# Nonlinear dynamic fracture problems with polynomial and strain-limiting
constitutive relations

Victoria Patel<EMAIL_ADDRESS>
###### Abstract.

We extend the framework of dynamic fracture problems with a phase-field
approximation to the case of a nonlinear constitutive relation between the
Cauchy stress tensor $\mathbf{T}$, linearised strain
$\boldsymbol{\varepsilon}(\boldsymbol{u})$ and strain rate
$\boldsymbol{\varepsilon}(\boldsymbol{u}_{t})$. The relationship takes the
form
$\boldsymbol{\varepsilon}(\boldsymbol{u}_{t})+\alpha\boldsymbol{\varepsilon}(\boldsymbol{u})=F(\mathbf{T})$
where $F$ satisfies certain $p$-growth conditions. We take particular care to
study the case $p=1$ of a ‘strain-limiting’ solid, that is, one in which the
strain is bounded a priori. We prove the existence of long-time, large-data
weak solutions of a balance law coupled with a minimisation problem for the
phase-field function and an energy-dissipation inequality, in any number $d$
of spatial dimensions. In the case of Dirichlet boundary conditions, we also
prove the satisfaction of an energy-dissipation equality.

###### Key words and phrases:

Nonlinear viscoelasticity, strain-limiting theory, evolutionary problem,
global existence, weak solution, dynamic fracture, phase-field.

###### 1991 Mathematics Subject Classification:

35M13, 35K99, 74D10, 74H20, 74R99.

V. Patel is supported by the UK Engineering and Physical Sciences Research
Council [EP/L015811/1].

Mathematical Institute, University of Oxford

Andrew Wiles Building, Woodstock Road

Oxford OX2 6GG, UK

##  1\. Introduction

In this work, we are interested in combining the theory of dynamic fracture
with the theory of nonlinear, implicit constitutive relations. At present,
mathematical analysis of dynamic fracture problems has only been performed in
the setting of linear elasticity to the author’s knowledge. In this work, we
aim to extend such fracture studies and examine time-dependent damage problems
with a constitutive relation from nonlinear viscoelasticity. Of particular
interest is the strain-limiting case, by which we mean that the linearised
strain
$\boldsymbol{\varepsilon}(\boldsymbol{u})=\frac{1}{2}(\nabla\boldsymbol{u}+(\nabla\boldsymbol{u})^{\mathrm{T}})$
is bounded a priori. A key motivation for studying such models is that such a
defining relation ensures that no singularity is experienced in the strain.
Indeed, we cannot contradict the small strain assumption, which is essential
to the construction of the model here. This contrasts the known results for
damage models with a linear constitutive relation where a singularity occurs
at the crack tip [37]. In this section, we discuss the theory of implicit
constitutive relations that was established by Rajagopal in his pioneering
work [34] and a series of later papers, followed by an introduction to the
theory of dynamic fracture based on the principles in [29, 27]. Then we
combine the two theories to obtain a fracture problem with a nonlinear
constitutive relation between the Cauchy stress tensor, linearised strain
tensor and strain rate. We particularly emphasise the work in Section 3 where
strain-limiting models are studied in the context of dynamic fracture
problems, the state of the art in this work.

A framework for nonlinear, implicit constitutive relations in continuum
mechanics was introduced by Rajagopal in his pioneering work [34] and a series
of later papers. Although such relations had been studied previously, this was
only in an ad hoc way. The novelty of [34] is the systematic and
thermodynamically consistent structure that is available for the family of
constitutive relations. These ideas are extended in [36] where nonlinear
relationships between the Cauchy stress tensor $\mathbf{T}$, strain tensor
$\boldsymbol{\varepsilon}$ and strain rate $\boldsymbol{\varepsilon}_{t}$ are
justified in the small strain setting. Namely, we may consider the defining
relation

(1.1)
$\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}+\alpha\boldsymbol{u})=F(\mathbf{T}),$

where $F$ is a nonlinear, monotonic function from $\mathbb{R}^{d\times
d}_{sym}$ to itself and $\alpha$ is a positive constant. We let
$\boldsymbol{\varepsilon}=\frac{1}{2}((\nabla\cdot)+(\nabla\cdot)^{\mathrm{T}})$
denote the symmetric gradient operator, $\boldsymbol{u}$ is the displacement
and $\mathbf{T}$ is the Cauchy stress tensor. The relation (1.1) is a
generalisation of the classical linear Kelvin–Voigt model for viscoelastic
bodies.

A subclass of these models that we take particular care to study here is that
of a strain-limiting solid. A strain-limiting constitutive relation ensures
that the strain is bounded a priori, irrespective of the magnitude of the
stress tensor. The motivation to study such models is as follows. It is well
known that brittle materials can fracture in the small strain range under
large stresses. Furthermore, under the assumption that the strain is small,
the mathematical model that results from the balance laws of momentum and mass
for the deformation of the body under investigation can be linearised. As we
are considering the motion of solids, it is sensible to assume that we are
working in the small strain setting. Indeed, solids damage and break under
sufficiently large strains. Moreover, the linearisation procedure as a result
of the small strain assumption simplifies the problem and makes it more
amenable to mathematical analysis. Hence, from now on, it is assumed that the
small displacement gradient assumption holds, that is,

(1.2) $\sup_{(t,x)\in Q}|\nabla\boldsymbol{u}(t,x)|=O(\delta),$

for a $\delta\ll 1$, where $Q$ is the space-time domain. It follows from (1.2)
that the linearised strain $\boldsymbol{\varepsilon}(\boldsymbol{u})$ as
defined above must be at most of order $\delta$.

Now, consider a body with a crack which is possibly growing due to an external
force acting on the material. In the case of a linear constitutive relation,
the strain has magnitude of order $r^{-\frac{1}{2}}$ where $r$ is the distance
to the crack tip [37]. The strain experiences a singularity at the crack tip,
contradicting the standing assumption that the strain is small. A more
reliable model could be one in which the magnitude of the strain is bounded a
priori so such contradictions cannot arise.

This motivates studying the constitutive relation

(1.3)
$\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}+\alpha\boldsymbol{u})=F(\mathbf{T}):=\frac{\mathbf{T}}{(1+|\mathbf{T}|^{a})^{\frac{1}{a}}},$

for fixed parameter $a>0$. Other choices for the function $F$ are available,
but for simplicity of the presentation, we make this specific choice as
suggested in [10, 35]. For further details of the possible generalisations, we
refer to the remark in Section 3.

The defining relation (1.3) is coupled with the balance law

(1.4) $\boldsymbol{u}_{tt}=\mathrm{div}(\mathbf{T})+\boldsymbol{f},$

where $\boldsymbol{f}$ represents the external force, alongside suitable
boundary and initial conditions. The balance law comes from considering the
balance of linear momentum under (1.2). The balance of angular momentum
reduces to $\mathbf{T}=\mathbf{T}^{\mathrm{T}}$, which is naturally included
in (1.3). Due to the assumption (1.2), we assume that the density is constant
in time and so we do not include a balance of mass. For further details on the
relevant kinematics and the derivation of this problem, we refer to [9] and
the references therein. We note that the strain-limiting problem (1.3), (1.4)
has been recently studied on domains without a crack. For the analysis of the
spatially periodic problem, we refer to [9] and for a general domain with
Dirichlet boundary conditions, we mention [8].

In this paper, we consider (1.3), (1.4) in the context of dynamic fracture.
Hence, before proceeding further, we give a brief introduction to the modern
theory of fracture. The principles that drive the mathematical analysis of
fracture in brittle bodies can be traced back to the seminal work of Griffith
[24]. In the quasi-static setting111By quasi-static setting, we mean that the
intertial term $\boldsymbol{u}_{tt}$ is negligible in the balance of momentum
equation (1.4)., he stipulated that a crack can only grow if the increase in
surface energy is exactly equal to the resulting decrease in elastic potential
energy. In other words, the newly created area is proportional to the loss of
stored energy. In the anti-plane setting222By anti-plane setting, we mean
that, in three spatial dimensions, the displacement $\boldsymbol{u}$ takes the
particular form $\boldsymbol{u}(x_{1},x_{2},x_{3})=(0,0,u(x_{1},x_{2}))$.
with a linear relationship between the stress and strain, we write this
mathematically as

(1.5) $\frac{\mu}{2}\int_{\Omega\setminus C(t)}|\nabla
u(t)|^{2}\,\mathrm{d}x+G_{c}\mathcal{H}^{d-1}(C(t))=\frac{\mu}{2}\int_{\Omega\setminus
C(0)}|\nabla u(0)|^{2}\,\mathrm{d}x+G_{c}\mathcal{H}^{d-1}(C(0)),$

where $C(t)\subset\Omega$ is the crack set, $\mu$ is the elasticity constant
and $G_{c}$ is the fracture toughness. We use $\mathcal{H}^{d-1}$ to denote
the $(d-1)$-dimensional Hausdorff measure. The first term in (1.5) corresponds
to the elastic energy and the second term to the surface energy. This
formulation was first introduced by Francfort and Marigo [22], based on the
Mumford–Shah functional that is used in image segmentation [30].

Alongside this energy balance, the balance of linear momentum in the quasi-
static setting must hold. We formulate this as a variational problem. Indeed,
the couple $(u(t),C(t))$ must satisfy

(1.6) $\mathcal{A}(u(t),C(t)):=\frac{\mu}{2}\int_{\Omega\setminus C(t)}|\nabla
u(t)|^{2}\,\mathrm{d}x+G_{c}\mathcal{H}^{d-1}(C(t))\leq\frac{\mu}{2}\int_{\Omega\setminus\tilde{C}}|\nabla\tilde{u}|^{2}\,\mathrm{d}x+G_{c}\mathcal{H}^{d-1}(\tilde{C}),$

for every suitable comparison couple $(\tilde{u},\tilde{C})$ with
$C(t)\subset\tilde{C}$. The principle of irreversibility is also required to
hold, that is, the crack is non-decreasing in time, so $C(s)\subset C(t)$ for
every $s<t$. Problems of the form (1.5), (1.6), where we work directly with
the crack set, are called sharp interface problems. Mathematical analysis of
this anti-plane shear problem took place in [22], with higher dimensional
settings treated in [17] and [21]. Both the crack set and displacement are
treated as unknowns in these works. This causes significant analytical
difficulties. Indeed, the analysis takes places on the set $\cup_{t\geq
0}\\{t\\}\times(\Omega\setminus C(t))$, which is not only a domain that
changes in time, but is a priori unknown because the crack set $(C(t))_{t\geq
0}$ is unknown.

An approach to avoid working on time dependent cracking domains is to
approximate the crack set by a phase-field function. The Ambrosio–Tortorelli
functional [1, 2] takes the following form in the context of quasi-static
fracture for linear elasticity:

(1.7)
$\mathcal{A}_{\epsilon}(u,v)=\frac{\mu}{2}\int_{\Omega}(v^{2}+\eta_{\epsilon})|\nabla
u|^{2}\,\mathrm{d}x+G_{c}\int_{\Omega}\epsilon|\nabla
v|^{2}+\frac{1}{4\epsilon}(1-v)^{2}\,\mathrm{d}x,$

where $\epsilon>0$ is an approximation parameter, $\eta_{\epsilon}$ is a
numerical stability parameter such that $0<\eta_{\epsilon}\ll\epsilon$, and
$v$ is the unknown phase-field function, also referred to in the literature as
the material parameter. The function $v$ approximates the crack set in the
sense that $v$ takes values close to $1$ away from the crack and values close
to $0$ near the crack. Although $\epsilon$ acts as a length scale for the
approximation, we must be careful not to assign a physical meaning to this
parameter [5]. For the mathematical analysis of this approximation in the
quasi-static setting, we refer to [26] and [23], for example.

The approximation (1.7) is a sensible choice because $\mathcal{A}_{\epsilon}$
converges to the sharp-interface functional $\mathcal{A}$ in the sense of
$\Gamma$-convergence. We refer to [6] for the details but the essence is as
follows. Let $((u^{\epsilon},v^{\epsilon}))_{\epsilon}$ be a sequence such
that $v^{\epsilon}$ minimises $v\mapsto\mathcal{A}_{\epsilon}(u^{\epsilon},v)$
over suitable test functions $v\leq v_{\epsilon}$ at every time. Then there
exists a subsequence in $\epsilon$ and a solution couple $(u,C)$ to (1.5),
(1.6) such that, for a.e. time $t$, we have

$\displaystyle v^{\epsilon}\nabla u^{\epsilon}(t)\rightarrow\nabla
u(t)\quad\text{ strongly in }L^{2}(\Omega)^{d},$
$\displaystyle\lim_{\epsilon\rightarrow
0+}\int_{\Omega}((v^{\epsilon})^{2}+\eta_{\epsilon})|\nabla
u^{\epsilon}|^{2}\,\mathrm{d}x=\int_{\Omega}|\nabla u(t)|^{2}\,\mathrm{d}x$
$\displaystyle\lim_{\epsilon\rightarrow 0+}\int_{\Omega}\epsilon|\nabla
v|^{2}+\frac{1}{4\epsilon}(1-v)^{2}\,\mathrm{d}x=\mathcal{H}^{d-1}(C(t)).$

Hence, the sequence of solutions for the approximate problem represents an
approximation to the solution of the sharp-interface functional.

We remark that there are other choices of approximation available. For
example, the fourth order functional

$\displaystyle\tilde{\mathcal{A}}_{\epsilon}(u,v)=\frac{\mu}{2}\int_{\Omega}(v^{2}+\eta_{\epsilon})|\nabla
u|^{2}\,\mathrm{d}x+\frac{G_{c}}{4}\int_{\Omega}\frac{1}{\epsilon}(1-v)^{2}+2\epsilon|\nabla
v|^{2}+\epsilon^{3}|\Delta v|^{2}\,\mathrm{d}x,$

is investigated in [4] and [31]. The functional $\Gamma$-converges to
$\mathcal{A}$. An advantage of this choice is that the higher order spatial
derivatives improve the convergence rate of numerical solutions. In
particular, when studying these fracture problems from the point of view of
computational analysis, we note that the Ambrosio–Tortorelli functional is not
necessarily the best choice of approximation for the sharp-interface
functional.

The aforementioned literature all concern the quasi-static problem. Here, we
are interested in the adaptation of Griffith’s theory to the dynamic problem,
that is, we do not assume that the term $\boldsymbol{u}_{tt}$ is negligible.
An extension was first introduced by Mott [29] and relies on the following
three principles:

  * (i)

elastodynamics, the balance of linear momentum should hold away from the crack
set;

  * (ii)

energy balance, an energy-dissipation balance that includes the kinetic energy
should hold;

  * (iii)

irreversibility, the crack cannot heal itself in time.

However, this is not sufficient for a well-posed problem. The case of a
stationary crack should be ruled out. The following additional principle was
introduced by Larsen [27]:

  * (iv)

maximal dissipation, that is, if the crack can grow while satisfying
(i)–(iii), then it should.

As with the quasi-static problem, the study of dynamic fracture problems
broadly follows one of two approaches: working directly in the sharp-interface
setting, or an approximation by means of a phase-field function, generally
based on the Ambrosio–Tortorelli functional. The focus in this paper is on the
use of a phase-field approximation, thus avoiding the technicalities of
working in a time dependent domain. However, for completeness, we first
discuss the known results in the sharp-interface case.

For sharp-interface problems, the analysis is significantly more complicated
compared to the quasi-static model due to the presence of the intertial term.
Hence, a first step in the analysis is to solve the elastodynamic equation for
an unknown displacement when the crack set $(C(t))_{t>0}$ is known a priori.
We look for a function $\boldsymbol{u}$ such that

(1.8)
$\boldsymbol{u}_{tt}=\mathrm{div}\big{(}\mathbb{E}(t,x)\nabla\boldsymbol{u}\big{)}+\boldsymbol{f}(t,x)\quad\text{
in }\Omega\setminus C(t),\quad t\in[0,T],$

where $\mathbb{E}$ is the elasticity tensor, given a sequence of subsets
$(C(t))_{t\geq 0}$ of $\Omega$ satisfying some regularity requirements. In an
attempt to better understand (1.8), in [14] for the scalar-valued case and
[38] for the vector-valued case, the wave equation

$\displaystyle\boldsymbol{u}_{tt}-\Delta\boldsymbol{u}-\gamma\Delta\boldsymbol{u}_{t}=\boldsymbol{f}$

is studied, where $\gamma\in\\{0,1\\}$. The system is damped if $\gamma=1$ and
undamped if $\gamma=0$. The viscous term from the damping provides a certain
smoothing effect. A result of this is that, if $\gamma=1$, uniqueness and an
energy-dissipation balance can be shown. However, uniqueness and satisfaction
of an appropriate energy-dissipation balance is an open problem in the
undamped setting. We note that in both [14] and [38], the existence proofs use
a discretisation in the time variable. This is the standard approach to tackle
such problems, and we use a similar approach here.

However, different approaches to prove the existence of solutions are
available under stronger regularity requirements on the crack set. In [32],
[16] and, for the vector-valued case, [11], a co-ordinate change is used to
transform the problem from one on $\cup_{t\geq 0}\\{t\\}\times(\Omega\setminus
C(t))$, a cracking domain, to one on the reference domain
$[0,\infty)\times(\Omega\setminus C(0))$. A solution is found on this
reference domain and it is shown that the existence of such a solution to the
transformed problem is equivalent to the existence of a solution to the
original problem on the cracking domain.

The next step in the analysis is to deal with the case where both the crack
set and displacement are a priori unknown. As far as the author is aware, the
only known result in the multi-dimensional setting is in [15]. Under some
assumptions on the regularity and geometry of the crack set, the authors show
that among all possible couples $(u(t),C(t))_{t\geq 0}$, there exists one
which satisfies the maximal dissipation condition. There is no damping term
present, which is an advantage of the analysis compared to the aforementioned
works. Although a significant amount remains unknown, this work suggests that
the maximal dissipation condition is the correct principle to impose on the
dynamic problem.

An important aspect of all of the aforementioned literature is that only
linear elasticity is considered, that is, a linear relationship between the
Cauchy stress tensor and linearised strain. This leads to a significant gap in
the literature. Indeed, to the author’s knowledge, there has been no study of
dynamic fracture problems in the context of nonlinear constitutive relations.
In this work, we aim to extend some of the known results to such models,
considering both a general case and that of a strain-limiting solid. To avoid
the technicalities that come with the sharp-interface model, we consider a
dynamic problem with a phase-field approximation based on the
Ambrosio–Tortorelli functional. A regularised model of the type that we study
in this work was first introduced in [5, 27, 28]. The formulation is based on
the principles of elastodynamics, energy-balance and irreversibility, but not
maximal dissipation as in [15]. This is due to the fact that maximal
dissipation is naturally guaranteed by the formulation. Indeed, the
minimisation problem (1.10) introduced below ensures that the ‘crack’ grows
whenever it is possible for it to do so.

In [28], the authors solve the equation

(1.9)
$\boldsymbol{u}_{tt}=\mathrm{div}\big{(}(v^{2}+\eta_{\epsilon})\mathbb{E}\boldsymbol{\varepsilon}(\boldsymbol{u}+\boldsymbol{u}_{t})\big{)}+\boldsymbol{f}\quad\text{
in }(0,T)\times\Omega,$

subject to the couple $(\boldsymbol{u},v)$ solving the minimisation problem

(1.10) $\displaystyle\mathcal{A}_{\epsilon}(\boldsymbol{u}(t),v(t))$
$\displaystyle:=\frac{1}{2}\int_{\Omega}(v(t)^{2}+\eta_{\epsilon})\mathbb{E}\boldsymbol{\varepsilon}(\boldsymbol{u}):\boldsymbol{\varepsilon}(\boldsymbol{u})\,\mathrm{d}x+\int_{\Omega}\frac{1}{4\epsilon}(1-v(t))^{2}+\epsilon|\nabla
v(t)|^{2}\,\mathrm{d}x$ $\displaystyle=\inf_{v\leq
v(t)}\mathcal{A}_{\epsilon}(\boldsymbol{u}(t),v).$

The first integral on the right-hand side of (1.10) represents an
approximation of the elastic energy and the second is an approximation of the
surface energy, with fixed approximation parameter $\epsilon$. The
minimisation problem is identical to that for the quasi-static model. In the
literature, it is sometimes referred to as unilateral minimality.

We note that an issue occurs as a result of the minimisation problem when
taking the limit in the approximation parameter $\epsilon$. As stated in [28],
“the unilateral minimality of the $v$ variable has no obvious counterpart in a
sharp-interface model.” This means that there is nothing ensuring maximal
dissipation in the limiting model. Another deficiency with the model (1.9),
(1.10) is the presence of the viscous term $\boldsymbol{u}_{t}$. When taking
the limit in the approximation parameter, the viscoelastic paradox can occur
[13], that is, the only possible solution of the corresponding sharp-interface
problem is constant in time. However, in such dynamic fracture models, our
interest lies in non-stationary solutions. Hence this paradox is a potential
deficiency of the damping term in (1.9). In the problems that we study here,
we include a damping term. It is out of the scope of the paper to remove this
term but will be the focus of future work.

Such a shortcoming is overcome in [12] where the author considers a variation
of (1.9), (1.10) that has no such damping term in (1.9). Rather than include a
dissipative term in (1.9), a rate-dependent term is included in the
minimisation problem (1.10). The author of [12] considers the balance law

(1.11)
$\boldsymbol{u}_{tt}=\mathrm{div}\big{(}(v^{2}+\eta_{\epsilon})\mathbb{E}\boldsymbol{\varepsilon}(\boldsymbol{u})\big{)}+\boldsymbol{f}$

coupled with the rate-dependent minimisation problem

(1.12)
$\mathcal{A}_{\epsilon}(\boldsymbol{u}(t),v(t))+(v_{t}(t),v(t))_{k,2}=\inf_{v\leq
v(t)}\Big{\\{}\mathcal{A}_{\epsilon}(\boldsymbol{u}(t),v)+(v_{t}(t),v)_{k,2}\Big{\\}},$

where $\mathcal{A}_{\epsilon}$ is as in (1.10) and $(\cdot,\cdot)_{k,2}$ is
the standard inner product in the Sobolev space $H^{k}(\Omega)$. For every
$k\in\mathbb{N}_{0}$, it is shown that a weak solution to (1.11), (1.12)
exists such that it satisfies an energy-dissipation inequality. However, if
$k>\frac{d}{2}$, it is possible to prove that the energy-dissipation
inequality is in fact an equality at every point in time. This is essentially
a consequence of the Sobolev embedding theorem, from which we know that
$H^{k}(\Omega)$ is continuously embedded into $C(\overline{\Omega})$ for
$k>\frac{d}{2}$.

The proof of the existence of solutions to the dynamic problem in both [28]
and [12] involves introducing a discretisation in the time variable. The time-
discrete formulation involves an alternating procedure. The elastodynamic
equation is solved for the displacement given the phase-field function from
the previous time step. The minimisation problem for an unknown phase-field
function is then solved, given the displacement function from the
aforementioned step. A similar procedure is used here.

A key novelty of the presentation in Section 2 is the formulation of the
problem and the extension of the analysis of the linear case to a model of
nonlinear viscoelasticity with a Kelvin–Voigt rheology, based on the implicit
constitutive framework discussed previously. In particular, we note the
derivation of bounds on the sequence of solutions to the time discrete
problem. This demonstrates the highly nonlinear nature of the problem.
However, the case of the strain-limiting material is the true highlight of
this work. We make a significant step towards the use of strain-limiting
constitutive relations in sharp-interface fracture models, which is the
ultimate goal.

The nonlinear nature of the problem adds significant analytical challenges to
problem, compared with the equivalent problem in the linear setting. Not only
does the problem require careful formulation, but we work hard to obtain
various a priori bounds and to improve the regularity of approximate solution
sequences. In the strain-limiting case, we have the added complication that
the stress tensor $\mathbf{T}$ is only bounded a priori in $L^{1}(Q)^{d\times
d}$. It is well known that this space has poor compactness properties. Indeed,
at best we can obtain weak-* convergence in the space of Radon measures on
$\overline{Q}$. However, we want to stay out of the space of measure-valued
solutions. We use careful manipulations to improve the regularity of the
approximations of the stress which allow the deduction of a pointwise
convergence result, despite the poor integrability. This is later improved to
strong convergence in $L^{1}(0,T;L^{1}_{loc}(\Omega)^{d\times d})$ and we
prove that the limit is exactly the solution that we are looking for.

There are two main results that we present in this work. The first, presented
in Section 2, concerns a constitutive relation of the form

(1.13)
$\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}+\alpha\boldsymbol{u})=F(\mathbf{T}):=|\mathbf{T}|^{p-2}\mathbf{T},$

where $p>1$. When $p\in(1,2]$, we prove the existence of a weak solution
$(\boldsymbol{u},v)$ to the elastodynamic equation (1.4) coupled with a
minimisation problem analogous to (1.10) and an energy-dissipation balance at
every point in time. For $p>2$, we also obtain an existence result. However,
the energy-dissipation balance only holds at almost every point in time. This
results from a lack of regularity of the phase-field function.

The second result, presented in Section 3, concerns strain-limiting solids
with the constitutive relation (1.3). The elastodynamic equation (1.4) is
coupled with the defining relation (1.3), a rate-dependent minimisation
problem of the type (1.12) and an energy-dissipation balance. We supplement
this with mixed Dirichlet–Neumann boundary conditions. We restrict ourselves
to $k>\frac{d}{2}+1$ to ensure good regularity of the phase-field function.
This allows the derivation of higher regularity estimates on approximations of
the displacement and stress tensor.

If the Neumann part of the boundary is non-empty, due to a lack of global
estimates on the stress tensor, the elastodynamic equation is shown to hold
weakly up to a penalisation term on the Neumann part of the boundary. Indeed,
the only global convergence result for the approximations of the stress tensor
is in the set of measures on $\overline{Q}$ due to the lack of weak
compactness in the Lebesgue space $L^{1}(Q)$. An important consequence of this
is that we are only able to show that an energy-dissipation inequality holds.
However, if we have a fully Dirichlet boundary condition, we have no such
penalisation term and are able to obtain a full existence result. Indeed, a
weak form of (1.4) holds, the constitutive relation is satisfied pointwise
a.e., the minimisation problem holds at every point in time and we have an
energy-dissipation equality. If additionally $a\in(0,\frac{2}{d})$, we also
have $\mathbf{T}\in L^{1+\delta}(0,T;L^{1+\delta}_{loc}(\Omega)^{d\times d})$
for every positive $\delta$ such that $a+\delta<\frac{2}{d}$.

The structure of the remainder of the paper is as follows. In Section 1.1, we
state the relevant notation and background results. In Section 2, we study the
nonlinear fracture problem for a constitutive relation of the form (1.13). The
case $p\in(1,2]$ is dealt with first. The existence of a solution is shown by
means of a discretisation in time and semi-discretisation in space. In
particular, we consider a Galerkin approximation in space with respect to the
displacement variable $\boldsymbol{u}$. Next, we adapt this work to the case
$p\in(2,\infty)$. We must use the monotonicity of the approximations of the
phase-field function to overcome the lack of uniform bounds on the time
derivatives of these functions. In Section 3, we study the strain-limiting
problem with constitutive relation (1.3). The proof uses a discretisation in
time with an elliptic regularisation. We obtain suitable weighted regularity
estimates on the sequence of approximations $(\mathbf{T}^{n})_{n}$ of the
stress tensor $(\mathbf{T}^{n})_{n}$ to overcome the poor integrability
properties of the sequence. These estimates are used to prove the pointwise
convergence of this sequence to a limit $\mathbf{T}$, which we show to be a
part of the required solution triple $(\boldsymbol{u},\mathbf{T},v)$. The
climax of this work in the proof of a full existence result under Dirichlet
boundary conditions and a partial existence result when the Neumann part of
the boundary is non-empty.

###  1.1. Notation and auxiliary results

Throughout this paper, $\Omega\subset\mathbb{R}^{d}$ denotes a bounded domain
with Lipschitz boundary $\partial\Omega$. The boundary has a Dirichlet part
$\Gamma_{D}$ and a Neumann part $\Gamma_{N}$. These are relatively open but
disjoint subsets of $\partial\Omega$ such that
$\overline{\Gamma_{D}\cup\Gamma_{N}}=\partial\Omega$. We always assume that
$\Gamma_{D}\neq\emptyset$. We let $\mathbf{n}$ denote the outward unit normal
to a point on the boundary $\partial\Omega$. We fix the spatial dimension
$d\geq 2$ and a final time horizon $T\in(0,\infty)$. We denote the space-time
domain by $Q=(0,T)\times\Omega$.

We use $W^{k,p}(\Omega)$ to denote the standard Sobolev space for
$k\in\mathbb{N}$, $p\in[1,\infty]$ with the usual norm $\|\cdot\|_{k,p}$. When
considering $W^{k,p}(\Omega_{0})$ for a subdomain $\Omega_{0}\subset\Omega$,
we write $\|\cdot\|_{W^{k,p}(\Omega_{0})}$. We write
$C^{\infty}_{D}(\overline{\Omega})$ for the set of functions $h\in
C^{\infty}(\overline{\Omega})$ such that $h=0$ on $\Gamma_{D}$ and let
$W^{k,p}_{D}(\Omega)$ denote the closure of
$C^{\infty}_{D}(\overline{\Omega})$ with respect to $\|\cdot\|_{k,p}$ for
$p\in[1,\infty)$. We define
$W^{k,\infty}_{D}(\Omega):=W^{k,\infty}(\Omega)\cap W^{k,1}_{D}(\Omega)$.

We define $C^{1}(\partial\Omega)^{d}$ to be the set of continuously
differentiable functions on $\partial\Omega$ such that there exists an
extension to the space $C^{1}(\overline{\Omega})^{d}$. The norm on
$C^{1}(\partial\Omega)^{d}$ is defined by

$\|\boldsymbol{w}\|_{C^{1}(\partial\Omega)}=\inf\big{\\{}\|\tilde{\boldsymbol{w}}\|_{1,\infty}\,:\,\tilde{\boldsymbol{w}}\in
C^{1}(\overline{\Omega})^{d},\,\tilde{\boldsymbol{w}}=\boldsymbol{w}\text{ on
}\partial\Omega\big{\\}}.$

We define $C^{1}_{0}(\Gamma_{N})^{d}$ to be the subset of those functions
$\boldsymbol{w}\in C^{1}(\partial\Omega)^{d}$ such that $\boldsymbol{w}$
vanishes on $\overline{\Gamma_{D}}$. By Theorem 1 of [19], we know that there
exists a linear map $E:C^{1}(\partial\Omega)\rightarrow
C^{1}(\overline{\Omega})$ such that $T\boldsymbol{w}=\boldsymbol{w}$ on
$\partial\Omega$ for every $\boldsymbol{w}\in C^{1}(\partial\Omega)^{d}$.
Furthermore, there exists a constant $C$, depending only on $\Omega$, such
that

(1.14) $\|T\boldsymbol{w}\|_{1,\infty}\leq
C\|\boldsymbol{w}\|_{C^{1}(\partial\Omega)}.$

The restriction of $T$ to the space $C^{1}_{D}(\overline{\Omega})^{d}$ maps
into $C^{1}_{0}(\Gamma_{N})^{d}$ by definition of the latter function space.
Moreover, for every $\boldsymbol{w}\in C^{1}_{0}(\Gamma_{N})^{d}$, there
exists an extension $\tilde{\boldsymbol{w}}\in
C^{1}_{D}(\overline{\Omega})^{d}$ such that (1.14) holds. This property is
vital when dealing with the penalisation term in Section 3 on the Neumann part
of the boundary.

If $\mathcal{F}$ is a function space of real-valued functions with norm
$\|\cdot\|_{\mathcal{F}}$, we let $\mathcal{F}^{d}$ denote the set of
$\mathbb{R}^{d}$-valued functions such that each co-ordinate is an element of
$\mathcal{F}$. The norm is given by $\||\cdot|\|_{\mathcal{F}}$, where
$|\cdot|$ is the Euclidean norm, but is denoted by $\|\cdot\|_{\mathcal{F}}$
with an abuse of notation. The meaning is to be understood from the context.
We define $\mathcal{F}^{d\times d}$ analogously, but with $|\cdot|$ meaning
the Frobenius norm.

For a Banach space $X$, we use $L^{q}(0,T;X)$ to denote the standard Bochner
space with norm denoted $\|\cdot\|_{L^{q}(0,T;X)}$. Furthermore,
$L^{\infty}_{w^{*}}(0,T;X^{*})$ denotes the set of functions
$h:[0,T]\rightarrow X^{*}$ such that $t\mapsto\langle
h(t),x\rangle_{(X^{*},X)}$ is measurable on $[0,T]$ for every element $x\in X$
and there exists a constant $c$, independent of $x$, such that

$|\langle h(t),x\rangle_{(X^{*},X)}|\leq c\|x\|_{X}\quad\text{ for a.e.
}t\in[0,T].$

By Theorem 10.1.16 of [33], for example, $L^{\infty}_{w^{*}}(0,T;X^{*})$ is
the dual space of $L^{1}(0,T;X)$. In particular, bounded subsets of
$L^{\infty}_{w^{*}}(0,T;X^{*})$ are weakly-* compact by the Banach–Alaoglu
theorem.

Let $H^{k}_{D}(\Omega)$ be the set of functions $h\in W^{k,2}(\Omega)$ such
that $h=0$ on $\Gamma_{D}$. We use $H^{k}_{D+1}(\Omega)$ to denote the set of
$h\in W^{k,2}(\Omega)$ such that $h=1$ on $\Gamma_{D}$. The duals of
$H^{1}_{D}(\Omega)$ and $W^{1,p}_{D}(\Omega)^{d}$ are denoted by
$H^{-1}_{D}(\Omega)$ and $W^{-1,p^{\prime}}_{D}(\Omega)^{d}$, respectively,
where $p^{\prime}=\frac{p}{p-1}$ is the Hölder conjugate of $p$.

We use lower case letters to denote scalar-valued functions, bold lower case
letters for vector-valued functions and bold upper case letters for tensor-
valued functions. The differential operator $\nabla$ denotes the gradient
operator so, if $\boldsymbol{u}$ is an $\mathbb{R}^{d}$-valued function and
$\mathbf{T}$ is an $\mathbb{R}^{d\times d}$-valued function, we have

$\nabla\boldsymbol{u}=(\partial_{i}\boldsymbol{u}_{j})_{i,j=1}^{d},\quad\nabla\mathbf{T}=(\partial_{k}T_{ij})_{i,j,k=1}^{d}.$

The divergence operator $\mathrm{div}$ is defined by

$\displaystyle\mathrm{div}(\boldsymbol{u})=\frac{\partial u_{i}}{\partial
x_{i}},\quad\mathrm{div}(\mathbf{T})=\Big{(}\frac{\partial T_{ij}}{\partial
x_{j}}\Big{)}_{j=1}^{d},$

assuming Einstein’s summation convention. Throughout this paper, we fix
positive problem parameters $\alpha$, $\epsilon$ and $\eta$, where $\alpha$ is
the inverse of the viscosity coefficient, $\epsilon$ is the approximation
level for the phase-field function and $\eta$ is a stability coefficient. In
practice, $\eta\ll\epsilon\ll 1$ but this is not required for the results
here.

We require certain properties of convex conjugate functions. In this paper, we
define the function $\varphi:\mathbb{R}^{d\times d}\rightarrow[0,\infty)$ by

(1.15) $\varphi(\mathbf{T})=\frac{|\mathbf{T}|^{p}}{p},\quad\text{ and
}\quad\varphi(\mathbf{T})=\int_{0}^{|\mathbf{T}|}\frac{t}{(1+t^{a})^{\frac{1}{a}}}\,\mathrm{d}t,$

in Section 2 and Section 3, respectively. Given one of these choices, we
further define

$\displaystyle
F(\mathbf{T})=\frac{\partial\varphi}{\partial\mathbf{T}}(\mathbf{T})=\Big{(}\frac{\partial\varphi}{\partial
T_{ij}}(\mathbf{T})\Big{)}_{i,j=1}^{d},\quad\varphi^{*}(\mathbf{T})=\sup_{\mathbf{S}\in\mathbb{R}^{d\times
d}}\Big{\\{}\mathbf{T}:\mathbf{S}-\varphi(\mathbf{S})\Big{\\}},$

the derivative and convex conjugate, respectively. On the image of $F$ over
$\mathbb{R}^{d\times d}$, we can show that
$\frac{\partial\varphi^{*}}{\partial\mathbf{T}}=F^{-1}$. For every
$\mathbf{T}\in\mathbb{R}^{d\times d}$ such that the statements make sense, we
have

(1.16)
$\mathbf{T}:F(\mathbf{T})=\varphi(\mathbf{T})+\varphi^{*}(F(\mathbf{T})),\quad\mathbf{T}:F^{-1}(\mathbf{T})=\varphi(F^{-1}(\mathbf{T}))+\varphi^{*}(\mathbf{T}).$

Given the choices in (1.15), we notice that
$F(\mathbf{T})=|\mathbf{T}|^{p-2}\mathbf{T}$ and
$F(\mathbf{T})=\frac{\mathbf{T}}{(1+|\mathbf{T}|^{a})^{\frac{1}{a}}}$,
respectively.

We require the following result in the analysis of the strain-limiting
problem. The proof can be found in [7].

######  Lemma 1.1.

For every $y\geq 0$ and $a>0$, we have

$\displaystyle\min\\{1,2^{-1+\frac{1}{a}}\\}(1+y)\leq(1+y^{a})^{\frac{1}{a}}\leq\max\\{1,2^{-1+\frac{1}{a}}\\}(1+y).$

A key tool in the deduction of various a priori estimates and uniform bounds
on the sequences of approximate solutions is the following variation of the
Korn–Poincaré inequality for a partly homogeneous Dirichlet boundary
condition. The result follows from Theorem 4 in [25], combined with an
application of the Poincaré inequality.

######  Theorem 1.1 (Korn–Poincaré inequality).

Let $\Omega$, $\Gamma_{D}$ and $\Gamma_{N}$ be as above with
$\Gamma_{D}\neq\emptyset$. For every $p\in(1,\infty)$, there exists a constant
$C$ depending only on $\Omega$, $\Gamma_{D}$ and $p$ such that, for every
$\boldsymbol{u}\in W^{1,p}_{D}(\Omega)^{d}$,

$\displaystyle\|\boldsymbol{u}\|_{1,p}\leq
C\|\boldsymbol{\varepsilon}(\boldsymbol{u})\|_{p}.$

##  2\. Nonlinear fracture problems with growth

In this section, we study nonlinear dynamic fracture problems with growth in
the constitutive relation, by which we mean that the strain is not a priori
bounded. This is the first step in the analysis of strain-limiting fracture
problems because we want to understand nonlinear dynamic fracture problems.
However, we start with this growth setting so that we gain some understanding
of the problem formulation without the additional difficulty of the lack of
integrability of the stress tensor that we have in the case that $F$ is
bounded.

We consider the following problem: find a triple
$(\boldsymbol{u},\mathbf{T},v):Q\rightarrow\mathbb{R}^{d}\times\mathbb{R}^{d\times
d}\times\mathbb{R}$ such that

(2.1) $\displaystyle\boldsymbol{u}_{tt}$
$\displaystyle=\mathrm{div}\big{(}b(v)\mathbf{T}\big{)}+\boldsymbol{f}$
$\displaystyle\text{ in }Q,$
$\displaystyle\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}+\alpha\boldsymbol{u})$
$\displaystyle=F(\mathbf{T}):=|\mathbf{T}|^{p-2}\mathbf{T}$
$\displaystyle\text{ in }Q,$ $\displaystyle\boldsymbol{u}\big{|}_{\Gamma_{D}}$
$\displaystyle=\mathbf{0},\,v\big{|}_{\Gamma_{D}}=1$ $\displaystyle\text{ on
}[0,T]\times\Gamma_{D},$ $\displaystyle b(v)\mathbf{T}\mathbf{n}$
$\displaystyle=\boldsymbol{g}$ $\displaystyle\text{ on
}[0,T]\times\Gamma_{N},$ $\displaystyle\boldsymbol{u}(0,\cdot)$
$\displaystyle=\boldsymbol{u}_{0},\boldsymbol{u}_{t}(0,\cdot)=\boldsymbol{u}_{1},\,v(0,\cdot)=v_{0}$
$\displaystyle\text{ in }\Omega,$

subject to the minimisation problem

(2.2)
$\mathcal{E}(\boldsymbol{u}(t),v(t))+\mathcal{H}(v(t))=\inf\Big{\\{}\mathcal{E}(\boldsymbol{u}(t),v)+\mathcal{H}(v)\,:\,v\in
H^{1}_{D+1}(\Omega),\,v\leq v(t)\Big{\\}},$

and the energy-dissipation equality

(2.3)
$\displaystyle\mathcal{F}(t;\boldsymbol{u}(t),\boldsymbol{u}_{t}(t),v(t))+\int_{0}^{t}\int_{\Omega}b(v)\big{[}F^{-1}(\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}+\alpha\boldsymbol{u}))-F^{-1}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}))\big{]}:\boldsymbol{\varepsilon}(\boldsymbol{u}_{t})+\boldsymbol{f}_{t}\cdot\boldsymbol{u}\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\quad+\int_{0}^{t}\int_{\Gamma_{N}}\boldsymbol{g}_{t}\cdot\boldsymbol{u}\,\mathrm{d}S\,\mathrm{d}s$
$\displaystyle=\mathcal{F}(0;\boldsymbol{u}_{0},\boldsymbol{u}_{1},v_{0}),$

for every $t\in[0,T]$, alongside the crack non-healing property $v_{t}\leq 0$.
The functionals $\mathcal{F}$, $\mathcal{E}$, $\mathcal{H}$ denote the total,
elastic and surface energy, respectively, and are defined by

$\displaystyle\mathcal{F}(t;\boldsymbol{u},\boldsymbol{w},v)$
$\displaystyle=\mathcal{K}(\boldsymbol{w})+\mathcal{E}(\boldsymbol{u},v)+\mathcal{H}(v)-\langle
l(t),\boldsymbol{u}\rangle,$ $\displaystyle\mathcal{E}(\boldsymbol{u},v)$
$\displaystyle=\int_{\Omega}\frac{b(v)}{\alpha}\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}))\,\mathrm{d}x,$
$\displaystyle\mathcal{H}(v)$
$\displaystyle=\frac{1}{4\epsilon}\|1-v\|_{2}^{2}+\epsilon\|\nabla
v\|_{2}^{2}.$

We define the functionals $\mathcal{K}$ and $l$ by

$\displaystyle\mathcal{K}(\boldsymbol{w})$
$\displaystyle=\frac{\|\boldsymbol{w}\|_{2}^{2}}{2},$ $\displaystyle\langle
l(t),\boldsymbol{u}\rangle$
$\displaystyle=\int_{\Omega}\boldsymbol{f}(t)\cdot\boldsymbol{u}\,\mathrm{d}x+\int_{\Gamma_{N}}\boldsymbol{g}(t)\cdot\boldsymbol{u}\,\mathrm{d}S,$

representing the kinetic energy and external force, respectively. To ensure
that the minimisation problem holds at the initial time $t=0$, we demand the
compatibility condition

(2.4)
$\mathcal{E}(\boldsymbol{u}_{0},v_{0})+\mathcal{H}(v_{0})=\inf\Big{\\{}\mathcal{E}(\boldsymbol{u}_{0},v)+\mathcal{H}(v)\,:\,v\in
H^{1}_{D+1}(\Omega),\,v\leq v_{0}\Big{\\}}.$

The function $b:\mathbb{R}\rightarrow\mathbb{R}$ is defined by

(2.5) $b(v)=v^{2}+\eta.$

The stability parameter $\eta$ is required so that we do not experience
degeneracy in the elastodynamic equation (2.1)1 at points where the phase-
field function $v$ vanishes.

We remark that the functional
$\mathcal{A}(v):=\mathcal{E}(\boldsymbol{u},v)+\mathcal{H}(v)$ is well-defined
on $H^{1}_{D+1}(\Omega)$, provided that we have $\boldsymbol{u}\in
W^{1,p^{\prime}}(\Omega)^{d}$, which we later show to be the case for
solutions of (2.1)–(2.3). However, the functional $\mathcal{A}$ may take the
value positive infinity when testing against functions that are not also
elements of $L^{\infty}(\Omega)$.

In this section we consider mixed Dirichlet–Neumann boundary conditions. We
assume a homogeneous boundary condition on the Dirichlet part for simplicity
of the presentation. However, it is not too difficult to adapt the arguments
to allow for inhomogeneous boundary conditions. We refer to [8], for example,
for details on how to do this.

Now we state the main results for this section and have a brief discussion
regarding the methods used in the proofs. There are two main results in this
section concerning the existence of weak energy solutions to (2.1)–(2.3), that
is, a weak solution satisfying an energy-dissipation balance. We differentiate
between the cases $p\in(1,2]$ and $p\in(2,\infty)$. This is due to a technical
argument that allows good time regularity of the phase-field function in the
former case, but not necessarily the latter. We remark that uniqueness remains
an open problem for dynamic fracture models and is out of the scope of this
paper. Our focus is on the existence of solutions. The two main results are as
follows.

######  Theorem 2.1.

Suppose that $p\in(1,2]$, $l\in C^{1}([0,T];W^{-1,p}_{D}(\Omega)^{d})$,
$\boldsymbol{u}_{0}$, $\boldsymbol{u}_{1}\in W^{1,p^{\prime}}(\Omega)^{d}$,
and $v_{0}\in H^{1}_{D+1}(\Omega)$ such that $v_{0}\in[0,1]$ a.e. in $\Omega$
and the compatibility condition (2.4) holds. There exists a weak energy
solution $(\boldsymbol{u},\mathbf{T},v)$ of problem (2.1)–(2.3) in the
following sense with

  * •

$\boldsymbol{u}\in W^{2,2}(0,T;L^{2}(\Omega)^{d})\cap
W^{1,\infty}(0,T;W^{1,p^{\prime}}_{D}(\Omega)^{d})$,

  * •

$\mathbf{T}\in L^{\infty}(0,T;L^{p}(\Omega)^{d\times d})$,

  * •

$v\in W^{1,\infty}(0,T;H^{1}(\Omega))$ with $v(t)\in H^{1}_{D+1}(\Omega)$ for
every $t\in[0,T]$.

The weak elastodynamic equation holds,

(2.6)
$\int_{\Omega}\boldsymbol{u}_{tt}(t)\cdot\boldsymbol{w}+b(v(t))\mathbf{T}(t):\boldsymbol{\varepsilon}(\boldsymbol{w})\,\mathrm{d}x=\langle
l(t),\boldsymbol{w}\rangle,$

for a.e. $t\in(0,T)$ and every $\boldsymbol{w}\in
W^{1,p^{\prime}}_{D}(\Omega)$, the constitutive relation
$\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}+\alpha\boldsymbol{u})=F(\mathbf{T})$
is satisfied pointwise a.e. in $Q$, the minimisation problem (2.2) holds for
every $t\in[0,T]$ and the energy-dissipation equality (2.3) is satisfied for
every $t\in[0,T]$. We have the non-healing property $v_{t}\leq 0$ pointwise
a.e. in $Q$. The initial conditions are satisfied in the sense that

$\displaystyle\lim_{t\rightarrow
0+}\big{[}\|\boldsymbol{u}(t)-\boldsymbol{u}_{0}\|_{1,p^{\prime}}+\|\boldsymbol{u}_{t}(t)-\boldsymbol{u}_{1}\|_{2}+\|v(t)-v_{0}\|_{1,2}\big{]}=0.$

######  Theorem 2.2.

Suppose that $p\in(2,\infty)$, $l\in C^{1}([0,T];W^{-1,p}_{D}(\Omega)^{d})$,
$\boldsymbol{u}_{0}$, $\boldsymbol{u}_{1}\in
W^{1,p^{\prime}}_{D}(\Omega)^{d}\cap L^{2}(\Omega)^{d}$, and $v_{0}\in
H^{1}_{D+1}(\Omega)$ such that $v_{0}\in[0,1]$ a.e. in $\Omega$ and the
compatibility condition (2.4) holds. There exists a weak energy solution
$(\boldsymbol{u},\mathbf{T},v)$ of (2.1)–(2.3) with regularity

  * •

$\boldsymbol{u}\in W^{2,2}(0,T;L^{2}(\Omega)^{d})\cap
W^{1,\infty}(0,T;W^{1,p^{\prime}}_{D}(\Omega)^{d})$,

  * •

$\mathbf{T}\in L^{\infty}(0,T;L^{p}(\Omega)^{d\times d})$,

  * •

$v:[0,T]\rightarrow H^{1}_{D+1}(\Omega)$ with $v\in
L^{\infty}(0,T;H^{1}(\Omega))$ and $v$ is continuous a.e. as a map from
$[0,T]$ to $L^{2}(\Omega)$,

solving the problem (2.1)–(2.3) in the following sense. The weak elastodynamic
equation (2.6) holds for a.e. $t\in(0,T)$ and every test function
$\boldsymbol{w}\in W^{1,p^{\prime}}_{D}(\Omega)^{d}\cap L^{2}(\Omega)^{d}$, we
have the constitutive relation
$\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}+\alpha\boldsymbol{u})=F(\mathbf{T})$
pointwise a.e. in $Q$, the minimisation problem (2.2) is satisfied for every
$t\in[0,T]$, and the energy-dissipation equality (2.3) holds for a.e.
$t\in[0,T]$. Furthermore, $v(t)\leq v(s)$ a.e. in $\Omega$, for every $0\leq
s\leq t\leq T$. The initial conditions hold in the sense that $v(0)=v_{0}$ and

$\displaystyle\lim_{t\rightarrow
0+}\big{[}\|\boldsymbol{u}(t)-\boldsymbol{u}_{0}\|_{1,p^{\prime}}+\|\boldsymbol{u}_{t}(t)-\boldsymbol{u}_{1}\|_{2}\big{]}=0.$

Both proofs follow the same steps initially. A two-level approximation is
introduced with parameters $(N,M)$, where $N$ refers to a discretisation in
space for the displacement variable and $M$ is a time discretisation
parameter. For notational simplicity, we denote $\beta=(N,M)$. We let
$M\rightarrow\infty$, followed by the limit as $N\rightarrow\infty$. It is in
the latter limit where we need to differentiate between the cases $p\in(1,2]$
and $p\in(2,\infty)$. Hence, we focus first on proving Theorem 2.1. In Section
2.5, we present the required changes for the proof of Theorem 2.2.

The two-level approximation suggests how the analysis could be adapted when
considering the problem in a numerical setting and using, for example, finite-
element approximations. However, we choose Galerkin approximations with very
high regularity to simplify the analysis when taking the limit as
$M\rightarrow\infty$. In particular, we are able to avoid technical arguments
such as those in Section 3.6 of [28].

We structure the proof of Theorem 2.1 in the following way. First, we
introduce the two-level approximation and in Theorem 2.3 prove the existence
of unique solutions to the time-discrete Galerkin approximation. In
Propositions 2.2 and 2.3, we show that the solution sequence satisfies
discrete counterparts of the minimisation problem (2.2) and the energy-
dissipation equality (2.3). Then we derive various $M$-independent bounds,
which allow us to take the limit as $M\rightarrow\infty$. We obtain a weak-
energy solution of a time-continuous Galerkin approximation of (2.1)–(2.3). We
derive various $N$-independent bounds on this solution. Due to the growth of
the function $F$, the solution of the time-continuous Galerkin approximation
experiences good integrability and regularity. Hence, we can take the limit as
$N\rightarrow\infty$ and use standard compactness arguments to obtain a weak
energy solution of (2.1)–(2.3) in the sense of Theorem 2.1. We remark that,
out of the time-discrete setting, we are unable to make any claims regarding
the uniqueness of solutions.

For the discretisation in space, let $(\boldsymbol{\phi}_{i})_{i=1}^{\infty}$
be an orthogonal basis of $W^{k,2}_{D}(\Omega)^{d}$ such that it is
orthonormal in $L^{2}(\Omega)^{d}$ and $k>\frac{d}{2}+1$. We make this choice
so that $W^{k,2}_{D}(\Omega)^{d}$ embeds continuously into
$C^{1}(\overline{\Omega})^{d}$, by the Sobolev embedding theorem. For details
of how to obtain such a basis, we refer to [20] and the references therein. We
define
$V_{N}=\mathrm{span}\\{\boldsymbol{\phi}_{1},\dots,\boldsymbol{\phi}_{N}\\}$
and let $P_{N}:L^{2}(\Omega)^{d}\rightarrow V_{N}$ denote the orthogonal
projection operator. By construction, there exists a constant $C>0$,
independent of $N$, such that

$\|P_{N}\boldsymbol{u}\|_{k,2}\leq C\|\boldsymbol{u}\|_{k,2}\quad\text{ for
every }\boldsymbol{u}\in W^{k,2}_{D}(\Omega)^{d}.$

In this finite-dimensional setting, for every $q\in[1,\infty]$,
$\|\cdot\|_{q}$ and $\|\cdot\|_{1,q}$ are norms on $V_{N}$ that are equivalent
to the usual norm $\|\cdot\|_{V_{N}}:=\|\cdot\|_{k,2}$, with the constants of
equivalence depending on $N$.

Now we define the time discrete, finite-dimensional problem. Let
$\beta=(N,M)\in\mathbb{N}^{2}$ with $V_{N}$ defined as above. We define the
time step $h=\frac{T}{M}$ and, for every $0\leq m\leq M$, we set
$t^{\beta}_{m}=mh$. For problem data $\boldsymbol{u}_{0}$,
$\boldsymbol{u}_{1}$, $v_{0}$, $\boldsymbol{f}$ and $\boldsymbol{g}$
satisfying the assumptions of Theorem 2.1, we define an initialisation for the
$\beta$-approximation of (2.1)–(2.3) by

(2.7)
$\boldsymbol{u}^{\beta}_{0}=\boldsymbol{u}_{0,N},\quad\boldsymbol{u}^{\beta}_{-1}=\boldsymbol{u}_{0,N}-h\boldsymbol{u}_{1,N},\quad
v^{\beta}_{0}=v^{N}_{0},$

where $v^{N}_{0}\in H^{1}_{D+1}(\Omega)$ is the solution of the minimisation
problem

(2.8)
$\mathcal{E}(\boldsymbol{u}_{0,N},v^{N}_{0})+\mathcal{H}(v^{N}_{0})=\inf\Big{\\{}\mathcal{E}(\boldsymbol{u}_{0,N},v)+\mathcal{H}(v):v\leq
v_{0},v\in H^{1}_{D+1}(\Omega)\Big{\\}}.$

We impose (2.8) so that, at the level of the approximate problem, the
minimisation problem is satisfied at $t=0$. The function $v^{N}_{0}$ exists
and is unique, applying the direct method in the calculus of variations and
strict convexity. The functions $\boldsymbol{u}_{0,N}$,
$\boldsymbol{u}_{1,N}\in W^{k,2}_{D}(\Omega)^{d}$ are approximations of the
initial data $\boldsymbol{u}_{0}$, $\boldsymbol{u}_{1}$, respectively, and are
chosen depending on the value of $p$.

In the case that $p\in(1,2]$, we choose the approximations such that
$\boldsymbol{u}_{0,N}$, $\boldsymbol{u}_{1,N}\in V_{N}$ and, for every $N$,

$\displaystyle
0=\lim_{\tilde{N}\rightarrow\infty}\|\boldsymbol{u}_{0}-\boldsymbol{u}_{0,\tilde{N}}\|_{1,p^{\prime}}+\|\boldsymbol{u}_{1}-\boldsymbol{u}_{1,\tilde{N}}\|_{1,p^{\prime}}\leq\|\boldsymbol{u}_{0}-\boldsymbol{u}_{0,N}\|_{1,p^{\prime}}+\|\boldsymbol{u}_{1}-\boldsymbol{u}_{1,N}\|_{1,p^{\prime}}\leq
C,$

where $C=C(\boldsymbol{u}_{0},\boldsymbol{u}_{1})$ depends only on the initial
data. Existence of such a construction is deduced as follows. The space
$\cup_{N\geq 1}V_{N}$ is dense in $W^{k,2}_{D}(\Omega)^{d}$ so, by the Sobolev
embedding theorem, it must be dense in $W^{1,p^{\prime}}_{D}(\Omega)^{d}$ for
any $p^{\prime}\in(1,\infty)$. Furthermore, $V_{N}$ is a closed subset of
$W^{1,p^{\prime}}_{D}(\Omega)^{d}$. Hence for every $\tilde{\boldsymbol{u}}\in
W^{1,p^{\prime}}_{D}(\Omega)^{d}$, there exists a
$\tilde{\boldsymbol{u}}_{N}\in V_{N}$ such that

$\|\tilde{\boldsymbol{u}}-\tilde{\boldsymbol{u}}_{N}\|_{1,p^{\prime}}=\inf_{\boldsymbol{v}_{N}\in
V_{N}}\|\tilde{\boldsymbol{u}}-\boldsymbol{v}_{N}\|_{1,p^{\prime}}\leq\|\tilde{\boldsymbol{u}}\|_{1,p^{\prime}}.$

By density, we must have

$\lim_{N\rightarrow\infty}\inf_{\boldsymbol{v}_{N}\in
V_{N}}\|\tilde{\boldsymbol{u}}-\boldsymbol{v}_{N}\|_{1,p^{\prime}}=0.$

Thus we conclude the existence of such approximation sequences for
$\boldsymbol{u}_{0}$ and $\boldsymbol{u}_{1}$ as above. Furthermore, we note
that because $p^{\prime}\geq 2$, it follows that the sequences
$(\boldsymbol{u}_{0,N})_{N}$ and $(\boldsymbol{u}_{1,N})_{N}$ are bounded in
$L^{2}(\Omega)^{d}$ and converge strongly in this space to
$\boldsymbol{u}_{0}$, $\boldsymbol{u}_{1}$, respectively, whenever
$\boldsymbol{u}_{0}$, $\boldsymbol{u}_{1}\in
W^{1,p^{\prime}}_{D}(\Omega)^{d}$.

In the case that $p>2$, we must choose the sequences in such a way so that
they approximate the data in both $L^{2}(\Omega)^{d}$ and
$W^{1,p^{\prime}}_{D}(\Omega)^{d}$. This is due to the fact that
$W^{1,p^{\prime}}_{D}(\Omega)$ is not necessarily embedded in $L^{2}(\Omega)$,
unlike the case that $p\leq 2$. We replace $\|\cdot\|_{1,p^{\prime}}$ in the
above by
$\|\tilde{\boldsymbol{u}}\|_{*}:=\|\tilde{\boldsymbol{u}}\|_{2}+\|\tilde{\boldsymbol{u}}\|_{1,p^{\prime}}$
and repeat the same arguments. Hence, given $\boldsymbol{u}_{0}$,
$\boldsymbol{u}_{1}\in L^{2}(\Omega)^{d}\cap
W^{1,p^{\prime}}_{D}(\Omega)^{d}$, we can find $\boldsymbol{u}_{0,N}$,
$\boldsymbol{u}_{1,N}\in V_{N}$ such that

$\displaystyle
0=\lim_{N\rightarrow\infty}\|\boldsymbol{u}_{i}-\boldsymbol{u}_{i,N}\|_{2}+\|\boldsymbol{u}_{i}-\boldsymbol{u}_{i,N}\|_{1,p^{\prime}}\
\leq\|\boldsymbol{u}_{i}\|_{2}+\|\boldsymbol{u}_{i}\|_{1,p^{\prime}}\quad\text{
for }i\in\\{1,2\\}.$

The $\beta$-approximation problem is defined recursively as follows. For every
$1\leq m\leq M$, we look for a function $\boldsymbol{u}^{\beta}_{m}\in V_{N}$
such that

(2.9)
$\int_{\Omega}\delta^{2}\boldsymbol{u}^{\beta}_{m}\cdot\boldsymbol{w}+b(v^{\beta}_{m-1})F^{-1}(\boldsymbol{\varepsilon}(\delta\boldsymbol{u}^{\beta}_{m}+\alpha\boldsymbol{u}^{\beta}_{m})):\boldsymbol{\varepsilon}(\boldsymbol{w})\,\mathrm{d}x=\langle
l^{\beta}_{m},\boldsymbol{w}\rangle,$

for every $\boldsymbol{w}\in V_{N}$, where $l^{\beta}_{m}:=l(t^{\beta}_{m})$.
Next we look for a function $v^{\beta}_{m}\in H^{1}_{D+1}(\Omega)$ that solves
the minimisation problem

(2.10)
$\mathcal{E}(\boldsymbol{u}^{\beta}_{m},v^{\beta}_{m})+\mathcal{H}(v^{\beta}_{m})=\inf\Big{\\{}\mathcal{E}(\boldsymbol{u}^{\beta}_{m},v)+\mathcal{H}(v)\,:\,v\in
H^{1}_{D+1}(\Omega),\,v\leq v^{\beta}_{m-1}\Big{\\}}.$

We use $\delta$ and $\delta^{2}$ to denote the first and second order
difference quotients, respectively, by which we mean

(2.11)
$\delta\boldsymbol{u}^{\beta}_{m}:=\frac{\boldsymbol{u}^{\beta}_{m}-\boldsymbol{u}^{\beta}_{m-1}}{h},\quad\delta^{2}\boldsymbol{u}^{\beta}_{m}:=\frac{\delta\boldsymbol{u}^{\beta}_{m}-\delta\boldsymbol{u}^{\beta}_{m-1}}{h}=\frac{\boldsymbol{u}^{\beta}_{m}-2\boldsymbol{u}^{\beta}_{m-1}+\boldsymbol{u}^{\beta}_{m-2}}{h^{2}}.$

We define $\delta v^{\beta}_{m}$ and $\delta l^{\beta}_{m}$ analogously.

We remark that the function $F$ is a bijection from $\mathbb{R}^{d\times d}$
to itself so we can formulate the problem in terms of only the displacement
and the phase-field function. We reintroduce the stress function later when
taking the limit as $N\rightarrow\infty$, to simplify the notation.

######  Theorem 2.3.

Suppose that $l\in C^{1}([0,T];W^{-1,p}_{D}(\Omega)^{d})$ with
$\boldsymbol{u}_{0}$, $\boldsymbol{u}_{1}\in
W^{1,p^{\prime}}_{D}(\Omega)^{d}\cap L^{2}(\Omega)^{d}$ and $v_{0}\in
H^{1}_{D+1}(\Omega)$ such that $v_{0}\in[0,1]$ a.e. in $\Omega$. For every
$\beta\in\mathbb{N}^{2}$, defining the initial data by (2.7), there exists a
unique sequence of solutions
$(\boldsymbol{u}^{\beta}_{m},v^{\beta}_{m})_{m=1}^{M}$ solving the time
discrete problem (2.10), (2.11).

Existence follows in a standard way from the Browder–Minty theorem for (2.10)
and the direct method in the calculus of variations for (2.11) so we do not
include the details. Uniqueness is a result of strict monotonicity and strict
convexity, respectively.

Before proceeding any further, we note the following convergence result for
the sequence $(v^{N}_{0})_{N}$. This is vital knowledge when we take the limit
as $N\rightarrow\infty$. It enables us to prove the satisfaction of the
energy-dissipation equality when taking the limit in $N$.

######  Proposition 2.1.

Let $v_{0}\in H^{1}_{D+1}(\Omega)$ with $v_{0}\in[0,1]$ a.e. in $\Omega$ and
$\boldsymbol{u}_{0}\in W^{1,p^{\prime}}_{D}(\Omega)^{d}\cap L^{2}(\Omega)^{d}$
such that the compatibility condition (2.4) holds. Let $v^{N}_{0}\in
H^{1}_{D+1}(\Omega)$ be the solution of the minimisation problem (2.8). Then
$v^{N}_{0}\in[0,1]$ a.e. in $\Omega$ and we have $v^{N}_{0}\rightarrow v_{0}$
strongly in $H^{1}(\Omega)$ as $N\rightarrow\infty$.

###### Proof.

First, we notice that, comparing $v$ with $\max\\{v,0\\}$, any minimiser of
(2.8) must be non-negative. By construction, we have $v_{0}^{N}\leq v_{0}\leq
1$. Hence $(v^{N}_{0})_{N}$ is uniformly bounded in $L^{\infty}(\Omega)$.
Next, by construction, we have

$\displaystyle C(\epsilon)\|v^{N}_{0}\|_{1,2}^{2}$
$\displaystyle\leq\mathcal{E}(\boldsymbol{u}_{0,N},v_{0}^{N})+\mathcal{H}(v^{N}_{0})$
$\displaystyle\leq\mathcal{E}(\boldsymbol{u}_{0,N},v_{0})+\mathcal{H}(v_{0})$
$\displaystyle\leq\frac{b(1)}{\alpha}\int_{\Omega}\varphi^{*}(\boldsymbol{\varepsilon}(\boldsymbol{u}_{0,N}))\,\mathrm{d}x+C(\epsilon)\|v_{0}\|_{1,2}^{2}$
$\displaystyle\leq
C\big{[}\|\boldsymbol{\varepsilon}(\boldsymbol{u}_{0,N})\|_{p^{\prime}}^{p^{\prime}}+\|v_{0}\|_{1,2}^{2}\big{]}$
$\displaystyle\leq
C\big{[}\|\boldsymbol{u}_{0}\|_{2}^{p^{\prime}}+\|\boldsymbol{u}_{0}\|_{1,p^{\prime}}^{p^{\prime}}+\|v_{0}\|_{1,2}^{2}\big{]}.$

It follows that $(v^{N}_{0})_{N}$ converges weakly in $H^{1}(\Omega)$ and
strongly in $L^{2}(\Omega)$ to a limit $\tilde{v}_{0}\in H^{1}_{D}(\Omega)$
with $\tilde{v}_{0}\leq v_{0}$, up to a subsequence that we do not relabel. It
follows from Fatou’s lemma and weak lower semi-continuity that

(2.12)
$\displaystyle\mathcal{E}(\boldsymbol{u}_{0},\tilde{v}_{0})+\mathcal{H}(\tilde{v}_{0})$
$\displaystyle\leq\lim_{N\rightarrow\infty}\Big{[}\mathcal{E}(\boldsymbol{u}_{0,N},v_{0}^{N})+\mathcal{H}(v^{N}_{0})\Big{]}$
$\displaystyle\leq\lim_{N\rightarrow\infty}\Big{[}\mathcal{E}(\boldsymbol{u}_{0,N},v_{0})+\mathcal{H}(v_{0})\Big{]}$
$\displaystyle=\mathcal{E}(\boldsymbol{u}_{0},v_{0})+\mathcal{H}(v_{0})$
$\displaystyle\leq\mathcal{E}(\boldsymbol{u}_{0},\tilde{v}_{0})+\mathcal{H}(\tilde{v_{0}}).$

The final inequality follows from the compatibility condition (2.4). Hence
$\tilde{v}_{0}$ is also a minimiser of the problem (2.4). By strict convexity,
we deduce that $v_{0}=\tilde{v}_{0}$.

Next, we notice that $(\mathcal{E}(\boldsymbol{u}_{0,N},v^{N}_{0}))_{N}$
converges to $\mathcal{E}(\boldsymbol{u}_{0},v_{0})$ as a result of the
dominated convergence theorem and the uniform boundedness of
$(b(v^{N}_{0}))_{N}$. Inserting this into (2.12), it follows that

$\displaystyle\lim_{N\rightarrow\infty}\mathcal{H}(v^{N}_{0})=\mathcal{H}(v_{0}).$

The functional $\mathcal{H}$ is equivalent to the usual norm on
$H^{1}(\Omega)$. Thus $(v^{N}_{0})_{N}$ converges weakly and in norm to
$v_{0}$ and so we have strong convergence. ∎

Now we prove that the sequence of solutions satisfies properties related to
the minimisation problem, and an inequality analogous to the energy-
dissipation equality for the time continuous solution. The first result is an
immediate consequence of the minimisation problem (2.11). However, the
formulation presented in Proposition 2.2 is more amenable when it comes to
taking the limit as $M\rightarrow\infty$. Corollary 2.1 follows from
Proposition 2.2, which is vital in proving uniform bounds on the sequence
$(\delta v^{\beta}_{m})_{m=1}^{M}$. In Proposition 2.3, we prove an energy-
dissipation inequality in the spirit of (2.3). We are not able to obtain an
equality in this setting due to the presence of numerical dissipation terms
resulting from the time discretisation. We later use this inequality to obtain
one direction of an energy-dissipation equality for the semi-discrete couple
that is constructed by taking the limit in $M$.

######  Proposition 2.2.

Let the assumptions of Theorem 2.3 hold. Then $v^{\beta}_{m}\in[0,1]$ and
$\delta v^{\beta}_{m}\leq 0$ a.e. in $\Omega$, for every $1\leq m\leq M$.
Furthermore, for every $\tilde{\chi}\in H^{1}_{D+1}(\Omega)$ such that
$\tilde{\chi}\leq v^{\beta}_{m-1}$, we have

(2.13) $\displaystyle 0$
$\displaystyle\leq\big{[}\partial_{v}\mathcal{E}(\boldsymbol{u}^{\beta}_{m},v^{\beta}_{m})+\mathcal{H}^{\prime}(v^{\beta}_{m})\big{]}(\tilde{\chi}-v^{\beta}_{m})$
$\displaystyle=\int_{\Omega}\frac{b^{\prime}(v^{\beta}_{m})}{\alpha}(\tilde{\chi}-v^{\beta}_{m})\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{\beta}_{m}))+\frac{1}{2\epsilon}(v^{\beta}_{m}-1)(\tilde{\chi}-v^{\beta}_{m})+2\epsilon\nabla
v^{\beta}_{m}\cdot\nabla(\tilde{\chi}-v^{\beta}_{m})\,\mathrm{d}x.$

In particular, for every $\chi\in H^{1}_{D}(\Omega)$ such that $\chi\leq 0$,

(2.14)
$0\leq\big{[}\partial_{v}\mathcal{E}(\boldsymbol{u}^{\beta}_{m},v^{\beta}_{m})+\mathcal{H}^{\prime}(v^{\beta}_{m})\big{]}(\chi).$

######  Corollary 2.1.

Let the assumptions of Theorem 2.3 hold. For every $1\leq m\leq M$, we have

$\displaystyle
0=\big{[}\partial_{v}\mathcal{E}(\boldsymbol{u}^{\beta}_{m},v^{\beta}_{m})+\mathcal{H}^{\prime}(v^{\beta}_{m})\big{]}(\delta
v^{\beta}_{m}).$

###### Proof.

Using (2.8), we see that (2.14) also holds for $m=0$. Reasoning as in the
proof of Lemma 3.2 of [28], we immediately obtain the following. Indeed, we
test in (2.13) against $\tilde{\chi}=v^{\beta}_{m-1}$ for “$\geq$” and against
$\tilde{\chi}=2v^{\beta}_{m}-v^{\beta}_{m-1}$ for the opposite direction
“$\leq$”. ∎

######  Proposition 2.3 (Discrete energy-dissipation inequality).

Let the assumptions of Theorem 2.3 hold. The sequence of solutions to (2.10),
(2.11) satisfies, for every $1\leq m\leq M$,

$\displaystyle\mathcal{F}(t^{\beta}_{m};\boldsymbol{u}^{\beta}_{m},\delta\boldsymbol{u}^{\beta}_{m},v^{\beta}_{m})+h\sum_{j=1}^{m}\langle\delta
l^{\beta}_{j},\boldsymbol{u}^{\beta}_{j}\rangle$
$\displaystyle\quad+h\sum_{j=1}^{m}\int_{\Omega}b(v^{\beta}_{j-1})\big{[}F^{-1}(\boldsymbol{\varepsilon}(\delta\boldsymbol{u}^{\beta}_{j}+\alpha\boldsymbol{u}^{\beta}_{j}))-F^{-1}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{\beta}_{m}))\big{]}:\boldsymbol{\varepsilon}(\delta\boldsymbol{u}^{\beta}_{m})\,\mathrm{d}x$
$\displaystyle\leq\mathcal{F}(0;\boldsymbol{u}_{0,N},\boldsymbol{u}_{1,N},v^{N}_{0}).$

###### Proof.

We test in the elastodynamic equation (2.10) against
$h\delta\boldsymbol{u}^{\beta}_{m}$. The interial term, that is, the one
involving $\delta^{2}\boldsymbol{u}^{\beta}_{m}$, can be rewritten as

(2.15) $\displaystyle\int_{\Omega}\delta^{2}\boldsymbol{u}^{\beta}_{m}\cdot
h\delta\boldsymbol{u}^{\beta}_{m}\,\mathrm{d}x$
$\displaystyle=\int_{\Omega}\big{(}\delta\boldsymbol{u}^{\beta}_{m}-\delta\boldsymbol{u}^{\beta}_{m-1}\big{)}\cdot\delta\boldsymbol{u}^{\beta}_{m}\,\mathrm{d}x$
$\displaystyle=\frac{1}{2}\Big{(}\|\delta\boldsymbol{u}^{\beta}_{m}\|_{2}^{2}-\|\delta\boldsymbol{u}^{\beta}_{m-1}\|_{2}^{2}+\|\delta\boldsymbol{u}^{\beta}_{m}-\delta\boldsymbol{u}^{\beta}_{m-1}\|_{2}^{2}\Big{)},$

as a result of the well-known relation

(2.16)
$\mathbf{a}\cdot(\mathbf{a}-\mathbf{b})=\frac{1}{2}\big{(}|\mathbf{a}|^{2}-|\mathbf{b}|^{2}+|\mathbf{a}-\mathbf{b}|^{2}\big{)}\quad\text{
for all }\mathbf{a},\,\mathbf{b}\in\mathbb{R}^{d}.$

For the term involving the external force, we write

(2.17) $\langle l^{\beta}_{m},h\delta\boldsymbol{u}^{\beta}_{m}\rangle=\langle
l^{\beta}_{m},\boldsymbol{u}^{\beta}_{m}\rangle-\langle
l^{\beta}_{m-1},\boldsymbol{u}^{\beta}_{m-1}\rangle+h\langle\delta
l^{\beta}_{m},\boldsymbol{u}^{\beta}_{m-1}\rangle.$

For the term involving the nonlinearity, we must be more careful. First, we
write

(2.18)
$\displaystyle\int_{\Omega}b(v^{\beta}_{m-1})F^{-1}(\boldsymbol{\varepsilon}(\delta\boldsymbol{u}^{\beta}_{m}+\alpha\boldsymbol{u}^{\beta}_{m})):\boldsymbol{\varepsilon}(h\delta\boldsymbol{u}^{\beta}_{m})\,\mathrm{d}x$
$\displaystyle=h\int_{\Omega}b(v^{\beta}_{m-1})\big{[}F^{-1}(\boldsymbol{\varepsilon}(\delta\boldsymbol{u}^{\beta}_{m}+\alpha\boldsymbol{u}^{\beta}_{m}))-F^{-1}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{\beta}_{m}))\big{]}:\boldsymbol{\varepsilon}(\delta\boldsymbol{u}^{\beta}_{m})\,\mathrm{d}x$
$\displaystyle\quad+\int_{\Omega}b(v^{\beta}_{m-1})F^{-1}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{\beta}_{m})):\boldsymbol{\varepsilon}(h\delta\boldsymbol{u}^{\beta}_{m})\,\mathrm{d}x.$

The first term on the right-hand side of (2.18) is in the form that we want.
For the second term, using that $F^{-1}$ is the derivative of $\varphi^{*}$,
we get

(2.19)
$\displaystyle\int_{\Omega}b(v^{\beta}_{m-1})F^{-1}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{\beta}_{m})):\boldsymbol{\varepsilon}(h\delta\boldsymbol{u}^{\beta}_{m})\,\mathrm{d}x$
$\displaystyle=\int_{\Omega}\int_{0}^{1}b(v^{\beta}_{m-1})F^{-1}(\alpha\boldsymbol{\varepsilon}(s\boldsymbol{u}^{\beta}_{m}+(1-s)\boldsymbol{u}^{\beta}_{m-1})):\boldsymbol{\varepsilon}(\boldsymbol{u}^{\beta}_{m}-\boldsymbol{u}^{\beta}_{m-1})\,\mathrm{d}s\,\mathrm{d}x$
$\displaystyle\quad+\int_{\Omega}\int_{0}^{1}b(v^{\beta}_{m-1})\big{[}F^{-1}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{\beta}_{m}))-F^{-1}(\alpha\boldsymbol{\varepsilon}(s\boldsymbol{u}^{\beta}_{m}+(1-s)\boldsymbol{u}^{\beta}_{m-1}))\big{]}:\boldsymbol{\varepsilon}(\boldsymbol{u}^{\beta}_{m}-\boldsymbol{u}^{\beta}_{m-1})\,\mathrm{d}s\,\mathrm{d}x$
$\displaystyle\geq\int_{\Omega}\int_{0}^{1}b(v^{\beta}_{m-1})F^{-1}(\alpha\boldsymbol{\varepsilon}(s\boldsymbol{u}^{\beta}_{m}+(1-s)\boldsymbol{u}^{\beta}_{m-1})):\boldsymbol{\varepsilon}(\boldsymbol{u}^{\beta}_{m}-\boldsymbol{u}^{\beta}_{m-1})\,\mathrm{d}s\,\mathrm{d}x$
$\displaystyle=\int_{\Omega}\frac{b(v^{\beta}_{m})}{\alpha}\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{\beta}_{m}))-\frac{b(v^{\beta}_{m-1})}{\alpha}\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{\beta}_{m-1}))+\frac{b(v^{\beta}_{m-1})-b(v^{\beta}_{m})}{\alpha}\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{\beta}_{m}))\,\mathrm{d}x$
$\displaystyle\geq\mathcal{E}(\boldsymbol{u}^{\beta}_{m},v^{\beta}_{m})-\mathcal{E}(\boldsymbol{u}^{\beta}_{m-1},v^{\beta}_{m-1})+\mathcal{H}(v^{\beta}_{m})-\mathcal{H}(v^{\beta}_{m-1}),$

where the transition to the final line follows from the minimisation problem
(2.10). Summing (2.15), (2.17), (2.18) and (2.19), we deduce that

$\displaystyle\mathcal{F}(t^{\beta}_{m};\boldsymbol{u}^{\beta}_{m},\delta\boldsymbol{u}^{\beta}_{m},v^{\beta}_{m})+h\langle\delta
l^{\beta}_{m},\boldsymbol{u}^{\beta}_{m-1}\rangle$
$\displaystyle\quad+h\int_{\Omega}b(v^{\beta}_{m-1})\big{[}F^{-1}(\boldsymbol{\varepsilon}(\delta\boldsymbol{u}^{\beta}_{m}+\alpha\boldsymbol{u}^{\beta}_{m}))-F^{-1}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{\beta}_{m}))\big{]}:\boldsymbol{\varepsilon}(\delta\boldsymbol{u}^{\beta}_{m})\,\mathrm{d}x$
$\displaystyle\leq\mathcal{F}(t^{\beta}_{m-1};\boldsymbol{u}^{\beta}_{m-1},\delta\boldsymbol{u}^{\beta}_{m-1},v^{\beta}_{m-1}).$

Using this inequality recursively, we obtain the required result. ∎

###  2.1. $M$-independent a priori bounds

Now we focus on finding estimates on the sequence of solutions to the time
discrete problem that are independent of $M$. This is so that we can take the
limit as $M\rightarrow\infty$ and obtain sufficiently strong convergence
results. All constants may depend on the data and parameters of the original
problem, but we specify the dependence on $N$ if it is present. The first
bound is an immediate corollary of the energy-dissipation inequality.

######  Lemma 2.1.

Let the assumptions of Theorem 2.3 hold and suppose that (2.4) holds. There
exists a constant $C=C(N)$, independent of $M$, such that

$\displaystyle\max_{1\leq m\leq
M}\|\boldsymbol{u}^{\beta}_{m}\|_{V_{N}}+\max_{1\leq m\leq
M}\|\delta\boldsymbol{u}^{\beta}_{m}\|_{V_{N}}+\max_{1\leq m\leq
M}\|v^{\beta}_{m}\|_{1,2}\leq C.$

###### Proof.

Rearranging the energy dissipation inequality and using monotonicity of $F$ to
eliminate terms, we see that

(2.20)
$\displaystyle\mathcal{K}(\delta\boldsymbol{u}^{\beta}_{m})+\mathcal{E}(\boldsymbol{u}^{\beta}_{m},v^{\beta}_{m})+\mathcal{H}(v^{\beta}_{m})$
$\displaystyle\leq\mathcal{K}(\delta\boldsymbol{u}^{\beta}_{0})+\mathcal{E}(\boldsymbol{u}^{\beta}_{0},v^{\beta}_{0})+\mathcal{H}(v^{\beta}_{0})-\langle
l^{\beta}_{0},\boldsymbol{u}^{\beta}_{0}\rangle-h\sum_{j=1}^{m}\langle\delta
l^{\beta}_{j},\boldsymbol{u}^{\beta}_{j-1}\rangle+\langle
l^{\beta}_{m},\boldsymbol{u}^{\beta}_{m}\rangle$
$\displaystyle\leq\frac{\|\boldsymbol{u}_{1,N}\|_{2}^{2}}{2}+\frac{b(1)}{\alpha}\int_{\Omega}\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}_{0,N}))\,\mathrm{d}x+\mathcal{H}(v^{N}_{0})$
$\displaystyle\quad+C_{T}\|l(0)\|_{-1,2}\|\boldsymbol{u}_{0,N}\|_{1,2}+\frac{h}{2}\sum_{j=1}^{m}\|\delta
l^{\beta}_{j}\|_{-1,2}^{2}+\frac{C_{T}h}{2}\sum_{j=0}^{m-1}\|\boldsymbol{u}^{\beta}_{j}\|_{1,2}^{2}$
$\displaystyle\quad+C_{T}\|l\|_{L^{\infty}(0,T;W^{-1,2}_{D}(\Omega))}\|\boldsymbol{u}^{\beta}_{m}\|_{1,2}.$

Using that
$\boldsymbol{u}^{\beta}_{m}=\boldsymbol{u}^{\beta}_{0}+\sum_{j=1}^{m}\delta\boldsymbol{u}^{\beta}_{j}$
and adding $\|\boldsymbol{u}^{\beta}_{m}\|_{2}^{2}$ to both sides of (2.20),
we apply Young’s inequality and the equivalence of norms in $V_{N}$ to deduce
that

$\displaystyle{\|\boldsymbol{u}^{\beta}_{m}\|_{2}^{2}}+{\|\delta\boldsymbol{u}^{\beta}_{m}\|_{2}^{2}}+\mathcal{E}(\boldsymbol{u}^{\beta}_{m},v^{\beta}_{m})+\mathcal{H}(v^{\beta}_{m})$
$\displaystyle\leq
C(N)\Big{[}\|\boldsymbol{u}_{1,N}\|_{1,2}^{2}+\|\boldsymbol{u}_{0,N}\|_{1,2}^{2}+\|\boldsymbol{u}_{0,N}\|_{1,p^{\prime}}^{p^{\prime}}+\sup_{N\in\mathbb{N}}\|v^{N}_{0}\|_{1,2}^{2}+\|l\|_{L^{\infty}(0,T;W^{-1,2}_{D}(\Omega))}^{2}$
$\displaystyle\quad+\|l_{t}\|_{L^{2}(0,T;W^{-1,2}_{D}(\Omega))}+h\sum_{j=1}^{m-1}\Big{(}\|\boldsymbol{u}^{\beta}_{j}\|_{2}^{2}+\|\delta\boldsymbol{u}^{\beta}_{j}\|_{2}^{2}\Big{)}\Big{]}.$

Applying the discrete Gronwall inequality and equivalence of norms again, we
maximise over the indices $1\leq m\leq M$ to obtain the stated result. ∎

Due to the regularity of the functions in $V_{N}$, Lemma 2.1 gives a very
strong bound on the solution sequence. Taking the limit in the time
discretisation first allows us to maintain these nice regularity properties as
long as possible. Without the compatibility condition (2.4), an
$M$-independent bound is still possible as the above result shows. However,
later $N$-independent bounds are not possible without the assumption, hence
why it is included in later results concerning $N$-independent bounds.

The phase-field function experiences a nonlinearity in the elastodynamic
equation, by which we mean the presence of the term $b(v^{\beta}_{m})$. Hence,
to identify the resulting limit, we require a strong convergence result for
$(v^{\beta}_{m})_{m=1}^{M}$. To this end, we look for a bound on the discrete
time derivatives $(\delta v^{\beta}_{m})_{m=1}^{M}$. Due to the nonlinearity
in the constitutive relation, we cannot argue as in the proof of Lemma 3.4
from [28]. In particular, we cannot ‘transfer’ the effects of the nonlinearity
between $\boldsymbol{\varepsilon}(\boldsymbol{u})$ and its derivative in time.
Instead we make use of the embedding $V_{N}\subset W^{k,2}_{D}(\Omega)\subset
C^{1}(\overline{\Omega})$, a consequence of the Sobolev embedding theorem for
$k>\frac{d}{2}+1$, and using Corollary 2.1.

######  Lemma 2.2.

Let the assumptions of Lemma 2.1 hold. There exists a constant $C=C(N)$
independent of $M$ such that

$\displaystyle\max_{1\leq m\leq M}\|\delta v^{\beta}_{m}\|_{1,2}\leq C.$

###### Proof.

Using Proposition 2.2 and Corollary 2.1, for every $1\leq m\leq M$, we have

(2.21)
$0=\big{[}\partial_{v}\mathcal{E}(\boldsymbol{u}^{\beta}_{m},v^{\beta}_{m})+\mathcal{H}^{\prime}(v^{\beta}_{m})\big{]}(\delta
v^{\beta}_{m}),$

and

(2.22)
$0\leq\big{[}\partial_{v}\mathcal{E}(\boldsymbol{u}^{\beta}_{m-1},v^{\beta}_{m-1})+\mathcal{H}^{\prime}(v^{\beta}_{m-1})\big{]}(\delta
v^{\beta}_{m}).$

Subtracting (2.21) from (2.22), we get

$\displaystyle\frac{1}{2\epsilon}\|\delta
v^{\beta}_{m}\|_{2}^{2}+2\epsilon\|\nabla\delta
v^{\beta}_{m}\|_{2}^{2}\leq\frac{1}{h}\int_{\Omega}\Big{[}\frac{b^{\prime}(v^{\beta}_{m-1})}{\alpha}\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{\beta}_{m-1}))-\frac{b^{\prime}(v^{\beta}_{m})}{\alpha}\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{\beta}_{m}))\Big{]}\cdot\delta
v^{\beta}_{m}\,\mathrm{d}x.$

We add and subtract
$\alpha^{-1}b(v^{\beta}_{m-1})\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{\beta}_{m}))$
to the right-hand side of the above and use that $b^{\prime}(v)=2v$ to deduce
that

$\displaystyle\frac{1}{2\epsilon}\|\delta
v^{\beta}_{m}\|_{2}^{2}+2\epsilon\|\nabla\delta
v^{\beta}_{m}\|_{2}^{2}+\frac{2}{\alpha}\int_{\Omega}|\delta
v^{\beta}_{m}|^{2}\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{\beta}_{m}))\,\mathrm{d}x$
$\displaystyle\leq\int_{\Omega}\int_{0}^{1}b^{\prime}(v^{\beta}_{m-1})\delta
v^{\beta}_{m}F^{-1}(\alpha\boldsymbol{\varepsilon}(s\boldsymbol{u}^{\beta}_{m-1}+(1-s)\boldsymbol{u}^{\beta}_{m})):\boldsymbol{\varepsilon}(\boldsymbol{u}^{\beta}_{m-1}-\boldsymbol{u}^{\beta}_{m})\,\mathrm{d}s\,\mathrm{d}x$
$\displaystyle\leq 2\|\delta
v^{\beta}_{m}\|_{2}\big{[}\|\boldsymbol{\varepsilon}(\boldsymbol{u}^{\beta}_{m-1})\|_{2}+\|\boldsymbol{\varepsilon}(\boldsymbol{u}^{\beta}_{m})\|_{2}\big{]}\cdot\big{[}\|\boldsymbol{\varepsilon}(\boldsymbol{u}^{\beta}_{m-1})\|_{\infty}^{p^{\prime}-1}+\|\boldsymbol{\varepsilon}(\boldsymbol{u}^{\beta}_{m})\|_{\infty}^{p^{\prime}-1}\big{]}$
$\displaystyle\leq\frac{\|\delta
v^{\beta}_{m}\|_{2}^{2}}{4\epsilon}+C(N)\Big{[}\max_{0\leq j\leq
M}\|\boldsymbol{u}^{\beta}_{j}\|_{V_{N}}^{2p^{\prime}}+\max_{1\leq j\leq
M}\|\delta\boldsymbol{u}^{\beta}_{j}\|_{V_{N}}^{2p^{\prime}}\Big{]},$

using Hölder’s inequality, followed by Young’s inequality. Applying Lemma 2.1,
the result follows. ∎

In the elastodynamic equation for weak energy solutions of (2.1)–(2.3), we
have the interial term $\boldsymbol{u}_{tt}$. Hence, we should look for a
bound on the sequence $(\delta^{2}\boldsymbol{u}^{\beta}_{m})_{m=1}^{M}$. The
finite-dimensional setting of $V_{N}$ simplifies this matter significantly.
Indeed, we need only use the weak elastodynamic equation (2.9), the
equivalence of norms in finite-dimensional spaces and the previously
determined bounds.

######  Lemma 2.3.

Let the assumptions of Lemma 2.1 hold. There exists a constant $C=C(N)$
independent of $M$ such that

$\displaystyle\max_{1\leq m\leq
M}\|\delta^{2}\boldsymbol{u}^{\beta}_{m}\|_{V_{N}}\leq C.$

###### Proof.

By the uniform bound on $(v^{\beta}_{m})_{m=0}^{M}$, the elastodynamic
equation (2.10) and Lemma 2.1, it follows that

$\displaystyle\max_{1\leq m\leq
M}\|\delta^{2}\boldsymbol{u}^{\beta}_{m}\|_{V_{N}^{*}}\leq C(N),$

where $V_{N}^{*}$ is the dual space. By equivalence of norms, the result
follows. ∎

###  2.2. The limit $M\rightarrow\infty$

Now we define interpolants of the solution sequence for the time discrete
problem. We derive bounds on these functions using the results of Lemmas 2.1,
2.2 and 2.3, and then take the limit as $M\rightarrow\infty$. The limiting
functions define a solution of a time continuous Galerkin approximation of
(2.1)–(2.3) with respect to $V_{N}$. We define the piecewise affine, backwards
and forwards interpolants, respectively, of the sequence
$(\boldsymbol{u}^{\beta}_{m})_{m=0}^{M}$ by

$\displaystyle\overline{\boldsymbol{u}}^{\beta}(t)$
$\displaystyle=\frac{t-t^{\beta}_{m-1}}{h}\boldsymbol{u}^{\beta}_{m}+\frac{t^{\beta}_{m}-t}{h}\boldsymbol{u}^{\beta}_{m-1}$
$\displaystyle\text{ for }t\in[t^{\beta}_{m-1},t^{\beta}_{m}],\,1\leq m\leq
M,$ $\displaystyle\boldsymbol{u}^{\beta,+}(t)$
$\displaystyle=\boldsymbol{u}^{\beta}_{m}$ $\displaystyle\text{ for
}t\in(t^{\beta}_{m-1},t^{\beta}_{m}],\,1\leq m\leq M,$
$\displaystyle\boldsymbol{u}^{\beta,-}(t)$
$\displaystyle=\boldsymbol{u}^{\beta}_{m-1}$ $\displaystyle\text{ for
}t\in[t^{\beta}_{m-1},t^{\beta}_{m}),\,1\leq m\leq M,$

where we extend $\boldsymbol{u}^{\beta,+}$ and $\boldsymbol{u}^{\beta,-}$
continuously to $t=0$ and $t=T$, respectively. Similarly, we denote by
$\overline{\boldsymbol{u}}^{\beta,\prime}$, $\boldsymbol{u}^{\beta,+,\prime}$,
$\boldsymbol{u}^{\beta,-,\prime}$ the interpolants of
$(\delta\boldsymbol{u}^{\beta}_{m})_{m=0}^{M}$,
$\boldsymbol{u}^{\beta,+,\prime\prime}$ the backwards interpolant of
$(\delta^{2}\boldsymbol{u}^{\beta}_{m})_{m=1}^{M}$, $\overline{v}^{\beta}$,
$v^{\beta,+}$, $v^{\beta,-}$ the interpolants of $(v^{\beta}_{m})_{m=1}^{M}$,
and $v^{\beta,+,\prime}$ the backwards interpolant of $(\delta
v^{\beta}_{m})_{m=1}^{M}$. We can identify

$\displaystyle\overline{\boldsymbol{u}}^{\beta}_{t}=\boldsymbol{u}^{\beta,+,\prime},\quad\overline{\boldsymbol{u}}^{\beta,\prime}_{t}=\boldsymbol{u}^{\beta,+,\prime\prime},\quad\overline{v}^{\beta}_{t}=v^{\beta,+,\prime}.$

By Lemmas 2.1, 2.2 and 2.3, we deduce that there exists a constant $C=C(N)$,
independent of $M$, such that

$\displaystyle\|\overline{\boldsymbol{u}}^{\beta}\|_{W^{1,\infty}(V_{N})}+\|\boldsymbol{u}^{\beta,\pm}\|_{L^{\infty}(V_{N})}+\|\overline{\boldsymbol{u}}^{\beta,\prime}\|_{W^{1,\infty}(V_{N})}+\|\boldsymbol{u}^{\beta,\pm,\prime}\|_{L^{\infty}(V_{N})}+\|\boldsymbol{u}^{\beta,+,\prime\prime}\|_{L^{\infty}(V_{N})}$
$\displaystyle\quad+\|\overline{v}^{\beta}\|_{W^{1,\infty}(H^{1})}+\|v^{\beta,\pm}\|_{W^{1,\infty}(H^{1})}+\|v^{\beta,+,\prime}\|_{L^{\infty}(H^{1})}$
$\displaystyle\leq C.$

With this bound and standard weak compactness results for Bochner spaces, we
obtain the convergence result in Lemma 2.4. Various strong convergence results
follow from application of the Aubin–Lions lemma and the compact embedding
$V_{N}\subset C^{1}(\overline{\Omega})^{d}$. We include only the ones that are
used in identifying the limiting couple as a solution of the problem of
interest. In particular, these strong convergence results are used for
identification of the nonlinear terms in the limit. We collect the
convergences in Lemma 2.4. We do not include the details of the proof as it
follows standard procedures. Then we prove various properties of the limiting
couple, denoted $(\boldsymbol{u}^{N},v^{N})$. In Proposition 2.4, we show that
a weak elastodynamic equation holds with respect to the Galerkin approximation
from $V_{N}$, and in Proposition 2.5 that an appropriate minimisation problem
is satisfied at every point in $[0,T]$. Finally, in Proposition 2.6, we prove
that an energy-dissipation equality holds.

######  Lemma 2.4.

Let the assumptions of Lemma 2.1 hold and denote $\beta=(N,M)$ where $N$ is
fixed. There exists a subsequence in $M$, independent of $N$, not relabelled,
and a limiting couple $(\boldsymbol{u}^{N},v^{N})$ such that
$\boldsymbol{u}^{N}\in W^{2,\infty}(0,T;V_{N})$ and $v^{N}\in
W^{1,\infty}(0,T;H^{1}(\Omega))$ with $v^{N}(t)\in H^{1}_{D+1}(\Omega)$ for
every $t\in[0,T]$, and the following convergence results hold as
$M\rightarrow\infty$:

  * •

$\overline{\boldsymbol{u}}^{(N,M)},\,\overline{\boldsymbol{u}}^{(N,M),\prime}\overset{\ast}{\rightharpoonup}\boldsymbol{u}^{N},\,\boldsymbol{u}^{N}_{t}$
weakly-* in $W^{1,\infty}(0,T;V_{N})$, respectively;

  * •

$\overline{\boldsymbol{u}}^{(N,M)},\,\overline{\boldsymbol{u}}^{(N,M),\prime}\rightarrow\boldsymbol{u}^{N},\,\boldsymbol{u}^{N}_{t}$
strongly in $C([0,T];C^{1}(\overline{\Omega})^{d})$, respectively;

  * •

$\boldsymbol{u}^{(N,M),\pm},\,\boldsymbol{u}^{(N,M),\pm,\prime}\overset{\ast}{\rightharpoonup}\boldsymbol{u}^{N},\,\boldsymbol{u}^{N}_{t}$
weakly-* in $L^{\infty}(0,T;V_{N})$, respectively;

  * •

$\overline{v}^{(N,M)}\overset{\ast}{\rightharpoonup}v^{N}$ weakly-* in
$W^{1,\infty}(0,T;H^{1}(\Omega))$;

  * •

$v^{(N,M),\pm}\overset{\ast}{\rightharpoonup}v^{N}$ weakly-* in
$L^{\infty}(0,T;H^{1}(\Omega))$.

Furthermore, $v^{N}\in[0,1]$ and $v^{N}_{t}\leq 0$ a.e. in $Q$.

######  Proposition 2.4.

Let the assumptions of Lemma 2.1 hold and let $(\boldsymbol{u}^{N},v^{N})$ be
the limiting couple constructed in Lemma 2.4. For a.e. $t\in(0,T)$ and every
$\boldsymbol{w}\in V_{N}$, we have

(2.23)
$\int_{\Omega}\boldsymbol{u}^{N}_{tt}(t)\cdot\boldsymbol{w}+b(v^{N}(t))F^{-1}(\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N})):\boldsymbol{\varepsilon}(\boldsymbol{w})\,\mathrm{d}x=\langle
l(t),\boldsymbol{w}\rangle.$

The initial conditions hold in the sense that

$\displaystyle\lim_{t\rightarrow
0+}\Big{[}\|\boldsymbol{u}^{N}(t)-\boldsymbol{u}_{0,N}\|_{V_{N}}+\|\boldsymbol{u}^{N}_{t}(t)-\boldsymbol{u}_{1,N}\|_{V_{N}}+\|v^{N}(t)-v^{N}_{0}\|_{1,2}\Big{]}=0.$

###### Proof.

For (2.23), rewriting (2.10) in terms of the interpolants, for every
$M\in\mathbb{N}$ and a.e. $t\in(0,T)$, we first notice that we have

$\displaystyle\int_{\Omega}\overline{\boldsymbol{u}}^{\beta,+,\prime}_{t}(t)\cdot\boldsymbol{w}+b(v^{\beta,-}(t))F^{-1}(\boldsymbol{\varepsilon}(\boldsymbol{u}^{\beta,+,\prime}+\alpha\boldsymbol{u}^{\beta,+})(t)):\boldsymbol{\varepsilon}(\boldsymbol{w})\,\mathrm{d}x=\langle
l^{\beta,+}(t),\boldsymbol{w}\rangle,$

for every $\boldsymbol{w}\in V_{N}$. By the convergence results in Lemma 2.4,
we deduce that, for every $t\in(0,T)$,

(2.24)
$\displaystyle\lim_{M\rightarrow\infty}\Big{\\{}\int_{\Omega}b(v^{\beta,-}(t))F^{-1}(\boldsymbol{\varepsilon}(\boldsymbol{u}^{\beta,+,\prime}+\alpha\boldsymbol{u}^{\beta,+})(t))):\boldsymbol{\varepsilon}(\boldsymbol{w})\,\mathrm{d}x-\langle
l^{\beta,+}(t),\boldsymbol{w}\rangle\Big{\\}}$
$\displaystyle=\int_{\Omega}b(v^{N}(t))F^{-1}(\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N})(t)):\boldsymbol{\varepsilon}(\boldsymbol{w})\,\mathrm{d}x-\langle
l(t),\boldsymbol{w}\rangle,$

and, for every $\psi\in C([0,T])$,

(2.25)
$\lim_{M\rightarrow\infty}\int_{Q}\overline{\boldsymbol{u}}^{\beta,+,\prime}_{t}(t)\cdot\boldsymbol{w}\psi\,\mathrm{d}x\,\mathrm{d}t=\int_{Q}\boldsymbol{u}^{N}_{tt}(t)\cdot\boldsymbol{w}\psi\,\mathrm{d}x\,\mathrm{d}t.$

Using (2.24), (2.25) and Lebesgue’s differentiation theorem, we obtain (2.23).
Satisfaction of the initial conditions follows from standard arguments and
regularity results for Bochner functions. (See [18, Chapter 5], for example.)
∎

######  Proposition 2.5.

Let the assumptions of Lemma 2.1 hold and let $(\boldsymbol{u}^{N},v^{N})$ be
the limiting couple constructed in Lemma 2.4. For every $t\in[0,T]$,
$v^{N}(t)$ solves the variational problem

(2.26)
$\mathcal{E}(\boldsymbol{u}^{N}(t),v^{N}(t))+\mathcal{H}(v^{N}(t))=\inf\Big{\\{}\mathcal{E}(\boldsymbol{u}^{N}(t),v)+\mathcal{H}(v)\,:\,v\in
H^{1}_{D+1}(\Omega),\,v\leq v^{N}(t)\Big{\\}}.$

###### Proof.

By the construction of $(v^{N}_{0})_{N}$, (2.26) holds at $t=0$. For
$t\in(0,T]$, it suffices to show that

(2.27)
$0\leq\int_{\Omega}\frac{b^{\prime}(v^{N}(t))}{\alpha}\chi\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N}(t)))+\frac{1}{2\epsilon}\big{(}v^{N}(t)-1\big{)}\chi+2\epsilon\nabla
v^{N}(t)\cdot\nabla\chi\,\mathrm{d}x,$

for every $\chi\in H^{1}_{D}(\Omega)$ with $\chi\leq 0$. However, by
Proposition 2.2, for every $\psi\in C([0,T])$ with $\psi\geq 0$, we have

(2.28)
$0\leq\int_{Q}\frac{b^{\prime}(v^{\beta,+})}{\alpha}\chi\psi\varphi^{*}_{n}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{\beta,+}))+\frac{1}{2\epsilon}(v^{\beta,+}-1)\chi\psi+2\epsilon\nabla
v^{\beta,+}\cdot\nabla(\chi\psi)\,\mathrm{d}x\,\mathrm{d}t.$

Letting $N\rightarrow\infty$ in (2.28) and using the weak convergence results,
an application of Lebesgue’s differentiation theorem gives (2.27) for a.e.
$t\in(0,T]$. Noticing that the right-hand side of (2.27) is continuous as a
function of $t$, we conclude that (2.27) holds for every $t\in[0,T]$. ∎

######  Proposition 2.6.

Let the assumptions of Lemma 2.1 hold and let $(\boldsymbol{u}^{N},v^{N})$be
the limiting couple constructed in Lemma 2.4. For every $t\in[0,T]$, the
following energy-dissipation equality holds:

(2.29)
$\displaystyle\mathcal{F}(t;\boldsymbol{u}^{N}(t),\boldsymbol{u}^{N}_{t}(t),v^{N}(t))+\int_{0}^{t}\langle
l_{t},\boldsymbol{u}^{N}\rangle\,\mathrm{d}s$
$\displaystyle\quad+\int_{0}^{t}\int_{\Omega}b(v^{N})\big{[}F^{-1}(\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N}))-F^{-1}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N}))\big{]}:\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t})\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle=\mathcal{F}(0;\boldsymbol{u}_{0,N},\boldsymbol{u}_{1,N},v^{N}_{0}).$

###### Proof.

Trivially the result holds for $t=0$. Letting $M\rightarrow\infty$ in the
discrete energy-dissipation inequality and using weak lower semi-continuity,
we deduce that (2.29) holds with “$=$” replaced by “$\leq$”, for a.e.
$t\in(0,T]$.

For the opposite inequality, we first notice that

$\displaystyle\mathcal{K}(\boldsymbol{u}^{N}_{t}(t))-\langle
l(t),\boldsymbol{u}^{N}(t)\rangle+\int_{0}^{t}\langle
l_{t},\boldsymbol{u}^{N}\rangle\,\mathrm{d}s+\int_{0}^{t}\int_{\Omega}b(v^{N})F^{-1}(\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N})):\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t})\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\quad-\mathcal{K}(\boldsymbol{u}_{1,N})+\langle
l(0),\boldsymbol{u}_{0,N}\rangle$
$\displaystyle=\int_{0}^{t}\int_{\Omega}\boldsymbol{u}^{N}_{tt}\cdot\boldsymbol{u}^{N}_{t}\,\mathrm{d}x\,\mathrm{d}s-\int_{0}^{t}\langle
l,\boldsymbol{u}^{N}_{t}\rangle\,\mathrm{d}s+\int_{0}^{t}\int_{\Omega}b(v^{N})F^{-1}(\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N})):\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t})\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle=0,$

using (2.23). Thus, it suffices to show that

(2.30)
$\displaystyle\mathcal{E}(\boldsymbol{u}^{N}(t),v^{N}(t))+\mathcal{H}(v^{N}(t))-\int_{0}^{t}\int_{\Omega}b(v^{N})F^{-1}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N})):\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t})\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\geq\mathcal{E}(\boldsymbol{u}_{0,N},v^{N}_{0})+\mathcal{H}(v^{N}_{0}).$

To prove that (2.30) holds, we mimic an argument from [28]. Fixing
$t\in(0,T]$, for each $K\in\mathbb{N}_{\geq 2}$ we define a time step
$h_{K}=\frac{t}{K}$ and the corresponding time-discrete approximate solution
sequence

$\displaystyle\boldsymbol{u}^{N,K}_{k}=\boldsymbol{u}^{N}(hk),\quad
v^{N,K}_{k}=v^{N}(hk)\quad\text{ for }0\leq k\leq K.$

Then we can write

(2.31)
$\mathcal{E}(\boldsymbol{u}^{N}(t),v^{N}(t))-\mathcal{E}(\boldsymbol{u}_{0,N},v^{N}_{0})=\sum_{k=1}^{K}\Big{[}\mathcal{E}(\boldsymbol{u}^{N,K}_{k},v^{N,K}_{k})-\mathcal{E}(\boldsymbol{u}^{N,K}_{k-1},v^{N,K}_{k-1})\Big{]}.$

Using the variational problem (2.26), we deduce that

(2.32)
$\displaystyle\mathcal{E}(\boldsymbol{u}^{N,K}_{k},v^{N,K}_{k})-\mathcal{E}(\boldsymbol{u}^{N,K}_{k-1},v^{N,K}_{k-1})$
$\displaystyle\geq\int_{\Omega}\frac{b(v^{N,K}_{k})}{\alpha}\Big{(}\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N,K}_{k}))-\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N,K}_{k-1}))\Big{)}\,\mathrm{d}x+\mathcal{H}(v^{N,K}_{k-1})-\mathcal{H}(v^{N,K}_{k})$
$\displaystyle=\frac{t}{K}\int_{\Omega}\int_{0}^{1}\frac{b(v^{N,K}_{k})}{\alpha}F^{-1}(\alpha\boldsymbol{\varepsilon}(s\boldsymbol{u}^{N,K}_{k}+(1-s)\boldsymbol{u}^{N,K}_{k-1})):\boldsymbol{\varepsilon}(\delta\boldsymbol{u}^{N,K}_{k})\,\mathrm{d}s\,\mathrm{d}x$
$\displaystyle\quad+\mathcal{H}(v^{N,K}_{k-1})-\mathcal{H}(v^{N,K}_{k}).$

Using the previous notation for interpolants of discrete solution sequences,
we insert (2.32) into (2.31) to obtain

(2.33)
$\displaystyle\mathcal{E}(\boldsymbol{u}^{N}(t),v^{N}(t))-\mathcal{E}(\boldsymbol{u}_{0,N},v^{N}_{0})-\mathcal{H}(v^{N}_{0})+\mathcal{H}(v^{N}(t))$
$\displaystyle\geq\int_{0}^{t}\int_{\Omega}\int_{0}^{1}\frac{b(v^{N,K,+})}{\alpha}F^{-1}(\alpha\boldsymbol{\varepsilon}(s\boldsymbol{u}^{N,K,+}+(1-s)\boldsymbol{u}^{N,K,-})):\boldsymbol{\varepsilon}(\overline{\boldsymbol{u}}^{N,K}_{t})\,\mathrm{d}s\,\mathrm{d}x\,\mathrm{d}\tau.$

Using that $\boldsymbol{u}^{N}\in W^{2,\infty}(0,T;V_{N})$ and $v^{N}\in
W^{1,\infty}(0,T;H^{1}(\Omega))\cap L^{\infty}(Q)$, we see that

  * •

$\boldsymbol{u}^{N,K,\pm}\rightarrow\boldsymbol{u}^{N}$ strongly in
$C([0,T];V_{N})$;

  * •

$\overline{\boldsymbol{u}}^{N,K}\rightarrow\boldsymbol{u}^{N}$ strongly in
$W^{1,\infty}(0,T;V_{N})$;

  * •

$v^{N,K,+}\rightarrow v^{N}$ strongly in $L^{\infty}(0,T;L^{2}(\Omega)^{d})$.

Letting $K\rightarrow\infty$ in (2.33), it follows that

$\displaystyle\mathcal{E}(\boldsymbol{u}^{N}(t),v^{N}(t))-\mathcal{E}(\boldsymbol{u}_{0,N},v^{N}_{0})-\mathcal{H}(v^{N}_{0})+\mathcal{H}(v^{N}(t))$
$\displaystyle\geq\int_{0}^{t}\int_{\Omega}\frac{b(v^{N})}{\alpha}F^{-1}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N})):\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t})\,\mathrm{d}x\,\mathrm{d}s.$

Hence the energy-dissipation equality holds for a.e. $t\in[0,T]$. Using the
continuity of the solution couple, we conclude that the equality must hold for
every $t\in[0,T]$. ∎

###  2.3. $N$-independent bounds

In this section, we derive $N$-independent bounds on the solution couple
$(\boldsymbol{u}^{N},v^{N})$ constructed in Lemma 2.4. This allows us to take
$N\rightarrow\infty$ and obtain a weak energy solution of the original problem
(2.1)–(2.3). We reintroduce the stress tensor at this stage to simplify the
representation. Also, we use the Korn–Poincaré inequality (Theorem 1.1)
throughout this section. The first two results provide bounds on the
displacement $\boldsymbol{u}$ and stress tensor $\mathbf{T}$. In Lemma 2.7, we
get a bound on the phase-field function $v$. Following this, we differentiate
between the cases $p\in(1,2]$ and $p\in(2,\infty)$. When $p\in(1,2]$ we are
able to improve the bounds on the phase-field function. Indeed, we obtain
bounds on the time derivative in the spirit of of Lemma 2.2. We are unable to
obtain such a result when $p>2$ for technical reasons. Instead, we make use of
the monotonicity of the phase-field functions as a function of the time
variable. These two cases are discussed in Section 2.4 and 2.5, respectively.

######  Lemma 2.5.

Let the assumptions of Lemma 2.4 hold and let
$(\boldsymbol{u}^{N},\mathbf{T}^{N},v^{N})$ be the solution triple constructed
there, where we define
$\mathbf{T}^{N}:=F^{-1}(\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N}))$.
There exists a constant $C$, independent of $N$, such that

$\displaystyle\sup_{t\in[0,T]}\|\boldsymbol{u}^{N}(t)\|_{2}+\sup_{t\in[0,T]}\|\boldsymbol{u}^{N}(t)\|_{1,p^{\prime}}+\sup_{t\in[0,T]}\|\boldsymbol{u}^{N}_{t}(t)\|_{2}+\int_{0}^{T}\|\mathbf{T}^{N}(t)\|_{p}^{p}+\|\boldsymbol{u}^{N}_{t}(t)\|_{1,p^{\prime}}^{p^{\prime}}\,\mathrm{d}t\leq
C.$

###### Proof.

For the first, third and fourth bounds, we test against
$\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N}$ in the elastodynamic
equation (2.23). Using the lower bound for $b$, we can argue as in [9] and so
do not include the details here. Using the constitutive relation and the
resulting memory kernel property, for every $t\in[0,T]$, we have

$\displaystyle|\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}(t))|\leq\mathrm{e}^{-\alpha
t}|\boldsymbol{\varepsilon}(\boldsymbol{u}_{0,N})|+\int_{0}^{t}\mathrm{e}^{\alpha(s-t)}|\mathbf{T}^{N}|^{p-1}\,\mathrm{d}s.$

Taking the $L^{p^{\prime}}(\Omega)$ norm of both sides, using the properties
of the approximation of the initial data, and applying the Korn–Poincaré
inequality, the $L^{\infty}(0,T;W^{1,p^{\prime}}(\Omega)^{d})$ bound on
$(\boldsymbol{u}^{N})_{N}$ follows. Using the bound on $\mathbf{T}^{N}$ with
the constitutive relation, we see that
$(\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N}))_{N}$
is bounded in $L^{p^{\prime}}(Q)^{d\times d}$. Putting this together with the
bound on $\boldsymbol{u}^{N}$ and applying the Korn–Poincaré inequality, the
final bound on $\boldsymbol{u}^{N}_{t}$ follows. ∎

######  Lemma 2.6.

Let the assumptions of Lemma 2.4 hold and let
$(\boldsymbol{u}^{N},\mathbf{T}^{N},v^{N})$ be the solution triple constructed
there. There exists a constant $C$ independent of $N$ such that

$\displaystyle\int_{Q}|\boldsymbol{u}^{N}_{tt}|^{2}\,\mathrm{d}x\,\mathrm{d}t+\sup_{t\in[0,T]}\Big{\\{}\|\mathbf{T}^{N}(t)\|_{p}+\|\boldsymbol{u}^{N}_{t}(t)\|_{1,p^{\prime}}\Big{\\}}\leq
C.$

###### Proof.

It suffices to prove the claim for the first two terms, repeating the
reasoning from the proof of Lemma 2.5 in order to obtain the bound on the
final term. We test in the elastodynamic equation (2.23) against
$\boldsymbol{u}^{N}_{tt}+\alpha\boldsymbol{u}^{N}_{t}$ to get

$\displaystyle\int_{\Omega}|\boldsymbol{u}^{N}_{tt}|^{2}\,\mathrm{d}x+\frac{\mathrm{d}}{\mathrm{d}t}\Big{(}\frac{\alpha}{2}\|\boldsymbol{u}^{N}_{t}\|_{2}^{2}\Big{)}+\int_{\Omega}b(v^{N})\mathbf{T}^{N}:F(\mathbf{T}^{N})_{t}\,\mathrm{d}x$
$\displaystyle=\langle
l,\boldsymbol{u}^{N}_{tt}+\alpha\boldsymbol{u}^{N}_{t}\rangle$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}t}\Big{(}\langle
l,\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N}\rangle\Big{)}-\langle
l_{t},\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N}\rangle.$

The term involving the nonlinearity can be rewritten as

$\displaystyle\int_{\Omega}b(v^{N})\mathbf{T}^{N}:F(\mathbf{T}^{N})_{t}\,\mathrm{d}x$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}t}\Big{(}\int_{\Omega}b(v^{N})\frac{p-1}{p}|\mathbf{T}^{N}|^{p}\,\mathrm{d}x\Big{)}-\int_{\Omega}b(v^{N})v^{N}_{t}\Big{[}\frac{p-1}{p}|\mathbf{T}^{N}|^{p}\Big{]}\,\mathrm{d}x$
$\displaystyle\geq\frac{\mathrm{d}}{\mathrm{d}t}\Big{(}\int_{\Omega}b(v^{N})\frac{p-1}{p}|\mathbf{T}^{N}|^{p}\,\mathrm{d}x\Big{)},$

recalling the non-positivity of $v^{N}_{t}$. Integrating over $(0,t)$ for an
arbitrary $t\in(0,T]$, we deduce that

(2.34)
$\displaystyle\int_{0}^{t}\int_{\Omega}|\boldsymbol{u}^{N}_{tt}|^{2}\,\mathrm{d}x\,\mathrm{d}s+\int_{\Omega}b(v^{N}(t))\frac{p-1}{p}|\mathbf{T}^{N}(t)|^{p}\,\mathrm{d}x$
$\displaystyle\leq\frac{\alpha}{2}\|\boldsymbol{u}_{1,N}\|_{2}^{2}+\int_{\Omega}b(v^{N}_{0})\frac{p-1}{p}|\mathbf{T}^{N}(0)|^{p}\,\mathrm{d}x+\langle
l(t),(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N})(t)\rangle$
$\displaystyle\quad-\langle
l(0),\boldsymbol{u}_{1,N}+\alpha\boldsymbol{u}_{0,N}\rangle-\int_{0}^{t}\langle
l_{t},\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N}\rangle\,\mathrm{d}s$
$\displaystyle\leq
C\Big{[}\|\boldsymbol{u}_{1,N}\|_{2}^{2}+\int_{\Omega}|F^{-1}(\boldsymbol{\varepsilon}(\boldsymbol{u}_{1,N}+\alpha\boldsymbol{u}_{0,N}))|^{p}\,\mathrm{d}x+\|l(t)\|_{-1,p}\|\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N})(t)\|_{p^{\prime}}$
$\displaystyle\quad+\|l(0)\|_{-1,p}\|\boldsymbol{\varepsilon}(\boldsymbol{u}_{1,N}+\alpha\boldsymbol{u}_{0,N})\|_{p^{\prime}}+\int_{0}^{t}\|l_{t}\|_{-1,p}\|\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N})\|_{p^{\prime}}\,\mathrm{d}s\Big{]}$
$\displaystyle\leq
C\Big{[}\|\boldsymbol{u}_{1,N}\|_{2}^{2}+\|\boldsymbol{\varepsilon}(\boldsymbol{u}_{1,N}+\alpha\boldsymbol{u}_{0,N})\|_{p^{\prime}}^{p^{\prime}}+\|l\|_{W^{1,\infty}(W^{-1,p})}\|\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N})(t)\|_{p^{\prime}}$
$\displaystyle\quad+\|l\|_{W^{1,\infty}(W^{-1,p})}\|\boldsymbol{\varepsilon}(\boldsymbol{u}_{1,N}+\alpha\boldsymbol{u}_{0,N})\|_{p^{\prime}}+\int_{0}^{t}\|l_{t}\|_{-1,p}^{p}+\|\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N})\|_{p^{\prime}}^{p^{\prime}}\,\mathrm{d}s\Big{]}.$

However, we have that

$\displaystyle|\mathbf{T}^{N}(t)|^{p}=|F^{-1}(\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N})(t))|^{p}=|\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N})|^{p^{\prime}}.$

Using this on the right-hand side of (2.34) and applying Young’s inequality,
we deduce that

$\displaystyle\int_{0}^{t}\int_{\Omega}|\boldsymbol{u}^{N}_{tt}|^{2}\,\mathrm{d}x\,\mathrm{d}s+\int_{\Omega}|\mathbf{T}^{N}(t)|^{p}+|\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N})(t)|^{p^{\prime}}\,\mathrm{d}x$
$\displaystyle\leq
C\Big{[}\|\boldsymbol{u}_{1,N}\|_{2}^{2}+\|\boldsymbol{\varepsilon}(\boldsymbol{u}_{1,N}+\alpha\boldsymbol{u}_{0,N})\|_{p^{\prime}}^{p^{\prime}}+\|l\|_{W^{1,\infty}(W^{-1,p})}^{p}+\int_{0}^{t}\|l_{t}\|_{-1,p}^{p}+\|\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{n})\|_{p^{\prime}}^{p^{\prime}}\,\mathrm{d}s\Big{]}$

Using Lemma 2.5 and the properties of the approximations of the initial data
(in particular, the boundedness in both $L^{2}(\Omega)^{d}$ and
$W^{1,p^{\prime}}_{D}(\Omega)^{d}$ irrespective of the value of $p$), the
right-hand side can be bounded above independent of $N$ so the required result
follows. ∎

######  Lemma 2.7.

Let the assumptions of Lemma 2.4 hold. Let
$(\boldsymbol{u}^{N},\mathbf{T}^{N},v^{N})$ be the solution triple constructed
there. Suppose additionally that the compatibility condition (2.4) holds.
There exists a constant $C$, independent of $N$, such that

$\displaystyle\sup_{t\in[0,T]}\|v^{N}(t)\|_{1,2}\leq C.$

###### Proof.

By the energy-dissipation equality, we have

$\displaystyle\mathcal{H}(v^{N}(t))$
$\displaystyle\leq\mathcal{F}(t;\boldsymbol{u}^{N}(t),\boldsymbol{u}^{N}_{t}(t),v^{N}(t))+\langle
l(t),\boldsymbol{u}^{N}(t)\rangle$
$\displaystyle\quad+\int_{0}^{t}\int_{\Omega}b(v^{N})\big{[}F^{-1}(\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N}))-F^{-1}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N}))\big{]}:\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t})\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle=\langle l(t),\boldsymbol{u}^{N}(t)\rangle-\int_{0}^{t}\langle
l_{t},\boldsymbol{u}^{N}\rangle\,\mathrm{d}s+\mathcal{F}(0;\boldsymbol{u}_{0,N},\boldsymbol{u}_{1,N},v^{N}_{0})$
$\displaystyle\leq
C\Big{[}\|l(t)\|_{-1,p}\|\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}(t))\|_{p^{\prime}}+\|\boldsymbol{u}_{1,N}\|_{2}^{2}+\|l(0)\|_{-1,p}\|\boldsymbol{\varepsilon}(\boldsymbol{u}_{0,N})\|_{p^{\prime}}$
$\displaystyle\quad+\int_{\Omega}b(v^{N}_{0})\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}_{0,N})))\,\mathrm{d}x+\|v^{N}_{0}\|_{1,2}^{2}+\int_{0}^{t}\|l_{t}(s)\|_{-1,p}\|\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}(s))\|_{p^{\prime}}\,\mathrm{d}s\Big{]}.$

Using the boundedness of $(v^{N}_{0})_{N}$ in $H^{1}_{D}(\Omega)$ and
$(\boldsymbol{u}_{0,N})_{N}$, $(\boldsymbol{u}_{1,N})_{N}$ in
$L^{2}(\Omega)^{d}\cap W^{1,p^{\prime}}_{D}(\Omega)^{d}$, alongside the bounds
in Lemma 2.5, the right-hand side can be bounded above independent of $N$ and
$t\in[0,T]$. Taking the supremum with respect to the time variable, we use the
definition of the functional $\mathcal{H}$ to yield the stated result. ∎

###  2.4. Limit in the case $p\in(1,2]$

To replicate the ideas of Lemma 2.2, we first need to show that testing in the
variational form of the minimisation problem against $v^{N}_{t}$ causes the
right-hand side of (2.27) to vanish. Namely, we need to prove the time
continuous analogue of Corollary 2.1.

######  Lemma 2.8.

Let the assumptions of Lemma 2.4 hold and let
$(\boldsymbol{u}^{N},\mathbf{T}^{N},v^{N})$ be the solution triple constructed
there. For a.e. $t\in(0,T)$, we have

(2.35)
$\big{[}\partial_{v}\mathcal{E}(\boldsymbol{u}^{N}(t),v^{N}(t))+\mathcal{H}^{\prime}(v^{N}(t))\big{]}(v^{N}_{t}(t))=0.$

###### Proof.

Differentiating the energy-dissipation equality from Proposition 2.6 with
respect to the time variable, we obtain

$\displaystyle 0$
$\displaystyle=\int_{\Omega}\boldsymbol{u}_{tt}^{N}\cdot\boldsymbol{u}_{t}^{N}\,\mathrm{d}x+\int_{\Omega}\frac{1}{2\epsilon}(v^{N}(t)-1)v^{N}_{t}(t)+2\epsilon\nabla
v^{N}(t)\cdot\nabla v^{N}_{t}(t)\,\mathrm{d}x$
$\displaystyle\quad+\int_{\Omega}\frac{b^{\prime}(v^{N})}{\alpha}v^{N}_{t}\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N}))+b(v^{N})F^{-1}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N})):\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t})\,\mathrm{d}x-\langle
l_{t},\boldsymbol{u}^{N}\rangle-\langle l,\boldsymbol{u}^{N}_{t}\rangle$
$\displaystyle\quad+\int_{\Omega}b(v^{N})\big{[}F^{-1}(\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N}))-F^{-1}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N}))\big{]}:\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t})\,\mathrm{d}x+\langle
l_{t},\boldsymbol{u}^{N}\rangle$
$\displaystyle=\Big{[}\int_{\Omega}\boldsymbol{u}_{tt}^{N}\cdot\boldsymbol{u}_{t}^{N}+F^{-1}(\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N})):\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t})\,\mathrm{d}x-\langle
l,\boldsymbol{u}^{N}_{t}\rangle\Big{]}$
$\displaystyle\quad+\Big{[}\int_{\Omega}\frac{b^{\prime}(v^{N})}{\alpha}v^{N}_{t}\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N}))+\frac{1}{2\epsilon}(v^{N}(t)-1)v^{N}_{t}(t)+2\epsilon\nabla
v^{N}(t)\cdot\nabla v^{N}_{t}(t)\,\mathrm{d}x\Big{]}.$

The terms in the first set of square brackets on the right-hand side vanishes
for a.e. $t\in(0,T)$, considering the elastodynamic equation with the test
function $\boldsymbol{u}^{N}_{t}$. Hence the terms in the second set of square
brackets must also vanish. However, this is exactly the left-hand side of
(2.35) and so we conclude the required result. ∎

######  Lemma 2.9.

Let the assumptions of Lemma 2.4 hold and let
$(\boldsymbol{u}^{N},\mathbf{T}^{N},v^{N})$ be the solution triple constructed
there. Suppose additionally that the compatibility condition (2.4) holds.
There exists a constant $C$, independent of $N$, such that

$\displaystyle\sup_{t\in[0,T]}\|v^{N}_{t}(t)\|_{1,2}\leq C.$

###### Proof.

Let $t\in(0,T]$ and fix $h>0$ sufficiently small such that $t-h>0$.
Subtracting (2.35), evaluated at time $t$, from the inequality

$\displaystyle
0\leq\big{[}\partial_{v}\mathcal{E}(\boldsymbol{u}^{N}(t-h),v^{N}(t-h))+\mathcal{H}^{\prime}(v^{N}(t-h))\big{]}(v^{N}_{t}(t)),$

which is a consequence of the proof of Proposition 2.5, we deduce that

$\displaystyle 0$
$\displaystyle\leq\int_{\Omega}\Big{[}\frac{b^{\prime}(v^{N}(t-h))}{\alpha}\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N}(t-h)))-\frac{b^{\prime}(v^{N}(t))}{\alpha}\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N}(t)))\Big{]}v^{N}_{t}(t)\,\mathrm{d}x$
$\displaystyle\quad+\frac{1}{2\epsilon}\int_{\Omega}\big{[}(v^{N}(t-h)-1)-(v^{N}(t)-1)\big{]}v^{N}_{t}(t)\,\mathrm{d}x$
$\displaystyle\quad+2\epsilon\int_{\Omega}\nabla\big{(}v^{N}(t-h)-v^{N}(t)\big{)}\cdot\nabla
v^{N}(t)\,\mathrm{d}x.$

Using the fundamental theorem of calculus in the first integral on the right-
hand side and dividing through by $h$, it follows that

(2.36)
$\displaystyle\frac{1}{2\epsilon}\int_{\Omega}\frac{v^{N}(t)-v^{N}(t-h)}{h}v^{N}_{t}(t)\,\mathrm{d}x+2\epsilon\int_{\Omega}\nabla\Big{(}\frac{v^{N}(t)-v^{N}(t-h)}{h}\Big{)}\cdot\nabla
v^{N}_{t}(t)\,\mathrm{d}x$
$\displaystyle\quad+\frac{2}{\alpha}\int_{\Omega}\frac{v^{N}(t)-v^{N}(t-h)}{h}v^{N}_{t}(t)\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N}(t-h)))\,\mathrm{d}x$
$\displaystyle\leq\int_{\Omega}\int_{0}^{1}\Big{\\{}\frac{b^{\prime}(v^{N}(t))}{\alpha}v^{N}_{t}(t)F^{-1}(\alpha\boldsymbol{\varepsilon}(s\boldsymbol{u}^{N}(t-h)+(1-s)\boldsymbol{u}^{N}(t)))$
$\displaystyle\quad:\boldsymbol{\varepsilon}\Big{(}\frac{\boldsymbol{u}^{N}(t-h)-\boldsymbol{u}^{N}(t)}{h}\Big{)}\Big{\\}}\,\mathrm{d}s\,\mathrm{d}x.$

Recalling that $\boldsymbol{u}^{N}\in C^{1}([0,T];V_{N})$ and $v^{N}\in
W^{1,2}(0,T;H^{1}(\Omega))$, taking the limit as $h\rightarrow 0+$ in (2.36)
yields the following, for a.e. $t\in(0,T)$:

(2.37) $\displaystyle\frac{1}{2\epsilon}\|v^{N}(t)\|_{2}^{2}+2\epsilon\|\nabla
v^{N}_{t}(t)\|_{2}^{2}+\frac{2}{\alpha}\int_{\Omega}|v^{N}_{t}(t)|^{2}\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N}(t)))\,\mathrm{d}x$
$\displaystyle\leq\int_{\Omega}\frac{b^{\prime}(1)}{\alpha}|v^{N}_{t}(t)||F^{-1}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N}(t)))||\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}(t))|\,\mathrm{d}x$
$\displaystyle=\int_{\Omega}\frac{b^{\prime}(1)}{\alpha}|v^{N}_{t}(t)||\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N}(t))|^{p^{\prime}-1}|\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}(t))|\,\mathrm{d}x$
$\displaystyle=\int_{\Omega}\frac{b^{\prime}(1)}{\alpha}\Big{(}|v^{N}_{t}(t)||\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N}(t))|^{\frac{p^{\prime}}{2}}\Big{)}\cdot\Big{(}|\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N}(t))|^{\frac{p^{\prime}-2}{2}}|\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}(t))|\Big{)}\,\mathrm{d}x$

Applying Hölder’s inequality to the final line of (2.37) and absorbing one of
the resulting terms into the left-hand side, we deduce that

$\displaystyle\frac{1}{2\epsilon}\|v^{N}(t)\|_{2}^{2}+2\epsilon\|\nabla
v^{N}_{t}(t)\|_{2}^{2}+\frac{1}{\alpha}\int_{\Omega}|v^{N}_{t}(t)|^{2}\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N}(t)))\,\mathrm{d}x$
$\displaystyle\leq
C\int_{\Omega}|\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}(t))|^{p^{\prime}-2}|\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}(t))|^{2}\,\mathrm{d}x.$

If $p=2$, we are done by the bound in Lemma 2.6. Otherwise, applying Hölder’s
inequality again but with parameters $q=\frac{p^{\prime}}{p^{\prime}-2}$ and
$q^{\prime}=\frac{p^{\prime}}{2}$, valid choices because $p^{\prime}>2$, we
obtain

(2.38) $\displaystyle\frac{1}{2\epsilon}\|v^{N}(t)\|_{2}^{2}+2\epsilon\|\nabla
v^{N}_{t}(t)\|_{2}^{2}+\frac{1}{\alpha}\int_{\Omega}|v^{N}_{t}(t)|^{2}\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N}(t)))\,\mathrm{d}x$
$\displaystyle\leq
C\|\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}(t))\|_{p^{\prime}}^{p^{\prime}-2}\|\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}(t))\|_{p^{\prime}}^{2}.$

By Lemma 2.6, the right-hand side of (2.38) is uniformly bounded above,
independent of $t$ and $N$, for a.e. $t\in(0,T)$. Hence the stated result
follows. ∎

######  Theorem 2.4.

Let $p\in(1,2]$. Suppose that $l\in C^{1}([0,T];W^{-1,p}_{D}(\Omega)^{d})$,
$\boldsymbol{u}_{0}$, $\boldsymbol{u}_{1}\in W^{1,p^{\prime}}(\Omega)^{d}$ and
$v_{0}\in H^{1}_{D+1}(\Omega)$ with $v_{0}\in[0,1]$ a.e. in $\Omega$, such
that the compatibility condition (2.4) holds. There exists a triple
$(\boldsymbol{u},\mathbf{T},v)$ that is a weak energy solution of (2.1)–(2.3)
in the sense of Theorem 2.1. Furthermore, there exists a subsequence in $N$,
not relabelled, such that if $(\boldsymbol{u}^{N},\mathbf{T}^{N},v^{N})$ is
the solution triple from Lemma 2.4, we have

  * •

$\boldsymbol{u}^{N}\rightharpoonup\boldsymbol{u}$ weakly in
$W^{2,2}(0,T;L^{2}(\Omega)^{d})$ and weakly-* in
$W^{1,\infty}(0,T;W^{1,p^{\prime}}(\Omega)^{d})$,

  * •

$\mathbf{T}^{N}\overset{\ast}{\rightharpoonup}\mathbf{T}$ weakly-* in
$L^{\infty}(0,T;L^{p}(\Omega)^{d\times d})$, and

  * •

$v^{N}\overset{\ast}{\rightharpoonup}v$ weakly-* in
$W^{1,\infty}(0,T;H^{1}(\Omega))$.

###### Proof.

The convergence results follow easily from Lemma 2.5, Lemma 2.6, Lemma 2.7,
Lemma 2.9 and standard compactness results for Bochner spaces. By the
Aubin–Lions lemma, we see that $\boldsymbol{u}^{N}\rightarrow\boldsymbol{u}$
strongly in $C^{1}([0,T];L^{p^{\prime}}(\Omega)^{d})$, and $v^{N}\rightarrow
v$ strongly in $C([0,T];L^{2}(\Omega))$, up to a possible further subsequence
that we do not relabel.

From the elastodynamic equation (2.23), for every $\boldsymbol{w}\in
C^{\infty}_{D}(\overline{\Omega})^{d}$ and every $\psi\in C([0,T])$, we have

$\displaystyle\int_{Q}\boldsymbol{u}_{tt}^{N}\cdot\big{(}\psi
P_{N}\boldsymbol{w}\big{)}+b(v^{N})\mathbf{T}^{N}:\boldsymbol{\varepsilon}(\psi
P_{N}\boldsymbol{w})\,\mathrm{d}x\,\mathrm{d}t=\int_{0}^{T}\langle l,\psi
P_{N}\boldsymbol{w}\rangle\,\mathrm{d}t.$

Using the weak convergence results and the strong convergence of
$(P_{N}\boldsymbol{w})_{N}$ in in $W^{k,2}_{D}(\Omega)^{d}$ to
$\boldsymbol{w}$ as $N\rightarrow\infty$, it follows that

$\displaystyle\int_{Q}\boldsymbol{u}_{tt}\cdot\big{(}\psi\boldsymbol{w}\big{)}+b(v)\mathbf{T}:\boldsymbol{\varepsilon}(\psi\boldsymbol{w})\,\mathrm{d}x\,\mathrm{d}t=\int_{0}^{T}\psi\langle
l,\boldsymbol{w}\rangle\,\mathrm{d}t.$

By Lebesgue’s differentiation theorem and a standard density argument, the
required elastodynamic equation follows for any test function
$\boldsymbol{w}\in W^{1,p^{\prime}}_{D}(\Omega)^{d}$. The satisfaction of the
initial conditions follows from the usual arguments, using the approximation
properties of the data $(\boldsymbol{u}_{0,N})_{N}$,
$(\boldsymbol{u}_{1,N})_{N}$ and $(v^{N}_{0})_{N}$.

To show that the constitutive relationship holds, we use a variant of Minty’s
method. First, we notice that

$\displaystyle\lim_{N\rightarrow\infty}\int_{Q}b(v^{N})\mathbf{T}^{N}:\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N})\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle=\lim_{N\rightarrow\infty}\int_{0}^{T}\Big{\\{}\langle
l,\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N}\rangle-\int_{\Omega}\boldsymbol{u}_{tt}^{N}\cdot(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N})\,\mathrm{d}x\Big{\\}}\,\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\Big{\\{}\langle
l,\boldsymbol{u}_{t}+\alpha\boldsymbol{u}\rangle-\int_{\Omega}\boldsymbol{u}_{tt}\cdot(\boldsymbol{u}_{t}+\alpha\boldsymbol{u})\,\mathrm{d}x\Big{\\}}\,\mathrm{d}t$
$\displaystyle=\int_{Q}b(v)\mathbf{T}:\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}+\alpha\boldsymbol{u})\,\mathrm{d}x\,\mathrm{d}t,$

using the strong convergence results in the transition to the second line. For
an arbitrary test function $\mathbf{S}\in L^{p}(Q)^{d\times d}$, it follows
that

$\displaystyle 0$
$\displaystyle\leq\lim_{N\rightarrow\infty}\int_{Q}b(v^{N})(\mathbf{T}^{N}-\mathbf{S}):(F(\mathbf{T}^{N})-F(\mathbf{S}))\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle=\int_{Q}b(v)(\mathbf{T}-\mathbf{S}):(\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}+\alpha\boldsymbol{u})-F(\mathbf{S}))\,\mathrm{d}x\,\mathrm{d}t.$

Replacing $\mathbf{S}$ with $\mathbf{T}\pm\gamma\mathbf{U}$ where $\gamma>0$
and $\mathbf{U}\in L^{\infty}(Q)^{d\times d}$, we get

$\displaystyle
0\leq\mp\int_{Q}b(v)\mathbf{U}:(\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}+\alpha\boldsymbol{u})-F(\mathbf{T}))\,\mathrm{d}x\,\mathrm{d}t,$

taking the limit as $\gamma\rightarrow 0+$. Choosing
$\mathbf{U}=\frac{\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}+\alpha\boldsymbol{u})-F(\mathbf{T})}{1+|\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}+\alpha\boldsymbol{u})-F(\mathbf{T})|}$
and recalling that $b$ is strictly positive, it follows that
$\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}+\alpha\boldsymbol{u})=F(\mathbf{T})$
pointwise a.e. in $Q$. Also, we deduce that

(2.39)
$\lim_{N\rightarrow\infty}\int_{Q}b(v^{N})|\mathbf{T}^{N}|^{p}\,\mathrm{d}x\,\mathrm{d}t=\int_{Q}b(v)|\mathbf{T}|^{p}\,\mathrm{d}x\,\mathrm{d}t.$

However, the sequence $(b(v^{N})^{\frac{1}{p}}\mathbf{T}^{N})_{N}$ converges
weakly in $L^{p}(Q)^{d\times d}$. The weak convergence with convergence of
norms (2.39) implies that $(b(v^{N})^{\frac{1}{p}}\mathbf{T}^{N})_{N}$
converges strongly in $L^{p}(Q)^{d\times d}$. Thus
$\mathbf{T}^{N}\rightarrow\mathbf{T}$ strongly in $L^{p}(Q)^{d\times d}$,
which in turn implies that
$(\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}+\alpha\boldsymbol{u}^{N}))_{N}$
converges strongly in $L^{p^{\prime}}(Q)^{d\times d}$ to
$\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}+\alpha\boldsymbol{u})$. Reasoning
as before, it follows that
$(\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}))_{N}$ converges strongly to
$\boldsymbol{\varepsilon}(\boldsymbol{u})$ in
$L^{\infty}(0,T;L^{p^{\prime}}(\Omega)^{d})$, and so, we deduce the pointwise
convergence of $(\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}))_{N}$ a.e. in
$Q$, up to a further subsequence in $N$. Pointwise convergence and strong
convergence in $L^{p^{\prime}}(Q)$ of
$(\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}))_{N}$ to
$\boldsymbol{\varepsilon}(\boldsymbol{u}_{t})$ then follows. Applying Fatou’s
lemma with the weak lower semi-continuity of norms to the energy-dissipation
equality for $(\boldsymbol{u}^{N},\mathbf{T}^{N},v^{N})$, we get

$\displaystyle\mathcal{F}(t;\boldsymbol{u}(t),\boldsymbol{u}_{t}(t),v(t))+\int_{0}^{t}\int_{\Omega}b(v)\big{[}\mathbf{T}-F^{-1}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}))\big{]}:\boldsymbol{\varepsilon}(\boldsymbol{u}_{t})\,\mathrm{d}x\,\mathrm{d}s+\int_{0}^{t}\langle
l_{t}(s),\boldsymbol{u}(s)\rangle\,\mathrm{d}s$
$\displaystyle\leq\liminf_{N\rightarrow\infty}\Big{\\{}\mathcal{F}(t;\boldsymbol{u}^{N}(t),\boldsymbol{u}^{N}_{t}(t),v^{N}(t))+\int_{0}^{t}\int_{\Omega}b(v^{N})\big{[}\mathbf{T}^{N}-F^{-1}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N}))\big{]}:\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}^{N})\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\quad+\int_{0}^{t}\langle
l_{t}(s),\boldsymbol{u}^{N}(s)\rangle\,\mathrm{d}s\Big{\\}}$
$\displaystyle=\liminf_{N\rightarrow\infty}\mathcal{F}(0;\boldsymbol{u}_{0,N},\boldsymbol{u}_{1,N},v^{N}_{0})$
$\displaystyle=\mathcal{F}(0;\boldsymbol{u}_{0},\boldsymbol{u}_{1},v_{0}).$

The final line follows from the choice of the approximations
$(\boldsymbol{u}_{0,N})_{N}$ and $(\boldsymbol{u}_{1,N})_{N}$, as well as
Proposition 2.1. For the opposite inequality, we reason as in the proof of
Proposition 2.6. To do so, we only need to prove that the limiting triple
satisfies the minimisation problem (2.2). For this, it suffices to show that

(2.40)
$0\leq\big{[}\partial_{v}\mathcal{E}(\boldsymbol{u}(t),v(t))+\mathcal{H}^{\prime}(v(t))\big{]}(\chi),$

for every $\chi\in H^{1}_{D}(\Omega)$. Using the fact that
$\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N}))\rightarrow\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}))$
strongly in $L^{1}(Q)$, by the strong convergence result for
$(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{N}))_{N}$, we obtain (2.40)
easily from taking the limit in (2.27). Hence we conclude that
$(\boldsymbol{u},\mathbf{T},v)$ is a solution in the required sense because we
have the weak elastodynamic equation, the minimisation problem, the energy-
dissipation equality and the satisfaction of the initial data. ∎

###  2.5. Limit in the case $p\in(2,\infty)$

In this case, we have $p^{\prime}\in(1,2)$. We cannot obtain a bound on
$(v^{N}_{t})_{N}$ by repeating the reasoning from Section 2.3. In particular,
the proof of Lemma 2.9 fails. Hence, at this stage $(v^{N})_{N}$ is at best
bounded in $L^{\infty}(0,T;H^{1}(\Omega))$ which implies that $(v^{N})_{N}$
converges weakly-* in $L^{\infty}(0,T;H^{1}(\Omega))$. However, we must find
the limit of the nonlinear term $(b(v^{N}))_{N}$ to take the limit in the
elastodynamic equation (2.23). Hence we need at least a pointwise convergence
result for the sequence $(v^{N})_{N}$. To obtain such a result, we make use of
the monotonicity of these functions with respect to the time variable. Indeed
the key is the following result from [12]. We combine this result with the
Rellich–Kondrachov compactness theorem in order to obtain a suitable strong
convergence result as stated in Corollary 2.2. Next we prove a continuity
result for the limiting function constructed in Corollary 2.2. This is vital
to prove one direction of the energy-dissipation inequality. Then we prove the
existence result for parameter values $p>2$.

######  Theorem 2.5.

Let $[a,b]\subset\mathbb{R}$ and $v^{m}:[a,b]\rightarrow L^{2}(\Omega)$,
$m\in\mathbb{N}$, a sequence of functions such that $v^{m}(s)\leq v^{m}(t)$ in
$\Omega$ for every $a\leq s\leq t\leq b$ and $m\in\mathbb{N}$. Suppose also
that there exists a constant $C$, independent of $m$, such that
$\|v^{m}(t)\|_{2}\leq C$ for every $t\in[a,b]$ and $m\in\mathbb{N}$. There
exists a subsequence in $m$, not relabelled, and a function
$v:[a,b]\rightarrow L^{2}(\Omega)$ such that, for every $t\in[a,b]$, we have
$v^{m}(t)\rightharpoonup v(t)$ in $L^{2}(\Omega)$ as $m\rightarrow\infty$.
Moreover, $\|v(t)\|_{2}\leq C$ for every $t\in[0,T]$ and $v(s)\leq v(t)$ in
$\Omega$, for every $a\leq s\leq t\leq b$.

######  Corollary 2.2.

Let $(v^{m})_{m\geq 1}$ be a sequence of functions defined on $[a,b]$ with
values in $H^{1}(\Omega)$ such that $v^{m}(s)\leq v^{m}(t)$ in $\Omega$ for
every $a\leq s\leq t\leq b$ and $m\in\mathbb{N}$. Suppose also that there
exists a constant $C$ independent of $m$ such that $\|v^{m}(t)\|_{1,2}\leq C$
for every $t\in[a,b]$ and $m\in\mathbb{N}$. There exists a subsequence in $m$,
not relabelled, and a function $v:[a,b]\rightarrow H^{1}(\Omega)$ such that,
for every $t\in[a,b]$, $v^{m}(t)\rightharpoonup v(t)$ weakly in
$H^{1}(\Omega)$ and strongly in $L^{2}(\Omega)$. Furthermore,
$\|v(t)\|_{1,2}\leq C$ for every $t\in[a,b]$ and $v(s)\leq v(t)$ in $\Omega$
for every $a\leq s\leq t\leq b$.

######  Proposition 2.7.

Let $(v^{m})_{m}$ be the subsequence constructed in Corollary 2.2, with
limiting function $v$. Then $v$ is almost everywhere continuous as a map from
$[a,b]$ into $L^{2}(\Omega)$.

###### Proof.

Fix a time $t\in(a,b)$ and, for $i\in\\{1,2\\}$, let $(\delta_{n,i})_{n\geq
1}$ be a sequence in $(0,\infty)$ such that $t+\delta_{n,i}\in(a,b]$ for every
$n$ and $\delta_{n,i}\rightarrow 0+$ as $n\rightarrow\infty$. The sequence
$(v(t+\delta_{n,i}))_{n}$ is bounded in $H^{1}(\Omega)$. Hence, up to a
subsequence that we do not relabel, we have that $(v(t+\delta_{n,i}))_{n}$
converges weakly in $H^{1}(\Omega)$ and strongly in $L^{2}(\Omega)$ to a limit
that we denote by $\tilde{w}_{i}$. Using the strong convergence result and the
monotonicity assumption, for every $n$, we have

$\displaystyle\|v(t+\delta_{n,1})\|_{2}\geq\lim_{m\rightarrow\infty}\|v(t+\delta_{m,2})\|_{2}=\|\tilde{w}_{2}\|_{2}.$

Letting $n\rightarrow\infty$, we deduce that
$\|\tilde{w}_{1}\|_{2}\geq\|\tilde{w}_{2}\|_{2}$. By symmetry, we deduce that
$\|\tilde{w}_{1}\|_{2}=\|\tilde{w}_{2}\|_{2}$. We conclude that every
subsequence of $(v(t+\delta))_{\delta>0}$ converges to the same limit as
$\delta\rightarrow 0+$. Hence the upper limit $v(t+)$ exists for every
$t\in(a,b)$. By analogous reasoning, the lower limit $v(t-)$ also exists.

By monotonicity, the function $v$ is discontinuous at the point $t$ in the
case that the upper and lower limits do not coincide. Indeed, we may write

$\displaystyle\\{t\in(a,b)\,:\,v\text{ is not continuous at }t\\}$
$\displaystyle=\cup_{\epsilon>0}\\{t\in(a,b)\,:\,\|v(t+)-v(t-)\|_{2}>\epsilon\\}$
$\displaystyle=\cup_{n\geq
1}\Big{\\{}t\in(a,b)\,:\,\|v(t+)-v(t-)\|_{2}>\frac{1}{n}\Big{\\}}.$

By monotonicity, we know that

$\displaystyle\|v(a)\|_{2}\leq\|v(t-)\|_{2}\leq\|v(t+)\|_{2}\leq\|v(b)\|_{2}.$

Hence, for every $n$, the set
$\\{t\in(a,b)\,:\,\|v(t+)-v(t-)\|_{2}>\frac{1}{n}\\}$ must be finite. Hence
the set of discontinuity points is countable and consequently $v$ is
continuous almost everywhere on $(a,b)$ as a map into $L^{2}(\Omega)$. ∎

######  Theorem 2.6.

Let $p\in(2,\infty)$. Suppose $l\in C^{1}([0,T];W^{-1,p}_{D}(\Omega)^{d})$,
$\boldsymbol{u}_{0}$, $\boldsymbol{u}_{1}\in
W^{1,p^{\prime}}_{D}(\Omega)^{d}\cap L^{2}(\Omega)^{d}$ and $v_{0}\in
H^{1}_{D+1}(\Omega)$ with $v_{0}\in[0,1]$ a.e. in $\Omega$, such that the
compatibility condition (2.4) holds. There exists a weak energy solution
$(\boldsymbol{u},\mathbf{T},v)$ of (2.1)–(2.3) in the sense of Theorem 2.2.
Furthermore, there exists a subsequence in $N$, not relabelled, such that if
$(\boldsymbol{u}^{N},\mathbf{T}^{N},v^{N})$ is the solution triple from Lemma
2.4, the following convergence results hold:

  * •

$\boldsymbol{u}^{N}\rightharpoonup\boldsymbol{u}$ weakly in
$W^{2,2}(0,T;L^{2}(\Omega)^{d})$ and weakly-* in
$W^{1,\infty}(0,T;W^{1,p^{\prime}}(\Omega)^{d})$;

  * •

$\mathbf{T}^{N}\overset{\ast}{\rightharpoonup}\mathbf{T}$ weakly-* in
$L^{\infty}(0,T;L^{p}(\Omega)^{d\times d})$;

  * •

$v^{N}(t)\rightarrow v(t)$ strongly in $L^{2}(\Omega)$ and weakly in
$H^{1}(\Omega)$ for every $t\in[0,T]$.

###### Proof.

Using the bounds in Section 2.3 with Corollary 2.2, the convergence results
are immediate. Considering the initial data for the approximate solution and
the corresponding convergence results, the claim for the initial conditions
follows. Combining the strong convergence result from Corollary 2.2 with the
uniform bound on $(v^{N})_{N}$ and Lebesgue’s dominated convergence theorem,
we deduce that $v^{N}\rightarrow v$ strongly in $L^{q}(Q)^{d\times d}$, for
every $q\in[1,\infty)$. Hence, reasoning as before, the elastodynamic equation
must be satisfied by the limiting triple. Using Minty’s method, it follows
that the constitutive relation holds. The strong convergence of
$(\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}))_{N}$ in
$L^{\infty}(0,T;L^{p^{\prime}}(\Omega)^{d\times d})$ and of
$(\boldsymbol{\varepsilon}(\boldsymbol{u}^{N}_{t}))_{N}$ in
$L^{p^{\prime}}(Q)^{d\times d}$ follows. From this, we get the “$\leq$”
direction of the energy-dissipation equality. Satisfaction of the minimisation
problem can be deduced exactly as in the proof of Theorem 2.4.

For the opposite direction “$\geq$” of the energy-dissipation equality, we can
repeat the reasoning from the proof of Proposition 2.6 and use the continuity
that follows from Proposition 2.7. Indeed, fix a point $t\in(0,T]$ such that
it is a point of continuity of $v$. Let $\mathcal{N}$ denote a set of null
measure such that $v$ is continuous on $[0,T]\setminus\mathcal{N}$.

Fix $K\in\mathbb{N}_{\geq 2}$ and define a time step $h_{K}=\frac{t}{K}$. For
each $0\leq k\leq K-1$, we fix a time point close to $hk$ at which $v$ is
continuous. Namely, we pick a
$t^{K}_{k}\in(hk-\frac{h_{K}}{4},hk+\frac{h_{K}}{4})\cap\big{(}(0,T)\setminus\mathcal{N}\big{)}$
such that $v$ is continuous at $t^{K}_{k}$ as a map into $L^{2}(\Omega)$.
Next, we define a sequence of approximations

$\displaystyle\boldsymbol{u}^{K}_{k}:=\boldsymbol{u}(t^{K}_{k}),\quad\mathbf{T}^{K}_{k}:=\mathbf{T}(t^{K}_{k}),\quad
v^{K}_{k}:=v(t^{K}_{k})\quad\text{ for }0\leq k\leq K.$

Then we have

$\displaystyle\mathcal{E}(\boldsymbol{u}(t),v(t))-\mathcal{E}(\boldsymbol{u}_{0},v_{0})=\sum_{k=1}^{K}\mathcal{E}(\boldsymbol{u}^{K}_{k},v^{K}_{k})-\mathcal{E}(\boldsymbol{u}^{K}_{k-1},v^{K}_{k-1}).$

Using the minimisation problem, we see that

$\displaystyle\mathcal{E}(\boldsymbol{u}^{K}_{k},v^{K}_{k})-\mathcal{E}(\boldsymbol{u}^{K}_{k-1},v^{K}_{k-1})$
$\displaystyle=\mathcal{E}(\boldsymbol{u}^{K}_{k},v^{K}_{k})-\mathcal{E}(\boldsymbol{u}^{K}_{k-1},v_{k}^{K})+\mathcal{E}(\boldsymbol{u}^{K}_{k-1},v_{k}^{K})-\mathcal{E}(\boldsymbol{u}^{K}_{k-1},v^{K}_{k-1})$
$\displaystyle\geq\int_{\Omega}\frac{b(v^{K}_{k})}{\alpha}\Big{(}\varphi^{*}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{K}_{k}))-\varphi^{*}_{n}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u}^{K}_{k-1}))\Big{)}\,\mathrm{d}x+\mathcal{H}(v^{K}_{k-1})-\mathcal{H}(v^{K}_{k})$
$\displaystyle=h_{K}\int_{\Omega}\int_{0}^{1}\frac{b(v^{K}_{k})}{\alpha}F^{-1}(\alpha\boldsymbol{\varepsilon}(s\boldsymbol{u}^{K}_{k}+(1-s)\boldsymbol{u}^{K}_{k-1})):\boldsymbol{\varepsilon}(\delta\boldsymbol{u}^{K}_{k})\,\mathrm{d}s\,\mathrm{d}x+\mathcal{H}(v^{K}_{k-1})-\mathcal{H}(v^{K}_{k}).$

As a result, we obtain the inequality

$\displaystyle\mathcal{E}(\boldsymbol{u}(t),v(t))-\mathcal{E}(\boldsymbol{u}_{0},v_{0})$
$\displaystyle\geq\mathcal{H}(v_{0})-\mathcal{H}(v(t))+h_{K}\sum_{k=1}^{K}\int_{\Omega}\int_{0}^{1}\frac{b(v^{K}_{k})}{\alpha}F^{-1}(\alpha\boldsymbol{\varepsilon}(s\boldsymbol{u}^{K}_{k}+(1-s)\boldsymbol{u}^{K}_{k-1})):\boldsymbol{\varepsilon}(\delta\boldsymbol{u}^{K}_{k})\,\mathrm{d}s\,\mathrm{d}x.$

For the sum on the right-hand side, we rewrite it in the following form:

$\displaystyle
h_{K}\sum_{k=1}^{K}\int_{\Omega}\int_{0}^{1}\frac{b(v^{K}_{k})}{\alpha}F^{-1}(\alpha\boldsymbol{\varepsilon}(s\boldsymbol{u}^{K}_{k}+(1-s)\boldsymbol{u}^{K}_{k-1})):\boldsymbol{\varepsilon}(\delta\boldsymbol{u}^{K}_{k})\,\mathrm{d}s\,\mathrm{d}x$
$\displaystyle=\int_{0}^{t}\int_{\Omega}\int_{0}^{1}\frac{b(v^{K,+})}{\alpha}F^{-1}(\alpha\boldsymbol{\varepsilon}(s\boldsymbol{u}^{K,+}+(1-s)\boldsymbol{u}^{K,-})):\boldsymbol{\varepsilon}(\overline{\boldsymbol{u}}^{K}_{t})\,\mathrm{d}s\,\mathrm{d}x\,\mathrm{d}\tau$
$\displaystyle=\int_{(0,t)\setminus\mathcal{N}}\int_{\Omega}\int_{0}^{1}\frac{b(v^{K,+})}{\alpha}F^{-1}(\alpha\boldsymbol{\varepsilon}(s\boldsymbol{u}^{K,+}+(1-s)\boldsymbol{u}^{K,-})):\boldsymbol{\varepsilon}(\overline{\boldsymbol{u}}^{K}_{t})\,\mathrm{d}s\,\mathrm{d}x\,\mathrm{d}\tau.$

Considering the limit as $K\rightarrow\infty$, we have

$\displaystyle b(v^{K}_{+})\rightarrow b(v)\quad\text{ strongly in
}L^{q}\big{(}\big{(}(0,t)\setminus\mathcal{N}\big{)}\times\Omega\big{)}\text{
for every }q\in[1,\infty),$
$\displaystyle\int_{0}^{1}F^{-1}(\alpha\boldsymbol{\varepsilon}(s\boldsymbol{u}^{K,+}+(1-s)\boldsymbol{u}^{K,-}))\,\mathrm{d}s\rightarrow
F^{-1}(\alpha\boldsymbol{\varepsilon}(\boldsymbol{u}))\quad\text{ strongly in
}L^{\infty}(0,t;L^{p}(\Omega)^{d\times d}).$

By the uniform boundedness of $v$ and the fact that $\mathcal{N}$ is null,
combining these convergences, we deduce that

$\displaystyle
b(v^{K}_{+})\int_{0}^{1}F^{-1}(\alpha\boldsymbol{\varepsilon}(s\boldsymbol{u}^{K,+}+(1-s)\boldsymbol{u}^{K,-}))\,\mathrm{d}s\rightarrow
b(v)F^{-1}(\alpha\boldsymbol{\varepsilon}(\boldsymbol{u}))\,\text{ strongly in
}L^{\infty}(0,t;L^{p}(\Omega)^{d\times d}).$

Suppose that
$(\boldsymbol{\varepsilon}(\overline{\boldsymbol{u}}^{K}_{t}))_{K}$ converges
weakly in $L^{p}((0,t)\times\Omega)^{d\times d}$ to
$\boldsymbol{\varepsilon}(\boldsymbol{u}_{t})$. Then we conclude that

$\displaystyle\lim_{K\rightarrow\infty}\int_{(0,t)\setminus\mathcal{N}}\int_{\Omega}\int_{0}^{1}\frac{b(v^{K,+})}{\alpha}F^{-1}(\alpha\boldsymbol{\varepsilon}(s\boldsymbol{u}^{K,+}+(1-s)\boldsymbol{u}^{K,-})):\boldsymbol{\varepsilon}(\overline{\boldsymbol{u}}^{K}_{t})\,\mathrm{d}s\,\mathrm{d}x\,\mathrm{d}\tau$
$\displaystyle=\int_{0}^{t}\int_{\Omega}b(v)F^{-1}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u})):\boldsymbol{\varepsilon}(\boldsymbol{u}_{t})\,\mathrm{d}x\,\mathrm{d}s.$

It follows that

$\displaystyle\mathcal{E}(\boldsymbol{u}(t),v(t))+\mathcal{H}(v(t))\geq\mathcal{E}(\boldsymbol{u}_{0},v_{0})+\mathcal{H}(v_{0})+\int_{0}^{t}\int_{\Omega}\frac{b(v)}{\alpha}F^{-1}(\boldsymbol{\varepsilon}(\alpha\boldsymbol{u})):\boldsymbol{\varepsilon}(\boldsymbol{u}_{t})\,\mathrm{d}x\,\mathrm{d}s.$

Reasoning as in Proposition 2.6, this is sufficient to show that the energy-
dissipation equality holds. It remains to prove the weak convergence result
for $(\boldsymbol{\varepsilon}(\overline{\boldsymbol{u}}^{K}_{t}))_{K}$. We
have

$\displaystyle\boldsymbol{\varepsilon}(\delta\boldsymbol{u}^{K}_{k})$
$\displaystyle=\frac{1}{h_{K}}\boldsymbol{\varepsilon}(\boldsymbol{u}^{K}_{k}-\boldsymbol{u}^{K}_{k-1})$
$\displaystyle=\frac{1}{h_{K}}\boldsymbol{\varepsilon}(\boldsymbol{u}(t_{k}^{K})-\boldsymbol{u}(t^{K}_{k-1}))$
$\displaystyle=\frac{t_{k}^{K}-t^{K}_{k-1}}{h_{K}}\fint_{t^{K}_{k-1}}^{t^{K}_{k}}\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}(s))\,\mathrm{d}s,$

where $\fint$ denotes the usual average integral operator. Noting that the
difference $\frac{t_{k}^{K}-t^{K}_{k-1}}{h_{K}}$ is uniformly bounded with
respect to $K$ and $k$, as a result of the construction of the points
$(t_{k}^{K})_{k}$, we deduce that

$\displaystyle\|\boldsymbol{\varepsilon}(\delta\boldsymbol{u}^{K}_{k})\|_{p^{\prime}}\leq
C\|\boldsymbol{\varepsilon}(\boldsymbol{u}_{t})\|_{L^{\infty}(L^{p^{\prime}})}.$

This implies that

$\|\boldsymbol{\varepsilon}(\overline{\boldsymbol{u}}^{K}_{t})\|_{L^{\infty}(L^{p^{\prime}})}\leq
C\|\boldsymbol{\varepsilon}(\boldsymbol{u}_{t})\|_{L^{\infty}(L^{p^{\prime}})}.$

By the weak-* compactness of bounded sets in
$L^{\infty}(0,T;L^{p^{\prime}}(\Omega)^{d\times d})$, the required convergence
result follows. ∎

##  3\. Strain-limiting fracture problems

In this section, we consider a strain-limiting dynamic fracture problem, by
which we mean that the strain is bounded a priori. Here, we present the state
of the art of this paper. Taking $p\rightarrow 1+$ in the constitutive
relation from Section 2, we obtain the relationship

$\displaystyle\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}+\alpha\boldsymbol{u})=\frac{\mathbf{T}}{|\mathbf{T}|}=:F_{1}(\mathbf{T}).$

However, the function $F_{1}$ is not an injection, nor is it continuous at the
origin. To avoid the issue at the origin, and to also have an injective
function in the constitutive relation, we regularise the denominator. This
motivates the study of the relationship (1.3) for a fixed parameter $a>0$,
that is,

$\displaystyle\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}+\alpha\boldsymbol{u})=\frac{\mathbf{T}}{(1+|\mathbf{T}|^{a})^{\frac{1}{a}}}=:F(\mathbf{T}).$

The parameter $a$ is fixed throughout this section so we do not indicate the
dependence of $F$ on $a$. We prove an existence result for a dynamic fracture
problem with a phase-field approximation and a strain-limiting relation,
independent of the value of $a$.

As discussed previously, the main issue that we need to overcome in the
analysis is the lack of integrability of the stress tensor $\mathbf{T}$.
Indeed, the approximations of the stress tensor are a priori bounded in the
non-reflexive space $L^{1}(Q)^{d\times d}$ and so are weakly-* compact only in