# Elastoplasticity of gradient-polyconvex materials

Martin Kružík  Czech Academy of Sciences, Institute of Information Theory and
Automation, Pod vodárenskou veží 4, 182 08 Prague, Czechia and Faculty of
Civil Engineering, Czech Technical University, Thákurova 7, 166 29 Prague,
Czechia<EMAIL_ADDRESS>http://staff.utia.cas.cz/kruzik and  Jiří Zeman
Institut für Mathematik, Universität Augsburg, D-86135 Augsburg, Germany
<EMAIL_ADDRESS>https://www.uni-
augsburg.de/de/fakultaet/mntf/math/prof/ana/arbeitsgruppe/jiri-zeman

###### Abstract.

We propose a model for rate-independent evolution in elastoplastic materials
under external loading, which allows large strains. In the setting of strain-
gradient plasticity with multiplicative decomposition of the deformation
gradient, we prove the existence of the so-called energetic solution. The
stored energy density function is assumed to depend on gradients of minors of
the deformation gradient which makes our results applicable to shape-memory
materials, for instance.

###### Key words and phrases:

elastoplasticity, energetic solution, gradient polyconvexity, rate-independent
model

##  1\. Introduction and notation

Elastoplasticity at large strains is an area of ongoing research, bringing
together contributions from modelling, analysis and numerical simulations. For
the mathematical analysis of elastoplastic models, it is often convenient to
use powerful tools from the calculus of variations, which are now able to
treat quasistatic evolutionary problems as well (see e.g. [49], [41] as
pioneering works). In principle, the existence of solutions could be ensured
by assuming some generalized convexity of the strain energy, but general
material behavior may contradict this assumption. For example, this is
manifested in shape-memory alloys (SMA) [4], magnetostrictive [14] and
ferroelectric materials [54], even if convexity is understood in a generalized
sense, such as polyconvexity (first defined in [1]).

As a remedy, one can then recourse to higher-gradient regularizations, where
the stored energy density $W$ also depends e.g. on the second gradient of the
deformation. From a mathematical point of view, this adds compactness to the
model, which is instrumental in proving the existence of solutions by the
direct method. [2] Materials with such constitutive equations are referred to
as non-simple and were introduced by Toupin [57, 58]. Since then, the concept
has been elaborated by many authors so that its thermodynamical side is also
better understood [25, 7, 50, 13, 18, 45, 19].

Gradient-polyconvex (GPC) materials form a special class of non-simple solids
and appeared in [5] for the first time. Their stored energy density is not a
function of the full second gradient $\nabla^{2}y$ of the deformation
$y\colon\Omega\to\mathbb{R}^{3}$, which maps the reference configuration
$\Omega$ to $\mathbb{R}^{3}$ , but it only depends, in a convex way, on the
weak gradients of ${\rm cof}\nabla y$ (and that of ${\rm det}\nabla y$, if
desirable). (See Section 3 for the definition of a cofactor matrix.) To
interpret the condition physically, note that since ${\rm det}\nabla y$
measures the local change in volume between the reference and current
configuration of the material and ${\rm cof}\nabla y$ describes the
transformation of area [27, p. 78], the stored energy $W$ depending on their
gradients offers a control of how abruptly these changes vary in space.
Gradient polyconvexity has since been applied to the evolution of SMA [35] and
a numerical implementation of GPC material models is also available [32]. GPC
allows us to consider stored energy densities without assuming any notion of
convexity in the deformation gradient variable. This makes it suitable for
modelling of shape-memory alloys, for instance, because the resulting
functional is lower semicontinuous in the underlying weak topology. An
alternative approach to energy functionals that are not lower semicontinuous
is relaxation [10], however, available results prevent us from considering
energies tending to infinity for extreme compression. In this article, we
study an elastoplastic model using gradient polyconvexity.

The idea of strain-gradient plasticity is similar to that of nonsimple
materials, as it also uses higher-order terms to prevent physical quantities
from unrealistic fine-scale oscillations. Incorporating gradients of plastic
variables in the constitutive equations is common in the engineering
literature [44, p. 250] and we refer an interested reader to [15, 26, 28, 36,
3, 48, 59] or [16] and the references therein. Gradient terms account for non-
local interactions of dislocations and we include them as well, as they offer
a suitable regularization to our model.

For our problem, we formulate the so-called energetic solution (let us name
[46] as an early reference; other related sources are cited in [44]). One
advantage of the energetic formulation is that it avoids derivatives of
constitutive equations and time derivatives of the solution itself. [41] The
variational nature of this solution concept also combines well with
homogenization and relaxation. Two conditions lie at the core of the energetic
formulation: a stability inequality, which couples minimization of the elastic
energy with a principle of maximum dissipation, and an energy balance. The two
requirements together imply that a usual plastic flow rule is satisfied for
sufficiently smooth solutions.

The plan of the paper is as follows: in Section 2 we review some basic facts
from the modelling of materials with internal variables, loosely following
[21] and [30]. For the sake of completeness, we also motivate the definition
of an energetic solution, although this has been done more thoroughly in
previous works of Mielke et al. The main part of this paper is Section 3,
where we study the rate-independent behavior of elastoplastic GPC materials
under external loads and prove the existence result. The section is concluded
with an example from crystal plasticity, which illustrates the usability of
our findings.

Our approach draws inspiration from [39], where the existence of energetic
solutions in large-strain elastoplasticity is proved in the presence of
plastic strain gradients and a polyconvex $W$. However, as our stored energy
is gradient polyconvex, our findings apply to the setting of multiwell
energies, as encountered e.g. in shape-memory alloys, cf. [6], [43] or [37]
for instance. We remark that energetic formulations in elastoplasticity have
also been propounded: in the linear framework [22], without strain gradients
[8], using a finite quasiconvex energy [33], involving a plastic Cauchy-Green
tensor [23, 24], for numerical computations [45], and elsewhere. The paper
[12] discusses different assumptions in quasistatic large-strain elastoplastic
evolutions. Lastly, in [34], rate-independent dislocation-free plasticity is
treated.

In what follows, $L^{\beta}(\Omega;\mathbb{R}^{n})$, $1\leq\beta<+\infty$
denotes the usual Lebesgue space of mappings $\Omega\to\mathbb{R}^{n}$ whose
modulus is integrable with the power $\beta$ and
$L^{\infty}(\Omega;\mathbb{R}^{n})$ is the space of measurable and essentially
bounded mappings $\Omega\to\mathbb{R}^{n}$. Further,
$W^{1,\beta}(\Omega;\mathbb{R}^{n})$ standardly represents the space of
mappings which live in $L^{\beta}(\Omega;\mathbb{R}^{n})$ and their gradients
belong to $L^{\beta}(\Omega;\mathbb{R}^{n\times n})$. Finally,
$W^{1,\beta}_{0}(\Omega;\mathbb{R}^{n})$ is a subspace of
$W^{1,\beta}(\Omega;\mathbb{R}^{n})$ of maps with a zero trace on
$\partial\Omega$. The weak convergence in $L^{\beta}(\Omega;\mathbb{R}^{n})$
is defined as follows: $y_{k}\to y$ weakly in
$L^{\beta}(\Omega;\mathbb{R}^{n})$ (weakly star for $\beta=+\infty$) if
$\int_{\Omega}y_{k}(x)\cdot\varphi(x)\,\mathrm{d}x\to\int_{\Omega}y(x)\cdot\varphi(x)\,\mathrm{d}x$
for all $\varphi\in L^{\beta^{\prime}}(\Omega;\mathbb{R}^{n})$ where
$\beta^{\prime}=\beta/(\beta-1)$ if $1<\beta<+\infty$, $\beta^{\prime}=1$ if
$\beta=+\infty$ and $\beta^{\prime}=+\infty$ for $\beta=1$. Weak convergence
of mappings and their gradients in $L^{\beta}$ then defines the weak
convergence in $W^{1,\beta}(\Omega;\mathbb{R}^{n})$. We also write
w-$\lim_{k\to+\infty}y_{k}=y$ or $y_{k}{\rightharpoonup}y$ to denote weak
convergence. Finally, $C(\Omega)$ or $C(\mathbb{R}^{n\times n})$ stand for
function spaces of functions continuous on $\Omega$ or $\mathbb{R}^{n\times
n}$, respectively.

If $f\colon\mathbb{R}^{n}\to\mathbb{R}$ is convex but possibly nonsmooth we
define its subdifferential at a point $x_{0}\in\mathbb{R}^{n}$ as the set of
all $v\in\mathbb{R}^{n}$ such that $f(x)\geq f(x_{0})+v\cdot(x-x_{0})$ for all
$x\in\mathbb{R}^{n}$. The subdifferential of $f$ will be denoted
$\partial^{\rm sub}f$ and its elements will be called subgradients of $f$ at
$x_{0}$.

##  2\. Motivation: modelling inelastic processes with internal variables

Consider $\Omega\subset\mathbb{R}^{n}$, a bounded Lipschitz domain
representing the so-called reference configuration of a solid body, and a
mapping $y\colon\Omega\to\mathbb{R}^{n}$, the deformation which the body is
subjected to. We explain the idea of elastoplasticity and the concept of
energetic solutions following freely the exposition in [21] and restricting
ourselves to simple materials.

According to Han and Reddy [30, p. 34], plastic deformation ‘is most
conveniently described in the framework of materials with internal variables’.
Those material models are not only governed by external (controllable)
variables, such as temperature or strain, but also incorporate a vector $z\in
Z\subset\mathbb{R}^{m}$ of internal variables, which describe e.g. an ongoing
chemical reaction, elastoplastic behavior or material damage. (Details can be
found in [29] or in more recent works on the subject listed in [30, p. 39].)
The hyperelastic stored energy density $\mathcal{W}$ then has the form
$\mathcal{W}=\mathcal{W}(F,z)$, if we consider a simple material, i.e. a
material with constitutive equations involving only the first gradient
$F=\nabla y$ of the deformation $y$.

The thermodynamic conception is usually that differentiating $\mathcal{W}$
with respect to $F$, we get the mechanical stress, whereas the derivative
$-\partial_{z}\mathcal{W}$ gives another stress-like variable – the so-called
thermodynamic force

$\displaystyle Q:=-\frac{\partial}{\partial z}\mathcal{W}(F,z)$ (2.1)

associated with the internal variable $z$. We can imagine that $Q$ tries to
restructure the material irreversibly, which would lead to changing the value
of $z$. The development of convex analysis [47] allowed quite a general
formulation of an evolution rule for the internal variable $z$. Assuming the
existence of a nonnegative convex potential of dissipative forces
$\delta=\delta(\dot{z})$, where $\dot{z}$ denotes the time derivative of $z$,
we write the flow law as

$\displaystyle Q(t)\in{\partial^{\rm sub}}\delta(\dot{z}(t))$ (2.2)

everywhere in $\Omega$. A common simplifying assumption is rate-independent
behavior. It is suitable for some particular materials or for the modelling of
processes with low rates of external loading. In simple terms, rate-
independence means that rescaling the loading in time only results in a
corresponding time-rescaling of the deformation and no additional viscous,
inertial or thermal effects arise. Rate-independence translates into positive
one-homogeneity of $\delta$, i.e.
$\delta(\alpha\dot{z})=\alpha\delta(\dot{z})$ for all $\alpha>0$.

######  Remark 2.1.

Since the subdifferential is monotone (see [51]), by (2.2) we have

$\displaystyle(Q(t)-\theta)\cdot(\dot{z}(t)-\xi)\geq 0.$ (2.3)

for all $\theta\in{\partial^{\rm sub}}\delta(\xi)$. Choosing $\xi=0$ and
observing that the one-homogeneity of $\delta$ implies
$\delta(\dot{z})=\omega\cdot\dot{z}$ for all $\omega\in{\partial^{\rm
sub}}\delta(\dot{z})$ we get

$\displaystyle\delta(\dot{z}(t))=Q(t)\cdot\dot{z}(t)\geq\theta\cdot\dot{z}(t)$
(2.4)

for all $\theta\in{\partial^{\rm sub}}\delta(0)$. Hence we derived the so-
called maximum dissipation principle (see e.g. [31] or [52, 53]) which states
that the plastic dissipation that takes place in reality is not less than any
possible dissipation due to thermodynamic forces “available” in the so-called
elastic domain ${\partial^{\rm sub}}\delta(0)$.

Hereafter, $\nu$ is the outer unit normal to $\partial\Omega$, and
$\partial\Omega\supset\Gamma_{0},\Gamma_{1}$ which are disjoint. Let
$f(t)\colon\Omega\to\mathbb{R}^{n}$ be the (volume) density of external body
forces and $g(t)\colon\Gamma_{1}\subset\partial\Omega\to\mathbb{R}^{n}$ be the
(surface) density of surface forces. The conservation of momentum yields the
equilibrium equations

$\displaystyle-{\rm div}\left(\frac{\partial}{\partial F}\mathcal{W}(\nabla
y(t),z(t))\right)=f(t)\mbox{ in $\Omega$},$ (2.5) $\displaystyle
y(t,x)=y_{0}(x)\mbox{ on $\Gamma_{0}$},$ (2.6)
$\displaystyle\frac{\partial}{\partial F}\mathcal{W}(\nabla
y(t),z(t))\nu(x)=g(t,x)\mbox{ on $\Gamma_{1}$}.$ (2.7)

If $z_{0}\in Z$ is an initial condition for the internal variable, the system
of equations (2.5)-(2.7) together with (2.8):

$\displaystyle-\frac{\partial}{\partial z}\mathcal{W}(\nabla
y(t),z(t))\in\partial\delta(\dot{z}(t)),\ z(0)=z_{0},$ (2.8)

governs the mechanical behavior characterized by the unknowns $y(t)$, $z(t)$.

Unfortunately, the system (2.5)-(2.7) is ill-posed in many situations. See the
works of Suquet [55] and Temam [56] for early analysis of this problem.

To get existence results, it seems necessary to include the gradient of the
internal variable, i.e. use the strain energy density

$\tilde{\mathcal{W}}(\nabla y,z,\nabla z):=\mathcal{W}(\nabla
y,z)+\epsilon|\nabla z|^{\alpha}$

for $\alpha\geq 1$ and $\varepsilon>0$. Let us briefly recall how an energy
balance, which appears in the energetic formulation of this regularized
problem, is obtained, under sufficient smoothness and integrability
assumptions on all the present mappings. For details, see [21, 41].

The functional

$\displaystyle\mathcal{I}(t,y(t),z(t)):=\int_{\Omega}\mathcal{W}(\nabla
y(t),z(t))\,\mathrm{d}x+\epsilon\int_{\Omega}|\nabla
z(t)|^{\alpha}\,\mathrm{d}x-L(t,y(t)),$ (2.9)

expresses the potential energy in our system, where the work done by external
forces is

$\displaystyle L(t,y(t)):=\int_{\Omega}f(t)\cdot
y(t)\,\mathrm{d}x+\int_{\Gamma_{1}}g(t)\cdot y(t)\,\mathrm{d}S.\ $ (2.10)

We also introduce the total dissipation along $z$,

${\rm
Diss}(z;[0,t]):=\int_{0}^{t}\int_{\Omega}\delta(\dot{z}(s))\,\mathrm{d}x\mathrm{d}s.$

Calculating the thermodynamic force $\partial_{z}\tilde{\mathcal{W}}$, using
the relation $\delta(\dot{z})=\omega\cdot\dot{z}$, $\omega\in{\partial^{\rm
sub}}\delta(\dot{z})$, mentioned above (2.4), we can deduce from (2.5)-(2.7)
the following energy balance:

$\mathcal{I}(t,y(t),z(t))+{\rm
Diss}(z;[0,t])=\mathcal{I}(0,y(0),z(0))+\int_{0}^{t}\dot{L}(s,y(s))\,\mathrm{d}s.$

Due to low regularity in time, a more general expression for ${\rm
Diss}(z;[0,t])$ must be used in practice, though – see (2.14) below.

To this end, we define a dissipation distance between two values
$z_{0},z_{1}\in{\mathbb{Z}}$ of the internal variable:

$\displaystyle
D(x,z_{0},z_{1}):=\inf_{z}\left\\{\int_{0}^{1}\delta(x,z(s),\dot{z}(s))\,\mathrm{d}s;\
z(0)=z_{0},\ z(1)=z_{1}\right\\},$ (2.11)

where $z\in C^{1}([0,1];{\mathbb{Z}})$, and set

$\displaystyle\mathcal{D}(z_{1},z_{2})=\int_{\Omega}D(x,z_{1}(x),z_{2}(x))\,\mathrm{d}x$
(2.12)

for $z_{1},z_{2}\in\mathbb{Z}:=\\{z\colon\Omega\to\mathbb{R}^{m};\ z(x)\in
Z\mbox{ a.e. in $\Omega$}\\}$, as in [41].

In order to find a quasistatic evolution of the system Mielke, Theil, and
Levitas [46] came up with the following definition of the energetic solution
which conveniently overcomes nonsmoothness mentioned above. Moreover, it fully
exploits the possible variational structure of the problem and allows for very
general energy and dissipation functionals. This concept has versatile
applications to many problems in continuum mechanics of solids. Additionally,
working with ${\mathcal{I}}$ and $\mathcal{D}$ directly allows us to include
also higher derivatives of $y$ into the model or to require integrability of
some functions of $\nabla y$ as also done in this contribution.

###  2.1. Energetic solution

Suppose that the evolution of $y(t)\in{\mathbb{Y}}$ and $z(t)\in{\mathbb{Z}}$
is studied during a time interval $[0,T]$. The following two properties
characterize the energetic solution due to Mielke et al. [46].

(i) Stability inequality:
$\forall t\in[0,T],\,\tilde{z}\in{\mathbb{Z}},\,\tilde{y}\in{\mathbb{Y}}$:

$\displaystyle\mathcal{I}(t,y(t),z(t))\leq\mathcal{I}(t,\tilde{y},\tilde{z})+\mathcal{D}(z(t),\tilde{z})$
(2.13)

(ii) Energy balance: $\forall\ 0\leq t\leq T$

$\displaystyle\mathcal{I}(t,y(t),z(t))+{\rm
Var}(\mathcal{D},z;[0,t])=\mathcal{I}(0,y(0),z(0))+\int_{0}^{t}\dot{L}(\xi,y(\xi))\,\mathrm{d}\xi,$
(2.14)

$\text{where }{\rm
Var}(\mathcal{D},z;[s,t]):=\sup\left\\{\sum_{i=1}^{N}\mathcal{D}(z(t_{i}),z(t_{i-1}));\
\\{t_{i}\\}\mbox{ partition of }[s,t]\right\\}.$

######  Definition 2.2.

The mapping $t\mapsto(y(t),z(t))\in{\mathbb{Y}}\times{\mathbb{Z}}$ is an
energetic solution to the problem $(\mathcal{I},\delta,L)$ if the stability
inequality and energy balance are satisfied for all $t\in[0,T]$.

######  Remark 2.3.

The mechanical idea behind stability inequality (i) is the following: imagine
first that $\tilde{z}:=z(t)$, then $\mathcal{D}(z(t),\tilde{z})$ vanishes,
since no change in the internal variables implies no dissipation.
Consequently, (i) simplifies to
$\mathcal{I}(t,y(t),z(t))\leq\mathcal{I}(t,\tilde{y},\tilde{z})$ for all
$\tilde{y}\in{\mathbb{Y}}$ and $y(t)$ is a global minimizer of
$\mathcal{I}(t,\cdot,z(t))$ over ${\mathbb{Y}}$. So we see that in this case,
(i) has the meaning of an elastic equilibrium. If $\tilde{z}\neq z(t)$, then
the amount of dissipated energy between the states $\tilde{z}$ and $z(t)$
must, by (i), at least compensate for, if not outweigh the associated loss in
the total energy, which is a version of the principle of maximum dissipation.
[41]

We will write $\mathbb{Q}:=\mathbb{Y}\times\mathbb{Z}$ and set $q:=(y,z)$.
Next let us define the set of stable states at time $t$ as

$\displaystyle\mathcal{S}(t):=\\{q\in\mathbb{Q};\
\forall\tilde{q}\in\mathbb{Q}:\
\mathcal{I}(t,q)\leq\mathcal{I}(t,\tilde{q})+\mathcal{D}(q,\tilde{q})\\}$
(2.15)

and

$\displaystyle\mathcal{S}_{[0,T]}:=\bigcup_{t\in[0,T]}\\{t\\}\times\mathcal{S}(t).$
(2.16)

Moreover, a sequence $\\{(t_{k},q_{k})\\}_{k\in{\mathbb{N}}}$ is called stable
if $q_{k}\in\mathcal{S}(t_{k})$.

##  3\. Applications to elastoplasticity

This section shows how the energetic approach can be applied to an
elastoplastic problem of gradient-polyconvex materials.

###  3.1. Gradient polyconvexity

Gradient polyconvexity was first defined in [5] in analogy with classical
polyconvexity [1]. The difference is that here we assume that the stored
energy density is convex in gradients of minors of the deformation gradient
but not in minors alone. More precisely, the following definition is taken
from [5]. We recall that for an invertible $F\in\mathbb{R}^{n\times n}$ we
define the cofactor of $F$, ${\rm cof}F=({\rm
det}F)F^{-\top}\in\mathbb{R}^{n\times n}$. As is shown in [5] there are maps
$y\in W^{1,1}(\Omega;\mathbb{R}^{3})$ such that ${\rm det}\nabla y$ and ${\rm
cof}\nabla y$ are Lipschitz continuous but $y\not\in
W^{2,1}(\Omega;\mathbb{R}^{3})$. The same conclusion can be reached for every
$n\geq 3$. On the other hand, if $n=2$ then $\nabla{\rm cof}\nabla y$ has the
same entries (up to the minus sign) as $\nabla^{2}y$.

######  Definition 3.1.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded open domain. Let
$\mathcal{W}_{1}\colon\mathbb{R}^{n\times n}\times\mathbb{R}^{n\times n\times
n}\to\mathbb{R}\cup\\{+\infty\\}$ be a lower semicontinuous function. The
functional

$\displaystyle J(y)=\int_{\Omega}\mathcal{W}_{1}(\nabla y(x),\nabla[{\rm
cof}\nabla y(x)])\,\mathrm{d}x,$ (3.1)

defined for any measurable function $y\colon\Omega\to\mathbb{R}^{n}$ for which
the weak derivatives $\nabla y$, $\nabla[{\rm cof}\nabla y]$ exist and are
integrable is called gradient polyconvex if the function
$\mathcal{W}_{1}(F,\cdot)$ is convex for every $F\in\mathbb{R}^{n\times n}$.

We assume that for some $c>0$, and numbers $\alpha>n-1$, $s>0$ it holds that
for every $F\in\mathbb{R}^{n\times n}$ and every $H\in\mathbb{R}^{n\times
n\times n}$ we have that

$\displaystyle\mathcal{W}_{1}(F,H)\geq\begin{cases}c\big{(}|F|^{\alpha}+({\rm
det}F)^{-s}+|H|^{\alpha/(n-1)}\big{)}&\text{ if }{\rm det}F>0,\\\
+\infty&\text{ otherwise,}\end{cases}$ (3.2)

where $|\cdot|$ denotes the Euclidean norm. These growth assumptions can be
surely weakened but we stick to them in order to simplify our presentation.

The idea behind condition (3.2) is that the energy blows up if the deformation
does not preserve orientation or if the measures of strain on the right-hand
side grow to extreme values. Coercivity conditions involving $|F|$ and ${\rm
det}F$ are commonly used in nonlinear elasticity (see e.g. [9]) and reflect
variations in volume or changes of the distances of points caused by the
deformation. The term $H$ which is a placeholder for $\nabla{\rm cof}\nabla y$
penalizes spatial changes of ${\rm cof}\nabla y$ and, consequently, it aims at
suppressing abrupt areal changes in the deformed configuration.

An important feature of gradient polyconvexity is that no convexity
assumptions on $\mathcal{W}_{1}$ are needed in the $F$-variable, so that very
general material laws can be considered including multiwell energy functions
[6] or the St. Venant–Kirchhoff energy density [9]. We call materials whose
stored $J$ energy obeys (3.1) also gradient-polyconvex.

###  3.2. Assumptions on problem data

As in [26] we will consider so-called separable materials, i.e. materials
where the elastoplastic energy density has the form

$\displaystyle\mathcal{W}(F_{\rm e},H,F_{p},\nabla F_{\rm p},p,\nabla
p):=\mathcal{W}_{1}(F_{\rm e},H)+\mathcal{W}_{2}(F_{p},\nabla F_{\rm
p},p,\nabla p).$ (3.3)

We will assume that $\mathcal{W}_{1}$ is continuous and satisfies (3.2) while
for $\mathcal{W}_{2}$ we require:
(i) The plastic part $\mathcal{W}_{2}$ is continuous in its all arguments.

(ii) Suppose that there are two constants $C,c>0$ so that the following
assumption holds for constants $c_{1}>0$, $\beta>n$, and $\omega>n$:

$\displaystyle C(1+|F_{\rm
p}|^{\beta}+|G|^{\beta}+|p|^{\omega}+|\pi|^{\omega})$
$\displaystyle\geq\mathcal{W}_{2}(F_{\rm p},G,p,\pi)$ $\displaystyle\geq
c(|F_{\rm p}|^{\beta}+|G|^{\beta}+|p|^{\omega}+|\pi|^{\omega})-c_{1}.$ (3.4)

(iii) There is $c_{2}>0$, $v^{*}\in\mathbb{R}^{m}$ and a modulus of continuity
$\hat{\omega}$ such that for all $\hat{\alpha}>0$ , $F_{\rm
p}\in\mathbb{R}^{n\times n}$, $G\in\mathbb{R}^{n\times n\times n}$,
$p\in\mathbb{R}^{m}$ and $\pi\in\mathbb{R}^{m\times n}$:

$\displaystyle|\mathcal{W}_{2}(F_{\rm
p},G,p+\hat{\alpha}v^{*},\pi)-\mathcal{W}_{2}(F_{\rm
p},G,p,\pi)|\leq\hat{\omega}(\hat{\alpha})(\mathcal{W}_{2}(F_{\rm
p},G,p,\pi)+c_{2}).$ (3.5)

Furthermore, let us suppose that for every $F_{\rm e}$, $F_{\rm
p}\in\mathbb{R}^{n\times n}$ and $p\in\mathbb{R}^{m}$, the functions
$\mathcal{W}_{1}(F_{\rm e},\cdot)$ and $\mathcal{W}_{2}(F_{\rm
p},\cdot,p,\cdot)$ are convex.

The dissipation distance
$\mathcal{D}\colon{\mathbb{Z}}\times{\mathbb{Z}}\to[0,+\infty]$ takes the form
(2.12) for a function $D\colon\Omega\times({\rm
SL}(n)\times\mathbb{R}^{m})^{2}$ and we only change the definition of
${\mathbb{Z}}$ to (3.11) below. We make the following assumptions on
$\mathcal{D}$:

(i) Lower semicontinuity:

$\displaystyle\mathcal{D}(z,\tilde{z})\leq\liminf_{k\to\infty}\mathcal{D}(z_{k},\tilde{z}_{k}),$
(3.6)

whenever $z_{k}{\rightharpoonup}z$ and
$\tilde{z}_{k}{\rightharpoonup}\tilde{z}$.

(ii) Positivity:

$\displaystyle\text{If }\\{z_{k}\\}\subset Z\text{ is bounded and
}\min\\{\mathcal{D}(z_{k},z),\mathcal{D}(z,z_{k})\\}\to 0\text{ then
}z_{k}{\rightharpoonup}z.$ (3.7)

(iii) For all $z_{1}$, $z_{2}\in\mathbb{Z}$: $\mathcal{D}(z_{1},z_{2})=0$ if
and only if $z_{1}=z_{2}$.

(iv) Triangle inequality:
$\mathcal{D}(z_{1},z_{3})\leq\mathcal{D}(z_{1},z_{2})+\mathcal{D}(z_{2},z_{3})$
for all $z_{1}$, $z_{2}$, $z_{3}\in\mathbb{Z}$.

We refer the reader to [39] to see that (ii) follows from (i) and (iii). After
stating Proposition 3.6, we specify further assumptions on $D$ ((3.16) or
(3.A)–(3.C)). Besides, it is naturally required that $D$ be such that (i)–(iv)
holds.

In order to prove the existence of a solution to (3.20) we must impose some
data qualifications. In what follows, we assume that

$\displaystyle f\in
C^{1}\left([0,T];L^{\tilde{d}}\left(\Omega;\mathbb{R}^{n}\right)\right),$
(3.8) $\displaystyle g\in
C^{1}\left([0,T];L^{\hat{d}}\left(\Gamma_{1};\mathbb{R}^{n}\right)\right),$
(3.9)

where $\tilde{d}\geq[nd/(n-d)]^{\prime}=nd/(nd-n+d)$ if $1\leq d<n$ or
$\tilde{d}>1$ otherwise. Similarly, we suppose that
$\hat{d}\geq[(nd-d)/(n-d)]^{\prime}=(nd-d)/(nd-n)$ if $d<n$ or $\hat{d}>1$
otherwise.

###  3.3. Formulation of the problem

From now on, $y\colon\Omega\to\mathbb{R}^{n}$ will represent the deformation
of a material body, whose reference configuration is a bounded Lipschitz
domain $\Omega\subset\mathbb{R}^{n}$. Since $y$ models both elastic and
plastic behaviour, we split the deformation gradient $F=\nabla y$ as $F=F_{\rm
e}F_{\rm p}$, where $F_{\rm e}$ stands for an elastic part and $F_{\rm
p}\in{\rm SL}(n):=\\{A\in\mathbb{R}^{n\times n};\ {\rm det}\ A=1\\}$ is a
plastic part, which irreversibly transforms the material. To capture e.g. back
stresses, we use the vector $p\in\mathbb{R}^{m}$ of hardening internal
variables. Written together, $z(x)=(F_{\rm p}(x),p(x))$ is a plastic variable,
lying in ${\rm SL}(n)\times\mathbb{R}^{m}$ for almost all $x\in\Omega$.

The energy functional ${\mathcal{I}}$ is given by

$\displaystyle{\mathcal{I}}(t,y(t),z(t)):=\int_{\Omega}\mathcal{W}(\nabla
yF_{\rm p}^{-1},\nabla[({\rm cof}\nabla y)F_{\rm p}^{\top}],F_{\rm p},\nabla
F_{\rm p},p,\nabla p)\,\mathrm{d}x-L(t,y(t)),$ (3.10)

with $L$ defined in (2.10).

Our stored energy density $\mathcal{W}$ does not explicitly depend on the
spatial variable $x$, but treating the inhomogeneous case would not need many
modifications.

Let us remark that $({\rm cof}\nabla y)F_{\rm p}^{\top}$ is the cofactor of
the elastic part $F_{\rm e}$, since by the product rule for cofactor matrices
[9, p. 4] and by $F_{\rm p}\in{\rm SL}(n)$, we have

${\rm cof}F_{\rm e}={\rm cof}(FF_{\rm p}^{-1})=({\rm cof}F){\rm cof}(F_{\rm
p}^{-1})=({\rm cof}F){\rm det}(F_{\rm p}^{-1})(F_{\rm p}^{-1})^{-\top}=({\rm
cof}F)F_{\rm p}^{\top}.$

The admissible deformations $y$ lie in

$\mathbb{Y}:=\\{y\in W^{1,d}(\Omega;\mathbb{R}^{n});\ y=y_{0}\mbox{ on
}\Gamma_{0}\\},$

where $\Gamma_{0}\subset\partial\Omega$ with a positive surface measure and
$y_{0}\in W^{1-1/d,d}(\Gamma_{0};\mathbb{R}^{n})$ is given. Assuming that
$\Gamma_{1}\subset\partial\Omega$ as in Section 2, we suppose
$\Gamma_{0}\cap\Gamma_{1}=\mathchar 31\relax$. For the internal states $z$ let
us define the set

$\displaystyle\mathbb{Z}:=\\{(F_{\rm p},p)\in
W^{1,\beta}(\Omega;\mathbb{R}^{n\times n})\times
W^{1,\omega}(\Omega;\mathbb{R}^{m}):\ F_{\rm p}(x)\in{\rm SL}(n)\mbox{ for
a.e.~{}$x\in\Omega$}\\}.$ (3.11)

For ease of notation, we write
$q=(y,z)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\in\mathbb{Q}=\mathbb{Y}\times\mathbb{Z}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
and understand $\mathcal{I}(t,\cdot)$, $L(t,\cdot)$ and $\mathcal{D}$ as
functions of $q$, that is:

$\displaystyle{\mathcal{I}}(t,q(t))=\int_{\Omega}\mathcal{W}(\nabla yF_{\rm
p}^{-1},\nabla[({\rm cof}\nabla y)F_{\rm p}^{\top}],F_{\rm p},\nabla F_{\rm
p},p,\nabla p)\,\mathrm{d}x-L(t,q(t)),$ $\displaystyle L(t,q(t):=L(t,y(t)),$
$\displaystyle\mathcal{D}(q_{1},q_{2}):=\mathcal{D}(z_{1},z_{2})$

if $q_{1}=(y_{1},z_{1})$ and $q_{2}=(y_{2},z_{2})$.

In order to prove the existence of an energetic solution to our problem we
will need the following results of technical nature.

###  3.4. Auxiliary results

We start this section by the following reverse Young inequality.

######  Lemma 3.2.

Suppose that $a>0$, $b>0$, $\delta>0$, $r>1$. Then

$\frac{a}{b}\geq
r\delta^{\frac{r}{r-1}}a^{\frac{1}{r}}-(r-1)\delta^{\frac{r^{2}}{(r-1)^{2}}}b^{\frac{1}{r-1}}.$

###### Proof.

Young’s inequality states that given a pair of positive numbers $\alpha,\beta$
and $1<p,q<+\infty$, $\frac{1}{p}+\frac{1}{q}=1$, then

$\alpha\beta\leq\frac{\alpha^{p}}{p}+\frac{\beta^{q}}{q}.$ (3.12)

Set $p=r$, $\alpha=a^{\frac{1}{r}}$, $\beta=\delta^{\frac{r}{r-1}}b$ in
(3.12). Then $q=\frac{r}{r-1}$ and Young’s inequality yields

$\frac{r-1}{r}\delta^{\frac{r^{2}}{(r-1)^{2}}}b^{\frac{r}{r-1}}+\frac{1}{r}a\geq\delta^{\frac{r}{r-1}}a^{\frac{1}{r}}b,$

which after multiplying by $\frac{r}{b}$ implies the desired result. ∎

It will also be useful to give a name to a kind of convergence which makes
${\mathcal{I}}(t,\cdot)$ lower semicontinuous.

######  Definition 3.3.

We say that the sequence $\\{q_{k}\\}_{k\in{\mathbb{N}}}\subset\mathbb{Q}$,
$q_{k}=(y_{k},z_{k})=(y_{k},F_{{\rm p}k},p_{k})$ _gpc-converges_ to
$q_{*}=(y_{*},z_{*})=(y_{*},F_{{\rm p}*},p_{*})$ if
$z_{k}{\rightharpoonup}z_{*}$ in $\mathbb{Z}$, $\nabla{\rm cof}(\nabla
y_{k}F_{{\rm p}k}^{-1}){\rightharpoonup}\nabla{\rm cof}(\nabla y_{*}F_{{\rm
p}*}^{-1})$ in $L^{\alpha/(n-1)}(\Omega;\mathbb{R}^{n\times n\times n})$ and
$\nabla y_{k}\to\nabla y_{*}$ in measure. We write
$q_{k}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny gpc}}}{\to}}q_{*}$ for
short.

######  Lemma 3.4.

Let $t_{k}\to t_{*}$ with $t_{k}$, $t_{*}\in[0,T]$, $k\in{\mathbb{N}}$, and
$q_{k}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny gpc}}}{\to}}q_{*}$,
$\\{q_{k}\\}_{k\in{\mathbb{N}}}\subset\mathbb{Q}$. Then
${\mathcal{I}}(t_{*},q_{*})\leq\liminf_{k\to\infty}{\mathcal{I}}(t_{k},q_{k})$.

###### Proof.

This is an immediate consequence of [17, Corollary 7.9]. Note that we can
construct a subsequence such that $\nabla y_{k_{j}}F_{{\rm
p}k_{j}}^{-1}\to\nabla y_{*}F_{{\rm p}*}^{-1}$ almost everywhere.∎

Even though we do not have the weak lower semicontinuity of ${\mathcal{I}}$ in
general, we can get it for a subsequence provided the respective values of
${\mathcal{I}}$ are bounded.

######  Lemma 3.5.

Provided that $\alpha^{-1}+\beta^{-1}\leq d^{-1}<(n-1)^{-1}$ and
$d>\frac{\beta(n-1)}{\beta-1}$, let $t_{k}\in[0,T]$, $k\in{\mathbb{N}}$, and
$q_{k}{\rightharpoonup}q_{*}$ in $\mathbb{Q}$. Suppose there is $C_{I}>0$ such
that for all $k\in{\mathbb{N}}$ the bound ${\mathcal{I}}(t_{k},q_{k})\leq
C_{I}$ holds true. Then there exists a subsequence
$\\{q_{k_{j}}\\}_{j\in{\mathbb{N}}}$ of $\\{q_{k}\\}_{k\in{\mathbb{N}}}$ that
gpc-converges to the same limit $q_{*}$.

###### Proof.

The proof goes the same way as in Proposition 3.8. To keep the flow of ideas
uninterrupted there, we postpone the presentation to that section. ∎

######  Proposition 3.6.

Let ${\mathcal{I}}$ be lower semicontinuous with respect to gpc-convergence
and let (3.8) and (3.9) hold. Let it for all
$(t_{*},q_{*})\in[0,T]\times\mathbb{Q}$ and all stable sequences
$\\{(t_{k},q_{k})\\}_{k\in{\mathbb{N}}}$ such that
w-$\lim_{k\to\infty}(t_{k},q_{k})=(t_{*},q_{*})$ be true that for all
$\tilde{q}\in\mathbb{Q}$ there is $\\{\tilde{q}_{k}\\}\subset\mathbb{Q}$ such
that

$\displaystyle\limsup_{k\to\infty}({\mathcal{I}}(t_{k},\tilde{q}_{k})+\mathcal{D}(q_{k},\tilde{q}_{k}))\leq{\mathcal{I}}(t_{*},\tilde{q})+\mathcal{D}(q_{*},\tilde{q})).$
(3.13)

Then for any stable sequence $\\{(t_{k},q_{k})\\}_{k\in{\mathbb{N}}}$ such
that w-$\lim_{k\to\infty}(t_{k},q_{k})=(t_{*},q_{*})$ and
${\mathcal{I}}(t_{k},q_{k})\leq C$ for some $C>0$, we have
$\lim_{k\to\infty}{\mathcal{I}}(t_{k},q_{k})={\mathcal{I}}(t_{*},q_{*})$ and
$q_{*}\in\mathcal{S}(t_{*})$.

Proof. We follow the proof of Prop. 4.3 in [39]. Take $\tilde{q}:=q_{*}$ in
(3.13), which yields a sequence $\\{\tilde{q}_{k}\\}_{k\in{\mathbb{N}}}$. Then
we get, by the stability of $q_{k}$,

$\displaystyle\limsup_{k\to\infty}{\mathcal{I}}(t_{k},q_{k})\leq\limsup_{k\to\infty}(({\mathcal{I}}(t_{k},\tilde{q}_{k})+\mathcal{D}(q_{k},\tilde{q}_{k}))\leq{\mathcal{I}}(t_{*},\tilde{q})+\mathcal{D}(q_{*},\tilde{q})={\mathcal{I}}(t_{*},q_{*}).$
(3.14)

The assumptions (3.8) and (3.9) on $f$ and $g$ further give

$\displaystyle\lim_{k\to\infty}|{\mathcal{I}}(t_{k},q_{k})-{\mathcal{I}}(t_{*},q_{k})|=\lim_{k\to\infty}|L(t_{k},q_{k})-L(t_{*},q_{k})|=0.$
(3.15)

Since ${\mathcal{I}}$ is lower semicontinuous with respect to gpc-convergence
(and by Lemma 3.5 we can pass to a gpc-convergent subsequence, without
relabeling it), we deduce by equation (3.15) that

$\liminf_{k\to\infty}{\mathcal{I}}(t_{k},q_{k})=\lim_{k\to\infty}({\mathcal{I}}(t_{k},q_{k})-{\mathcal{I}}(t_{*},q_{k}))+\liminf_{k\to\infty}{\mathcal{I}}(t_{*},q_{k})\geq{\mathcal{I}}(t_{*},q_{*}).$

This combined with (3.14) establishes the weak continuity along a stable
sequence: ${\mathcal{I}}(t_{k},q_{k})\to{\mathcal{I}}(t_{*},q_{*})$. In the
end, pick a $\tilde{q}\in\mathbb{Q}$ and apply (3.13) to it:

${\mathcal{I}}(t_{*},q_{*})=\lim_{k\to\infty}{\mathcal{I}}(t_{k},q_{k})\leq\liminf_{k\to\infty}({\mathcal{I}}(t_{k},\tilde{q}_{k})+\mathcal{D}(q_{k},\tilde{q}_{k}))\leq{\mathcal{I}}(t_{*},\tilde{q})+\mathcal{D}(q_{*},\tilde{q});$

hence, the stability of $q_{*}$ is proved. $\Box$

A natural question is how to ensure the validity of (3.13). If
$\mathcal{D}\colon\mathbb{Q}\times\mathbb{Q}\to[0,+\infty)$, i.e. no
irreversibility constraint is imposed on plastic processes, then it is
sufficient if $D$ from (2.12) satisfies

$\displaystyle\begin{aligned} &D\text{ is a Carathéodory mapping and}\\\
&D(x,z_{1},z_{2})\leq c(x)+C(|F_{{\rm p}1}|^{\beta^{*}-\epsilon}+|F_{{\rm
p}2}|^{\beta^{*}-\epsilon}+|p_{1}|^{\omega^{*}-\epsilon}+|p_{2}|^{\omega^{*}-\epsilon}),\end{aligned}$
(3.16)

where $\epsilon>0$ is small enough and $\beta^{*}:=n\beta/(n-\beta)$ if
$n>\beta$ and $\beta^{*}>1$ if $\beta\geq n$. Similarly,
$\omega^{*}:=n\omega/(n-\omega)$ if $n>\omega$ and $\omega^{*}>1$ if
$\omega\geq n$. Then the compact embedding provides the continuity of
$\mathcal{D}$. This shows that (3.13) is valid with a constant sequence
$\tilde{q}_{k}=\tilde{q}$.

If $\mathcal{D}\colon\mathbb{Q}\times\mathbb{Q}\to[0,+\infty]$, the
assumptions are more elaborate. Following [39] we impose the following
sufficient conditions on $D$ from (2.11):
(3.A) $D(x,\cdot,\cdot)\colon\mathbb{D}(x)\to[0,+\infty)$ is continuous, where
$\mathbb{D}(x):=\\{(z_{1},z_{2});\ D(x,z_{1},z_{2})<+\infty\\}$,
(3.B) For every $R>0$ there is $K>0$ such that for almost all $x\in\Omega$:
$D(x,z_{1},z_{2})<K$ if $z_{1},z_{2}\in\mathbb{D}(x)$ and $|z_{1}|,|z_{2}|<R$,
and
(3.C) The direction $v^{*}\in\mathbb{R}^{m}$ from (3.5) has the property that
for all $\alpha,R>0$ there is $\rho>0$ such that for almost every $x\in\Omega$
and every $z,z_{0},z_{1}$:

$|z-z_{0}|<\rho\mbox{, }(z_{0},z_{1})\in\mathbb{D}(x)\mbox{ and
}|z_{0}|,|z_{1}|<R\mbox{ implies }(z,z_{1}+(0,\alpha v^{*}))\in\mathbb{D}(x).$

######  Proposition 3.7.

Let $\beta,\omega>n$. Let $D$ satisfy (3.A)–(3.C). Then (3.13) holds.

Proof. The reasoning follows the lines of [39]. If
$\mathcal{D}(q_{*},\tilde{q})=+\infty$ in (3.13), the proof is finished. So,
we assume that

$\mathcal{D}(q_{*},\tilde{q})\in\mathbb{R}.$

If $q_{k}{\rightharpoonup}q_{*}$, we observe that

$\displaystyle\rho_{k}:=\|F_{{\rm p}k}-F_{{\rm
p}*}\|_{C(\bar{\Omega};\mathbb{R}^{n\times
n})}+\|p_{k}-p_{*}\|_{C(\bar{\Omega};\mathbb{R}^{m})}\to 0.$ (3.17)

by the compact embedding. Then $|z_{k}|+|z_{*}|+|\tilde{z}|<R$ for some $R>0$
if $k$ is large enough. Define $\tilde{z}_{k}:=(\tilde{F}_{\rm
p},\tilde{p}+\alpha_{k}v^{*})$ where $\alpha_{k}\to 0$ and relates to
$\rho_{k}$ as in (3.C) (we may need to redefine the $\rho_{k}$ from (3.17) by
passing to a subsequence, which is without loss of generality). Thus,
$(z_{k},\tilde{z}_{k})\in\mathbb{D}(x)$ a.e. in $\Omega$ and we have
$|z_{k}|$, $|\tilde{z}_{k}|<R$. The continuity of $D$ gives the convergence of
$D(x,z_{k},\tilde{z}_{k})\to D(x,z_{*},\tilde{z})$ pointwise so that
$\mathcal{D}(q_{k},\tilde{q}_{k})\to\mathcal{D}(q_{*},\tilde{q})$ by condition
(3.B) and the dominated convergence theorem. Furthermore, properties of
$\mathcal{W}_{2}$ and $L$ (assumptions (3.8), (3.9)) imply that
${\mathcal{I}}(t_{k},\tilde{q}_{k})\to{\mathcal{I}}(t_{*},\tilde{q}_{*})$.
Summing up, we deduce that (3.13) is satisfied with equality. $\Box$

###  3.5. Incremental problems

Next, we define the following sequence of incremental problems. We consider a
stable initial condition $q^{0}_{\tau}:=q^{0}\in\mathbb{Q}$.

Let us take $\tau>0$, a time step, chosen in the way that
$N=T/\tau\in{\mathbb{N}}$. For $1\leq k\leq N$, $t_{k}:=k\tau$, find
$q_{\tau}^{k}\in\mathbb{Q}$ such that $q^{k}_{\tau}$ solves

$\displaystyle\left.\begin{array}[]{ll}\mbox{minimize
}&{\mathcal{I}}(t_{k},q)+\mathcal{D}(q_{\tau}^{k-1},q)\\\ \mbox{subject to
}&q^{k}_{\tau}\in\mathbb{Q}.\end{array}\right\\}\ $ (3.20)

######  Proposition 3.8.

Let $\alpha^{-1}+\beta^{-1}\leq
d^{-1}<\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(n-1)^{-1}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}d>\frac{\beta(n-1)}{\beta-1}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$.
Let the assumptions on $\mathcal{W}$ and $\mathcal{D}$ be satisfied. Let
further (3.8) and (3.9) be satisfied. Then the problem (3.20) has a solution
for all $k=1,\ldots,T/\tau$. In addition, for the solution
$q_{\tau}^{k}=(y_{\tau}^{k},z_{\tau}^{k})$ we get ${\rm det}\nabla
y_{\tau}^{k}>0$ a.e. in $\Omega$.

Proof. Given $q_{\tau}^{k-1}\in\mathbb{Q}$ from the previous time step,
suppose that $\\{q_{j}\\}\subset\mathbb{Q}$ is a minimizing sequence for
$q\mapsto{\mathcal{I}}(t_{k},q)+\mathcal{D}(q_{\tau}^{k-1},q)$. The assumption
(3.2) implies that $\\{z_{j}\\}$ is uniformly bounded in
$W^{1,\beta}(\Omega;\mathbb{R}^{n\times n})\times
W^{1,\omega}(\Omega;\mathbb{R}^{m})$. Hence, as $\beta,\omega>1$ we can
extract a weakly converging subsequence (not relabeled)
$z_{j}{\rightharpoonup}z$ in $W^{1,\beta}(\Omega;\mathbb{R}^{n\times n})\times
W^{1,\omega}(\Omega;\mathbb{R}^{m})$. The strong convergence of $z_{j}\to
z:=(F_{\rm p},p)$ in $L^{\beta}(\Omega;\mathbb{R}^{n\times n})\times
L^{\omega}(\Omega;\mathbb{R}^{m})$ ensures that $F_{\rm p}(x)\in{\rm SL}(n)$
almost everywhere. Write $z_{j}=(F_{\rm p}^{j},p^{j})$, $q_{j}=(y^{j},z_{j})$.
Exploiting the submultiplicativity of the Euclidean norm, we estimate using
Lemma 3.2

$\displaystyle\int_{\Omega}|\nabla y^{j}(x)(F_{\rm
p}^{j})^{-1}|^{\alpha}\,\mathrm{d}x$ $\displaystyle\geq\frac{\|\nabla
y^{j}\|_{L^{d}(\Omega;\mathbb{R}^{n\times n})}^{\alpha}}{\|F^{j}_{\rm
p}\|_{L^{\beta}(\Omega;\mathbb{R}^{n\times n})}^{\alpha}}$
$\displaystyle\geq\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{\alpha}{d}\delta^{\alpha/(\alpha-d)}\|\nabla
y^{j}\|_{L^{d}(\Omega;\mathbb{R}^{n\times
n})}^{d}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-\frac{\alpha-d}{d}\delta^{\alpha^{2}/(\alpha-d)^{2}}\|F^{j}_{\rm
p}\|_{L^{\beta}(\Omega;\mathbb{R}^{n\times n})}^{\beta}.$ (3.21)

The $L^{d}$-term on the right hand side is bounded due to (3.2) and the
boundedness of $\\{y^{j}\\}$ in $W^{1,d}(\Omega;\mathbb{R}^{n})$ follows by
the Poincaré inequality if $\delta>0$ is taken small. Hence
$y^{j}{\rightharpoonup}y$ in $W^{1,d}(\Omega;\mathbb{R}^{n})$ (up to a
subsequence). This then also implies that ${\rm cof}\nabla
y^{j}{\rightharpoonup}{\rm cof}\nabla y$ in
$L^{d/(n-1)}(\Omega;\mathbb{R}^{n\times n})$; cf. [9]. Due to reflexivity of
$W^{1,\alpha/(n-1)}(\Omega;\mathbb{R}^{n\times n})$ we get for a non-
relabelled subsequence that $({\rm cof}\nabla y^{j})F^{j\top}_{\rm
p}{\rightharpoonup}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\,\Xi\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
in $W^{1,\alpha/(n-1)}(\Omega;\mathbb{R}^{n\times n})$ for some $\Xi$ .
However, the weak convergence of $\\{{\rm cof}\nabla y^{j}\\}_{j}$ and the
strong convergence of $\\{F_{\rm p}^{j}\\}$ allow us to identify
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Xi\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}=({\rm
cof}\nabla y)F^{\top}_{\rm p}$. Moreover, the growth condition (3.2) implies
that ${\rm det}\nabla y>0$ a.e. in $\Omega$ (see [5]). Cramer’s rule together
with ${\rm det}F_{\rm p}^{j}=1$ and ${\rm det}^{n-1}(\nabla y^{j}(F^{j}_{\rm
p})^{-1})={\rm det}({\rm
cof}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}[\nabla
y^{j}(F^{j}_{\rm
p})^{-1}]\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})={\rm
det}({\rm cof}\nabla y^{j})$ give

$\frac{{\rm cof}[\nabla y^{j}(F_{\rm p}^{j})^{-1}]}{{\rm det}\nabla
y^{j}}=(\nabla y^{j}(F_{\rm p}^{j})^{-1})^{-\top}$

and the transpose of the left-hand side converges pointwise (for a subsequence
again) to $(\nabla y(F_{\rm p})^{-1})^{-1}$ a.e. in $\Omega$. This, together
with the fact that ${\rm det}\nabla y>0$ a.e., implies that $\nabla
y^{j}(F^{j}_{\rm p})^{-1}\to\nabla y(F_{\rm p})^{-1}$ a.e. in $\Omega$. Then
by[17, Corollary 7.9] we see that $\mathcal{I}$ is weakly lower
semicontinuous.

The assumptions on $\mathcal{D}$ ensure that it is sequentially weakly lower
semicontinuous on $\mathbb{Q}$, too. By the direct method of the calculus of
variations, we conclude that a minimizer $q_{\tau}^{k}$ exists. $\Box$

######  Remark 3.9.

Note that in fact, we proved that a minimizing sequence has a gpc-convergent
subsequence.

###  3.6. Interpolation in time

We denote by $q_{\tau}$ a piecewise constant interpolation of
$q_{\tau}^{k}=:(y_{\tau}^{k},z^{k}_{\tau})$, i.e. $q_{\tau}(t)=q_{\tau}^{k}$
if $t\in[k\tau,(k+1)\tau)$ and $k=0,\ldots,T/\tau-1$. Finally,
$q_{\tau}(T):=q_{\tau}^{N}$. Analogously,
$L_{\tau}(t,q_{\tau}(k\tau)):=L(k\tau,q_{\tau}(k\tau))$ is a piecewise
constant interpolation of $L$ and
${\mathcal{I}}_{\tau}(t,q_{\tau}(k\tau)):={\mathcal{I}}(k\tau,q_{\tau}(k\tau))$
is a piecewise constant interpolation of ${\mathcal{I}}$.

######  Proposition 3.10.

Under the assumptions of Proposition 3.8, problem (3.20) has a solution
$q_{\tau}(t)$ which is stable, i.e., for all $t\in[0,T]$ and for every
$q\in\mathbb{Q}$,

$\mathcal{I}_{\tau}(t,q_{\tau}(t))\leq\mathcal{I}_{\tau}(t,q)+\mathcal{D}\left(q_{\tau}(t),q\right).$
(3.22)

Moreover, for all $t_{I}\leq t_{I\\!I}$ from the set $\\{k\tau\\}_{k=0}^{N}$,
the following discrete energy inequalities hold if one extends the definition
of $q_{\tau}(t)$ by setting $q_{\tau}(t):=q^{0}$ if $t<0$:

$-\int_{t_{I}}^{t_{I\\!I}}\dot{L}\left(t,q_{\tau}(t-\tau)\right)\,\mathrm{d}t\leq\mathcal{I}\left(t_{I\\!I},q_{\tau}(t_{I\\!I})\right)+{\rm
Var}\left(\mathcal{D},q_{\tau};\left[t_{I},t_{I\\!I}\right]\right)-\mathcal{I}\left(t_{I},q_{\tau}(t_{I})\right)\\\
\leq-\int_{t_{I}}^{t_{I\\!I}}\dot{L}\left(t,q_{\tau}(t)\right)\,\mathrm{d}t.$
(3.23)

Proof. In Proposition 3.8 we proved the existence of a solution to (3.20).

To show the stability estimate (3.22), we use the minimizing property of
$q_{\tau}^{k}$ and a triangle inequality for $\mathcal{D}$. Indeed, since
$q_{\tau}^{k}$ is a minimizer,

$\mathcal{I}\left(k\tau,q^{k}_{\tau}\right)+\mathcal{D}\left(q^{k-1}_{\tau},q^{k}_{\tau}\right)\leq\mathcal{I}\left(k\tau,q\right)+\mathcal{D}\left(q^{k-1}_{\tau},q\right),$
(3.24)

from which we infer that

$\mathcal{I}\left(k\tau,q^{k}_{\tau}\right)\leq\mathcal{I}\left(k\tau,q\right)+\mathcal{D}\left(q^{k-1}_{\tau},q\right)-\mathcal{D}\left(q^{k-1}_{\tau},q^{k}_{\tau}\right).$

However, the structure of the metric implies that

$\mathcal{D}\left(q^{k-1}_{\tau},q\right)-\mathcal{D}\left(q^{k-1}_{\tau},q^{k}_{\tau}\right)\leq\mathcal{D}\left(q^{k}_{\tau},q\right),$

from which (3.22) follows.

The next step is to verify energy inequality (3.23), following the thoughts of
[46]. Testing the stability of $q^{k-1}_{\tau}$ with $q:=q^{k}_{\tau}$, we get

$\displaystyle\mathcal{I}\left((k-1)\tau,q^{k-1}_{\tau}\right)$
$\displaystyle\leq\mathcal{I}\left((k-1)\tau,q^{k}_{\tau}\right)+\mathcal{D}\left(q^{k-1}_{\tau},q^{k}_{\tau}\right)$
(3.25)
$\displaystyle=\mathcal{I}\left(k\tau,q^{k}_{\tau}\right)+L\left(k\tau,q^{k}_{\tau}\right)-L\left((k-1)\tau,q^{k}_{\tau}\right)+\mathcal{D}\left(q^{k-1}_{\tau},q^{k}_{\tau}\right).$

For nonnegative integers $k_{1}$, $k_{2}$ with $k_{1}\leq k_{2}\leq N$, let
$t_{I}=k_{1}\tau$ and $t_{I\\!I}=k_{2}\tau$. Summing (3.25) over
$k=k_{1}+1,\dots,k_{2}$ gives

$\displaystyle\sum_{k=k_{1}+1}^{k_{2}}\left[L\left((k-1)\tau,q^{k}_{\tau}\right)-L\left(k\tau,q^{k}_{\tau}\right)\right]$
$\displaystyle\leq$
$\displaystyle\mathcal{I}\left(k_{2}\tau,q^{k_{2}}_{\tau}\right)-\mathcal{I}\left(k_{1}\tau,q^{k_{1}}_{\tau}\right)$
$\displaystyle+$
$\displaystyle\sum_{k=k_{1}+1}^{k_{2}}\mathcal{D}\left(q^{k-1}_{\tau},q^{k}_{\tau}\right).$

Replacing $q_{\tau}^{k_{1}}$, $q_{\tau}^{k_{2}}$ with $q_{\tau}$ evaluated at
suitable time points, we discover the first inequality in (3.23),

$\displaystyle-\int_{t_{I}}^{t_{I\\!I}}\dot{L}\left(t,q_{\tau}(t-\tau)\right)\,\mathrm{d}t$
$\displaystyle\leq\mathcal{I}\left(k_{2}\tau,q^{k_{2}}_{\tau}\right)-\mathcal{I}\left(k_{1}\tau,q^{k_{1}}_{\tau}\right)+\sum_{k=k_{1}+1}^{k_{2}}\mathcal{D}\left(q^{k-1}_{\tau},q^{k}_{\tau}\right)$
$\displaystyle=\mathcal{I}\left(t_{I\\!I},q^{k_{2}}_{\tau}\right)-\mathcal{I}\left(t_{I},q^{k_{1}}_{\tau}\right)+{\rm
Var}\left(\mathcal{D},q_{\tau};\left[t_{I},t_{I\\!I}\right]\right)$

(for a step function, calculating ${\rm
Var}\left(\mathcal{D},q_{\tau};\left[t_{I},t_{I\\!I}\right]\right)$ is easy).
The proof of the second inequality in (3.23) is similar. Starting from the
minimality of $q^{k}_{\tau}$, when compared with $q^{k-1}_{\tau}$ in (3.24),
yields

$\mathcal{I}\left(k\tau,q^{k}_{\tau}\right)+\mathcal{D}\left(q^{k-1}_{\tau},q^{k}_{\tau}\right)\leq\mathcal{I}\left(k\tau,q^{k-1}_{\tau}\right)\\\
=\mathcal{I}\left((k-1)\tau,q^{k-1}_{\tau}\right)+L\left((k-1)\tau,q^{k-1}_{\tau}\right)-L\left(k\tau,q^{k-1}_{\tau}\right).$

We sum again over $k=k_{1}+1,\dots,k_{2}$ to obtain

$\displaystyle\mathcal{I}\left(k_{2}\tau,q^{k_{2}}_{\tau}\right)$
$\displaystyle-$
$\displaystyle\mathcal{I}\left(k_{1}\tau,q^{k_{1}}_{\tau}\right)+\sum_{k=k_{1}+1}^{k_{2}}\mathcal{D}\left(q^{k-1}_{\tau},q^{k}_{\tau}\right)$
$\displaystyle\leq$
$\displaystyle\sum_{k=k_{1}+1}^{k_{2}}\left[L\left((k-1)\tau,q^{k-1}_{\tau}\right)-L\left(k\tau,q^{k-1}_{\tau}\right)\right],$

so we find

$\mathcal{I}\left(k_{2}\tau,q^{k_{2}}_{\tau}\right)-\mathcal{I}\left(k_{1}\tau,q^{k_{1}}_{\tau}\right)+{\rm
Var}\left(\mathcal{D},q_{\tau};\left[t_{I},t_{I\\!I}\right]\right)\leq-\int_{t_{I}}^{t_{I\\!I}}\dot{L}\left(t,q_{\tau}(t)\right)\,\mathrm{d}t$

and the second inequality in (3.23) is shown. $\Box$

We would like to pass with the step size $\tau$ to zero and for this we need
certain _a priori_ bounds.

######  Proposition 3.11.

Let (3.8) and (3.9) be satisfied. Then there is $\kappa\in\mathbb{R}$ such
that for any $\tau>0$:

$\left\lVert
y_{\tau}\right\rVert_{L^{\infty}\left(0,T;W^{1,d}(\Omega;\mathbb{R}^{n})\right)}<\kappa,$
(3.27) ${\rm Var}(\mathcal{D},q_{\tau};[0,T]))<\kappa,$ (3.28) $\left\lVert
z_{\tau}\right\rVert_{L^{\infty}\left(0,T;W^{1,\alpha}(\Omega;\mathbb{R}^{n\times
n})\times W^{1,\beta}(\Omega;\mathbb{R}^{m})\right)}<\kappa.$ (3.29)

Proof. For $q=(y,F_{\rm p},p)\in\mathbb{Q}$, set

$V(q)=\int_{\Omega}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{W}(\nabla
y(x)F_{\rm p}^{-1},\nabla[({\rm cof}\nabla y(x))F_{\rm p}^{\top}(x)],F_{\rm
p}(x),\nabla F_{\rm p},p(x),\nabla
p(x))\,\mathrm{d}x\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>
Like in the proof of Proposition 3.8, we can show by the growth conditions on
$\mathcal{W}$ that for any $q$ with $V(q)<+\infty$ and some constants $C_{0}$,
$C>0$,

$\displaystyle-
C_{0}+C\left(\|y\|^{d}_{W^{1,d}(\Omega;\mathbb{R}^{n})}+\|F_{\rm
p}\|^{\beta}_{W^{1,\beta}(\Omega;\mathbb{R}^{n\times
n})}+\|p\|^{\omega}_{W^{1,\omega}(\Omega;\mathbb{R}^{m})}\right)\leq V(q).$
(3.30)

Using a lower energy estimate from the proof of (3.23) for $k_{1}=0$ we get

$V(q^{k_{2}}_{\tau})-L(k_{2}\tau,q^{k_{2}}_{\tau})-V(q^{0}_{\tau})+L(0,q^{0}_{\tau})\leq\sum_{k=1}^{k_{2}}\left[L\left((k-1)\tau,q^{k-1}_{\tau}\right)-L\left(k\tau,q^{k-1}_{\tau}\right)\right].$

Hence for a constant $C$ which only depends on the data and the initial
condition,

$\displaystyle
V(q^{k_{2}}_{\tau})\leq\sum_{k=1}^{k_{2}}\left[L\left((k-1)\tau,q^{k-1}_{\tau}\right)-L\left(k\tau,q^{k-1}_{\tau}\right)\right]+L(k_{2}\tau,q^{k_{2}}_{\tau})+C.$
(3.31)

So, denoting $Y_{\tau}^{d}:=\max_{1\leq\ell\leq
T/\tau}\|y^{\ell}_{\tau}\|^{d}_{W^{1,d}(\Omega;\mathbb{R}^{n})}$ and taking
into account (3.30) with $q:=q^{k_{2}}_{\tau}$ we have

$\displaystyle Y_{\tau}^{d}\leq
C\sum_{k=1}^{k_{2}}\left[L\left((k-1)\tau,q^{k-1}_{\tau}\right)-L\left(k\tau,q^{k-1}_{\tau}\right)\right]+C(q^{k_{2}}_{\tau}).$
(3.32)

This gives the bound (3.27), because $Y_{\tau}$ is raised to the power $d>1$
on the left hand side, whereas the right hand side is merely linear in
$y_{\tau}^{\ell}$. We can proceed similarly for (3.29). The upper bound on the
total dissipation can be derived by algebraic manipulations as in [21]. $\Box$

###  3.7. Limit passage

The following lemma is proved in [38].

######  Lemma 3.12.

Let $\mathcal{D}\colon\mathbb{Z}\times\mathbb{Z}\to[0,+\infty]$ satisfy (3.6)
and (3.7). Let $\mathcal{K}$ be a weakly sequentially compact subset of
$\mathbb{Z}$. Then for every sequence $\\{z_{k}\\}_{k\in{\mathbb{N}}}$,
$z_{k}\colon[0,T]\to\mathcal{K}$ for which $\sup_{k\in{\mathbb{N}}}\,{\rm
Var}(\mathcal{D},z_{k};[0,T])<C$, $C>0$, there exists a subsequence (not
relabelled), a function $z\colon[0,T]\to\mathcal{K}$, and a function
$\Delta\colon[0,T]\to[0,C]$ such that:
(i) ${\rm Var}(\mathcal{D},z_{k};[0,t])\to\Delta(t)$ for all $t\in[0,T]$,
(ii) $z_{k}{\rightharpoonup}z$ in $\mathbb{Z}$ for all $t\in[0,T]$, and
(iii) ${\rm Var}(\mathcal{D},z;[t_{0},t_{1}])\leq\Delta(t_{1})-\Delta(t_{0})$
for all $0\leq t_{0}<t_{1}\leq T$.

Let us denote $\mathbb{X}:=L^{\beta}(\Omega;\mathbb{R}^{n\times n})\times
L^{\omega}(\Omega;\mathbb{R}^{m})$. Finally, we proved the existence of an
energetic solution.

######  Theorem 3.13.

Let $\alpha^{-1}+\beta^{-1}\leq
d^{-1}<\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(n-1)^{-1}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}d>\frac{\beta(n-1)}{\beta-1}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$.
Let $q^{0}\in\mathbb{Q}$ be a stable initial condition. Let the assumptions on
$\mathcal{W}$, $\mathcal{D}$, $f$ and $g$ from Section 3.2 hold. Let further
(3.16) or (3.A), (3.B), (3.C) hold. Then there is a process
$q\colon[0,T]\to\mathbb{Q}$ with $q(t)=(y(t),z(t))$ such that $q$ is an
energetic solution according to Definition 2.2. The following convergence
statements also hold:
(i) for a $t$-dependent (not relabelled) subsequence w-$\lim_{\tau\to
0}y_{\tau}(t)=y(t)$ in $W^{1,d}(\Omega;\mathbb{R}^{n})$ for all $t\in[0,T]$,
(ii) for a (not relabelled) subsequence $\lim_{\tau\to 0}z_{\tau}(t)=z(t)$ in
$\mathbb{X}$ for all $t\in[0,T]$,
(iii) for a (not relabelled) subsequence $\lim_{\tau\to
0}{\mathcal{I}}_{\tau}(t,q_{\tau}(t))={\mathcal{I}}(t,q(t))$ for all
$t\in[0,T]$, and
(iv) for a (not relabelled) subsequence $\lim_{\tau\to 0}{\rm
Var}(\mathcal{D},q_{\tau};[0,t])={\rm Var}(\mathcal{D},q;[0,t])$ for all
$t\in[0,T]$.

###### Proof.

We have adapted the proof from [21, 42] and divided it into three steps.

Step 1: Assertion (i) follows from the a priori estimate in Proposition 3.11
and (ii) results from Lemma 3.12. Hence we have a limit $q(t)=(y(t),z(t))$. We
easily get that $q(t)\in\mathbb{Q}$ for all $t\in[0,T]$.

Put $S(t,\tau):=\max_{k\in{\mathbb{N}}\cup\\{0\\}}\\{k\tau;\ k\tau\leq t\\}$.
Then $\lim_{\tau\to 0}S(t,\tau)=t$ and by definition
$q_{\tau}(t):=q_{\tau}(S(t,\tau))\in\mathcal{S}(S(t,\tau))$. As we have seen,
$\mathcal{D}$ can be assumed such that (3.13) holds. Hence by Proposition 3.6,
the limit is stable, i.e. $q(t)\in\mathcal{S}(t)$ (thanks to our a priori
estimates, we can pass to a gpc-convergent subsequence to get lower
semicontinuity).

Step 2: Notice that $\theta_{\tau}(t):=\frac{\partial L}{\partial
t}(t,q_{\tau}(t))$ is bounded in $L^{\infty}(0,T)$ by (3.8), (3.9) and (3.27),
so we deduce a weak*-convergent subsequence, which we do not relabel, with the
limit $\theta$. By Fatou’s lemma $\theta_{\rm i}\leq\theta$.

Our interpolation satisfies $q_{\tau}(t)=q_{\tau}(k\tau)$ for $0\leq
t-k\tau<\tau$. Successively using (3.23), (3.8), (3.9) and (3.27), we get for
some $C$, $C_{1}>0$

$\displaystyle{\mathcal{I}}(t,q_{\tau}(t))+{\rm
Var}(\mathcal{D},q_{\tau};[0,t])$ $\displaystyle\leq$
$\displaystyle{\mathcal{I}}(k\tau,q_{\tau}(k\tau))+{\rm
Var}(\mathcal{D},q_{\tau};[0,k\tau])+C\tau$ $\displaystyle\leq$
$\displaystyle{\mathcal{I}}(0,q_{\tau}(0))-\int_{0}^{k\tau}\dot{L}(s,q_{\tau}(s))\,\mathrm{d}s+C\tau$
$\displaystyle\leq$
$\displaystyle{\mathcal{I}}(0,q_{\tau}(0))-\int_{0}^{t}\dot{L}(s,q_{\tau}(s))\,\mathrm{d}s+C_{1}\tau.$

On account of Lemma 3.12 (i) and Proposition 3.6 we find that for $\tau\to 0$

$\displaystyle{\mathcal{I}}(t,q(t))+\Delta(t)$ $\displaystyle\leq$
$\displaystyle{\mathcal{I}}(0,q(0))-\int_{0}^{t}\theta(s)\,\mathrm{d}s.$

Following [42] we set $\theta_{\rm i}(t):=\liminf_{\tau\to
0}\theta_{\tau}(t)$. Since $\Delta(t)\geq{\rm Var}(\mathcal{D},q;[0,t])$ and
by Fatou’s lemma
$\int_{0}^{t}\theta(s)\,\mathrm{d}s\geq\int_{0}^{t}\theta_{\rm
i}(s)\,\mathrm{d}s$, for a.a. $t\in[0,T]$ we get

$\displaystyle{\mathcal{I}}(t,q(t))+{\rm Var}(\mathcal{D},q;[0,t])$
$\displaystyle\leq$
$\displaystyle{\mathcal{I}}(0,q(0))-\int_{0}^{t}\theta_{\rm
i}(s)\,\mathrm{d}s.$

The convergence $\theta_{\tau}(s)=\dot{L}(s,q_{\tau}(s))\to\dot{L}(s,q(s))$
and extracting an $s$-dependent subsequence of $\\{\theta_{\tau}(s)\\}$
converging to $\theta_{\rm i}(s)$ show that $\theta_{\rm
i}(s)=\dot{L}(s,q(s))$. We finally get the upper energy estimate

$\displaystyle{\mathcal{I}}(t,q(t))+{\rm Var}(\mathcal{D},q;[0,t])$
$\displaystyle\leq$
$\displaystyle{\mathcal{I}}(0,q(0))-\int_{0}^{t}\dot{L}(s,q(s)\,\mathrm{d}s.$
(3.33)

Step 3: The fact that $q(t)$ is stable for all $t\in[0,T]$ will be useful in
proving the lower estimate. Consider a partition of a time interval
$[t_{I},t_{I\\!I}]\subset[0,T]$ such that
$t_{I}=\vartheta_{0}^{M}<\vartheta_{1}^{M}<\vartheta_{2}^{M}<\cdots<\vartheta_{K_{M}}^{M}=t_{I\\!I}$
and $\max_{i}(\vartheta_{i}^{M}-\vartheta_{i-1}^{M})=:\vartheta^{M}\to 0$ as
$M\to\infty$. Let us test the stability of $q(\vartheta_{k-1}^{M})$ with
$q(\vartheta_{k}^{M})$, $k=1,\ldots,K_{M}$. Similarly to (3.6) we get

$\displaystyle\sum_{k=1}^{K_{M}}\left[L\left((\vartheta_{k-1}^{M},q(\vartheta_{k}^{M})\right)-L\left(\vartheta_{k}^{M},q(\vartheta_{k}^{M})\right)\right]$
$\displaystyle\leq$
$\displaystyle\mathcal{I}\left(t_{I\\!I},q(t_{I\\!I})\right)-\mathcal{I}\left(t_{I},q(t_{I})\right)$
$\displaystyle+$
$\displaystyle\sum_{k=1}^{K_{M}}\mathcal{D}\left(q(\vartheta_{k-1}^{M}),q(\vartheta_{k}^{M})\right).$

Thus,

$\displaystyle\sum_{k=1}^{K_{M}}-\int_{\vartheta_{k-1}^{M}}^{\vartheta_{k}^{M}}\dot{L}(s,q(\vartheta_{k}^{M}))\,\mathrm{d}s$
$\displaystyle\leq$
$\displaystyle\mathcal{I}\left(t_{I\\!I},q(t_{I\\!I}\right)-\mathcal{I}\left(t_{I},q(t_{I})\right)$
(3.35) $\displaystyle+$ $\displaystyle{\rm
Var}(\mathcal{D},q;[t_{I},t_{I\\!I}]).$

We finally rearrange the terms as

$\displaystyle\sum_{k=1}^{K_{M}}\int_{\vartheta_{k-1}^{M}}^{\vartheta_{k}^{M}}\dot{L}(s,q(\vartheta_{k}^{M}))\,\mathrm{d}s$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{K_{M}}\dot{L}(\vartheta_{k}^{M},q(\vartheta_{k}^{M}))(\vartheta_{k}^{M}-\vartheta_{k-1}^{M})$
(3.36) $\displaystyle+$
$\displaystyle\sum_{k=1}^{K_{M}}\int_{\vartheta_{k-1}^{M}}^{\vartheta_{k}^{M}}(\dot{L}(s,q(\vartheta_{k}^{M}))-\dot{L}(\vartheta_{k}^{M},q(\vartheta_{k}^{M})))\,\mathrm{d}s.$

The last sum on the right-hand side of (3.36) goes to zero with
$\vartheta^{M}\to 0$ because the time derivative of the external loading is
uniformly continuous in time by (3.8) and (3.9). The first sum on the right-
hand side converges to $\int_{t_{I}}^{t_{I\\!I}}\dot{L}(s,q(s))\,\mathrm{d}s$
by [11, Lemma 4.12].

Altogether, the upper and lower estimates give us the energy balance

$\displaystyle{\mathcal{I}}(t,q(t))+{\rm Var}(\mathcal{D},q;[0,t])$
$\displaystyle=$
$\displaystyle{\mathcal{I}}(0,q(0))-\int_{0}^{t}\dot{L}(s,q(s)\,\mathrm{d}s.$
(3.37)

Moreover,

$\displaystyle{\mathcal{I}}(0,q(0))-\int_{0}^{t}\theta_{\rm
i}(s)\,\mathrm{d}s\leq{\mathcal{I}}(t,q(t))+{\rm
Var}(\mathcal{D},q;[0,t]))\leq{\mathcal{I}}(t,q(t))+\Delta(t)$ (3.38)
$\displaystyle\leq$
$\displaystyle{\mathcal{I}}(0,q(0))-\int_{0}^{t}\theta(s)\,\mathrm{d}s\leq{\mathcal{I}}(0,q(0))-\int_{0}^{t}\theta_{\rm
i}(s)\,\mathrm{d}s.$

Thus in fact, all inequalities in (3.38) are equalities and we get (iv).
Proposition 3.6 also implies (iii). ∎

Finally, we include a simple example from single-crystal plasticity, covered
by our approach, in the spirit of [26] and [40].

######  Example 3.1 (single slip without hardening).

Consider two orthonormal vectors $a$, $b\in\mathbb{R}^{3}$, describing the
motion of edge dislocations in such a way that $a$ is the glide direction and
$b$ is the slip-plane normal. Further we focus on a particular case of the so-
called separable material where

$\mathcal{W}(F_{\rm e},H,z)=\mathcal{W}_{1}(F_{\rm e})+c|H|^{2}+c{\rm
det}(F_{\rm e})^{-s}+\varepsilon|F_{\rm p}|^{6}+\varepsilon|\nabla F_{\rm
p}|^{6},\;c,s,\varepsilon>0,$

with $F_{\rm p}(x,t)=\mathbb{I}+\gamma(x,t)a\otimes b$ where $\gamma$ is the
plastic slip and we can choose $\mathcal{W}_{1}$ to be e.g. the stored energy
density of the Saint Venant–Kirchhoff material, i.e. $\mathcal{W}_{1}(F_{\rm
e})=\frac{1}{8}(\textstyle{F_{\rm e}}F_{\rm
e}-\mathbb{I}):\mathbb{C}:(\textstyle{F_{\rm e}}F_{\rm e}-\mathbb{I})$ so that
$\mathcal{W}_{1}(F_{\rm e})\geq c(|F_{e}|^{4}-1)$. The vectors $a$, $b$ are
not fixed in the reference configuration in general (more precisely, they lie
in an intermediate lattice space, see [27]). The slip-plane normal $\tilde{b}$
in the reference configuration is given by $\tilde{b}=(F_{\rm p})^{\top}b$.
However, in the special case of a single slip we obtain $\tilde{b}=b$ and the
slip-plane normal remains unchanged during plastic transformations.

As $F_{\rm p}$ is completely described by $\gamma$, we identify $z:=\gamma$
and use the dissipation potential

$\delta(\dot{\gamma})=\kappa|\dot{\gamma}|,$

where $\kappa>0$ represents the resistance to the slip.

This corresponds to the dissipation distance

$\mathcal{D}(\gamma_{1},\gamma_{2})=\int_{\Omega}\kappa|\gamma_{1}(x)-\gamma_{2}(x)|\mathrm{d}x.$

An extension to multi-slip, described by several glide directions
$\\{a_{i}\\}$ and slip-plane normals $\\{b_{i}\\}$, $1\leq i\leq N$, would be
feasible, cf. [40].

As a last remark, note that our stored energy $\mathcal{W}$ could also depend
on $F_{\rm e}^{-1}$, cf. [5].

Acknowledgements. This work was supported by the GAČR project 18-03834S (MK &
JZ) and by the DFG Priority Programme (SPP) 2256 (JZ).

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