an isometric isomorphism, the space of Borel measures), the continuously differentiable functionsC¹( ¯Ω;R^k), the Lebesgue space L^p(Ω;R^k), the Sobolev space W^1,p(Ω;R^k), and the Bochner space of X-valued Bochner measurable p-integrable functionsL^p(I;X). IfX = (X)^∗, the notationL^∞_w∗(I;X) stands for space of weakly* measurable functions I →X; this space is dual to the spaceL¹(I;X) and, in general, is not equal toL^∞(I;X). If X is separable reflexive, thenL^∞(I;X) =L^∞_w∗(I;X) by Pettis’ theorem, however. Moreover, we denote byB( ¯I;X), B_w∗( ¯I;X), BV( ¯I;X) orC_w( ¯I;X) the Banach space of functions ¯I→X that are bounded Bochner measurable, bounded weakly* measurable, have a bounded variation or are weakly continuous, respectively; note that all these functions are defined everywhere on ¯I. We will use the notationq=q/(q−1) for the conjugate exponent toq. Instead ofu(t,·) orπ(t,·) orη(t,·) orw(t,·), we will write brieflyu(t) orπ(t) orη(t) orw(t), respectively.

Definition 3.1 (energetic solution). Assuming (2.19)–(3.10), we call a quadruple (u, π, η, w) with

u∈C_w(I;W^1,2(Ω;R^d)), (3.6a)

∂u

∂t ∈L²(I;W^1,2(Ω;R^d))∩W^1,2(I;W^1,2(Ω;R^d)^∗), (3.6b) π∈B( ¯I;L²(Ω;R^d×d_dev))∩BV( ¯I;L¹(Ω;R^d×d_dev)), (3.6c)

η∈B( ¯I;L²(Ω))∩BV( ¯I;L¹(Ω)), (3.6d)

w∈L^r(I;W^1,r(Ω)) ∩ L^∞(I;L¹(Ω)) ∩ B_w∗( ¯I;M( ¯Ω)) with any 1≤r < d+ 2

d+ 1, (3.6e)

∂w

∂t ∈ M( ¯I;W^1+d,2(Ω)^∗) (3.6f)

an energetic solution to (3.3) with the initial/boundary conditions (3.5) and (4.3) if the following five conditions hold:

(i) The weakly formulated momentum-equilibrium equation (3.3a) with (4.3a), (4.3b) holds, i.e. for all v∈C¹( ¯Q;R^d) such thatv|Σ0=0,

Ω

∂u

∂t(T)·v(T) dx+

Q

De

∂u

∂t

+C

e(u)−π−EΘ(w)

:e(v)−∂u

∂t ·∂v

∂tdxdt

=

Σ

g·vdSdt+

Ω

u˙₀·v(0) dx. (3.7a) (ii) The weakly formulated enthalpy equation (3.3c) with (4.3c) holds,i.e. for allv∈C¹( ¯Q) withv(T) = 0,

Q

K(w)∇w· ∇v−w∂v

∂t −Θ(w)E:Ce

∂u

∂t

v−De

∂u

∂t

:e

∂u

∂t

vdxdt

=

Q¯

vhπ,η(dxdt) +

Ω

w₀v(0) dx+

Σ

f vdSdt (3.7b)

wherew₀=h₀(θ₀) andhπ,η is a measure (= heat produced by rate-independent dissipation) defined by prescribing its values for every closed set of the typeA:= [t₁, t₂]×B withB a Borel subset of ¯Ω by

hπ,η(A) := Var_ζ1

(π, η)|B;t₁, t₂ with Var_ζ1(z;t₁, t₂) := sup

k i=1

Ω

ζ₁

z(s_i, x)−z(s_i−1, x)

dx (3.7c)

where the supremum is taken over all partitions of the typet₁≤s₀< ... < s_k≤t₂,k∈N.

(iii) The total energy equality holds,i.e. with Φ andT_kin from (2.12) and (2.13), T_kin

∂u

∂t(T)

+ Φ

u(T), π(T), η(T) +

Ω¯

w(T,dx)

=T_kin

˙ u₀

+ Φ

u₀, π₀, η₀ +

Ω

h₀(θ₀) dx+

Σ

g·∂u

∂t +fdSdt. (3.7d) (iv) The “semistability” holds for any ˜π∈L²(Ω;R^d×d_dev) and ˜η∈L²(Ω) and for allt∈[0, T],i.e.

Φ

u(t), π(t), η(t)

≤Φ

u(t),π,˜ η˜ +

Ω

ζ₁(˜π−π(t),η˜−η(t)) dx. (3.7e) (v) The initial conditionsu(0) =u₀,π(0) =π₀, andη(0) =η₀ hold.

Note also that (3.6f) makes values ofw(t) well defined in the sense ofW^1+n,2(Ω)^∗ and (3.6e) further shows that evenw(t)∈ M( ¯Ω), which has been exploited in (3.7d) for the time t=T. It should be emphasized that t → w(t) cannot be expected to be continuous in any sense because, since ζ₁ is homogeneous degree-1, the measurehπ,η may concentrate at particular time instances.

In addition to (2.19) which guarantees thatC and Dare positive definite, we will assume throughout this article that

≥0. (3.8)

Other assumptions are on nonlinearitiesc_v andK, namely we assume:

c_v: [0,+∞)→R⁺ continuous, (3.9a)

∃ω₁≥ω≥1, c₁≥c₀>0∀θ∈R⁺: c₀(1 +θ)^ω−1≤c_v(θ)≤c₁(1 +θ)^ω1−1, (3.9b) K:R→R^d×d bounded, continuous, and inf

(w,ξ)∈R×R^d,|ξ|=1K(w)ξ·ξ >0 withK from (3.2) below; (3.9c) later in (4.26) we impose further restrictions onω. As far as the loading qualification concerns, we assume

g∈L²(I;L^q(Γ;R^d)), q≥2−2/d (orq >1 ifd≤2), (3.10a)

f ∈L¹(Σ), f ≥0, (3.10b)

u₀∈W^1,2(Ω;R^d), (3.10c)

˙

u₀∈L²(Ω;R^d), (3.10d)

π₀∈L²(Ω;R^d×d_dev), (3.10e)

η₀∈L²(Ω), η₀>0, (3.10f)

θ₀∈L^ω(Ω), θ₀≥0, (3.10g)

where we denoted Σ :=I×Γ in (3.10b).

4. Discretization and numerical analysis

To prove stability of any discrete scheme needed for possible convergence, one must inevitably deal with executing fine a-priori estimates. This is not entirely easy in coupled systems with super-linear growth of nonmonotone terms, as it is typically the case of thermodynamically consistent continuum-mechanical problems.

It has to be done essentially in two steps: first the physical internal and kinetic energy is to be estimated uniformly in time using (and proving) also non-negativity of the enthalpy, and, from this, some additional finer estimates by using Gagliardo-Nirenberg interpolation several times. In this second step, one estimates especially the gradient of enthalpy and also total dissipated energy, cf. also [29] and, for the plasticity, especially [31].

Further important phenomenon here is that, proving existence of a solution, we need to pass to the limit in the non-linear Nemytski˘ı operators induced by the dissipation heat ξ. Another peculiarity is that, due to degree-1 homogeneity of ζ₁, the heat equation has its right-hand side not only in L¹(Q) (as it would be in case of higher-degree homogeneity of dissipative-force potential) but even in measures. For this, the key trick is to recover the exact energy balance in the limit. In addition to [31], there are some further peculiarities related with spatial discretisation. In particular, it seems difficult to make spatial discretization of the term

−div(K(w)∇w) compatible with the maximum principle even on acute triangulations ifK is nonconstant (and the qualification (3.9b), (3.9d) with (4.26) below exclude constantKifd≥2).

Therefore, a design of a convergent numerical scheme is technically rather delicate. Following [31], we will use afully implicit time-discretization with a constant time-stepτ >0, assumingK_τ =T /τ∈Nand defining thebackward difference operatorby

D_tφ^k:= φ^k−φ^k−1

τ (4.1)

for any sequence {φ^k}_k≥0, combined with a regularization of the momentum equation and of the flow rule, (using as a parameter just the time-stepτ >0) and of the enthalpy equation (using a parameterε=ε(τ)>0 whose dependence onτ will be implicitly specified later in (4.34)). More specifically, we consider the following

recursive increment formula D²_tu^k_τ−div

De D_tu^k_τ

+C

e(u^k_τ)−π^k_τ−EΘ(w^k_τ)

+τe(u^k_τ)^γ−2e(u^k_τ)

= 0, (4.2a)

∂ζ₁

D_tπ^k_τ,D_tη^k_τ

+ Cπ^k_τ+Hπ_τ^k bη_τ^k

+τS π^k_τ η_τ^k

Ce(u^k_τ) 0

, (4.2b)

D_tw_τ^k−div

K(w^k_τ)∇w_τ^k

+ε(τ)|w^k_τ|^β−2w^k_τ =ζ₁

D_tπ_τ^k,D_tη^k_τ +De

D_tu^k_τ :e

D_tu^k_τ

+ Θ(w^k_τ)E:Ce D_tu^k_τ

, (4.2c) fork= 1, . . . , K_τ=T /τ with the corresponding boundary conditions

De D_tu^k_τ

+C

e(u^k_τ)−π^k_τ−EΘ(w^k_τ)

+τe(u^k_τ)^γ−2e(u^k_τ)

ν =g^k_τ, (4.3a)

K(w^k_τ)∇w^k_τ

·ν =f_τ^k (4.3b)

on Γ, starting fork= 1 by using

u⁰_τ =u_0,τ, u⁻¹_τ =u_0,τ−τu˙₀, π_τ⁰=π_0,τ, η_τ⁰=η_0,τ, w_τ⁰=w_0,τ, (4.4) where

g_τ^k(t, x) := 1 τ

_kτ

(k−1)τg(t, x) dt and f_τ^k := 1 τ

_kτ

(k−1)τ

f˜_τ(t, x) dt. (4.5)

Note that, in (4.4) and (4.5), we regularized the initial values and the boundary flux u₀, z₀, w₀, and f by u_0,τ,z_0,τ,w_0,τ, and ˜f_τ, respectively,cf.(4.8) below. Moreover,S in (4.2b) is a regularizing selfadjoint positive definite linear operator having the quadratic potential

1 2z²

W^a,2(Ω) with some 0< a <1/2 (4.6)

applied component-wise in (4.2b), with| · |_W^a,2_(Ω) meaning the standard seminorm in the Sobolev-Slobodetski˘ı fractional-derivative space. Later we can also use its square rootS^1/2, defined as a selfadjoint positive definite operator such that S^1/2◦ S^1/2=S. Thus

S :W^a,2(Ω)→W^a,2(Ω)^∗ and S^1/2:W^a,2(Ω)→L²(Ω). (4.7) Note that we regularized also the initial state for the mechanical part (but not the initial velocity).

Let us comment the purpose of the regularizing terms. The “γ-term” in (4.2a) and the “β-term” in (4.2c) are to compensate the superlinear growth of the right-hand-side terms in the heat equation; the former one has already been used in [31] for a mere time discretization, while the latter one is here needed because, in the spatially discretized scheme, we will not be able to test by nonlinear functions of w, in contrast with the spatially continuous case in [31]. Eventually, the “S-term” in (4.2b) helps to make a limit passage in space discretization without using numerical integration formulae,cf.(4.20) below.

As far as the (regularized) initial and boundary conditions and the loading concerns, we assume u_0,τ ∈W^1,γ(Ω;R^d), lim

τ↓0

√γ

τe(u_0,τ)

L^γ(Ω;R^d×d)= 0, lim

τ↓0u_0,τ =u₀ inW^1,2(Ω;R^d), (4.8a) π_0,τ ∈W^a,2(Ω;R^d×d+1), lim

τ↓0

√τπ_0,τ

W^a,2(Ω;R^d×d+1)= 0, lim

τ↓0π_0,τ =π₀in L²(Ω;R^d×d), (4.8b) η_0,τ ∈W^a,2(Ω;R^d×d+1), lim

τ↓0

√τη_0,τ

W^a,2(Ω;R^d×d+1)= 0, lim

τ↓0η_0,τ =η₀in L²(Ω), (4.8c) w_0,τ ∈L²(Ω), lim

τ↓0

√τw_0,τ

L²(Ω)= 0, lim

τ↓0w_0,τ =w₀:=h₀(θ₀) inL¹(Ω), (4.8d) f˜_τ ∈L^∞(Σ), f˜_τ ≥0, lim

τ↓0

√τf˜_τ

L²(I;L^4/3(Γ))= 0, lim

τ↓0f˜_τ =f in L¹(Σ). (4.8e) We will further make a spatial discretization. For this, we assume that we are given a sequence of tri- angulations {T_h}_h>0 of the polyhedral domain Ω without hanging nodes but otherwise entirely general. We suppose thath >0 range over countable sets of positive real numbers with accumulation points at 0, and that max_E∈Thdiam(E)≤h.

We considerC⁰-conforming P1-elements for the approximation ofuandwand P0-elements for the approxi- mation ofπandη. The finite-dimensional subspaces ofL²(Ω) andW^1,2(Ω) related to P0- and P1-elements and subordinate to the triangulationTh respectively byV_0,handV_1,h.

Forj= 0,1, theL²orthogonal projection ontoV_j,h is denoted byP_j,h. We have the following approximation property at our disposal for any 1≤γ <∞:

∀v∈L²(Ω) : P_0,hv→v inL²(Ω), (4.9a)

∀v∈W^1,γ(Ω) : P_1,hv→v inW^1,γ(Ω), (4.9b)

as follows from approximation results in [8].

Then we devise the Galerkin scheme as follows. We seek (u^k_{τ h}, π^k_{τ h}, η_{τ h}^k , w^k_{τ h}) ∈V_1,h^d ×V_0,h^d×d×V_0,h×V_1,h, withπ^k_{τ h}∈R^d×d_dev a.e. on Ω, satisfying

Ω

D²_tu^k_{τ h}·v+ De

D_tu^k_{τ h} +C

e(u^k_{τ h})−π^k_{τ h}−EΘ(w^k_{τ h})

+τe(u^k_{τ h})^γ−2e(u^k_{τ h})

:e(v) dx

=

Γ

g^k_τ·vdS for allv∈V_1,h^d , (4.10a)

Ω

ζ₁(˜π,η) + (Cπ˜ _{τ h}^k −Ce(u^k_{τ h}) +Hπ^k_{τ h}) : (˜π−Dtπ_{τ h}^k ) +bη^k_{τ h}(˜η−Dtη^k_{τ h}) +τS^1/2π^k_{τ h}:S^1/2(˜π−Dtπ_{τ h}^k ) +τS^1/2η^k_{τ h}S^1/2(˜η−Dtη^k_{τ h}) dx≥

Ω

ζ₁(D_tπ_{τ h}^k ,D_tη^k_{τ h}) dx for all (˜π,η)˜ ∈V_1,h^d×d×V_1,h, (4.10b)

Ω

D_tw^k_{τ h}+ε(τ)|w^k_{τ h}|^β−2w^k_{τ h}

v+K(w_{τ h}^k )∇w^k_{τ h}· ∇v−ζ₁

D_tπ^k_{τ h},D_tη_{τ h}^k v

−De D_tu^k_{τ h}

:e D_tu^k_{τ h}

vdx=

Ω

Θ(w_{τ h}^k )E:Ce D_tu^k_{τ h}

vdx+

Γ

f_τ,h^k vdS for allv∈V_1,h. (4.10c) Let us define the piecewise affine interpolant (u_{τ h}, π_{τ h}, η_{τ h}, w_{τ h}) by

u_{τ h}, π_{τ h}, η_{τ h}, w_{τ h}

(t) := t−(k−1)τ τ

u^k_{τ h}, π_{τ h}^k , η^k_{τ h}, w_{τ h}^k

+kτ−t τ

u^k−1_{τ h} , π_{τ h}^k−1, η_{τ h}^k−1, w^k−1_{τ h}

for t∈[(k−1)τ, kτ] with k= 0, . . . , K_τ :=T /τ . (4.11)

Besides, we define also the back-ward piecewise constant interpolant (¯u_{τ h},¯π_{τ h},η¯_{τ h},w¯_{τ h}) by u¯_{τ h},π¯_{τ h},η¯_{τ h},w¯_{τ h}

(t) :=

u^k_{τ h}, π_{τ h}^k , η^k_{τ h}, w_{τ h}^k

for (k−1)τ < t≤kτ, k= 1, . . . , K_τ. (4.12) Similarly, we will later useu_τ, ¯u_τ, etc. We will also use the notation ¯g_τ and ¯f_τ defined by ¯g_τ|((k−1)τ,kτ] =g_τ^k and ¯f_τ|((k−1)τ,kτ]=f_τ^k fork= 1, . . . , K_τ.

Lemma 4.1 (existence and estimates of discrete solutions). Let(2.19), (3.9), (3.10), and(4.8)hold. Moreover, let

β >2, γ >max

4, 2ω ω−1

, and ω >1. (4.13)

Then there exists a solution (u^k_{τ h}, π^k_{τ h}, η_{τ h}^k , w_{τ h}^k )∈V_1,h^d ×V_0,h^d×d×V_0,h×V_1,h, with π(·)∈R^d×d_dev a.e. onΩ, for the system (4.10). Moreover,

u_{τ h}

W^1,∞(I;W^1,γ(Ω;R^d))≤C_τ, (4.14a)

π_{τ h}

W^1,∞(I;W^a,2(Ω;R^d×d_dev)) ≤C_τ, (4.14b) η_{τ h}^k

W^1,∞(I;W^a,2(Ω)) ≤C_τ, (4.14c)

w_{τ h}^k

W^1,∞(I;W^1,2(Ω))≤C_τ, (4.14d)

with some C_τ independent of hand witha∈(0,1/2) referring to (4.6).

Sketch of the proof. We can see existence of a solution to (4.10) by a standard argument for coercive pseu- domonotone set-valued operators; cf. e.g. [17] for a general concept or, here, [28], Section 5.3, for inclusions with pseudomonotone operators whose set-valued part has a convex potential. The coercivity of the underlying operator can be shown by testing (4.10a)–(4.10c) by u^k_{τ h} ∈ V_1,h^d , π^k_{τ h} ∈ V_0,h^d×d, η_{τ h}^k ∈ V_0,h, and w_{τ h}^k ∈ V_1,h, respectively. Note that these test-functions live in the corresponding finite-dimensional spaces and are thus legal for this test. It is important that the right-hand sides of (4.10a), (4.10c) have the growth that can be dominated by the growth of the coercive terms in the left-hand sides; this is ensured by having takenβ andγ large enough and by the assumption (3.9b) which ensures a sublinear growth of Θ, namely

Θ(w)≤

w ωc₀ + 1

_1/ω

−1≤

w ωc₀

_1/ω

(4.15)

because obviouslyh₀(θ)≥ωc₀(1+θ)^ω−ωc₀,cf.the definition (3.2). Realize that the coercivity can be estimated (up to multiplicative constants) as |e|^γ+|π|²+|η|²+|w|^β which indeed dominates the growth of the “right- hand-side terms” is of the type|w|^1/ω|e|+ (|π|+|η|)|w|+|e|²|w|+|w|^1+1/ω|e|. The term (|π|+|η|)|w| bears the estimation by δ|π|²+δ|η|²+δ|w|^β+C_δ and similarly |e|²|w| ≤ δ|e|^γ +δ|w|² +C_δ with any δ > 0 and someC_δ; here β >2 andγ >4 have respectively been used. The last term can be estimated as|w|^1+1/ω|e| ≤

1

γ|e|^γ+|w|(1+1/ω)γ/(γ−1)≤_γ¹|e|^γ+¹₂|w|²+C_γ for someC_γ ∈R; here the conditionγ >2ω/(ω−1) has originated.

The a-priori estimates (4.14) then follows from the above test by standard procedure, i.e. by using the

H¨older, the Young, and the discrete Gronwall inequalities.

Lemma 4.2 (convergence for h↓0). There is a subsequence of {(uτ h, π_{τ h}, η_{τ h}, w_{τ h})}h>0 converging for h↓0 weakly* in the topologies indicated in (4.14) to some(u_τ, π_τ, η_τ, w_τ) and each quadruple obtained by such way