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Comput. Methods Appl. Mech. Engrg. 196 (2007) 2377–2389

Unstable perturbation of the equilibrium under Coulomb friction:Nonlinear eigenvalue analysis

Riad Hassani a, Ioan R. Ionescu b,*, Nour-Dine Sakki c

a Laboratoire de Geophysique Interne et Tectonophysique, Universite de Savoie and CNRS, 73376 Le Bourget du Lac, Franceb Laboratoire de Mathematiques, Universite de Savoie and CNRS, 73376 Le Bourget du Lac, France

c Laboratoire de Mathematiques de Besancon, Universite de Franche-Comte and CNRS, 16 route de Gray, 25030 Besancon, France

Received 11 April 2006; received in revised form 21 November 2006; accepted 28 November 2006

Abstract

Equilibrium configurations of elastic structures under Coulomb friction are considered. We use a nonlinear eigenvalue analysis to seewhen an equilibrium configuration can be (dynamically) unstable or when it admits (quasi-static) bifurcations. We prove that if the non-linear eigenvalue problem has a smooth solution then there exists a family of (dynamic) perturbations of the equilibrium which have anexponential growth in time. The above results are obtained under some conditions on the boundary between the free, stick and slippingzones of the contact boundary. These conditions are specific to the continuous problem and they are not necessary when we reformulatethe same result for the discrete (finite element) problem.

We propose a specific numerical algorithm to compute the eigenvalue, and the corresponding eigenfunction for a mixed finite elementapproach of the nonlinear eigenvalue problem. It makes use of the successive iterates of a nonlinear operator. We proved that the algo-rithm is consistent (i.e. if it is convergent then its limit is a solution), but we cannot give a mathematical proof of its convergence. We givesome numerical experiments to examine the convergence of the proposed algorithm. Other examples illustrate how from the nonlineareigenvalue analysis we get information on the stability of the equilibrium configurations.� 2007 Elsevier B.V. All rights reserved.

Keywords: Coulomb friction; Linear elasticity; Equilibrium configurations; Non-linear eigenvalue analysis; Instability; Non-uniqueness; Bifurcation;Mixed finite elements

1. Introduction

Equilibrium configurations of elastic structures underCoulomb friction are of great interest in science andengineering. Applications can be founded in contactmechanics, geophysics (tectonic plates, etc.) but also inproblems associated with automated assembly and manu-facturing processes. We do not want to discuss here inwhich conditions equilibrium configurations exist (see[1,5] for wedged configuration) and we focus on their sta-bility analysis. In some cases, as for instance earthquakesphenomena, the unstable perturbations are seen as a disas-

0045-7825/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2006.11.020

* Corresponding author. Tel.: +33 479 75 8642; fax: +33 479 75 8142.E-mail address: [email protected] (I.R. Ionescu).

ter. In other cases the fact that an equilibrium configura-tion is unstable tells us that the equilibrium cannot bemaintained, i.e. the configuration is not physicallyacceptable.

The general method used in analyzing the stability of asolution uref for (dynamic or quasi-static) elastic problemunder Coulomb friction (see for instance Cho and Barber[2], Nguyen [10], Vola et al. [14]) consists in constructingthe ‘‘tangent problem’’. Then they deduce the ‘‘linear sta-bility’’ through an eigenvalue analysis of this last linearproblem. In the quasi-static case Hassani et al. [4,6] havegiven sufficient conditions for non-uniqueness (or bifurca-tion) by considering the friction coefficient as an eigen-value. However, when constructing the tangent problemone has to suppose that the slip rate _uref

t of the reference

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2378 R. Hassani et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 2377–2389

solution is not vanishing. In this case there are no unilateralconstraints to be imposed for the tangent problem and theunloading of the frictional ratio (the absolute value of thetangential stress divided by the normal stress) does notoccur. This analysis is not possible when the reference solu-tion uref is an equilibrium state, ueq, for which the slip rateis vanishing. To see that, suppose that we focus on onepoint x of the frictional boundary where the frictional ratiois equal to the friction coefficient. If we want to constructthe tangent problem for ueq then we have two possibilities.The first one consists in sliding in the opposite sense of thetangential stress and the second one consists in unloadingthe frictional ratio with a vanishing slip rate. Hence, ineach point, we have two choices in the linearization pro-cess. Moreover, locally in time, each tangent problem con-tains unilateral constraints. That is why we cannot use alinear eigenvalue analysis in the study of the stability ofthe equilibrium configurations and we have to introducea nonlinear eigenvalue problem.

The specific nonlinear eigenvalue problem in linear elas-ticity under Coulomb friction was first introduced by Mar-tins et al. [9] in a finite element context, to characterizeunstable dynamic (or quasi-static) processes. They calledit (linear) eigenvalue problem with complementary inequal-ities (conditions). In order to approach numericallythe nonlinear eigenvalue problem Pinto Da Costa et al.[12,13] have rewritten it as a mixed complementary prob-lem and then solved with a general algorithm (PATH [3]).

Another type of nonlinear eigenvalue problem involvingtwo linear tangent problems with unilateral constrains wasintroduced in [7,8] to model the interacting of faults inearthquake nucleation. Moreover, Ionescu and Wolf [8]have proposed a specific algorithm for this type of nonlin-ear eigenvalue problem. In contrast with the problem stud-ied there, for the Coulomb friction the nonlinear eigenvalueproblem cannot be associated with an optimization prob-lem. This is due to the non-associative character of theCoulomb friction law.

In this paper, we have two objectives. First, we want touse the above nonlinear eigenvalue problem to see when anequilibrium configuration of an elastic body with Coulombfriction can be (dynamically) unstable or when it admits(quasi-static) bifurcations. Second, we want to proposea specific algorithm to solve the nonlinear eigenvalueproblem.

Let us outline the content of the paper. In the secondsection we define an equilibrium configuration of an elasticbody with Coulomb friction and we give the setting of thedynamic or quasi-static problems in a continuous context(i.e. before the finite element discretization). These lasttwo problems will be seen as perturbations of the equilib-rium. In the next section we introduce the (continuous)nonlinear eigenvalue problem and we give its variationalformulation. We give here sufficient conditions for the‘‘local’’ instability of the equilibrium. More precisely, weprove that if the nonlinear eigenvalue problem has asmooth solution (eigenvalue/mode) then there exists a fam-

ily of dynamic perturbations of the equilibrium, corre-sponding to that mode, which has an exponential growthin time. In the quasi-static case the eigenvalue has to vanishand we deal with a bifurcation of the solution. In order tostate them we have to impose some conditions on theboundary between the free, the stick and the slipping zones.These conditions are specific to the continuous problemand they are not necessary for the discrete (finite element)problem.

In Section 4 we give the mixed finite element approachof the above problems and in Section 5 we introduce thediscrete nonlinear eigenvalue problem and we state the sta-bility results. In Section 6 we propose a numerical algo-rithm to compute the eigenvalue, and the correspondingeigenfunction. It makes use of the successive iterates of anonlinear operator. For linear operators this algorithm isconvergent to the largest eigenvalue. The main idea is thesame as the one used in [8] for the potential operatorswhich appears in modeling the earthquake nucleation.Even if for the Coulomb friction problem, studied here,there exists no energy to be associated the algorithm is stillworking very well. We proved that the algorithm is consis-tent (i.e. if it’s convergent then its limit is a solution), butwe cannot give a mathematical proof of its convergence.

In the last section we show some numerical experimentsto examine the convergence of the proposed algorithm. Wegive some examples to illustrate how from the nonlineareigenvalue analysis we get information on the stability ofthe equilibrium configurations.

2. Setting of the continuous problems

2.1. The equilibrium problem

We consider the in-plane deformation of an elastic bodyoccupying, in the initial unconstrained configuration, adomain X in R2 where plane strain assumptions areassumed. The Lipschitz boundary oX of X consists of CD,CN and CC where the measure of CD does not vanish.The body X is submitted to given displacements U on CD

and subjected to surface traction forces F on CN; the bodyforces are denoted f. In the initial configuration, the partCC is considered as the candidate contact surface on a rigidfoundation. The contact is assumed to be frictional and thestick, slip and separation zones on CC are not known inadvance. We denote by l > 0 the given friction coefficienton CC. The unit outward normal and tangent vectors ofoX are n ¼ ðn1; n2Þ and t ¼ ð�n2; n1Þ, respectively. After-wards we adopt the following notation for any displace-ment field u and for any density of surface forces rðuÞndefined on CC:

u ¼ unnþ utt and rðuÞn ¼ rnðuÞnþ rtðuÞt:

We say that ueq : X! R2 is an equilibrium (solution) ofthe Coulomb’s problem if it satisfies (1)–(5):

divrðueqÞ þ f ¼ 0 in X; ð1Þ

Ω

ΓΓ

ΓΓΓ

ΓΓ

ND

C

C

CC

Cfree

free

stick stick

0

Fig. 1. A schematic partition of the contact boundary CC involving CfreeC

the free (no contact) zone, CstickC the ‘‘stick’’ zone and C0

C the ‘‘potentialslip’’ zone.

R. Hassani et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 2377–2389 2379

rðueqÞ ¼ CeðueqÞ in X; ð2Þueq ¼ U on CD; ð3ÞrðueqÞn ¼ F on CN ; ð4Þueq

n 6 0; rnðueqÞ 6 0; rnðueqÞueqn ¼ 0;

jrtðueqÞj 6 �lrnðueqÞ: ð5Þ

The notation rðueqÞ : X!S2 represents the stress tensorfield lying in S2, the space of second order symmetrictensors on R2. The linearized strain tensor field iseðueqÞ ¼ ð$ueq þ $T ueqÞ=2 and C is the fourth order sym-metric and elliptic tensor of linear elasticity. On CC, thefour conditions represent the equilibrium of unilateral con-tact under Coulomb friction.

2.2. The dynamic and quasi-static problems

The dynamic problem with Coulomb’s friction law con-sists of finding the displacement field u : ½0; T � � X! R2

satisfying (6)–(12):

divrðuðtÞÞ þ f ¼ q€uðtÞ in X; ð6ÞrðuðtÞÞ ¼ CeðuðtÞÞ in X; ð7ÞuðtÞ ¼ U on CD; ð8ÞrðuðtÞÞn ¼ F on CN ; ð9Þ

where q > 0 is the density and the dots represent the timederivative. On CC, the three conditions representing theunilateral contact under Coulomb friction are given by

unðtÞ 6 0; rnðuðtÞÞ 6 0; rnðuðtÞÞunðtÞ ¼ 0; ð10Þ_utðtÞ ¼ 0) jrtðuðtÞÞj 6 �lrnðuðtÞÞ;_utðtÞ 6¼ 0) rtðuðtÞÞ ¼ lrnðuðtÞÞ _utðtÞ

j _utðtÞj :

(ð11Þ

To state the dynamic problem we have to complete theabove equations and boundary condition with the initialconditions

uð0Þ ¼ u0; _uð0Þ ¼ u1; ð12Þ

which are small perturbations of the equilibrium, i.e.ju0 � ueqj and ju1j are small.

The quasi-static problem with Coulomb’s friction lawconsists of finding the displacement field u : ½0; T ��X! R2 satisfying

divrðuðtÞÞ þ f ¼ 0; ð13Þ

and (7)–(11).Let us remark that uðtÞ ¼ ueq is a solution of both

dynamic and quasi-static problems.

3. The nonlinear eigenvalue problem. Continuous approach

3.1. Problem statement

Let ueq be a smooth solution of the equilibrium problem,called in the following the (reference) equilibrium solu-tion. The free (no contact) zone, the ‘‘stick’’ zone and the

‘‘potential slip’’ zone of CC (see Fig. 1) are defined asfollows:

CfreeC ¼ fx 2 CC; ueq

n ðxÞ < 0g; ð14Þ

CstickC ¼ fx 2 CC; ueq

n ðxÞ ¼ 0;

jrtðueqÞðxÞj < �lrnðueqÞðxÞg; ð15Þ

C0C ¼ fx 2 CC; ueq

n ðxÞ ¼ 0;

jrtðueqÞðxÞj ¼ �lrnðueqÞðxÞ > 0g: ð16Þ

We suppose in all the paper that the three zones define apartition of CC

CC ¼ CfreeC [ Cstick

C [ C0C;

i.e. the zone fx 2 CC; ueqn ðxÞ ¼ 0; rtðueqÞðxÞ ¼ rnðueqÞðxÞ ¼

0g has an empty interior (with respect to the topology in-duced by CC). We denote by oCfree

C , oCstickC and oC0

C theboundary of Cfree

C , CstickC and C0

C, respectively, related tothe (one-dimensional) topology induced by CC.

We give now a new partition of C ¼ C0D [ C0

N [ C0C

through

C0D ¼ CD [ Cstick

C ; C0N ¼ CN [ Cfree

C ;

and we define v : C0C ! f�1; 1g by

vðxÞ ¼ �rtðueqÞðxÞ=jrtðueqÞðxÞj:

The general method in analyzing the stability of the non-linear evolution problem (6)–(12) consists in constructingthe ‘‘tangent problem’’ of the reference solution (in ourcase the equilibrium ueq). Then we deduce the ‘‘linear sta-bility’’ through an eigenvalue analysis of this last linearproblem. Let us see now how the tangent problem can beconstructed in our case in each point x of CC if we supposethat uðtÞ is a smooth and small perturbation of ueq. Ifx 2 Cfree

C then at least locally in time unðt; xÞ < 0 hence x isstill in a free zone. For x 2 Cstick

C then at least locally in timejrtðuÞðt; xÞj < �lrnðuÞðt; xÞ, i.e. x is still in a stick zone (thepoint S in Fig. 2). If x 2 C0

C (the point A in Fig. 2) then wehave two possibilities. The first one consists in sliding in theopposite sense of the tangential stress, i.e. v _utðt; xÞP 0,

slip rate

Equilibrium on a sliding zone

Equilibrium on a stick zone

Other reference solution on a sliding zone

Nonlinear eigenproblem Linear eigenproblem

Frictional ratio

Frictioncoefficient

AB

S

Fig. 2. The frictional ratio jrt jjrn j versus the slip rate v _ut.


which means that the point x is still in a sliding zone fort > 0. The second one consists in unloading the frictional

ratio jrtðt;xÞjjrnðt;xÞj with a vanishing slip rate _utðt; xÞ ¼ 0. In this

case we deal with a change of status: the point x will belongto the stick zone for t > 0.

Following the above analysis we can introduce now thenonlinear eigenvalue problem, which consists of findingk2 and the non-vanishing displacement field U : X! R2

satisfying

divrðUÞ ¼ k2qU in X; ð17ÞrðUÞ ¼ CeðUÞ in X; ð18ÞU ¼ 0 on C0

D; ð19ÞrðUÞn ¼ 0 on C0

N ; ð20ÞUn ¼ 0; vUt P 0; vrtðUÞ � lrnðUÞP 0;

ðvrtðUÞ � lrnðUÞÞUt ¼ 0: ð21Þ

Notice that only few information from the referenceequilibrium state ueq are needed. Indeed, only some geo-metrical features, as the partition of the contact surfaceCC, and the directions of the tangential stress, involved inv, are present in the formulation of the nonlinear eigen-value problem. The above problem, introduced by Martinset al. [9] (see also [13,12]) in a finite element context, is alsocalled (linear) eigenvalue problem with complementary(conditions) inequalities.

Let us discuss now the relation with the linear eigen-value problem studied in [4,6]. The hypothesis consideredby Hassani et al. (see Corollary 3.3 in [4] or Assumption(3.1) in [6]) imply that the slip rate of the reference config-uration _ut

refð0; xÞ 6¼ 0 is not vanishing (point B in Fig. 2). Inthis case there are no unilateral constraints to be imposedon the tangent problem since the unloading of the frictionalratio is not present. This analysis is not possible when thereference configuration is an equilibrium state for which_ut

eqð0; xÞ � 0. That is why we cannot use a linear eigenvalue

analysis in the study of the stability of the equilibriumconfigurations.

3.2. Variational formulation of the eigenvalue problem

The variational formulation of problem (17)–(21) con-sists of finding U 2 eK and k verifying:

aðU; v�UÞ þ k2bðU; v�UÞP ljðU; v�UÞ 8v 2 K;

ð22Þwhere

aðu; vÞ ¼Z

XðCeðuÞÞ : eðvÞ dX; bðu; vÞ ¼

ZX

qu � v dX:

In (22), K denotes the closed convex set of admissible dis-placement fields:

K ¼ fv 2 ðH 1ðXÞÞ2 : v ¼ 0 on C0D; vn 6 0 on C0

C;

vvt P 0 on C0Cg:

and the functional jð:; :Þ, given by

jðu; vÞ ¼Z

C0C

rnðuÞvvt dC; ð23Þ

is defined for any v in ðH 1ðXÞÞ2 but more regularity is re-quired for u. To handle this we introduceeV ¼ fv 2 ðH 1ðXÞÞ2 : divrðvÞ 2 ðL2ðXÞÞ2g; eK ¼ K \ eV:If u 2 eV then rðuÞ belongs to Hðdiv;XÞ and rnðuÞ is an ele-ment of H�

12ðCÞ (i.e. the dual of H

12ðCÞ). Since H�

12ðCÞjC0

Cis

different from H�12ðC0

CÞ we have to suppose in addition thatrnðuÞ 2 H�

12ðC0

CÞ. With this assumption, (23) makes sense ifwe replace the integral term by the duality product. For amore precise formulation involving the convenient Sobolevspaces and the set of non-negative Radon measures, adetailed study can be found in [11].


3.3. Unstable perturbations of the equilibrium

The following theorem gives sufficient conditions for the‘‘local’’ instability of the equilibrium. More precisely, weprove that if the nonlinear eigenvalue problem (17)–(21)has a smooth solution (U,k), with k P 0, then there exista family of (dynamic) perturbations of the equilibriumueq, corresponding to the U mode, which has an exponen-tial growth in time (see Fig. 3). In the quasi-static case wedeal with a bifurcation (loss of uniqueness) from the equi-librium, in the direction of the U mode but its time varia-tion is arbitrary. This behavior is ‘‘local’’ in time, i.e. it isvalid at least for a first time interval.

Theorem 1. Let k P 0 and U : X! R2 be a smooth solution

of the nonlinear eigenvalue problem (17)–(21). We suppose

that

lim infx!y;x2Cfree

C

UnðxÞueq

n ðxÞ> �1 for all y 2 oCfree

C ; ð24Þ

lim infx!y;x2Cstick

C

jrtðUÞðxÞj þ lrnðUÞðxÞjrtðueqÞðxÞj þ lrnðueqÞðxÞ > �1

for all y 2 oCstickC ; ð25Þ

lim infx!y;x2C0

C

rnðUÞðxÞrnðueqÞðxÞ > �1;

lim infx!y;x2C0

C

rtðUÞðxÞrtðueqÞðxÞ > �1 for all y 2 oC0

C: ð26Þ

(i) If k > 0 then

uðtÞ :¼ ueq þ �0 coshðktÞ þ �1

sinhðktÞk

� �U ð27Þ

is a solution of the dynamic problem (6)–(11) for all �0 P 0and �1T P 0 small enough.

slip rate

timeEquilibrium

Dynamic : exponential growth

Dynamic : linear growth

Quasi-static

1ε

Fig. 3. The exponential (or linear growth) in time of the dynamicperturbations of the equilibrium ueq, corresponding to the U mode fork > 0 (or k ¼ 0). Bifurcation (loss of uniqueness) of the quasi-staticsolution from the equilibrium, in the direction of the U mode.

(ii-a) If k ¼ 0 then

uðtÞ :¼ ueq þ ð�0 þ �1tÞU ð28Þis a solution of the dynamic problem (6)–(11) for all �0 P 0and �1T P 0 small enough.

(ii-b) If k ¼ 0 then for all a : ½0; T � ! ½0;þ1Þ of class C1

with _aðtÞ > 0

uðtÞ :¼ ueq þ aðtÞU ð29Þis a solution of the quasi-static problem 13, 7, 8, 9, 10, 11 or

að0Þ and T small enough.

Let us discuss first the conditions (24)–(26) which relateueq and U at the boundary between the three zones of CC.These conditions are specific to the continuous problemand they are not necessary when we want to formulatethe same result for the discrete (finite element) problem(see the next sections). As a mater of fact all the abovelimits concerns the case 0

0. Indeed, if y 2 oCfree

C then either

y 2 CfreeC \ Cstick

C or y 2 CfreeC \ C0

C. In both cases from (15),(16) and (19), (21) we deduce

ueqn ðyÞ ¼ UnðyÞ ¼ 0 for all y 2 oCfree

C : ð30ÞLet y be in oCstick

C . If y 2 CstickC \ Cfree

C from (5) and (20) wededuce rnðueqÞðyÞ ¼ rtðueqÞðyÞ ¼ rnðUÞðyÞ ¼ rtðUÞðyÞ ¼ 0.

If analyze now the case y 2 CstickC \ C0

C from (16) we get that

jrtðueqÞðyÞj þ lrnðueqÞðyÞ ¼ 0 for all y 2 oCstickC : ð31Þ

On the other hand for all y in C0C \ Cfree

C � oC0C we have

rnðueqÞðyÞ ¼ rnðUÞðyÞ ¼ 0; rtðueqÞðyÞ ¼ rtðUÞðyÞ ¼ 0:

ð32ÞWe give here an explanation of the fact that we are look-

ing only on the real eigenvalues k P 0. The complex eigen-values with positive real part and non-vanishing imaginarypart are associated with flutter instability. If the referenceconfiguration has a non-vanishing slip rate then the (flut-ter) unstable solution will oscillate (see Fig. 4) with anexponential growth in the amplitude. The analysis is validas long as the slip rate is not vanishing (t 2 ½0; T � inFig. 4). After that, the tangent problem becomes quite dif-ferent which implies that its spectrum can change a lot.When the reference configuration is an equilibrium, the sliprate is zero which means that the flutter instability canoccur only on a half period of oscillation (i.e. for T verysmall). As we can see in Fig. 3 that is not the case if theimaginary part is vanishing.

Proof of the theorem. First we check that u given by (27) orby (27) verifies (6) and if u is given by (29) then it verifies(13). Since the other boundary conditions are the same forthe dynamic and quasi-static problems we shall consider inthe following only the case (29) which includes the othertwo. The affine boundary conditions (8) and (9) are verifiedand we have only to check the nonlinear conditions on CC.

On CfreeC we have ueq

n < 0 and from (24) we deduce thatunðtÞ ¼ ueq

n þ aðtÞUn < 0 on CfreeC for 1=aðtÞ large enough.

T

slip rate

time

Reference slip rate

Flutter instability

Fig. 4. The flutter instability of a sliding reference solution associated withan eigenvalue with positive real part and non-vanishing imaginary part.The stability analysis is valid as long as the slip rate is not vanishing(t 2 ½0; T �).


Since rðueqÞn ¼ 0 we get that rðuðtÞÞn ¼ rðueqÞn þaðtÞrðUÞn ¼ 0, i.e. Cfree

C is stress free.On Cstick

C we get unðtÞ ¼ ueqn þ aðtÞUn ¼ 0. For 1=aðtÞ

large enough from (25) we deduce 0 > aðtÞðjrtðUÞj þlrnðUÞÞ þ jrtðueqÞj þ lrnðueqÞPj rtðuðtÞÞ j þlrnðuðtÞÞ. Onthe other hand _utðtÞ ¼ _aðtÞUt ¼ 0, i.e. the body is stick onCstick

C .We analyze now the boundary conditions on C0

C. SincernðueqÞ < 0 and vrtðueqÞ ¼ �jrtðueqÞj < 0 from (26) wededuce rnðuðtÞÞ < 0 and vrtðuðtÞÞ ¼ �jrtðuðtÞÞj for 1=aðtÞlarge enough. We have also unðtÞ ¼ ueq

n þ aðtÞUn ¼ 0 andv _utðtÞ ¼ _aðtÞvUt P 0.

If UtðxÞ 6¼ 0 then rtðUÞðxÞ ¼ lvðxÞrnðUÞðxÞ. On theother hand rtðuðtÞÞðxÞ ¼ rtðueqÞðxÞ þ aðtÞrtðUÞðxÞ ¼ �vðxÞjrtðueqÞðxÞj þ aðtÞlvðxÞrnðUÞðxÞ ¼ lrnðuðtÞÞðxÞvðxÞ and since_utðx; tÞ=j _utðx; tÞj ¼ vðxÞ we get (10), i.e. the body is slidingon C0

C.If UtðxÞ ¼ 0 then vðxÞrtðUÞðxÞP lrnðUÞðxÞ and we

have jrtðuðtÞÞðxÞj¼�vðxÞrtðuðtÞÞðxÞ¼�vðxÞrtðueqÞðxÞ�aðtÞ-vrtðUÞðxÞ¼ jrtðueqÞðxÞj�aðtÞvðxÞrtðUÞðxÞ6�lrnðueqÞðxÞ�aðtÞlrnðUÞðxÞ¼�lrnðuðtÞÞðxÞ. Bearing in mind that_utðx;tÞ¼0 we obtain that the body is stick on C0

C and(10) follows. h

One can choose to compute ðU; lÞ for each k2, ratherthan ðU; k2Þ for each l. In this case the above problembecomes:

For a given k2 P 0, find U 2 eK and l P 0 such that(17)–(21) (or (22)) hold.

Note that in this way we can formulate both quasi-staticand dynamic problems in one nonlinear eigenvalue prob-lem (you just have to put k2 ¼ 0 for the quasi-static case).

4. Mixed finite element approach

The body X is discretized by using a family of triangula-tions ðThÞh made of finite elements of degree k P 1 where

h > 0 is the discretization parameter representing the larg-est diameter of a triangle in Th. We denote by Vh the space:

Vh ¼ fvh; vh 2 ðCðXÞÞ2; vhjT 2 ðP kðT ÞÞ28T 2Th;

vh ¼ 0 on CDg;

where CðXÞ stands for the space of continuous functions onX and P kðT Þ represents the space of polynomial functionsof degree k on T. Let us mention that we focus on the dis-crete problem and that any discussion concerning the con-vergence of the finite element problem towards thecontinuous model is out of the scope of this paper.

On the boundary of X, we still keep the notationvh ¼ vhnnþ vhtt for every vh 2 Vh and we denote by ðT hÞhthe family of monodimensional meshes on CC inheritedby ðThÞh. Set

W h ¼ fm; m ¼ vhjCC:n; vh 2 Vhg;

which is included in the space of continuous functions onCC which are piecewise of degree k on ðT hÞh and coincideswith the latter space when CC \ CN ¼ ;.

We denote by p the dimension of W h and by wi; 1 6i 6 p the corresponding canonical finite element basis func-tions of degree k. For all m 2 W h we shall denote by F ðmÞ ¼ðF iðmÞÞ16i6p the generalized loads at the nodes of CC:

F iðmÞ ¼Z

CC

mwi; 81 6 i 6 p:

We next introduce the sets of Lagrange multipliers:

Mhn ¼ fm; m 2 W h; F iðmÞ 6 0 81 6 i 6 pg

and, for any g 2 �Mhn

MhtðgÞ ¼ fm; m 2 W h; jF iðmÞj 6 F iðgÞ 81 6 i 6 pg:

Hence, the discrete problem issued from (1)–(5) becomes:

4.1. The equilibrium problem

Find ðueqh ; s

eqhn; s

eqht Þ 2 Uad;h �Mhn �Mhtð�lseq

hnÞ ¼ Uad;h�Mhð�lseq

hnÞ such that

aðueqh ; vhÞ �

RCC

seqhnvhndC�

RCC

seqht vht dC¼ LðvhÞ 8vh 2 Vh;R

CCðmhn � seq

hnÞueqhn dC P 0 8mhn 2Mhn;

(ð33Þ

where

LðvhÞ ¼Z

Xf � vh dXþ

ZCN

F � vh dC;

Uad;h ¼ fvh; vh 2 ðCðXÞÞ2; vhjT 2 ðP kðT ÞÞ2 8T 2Th;

vh ¼ Uh on CDg

and Uh denotes a convenient approximation of U on CD.Let ðueq

hnÞi and ðueqht Þi; 1 6 i 6 p denote the nodal values

on CC of ueqhn and ueq

ht , respectively. It can be easily checkedthat the vector formulation of the frictional contact condi-tions incorporated in the inequality of (33) are:


F iðseqhnÞ 6 0; ðueq

hnÞi 6 0; F iðseqhnÞðu

eqhnÞi ¼ 0;

jF iðseqht Þj 6 �lF iðseq

hnÞ; ð34Þ

for all 1 6 i 6 p.The discrete problem issued from the dynamic problem

(6)–(12) becomes:

4.2. The dynamic problem

Find ðuh; shn; shtÞ : ½0; T � ! Uad;h �Mhð�lshnÞ such thatfor all vh 2 Vh and 8ðmhn; mhtÞ 2Mhð�lshnðtÞÞ we have:

bð€uhðtÞ; vhÞ þ aðuhðtÞ; vhÞ �R

CCshnðtÞvhn dC

�R

CCshtðtÞvht dC ¼ LðvhÞ;R

CCðmhn � shnðtÞÞuhnðtÞ dCþ

RCCðmht � shtðtÞÞ _uhtðtÞ dC P 0:

8><>:ð35Þ

The vector formulation of the frictional contact conditionsincorporated in the above inequality are:

F iðshnðtÞÞ 6 0; ðuhnðtÞÞi 6 0; F iðshnðtÞÞðuhnðtÞÞi ¼ 0;

1 6 i 6 p; ð36ÞjF iðshtðtÞÞj 6 �lF iðshnðtÞÞ; F iðshtðtÞÞð _uhtðtÞÞi 6 0;

1 6 i 6 p; ð37ÞjF iðshtðtÞÞj < �lF iðshnðtÞÞ ) ð _uhtðtÞÞi ¼ 0;

1 6 i 6 p; ð38Þ

where ðuhnðtÞÞi and ðuhtðtÞÞi; 1 6 i 6 p denote the vectors ofcomponents the nodal values on CC of uhnðtÞ and uhtðtÞ,respectively.

We have to complete the above equations with the initialconditions

uhð0Þ ¼ u0h; _uhð0Þ ¼ u1h;

where u0h; u1h denotes a convenient approximation of u0

and u1 on X.The discrete problem issued from the quasi-static prob-

lem (13), (7)–(11) is:

4.3. The quasi-static problem

Find ðuh; shn; shtÞ : ½0; T � ! Uad;h �Mhð�lshnÞ such thatfor all vh 2 Vh and all ðmhn; mhtÞ 2Mhð�lshnðtÞÞ we have:

aðuhðtÞ; vhÞ �R

CCshnðtÞvhn dC�

RCC

shtðtÞvht dC ¼ LðvhÞ;RCCðmhn � shnðtÞÞuhnðtÞ dCþ

RCCðmht � shtðtÞÞ _uhtðtÞ dC P 0;

(ð39Þ

and the vector formulation of the frictional contact condi-tions are the same as in the dynamic case.

5. Discrete nonlinear eigenvalue problem

Let us consider a solution ðueqh ; s

eqhn; s

eqht Þ 2 Vh�

Mhð�lshnÞ of the discrete equilibrium Coulomb frictionalcontact problem (33). Then we denote by If, Is and Ic theset of nodes of CC which are currently free (separated from

the rigid foundation), the set of nodes of CC which arestuck to the rigid foundation, and the set of nodes of CC

which are currently in contact but are candidate to slip,respectively. In other words, if p ¼ dimðW hÞ denotes thenumber of nodes belonging to CC, we can write

If ¼ fi 2 ½1; p�; ðueqhnÞi < 0g;

Is ¼ fi 2 ½1; p�; ðueqhnÞi ¼ 0; jF iðseq

ht Þj < �lF iðseqhnÞg;

Ic ¼ fi 2 ½1; p�; ðueqhnÞi ¼ 0; jF iðseq

ht Þj ¼ �lF iðseqhnÞ > 0g:

Henceforth, we assume that in all the nodes of Ic the tan-gential and normal stresses are not vanishing, i.e. fi 2½1; p�; ðueq

hnÞi ¼ 0; jF iðseqht Þj ¼ �lF iðseq

hnÞ ¼ 0g is empty. Fol-lowing this assumption for all node i of Ic we denote byvi the sign of the tangential stress

vi ¼ �F iðseq

ht ÞjðF iðseq

ht ÞÞj8i 2 Ic:

Next we consider the following discrete nonlinear eigen-value problem:

5.1. Nonlinear eigenvalue problem

Find the eigenvalue kh P 0 and the correspondingeigenfunction(s) ðUh; hhn; hhtÞ 2 Kh � W h � W h such that

aðUh; vhÞ þ k2hbðUh; vhÞ �

RCC

hhnvhn dC

�R

CChhtvht dC ¼ 0 8vh 2 Vh;

F iðhhnÞ ¼ F iðhhtÞ ¼ 0 8i 2 If ;RCC

hhtðvht � UhtÞ dC

P lR

CChhnðT vðvhtÞ � T vðUhtÞÞ dC 8vh 2 Kh

8>>>>>>><>>>>>>>:ð40Þ

where T v : W h ! W h is the linear operator given by

ðT vðmhÞÞi ¼ viðmhÞi; if i 2 Ic; ðT vðmhÞÞi ¼ ðmhÞi; if i 62 Ic

and

Kh ¼ fvh 2 Vh; ðvhnÞi ¼ ðvhtÞi ¼ 0; 8i 2 Is; ðvhnÞi ¼ 0; viðvhtÞiP 0; 8i 2 Icg:

The vector formulation of the frictional contact conditionsincorporated in the eigenvalue inequality are:

ðUhnÞi ¼ 0; viðUhtÞi P 0; viF iðhhtÞ � lF iðhhnÞP 0;

ðviF iðhhtÞ � lF iðhhnÞÞðUhtÞi ¼ 0; ð41Þ

for all i 2 Ic.The following theorem explains how the existence of a

solution for the nonlinear eigenvalue problem gives suffi-cient conditions for the unstable behavior of the equilib-rium. This theorem, which is the discrete version ofTheorem 1, constructs solutions ðuh; shn; shtÞ of the dynamicand quasi-static problems which are as close to the equilib-rium ueq

h at t ¼ 0 but have an exponential growth, followingthe direction of the mode ðUh; hhn; hhtÞ, during a first timeinterval. More precisely we have:


Theorem 2. Let kh P 0 and ðUh; hhn; hhtÞ 2 Kh � W h � W h

be a solution of the nonlinear eigenvalue problem (40). For a

given a : ½0; T � ! ½0;þ1Þ we denote by

uhðtÞ :¼ ueqh þ aðtÞUh; shnðtÞ :¼ seq

hn þ aðtÞhhn;

shtðtÞ :¼ seqht þ aðtÞhht: ð42Þ

(i) If kh > 0 and we put

aðtÞ ¼ �0 coshðkhtÞ þ �1sinhðkhtÞ

khð43Þ

then ðuh; shn; shtÞ : ½0; T � ! Uad;h �Mhð�lshnÞ is a

solution of the dynamic problem (35) for all �0 P 0and �1T P 0 small enough.

(ii-a) If kh ¼ 0 and we put

aðtÞ ¼ �0 þ �1t; ð44Þthen ðuh; shn; shtÞ : ½0; T � ! Uad;h �Mhð�lshnÞ is a

solution of the dynamic problem (35) for all �0 P 0and �1T P 0 small enough.

(ii-b) If kh ¼ 0 then for all a : ½0; T � ! ½0;þ1Þ of class C1

with _aðtÞ > 0 the function ðuh; shn; shtÞ : ½0; T � !Uad;h �Mhð�lshnÞ is a solution of the quasi-static

problem (39), for að0Þ and T small enough.

Proof. The proof is very similar with the continuous case.We make use the vector formulation of the frictional con-tact conditions. We deal here with a finite number of nodeswhich implies that we do not need extra condition, as (24)–(26), at the boundary between the stick, free and slippingzones of CC. h

6. A numerical algorithm for solving the nonlinear eigenvalue

problem

As in the continuous case one can choose to computeðUh; hhn; hhtÞ and lh P 0 for each k2, rather than ðUh;hhn; hhtÞ and kh P 0 for each l. In this case the nonlineareigenvalue problem becomes:

For a given k2 P 0; find ðUh; hhn; hhtÞ and

lh P 0 such that ð40Þ holds: ð45Þ

Note that in this way we can formulate both quasi-staticand dynamic problems in one nonlinear eigenvalue prob-lem : we just have to put k2 ¼ 0 for the quasi-static case.

In order to give the formulation of (45) as an eigenvalueproblem for a nonlinear operator we introduce

W 0h ¼ fmh 2 W h; F iðmhÞ ¼ 0 8i 2 Ifg:

For all mh 2 W 0h we denote by wh ¼ P hðmhÞ 2 Kh the

unique solution of the following variational inequality

k2bðwh; vh � whÞ þ aðwh; vh � whÞ

PZ

CC

mhðT vðvhtÞ � T vðwhtÞÞ dC; ð46Þ

for all vh 2 Kh. One can easily check that wh belongs to

V0h ¼ fvh 2 Vh; k

2bðvh; uhÞ þ aðvh; uhÞ ¼ 0 8uh 2 Vh;

uh ¼ 0 on CCg:

Let us remark that P h : W 0h ! V0

h is a positively homog-enous (i.e. P hðtmhÞ ¼ tP hðmhÞ for all t P 0) but is not a linearoperator.

For all wh 2 V0h we denote by QðwhÞ ¼ ðQnðwhÞ;

QtðwhÞÞ 2 W h � W h the normal and tangential stressesassociated with wh by

k2bðwh; vhÞ þ aðwh; vhÞ

¼Z

CC

QnðwhÞvhn dCþZ

CC

QtðwhÞvht dC 8vh 2 Vh:

Let us define now the operator Rh : W 0h ! W h by

RhðmhÞ ¼ QnðP hðmhÞÞ 8mh 2 W 0h:

After we check that if wh ¼ P hðmhÞ then F iðQnðwhÞÞ ¼F iðQtðwhÞÞ ¼ 0, for all i 2 If we can conclude that Rh :W 0

h ! W 0h and we have the following characterization of

the nonlinear eigenvalue problem in terms of operatortheory.

Proposition 1. The pair ðgh; mhÞ 2 Rþ � W 0h is an eigenvalue/

eigenfunction of the operator Rh, i.e.

RhðmhÞ ¼ ghmh ð47Þif and only if lh ¼ 1=gh;Uh ¼ P hðmhÞ; shn ¼ ghmh; sht ¼ QtðUhÞis a solution of the nonlinear eigenvalue problem (45).

We give in the following a numerical algorithm to com-pute the eigenvalue lh of (45), and the corresponding eigen-function U0h. For this we consider the lagrangian L : Wh�W h ! R given by

Lðvh; phÞ :¼ 1

2aðvh; vhÞ þ

1

2k2bðvh; vhÞ �

ZC

phT vðvhtÞ dr

ð48Þ

Algorithm. The algorithm starts with an arbitrary p0h 2 W 0

h.

At iteration n + 1, having pnh 2 W 0

h, we compute wnþ1h 2 Kh

solution of

Lðwnþ1h ; pn

hÞ 6Lðvh; pnhÞ; for all vh 2 Kh: ð49Þ

Then we update:

pnþ1h ¼ Qnðwnþ1

h Þ; lnþ1h ¼

kpnhkL2ðCCÞ

kpnþ1h kL2ðCCÞ

; ð50Þ

and we normalize

Unþ1h ¼ 1

kpnþ1h kL2ðCCÞ

wnþ1h ; snþ1

hn ¼1

kpnþ1h kL2ðCCÞ

pnþ1h ;

snþ1ht ¼ QtðUnþ1

h Þ: ð51Þ

The algorithm stops when jlnþ1h � ln

hj þ kUnþ1h � Un

hk is smallenough.


Let us prove now that the algorithm is consistent, i.e. ifit is convergent to some vector field then this limit is a solu-tion of the nonlinear eigenvalue problem (40). From thedefinition of the operator Q we have

aðUnþ1h ; vhÞ þ k2

hbðUnþ1h ; vhÞ �

ZCC

hnþ1hn vhn dC

�Z

CC

hnþ1ht vht dC ¼ 0 8vh 2 Vh;

and from (49) we deduce

k2bðwnþ1h ; vh � wnþ1

h Þ þ aðwnþ1h ; vh � wnþ1

h Þ

PZ

CC

pnhðT vðvhtÞ � T vðwnþ1

ht ÞÞ dC 8vh 2 Kh;

which imply that F iðhnþ1hn Þ ¼ F iðhnþ1

ht Þ ¼ 0; 8i 2 If ; andZCC

hnþ1ht ðvht � Unþ1

ht Þ dC

P lnþ1h

ZCC

hnhnðT vðvhtÞ � T vðUnþ1

ht ÞÞ dC 8vh 2 Kh:

We can see now that if the convergence of the algorithm isassured, i.e. mn

h ! mh;Unh ! Uh; h

nhn ! hhn; h

nht ! hht, then lh

and ðUh; hhn; hhtÞis a solution of the nonlinear eigenvalueproblem (40).

Let us relate now the above algorithm to the operatorialformulation (47) of the nonlinear eigenvalue problem.For this, let us remark that (49) is equivalent withwnþ1

h ¼ P hðpnhÞ, which means that

pnþ1h ¼ Rðpn

hÞ; lnþ1h ¼

kpnhkL2ðCCÞ

kpnþ1h kL2ðCCÞ

: ð52Þ

Moreover, if we denote qnh ¼ pn

h=kpnhkL2ðCCÞ, then we can in-

clude the normalization into the algorithm (52) to get

qnþ1h ¼ lnþ1

h RhðqnhÞ; kqnþ1

h kL2ðCCÞ ¼ 1

and from the update we get the following formulae relatedto (50)–(51):

Unþ1h ¼ lnþ1

h P hðqnhÞ; snþ1

ht ¼ QtðUnþ1h Þ; snþ1

hn ¼ QnðUnþ1h Þ:ð53Þ

We see now that the algorithm makes use of the succes-sive iterates of the nonlinear operator R : the ratio of themoduli of two consecutive iterates should convergetowards the eigenvalue gh ¼ 1

lhof Rh. For linear operators

this algorithm is called the ‘‘power method’’ and it is con-vergent to the largest eigenvalue. For nonlinear operatorsthere exists, as far as we know, no convergence result. Thismethod was also used (see [8]) with success for the potentialoperators which appears in modelling the nucleation phaseof an earthquake. In this case they proved the existence ofthe ‘‘Rayleigh quotient’’ which is associated to the largesteigenvalue. For the Coulomb friction problem studiedhere, there exists no energy to be associated (i.e. the oper-ator R is not potential), and we expect that the algorithm

converge to different limits for different choices of the ini-tial guess p0

h.To minimize Lð�; pn

hÞ in (49), we use the Uzawa algo-rithm. To formulate it we denote by

Wh ¼ fvh 2 Vh; ðvhnÞi ¼ ðvhtÞi ¼ 0 8i 2 Is; ðvhnÞi ¼ 0 8i 2 Icg;

and we introduce the modified Lagrangian

L�ðvh; ph; sÞ ¼Lðvh; phÞ �Xi2Ic

visiðvhtÞi:

We start with an arbitrary s0h 2 R

pcþ , where pc is the cardinal

of Ic. At iteration k þ 1, having skh 2 R

pcþ , we compute

wn;kþ1h 2Wh the solution of

L�ðwn;kþ1h ; pn

h; skhÞ 6L�ðvh; pn

h; skhÞ for all vh 2Wh ð54Þ

without any difficulty since we minimize here a quadraticfunctional without constraints. Then, we update

ðskþ1h Þi ¼ ½ðsk

hÞi � rviðwn;kþ1ht Þi�þ; for all i 2 Ic;

with some r > 0 (here ½�þ denotes the positive part). Thealgorithm stops when kskþ1

h � skhk þ kw

n;kþ1h � wn;k

h k is smallenough.

7. Numerical results

7.1. Convergence in the linear case

First we examine the convergence of the proposed algo-rithm (49)–(51) in detecting the solutions of the nonlineareigenvalue problem. For this we have chosen the squaregeometry depicted in Fig. 5 and isotropic elastic materialwith a Poisson ratio m ¼ 0:3. The partition of C is as fol-lows: CC the bottom side, CD the top side and CN the ver-tical sides. We considered an equilibrium solution ueq

which has all the points of CC in sliding at the left (i.e.v ¼ 1 on C0

C ¼ CC). We try to find the friction coefficientl* which corresponds to the quasi-static case (i.e. k2 ¼ 0).Since in this case the eigenvalue k2 is an increasing functionof l the computed value l* represents the critical coeffi-cient, i.e. the system is (dynamically) unstable for l > l�.

For this geometry we expect that the tangential slip iseverywhere positive, which means that the solution of thenonlinear eigenvalue problem coincides with the solutionof the linear eigenvalue problem described in [4,6]. Sincethe constraints present in Kh are not active the algorithm(49)–(51) is nothing else than an iterative method of findingthe largest eigenvalue of a matrix.

Concerning the mixed finite element discretization wehave chosen triangle finite elements of degree k ¼ 1 on anuniform mesh. On the left side of Fig. 6 we have plottedthe computed friction coefficient at different iterations n.We remark that the algorithm is convergent tol�h ¼ 1:945 the approximate limit obtained in [6] for the lin-ear eigenvalue, using a different method (the ARPACKlibrary). On the right side of Fig. 6 we have plotted thecomputed critical frictional coefficient (the eigenvalue) lh

*

Fig. 5. First example. The geometry of X and the partition of the boundary C. The computed eigenfunction U�h. Left: the distribution of the Von-Misesstress (jr0ðU�hÞj). Right: the deformed mesh.

0.5

1.0

1.5

2.0

μ

0 10 20 30 40 50 60Number of iterations

1.944

1.945

1.946

1.947

1.948

1.949

1.950

μ

10 20 30 40 50 60 70 80 90 1001/h

Fig. 6. First example: The convergence of the critical friction coefficient lnh towards l�h with respect to n the number of iterations (left); the convergence of

the critical friction coefficient l�h, with respect to the discretization parameter 1/h (right).


for different values of the discretization parameter 1=h. Weremark a good convergence to the ‘‘continuous’’ value formore than 60 nodes on each edge of the square.

7.2. Nonlinear features

In this subsection we want to point out the nonlinearcharacter of our eigenproblem. For this we have chosen

Fig. 7. Second example. The geometry of X and the partition of the bounl ¼ l� ¼ 1:75. Left: the distribution of the Von-Mises stress (jr0ðU�hÞj). Right:

the trapezoidal geometry depicted in Fig. 7 and isotropicelastic material with a Poisson ratio m ¼ 0:3. The partitionof C is as follows: CC the bottom and top sides, CD ¼ ; andCN the vertical sides. The equilibrium solution ueq has allthe points of CC in sliding at the left on the bottom side,while on the top side the sliding is on the right. Havingin mind the orientation of the boundary of X we putv ¼ �1 on C0

C ¼ CC.

dary C. The computed eigenfunction U�h corresponding to k2 ¼ 0 andthe deformed mesh.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

χutχτt - μτn

x

Fig. 8. Second example. The computed eigenfunction U�h correspondingto k2 ¼ 0 and l ¼ l� ¼ 1:75. The distribution of the normalized tangentialslip x! vU�t ðxÞ=U (in blue) and of the normalized stress dropx! ðvrtðU�ÞðxÞ � l�rnðU�ÞðxÞÞ=S (in red) on the top side. (For interpre-tation of the references to color in this figure legend, the reader is referredto the web version of this article.)

0

1

2

3

1.7 1.8 1.9 2.0 2.1

μ = 2.0259μ* = 1.75 μm = 2.1

Fig. 9. Second example. The computed nonlinear eigenvalues k versus thefriction coefficient for l P l�.


First, we are looking for l* the friction coefficient corre-sponding to k2 ¼ 0. As before, the eigenvalue k2 is anincreasing function of l in the neighborhood of l* whichrepresents a critical friction coefficient (i.e. the system isunstable for l > l�). We founded that l� ¼ 1:75 and thecomputed nonlinear eigenfunction U�h is plotted in Fig. 7.

Fig. 10. Second example. The computed eigenfunctions U for l ¼ 2:0259eigenfunction U1 corresponding to k ¼ 1:0.

In Fig. 8 we have plotted the distribution of the normal-ized tangential slip (blue line) x! vU�t ðxÞ=U , withU ¼ maxfvU�t ðxÞg, on the top side of CC. We have plottedalso the normalized stress drop x! ðvrtðU�ÞðxÞ�l�rnðU�ÞðxÞÞ=S, here S ¼ maxfvrtðU�ÞðxÞ � l�rnðU�ÞðxÞg,on the same side (red line). Remark that the nonlinear con-strains (i.e. Signorini type inequalities) are satisfied but ineach point x we have a different tangent problem. Indeed,at left (i.e. for x < 0:3) the tangential slip rate is vanishingand we deal with an unloading the frictional ratiojrtðxÞj=jrnðxÞj under l*. At the right side (i.e. for x > 0:3)the slip rate is positive and the frictional ratio jrtðxÞj=jrnðxÞj is equal to l*.

To see the dependence of the k with respect to l we haveplotted in Fig. 9 the computed nonlinear eigenvalues k ver-sus the friction coefficient for l P l�. As a matter of fact,following the algorithm proposed in the previous sectionwe have computed l for different positive values of k2.We remark that the dependence of the eigenvalues on thefriction coefficient l exhibits a returning point. In theneighborhood of l* the algorithm finds one or two eigen-values k. For l 2 ½l�; lm�, here lm ¼ 2:1, an eigenvalue,denoted k0, is very small but still positive. Even if we can-not see it in Fig. 9 the function l! k0ðlÞ is increasing.Moreover, there exists another one, denoted k1, which cor-responds to a larger exponential time explosion of the per-turbation. For instance, for l ¼ 2:0259 we foundedk0 ¼ 0:0189 and k1 ¼ 1:0.

To see the difference between two different eigenfunc-tions we have plotted in Fig. 10 the computed eigenfunc-tions U0 and U1, corresponding to k ¼ 0:0189 and k ¼1:0, respectively. We remark that the top left corner isloaded for U0 but completely unloaded for U1. This ismaybe due to different repartitions of the points x wherethe two (linear) tangent problems are valid.

7.3. Unstable wedged configuration

In this example we examine the stability of an equilib-rium configuration ueq under vanishing loads, called

. Left: the eigenfunction U0 corresponding to k ¼ 0:0189. Right: the

Fig. 11. Third example. The computed equilibrium ueqh (wedged configuration). Left: the distribution of the Von-Mises stress jr0ðueq

h Þj (color scale). Right:the deformed mesh corresponding to the displacement ueq

h . (For interpretation of the references to color in this figure legend, the reader is referred to theweb version of this article.)

0.00

0.05

0.10

0.15

0.20

1.4 1.5 1.6 1.7 1.8 1.9

μw = 1.596μ* = 1.424

μ

Fig. 12. Third example. The computed nonlinear eigenvalues k versus thefriction coefficient for l P l�.


wedged configuration, computed in [5] using a genetic algo-rithm approach. The geometry is plotted in Fig. 11, withthe surface CC represented by the solid line and the otherpart of the boundary is stress free. For all friction coeffi-cients l greater than the wedged frictional coefficient,founded to be lw ¼ 1:59627, there exists a wedged configu-ration ueq, plotted in Fig. 11.

Following the computations of [5] for l ¼ lw ¼ 1:59627the partition of CC has to be as follows: Cstick

C reduces tothree nodes (in the middle of the left side and on the leftand right corners of the bottom side) and Cfree

C is empty.The corresponding sliding function v is �1 on the left sideof C0

C and 1 on the bottom side of C0C. With this choices the

nonlinear eigenproblem is well defined. In Fig. 12 we haveplotted the computed nonlinear eigenvalues k versus thefriction coefficient for l P l�. Here, as before, l* is thesmallest friction coefficient l for which there exists a posi-tive eigenvalue k. In this example the dependence of k withrespect to l is monotonic and it does not present anyreturning points as in the second example.

For l ¼ lw ¼ 1:59627 we found that the correspondingeigenvalue is k ¼ 0:065, that means that analyzed equilib-

Fig. 13. Third example. The computed eigenfunction Uh for l ¼ lw ¼ 1:5962stress (jr0ðUhÞj). Right: the deformed mesh.

rium configuration ueq is not stable. The perturbations ofthe equilibrium will have an exponential time growth inthe direction of the corresponding eigenfunction Uh, whichis plotted in Fig. 13. As we can see in this figure the elasticbody will unload and will reach the reference undeformedconfiguration.

7 and corresponding to k ¼ 0:065. Left: the distribution of the Von-Mises


Acknowledgment

We want to thank the anonymous reviewer for his care-ful reading of the paper and his comments.

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http://dx.doi.org/10.1016/j.ijsolstr.2007.02.020

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