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

Continuum theory of nematic liquid crystals

Publisher Summary This chapter emphasizes the fact that, mathematically, elasticity theory has much in common with modern theories of thin elastic shells, plates, or rods; static theories of liquid crystals; and various other local theories associated with the most common types of variational principles. This chapter discusses that the interest in elasticity theory lies in improving the theory underlying the somewhat mystical process whereby definite forms of constitutive equations are selected. The domain of constitutive equations may include subdomains that are, in principle, inaccessible to the experimentist. The operationalist holds a contrary view. The chapter highlights that the elasticity theory can predict the effects that are not commonly thought of as being associated with the adjective “elastic.” In such cases, elasticity theory enters into free competition with other theories capable of describing the effect at hand.

Special Topics in Elastostatics

Nilpotent energies in liquid crystal theory

Constitutive theory for some constrained elastic crystals

In recent years, considerable effort has been expended in attempting to gain an understanding of the behavior of non-Newtonian fluids in shear. MARKOVITZ [1] recently attempted to collect and organize experimental data and to compare it with predictions of existing theories. One difficulty is that conclusions which can be drawn from different types of experiments seem to disagree. Theoretical work on the steady flow of non-Newtonian fluids has brought to light a new phenomenon. According to the linear theory of viscous fluids, it is always possible for a fluid flowing through a cylindrical tube, to which it adheres, to undergo steady rectilinear motion, each particle moving with constant speed in a straight line parallel to the generators of the cylinder. Analyses made by ERICKSEN [2] and by GREEN • RIVLm [31 show that, for many ideal fluids described by the REINER [41 -RIVLIN [5] theory, this simple type of motion is possible only for very special shapes of tubes and that, in general, it is replaced by a flow consisting of such a rectilinear motion combined with a secondary flow in cross-sectional planes. Using the more general theory of fluids proposed by RIVLIN & ERICKSEN [61, LANGLOIS [7] has obtained solutions involving secondary flows in tubes and other types of "boundary geometries. As has been discussed by RIVLIN [8], similar secondary flows have been observed in the mean flow of fluids through non-circular tubes when the motion is turbulent. To our knowledge, it is not known whether secondary flows occur in the steady flow of any real fluid through tubes. Our analysis indicates that, if the conclusions which ROBERTS [9] drew from his experiments ai-e correct, the fluids which he observed should be capable of undergoing rectilinear motionthrough tubes of any shape. MARKOVITZ [11 draws self-consistent conclusions from other experimental data which contradict some of ROBERTS' conclusions. Calculations made in accordance with these conclusions suggest that secondary flows are to be expected in some shapes of tubes. We offer a partial proof that if secondary flows do not occur in elliptical tubes, they will not occur in tubes of any other shape.

J. L. Ericksen

Papers

Continuum theory of nematic liquid crystals

Special Topics in Elastostatics

Nilpotent energies in liquid crystal theory

Constitutive theory for some constrained elastic crystals

Steady shear flow of non-Newtonian fluids