Michel Saint Jean

The non-smooth contact dynamics method

The main features of the Non-Smooth Contact Dynamics method are presented in this paper, the use of the dynamical equation, the non-smooth modelling of unilateral contact and Coulombs law, fully implicit algorithms to solve the dynamical frictional contact problem for systems with numerous contacting points, in particular large collections of rigid or deformable bodies. Emphasis is put on contact between deformable bodies. Illustrating numerical simulation examples are given for granular materials, deep drawing and buildings made of stone blocks.

Consistent time discretization for a dynamical frictional contact problem and complementarity techniques

Abstract.The dynamical problem of a viscoelastic body involving unilateral contact and Coulomb friction is set so as to take into account accurately eventual discontinuities of the relative velocit...

A visco-hyperelastic model with damage for the knee ligaments under dynamic constraints.

The aim of this study was to identify the behaviour laws governing the knee ligaments, accounting for the damage incurred by the structure under dynamic constraints. The model is developed using a thermodynamic formulation based on the coupling between a viscoelastic model and a damage model. Identification is carried out using the results of dynamic traction tests performed on a bone ligament/bone complex to which traction velocities of around 1.98 m/s were applied. The results show the ability of the model to account for the brittle and ductile failure processes occurring in the cruciate and lateral ligaments, respectively.

Image charges revisited: Beyond classical electrostatics

The method of images in electrostatics seems to lead to a paradox: The charges induced on the surface of a grounded conductor by an external point charge do not spread out although the electrostatic field is normal to the surface. This effect can be understood if one takes into account the spreading of the charges inside the conductor due to the kinetic energy (either thermal or quantum mechanical) of the electrons. Corrections to the usual electrostatics of images are then calculated (in particular, it is shown that the conductor surface is not an exact equipotential) when the distance of the point charge to the surface is much larger than the screening length. Applications are discussed.

Interaction, coalescence, and collapse of localized patterns in a quasi-one-dimensional system of interacting particles

We study the path toward equilibrium of pairs of solitary wave envelopes (bubbles) that modulate a regular zigzag pattern in an annular channel. We evidence that bubble pairs are metastable states, which spontaneously evolve toward a stable single bubble. We exhibit the concept of topological frustration of a bubble pair. A configuration is frustrated when the particles between the two bubbles are not organized in a modulated staggered row. For a nonfrustrated (NF) bubble pair configuration, the bubbles interaction is attractive, whereas it is repulsive for a frustrated (F) configuration. We describe a model of interacting solitary wave that provides all qualitative characteristics of the interaction force: It is attractive for NF systems and repulsive for F systems and decreases exponentially with the bubbles distance. Moreover, for NF systems, the bubbles come closer and eventually merge as a single bubble, in a coalescence process. We also evidence a collapse process, in which one bubble shrinks in favor of the other one, overcoming an energetic barrier in phase space. This process is relevant for both NF systems and F systems. In NF systems, the coalescence prevails at low temperature, whereas thermally activated jumps make the collapse prevail at high temperature. In F systems, the path toward equilibrium involves a collapse process regardless of the temperature.

Stability, diffusion and interactions of nonlinear excitations in a many body system

When repelling particles are confined in a quasi-one-dimensional trap by a transverse potential, a configurational phase transition happens. All particles are aligned along the trap axis at large confinement, but below a critical transverse confinement they adopt a staggered row configuration (zigzag phase). This zigzag transition is a subcritical pitchfork bifurcation in extended systems and in systems with cyclic boundary conditions in the longitudinal direction. Among many evidences, phase coexistence is exhibited by localized nonlinear patterns made of a zigzag phase embedded in otherwise aligned particles. We give the normal form at the bifurcation and we show that these patterns can be described as solitary wave envelopes that we call bubbles. They are stable in a large temperature range and can diffuse as quasi-particles, with a diffusion coefficient that may be deduced from the normal form. The potential energy of a bubble is found to be lower than that of the homogeneous bifurcated phase, which explains their stability. We observe also metastable states, that are pairs of solitary wave envelopes which spontaneously evolve toward a stable single bubble. We evidence a strong effect of the discreteness of the underlying particles system and introduce the concept of topological frustration of a bubble pair. A configuration is frustrated when the particles between the two bubbles are not organized in a modulated staggered row. For a nonfrustrated (NF) bubble pair configuration, the bubbles interaction is attractive so that the bubbles come closer and eventually merge as a single bubble. In contrast, the bubbles interaction is found to be repulsive for a frustrated (F) configuration. We describe a model of interacting solitary wave that provides all qualitative characteristics of the interaction force: it is attractive for NF-systems, repulsive for F-systems, and decreases exponentially with the bubbles distance.

Hysteretic and intermittent regimes in the subcritical bifurcation of a quasi-one-dimensional system of interacting particles.

In this article, we study the effects of white Gaussian additive thermal noise on a subcritical pitchfork bifurcation. We consider a quasi-one-dimensional system of particles that are transversally confined, with short-range (non-Coulombic) interactions and periodic boundary conditions in the longitudinal direction. In such systems, there is a structural transition from a linear order to a staggered row, called the zigzag transition. There is a finite range of transverse confinement stiffnesses for which the stable configuration at zero temperature is a localized zigzag pattern surrounded by aligned particles, which evidences the subcriticality of the bifurcation. We show that these configurations remain stable for a wide temperature range. At zero temperature, the transition between a straight line and such localized zigzag patterns is hysteretic. We have studied the influence of thermal noise on the hysteresis loop. Its description is more difficult than at T=0 K since thermally activated jumps between the two configurations always occur and the system cannot stay forever in a unique metastable state. Two different regimes have to be considered according to the temperature value with respect to a critical temperature T_{c}(τ_{obs}) that depends on the observation time τ_{obs}. An hysteresis loop is still observed at low temperature, with a width that decreases as the temperature increases toward T_{c}(τ_{obs}). In contrast, for T>T_{c}(τ_{obs}) the memory of the initial condition is lost by stochastic jumps between the configurations. The study of the mean residence times in each configuration gives a unique opportunity to precisely determine the barrier height that separates the two configurations, without knowing the complete energy landscape of this many-body system. We also show how to reconstruct the hysteresis loop that would exist at T=0 K from high-temperature simulations.

Stability in Unilateral Contact Problems with Dry Friction

We discuss stability in the case of systems with unilateral contact and Coulomb friction. Classical stability results for dynamical systems concern perturbations of the initial data in a classical phase space. Here we establish results concerning the trajectories issued from a perturbation of the external forces. With such a notion of stability a conjecture is given that we back up in detail by analytical computations in the case of a simple model and that we begin to extend to more complex systems.

Constitutive Laws and Failure Models for Compact Bones Subjected to Dynamic Loading

Many biological tissues, such as bones and ligaments, are fibrous. The geometrical structure of these tissues shows that they exhibit a similar hierarchy in their ultra- and macro-structures. The aim of this work is to develop a model to study the failure of fibrous structures subjected to dynamic loading. The important feature of this model is that it describes failure in terms of the loss of cohesion between fibres. We have developed a model based on the lamellar structure of compact bone with fibres oriented at 0, 45 and 90° to the longitudinal axis of the bone and have studied the influence of the model parameters on the failure process. Bone porosity and joint stress force at failure were found to be the most significant parameters. Using least square resolution, we deduced a phenomenological model of the lamellar structure. Finally, experimental results were found to be comparable with our numerical model.