Existence and uniqueness of solutions to dynamical unilateral contact problems with Coulomb friction: the case of a collection of points

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Existence and uniqueness of solutions to dynamical unilateral contact problems with Coulomb friction: the

case of a collection of points

Alexandre Charles, Patrick Ballard

To cite this version:

Alexandre Charles, Patrick Ballard. Existence and uniqueness of solutions to dynamical unilateral contact problems with Coulomb friction: the case of a collection of points. ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2014, 48 (1), pp.1-25. �10.1051/m2an/2013092�.

�hal-00934944�

EXISTENCE AND UNIQUENESS OF SOLUTIONS TO DYNAMICAL UNILATERAL CONTACT PROBLEMS WITH COULOMB FRICTION:

THE CASE OF A COLLECTION OF POINTS

Alexandre Charles

and Patrick Ballard

Abstract. This study deals with the existence and uniqueness of solutions to dynamical problems of finite freedom involving unilateral contact and Coulomb friction. In the frictionless case, it has been established [P. Ballard, Arch. Rational Mech. Anal.154(2000) 199–274] that the existence and uniqueness of a solution to the Cauchy problem can be proved under the assumption that the data are analytic, but not if they are assumed to be only of class C^∞. Some years ago, this finding was extended [P. Ballard and S. Basseville,Math. Model. Numer. Anal.39(2005) 59–77] to the case where Coulomb friction is included in a model problem involving a single point particle. In the present paper, the existence and uniqueness of a solution to the Cauchy problem is proved in the case of a finite collection of particles in (possibly non-linear) interactions.

1. Introduction

The question of the existence and uniqueness of a solution to dynamical problems involving unilateral contact and dry friction is still largely open.

These problems are of two kinds:

• continuum problems, on which very few results are available (for a recent survey, see [5]),

• finite freedom systems, such as a collection of rigid bodies or particles, or a system resulting from the spatial discretization of a continuum.

Different sets of equations are usually written to account for these two kinds of problems. In the case of (three- dimensional, elastic or visco-elastic) continua, the usual elastodynamic equations are simply combined with boundary conditions of the Signorini type. In the case of finite freedom unilateral dynamical systems, such a set of equations is known to introduce some indetermination. The reason for this indetermination is that the bouncing of a real body against an obstacle is governed by the waves crossing the deformable body (and the obstacle). Since the dynamic theory of rigid bodies can not describe these waves, an indetermination arises, that

Keywords and phrases.Unilateral dynamics with friction, frictional dynamical contact problems, existence and uniqueness.

1 Laboratoire de M´ecanique et d’Acoustique, CNRS, 31, chemin Joseph Aiguier, 13402 Marseille Cedex 20, France.

ballard@lma.cnrs-mrs.fr

has to be solved by introducing an additional constitutive law: the impact constitutive law. One well-known example of an impact constitutive law is that based on Newton’s famous restitution coefficient.

The well-posedness of the dynamics of discrete (synonymously, finite freedom) systems with unilateral con- straints (without friction) was first investigated by Schatzman [14], who proved the existence of a solution using a penalization technique in the case of the elastic impact law (that is, with a restitution coefficient equal to 1).

The latter author also gave a striking counter-example showing that, even in the cases where the data show regularityC^∞, one cannot generally expect the solution to be unique. Percivale then commented in [13] that, in the case of the (necessarily frictionless) one degree-of-freedom problem with an external force depending only on time, the uniqueness of the solution can be restored by taking the external force to be ananalytic function of time (instead ofC^∞). Schatzman [15] subsequently extended this finding about the uniqueness under analyt- icity assumptions, still in the context of the one degree-of-freedom problem, to the more general case where the external force is allowed to depend not only on time but also on the current position and velocity. However, her proof was specific to the one degree-of-freedom problem. A simpler proof was given by Ballard [2] who, then, extended these results to the general case involving an arbitrary number of degrees of freedom and unilateral constraints, but only in the case of frictionless situations.

The case of dry friction was first addressed by Monteiro Marques [7], in the case of a single smooth unilateral constraint and a completely inelastic impact law (that is, with a restitution coefficient equal to 0). Using the time-stepping algorithm introduced by Moreau [9,10] (which is, roughly speaking, an adaptation of the implicit Euler scheme to the non-smooth situation under consideration) to build a sequence of approximants, Monteiro Marques succeeded in passing to the limit by extracting a subsequence using a compactness argument, resulting in a proof of existence of a solution. The issue of uniqueness was first addressed by Ballard and Basseville in [3]

in the case of the simplest system involving unilateral contact and Coulomb friction—a system which was first introduced by Klarbring [6]. By extending Ballard’s original method [2], these authors proved the uniqueness (under analyticity assumptions) of the solution also, in the case of this model problem involving dynamical unilateral contact and Coulomb friction. In the study by Ballard and Basseville, it was stated that the use of this method could be extended to more general situations than this model problem, but it was proposed to investigate in a future study how this could be achieved. The aim of the present paper is therefore to identify the more general situations in which existence and uniqueness can be proved using the method developed in [3].

In cases where a finite-degree-of-freedom system contains a rigid body that do not reduce to a point, examples of non-uniqueness or non-existence of solution to the Cauchy problem, which are known as Painlev´e paradoxes, were exhibited by Painlev´e more than one century ago [12]. The present analysis therefore focuses on the case of a finite collection of point particles. The case of rigid bodies that do not reduce to points requires, in our opinion, a re-visitation of the formulation of the evolution problem. This point is currently under investigation and will be extensively discussed in a forthcoming paper.

2. Statement of the problem and results

Let us take a finite collection of n(n≥1) point particles in^Êd (d= 2,3). Some of these particles (for the sake of simplification, we will impose this constraint on all of them, without any loss of generality) have to remain on one side of a given obstacle associated with that particle. To simplify the writing slightly, it will be assumed in the first step, that each of these obstacles is straight, that is, it can be defined by a hyperplane. It is only in the second step (in Sect.5) that we will deal with how the results can be extended to the case of possibly curved obstacles. For each particle, an affine orthonormal coordinate system, based on the corresponding straight obstacle, is introduced so that the position of the particleiis described by:

uⁱ=

uⁱ_n∈^Ê uⁱ_t∈^Êd−1

∈^Êd,

(a) straight obstacles. (b) curved obstacles.

Figure 1. Geometry of the problem.

and the unilateral constraint reduces touⁱ_n≤0. A configuration of the system of particles will be described by:

U=

⎛

⎜⎝

u¹ ∈^Êd ... uⁿ ∈^Êd

⎞

⎟⎠∈^Ênd.

We take:

U_n=

⎛

⎜⎝

u¹_n∈^Ê ... uⁿ_n∈^Ê

⎞

⎟⎠∈^Ên,

to denote the vector formed by the normal components of each particle. A similar vectorU_t∈^Ên(d−1)formed by the tangential components will also be used below.

It is now proposed to formulate a Cauchy problem associated with the dynamics of this system of particles subjected to internal and external forces and unilateral contact with Coulomb friction with respect to the obstacles. This requires introducing a reaction force rⁱ ∈ ^Êd for the particle having the index i. The normal component of this reaction force, which will be denoted byr_nⁱ, will be required to be non-positive (non-adhesive unilateral contact). The reaction force at work in the system will be denoted byR∈^Ênd(of normal component R_n∈^Ên) and is built with therⁱin the same way asUis based on theuⁱ. Given an arbitrary time intervalI, a motion of the system will be described by a functionU(t) (t∈I). Due to the obstacles, we must expect velocity jumps, that is, impacts. Hence, the acceleration ¨Uwill have to be understood in the sense of distributions and not in the classical sense. Since the distributionR:I→^Ênd is required to take values in a cone, it classically entails that it is a measure on I with values in ^Ênd. Hence, the acceleration will be a measure (notation U¨ ∈ M(I,^Ênd)) and the search for the motion will be conducted in the space MMA(I,^Ênd), consisting of all the distributions onIwhose twice-derivative is a measure. This space was first introduced by Schatzman in [14].

It contains only continuous functions, having left and right derivatives ˙U⁻(t) and ˙U⁺(t) (in the classical sense) at each t in the interior of I. The functions ˙U⁻(t) and ˙U⁺(t) are functions with locally bounded variation.

Given an arbitraryT >0, the Cauchy problem can now be formulated in line with Moreau [8,10].

ProblemP_u. FindU∈MMA

[0, T] ;^Ênd

andR∈ M

[0, T] ;^Ênd

such that:

• U(0) =U₀ ; U˙⁺(0) =V₀ (initial condition),

• U¨ =F(t,U,U) +˙ R, in [0, T] (motion equation),

• ∀i∈ {1,2, . . . , n}, ∀t∈]0, T[,

uⁱ_n≤0, rⁱ_n≤0, uⁱ_nrⁱ_n= 0 (unilateral contact),

• ∀i∈ {1,2, . . . , n}, ∀v∈C⁰

[0, T];^Êd−1 ,

[0,T]

rⁱ_t·

v−u˙ⁱ⁺_t

−μ_irⁱ_n

v − u˙ⁱ⁺_t

≥0, (Coulomb friction),

• ∀i∈ {1,2, . . . , n}, ∀t∈]0, T[,

uⁱ_n(t) = 0 =⇒ u˙ⁱ⁺_n (t) =−eiu˙ⁱ⁻_n (t), (impact law), where the data are as follows.

• The initial condition (U₀,V₀)∈^Ênd×^Êndis assumed to be compatible with the unilateral constraint:

U_0n≤0 and uⁱ_0n= 0 =⇒ vⁱ_0n≤0,

where, by convention,v≤0, forv∈^Ên, means that all its components are non-positive.

• The force mapping F(t,U,U) is a given mapping˙ F: [0, T]×^Ênd×^Ênd→^Ênd. The force acting on the particle having the indexiwill be denoted byfⁱ(t,U,U), so that˙ Fis obtained from thefⁱin the same way asUis based on theuⁱ.

• Theμ_i≥0 are the friction coefficients of each particle, and thee_i ∈[0,1] are the restitution coefficients.

A weak formulation for the Coulomb friction law has been used here because the reaction force is generally a measure. In cases where this measure reduces to some continuous function, it can easily be checked that this weak formulation is equivalent to the usual pointwise formulation. The following theorem will be proved below.

Theorem 2.1. Let us assume that the functionF: [0, T]×^Ênd×^Ênd→^Êndsatisfies the following hypotheses:

(i) the function(U,V) →F(t,U,V)satisfies a Lipschitz condition of modulusκ≥0, for allt∈[0, T]:

∀t∈[0, T], ∀(U₁,V₁),(U₂,V₂)∈^Ênd×^Ênd,

F(t,U₁,V₁)−F(t,U₂,V₂)≤κ(U₁,V₁)−(U₂,V₂), where · stands for the Euclidean norm in ^Ênd×^Ênd;

(ii) the function(t,U,V) →F(t,U,V)is analytic;

(iii) eachfⁱ does not depend on theu˙^j whenj=i.

Then, problem Pu has one and only one solution.

The hypotheses adopted in Theorem2.1 do not all have the same status. Hypothesis (i) is simply used for convenience and is not essential. Dropping it would permit the solution to blow up at a finite time, so that the conclusion in Theorem2.1would have to be replaced by: there exists one and only one maximal solution (in the sense that any other solution is a restriction of that maximal solution to a smaller time interval). Hypothesis(ii) was adopted because even in the frictionless situations, multiple solutions can be encountered with F∈C^∞. Hypothesis (ii) was also adopted in the preliminary analysis of a model problem in [3]. Hypothesis (iii) is something else. It turns out that the strategy on which the proof in [3] was based can be used to deal with

cases where the force mappingF(t,U) depends arbitrarily on the configuration but not on the velocity. We were surprised to discover that this strategy does not seem to be valid in cases where the force mappingF(t,U,U)˙ depends arbitrarily on the velocity ˙U, and we have no idea so far how to extend the result to that case, or even if it is true. In concrete terms, hypothesis(iii)excludes viscous interactions between particles undergoing unilateral constraints. This is a severe restriction which requires better understanding in the future. Note however, that for a system consisting of a single particle, hypothesis(iii)is automatically satisfied and is no restriction at all.

The strategy used to prove Theorem2.1will be the same as that used in [3]. First, the unilateral constraints will be frozen into bilateral constraints. Due to Coulomb friction, the bilateral problem is not governed by an ordinary differential equation as in the frictionless situation studied in [2]. In Section3.1, the bilateral problem is expressed in the form of a differential inclusion which is solved in Section 3.2using standard monotonicity techniques as in [3]. It is then proved in Section3.3that the restriction of the solution to some right-neighborhood of the time origin is analytic, as long as the external force is analytic. The analysis of the bilateral problem, as performed in Section3, is used in Section4.1 to build a local analytic solution to the unilateral problem with an analytic external force. To ensure well-posedness of the unilateral problem, it then suffices to prove that there cannot exist any other local solution inMMA, which differs from the local analytic one. This is done in Section4.2 by adapting Ballard’s strategy [2] to the situation under consideration.

Theorem 2.1 extends to those cases where the obstacles may be curved instead of being straight. Details about how this extension is achieved are given in Section5.

3. The bilateral problem

In this section, we study the simpler situation where some of the unilateral constraints are frozen into bilateral constraints and the other unilateral constraints are simply dropped.

3.1. Formulation of the bilateral problem

We are given two setsB,Iof indices such thatB ∩I =∅andB ∪I={1,2. . . , n}, and the unilateral contact condition in problemP_uis replaced by:

∀i∈ B, uⁱ_n≡0, ∀i∈ I, rⁱ_n≡0.

We shall therefore consider the following evolution problem.

ProblemPb. FindU∈MMA

[0, T] ;^Ênd

andR∈ M

[0, T] ;^Ênd

such that:

• U(0) =U₀ ; U˙⁺(0) =V₀ (initial condition),

• U¨ =F(t,U,U) +˙ R, in [0, T] (motion equation),

• ∀i∈ B, uⁱ_n≡0, (bilateral constraints),

• ∀i∈ I, rⁱ_n≡0, (inactive constraints),

• ∀i∈ {1,2, . . . , n}, ∀v∈C⁰

[0, T];^Êd−1 ,

[0,T]

rⁱ_t·

v−u˙ⁱ⁺_t

+μ_i|rⁱ_n|

||v|| − u˙ⁱ⁺_t

≥0, (Coulomb friction),

where the initial condition (U₀,V₀)∈^Ênd×^Êndis assumed to be compatible with the bilateral constraint:

∀i∈ B, uⁱ_0n= 0 and v_0nⁱ = 0.

From now on, the function Fis supposed to satisfy both condition(i)of page4 and:

(ii’) For all (U,V)∈^Ênd×^Ênd, the functiont →F(t,U,V) is integrable over [0, T].

These hypotheses are weaker than those in Theorem2.1. IfU denotes an arbitrary solution of problemP_b, then U is absolutely continuous and ˙U has bounded variation, so that they are both integrable functions.

Hence, the hypotheses made aboutFentail that the (almost everywhere defined) functiont →F_n(t,U(t),U(t))˙ is integrable. Furthermore the normal components of the motion equation read as follows:

U¨_n=F_n(t,U,U) +˙ R_n, so that:

r_nⁱ =−f_nⁱ(t,U,U),˙ ifi∈ B,

r_nⁱ = 0, ifi∈ I.

Therefore, the measuresrⁱ_nare absolutely continuous with respect to the Lebesgue measure. But the Coulomb friction law entails the following inequality between the absolute value measures:

∀i∈ {1,2, . . . , n}, rⁱ_t≤μ_ir_nⁱ,

which shows that therⁱ_tare also absolutely continuous with respect to the Lebesgue measure, that is, they are integrable functions. Hence, any solution (U,R) of problemPb must actually belong toW^2,1×L¹.

Next, denote byR(t,U,U) the vector in˙ ^Ên with components:

Rⁱ(t,U,U) =˙

μ_if_nⁱ(t,U,U),˙ ifi∈ B,

0, ifi∈ I.

The vector R(t,U,U) contains the friction thresholds for each particle. Note that the regularity hypotheses˙ adopted about the functionFalso hold true for the functionR(t,U,U). Let˙ Bthe closed unit ball in^Êd−1. The support function of a non-empty closed convex set K will be denoted byS_K and its sub-differential by∂S_K. With these notations, the following differential inclusion:

¨

uⁱ_t(t)−f_tⁱ

t,U(t),U(t)˙

∈∂S_|Ri(t,U(t),U(t))|˙ B

−u˙ⁱ_t(t) ,

holds true, for almost allt∈[0, T] and alli∈ {1,2. . . , n}. ForR ∈^Ên, we will use the short-hand notation:

R·B^def= n

i=1

RⁱB⊂^Ên(d−1), where the

stands for the Cartesian product. Hence, forR ∈^Ên andX∈^Ên(d−1), one has the identity:

∂S_|R|·B X

= n i=1

∂S_|Ri|B

Xⁱ ,

and therefore, the following differential inclusion:

U¨_t(t)−F_t

t,U(t),U(t)˙

∈∂S_{|R(t,U(t),U(t))|·B˙}

−U˙_t(t) , holds true, for almost allt∈[0, T]. For the normal components, note that:

¨

uⁱ_n= 0,ifi∈ B, r_nⁱ = 0, ifi∈ I, so that, defining a functionF(t,U,V) such thatFt≡F_t and:

F_nⁱ ≡0, i∈ B, F_nⁱ ≡f_nⁱ, i∈ I,

we obtain the following differential inclusion obeyed byU:

U(t)¨ − F

t,U(t),U(t)˙

∈∂S_{|R(t,U(t),U(t))|·C˙}

−U(t)˙ ,

with the notation C ^def= {0} ×B ⊂ ^Êd. Finally, problem Pb can be handled using the following simpler but equivalent formulation.

ProblemPb. FindU∈W^2,1(0, T;^Ênd) such that:

• U(0) =U₀ ; U(0) =˙ V₀ (initial condition),

• U(t)¨ − F

t,U(t),U(t)˙

∈∂S_{|R(t,U(t),U(t))|·C˙}

−U(t)˙

, for a.a.t,

where the functions F(t,U,V) andR(t,U,V) satisfy hypotheses(i)and(ii’) adopted aboutF(t,U,V).

3.2. Existence and uniqueness of a solution to the bilateral problem This section is devoted to proving the following proposition.

Proposition 3.1. Problem Pb admits a unique solutionU∈W^2,1(0, T;^Ênd).

Proof.

Step 1.Proposition3.1 holds true in the particular case where the functionRis constant.

In the case where the function R is constant and the function F is simply an integrable function of time, Proposition3.1holds true by virtue of proposition 3.4, p. 69 of [4]. In the case whereF also depends on (U,U),˙ the method of successive approximations will be used. Let (V^k) be the sequence of functions inW^1,1(0, T;^Ênd) defined by V⁰ ≡V₀ and the following induction. If V^k is given, standard theorems on differential inclusions associated with the subdifferential of a lower semi-continuous function (see, for example, Prop. 3.4, p. 69 of [4]) provide a unique solutionV^k+1 ∈W^1,1(0, T;^Ênd) of the evolution problem:

• V^k+1(0) =V₀,

• V˙^k+1(t)− F

t,U₀+_t

0V^k,V^k(t)

∈∂S_|R|·C

−V^k+1(t)

, for a.a.t.

We are now going to prove the convergence of the sequence (V^k) and establish that the limit solves the expected evolution problem. Based on the monotonicity of the sub-differential, we have:

V^k+1(t)−V^k(t) ·

V˙^k+1(t)−V˙^k(t)− F

t,U₀+_t

0V^k,V^k(t)

+F

t,U₀+_t

0V^k−1,V^k−1(t) ≤0,

for almost allt∈[0, T]. Integrating:

1

2V^k+1(t)−V^k(t)²≤ _t

0

V^k+1(s)−V^k(s).F

s,U₀+_s

0 V^k,V^k(s) −F

s,U₀+_s

0 V^k−1,V^k−1(s)ds, and, from Lemma A.5², p. 157 of [4]:

∀t∈[0, T], V^k+1(t)−V^k(t)≤ _t

0

F

s,U₀+_s

0 V^k,V^k(s) − F

s,U₀+_s

0 V^k−1,V^k−1(s)ds.

2This lemma is the following assertion. Letabe nonnegative real constant,mbe a nonnegative locally integrable function andφ a continuous function satisfying:

∀t≥0, 1

2φ²(t)≤1 2a²+

_t

0 m(s)φ(s) ds.

Then:

∀t≥0, φ(t)≤a+ _t

0 m(s)φ(s) ds.

Usingtheassumptionthat F isLipschitz-continuouswithrespectto (U,V),weobtain:

V^k+1(t)−V^k(t)≤κ _t

0

V^k(s)−V^k−1(s)+_s

0V^k−V^k−1ds,

≤κ(1 +T) _t

0

V^k(s)−V^k−1(s)ds.

By induction, we obtain:

∀k∈^Æ, ∀t∈[0, T], V^k+1(t)−V^k(t)≤

κt(1 +T)_k

k! V¹−V⁰

C⁰,

which entails that, since the sequence (V^k) is a Cauchy sequence, it converges uniformly in [0, T] towards some limit V ∈ C⁰([0, T];^Ênd). Now, let W ∈ W^1,1(0, T;^Ênd) be the unique solution of the following evolution problem:

• W(0) =V₀,

• W(t)˙ − F

t,U₀+_t

0V,V(t)

∈∂S_|R|·C

−W(t)

, for a.a.t.

Taking the difference between the latter differential inclusion and that definingV^k+1, multiplying byW−V^k+1, and given the monotonicity of the sub-differential and then Lemma A.5, p. 157 of [4], we obtain:

∀k∈^Æ, ∀t∈[0, T], W(t)−V^k+1(t)≤κ(1 +T) _t

0

V−V^k.

Taking the limitk→ ∞, it can be seen that the sequence (V^k) converges uniformly in [0, T] towardsW=V∈ W^1,1(0, T;^Ênd), so that:

• V(0) =V₀,

• V(t)˙ − F

t,U₀+_t

0V,V(t)

∈∂S_|R|·C

−V(t)

, for a.a.t, and the functionU(t)^def= U₀+_t

0Vis a solution of the evolution problem under consideration.

To establish that this is the only solution, take another one, sayU. Using the same algebra as above, based on the monotonicity of the sub-differential, we obtain:

∀t∈[0, T], U(t)˙ −U˙(t)≤κ(1 +T) _t

0

U˙ −U˙, and Gronwall’s lemma yields U≡U.

Step 2.Proposition3.1holds true in the particular case where the functionRis simply an integrable function of time.

In this step, it is proposed to extend the result proved in step 1 for the case whereRis a constant function to the case where it is an integrable function of time t alone. Note that since the proof of uniqueness part of step 1 is also valid in this extended case, only the existence of a solution has to be proved.

First, we take a sequence (R^k) of piecewise constant functions that converges strongly inL¹(0, T;^Ên) towards the integrable functionR. The sequence (R^k) can be chosen so that all the components with indexi∈ I vanish.

Based on step 1, there exists a sequence (U^k) in W^2,1(0, T;^Ênd) such that:

• U^k(0) =U₀ ; U˙^k(0) =V₀ (initial condition),

• U¨^k(t)− F

t,U^k(t),U˙^k(t)

∈∂S_|Rk(t)|·C

−U˙^k(t)

, for a.a.t.

From the definition of the sub-differential, forf ∈∂S_rB[x] andf∈∂S_rB[x]:

rx ≥rx+f· x−x

, rx ≥rx+f·

x−x ,

and summing up, we obtain:

f−f

· x−x

≥ r−r

x − x .

Hence:

U¨^p+q(t)−U¨^p(t) ·

U˙^p+q(t)−U˙^p(t)− F

t,U^p+q(t),U˙^p+q(t) +F

t,U^p(t),U˙^p(t)

=

U¨^p+q_t (t)−U¨^p_t(t) ·

U˙^p+q_t (t)−U˙^p_t(t)− F_t

t,U^p+q(t),U˙^p+q(t) +F_t

t,U^p(t),U˙^p(t) ,

≤ − n i=1

R^p+q,i(t)−R^p,i(t)u˙^p+q,i(t)−u˙^p,i(t),

≤ n

i=1

R^p+q,i(t)− R^p,i(t) U˙^p+q(t)−U˙^p(t),

where R^p,i denotes the ith component of R^p. Integrating and then, using the Lipschitz property for F and Lemma A.5, p. 157 of [4], we obtain:

U˙^p+q(t)−U˙^p(t)≤ _t

0

n i=1

R^p+q,i− R^p,i+κ _t

0

U^p+q−U^p+κ _t

0

U˙^p+q−U˙^p,

≤ _t

0

n i=1

R^p+q,i− R^p,i+κ(1 +T) _t

0

U˙^p+q−U˙^p

Based on the Gronwall−Bellman lemma (Lem. A.4, p. 156 of [4]):

∀t∈[0, T], U˙^p+q(t)−U˙^p(t)≤e^κT(1+T) _T

0

_n

i=1

R^p+q,i− R^p,i .

Hence, the sequence ( ˙U^k) is Cauchy in the Banach spaceC⁰([0, T],^Ênd) and therefore converges in that space towards some limitV∈C⁰. DefiningU=U₀+_t

0V, it can be readily checked that (U^k) converges uniformly on [0, T] towardsUand that ˙U=V.

We shall now prove thatU∈W^2,1and is the expected solution of the evolution problem under consideration.

Take 0≤t₁≤s≤t₂≤T. Based on the definition of the sub-differential:

U˙^k(s)−U˙^k(t₁) ·

U¨^k(s)− F

s,U^k(s),U˙^k(s)

≤ n i=1

R^k,i(s)u˙^k,i(t₁)−u˙^k,i(s),

≤ _n

i=1

R^k,i(s)

U˙^k(s)−U˙^k(t₁),

andintegratingover[t₁, t]:

1

2U˙^k(t)−U˙^k(t₁)²≤ _t

t1

n i=1

R^k,i(s)U˙^k(s)−U˙^k(t₁)+ _t

t1

F

s,U^k(s),U˙^k(s)

·

U˙^k(s)−U˙^k(t₁)

,

≤ _t

t1

F

s,U₀,V₀+ n i=1

R^k,i(s)

U˙^k(s)−U˙^k(t₁) +κ

_t

t1

U^k(s)−U₀+U˙^k(s)−V₀·U˙^k(s)−U˙^k(t₁).

Again using Lemma A.5, p. 157 of [4], we obtain the estimate:

U˙^k(t₂)−U˙^k(t₁)≤ _t2

t1

F

s,U₀,V₀+ n i=1

R^k,i(s)+κU^k(s)−U₀+κU˙^k(s)−V₀. (1) Taking the limitk→ ∞, we finally obtain:

U(t˙ ₂)−U(t˙ ₁)≤ _t2

t1

F

s,U₀,V₀+ n i=1

Rⁱ(s)+ 2κU

C⁰+ 2κU˙

C⁰.

which shows that the function ˙U is absolutely continuous: ˙U ∈ W^1,1(0, T;^Ênd). Furthermore, going back to estimate (1), we can write:

U¨^k(t)≤F

t,U₀,V₀+ n i=1

R^k,i(t)+κU^k(t)−U₀+κU˙^k(t)−V₀, for almost allt∈[0, T] and allk∈^Æ. Therefore:

U¨^k

L¹ ≤M,

for some constant independent of k. Therefore, extracting a sub-sequence if necessary, the sequence ( ¨U^k) con- verges inM([0, T];^Ênd) weak-∗. Its limit is necessarily ¨U. Furthermore, the differential inclusion under consid- eration can be given the following equivalent form.

∀i∈ {1, . . . , n}, ∀v∈C⁰

[0, T] ;^Êd−1 _T ,

0

R^k,i· v ≥ _T

0

R^k,i·u˙^k,i_t + _T

0

¨

u^k,i_t − Fⁱ

s,U^k,U˙^k

·(v+ ˙u^k,i_t ).

Thanks to the convergence properties of all the sequences involved, we can take the limit as k → ∞ in this inequality. We deduce thatUis a solution of the evolution problem under consideration.

Step 3.Proof of Proposition3.1in the general case.

Let (U^k) be the sequence in W^2,1(0, T;^Ênd) defined by its first term:

U=U₀+tV₀,

and the following induction: givenU^k, the function U^k+1 is defined as the unique solution, given by step 2, of the following evolution problem.

• U^k+1(0) =U₀ ; U˙^k+1(0) =V₀ (initial condition),

• U¨^k+1(t)− F

t,U^k+1(t),U˙^k+1(t)

∈∂S_|R(t,Uk(t),U˙^k(t))|·C

−U˙^k+1(t)

, for a.a.t.

Using the same algebra as at the beginning of step 2 and applying Lemma A.5, p 157 of [4] gives:

U˙^k+1(t)−U˙^k(t)≤ _t

0

F

s,U^k+1,U˙^k+1

− F

s,U^k,U˙^k+ _t

0

R

s,U^k,U˙^k

− R

s,U^k−1,U˙^k−1,

≤κ(1 +T) _t

0

U˙^k+1−U˙^k+κ(1 +T) _t

0

U˙^k−U˙^k−1,

whereκ =κmax_iμ_iis the Lipschitz modulus of the functionR. Using the Gronwall−Bellman lemma (Lem. A.4, p. 156 of [4]):

∀t∈[0, T], U˙^k+1(t)−U˙^k(t)≤e^κT(1+T)κ(1 +T) _t

0

U˙^k−U˙^k−1,

≤

κt(1 +T)e^κT(1+T)_k

k! U˙¹−U˙⁰

C⁰,

which shows that since ( ˙U^k) is a Cauchy sequence in the Banach space C⁰([0, T];^Ênd), it converges in that space towards a limit denoted byV. Setting U ^def= U₀+_t

0V, it can be readily checked that (U^k) converges towards U in C⁰ and that ˙U =V. Based on exactly the same reasoning as at the end of step 1, we deduce thatU solves the evolution problem under consideration and that it is the only solution.

3.3. The bilateral problem with analytic force

The aim of this section is to prove that if the functionsF(t,U,V) andR(t,U,V) are assumed in addition to beanalytic, then the solution of problemP_bgiven by Proposition3.1, is also analytic on a right-neighborhood oft= 0.

From now on, we assume that the functionF: [0, T]×^Ênd×^Ênd→^Êndsatisfies the following hypotheses:

(i) the function (U,V) →F(t,U,V) satisfies a Lipschitz condition of modulusκ≥0, for allt∈[0, T]:

∀t∈[0, T], ∀(U1,V₁),(U₂,V₂)∈^Ênd×^Ênd,

F(t,U₁,V₁)−F(t,U₂,V₂)≤κ(U₁,V₁)−F(t,U₂,V₂). (ii) the function (t,U,V) →F(t,U,V) is analytic,

These hypotheses are also satisfied by the two functions F(t,U,V) andR(t,U,V). With this additional hy- pothesis, the solution of problemPbfulfils the following regularity result.

Proposition 3.2. There exists η >0such that the restriction to[0, η[of the solutionU∈W^2,1([0, T];^Ênd)to problem Pb, resulting from Proposition3.1, is analytic.

Proof. The strategy on which the proof will be based is as follows. We will look for a solution as a formal power seriesS=_∞

k=0Ukt^kand will attempt to identify one term after another. Since the zeroes of an analytic function of time cannot accumulate, the same is expected to be true of the switches between the stick and slip status of each particle in the system. Therefore, the analysis will consist in determining the status of each particle (stick or slip) in a right-neighborhood of the time origin. The status of each particle is naturally given by the first non-zero term in the formal power expansion of its displacement, but higher order analysis is necessary because of the couplings between particles. Once the status of each particle (stick or slip) has been determined, the evolution problem can be written in the form of an ordinary differential equation, and the analyticity of its solution is given by classical theorems. To conclude, it will then suffice to prove that a restriction of this analytic function to a right-neighborhood of the time origin yields a restriction of the unique solution to problemP_b.

Step 1.Calculating theUk.

Initializing.The first two termsU0 andU1 in the formal power seriesS=_∞

k=0Ukt^k are obviously given by the initial condition:

U0=U₀, U1=V₀.

The calculation of U2 runs as follows. The vector U2 ∈ ^Ênd is obtained from the contributions uⁱ₂ ∈ ^Êd (i∈ {1, . . . , n}) of each particle, so that identifyingU2 is equivalent to determining each of theuⁱ₂.

• Withi ∈ I, the corresponding particles are not in active contact with an obstacle so that their reaction force vanishes and the equation of motion gives:

∀i∈ I, 2uⁱ₂=Fⁱ

0,U₀,V₀

=fⁱ

0,U₀,V₀ .

• Withi ∈ G0 def

=

i∈ B | vⁱ₀ =0

, the corresponding particles are slipping in a right-neighborhood of the time origin. Theuⁱ₂are given by:

∀i∈ G0, 2uⁱ₂=Fⁱ

0,U₀,V₀

−Rⁱ

0,U₀,V₀ v₀ⁱ v₀ⁱ·

• Withi ∈ A0 def

=

i ∈ B | vⁱ₀ =0andFⁱ(0,U₀,V₀) <|Rⁱ(0,U₀,V₀)|

, the corresponding particles are sticking in a right-neighborhood of the time origin and:

∀i∈ A0, uⁱ₂=0=uⁱ_k (k∈^Æ).

• Withi∈ B \(G0∪ A0), the particle has vanishing velocity and its associated reaction force is exactly at the friction threshold. Upon examining the motion equation, it can be readily seen that:

∀i∈ B \(G0∪ A0), uⁱ₂=0,

and the status of these particles (stick or slip) must be determined by higher order analysis.

Induction.We assume that the expected formal power expansion has been calculated up to orderp+ 2:

S_p+2=

p+2

k=0

U_kt^k,

and that it is associated with two non-intersecting subsets Gp and Ap of B. These particles in Gp are those particles that have been determined as being slipping particles because the formal power series expansion of their displacement has a non-zero term of an order no larger than p+ 2. These particles inA_p are those that have been determined as being sticking particles because the reaction force associated with the partial (up to order p) formal power series expansion of their displacement is not at the friction threshold. The particles in B \(Gp∪ Ap) are those whose status (stick or slip) is still undetermined at order p+ 2, because they have a vanishing velocity at orderp+ 2 and the corresponding reaction force at orderpis at the friction threshold.

Given a formal power series, we introduce the following notation for extracting the coefficient of orderp:

_∞

k=0

a_kt^k

p

=a_p. The calculation ofUp+3 runs as follows.

• Withi ∈ I, the corresponding particles are not in active contact with an obstacle so that their reaction force vanishes and the equation of motion gives:

∀i∈ I, (p+ 2)(p+ 3)uⁱ_p+3= Fⁱ

t,S_p+2(t),S˙_p+2(t)

p+1=

fⁱ

t,S_p+2(t),S˙_p+2(t)

p+1.