# Singular limits of a coupled elasto-plastic damage system
as viscosity and hardening vanish

Vito Crismale, Giuliano Lazzaroni and Riccarda Rossi  V. Crismale,
Dipartimento di Matematica “Guido Castelnuovo”, Sapienza Università di Roma,
Piazzale Aldo Moro 2, I-00185 Roma, Italy  vito.crismale @ uniroma1.it G.
Lazzaroni, Dipartimento di Matematica e Informatica “Ulisse Dini”, Università
degli Studi di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy
giuliano.lazzaroni @ unifi.it R. Rossi, DIMI, Università degli studi di
Brescia, via Branze 38, 25133 Brescia, Italy  riccarda.rossi @ unibs.it

###### Abstract.

The paper studies the asymptotic analysis of a model coupling elastoplasticity
and damage depending on three parameters – governing viscosity, plastic
hardening, and convergence rate of plastic strain and displacement to
equilibrium – as they vanish in different orders. The notion of limit
evolution obtained is proven to coincide in any case with a notion introduced
by Crismale and Rossi in [CR19]; moreover, such solutions are closely related
to those obtained in the vanishing-viscosity limit by Crismale and Lazzaroni
in [CL16], for the analogous model where only the viscosity parameter was
present.

The authors have been funded by the Italian Ministry of University and
Research through two different projects: MIUR - PRIN project 2017BTM7SN
_Variational Methods for stationary and evolution problems with singularities
and interfaces_ , MIUR - PRIN project 2017TEXA3H _Gradient Flows, Optimal
Transport and Metric Measure Structures_. G.L. and R.R. have been partially
supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e
le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica
(INdAM)

2020 Mathematics Subject Classification: 35A15, 35Q74, 74C05.

Key words and phrases: rate-independent systems, variational models, vanishing
viscosity and hardening, Balanced Viscosity solutions, damage, elasto-
plasticity.

##  1\. Introduction

Rate-independent processes model evolutionary phenomena where the external
loading is much slower than the internal oscillations of materials, while
viscosities may be neglected. Despite a wide literature on the subject (see
[MR15] and references therein), a further understanding is needed of the
relations between the different notions of solution that have been proposed.
In particular, a problem of interest in applications is to determine which
kind of solution captures the limiting behavior of dynamical systems for small
viscosity or inertia. For this reason, in this paper we compare different
notions of solution, obtained with different approximation methods as
viscosities tend to zero at different rates (but with no inertia).

We focus on a rate-independent system modeling damage in an elasto-plastic
body occupying a bounded Lipschitz domain $\Omega\subset\mathbb{R}^{n}$,
$n\geq 2$. The model was advanced and first studied in [AMV14, AMV15], while
the existence of _globally minimizing quasistatic evolutions_ (or,
equivalently, _Energetic solutions_) was first proved in [Cri16]. In [CL16,
CR19], the _vanishing-viscosity_ approach was instead adopted to find the so-
called _Balanced Viscosity_ solutions, obtaining the rate-independent system
as the limit of a viscously perturbed system. We refer to the pioneering
[EM06], and the subsequent [MRS12a, MRS16a], for the definition and properties
of such solutions in the context of an ‘abstract’ rate-independent system. The
vanishing-viscosity technique has also been adopted in various concrete
applications, ranging from plasticity (cf., e.g., [DMDMM08, DMDS11, BFM12,
FS13, Sol14]), to damage, fracture, and fatigue (see for instance [KMZ08,
LT11, KRZ13, Alm17, CL17, ACO19]).

In this paper we aim to gain further insight into the different ways of
constructing _Balanced Viscosity_ solutions to the model for damage and
plasticity from [AMV15], which were explored in [CL16] and [CR19]. We show
that these notions of solutions essentially coincide if the hardening vanishes
together with viscosities, while they retain different features if the
hardening parameter is positive. In particular, it turns out that perfect
plasticity coupled with damage may equivalently be approximated by means of
processes where viscosity is confined to the flow rule for damage, or with
viscosity also in the momentum equation and in the plastic flow rule.

### The model

The rate-independent process we are going to address describes the evolution,
in the time interval $(0,T)$, of the _displacement_
$u:(0,T)\times\Omega\to\mathbb{R}^{n}$, of the plastic strain
$p:(0,T)\times\Omega\to{\mathbb{M}}^{n{\times}n}_{\mathrm{D}}$, and of the
damage variable $z:(0,T)\times\Omega\to[0,1]$ that describes the soundness of
the material: for $z(t,x)=1$ (respectively, $z(t,x)=0$) the material is in the
undamaged (fully damaged, resp.) state, at the time $t\in(0,T)$ and ‘locally’
around the point $x\in\Omega$. In fact, the related PDE system consists of

  * -

the momentum balance

$\displaystyle-\mathrm{div}\,\sigma=f\quad\text{ in
}\Omega\times(0,T)\,,\qquad\quad\sigma{\mathrm{n}}=g\text{ on
}\Gamma_{\mathrm{Neu}}\times(0,T),$ (1.1a)

with $f,\,g$ some external forces, ${\mathrm{n}}$ the outer unit normal vector
to $\Omega$, $\sigma$ the _stress_ tensor

$\sigma=\mathbb{C}(z)e\quad\text{ in }\Omega\times(0,T),$ (1.1b)

$\mathbb{C}$ the elastic stress tensor, and
$e:(0,T)\times\Omega\to{\mathbb{M}^{n\times n}_{\mathrm{sym}}}$ the elastic
strain; together with the plastic strain $p$, the elastic strain $e$ concurs
to the kinematic admissibility condition for the _strain_
${\mathrm{E}}(u)=\frac{\nabla u+\nabla u^{T}}{2}$, i.e.

${\mathrm{E}}(u)=e+p\quad\text{ in }\Omega\times(0,T);$ (1.1c)

  * -

the flow rule for the damage variable $z$

$\displaystyle\partial\mathrm{R}(\dot{z})+A_{\mathrm{m}}(z)+W^{\prime}(z)\ni-\tfrac{1}{2}\mathbb{C}^{\prime}(z)e:e\quad\text{
in }\Omega\times(0,T),$ (1.1d)

where $\partial\mathrm{R}:\mathbb{R}\rightrightarrows\mathbb{R}$ denotes the
convex analysis subdifferential of the density of dissipation potential

$\mathrm{R}:\mathbb{R}\to[0,+\infty]\ \text{ defined by }\
\mathrm{R}(\eta):=\left\\{\begin{array}[]{ll}\kappa|\eta|&\text{ if }\eta\leq
0,\\\ +\infty&\text{ otherwise},\end{array}\right.$

encompassing the unidirectionality in the evolution of damage,
$A_{\mathrm{m}}$ is the $\mathrm{m}$-Laplacian operator, with
$\mathrm{m}>\tfrac{n}{2}$, and $W$ is a suitable nonlinear, possibly
nonsmooth, function;

  * -

the flow rule for the plastic tensor

$\displaystyle\partial_{\dot{p}}\mathrm{H}(z,\dot{p})\ni\sigma_{\mathrm{D}}\quad\text{
in }\Omega\times(0,T),$ (1.1e)

with $\sigma_{\mathrm{D}}$ the deviatoric part of the stress tensor $\sigma$
and $\mathrm{H}(z,\cdot)$ the density of the plastic dissipation potential.

System (1.1a)–(1.1e) is complemented by the boundary conditions

$u=w\text{ on
}\Gamma_{\mathrm{Dir}}\times(0,T),\qquad\sigma{\mathrm{n}}=g\text{ on
}\Gamma_{\mathrm{Neu}}\times(0,T),\qquad\partial_{{\mathrm{n}}}z=0\text{ on
}\partial\Omega\times(0,T),$ (1.1f)

where $\Gamma_{\mathrm{Dir}}$ is the Dirichlet part of the boundary
$\partial\Omega$ and $w$ a time-dependent Dirichlet loading, while
$\Gamma_{\mathrm{Neu}}$ is the Neumann part of $\partial\Omega$ and $g$ an
assigned traction.

Alternative models for damage and plasticity have been analyzed in, e.g.,
[RV16, RV17, DRS19], albeit from a different perspective. In fact, those
papers address the rate-independent evolution of the damage and plastic
processes coupled with a _rate-dependent_ momentum balance, featuring
viscosity and even inertial terms. Therefore, the resulting system has a
_mixed_ rate-dependent/independent character and is formulated in terms of a
weak, energetic-type notion of solution. Instead, both in [CL16] and [CR19],
(two distinct) viscous regularization procedures, described below, were
advanced to construct Balanced Viscosity solutions to the fully rate-
independent system (1.1).

### Balanced Viscosity solutions

In [CL16] the vanishing-viscosity approximation of system (1.1) was carried
out by perturbing the damage flow rule by a viscous term, which led to the
_viscously regularized_ system

$\displaystyle-\mathrm{div}\,\sigma=f\quad\text{with $\sigma=\mathbb{C}(z)e$}$
$\displaystyle\text{ in }\Omega\times(0,T),$ (1.2a)
$\displaystyle\partial\mathrm{R}(\dot{z})+\varepsilon\dot{z}+A_{\mathrm{m}}(z)+W^{\prime}(z)\ni-\tfrac{1}{2}\mathbb{C}^{\prime}(z)e:e$
$\displaystyle\text{ in }\Omega\times(0,T),$ (1.2b)
$\displaystyle\partial_{\dot{p}}\mathrm{H}(z,\dot{p})\ni\sigma_{\mathrm{D}}$
$\displaystyle\text{ in }\Omega\times(0,T),$ (1.2c) supplemented by the
boundary conditions $u=w\text{ on
}\Gamma_{\mathrm{Dir}}\times(0,T),\quad\sigma{\mathrm{n}}=g\text{ on
}\Gamma_{\mathrm{Neu}}\times(0,T),\quad\partial_{\mathrm{n}}z=0\text{ on
}\partial\Omega\times(0,T).$ (1.2d)

Passing to the limit in a reparameterized version of (1.2) led to a first
construction of $\mathrm{BV}$ solutions to system (1.1). We shall illustrate
the notion of _parameterized_ Balanced Viscosity solution thus obtained in the
forthcoming Section 3.2. In what follows, for quicker reference we will call
the $\mathrm{BV}$ solutions from [CL16] _$\mathrm{BV}_{0}$ solutions to system
(1.1)_, where the subscript 0 indicates that the solutions are obtained in the
limit as $\varepsilon\downarrow 0$.

In [CR19], a different construction of $\mathrm{BV}$ solutions to system (1.1)
was proposed, based on a viscous regularization of the momentum balance and of
the plastic flow rule, _in addition_ to that of the damage flow rule. This
alternative approach was first proposed in [MRS16b] in a finite-dimensional
context, and extended to infinite-dimensional systems in the recent [MR21].
Both papers address the vanishing-viscosity analysis of an abstract
evolutionary system, that can be thought of as prototypical of rate-dependent
systems in solid mechanics, governing the evolution of an elastic variable
$\mathsf{u}$ and of an internal variable $\mathsf{z}$. In those papers,
$\mathsf{z}$ has relaxation time $\varepsilon$, while $\mathsf{u}$ has a
viscous damping with relaxation time $\varepsilon^{\alpha}$, $\alpha>0$. To
emphasize the occurrence of these three time scales (the time scale
$\varepsilon^{0}=1$ of the external loading, the relaxation time $\varepsilon$
of $\mathsf{z}$, and the - possibly different - relaxation time
$\varepsilon^{\alpha}$ of $\mathsf{u}$), the term ‘multi-rate system’ was used
in [MRS16b]. Therein, as well as in [MR21], it was shown that, in the three
cases $\alpha\in(0,1)$, $\alpha=1$ and $\alpha>1$, the vanishing-viscosity
analysis leads to different notions of Balanced-Viscosity solutions, including
in particular different descriptions of the jump behavior developing in the
limit $\varepsilon\downarrow 0$.

Thus, along the lines of [MRS16b], in [CR19] the authors addressed the
following, alternative, viscous regularization of system (1.1):

$\displaystyle-\mathrm{div}(\varepsilon\nu\mathbb{D}{\mathrm{E}}{(\dot{u})}+\sigma)=f$
$\displaystyle\text{ in }\Omega\times(0,T),$ (1.3a)
$\displaystyle\partial\mathrm{R}(\dot{z})+\varepsilon\dot{z}+A_{\mathrm{m}}(z)+W^{\prime}(z)\ni-\tfrac{1}{2}\mathbb{C}^{\prime}(z)e:e$
$\displaystyle\text{ in }\Omega\times(0,T),$ (1.3b)
$\displaystyle\partial_{\dot{p}}\mathrm{H}(z,\dot{p})+\varepsilon\nu\dot{p}+{\mu}p\ni\sigma_{\mathrm{D}}$
$\displaystyle\text{ in }\Omega\times(0,T),$ (1.3c) supplemented by the
boundary conditions $u=w\text{ on
}\Gamma_{\mathrm{Dir}}\times(0,T),\quad(\varepsilon\nu\mathbb{D}{\mathrm{E}}{(\dot{u})}+\sigma){\mathrm{n}}=g\text{
on }\Gamma_{\mathrm{Neu}}\times(0,T),\quad\partial_{\mathrm{n}}z=0\text{ on
}\partial\Omega\times(0,T),$ (1.3d)

where $\mathbb{D}$ is a positive-definite fourth-order tensor. System (1.3)
features a viscous regularization _both_ in the damage flow rule _and_ in the
displacement equation and the plastic flow rule. Let us now illustrate the
role of the various parameters appearing therein, namely the

  * -

(vanishing-)viscosity parameter $\varepsilon>0$,

  * -

(vanishing-)hardening parameter $\mu>0$,

  * -

additional parameter $\nu>0$, which was required to fulfill $\nu\leq\mu$ in
order to get suitable a priori estimates. We have referred to $\nu$ as a
_rate_ parameter, since it sets the mutual rate at which, on the one hand, the
displacement and the plastic strain converge to equilibrium and rate-
independent evolution, and, on the other hand, the damage parameter converges
to rate-independent evolution. More precisely, if $\nu>0$ stays fixed then $u$
and $p$ converge at the same rate as $z$, while their convergence occurs at a
faster rate if $\nu\downarrow 0$. This is clear if one chooses e.g.
$\nu=\mu=\varepsilon$, so that the viscous terms ${\mathrm{E}}{(\dot{u})}$ and
$\dot{p}$ in (1.3a) and (1.3c) are modulated by the coefficient
$\varepsilon^{2}$, as opposed to the coefficient $\varepsilon$ in the damage
flow rule. Observe that, upon taking the vanishing-hardening limit
$\mu\downarrow 0$, the constraint $\nu\leq\mu$ forces the joint vanishing-
viscosity and vanishing-hardening limit to occur at a faster rate for $u$ and
$p$ than for $z$.

We will refer to the vanishing-viscosity analyses in system (1.3) as _full_ ,
as opposed to the _partial_ vanishing-viscosity approximation provided by
system (1.2), where only the damage flow rule is regularized.

Indeed, in [CR19] three _full_ vanishing-viscosity analyses have been carried
out for system (1.3), leading to three different notions of solution for
system (1.1), possibly regularized by a hardening term. Let us briefly
illustrate them.

(1) $\mathrm{BV}_{0}^{\mu,\nu}$ solutions:

The limit passage in a (reparameterized) version of (1.3) as
$\varepsilon\downarrow 0$, while the positive parameters $\mu$ and $\nu$
stayed fixed, has led to $\mathrm{BV}$ solutions for a variant of the system
(1.1), where the plastic flow rule was regularized by the hardening term $\mu
p$, i.e. for the rate-independent system with hardening

$\displaystyle-\mathrm{div}(\mathbb{C}(z)e)=f$ $\displaystyle\text{ in
}\Omega\times(0,T),$ (1.4a)
$\displaystyle\partial\mathrm{R}(\dot{z})+A_{\mathrm{m}}(z)+W^{\prime}(z)\ni-\tfrac{1}{2}\mathbb{C}^{\prime}(z)e:e$
$\displaystyle\text{ in }\Omega\times(0,T),$ (1.4b)
$\displaystyle\partial_{\dot{p}}\mathrm{H}(z,\dot{p})+{\mu}p\ni\sigma_{\mathrm{D}}$
$\displaystyle\text{ in }\Omega\times(0,T),$ (1.4c) coupled with the boundary
conditions $u=w\text{ on
}\Gamma_{\mathrm{Dir}}\times(0,T),\quad\sigma{\mathrm{n}}=g\text{ on
}\Gamma_{\mathrm{Neu}}\times(0,T)\quad\qquad\partial_{\mathrm{n}}z=0\text{ on
}\partial\Omega\times(0,T).$ (1.4d)

The $\mathrm{BV}$ solutions to system (1.4) thus constructed reflect their
origin from a system viscously regularized in all of the three variables $u$,
$z$, $p$. In fact, in the jump regime the system may switch to viscous
behavior in the three variables $u$, $z$, and $p$. Since the convergence of
$u$, $z$, and $p$ to elastic equilibrium and rate-independent evolution has
occurred at the _same rate_ (as $\nu>0$ stayed fixed), viscous behavior in
$u$, $z$, and $p$ may _equally_ intervene in the jump regime. We shall refer
to such solutions as $\mathrm{BV}_{0}^{\mu,\nu}$ _solutions to system_ (1.4).
The subscript $0$ suggests that they have been obtained in the vanishing-
viscosity limit $\varepsilon\downarrow 0$, while the occurrence of the
parameter $\mu$ keeps track of the presence of hardening. Also the parameter
$\nu$ appears in the notation, since it still features in the limiting
evolution as a coefficient of the viscous terms in the displacement equation
and in the plastic flow rule, which may be active in the jump regime.

(2) $\mathrm{BV}_{0}^{\mu,0}$ solutions:

The limit passage in a (reparameterized) version of (1.3) as
$\varepsilon\downarrow 0$ simultaneously with $\nu\downarrow 0$, while $\mu>0$
stayed fixed, has again led to $\mathrm{BV}$ solutions for the rate-
independent elasto-plastic damage system with hardening (1.4). These solutions
still have the feature that, in the jump regime, the system may switch to
viscous behavior in $u$, $z$, and $p$. However, the $\mathrm{BV}$ solutions
thus obtained reflect the fact that the convergence of $u$ and $p$ to elastic
equilibrium and rate-independent evolution has occurred at a _faster rate_ (as
$\nu\downarrow 0$) than that for $z$. To emphasize this, such solutions were
termed _$\mathrm{BV}$ solutions to the multi-rate system for damage with
hardening_. We will refer to them as _$\mathrm{BV}_{0}^{\mu,0}$ solutions to
system (1.1)_. In this notation, the double occurrence of $0$ relates to the
fact that such solutions were obtained in the limit $\varepsilon,\nu\downarrow
0$, as opposed to the $\mathrm{BV}_{0}$ solutions from [CL16] (arising in the
limit of system (1.3) as $\varepsilon\downarrow 0$ and $\mu=\nu=0$).

(3) $\mathrm{BV}_{0}^{0,0}$ solutions:

The limit passage in a (reparameterized) version of (1.3) as
$\varepsilon,\,\mu,\,\nu\downarrow 0$ _jointly_ led to _$\mathrm{BV}$
solutions to the multi-rate system for damage and perfect plasticity_ (1.1),
again reflecting the fact that the convergence of $u$ and $p$ to elastic
equilibrium and rate-independent, _perfectly plastic_ evolution happened at a
rate faster than that for $z$. The vanishing-viscosity solutions arising from
this joint limit will receive specific attention in this paper. In what
follows, we will refer to them as _$\mathrm{BV}_{0}^{0,0}$ solutions to system
(1.1)_. Here, the triple occurrence of $0$ relates to the fact that such
solutions were obtained in the limit $\varepsilon,\,\mu,\,\nu\downarrow 0$ and
thus immediately suggests the comparison with the $\mathrm{BV}_{0}$ solutions
from [CL16].

### Our results

The aim of this paper is twofold:

  * (i)

We propose to gain further insight into $\mathrm{BV}_{0}^{0,0}$ solutions to
system (1.1) (cf. item $\\#3$ in the above list). More precisely, first of all
we shall provide a _differential characterization_ of such solutions, cf.
Proposition 3.11 ahead. Relying on that, in Theorem 4.1 we will subsequently
prove that, after an initial phase in which $z$ is constant while $u$ and $p$,
evolving by viscosity, relax to elastic equilibrium and to rate-independent
evolution, respectively, it turns out that $u$ never leaves the equilibrium,
and $p$ the rate-independent regime. Afterwards, the evolution of system (1.1)
is captured by the notion of $\mathrm{BV}_{0}$ solution as obtained in [CL16]
by taking the vanishing-viscosity limit as $\varepsilon\downarrow 0$ of system
(1.2). In other words, viscosity in $u$ and $p$ (may) intervene only in an
initial phase in the reparameterized time scale, corresponding to a time
discontinuity in the time scale of the loading. After this initial phase, the
$\mathrm{BV}_{0}^{0,0}$ solutions to the perfectly plastic system for damage
(1.1) arising by the _full_ vanishing-viscosity approach of [CR19] comply with
the same notion of solution of [CL16], where viscosity for $u$ and $p$ was
neglected.

We point out that an analogous characterization can be proved for the
$\mathrm{BV}_{0}^{\mu,0}$ solutions to the multi-rate system with _fixed_
hardening parameter $\mu>0$, obtained in the limit passage $\\#2$ of the above
list; see Remark 5.2 ahead.

  * (ii)

We aim to ‘close the circle’ in the analysis of the singular limits of system
(1.3), by showing that, for two given sequences
$(\mu_{k})_{k},\,(\nu_{k})_{k}\subset(0,+\infty)$ with
$0<\nu_{k}\leq\mu_{k}\downarrow 0$ as $k\to\infty$,

    1. (1)

$\mathrm{BV}_{0}^{\mu_{k},\nu_{k}}$ solutions to the _single_ -rate system
with hardening converge as $k\to\infty$ to a $\mathrm{BV}_{0}^{0,0}$ solution
of the perfectly plastic damage system, which will be shown in Theorem 5.6
ahead;

    2. (2)

$\mathrm{BV}_{0}^{\mu_{k},0}$ solutions to the _multi_ -rate system with
hardening converge as $k\to\infty$ to a $\mathrm{BV}_{0}^{0,0}$ solution, cf.
Theorem 5.3.

In particular, we will prove that the diagram in Figure 1 commutes.

${\begin{array}[]{c}\mathrm{BV}_{0}^{\mu,\nu}\\\
\mu\geq\nu>0\end{array}}$${\begin{array}[]{c}\mathrm{V}_{\varepsilon}^{\mu,\nu}\\\
\varepsilon,\nu,\mu>0\end{array}}$${\mathrm{BV}_{0}^{0,0}}$${\begin{array}[]{c}\mathrm{BV}_{0}^{\mu,0}\\\
\mu>0\end{array}}$

$\varepsilon\to 0$

[CR19,§6.1]

$\varepsilon,\nu,\mu\to 0$[CR19,§7]

[CR19,§6.2]

$\varepsilon,\nu\to 0$

$\nu\leq\mu\to 0$

(Theorem 5.6)

$\mu\to 0$

(Theorem 5.3)

Figure 1. The diagram displays the asymptotic relations between different
notions of solution. The symbol $\mathrm{V}_{\varepsilon}^{\mu,\nu}$ indicates
solutions to the viscous system (1.3). Solid lines represent convergences
(along sequences) to the limiting solutions of type
$\mathrm{BV}_{0}^{\mu,\nu}$, $\mathrm{BV}_{0}^{\mu,0}$,
$\mathrm{BV}_{0}^{0,0}$, already proved in [CR19, Sections 6.1, 6.2, 7].
Dashed lines represent convergences proved in the present paper, the
corresponding theorems being referred to in the diagram. Starting from
$\mathrm{V}_{\varepsilon}^{\mu,\nu}$ one may either pass to the limit as
$\varepsilon\downarrow 0$ and then as $\mu,\,\nu\downarrow 0$; or pass to the
limit as $\varepsilon,\,\nu\downarrow 0$ and then as $\mu\downarrow 0$. Since
there is no uniqueness, it is not guaranteed that one gets the very same
solution found in the joint limit $\varepsilon,\,\mu,\,\nu\downarrow 0$.
However, we prove that through the three different procedures one finds
evolutions satisfying the same _notion_ of solution. In this sense we may say
that the diagram commutes.

We emphasize that these results establish asymptotic relations between
$\mathrm{BV}$ solutions, already obtained as vanishing-viscosity limits. These
convergence analyses show that $\mathrm{BV}_{0}^{0,0}$ solutions are robust
enough to capture the asymptotic behavior of a wide class of $\mathrm{BV}$
solutions depending on different parameters. However, $\mathrm{BV}_{0}^{0,0}$
solutions reduce to $\mathrm{BV}_{0}$ solutions after an initial phase in
which $u$ and $p$ converge to elastic equilibrium and stability, respectively;
in particular, if the initial conditions are at equilibrium, then the two
notions of solution coincide. This feature may be traced back to the convex
character of perfect plasticity and to the multi-rate character inherent to
the system, since with $\nu\downarrow 0$ we have forced faster convergence to
equilibrium in $u$ and stability in $p$.

##### Plan of the paper.

In Section 2 we detail the setup of the problem, list our assumptions, and
provide some preliminary results. In Section 3 we illustrate the notion of
$\mathrm{BV}_{0}$ solution to system (1.1) arising from the _partial_
vanishing-viscosity approach of [CL16], and that of $\mathrm{BV}_{0}^{0,0}$
solution via the _full_ vanishing-viscosity analysis in [CR19]. In Section 4
we establish Theorem 4.1, providing a complete characterization of
$\mathrm{BV}_{0}^{0,0}$ solutions. Section 5 is devoted to the vanishing-
hardening analysis of $\mathrm{BV}_{0}^{\mu,0}$ and
$\mathrm{BV}_{0}^{\mu,\nu}$ solutions to the system with hardening. The proofs
of Theorems 5.3 and 5.6 rely on some technical results collected in the
Appendix.

##  2\. Setup and preliminaries

Throughout the paper we will use the following

######  Notation 2.1 (General notation and preliminaries).

Let $X$ be a Banach space. By
$\sideset{}{{}_{X}}{\mathop{\langle\cdot,\cdot\rangle}}$ we denote the duality
between $X^{*}$ and $X$ or between $(X^{n})^{*}$ and $X^{n}$ (whenever $X$ is
a Hilbert space, $\sideset{}{{}_{X}}{\mathop{\langle\cdot,\cdot\rangle}}$ will
be the inner product), while $\|\cdot\|_{X}$ stands for the norm in $X$ or in
$X^{n}$. The inner Euclidean product in $\mathbb{R}^{n}$, $n\geq 1$, is
denoted by $\sideset{}{}{\mathop{\langle\cdot,\cdot\rangle}}$ and the
Euclidean norm in $\mathbb{R}^{n}$ by $|\cdot|$. The symbol $B_{r}(0)$ stands
for the open ball in $\mathbb{R}^{n}$ with radius $r$ and center $0$.

We write $\|\cdot\|_{L^{p}}$ for the $L^{p}$-norm on the space
$L^{p}(O;\mathbb{R}^{d})$, with $O$ a measurable subset of $\mathbb{R}^{n}$
and $1\leq p<+\infty$, and similarly $\|\cdot\|_{H^{\mathrm{m}}}$ for the norm
of the Sobolev-Slobodeskij space $H^{\mathrm{m}}(O)$, for $0\leq{\rm
m}\in\mathbb{R}$. The symbol $\mathrm{M}_{\mathrm{b}}(O;\mathbb{R}^{d})$
stands for the space of $\mathbb{R}^{d}$-valued bounded Radon measures in $O$.

The space of symmetric $(n{\times}n)$-matrices is denoted by
${\mathbb{M}^{n\times n}_{\mathrm{sym}}}$, while the subspace of the
deviatoric matrices with null trace is denoted by
${\mathbb{M}}^{n{\times}n}_{\mathrm{D}}$. One has ${\mathbb{M}^{n\times
n}_{\mathrm{sym}}}={\mathbb{M}}^{n{\times}n}_{\mathrm{D}}\oplus\mathbb{R}I$,
where $I$ is the identity matrix, i.e. any $\eta\in{\mathbb{M}^{n\times
n}_{\mathrm{sym}}}$ can be decomposed as
$\eta=\eta_{\mathrm{D}}+\frac{\mathrm{tr}(\eta)}{n}I$, where
$\eta_{\mathrm{D}}$ is the orthogonal projection of $\eta$ onto
${\mathbb{M}}^{n{\times}n}_{\mathrm{D}}$. The latter is called the _deviatoric
part_ of $\eta$. The symbol
$\mathrm{Sym}({\mathbb{M}}^{n{\times}n}_{\mathrm{D}};{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})$
stands for the set of symmetric endomorphisms on
${\mathbb{M}}^{n{\times}n}_{\mathrm{D}}$.

Given a function $v:\Omega\times(0,T)\to\mathbb{R}$ differentiable, w.r.t.
time a.e. on $\Omega\times(0,T)$, its (almost everywhere defined) partial time
derivative is indicated by $\dot{v}:\Omega\times(0,T)\to\mathbb{R}$. A
different notation will be employed when considering $v$ as a (Bochner)
function, from $(0,T)$ with values in a Lebesgue or Sobolev space $X$ (with
the Radon-Nikodým property): if $v\in\mathrm{AC}([0,T];X)$, then its (almost
everywhere defined) time derivative is indicated by $v^{\prime}:(0,T)\to X$.

The symbols $c,\,c^{\prime},\,C,\,C^{\prime}$ will denote positive constants
whose precise value may vary from line to line (or within the same line). We
will sometimes employ the symbols $I_{i}$, $i=0,1,...$, as place-holders for
terms appearing in inequalities: also in this case, such symbols may appear in
different proofs with different meaning.

#### Function of bounded deformation

The state space for the displacement variable for the systems with hardening
will be

$H_{\mathrm{Dir}}^{1}(\Omega;\mathbb{R}^{n}):=\\{u\in
H^{1}(\Omega;\mathbb{R}^{n})\,:\ u=0\text{ on }\Gamma_{\mathrm{Dir}}\\}$

(recall that $\Gamma_{\mathrm{Dir}}$ is the Dirichlet part of
$\partial\Omega$, cf. (2.$\Omega$) ahead).

For the perfectly plastic damage system, displacements will belong to the
space of _functions of bounded deformations_ , defined by

$\mathrm{BD}(\Omega):=\\{u\in L^{1}(\Omega;\mathbb{R}^{n})\,:\
\mathrm{E}(u)\in\mathrm{M}_{\mathrm{b}}(\Omega;{\mathbb{M}^{n\times
n}_{\mathrm{sym}}})\\},$

with $\mathrm{M}_{\mathrm{b}}(\Omega;{\mathbb{M}^{n\times n}_{\mathrm{sym}}})$
the space of ${\mathbb{M}^{n\times n}_{\mathrm{sym}}}$-valued bounded Radon
measures on $\Omega$. We recall that
$\mathrm{M}_{\mathrm{b}}(\Omega;{\mathbb{M}^{n\times n}_{\mathrm{sym}}})$ can
be identified with the dual of the space
$\mathrm{C}^{0}_{0}(\Omega;{\mathbb{M}^{n\times n}_{\mathrm{sym}}})$ of
continuous ${\mathbb{M}^{n\times n}_{\mathrm{sym}}}$-valued functions
vanishing at the boundary of $\Omega$. The space $\mathrm{BD}(\Omega)$ has a
Banach structure if equipped with the norm

$\|u\|_{\mathrm{BD}(\Omega)}:=\|u\|_{L^{1}(\Omega;\mathbb{R}^{n})}+\|\mathrm{E}(u)\|_{\mathrm{M}_{\mathrm{b}}(\Omega;{\mathbb{M}^{n\times
n}_{\mathrm{sym}}})}.$

Indeed, $\mathrm{BD}(\Omega)$ is the dual of a normed space, cf. [TS80], and
such duality provides a weak∗ convergence on $\mathrm{BD}(\Omega)$: a sequence
$(u_{k})_{k}$ converges to $u$ weakly∗ in $\mathrm{BD}(\Omega)$ if
$u_{k}\rightharpoonup u$ in $L^{1}(\Omega;\mathbb{R}^{n})$ and
$\mathrm{E}(u_{k})\overset{*}{\rightharpoonup}\mathrm{E}(u)$ in
$\mathrm{M}_{\mathrm{b}}(\Omega;{\mathbb{M}^{n\times n}_{\mathrm{sym}}})$. It
holds $\mathrm{BD}(\Omega)\subset L^{n/(n{-}1)}(\Omega;\mathbb{R}^{n})$. Up to
subsequences, every bounded sequence in $\mathrm{BD}(\Omega)$ converges
weakly∗, weakly in $L^{n/(n{-}1)}(\Omega;\mathbb{R}^{n})$, and strongly in
$L^{p}(\Omega;\mathbb{R}^{n})$ for any $1\leq p<\frac{n}{n-1}$. Finally, we
recall that the trace $u|_{\partial\Omega}$ of a function
$u\in\mathrm{BD}(\Omega)$ is well defined and is an element in
$L^{1}(\partial\Omega;\mathbb{R}^{n})$.

#### A divergence operator

First of all, we observe that any $\sigma\in{L^{2}(\Omega;{\mathbb{M}^{n\times
n}_{\mathrm{sym}}})}$ such that $\mathrm{div}\,\sigma\in
L^{2}(\Omega;\mathbb{R}^{n})$ induces the distribution $[\sigma{\mathrm{n}}]$
defined by

$\langle[\sigma{\mathrm{n}}],\psi\rangle_{\partial\Omega}:=\langle\mathrm{div}\,\sigma,\psi\rangle_{L^{2}}+\langle\sigma,{\mathrm{E}}(\psi)\rangle_{L^{2}}\qquad\text{for
every }\psi\in H^{1}(\Omega;\mathbb{R}^{n}).$ (2.1)

By [KT83, Theorem 1.2] and [DMDM06, (2.24)] we have that
$[\sigma{\mathrm{n}}]\in H^{-1/2}(\partial\Omega;\mathbb{R}^{n})$; moreover,
if $\sigma\in\mathrm{C}^{0}(\overline{\Omega};{\mathbb{M}^{n\times
n}_{\mathrm{sym}}})$, then the distribution $[\sigma{\mathrm{n}}]$ fulfills
$[\sigma{\mathrm{n}}]=\sigma{\mathrm{n}}$, where the right-hand side is the
standard pointwise product of the matrix $\sigma$ and the normal vector
${\mathrm{n}}$ in $\partial\Omega$.

For the treatment of the perfectly plastic system for damage it will be
crucial to work with the space

$\Sigma(\Omega):=\\{\sigma\in{L^{2}(\Omega;{\mathbb{M}^{n\times
n}_{\mathrm{sym}}})}\colon\mathrm{div}\,\sigma\in
L^{n}(\Omega;\mathbb{R}^{n}),\ \sigma_{\mathrm{D}}\in
L^{\infty}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})\\}.$ (2.2)

Furthermore, our choice of external loadings (see (2.8a)) will ensure that the
stress fields $\sigma$ that we consider, at equilibrium, have the additional
property that $[\sigma{\mathrm{n}}]\in L^{\infty}(\Omega;\mathbb{R}^{n})$ and
$\sigma\in\Sigma(\Omega)$ (cf. Lemma 3.1). Therefore, any of such fields
induces a functional $-\mathrm{Div}\,\sigma\in\mathrm{BD}(\Omega)^{*}$ via

$\langle-\mathrm{Div}\,\sigma,v\rangle_{\mathrm{BD}(\Omega)}:=\langle-\mathrm{div}\,\sigma,v\rangle_{L^{\frac{n}{n-1}}(\Omega;\mathbb{R}^{n})}+\langle[\sigma{\mathrm{n}}],v\rangle_{L^{1}(\Gamma_{\mathrm{Neu}};\mathbb{R}^{n})}\qquad\text{for
all }v\in\mathrm{BD}(\Omega).$ (2.3)

With slight abuse of notation, we shall denote by $-\mathrm{Div}\,\sigma$ also
the restriction of the above functional to $H^{1}(\Omega;\mathbb{R}^{n})^{*}$.

#### The $A_{\mathrm{m}}$-Laplacian

The damage flow rule features a gradient regularizing contribution in terms of
the $A_{\mathrm{m}}$-Laplacian operator, that is defined from the bilinear
form

$\displaystyle a_{\mathrm{m}}:H^{\mathrm{m}}(\Omega)\times
H^{\mathrm{m}}(\Omega)\to\mathbb{R},\quad
a_{\mathrm{m}}(z_{1},z_{2}):=\int_{\Omega}\int_{\Omega}\frac{\big{(}\nabla
z_{1}(x)-\nabla z_{1}(y)\big{)}\cdot\big{(}\nabla z_{2}(x)-\nabla
z_{2}(y)\big{)}}{|x-y|^{n+2(\mathrm{m}-1)}}\;\\!\mathrm{d}x\;\\!\mathrm{d}y\text{
with }\mathrm{m}>\frac{n}{2}.$

Then,

$A_{\mathrm{m}}:H^{\mathrm{m}}(\Omega)\to H^{\mathrm{m}}(\Omega)^{*}\text{ is
defined by }\langle
A_{\mathrm{m}}(z),w\rangle_{H^{\mathrm{m}}(\Omega)}:=a_{\mathrm{m}}(z,w)\quad\text{for
every $z,\,w\in H^{\mathrm{m}}(\Omega)$.}$

The inner product $\sideset{}{{}_{H^{\mathrm{m}}(\Omega)}}{\mathop{\langle
z_{1},z_{2}\rangle}}:=\int_{\Omega}z_{1}z_{2}\;\\!\mathrm{d}x+a_{\mathrm{m}}(z_{1},z_{2})$
makes $H^{\mathrm{m}}(\Omega)$ a Hilbert space. Throughout the paper we shall
assume $\mathrm{m}>\tfrac{n}{2}$ and rely on the compact embedding
$H^{\mathrm{m}}(\Omega)\Subset\mathrm{C}^{0}(\overline{\Omega})$.

###  2.1. Assumptions and preliminary results

This section and Sec. 2.2 collect all our assumptions on the constitutive
functions of the model and on the problem data. We will omit to invoke them
explicitly in the statement of the various results.

#### The reference configuration

In what follows we will assume that $\Omega\subset\mathbb{R}^{n}$,
$n\in\\{2,3\\}$, is a bounded Lipschitz domain satisfying the so-called _Kohn-
Temam condition_

$\begin{gathered}\partial\Omega=\Gamma_{\mathrm{Dir}}\cup\Gamma_{\mathrm{Neu}}\cup\Sigma\quad\text{
with $\Gamma_{\mathrm{Dir}},\,\Gamma_{\mathrm{Neu}},\,\Sigma$ pairwise
disjoint,}\\\ \text{ $\Gamma_{\mathrm{Dir}}$ and $\Gamma_{\mathrm{Neu}}$
relatively open in $\partial\Omega$, and
$\partial\Gamma_{\mathrm{Dir}}=\partial\Gamma_{\mathrm{Neu}}=\Sigma$ their
relative boundary in $\partial\Omega$,}\\\ \text{ with $\Sigma$ of class
$\mathrm{C}^{2}$ and ${\mathcal{H}}^{n-1}(\Sigma)=0$, and with
$\partial\Omega$ Lipschitz and of class $\mathrm{C}^{2}$ in a neighborhood of
$\Sigma$.}\end{gathered}$ (2.$\Omega$)

#### The elasticity and viscosity tensors

We assume that the elastic tensor
$\mathbb{C}:[0,+\infty)\to\mathrm{Lin}({\mathbb{M}^{n\times
n}_{\mathrm{sym}}};{\mathbb{M}^{n\times n}_{\mathrm{sym}}})$ fulfills the
following conditions

$\displaystyle\mathbb{C}\in\mathrm{C}^{1,1}([0,+\infty);\mathrm{Lin}({\mathbb{M}^{n\times
n}_{\mathrm{sym}}};{\mathbb{M}^{n\times n}_{\mathrm{sym}}})),$
($2.\mathbb{C}_{1}$) $\displaystyle z\mapsto\mathbb{C}(z)\xi:\xi\ \text{is
nondecreasing for every}\ \xi\in{\mathbb{M}^{n\times n}_{\mathrm{sym}}},$
($2.\mathbb{C}_{2}$) $\displaystyle\exists\,\gamma_{1},\,\gamma_{2}>0\ \
\forall\,z\in[0,+\infty)\ \forall\,\xi\in{\mathbb{M}^{n\times
n}_{\mathrm{sym}}}\,:\quad\gamma_{1}|\xi|^{2}\leq\mathbb{C}(z)\xi:\xi\leq\gamma_{2}|\xi|^{2}.$
($2.\mathbb{C}_{3}$)

For the viscosity tensor $\mathbb{D}$ we require that

$\displaystyle\mathbb{D}\in\mathrm{C}^{0}(\overline{\Omega};\mathrm{Sym}({\mathbb{M}}^{n{\times}n}_{\mathrm{D}};{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})),\text{
and }$ ($2.\mathbb{D}_{1}$) $\displaystyle\
\exists\,\delta_{1},\,\delta_{2}>0\ \forall x\in\Omega\ \ \forall
A\in\mathbb{M}_{\mathrm{sym}}^{d\times d}\,:\quad\
\delta_{1}|A|^{2}\leq\mathbb{D}(x)A:A\leq\delta_{2}|A|^{2}.$
($2.\mathbb{D}_{2}$)

Thus, $\mathbb{D}$ induces an equivalent (by a Korn-Poincaré-type inequality)
Hilbert norm on $H_{\mathrm{Dir}}^{1}(\Omega;\mathbb{R}^{n})$, i.e.

$\|u\|_{H^{1},\mathbb{D}}:=\left(\int_{\Omega}\mathbb{D}\mathrm{E}(u):\mathrm{E}(u)\;\\!\mathrm{d}x\right)^{1/2}\quad\text{
and }\quad\exists\,K_{\mathbb{D}}>0\ \forall\,u\in
H_{\mathrm{Dir}}^{1}(\Omega;\mathbb{R}^{n}):\ \|u\|_{H^{1},\mathbb{D}}\leq
K_{\mathbb{D}}\|\mathrm{E}(u)\|_{L^{2}}\,.$ (2.4)

The related ‘dual norm’ is

$\|\eta\|_{(H^{1},\mathbb{D})^{*}}:=\left(\int_{\Omega}\mathbb{D}^{-1}\xi:\xi\right)^{1/2}\quad\text{for
all }\eta\in H_{\mathrm{Dir}}^{1}(\Omega;\mathbb{R}^{n})^{*}\text{ with
}\eta=\mathrm{Div}\,\xi\text{ for some }\xi\in\widetilde{\Sigma}(\Omega).$
(2.5)

#### The potential energy for the damage variable

In addition to the regularizing, nonlocal gradient contribution featuring the
bilinear form $a_{\mathrm{m}}$, the $z$-dependent part of the mechanical
energy functional shall feature a further term with density $W$ satisfying

$\displaystyle
W\in{\mathrm{C}}^{2}((0,+\infty);\mathbb{R}^{+})\cap{\mathrm{C}}^{0}([0,+\infty);\mathbb{R}^{+}{\cup}\\{+\infty\\}),$
($2.W_{1}$) $\displaystyle s^{2n}W(s)\rightarrow+\infty\text{ as }s\rightarrow
0^{+},$ ($2.W_{2}$)

where $W\in{\mathrm{C}}^{0}([0,+\infty);\mathbb{R}^{+}\cup\\{+\infty\\})$
means that $W(0)=\infty$ and $W(z)\to+\infty$ if $z\downarrow 0$, in
accordance with ($2.W_{2}$).

Indeed, the energy contribution involving $W$ forces $z$ to be strictly
positive; consequently, the material never reaches the most damaged state at
any point.

#### The plastic and the damage dissipation densities

The plastic dissipation potential shall reflect the requirement that the
admissible stresses belong to given constraint sets which, in turn, depend on
the damage variable $z$. More precisely, as in [CL16] we ask that the
constraint sets $(K(z))_{z\in[0,+\infty)}$ fulfill

$\displaystyle K(z)\subset{\mathbb{M}}^{n{\times}n}_{\mathrm{D}}\text{ is
closed and convex for all }z\in[0,+\infty),$ ($2.K_{1}$)
$\displaystyle\exists\,0<\bar{r}<\bar{R}\quad\forall\,0\leq z_{1}\leq
z_{2}\,:\qquad B_{\bar{r}}(0)\subset K(z_{1})\subset K(z_{2})\subset
B_{\bar{R}}(0),$ ($2.K_{2}$)
$\displaystyle\exists\,C_{K}>0\quad\forall\,z_{1},\,z_{2}\in[0,+\infty)\,:\qquad
d_{\mathscr{H}}(K(z_{1}),K(z_{2}))\leq C_{K}|z_{1}{-}z_{2}|,$ ($2.K_{3}$)

with $d_{\mathscr{H}}$ the Hausdorff distance between two subsets of
${\mathbb{M}}^{n{\times}n}_{\mathrm{D}}$, defined by

$d_{\mathscr{H}}(K_{1},K_{2}):=\max\left(\sup_{x\in
K_{1}}\mathrm{dist}(x,K_{2}),\,\sup_{x\in
K_{2}}\mathrm{dist}(x,K_{1})\right).$

The associated support function
$H:[0,+\infty)\times{\mathbb{M}}^{n{\times}n}_{\mathrm{D}}\to[0,+\infty)$,
defined by

$H(z,\pi):=\sup_{\sigma\in K(z)}\sigma:\pi\qquad\text{for all
}(z,\pi)\in[0,+\infty)\times{\mathbb{M}}^{n{\times}n}_{\mathrm{D}},$ (2.6)

will act as density function for the plastic dissipation potential, cf. (2.14)
later on.

We choose as damage dissipation density the function
$\mathrm{R}:\mathbb{R}\to[0,+\infty]$ given by

$\mathrm{R}(\zeta):=\begin{cases}-\kappa\zeta&\text{if }\zeta\leq 0,\\\
+\infty&\text{otherwise,}\end{cases}$ (2.7)

with $\kappa>0$ a constant related to the toughness of the material.

#### Body and surface forces, Dirichlet loading, and initial data

We assume that the volume force $f$ and the assigned traction $g$ fulfill

$f\in H^{1}(0,T;L^{n}(\Omega;\mathbb{R}^{n})),\quad g\in
H^{1}(0,T;L^{\infty}(\Gamma_{\mathrm{Neu}};\mathbb{R}^{n})).$ (2.8a) The
induced total load is the function
$F\colon[0,T]\to\mathrm{BD}(\Omega)^{*},\qquad\langle
F(t),v\rangle_{\mathrm{BD}(\Omega)}:=\langle
f(t),v\rangle_{L^{n/(n{-}1)}(\Omega;\mathbb{R}^{n})}+\langle
g(t),v\rangle_{L^{1}(\Gamma_{\mathrm{Neu}};\mathbb{R}^{n})}\quad\text{for all
}v\in\mathrm{BD}(\Omega).$ Furthermore, as customary for perfect plasticity,
we shall impose a _uniform safe load condition_ , namely that there exists
$\rho\in H^{1}(0,T;{L^{2}(\Omega;{\mathbb{M}^{n\times
n}_{\mathrm{sym}}})})\quad\text{ with }\quad\rho_{\mathrm{D}}\in
H^{1}(0,T;{L^{\infty}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})})$ (2.8b)
and there exists $\alpha>0$ such that for every $t\in[0,T]$ (recall (2.1))
$-\mathrm{div}\,\varrho(t)=f(t)\text{ a.e.\ on
}\Omega,\quad\qquad[\varrho(t){\mathrm{n}}]=g(t)\text{ on
}\Gamma_{\mathrm{Neu}},$ (2.8c) $\rho_{\mathrm{D}}(t,x)+\xi\in
K\qquad\text{for a.a.}\ x\in\Omega\ \text{and for every}\
\xi\in{\mathbb{M}^{n\times n}_{\mathrm{sym}}}\ \text{s.t.}\ |\xi|\leq\alpha.$
(2.8d) Observe that, combining (2.8a) with (2.8b)–(2.8d) yields
$-\mathrm{Div}\,\varrho(t)=F(t)$ for all $t\in[0,T]$.

As for the time-dependent Dirichlet boundary condition $w$, we assume that

$w\in H^{1}(0,T;H^{1}(\mathbb{R}^{n};\mathbb{R}^{n})).$ (2.8e)

Finally, we shall consider initial data $q_{0}=(u_{0},z_{0},p_{0})$ with

$u_{0}\in H_{\mathrm{Dir}}^{1}(\Omega;\mathbb{R}^{n}),\qquad z_{0}\in
H^{\mathrm{m}}(\Omega)\text{ with }W(z_{0})\in L^{1}(\Omega)\text{ and
}z_{0}\leq 1\text{ in }\overline{\Omega},\qquad p_{0}\in
L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}}).$ (2.8f)

#### The stress-strain duality

For the treatment of the perfectly plastic damage system it is essential to
resort to a suitable notion of stress-strain duality that we borrow from
[KT83, DMDM06], also relying on the more recent extension to Lipschitz
boundaries from [FG12], to which we refer for the properties mentioned below.
Following [DMDM06] we introduce the class $A(w)$ of _admissible displacements
and strains_ associated with a function $w\in
H^{1}(\mathbb{R}^{n};\mathbb{R}^{n})$, that is

$\begin{split}A(w):=\\{(u,e,p)\in\,&\mathrm{BD}(\Omega)\times
L^{2}(\Omega;{\mathbb{M}^{n\times
n}_{\mathrm{sym}}})\times{\mathrm{M}_{\mathrm{b}}(\Omega\cup\Gamma_{\mathrm{Dir}};{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}\colon\\\
&{\mathrm{E}}(u)=e+p\text{ in
}\Omega,\,p=(w-u){\,\odot\,}{\mathrm{n}}\,\mathscr{H}^{n-1}\text{ on
}\Gamma_{\mathrm{Dir}}\\},\end{split}$

where $\mathrm{n}$ denotes the normal vector to $\partial\Omega$ and $\odot$
the symmetrized tensorial product. The _space of admissible plastic strains_
is

$\begin{split}\Pi(\Omega):=\\{p\in{\mathrm{M}_{\mathrm{b}}(\Omega\cup\Gamma_{\mathrm{Dir}};{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}\colon\exists\,(u,w,e)\in\mathrm{BD}(\Omega)\times
H^{1}(\mathbb{R}^{n};\mathbb{R}^{n})\times L^{2}(\Omega;{\mathbb{M}^{n\times
n}_{\mathrm{sym}}})\,\text{ s.t.}\,(u,e,p)\in A(w)\\}.\end{split}$

Given $\sigma\in\Sigma(\Omega)$ (cf. (2.2)), $p\in\Pi(\Omega)$, and $u,\,e$
such that $(u,e,p)\in A(w)$ we define

$\langle[\sigma_{\mathrm{D}}:p],\varphi\rangle:=-\int_{\Omega}\varphi\sigma\cdot(e{-}{\mathrm{E}}(w))\,\mathrm{d}x-\int_{\Omega}\sigma\cdot[(u{-}w)\odot\nabla\varphi]\,\mathrm{d}x-\int_{\Omega}\varphi\,(\mathrm{div}\,\sigma)\cdot(u{-}w)\,\mathrm{d}x$
(2.9)

for every $\varphi\in\mathrm{C}^{\infty}_{c}(\mathbb{R}^{n})$; in fact, this
definition is independent of $u$ and $e$. Under these assumptions, $\sigma\in
L^{r}(\Omega;{\mathbb{M}^{n\times n}_{\mathrm{sym}}})$ for every $r<\infty$,
and $[\sigma_{\mathrm{D}}:p]$ is a bounded Radon measure with
$\|[\sigma_{\mathrm{D}}:p]\|_{1}\leq\|\sigma_{\mathrm{D}}\|_{L^{\infty}}\|p\|_{1}$
in $\mathbb{R}^{n}$. Restricting such measure to
$\Omega\cup\Gamma_{\mathrm{Dir}}$, we set

$\langle\sigma_{\mathrm{D}}\,|\,p\rangle:=[\sigma_{\mathrm{D}}:p](\Omega\cup\Gamma_{\mathrm{Dir}}).$
(2.10)

By (2.$\Omega$) and (2.9), since $u\in\mathrm{BD}(\Omega)\subset
L^{\frac{n}{n-1}}(\Omega;\mathbb{R}^{n})$, we get the following integration by
parts formula, valid if the distribution $[\sigma{\mathrm{n}}]$ defined in
(2.1) belongs to $\in L^{\infty}(\Gamma_{\mathrm{Neu}};\mathbb{R}^{n})$:

$\langle\sigma_{\mathrm{D}}\,|\,p\rangle=-\langle\sigma,e-{\mathrm{E}}(w)\rangle_{{L^{2}(\Omega;{\mathbb{M}^{n\times
n}_{\mathrm{sym}}})}}+\langle-\mathrm{div}\,\sigma,u-w\rangle_{L^{\frac{n}{n-1}}(\Omega;\mathbb{R}^{n})}+\langle[\sigma{\mathrm{n}}],u-w\rangle_{L^{1}(\Gamma_{\mathrm{Neu}};\mathbb{R}^{n})}$

for every $\sigma\in\Sigma(\Omega)$ and $(u,e,p)\in A(w)$.

###  2.2. Energetics

A key ingredient for the construction of $\mathrm{BV}$ solutions to the rate-
independent systems (1.1) (damage with perfect plasticity) and (1.4) (damage
and plasticity with hardening) is the observation that their rate-dependent
regularizations (1.2) and (1.3) have a _gradient-system structure_. Namely,
they can be reformulated in terms of the generalized gradient flow

$\partial\Psi(q^{\prime}(t))+{\mathrm{D}}{\mathcal{E}}(t,q(t))\ni
0\qquad\text{in }{\mathbf{Q}}^{*},\ \text{a.e.\ in }\,(0,T),$

for suitable choices of

  * -

the state space ${\mathbf{Q}}$ for the triple $q=(u,z,p)$;

  * -

the driving energy functional
${\mathcal{E}}:(0,T)\times{\mathbf{Q}}\to\mathbb{R}\cup\\{+\infty\\}$;

  * -

the dissipation potential $\Psi:{\mathbf{Q}}\to[0,+\infty]$, with convex
analysis subdifferential
$\partial\Psi:{\mathbf{Q}}\rightrightarrows{\mathbf{Q}}^{*}$,

as rigorously proved in [CR19]. Observe that also the rate-independent systems
(1.1) and (1.4) have a gradient structure that is, however, only formal due to
the fact that the functions $u$, $z$, and $p$ may have jumps as functions of
time. Nonetheless, for our analysis it is crucial to detail the energetics
underlying both the rate-dependent and the rate-independent systems.

#### The state spaces

The state space for the rate-dependent/independent damage systems with
hardening is

${\mathbf{Q}}_{\scriptstyle\tiny\mathrm{H}}:=H^{1}_{\mathrm{Dir}}(\Omega;\mathbb{R}^{n})\times
H^{\mathrm{m}}(\Omega)\times
L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}}).$

For the rate-independent damage system with perfect plasticity, the
displacements are just functions of bounded deformation and the plastic
strains are only bounded Radon measures on $\Omega\cup\Gamma_{\mathrm{Dir}}$,
so that the associated state space is

$\displaystyle{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}}:=\\{q=(u,z,p)\in$
$\displaystyle\,\mathrm{BD}(\Omega)\times
H^{\mathrm{m}}(\Omega)\times{\mathrm{M}_{\mathrm{b}}(\Omega\cup\Gamma_{\mathrm{Dir}};{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}\,:$
(2.11) $\displaystyle\,e:=\mathrm{E}(u)-p\in{L^{2}(\Omega;{\mathbb{M}^{n\times
n}_{\mathrm{sym}}})},\ u\odot\mathrm{n}\,\mathscr{H}^{n-1}+p=0\text{ on
}\Gamma_{\mathrm{Dir}}\\}.$

Observe that in the definition of
${\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}}$ it is in fact required that
$(u,e,p)\in A(0)$: indeed, the condition
$u\odot\mathrm{n}\,\mathscr{H}^{n-1}+p=0$ is a relaxation of the homogeneous
Dirichlet condition $u=0$ on $\Gamma_{\mathrm{Dir}}$.

#### The energy functionals

The energy functional governing the rate-dependent and rate-independent
systems _with hardening_ (1.3) and (1.4), respectively, consists of

  1. (1)

a contribution featuring the elastic energy

${\mathcal{Q}}(z,e)=\int_{\Omega}\tfrac{1}{2}\mathbb{C}(z)e:e\;\\!\mathrm{d}x;$

  2. (2)

the potential energy for the damage variable and for the hardening term;

  3. (3)

the time-dependent volume and surface forces.

Namely, for $\mu>0$ given,
${\mathcal{E}}_{\mu}\colon[0,T]\times{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{H}}\to\mathbb{R}\cup\\{+\infty\\}$
is defined by

$\displaystyle{\mathcal{E}}_{\mu}(t,u,z,p):={\mathcal{Q}}(z,\mathrm{E}(u{+}w(t)){-}p)+\int_{\Omega}\left(W(z){+}\frac{\mu}{2}|p|^{2}\right)\,\mathrm{d}x+\frac{1}{2}a_{\mathrm{m}}(z,z)-\langle
F(t),u+w(t)\rangle_{H^{1}(\Omega;\mathbb{R}^{n})}.$

In what follows, we will use the short-hand notation

$e(t):=\mathrm{E}(u{+}w(t)){-}p$

for the elastic part of the strain tensor.

The energy functional for the rate-independent perfectly plastic damage system
(1.1) is
${\mathcal{E}}_{0}:[0,T]\times{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}}\to\mathbb{R}{\cup}\\{+\infty\\}$
given by

$\displaystyle{\mathcal{E}}_{0}(t,u,z,p):={\mathcal{Q}}(z,e(t))+\int_{\Omega}W(z)\,\mathrm{d}x+\frac{1}{2}a_{\mathrm{m}}(z,z)-\langle
F(t),u+w(t)\rangle_{\mathrm{BD}(\Omega)}.$ (2.12)

Observe that

$\mathrm{dom}({\mathcal{E}}_{0})=[0,T]\times\mathrm{D}\qquad\text{with
}\mathrm{D}=\\{q\in{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}}\,:\ W(z)\in
L^{1}(\Omega)\\}.$ (2.13)

#### The dissipation potentials

The dissipation density $\mathrm{R}$ from (2.7) clearly induces an integral
functional ${\mathcal{R}}:L^{1}(\Omega)\to[0,+\infty]$. However, since the
damage flow rule will be posed in $H^{\mathrm{m}}(\Omega)^{*}$ (cf. (2.20)
ahead), we will restrict ${\mathcal{R}}$ to the space
$H^{\mathrm{m}}(\Omega)$, denoting the restriction by the same symbol. Hence,
we will work work with the functional

${\mathcal{R}}:H^{\mathrm{m}}(\Omega)\to[0,+\infty],\quad{\mathcal{R}}(\zeta)=\int_{\Omega}\mathrm{R}(\zeta(x))\;\\!\mathrm{d}x$
(2.14)

and with its convex analysis subdifferential
$\partial{\mathcal{R}}:H^{\mathrm{m}}(\Omega)\rightrightarrows
H^{\mathrm{m}}(\Omega)^{*}$. We will often use the following characterization
of $\partial{\mathcal{R}}$, due to the $1$-homogeneity of the potential
${\mathcal{R}}$:

$\chi\in\partial{\mathcal{R}}(\zeta)\quad\Leftrightarrow\quad\begin{cases}{\mathcal{R}}(\tilde{\zeta})\geq\sideset{}{{}_{H^{\mathrm{m}}}}{\mathop{\langle\chi,\tilde{\zeta}\rangle}}\text{
for all }\tilde{\zeta}\in H^{\mathrm{m}}(\Omega)\\\
{\mathcal{R}}(\zeta)\leq\sideset{}{{}_{H^{\mathrm{m}}}}{\mathop{\langle\chi,\zeta\rangle}}\end{cases}\,.$
(2.15)

For the systems with hardening (i.e. (1.3) and (1.4)), the _plastic
dissipation potential_
$\mathcal{H}\colon{\mathrm{C}}^{0}(\overline{\Omega};[0,+\infty))\times
L^{1}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})\to\mathbb{R}$ is defined
by

$\mathcal{H}(z,\pi):=\int_{\Omega}H(z(x),\pi(x))\;\\!\mathrm{d}x,$ (2.16)

where $H$ is given by (2.6) and $\pi$ is a place-holder for the plastic rate
$\dot{p}$. Its convex analysis subdifferential
$\partial_{\pi}{\mathcal{H}}\colon{\mathrm{C}}^{0}(\overline{\Omega};[0,+\infty))\times
L^{1}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})\rightrightarrows
L^{\infty}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})$, w.r.t. the second
variable $\pi$, given by

$\omega\in\partial_{\pi}{\mathcal{H}}(z,\pi)\quad\text{ if and only if
}\quad{\mathcal{H}}(z,\varrho)-{\mathcal{H}}(z,\pi)\geq\int_{\Omega}\omega(\varrho-\pi)\;\\!\mathrm{d}x\quad\text{for
all }\varrho\in L^{1}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}}),$

fulfills

$\omega\in\partial_{\pi}{\mathcal{H}}(z,\pi)\quad\text{ if and only if
}\quad\omega(x)\in\partial_{\pi}H(z(x),\pi(x))\quad\text{for
a.a.}\,x\in\Omega.$ (2.17)

A characterization analogous to (2.17) holds for the subdifferential
$\partial_{\pi}{\mathcal{H}}(z,\cdot)\colon
L^{1}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})\rightrightarrows
L^{\infty}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})$.

In order to handle the perfectly plastic system for damage, the plastic
dissipation potential $\mathcal{H}(z,\cdot)$ has to be extended to the space
${\mathrm{M}_{\mathrm{b}}(\Omega\cup\Gamma_{\mathrm{Dir}};{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}$.
We define
$\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}\colon{\mathrm{C}}^{0}(\overline{\Omega};[0,1])\times{\mathrm{M}_{\mathrm{b}}(\Omega\cup\Gamma_{\mathrm{Dir}};{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}\to\mathbb{R}$
by

$\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}(z,\pi):=\int_{\Omega\cup\Gamma_{\mathrm{Dir}}}H\biggl{(}z(x),\frac{\mathrm{d}\pi}{\mathrm{d}\mu}(x)\biggr{)}\,\mathrm{d}\mu(x),$

where
$\mu\in{\mathrm{M}_{\mathrm{b}}(\Omega\cup\Gamma_{\mathrm{Dir}};{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}$
is a positive measure such that $\pi\ll\mu$ and
$\frac{\mathrm{d}\pi}{\mathrm{d}\mu}$ is the Radon-Nikodým derivative of $p$
with respect to $\mu$; by one-homogeneity of $H(z(x),\cdot)$, the definition
of $\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}$ does not depend of $\mu$. For
the theory of convex functions of measures we refer to [GS64]. By [AFP05,
Proposition 2.37], for every $z\in{\mathrm{C}}^{0}(\overline{\Omega};[0,1])$
the functional $p\mapsto\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}(z,p)$ is
convex and positively one-homogeneous. We recall that by Reshetnyak’s lower
semicontinuity theorem, if
$(z_{k})_{k}\subset{\mathrm{C}}^{0}(\overline{\Omega};[0,1])$ and
$(\pi_{k})_{k}\subset{\mathrm{M}_{\mathrm{b}}(\Omega\cup\Gamma_{\mathrm{Dir}};{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}$
are such that $z_{k}\rightarrow z$ in ${\mathrm{C}}^{0}(\overline{\Omega})$
and $\pi_{k}\rightharpoonup\pi$ weakly∗ in
${\mathrm{M}_{\mathrm{b}}(\Omega\cup\Gamma_{\mathrm{Dir}};{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}$,
then

$\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}(z,\pi)\leq\liminf_{k\rightarrow+\infty}\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}(z_{k},\pi_{k}).$

Finally, from [FG12, Proposition 3.9] it follows that for every
$\sigma\in\mathcal{K}_{z}(\Omega)$

$H\biggl{(}z,\frac{\mathrm{d}p}{\mathrm{d}|p|}\biggr{)}|p|\geq[\sigma_{\mathrm{D}}:p]\quad\text{as
measures on }\Omega\cup\Gamma_{\mathrm{Dir}}.$ (2.18)

In particular, we have

$\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}(z,p)\geq\sup_{\sigma\in\mathcal{K}_{z}(\Omega)}\langle\sigma_{\mathrm{D}}\,|\,p\rangle\,\qquad\text{for
every $p\in\Pi(\Omega)$.}$ (2.19)

The (partially) viscously regularized system (1.2) also features the
$2$-homogeneous dissipation potential

${\mathcal{R}}_{2}(\zeta):=\int_{\Omega}\tfrac{1}{2}|\zeta(x)|^{2}\;\\!\mathrm{d}x,$

while the (fully) viscously regularized system (1.3) additionally involves the
quadratic potentials

${\mathcal{V}}_{2,\nu}(v):=\int_{\Omega}\tfrac{\nu}{2}\mathbb{D}\mathrm{E}(v):\mathrm{E}(v)\;\\!\mathrm{d}x,\qquad{\mathcal{H}}_{2,\nu}(\pi)=\int_{\Omega}\tfrac{\nu}{2}|\pi|^{2}\;\\!\mathrm{d}x.$

#### The gradient structure for system (1.3)

It was proved in [CR19, Lemma 3.3] that for every $t\in[0,T]$ the functional
$(u,z,p)=q\mapsto{\mathcal{E}}_{\mu}(t,q)$ is Fréchet differentiable on its
domain $[0,T]\times\mathbf{D}$, with
$\mathbf{D}=\\{(u,z,p)\in{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{H}}\,:\
z>0\text{ in }\overline{\Omega}\\}$, and that for all
$q\in{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{H}}$ the function
$t\mapsto{\mathcal{E}}_{\mu}(t,q)$ belongs to $H^{1}(0,T)$. Relying on this,
it was shown that system (1.3) reformulates as the generalized gradient system

$\partial_{q^{\prime}}\Psi_{\varepsilon,\nu}(q(t),q^{\prime}(t))+{\mathrm{D}}_{q}{\mathcal{E}}_{\mu}(t,q(t))\ni
0\qquad\text{in }{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{H}}^{*}\quad\text{for
a.a.}\,t\in(0,T),$ (2.20)

involving the overall dissipation potential

$\Psi_{\nu}(q,q^{\prime}):={\mathcal{V}}_{2,\nu}(u^{\prime})+{\mathcal{R}}(z^{\prime})+{\mathcal{R}}_{2}(z^{\prime})+{\mathcal{H}}(z,p^{\prime})+{\mathcal{H}}_{2,\nu}(p^{\prime})$

and its rescaled version

$\Psi_{\varepsilon,\nu}(q,q^{\prime}):=\frac{1}{\varepsilon}\Psi_{\nu}(q,\varepsilon
q^{\prime})=\int_{\Omega}\tfrac{\varepsilon\nu}{2}\mathbb{D}\mathrm{E}(u^{\prime}){:}\mathrm{E}(u^{\prime})\;\\!\mathrm{d}x+{\mathcal{R}}(z^{\prime})+\int_{\Omega}\tfrac{\varepsilon}{2}|z^{\prime}|^{2}\;\\!\mathrm{d}x+{\mathcal{H}}(z,p^{\prime})+\int_{\Omega}\tfrac{\varepsilon\nu}{2}|p^{\prime}|^{2}\;\\!\mathrm{d}x.$

##  3\. The partial versus the full vanishing-viscosity approach

In this section we aim to gain further insight into the notion of
$\mathrm{BV}_{0}^{0,0}$ solution to system (1.1) arising from the _full_
vanishing-viscosity approach of [CR19] and compare it with the
$\mathrm{BV}_{0}$ concept from [CL16]. In order to properly introduce both
notions, it can be useful to recall the reparameterized energy-dissipation
balance where one passes to the limit to obtain Balanced Viscosity solutions.

At this heuristic stage, we will treat the partial vanishing-viscosity
approximation of [CL16] and the full approximation of [CR19] in a unified way,
although at the level of the viscous approximation in [CL16] there is the
significant difference that the plastic strain evolves rate-independently.
Still, we will disregard this and instead focus on the similarities between
the _rate-dependent_ systems (1.3) (wherefrom the $\mathrm{BV}_{0}^{0,0}$
solutions of [CR19] stem), and (1.2) (wherefrom the $\mathrm{BV}_{0}$
solutions of [CL16]). In fact, (1.2) can be formally obtained from [CR19] by
setting $\nu=\mu=0$. That is why, in what follows to fix the main ideas we
will illustrate the limit passage in the energy-dissipation balance associated
with system (1.3).

Throughout this section and the remainder of the paper, we will suppose that
the assumptions of Sec. 2 are in force without explicitly invoking them.

### The energy-dissipation balance for the viscous system

As observed in [CR19, Proposition 3.4], a curve $q=(u,z,p)\in
H^{1}(0,T;{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{H}})$ is a solution to (1.3),
namely to the generalized gradient system (2.20), if and only if it satifies
the energy-dissipation balance

$\int_{0}^{t}\left(\Psi_{\varepsilon,\nu}(q(r),q^{\prime}(r))+\Psi_{\varepsilon,\nu}^{*}(q(r),{-}{\mathrm{D}}_{q}{\mathcal{E}}_{\mu}(r,q(r)))\right)\;\\!\mathrm{d}r+{\mathcal{E}}_{\mu}(t,q(t))={\mathcal{E}}_{\mu}(0,q(0))+\int_{0}^{t}\partial_{t}{\mathcal{E}}_{\mu}(r,q(r))\;\\!\mathrm{d}r$
(3.1)

for every $t\in[0,T]$. Let us now consider a family
$(q_{\varepsilon,\nu}^{\mu})$ of solutions to (2.20). In [CR19] suitable a
priori estimates, uniform w.r.t. the parameters $\varepsilon,\nu,\mu>0$, for
the length (measured in an appropriate norm) of the curves
$q_{\varepsilon,\nu}^{\mu}$ were derived. Based on such estimates it is
possible to reparameterize such curves, obtaining _parameterized curves_
defined on an ‘artificial’ time interval $[0,S]$

$(\mathsf{t}_{\varepsilon,\nu}^{\mu},\mathsf{q}_{\varepsilon,\nu}^{\mu}):[0,S]\to[0,T]\times{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{H}}\quad\text{with
}\mathsf{t}_{\varepsilon,\nu}^{\mu}:=(\mathsf{s}_{\varepsilon,\nu}^{\mu})^{-1},\quad\mathsf{q}_{\varepsilon,\nu}^{\mu}:=q_{\varepsilon,\nu}^{\mu}\circ\mathsf{t}_{\varepsilon,\nu}^{\mu}$

and $\mathsf{s}_{\varepsilon,\nu}^{\mu}$ (suitable) arclength functions
associated with the curves $(q_{\varepsilon,\nu}^{\mu})$. Now, in terms of the
parameterized curves
$(\mathsf{t}_{\varepsilon,\nu}^{\mu},\mathsf{q}_{\varepsilon,\nu}^{\mu})$,
(3.1) translates into the _reparameterized_ energy-dissipation balance

$\displaystyle{\mathcal{E}}(\mathsf{t}_{\varepsilon,\nu}^{\mu}(s),\mathsf{q}_{\varepsilon,\nu}^{\mu}(s))+\int_{0}^{s}{\mathcal{M}}_{\varepsilon}^{\mu,\nu}(\mathsf{t}_{\varepsilon,\nu}^{\mu}(r),\mathsf{q}_{\varepsilon,\nu}^{\mu}(r),(\mathsf{t}_{\varepsilon,\nu}^{\mu})^{\prime}(r),(\mathsf{q}_{\varepsilon,\nu}^{\mu})^{\prime}(r))\;\\!\mathrm{d}r$
(3.2)
$\displaystyle={\mathcal{E}}(\mathsf{t}_{\varepsilon,\nu}^{\mu}(0),\mathsf{q}_{\varepsilon,\nu}^{\mu}(0))+\int_{0}^{s}\partial_{t}{\mathcal{E}}(\mathsf{t}_{\varepsilon,\nu}^{\mu}(r),\mathsf{q}_{\varepsilon,\nu}^{\mu}(r))\,(\mathsf{t}_{\varepsilon,\nu}^{\mu})^{\prime}(r)\;\\!\mathrm{d}r$

featuring the functional
${\mathcal{M}}_{\varepsilon}^{\mu,\nu}:[0,T]\times{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{H}}\times(0,+\infty)\times{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{H}}\to[0,+\infty]$

$\displaystyle{\mathcal{M}}_{\varepsilon}^{\mu,\nu}(t,q,t^{\prime},q^{\prime}):={\mathcal{R}}(z^{\prime})+{\mathcal{H}}(z,p^{\prime})+{\mathcal{M}}_{\varepsilon,\mathrm{red}}^{\mu,\nu}(t,q,t^{\prime},q^{\prime})$
(3.3) with the reduced functional
$\displaystyle{\mathcal{M}}_{\varepsilon,\mathrm{red}}^{\mu,\nu}\text{ defined
by }$
$\displaystyle{\mathcal{M}}_{\varepsilon,\mathrm{red}}^{\mu,\nu}(t,q,t^{\prime},q^{\prime}):=\
\frac{\varepsilon}{2t^{\prime}}\mathcal{D}_{\nu}(q^{\prime})^{2}+\frac{t^{\prime}}{2\varepsilon}(\mathcal{D}_{\nu}^{*,\mu}(t,q))^{2},$

and

$\displaystyle\mathcal{D}_{\nu}({q}^{\prime}):=\sqrt{\nu\|{u}^{\prime}(t)\|^{2}_{H^{1},\mathbb{D}}{+}\|{z}^{\prime}(t)\|_{L^{2}}^{2}{+}\nu\|{p}^{\prime}(t)\|_{L^{2}}^{2}},$
(3.4)
$\displaystyle\mathcal{D}_{\nu}^{*,\mu}({t},{q}):=\sqrt{\frac{1}{\nu}\,\|{-}\mathrm{D}_{u}{\mathcal{E}}_{\mu}({t},{q})\|^{2}_{(H^{1},\mathbb{D})^{*}}+\widetilde{d}_{L^{2}}({-}\mathrm{D}_{z}{\mathcal{E}}_{\mu}({t},{q}),\partial{\mathcal{R}}(0))^{2}+\frac{1}{\nu}\,d_{L^{2}}({-}\mathrm{D}_{p}{\mathcal{E}}_{\mu}({t},{q}),\partial_{\pi}{\mathcal{H}}({z},0))^{2}}\,.$

In (3.4), $\|\cdot\|_{H^{1},\mathbb{D}}$ and
$\|\cdot\|_{(H^{1},\mathbb{D})^{*}}$ are the norms introduced in (2.4) and
(2.5), while the distance functional
$\widetilde{d}_{L^{2}(\Omega)}(\cdot,\partial{\mathcal{R}}(0)):H^{\mathrm{m}}(\Omega)^{*}\to[0,+\infty]$
is defined by

$\widetilde{d}_{L^{2}(\Omega)}({\chi},\partial{\mathcal{R}}(0))^{2}:=\min_{\gamma\in\partial{\mathcal{R}}(0)}\mathfrak{f}_{2}({\chi}{-}\gamma)\quad\text{with
}\mathfrak{f}_{2}(\beta):=\begin{cases}\|\beta\|_{L^{2}(\Omega)}^{2}&\text{if
}\beta\in L^{2}(\Omega),\\\ +\infty&\text{if }\beta\in
H^{\mathrm{m}}(\Omega)^{*}\setminus L^{2}(\Omega).\end{cases}$ (3.5)

Clearly, the functional ${\mathcal{M}}_{\varepsilon}^{\mu,\nu}$ from (3.3)
encompasses, in the energy-dissipation balance (3.2), the competition between
viscous dissipation and tendency to relax towards equilibrium & rate-
independent behavior. In fact, viscous dissipation is encoded in the term
$\mathcal{D}_{\nu}({q}^{\prime})$, which is modulated by the viscosity
parameter $\varepsilon$. In turn, the relaxation to rate-independent behavior
is encompassed in the term $\mathcal{D}_{\nu}^{*,\mu}({t},{q})$, modulated by
$\tfrac{1}{\varepsilon}$.

Now, the concepts of $\mathrm{BV}_{0}$ and $\mathrm{BV}_{0}^{0,0}$ solutions
to system (1.1) are defined in terms of parameterized energy-dissipation
balances akin to (3.2). These energy identities involve a (positive)
_vanishing-viscosity potential_ ${\mathcal{M}}$ that is defined on
$[0,T]\times{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}}\times[0,+\infty)\times{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}}$
(recall that ${\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}}$ is the state space
for the perfectly plastic damage system, cf. (2.11)). At least formally,
${\mathcal{M}}$ arises as $\Gamma$-limit

  * -

of the functionals $({\mathcal{M}}_{\varepsilon}^{0,0})_{\varepsilon}$ as
$\varepsilon\downarrow 0$, in the case of the vanishing-viscosity analysis in
[CL16] (recall that system (1.2) is formally a particular case of (1.3), with
$\nu=\mu=0$);

  * -

of $({\mathcal{M}}_{\varepsilon}^{\mu,\nu})_{\varepsilon,\nu,\mu}$ as
$\varepsilon,\,\nu,\,\mu\downarrow 0$, in the case of the joint vanishing-
viscosity and hardening analysis in [CR19].

However, in order to _rigorously_ define the vanishing-viscosity contact
potentials ${\mathcal{M}}$ relevant for the two concepts of $\mathrm{BV}$
solutions we will need to provide some technical preliminaries in the
following section.

###  3.1. Preliminary definitions

The vanishing-viscosity potentials at the core of the definitions of
$\mathrm{BV}_{0}$ solutions in [CL16] and $\mathrm{BV}_{0}^{0,0}$ solutions in
[CR19] have a structure akin to that of the functional
${\mathcal{M}}_{\varepsilon}^{\mu,\nu}$ from (3.3), but they are tailored to
the driving energy ${\mathcal{E}}_{0}$ (2.12) for the perfectly plastic
system. Recalling the definition of ${\mathcal{M}}_{\varepsilon}^{\mu,\nu}$,
it is thus clear that, in order to define such vanishing-viscosity potentials,
one needs suitable surrogates of the $(\mathbb{D},H^{1})^{*}$-norm of
${\mathrm{D}}_{u}{\mathcal{E}}_{0}(t,q)$, and of the $L^{2}$-distance of
${\mathrm{D}}_{p}{\mathcal{E}}_{0}(t,q)$ from the stable set
$\partial_{\pi}\mathcal{H}(z,0)$, cf. (3.4). Indeed, such quantities are no
longer well defined for the functional ${\mathcal{E}}_{0}$ at every
$(t,q)\in[0,T]\times{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}}$; instead,
note that the $L^{2}$-distance, in the sense of (3.5), of
${\mathrm{D}}_{z}{\mathcal{E}}_{0}(t,q)$ from the stable set
$\partial{\mathcal{R}}(0)$ still makes sense. Following [CR19, Sec. 7], we
will set at every
$(t,q)\in\mathrm{dom}({\mathcal{E}}_{0})=[0,T]\times\mathbf{D}$ (cf. (2.13))

$\displaystyle{\mathcal{S}}_{u}{\mathcal{E}}_{0}(t,q):=\sup_{\begin{subarray}{c}\eta_{u}\in
H_{\mathrm{Dir}}^{1}(\Omega;\mathbb{R}^{n})\\\
\|\eta_{u}\|_{(H^{1},\mathbb{D})}\leq
1\end{subarray}}\langle-\mathrm{Div}\,\sigma(t)-F(t),\eta_{u}\rangle_{H_{\mathrm{Dir}}^{1}(\Omega;\mathbb{R}^{n})},$
(3.6a)
$\displaystyle\mathcal{W}_{p}{\mathcal{E}}_{0}(t,q):=\sup_{\begin{subarray}{c}\eta_{p}\in
L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})\\\
\|\eta_{p}\|_{L^{2}}\leq
1\end{subarray}}\left(\langle\sigma_{\mathrm{D}}(t),\eta_{p}\rangle_{L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}-{\mathcal{H}}(z,\eta_{p})\right).$
(3.6b)

with $\sigma(t)=\mathbb{C}(z)e(t)$ and $e(t)={\mathrm{E}}(u+w(t))-p$. Observe
that the expressions in (3.6a) and (3.6b) are well defined since $e(t)$ and, a
fortiori, $\sigma(t)$ are elements in $L^{2}(\Omega;{\mathbb{M}^{n\times
n}_{\mathrm{sym}}})$ whenever
$(u,z,p)\in{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}}$. Formulae (3.6a) and
(3.6b) have been inspired by the obvious fact that

$\|\zeta\|_{(H^{1},\mathbb{D})^{*}}=\sup_{\begin{subarray}{c}\eta_{u}\in
H_{\mathrm{Dir}}^{1}(\Omega;\mathbb{R}^{n})\\\
\|\eta_{u}\|_{(H^{1},\mathbb{D})}\leq
1\end{subarray}}\langle\zeta,\eta_{u}\rangle_{H_{\mathrm{Dir}}^{1}(\Omega;\mathbb{R}^{n})}\qquad\text{for
every }\zeta\in H_{\mathrm{Dir}}^{1}(\Omega;\mathbb{R}^{n})^{*},$

and by the formula

$d_{L^{2}}(\varsigma,\partial_{\pi}\mathcal{H}(z,0))=\sup_{\begin{subarray}{c}\eta\in
L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})\\\ \|\eta\|_{L^{2}}\leq
1\end{subarray}}\left(\langle\varsigma,\eta\rangle_{L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}-{\mathcal{H}}(z,\eta)\right)\qquad\text{for
every }\varsigma\in L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}}),$

which was proved in [CR19, Lemma 3.6]. In particular, we note that for every
$q\in\mathrm{D}$

$\displaystyle\mathcal{W}_{p}{\mathcal{E}}_{0}(t,q)=d_{L^{2}}(\sigma_{\mathrm{D}}(t),\partial_{\pi}\mathcal{H}(z,0)),$
(3.7a) while
$\displaystyle{\mathcal{S}}_{u}{\mathcal{E}}_{0}(t,q)=\|-\mathrm{Div}\,\sigma(t)-F(t)\|_{(H^{1},\mathbb{D})^{*}}\qquad\text{if
}u\in H_{\mathrm{Dir}}^{1}(\Omega;\mathbb{R}^{n}).$ (3.7b)

For later use, we also record here the following result.

######  Lemma 3.1.

For every $(t,q)\in[0,T]\times\mathbf{D}$ there holds

$\begin{gathered}{\mathcal{S}}_{u}{\mathcal{E}}_{0}(t,q)=\mathcal{W}_{p}{\mathcal{E}}_{0}(t,q)=0\quad\Longleftrightarrow\\\
\sigma(t)\in{\mathcal{K}}_{z}(\Omega)\,,\quad-\mathrm{div}\,\sigma(t)=f(t)\text{
a.e.\ in }\Omega\,,\quad[\sigma(t){\mathrm{n}}]=g(t)\
\mathscr{H}^{n-1}\text{-a.e.\ on }\Gamma_{\mathrm{Neu}},\end{gathered}$ (3.8)

with $\sigma(t)=\mathbb{C}(z)e(t)$ and $e(t)=\mathrm{E}(e+w(t))-p$.

###### Proof.

The implication $\Rightarrow$ was proved in [CR19, Lemma 7.4]. A close perusal
of the proof, also mimicking the convexity arguments from [DMDM06, Prop. 3.5],
also yields the converse implication. ∎

We then set

$\mathcal{D}^{*}(t,q):=\sqrt{({\mathcal{S}}_{u}{\mathcal{E}}_{0}(t,q))^{2}+(\mathcal{W}_{p}{\mathcal{E}}_{0}(t,q))^{2}}.$
(3.9)

We are now in a position to define the vanishing-viscosity contact potentials
involved in the definitions of $\mathrm{BV}$ solutions in [CL16] and [CR19].
We will use the notation $I_{\\{0\\}}$ for the indicator function of the
singleton $\\{0\\}$, namely

$I_{\\{0\\}}(\xi)=\begin{cases}0&\text{if }\xi=0,\\\
+\infty&\text{otherwise}\end{cases}\qquad\text{for all }\xi\in\mathbb{R}.$

###  3.2. $\mathrm{BV}_{0}$ solutions via the partial vanishing-viscosity
approach

The vanishing-viscosity contact potential for the $\mathrm{BV}_{0}$ solutions
from [CL16] is the functional

${\mathcal{M}}_{0}:[0,T]\times{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}}\times[0,+\infty)\times{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}}\to[0,+\infty],\qquad{\mathcal{M}}_{0}(t,q,t^{\prime},q^{\prime})={\mathcal{R}}(z^{\prime})+\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}(z,p^{\prime})+{\mathcal{M}}_{0,\mathrm{red}}(t,q,t^{\prime},q^{\prime}),$
(3.10a) where for $q=(u,z,p)$ and
$q^{\prime}=(u^{\prime},z^{\prime},p^{\prime})$ we have
${\mathcal{M}}_{0,\mathrm{red}}(t,q,t^{\prime},q^{\prime})=I_{\\{0\\}}(\mathcal{D}^{*}(t,q))+\widetilde{{\mathcal{M}}}_{0,\mathrm{red}}(t,q,t^{\prime},q^{\prime})$
(3.10b) and $\displaystyle\text{if
}t^{\prime}>0,\quad\widetilde{{\mathcal{M}}}_{0,\mathrm{red}}(t,q,t^{\prime},q^{\prime}):=I_{\\{0\\}}(\widetilde{d}_{L^{2}}({-}{\mathrm{D}}_{z}{\mathcal{E}}_{0}(t,q),\partial{\mathcal{R}}(0))),$
(3.10c) $\displaystyle\text{if
}t^{\prime}=0,\quad\widetilde{{\mathcal{M}}}_{0,\mathrm{red}}(t,q,t^{\prime},q^{\prime}):=\begin{cases}\|z^{\prime}\|_{L^{2}}\,\widetilde{d}_{L^{2}}({-}\mathrm{D}_{z}{\mathcal{E}}_{0}(t,q),\partial{\mathcal{R}}(0))&\text{if
}\widetilde{d}_{L^{2}}({-}\mathrm{D}_{z}{\mathcal{E}}_{0}(t,q),\partial{\mathcal{R}}(0))<+\infty,\\\
+\infty&\text{otherwise}\end{cases}\,.$

Observe that the definition of ${\mathcal{M}}_{0}(t,q,0,q^{\prime})$ again
reflects, in the jump regime, the tendencies of the system to evolve viscously
and to relax towards equilibrium and rate-independent evolution. However, here
viscous dissipation only affects the damage variable. Likewise, rate-
independent behavior is solely encompassed by the relation
$\widetilde{d}_{L^{2}}({-}\mathrm{D}_{z}{\mathcal{E}}_{0}(t,q),\partial{\mathcal{R}}(0))=0$.

Let us now detail the properties of the parameterized curves providing
$\mathrm{BV}_{0}$ solutions.

######  Definition 3.2.

We call a parameterized curve
$(\mathsf{t},\mathsf{q})=(\mathsf{t},\mathsf{u},\mathsf{z},\mathsf{p})\colon[0,S]\to[0,T]\times{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}}$
_admissible_ if it satisfies

$\mathsf{t}\colon[0,S]\to[0,T]$ is nondecreasing, (3.11a)
$\displaystyle(\mathsf{t},\mathsf{u},\mathsf{z},\mathsf{p})\in\mathrm{AC}\big{(}[0,S];[0,T]{\times}\mathrm{BD}(\Omega){\times}H^{\mathrm{m}}(\Omega){\times}{\mathrm{M}_{\mathrm{b}}(\Omega\cup\Gamma_{\mathrm{Dir}};{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}\big{)},$
(3.11b)
$\displaystyle\mathsf{e}=\mathrm{E}(\mathsf{u}+w(\mathsf{t}))-\mathsf{p}\in\mathrm{AC}([0,S];{L^{2}(\Omega;{\mathbb{M}^{n\times
n}_{\mathrm{sym}}})}),$ (3.11c) $\displaystyle\mathsf{t}\text{ is constant in
every connected component of
}A^{\circ}=\\{s\in(0,S)\colon\widetilde{d}_{L^{2}}({-}\mathrm{D}_{z}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s)),\partial{\mathcal{R}}(0))>0\\}.$
(3.11d)

We will denote by
$\mathcal{A}(0,S;[0,T]{\times}{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}})$
the class of admissible parameterized curves from $[0,S]$ to
$[0,T]\times{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}}$.

We are now in a position to give the definition of Balanced Viscosity solution
in the sense of [CL16].

######  Definition 3.3.

We call an admissible parameterized curve
$(\mathsf{t},\mathsf{q})=(\mathsf{t},\mathsf{u},\mathsf{z},\mathsf{p})\in\mathcal{A}(0,S;[0,T]{\times}{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}})$
a Balanced Viscosity solution for the perfectly plastic damage system (1.1) in
the sense of [CL16] (a _$\mathrm{BV}_{0}^{0,0}$ solution_, for short), if it
satisfies the energy-dissipation balance

${\mathcal{E}}_{0}(\mathsf{t}(s_{2}),\mathsf{q}(s_{2}))+\int_{s_{1}}^{s_{2}}{\mathcal{M}}_{0}(\mathsf{t}(r),\mathsf{q}(r),\mathsf{t}^{\prime}(r),\mathsf{q}^{\prime}(r))\;\\!\mathrm{d}r={\mathcal{E}}_{0}(\mathsf{t}(s_{1}),\mathsf{q}(s_{1}))+\int_{s_{1}}^{s_{2}}\partial_{t}{\mathcal{E}}_{0}(\mathsf{t}(r),\mathsf{q}(r))\;\\!\mathrm{d}r$
(3.12)

for every $0\leq s_{1}\leq s_{2}\leq S$.

The existence of $\mathrm{BV}_{0}$ solutions was proved in [CL16, Thm. 5.4]
under the condition that the initial data $q_{0}=(u_{0},z_{0},p_{0})$ for the
perfectly plastic damage system (1.1) fulfill (2.8f) and the additional
condition that

$\displaystyle{\mathrm{D}}_{q}{\mathcal{E}}_{0}(0,q_{0})=\left(-\mathrm{Div}\,\sigma_{0}-F(0),A_{\mathrm{m}}(z_{0})+W^{\prime}(z_{0})+\tfrac{1}{2}\mathbb{C}^{\prime}(z_{0})e_{0}:e_{0},{\mu}p_{0}-(\sigma_{0})_{\mathrm{D}}\right)\in
L^{2}(\Omega;\mathbb{R}^{n}{\times}\mathbb{R}{\times}{\mathbb{M}}^{n{\times}n}_{\mathrm{D}}).$
(3.13)

###  3.3. A differential characterization for $\mathrm{BV}_{0}$ solutions

We now aim to provide a _differential characterization_ for the notion of
$\mathrm{BV}$ solution from Definition 3.2, in terms of a suitable system of
subdifferential inclusions for the displacement variable (which in fact shall
satisfy the elastic equilibrium equation), the damage variable, and the
plastic strain. In order to properly formulate the flow rule governing the
latter, we need the following result; the proof of one implication can be
found [CL16], in turn based on arguments from [DMDM06].

######  Lemma 3.4.

Let an admissible parameterized curve
$(\mathsf{t},\mathsf{q})=(\mathsf{t},\mathsf{u},\mathsf{e},\mathsf{p})\in\mathcal{A}(0,S;[0,T]{\times}{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}})$
satisfy

$\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}(\mathsf{z}(s),\mathsf{p}^{\prime}(s))=\langle{\sansmath\sigma}_{\mathrm{D}}(s)\,|\,\mathsf{p}^{\prime}(s)\rangle\qquad\text{for
a.a.}\,s\in(0,S)$ (3.14)

with ${\sansmath\sigma}(s)=\mathbb{C}(\mathsf{z}(s))\mathsf{e}(s)$,
$\mathsf{e}(s)=e(\mathsf{t}(s))$, and $\langle\cdot\,|\,\cdot\rangle$ the
stress-strain duality from (2.10).

Then, $(\mathsf{t},\mathsf{q})$ satisfies _Hill’s maximum work principle_

$H\left(\mathsf{z}(s),\frac{\;\\!\mathrm{d}{\mathsf{p}}^{\prime}(s)}{\;\\!\mathrm{d}|\mathsf{p}^{\prime}(s)|}\right)|\mathsf{p}^{\prime}(s)|=[{\sansmath\sigma}_{\mathrm{D}}(s):\mathsf{p}^{\prime}(s)]\qquad\text{for
a.a.}\,s\in(0,S),$ (3.15)

where the above equality holds in the sense of measures on
$\Omega\cup\Gamma_{\mathrm{Dir}}$, with
$[{\sansmath\sigma}_{\mathrm{D}}(s):\mathsf{p}^{\prime}(s)]$ the distribution
defined in (2.9). Furthermore, defining
$\mu(s):=\mathcal{L}^{n}+|\mathsf{p}^{\prime}(s)|$ for every $s\in[0,S]$,
there exists $\widehat{{\sansmath\sigma}}_{\mathrm{D}}(s)\in
L_{\mu(s)}^{\infty}(\Omega{\cup}\Gamma_{\mathrm{Dir}};{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})$
such that for almost all $s\in(0,S)$ the following properties hold:

$\displaystyle\widehat{{\sansmath\sigma}}_{\mathrm{D}}(s)={\sansmath\sigma}_{\mathrm{D}}(s)$
$\displaystyle\mathcal{L}^{n}\text{-a.e.\ on }\Omega,$ (3.16a)
$\displaystyle[{\sansmath\sigma}_{\mathrm{D}}(s):\mathsf{p}^{\prime}(s)]=\left(\widehat{{\sansmath\sigma}}_{\mathrm{D}}(s){:}\frac{\;\\!\mathrm{d}\mathsf{p}^{\prime}(s)}{\;\\!\mathrm{d}|\mathsf{p}^{\prime}(s)|}\right)|\mathsf{p}^{\prime}(s)|$
$\displaystyle\text{on }\Omega\cup\Gamma_{\mathrm{Dir}},$ (3.16b)
$\displaystyle\partial_{\pi}H\left(\mathsf{z}(s),\frac{\;\\!\mathrm{d}\mathsf{p}^{\prime}(s)}{\;\\!\mathrm{d}|\mathsf{p}^{\prime}(s)|}\right)\ni\widehat{{\sansmath\sigma}}_{\mathrm{D}}(s)$
$\displaystyle\text{for }|\mathsf{p}^{\prime}(s)|\text{-a.e.
}x\in\Omega\cup\Gamma_{\mathrm{Dir}}.$ (3.16c)

Conversely, (3.16) imply (3.15) which, in turn, gives (3.14).

###### Proof.

We refer to [CL16, Prop. 6.5] for the proof of the fact that (3.14) implies
(3.15) and (3.16). In turn, recalling (2.15), from (3.16c) we infer that

$H\left(\mathsf{z}(s),\frac{\;\\!\mathrm{d}\mathsf{p}^{\prime}(s)}{\;\\!\mathrm{d}|\mathsf{p}^{\prime}(s)|}\right)=\left(\widehat{{\sansmath\sigma}}_{\mathrm{D}}(s){:}\frac{\;\\!\mathrm{d}\mathsf{p}^{\prime}(s)}{\;\\!\mathrm{d}|\mathsf{p}^{\prime}(s)|}\right)\qquad|\mathsf{p}^{\prime}(s)|\text{-a.e.\
in}\,\Omega\cup\Gamma_{\mathrm{Dir}}.$

Combining this with (3.16a) and (3.16b) we conclude (3.15), which yields
(3.14) in view of the definition (2.16) of
$\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}$, and of the definition (2.10) of
the stress-strain duality product. ∎

It is in the sense of (3.16) that we need to understand the (formally written)
inclusion
$\partial_{\pi}H(\mathsf{z},\mathsf{p}^{\prime}\ni{\sansmath\sigma}_{\mathrm{D}}$.

We are now in a position to prove the following differential characterization
for the concept of $\mathrm{BV}$ solution from [CL16]. We mention in advance
that, for notational simplicity, in (3.18b) we have simply written
$\mathsf{z}^{\prime}(s)$ in place of $J(\mathsf{z}^{\prime}(s))$, with
$J:L^{2}(\Omega)\to H^{\mathrm{m}}(\Omega)^{*}$ the Riesz operator.

######  Proposition 3.5.

An admissible parameterized curve
$(\mathsf{t},\mathsf{q})\in\mathcal{A}(0,S;[0,T]{\times}{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}})$
is a $\mathrm{BV}_{0}$ solution to system (1.1) if and only if there exists a
measurable function $\lambda_{\mathsf{z}}:[0,S]\to[0,1]$ such that

$\displaystyle\mathsf{t}^{\prime}(s)\lambda_{\mathsf{z}}(s)=0$
$\displaystyle\text{for a.a.}\ s\in(0,S),$ (3.17)

and $(\mathsf{t},\mathsf{q})$ satisfies for a.a. $s\in(0,S)$

$\displaystyle{\sansmath\sigma}(s)\in{\mathcal{K}}_{\mathsf{z}(s)}(\Omega),\quad-\mathrm{div}\,{\sansmath\sigma}(s)=f(\mathsf{t}(s))\text{
a.e.\ in
}\Omega,\quad[{\sansmath\sigma}(\mathsf{t}(s)){\mathrm{n}}]=g(\mathsf{t}(s))\
\mathscr{H}^{n-1}\text{-a.e.\ on }\Gamma_{\mathrm{Neu}},$ (3.18a)
$\displaystyle(1{-}\lambda_{\mathsf{z}}(s))\,\partial\mathcal{R}(\mathsf{z}^{\prime}(s))+\lambda_{\mathsf{z}}(s)\,\mathsf{z}^{\prime}(s)+(1{-}\lambda_{\mathsf{z}}(s))\,\left[A_{\mathrm{m}}\mathsf{z}(s){+}W^{\prime}(\mathsf{z}(s)){+}\tfrac{1}{2}\mathbb{C}^{\prime}(\mathsf{z}(s))\mathsf{e}(s):\mathsf{e}(s)\right]\ni
0\quad\text{in }H^{\mathrm{m}}(\Omega)^{*},$ (3.18b)
$\displaystyle\partial_{\pi}\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}(\mathsf{z}(s),\mathsf{p}^{\prime}(s))\ni{\sansmath\sigma}_{\mathrm{D}}(s)\quad\text{in
the sense of \eqref{eq:lambda-CL-3-precise}}.$ (3.18c)

In fact, (3.18a) holds at every $s\in(0,S]$.

######  Remark 3.6.

System (3.18) illustrates in a clear way how viscous behavior, _only w.r.t.
the variable $z$_, may arise in the jump regime, namely when the system still
evolves while $\mathsf{t}^{\prime}=0$ (i.e. the external time, recorded by the
function $\mathsf{t}$, is frozen). In that case, the parameter
$\lambda_{\mathsf{z}}$ may be non-zero, thus activating the viscous
contribution to the flow rule (3.18b).

Prior to carrying out the proof of Proposition 3.5, we record the following
key chain-rule inequality, whose proof may be immediately inferred from that
of [CR19, Lemma 7.6] (cf. also Lemma 3.13 ahead).

######  Lemma 3.7.

Along any admissible parameterized curve

$\displaystyle(\mathsf{t},\mathsf{q})\in\mathcal{A}(0,S;[0,T]{\times}{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}})\text{
s.t.
}\mathcal{M}_{0}(\mathsf{t},\mathsf{q},\mathsf{t}^{\prime},\mathsf{q}^{\prime})<+\infty\
\text{a.e.\ in }\,(0,S),$ we have that
$s\mapsto{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s))$ is absolutely
continuous on $[0,S]$ and there holds $\displaystyle\begin{aligned}
&-\frac{\;\\!\mathrm{d}}{\;\\!\mathrm{d}s}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s))+\partial_{t}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s))\,\mathsf{t}^{\prime}(s)\\\
&=-\langle{\mathrm{D}}_{z}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s)),\mathsf{z}^{\prime}(s)\rangle_{H^{\mathrm{m}}}+\langle{\sansmath\sigma}_{\mathrm{D}}(s)\,|\,\mathsf{p}^{\prime}(s)\rangle+\langle\mathrm{Div}\,{\sansmath\sigma}(s){+}F(\mathsf{t}(s)),\mathsf{u}^{\prime}(s)\rangle_{\mathrm{BD}}\\\
&\leq\mathcal{M}_{0}(\mathsf{t}(s),\mathsf{q}(s),\mathsf{t}^{\prime}(s),\mathsf{q}^{\prime}(s))\qquad\text{for
a.a.}\,s\in(0,S).\end{aligned}$ (3.19)

As an immediate consequence of Lemma 3.7 we have the following
characterization of the $\mathrm{BV}_{0}$ solutions from Definition 3.2, which
complements the other characterizations provided in [CL16, Prop. 5.3].

######  Corollary 3.8.

An admissible parameterized curve
$(\mathsf{t},\mathsf{q})\in\mathcal{A}(0,S;[0,T]{\times}{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}})$
is a $\mathrm{BV}_{0}$ solution to system (1.1) if and only if the function
$[0,S]\ni s\mapsto{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s))$ is
absolutely continuous and there holds

$\displaystyle-\frac{\;\\!\mathrm{d}}{\;\\!\mathrm{d}s}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s))+\partial_{t}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s))\,\mathsf{t}^{\prime}(s)$
(3.20)
$\displaystyle=-\langle{\mathrm{D}}_{z}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s)),\mathsf{z}^{\prime}(s)\rangle_{H^{\mathrm{m}}}+\langle{\sansmath\sigma}_{\mathrm{D}}(s)\,|\,\mathsf{p}^{\prime}(s)\rangle+\langle\mathrm{Div}\,{\sansmath\sigma}(s){+}F(\mathsf{t}(s)),\mathsf{u}^{\prime}(s)\rangle_{\mathrm{BD}}=\mathcal{M}_{0}(\mathsf{t}(s),\mathsf{q}(s),\mathsf{t}^{\prime}(s),\mathsf{q}^{\prime}(s))$

for almost all $s\in(0,S)$.

We are now in a position to carry out the

###### Proof of Proposition 3.5.

By Corollary 3.8, $\mathrm{BV}_{0}$ solutions in the sense of Def. 3.2 can be
characterized in terms of (3.20), whence we deduce that
$\mathcal{M}_{0}(\mathsf{t}(s),\mathsf{q}(s),\mathsf{t}^{\prime}(s),\mathsf{q}^{\prime}(s))<\infty$
for almost all $s\in(0,S)$. Therefore,

$\mathcal{D}^{*}(\mathsf{t}(s),\mathsf{q}(s))=0\qquad\text{for
a.a.}\,s\in(0,S).$ (3.21)

In turn, (3.21) is equivalent to the validity of (3.18a). Observe that (3.18a)
extends to every $s\in[0,S]$ by the continuity of the functions
$s\mapsto{\sansmath\sigma}(s)$, $s\mapsto f(\mathsf{t}(s))$, $s\mapsto
g(\mathsf{t}(s))$ and by the continuity w.r.t. the Hausdorff distance of
$s\mapsto{\mathcal{K}}_{\mathsf{z}(s)}(\Omega)$ thanks to ($2.K_{3}$).

In view of (3.21) and recalling the definition (3.10) of ${\mathcal{M}}_{0}$,
(3.20) can be then rewritten as

$-\langle{\mathrm{D}}_{z}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s)),\mathsf{z}^{\prime}(s)\rangle_{H^{\mathrm{m}}}-{\mathcal{R}}(\mathsf{z}^{\prime}(s))-\widetilde{{\mathcal{M}}}_{0,\mathrm{red}}(\mathsf{t}(s),\mathsf{q}(s),\mathsf{t}^{\prime}(s),\mathsf{q}^{\prime}(s))=\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}(\mathsf{z}(s),\mathsf{p}^{\prime}(s))-\langle{\sansmath\sigma}_{\mathrm{D}}(s)\,|\,\mathsf{p}^{\prime}(s)\rangle$

for a.a. $s\in(0,S)$. Now, on the one hand the right-hand side of the above
equality is positive thanks to (2.19). On the other hand, let
$\omega\in\partial{\mathcal{R}}(0)$ satisfy
$\widetilde{d}_{L^{2}}({-}\mathrm{D}_{z}{\mathcal{E}}_{0}(t,q),\partial{\mathcal{R}}(0))=\|{-}{\mathrm{D}}_{z}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s)){-}\omega\|_{L^{2}}$.
Then, we may estimate the left-hand side of the above equality via

$\displaystyle\langle{-}{\mathrm{D}}_{z}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s)),\mathsf{z}^{\prime}(s)\rangle_{H^{\mathrm{m}}}-{\mathcal{R}}(\mathsf{z}^{\prime}(s))-\widetilde{{\mathcal{M}}}_{0,\mathrm{red}}(\mathsf{t}(s),\mathsf{q}(s),\mathsf{t}^{\prime}(s),\mathsf{q}^{\prime}(s))$
$\displaystyle=\begin{array}[t]{c}\begin{array}[t]{c}\underbrace{\langle{-}{\mathrm{D}}_{z}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s)){-}\omega,\mathsf{z}^{\prime}(s)\rangle_{H^{\mathrm{m}}}-\widetilde{{\mathcal{M}}}_{0,\mathrm{red}}(\mathsf{t}(s),\mathsf{q}(s),\mathsf{t}^{\prime}(s),\mathsf{q}^{\prime}(s))}\vspace*{.6em}\end{array}\vspace*{-1em}\\\
{}_{\text{\footnotesize$T_{1}$}}\end{array}+\begin{array}[t]{c}\begin{array}[t]{c}\underbrace{\langle\omega,\mathsf{z}^{\prime}(s)\rangle_{H^{\mathrm{m}}}-{\mathcal{R}}(\mathsf{z}^{\prime}(s))}\vspace*{.6em}\end{array}\vspace*{-1em}\\\
{}_{\text{\footnotesize$T_{2}$}}\end{array}\leq 0,$

where we have used that $T_{1}\leq 0$ by the very definition of
$\widetilde{{\mathcal{M}}}_{0,\mathrm{red}}$ (cf. (3.10c)), while $T_{2}\leq
0$ thanks to (2.15). Therefore, (3.20) is ultimately _equivalent_ to

$\displaystyle\langle{-}{\mathrm{D}}_{z}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s)){-}\omega,\mathsf{z}^{\prime}(s)\rangle_{H^{\mathrm{m}}}=\widetilde{{\mathcal{M}}}_{0,\mathrm{red}}(\mathsf{t}(s),\mathsf{q}(s),\mathsf{t}^{\prime}(s),\mathsf{q}^{\prime}(s))$
$\displaystyle\text{for a.a.}\,s\in(0,S),$ (3.22a)
$\displaystyle\langle\omega,\mathsf{z}^{\prime}(s)\rangle_{H^{\mathrm{m}}}={\mathcal{R}}(\mathsf{z}^{\prime}(s))$
$\displaystyle\text{for a.a.}\,s\in(0,S),$ (3.22b)
$\displaystyle\langle{\sansmath\sigma}_{\mathrm{D}}(s)\,|\,\mathsf{p}^{\prime}(s)\rangle=\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}(\mathsf{z}(s),\mathsf{p}^{\prime}(s))$
$\displaystyle\text{for a.a.}\,s\in(0,S)$ (3.22c)

for every $\omega\in\partial{\mathcal{R}}(0)$ such that
$\|{-}{\mathrm{D}}_{z}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s)){-}\omega\|_{L^{2}}=\widetilde{d}_{L^{2}(\Omega)}({-}\mathrm{D}_{z}{\mathcal{E}}_{0}(t,q),\partial{\mathcal{R}}(0))$.

Now, the same argument as in, e.g., [KRZ13, Prop. 5.1] allows us to infer that
(3.22a) and (3.22b) are _equivalent_ to (3.18b), together with (3.17). In
turn, by Lemma 3.4 it follows that (3.22c) is _equivalent_ to (3.16). All in
all, we have shown the equivalence between (3.20) and (3.17)&(3.18). This
concludes the proof. ∎

###  3.4. $\mathrm{BV}$ solutions via the full vanishing-viscosity approach

The vanishing-viscosity contact potential for the $\mathrm{BV}$ solutions from
[CR19] is the functional
${\mathcal{M}}^{0,0}_{0}:[0,T]\times{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}}\times[0,+\infty)\times{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}}\to[0,+\infty]$
defined via

${\mathcal{M}}^{0,0}_{0}(t,q,t^{\prime},q^{\prime}):={\mathcal{R}}(z^{\prime})+\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}(z,p^{\prime})+{\mathcal{M}}_{0,\mathrm{red}}^{0,0}(t,q,t^{\prime},q^{\prime})$
(3.23a) where for $q=(u,z,p)$ and
$q^{\prime}=(u^{\prime},z^{\prime},p^{\prime})$ we have $\displaystyle\text{if
}t^{\prime}>0,\quad{\mathcal{M}}_{0,\mathrm{red}}^{0,0}(t,q,t^{\prime},q^{\prime}):=\begin{cases}0&\text{if
}\begin{cases}{\mathcal{S}}_{u}{\mathcal{E}}_{0}(t,q)=0,\\\
\widetilde{d}_{L^{2}}({-}{\mathrm{D}}_{z}{\mathcal{E}}_{0}(t,q),\partial{\mathcal{R}}(0))=0,\
\text{and}\\\ {\mathcal{W}}_{p}{\mathcal{E}}_{0}(t,q)=0,\end{cases}\\\
+\infty&\text{otherwise,}\end{cases}$ (3.23b) $\displaystyle\text{if
}t^{\prime}=0,\quad{\mathcal{M}}_{0,\mathrm{red}}^{0,0}(t,q,t^{\prime},q^{\prime}):=\begin{cases}\mathcal{D}(u^{\prime},p^{\prime})\,\mathcal{D}^{*}(t,q)&\text{if
}z^{\prime}=0,\\\
\|z^{\prime}\|_{L^{2}}\widetilde{d}_{L^{2}}({-}\mathrm{D}_{z}{\mathcal{E}}_{0}(t,q),\partial{\mathcal{R}}(0))&\text{if
}\mathcal{D}^{*}(t,q)=0\\\ &\text{ and
}\widetilde{d}_{L^{2}}({-}\mathrm{D}_{z}{\mathcal{E}}_{0}(t,q),\partial{\mathcal{R}}(0))<+\infty,\\\
+\infty&\text{otherwise},\end{cases}$ (3.23c) with
$\mathcal{D}(u^{\prime},p^{\prime}):=\sqrt{\|u^{\prime}\|_{H^{1},\mathbb{D}}^{2}{+}\|p^{\prime}\|_{L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}^{2}}\quad\text{
and }\quad\mathcal{D}^{*}\text{ from \eqref{right-Dstar}.}$ (3.23d)

As previously remarked, for every
$(t,q)\in[0,T]\times{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}}$ we have that
${\mathcal{S}}_{u}{\mathcal{E}}_{0}(t,q)<+\infty$ and
${\mathcal{W}}_{p}{\mathcal{E}}_{0}(t,q)<+\infty$, hence
$\mathcal{D}^{*}(t,q)<+\infty$ and the product
$\mathcal{D}(u^{\prime},p^{\prime})\,\mathcal{D}^{*}(t,q)$ is well defined as
soon as $u^{\prime}\in H^{1}(\Omega;\mathbb{R}^{n})$ and $p^{\prime}\in
L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})$; otherwise, we mean
$\mathcal{D}(u^{\prime},p^{\prime})\,\mathcal{D}^{*}(t,q)=+\infty$.

The (reduced) vanishing-viscosity contact potentials
${\mathcal{M}}_{0,\mathrm{red}}$ from (3.10b) and
${\mathcal{M}}^{0,0}_{0,\mathrm{red}}$ differ from each other, both in their
definition for $t^{\prime}>0$ and for $t^{\prime}=0$. For $t^{\prime}>0$,
${\mathcal{M}}^{0,0}_{0,\mathrm{red}}$ has to additionally enforce elastic
equilibrium (i.e. ${\mathcal{S}}_{u}{\mathcal{E}}_{0}(t,q)=0$) and the
_stability_ constraint that
$\sigma_{\mathrm{D}}\in\partial_{\pi}\mathcal{H}(z,0)$ (i.e.
${\mathcal{W}}_{p}{\mathcal{E}}_{0}(t,q)=0$) since, for the fully rate-
dependent viscous systems, these costraints are no longer fulfilled.
Accordingly, viscous behavior in $u$ and $p$ may intervene in the jump regime
of the rate-independent limit system. This is encoded in the new term
$\mathcal{D}(u^{\prime},p^{\prime})\,\mathcal{D}^{*}(t,q)$ featuring in
(3.23c), which appears in the energy-dissipation balance at jumps (i.e. for
$t^{\prime}=0$), when $z^{\prime}=0$.

The following definition specifies the properties of the parameterized curves
that are $\mathrm{BV}_{0}^{0,0}$ solutions and is to be compared with
Definition 3.2 of admissible parameterized curves in the sense of [CL16].

######  Definition 3.9.

We call a parameterized curve
$(\mathsf{t},\mathsf{q})=(\mathsf{t},\mathsf{u},\mathsf{z},\mathsf{p})\colon[0,S]\to[0,T]\times{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}}$
_admissible in an enhanced sense_ (‘enhanced admissible’ for short) if it
satisfies (3.11a), (3.11b), (3.11c) and, in addition,

$\displaystyle(\mathsf{u},\mathsf{p})\in\mathrm{AC}_{\mathrm{loc}}(B^{\circ};H^{1}(\Omega;\mathbb{R}^{n}){\times}L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})),\text{
where
}B^{\circ}:=\\{s\in(0,S)\colon\mathcal{D}^{*}(\mathsf{t}(s),\mathsf{q}(s))>0\\},$
(3.24a) $\displaystyle\mathsf{t}\text{ is constant in every connected
component of }B^{\circ}.$ (3.24b)

We will denote by
$\mathcal{EA}(0,S;[0,T]{\times}{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}})$
the class of enhanced admissible parameterized curves from $[0,S]$ to
$[0,T]\times{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}}$.

Hence, enhanced admissible curves enjoy better spatial regularity, with
$\mathsf{u}(\cdot)\in H^{1}(\Omega;\mathbb{R}^{n})$ and $\mathsf{p}(\cdot)\in
L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})$, in the set in which
either
${\mathcal{S}}_{u}{\mathcal{E}}_{0}(\mathsf{t}(\cdot),\mathsf{q}(\cdot))>0$ or
${\mathcal{W}}_{p}{\mathcal{E}}_{0}(\mathsf{t}(\cdot),\mathsf{q}(\cdot))>0$.
With that definition at hand, we are now in a position to give the definition
of $\mathrm{BV}$ solution in the sense of [CR19].

######  Definition 3.10.

We call an enhanced admissible parameterized curve
$(\mathsf{t},\mathsf{q})=(\mathsf{t},\mathsf{u},\mathsf{z},\mathsf{p})\in\mathcal{EA}(0,S;[0,T]{\times}{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}})$
a Balanced Viscosity solution for the perfectly plastic damage system (1.1) in
the sense of [CR19] (a _$\mathrm{BV}_{0}^{0,0}$ solution_, for short), if it
satisfies the energy-dissipation balance

${\mathcal{E}}_{0}(\mathsf{t}(s_{2}),\mathsf{q}(s_{2}))+\int_{s_{1}}^{s_{2}}{\mathcal{M}}^{0,0}_{0}(\mathsf{t}(r),\mathsf{q}(r),\mathsf{t}^{\prime}(r),\mathsf{q}^{\prime}(r))\;\\!\mathrm{d}r={\mathcal{E}}_{0}(\mathsf{t}(s_{1}),\mathsf{q}(s_{1}))+\int_{s_{1}}^{s_{2}}\partial_{t}{\mathcal{E}}_{0}(\mathsf{t}(r),\mathsf{q}(r))\;\\!\mathrm{d}r$
(3.25)

for every $0\leq s_{1}\leq s_{2}\leq S$.

The existence of $\mathrm{BV}_{0}^{0,0}$ solutions to system (1.1) was proved
in [CR19, Thm. 7.9] for initial data $q_{0}=(u_{0},z_{0},p_{0})$ complying
with (2.8f) and (3.13).

###  3.5. A differential characterization for $\mathrm{BV}_{0}^{0,0}$
solutions

In this section we provide a differential characterization for
$\mathrm{BV}_{0}^{0,0}$ solutions. Preliminarily, we need to make precise in
which sense we are going to understand the subdifferential inclusions
governing the evolution of the reparameterized displacement and of the plastic
variables. Indeed, by formally writing

$-\lambda\mathrm{div}\,\mathbb{D}\mathrm{E}(\mathsf{u}^{\prime})-(1{-}\lambda)\mathrm{Div}\,{\sansmath\sigma}=(1{-}\lambda)F(\mathsf{t})\quad\text{a.e.\
in }\,(0,S),$

with $\lambda:[0,S]\to[0,1]$ a measurable function (below we will have
$\lambda=\lambda_{\mathsf{u},\mathsf{p}}$), we shall mean

$\displaystyle-\mathrm{Div}\,{\sansmath\sigma}(s)=F(\mathsf{t}(s))\qquad\text{in
}\mathrm{BD}(\Omega)^{*}$ $\displaystyle\text{if }\lambda(s)=0,$ (3.26a)
$\displaystyle\begin{cases}\mathsf{u}^{\prime}(s)\in
H_{\mathrm{Dir}}^{1}(\Omega;\mathbb{R}^{n}),\\\
-\lambda(s)\mathrm{div}\,\mathbb{D}\mathrm{E}(\mathsf{u}^{\prime}(s))-(1{-}\lambda(s))\mathrm{Div}\,{\sansmath\sigma}(s)=(1{-}\lambda(s))F(\mathsf{t}(s))\qquad\text{in
}H^{1}(\Omega;\mathbb{R}^{n})^{*}\end{cases}$ $\displaystyle\text{if
}\lambda(s)>0,$ (3.26b)

where in (3.26) $\mathrm{Div}\,{\sansmath\sigma}(s)$ denotes the restriction
of the functional from (2.3) to $H^{1}(\Omega;\mathbb{R}^{n})$. In particular,
let us emphasize that, when $\lambda>0$ the displacement variable enjoys
additional spatial regularity, and the quasistatic momentum balance (3.26b)
allows for test functions in $H^{1}(\Omega;\mathbb{R}^{n})$. Likewise, by
writing

$(1{-}\lambda)\partial_{\pi}\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}(\mathsf{z},\mathsf{p}^{\prime})+\lambda\mathsf{p}^{\prime}\ni(1{-}\lambda){\sansmath\sigma}_{\mathrm{D}}$

with $\lambda:[0,S]\to[0,1]$ a measurable function, we shall mean

$\displaystyle\partial_{\pi}\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}(\mathsf{z}(s),\mathsf{p}^{\prime}(s))\ni{\sansmath\sigma}_{\mathrm{D}}(s)\quad\text{in
the sense of \eqref{eq:lambda-CL-3-precise} }$ $\displaystyle\text{if
}\lambda(s)=0,$ (3.27a) $\displaystyle\begin{cases}\mathsf{p}^{\prime}(s)\in
L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}}),\\\
(1{-}\lambda(s))\partial_{\pi}\mathcal{H}(\mathsf{z}(s),\mathsf{p}^{\prime}(s))+\lambda(s)\mathsf{p}^{\prime}(s)\ni(1{-}\lambda(s)){\sansmath\sigma}_{\mathrm{D}}(s)\quad\text{a.e.\
in }\,\Omega\end{cases}$ $\displaystyle\text{if }\lambda(s)>0.$ (3.27b)

Namely, the plastic flow rule improves to a pointwise-in-space formulation in
the set $\\{\lambda>0\\}$, whereas in the set $\\{\lambda=0\\}$ it only holds
in the weak form (3.16).

We are now in a position to state our differential characterization of
$\mathrm{BV}_{0}^{0,0}$ solutions. Observe that the definition of enhanced
admissible curve is tailored to the subdifferential inclusions (3.29).

######  Proposition 3.11.

An enhanced admissible parameterized curve
$(\mathsf{t},\mathsf{q})\in\mathcal{EA}(0,S;[0,T]{\times}{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}})$
is a $\mathrm{BV}_{0}^{0,0}$ solution to system (1.1) if and only if there
exist two measurable functions
$\lambda_{\mathsf{u},\mathsf{p}}\,\lambda_{\mathsf{z}}:[0;S]\to[0;1]$ such
that

$\displaystyle\mathsf{t}^{\prime}(s)\lambda_{\mathsf{u},\mathsf{p}}(s)=\mathsf{t}^{\prime}(s)\lambda_{\mathsf{z}}(s)=0$
$\displaystyle\text{for a.a.}\ s\in(0,S),$ (3.28a)
$\displaystyle\lambda_{\mathsf{u},\mathsf{p}}(s)(1{-}\lambda_{\mathsf{z}}(s))=0$
$\displaystyle\text{for a.a.}\ s\in(0,S),$ (3.28b)

and the curves $(\mathsf{t},\mathsf{q})$ satisfy for a.a. $s\in(0,S)$ the
system of subdifferential inclusions

$\displaystyle-\lambda_{\mathsf{u},\mathsf{p}}(s)\mathrm{div}\,\mathbb{D}\mathrm{E}(\mathsf{u}^{\prime}(s))-(1{-}\lambda_{\mathsf{u},\mathsf{p}}(s))\mathrm{Div}\,{\sansmath\sigma}(s)=(1{-}\lambda_{\mathsf{u},\mathsf{p}}(s))F(\mathsf{t}(s)),$
(3.29a)
$\displaystyle(1{-}\lambda_{\mathsf{z}}(s))\,\partial\mathcal{R}(\mathsf{z}^{\prime}(s))+\lambda_{\mathsf{z}}(s)\,\mathsf{z}^{\prime}(s)+(1{-}\lambda_{\mathsf{z}}(s))\,\left(A_{\mathrm{m}}\mathsf{z}(s){+}W^{\prime}(\mathsf{z}(s)){+}\tfrac{1}{2}\mathbb{C}^{\prime}(\mathsf{z}(s))\mathsf{e}(s):\mathsf{e}(s)\right)\ni
0\quad\text{in }H^{\mathrm{m}}(\Omega)^{*},$ (3.29b)
$\displaystyle(1{-}\lambda_{\mathsf{u},\mathsf{p}}(s))\,\partial_{\pi}\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}(\mathsf{z}(s),\mathsf{p}^{\prime}(s))+\lambda_{\mathsf{u},\mathsf{p}}(s)\,\mathsf{p}^{\prime}(s)\ni(1{-}\lambda_{\mathsf{u},\mathsf{p}}(s))\,{\sansmath\sigma}_{\mathrm{D}}(s),$
(3.29c)

where (3.29a) and (3.29c) need to be interpreted as (3.26) and (3.27),
respectively.

######  Remark 3.12.

In comparison to the differential characterization for $\mathrm{BV}_{0}$
solutions provided by system (3.5), system (3.29) features _two_ parameters,
instead of one. Both $\lambda_{\mathsf{z}}$ and
$\lambda_{\mathsf{u},\mathsf{p}}$ have the role of activating the viscous
contributions to the damage flow rule, and to the displacement
equation/plastic flow rule, respectively, in the jump regime (i.e. when
$\mathsf{t}^{\prime}=0$). In fact, the viscous terms in (3.29a) and (3.29c)
are modulated by the same parameter, which reflects the fact that viscous
behavior intervenes for the variables $\mathsf{u}$ and $\mathsf{p}$ equally
(or, in other terms, that $\mathsf{u}$ and $\mathsf{p}$ relax to elastic
equilibrium and rate-independent evolution at the same rate, faster than
$\mathsf{z}$).

As in the case of Prop. 3.5, the proof of Prop. 3.11 will rely on a suitable
chain-rule inequality, which we recall below.

######  Lemma 3.13.

[CR19, Lemma 7.6] Along any _enhanced admissible_ parameterized curve

$\displaystyle(\mathsf{t},\mathsf{q})\in\mathcal{EA}(0,S;[0,T]{\times}{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}})\text{
s.t.
}\mathcal{M}^{0,0}_{0}(\mathsf{t},\mathsf{q},\mathsf{t}^{\prime},\mathsf{q}^{\prime})<+\infty\quad\text{a.e.\
in }\,(0,S)$

we have that

$\displaystyle s\mapsto{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s))\text{ is
absolutely continuous on }[0,S]\text{ and there holds }\text{for
a.a.}\,s\in(0,S)$ (3.30)
$\displaystyle-\frac{\;\\!\mathrm{d}}{\;\\!\mathrm{d}s}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s))+\partial_{t}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s))\,\mathsf{t}^{\prime}(s)$
$\displaystyle=-\langle{\mathrm{D}}_{z}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s)),\mathsf{z}^{\prime}(s)\rangle_{H^{\mathrm{m}}}+\langle{\sansmath\sigma}_{\mathrm{D}}(s)\,|\,\mathsf{p}^{\prime}(s)\rangle+\langle\mathrm{Div}\,{\sansmath\sigma}(s){+}F(\mathsf{t}(s)),\mathsf{u}^{\prime}(s)\rangle_{\mathrm{BD}}$
$\displaystyle\leq\mathcal{M}^{0,0}_{0}(\mathsf{t}(s),\mathsf{q}(s),\mathsf{t}^{\prime}(s),\mathsf{q}^{\prime}(s)).$

For later use, we also record the following consequence of the chain-rule
inequality, cf. [CR19, Proposition 7.7].

######  Corollary 3.14.

An enhanced admissible parameterized curve
$(\mathsf{t},\mathsf{q})=(\mathsf{t},\mathsf{u},\mathsf{z},\mathsf{p})\in\mathcal{EA}(0,S;[0,T]{\times}{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}})$
is a $\mathrm{BV}_{0}^{0,0}$ solution if and only if it satisfies one of the
following _equivalent_ conditions:

  1. (1)

the energy-dissipation balance (3.25) holds as the inequality $\leq$;

  2. (2)

$(\mathsf{t},\mathsf{q})$ fulfills (3.30) as a chain of _equalities_ , i.e.

$\displaystyle-\frac{\;\\!\mathrm{d}}{\;\\!\mathrm{d}s}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s))+\partial_{t}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s))\,\mathsf{t}^{\prime}(s)$
(3.31)
$\displaystyle=-\langle{\mathrm{D}}_{z}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s)),\mathsf{z}^{\prime}(s)\rangle_{H^{m}}+\langle{\sansmath\sigma}_{\mathrm{D}}(s)\,|\,\mathsf{p}^{\prime}(s)\rangle+\langle\mathrm{Div}\,{\sansmath\sigma}(s){+}F(\mathsf{t}(s)),\mathsf{u}^{\prime}(s)\rangle_{\mathrm{BD}}$
$\displaystyle=\mathcal{M}^{0,0}_{0}(\mathsf{t}(s),\mathsf{q}(s),\mathsf{t}^{\prime}(s),\mathsf{q}^{\prime}(s))\qquad\text{for
a.a.}\,s\in(0,S).$

We are now in a position to carry out the

###### Proof of Proposition 3.11.

We exploit the characterization of $\mathrm{BV}_{0}^{0,0}$ solutions in terms
of the chain of equalities (3.31). Now, we shall distinguish three cases:
_Case $1$: $\mathsf{t}^{\prime}(s)>0$._ Then, by the definition (3.23) of
${\mathcal{M}}^{0,0}_{0}$, from
$\mathcal{M}^{0,0}_{0}(\mathsf{t}(s),\mathsf{q}(s),\mathsf{t}^{\prime}(s),\mathsf{q}^{\prime}(s))<\infty$
we infer
$\widetilde{d}_{L^{2}}({-}{\mathrm{D}}_{z}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s)),\partial{\mathcal{R}}(0))=0$
and

${\mathcal{S}}_{u}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s))={\mathcal{W}}_{p}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s))=0.$

By (3.8), The latter property is equivalent to (3.18a), hence we find the
validity of (3.29a) with $\lambda_{\mathsf{u},\mathsf{p}}(s)=0$. All in all,
identity (3.31) reduces to

$\langle{\sansmath\sigma}_{\mathrm{D}}(s)\,|\,\mathsf{p}^{\prime}(s)\rangle-\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}(\mathsf{z}(s),\mathsf{p}^{\prime}(s))=\langle{\mathrm{D}}_{z}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s)),\mathsf{z}^{\prime}(s)\rangle_{H^{\mathrm{m}}}+{\mathcal{R}}(\mathsf{z}^{\prime}(s)).$
(3.32)

Now, since ${\sansmath\sigma}(s)\in\mathcal{K}_{\mathsf{z}(s)}$ for a.a.
$s\in(0,S)$, by (2.18) the above left-hand-side is negative. On the other
hand, from
$\widetilde{d}_{L^{2}}({-}{\mathrm{D}}_{z}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s)),\partial{\mathcal{R}}(0))=0$
we infer that
${\mathcal{R}}(v)\geq\langle{-}{\mathrm{D}}_{z}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s)),v\rangle_{H^{\mathrm{m}}}$
for every $v\in H^{\mathrm{m}}(\Omega)$. Therefore, the above right-hand side
is positive. Hence, both sides are equal to zero. From

$\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}(\mathsf{z}(s),\dot{\mathsf{p}}^{\prime}(s))=\langle{\sansmath\sigma}_{\mathrm{D}}(s)\,|\,\mathsf{p}^{\prime}(s)\rangle$
(3.33)

we infer (recall Lemma 3.4) that (3.29c) holds with
$\lambda_{\mathsf{u},\mathsf{p}}(s)=0$. Likewise, from

${\mathcal{R}}(\mathsf{z}^{\prime}(s))=-\langle{\mathrm{D}}_{z}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s)),\mathsf{z}^{\prime}(s)\rangle_{H^{\mathrm{m}}},$
(3.34)

recalling (2.15) we deduce that
$-{\mathrm{D}}_{z}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s)\in\partial{\mathcal{R}}(\mathsf{z}^{\prime}(s))$,
i.e. the validity of (3.29b) with $\lambda_{\mathsf{z}}(s)=0$.

Conversely, from (3.29b) with $\lambda_{\mathsf{z}}(s)=0$ and (3.29c) with
$\lambda_{\mathsf{u},\mathsf{p}}(s)=1$ we deduce (3.33) and (3.34),
respectively, hence (3.32) which, in this case, is equivalent to (3.31).

_Case $2$: $\mathsf{t}^{\prime}(s)=0$ and $s\in B^{\circ}$_ (with $B^{\circ}$
the set from (3.24a)). Then, from
$\mathcal{M}^{0,0}_{0}(\mathsf{t}(s),\mathsf{q}(s),\mathsf{t}^{\prime}(s),\mathsf{q}^{\prime}(s))<\infty$
we infer that

$\mathsf{z}^{\prime}(s)=0,$ (3.35)

and (3.31) reduces to

$\displaystyle\langle{\sansmath\sigma}_{\mathrm{D}}(s),\mathsf{p}^{\prime}(s)\rangle_{L^{2}}+\langle\mathrm{Div}\,{\sansmath\sigma}(s){+}F(\mathsf{t}(s)),\mathsf{u}^{\prime}(s)\rangle_{H^{1}}$
(3.36)
$\displaystyle={\mathcal{H}}(\mathsf{z}(s),\mathsf{p}^{\prime}(s))+\mathcal{D}(\mathsf{u}^{\prime}(s),\mathsf{p}^{\prime}(s))\,\mathcal{D}^{*}(\mathsf{t}(s),\mathsf{q}(s))$
$\displaystyle={\mathcal{H}}(\mathsf{z}(s),\mathsf{p}^{\prime}(s))+\sqrt{\|\mathsf{u}^{\prime}(s)\|_{H^{1},\mathbb{D}}^{2}{+}\|\mathsf{p}^{\prime}(s)\|_{L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}^{2}}\sqrt{\|\mathrm{Div}\,{\sansmath\sigma}(s){+}F(\mathsf{t}(s))\|_{(H^{1},\mathbb{D})^{*}}^{2}{+}d_{L^{2}}({\sansmath\sigma}_{\mathrm{D}}(s),\partial_{\pi}\mathcal{H}(z,0))^{2}}\,.$

For (3.36), we have used that, on the left-hand side, the duality pairing
involving $\mathsf{u}^{\prime}$ is between $H^{1}(\Omega;\mathbb{R}^{n})$ and
$H^{1}(\Omega;\mathbb{R}^{n})^{*}$ since the admissible curve
$(\mathsf{t},\mathsf{q})$ enjoys the enhanced spatial regularity
$\mathsf{u}^{\prime}\in H^{1}(\Omega;\mathbb{R}^{n})$ on the set $B^{\circ}$.
In turn, the right-hand side of (3.36) has been rewritten in view of (3.7).
Likewise,
$\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}(\mathsf{z}(s),\mathsf{p}^{\prime}(s))={\mathcal{H}}(\mathsf{z}(s),\mathsf{p}^{\prime}(s))$
and the stress-strain duality reduces to the scalar product in $L^{2}$ because
$\mathsf{p}^{\prime}(s)\in L^{2}(\Omega;{\mathbb{M}^{n\times
n}_{\mathrm{sym}}})$. Let us now consider a measurable selection
$s\mapsto\zeta(s)\in\partial_{\pi}{\mathcal{H}}(\mathsf{z}(s),0)$ such that

$d_{L^{2}}({\sansmath\sigma}_{\mathrm{D}}(s),\partial_{\pi}{\mathcal{H}}(z,0))=\|{\sansmath\sigma}_{\mathrm{D}}(s){-}\zeta(s)\|_{L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}.$

Then, (3.36) rewrites as

$\displaystyle\langle{\sansmath\sigma}_{\mathrm{D}}(s)-\zeta(s),\mathsf{p}^{\prime}(s)\rangle_{L^{2}}+\langle\mathrm{Div}\,{\sansmath\sigma}(s){+}F(\mathsf{t}(s)),\mathsf{u}^{\prime}(s)\rangle_{H^{1}}$
$\displaystyle\quad-\sqrt{\|\mathsf{u}^{\prime}(s)\|_{H^{1},\mathbb{D}}^{2}{+}\|\mathsf{p}^{\prime}(s)\|_{L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}^{2}}\sqrt{\|\mathrm{Div}\,{\sansmath\sigma}(s){+}F(\mathsf{t}(s))\|_{(H^{1},\mathbb{D})^{*}}^{2}{+}\|{\sansmath\sigma}_{\mathrm{D}}(s){-}\zeta(s)\|_{L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}^{2}}$
$\displaystyle={\mathcal{H}}(\mathsf{z}(s),\mathsf{p}^{\prime}(s))-\langle\zeta(s),\mathsf{p}^{\prime}(s)\rangle_{L^{2}}.$

While the left-hand side is negative by Cauchy-Schwarz inequality, the right-
hand side is positive since
$\zeta(s)\in\partial_{\pi}{\mathcal{H}}(\mathsf{z}(s),0)$, cf. (2.15). All in
all, we conclude that both sides are equal to zero. Now, combining the fact
that $\zeta(s)\in\partial_{\pi}{\mathcal{H}}(\mathsf{z}(s),0)$ with the
identity

${\mathcal{H}}(\mathsf{z}(s),\mathsf{p}^{\prime}(s))=\langle\zeta(s),\mathsf{p}^{\prime}(s)\rangle_{L^{2}}$

and again resorting to (2.15), we find that

$\zeta(s)\in\partial_{\pi}{\mathcal{H}}(\mathsf{z}(s),\mathsf{p}^{\prime}(s)).$
(3.37)

From the equality

$\displaystyle\sqrt{\|\mathsf{u}^{\prime}(s)\|_{H^{1},\mathbb{D}}^{2}{+}\|\mathsf{p}^{\prime}(s)\|_{L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}^{2}}\sqrt{\|\mathrm{Div}\,{\sansmath\sigma}(s){+}F(\mathsf{t}(s))\|_{(H^{1},\mathbb{D})^{*}}^{2}{+}\|{\sansmath\sigma}_{\mathrm{D}}(s){-}\zeta(s)\|_{L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}^{2}}$
$\displaystyle=\langle{\sansmath\sigma}_{\mathrm{D}}(s)-\zeta(s),\mathsf{p}^{\prime}(s)\rangle_{L^{2}}+\langle\mathrm{Div}\,{\sansmath\sigma}(s){+}F(\mathsf{t}(s)),\mathsf{u}^{\prime}(s)\rangle_{H^{1}}$

we infer that

$\displaystyle-\tilde{\lambda}(s)\mathrm{div}\,\mathbb{D}\mathrm{E}(\mathsf{u}^{\prime}(s))=\mathrm{Div}\,{\sansmath\sigma}(s)+F(\mathsf{t}(s))$
$\displaystyle\text{in}\ H_{\mathrm{Dir}}^{1}(\Omega;\mathbb{R}^{n})^{*},$
(3.38a)
$\displaystyle\tilde{\lambda}(s)\mathsf{p}^{\prime}(s)={\sansmath\sigma}_{\mathrm{D}}(s){-}\zeta(s)$
$\displaystyle\text{a.e.\ in }\,\Omega\,,$ (3.38b) $\displaystyle\text{with
}\tilde{\lambda}(s)=\frac{\sqrt{\|\mathrm{Div}\,{\sansmath\sigma}(s){+}F(\mathsf{t}(s))\|_{(H^{1},\mathbb{D})^{*}}^{2}{+}\|{\sansmath\sigma}_{\mathrm{D}}(s){-}\zeta(s)\|_{L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}^{2}}}{\sqrt{\|\mathsf{u}^{\prime}(s)\|_{H^{1},\mathbb{D}}^{2}{+}\|\mathsf{p}^{\prime}(s)\|_{L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}^{2}}}\,.$
(3.38c)

Combining (3.35), (3.37), and (3.38) we deduce the validity of system (3.29)
with

$\lambda_{\mathsf{z}}(s)=1\quad\text{ and
}\quad\lambda_{\mathsf{u},\mathsf{p}}(s)=\frac{\tilde{\lambda}(s)}{1+\tilde{\lambda}(s)}\in(0,1).$

Conversely, it can be easily checked that, if $\mathsf{t}^{\prime}(s)=0$ and
$s\in B^{\circ}$, the validity of system (3.29) yields (3.36), hence (3.31).

_Case $3$: $\mathsf{t}^{\prime}(s)=0$ and $s\notin B^{\circ}$._ Hence,
$\mathcal{D}^{*}(\mathsf{t}(s),\mathsf{q}(s))=0$, yielding

$-\mathrm{Div}\,{\sansmath\sigma}(s)=F(\mathsf{t}(s)),$

i.e. (3.29a) with $\lambda_{\mathsf{u},\mathsf{p}}(s)=0$. Then, (3.31) reduces
to

$\displaystyle\langle{-}{\mathrm{D}}_{z}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s)),\mathsf{z}^{\prime}(s)\rangle_{H^{\mathrm{m}}}+\langle{\sansmath\sigma}_{\mathrm{D}}(s)\,|\,\mathsf{p}^{\prime}(s)\rangle$
(3.39)
$\displaystyle={\mathcal{R}}(\mathsf{z}^{\prime}(s))+\|\mathsf{z}^{\prime}(s)\|_{L^{2}}\widetilde{d}_{L^{2}}({-}\mathrm{D}_{z}{\mathcal{E}}_{0}(\mathsf{t}(s),\mathsf{q}(s)),\partial{\mathcal{R}}(0))+\mathcal{H}_{\scriptstyle\tiny\mathrm{PP}}(\mathsf{z}(s),\mathsf{p}^{\prime}(s))$

and, arguing in the same way as in the proof of Proposition 3.5 we conclude
the validity of (3.29b) and (3.29c) with
$\lambda_{\mathsf{u},\mathsf{p}}(s)=0$ and some $\lambda_{\mathsf{z}}(s)\geq
0$.

Conversely, it can be proved that (3.29b) and (3.29c) with
$\lambda_{\mathsf{u},\mathsf{p}}(s)=0$ and some $\lambda_{\mathsf{z}}(s)\geq
0$ yield (3.39).

With this, we conclude the proof. ∎

##  4\. A complete characterization of $\mathrm{BV}_{0}^{0,0}$ solutions

The main result of this section is the following theorem, whose proof adapts
that of [MRS16b, Prop. 5.5].

######  Theorem 4.1.

Let
$(\mathsf{t},\mathsf{q})\in\mathcal{EA}(0,S;[0,T]{\times}{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}})$
be a $\mathrm{BV}_{0}^{0,0}$ solution to the perfectly plastic rate-
independent system for damage (1.1). Suppose that $(\mathsf{t},\mathsf{q})$ is
_non-degenerate_ , namely that

$\mathsf{t}^{\prime}(s)+\|\mathsf{u}^{\prime}(s)\|_{H^{1}(\Omega)}+\|\mathsf{z}^{\prime}(s)\|_{H^{\mathrm{m}}(\Omega)}+\|\mathsf{p}^{\prime}(s)\|_{L^{2}(\Omega)}>0\qquad\text{for
a.a.}\,s\in(0,S).$ (4.1)

Set

$\mathfrak{S}:=\\{s\in[0,S]\,:\
\mathcal{D}^{*}(\mathsf{t}(s),\mathsf{q}(s))=0\\}.$

Then, $\mathfrak{S}$ is either empty or it has the form $[s_{*},S]$ for some
$s_{*}\in[0,S]$.

  * (a)

Assume $s_{*}>0$, then for $s\in[0,s_{*})=[0,S]\setminus\mathfrak{S}$ we have
$\mathsf{t}(s)\equiv\mathsf{t}(0)$ and $\mathsf{z}(s)\equiv\mathsf{z}(0)$,
whereas

$\displaystyle\mathcal{D}^{*}(\mathsf{t}(0),\mathsf{q}(0))>0$ (4.2a) and
$(\mathsf{u},\mathsf{p})$ is a solution to the system
$\displaystyle\begin{aligned}
&-\lambda_{\mathsf{u},\mathsf{p}}(s)\mathrm{div}\,\mathbb{D}\mathrm{E}(\mathsf{u}^{\prime}(s))-(1{-}\lambda_{\mathsf{u},\mathsf{p}}(s))\mathrm{Div}\,\mathbb{C}(\mathsf{z}(0))\mathsf{e}(s)=(1{-}\lambda_{\mathsf{u},\mathsf{p}}(s))F(\mathsf{t}(s))\,,\\\
&(1{-}\lambda_{\mathsf{u},\mathsf{p}}(s))\,\partial_{\pi}\mathcal{H}(\mathsf{z}(0),\mathsf{p}^{\prime}(s))+\lambda_{\mathsf{u},\mathsf{p}}(s)\,\mathsf{p}^{\prime}(s)\ni(1{-}\lambda_{\mathsf{u},\mathsf{p}}(s))\,(\mathbb{C}(\mathsf{z}(0))\mathsf{e}(s))_{\mathrm{D}}.\end{aligned}$
(4.2b)

  * (b)

Suppose that $s_{*}<S$. Then,
$\mathcal{D}^{*}(\mathsf{t}(s),\mathsf{q}(s))\equiv 0$ for every
$s\in\mathfrak{S}=[s_{*},S],$ and the curve $(\mathsf{t},\mathsf{q})$ is a
$\mathrm{BV}$ solution to system (1.1) in the sense of Definition 3.3.

Thus, Theorem 4.1 provides a complete characterization of (non-degenerate)
$\mathrm{BV}_{0}^{0,0}$ solutions. It asserts that, if a
$\mathrm{BV}_{0}^{0,0}$ solution $(\mathsf{t},\mathsf{q})$ starts from an
unstable datum $q_{0}$ with $\mathcal{D}^{*}(0,q_{0})>0$, then during an
initial interval the damage variable $\mathsf{z}$ is frozen and the pair
$(\mathsf{u},\mathsf{p})$ evolves, possibly governed by viscosity in both
variables. If it reaches, at some time $s^{*}$, the state in which elastic
equilibrium
(${\mathcal{S}}_{u}{\mathcal{E}}_{0}(\mathsf{t}(s^{*}),\mathsf{q}(s^{*}))=0$)
and the plastic constraint
(${\mathcal{W}}_{p}{\mathcal{E}}_{0}(\mathsf{t}(s^{*}),\mathsf{q}(s^{*}))=0$)
are fulfilled, then it never leaves that state afterwards, and subsequently
$(\mathsf{t},\mathsf{q})$ behaves as a $\mathrm{BV}_{0}$ solution.

###### Proof.

Step $1$: As in the proof of [MRS16b, Prop. 5.5], we start by analyzing the
behavior of a $\mathrm{BV}_{0}^{0,0}$ solution $(\mathsf{t},\mathsf{q})$ on an
interval $(s_{1},s_{2})\subset[0,S]\setminus\mathfrak{S}$. Since
$\mathcal{D}^{*}(\mathsf{t}(s),\mathsf{q}(s))>0$ for all $s\in(s_{1},s_{2})$,
we read from (3.29a) and (3.29c) that $\lambda_{\mathsf{u},\mathsf{p}}>0$ on
$(s_{1},s_{2})$. Thus, (3.28a) yields $\mathsf{t}^{\prime}\equiv 0$ on
$(s_{1},s_{2})$, so that $\mathsf{t}(s)\equiv\mathsf{t}(s_{1})$ for all
$s\in[s_{1},s_{2}]$. Furthermore, (3.28b) gives $\lambda_{\mathsf{z}}\equiv 1$
on $(s_{1},s_{2})$. Combining this with (3.29b) we gather that
$\mathsf{z}^{\prime}\equiv 0$ on $(s_{1},s_{2})$, so that
$\mathsf{z}(s)\equiv\mathsf{z}(s_{1})$ for all $s\in[s_{1},s_{2}]$. From (4.1)
we conclude that

$\mathsf{u}^{\prime}(s)\neq 0\text{ or }\mathsf{p}^{\prime}(s)\neq
0\qquad\text{for a.a.}\,s\in(s_{1},s_{2}),$ (4.3a) and, therefore, from
(3.29a) and (3.29c) we infer that $\lambda_{\mathsf{u},\mathsf{p}}(s)<1$ for
almost all $s\in(s_{1},s_{2})$. Hence, the evolution of
$(\mathsf{t},\mathsf{q})$ in $(s_{1},s_{2})$ is characterized by property
(4.3a), joint with the previously found
$\displaystyle\mathsf{t}(s)\equiv\mathsf{t}(s_{1}),\ \
\mathsf{z}(s)\equiv\mathsf{z}(s_{1})$ $\displaystyle\text{ for all
}s\in[s_{1},s_{2}],$ (4.3b)
$\displaystyle-\mathrm{div}\,\mathbb{D}\mathrm{E}(\mathsf{u}^{\prime}(s))-\widehat{\lambda}(s)\mathrm{Div}\,{\sansmath\sigma}(s)=\widehat{\lambda}(s)F(\mathsf{t}(s_{1}))$
$\displaystyle\text{for a.a.}\,s\in(s_{1},s_{2}),$ (4.3c)
$\displaystyle\widehat{\lambda}(s)\zeta(s)+\mathsf{p}^{\prime}(s)\ni\widehat{\lambda}(s){\sansmath\sigma}_{\mathrm{D}}(s)$
$\displaystyle\text{for a.a.}\,s\in(s_{1},s_{2})\,,$ (4.3d)
$\displaystyle\text{with
}\widehat{\lambda}:=\tfrac{1{-}\lambda_{\mathsf{u},\mathsf{p}}}{\lambda_{\mathsf{u},\mathsf{p}}}=\frac{\sqrt{\|\mathsf{u}^{\prime}\|_{H^{1},\mathbb{D}}^{2}{+}\|\mathsf{p}^{\prime}\|_{L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}^{2}}}{\mathcal{D}^{*}(\mathsf{t},\mathsf{q})}$

(observe that $\widehat{\lambda}=\tilde{\lambda}^{-1}$ with $\tilde{\lambda}$
from, (3.38)). In turn, it can be easily checked that, for a _given_ function
$\widehat{\lambda}:(s_{1},s_{2})\to[0,+\infty)$, the Cauchy problem for system
(4.3b)—(4.3d) does admit a unique solution.

Step $2$: Since the function $[0,S]\ni
s\mapsto\mathcal{D}^{*}(\mathsf{t}(s),\mathsf{q}(s))$ is lower semicontinuous
by [CR19, Lemma 7.8], the set $\mathfrak{S}$ is closed, hence its complement
$[0,S]\setminus\mathfrak{S}$ is relatively open, and thus it is the finite or
countable union of disjoint intervals. Its connected components are of the
form $(s_{1},S]$, $(s_{2},s_{3})$, $[0,s_{4})$ or $[0,S]$ (if
$\mathfrak{S}=\mathchar 31\relax$). By the lower semicontinuity of
$s\mapsto\mathcal{D}^{*}(\mathsf{t}(s),\mathsf{q}(s))$, it is immediate to
check that $s_{j}\in\mathfrak{S}$ for all $j\in\\{1,\ldots,4\\}$.

Now, we aim to show that connected components of the type $(s_{1},S]$ and
$(s_{2},s_{3})$ cannot occur. To this end, let us study the properties of the
$\mathrm{BV}_{0}^{0,0}$ solution $(\mathsf{t},\mathsf{q})$ on an interval of
the type $(s_{1},S]$ or $(s_{2},s_{3})$, with $s_{1},\,s_{2}\in\mathfrak{S}$.
Since $\mathcal{D}^{*}(\mathsf{t}(s_{j}),\mathsf{q}(s_{j}))=0$, we have that
${\mathcal{S}}_{u}{\mathcal{E}}_{0}(\mathsf{t}(s_{j}),\mathsf{q}(s_{j}))=\mathcal{W}_{p}{\mathcal{E}}_{0}(\mathsf{t}(s_{j}),\mathsf{q}(s_{j}))=0$
for $j=1,2$. As shown in Step $1$, the evolution on the intervals $(s_{1},S)$
and $(s_{2},s_{3})$ is characterized by (4.3). Recall that system
(4.3b)–(4.3d) admits a unique solution. Now, since
${\mathcal{S}}_{u}{\mathcal{E}}_{0}(\mathsf{t}(s_{j}),\mathsf{q}(s_{j}))=\mathcal{W}_{p}{\mathcal{E}}_{0}(\mathsf{t}(s_{j}),\mathsf{q}(s_{j}))=0$
for $j=1,2$, it is immediate to check that the constant functions
$\widehat{\mathsf{u}}(s)\equiv\mathsf{u}(s_{j})$ and
$\widehat{\mathsf{p}}(s)\equiv\mathsf{p}(s_{j})$ provide the unique solution
to (4.3b)–(4.3d). Thus, we conclude that $\mathrm{BV}_{0}^{0,0}$ solution
$(\mathsf{t},\mathsf{q})$ on an interval of the type $(s_{1},S]$ or
$(s_{2},s_{3})$ must be constant, which is a contradiction to (4.3a).

Therefore, $[0,S]\setminus\mathfrak{S}$ does not possess connected components
of the form $(s_{1},S]$ or $(s_{2},s_{3})$. Hence, either
$\mathfrak{S}=\mathchar 31\relax$, or $\mathfrak{S}=[s_{*},S]$ for some
$s_{*}>0$. In the latter case, the calculations from Step $1$ show that on
$[0,S]\setminus\mathfrak{S}=[0,s_{*})$ the evolution of
$(\mathsf{t},\mathsf{q})$ is characterized by (4.2).

Step $3$: Suppose that $s_{*}<S$. Clearly,
$\mathcal{D}^{*}(\mathsf{t}(s),\mathsf{q}(s))\equiv 0$ for every
$s\in\mathfrak{S}=[s_{*},S]$. Hence it satisfies system (3.29) with
$\lambda_{\mathsf{u},\mathsf{p}}(s)\equiv 0$ for every $s\in[s_{*},S]$, which
coincides with system (3.18). This concludes the proof. ∎

##  5\. Vanishing-hardening limit of $\mathrm{BV}$ solutions

In this section we carry out the asymptotic analysis as the hardening
parameter $\mu$ tends to 0 for the $\mathrm{BV}$ solutions to system (1.4),
both in the _single-rate_ case (i.e., for $\mathrm{BV}_{0}^{\mu,\nu}$
solutions, with $0<\nu\leq\mu$), and in the _multi-rate_ case (i.e, for
$\mathrm{BV}_{0}^{\mu,0}$ solutions). Indeed, we first address the latter case
in Section 5.1 ahead, while the former will be sketched in Sec. 5.2. For both
analyses, we will resort to some technical results collected in the Appendix.

###  5.1. Vanishing-hardening analysis for _multi-rate_ solutions

As recalled in the Introduction, $\mathrm{BV}$ solutions to the _multi-rate_
system with hardening have been constructed in [CR19] (cf. Theorem 6.13
therein) by passing to the limit in the (reparameterized) version of (1.3) as
the viscosity parameter $\varepsilon$ tends to $0$ simultaneously with the
rate parameter $\nu\downarrow 0$, while the hardening parameter $\mu>0$ stayed
fixed. The solutions accordingly obtained, hereafter referred to as
$\mathrm{BV}_{0}^{\mu,0}$ solutions, thus account for multiple rates in the
system with hardening. In particular, like for $\mathrm{BV}_{0}^{0,0}$
solutions, the way in which viscous behavior in $u$, $z$, and $p$ manifests
itself in the jump regime reflects the fact that the convergence of $u$ and
$p$ to elastic equilibrium and rate-independent evolution has occurred at a
_faster rate_ (as $\nu\downarrow 0$) than that for $z$, cf. Remark 5.2 ahead.

In order to recall the definition of $\mathrm{BV}_{0}^{\mu,0}$ solutions for
fixed $\mu>0$, we need to introduce the related vanishing-viscosity contact
potential

$\displaystyle{\mathcal{M}}_{0}^{\mu,0}:[0,T]\times{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{H}}\times[0,+\infty)\times{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{H}}\to[0,+\infty],$
$\displaystyle{\mathcal{M}}_{0}^{\mu,0}(t,q,t^{\prime},q^{\prime}):={\mathcal{R}}(z^{\prime})+{\mathcal{H}}(z,p^{\prime})+{\mathcal{M}}_{0,\mathrm{red}}^{\mu,0}(t,u,z,p,t^{\prime},u^{\prime},z^{\prime},p^{\prime})$

where

$\displaystyle\text{if }t^{\prime}>0,\quad$
$\displaystyle{\mathcal{M}}_{0,\mathrm{red}}^{\mu,0}(t,q,t^{\prime},q^{\prime}):=\begin{cases}0&\text{if
}\begin{cases}-\mathrm{D}_{u}{\mathcal{E}}_{\mu}(t,q)=0,\\\
-\mathrm{D}_{z}{\mathcal{E}}_{\mu}(t,q)\in\partial{\mathcal{R}}(0),\
\text{and}\\\
-\mathrm{D}_{p}{\mathcal{E}}_{\mu}(t,q)\in\partial_{\pi}{\mathcal{H}}(z,0),\end{cases}\\\
+\infty&\text{otherwise,}\end{cases}$ (5.1a) $\displaystyle\text{if
}t^{\prime}=0,\quad$
$\displaystyle{\mathcal{M}}_{0,\mathrm{red}}^{\mu,0}(t,q,t^{\prime},q^{\prime}):=\begin{cases}\mathcal{D}(u^{\prime},p^{\prime})\,\mathcal{D}^{*,\mu}(t,q)&\text{if
}z^{\prime}=0,\\\
\|z^{\prime}\|_{L^{2}}\,\widetilde{d}_{L^{2}}({-}\mathrm{D}_{z}{\mathcal{E}}_{\mu}(t,q),\partial{\mathcal{R}}(0))&\text{if
}\mathcal{D}^{*,\mu}(t,q)=0\\\ &\text{ and
}\widetilde{d}_{L^{2}}({-}\mathrm{D}_{z}{\mathcal{E}}_{\mu}(t,q),\partial{\mathcal{R}}(0))<+\infty,\\\
+\infty&\text{otherwise}.\end{cases}$ (5.1b) In (5.1) we have employed the
notation
$\displaystyle\mathcal{D}(u^{\prime},p^{\prime}):=\sqrt{\|u^{\prime}\|_{H^{1},\mathbb{D}}^{2}{+}\|p^{\prime}\|_{L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}^{2}},$
(5.1c)
$\displaystyle\mathcal{D}^{*,\mu}(t,q):=\sqrt{\|{-}\mathrm{D}_{u}{\mathcal{E}}_{\mu}(t,q)\|^{2}_{(H^{1},\mathbb{D})^{*}}{+}d_{L^{2}}({-}\mathrm{D}_{p}{\mathcal{E}}_{\mu}(t,q),\partial_{\pi}{\mathcal{H}}(z,0))^{2}}.$

We are now in a position to recall the notion of solution to system (1.4) from
[CR19, Def. 6.10]. Observe that it involves (reparameterized) curves that are
absolutely continuous on the _whole_ interval $[0,S]$, with values in
${\mathbf{Q}}_{\scriptstyle\tiny\mathrm{H}}$.

######  Definition 5.1.

We call a parameterized curve
$(\mathsf{t},\mathsf{q})=(\mathsf{t},\mathsf{u},\mathsf{z},\mathsf{p})\in\mathrm{AC}([0,S];[0,T]{\times}{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{H}})$
a _(parameterized) Balanced Viscosity_ solution to the _multi-rate_ system
with hardening (1.4) (a $\mathrm{BV}_{0}^{\mu,0}$ solution, for short), if
$\mathsf{t}\colon[0,S]\to[0,T]$ is nondecreasing and $(\mathsf{t},\mathsf{q})$
fulfills for all $0\leq s\leq S$ the energy-dissipation balance

${\mathcal{E}}_{\mu}(\mathsf{t}(s),\mathsf{q}(s))+\int_{0}^{s}{\mathcal{M}}_{0}^{\mu,0}(\mathsf{t}(\tau),\mathsf{q}(\tau),\mathsf{t}^{\prime}(\tau),\mathsf{q}^{\prime}(\tau))\;\\!\mathrm{d}\tau={\mathcal{E}}_{\mu}(\mathsf{t}(0),\mathsf{q}(0))+\int_{0}^{s}\partial_{t}{\mathcal{E}}_{\mu}(\mathsf{t}(\tau),\mathsf{q}(\tau))\,\mathsf{t}^{\prime}(\tau)\;\\!\mathrm{d}\tau.$
(5.2)

We say that $(\mathsf{t},\mathsf{q})$ is _non-degenerate_ if it fulfills
(4.1).

######  Remark 5.2.

In [CR19, Prop. 6.11] a characterization was provided for
$\mathrm{BV}_{0}^{\mu,0}$ solutions in terms of a subdifferential system that
features two positive parameters $\lambda_{\mathsf{u},\mathsf{p}}$ and
$\lambda_{\mathsf{z}}$ activating viscous terms in the displacement equation &
plastic flow rule, and in the damage flow rule, respectively. We refrain from
recalling that system because it is completely analogous to (3.29) (with the
same _switching conditions_ (3.28)).

Accordingly, repeating the very same arguments as in the proof of Theorem 4.1,
it is possible to provide an additional characterization of
$\mathrm{BV}_{0}^{\mu,0}$ solutions completely analogous to that from the
latter result. In particular, if a $\mathrm{BV}_{0}^{\mu,0}$ solution
$(\mathsf{t},\mathsf{q})$ originates from an unstable datum $q_{0}$ with
$\mathcal{D}^{*,\mu}(0,q_{0})>0$, then, during an initial interval the damage
variable $\mathsf{z}$ is frozen and the pair $(\mathsf{u},\mathsf{p})$
evolves, possibly in a viscous way. If $(\mathsf{t},\mathsf{q})$ reaches, at
some time $s^{*}$, the state in which elastic equilibrium
($-\mathrm{D}_{u}{\mathcal{E}}_{\mu}(\mathsf{t},\mathsf{q})=0$) and stability
($-\mathrm{D}_{p}{\mathcal{E}}_{\mu}(\mathsf{t},\mathsf{q})\in\partial_{\pi}{\mathcal{H}}(z,0)$)
are fulfilled, then it never leaves that state afterwards, and subsequently
behaves as a Balanced Viscosity solution with viscous behavior in the variable
$\mathsf{z}$, only (namely, the counterpart, for the system with hardening, of
the $\mathrm{BV}_{0}$ concept).

We mention in advance that the analogue of Theorem 4.1 does not hold, instead,
for $\mathrm{BV}_{0}^{\mu,\nu}$ solutions.

We now consider a vanishing sequence $(\mu_{k})_{k}$ and set $\mu=\mu_{k}$. By
[CR19, Theorem 6.13], under the assumptions of Section 2 and (3.13) (cf. also
Remark 5.4 below), for any fixed $k$ there exists a parameterized Balanced
Viscosity solution $(\mathsf{t}_{k},\mathsf{q}_{k})$ in the sense of the
previous definition. Moreover, for the sequence
$(\mathsf{t}_{k},\mathsf{q}_{k})_{k}$ we may assume the validity of the
following a priori estimates:

$\displaystyle\exists\,C>0\ \ \forall\,k\in\mathbb{N}\,\ \text{for
a.a.}\,s\in(0,S)\,:$
$\displaystyle{\mathsf{t}}_{k}^{\prime}(s){+}\|\mathsf{u}_{k}^{\prime}(s)\|_{W^{1,1}(\Omega)}{+}\|\mathsf{z}_{k}^{\prime}(s)\|_{H^{\mathrm{m}}(\Omega)}{+}\|\mathsf{p}_{k}^{\prime}(s)\|_{L^{1}(\Omega)}{+}\sqrt{\mu_{k}}\,\|\mathsf{p}_{k}^{\prime}(s)\|_{L^{2}(\Omega)}$
(5.3)
$\displaystyle\quad{+}\|\mathsf{e}_{k}^{\prime}(s)\|_{L^{2}(\Omega)}{+}\mathcal{D}(\mathsf{u}_{k}^{\prime}(s),\mathsf{p}_{k}^{\prime}(s))\,\mathcal{D}^{*,\mu_{k}}(\mathsf{t}_{k}(s),\mathsf{q}_{k}(s))\leq
C.$

Indeed, the existence of a sequence $(\mathsf{t}_{k},\mathsf{q}_{k})_{k}$
enjoying the bounds (5.3) follows by time-discretization, cf. [CR19, Prop.
4.4], and by a reparameterization argument. Up to a further time
reparameterization we may also assume that the solutions are non-degenerate,
cf. (4.1) and [CR19, Remark 6.9].

######  Theorem 5.3.

Let $(\mu_{k})_{k}$ be a vanishing sequence and
$(\mathsf{t}_{k},\mathsf{q}_{k})_{k}$ be a sequence of
$\mathrm{BV}_{0}^{\mu_{k},0}$ solutions to system (1.4), such that estimate
(5.3) holds.

Then, there exist a (not relabeled) subsequence and a curve
$(\mathsf{t},\mathsf{q})=(\mathsf{t},\mathsf{u},\mathsf{z},\mathsf{p})\in\mathcal{EA}(0,S;[0,T]{\times}{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}})$
such that

  1. (1)

for all $s\in[0,S]$ the following convergences hold as $k\to+\infty$

$\begin{split}&\mathsf{t}_{k}(s)\to\mathsf{t}(s),\quad\mathsf{u}_{k}(s)\overset{*}{\rightharpoonup}\mathsf{u}(s)\text{
in
}\mathrm{BD}(\Omega),\quad\mathsf{z}_{k}(s)\rightharpoonup\mathsf{z}(s)\text{
in }H^{\mathrm{m}}(\Omega),\\\
&\mathsf{e}_{k}(s)\rightharpoonup\mathsf{e}(s)\text{ in
}{L^{2}(\Omega;{\mathbb{M}^{n\times
n}_{\mathrm{sym}}})},\quad\mathsf{p}_{k}(s)\overset{*}{\rightharpoonup}\mathsf{p}(s)\text{
in
}{\mathrm{M}_{\mathrm{b}}(\Omega\cup\Gamma_{\mathrm{Dir}};{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})};\end{split}$
(5.4)

  2. (2)

there exists $\overline{C}>0$ such that for a.e. $s\in(0,S)$ there holds

$\displaystyle{\mathsf{t}}^{\prime}(s){+}\|\mathsf{u}^{\prime}(s)\|_{\mathrm{BD}(\Omega)}{+}\|\mathsf{z}^{\prime}(s)\|_{H^{\mathrm{m}}(\Omega)}{+}\|\mathsf{p}^{\prime}(s)\|_{{\mathrm{M}_{\mathrm{b}}(\Omega\cup\Gamma_{\mathrm{Dir}};{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}}$
(5.5)
$\displaystyle{+}\|\mathsf{e}^{\prime}(s)\|_{{L^{2}(\Omega;{\mathbb{M}^{n\times
n}_{\mathrm{sym}}})}}{+}\mathcal{D}(\mathsf{u}^{\prime}(s),\mathsf{p}^{\prime}(s))\,\mathcal{D}^{*}(\mathsf{t}(s),\mathsf{q}(s))\leq\overline{C};$

  3. (3)

$(\mathsf{t},\mathsf{q})$ is a Balanced Viscosity solution for the perfectly
plastic damage system (1.1) in the sense of Definition 3.10.

######  Remark 5.4.

The validity of Theorem 5.3 extends to sequences
$(\mathsf{t}_{k},\mathsf{q}_{k})_{k}$ originating from initial data
$q_{0}^{k}=(u_{0}^{k},z_{0}^{k},p_{0}^{k})_{k}$ fulfilling

$\|u_{0}^{k}\|_{H_{\mathrm{Dir}}^{1}(\Omega)}+\|z_{0}^{k}\|_{H^{\mathrm{m}}(\Omega)}+\|W(z_{0}^{k})\|_{L^{1}(\Omega)}+\|p_{0}^{k}\|_{L^{2}(\Omega)}+\|{\mathrm{D}}_{q}{\mathcal{E}}_{0}(0,q_{0}^{k})\|_{L^{2}(\Omega)}\leq
C$

with $C$ independent of $k$.

In particular, the last condition yields $\mu_{k}\,p_{0}^{k}\to 0$ in
$L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})$, as needed for (5.10)
below.

###### Proof.

The proof is divided in two steps.

##### Step $1$: Compactness.

For later use, we observe that, due to estimate (5.3),

$\exists\,R>0\ \ \forall\,k\in\mathbb{N}\ \
\forall\,s\in[0,S]\,:\quad\|\mathsf{u}_{k}(s)\|_{\mathrm{BD}(\Omega)}+\|\mathsf{z}_{k}(s)\|_{H^{\mathrm{m}}(\Omega)}+\|\mathsf{p}_{k}(s)\|_{\mathrm{M}_{\mathrm{b}}(\Omega{\cup}\Gamma_{\mathrm{Dir}})}\leq
R.$ (5.6)

By the assumptions on initial data and external loading and by (5.3), we have
$\sup_{s\in[0,S]}{\mathcal{E}}_{\mu_{k}}(\mathsf{t}_{k}(s),\mathsf{q}_{k}(s))\leq
C$ for a constant independent of $k$. In particular, this implies that, as
$k\to\infty$,

${\mu_{k}}\,\mathsf{p}_{k}(s)\to 0\quad\text{in
}L^{2}(\Omega;{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})\text{ for every
$s\in[0,S]$.}$ (5.7)

In view of (5.3), we find a (not relabeled) subsequence and a Lipschitz curve
$(\mathsf{t},\mathsf{q})\in$
$W^{1,\infty}([0,S];[0,T]{\times}{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}})$
such that the following convergences hold as $k\to\infty$

$\displaystyle\mathsf{t}_{k}\overset{*}{\rightharpoonup}\mathsf{t}\quad\text{in
}W^{1,\infty}(0,S;[0,T]),$
$\displaystyle\mathsf{u}_{k}\overset{*}{\rightharpoonup}\mathsf{u}\quad\text{in
}W^{1,\infty}(0,S;\mathrm{BD}(\Omega)),$ (5.8a)
$\displaystyle\mathsf{z}_{k}\overset{*}{\rightharpoonup}\mathsf{z}\quad\text{in
}W^{1,\infty}(0,S;H^{\mathrm{m}}(\Omega)),$ (5.8b)
$\displaystyle\mathsf{e}_{k}\overset{*}{\rightharpoonup}\mathsf{e}\quad\text{in
}W^{1,\infty}(0,S;{L^{2}(\Omega;{\mathbb{M}^{n\times n}_{\mathrm{sym}}})}),$
$\displaystyle\mathsf{p}_{k}\overset{*}{\rightharpoonup}\mathsf{p}\quad\text{in
}W^{1,\infty}(0,S;{\mathrm{M}_{\mathrm{b}}(\Omega\cup\Gamma_{\mathrm{Dir}};{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}),$
(5.8c)

where $\mathsf{e}=\mathrm{E}(\mathsf{u}{+}w(\mathsf{t}))-\mathsf{p}$.
Furthermore, an argument based on the Ascoli-Arzelà theorem (cf. [AGS08, Prop.
3.3.1]) also yields

$\displaystyle\mathsf{u}_{k}\to\mathsf{u}$ $\displaystyle\text{in
}\mathrm{C}^{0}([0,S];\mathrm{BD}(\Omega)_{\mathrm{w}^{*}}),$ (5.9a)
$\displaystyle\mathsf{e}_{k}\to\mathsf{e}$ $\displaystyle\text{in
}\mathrm{C}^{0}([0,S];L^{2}(\Omega;{\mathbb{M}^{n\times
n}_{\mathrm{sym}}})_{\mathrm{w}}),$ (5.9b)
$\displaystyle\mathsf{z}_{k}\to\mathsf{z}$ $\displaystyle\text{in
}\mathrm{C}^{0}([0,S];H^{\mathrm{m}}(\Omega)_{\mathrm{w}}),$ (5.9c)
$\displaystyle\mathsf{p}_{k}\to\mathsf{p}$ $\displaystyle\text{in
}\mathrm{C}^{0}([0,S];{\mathrm{M}_{\mathrm{b}}(\Omega\cup\Gamma_{\mathrm{Dir}};{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}_{\mathrm{w}^{*}}).$
(5.9d)

Indeed, convergences (5.9a) and (5.9d) are to be intended in the spaces
$\mathrm{C}^{0}([0,S];(\mathrm{BD}(\Omega),d_{\mathrm{BD},\mathrm{weak}^{*}}))$
and in
$\mathrm{C}^{0}([0,S];({\mathrm{M}_{\mathrm{b}}(\Omega\cup\Gamma_{\mathrm{Dir}};{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})},d_{\mathrm{M}_{\mathrm{b}},\mathrm{weak}^{*}}))$,
where $d_{\mathrm{BD},\mathrm{weak}^{*}}$ and
$d_{\mathrm{M}_{\mathrm{b}},\mathrm{weak}^{*}}$ metrize the weak∗ topologies
of $\mathrm{BD}(\Omega)$ and
${\mathrm{M}_{\mathrm{b}}(\Omega\cup\Gamma_{\mathrm{Dir}};{\mathbb{M}}^{n{\times}n}_{\mathrm{D}})}$,
respectively, on the balls of radius $R$ that contain $(\mathsf{u}_{k})_{k}$
and $(\mathsf{p}_{k})_{k}$, resp. (cf. (5.6); here we use that
$\mathrm{BD}(\Omega)$ is the dual of a separable space). The second and third
convergences have an analogous meaning. Hence, (5.4) follows.

With the very same arguments as in the proof of [CR19, Prop. 7.9], based on
the estimate

$\exists\,C>0\ \forall\,k\in\mathbb{N}\ \text{for a.a.}\,s\in(0,S)\,:\
\mathcal{D}(\mathsf{u}_{k}^{\prime}(s),\mathsf{p}_{k}^{\prime}(s))\,\mathcal{D}^{*,\mu_{k}}(\mathsf{t}_{k}(s),\mathsf{q}_{k}(s))\leq
C,$

we also prove the enhanced regularity
$(\mathsf{t},\mathsf{q})\in\mathcal{EA}(0,S;[0,T]{\times}{\mathbf{Q}}_{\scriptstyle\tiny\mathrm{PP}})$.

##### Step $2$: energy-dissipation upper estimate.

By Corollary 3.14, it is sufficient to show that the pair
$(\mathsf{t},\mathsf{q})$ complies with the energy-dissipation inequality