Received 11 May 2015
Accepted 27 September 2015

Elastic material; frictional contact; normal compliance; unilateral constraint; wear; weak solution; penalization

74M15; 74G25; 74G30; 49J40

Contact processes between deformable bodies or between a deformable body and a foundation abound in industry and everyday life. Their modeling is rather complex and, usually, leads to strongly nonlinear boundary value problems. Basic reference in the field includes [1-5] and, more recently.[69] There, the mathematical analysis of various models of contact is provided, including existence and uniqueness results of the solution. The references [2,3,7] deal also with the numerical analysis of various models of contact, including the study of fully discrete schemes, error estimates and numerical simulations.

Contact processes are accompanied by a number of phenomena among which the main one is the friction. Nevertheless, more is involved in contact than just friction. Indeed, during a contact process elastic or plastic deformations of the surface asperities may happen. Also, some or all of the following may take place: squeezing of oil or other fluids, breaking of the asperities’ tips and production of debris, motion of the debris, formation or welding of junctions, creeping, fracture, etc. Moreover, frictional contact is associated with heat generation, material damage, wear and adhesion of contacting surfaces.

As the contact process evolves, the contacting surfaces evolve too, via their wear. Wear in sliding systems is often very slow but it is persisting, continuous and cumulative. There may be increase in the conformity of the surfaces and their smoothness, or increase in the surface roughness, fogging of the surface, generation of scratches and grooves, initiation of cracks and generation of debris
which may change the contact characteristics. Asperities under large contact stresses may deform plastically or break. In the first case, the surface morphology changes and, therefore, both the contact stress and the friction traction are affected. These may be incorporated into a history or memorydependent friction coefficient. In the second case, when asperities break, the surfaces wear out, debris are produced, and again the surface structure changes over time. This must be taken into account if the long time behavior of the system is to be realistically predicted.

To model the wear of the contacting surfaces the wear function $w=w(\boldsymbol{x},t)$ is introduced, measuring the depth, in the normal direction, of the removed material. Therefore, it measures the change in the surface geometry, and represents the cumulative amount of material removed, per unit surface area, in the neighborhood of the point $\boldsymbol{x}$ up to time $t$. Since the amounts of material removed are small, as an approximation, one may treat it as a change in the gap. It is usually assumed that the rate of wear of the surface is proportional to the contact pressure and to the relative slip rate, that is to the dissipated frictional power. This leads to the rate form of Archard’s law of surface wear,

where $k$ is the wear coefficient, a very small positive constant in practice. Also, $\sigma_{v}$ represents the normal stress on the contact surface and $\|\boldsymbol{v}\|$ denotes the relative slip rate. The initial condition is $w(\boldsymbol{x},0)=w_{0}(\boldsymbol{x})$, and $w_{0}(\boldsymbol{x})=0$ when the surface is new or the initial shape is used as the reference configuration. The wear implies the evolution of contacting surfaces and these changes affect the contact process. Thus, due to its crucial role, there exists a large engineering and mathematical literature devoted to this topic. We resume to mention here the references [8,10-22], among others.

A mathematical model which describes the equilibrium of an elastic body in frictional contact with a moving foundation was recently considered in [23]. There, the contact was modeled with a normal compliance condition with unilateral constraints, associated to a sliding version of Coulomb’s law of dry friction. The unique weak solvability of the model was proved, by using arguments of elliptic quasivariational inequalities. The current paper represents a continuation of [23]. Here, we complete the model studied in [23] by taking into account the wear of the foundation. We model the wear process with a version of Archard’s law (1.1), as is customary in the mathematical literature. This leads to a new and interesting mathematical model which, in contrast to the model in [23], is evolutionary. Providing the variational analysis of this new model represents the main aim of this paper.

The rest of the manuscript is structured as follows. In Section 2, we present the notation and some preliminary material. In Section 3, we introduce the model of sliding frictional contact with wear, list the assumptions on the data and derive its variational formulation. The unique weak solvability of the contact problem is presented in Section 4. There, we state and prove our main existence and uniqueness result, Theorem 4.1. The proof is based on arguments on time-dependent variational inequalities and fixed point. Finally, in Section 5 we present our second result, Theorem 5.1. It states the convergence of the solution of a penalized frictional contact problem with wear to the solution of the contact model considered in Section 3, as the penalization parameter converges to zero.

In this section, we present the notation we shall use and some preliminary material. 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 $d\in\mathbb{N}$, we denote by $\mathbb{S}^{d}$ the space of second-order symmetric tensors on $\mathbb{R}^{d}$. Moreover, the inner product and norm on $\mathbb{R}^{d}$ and $\mathbb{S}^{d}$ are defined by

Let $\Omega\subset\mathbb{R}^{d}(d=1,2,3)$ 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 $\Gamma$. Also, we use standard notation for the Lebesgue and Sobolev spaces associated to $\Omega$ and $\Gamma$. In particular, we recall that the inner products on the Hilbert spaces $L^{2}(\Omega)^{d}$ and $L^{2}(\Gamma)^{d}$ are given by

and the associated norms will be denoted by $\|\cdot\|_{L^{2}(\Omega)^{d}}$ and $\|\cdot\|_{L^{2}(\Gamma)^{d}}$, respectively. 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 $\boldsymbol{\varepsilon}$ is the deformation operator given by

Recall that the completeness of the space $\left(V,\|\cdot\|_{V}\right)$ follows from the assumption meas $\left(\Gamma_{1}\right)>0$, 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 $v_{\nu}=\boldsymbol{v}\cdot\boldsymbol{v},\quad\boldsymbol{v}_{\tau}=\boldsymbol{v}-\boldsymbol{v}_{\nu}\boldsymbol{v}$. 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 $\boldsymbol{\sigma}:\boldsymbol{\Omega}\cup\Gamma\rightarrow\mathbb{S}^{d}$ we denote by $\sigma_{\nu}$ and $\boldsymbol{\sigma}_{\tau}$ the normal and the tangential components of the vector $\boldsymbol{\sigma}\boldsymbol{v}$ on $\Gamma$, respectively, and we recall that $\sigma_{\nu}=\boldsymbol{\sigma}\boldsymbol{v}\cdot\boldsymbol{v}$ and $\boldsymbol{\sigma}_{\tau}=\boldsymbol{\sigma}\boldsymbol{v}-\sigma_{\nu}\boldsymbol{v}$. Moreover, the following Green’s formula holds:

We end this section with two abstract results which will be used in the rest of the paper. The first one represents an existence, uniqueness, and convergence result for elliptic variational inequalities. To introduce it, we consider a real Hilbert space $X$ endowed with the inner product $(\cdot,\cdot)_{X}$ and the associated norm $\|\cdot\|_{X}$. We assume that:

With these given data we consider the problem of finding an element $u$ such that

and, for each $\rho>0$, we consider the problem of finding an element $u_{\rho}$ such that

The following result, proved in [9], will be used in Sections 4 and 5 of this paper.
Theorem 2.1: Let $X$ be a Hilbert space and assume that (2.3)-(2.5) hold. Then:
(1) The variational inequality (2.7) has a unique solution.
(2) For each $\rho>0$ there exists a unique element $u_{\rho}$ which solves the nonlinear Equation (2.8).
(3) The solution $u_{\rho}$ of (2.8) converges strongly to the solution $u$ of (2.7), i.e.

The second abstract result we need is a fixed point result. To introduce it we consider a Banach space $X$. We use the notation $C\left(\mathbb{R}_{+};X\right)$ for the space of continuous functions defined on $\mathbb{R}_{+}$with values in $X$ and, 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 $K$. We also use the notation $C^{1}\left(\mathbb{R}_{+};X\right)$ for the space of continuous differentiable functions defined on $\mathbb{R}_{+}$with values in $X$.

The following fixed-point result will be used in Section 5 of the paper.
Theorem 2.2: Let $\left(X,\|\cdot\|_{X}\right)$ be a real Banach space and let $\Lambda:C\left(\mathbb{R}_{+};X\right)\rightarrow C\left(\mathbb{R}_{+};X\right)$ be a nonlinear operator with the following property: for each $n\in\mathbb{N}$ there exists $c_{n}>0$ such that

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

Theorem 2.2 represents a simplified version of Corollary 2.5 in [24]. We underline that in (2.10) and below, the notation $\Lambda\eta(t)$ represents the value of the function $\Lambda\eta$ at the point $t$, i.e. $\Lambda\eta(t)=(\Lambda\eta)(t)$.

In this section we introduce the contact problem, list the assumptions on the data and derive its variational formulation.

The physical setting is as follows. An elastic body occupies a bounded domain $\Omega\subset\mathbb{R}^{d}(d=2,3)$ with a Lipschitz continuous boundary $\Gamma$, divided into three measurable parts $\Gamma_{1},\Gamma_{2}$, and $\Gamma_{3}$ such that meas $\left(\Gamma_{1}\right)>0$ and, in addition, $\Gamma_{3}$ is plane. The body is subject to the action of body forces of density $\boldsymbol{f}_{0}$. It is fixed on $\Gamma_{1}$ and surfaces tractions of density $\boldsymbol{f}_{2}$ act on $\Gamma_{2}$. On $\Gamma_{3}$, the body is in frictional contact with a moving obstacle, the so-called foundation. We denote by $\boldsymbol{v}^{*}$ the velocity of the foundation, which is supposed to be a non-vanishing time-dependent function in the plane of $\Gamma_{3}$. The friction implies the wear of the foundation that we model with a surface variable, the wear function. Its evolution is governed by a simplified version of Archard’s law that we shall describe below. Moreover, we assume that the foundation is deformable and, therefore, its penetration is allowed. We model the contact with a normal compliance condition with unilateral constraint, which takes into account the wear of the foundation. We associate this condition to a sliding version of Coulomb’s law of dry friction. We adopt the framework of the small strain theory and we assume
that the contact process is quasistatic and it is studied in the interval of time $\mathbb{R}_{+}=[0,\infty)$. Then, the classical formulation of the contact problem under consideration is the following.
Problem $\mathcal{P}$. Find a stress field $\boldsymbol{\sigma}:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{S}^{d}$, a displacement field $\boldsymbol{u}:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{R}^{d}$, and a wear function $w:\Gamma_{3}\times\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$such that

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

Here and below, for simplicity, we do not indicate explicitly the dependence of various functions on the spatial variable $\boldsymbol{x}$. Moreover, the functions $\boldsymbol{n}^{*}$ and $\alpha$ are given by

where $k$ represents the wear coefficient.
We now provide a brief explanation for the equations and conditions in Problem $\mathcal{P}$. First, Equation (3.1) represents the elastic constitutive law of the material in which $\mathcal{F}$ denotes a given nonlinear operator. Equation (3.2) is the equilibrium equation in which Div represents the divergence operator for tensor-valued functions. Conditions (3.3) and (3.4) are the displacement and traction boundary conditions, respectively.

Next, condition (3.5) represents the contact condition in which $g>0$ and $p$ is a positive Lipschitz continuous increasing function which vanishes for a negative argument. This condition can be derived in the following way. First, we assume that the obstacle is made of a hard material covered by a layer of soft material of thickness $g$. Thus, at each moment $t$, the normal stress has an additive decomposition of the form

in which the function $\sigma_{v}^{R}(t)$ describes the reaction to penetration of the hard material and $\sigma_{v}^{S}(t)$ describes the reaction of the soft material. The hard material does not wear and is perfectly rigid. Therefore, the penetration is limited by the bound $g$ and $\sigma_{\nu}^{R}$ satisfies the Signorini condition in the form with a gap function, i.e.

The soft material is elastic and could wear. Therefore, we assume that $\sigma_{\nu}^{S}(t)$ satisfies a normal compliance contact condition with wear, that is

This condition shows that at each moment $t$, the reaction of the soft layer depends on the current value of the penetration, represented by $u_{v}(t)-w(t)$. Indeed, we assume that a wear process of the
soft layer of the foundation takes place and the debris are immediately removed from the system. Thus, the penetration becomes $u_{\nu}(t)-w(t)$, instead of $u_{\nu}(t)$ as in the case without wear. Condition (3.12) describes the fact that the surface geometry of foundation is affected by wear, see [8] for details. We now combine (3.10) and (3.12) to see that

Then we substitute equality (3.13) in (3.11) to obtain the contact condition (3.5).
We now describe the frictional contact condition (3.6). First, we recall the classical Coulomb law of dry friction,

Here $\mu$ represents the friction coefficient, $\dot{\boldsymbol{u}}_{\tau}(t)$ is the tangential velocity, and $\dot{\boldsymbol{u}}_{\tau}(t)-\boldsymbol{v}^{*}(t)$ represents the relative tangential velocity or the relative slip rate. We assume that at each moment $t$ the velocity of the foundation, $\boldsymbol{v}^{*}(t)$, is large in comparison with the tangential velocity $\dot{\boldsymbol{u}}_{\tau}(t)$ and, for this reason, we approximate the relative slip rate by $\boldsymbol{v}^{*}(t)$. Therefore, using the approximations $\dot{\boldsymbol{u}}_{\tau}(t)-\boldsymbol{v}^{*}(t)\approx-\boldsymbol{v}^{*}(t)\neq\mathbf{0},\left\|\dot{\boldsymbol{u}}_{\tau}(t)-\boldsymbol{v}^{*}(t)\right\|\approx\left\|\boldsymbol{v}^{*}(t)\right\|$, the friction law (3.14) implies that

Therefore, using the definition (3.9) of the vector $\boldsymbol{n}^{*}(t)$ yields

Next, we note that as far as the contact of the elastic body is in the status of normal compliance (i.e. $u_{\nu}(t)<g$ ), condition (3.5) shows that

and, therefore, substituting this equality in (3.15) we deduce that (3.6) holds. We extend this condition to the case when the contact is unilateral, i.e. when $u_{\nu}(t)=g$. In this way, we fully justify the friction law (3.6).

Next, to obtain the differential Equation (3.7) we start from the Archard’s law, (1.1), i.e.

Then, using again the approximation $\left\|\dot{\boldsymbol{u}}_{\tau}(t)-\boldsymbol{v}^{*}(t)\right\|\approx\left\|\boldsymbol{v}^{*}(t)\right\|$, Equation (3.17) leads to

We now use the definition (3.9) of the function $\alpha$ to obtain

Next, we note that as far as the contact of the elastic body is in the status of normal compliance (3.16) holds and, therefore, substituting this equality in (3.18) we deduce (3.7). We extend this equality in the case of the unilateral contact, i.e. in the case when $u_{\nu}(t)=g$. In this way we fully justify the differential Equation (3.7) which governs the evolution of the wear function.

Finally (3.8) represents the initial condition for the wear function, which shows that at the initial moment the foundation is new.

We note that considering an arbitrary contact surface $\Gamma_{3}$ and a thickness $g=g(\boldsymbol{x})$ depending on the spatial variable does not cause additional mathematical difficulties in the analysis of Problem $\mathcal{P}$. Nevertheless, we decided to assume that $\Gamma_{3}$ is plane and $g$ is a constant since these assumptions arise in a large number of the industrial process and lead to a simple geometry which helps the reader to better understand the wear phenomenon.

We now turn to the variational formulation of Problem $\mathcal{P}$ and, to this end, we list the assumptions on the data. First, we assume that the elasticity operator $\mathcal{F}$ and the normal compliance function satisfy the following condition.

The densities of body forces and surface tractions have the regularity

Finally, the friction coefficient, the wear coefficient, and the foundation velocity verify

Note that assumption (3.24) is compatible with the physical setting described above since, at each time moment, the velocity of the foundation is assumed to be large enough. In addition, (3.9), (3.23), and (3.24) imply that

and, moreover,

Next, we introduce the set of admissible displacements fields defined by

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

for all $\boldsymbol{v}\in V$ and $t\in\mathbb{R}_{+}$. It follows from assumption (3.21) that $\boldsymbol{f}$ has the regularity

Assume in what follows that ( $\boldsymbol{\sigma},\boldsymbol{u},w$ ) are sufficiently regular functions which satisfy (3.1)-(3.8) and let $\boldsymbol{v}\in U$ and $t>0$ be given. We use Green formula (2.2) and the equilibrium Equation (3.2) to obtain

Next, we split the boundary integral over $\Gamma_{1},\Gamma_{2}$, and $\Gamma_{3}$. Since $\boldsymbol{v}-\boldsymbol{u}(t)=\mathbf{0}$ on $\Gamma_{1},\boldsymbol{\sigma}(t)\boldsymbol{v}=\boldsymbol{f}_{2}(t)$ on $\Gamma_{2}$, taking into account (3.28) we deduce that

Note that

and, using contact condition (3.5) and the definition (3.27) of the set $U$, we have

Therefore, taking into account identity (3.31), inequality (3.32), and the friction law (3.6) we obtain that

We now combine (3.30) and (3.33) to obtain that

In addition, we note that the boundary condition (3.3), the first inequality in (3.5) and (3.27) imply that $\boldsymbol{u}(t)\in U$. Finally, we integrate the differential Equation (3.7) with the initial condition (3.8) to obtain that

We now gather the constitutive law (3.1), the variational inequality (3.34), and the integral Equation (3.35) to obtain the following variational formulation of the contact problem $\mathcal{P}$.

Problem $\mathcal{P}^{V}$. Find a stress field $\boldsymbol{\sigma}:\mathbb{R}_{+}\rightarrow Q$, a displacement field $\boldsymbol{u}:\mathbb{R}_{+}\rightarrow U$, and a wear function $w:\mathbb{R}_{+}\rightarrow L^{2}\left(\Gamma_{3}\right)$ such that

for all $t\in\mathbb{R}_{+}$.
The unique solvability of Problem $\mathcal{P}^{V}$ will be proved in the next section. A triple ( $\boldsymbol{\sigma},\boldsymbol{u},w$ ) which satisfy (3.36)-(3.38) is called weak solution of Problem $\mathcal{P}$.

We end this section with some additional comments on our contact model. Assume that (3.1)(3.8) has a classical solution. Then, since $\alpha$ and $p$ are positive functions, it follows from (3.7) what $\dot{w}(t)\geq 0$ for all $t$, i.e. the wear is increasing, in each point of the contact surface. Moreover, if at a moment $t_{0}$ we have $w\left(t_{0}\right)=g$, then, using Equation (3.7) and the properties (3.20) of the function $p$, it can be easily proved that $w\left(t_{0}\right)=g$ for all $t\geq t_{0}$. This behavior shows that the wear of the foundation is limited by the constraint $w(t)\leq g$, which fits with the assumption that rigid layer of the foundation does not wear.

In this section, we state and prove the following existence and uniqueness result.
Theorem 4.1: Assume that (3.19)-(3.24) hold. Then there exists a constant $\mu_{0}$ which depends only on $\Omega,\Gamma_{1},\Gamma_{3},\mathcal{F}$, and $p$ such that, if

then Problem $\mathcal{P}^{V}$ has a unique solution. Moreover, the solution has the regularity

and, in addition,

We conclude from above that Problem $\mathcal{P}$ has a unique weak solution, provided that assumptions of Theorem 4.1 are satisfied. In addition, note that condition (4.1) represents a smallness condition on the coefficient of friction which is frequently needed in the study of static or quasistatic frictional contact problems with elastic materials. The question if this condition describes an intrinsic feature of the frictional contact process or it represents a limitation of our mathematical tools represents an open question which, clearly, has to be investigated in the future.

The proof of Theorem 4.1 will be carried out in several steps. We assume in the rest of this section that (3.19)-(3.24) hold. In the first step, we consider a given wear function $w\in C\left(\mathbb{R}_{+};L^{2}\left(\Gamma_{3}\right)\right)$ and we construct the following intermediate variational problem.

Problem $\mathcal{P}_{w}^{V}$. Find a displacement field $\boldsymbol{u}_{w}:\mathbb{R}_{+}\rightarrow U$ such that

for all $t\in\mathbb{R}_{+}$.
In the study of Problem $\mathcal{P}_{w}^{V}$, we have the following existence and uniqueness result.
Lemma 4.2: There exists a constant $\mu_{0}$ which depends only on $\Omega,\Gamma_{1},\Gamma_{3},\mathcal{F}$, and $p$ such that, if (4.1) holds, then there exists a unique solution to Problem $\mathcal{P}_{w}^{V}$ which satisfies $\boldsymbol{u}_{w}\in C\left(\mathbb{R}_{+};U\right)$.
Proof: Let $t\in\mathbb{R}_{+}$and consider the operator $A_{wt}:V\rightarrow V$ defined by

We use assumptions (3.19), (3.20), (3.22), and inequality (2.1) to see that the operator $A_{wt}$ is Lipschitz continuous, i.e. it verifies the inequality

for all $\boldsymbol{u}_{1},\boldsymbol{u}_{2}\in V$. Next, we introduce the constant $\mu_{0}$ defined by

and note that it depends only on $\Omega,\Gamma_{1},\Gamma_{3},\mathcal{F}$, and $p$. Assume that (4.1) holds. Then, we obtain

We use again assumptions (3.19), (3.20), and (3.20) and inequalities (2.1) and (4.8) to deduce that the operator $A_{wt}$ is strongly monotone, i.e. it satisfies the inequality

for all $\boldsymbol{u}_{1},\boldsymbol{u}_{2}\in V$.
Using these ingredients, by Theorem 2.1(1), we deduce that there exists a unique element $\boldsymbol{u}_{wt}\in U$ such that