Second order tensorial framework for 2D medium with open and closed cracks

International audienceThe tensorial nature of crack density of an initially isotropic 2D medium with open and closed cracks is studied by means of polar decomposition rewriting of standard micro-mechanics results. The question of both indicial and constitutive symmetries of different crack density tensors is addressed: for instance the standard fourth order crack density tensor is rari-constant (totally symmetric) and the fourth order closed cracks density tensor by which closed cracks are acting is found to have the square symmetry. The effect of cracks closure and sliding is accordingly shown to be represented by a second order tensor so that only two second order crack density tensors, are needed for 2D medium with open and closed sliding cracks. Similarly to the open cracks case, any arbitrary closed crack system is shown to be represented by only two non orthogonal families of cracks. The question of macroscopic cracks closure conditions is finally studied. Present study leads to an approximate framework in which the only internal variable representative of physical cracks, open and closed, is second order cracks density tensor. Proposed second order tensorial framework is shown to be exact in the case of two orthogonal arrays of cracks, open and/or closed, it is approximate in the general case of many arrays of cracks, open and/or closed

Structural rigidity optimization with an initial design dependent stress field. Application to thermo-elastic stress loads

International audienceThis paper presents a compliance optimization methodology considering an initial stress field. Assuming a local dependency of the initial stress with the optimization parameters, the obtained numerical procedure is composed of finite element stress calculations and local minimization problems that are solved analytically or using a simple bisection procedure. This optimization algorithm is then found to be numerically very efficient. Two optimization problems that fall in the scope of the proposed methodology are considered: topology optimization of the isotropic mixture of two isotropic linear elastic materials (one of the material possibly being void) and orientational optimization (i.e. distribution of anisotropy) with an orthotropic linear elastic material. Numerical examples illustrating the optimization methodology in the case of thermo-elastic stress loads are presented

Structural rigidity optimization with frictionless unilateral contact

AbstractThe problem of maximization the global rigidity (measured by the compliance) of an elastic structure with frictionless unilateral contact is considered in the framework of topology optimization. The frictionless unilateral contact is introduced in the continuous formulation of the elastic problem (under the assumption of small strains and small displacements) in the regularized form of an interface with an asymmetric behavior law relating the normal component of the stress vector transmitted through the contact surface to the normal displacement (in the case of contact with a rigid foundation) or the jump of normal displacement (in the case of internal contact of two surfaces of the elastic medium). Using the concept of homogeneous thermodynamical potentials, we extend a convergent and numerically efficient optimization algorithm introduced in the framework of linear elasticity to this nonlinear case of an elastic structure with unilateral contact. Numerical examples in two-dimensional elasticity are presented

Reduced algebraic conditions for plane or 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 [Formula: see text]. These conditions are effective and of degree [Formula: see text] (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. </jats:p

Generic separating sets for three-dimensional elasticity tensors

We define a generic separating set of invariant functions (a.k.a. a weak functional basis ) for tensors. We then produce two generic separating sets of polynomial invariants for three-dimensional elasticity tensors, one consisting of 19 polynomials and one consisting of 21 polynomials (but easier to compute), and a generic separating set of 18 rational invariants. As a by-product, a new integrity basis for the fourth-order harmonic tensor is provided. </jats:p