Danielle Fortuné

Incremental behaviour of hyperelastic dielectrics and piezoelectric crystals

Abstract. The equations describing the behaviour of a hyperelastic dielectric under pre-existing mechanical and electrical fields are derived. The associated differential system is shown to be self-adjoint. This property, in turn, is used to establish the equivalence of linear static and dynamic stability criteria.

Nonlinear isotropic constitutive laws : Choice of the three invariants, convex potentials and constitutive inequalities

The aim of this paper is a new formulation of nonlinear isotropic constitutive laws. Our main hypothesis claims that the eigenvalues of stress and strain tensors are classified in the same order (the eigenvector associated to the highest eigenvalue of the stress tensor is also associated to the highest eigenvalue of the strain tensor, etc.). Further, we assume the existence of a differentiable convex isotropic potential. By introducing three new invariants for each tensor (called X, Y, Z for the stress tensor S and x, y, z for the strain tensor E) a constitutive law is revealed to be a simple duality between the chosen invariants: (x, y, z) and (X, Y, Z) look like Cartesian coordinates of E and S. We look at several potentials chosen as polynomials of these invariants. Finally, first and third order isotropic elasticity laws are reviewed and convexity of the potentials is discussed.

Bianchi identities in the case of large deformations

Explicit relations are given for the conditions satisfied by the tensors involved in the compatibility equations of three-dimensional continua undergoing large deformations. These relations are Bianchi identities: they reflect the dependence between compatibility equations drawn up by performing the polar decomposition of the transformation gradient.

Torsion Equation in Anisotropic Elasto-Plastic Materials with Continuously Distributed Dislocations

Within the constitutive framework of finite elasto-plasticity we formulate compatibility conditions in a general problem, in which no potentiality condition for an invertible second-order tensor field F has been assumed, due to the signification of F to be a plastic distortion. In the problem the affine connection F, with vanishing Riemann curvature, is thought to be a plastic connection. Consequently, in order to ensure the existence of the continuously distributed dislocation, the Cartan torsion attached to the connection is supposed to be non-zero. The principal result concerns the compatibility conditions, which are viewed, for a given symmetric and positive definite tensor (the metric tensor), as partial differential equations for the torsion (defined in terms of the second-order torsion tensor). In our problem, the non-zero torsion is essential, while in finite elasticity the compatibility conditions are formulated in terms of zero torsion. The following implications of the theorem relative to the torsion, concerning the evolution equations for the pair of the plastic distortion and plastic connections, can be drawn: 1. only the evolution equation for the plastic metric has to be defined, because the torsion could be defined as a solution of the appropriate partial differential equations, within the approach to finite elastoplasticity with plastic connection having a zero fourth-order curvature tensor, 2. if no relationships between the plastic metric and (plastic) torsion have been imposed then the evolution equations could be defined for plastic distortion as well as for plastic connection.

Pontryagin Calculus in Riemannian Geometry

In this contribution, we study systems with a finite number of degrees of freedom as in robotics. A key idea is to consider the mass tensor associated to the kinetic energy as a metric in a Riemannian configuration space. We apply Pontryagin’s framework to derive an optimal evolution of the control forces and torques applied to the mechanical system. This equation under covariant form uses explicitly the Riemann curvature tensor.

SUBDIFFERENTIAL OF THE LARGEST EIGENVALUE OF A SYMMETRICAL MATRIX APPLICATION OF DIRECT PROJECTION METHODS

In many physical circumstances, for example, in studying the linear vibrations of a mechanical or acoustical system, a key tool is to determine numerically the components of the eigenvectors associated with the largest eigenvalue of a symmetrical matrix with real coefficients. To find out the largest eigenvalue λ1(S) of such a symmetrical n × n matrix S, the well-known Rayleighs method consists in maximizing the quotient (VTSV)/(VTV) among all the nonvanishing vectors V of ℝn. When the eigenvalue λ1(S) is simple, the maximum is attained for vectors V colinear to a unit eigenvector N, and the function λ1 is differentiable with the projector NNT over the direction N as a gradient. When the largest eigenvalue is not simple, the function λ1 is no longer differentiable; it remains convex, but the subdifferential ∂λ1(S) is not reduced to a single gradient. This paper is devoted to determine the subgradients therein ∂λ1(S) by direct methods that do not require the preliminary determination of λ1(S).

Modelling of Sliding Wave Phenomenon, on the Contact Boundary between Two Bodies, by the Boundary Integral Element Method: Numerical Visualization of Isochroms

The aim of the present numerical study is to model a phenomenon of propagating stress waves on the contact boundary between two bodies. The photoelasticimetry studies achieved by PROGRI R, VILLECHAISE B (1984,1984), MOUWAKEH M (1989), ZEGHLOUL T (1992) show this phenomenon. In particular, it is displayed that the global sliding is due to the crossing of a localised perturbation of the isochroms field over all the contact zone. The tools of this numerical modelling are: an incremental formulation of the evolution problem, the boundary integral element method to solve the problem, the Coulomb friction law or non linear law, non local law with a variable coefficient of friction.