Boris Desmorat

Compliance optimization with nonlinear elastic materials: Application to constitutive laws dissymmetric in tension–compression

Abstract The compliance as a measure of the global rigidity of a structure is classically used in optimization in the case of linear elasticity. Under the assumption of small strains and small displacements, the aim of the paper is to extend a classical compliance optimization algorithm to nonlinear elastic materials with proportional dual thermodynamics potentials. A systematic method to prove the dual potentials proportionality is developed. It is used for different constitutive laws dissymmetric in tension–compression, and especially in the case of long fiber composite membranes.

Stiffness and Strength Optimization of the Anisotropy Distribution for Laminated Structures

In this paper, we deal with the problem of optimizing the anisotropy distribution of a laminated structure in order to maximize, simultaneously, its stiffness and strength. For these two objectives, two functionals are considered: the compliance as a measure of the plate stiffness and a laminate-level failure index as a measure of the strength. To solve this optimization problem we used a two-step hierarchical strategy: in the first step the aim is to find the best distribution of the stiffness and strength anisotropic tensors of an equivalent homogenized plate, while in the second one the objective is to find at least one laminate lay-up satisfying the optimal properties obtained as result of the first step. The polar formalism has been used to represent both the stiffness and strength tensors and allowed us to find analytical solutions to the local minimizations of the two functionals. A test case is finally presented to show the effectiveness of the proposed strategy.

Harmonic factorization and reconstruction of the elasticity tensor

In this paper, we study anisotropic Hooke’s tensor: we propose a factorization of its fourth-order harmonic part into second-order tensors. We obtain moreover explicit equivariant reconstruction formulas, using second-order covariants, for transverse isotropic and orthotropic fourth-order harmonic tensors, and for trigonal and tetragonal fourth-order harmonic tensors up to a cubic fourth order covariant remainder.

Micromechanics based framework with second-order damage tensors

The harmonic product of tensors---leading to the concept of harmonic factorization---has been defined in a previous work (Olive et al, 2017). In the practical case of 3D crack density measurements on thin or thick walled structures, this mathematical tool allows us to factorize the harmonic (irreducible) part of the fourth-order damage tensor as an harmonic square: an exact harmonic square in 2D, an harmonic square over the set of so-called mechanically accessible directions for measurements in the 3D case. The corresponding micro-mechanics framework based on second---instead of fourth---order damage tensors is derived. An illustrating example is provided showing how the proposed framework allows for the modeling of the so-called hydrostatic sensitivity up to high damage levels.

Design Problems of Anisotropic Structures: Some Recent Results

We show in this paper a general procedure to tackle the problem of designing anisotropic laminates corresponding to given criteria and optimizing a given objective function. The method is based, on one side, on the use of tensor invariants for the description of the anisotropic properties and, on the other side, on a free-material approach for the determination, first, of the optimal fields of anisotropic properties and, then, using numerical metaheuristics, of a suitable stacking sequence. Some general results concerning the method are introduced, along with a discussion of the influence of anisotropy on the optimal design of laminates. Some different problems are dealt with for showing the effectiveness of the approach.

Topology optimization of elasto-plastic structures in damage governed low cycle fatigue

The aim of the work is to apply topology optimization (i.e. to optimize the distribution of material and void in a fixed design domain) in order to maximize the low cycle fatigue life of a structure made of an elasto-plastic material. Linear kinematic hardening is considered in order to model the fatigue stabilized cycle. With the assumption of proportional loading, the 3D total strain and stress amplitudes are related by a behavior law that derives from a state potential. To increase the fatigue life of a structure, we will minimize this state potential over the entire structure with respect to material and void distribution.The proposed approach allows to minimize globally the defined criterion, but thanks to local minimization steps, it also minimizes the maximal increment of damage over one cycle in the structure. It is then possible with the use of successive optimizations to obtain the minimum volume and the distribution of material that achieves a given number of cycles to crack initiation for the structure.