Research Article | Open Access
Theoretical Study of a Chain Sliding on a Fixed Support
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 , 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 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 𝜎
(b) The graph 𝛽
According to , 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:
3. Transformations of Equations
Let us assume that the external forcing are known.
3.1. Dynamical Case
We assume in this section that
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
where is the identity of and for , , with ,
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
Reactions and displacement can be determined thanks to
As in Section 3.1.1, let us set
and for , , with
3.2. Semi-Dynamical Case
In this section, we assume that
Equation (3.2b) implies
where is the convex indicatrix function of the convex domain . Thus, (3.17a) is equivalent to
Similarly, (3.2a) gives
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
Under the assumption
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
Reactions and can be determined thanks to
then we can calculate thanks to (3.27). Successively, , , , and are defined by
3.2.2. Clamped-Free Mechanical System
The reaction and the displacement can be determined thanks to (3.36a) and
3.3. Quasistatic Case
In this section, we assume that
3.3.1. Clamped Mechanical System
Thus, we have Under assumption
the matrix is symmetric definite positive (see proof in Lemma B.1), so that
Reactions and can be determined thanks to
3.3.2. Clamped-Free Mechanical System
(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
According to previous remark, assumption
and Lemma B.1 ensure that matrix is symmetric definite positive. Thus, like previously, the system is equivalent to
and defined by
4. Existence of Uniqueness Results
Thus, as proved in , 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
with time step , discretized time , and approximations of the exact solution provided by the numerical scheme. Previous studies  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 ). According to , 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 ). 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 , is the solution of minimization problem: considering the norm define by the inner product given by (A.4)
and such problem can be solved in practice following efficient algorithms .
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.
A. A Few Theoretical Reminders about the Class of Maximal Monotone Differential Equations Used
The reader is referred to . 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 ,
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 .
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Copyright © 2009 Jérôme Bastien and Claude-Henri Lamarque. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.