Generic separating sets for 3D elasticity tensors

We define what is a generic separating set of invariant functions (a.k.a. a weak functional basis) for tensors. We produce then two generic separating sets of polynomial invariants for 3D elasticity tensors, one made of 19 polynomials and one made of 21 polynomials (but easier to compute) and a generic separating set of 18 rational invariants. As a byproduct, a new integrity basis for the fourth-order harmonic tensor is provided

Reduced algebraic conditions for plane/axial tensorial symmetries

In this article, we formulate necessary and sufficient polynomial equations for the existence of a symmetry plane or an order-two axial symmetry for a totally symmetric tensor of order n ≥ 1. These conditions are effective and of degree n (the tensor's order) in the components of the normal to the plane (or the direction of the axial symmetry). These results are then extended to obtain necessary and sufficient polynomial conditions for the existence of such symmetries for an Elasticity tensor, a Piezo-electricity tensor or a Piezo-magnetism pseudo-tensor

Computation of minimal covariants bases for 2D coupled constitutive laws

We produce minimal integrity bases for both isotropic and hemitropic invariant algebras (and more generally covariant algebras) of most common bidimensional constitutive tensors and -- possibly coupled -- laws, including piezoelectricity law, photoelasticity, Eshelby and elasticity tensors, complex viscoelasticity tensor, Hill elasto-plasticity, and (totally symmetric) fabric tensors up to twelfth-order. The concept of covariant, which extends that of invariant is explained and motivated. It appears to be much more useful for applications. All the tools required to obtain these results are explained in detail and a cleaning algorithm is formulated to achieve minimality in the isotropic case. The invariants and covariants are first expressed in complex forms and then in tensorial forms, thanks to explicit translation formulas which are provided. The proposed approach also applies to any $n$-uplet of bidimensional constitutive tensors

Modeling microdefects closure effect with isotropic/anisotropic damage

International audienceContinuum damage mechanics (CDM) for metals is often written in terms of an isotropic (scalar) damage. In this case, solutions have been proposed to represent the differences of behavior in tension and in compression also called quasi-unilateral (QU) conditions or microdefects closure effect. A recent anisotropic damage model has been developed to take into account the damage orthotropy induced by plasticity (Lemaitre, J., Demorat R. and Sauzay, M. (2000). Anisotropic Damage Law of Evolution, Eur. J. Mech. A/Solids, 19: 513--524). The purposes here are then two. First, a unified framework for isotropic and anisotropic damage is proposed. Then, it is to extend Ladevèze and Lemaitre's framework (Ladevèze, P. and Lemaitre, J. (1984). Damage Effective Stress in Quasi Unilateral Conditions, In: Proceedings of the 16th International Congress of Theoretical and Applied Mechanics, Lyngby, Denmark) for the QU conditions to anisotropic damage induced by plasticity. Yield surfaces and damage versus accumulated plastic strain curves, drawn for different loading, illustrate the effect of the QU conditions on the damage evolution

Souriau's Relativistic general covariant formulation of hyperelasticity revisited

We present and modernize Souriau's 1958 geometric framework for Relativistic continuous media, and enlighten the necessary and the ad hoc modeling choices made since, focusing as much as possible on the Continuum Mechanics point of view. We describe the general covariant formulation of Hyperelasticity in General Relativity, and then in the particular case of a static spacetime. Finally, we apply this formalism for the Schwarzschild's metric, and recover the Classical Galilean Hyperelasticity with gravity, as the Newton-Cartan infinite light speed limit of this formulation