Optimal shape design of structural elements based on boundary variations results in final designs that are topologically equivalent to the initial choice of design, and general, stable computational schemes for this approach often require some kind of remeshing of the finite element approximation of the analysis problem. This paper presents a methodology for optimal shape design where both these drawbacks can be avoided. The method is related to modern production techniques and consists of computing the optimal distribution in space of an anisotropic material that is constructed by introducing an infimum of periodically distributed small holes in a given homogeneous, i~otropic material, with the requirement that the resulting structure can carry the given loads as well as satisfy other design requirements. The computation of effective material properties for the anisotropic material is carried out using the method of homogenization. Computational results are presented and compared with results obtained by boundary variations.

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Generating optimal topologies in structural design using a homogenization method

Micro-structure in linear elasticity

The Classical Field Theories

Matter is commonly found in the form of materials. Analytical mechanics turned its back upon this fact, creating the centrally useful but abstract concepts of the mass point and the rigid body, in which matter manifests itself only through its inertia, independent of its constitution; “modern” physics likewise turns its back, since it concerns solely the small particles of matter, declining to face the problem of how a specimen made up of such particles will behave in the typical circumstances in which we meet it. Materials, however, continue to furnish the masses of matter we see and use from day to day: air, water, earth, flesh, wood, stone, steel, concrete, glass, rubber, ... All are deformable. A theory aiming to describe their mechanical behavior must take heed of their deformability and represent the definite principles it obeys.

The non-linear field theories of mechanics

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Elastic materials with couple-stresses R. Toupin

https://hal.archives-ouvertes.fr/hal-00852443/document

Elastic materials with couple-stresses

Nous avons derive trois solutions des equations de l'elasticite non lineaire. Nous avons determine les deformations qui sont possibles dans un solide isotrope et incompressible quelconque.

Deformations possible in every isotropic, incompressible, perfectly elastic body

Generalized Plate Models and Optimal Design.- The Effective Dielectric Coefficient of a Composite Medium: Rigorous Bounds From Analytic Properties.- Variational Bounds on Darcy's Constant.- Micromodeling of Void Growth and Collapse.- On Bounding the Effective Conductivity of Anisotropic Composites.- Thin Plates with Rapidly Varying Thickness, and Their Relation to Structural Optimization.- Modelling the Properties of Composites by Laminates.- Waves in Bubbly Liquids.- Some Examples of Crinkles.- Microstructures and Physical Properties of Composites.- Remarks on Homogenization.- Variational Estimates for the Overall Response of an Inhomogeneous Nonlinear Dielectric.- Information About Other Volumes in this Program.

Homogenization and effective moduli of materials and media

Some phase transitions in crystals

: Commonly, hypotheses introduced by Cauchy or Born are used to relate macroscopic deformation to atomic motions, in molecular theories of elasticity. Our purpose is to discuss the applicability of these to crystal-crystal phase transformations and the ambiguities which are involved in estimating the deformation from observations of lattice vectors. (Author)

The Cauchy and Born Hypotheses for Crystals.

Part 1 Generalities: energy, heat and power temperature and entropy thermodynamic equilibrium. Part 2 Constitutive theory of heat transfer for bars and plates: thermodynamics of rigid bars constitutive theory for rigid bars constitutive theory for thermoelastic bars constitutive theory for shearing of plates thermodynamic experiments. Part 3 Equilibrium theory of bars: equilibrium of bars subjected to dead loads equilibrium theory of bars and hard devices. Part 4 Equilibrium theory of plates: martensic transformations bifurcation diagrams. Part 5 Balloon problems: equilibrium of spherical balloons at fixed pressure equilibrium of balloons containing a fixed mass of gas. Part 6 Biaxial stretch of rubber sheets: the idealized problem the Treolar instability. Part 7 Moving discontinuities: shock waves in bars breaking bars a peeling problem another peeling problem. Part 8 Mixture theory: general remarks elementary theory a solid in an ideal gas. Part 9 Equilibrium of liquid crystals and rods: liquid crystal energies orientation by fields and walls measurement of moduli elastica theory. Part 10 Reconsideration of generalities: systems, energy and temperature general processes static and reversible processes cyclic processes and cycles states laws of classical thermodynamics.

J. L. Ericksen

Papers

Deformations possible in every isotropic, incompressible, perfectly elastic body

Homogenization and effective moduli of materials and media

Some phase transitions in crystals

The Cauchy and Born Hypotheses for Crystals.

Introduction to the thermodynamics of solids