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.

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 [

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

We refer to previous works for description of some rheological models (see for example [

We consider the model of Figure

The reader is referred to Appendix

Let

Two useful multivalued maximal monotone graphs.

The graph

The graph

The studied model with mechanical components.

Let us assume (see Figure

The studied model with external forces

This mechanical system is submitted to external forces

For

For

For

For

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:

We can observe that (

Now, we study systems (

Now, as in [

Let us assume that the external forcing

We assume in this section that

Equations (

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

We set, for all

Set

Reciprocally, if (

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

As in Section

As in Section

In this section, we assume that

Equation (

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

Reactions

Reciprocally, as in Section

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 (

The calculus are similar to those of Section

In this section, we assume that

As it has been previously noticed, (

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

As in Section

Thus, we have

Reactions

We assume that our mechanical system is clamped at its left extremity so that we can write the boundary condition (

The calculus is similar to those of [

We assume that the displacement

We introduce

Reactions

We assume that external forcing

The calculus are similar to the previous case.

Equation (

So, (

As in [

According to previous remark, assumption

Thus, as proved in [

The dimension of the system, the convex function and the symmetric positive definite matrix used for the above described mechanical models.

System | function | matrix | |

( | |||

( | |||

( | |||

( | |||

( | |||

( | |||

( |

All the models examined here can be written under the form (

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

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:

all the problems are well posed;

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.

The reader is referred to [

If

We observe that if

We give now the general mathematical formulation of our problem. We assume that

If the matrix

Proof of this result can be found in [

Under assumption

We have, for all