Analytical bounds for damage induced planar anisotropy

The damage of a planar, elastic and initially isotropic material is considered in the framework of a classical approach where the damaged elasticity tensor is ruled by a fourth-rank symmetric damage tensor. The analysis is completely carried on using the so-called polar method for the invariant representation of tensors in 2 . The final elastic behavior, induced by damage, can be anisotropic: all the possible situations of elastic symmetries are considered, and for each one an analytical expression for the bounds on the invariants of the damaged elastic tensor and of the damage tensor is given. An admissible domain for the damage invariants and for the damaged elastic invariants is so provided, the convexity of these domains is also proved

Minimal functional bases for elasticity tensor symmetry classes

Functional bases, synonymous with separating sets, are usually formulated for an entire vector space, such as the space Ela of elasticity tensors. We propose here to define functional bases limited to symmetry strata, i.e. sets of tensors of the same symmetry class. We provide such lowcardinal minimal bases for tetragonal, trigonal, cubic or transversely isotropic symmetry strata of the elasticity tensor

Optimisation de structures non résistantes à la traction: application à l'étude des arcs-boutants.

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Compliance optimization with frictionless unilateral contact

We consider the problem of topological optimization of a structure with frictionless unilateral contact. The boundary conditions (zero displacements, loads) and the geometry of the contact surfaces being known, we consider the problem of minimization of the compliance of the structure in order to maximize its global rigidity. Within the assumption of small strains and small displacements, we assume that the available distributed material has a behavior law deriving from a degree 2 positively homogeneous thermodynamical potential. The contact is modeled as an interface on which the tangential component of the stress vector is imposed to zero on both surfaces and with a behavior law relating the normal component of the stress vector to the normal displacement jump with a high/small rigidity in the respective cases of effective contact/noncontact. The contact behavior law is chosen to derive from a degree 2 positively homogeneous potential. The variationnal formulation and the complementary energy theorem are written. With the assumption of the degree 2 homogeneity for each potential, the compliance is equal to the double of the complementary energy. The minimization of the sum of the compliance and a cost term is then put into the form of a double minimization and solved by successive FEM calculations and local minimizations with respect to the design parameters. Numerical examples with the SIMP approach are presented. 2. Keywords: Compliance, topology optimization, unilateral contact, homogeneous potential, nonlinear behavior law

Stiffness optimization in nonlinear pantographic structures

Mechanical metamaterials are microstructured mechanical systems showing an overall macroscopic behaviour that depends mainly on their microgeometry and microconstitutive properties. Moreover, their exotic properties are very often extremely sensitive to small variations of mechanical and geometrical properties in their microstructure. Clearly, the methods of structural optimization, once combined with the techniques used to describe multiscale systems, are expected to determine a dramatic improvement in the quality of newly designed metamaterials. In this paper, we consider, only as a demonstrative example, planar pantographic structures which have proved to be extremely tough in extension, To describe pantographic structure behaviour in an efficient way, it has been proposed to use Piola–Hencky-type Lagrangian models, in which the understanding of the mechanics of involved microdeformation processes allows for the formulation of efficient numerical codes. In this paper, we prove that it is possible, via a suitable choice of the macroscopic shear stiffness, to increase the maximal elongation of pantographic structures, in the standard bias test, before the occurrence of rupture phenomena. The basic tool employed to this aim is a constrained optimization algorithm, which uses the numerical tool, previously developed for determining equilibrium shapes, as a subroutine. Actually, one looks for the shear stiffness distribution, which, given the imposed elongation of the pantographic structure and the force applied to it by the used hard device, minimizes the total elongation energy. The so-optimized shear stiffness distribution does prove able to extend the range of imposed elongations that the specimen can experience while remaining undamaged

Analytical strain localization analysis of isotropic and anisotropic damage models for quasi-brittle materials

Strain and damage localization are usually precursors of rupture. We present a three-dimensional method dedicated to quasi-brittle materials based on the works of Bigoni & Hueckel, and Jirásek and coworkers, aiming at simplifying the analysis of the localization properties of continuous damage models of a general form, and possibly anisotropic. The method reformulates the localization problem as a two-variable polynomial maximization problem, a strategy commonly used in softening plasticity models, but not so much in Continuum Damage mechanics. The quasi-brittle hypothesis is exploited to render the problem solvable in a fully analytical way, and a post-analysis criterion for the validity of the analysis is also exhibited. In this work, the method is fully established from a theoretical viewpoint, and examples illustrating its use are provided. Multiaxial calculations are performed for four continuous damage models (two isotropic and two anisotropic ones). The method applies to induced anisotropy and constitutive models representing isotropic linear elasticity before damage growth, and remains accurate when models display immediate softening after the elastic limit (and thus to multiaxial tensile cases). The analytical method is, however, entirely general and allows for the calculation of (i) the orientation of a potential localization plane, (ii) the mode angle of the weak discontinuity, and (iii) the validity domain of such a simplified analysis