Processes of contact with adhesion are important in many industrial settings where parts, usually nonmetallic, are glued together. For this reason, a considerable effort has been made in their modeling, analysis, numerical analysis, and numerical simulations and, as a result, the engineering and computational literature on this related topics is extensive. Moreover, the mathematical literature devoted to the analysis of adhesive contact process is rapidly growing.

General models with adhesion can be found in [7, 8, 9, 15, 18, and the references therein. In particular, a description of the derivation of various equations and conditions related to the adhesive contact can be found in [9]. The analysis of various contact models with adhesion, including existence and uniqueness results for weak solutions, can be found in [1, 2, 3, 4, 20, 26, 25, for instance. In carrying out this analysis, a systematic use of results on elliptic and evolutionary variational inequalities, convex analysis, nonlinear equations with monotone operators, and fixed points of operators was made. The numerical analysis of quasistatic and dynamic models of adhesive contact can be found in [20]. There, fully discrete schemes were considered and error estimates were derived. Moreover, an application of the theory in the medical field of prosthetic limbs was described in [16, 17. There, the bonding arises between the artificial limb and the tissue and is of considerable importance, since debonding may lead to decrease in the person’s ability to use the limb. The main ingredient in the models presented in all the above mentioned papers is the introduction of a surface internal variable, the bonding field, which describes the fractional density of active bonds on the contact surface. As a fraction its values are restricted to $0\leq\beta\leq 1$. When $\beta=1$ at a point of the contact surface, the adhesion is complete and all the bonds are active; when $\beta=0$ all the bonds are inactive, severed, and there is no adhesion; finally, when $0<\beta<1$ the adhesion is partial and only a fraction $\beta$ of the bonds is active.

In this paper, we cover the modelling and variational analysis of a new contact problem with adhesion within the infinitesimal strain theory. The evolution of the bonding field is described by a general first order ordinary differential equation, already used in the previously cited papers. Nevertheless, we introduce three novelties in the contact model, which make the difference with our previous papers. First, we describe material’s behavior by a viscoelastic constitutive law with long memory. Second, we model the adhesive contact with a normal compliance condition with unilateral penetration which takes into account the memory effect of the surfaces. A similar condition was introduced in [23, 24] in the study of frictionless contact process without adhesion. Also, a contact condition with normal compliance, unilateral constraint and adhesion was used in [11. There, in contrast with this paper, the memory effects of the foundation were neglected, the process was assumed to be dynamic and the material’s behavior was described with an elastic-visco-plastic constitutive law. The third novelty arises in the fact that, unlike a large number of references, the adhesive contact problem considered in this paper are formulated on the unbounded interval of time $\mathbb{R}_{+}=[0,\infty)$. This implies the use of the framework of Fréchet spaces of continuous functions, instead of that of the classical Banach spaces of continuous functions defined on a bounded interval of time, used in our previous papers.

The rest of the paper is organized as follows. In Section 2 we introduce some notations and preliminary material. In Section 3 we provide a detailed description of the model of adhesive contact. Then, in Section 4, we list the assumptions on the data, derive the variational formulation of the problem and state our main existence
and uniqueness result, Theorem 4.1. The proof is provided in Section 5. It is based on arguments of variational inequalities and fixed point.

In this short section we present the notations we shall use and some preliminary material. For further details we refer the reader to [10, 18, 22]. Everywhere in this paper we use the notation $\mathbb{N}$ for the set of positive integers and $\mathbb{R}_{+}$will represent the set of nonnegative real numbers, i.e. $\mathbb{R}_{+}=[0,+\infty)$. 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 $i,j,k,l$ run between 1 and $d$ and, unless stated otherwise, the summation convention over repeated indices is used.

Let $\Omega$ be a bounded domain $\Omega\subset\mathbb{R}^{d}(d=1,2,3)$ with a Lipschitz continuous boundary $\Gamma$ and let $\Gamma_{1}$ be a measurable part 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 $\boldsymbol{\nu}=\left(\nu_{i}\right)$ 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

Completeness of the space ( $V,\|\cdot\|_{V}$ ) follows from the assumption meas ( $\Gamma_{1}$ ) $>0$, which allows the use of Korn’s inequality.

For an element $\boldsymbol{v}\in V$ we still write $\boldsymbol{v}$ for the trace of $\boldsymbol{v}$ on the boundary $\Gamma$. We denote by $v_{\nu}$ and $\boldsymbol{v}_{\tau}$ the normal and the tangential component of $\boldsymbol{v}$ on $\Gamma$, respectively,
defined by $v_{\nu}=\boldsymbol{v}\cdot\boldsymbol{\nu},\quad\boldsymbol{v}_{\tau}=\boldsymbol{v}-v_{\nu}\boldsymbol{\nu}$. Let $\Gamma_{3}$ be a measurable part of $\Gamma$. Then, by the Sobolev trace theorem, there exists a positive constant $c_{0}$ which depends on $\Omega$, $\Gamma_{1}$ and $\Gamma_{3}$ such that

For a regular function $\boldsymbol{\sigma}:\Omega\cup\Gamma\rightarrow\mathbb{S}^{d}$ we denote by $\sigma_{\nu}$ and $\boldsymbol{\sigma}_{\tau}$ the normal and the tangential components of the vector $\boldsymbol{\sigma}\boldsymbol{\nu}$ on $\Gamma$, respectively, and we recall that $\sigma_{\nu}=\boldsymbol{\sigma}\boldsymbol{\nu}\cdot\boldsymbol{\nu}$ and $\boldsymbol{\sigma}_{\tau}=\boldsymbol{\sigma}\boldsymbol{\nu}-\sigma_{\nu}\boldsymbol{\nu}$.

We also 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, a simple calculation shows that

Given a normed space ( $X,\|\cdot\|_{X}$ ) we use the notation $C\left(\mathbb{R}_{+};X\right)$ for the space of continuous functions defined on $\mathbb{R}_{+}$with values in $X$. It is well known that, if $X$ is a Banach space, then $C\left(\mathbb{R}_{+};X\right)$ can be organized in a canonical way as a Fréchet space, i.e., as a complete metric space in which the corresponding topology is induced by a countable family of seminorms. Details can be found in [5] and [14], for instance. Also, for a subset $K\subset X$ we still use the symbol $C\left(\mathbb{R}_{+};K\right)$ for the set of continuous functions defined on $\mathbb{R}_{+}$with values in $K$. Finally, for $n\in\mathbb{N}$ and $K\subset X$ we denote by $C([0,n];K)$ the set of continuous functions defined on $[0,n]$ with values in $K$.

We end this section with the following result which will be used in Section 5 of the paper.

Theorem 2.1 Let ( $X,\|\cdot\|_{X}$ ) be a Banach space, $K$ a nonempty closed subset of $X$ and let $\Lambda:C\left(\mathbb{R}_{+};K\right)\rightarrow C\left(\mathbb{R}_{+};K\right)$ be a nonlinear operator. Assume that there exists $k\in\mathbb{N}$ with the following property: for all $n\in\mathbb{N}$ there exist two constants $c_{n}\geq 0$ and $d_{n}\in[0,1)$ such that

for all $u,v\in C\left(\mathbb{R}_{+};K\right)$ and for all $t\in[0,n]$. Then the operator $\Lambda$ has a unique fixed point $\eta^{*}\in C\left(\mathbb{R}_{+};K\right)$.

Theorem 2.1 was proved in [19] in the case when $K=X$. Nevertheless, a careful examination of the proof shows that the theorem is still valid for operators
$\Lambda:C\left(\mathbb{R}_{+};K\right)\rightarrow C\left(\mathbb{R}_{+};K\right)$, provided that $K$ is a nonempty closed part of $X$. The modification in proof are straightforward and, therefore, we do not provide the details. We only mention that the difference consists in the use of the Banach fixed point argument for contractive maps defined on the set $C([0,n];K)$ with values on $C([0,n];K)$, for all $n\in\mathbb{N}$, instead of contractive maps defined on the space $C([0,T];X)$ with values in $C([0,T];X)$.

The physical setting is as follows. A viscoelastic body occupies a bounded domain $\Omega\subset\mathbb{R}^{d}(d=1,2,3)$ with a Lipschitz continuous boundary $\Gamma$, divided into three measurable parts $\Gamma_{1},\Gamma_{2}$ and $\Gamma_{3}$, such that meas $\left(\Gamma_{1}\right)>0$. The body is subject to the action of body forces of density $\boldsymbol{f}_{0}$, is fixed on $\Gamma_{1}$ and is submitted to the action of surface tractions of density $\boldsymbol{f}_{2}$ on $\Gamma_{2}$. Moreover, the body is in adhesive contact on $\Gamma_{3}$ with an obstacle, the so-called foundation. We adopt the framework of the small strain theory, we assume that the contact process is quasistatic and we study it in the interval of time $\mathbb{R}_{+}=[0,\infty)$. To derive a mathematical model which corresponds to this physical setting we need to precise the constitutive law of the material, the balance equations and the boundary conditions, as well.

In this paper we assume that the material’s behavior follows a viscoelastic constitutive law with long memory of the form

where, here and below, $\boldsymbol{u}$ denotes the displacement field, $\boldsymbol{\sigma}$ represents the stress field, $\boldsymbol{\varepsilon}(\boldsymbol{u})$ is the linearized strain tensor and $t\in\mathbb{R}_{+}$represents the time variable. Also, $\mathcal{A}$ and $\mathcal{B}$ represent the elasticity operator and the relaxation tensor, respectively, and are assumed to verify 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 $\boldsymbol{x}$. Various examples and mechanical interpretation concerning viscoelastic constitutive laws of the from (3.1) can be found in [6, 21, 22,

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

where $\operatorname{Div}$ denotes the divergence operator for tensor valued functions, i.e. $\operatorname{Div}\boldsymbol{\sigma}=\left(\sigma_{ij,j}\right)$. This equation shows that at each time moment the external forces are balanced by the internal stresses. Moreover, since the body is fixed on $\Gamma_{1}$ and given tractions are acting on $\Gamma_{2}$ we impose the following displacement-traction conditions:

In addition, we assume that the densities of body forces and surface tractions have regularity

We now turn to the description on the adhesive contact conditions on the surface $\Gamma_{3}$ in which our main interest is. First, we assume that the penetration is limited by a bound $g>0$ and, therefore, at each time moment $t\in\mathbb{R}_{+}$, the normal displacement satisfies the inequality

Next, we assume that the normal stress has an additive decomposition of the form

where the functions $\sigma_{\nu}^{D}(t),\sigma_{\nu}^{R}(t),\sigma_{\nu}^{A}(t)$ and $\sigma_{\nu}^{M}(t)$ describe the deformability, the rigidity, the adhesive and the surface memory properties of the foundation. We assume that $\sigma_{\nu}^{D}(t)$ satisfies the normal compliance contact condition, that is

Here $p_{\nu}$ is a given function which satisfies

The part $\sigma_{\nu}^{R}(t)$ of the normal stress satisfies the Signorini condition in the form with a gap function, i.e.

Details on the Signorini condition and normal compliance function can be found in [10, 12, 13, 18, for instance. Here we restrict ourselves to recall that the normal compliance condition describes the contact with a deformable foundation and the Signorini contact condition describes the contact with a perfectly rigid foundation.

The function $\sigma_{\nu}^{M}(t)$ satisfies the condition

where $b$ is a surface memory function which verifies

Details on this condition can be found in [24.
Finally, the contribution of the bonding to the normal traction, $\sigma_{\nu}^{A}(t)$, satisfies

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

Here and below $L>0$ is the characteristic length of the bond, beyond which it stretches without offering any additional resistance (see, e.g., [15]) and $\gamma_{\nu}$ represents an adhesion coefficient. More details on this condition can be found in [20.

We combine (3.9), (3.10), (3.15) and denote $-\sigma_{\nu}^{M}(t)=\xi(t)$ to see that

Then we substitute equality (3.17) in (3.12) and use (3.8), (3.13) and (3.14) to obtain
the following contact condition

To complete our model we assume that the resistance to tangential motion is generated mainly by the glue, and the frictional traction can be neglected. In particular, when all the adhesive bonds are inactive, or broken, the motion is frictionless. Thus, the tangential traction depends on the intensity of adhesion and on the tangential displacement, but only up to the bond length $L$, that is

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

Then, $p_{\tau}(\beta)$ acts as the stiffness or spring constant, and the traction is in direction opposite to the displacement. The tangential function $p_{\tau}$ satisfies

We follow [7, 8, 20] and assume that the bonding field satisfies the unilateral constraint

Moreover, its evolution is governed by the differential equation

in which $\varepsilon_{a}$ represents the Dupré energy and $R$ is the truncation operator given by

We complete this differential equation with the initial condition

and we assume that the adhesion coefficient, $\gamma_{\nu}$, the Dupré energy $\varepsilon_{a}$, and initial bonding field, $\beta_{0}$, satisfy the conditions

We gather the above equations and conditions to obtain the following formulation of the mechanical problem of quasistatic adhesive contact with normal compliance, unilateral constraint and surface memory term.

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

for all $t\in\mathbb{R}_{+}$, there exists $\xi:\Gamma_{3}\times\mathbb{R}_{+}\rightarrow\mathbb{R}$ which satisfies

for all $t\in\mathbb{R}_{+}$and, moreover,

The unique weak solvability of the contact problem $\mathcal{P}$ will be stated in Section 4 and proved in Section 5. We end this current section with some additional comments of the contact condition (3.34), which represents one of the novelties of this paper.

First, we recall that (3.34) describes a condition with unilateral constraint, since inequality (3.8) holds at each time moment. The rest of the comments in this paragraph, together with the corresponding equalities and inequalities, are valid for a given point $\boldsymbol{x}$ on the contact surface $\Gamma_{3}$. Nevertheless, we recall that, for simplicity, we skip the dependence of various functions on $\boldsymbol{x}$. Assume that at a given moment $t$ there is penetration which did not reach the bound $g$, i.e. $0<u_{\nu}(t)<g$. Then, (3.34) yields

This equality shows that at the moment $t$, the reaction of the foundation depends both on the current value of the penetration (represented by the term $p_{\nu}\left(u_{\nu}(t)\right)$ ) and on the history of the penetration (represented by the integral term in (3.36)). Assume now that at a given moment $t$ there is separation between the body and the foundation, i.e. $u_{\nu}(t)<0$. Then, (3.34) shows that

i.e. 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 $L$. Once it exceeds it the normal traction remains constant and $\left|\sigma_{\nu}(t)\right|\leq\gamma_{\nu}L$.

In conclusion, condition (3.34) shows that delimitation takes place when there is separation. When there is penetration the contact stress is given by a normal compliance condition with memory term of the form (3.36) but up to the limit $g$. When the limit $g$ is reached, the stress is given by a Signorini-type unilateral condition. This condition can be interpreted physically as follows. The foundation is assumed to be made of a hard material covered with a thin layer made of a soft adhesive material with thickness $g$. The layer has adhesive viscoelastic behavior, i.e. is deformable, allows penetration and develops memory effects. The hard material is perfectly rigid and, therefore, it does not allow penetration. To summarize, the foundation 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.

We now turn to the variational formulation of Problem $\mathcal{P}$ and, to this end, we assume in what follows that ( $\boldsymbol{u},\boldsymbol{\sigma},\boldsymbol{\beta}$ ) represent a triple of regular functions which satisfy (3.28)-(3.35). We introduce the set of admissible displacements and the set of admissible bonding fields, respectively, defined by

Let $\boldsymbol{v}\in U$ and $t\in\mathbb{R}_{+}$be given. We use the Green’s formula to see that

and, combining this equality with the equilibrium equation (3.29), we find that

Then, we split the surface integral over $\Gamma_{1},\Gamma_{2}$ and $\Gamma_{3}$ and, since $\boldsymbol{v}-\boldsymbol{u}(t)=\mathbf{0}$ a.e. on $\Gamma_{1},\boldsymbol{\sigma}(t)\boldsymbol{\nu}=\boldsymbol{f}_{2}(t)$ on $\Gamma_{2}$, we deduce that

Moreover, since

taking into account condition (3.32) we obtain

We write now

then we use the contact condition (3.34) and definition (4.1) of the set $U$ to see that

We use (3.34), again, and the hypothesis (3.14) on function $b$ to deduce that

Then we add the inequalities (4.4) and (4.5) and integrate the result on $\Gamma_{3}$ to find that

Finally, we combine (4.3) and (4.6) to obtain that