Báo cáo hóa học: "VISCOELASTIC FRICTIONLESS CONTACT PROBLEMS WITH ADHESION" pot

In the first problem the contact is modelled with Signorini’s conditions and in the second one is modelled with normal compliance.. In both problems the adhesion of the contact surfaces

PROBLEMS WITH ADHESION

MOHAMED SELMANI AND MIRCEA SOFONEA

Received 16 December 2005; Revised 8 March 2006; Accepted 9 March 2006

We consider two quasistatic frictionless contact problems for viscoelastic bodies with longmemory In the first problem the contact is modelled with Signorini’s conditions and

in the second one is modelled with normal compliance In both problems the adhesion

of the contact surfaces is taken into account and is modelled with a surface variable,the bonding field We provide variational formulations for the mechanical problems andprove the existence of a unique weak solution to each model The proofs are based onarguments of time-dependent variational inequalities, differential equations, and a fixedpoint theorem Moreover, we prove that the solution of the Signorini contact problem can

be obtained as the limit of the solutions of the contact problem with normal compliance

as the stiffness coefficient of the foundation converges to infinity

Copyright © 2006 M Selmani and M Sofonea This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, dis-tribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

The adhesive contact between deformable bodies, when a glue is added to prevent tive motion of the surfaces, has received recently increased attention in the mathematicalliterature Basic modelling can be found in [7–9,12,17] Analysis of models for adhesivecontact can be found in [1–6,10] and in the recent monographs [15,16] An application

rela-of the theory rela-of adhesive contact in the medical field rela-of prosthetic limbs was considered

in [13,14]; there, the importance of the bonding between the bone-implant and the sue was outlined, since debonding may lead to decrease in the persons ability to use theartificial limb or joint

tis-The novelty in all the above papers is the introduction of a surface internal variable, the

bonding field, denoted in this paper by β; it describes the pointwise fractional density of active bonds on the contact surface, and sometimes referred to as the intensity of adhesion.

Following [7,8], the bonding field satisfies the restrictions 0≤ β ≤1; whenβ =1 at apoint of the contact surface, the adhesion is complete and all the bonds are active; when

Hindawi Publishing Corporation

Journal of Inequalities and Applications

Volume 2006, Article ID 36130, Pages 1 22

DOI 10.1155/JIA/2006/36130

β =0 all the bonds are inactive, severed, and there is no adhesion; when 0< β < 1 the

adhesion is partial and only a fractionβ of the bonds is active We refer the reader to the

extensive bibliography on the subject in [9,12,13,15,16]

The aim of this paper is to continue the study of adhesive problems begun in [3,4,6].There, models for dynamic or quasistatic process of frictionless adhesive contact between

a deformable body and a foundation have been analyzed and simulated; the contact wasdescribed with normal compliance or was assumed to be bilateral, and the behavior ofthe material was modelled with a nonlinear Kelvin-Voigt viscoelastic constitutive law;the models included the bonding field as an additional dependent variable, defined andevolving on the contact surface; the existence of a unique weak solution to the modelshas been obtained by using arguments of evolutionary equations in Banach spaces and afixed point theorem

In this paper we study two quasistatic problems of frictionless adhesive contact Thenovelty with respect to the papers referred to in the previous paragraph consists in thefact that here we model the material’s behavior with a viscoelastic constitutive law withlong memory and the contact with Signorini’s conditions or with normal compliance

We derive a variational formulation of the problems and prove the existence of a uniqueweak solution to each one To this end, we use similar arguments as in [3,4,6] but with

a different choice of functionals and operators, since the constitutive law and the contactboundary conditions, here and in the above-mentioned papers, are different Moreover,

we study the behavior of the solutions of the problem with normal compliance as thestiffness coefficient of the foundation tends to infinity

The paper is structured as follows InSection 2, we present some notation and inary material InSection 3, we state the mechanical models of viscoelastic frictionlesscontact with adhesion, list the assumptions on the data, and derive their variational for-mulation In Sections4 and5, we present our main existence and uniqueness results,Theorems4.1and5.1, which state the unique weak solvability of the adhesive frictionlesscontact problem with Signorini and normal compliance conditions, respectively Finally,

prelim-inSection 6we prove a convergence result,Theorem 6.1; it states that the solution of theadhesive contact problem with normal compliance converges to the solution of the adhe-sive Signorini contact problem as the stiffness coefficient of the foundation converges toinfinity

2 Notations and preliminaries

Everywhere in this paper we denote bySdthe space of second-order symmetric tensors on

Rd(d =1, 2, 3), while “·” and · represent the inner product and the Euclidean norm

onRdandSd, respectively Thus, for every u, v∈ R dandσ,τ ∈ S dwe have

u·v= u i v i, v =(v·v)1/2, σ · τ = σ i j τ i j,  τ  =(τ · τ)1/2 (2.1)

Here and below, the indicesi, j, k, l run between 1 and d and the summation convention

over repeated indices is adopted

LetΩ⊂ R d be a bounded domain with a Lipschitz continuous boundaryΓ In whatfollows we use the standard notation for theL pand Sobolev spaces associated toΩ and Γ,

and the index that follows a comma indicates a derivative with respect to the

correspond-ing component of the spatial variable x∈ Ω We also use the spaces

H1= H1(Ω)d =u=u i

:u i ∈ H1(Ω),

Q =σ =σ i j

:σ i j = σ ji ∈ L2(Ω),

Since the boundaryΓ is Lipschitz continuous, the unit outward normal vector ν on

the boundary is defined almost everywhere For every vector field v∈ H1 we use the

notation v for the trace of v onΓ and we denote by vνand vτ the normal and the tangential

components of v on the boundary, given by

For a regular (sayC1) stress fieldσ, the application of its trace on the boundary to ν is the

Cauchy stress vectorσν We define, similarly, the normal and tangential components of

the stress on the boundary by the formulas

σ ν =(σν) · ν, σ τ = σν − σ ν ν, (2.6)and we recall that the following Green’s formula holds:

spacesL p(0,T;X) and W k,p(0,T;X) where 1 ≤ p ≤+∞,k =1, 2, , and we denote by

C([0,T];X) the space of continuous functions on [0,T] with values on X, with the norm

clamped onΓ1×(0,T) and, therefore, the displacement field vanishes there A volume

force of density f0acts inΩ×(0,T) and surface tractions of density f2act onΓ2×(0,T).

The body is in an adhesive frictionless contact with an obstacle, the so-called tion, over the potential contact surfaceΓ3 Moreover, the process is quasistatic, that is,the inertial terms are neglected in the equation of motion We use a linearly viscoelas-tic constitutive law with long memory to model the material’s behavior and an ordinarydifferential equation to describe the evolution of the bonding field

founda-For the first problem, we consider here the contact is modelled with Signorini’s ditions with adhesion Thus, the classical model for the process is the following

con-Problem 3.1 Find a displacement field u :Ω×[0,T] → R d, a stress fieldσ : Ω ×[0,T] →

Sd, and a bonding fieldβ : Γ3×[0,T] →[0, 1] such that, for allt ∈[0,T],

We now describe the equations and conditions involved in our model above

First, (3.1) represent the viscoelastic constitutive law with memory, in whichᏭ and

Ꮾ denote the elasticity and the relaxation fourth-order tensors, respectively Equation(3.2) is the equilibrium equation while (3.3) and (3.4) are the displacement and tractionboundary conditions, respectively

Conditions (3.5) represent the Signorini conditions with adhesion whereγ νis a givenadhesion coefficient and Rνis the truncation operator defined by

HereL > 0 is the characteristic length of the bond beyond which it does not offer any

additional traction The introduction of the operatorR ν, together with the operator Rτ

defined below, is motivated by the mathematical arguments but it is not restrictive for theapplied point of view, since no restriction on the size of the parameterL is made in what

follows Thus, by choosingL very large, we can assume that R ν(u ν)= − u νand, therefore,from (3.5) we recover the contact conditions

u ν ≤0, σ ν+γ ν u ν β2≤0, 

σ ν+γ ν u ν β2 

u ν =0 on Γ3×(0,T). (3.10)These conditions were used in [5,12] to model the unilateral adhesive contact It followsfrom (3.5) that there is no penetration between the body and the foundation, sinceu ν ≤0during the process Also, note that when the bonding field vanishes, then the contactconditions (3.5) become the classical Signorini contact conditions with zero gap function,that is

u ν ≤0, σ ν ≤0, σ ν u ν =0 onΓ3×(0,T). (3.11)Condition (3.6) represents the adhesive contact condition on the tangential plane inwhichp τis a given function and Rτis the truncation operator given by

Next, (3.7) represents the ordinary differential equation which describes the evolution

of the bonding field and it was already used in [3,5], see also [15,16] for more details.Here, besidesγ ν, two new adhesion coefficients are involved, γτ and  a, and R ν(s)2 is

a short notation for (R ν( s))2, that is, R ν( s)2=(R ν( s))2 Notice that in this model oncedebonding occurs, bonding cannot be reestablished since, as it follows from (3.7), ˙β ≤0.Finally, (3.8) represents the initial condition in whichβ0is the given initial bondingfield

For the second problem, we study in this paper that the contact is modelled with mal compliance and adhesion, and therefore the classical model for the process is thefollowing

nor-Problem 3.2 Find a displacement field u :Ω×[0,T] → R d, a stress fieldσ : Ω ×[0,T] →

Sd, and a bonding fieldβ : Γ3×[0,T] →[0, 1] such that, for allt ∈[0,T],

mean-be descrimean-bed mean-below In this condition the interpenetrability mean-between the body and thefoundation is allowed, that is,u νcan be positive onΓ3 The contribution of the adhesivetraction to the normal one is represented by the termγ ν β2R ν(u ν); the adhesive traction

is tensile and is proportional, with proportionality coefficient γν, to the square of the tensity of adhesion and to the normal displacement, but as long as it does not exceed thebond lengthL The maximal tensile traction is γ ν L The contact condition (3.17) was used

in-in various papers; see, for example, [3,4,15,16] and the references therein

We turn to the variational formulation of the mechanical Problems3.1and3.2 To thisend, for the displacement field we need the closed subspace ofH1defined by

V =v∈ H1|v=0 onΓ1



Since measΓ1> 0, Korn’s inequality holds; thus, there exists a constant c K > 0, that

de-pends only onΩ and Γ1, such that

ε(v)

A proof of Korn’s inequality may be found in [11, page 79] OnV we consider the inner

product and the associated norm given by

(u, v)V =ε(u),ε(v)Q, v V =ε(v)

Q ∀u, v∈ V. (3.23)

It follows from Korn’s inequality that ·  H1and ·  V are equivalent norms onV and

therefore (V,  ·  V) is a real Hilbert space Moreover, by the Sobolev trace theorem there

exists a constantc0, depending only onΩ, Γ1, andΓ3, such that

v L2 (Γ 3 )d ≤ c0v V ∀v∈ V. (3.24)For the bonding field we will use the set

ᏽ=θ : [0,T] −→ L2

Γ3

: 0≤ θ(t) ≤1∀ t ∈[0,T], a.e on Γ3



. (3.25)Finally, we consider the space of fourth-order tensor fields:

Q∞=Ᏹ=Ᏹi jkl:Ᏹi jkl=Ᏹjikl=Ᏹkli j∈ L ∞(Ω), 1≤ i, j,k,l ≤ d

∃ M τ > 0 such thatp τ(x,β)  ≤ M τ ∀ β ∈ R, a.e x∈Γ3, (3.30c)

the mapping x−→ p τ(x,β) is measurable on Γ3, for anyβ ∈ R, (3.30d)

the mapping x−→ p τ(x, 0) belongs toL2 

Γ3



We also suppose that the body forces and surface tractions have the regularity

f0∈ C[0,T];L2(Ω)d

Γ3

, γ ν,γ τ, a ≥0, a.e onΓ3 (3.32)and, finally, the initial bonding field satisfies

β0∈ L2 

Γ3 , 0≤ β0≤1, a.e onΓ3. (3.33)

Next, we denote by f : [0,T] → V the function defined by

Problem 3.3 Find a displacement field u : [0, T] → V and a bonding field β : [0,T] →

Note that the variational Problems3.3and3.4are formulated in terms of displacement

and bonding fields, since the stress field was eliminated However, if the solution (u,β) of

these variational problems is known, then the corresponding stress fieldσ can be easily

obtained by using the linear viscoelastic constitutive law (3.1) or (3.13)

Remark 3.5 We also note that, unlike in Problems3.1and3.2, in the variational Problems3.3and3.4we do not need to impose explicitly the restriction 0≤ β ≤1 Indeed, (3.40)and (3.43) guarantee thatβ(x,t) ≤ β0(x) and, therefore, assumption (3.33) shows that

β(x,t) ≤1 fort ≥0, a.e x∈Γ3 On the other hand, ifβ(x,t0)=0 at timet0, then it followsfrom (3.40) and (3.43) that ˙β(x,t) =0 for allt ≥ t0and, therefore,β(x,t) =0 for allt ≥ t0,

a.e x∈Γ3 We conclude that 0≤ β(x,t) ≤1 for allt ∈[0,T], a.e x ∈Γ3

The well-posedness of Problems3.3and3.4will be provided in Sections4and5, spectively In the proofs we use a number of inequalities involving the functionalsjadand

re-jnc that we present in what follows Below in this sectionβ, β1,β2 denote elements of

L2(Γ3) such that 0≤ β, β1,β2≤1 a.e onΓ3, u1, u2, and v represent elements ofV and

c > 0 represent generic constants which may depend on Ω, Γ1,Γ3,p ν,p τ,γ ν,γ τ, andL.

First, we notice that jadand jncare linear with respect to the last argument and fore

there-jad



β,u, −v

= − jad(β,u,v), jnc(u,−v)= − jnc(u, v). (3.45)

Next, using (3.37), the properties of the truncation operators R ν and Rτ as well asassumption (3.30) on the functionp τ, after some calculus we find

jad



β1, u1, u2−u1

+jad

jad



β1, u1, u2−u1

+jad

Take u1=v and u2=0 in the previous inequality and use (3.29e), (3.45) to obtain

The inequalities (3.47)–(3.55) combined with equalities (3.45) will be used in variousplaces in the rest of the paper

4 Analysis of the Signorini contact problem

The main result in this section is the following existence and uniqueness result

Theorem 4.1 Assume that ( 3.28a )–( 3.28b ) and ( 3.30 )–( 3.33 ) hold Then Problem 3.3 has

a unique solution (u, β) which satisfies

u∈ C[0,T];V

Sig-σ ∈ C[0,T];Q1



Indeed, it follows from (3.39) that Divσ(t) + f0(t) =0 for allt ∈[0,T] and, therefore, the

regularity (4.1) of u, combined with (3.28a), (3.28b), and the regularity of f2in (3.31),implies (4.3)

We turn now to the proof ofTheorem 4.1which will be carried out in several steps Tothis end, we assume in the following that (3.28a), (3.28b), and (3.30)–(3.33) hold; below,

c denotes a generic positive constant which may depend on Ω, Γ1,Γ3,Ꮽ, Ꮾ, γν,γ τ,p τ,L,

andT but does not depend on t nor on the rest of the input data, and whose value may

change from place to place Moreover, for the sake of simplicity, we suppress, in what

follows, the explicit dependence of various functions on x∈Ω∪Γ3

Letᐆ denote the closed subset of the space C([0,T];L2(Γ3)) defined by

Problem 4.2 Find a displacement field u β: [0,T] → V such that