#### Abstract

A chain sliding on a fixed support, made out of some elementary rheological models (dry friction element and linear spring) can be covered by the existence and uniqueness theory for maximal monotone operators. Several behavior from quasistatic to dynamical are investigated. Moreover, classical results of numerical analysis allow to use a numerical implicit Euler scheme.

#### 1. Introduction

This paper is the next step of a series of previous works dealing with modelling of discrete mechanical systems with finite number of degrees of freedom involving assemblies of classical smooth constitutive elements (in the mechanical point of view they correspond to linear or non linear springs, dashpots) and nonsmooth ones mainly based on St-Venant Elements. Let us cite basic rheological models [1], with different applications and developpements [2–7]. Delay or stochastic frame have also been investigated in [8–10].

In this paper we examine a new model: it can be associated with motion of a discretized beam “sliding” on soil. We do not give more details on this discretization.

This paper is organized as follows in Section 2, the model is described. In Section 3, the general model is adapted to different dynamical, semi-dynamical or quasistatic cases. In Section 4, existence and uniqueness is addressed. In Section 5, numerical scheme is described and its convergence obtained.

#### 2. Description of the Model

We refer to previous works for description of some rheological models (see for example [1, 6]).

We consider the model of Figure 2. (with ) correspond to masses, to stiffness, and to St-Venant elements thresholds.

The reader is referred to Appendix A.

Let be the multivalued graph sign defined by (see Figure 1(a)).

(a) The graph |

(b) The graph |

According to [11], this graph is maximal monotone. Therefore:

Let us assume (see Figure 3) the following

These two last notations are justified by the study of particular cases in the next sections.

The different equations of the model are successively given by the fundamental Newton law:

by the constitutive laws of linear springs:

by the constitutive of laws St-Venant elements:

by the geometrical connexions:

and finally by the boundary conditions:

We can observe that (2.3d)–(2.3e) are equivalent to

Now, we study systems (2.3a), (2.3b), (2.3c), (2.3f), (2.3g), and (2.4a)–(2.4c).

#### 3. Transformations of Equations

Now, as in [1, 6], we transform system (2.3a)-(2.3b)-(2.3c)-(2.3f)-(2.3g)-(2.4b)-(2.4c) to rewrite it under the usual form (A.7) according to different kinds of problem and of boundary conditions.

Let us assume that the external forcing are known.

##### 3.1. Dynamical Case

We assume in this section that

Equations (2.3a)-(2.3b)-(2.3c)-(2.4a)–(2.4c) imply

###### 3.1.1. Clamped Mechanical System

We assume that our mechanical system is clamped at its two extremities so that we can write the boundary conditions:

We set, for all ,

Thus, by setting

and defining the maximal monotone operator by

equations (3.2a)–(3.2c) imply the system of equations

Reactions and can be determined thanks to (2.3f)-(2.3g) which give

Set

where is the identity of and for , , with ,

Then, the system (3.6) is equivalent to (A.7) (see Appendix A).

Reciprocally, if (3.6) and (3.7a)-(3.7b) hold, we define , , , , and successivelly by

Then, we can deduce (2.3a), (2.3b), (2.3c), (2.3f), (2.3g), (2.4b), and (2.4c).

###### 3.1.2. Clamped-Free Mechanical System

We assume that our mechanical system is clamped at its left extremity and free at its right extremity so that we can write the boundary condition:

As in Section 3.1.1, by setting

we can prove that equations (3.2a)–(3.2c) imply the system of equations

Reactions and displacement can be determined thanks to

As in Section 3.1.1, let us set

and for , , with

Then, system (3.12) is equivalent to (A.7).

As in Section 3.1.1, reciprocally, if (3.12) and (3.13a)-(3.13b) hold, we define , , , , and successivelly by

Then, we can deduce (2.3a), (2.3b), (2.3c), (2.3f), (2.3g), (2.4b), and (2.4c).

##### 3.2. Semi-Dynamical Case

In this section, we assume that

Equation (3.2b) implies

with

As in [6, 7], we introduce , the inverse graph of (in the sens of [11], see Figure 1(b)):

We have

where is the convex indicatrix function of the convex domain . Thus, (3.17a) is equivalent to

Similarly, (3.2a) gives

with

and (3.2c) gives

###### 3.2.1. Clamped Mechanical System

We assume that our mechanical system is clamped at its two extremities so that we can write the boundary conditions (3.3a)-(3.3b). As in [6, 7], let us set where is defined by (3.4). Thus, according to (3.17b)–(3.21b), we have

and from (3.17a)–(3.21a) we can write

Under the assumption

the matrix is symmetric definite positive (see proof in Lemma B.1 of Appendix B), so that

and (3.25) gives

which is equivalent to

For integer and vector of , we denote by

the th component of . Equation (3.22) gives

which can be rewritten under the following form:

Let be the vector of defined by

Note that

We set and for all , , , Then, system (3.29)–(3.32) is equivalent to (A.7).

Reactions and can be determined thanks to

Reciprocally, as in Section 3.1.1, if (3.29)–(3.32) hold, we can determine and thanks to

then we can calculate thanks to (3.27). Successively, , , , and are defined by

Then, we can deduce (2.3a), (2.3b), (2.3c), (2.3f), (2.3g), (2.4b), and (2.4c).

###### 3.2.2. Clamped-Free Mechanical System

We assume that our mechanical system is clamped at its left extremity and free at its right extremity so that we can write boundary condition (3.10a)–(3.10d).

The calculus are similar to those of Section 3.2.1; Equation (3.29) holds and (3.31) is replaced by

Using notations (3.23a)–(3.23f), we obtain the system (A.7), where we set and for all , , , ,

The reaction and the displacement can be determined thanks to (3.36a) and

##### 3.3. Quasistatic Case

In this section, we assume that

As it has been previously noticed, (3.17a)-(3.17b) and (3.21a)-(3.21b) are not modified, and (3.22) gives

with

###### 3.3.1. Clamped Mechanical System

We assume that our mechanical system is clamped at its two extremities so that we can write the boundary conditions (3.3a)-(3.3b).

As in Section 3.2.1, following [6, 7], we set where is defined by (3.4).

Thus, we have Under assumption

the matrix is symmetric definite positive (see proof in Lemma B.1), so that

We set and, for all , for all Then, the system (3.48) is equivalent to (A.7).

Reactions and can be determined thanks to

###### 3.3.2. Clamped-Free Mechanical System

We assume that our mechanical system is clamped at its left extremity so that we can write the boundary condition (3.10a) and (3.10b). Boundary conditions for its right extremity is given later.

The calculus is similar to those of [6, 7].

*(i) First Case: Displacement-Force Model*

We assume that the displacement is known and that the force is unknown.

We introduce , , and matrix defined by (3.44a), (3.44b), (3.44d), and (3.44e) and defined by

and we obtain, as in Section 3.3.1,
By setting
and, for all , for all ,
we remark that system (3.53) is equivalent to (A.7).

Reactions and can be determined thanks to

*(ii) Second Case: Force-Displacement Model*

We assume that external forcing are known and displacement is unknown.

The calculus are similar to the previous case.

Equation (3.43b) is replaced by

Following the same method, we introduce , , and defined by (3.44a)-(3.44b)-(3.44d), and matrix defined by (3.11c). Vector is defined by

So, (3.52) is replaced by

and (3.48) is replaced by

*Remark 3.1. *As in [6], let us notice that matrix defined by (3.11c) for force-displacement model corresponds to matrix for displacement-force model defined by (3.4) with

According to previous remark, assumption

and Lemma B.1 ensure that matrix is symmetric definite positive. Thus, like previously, the system is equivalent to

By giving , defined by (3.49a)–(3.49d), defined by for all , for all ,

and defined by

we remark that system (3.62) is equivalent to (A.7). Reactions and displacement can be determined thanks to

#### 4. Existence of Uniqueness Results

Thus, as proved in [1], all the systems of Section 3 can be written under the form (A.7) and, according to Proposition A.1 (see Appendix A), have a unique solution. For all systems, Table 1 provides the corresponding integer , function , and matrix . It is easy to prove that is convex proper and lower semi-continuous function on and that is symmetric positive definite.

#### 5. Convergence of Numerical Scheme

All the models examined here can be written under the form (A.7). Based on [1, 12], general writing of the implicit Euler scheme corresponds to

with time step , discretized time , and approximations of the exact solution provided by the numerical scheme. Previous studies [12] ensure that this numerical scheme is convergent with order (systems (3.6), (3.12), (3.29)–(3.32), and (3.29)–(3.39)) or 1 (systems (3.48), (3.53), and (3.62)).

In practice for computation of solutions, three cases can be distinguished, based on further expression of :

where is the identity and is the inverse of the graph (see [11]). According to [11], is a monovalued operator, providing a unique solution . In the first case, effective computations of associated with diagonal matrix is explicit: this situation corresponds to systems (3.6) and (3.12). In the second case, is defined as the indicatrix function of a closed convex set: this situation corresponds to systems (3.48), (3.53), and (3.62). Effective computation of is given by the projection of a given vector on a closed convex set (see [6]). In the third case (for systems (3.29)–(3.32) and (3.29)–(3.39)), is involving indicatrix function of a closed convex set and a norm function. In such case, computation of leads to the following problem: according to [11], is the solution of minimization problem: considering the norm define by the inner product given by (A.4)

solve

and such problem can be solved in practice following efficient algorithms [13].

#### 6. Conclusion

In this paper, a mechanical system involving finite degrees of freedom and nonsmooth terms have been investigated from the mechanical point of view. Dynamical, semi-dynamical, and quasistatic modeling have been established. The main results are theoretical ones:

(i)all the problems are well posed;(ii)it has been explained how a numerical approximation of solutions can be effectively computed.All the mechanical systems have been considered in a deterministic frame. Theoretical results and corresponding effective computations could be extended to the stochastic frame.

#### Appendices

#### A. A Few Theoretical Reminders about the Class of Maximal Monotone Differential Equations Used

The reader is referred to [11]. Let be scalar product on . If is a convex proper and lower semi-continuous function from to , we can define its subdifferential by

Moreover, is a maximal monotone graph in .

If is a closed convex nonempty subset of , we denote by the indicatrix of defined by

In this particular case, , which is the subdifferential of , is given by

The domain of the maximal monotone operator is equal to .

We observe that if is equipped with its canonical scalar product , and with another scalar product,

where is symmetric positive definite, then we can relate the subdifferential of relatively to the canonical scalar product and the subdifferential relatively to by

We give now the general mathematical formulation of our problem. We assume that is strictly positive and that is a function from to which is Lipschitz continuous with respect to its second argument, that is, there exists such that

Moreover, we assume that

Proposition A.1. *If the matrix is symmetric positive definite and is convex proper and lower semicontinuous on , under assumptions (A.6a)-(A.6b), for all , there exists a unique function in such that
**
where the differential inclusion can be written as an inequality: for almost every in ,
*

Proof of this result can be found in [1, Proposition ], based on [11, Proposition , page 107] and (A.5).

#### B. Defined by (3.4) Is Symmetric Definite Positive

Lemma B.1. *Under assumption
**
matrix defined by (3.4) is symmetric definite positive.*

*Proof. *We have, for all ,
Under assumption (B.1), then implies .