J. A. C. Martins

Instability and ill-posedness in some friction problems

The steady frictional sliding of an elastic half-space in contact with a rigid flat surface is considered. Perturbed solutions in the elastic body are sought which correspond to surface (flutter) instabilities of the steady sliding state. The behavior of these surface solutions for vanishing small values of their arbitrary wavelengths is made clear. The effect of introducing intrinsic length scales in the problem by means of the earlier proposed non-local friction law and normal compliance law or by means of viscous dissipation in the deformable body is discussed.

Dissipative graph solutions for a 2 degree-of-freedom quasistatic frictional contact problem

Abstract In this paper we discuss the nature of the quasitatic purely elastic limit to the dynamic viscoelastic solutions to a 2 degree-of-freedom (d.f.) frictional contact problem. In a significant situation for which a continuous equilibrium path cannot exist in the limit, we show that, when the mass and viscosity coefficients are decreased to zero in an appropriate manner, a connected graph is approached in the 3-dimensional space of the displacement components plus the time (load parameter) variable. This graph solution contains an instantaneous portion, a displacement discontinuity with respect to time, along which the total energy dissipated into some external sink is non-negative. These (possibly oscillatory and non-equilibrated) instantaneous paths and some of their qualitative features are discussed and compared with those resulting from quasistatic viscoelastic approximation procedures. For a particular class of graph solutions which may be approached by sequences of either dynamic or quasistatic viscoelastic solutions, we show that a near future evolution of the system does always exist and that direction of that evolution for the initiation or the continuation of an instantaneous path is uniquely defined.

Bifurcations and Instabilities in Some Finite Dimensional Frictional Contact Problems

The present work is part of a research effort devoted to the study of bifurcation and instability phenomena in frictional contact problems. Several situations have been considered in these studies: (i) the occurrence of bifurcations in quasi-static paths; this is a stiffness and friction induced phenomenon of non-uniqueness of quasi-static solutions; (ii) the initiation of dynamic solutions at equilibrium positions, with no initial perturbations, but with initial acceleration and reaction discontinuities; this is a mass and friction induced phenomenon of non-uniqueness of dynamic solutions; (iii) the existence of smooth non-oscillatory growing dynamic solutions with perturbed initial conditions arbitrarily close to equilibrium configurations, i.e. the divergence instability of equilibrium states; (iv) the existence of non-oscillatory or oscillatory growing dynamic solutions with perturbed initial conditions arbitrarily close to portions of quasi-static paths; (v) the occurrence of non-oscillatory (divergence) or oscillatory (flutter) instabilities of steady sliding equilibrium states.

(In)stability of quasi-static paths of some finite dimensional smooth or elastic-plastic systems

In this paper we discuss some mathematical issues related to the stability of quasistatic paths of finite dimensional mechanical systems that have a smooth or an elastic-plastic behavior. The concept of stability of quasi-static paths used here is essentially a continuity property relatively to the size of the initial perturbations (as in Lyapunov stability) and to the smallness of the rate of application of the external forces (which here plays the role of the small parameter in singular perturbation problems). A related concept of attractiveness is also proposed. Sufficient conditions for attractiveness or for instability of quasi-static paths of smooth systems are presented. The Ziegler column and other examples illustrate these situations. Mathematical formulations (plus existence and uniqueness results) for dynamic and quasi-static elastic-plastic problems with linear hardening are recalled. A stability result is proved for the quasi-static evolution of these systems.

Friction and Instabilities: Stress Waves in a Sliding Contact

In their accurate experimental works, Villechaise, Mouwakeh, Zeghloul (1992), have shown the occurrence of isolated fast stress waves in the contact area of a deformable block sliding on a rigid plane. Each wave is coupled with a jump in the total tangential force.

MATHEMATICAL RESULTS ON THE STABILITY OF QUASI-STATIC PATHS OF ELASTIC-PLASTIC SYSTEMS WITH HARDENING

In this paper, existence and uniqueness results for a class of dynamic and quasistatic problems with elastic-plastic systems are recalled, and a stability result is obtained for the quasi-static paths of those systems. The studied elasticplastic systems are continuum 1D (bar) systems that have linear hardening, and the concept of stability of quasi-static paths used here takes into account the existence of fast (dynamic) and slow (quasi-static) times scales in the system. That concept is essentially a continuity property relatively to the size of the initial perturbations (as in Lyapunov stability) and relatively to the smallness of the rate of application of the forces (which plays here the role of the small parameter in singular perturbation problems).

Modeling of Passive Behavior of Soft Tissues Including Viscosity and Damage

The mechanical properties of soft tissues depend strongly on the orientation of their fibers, and usually they have a highly nonlinear behavior: their stiffness increases as they are stretched. We are interested here in the passive behavior of soft tissues, when subjected to significant stretches, possibly leading to damage. In this paper we develop a model for a transversely isotropic material that has a damageable viscoelastic behavior. This model is then used to simulate the damage evolution of the tissue. The model is developed with the underlying framework of hyperelasticity, and the corresponding strain energy has different parts associated to different contributions to the material behavior: volumetric, isotropic, anisotropic and dissipative contributions. Since soft tissues are almost incompressible we use a multiplicative split of the deformation gradient into a volume preserving part and a part with (small) volume changes. The anisotropic behavior is characterized by the existence of a family of fiber directions within the tissue. The viscoelastic behavior associated with the non-equilibrium stress is treated as a standard solid material with M Maxwell elements simulating the fact that the response of soft tissues is almost independent of the loading frequency. The total damage is modeled by splitting the energy degradation into one isotropic part and one anisotropic part. That is, we can have fiber degradation independently of the damage of the surrounding matrix.

Arc-Length Method for Frictional Contact with a Criterion of Maximum Dissipation of Energy

In this paper, we propose an arc-length equilibrium path-following method for quasi-static frictional contact problems incorporating a criterion of maximum dissipation of energy, which is applicable to cases in which there exist critical points along the equilibrium path. The Coulomb friction law and the unilateral contact condition are considered.

On the stability of quasi-static paths of a linearly elastic system with friction

In this paper we discuss the stability of quasi-static paths of a single degree of freedom linearly elastic system with Coulomb friction and known normal force. A common and useful approximation for the equations that govern the slow evolution of many mechanical systems is to neglect inertia effects in the dynamic balance equations, and replace them by static equilibrium equations. Slow evolutions calculated with this approximation are called quasi-static evolutions. The relationship of this issue with the theory of singular perturbations has been established in [1], where the existence of fast (dynamic) and slow (quasi-static) time scales was recognized: a change of variables is performed that replaces the (fast) physical time t by a (slow) loading parameter λ, whose rate of change with respect to time, e = dλ/dt, is decreased to zero. This change of variables leads to a system of dynamic differential equations or inclusions that defines a singular perturbation problem: the small parameter e multiplies some of the highest order derivatives in the system. The concept of stability of quasi-static paths used here is essentially a continuity property relatively to the size of the initial perturbations (as in Lyapunov stability) and to the smallness of the rate of application of the external forces, e (as in singular perturbation problems). This study applies for the first time to a nonsmooth context, the definition of stability of quasi-static paths, recently proposed by Martins et al. ([2], [3]).

Some results on the stability of quasi-static paths of elastic-plastic systems with hardening

In this paper, a concept of stability of quasi-static paths is discussed. This takes into account the existence of fast (dynamic) and slow (quasi-static) times scales in the mechanical systems that have an elastic-plastic behavior with linear hardening. The proposed concept is essentially a continuity property relative to the size of the initial perturbations (as in Lyapunov stability) and relative to the smallness of the rate of application of the forces (which here plays the role of the small parameter in singular perturbation problems). Existence and uniqueness results for the dynamic and quasi-static problems are recalled and the stability of quasi-static paths for elastic-plastic systems with hardening is obtained.