variational inequality; regularization; weak solution; normal compliance

In this paper we introduce two novelties that we describe in what follows. First, we state and prove an abstract regularization result for a class of history-dependent variational inequalities in Hilbert spaces. More exactly, we continue the analysis provided in [1, Section 2.2.3], where regularization arguments are used to prove the unique solvability of the following variational inequality

Here $\left(X,(\cdot,\cdot)_{X},\|\cdot\|_{X}\right)$ denotes a real Hilbert space, $A:X\rightarrow X$ is a Lipschitz continuous and strongly monotone operator, $j:X\rightarrow\mathbb{R}$ is a convex, lower-semicontinuous functional and $f\in X$. With respect to [1] we study the unique solvability of the following history-dependent variational inequality

using a regularization method involving Gâteaux differentiable functionals. Here, both the data and the solution $u$ depend on the time variable $t\in[0,T]$, where $T>0$. Moreover, concerning the operator $\mathcal{S}$, the current value $\mathcal{S}u(t)$ at moment $t$ depends on the history of the values of $u$ at moments $0\leq s\leq t$. Its presence implies the use of Gronwall’s inequality or Lebesgue’s convergence theorem.

We consider a contact problem which describes the frictionless contact between a viscoelastic body and a deformable foundation. We model the material’s behaviour with a constitutive law with long memory of the

form

where $\boldsymbol{u}$ denotes the displacement field, $\boldsymbol{\sigma}$ represents the stress tensor, $\boldsymbol{\varepsilon}(\boldsymbol{u})$ is the linearized strain tensor, $\mathcal{A}$ is the elasticity operator and $\mathcal{B}$ represents the relaxation tensor. Moreover, we assume that the contact process is quasistatic and we study it in the interval of time $[0,T]$. Finally, we use a multivalued normal compliance condition to describe the contact process in normal direction.

This mathematical model was considered in [2]. There, the variational formulation was derived and unique weak solvability was proved. Moreover, the weak solution was approximated using a penalty method. Also, in [3] the dependence of the solution with respect to the data was studied and a fully discrete scheme was introduced. In addition, an optimal order error estimate was derived and numerical simulations were provided for a two-dimensional test problem.

The second novelty of the paper arises from the fact that we analyse the weak solvability of contact problem using regularization arguments. With respect to [2,3] we introduce a regularized contact problem where a single-valued normal compliance condition, including a regularization parameter, is considered. We prove that the weak solution of the regularized problem converges to the weak solution of the original contact problem, as the regularization parameter goes to zero. To this end, we apply the abstract regularization results obtained in the study of (1.1).

Next, we recall that general results on analysis of various classes of variational inequalities, including existence, uniqueness and regularity, can be found in [4-9]. The numerical analysis of variational inequalities, including solution algorithms and error estimates, was treated in [10, 11] and also in [12, 13]. Additional results on optimal control of variational inequalities can be found in [14]. An early attempt to study contact problems within the framework of variational inequalities was made in [15]. The variational analysis of contact models is given in $[1,9,13,16,17]$. There, the mathematical analysis of contact problems is provided, including existence and uniqueness results of the weak solution. Numerical analysis of contact models, including the study of fully discrete scheme, error estimates and numerical simulations, can be found in [12, 13, 17, 18]. An excellent reference in the study of frictional contact associated with heat generation, material damage, wear and adhesion of contacting surfaces is [19].

The rest of the paper is structured as follows. In Section 2 we describe the classical formulation of the model and list the assumptions on the data. In Section 3 we introduce the regularized problem and derive the weak formulations. Then, we state our main existence, uniqueness and convergence result, Theorem 3.1. In Section 4 we provide the abstract problem and recall its unique solvability obtained in [20]. Then we consider the associated regularized problem and state our main abstract result, Theorem 4.3. The proof of this theorem is given in Section 5. Finally, in Section 6 we illustrate the use of abstract arguments in the proof of Theorem 3.1.

In this section we introduce the classical formulation of the contact problem and list the assumptions on the data. First of all, we present the notation we shall use and preliminaries related to the contact model. More exactly, we denote by $\mathbb{S}^{d}(d=1,2,3)$ the space of second-order symmetric tensors on $\mathbb{R}^{d}$ and we define the following inner products and norms

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain with Lipschitz continuous boundary $\Gamma$ and let $\Gamma_{1},\Gamma_{2}$ and $\Gamma_{3}$ be three measurable parts of $\Gamma$ such that meas $\left(\Gamma_{1}\right)>0$. We use the notation $\boldsymbol{x}=\left(x_{i}\right)$ for a typical point in $\Omega\cup\Gamma$ and we denote by $\boldsymbol{v}=\left(v_{i}\right)$ the outward unit normal at $\Gamma$. The indices $i,j,k,l$ run between 1 and $d$ and the summation convention over repeated indices is used. An index that follows a comma represents the partial derivative with respect to the corresponding component of the spatial variable, e.g. $u_{i,j}=\partial u_{i}/\partial x_{j}$. Moreover, we consider the Hilbert spaces

endowed with the inner products

and the associated norms $\|\cdot\|_{V}$ and $\|\cdot\|_{Q}$, respectively. Here $\boldsymbol{\varepsilon}$ represents the deformation operator given by

For an element $\boldsymbol{v}\in V$ we still write $\boldsymbol{v}$ for the trace of $\boldsymbol{v}$ on the boundary and we denote by $v_{\nu}$ and $\boldsymbol{v}_{\tau}$ the normal and tangential components of $\boldsymbol{v}$ on $\Gamma$, given by $v_{\nu}=\boldsymbol{v}\cdot\boldsymbol{v},\boldsymbol{v}_{\tau}=\boldsymbol{v}-v_{\nu}\boldsymbol{v}$. We recall that there exists a positive constant $c_{0}$ such that

For a function $\boldsymbol{\sigma}\in Q$ we use the notation $\sigma_{\nu}$ and $\boldsymbol{\sigma}_{\tau}$ for the normal and tangential traces, i.e. $\sigma_{\nu}=(\boldsymbol{\sigma}\boldsymbol{v})\cdot\boldsymbol{v}$ and $\boldsymbol{\sigma}_{\tau}=\boldsymbol{\sigma}\boldsymbol{v}-\sigma_{\nu}\boldsymbol{v}$.

We denote by $\mathbf{Q}_{\infty}$ the space of fourth-order tensor fields given by

and it is a real Banach space with the norm

Moreover,

We present the physical setting of the problem. A viscoelastic body occupies the bounded domain $\Omega$ described above. The body is subjected to the action of body forces of density $\boldsymbol{f}_{0}$. We also assume that it is fixed on $\Gamma_{1}$ and surface tractions of density $\boldsymbol{f}_{2}$ act on $\Gamma_{2}$. On $\Gamma_{3}$ the body is in frictionless contact with a deformable foundation. The contact process is quasistatic and we study it in the interval of time $[0,T]$.

The classical formulation of the contact problem is the following.
Problem $\mathcal{Q}$. Find a displacement field $\boldsymbol{u}:\Omega\times[0,T]\rightarrow\mathbb{R}^{d}$ and a stress field $\boldsymbol{\sigma}:\Omega\times[0,T]\rightarrow\mathbb{S}^{d}$ such that

and there exists $\xi:\Omega\times[0,T]\rightarrow\mathbb{R}$ which satisfies

for all $t\in[0,T]$.
Next, we give a short description of Problem $\mathcal{Q}$ and list the assumptions on the data. Here and below we do not indicate explicitly the dependence of various functions on the spatial variable $\boldsymbol{x}$. Equation (2.3) represents
the viscoelastic constitutive law with long memory introduced in Section 1. The operators $\mathcal{A}$ and $\mathcal{B}$ verify the following conditions

(a) $\mathcal{A}: \Omega \times \mathbb{S}^{d} \rightarrow \mathbb{S}^{d}$.
(b) There exists $L_{\mathcal{A}}>0$ such that
    $\left\|\mathcal{A}\left(\boldsymbol{x}, \boldsymbol{\varepsilon}_{1}\right)-\mathcal{A}\left(\boldsymbol{x}, \boldsymbol{\varepsilon}_{2}\right)\right\| \leq L_{\mathcal{A}}\left\|\boldsymbol{\varepsilon}_{1}-\boldsymbol{\varepsilon}_{2}\right\|$
        for all $\boldsymbol{\varepsilon}_{1}, \boldsymbol{\varepsilon}_{2} \in \mathbb{S}^{d}$, almost every $\boldsymbol{x} \in \Omega$.
(c) There exists $m_{\mathcal{A}}>0$ such that
    $\left(\mathcal{A}\left(\boldsymbol{x}, \boldsymbol{\varepsilon}_{1}\right)-\mathcal{A}\left(\boldsymbol{x}, \boldsymbol{\varepsilon}_{2}\right)\right) \cdot\left(\boldsymbol{\varepsilon}_{1}-\boldsymbol{\varepsilon}_{2}\right) \geq m_{\mathcal{A}}\left\|\boldsymbol{\varepsilon}_{1}-\boldsymbol{\varepsilon}_{2}\right\|^{2}$
        for all $\boldsymbol{\varepsilon}_{1}, \boldsymbol{\varepsilon}_{2} \in \mathbb{S}^{d}$, almost every $\boldsymbol{x} \in \Omega$.
(d) The mapping $\boldsymbol{x} \mapsto \mathcal{A}(\boldsymbol{x}, \boldsymbol{\varepsilon})$ is measurable on $\Omega \quad$ for all $\boldsymbol{\varepsilon} \in \mathbb{S}^{d}$.
(e) The mapping $\boldsymbol{x} \mapsto \mathcal{A}(\boldsymbol{x}, \mathbf{0})$ belongs to $Q$.
        $\mathcal{B} \in C\left([0, T] ; \mathbf{Q}_{\infty}\right)$.

Various examples and mechanical interpretation concerning constitutive laws in solid mechanics can be found in [1, 16, 19]. Equation (2.4) represents the equation of equilibrium in which Div denotes the divergence operator for tensor-valued functions

Conditions (2.5) and (2.6) are the displacement boundary condition and traction boundary condition, respectively. We assume that the densities of body forces and surface tractions have regularity

Equation (2.7) represents the frictionless condition. Frictionless contact problems were considered, for example, in [1, 19, 21].

Next, (2.8) is the contact condition in which $\sigma_{\nu}$ denotes the normal stress and $u_{\nu}$ is the normal displacement. Moreover, the functions $p$ and $F$ satisfy
(a) $p:\mathbb{R}\rightarrow\mathbb{R}_{+}$.
(b) There exists $L_{p}>0$ such that

(c) $\left(p\left(r_{1}\right)-p\left(r_{2}\right)\right)\left(r_{1}-r_{2}\right)\geq 0\quad$ for all $r_{1},r_{2}\in\mathbb{R}$.
(d) $p(r)=0$ iff $r\leq 0$.

This condition can be derived in the following way. Let $t\in[0,T]$. We consider that the normal stress has an additive decomposition of the form

in which $\sigma_{\nu}^{D}(t)$ describes the deformability of foundation and $\sigma_{\nu}^{M}(t)$ describes the rigidity. We assume that $\sigma_{\nu}^{D}(t)$ satisfies a normal compliance contact condition

We recall that the normal compliance contact condition was first used in [22] and the term normal compliance was first introduced in [23,24].

Finally, $\sigma_{v}^{M}(t)$ satisfies

We combine (2.14)-(2.16) and write $-\sigma_{v}^{M}(t)=\xi(t)$ to obtain (2.8). A version of this condition including unilateral constraints was used in [2,3].

We present additional details of the contact condition (2.8) which is depicted in Figure 1. We assume that at a given moment $t$ there is a separation between the body and the foundation, i.e. $u_{v}(t)<0$. Then, (2.12) part (d) and (2.8) show that $\sigma_{\nu}(t)=0$, i.e. the reaction of foundation vanishes. Assume now that at the moment $t$ there is penetration, i.e. $u_{\nu}(t)>0$. Then, (2.8) yields

This equality implies that, at the moment $t$, the reaction of the foundation depends on the penetration and represents a normal compliance-type condition. Moreover, (2.8) shows that if at the moment $t$ we have penetration, then $-\sigma_{\nu}(t)\geq F$. Indeed, if $u_{\nu}(t)>0$, then (2.17) holds and this implies that $-\sigma_{\nu}(t)\geq F$. We conclude that if $-\sigma_{\nu}(t)<F$, then there is no penetration and $F$ represents a yield limit of the normal pressure, under which the penetration is not possible. This kind of behaviour characterizes a rigid-elastic foundation.

In conclusion, condition (2.8) shows that when there is a separation between the body’s surface and the foundation then the normal stress vanishes; the penetration occurs only if the normal stress reaches the critical value $F$; when there is penetration, the contact follows a multivalued normal compliance contact condition. It can be interpreted physically as follows. The foundation is assumed to be made of a rigid-elastic material which allows penetration, but only if the normal stress arrives to the yield limit $F$.

In this section we derive the variational formulation of Problem $\mathcal{Q}$. Moreover, we consider a regularized problem and state an existence, uniqueness and convergence result. To this end, let $t\in[0,T],\rho>0$ and $\boldsymbol{v}\in V$ be given. We assume in what follows that ( $\boldsymbol{u},\boldsymbol{\sigma}$ ) are sufficiently regular functions that satisfy (2.3)-(2.8). We recall the following Green’s formula

Next, we take $\boldsymbol{v}:=\boldsymbol{v}-\boldsymbol{u}(t)$ in (3.1) and use (2.4) to see that

We split the surface integral over $\Gamma_{1},\Gamma_{2}$ and $\Gamma_{3}$ and taking into account (2.7) we obtain

We use (2.8) and assumption (2.13) to deduce that

where $r^{+}=\max\{r,0\}$ for all $r\in\mathbb{R}$. We again use (2.8) and (3.3) to find that

We apply Riesz representation theorem to define the function $\boldsymbol{f}:[0,T]\rightarrow V$ by

It follows from (2.11) that this function has regularity

Finally, we combine equality (3.2), inequality (3.4), constitutive law (2.3) and (3.5) to obtain the following variational formulation of Problem $\mathcal{Q}$.

Problem $\mathcal{Q}^{V}$. Find a displacement field $\boldsymbol{u}:[0,T]\rightarrow V$ such that, for all $t\in[0,T]$, the following inequality holds:

The previous variational formulation leads to a history-dependent variational inequality involving a nondifferentiable term. To avoid this difficulty, we use a regularity procedure. Moreover, condition (2.8) can be written in the following equivalent form

Replacing the non-differentiable absolute value $\left|u_{v}(t)\right|$ we obtain the following regularized contact problem.

Problem $\mathcal{Q}_{\rho}$. Find a displacement field $\boldsymbol{u}_{\rho}:\Omega\times[0,T]\rightarrow\mathbb{R}^{d}$ and a stress field $\boldsymbol{\sigma}_{\rho}:\Omega\times[0,T]\rightarrow\mathbb{S}^{d}$ such that

for all $t\in[0,T]$.
Note that here and below $u_{\rho\nu}$ is the normal component of the displacement field $\boldsymbol{u}_{\rho}$ and $\sigma_{\rho\nu},\boldsymbol{\sigma}_{\rho\tau}$ represent the normal and tangential components of the stress tensor $\boldsymbol{\sigma}_{\rho}$, respectively. The equations and boundary conditions in problem (3.9)-(3.14) have a similar interpretation as those in problem (2.3)-(2.8). The difference arises in the fact that here we replace the multivalued normal compliance contact condition (2.8) with the single-valued normal compliance condition (3.14). In this condition $\rho$ represents a regularization parameter. We recall that regularizations of Coulomb friction law were considered in [16,25].

Using similar arguments as those used in the case of Problem $\mathcal{Q}$, we obtain the following variational formulation of Problem $\mathcal{Q}_{\rho}$.
Problem $\mathcal{Q}_{\rho}^{V}$. Find a displacement field $\boldsymbol{u}_{\rho}:[0,T]\rightarrow V$ such that, for all $t\in[0,T]$, the following equality holds:

We have the following existence, uniqueness and convergence result.
Theorem 3.1 Assume that (2.9)-(2.13) hold. Then:
(1) Problem $\mathcal{Q}^{V}$ has a unique solution $\boldsymbol{u}\in C([0,T];V)$;
(2) for each $\rho>0$ Problem $\mathcal{Q}_{\rho}^{V}$ has a unique solution $\boldsymbol{u}_{\rho}\in C([0,T];V)$;
(3) the solution of Problem $\mathcal{Q}_{\rho}^{V}$ converges to the solution of Problem $\mathcal{Q}^{V}$, that is

for all $t\in[0,T]$.
The convergence result (3.16) is important from a mechanical point of view. More exactly, it shows that the weak solution of viscoelastic contact problem with multivalued normal compliance contact condition may be approached as closely as one wishes by the solution of a viscoelastic contact problem with single-valued normal compliance condition, with a sufficiently small regularization parameter. Note that convergence (3.16) above is understood in the following sense: for each $t\in[0,T]$ and for every sequence $\left\{\rho_{n}\right\}\subset\mathbb{R}_{+}$converging to zero as $n\rightarrow\infty$ we have $\boldsymbol{u}_{\rho_{n}}(t)\rightarrow\boldsymbol{u}(t)$ as $n\rightarrow\infty$. The proof of Theorem 3.1 is given in Section 6. To this end, we use the abstract result introduced in Section 4 and proved in Section 5.

In this section we state our main abstract result, Theorem 4.3. It represents an extension of Theorem 2.12 of [1] to a class of history-dependent variational inequalities. We consider a real Hilbert space $X$ with inner product $(\cdot,\cdot)_{X}$ and associated norm $\|\cdot\|_{X}$ and we use notation $C([0,T];X)$ for the space of continuous functions defined on $[0,T]$ with values in $X$. Moreover, let $A:X\rightarrow X,\mathcal{S}:C([0,T];X)\rightarrow C([0,T];X)$ be two operators, let the functional $j:X\rightarrow\mathbb{R}$ and the function $f:[0,T]\rightarrow X$. We assume in what follows that $A$ is a strongly monotone and Lipschitz continuous operator.

Moreover, we assume that operator $\mathcal{S}$ satisfies the following condition.

Finally, we suppose that

and

We consider the following problem.
Problem $\mathcal{P}$. Find a function $u:[0,T]\rightarrow X$ such that, for all $t\in[0,T]$, the following inequality holds:

Following the terminology introduced in $[1,20]$ we refer to operator $\mathcal{S}$, which satisfies (4.2), as a historydependent operator. In addition, we refer to (4.5) as a history-dependent variational-inequality.

The unique solvability of Problem $\mathcal{P}$ is provided by the following existence and uniqueness result.
Theorem 4.1 Let $X$ be a Hilbert space and assume that (4.1)-(4.4) hold. Then, Problem $\mathcal{P}$ has a unique solution $u\in C([0,T];X)$.

Theorem 4.1 was proved in [20] by using fixed-point arguments. It represents a crucial tool in studying the weak solvability of a large number of contact problems. We send the reader to [1] for more details.

In the particular case $j\equiv 0$ we have the following consequence of Theorem 4.1.
Corollary 4.2 Let $X$ be a Hilbert space and assume that (4.1)-(4.2) and (4.4) hold. Then, there exists a unique function $u\in C([0,T];X)$ which satisfies the following equality

Let $\rho>0$ be a parameter. In order to formulate the regularized problem associated to Problem $\mathcal{P}$ we consider the following family of functionals $\left(j_{\rho}\right)$ which satisfies

The assumption made in (4.7) requires the functional $j$ to be approached by a sequence of more regular functionals ( $j_{\rho}$ ), since they are Gateaux differentiable. Next, we consider the following problem.
Problem $\mathcal{P}_{\rho}$. Find $u_{\rho}\in C([0,T];X)$ such that, for all $t\in[0,T]$, equality below holds

We have the following existence, uniqueness and convergence result.
Theorem 4.3 Let $X$ be a Hilbert space and assume that (4.1)-(4.4), (4.6) and (4.7) hold. Then:
(1) for each $\rho>0$ Problem $\mathcal{P}_{\rho}$ has a unique solution;
(2) the solution of Problem $\mathcal{P}_{\rho}$ converges to the solution of Problem $\mathcal{P}$, that is

for all $t\in[0,T]$.

The main feature of Theorem 4.3 consists of showing that the solution of an ’irregular’ history-dependent inequality may be approached as limit of the solutions of ’regular’ history-dependent variational equalities.

In this section we give in several steps the proof of Theorem 4.3. First of all, we provide the unique solvability of (4.8).

Lemma 5.1 For each $\rho>0$ there exists a unique function $u_{\rho}\in C([0,T];X)$ which satisfies (4.8) for all $t\in[0,T]$.

Proof. Let $\rho>0$. Taking into account assumption (4.6) we have that $\nabla j_{\rho}$ is a monotone operator on $X$. Moreover, using (4.1) and (4.6) we deduce that the operator $A_{\rho}:X\rightarrow X$ defined by

is a strongly monotone and Lipschitz continuous operator on $X$. Lemma 5.1 is now a consequence of Corollary 4.2.

Next, we consider the following intermediate problem.
Problem $\widetilde{\mathcal{P}}_{\rho}$. Find $\widetilde{u}_{\rho}:[0,T]\rightarrow X$ such that, for all $t\in[0,T]$, the following equality holds:

Note that the difference between (4.8) and (5.2) arises in the fact that in (5.2) the operator $S$ is applied to a known function $u$, which is the solution of (4.5). Thus, equality (5.2) is a time-dependent variational equality. In contrast, (4.8) is a history-dependent variational equality, since the operator $\mathcal{S}$ is applied to the unknown function $u_{\rho}$.

We have the following existence and uniqueness result.
Lemma 5.2 For each $\rho>0$ there exists a unique function $\widetilde{u}_{\rho}\in C([0,T];X)$ which satisfies (5.2) for all $t\in[0,T]$.

Proof. In addition to the operator $A_{\rho}$, given in (5.1), we define the function $\widetilde{f}:[0,T]\rightarrow X$ by equality

We use assumptions (4.2), (4.4) and Corollary 4.2 to conclude the proof.

The next step is provided by the following weak convergence result.
Lemma 5.3 For each $t\in[0,T]$ the sequence $\left\{\widetilde{u}_{\rho}(t)\right\}$ converges weakly to $u(t)$, i.e.

Proof. Let $\rho>0$ and $t\in[0,T]$. We use (5.2) with $v:=v-\widetilde{u}_{\rho}(t)$ to obtain

Moreover, taking into account assumption (4.6) we deduce that

We now take $v:=0_{X}$ in (5.5) and using assumption (4.7) we have

Assumption (4.3) implies that the functional $j$ is bounded below by an affine function, i.e there exists $w\in X$ and $\alpha\in\mathbb{R}$, which do not depend on $t$, such that

Therefore, we deduce that

Using assumption (4.1) and Cauchy-Schwarz inequality it follows that