Phenomena of contact between deformable bodies are important in industry and everyday life. Their mathematical analysis was developed in a large number of works. Thus, various existence and uniqueness results, examples, numerical analysis and mechanical interpretation in the study of contact problems can be found in $[1-10]$. To be accurate mathematical models need to take into consideration the additional phenomena as friction, heat generation, wear or adhesion. The analysis of such models leads to a weak formulation. In many cases, it is given in a form of a system which couples a time-dependent variational inequality and integral equations, as illustrated in [ $3,8,11$ ]. Moreover, the weak form of a large number of contact problems with unilateral constraints can be cast in a mixed-variational formulation with Lagrange multipliers. Their study is based on arguments on saddle points theory, fixed point, and duality. We recall that the use of Lagrange multipliers represents a mathematical tool to remove the unilateral constraints. Concerning the literature in the field, see for instance [12-15] and recent papers [16,17].

In this paper, we analyze the weak solvability of a contact problem with adhesion and memory effects. The model was introduced in [11]. There, the contact in normal direction was modeled with normal compliance condition, unilateral constraint, and adhesion. In addition, memory effect of the surfaces was introduced. A similar contact condition was considered in [18] in the study of frictionless contact process without adhesion. A variational formulation of the problem was derived, in a form of the system which couples a history-dependent quasi-variational inequality for displacement field, and an integral equation for adhesion field. The unique solvability of the weak problem was proved in two steps, using arguments on history-dependent quasi-variational inequalities and fixed point.

In the current paper, we introduce two main novelties. The first one concerns the constitutive law since, in contrast with [11], the material’s behavior is modeled with a viscoplastic constitutive law. The second novelty consists in the fact that we derive a mixed-variational formulation of the problem. The unknowns are the displacement field, the stress field, the adhesion function, and the Lagrange multiplier. We define two history-dependent operators to obtain an equivalent formulation in which the unknowns are only the displacement field and the Lagrange multiplier. Next, we prove its unique solvability using an abstract result provided in [17].

The rest of the paper is structured as follows. In Section 2, we present the geometrical configuration of the contact problem. We also introduce some notations and function spaces and we recall some preliminary results as Green’s formula. In Section 3, we describe our mathematical model of contact and list the assumptions on the data. Then, in Section 4 we derive the mixed-variational formulation of the problem and state our main existence and uniqueness result in Theorem 4.1. In Section 5, we prove Theorem 4.1. The proof is provided through two steps. More exactly, in the first part an equivalence result, Lemma 5.1, is given. In the second part an existence and uniqueness result, Lemma 5.3, is proved.

In this section we present the notations, the geometrical configuration of the problem and some preliminary material. For a given $r\in\mathbb{R}$ we denote by $r^{+}$its positive part, i.e. $r^{+}=\max\{r,0\}$. Let $d\in\mathbb{N}$. Then, we denote by $\mathbb{S}^{d}$ the space of second-order symmetric tensors on $\mathbb{R}^{d}$. The inner product and norm on $\mathbb{R}^{d}$ and $\mathbb{S}^{d}$ are defined by

Here and below the indices $i,j,k$, and $l$ run between 1 and $d$ and the summation convention over repeated indices is used. The physical setting of the contact problem is as follows. A viscoplastic body occupies in its reference configuration a bounded domain $\Omega\in\mathbb{R}^{d}$ ( $d=2,3$ in applications) with a Lipschitz-continuous boundary $\Gamma$. Body forces of density $\boldsymbol{f}_{0}$ act in $\Omega$. The boundary $\Gamma$ is divided into three disjoint measurable parts $\Gamma=\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{3}$, such that meas $\left(\Gamma_{1}\right)>0$, see Figure 1 for more details. The body is fixed on $\Gamma_{1}$ and surface tractions of density $f_{2}$ act on $\Gamma_{2}$. On the part $\Gamma_{3}$, the body is in contact with a foundation. The foundation is assumed to be made of a hard material covered with a thin layer made of a soft adhesive material with thickness $g>0$. It has a rigid adhesive viscoelastic behavior; its adhesive viscoelastic behavior is caused by the layer of the soft material while its rigid behavior is caused by the hard material. Thus, in normal direction the contact is modeled with normal compliance condition, unilateral constraint, memory effects, and adhesion.

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 $\Gamma$. Also, an index that follows a comma represents the partial derivative with respect to the corresponding component of the spatial variable, e.g. $u_{i,j}=\partial u_{i}/\partial x_{j}$. We use standard notations for the Lebesgue and Sobolev spaces associated to $\Omega$ and $\Gamma$ and, moreover, we consider the spaces

These are real 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 $\Gamma$. The normal and tangential components of $\boldsymbol{v}$ on $\Gamma$ are defined by $v_{\nu}=\boldsymbol{v}\cdot\boldsymbol{v},\quad\boldsymbol{v}_{\tau}=\boldsymbol{v}-v_{\nu}\boldsymbol{v}$. We introduce the following subset of the Hilbert space $V$

We note that $U$ is a closed, convex subset of $V$ such that $\mathbf{0}_{V}\in U$. We recall that there exists a positive constant $c_{0}$ which depends on $\Omega,\Gamma_{1}$, and $\Gamma_{3}$ such that

The previous inequality and the constant $c_{0}$ will be used in Section 5.
As in [17] we consider the set

where $\left.\boldsymbol{v}\right|_{\Gamma_{3}}$ denotes the restriction of trace of the element $\boldsymbol{v}\in V$ to $\Gamma_{3}$. We recall that $W$ can be organized as a Hilbert space. We denote by $D$ its dual and $\langle\cdot,\cdot\rangle_{\Gamma_{3}}$ represents the duality pairing between $D$ and $W$. For simplicity, we write $\langle\boldsymbol{\mu},\boldsymbol{v}\rangle_{\Gamma_{3}}$ instead of $\left\langle\boldsymbol{\mu},\left.\boldsymbol{v}\right|_{\Gamma_{3}}\right\rangle_{\Gamma_{3}}$, when $\boldsymbol{\mu}\in D$ and $\boldsymbol{v}\in V$.

For a regular function $\sigma:\Omega\cup\Gamma\rightarrow\mathbb{S}^{d}$ the normal and the tangential components of the vector $\boldsymbol{\sigma}\boldsymbol{v}$ on $\Gamma$ are given by $\sigma_{\nu}=\boldsymbol{\sigma}\boldsymbol{v}\cdot\boldsymbol{v}$ and $\boldsymbol{\sigma}_{\tau}=\boldsymbol{\sigma}\boldsymbol{v}-\sigma_{\nu}\boldsymbol{v}$.
We introduce the space of fourth-order tensor fields given by

and we recall that $\mathbf{Q}_{\infty}$ is a real Banach space with the norm

Moreover,

Finally, we recall the following Green’s formula

where $\operatorname{Div}$ denotes the divergence operator for tensor valued functions, i.e. $\operatorname{Div}\sigma=\left(\sigma_{ij,j}\right)$. The previous formula will be used to determine the mixed-weak formulation of our contact problem.

In this section, we derive the mathematical model which corresponds to the physical setting intro- duced in Section 2. More exactly, we precise the constitutive law of the material, the balance equation and the boundary and initial conditions, as well. Moreover, we define the evolution equation of adhesion field. To this end, we introduce the following notations. Thus, $\boldsymbol{u}$ denotes the displacement field, $\boldsymbol{\varepsilon}(\boldsymbol{u})$ represents the linearized strain tensor, $\boldsymbol{\sigma}$ is the stress field, $\beta$ is the adhesion field, and $t\in\mathbb{R}_{+}=[0,+\infty)$ represents the time variable.

We assume that the material’s behavior follows a viscoplastic constitutive law of the form

In (3.1) and everywhere in this paper the dot above a variable represents derivative with respect to time variable $t$. Here $\mathcal{E}$ and $\mathcal{G}$ represent the elasticity tensor and a nonlinear constitutive function, respectively, and are assumed to satisfy the following conditions

Note that in (3.1) and below, in order to simplify the notation, we do not indicate explicitly the dependence of various functions on the spatial variable $\boldsymbol{x}$. To complete the constitutive law (3.1), we assume the following initial conditions

In (3.4) $\boldsymbol{u}_{0}$ and $\boldsymbol{\sigma}_{0}$ denote the initial displacement and the initial stress field, respectively. We assume that

A general description of various constitutive relations in solid mechanics can be found in [3,8] or [9].
Integrating (3.1) with initial condition (3.4), we find that

Next, since the process is quasistatic, we use the equilibrium equation

Moreover, taking into account the physical setting introduced in Section 2, we impose the following displacement-traction conditions:

We assume that the densities of body forces and surface tractions have regularity

In addition, we use Riesz representation theorem to define the function $f:\mathbb{R}_{+}\rightarrow V$ by equality

for all $\boldsymbol{v}\in V$ and $t\in\mathbb{R}_{+}$. It follows from (3.10) that this function has regularity

On the part $\Gamma_{3}$, the evolution of bonding field is governed by the differential equation

in which $\gamma_{\nu}$ is the adhesion coefficient, $\varepsilon_{a}$ represents the Dupré energy, and $R$ is the truncation operator given by

where $L>0$. The initial condition for differential equation (3.13) is

and we assume that

We note that (3.13) does not allow for rebonding, once debonding takes place, since $\dot{\beta}\leq 0$. Moreover, the values of bonding field are restricted to $0\leq\beta\leq 1$. When $\beta=0$ at a point of contact surface, there are no active bonds; when $\beta=1$ the adhesion is complete and all the bonds are active; finally, when $0<\beta<1$ partial adhesion takes place. A comprehensive treatment of several models for contact problems with adhesion can be found in [7-9,19-21], and, more recently, in [22]. An application of the theory to rocks and in the medical field was described in [23-25].

Finally, we introduce the normal and tangential conditions on the surface $\Gamma_{3}$. In the normal direction, we have the following contact condition with normal compliance, memory effect, unilateral
constraint, and adhesion.

It was introduced and justified in [11], and for this reason we give here only a short description. The condition was obtained by assuming an additive decomposition of the normal stress into four components, which describe the deformability, the rigidity, the adhesive, and the surface memory properties of the foundation. The first inequality in (3.18) shows that the penetration is limited by the bound $g$ and describes a condition with unilateral constraint. In the case $0<u_{\nu}(t)<g$, i.e. there is penetration which did not reach the bound $g$, (3.18) yields

Here $p_{v}$ and $b$ are given functions which satisfy

Equality (3.19) shows that at the moment $t$ the reaction of foundation depends both on the current value of the penetration (represented by the term $p_{v}\left(u_{v}(t)\right)$ ) and on the history of the penetration (represented by the integral term). The term $p_{v}\left(u_{v}(t)\right)$ represents the so-called normal compliance condition and describes the deformability of foundation. It assigns a reactive normal pressure that depends on the interpenetration of the asperities on the bodys surface and those of the foundation. We recall that the normal compliance contact condition was first used in [26] and the term normal compliance was first introduced in [27,28]. A commonly example of the normal compliance function $p_{\nu}$ is

The constant $c_{\nu}>0$ is the surface stiffness coefficient. An idealization of the normal compliance condition, which is used often in engineering literature, and can also be found in mathematical publications, is the Signorini contact condition, in which the foundation is assumed to be perfectly rigid. This condition was first introduced in [29].

In the case $u_{\nu}(t)<0$, i.e. there is separation between the body and the foundation, (3.18) yields

where the truncation operator $\widetilde{R}$ is given by

The reaction of the foundation is nonnegative and depends on adhesion coefficient, on the square of intensity of adhesion and on the normal displacement, but as it does not exceed the bound length $L$. The maximal normal traction is $\gamma_{\nu}L$. Next, we recall a physical interpretation, given in [18], of the integral term in (3.18). More exactly, assume that in the time interval $[0,t]$ there is only penetration (i.e. $u_{\nu}(s)\geq 0$ for all $s\in[0,t]$ ). Then, taking into account assumption (3.21) we deduce that the reaction of the foundation is larger than that given by the term $p_{\nu}\left(u_{\nu}(t)\right)$ and we conclude that equality (3.19) models the hardening phenomena of the surface. Also, if in the time interval $[0,t]$ there is separation (i.e. $u_{\nu}(s)<0$ for all $s\in[0,t]$ ) then the integral term vanishes. Now, assume a situation in which $u_{\nu}$ is positive in interval $\left[0,t_{0}\right]$ and negative on the interval $\left[t_{0},t\right]$. Then,

In addition, the support of the function $b$ is included in the interval $[0,\delta]$ with $\delta>0$. Two possibilities arise. First, if $t-t_{0}>\delta$ it follows that $b(t-s)=0$ for all $s\in\left[0,t_{0}\right]$ and (3.25) shows that the integral term vanishes. Second, if $t-t_{0}\leq\delta$ (3.25) implies that a residual pressure exists at the moment $t$ on the body’s surface. In the rest of the paper, we assume that there are no cycles of contact and separation during time interval of interest.

In conclusion, condition (3.18) shows that when there is penetration the contact stress is given by a normal compliance condition with memory term of the form (3.19) but up to the limit $g$. When the limit $g$ is reached, the stress is given by a Signorini-type unilateral condition. Moreover, adhesion takes place when there is separation.

As in [19] or [11], we assume that the tangential traction is given by

where the truncation operator $\boldsymbol{R}^{*}$ is defined by

and the function $p_{\tau}$ satisfies

More exactly, (3.26) shows that the resistance to tangential motion is generated mainly by the glue, and frictional traction is neglected. In the particular case, when all the adhesive bonds are inactive the motion is frictionless. Definition (3.27) of operator $\boldsymbol{R}^{*}$ implies that the tangential traction depends on the tangential displacement, but only up to the bond length $L$. Moreover, $p_{\tau}(\beta)$ acts as the stiffness or spring constant, and the traction is in direction opposite to the displacement.

The classical formulation of the contact problem is the following.

Problem $\mathcal{P}$. Find a displacement field $\boldsymbol{u}:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{R}^{d}$, a stress field $\sigma:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{S}^{d}$, and an adhesion field $\beta:\Gamma_{3}\times\mathbb{R}_{+}\rightarrow[0,1]$ such that

We end this section with the following assumption. More exactly, we assume that there exists

This additional assumption concerns only the geometry of the problem and was introduced in [30]. Examples in two- and three-dimensional cases are presented in [31].

In this section, we derive a mixed-variational formulation of problem $\mathcal{P}$. To this end, we assume in what follows that ( $\boldsymbol{\sigma},\boldsymbol{u},\beta$ ) represents a triple of sufficiently regular functions which satisfies (3.29)(3.37). We define the sets $K\subset V$ and $\Lambda\subset D$ in the following way

and we introduce the set of admissible bonding fields

Let $t\in\mathbb{R}_{+},\boldsymbol{v}\in V$, and $\boldsymbol{\mu}\in\Lambda$. We integrate (3.35) with initial condition (3.36) to find that

The assumptions (3.16) and (3.17) combined with (4.3) yield

see [9,19] or [11] for more details. Next, we introduce the bilinear form $\widetilde{b}:V\times D\rightarrow\mathbb{R}$ by

We use Green formula (2.7), equilibrium equation (3.30), boundary conditions (3.31), and (3.32), definition (3.11) and

to obtain that

Let $\lambda(t)\in D$ be the Lagrange multiplier given by

Using (3.33), (4.1), and (4.2) we deduce that $\lambda(t)\in\Lambda$.
Taking into account (4.6), we can write

and, combining this equality with (4.7) and tangential condition (3.34) we obtain that

for all $\boldsymbol{v}\in V$. The previous equality shows the importance of Lagrange multiplier $\lambda(t)$ defined by duality in (4.8). More exactly, it removes the unilateral constraint and concerning the test functions we can use the entire space $V$. In this case, it plays the role of a normal force.

We introduce the element $\boldsymbol{h}\in V$ by

Using assumption (3.38), definitions (4.1), (4.2), (4.6), and contact condition (3.33) we deduce that

We gather equalities (3.6), (4.4) and (4.10) and inequality (4.12) to obtain the following mixed variational formulation of problem $\mathcal{P}$.

Problem $\mathcal{P}^{V}$. Find a stress field $\boldsymbol{\sigma}:\mathbb{R}_{+}\rightarrow Q$, a displacement field $\boldsymbol{u}:\mathbb{R}_{+}\rightarrow V$, a bonding field $\beta:\mathbb{R}_{+}\rightarrow Z$ and a Lagrange multiplier $\lambda:\mathbb{R}_{+}\rightarrow\Lambda$ such that

for all $\boldsymbol{\mu}\in\Lambda,\boldsymbol{v}\in V,t\in\mathbb{R}_{+}$.
Problem $\mathcal{P}^{V}$ represents a mixed-variational formulation which includes two implicit integral equations for the stress field and bonding field, respectively. A history-dependent variational equation for displacement field and a first-order time-dependent variational inequality for the Lagrange multiplier complete the problem.

We have the following existence and uniqueness result.
Theorem 4.1: Assume that (3.2), (3.3), (3.5), (3.10), (3.16), (3.17), (3.20), (3.21), and (3.28) hold. There exists $e_{0}>0$ which depends only on $\mathcal{E},p_{v},\Omega,\Gamma_{1}$, and $\Gamma_{3}$ such that if

then Problem $\mathcal{P}^{V}$ has a unique solution ( $\boldsymbol{\sigma},\boldsymbol{u},\beta,\boldsymbol{\lambda}$ ). Moreover, the solution satisfies

For the convenience of the readers, we recall that $M_{\tau}$ is the constant introduced in (3.28) (c). The proof of Theorem 4.1 will be carried out in two steps in the next section. In the first step, an equivalence result, Lemma 5.1, is established. Second step includes an existence and uniqueness result for an intermediate problem, Lemma 5.3.

In this section, we present the proof of Theorem 4.1. To this end, in what follows we assume that (3.2), (3.3), (3.5), (3.10), (3.16), (3.17) (3.20), (3.21), and (3.28) hold.

In order to provide an equivalence form of Problem $\mathcal{P}^{V}$, we introduce two history-dependent operators. More exactly, we define operator $\mathcal{S}_{1}:C\left(\mathbb{R}_{+};V\right)\rightarrow C\left(\mathbb{R}_{+};Q\right)$ by

for all $t\in\mathbb{R}_{+}$. The unique existence of $\mathcal{S}_{1}$ was proved in [32] using assumptions (3.2), (3.3), (3.5), and fixed point arguments.

We use Riesz’s representation theorem to define operator $\mathcal{S}_{2}:C\left(\mathbb{R}_{+};V\right)\rightarrow C\left(\mathbb{R}_{+};V\right)$ by

Finally, let $\mathcal{S}:C\left(\mathbb{R}_{+};V\right)\rightarrow C\left(\mathbb{R}_{+};V\right)$ be the operator defined by

In [33] was proved that $\mathcal{S}$ is a history-dependent operator, i.e. $\forall n\in\mathbb{N}$ there exists $s_{n}>0$ such that