International Journal of Analysis and Applications

Volume 18, Number 2 (2020), 161-171

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-18-2020-161

ANALYSIS OF QUASISTATIC FRICTIONAL CONTACT PROBLEM WITH

SUBDIFFERENTIAL FORM, UNILATERAL CONDITION AND LONG-TERM

MEMORY

A. OURAHMOUN1,∗, B. BOUDERAH1, T. SERRAR2

1University of M’sila 28000, M’sila LMPA Laboratory, Algeria

2University of Setif 1, 19000, Algeria

∗Corresponding author: ourahmounabbes@yahoo.fr

Abstract. We consider a quasistatic problem which models the contact between a deformable body and

an obstacle called foundation. The material is assumed to have a viscoelastic behavior that we model with

a constitutive law with long-term memory, thus at each moment of time, the stress tensor depends not

only on the present strain tensor, but also on its whole history. In Contact Mechanics, history-dependent

operators could arise both in the constitutive law of the material and in the frictional contact conditions. The

mathematical analysis of contact models leads to the study of variational and hemivariational inequalities.

For this reason a large number of contact problems lead to inequalities which involve history dependent

operators, called history dependent inequalities. Such inequalities could be variational or hemivariational

and variational hemivariational.

In this paper we derive a weak formulation of the problem and, under appropriate regularity hypotheses,

we stablish an existence and uniqueness result. The proof of the result is based on arguments of variational

inequalities monotone operators and Banach fixed point theorem.

1. Introduction

Contact mechanics still remain a rich domain of research, and the literature devoted to various aspects of

the subject is growing. An early attempt at the study of contact problems for elastic viscoelastic materials

Received 2019-12-31; accepted 2020-02-03; published 2020-03-02.

2000 Mathematics Subject Classification. Primary 74M10, 74M15, 49J40.

Key words and phrases. elastic material; frictional contact; unilateral condition; weak solution.

c©2020 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

161

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-161


Int. J. Anal. Appl. 18 (2) (2020) 162

within the mathematical analysis framework was introduced in the pioneering reference works [6, 7, 15]. Fur-

ther extensions to non convex contact conditions with non-monotone and possible multi-valued constitutive

laws led to the active domain of non-smooth mechanic within the framework of the so-called hemivariational

inequalities, for a mathematical as well as mechanical treatment we refer to [10]. There is a growing in-

terest in the study of history-dependent inequalities. For instance, a class of variational inequalities with

history dependent operators was considered in [15], where abstract existence, uniqueness and regularity

results were proved. These results were extended in[18] to a more general class of variational inequalities

and were completed in [6] with error estimate and convergence results. Various results on hemivariational

and variational-hemivariational inequalities with history dependent operators, formulated in Sobolev-type

spaces, could befound in [7, 9].

We introduce a new model of frictional contact for viscoelastic materials and to illustrate the use of history

dependent variational hemivariational inequality in its variational analysis. Thus, in Section 2 we introduce

the contact problem, in which the material’s behavior is modeled by a nonlinear viscoelastic constitutive

law with long memory, the process is quasistatic, the contact is frictional and the contact conditions are

in a subdifferential form with unilateral conditions for the displacement. Then, in Section 3 we list the

assumptions on the data and derive the variational formulation of the problem. It is in a form of a history-

dependent variational-hemivariational inequality in which the unknown is the displacement field.Next in

Section 4 we state our main existence and uniqueness result, Theorem (4.2) the proof of the theorem is

obtained by using arguments of elliptic variational-hemivariational inequalities and a fixed point result for

history dependent operators.

2. The Contact Model

The physical setting we consider is the following. A deformable body occupies a domain Ω ⊂ Rd (d =

1, 2, 3 in applications) with outer Lipschitz surface Γ that is divided into three disjoint measurable parts Γi

(i = 1, 2, 3) such that meas(Γ1) > 0. Let [0,T] be the time interval of interest, where T > 0. The body

is clamped on Γ1 × (0,T) and therefore the displacement field vanishes there. A volume force of density f0

acts in Ω ×(0,T) and surface tractions of density f1 act on Γ2 ×(0,T).

The body is in contact on Γ3 ×(0,T) with a rigid obstacle, the so-called foundation is in frictional contact.

We assume that the process is quasistatic with long term memory and we use (1) as constitutive law. We

denote by u, σ and ε(u) the displacement field, the stress field and the linearized strain tensor, respectively,

and let v be the unit outward normal vector to Γ. Here and below, we sometimes do not indicate explicitly

the dependence of various functions on the spatial variable x ∈ Ω ∪ Γ. For a vector field u, we use notation

uv = u ·v and uτ = u − uv v for the normal and tangential components of u on Γ. Similarly, for the stress



Int. J. Anal. Appl. 18 (2) (2020) 163

field σ, its normal and tangential components on the boundary are defined by equalities σv = (σv)·v and

στ = σv −σvv, respectively.

Finally, we use Sd for the space of second order symmetric tensors on Rd and “·” will represent the

canonical inner product and the Euclidean norm on the spaces Rd and Sd, respectively. We also use the

following notation:

H =
(
L2(Ω)

)d
,H =

{
σ = (σij) | σij = σji ∈ L2(Ω), 1 ≤ i ≤ j ≤ d

}

H1 = {u ∈ H : ε(u) ∈H} ; H1 = {σ ∈H | Divσ ∈ H}

Here ε : H1 → H and Div : H1 → H are the deformation and the divergence operators, respectively,

defined by:

ε(u) = (εij(u)) ; εij(u) =
1

2
(ui,j + uj,i) ; Divσ = (σij,j)

The spaces H, H, H1 and H1 are real Hilbert spaces endowed with the canonical inner products given by:

(u,v)H =

∫
Ω

ui.vidx ,(σ,τ)H =

∫
Ω

σij.τijdx

(u,v)H1 = (u,v)H + (ε(u),ε(v))H

(σ,τ)H1 = (σ,τ)H + (Divσ,Divτ)H

We recall that C denotes the class of continuous functions; and Cm, m ∈ N∗ the set of m times continuously

differentiable functions.

Finally D(Ω) denotes the set of infinitely differentiable real functions with compact support in Ω; and W

m,p ,m ∈ N, 1 ≤ p ≤ +∞ for the classical Sobolev spaces; and

Hm0 (Ω) := {w ∈ W
m,2(Ω),w = 0 on Γ},m ≥ 1.

With these assumptions, the classical formulation or mathematical model which describes the equilibrium

of the body in the physical setting above is the following.

Problem P. Find a displacement field u : Ω×R+ → Rd, a stress field σ : Ω×R+ →Sd and two interface

forces ηv : Γ3 ×R+ → R and ξv : Γ3 ×R+ → R such that

σ(t) = Aε(u(t)) +
t∫

0

B(t−s)ε(u(s))ds in Ω × (0,T) (2.1)



Int. J. Anal. Appl. 18 (2) (2020) 164

Div(σ(t)) + f0(t) = 0 in Ω × (0,T) (2.2)

u(t) = 0 on Γ1 × (0,T) (2.3)

σ(t)v = f2(t) on Γ2 × (0,T) (2.4)




uv(t) ≤ g , σv(t) + ξv(t) + ηv(t) ≤ 0

(uv(t) −g)(σv(t) + ξv(t) + ηv(t)) = 0

|uv(t)| ≤ Fm


 t∫

0

u+v (s)ds




ηv(t) =




0 if uv(t) ≺ 0

Fm


 t∫

0

u+v (s)ds


 if uv(t) ≥ 0

ξv(t) ∈ ∂jv(uv(t)) on Γ3 × (0,T)

on Γ3 × (0,T) (2.5)




‖στ (t)‖≤ Fb(uv(t))

−στ (t) = Fb(uv(t))
uτ (t)

‖uτ (t)‖
if uτ (t) 6= 0

on Γ3 × (0,T) (2.6)

First, Eq.(2.1) is the constitutive law for viscoelastic materials in which A represent the elasticity operator

and B represents the relaxation tensor. Various comments and mechanical interpretation related to such

kind of equations could be found in [8, 16]. Equation (2.2) is the equilibrium equation that we use here since

we assume that the process is quasistatic. Conditions (2.3) and (2.4) represent the displacement and traction

conditions, respectively. Condition (2.5) represents the contact condition in which g > 0, jv and Fm are

given functions and ∂jv represents the Clarke subdifferential of jv. Finally, relations (2.7) represent the static

version of Coulomb’s law of dry friction. Here Fb denotes a positive function, the friction bound assumed

to depend on the normal displacement uv. The contact condition (2.5) represents the trait of novelty of our

model. Note that this condition models the contact with a foundation made of a rigid body covered by a

layer made of soft material and a thin crust with memory effects.

3. Variational analysis

To derive a variational formulation of the problem we use the spaces for the displacement field we use the

space

V =
{
v = (vi) ∈ H1(Ω) | v = 0 on Γ1

}
(3.1)



Int. J. Anal. Appl. 18 (2) (2020) 165

which is a real Hilbert space with inner product (u,v)V = (ε(u),ε(v))H where

(u,v)V =

∫
Ω

ε(u) ·ε(v)dx

and associated norm ‖·‖V .

we consider the space of fourth-order tensor fields

Q∞ =
{
E = (Eijkl) | Eijkl = Eklij = Ejikl ∈ L

∞(Ω)
}

which is a real Banach space with norm

‖E‖Q∞ = max0≤i,j,k,l≤d
‖Eijkl‖L∞(Ω)

Finally, we use N for the set of positive integers and R+ for the set of nonnegative real numbers. For a

normed space X, we use the notation C(R+; X) for the space of continuous functions defined on R+ with

values in X.

By the Sobolev trace theorem, we have

‖v‖L2(Γ3,Rd) ≤‖γ‖‖v‖V , ∀v ∈ V (3.2)

‖γ‖ being the norm of the trace operator γ : V → L2(Γ3,Rd)

We now list the assumptions on the data and we assume that

1. the elasticity operator A : Ω ×Sd →Sd satisfies the following properties




(a) There exists LA > 0 such that for all ε1,ε2 ∈Sd,a.e.x ∈ Ω,

‖A(x,ε1) −A(x,ε2)‖≤ LA |ε1 − ε2|

(b) There exists m > 0 such that for all ε1,ε2 ∈Sd,a.e.x ∈ Ω,

(A(x,ε1) −A(x,ε2)) . (ε1 − ε2) ≥ m‖ε1 − ε2‖
2

A(·,ε1) is measurable on Ω for all ε ∈Sd

A(x, 0) = 0 for a.e.x ∈ Ω

(3.3)

2. The relaxation tensor B is such that

B ∈C(R+,Q∞) (3.4)



Int. J. Anal. Appl. 18 (2) (2020) 166

3. the potential function jv : Γ3 × R → R,assumed to satisfy the following conditions




(a) jv(·,r) is measurable on Γ3for all r ∈ R and there

exists
−
e ∈ L2(Γ3) such that jv(·,

−
e) ∈ L1(Γ3)

(b) jv(x, ·) is locally Lipschitz on R for a.e.x ∈ Γ3

(c) | jv(x,r)| ≤
−
c0 +

−
c1 |r| for a.e.x ∈ Γ3 and

for all r ∈ R with
−
c0,
−
c1 ≥ 0

(3.5)

Next, we assume that the penetration bound g : Γ3 → R, the memory function Fm :Γ3 × R → R+ and

the friction bound Fb : Γ3 × R → R are assumed to satisfy the following conditions.

g ∈ L2(Γ3) , g(x) ≥ 0 a.e.on Γ3. (3.6)

And




(a)There exists LFm > 0 such that |Fm(x,r1) −Fm(x,r1)| ≤ LFm |r1 −r2|

for all r1,r2 ∈ R, a.e.x ∈ Γ3

(b) Fm(·,r) is measurable on Γ3 for all r ∈ R

(c) x → Fm(x, 0) ∈ L2(Γ3)

(3.7)




(a)There exists LFb > 0 such that |Fb(x,r1) −Fb(x,r1)| ≤ LFb |r1 −r2|

for all r1,r2 ∈ R,a.e.x ∈ Γ3

(b) Fb(·,r) is measurable on Γ3, for all r1,r2 ∈ R,a.e.x ∈ Γ3

(c) Fb(x,r) = 0 for r ≤ 0 ,Fb(x,r) > 0 for all r > 0 a.e.x ∈ Γ3

(3.8)

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

f0 ∈ C(R+; L2(Ω; Rd)) , f2 ∈ C(R+; L2(Γ2; Rd)) (3.9)

and, finally, we assume the smallness condition

LFb ‖γ‖ + α jv < mF (3.10)

We now introduce the set of the admissible displacement fields U ⊂ V and the function f : R+ → V
′

defined by


 U = {v ∈ V | vv ≤ g on Γ3}〈f(t), v〉V ′×V = (f0(t), v)(L2(Ω),Rd) + (f2(t), v)(L2(Γ3),Rd), for all v ∈ V, t ∈ R+ (3.11)



Int. J. Anal. Appl. 18 (2) (2020) 167

Assume now that (u,σ) represents a couple of regular functions which satisfy (2.1)−(2.6) and let t ∈ R+,

v ∈ U. We perform an integration by parts, split the surface integral on three integrals on Γ1, Γ2 and Γ3,

and use the equalities (2.2) − (2.4)to deduce that

∫
Ω

σ(t) · (ε(v) −ε(u(t)))dx =
∫
Ω

f0(t) · (v −u(t))dx

+

∫
Γ2

f2(t) · (v −u(t))dΓ +
∫
Γ3

σv(t)(vv −uv(t))dΓ +
∫
Γ3

στ (t)(vτ −uτ (t))dΓ (3.12)

Next, we use the contact boundary condition (2.5), the definition (3.12) and the definition of the Clarke

subdifferential to obtain that

∫
Γ3

σv(t)(vv −uv(t))dΓ +
∫
Γ3

Fm


 t∫

0

u+v (s)ds


 (v+v −u+v (t))dΓ

+

∫
Γ3

j0v(uv(t), vv −uv(t))dΓ ≥ 0 (3.13)

Note that here and below we use notation j0v(r1; r2) for the generalized directional derivative of jv at r1

in the direction r2, see [1, 2] for details.

On the other hand, the friction law (2.6) yields

∫
Γ3

στ (t)(vτ −uτ (t))dΓ +
∫
Γ3

Fb(uv(t) (‖vτ‖−‖uτ (t)‖) dΓ ≥ 0 (3.14)

We now combine equality (3.13) with inequalities (3.14), (3.15) to deduce that

∫
Ω

σ(t) · (ε(v) −ε(u(t)))dx +
∫
Γ3

Fb(uv(t) (‖vτ‖−‖uτ (t)‖) dΓ+

∫
Γ3

Fm


 t∫

0

u+v (s)ds


 (v+v −u+v (t))dΓ + ∫

Γ3

j0v(uv(t), vv −uv(t))dΓ

≥
∫
Ω

f0(t) · (v −u(t))dx +
∫
Γ2

f2(t) · (v −u(t))dΓ (3.15)

Finally, we substitute the consitutive law (2.1) in (3.15) and use notation (3.12) to obtain the following

variational formulation of Problem P, in terms of displacement.

Problem PV Find a displacement field u : R+ → U such that

The unique solvability of Problem PV is given by the following existence and uniqueness result, that we

state here and prove in the next section.

Theorem 3.1 Assume that (3.7)–(3.11) hold. Then, Problem PV has a unique solution u ∈ C(R+; U).



Int. J. Anal. Appl. 18 (2) (2020) 168

We end this section with some remarks on the weak solvability of the contact problem P.

First, a couple of functions (u,σ) defined on the positive real line R+ with values on the product space

V ×Q is called a weak solution to Problem P if u is a solution of the variational problem PV and σ satisfies

the constitutive law (2.1).

We conclude that, under the assumption of Theorem 8.1, Problem P has a unique weak solution. Moreover,

the solution has the regularity u ∈ C(R+; U) and σ ∈ C(R+; Q).

Next, recall that Theorem 8.1 provides the weak solvability of the contact problem P under the smallness

assumption (24) involving the friction bound Fb, and the normal compliance potential jv. Finally, note that

the unknowns ηv and ξv of Problem P cannot be recovered since they cannot be computed when the solution

u of Problem P is known.Actually, these unknowns represent interface forces and, as usual in solving contact

problems with unilateral constraints, we do not have information neither on the uniqueness of these functions

and on their regularity.

4. An Existence and Uniqueness Result

We present in this section an abstract result on history-dependent variational-hemivariational inequalities

that we shall use to prove the unique solvability of Problem PV. For more details on the material presented

in this section, we send the reader to [1, 2].

Theorem 4.1.Let X be a reflexive Banach space and Y be a normed space. We denote by X
′

the dual

of X and by 〈·, ·〉 X′×X the duality pairing of X and X
′
. Let K be a subset of X and A : X → X

′
, Ψ :

C(R+; X) → C(R+; Y ) be given operators ,consider also a function φ : Y ×K ×K → R, a locally Lipschitz

function j : X → R and a function f : R+ → X
′
. With these data we consider the problem of finding a

function u : R+ → U such that, for each t ∈ R+, the following inequality holds:

〈Au(t),v −u(t))〉 + φ((Ψu)(t),u(t),v) −φ((Ψu)(t),u(t),u(t)) (4.1)

+j0(u(t),v −u(t)) ≥〈f(t),v −u(t)〉 , for all v ∈ K

In the study of (4.1), we assume the following hypotheses.

K is a nonempty, closed and convex subset of X.

A : X → X
′

is an operator such that




(a)A is pseudomonotone and there exist

αA > 0,βA,γA ∈ R and u0 ∈ K such that:

〈Av,v −u0〉≥ αA‖v‖
2
X −βA‖v‖

2
X −γA for all v ∈ X.

(b)A is strongly monotone,i.e.,there exists mA > 0 such that

〈Av1 −Av2,v1 −v2〉≥ mA‖v1 −v2‖
2
X , ∀v1,v2 ∈ X

(4.2)



Int. J. Anal. Appl. 18 (2) (2020) 169

φ : Y ×K ×K → R is a function such that




(a) φ(y,u, ·) is convex and l.s.c. on K, for all y ∈ Y,u ∈ K

(b) there exist αφ , βφ > 0 such that

φ(y1,u1,v2) −φ(y1,u1,v1) + φ(y2,u2,v1) −φ(y2,u2,v2) ≤

αφ‖u1 −u2‖X ‖v1 −v2‖X + βφ‖y1 −y2‖Y ‖v1 −v2‖X
for all y1,y2 ∈ Y , u1,u2,v1,v2 ∈ K

(4.3)

j : X → R is a function such that




(a) j is locally Lipschitz

(b) ‖∂j(v)‖X′ ≤ c0 + c1 ‖v‖X , for all v ∈ V , c0 ,c1 ≥ 0

(c) there exists αj > 0 such that

j0(v1,v2 −v1) − j0(v2,v1 −v2) ≤ αj ‖v1 −v2‖
2
X , for all v1,v2 ∈ X

(4.4)




For any n ∈ N , there exists sn > 0 such that

‖(Ψu1) (t) − (Ψu2) (t)‖Y ≤ sn
∫ t

0
‖u1(s) −u2(s)‖ds

for all u1,u2 ∈ C(R+; X), for all t ∈ [0,n].

(4.5)

αϕ + αj < mA ; αϕ < αj (4.6)

f ∈ C(R+; X∗) (4.7)

Note that an operator Ψ which satisfies condition (4.5) is called a history dependent operator. Inequality

(4.1) is governed both by the function φ which is assumed to be convex with respect its second argument

and by the function j which is locally Lipschitz and could be nonconvex. Therefore, this inequality is a

variational-hemivariational inequality. In addition, the function φ in (4.1) depends on the operator Ψ ,

assumed to be history-dependent.For this problem we have the following existence and uniqueness result.

Theorem 4.2. Let X be a reflexive Banach space, Y a normed space, and assume that (4.2)–(4.7) hold.

Then, inequality (4.1) has a unique solution u ∈ C(R+; K).

The proof of is obtained by using arguments of elliptic variational-hemivariational inequalities and a

fixed point result for history dependent operators.

Proof (Theorem 4.1) We start by defining the operators A : V → V
′
,F : C(R+; V ) → C(R+; Q×L2(Γ3))

and the functions φ : L2(Γ3) ×V ×V → R and j : V → R by

(Au, v) =

∫
Ω

Fε(u) ·ε(v)dx for all u, v ∈ V (4.8)



Int. J. Anal. Appl. 18 (2) (2020) 170

(Fu)(t) =


∫

Ω

B(t−s)ε(u(s))ds,Fm


 t∫

0

u+v (s)ds




 (4.9)

for all u ∈ C(R+; V ), t ∈ R+

ϕ(ξ, u, v) = (ξ1,ε(v))Q +
(
ξ2,v

+
v

)
L2(Γ3)

+ (Fb(uv),‖vτ‖)L2(Γ3) (4.10)

for all ξ = (ξ1,ξ2) ∈ Q×L
2(Γ3), u, v ∈ V

j(v) =

∫
Γ3

jv(vv)dΓ ; for all v ∈ V. (4.11)

Then, it is easy to see that Problem PV is equivalent to the problem of finding a function u : R+ → U such

that for each t ∈ R+, the following inequality holds:

〈Au(t), v − u(t)〉 + ϕ((Fu)(t), u(t), v)−ϕ((Fu)(t), u(t), u)+ (4.12)

j0(u(t), v − u(t)) ≥〈f(t), v − u(t))〉 , for all v ∈ V

To solve this problem, we use Theorem 4.1 with X = V , Y = L2(Γ3) and K = U and, to this end, we check

in what follows that assumptions (4.2)–(4.7) hold. We use arguments similar to those used in our previous

works [8, 9] and, for this reason, we skip the details and we resume the proof as follows. First, we note

that assumption (3.7) and definition (3.12) imply (4.2). Next, a simple calculation based on the definition

(4.5) of the operator A and the properties (3.4) of the elasticity operator show that (4.2) holds with m A

= αA = mF. Moreover, using assumption (3.9) and the trace inequality (3.2), it is easy to see that the

function φ defined by (4.5) satisfies condition (4.3) with αφ = L Fb ‖γ‖. On the other hand, assumption

(3.6) on the function jv and definition (4.8) show that condition (4.4) holds with α j = α jv . And, a simple

calculation based on assumptions (3.5), (3.9) imply that the operator (4.9) is a history-dependent operator,

i.e., it satisfies condition (4.5). Now, keeping in mind that m A = αA = mF,α φ =L Fb ‖γ‖ and αj = α

jv , we easily deduce that the smallness assumption (3.11) shows that conditions (4.6) hold, too. Finally, we

note that regularity (3.9) on the densites of the body forces and tractions combined with definition (3.12)

show that condition (4.7) is satisfied. We are now in a position to use Theorem 4.2 to deduce the existence

of a unique function u ∈ C(R+; U) such that (4.12) holds, for each t ∈ R+. And, using notation (4.8)–(4.12),

we deduce that u is the unique solution to Problem PV which concludes the proof.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.



Int. J. Anal. Appl. 18 (2) (2020) 171

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	1. Introduction
	2. The Contact Model
	3. Variational analysis
	4. An Existence and Uniqueness Result
	References