A Well-Posed Evolutionary Inclusion in Mechanics

Abstract

We consider an evolutionary inclusion associated with a time-dependent convex in an abstract Hilbert space. We recall a unique solvability result obtained based on arguments of nonlinear equations with maximal monotone operators combined with a penalty method. Then, we state and prove two well-posedness results. Next, we provide three examples of such inclusions that arise in mechanics. The first one concerns an elastic–perfectly plastic constitutive law, while the last two examples are mathematical models that describe the equilibrium of an elastic body and an elastic–perfectly plastic body, respectively, in frictional contact with an obstacle. The contact is bilateral and the friction is modeled with the Tresca friction law. We use our abstract results in the study of these examples to provide the convergence of the solution with respect to the data.

1. Introduction

Various mathematical models in mechanics lead to evolutionary variational or hemivariational inequalities. Motivated by various applications in industry, the theory of variational and hemivariational inequalities has received particular attention in the mathematical literature. General existence and uniqueness results for various classes of stationary, time-dependent and evolutionary inequalities, including their applications in the study of mathematical models of contact, can be found in [1,2,3]. The numerical analysis of these models can be found in [4,5], for instance. References for evolutionary hemivariational inequalities with applications in contact mechanics include [6,7]. There, results on unique solvability have been obtained, by using arguments of pseudomonotonicity for multivalued operators. Comprehensive references for the numerical approximation of hemivariational inequalities are [8,9]. Most of the inequality problems studied in the above mentioned references involve unilateral constraints and are governed by a nondifferentiable function that could be convex or nonconvex. Expressed in terms of the subgradient in the sense of the convex analysis or the sense of Clarke, these inequalities lead to various classes of stationary or evolutionary inclusions, as shown in [10].

Inspired by some mathematical models in solid and contact mechanics, in this paper, we consider an abstract evolutionary inclusion of the form

$$\begin{array}{ccc}& & A\dot{\sigma }(t)+\partial {\psi }_{K\cap \mathsf{\Sigma }(t)}\sigma (t)\ni f(t)\mathrm{a}.\mathrm{e}.t\in (0,T),\hfill \end{array}$$

(1)

$$\begin{array}{ccc}& & \sigma (0)={\sigma }_0.\hfill \end{array}$$

(2)

Here and below, X is a real Hilbert space, $T>0$, $A:X\to X$, K and $\mathsf{\Sigma }(t)$ are a fixed and a time-dependent subsets of X, respectively, ${\psi }_{K\cap \mathsf{\Sigma }(t)}$ represents the indicator function of the set $K\cap \mathsf{\Sigma }(t)$ and $\partial {\psi }_{K\cap \mathsf{\Sigma }(t)}$ denotes its convex subdifferential. Moreover, $f:[0,T]\to X$ is a given function; ${\sigma }_0\in X$ and the dot above denotes the derivative with respect to the time variable t.

Note that in the particular case when $\mathsf{\Sigma }(t)=X$, the inclusion (1) becomes

$$A\dot{\sigma }(t)+\partial {\psi }_K\sigma (t)\ni f(t)\mathrm{a}.\mathrm{e}.t\in (0,T).$$

(3)

Such inclusions model the bahavior of elastic–perfectly plastic materials. There, the unknown $\sigma$ represents the stress field, A is related to the elasticity tensor, and f represents the rate of the deformation. Moreover, the set K represents the convex of elasticity. References in the field are [11,12,13].

On the other hand, when $K=X$, the inclusion (1) becomes

$$A\dot{\sigma }(t)+\partial {\psi }_{\mathsf{\Sigma }(t)}\sigma (t)\ni f(t)\mathrm{a}.\mathrm{e}.t\in (0,T).$$

(4)

Such inclusions arise in the study of quasistatic frictional contact problems with elastic materials. There, the set $\mathsf{\Sigma }(t)$ is determined by the body forces, surface tractions and frictional condition and, again, the unknown $\sigma$ represents the stress field. References in the field are [14,15], where existence and uniqueness results for Cauchy problems of the form (4), (2) have been provided.

The aim of this current paper is two-fold. The first one is to study the well-posedness of the Cauchy problem (1) and (2). Well-posedness concepts originate in the papers of Tykhonov [16] and Levitin–Polyak [17], where the well-posedness of minimization problems was considered. The well-posedness of variational inequalities was studied for the first time in [18,19]. Relevant references in the field are [20,21]. The novelty of our current paper consists in the fact that here, we extend the well-posedness analysis to the Cauchy problem (1) and (2). Our second aim is to illustrate the abstract well-posedness results in the study of three models arising in mechanics: a constitutive model that describes the elastic–perfectly plastic behavior, a quasistatic elastic contact model, and a quasistatic contact model with elastic–perfectly plastic materials. For all these models, we provide well-posedness and convergence results, which represent the second novelty of the current paper.

Our manuscript is structured as follows. In Section 2, we introduce some notation and preliminary results. In Section 3, we present two examples of problems that lead to evolutionary inclusions of the form (3), (2) and (4), (2), respectively, in a specific functional framework. In Section 4, we deal with the evolutionary inclusion (1), (2), in the abstract framework of Hilbert spaces. There, we start by recalling the existence and uniqueness result in [15]; then, we prove two well-posedness results for this inclusion. In Section 5, we apply these abstract results in the study of the inclusions introduced in Section 3. In Section 6, we apply our abstract results in the study of a regularized perfectly plastic frictional contact problem. Finally, in Section 7, we present some conclusions and problems for further research.

2. Notation and Preliminaries

In this section, we present notation and preliminary results we use everywhere in the rest of the manuscript. They concern the abstract framework, the space of second-order tensors and the function spaces associated to a domain $\mathsf{\Omega }\subset {\mathbb{R}}^d$.

The abstract framework. Everywhere in this paper, X represents a real Hilbert space with the inner product ${(·,·)}_X$ and its norm ${\parallel ·\parallel }_X$. The set of parts of X will be denoted by $2^X$ and notation $0_X$, $I_X$ will represent the zero element and the identity operator on X, respectively. For a sequence $\{{\theta }_n\}\subset {\mathrm{I}\mathrm{R}}_{+}$ that converges to zero, we use the shorthand notation $0\le {\theta }_n\to 0$.

Let $M\subset X$. We denote by ${\psi }_M:X\to [0,+\infty )$, the indicator function of M, by $\partial {\psi }_M:X\to 2^X$, its convex subdifferential and by $d(\sigma ,M)$, the distance between an element $\sigma \in X$ and the set M. Then, the following relations hold:

$$\begin{array}{ccc}& & {\psi }_K(\tau )=\left\{\begin{array}{c}0\mathrm{if}\tau \in M,\hfill \\ +\infty \mathrm{otherwise}.\hfill \end{array}\right.\hfill \end{array}$$

(5)

$$\begin{array}{ccc}& & \xi \in \partial {\psi }_M(\sigma )⟺\sigma \in M,{(\xi ,\tau -\sigma )}_X\le 0\forall \tau \in M.\hfill \end{array}$$

(6)

$$\begin{array}{ccc}& & d(\sigma ,M)=\underset{\tau \in M}{inf}{\parallel \sigma -\tau \parallel }_X.\hfill \end{array}$$

(7)

Note that relation (6) will be used in various places below, in order to establish the equivalence between inclusions and variational inequalities. Moreover, if $M\subset X$ is a nonempty closed subset, we use notation $P_M$ for the projection operator on M. We recall that $P_M:X\to M$ and, in addition,

$$d(\sigma ,M)=\parallel \sigma -P_M{\sigma \parallel }_X\forall \sigma \in X.$$

(8)

For a given $T>0$, we use the space

$$C([0,T];X)=\{\tau :[0,T]\to X:\tau \mathrm{is} \mathrm{continuous}\}$$

equipped with the norm

$${\parallel \tau \parallel }_{C([0,T];X)}=\underset{t\in [0,T]}{max}{\parallel \tau (t)\parallel }_X.$$

For simplicity, we use the shorthand notation $C([0,T])$ when $X=\mathbb{R}$, that is, $C([0,T])=C([0,T],\mathbb{R})$. Moreover, we use the standard notation for the function spaces $L^p(0,T;X)$ and $W^{k,p}(0,T;X)$.

The space of second order symmetric tensors. Let $d\in \{1,2,3\}$. We denote by ${\mathbb{S}}^d$ the space of second-order symmetric tensors on ${\mathbb{R}}^d$ endowed with the inner product and norm

$$\mathsf{\sigma }·\mathsf{\tau }={\sigma }_{ij}{\tau }_{ij},\parallel \mathsf{\tau }\parallel ={(\mathsf{\tau }·\mathsf{\tau })}^{1/2}\forall \mathsf{\sigma }=({\sigma }_{ij}),\mathsf{\tau }=({\tau }_{ij})\in {\mathbb{S}}^d.$$

Here and below, the indices i, j, k, l run between 1 and d. Moreover, for simplicity, we denote by 0 the zero element of the spaces ${\mathbb{R}}^d$ and ${\mathbb{S}}^d$ and we keep the notation “$·$” and $\parallel ·\parallel$ for the inner product and the Euclidean norm on the space ${\mathbb{R}}^d$. In addition, we use ${\mathsf{\tau }}^D$ for the deviatoric part of the tensor $\mathsf{\tau }$, that is,

$${\mathsf{\tau }}^D=\mathsf{\tau }-{\displaystyle \frac{1}{d}}(\mathrm{tr}\mathsf{\tau })I_d$$

where $\mathrm{tr}\mathsf{\tau }={\tau }_{ii}$ denotes the trace of the tensor $\mathsf{\tau }=({\tau }_{ij})\in {\mathbb{S}}^d$ and $I_d\in {\mathbb{S}}^d$ represents the identity tensor. A simple estimate shows that

$$\parallel {\mathsf{\tau }}^D\parallel \le {\omega }_d\parallel \mathsf{\tau }\parallel \forall \mathsf{\tau }\in {\mathbb{S}}^d$$

(9)

with ${\omega }_d=1+\sqrt{d}$. We shall use this inequality in Section 5 and Section 6 of this paper.

Function spaces. Let $\mathsf{\Omega }\subset {\mathbb{R}}^d$ ($d\in \{2,3\}$) be a domain with smooth boundary $\mathsf{\Gamma }$, divided into three parts ${\overline{\mathsf{\Gamma }}}_1$, ${\overline{\mathsf{\Gamma }}}_2$ and ${\overline{\mathsf{\Gamma }}}_3$, with ${\mathsf{\Gamma }}_1$, ${\mathsf{\Gamma }}_2$ and ${\mathsf{\Gamma }}_3$ being relatively open and mutually disjoint. Moreover, assume that the $d-1$ measure of ${\mathsf{\Gamma }}_1$, denoted by $meas({\mathsf{\Gamma }}_1)$, is positive. A generic point in $\mathsf{\Omega }\cup \mathsf{\Gamma }$ will be denoted by $x=(x_i)$ and $\mathsf{\nu }$ will represent the unit outward normal to $\mathsf{\Gamma }$. For an element $\mathit{v}\in H^1{(\mathsf{\Omega })}^d$, we still write v for the trace of v to $\mathsf{\Gamma }$ and we denote by $v_{\nu }$ and ${\mathit{v}}_{\tau }$ its normal and tangential components on $\mathsf{\Gamma }$, given by $v_{\nu }=\mathit{v}·\mathsf{\nu }$ and ${\mathit{v}}_{\tau }=\mathit{v}-v_{\nu }\mathsf{\nu }$. In addition, for a tensor-valued field $\mathsf{\sigma }:\mathsf{\Omega }\to {\mathbb{S}}^d$, we denote ${\sigma }_{\nu }=\mathsf{\sigma }\mathsf{\nu }·\mathsf{\nu }$ and ${\mathsf{\sigma }}_{\tau }=\mathsf{\sigma }\mathsf{\nu }-{\sigma }_{\nu }\mathsf{\nu }$.

Next, we consider the spaces V and Q, defined as follows:

$$\begin{array}{ccc}& & V=\{\mathit{v}\in H^1{(\mathsf{\Omega })}^d:\mathit{v}=\mathbf{0}\mathrm{on}{\mathsf{\Gamma }}_1,v_{\nu }=0\mathrm{on}{\mathsf{\Gamma }}_3\},\hfill \\ & & Q=\{\mathsf{\sigma }=({\sigma }_{ij}):{\sigma }_{ij}={\sigma }_{ji}\in L^2(\mathsf{\Omega })\forall i,j=1,\dots ,d\}.\hfill \end{array}$$

The spaces V and Q are real Hilbert spaces endowed with the inner products

$${(\mathit{u},\mathit{v})}_V={\int }_{\mathsf{\Omega }}\mathsf{\epsilon }(\mathit{u})·\mathsf{\epsilon }(\mathit{v})dx,{(\mathsf{\sigma },\mathsf{\tau })}_Q={\int }_{\mathsf{\Omega }}\mathsf{\sigma }·\mathsf{\tau }dx.$$

(10)

Here and below,

$$\mathsf{\epsilon }(\mathit{u})=({\epsilon }_{ij}(\mathit{u})),{\epsilon }_{ij}(\mathit{u})={\displaystyle \frac{1}{2}}(u_{i,j}+u_{j,i}),$$

and $u_{i,j}={\displaystyle \frac{\partial u_i}{\partial x_j}}$, $u_{j,i}={\displaystyle \frac{\partial u_j}{\partial x_i}}$. The associated norms on these spaces will be denoted by ${\parallel ·\parallel }_V$ and ${\parallel ·\parallel }_Q$, respectively. Note also that

$${(\mathit{u},\mathit{v})}_V={(\mathsf{\epsilon }(\mathit{u}),\mathsf{\epsilon }(\mathit{v}))}_Q{,\parallel \mathit{v}\parallel }_V={\parallel \mathsf{\epsilon }(\mathit{v})\parallel }_Q\forall \mathit{u},v\in V.$$

(11)

These equalities are useful in the rest of the manuscript.

3. Two Examples

In this section, we present two examples of evolutionary inclusions of the form (3) and (4), respectively: an elastic–plastic constitutive law and an elastic contact problem.

An elastic–plastic constitutive law. The constitutive law we consider is based on rheological arguments that can be found in [22], for instance. Thus, we consider a connexion in serial of an elastic rheological element with a rigid-plastic rheological element. Therefore, at each time moment $t\in [0,T]$, the total strain $\mathsf{\epsilon }(t)\in {\mathbb{S}}^d$ satisfies the equality

$$\mathsf{\epsilon }(t)={\mathsf{\epsilon }}^E(t)+{\mathsf{\epsilon }}^{RP}(t).$$

(12)

Here, ${\mathsf{\epsilon }}^E(t)$ is the strain in the elastic element and ${\mathsf{\epsilon }}^{RP}(t)$ is the strain in the rigid-plastic element. We denote by $\mathsf{\sigma }(t)\in {\mathbb{S}}^d$ the stress tensor and we recall that, since the connexion is in serial, the stress tensor is the same in the two rheological components we consider.

For the elastic element, we assume that

$${\mathsf{\epsilon }}^E(t)=A\mathsf{\sigma }(t)\forall t\in [0,T],$$

(13)

in which $A=(A_{ijkl}):{\mathbb{S}}^d\to {\mathbb{S}}^d$. Moreover, for the rigid-plastic element, we assume that

$${\dot{\mathsf{\epsilon }}}^{RP}(t)\in \partial {\psi }_K\mathsf{\sigma }(t)\forall t\in [0,T],$$

(14)

where $K\subset {\mathbb{S}}^d$ is a given set. For a stress fields $\mathsf{\sigma }(t)$ such that $\mathsf{\sigma }(t)\in intK$, we have $\partial {\psi }_K\mathsf{\sigma }(t)=0$ and, therefore, Equations (14) and (12) imply that $\mathsf{\epsilon }(t)=A\mathsf{\sigma }(t)$. We conclude that in this case, Equation (12) describes a linearly elastic behavior. For stress fields $\mathsf{\sigma }(t)\in K-intK$, we could have ${\mathsf{\epsilon }}^{RP}(t)\ne 0$ and, therefore, (12) shows that a plastic deformation could appear. It follows from above that the constitutive model we consider describes an elastic–plastic behavior.

We now derive equalities (12) and (13) and use the inclusion (14) to see that

$$A\dot{\mathsf{\sigma }}(t)+\partial {\psi }_K\mathsf{\sigma }(t)\ni \dot{\mathsf{\epsilon }}(t)\forall t\in [0,T].$$

(15)

Here, the strain function $\mathsf{\epsilon }:[0,T]\to {\mathbb{S}}^d$ is assumed to be given. We complete this inclusion with the initial condition $\mathsf{\sigma }(0)={\mathsf{\sigma }}_0$, where ${\mathsf{\sigma }}_0\in {\mathbb{S}}^d$ is given, too. In this way, we obtain the following problem.

Problem 1. 

Find a stress function $\mathsf{\sigma }:[0,T]\to {\mathbb{S}}^d$ such that

$$\begin{array}{ccc}& & A\dot{\mathsf{\sigma }}(t)+\partial {\psi }_K\mathsf{\sigma }(t)\ni \dot{\mathsf{\epsilon }}(t)\mathrm{a}.\mathrm{e}.t\in (0,T),\hfill \end{array}$$

(16)

$$\begin{array}{ccc}& & \mathsf{\sigma }(0)={\mathsf{\sigma }}_0.\hfill \end{array}$$

(17)

Note that Problem 1 is an inclusion of the form (3), (2) in the space $X={\mathbb{S}}^d$. Its well-posedness will be discussed in Section 5 below.

We end the description of this example with the remark that a relevant constitutive law of the form (16) is obtained by considering the von Mises convex, used in [23,24], for instance. It is given by

$$K=\{\mathsf{\tau }\in {\mathbb{S}}^d:\parallel {\mathsf{\tau }}^D\parallel \le k\},$$

(18)

where ${\mathsf{\tau }}^D$ denotes the deviatoric part of $\mathsf{\tau }$ and k is a given yield limit. Then, for stress fields $\mathsf{\sigma }(t)$ such that $\parallel {\mathsf{\sigma }}^D(t)\parallel \le k$, the material described by the constitutive law (15), (18) behaves elastically. For stress fields $\mathsf{\sigma }(t)$ such that $\parallel {\mathsf{\sigma }}^D(t)\parallel =k$, the material could have plastic deformations. Since k is fixed, we refer to the corresponding constitutive law as a perfectly plastic constitutive law.

An elastic contact problem. Consider an elastic body that occupies a bounded domain $\mathsf{\Omega }\subset {\mathbb{R}}^d$ ($d\in \{2,3\}$). As in Section 2, the boundary of the domain is denoted by $\mathsf{\Gamma }$, is assumed to be regular and is divided into three parts, ${\overline{\mathsf{\Gamma }}}_1$, ${\overline{\mathsf{\Gamma }}}_2$ and ${\overline{\mathsf{\Gamma }}}_3$, with ${\mathsf{\Gamma }}_1$, ${\mathsf{\Gamma }}_2$ and ${\mathsf{\Gamma }}_3$ being relatively open and mutually disjoint and, moreover, $meas({\mathsf{\Gamma }}_1)>0$. The body is fixed on the part ${\mathsf{\Gamma }}_1$ of its boundary, is in frictional contact with an obstacle on ${\mathsf{\Gamma }}_3$, the so-called foundation, and is in equilibrium under the action of time-dependent body forces and surface tractions.

We denote by $[0,T]$ the time interval of interest with $T>0$, and, for simplicity, we sometimes skip the dependence of various functions on $x\in \mathsf{\Omega }\cup \mathsf{\Gamma }$. Then, using the notation in Section 2, the elastic contact model we consider is as follows.

Problem 2. 

Find a displacement field $\mathit{u}:\mathsf{\Omega }\times [0,T]\to {\mathbb{R}}^d$ and a stress field $\mathsf{\sigma }:\mathsf{\Omega }\times [0,T]\to {\mathbb{S}}^d$ such that

$$\begin{array}{cccc}\hfill \mathsf{\epsilon }(\mathit{u}(t))& =\mathcal{A}\mathsf{\sigma }(t)\hfill & \hfill \mathrm{in}& \mathsf{\Omega },\hfill \end{array}$$

(19)

$$\begin{array}{cccc}\hfill \mathrm{Div}\mathsf{\sigma }(t)+f_0(t)& =0\hfill & \hfill \mathrm{in}& \mathsf{\Omega },\hfill \end{array}$$

(20)

$$\begin{array}{cccc}\hfill \mathit{u}(t)& =0\hfill & \hfill \mathrm{on}& {\mathsf{\Gamma }}_1,\hfill \end{array}$$

(21)

$$\begin{array}{cccc}\hfill \mathsf{\sigma }(t)\mathsf{\nu }& =f_2(t)\hfill & \hfill \mathrm{on}& {\mathsf{\Gamma }}_2,\hfill \end{array}$$

(22)

$$\begin{array}{cccc}\hfill u_{\nu }(t)=0,-{\mathsf{\sigma }}_{\tau }(t)& =g{\displaystyle \frac{{\dot{\mathit{u}}}_{\tau }(t)}{\parallel {\dot{\mathit{u}}}_{\tau }(t)\parallel }}\mathrm{if}{\dot{\mathit{u}}}_{\tau }(t)\ne 0\hfill & \hfill \mathrm{on}& {\mathsf{\Gamma }}_3\hfill \end{array}$$

(23)

for all $t\in [0,T]$ and, in addition,

$$\mathit{u}(0)={\mathit{u}}_0\mathrm{in}\mathsf{\Omega }.$$

(24)

Note that equation (19) is the linear elastic constitutive law in which $\mathcal{A}$ is a fourth-order tensor. Equation (20) is the equilibrium equation in which $f_0$ denotes the density of the body forces. Condition (21) represents the displacement boundary condition. Condition (22) is the traction condition, which shows that surface tractions of density $f_2$, assumed to be time-dependent, act on ${\mathsf{\Gamma }}_2$. Finally, condition (23) is the frictional contact condition. Equality $u_{\nu }=0$ shows that there is no separation between the body and the foundation and, therefore, the contact is bilateral. The second condition on ${\mathsf{\Gamma }}_3$ represents the quasistatic version of Tresca friction law, in which g is the friction bound, i.e., the magnitude of the limiting friction traction at which slip begins. Finally, (24) represents the initial condition in which $u_0$ is a given displacement field.

In the study of Problem 2, we assume that the tensor $\mathcal{A}$, the density of body forces, the friction bound and the initial displacement satisfy the following conditions.

$$\begin{array}{ccc}& & \left\{\begin{array}{c}\mathcal{A}=({\mathcal{A}}_{ijkl}):\mathsf{\Omega }\times {\mathbb{S}}^d\to {\mathbb{S}}^d\mathrm{is} \mathrm{such} \mathrm{that}\hfill \\ (\mathrm{a}){\mathcal{A}}_{ijkl}={\mathcal{A}}_{klij}={\mathcal{A}}_{jikl}\in L^{\infty }(\mathsf{\Omega }),1\le i,j,k,l\le d,\hfill \\ (\mathrm{b})\mathrm{there}\mathrm{exists}{m}_{\mathcal{A}}>0\mathrm{such}\mathrm{that}\hfill \\ \mathcal{A}\mathsf{\tau }·\mathsf{\tau }\ge m_{\mathcal{A}}{\parallel \mathsf{\tau }\parallel }^2\mathrm{for} \mathrm{all}\mathsf{\tau }\in {\mathbb{S}}^d,\mathrm{a}.\mathrm{e}.\mathrm{in}\mathsf{\Omega }.\hfill \end{array}\right.\hfill \end{array}$$

(25)

$$\begin{array}{ccc}& & f_0\in W^{1,\infty }(0,T;L^2{(\mathsf{\Omega })}^d),f_2\in W^{1,\infty }(0,T;L^2{({\mathsf{\Gamma }}_2)}^d).\hfill \end{array}$$

(26)

$$\begin{array}{ccc}& & g>0.\hfill \end{array}$$

(27)

$$\begin{array}{ccc}& & {\mathit{u}}_0\in V.\hfill \end{array}$$

(28)

We now introduce the operator $A:Q\to Q$, the functions $j:V\to \mathbb{R}$, $f:[0,T]\to V$ and the time-dependent set $\mathsf{\Sigma }(t)$ defined as follows:

$$\begin{array}{ccc}& & {(A\mathsf{\sigma },\mathsf{\tau })}_Q={\int }_{\mathsf{\Omega }}\mathcal{A}\mathsf{\sigma }·\mathsf{\tau }dx\forall \mathsf{\sigma },\mathsf{\tau }\in Q,\hfill \end{array}$$

(29)

$$\begin{array}{ccc}& & j(\mathit{v})={\int }_{{\mathsf{\Gamma }}_3}\parallel {\mathit{v}}_{\tau }\parallel da,\forall \mathit{v}\in V,\hfill \end{array}$$

(30)

$$\begin{array}{ccc}& & {\displaystyle {(f(t),\mathit{v})}_V={\int }_{\mathsf{\Omega }}{f}_0(t)·\mathit{v}dx+{\int }_{{\mathsf{\Gamma }}_2}{f}_2(t)·\mathit{v}da\forall \mathit{v}\in V,t\in [0,T],}\hfill \end{array}$$

(31)

$$\begin{array}{ccc}& & \mathsf{\Sigma }(t)=\left\{\mathsf{\tau }\in Q:{(\mathsf{\tau },\mathsf{\epsilon }(\mathit{v}))}_Q+gj(\mathit{v})\ge {(f(t),\mathit{v})}_V\forall \mathit{v}\in V\right\}\forall t\in [0,T].\hfill \end{array}$$

(32)

Next, we turn to derive a variational formulation of the mechanical problem (19)–(24). To this end, we assume that $(\mathit{u},\mathsf{\sigma })$ are regular functions satisfying (19)–(24). Let $\mathit{v}\in V$ be arbitrary, and let $t\in [0,T]$. Then, using standard arguments we obtain that

$${(\mathsf{\sigma }(t),\mathsf{\epsilon }(\mathit{v})-\mathsf{\epsilon }(\dot{\mathit{u}}(t)))}_Q+gj(\mathit{v})-gj(\dot{\mathit{u}}(t))\ge {(f(t),\mathit{v}-\dot{\mathit{u}}(t))}_V.$$

(33)

We now test in (33) with $\mathit{v}=2\dot{\mathit{u}}(t)$ and $\mathit{v}=0_V$ to see that

$${(\mathsf{\sigma }(t),\mathsf{\epsilon }(\dot{\mathit{u}}(t)))}_Q+gj(\dot{\mathit{u}}(t))={(f(t),\dot{\mathit{u}}(t))}_V.$$

(34)

Therefore, using (33) and (34), we find that

$${(\mathsf{\sigma }(t),\mathsf{\epsilon }(\mathit{v}))}_Q+gj(\mathit{v})\ge {(f(t),\mathit{v})}_V\forall \mathit{v}\in V,$$

which shows that

$$\mathsf{\sigma }(t)\in \mathsf{\Sigma }(t).$$

(35)

On the other hand, by the equality (34) and definition (32), since $\dot{\mathit{u}}(t)\in V$, we find that

$${(\mathsf{\tau }-\mathsf{\sigma }(t),\mathsf{\epsilon }(\dot{\mathit{u}}(t)))}_Q\ge 0\forall \mathsf{\tau }\in \mathsf{\Sigma }(t).$$

Therefore, using the constitutive law (19) and definition (29), we deduce that

$${(A\dot{\mathsf{\sigma }}(t),\mathsf{\tau }-\mathsf{\sigma }(t))}_Q\ge 0\forall \mathsf{\tau }\in \mathsf{\Sigma }(t).$$

(36)

Note that the operator A defined by (29) is invertible and denote by $A^{-1}:Q\to Q$ its inverse. Therefore, since

$${(\mathsf{\epsilon }(\mathit{u}(t)),\mathsf{\tau })}_Q={(A\mathsf{\sigma }(t),\mathsf{\tau })}_Q\forall \mathsf{\tau }\in Q,t\in [0,T],$$

the initial condition (24) implies that

$$\mathsf{\sigma }(0)={\mathsf{\sigma }}_0,$$

(37)

with ${\mathsf{\sigma }}_0$ being the element given by ${\mathsf{\sigma }}_0=A^{-1}\mathsf{\epsilon }({\mathit{u}}_0)$. We now combine (35), (36), (6) and (37) to find the following formulation of the problem (19)–(24), in terms of the stress.

Problem 3. 

Find a stress field $\mathsf{\sigma }:[0,T]\to Q$ such that

$$\begin{array}{ccc}& & A\dot{\mathsf{\sigma }}(t)+\partial {\psi }_{\mathsf{\Sigma }(t)}\mathsf{\sigma }(t)\ni {\mathbf{0}}_Q\mathrm{a}.\mathrm{e}.t\in (0,T),\hfill \\ & & \mathsf{\sigma }(0)={\mathsf{\sigma }}_0.\hfill \end{array}$$

Note that Problem 3 is an inclusion of the form (4), (2) in the space $X=Q$. Its well-posedness will be discussed in Section 5 below.

4. Well-Posedness Results

The well-posedness concepts of nonlinear problems provide a tool which allows to prove both convergence results and the link between problems with different structure. A well-posedness concept requires the existence of a unique solution to the corresponding problem and the convergence to it of the so-called approximating sequences.

In this section, we deal with two well-posedness concepts in the study of the Cauchy problem (1) and (2). To this end, we start by using the equivalence (6) in order to see that this problem is equivalent with the following one.

Problem 4. 

Find a function $\sigma :[0,T]\to X$ such that

$$\begin{array}{ccc}& & \sigma \in K\cap \mathsf{\Sigma }(t)\forall t\in [0,T],\hfill \end{array}$$

(38)

$$\begin{array}{ccc}& & (A\dot{\sigma }(t),\tau -\sigma {(t)}_X\ge {(f(t),\tau -\sigma (t))}_X\forall \tau \in K\cap \mathsf{\Sigma }(t),\mathrm{a}.\mathrm{e}.t\in (0,T),\hfill \end{array}$$

(39)

$$\begin{array}{ccc}& & \sigma (0)={\sigma }_0.\hfill \end{array}$$

(40)

In the study of this problem, we consider the following assumption on the data K, A and f.

$$\begin{array}{ccc}& & \mathrm{K} \mathrm{i}\mathrm{s} \mathrm{a}\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{x} \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{s}\mathrm{e}\mathrm{t} \mathrm{o}\mathrm{f}X.\hfill \end{array}$$

(41)

$$\begin{array}{ccc}& & \left\{\begin{array}{c}A:X\to X\mathrm{i}\mathrm{s} \mathrm{a} \mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{y} \mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e} \mathrm{s}\mathrm{y}\mathrm{m}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c} \mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}, \mathrm{i}.\mathrm{e}.,:\hfill \\ (\mathrm{a})\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} \mathrm{e}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{s}{m}_A>0\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h} \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}{(A\tau ,\tau )}_X\ge m_A{\parallel \tau \parallel }_X^2\forall \tau \in X.\hfill \\ (\mathrm{b}){(A\sigma ,\tau )}_X={(\sigma ,A\tau )}_X\forall \sigma ,\tau \in X.\hfill \end{array}\right.\hfill \end{array}$$

(42)

$$\begin{array}{ccc}& & f\in L^2(0,T;X).\hfill \end{array}$$

(43)

Moreover, concerning the data $\mathsf{\Sigma }(t)$ and ${\sigma }_0$, we assume that there exists a set ${\mathsf{\Sigma }}_0$, a function $\chi$ and a constant $\delta >0$ such that

$$\begin{array}{ccc}& & {\mathsf{\Sigma }}_0 \mathrm{i}\mathrm{s} \mathrm{a} \mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{x} \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{s}\mathrm{e}\mathrm{t} \mathrm{o}\mathrm{f}X.\hfill \end{array}$$

(44)

$$\begin{array}{ccc}& & \chi \in W^{1,\infty }(0,T;X).\hfill \end{array}$$

(45)

$$\begin{array}{ccc}& & \mathsf{\Sigma }(t)={\mathsf{\Sigma }}_0+\chi (t)\forall t\in [0,T].\hfill \end{array}$$

(46)

$$\begin{array}{ccc}& & \chi (t)+z\in K\forall t\in [0,T],\forall z\in {X\mathrm{such}\mathrm{that}\parallel z\parallel }_X\le \delta .\hfill \end{array}$$

(47)

$$\begin{array}{ccc}& & {\sigma }_0=\chi (0)\in \mathsf{\Sigma }(0).\hfill \end{array}$$

(48)

Note that assumption (42) implies that $A:X\to X$ is linear and continuous, that is, there exists $M_A>0$ such that

$${\parallel A\tau \parallel }_X\le M_A{\parallel \tau \parallel }_X\forall \tau \in X.$$

This allows us to consider the energetic norm on the space X defined by

$${\parallel \tau \parallel }_A={(A\tau ,\tau )}^{{\displaystyle \frac{1}{2}}}\forall \tau \in X.$$

(49)

This norm is equivalent to the norm ${\parallel ·\parallel }_X$ on X, since

$$m_A{\parallel \tau \parallel }_X^2\le {\parallel \tau \parallel }_A^2\le M_A{\parallel \tau \parallel }_X^2\forall \tau \in X.$$

(50)

Moreover, note that assumption (44)–(46) describe a particular structure of the time-dependent set $\mathsf{\Sigma }(t)$. Indeed, these assumptions show that, at each moment $t\in [0,T]$, the set $\mathsf{\Sigma }(t)$ is obtained by translating a time-independent set ${\mathsf{\Sigma }}_0$ with a time-dependent vector $\chi (t)$. In addition, conditions (47) and (48) represent a compatibility condition between the data K, $\mathsf{\Sigma }(t)$ and ${\sigma }_0$. Combined with (46), these assumptions show that $0_X\in {\mathsf{\Sigma }}_0$ and, moreover,

$$\chi (t)\in K\cap \mathsf{\Sigma }(t)\forall t\in [0,T].$$

(51)

We conclude from here that the set $K\cap \mathsf{\Sigma }(t)$ is nonempty, for all $t\in [0,T]$.

We have the following existence and uniqueness result.

Theorem 1. 

Assume that (41)–(48) hold. Then, there exists a unique solution $\sigma \in W^{1,2}(0,T;X)$ to the Cauchy problem (38)–(40).

A proof of Theorem 1 can be found in [15], based on arguments on evolutionary equations with multivalued operators combined with a penalty method.

We now proceed with an estimate result that will be useful in the next sections.

Lemma 1. 

Assume that (41)–(48) hold and denote by σ the solution of Problem (38)–(40). Then, there exists a constant $D>0$ that depends on A, χ, f and T such that

$${\parallel \sigma (s)\parallel }_X\le D\forall s\in [0,T].$$

(52)

Proof. 

We use the inclusion (51) and take $\tau =\chi (t)$ in (39) to see that

$${(A\dot{\sigma }(t),\chi (t)-\sigma (t))}_X\ge {(f(t),\chi (t)-\sigma (t))}_X\mathrm{a}.\mathrm{e}.t\in (0,T),$$

which implies that

$$\begin{array}{ccc}& & {(A\dot{\sigma }(t)-A\dot{\chi }(t),\sigma (t)-\chi (t))}_X\le {(f(t)-A\dot{\chi }(t),\sigma (t)-\chi (t))}_X\hfill \\ & & \le \parallel f(t)-A\dot{\chi }{(t)\parallel }_X{\parallel \sigma (t)-\chi (t)\parallel }_X\hfill \\ & & \le {\displaystyle \frac{1}{2}}\parallel f(t)-A\dot{\chi }{(t)\parallel }_X^2+{\displaystyle \frac{1}{2}}{\parallel \sigma (t)-\chi (t)\parallel }_X^2\mathrm{a}.\mathrm{e}.t\in (0,T).\hfill \end{array}$$

Let $s\in [0,T]$. We integrate this inequality on $[0,s]$ and use (49) to find that

$${\parallel \sigma (s)-\chi (s)\parallel }_A^2\le {\parallel \sigma (0)-\chi (0)\parallel }_A^2+\parallel f-A\dot{\chi }{\parallel }_{L^2(0,T;X)}^2+{\int }_0^s{\parallel \sigma (t)-\chi (t)\parallel }_X^{2}dt.$$

Then, using (48) and (50), we deduce that

$${\parallel \sigma (s)-\chi (s)\parallel }_X^2\le {\displaystyle \frac{1}{m_A}}\parallel f-A\dot{\chi }{\parallel }_{L^2(0,T;X)}^2+{\displaystyle \frac{1}{m_A}}{\int }_0^s{\parallel \sigma (t)-\chi (t)\parallel }_X^{2}dt.$$

We now apply the Gronwall argument to see that

$${\parallel \sigma (s)-\chi (s)\parallel }_X^2\le {\displaystyle \frac{1}{m_A}}{\parallel f-A\dot{\chi }\parallel }_{L^2(0,T;X)}^2{e}^{{\displaystyle \frac{s}{m_A}}},$$

which implies the bound (52) with

$$D={\parallel \chi \parallel }_{C([0,T];X)}+{\left({\displaystyle \frac{1}{m_A}}{\parallel f-A\dot{\chi }\parallel }_{L^2(0,T;X)}^2{e}^{{\displaystyle \frac{T}{m_A}}}\right)}^{{\displaystyle \frac{1}{2}}}$$

(53)

and concludes the proof. □

We now introduce the following well-posedness concept, inspired by Tykhonov [16].

Definition 1. 

(a) 

A sequence $\{{\sigma }_n\}\subset W^{1,2}(0,T;X)$ is said to be an approximating sequence for Problem 4 if there exists a sequence $0\le {\theta }_n\to 0$ such that, for any $n\in \mathbb{N}$, the following hold:

$$\begin{array}{ccc}& & {\sigma }_n\in K\cap \mathsf{\Sigma }(t)\forall t\in [0,T],\hfill \end{array}$$

(54)

$$\begin{array}{ccc}& & {(A{\dot{\sigma }}_n(t),\tau -{\sigma }_n(t))}_X+{\theta }_n{\parallel \tau -{\sigma }_n\parallel }_X\ge {(f(t),\tau -{\sigma }_n(t))}_X\hfill \\ & & \forall \tau \in K\cap \mathsf{\Sigma }(t),\mathrm{a}.\mathrm{e}.t\in (0,T),\hfill \end{array}$$

(55)

$$\begin{array}{ccc}& & \parallel {\sigma }_n(0)-{\sigma }_0{\parallel }_X\le {\theta }_n.\hfill \end{array}$$

(56)

(b) 

Problem 4 is said to be well-posed in the sense of Tykhonov if it has a unique solution $\sigma \in W^{1,2}(0,T;X)$ and any approximating sequence converges in $C([0,T];X)$ to σ.

Our first result in this section is the following.

Theorem 2. 

Assume that (41)–(48) hold. Then, Problem 4 is well-posed in the sense of Tykhonov.

Proof. 

The existence and uniqueness of the solution are guaranteed by Theorem 1. Now, let $\{{\sigma }_n\}$ be an approximating sequence. We fix $n\in \mathbb{N}$ and $s\in [0,T]$. Then, we take $\tau ={\sigma }_n(t)$ in (39), $\tau =\sigma (t)$ in (55) and add the resulting inequalities to find that

$$\begin{array}{ccc}& & {(A{\dot{\sigma }}_n(t)-A\dot{\sigma }(t),{\sigma }_n(t)-\sigma (t))}_X\le {\theta }_n{\parallel {\sigma }_n(t)-\sigma (t)\parallel }_X\hfill \\ & & \le {\displaystyle \frac{{\theta }_n^2}{2}}+{\displaystyle \frac{1}{2}}{\parallel {\sigma }_n(t)-\sigma (t)\parallel }_X^2\mathrm{a}.\mathrm{e}.t\in (0,T).\hfill \end{array}$$

We now integrate this inequality on $[0,s]$ and use (49), (40) to see that

$$\parallel {\sigma }_n{(s)-\sigma (s)\parallel }_A^2\le \parallel {\sigma }_n(0)-{\sigma }_0{\parallel }_A^2+T{\theta }_n^2+{\int }_0^s{\parallel {\sigma }_n(t)-\sigma (t)\parallel }_X^{2}dt.$$

Then, using inequalities (50) and (56), we deduce that

$$\parallel {\sigma }_n{(s)-\sigma (s)\parallel }_X^2\le {\displaystyle \frac{M_A+T}{m_A}}{\theta }_n^2+{\displaystyle \frac{1}{m_A}}{\int }_0^s{\parallel {\sigma }_n(t)-\sigma (t)\parallel }_X^{2}dt.$$

We now use the Gronwall’s lemma to see that

$$\parallel {\sigma }_n{(s)-\sigma (s)\parallel }_X\le {\theta }_n\sqrt{{\displaystyle \frac{M_A+T}{m_A}}{e}^{{\displaystyle \frac{s}{m_A}}}}.$$

Therefore, since ${\theta }_n\to 0$, we find that ${\sigma }_n\to \sigma$ in $C([0,T];X)$. □

The solution provided by Theorem 2 depends on the data f, that is $\sigma =\sigma (f)$. Next, we assume that (41), (42), (44)–(48) hold. We reinforce assumption (43) by considering that $f\in L^{\infty }(0,T;X$); for each $n\in \mathbb{N}$, we consider a function $f_n\in L^{\infty }(0,T;X)$, and we denote by ${\sigma }_n$ the solution to Problem 4 with the data $f_n$, that is, ${\sigma }_n=\sigma (f_n)$. We have the following result.

Corollary 1. 

Under the previous assumptions, if $f_n\to f$ in $L^{\infty }(0,T;X)$, then ${\sigma }_n\to \sigma$ in $C([0,T];X)$.

Proof. 

Let $n\in \mathbb{N}$. Then, it follows that

$$\begin{array}{ccc}& & {\sigma }_n(t)\in K\cap \mathsf{\Sigma }(t)\forall t\in [0,T],\hfill \end{array}$$

(57)

$$\begin{array}{ccc}& & (A{\dot{\sigma }}_n(t),\tau -{\sigma }_n(t))\ge {(f_n(t),\tau -{\sigma }_n(t))}_Q\forall \tau \in K\cap \mathsf{\Sigma }(t),\mathrm{a}.\mathrm{e}.t\in (0,T),\hfill \end{array}$$

(58)

$$\begin{array}{ccc}& & {\sigma }_n(0)={\sigma }_0.\hfill \end{array}$$

(59)

Using (58), we deduce that

$$\begin{array}{ccc}& & {(A{\dot{\sigma }}_n(t),\tau -{\sigma }_n(t))}_X+{(f(t)-f_n(t),\tau -{\sigma }_n(t))}_X\hfill \\ & & \ge {(f(t),\tau -{\sigma }_n(t))}_X\forall \tau \in K\cap \mathsf{\Sigma }(t),\mathrm{a}.\mathrm{e}.t\in (0,T),\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}& & {(A{\dot{\sigma }}_n(t),\tau -{\sigma }_n(t))}_X+\parallel f_n{(t)-f(t)\parallel }_X\parallel \tau -{\sigma }_n{(t\parallel }_X\hfill \\ & & \ge {(f(t),\tau -{\sigma }_n(t))}_X\forall \tau \in K\cap \mathsf{\Sigma }(t),\mathrm{a}.\mathrm{e}.t\in (0,T).\hfill \end{array}$$

(60)

Denote

$${\theta }_n={\parallel f_n-f\parallel }_{L^{\infty }(0,T;X)}.$$

(61)

and note that

$${\theta }_n\to 0.$$

(62)

We now combine (57), (60)–(62) to deduce that $\{{\sigma }_n\}$ is an approximating sequence for Problem 4. Corollary 1 is now a direct consequence of Theorem 2 and Definition 1 (b). □

We now proceed by introducing a second well-posedness concept for Problem 4, inspired by Levitin–Polyak [17].

Definition 2. 

(a) 

A sequence $\{{\sigma }_n\}\subset W^{1,2}(0,T;X)$ is said to be an $LP$-approximating sequence for Problem 4 if there exists a sequence $0\le {\theta }_n\to 0$ such that

$$\begin{array}{ccc}& & d({\sigma }_n(t),K\cap \mathsf{\Sigma }(t))\le {\theta }_n\forall t\in [0,T],\hfill \end{array}$$

(63)

$$\begin{array}{ccc}& & {(A{\dot{\sigma }}_n(t),\tau -{\sigma }_n(t))}_X+{\theta }_n(1+\parallel \tau -{\sigma }_n{\parallel }_X)\ge {(f(t),\tau -{\sigma }_n(t))}_X\hfill \\ & & \forall \tau \in K\cap \mathsf{\Sigma }(t),\mathrm{a}.\mathrm{e}.t\in (0,T),\hfill \end{array}$$

(64)

$$\begin{array}{ccc}& & \parallel {\sigma }_n(0)-{\sigma }_0{\parallel }_X\le {\theta }_n.\hfill \end{array}$$

(65)

(b) 

Problem 4 is said to be well-posed in the sense of Levitin–Polyak if it has a unique solution $\sigma \in W^{1,2}(0,T;X)$ and any LP-approximating sequence converges in $C([0,T];X)$ to σ.

It is easy to see that any approximating sequence for Problem 4 is an $LP$-approximating sequence. The converse is not true, as it follows from the one-dimensional example below.

Example 1. 

Consider Problem 4 in the particular case when $X=\mathbb{R}$, $T={\displaystyle \frac{1}{4}}$, $A\tau =\tau$ for all $\tau \in \mathbb{R}$, $K=[0,1]$, $\mathsf{\Sigma }(t)=\left[{\displaystyle \frac{1}{2}}-t,{\displaystyle \frac{3}{2}}-t\right]$, $f(t)=-t-1$, ${\sigma }_0={\displaystyle \frac{1}{2}}$. Note that in this particular case, Problem 4 consists of finding a function $\sigma :[0,T]\to \mathbb{R}$ such that, for all $t\in [0,T]$, the following hold:

$$\begin{array}{ccc}& & \sigma (t)\in \left[{\displaystyle \frac{1}{2}}-t,1\right],\hfill \end{array}$$

(66)

$$\begin{array}{ccc}& & \dot{\sigma }(t)(\tau -\sigma (t))\ge (-t-1)(\tau -\sigma (t))\forall \tau \in \left[{\displaystyle \frac{1}{2}}-t,1\right],\hfill \end{array}$$

(67)

$$\begin{array}{ccc}& & \sigma (0)={\displaystyle \frac{1}{2}}.\hfill \end{array}$$

(68)

It is easy to see that conditions (41)–(48) are satisfied with ${\mathsf{\Sigma }}_0=[0,1],\chi (t)={\displaystyle \frac{1}{2}}-t$ for all $t\in [0,{\displaystyle \frac{1}{4}}]$ and $\delta ={\displaystyle \frac{1}{4}}$. Moreover, the unique solution to problem (66)–(68) is the function given by $\sigma (t)={\displaystyle \frac{1}{2}}-t$ for all $t\in [0,{\displaystyle \frac{1}{4}}]$.

Consider now the sequence $\{{\sigma }_n\}\in W^{1,2}(0,T;X)$ defined by

$${\sigma }_n(t)={\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n}}-t\forall t\in \left[0,{\displaystyle \frac{1}{4}}\right],n\in \mathbb{N}.$$

Then, it is easy to see that $\{{\sigma }_n\}$ is an $LP$-approximating sequence for problem (66)–(68), with ${\theta }_n={\displaystyle \frac{1}{n}}$, for all $n\in \mathbb{N}$. Nevertheless, condition ${\sigma }_n(t)\in \left[{\displaystyle \frac{1}{2}}-t,1\right]$ is not satisfied and, therefore, $\{{\sigma }_n\}$ is not an approximating sequence for problem (66)–(68). We conclude from here that there exist $LP$-approximating sequences that are not approximating sequences, as claimed. As a consequence, we conclude that the well-posedness concept in the sense of Levitin–Polyak of Problem 4 is more general than its well-posedness concept in the sense of Tykhonov.

Our second result in this section is the following.

Theorem 3. 

Assume that (41)–(48) hold. Then, Problem 4 is well-posed in the sense of Levitin–Polyak.

Proof. 

The unique solvability is guaranteed by Theorem 1. Let $\{{\sigma }_n\}$ be an $LP$-approximating sequence. We fix $n\in \mathbb{N}$ and $s\in [0,T]$ and define the functions $v_n:[0,T]\to X$, $w_n:[0,T]\to X$ by equalities

$$v_n(t)=P_{K\cap \mathsf{\Sigma }(t)}{\sigma }_n(t),w_n(t)={\sigma }_n(t)-P_{K\cap \mathsf{\Sigma }(t)}{\sigma }_n(t)\forall t\in [0,T],$$

with $P_{K\cap \mathsf{\Sigma }(t)}:X\to K\cap \mathsf{\Sigma }(t)$ being the projection operator on set $K\cap \mathsf{\Sigma }(t)$. Note that this definition is possible since $K\cap \mathsf{\Sigma }(t)$ is a nonempty closed convex of X. Then, it is easy to see that

$${\sigma }_n(t)=v_n(t)+w_n(t)\forall t\in [0,T]$$

(69)

and, since (8) shows that $\parallel w_n{(t)\parallel }_X=d({\sigma }_n(t),K\cap \mathsf{\Sigma }(t))$, condition (63) implies that

$$\parallel w_n{(t)\parallel }_X\le {\theta }_n\forall t\in [0,T].$$

(70)

We now take $\tau =v_n(t)$ in (39), $\tau =\sigma (t)$ in (64), both in $K\cap \mathsf{\Sigma }(t)$; then, we add the resulting inequalities to find that

$$\begin{array}{ccc}& & {(A{\dot{\sigma }}_n(t),{\sigma }_n(t)-\sigma (t))}_X+{(A\dot{\sigma }(t),\sigma (t)-v_n(t))}_X\le {\theta }_n(1+\parallel {\sigma }_n(t)-\sigma (t){\parallel }_X)\hfill \\ & & +{(f(t),{\sigma }_n(t)-\sigma (t))}_X+{(f(t),\sigma (t)-v_n(t))}_X\mathrm{a}.\mathrm{e}.t\in (0,T).\hfill \end{array}$$

Next, using (69), we find that

$$\begin{array}{ccc}& & {(A{\dot{\sigma }}_n(t)-A\dot{\sigma }(t),{\sigma }_n(t)-\sigma (t))}_X\le -{(A\dot{\sigma }(t),w_n(t))}_X\hfill \\ & & +{\theta }_n(1+\parallel {\sigma }_n(t)-\sigma (t){\parallel }_X)+{(f(t),w_n(t))}_X\mathrm{a}.\mathrm{e}.t\in (0,T).\hfill \end{array}$$

We now use inequality (70) to see that

$$\begin{array}{ccc}& & {(A{\dot{\sigma }}_n(t)-A\dot{\sigma }(t),{\sigma }_n(t)-\sigma (t))}_X\le \hfill \\ & & +{\theta }_n(1+\parallel A\dot{\sigma }{(t)\parallel }_X+{\parallel f(t)\parallel }_X)+{\theta }_n{\parallel {\sigma }_n(t)-\sigma (t)\parallel }_X\mathrm{a}.\mathrm{e}.t\in (0,T).\hfill \end{array}$$

(71)

Then, we use the elementary inequalities

$$1+a+b\le 2+{\displaystyle \frac{1}{2}}(a^2+b^2),ab\le {\displaystyle \frac{1}{2}}(a^2+b^2),$$

to find that

$$\begin{array}{ccc}& & 1+\parallel A\dot{\sigma }{(t)\parallel }_X+{\parallel f(t)\parallel }_X\le 2+{\displaystyle \frac{1}{2}}(\parallel A\dot{\sigma }{(t)\parallel }_X^2+{\parallel f(t)\parallel }_X^2),\hfill \end{array}$$

(72)

$$\begin{array}{ccc}& & {\theta }_n\parallel {\sigma }_n{(t)-\sigma (t)\parallel }_X\le {\displaystyle \frac{{\theta }_n^2}{2}}+{\displaystyle \frac{1}{2}}{\parallel {\sigma }_n(t)-\sigma (t)\parallel }_X^2.\hfill \end{array}$$

(73)

We substitute the inequalities (72) and (73) in (71); then, we integrate the resulting inequality on $[0,s]$ and use definition (49) together with the initial condition (40) to see that

$${\displaystyle \frac{1}{2}}\parallel {\sigma }_n{(s)-\sigma (s)\parallel }_A^2\le {\displaystyle \frac{1}{2}}\parallel {\sigma }_n(0)-{\sigma }_0{\parallel }_A^2+C_1{\theta }_n+C_2{\theta }_n^2+{\int }_0^s{\parallel {\sigma }_n(t)-\sigma (t)\parallel }_X^{2}dt$$

where here and below, $C_i$ denote various positive constants that do not depend on n and s. Next, employing inequalities (50) and (56), we deduce that

$$\parallel {\sigma }_n{(s)-\sigma (s)\parallel }_X^2\le C_3{\theta }_n+C_4{\theta }_n^2+C_5{\int }_0^s{\parallel {\sigma }_n(t)-\sigma (t)\parallel }_{X}dt.$$

We now use the Gronwall argument and the convergence ${\theta }_n\to 0$, to see that ${\sigma }_n\to \sigma$ in $C([0,T];X)$, which concludes the proof. □

It follows from Definition 2 that conditions (63) and (65) with ${\theta }_n\to 0$ are necessary conditions for the well-posedness (in the sense of Levitin–Polyak) of Problem 4. Moreover, Theorem 3 guarantees that, combining these conditions with inequality (64), we obtain a package of three sufficient conditions that guarantee the well-posedness (in the sense of Levitin–Polyak) of this problem. The question of whether the condition (64) is a necessary condition for the well-posedness (in the sense of Levitin–Polyak) of Problem 4 is an open question that deserves to be studied in the future.

5. Well-Posedness of the Problems in Section 3

In this section, we provide some applications of Theorem 3 in the study of the mathematical models introduced in Section 3. In this way, we obtain existence, uniqueness and convergence results in the study of the corresponding models. The results we present here show the importance of the well-posedness concept we introduced in Section 4.

An elastic–plastic constitutive law. We start with Problem 1, which we restate in the following equivalent form.

Problem 5. 

Find a stress function $\mathsf{\sigma }:[0,T]\to {\mathbb{S}}^d$ such that

$$\begin{array}{ccc}& & \mathsf{\sigma }(t)\in K\forall t\in [0,T],\hfill \\ & & A\dot{\mathsf{\sigma }}(t)·(\mathsf{\tau }-\mathsf{\sigma }(t))\ge \dot{\mathsf{\epsilon }}(t)·(\mathsf{\tau }-\mathsf{\sigma }(t))\forall \mathsf{\tau }\in K,\mathrm{a}.\mathrm{e}.t\in (0,T),\hfill \\ & & \mathsf{\sigma }(0)={\mathsf{\sigma }}_0.\hfill \end{array}$$

In the study of this problem, we assume the following.

$$\begin{array}{ccc}& & K \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{v}\mathrm{o}\mathrm{n} \mathrm{M}\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{s} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{x} \mathrm{g}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n} \mathrm{b}\mathrm{y} (18) \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} \mathrm{k}>0.\hfill \end{array}$$

(74)

$$\begin{array}{ccc}& & A:{\mathbb{S}}^d\to {\mathbb{S}}^d \mathrm{i}\mathrm{s} \mathrm{a} \mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{y} \mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d} \mathrm{s}\mathrm{y}\mathrm{m}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c} \mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{o}\mathrm{r}.\hfill \end{array}$$

(75)

$$\begin{array}{ccc}& & \mathsf{\epsilon }\in W^{1,2}(0,T;{\mathbb{S}}^d).\hfill \end{array}$$

(76)

$$\begin{array}{ccc}& & {\mathsf{\sigma }}_0\in {\mathbb{S}}^d\mathrm{and}\parallel {\mathsf{\sigma }}_0\parallel \le {\displaystyle \frac{\tilde{k}}{{\omega }_d}}\mathrm{with}\mathrm{some}\tilde{k}<k.\hfill \end{array}$$

(77)

Recall that, here and below, ${\omega }_d$ is the positive constant in (9).

Our first result in the study of Problem 5 is the following.

Theorem 4. 

Assume that (74)–(77) hold. Then, Problem 5 is well-posed in the sense of Levitin–Polyak.

Proof. 

We check the validity of the conditions (41)–(48) on the space $X={\mathbb{S}}^d$, with $f=\dot{\mathsf{\epsilon }}$ and $\mathsf{\Sigma }(t)=X={\mathbb{S}}^d$. First, we note that conditions (41)–(43) are guaranteed by assumptions (74)–(76). Next, it is easy to see that conditions (44)–(46) and (48) are satisfied with ${\mathsf{\Sigma }}_0={\mathbb{S}}^d$ and $\chi (t)={\mathsf{\sigma }}_0$ for all $t\in [0,T]$. Assume now that $z\in {\mathbb{S}}^d$ is such that $\parallel z\parallel \le \delta$ with $\delta ={\displaystyle \frac{k-\tilde{k}}{{\omega }_d}}>0$. Then, using (9) and (77), we have

$$\parallel {({\mathsf{\sigma }}_0+z)}^D\parallel \le \parallel {\mathsf{\sigma }}_0^D\parallel +\parallel z^D\parallel \le \tilde{k}+{\omega }_d\delta \le k,$$

which implies that condition (47) is satisfied, too. Theorem 4 is now a direct consequence of Theorem 3. □

Note that Theorem 4 combined with Definition 2 guarantees the existence of a unique solution to Problem 5, provided that conditions (74)–(77) hold. The solution depends on the data k, $\mathsf{\epsilon }$ en ${\mathsf{\sigma }}_0$, that is, $\mathsf{\sigma }=\mathsf{\sigma }(k,\mathsf{\epsilon },{\mathsf{\sigma }}_0)$.

Next, we assume that (74), (75), (77) hold and we reinforce assumption (76) by considering that $\mathsf{\epsilon }\in W^{1,\infty }(0,T;{\mathbb{S}}^d$), Then, for each $n\in \mathbb{N}$, we consider a constant $k_n>0$, a function ${\mathsf{\epsilon }}_n\in W^{1,\infty }(0,T;{\mathbb{S}}^d)$ and a tensor ${\mathsf{\sigma }}_{0n}$, which satisfies the conditions

$${\mathsf{\sigma }}_{0n}\in {\mathbb{S}}^d\mathrm{and}\parallel {\mathsf{\sigma }}_{0n}\parallel \le {\displaystyle \frac{\widetilde{k_n}}{{\omega }_d}}\mathrm{with}\mathrm{some}{\tilde{k}}_n<k_n.$$

(78)

Moreover, we denote by ${\mathsf{\sigma }}_n$ the solution of Problem 5 with the data $k_n$, ${\mathsf{\epsilon }}_n$ and ${\mathsf{\sigma }}_{0n}$, that is, ${\mathsf{\sigma }}_n=\mathsf{\sigma }(k_n,{\mathsf{\epsilon }}_n,{\mathsf{\sigma }}_{0n})$. We have the following result.

Corollary 2. 

Under the previous assumptions, if $k_n\ge k$ for all $n\in \mathbb{N}$ and

$$k_n\to k,{\mathsf{\epsilon }}_n\to \mathsf{\epsilon }\mathrm{in}{W}^{1,\infty }(0,T;{\mathbb{S}}^d),{\mathsf{\sigma }}_{0n}\to {\mathsf{\sigma }}_0\mathrm{in}{\mathbb{S}}^d,$$

(79)

then

$${\mathsf{\sigma }}_n\to \mathsf{\sigma }\mathrm{in}C([0,T];{\mathbb{S}}^d).$$

(80)

Proof. 

Let $n\in \mathbb{N}$. Then, it follows that

$$\begin{array}{ccc}& & {\mathsf{\sigma }}_n(t)\in K_n\forall t\in [0,T],\hfill \end{array}$$

(81)

$$\begin{array}{ccc}& & A{\dot{\mathsf{\sigma }}}_n(t)·(\mathsf{\tau }-{\mathsf{\sigma }}_n(t))\ge {\dot{\mathsf{\epsilon }}}_n(t)·(\mathsf{\tau }-{\mathsf{\sigma }}_n(t))\forall \mathsf{\tau }\in K_n,\mathrm{a}.\mathrm{e}.t\in (0,T),\hfill \end{array}$$

(82)

$$\begin{array}{ccc}& & {\mathsf{\sigma }}_n(0)={\mathsf{\sigma }}_{0n}\hfill \end{array}$$

(83)

where recall, the set $K_n$ is given by

$$K_n=\{\mathsf{\tau }\in {\mathbb{S}}^d:\parallel {\mathsf{\tau }}^D\parallel \le k_n\}.$$

(84)

Let ${\tilde{\mathsf{\sigma }}}_n:[0,T]\to {\mathbb{S}}^d$ the function defined by

$${\tilde{\mathsf{\sigma }}}_n(t)={\displaystyle \frac{k}{k_n}}{\mathsf{\sigma }}_n(t)\forall t\in [0,T].$$

(85)

Let $t\in [0,T]$. Then, it easy to see that ${\tilde{\mathsf{\sigma }}}_n(t)\in K$ for all $t\in [0,T]$ and, therefore, since $k_n\ge k$, we find that

$$d({\mathsf{\sigma }}_n(t),K)\le \parallel {\mathsf{\sigma }}_n(t)-{\tilde{\mathsf{\sigma }}}_n(t)\parallel =\parallel {\mathsf{\sigma }}_n(t)-{\displaystyle \frac{k}{k_n}}{\mathsf{\sigma }}_n(t)\parallel \le {\displaystyle \frac{k_n-k}{k}}\parallel {\mathsf{\sigma }}_n(t)\parallel .$$

(86)

On the other hand, Lemma 1, equality (53) and the convergence ${\mathsf{\epsilon }}_n\to \mathsf{\epsilon }\in W^{1,\infty }(0,T;{\mathbb{S}}^d)$ show that $\parallel {\mathsf{\sigma }}_n{(t)\parallel }_X\le D$ with a constant D which does not depend on n and t. We conclude form (86) that

$$d({\mathsf{\sigma }}_n(t),K)\le D{\displaystyle \frac{k_n-k}{k}}.$$

(87)

Let $\mathsf{\tau }\in K$. Then, since $k_n\ge k$, it follows that $\mathsf{\tau }\in K_n$ and, using (82) we deduce that

$$\begin{array}{ccc}& & A{\dot{\mathsf{\sigma }}}_n(t)·(\mathsf{\tau }-{\mathsf{\sigma }}_n(t))\ge {\dot{\mathsf{\epsilon }}}_n(t)·(\mathsf{\tau }-{\mathsf{\sigma }}_n(t))=({\dot{\mathsf{\epsilon }}}_n(t)-\dot{\mathsf{\epsilon }}(t))·(\mathsf{\tau }-{\mathsf{\sigma }}_n(t))\hfill \\ & & +\dot{\mathsf{\epsilon }}(t)·(\mathsf{\tau }-{\mathsf{\sigma }}_n(t))\mathrm{a}.\mathrm{e}.t\in (0,T),\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}& & A{\dot{\mathsf{\sigma }}}_n(t)·(\mathsf{\tau }-{\mathsf{\sigma }}_n(t))+\parallel {\dot{\mathsf{\epsilon }}}_n(t)-\dot{\mathsf{\epsilon }}(t)\parallel \parallel \mathsf{\tau }-{\mathsf{\sigma }}_n(t)\parallel \hfill \\ & & \ge \dot{\mathsf{\epsilon }}(t)·(\mathsf{\tau }-{\mathsf{\sigma }}_n(t))\mathrm{a}.\mathrm{e}.t\in (0,T).\hfill \end{array}$$

(88)

Finally, (83) yields

$$\parallel {\mathsf{\sigma }}_n(0)-{\mathsf{\sigma }}_0\parallel =\parallel {\mathsf{\sigma }}_{0n}-{\mathsf{\sigma }}_0\parallel .$$

(89)

Denote

$${\theta }_n=max\left\{D{\displaystyle \frac{k_n-k}{k}},\parallel {\dot{\mathsf{\epsilon }}}_n-\dot{\mathsf{\epsilon }}{\parallel }_{L^{\infty }(0,T;{\mathbb{S}}^d)},\parallel {\mathsf{\sigma }}_{0n}-{\mathsf{\sigma }}_0\parallel \right\}$$

(90)

and note that the assumptions (79) imply that

$${\theta }_n\to 0.$$

(91)

We now combine (87)–(91) to deduce that $\{{\mathsf{\sigma }}_n\}$ is an $LP$-approximating sequence for Problem 5. The convergence (80) is now a direct consequence of Theorem 4 and Definition 2 (b)). □

An elastic contact problem. We now proceed with the analysis of Problem 3. To this end, we assume (25)–(27), use notation (29)–(32) and, in addition, we consider the compatibility condition

$${\mathsf{\sigma }}_0=\mathsf{\epsilon }(\mathit{f}(0)).$$

(92)

Next, using (6), we restate Problem 3 in the following equivalent form.

Problem 6. 

Find a stress field $\mathsf{\sigma }:[0,T]\to Q$ such that

$$\begin{array}{ccc}& & \mathsf{\sigma }\in \mathsf{\Sigma }(t)\forall t\in [0,T],\hfill \end{array}$$

(93)

$$\begin{array}{ccc}& & {(A\dot{\mathsf{\sigma }}(t),\mathsf{\tau }-\mathsf{\sigma }(t))}_Q\ge 0\forall \mathsf{\tau }\in \mathsf{\Sigma }(t),\mathrm{a}.\mathrm{e}.t\in (0,T),\hfill \end{array}$$

(94)

$$\begin{array}{ccc}& & \mathsf{\sigma }(0)={\mathsf{\sigma }}_0.\hfill \end{array}$$

(95)

Our first result in the study of Problem 6 is the following.

Theorem 5. 

Assume (25)–(27 and (92). Then, Problem 6 is well-posed in the sense of Levitin–Polyak.

Proof. 

We check the validity of the conditions (41)–(48) on the space $X=Q$, with $\mathit{f}=0_Q$ and $K=X=Q$. First, we note that conditions (41), (43) and (47) are obviously satisfied, while condition (42) follows from assumption (25). Next, we consider the set ${\mathsf{\Sigma }}_0$ and the function $\mathsf{\chi }$ defined by

$$\begin{array}{ccc}& & {\mathsf{\Sigma }}_0=\left\{\mathsf{\tau }\in Q:{(\mathsf{\tau },\mathsf{\epsilon }(v))}_Q+gj(v)\ge 0\forall v\in V\right\},\hfill \end{array}$$

(96)

$$\begin{array}{ccc}& & \mathsf{\chi }(t)=\mathsf{\epsilon }(\mathit{f}(t))\forall t\in [0,T].\hfill \end{array}$$

(97)

It is easy to see that condition (44) is satisfied and, using assumptions (26) combined with notation (31) and equalities (10), it follows that conditions (45) and (46) hold, too. Moreover, assumption (92) guarantees that condition (48) holds. Theorem 5 is now a direct consequence of Theorem 3. □

Note that Theorem 5 combined with Definition 2 guarantees the existence of a unique solution to Problem 6, provided that conditions (25)–(27) and (92) hold. The solution depends on g, that is, $\mathsf{\sigma }=\mathsf{\sigma }(g)$. Next, we assume that (25)–(27) hold; for each $n\in \mathbb{N}$, we consider a constant $g_n>0$ and we denote by ${\mathsf{\sigma }}_n$ the solution of Problem 6 with the data $g_n$, that is, ${\mathsf{\sigma }}_n=\mathsf{\sigma }(g_n)$. Then, we have the following result.

Corollary 3. 

Under the previous assumptions, if $g_n\ge g$ for all $n\in \mathbb{N}$ and $g_n\to g$, then, ${\mathsf{\sigma }}_n\to \mathsf{\sigma }$ in $C([0,T];Q)$.

Proof. 

Let $n\in \mathbb{N}$. Then, it follows that

$$\begin{array}{ccc}& & {\mathsf{\sigma }}_n(t)\in {\mathsf{\Sigma }}_n(t)\forall t\in [0,T],\hfill \end{array}$$

(98)

$$\begin{array}{ccc}& & {(A{\dot{\mathsf{\sigma }}}_n(t),\mathsf{\tau }-{\mathsf{\sigma }}_n(t))}_Q\ge 0\forall \mathsf{\tau }\in {\mathsf{\Sigma }}_n(t),\mathrm{a}.\mathrm{e}.t\in (0,T),\hfill \end{array}$$

(99)

$$\begin{array}{ccc}& & {\mathsf{\sigma }}_n(0)={\mathsf{\sigma }}_0\hfill \end{array}$$

(100)

where recall, the set ${\mathsf{\Sigma }}_n(t)$ is given by

$${\mathsf{\Sigma }}_n(t)=\left\{\mathsf{\tau }\in Q:{(\mathsf{\tau },\mathsf{\epsilon }(\mathit{v}))}_Q+g_{n}j(\mathit{v})\ge {(\mathit{f}(t),\mathit{v})}_V\forall \mathit{v}\in V\right\}.$$

(101)

We now consider the function ${\tilde{\mathsf{\sigma }}}_n:[0,T]\to {\mathbb{S}}^d$ defined by

$${\tilde{\mathsf{\sigma }}}_n(t)={\displaystyle \frac{g}{g_n}}{\mathsf{\sigma }}_n(t)-{\displaystyle \frac{g}{g_n}}\mathsf{\epsilon }(\mathit{f}(t))+\mathsf{\epsilon }(\mathit{f}(t))\forall t\in [0,T].$$

(102)

Let $\mathit{v}\in V$ and $t\in [0,T]$. Then, using (102) and (11), we have

$$\begin{array}{ccc}& & {({\tilde{\mathsf{\sigma }}}_n(t),\mathsf{\epsilon }(\mathit{v}))}_Q+gj(\mathit{v})-{(\mathit{f}(t),\mathit{v})}_V\hfill \\ & & ={\displaystyle \frac{g}{g_n}}\left[{({\mathsf{\sigma }}_n(t),\mathsf{\epsilon }(\mathit{v}))}_Q-{(\mathit{f}(t),\mathit{v})}_V\right]+gj(\mathit{v})\hfill \end{array}$$

(103)

and, since ${\mathsf{\sigma }}_n(t)\in {\mathsf{\Sigma }}_n(t)$, we find that

$${({\mathsf{\sigma }}_n(t),\mathsf{\epsilon }(\mathit{v}))}_Q-{(\mathit{f}(t),\mathit{v})}_V\ge -g_{n}j(\mathit{v}).$$

(104)

We conclude from (103) and (104) that

$${({\tilde{\mathsf{\sigma }}}_n(t),\mathsf{\epsilon }(\mathit{v}))}_Q+gj(\mathit{v})-{(\mathit{f}(t),\mathit{v})}_V\ge 0,$$

which implies that

$${\tilde{\mathsf{\sigma }}}_n(t)\in \mathsf{\Sigma }(t).$$

(105)

This inclusion combined with (11) and inequality $g_n\ge g$ shows that

$$d({\mathsf{\sigma }}_n(t),\mathsf{\Sigma }(t))\le {\parallel {\mathsf{\sigma }}_n(t)-{\tilde{\mathsf{\sigma }}}_n(t)\parallel }_Q\le {\displaystyle \frac{g_n-g}{g}}\left(\parallel {\mathsf{\sigma }}_n{(t)\parallel }_Q+{\parallel \mathit{f}(t)\parallel }_V\right).$$

We now use (52), (53) to see that there exists $D>0$, which does not depend on n and t such that

$$d({\mathsf{\sigma }}_n(t),\mathsf{\Sigma }(t))\le {\displaystyle \frac{g_n-g}{g}}\left(D+{\parallel \mathit{f}\parallel }_{C([0,T];V)}\right).$$

(106)

Let $\mathsf{\tau }\in \mathsf{\Sigma }(t)$. Since $g_n\ge g$, it follows that $\mathsf{\tau }\in {\mathsf{\Sigma }}_n(t)$ and, using (99) we deduce that

$${(A{\dot{\mathsf{\sigma }}}_n(t),\mathsf{\tau }-{\mathsf{\sigma }}_n(t))}_Q\ge 0\forall \mathsf{\tau }\in \mathsf{\Sigma }(t),\mathrm{a}.\mathrm{e}.t\in (0,T).$$

(107)

Denote

$${\theta }_n={\displaystyle \frac{g_n-g}{g}}\left(D+{\parallel \mathit{f}\parallel }_{C([0,T];V)}\right)$$

(108)

and note that the assumption $g_n\to g$ implies that

$${\theta }_n\to 0.$$

(109)

We now combine (106)–(109) to deduce that $\{{\mathsf{\sigma }}_n\}$ is an $LP$-approximating sequence for Problem 6. Corollary 3 follows from Theorem 5 and Definition 2 (b)). □

6. A Regularized Contact Problem

In this section, we assume (25)–(27), (92) and use the notation (29)–(32). Our aim is to extend the results obtained in the study of Problem 3 to the following Cauchy problem.

Problem 7. 

Find a stress field $\mathsf{\sigma }:[0,T]\to Q$ such that

$$\begin{array}{ccc}& & A\dot{\mathsf{\sigma }}(t)+\partial {\psi }_{\mathcal{K}\cap \mathsf{\Sigma }(t)} \mathsf{\sigma }(t)\ni 0_Q\mathrm{a}.\mathrm{e}.t\in (0,T),\hfill \\ & & \mathsf{\sigma }(0)={\mathsf{\sigma }}_0.\hfill \end{array}$$

Here and everywhere in this section, $\mathcal{K}$ denotes the set given by

$$\mathcal{K}=\{\mathsf{\tau }\in Q:\parallel {\mathsf{\tau }}^D{\parallel }_Q\le k\}$$

(110)

where recall, ${\mathsf{\tau }}^D$ represents the deviator of the stress field $\mathsf{\tau }\in Q$ and $k>0$. Using (6), we restate Problem 7 in the following equivalent form.

Problem 8. 

Find a stress field $\mathsf{\sigma }:[0,T]\to Q$ such that

$$\begin{array}{ccc}& & \mathsf{\sigma }(t)\in \mathcal{K}\cap \mathsf{\Sigma }(t)\forall t\in [0,T],\hfill \end{array}$$

(111)

$$\begin{array}{ccc}& & {(\mathcal{A}\dot{\mathsf{\sigma }}(t),\mathsf{\tau }-\mathsf{\sigma }(t))}_Q\ge 0\forall \mathsf{\tau }\in \mathcal{K}\cap \mathsf{\Sigma }(t),\mathrm{a}.\mathrm{e}.t\in (0,T),\hfill \end{array}$$

(112)

$$\begin{array}{ccc}& & \mathsf{\sigma }(0)={\mathsf{\sigma }}_0.\hfill \end{array}$$

(113)

We now use the constant ${\omega }_d$ in (9) and consider the following assumption:

$${\parallel \mathit{f}(t)\parallel }_V\le {\displaystyle \frac{\tilde{k}}{{\omega }_d}}\forall t\in [0,T]\mathrm{with}\mathrm{some}\tilde{k}<k.$$

(114)

Our first result in the study of Problem 8 is the following.

Theorem 6. 

Assume (25)–(27), (92) and (114). Then, Problem 8 is well-posed in the sense of Levitin-Polyak.

Proof. 

We check the validity of the conditions (41)–(48) on the space $X=Q$, with $f=0_Q$ and $K=\mathcal{K}$. First, we note that condition (41) is obviously satisfied and conditions (42)–(46), (48) have been checked in the proof of Theorem 5. They are valid with the set ${\mathsf{\Sigma }}_0$ and the function $\mathsf{\chi }$ given by (96) and (97), respectively. Assume now that $z\in Q$ is such that ${\parallel z\parallel }_Q\le \delta$ with $\delta ={\displaystyle \frac{k-\tilde{k}}{{\omega }_d}}>0$ and let $t\in [0,T]$. Then, using (9), (114) and (97), we have

$${\parallel (\mathsf{\chi }(t)+\mathit{z})}^D{\parallel }_Q{\le \parallel \mathsf{\epsilon }(\mathit{f}(t))}^D{\parallel }_Q+\parallel z^D{\parallel }_Q\le {\omega }_d{\parallel \mathit{f}(t)\parallel }_V+{\omega }_d{\parallel \mathit{z}\parallel }_Q\le \tilde{k}+{\omega }_d\delta \le k.$$

This shows that $\mathsf{\chi }(t)+\mathit{z}\in \mathcal{K}$ and, therefore, condition (47) holds. Theorem 6 is now a direct consequence of Theorem 3. □

Note that Theorem 6 combined with Definition 2 guarantees the existence of a unique solution $\mathsf{\sigma }$ to Problem 8, provided that conditions (25)–(27), (92) and (114) hold. The solution depends on k and g, that is, $\mathsf{\sigma }=\mathsf{\sigma }(k,g)$. Next, we assume (25)–(27), (92), (114) and, for each $n\in \mathbb{N}$, we consider two constants $k_n$ and $g_n$ such that

$$g_n>g,k_n\ge k,{\displaystyle \frac{g_{n}k-g{k}_n}{g_n-g}}>\tilde{k}$$

(115)

with $\tilde{k}$ being the constant in assumption (114). We denote by ${\mathsf{\sigma }}_n$ the solution to Problem 8 with the data $k_n$ and $g_n$, that is, ${\mathsf{\sigma }}_n=\mathsf{\sigma }(k_n,g_n)$.

Theorem 7. 

Under the previous assumptions, if the convergence $g_n\to g$ holds, ${\mathsf{\sigma }}_n\to \mathsf{\sigma }$ in $C([0,T];Q)$.

Proof. 

Let $n\in \mathbb{N}$. Then, it follows that

$$\begin{array}{ccc}& & {\mathsf{\sigma }}_n(t)\in {\mathcal{K}}_n\cap {\mathsf{\Sigma }}_n(t)\forall t\in [0,T],\hfill \end{array}$$

(116)

$$\begin{array}{ccc}& & {(A{\dot{\mathsf{\sigma }}}_n(t),\mathsf{\tau }-{\mathsf{\sigma }}_n(t))}_Q\ge 0\forall \mathsf{\tau }\in {\mathcal{K}}_n\cap {\mathsf{\Sigma }}_n(t),\mathrm{a}.\mathrm{e}.t\in (0,T),\hfill \end{array}$$

(117)

$$\begin{array}{ccc}& & {\mathsf{\sigma }}_n(0)={\mathsf{\sigma }}_0,\hfill \end{array}$$

(118)

where

$${\mathcal{K}}_n=\{\mathsf{\tau }\in Q:\parallel {\mathsf{\tau }}^D{\parallel }_Q\le k_n\}$$

(119)

and ${\mathsf{\Sigma }}_n(t)$ is the set given by (101).

Let ${\tilde{\mathsf{\sigma }}}_n:[0,T]\to {\mathbb{S}}^d$ the function defined by (102), let $t\in [0,T]$ and recall that the inclusion (105) holds. On the other hand, using inequalities (9), (114) and assumption (115), it follows that

$$\parallel {\tilde{\mathsf{\sigma }}}_n^D{(t)\parallel }_Q\le {\displaystyle \frac{g}{g_n}}\parallel {\mathsf{\sigma }}_n^D{(t)\parallel }_Q+\left(1-{\displaystyle \frac{g}{g_n}}\right){\parallel \mathsf{\epsilon }{(\mathit{f}(t))}^D\parallel }_Q\le {\displaystyle \frac{g}{g_n}}{k}_n+\left(1-{\displaystyle \frac{g}{g_n}}\right)\tilde{k}\le k,$$

(120)

which shows that ${\tilde{\mathsf{\sigma }}}_n^D(t)\in \mathcal{K}$. We conclude from (120) that ${\tilde{\mathsf{\sigma }}}_n^D(t)\in \mathcal{K}\cap \mathsf{\Sigma }(t)$. This inclusion combined with (11) and inequality $g_n\ge g$ implies that

$$d({\mathsf{\sigma }}_n(t),\mathcal{K}\cap \mathsf{\Sigma }(t))\le {\parallel {\mathsf{\sigma }}_n(t)-{\tilde{\mathsf{\sigma }}}_n(t)\parallel }_Q\le {\displaystyle \frac{g_n-g}{g}}\left(\parallel {\mathsf{\sigma }}_n{(t)\parallel }_Q+{\parallel \mathit{f}(t)\parallel }_V\right).$$

We now use the bound (52) to see that there exists $D>0$, which does not depend on n and t such that

$$d({\mathsf{\sigma }}_n(t),\mathcal{K}\cap \mathsf{\Sigma }(t))\le {\displaystyle \frac{g_n-g}{g}}\left(D+{\parallel \mathit{f}\parallel }_{C([0,T];V)}\right).$$

(121)

Since $k_n\ge k$ and $g_n>g$, it follows that $\mathcal{K}\cap \mathsf{\Sigma }(t)\subset {\mathcal{K}}_n\cap {\mathsf{\Sigma }}_n(t)$ and, using (117), we deduce that

$${(A{\dot{\mathsf{\sigma }}}_n(t),\mathsf{\tau }-{\mathsf{\sigma }}_n(t))}_Q\ge 0\forall \mathsf{\tau }\in \mathcal{K}\cap \mathsf{\Sigma }(t),\mathrm{a}.\mathrm{e}.t\in (0,T).$$

(122)

Denote

$${\theta }_n={\displaystyle \frac{g_n-g}{g}}\left(D+{\parallel \mathit{f}\parallel }_{C([0,T];V)}\right)$$

(123)

and note that the assumption $g_n\to g$ implies that

$${\theta }_n\to 0.$$

(124)

We now combine (121)–(124) to deduce that $\{{\mathsf{\sigma }}_n\}$ is an $LP$-approximating sequence for Problem 8. The convergence ${\mathsf{\sigma }}_n\to \mathsf{\sigma }$ in $C([0,T];Q)$ is now a direct consequence of Theorem 6 and Definition 2 (b)). □

We now present some comments and results which show the link between Problems 8 and 6, on the one hand, as well as the link between Problem 8 and an elastic–perfectly plastic contact problem, on the other hand.

The link between Problems 8 and 6. The following result shows that for a large k, the solution of Problem 8 is the solution of Problem 6, too.

Theorem 8. 

Assume (25)–(28), (92) and denote by $\mathsf{\sigma }\in W^{1,2}(0,T;Q)$ the solution of Problem 6 obtained in Theorem 5. Then, there exists a constant $k_D>0$ that depends on $\mathcal{A}$, f and T such that $\mathsf{\sigma }$ is the unique solution of Problem 8, for any $k>k_D$.

Proof. 

We use Lemma 1 to deduce that there exists a constant D that depends on $\mathcal{A}$, f and T such that

$${\parallel \mathsf{\sigma }(t)\parallel }_Q\le D\forall t\in [0,T].$$

Therefore, using inequality (9), we find that

$$\parallel {\mathsf{\sigma }}^D{(t)\parallel }_Q\le {\omega }_{d}D\forall t\in [0,T].$$

(125)

Let

$$k_D=max\left\{{\omega }_{d}D,{\omega }_d{\parallel \mathit{f}\parallel }_{C([0,T];V)}\right\}$$

(126)

and let $k>k_D$. Then, there exists $\tilde{k}$ such that condition (114) is satisfied and, using Theorem 6, we deduce that Problem 8 has a unique solution, $\tilde{\mathsf{\sigma }}\in W^{1,2}(0,T;Q)$.

On the other hand, inequality (125) and notation (126), (110) imply that $\mathsf{\sigma }(t)\in \mathcal{K}$, for all $t\in [0,T]$. We now combine this inclusion with (93)–(95) to see that $\mathsf{\sigma }$ satisfies relations (111)–(113), too. This shows that $\mathsf{\sigma }$ is a solution of Problem 8 and, by the unicity of the solution, we find that $\mathsf{\sigma }=\tilde{\mathsf{\sigma }}$, which concludes the proof. □

An elastic–perfectly plastic contact problem. Let $k^{\prime }>0$; denote by $K^{\prime }$ the set

$$K^{\prime }=\{\mathsf{\tau }\in {\mathbb{S}}^d:\parallel {\mathsf{\tau }}^D\parallel \le k^{\prime }\},$$

and consider the following contact problem.

Problem 9. 

Find a displacement field $\mathit{u}:\mathsf{\Omega }\times [0,T]\to {\mathbb{R}}^d$ and a stress field $\mathsf{\sigma }:\mathsf{\Omega }\times [0,T]\to {\mathbb{S}}^d$ such that

$$\begin{array}{cccc}\hfill \mathcal{A}\dot{\mathsf{\sigma }}(t)+\partial {\psi }_{K^{\prime }}\mathsf{\sigma }(t)& \ni \mathsf{\epsilon }(\dot{\mathit{u}}(t))\hfill & \hfill \mathrm{in}& \mathsf{\Omega },\hfill \end{array}$$

(127)

$$\begin{array}{cccc}\hfill \mathrm{Div}\mathsf{\sigma }(t)+{\mathit{f}}_0(t)& =0\hfill & \hfill \mathrm{in}& \mathsf{\Omega },\hfill \end{array}$$

(128)

$$\begin{array}{cccc}\hfill \mathit{u}(t)& =0\hfill & \hfill \mathrm{on}& {\mathsf{\Gamma }}_1,\hfill \end{array}$$

(129)

$$\begin{array}{cccc}\hfill \mathsf{\sigma }(t)\mathsf{\nu }& ={\mathit{f}}_2(t)\hfill & \hfill \mathrm{on}& {\mathsf{\Gamma }}_2,\hfill \end{array}$$

(130)

$$\begin{array}{cccc}\hfill {\mathit{u}}_{\nu }(t)=0,-{\mathsf{\sigma }}_{\tau }(t)& =g{\displaystyle \frac{{\dot{\mathit{u}}}_{\tau }(t)}{\parallel {\dot{\mathit{u}}}_{\tau }(t)\parallel }}\mathrm{if}{\dot{\mathit{u}}}_{\tau }(t)\ne 0\hfill & \hfill \mathrm{on}& {\mathsf{\Gamma }}_3\hfill \end{array}$$

(131)

for all $t\in [0,T]$ and, in addition,

$$\mathit{u}(0)={\mathit{u}}_0,\mathsf{\sigma }(0)={\mathsf{\sigma }}_0\mathrm{in}\mathsf{\Omega }.$$

(132)

Note that Problem 9 is similar to Problem 2, the difference arising being that the elastic constitutive law (19) was replaced by the elastic–perfectly plastic constitutive law (127) and, therefore, the initial condition (24) was replaced by the initial conditions (132). Let ${\mathcal{K}}^{\prime }$ be the set

$${\mathcal{K}}^{\prime }=\{\mathsf{\tau }\in Q:\parallel {\mathsf{\tau }}^D\parallel \le k^{\prime }\mathrm{a}.\mathrm{e}.\mathrm{in}\mathsf{\Omega }\}.$$

(133)

Then, arguments similar to those used in Section 3 allows us to deduce the following variational formulation of Problem 9.

Problem 10. 

Find a stress field $\mathsf{\sigma }:[0,T]\to Q$ such that

$$\begin{array}{c}\mathsf{\sigma }(t)\in {\mathcal{K}}^{\prime }\cap \mathsf{\Sigma }(t)\forall t\in [0,T],\end{array}$$

(134)

$$\begin{array}{c}{(\mathcal{A}\dot{\mathsf{\sigma }}(t),\mathsf{\tau }-\mathsf{\sigma }(t))}_Q\ge 0\forall \mathsf{\tau }\in {\mathcal{K}}^{\prime }\cap \mathsf{\Sigma }(t),\mathrm{a}.\mathrm{e}.t\in (0,T),\end{array}$$

(135)

$$\begin{array}{c}\mathsf{\sigma }(0)={\mathsf{\sigma }}_0.\end{array}$$

(136)

The unique solvability of problem (134)–(136) was proved in [15], under a so-called “safe loading” assumption, already used in the ([11,12,13]).

Consider now the set $\mathcal{K}$ defined by (110) with $k=k^{\prime }{(meas(\mathsf{\Omega })) }^{{\displaystyle \frac{1}{2}}}$. Then, it is easy to see that ${\mathcal{K}}^{\prime }\subset \mathcal{K}$ and, therefore, if the solution $\mathsf{\sigma }$ of Problem 8 has the regularity $\mathsf{\sigma }(t)\in {\mathcal{K}}^{\prime }$ for all $t\in [0,T]$, then $\mathsf{\sigma }$ is also the solution of problem (134)–(136). For this reason, we refer to Problem 8 as a regularized version of Problem 10 or, alternatively, a regularized variational formulation of the contact problem (127)–(132). Moreover, we shall refer to the solution of Problem 8 as the weak solution of the elastic–perfectly plastic contact problem (127)–(132).

Adopting this terminology, we conclude that Theorem 7 provides continuous dependence results of the weak solution of the contact problem (127)–(132) with respect to the friction bound. Moreover, Theorem 8 shows that the weak solution of the elastic contact problem (19)–(24) is the weak solution of the elastic–perfectly plastic contact problem (127)–(132), for a sufficiently large yield limit.

7. Conclusions

In this paper, we considered an evolutionary inclusion associated to a time-dependent convex set. We chose this problem since, as shown in the previous sections, we have good examples arising in mechanics. Even if the problem is formulated in an abstract real Hilbert space, its statement and the assumptions on the data are widely inspired by some relevant examples in mechanics. We introduced two well-posedness concepts in the study of the inclusion and proved the corresponding well-posedness results. Then, we applied these results in the study of three relevant examples arising in elasticity, plasticity and contact mechanics. In this way, we obtained various convergence results and provided the corresponding mechanical interpretations.

The results presented here could be extended in various directions. Here, we restrict ourselves to present only three of them. The first one would be to establish necessary and sufficient conditions for the convergence of an arbitrary sequence $\{{\sigma }_n\}$ to the solution $\sigma$ of the evolutionary inclusion (1) and (2). This would allow to introduce a well-posedness concept in which all the sequences converging to the solution are approximating sequences and, therefore, would lead to the best well-posedness concept in the study of the inclusion (1) and (2). A second direction would be to construct numerical schemes in the study of the mechanical models described in this paper, to implement them in a computer and to obtain the corresponding numerical simulations. On this matter, we mention that the numerical analysis of inequality problems similar to those presented in the paper can be found in [4]. There, error estimates for semidiscrete and fully discrete schemes have been presented, and numerical simulations have been performed. A third direction would be to extend the results presented here to systems coupling an inclusion of the form (1), (2) with a differential or integral equation. There is a strong indication that, under appropriate assumptions, Theorems 1 and 2 can be extended to such problems.