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1986 | Book

Generalized Plate Models and Optimal Design Read chapter Read first chapter

Homogenization and Effective Moduli of Materials and Media

Editors: J. L. Ericksen, David Kinderlehrer, Robert Kohn, J.-L. Lions

Publisher: Springer New York

Book Series : The IMA Volumes in Mathematics and its Applications

Included in: Professional Book Archive

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About this book

This IMA Volume in Mathematics and its Applications Homogenization and Effective Moduli of Materials and Media represents the proceedings of a workshop which was an integral part of the 19R4-R5 IMA program on CONTINUUM PHYSICS AND PARTIAL DIFFERENTIAL EQUATIONS. We are grateful to the Scientific Committee: J . L. Ericksen D. Kinderlehrer H. Brezis C. Dafermos for their dedication and hard work in rleveloping an imaginative, stimulating, and productive year-long program. George R. Sell Hans Weinherger PREFACE The papers in this volume were presented at a workshop on homogenization of differential equations and the determination of effective moduli of materials and media, primarily in the context of continuum theory. These areas are closely linked to a variety of phenomena, such as the elastic and dielectric responses of composites, and the effective properties of shales and soils. For instance, the ability to predict the effective stiffness response of a composite across a broad range of frequencies allows its performance under given circumstances to be assessed by means of nondestructive testing. A fundamental mathematical tool is homogenization, the study of partial differential equations with rapidly varying coefficients or boundary conditions. The recent alliance of homogenization with optimal design has stimulated the development of both fields. The presentations at the workshop emphasized recent advances and open questions.

Table of Contents

Frontmatter
Generalized Plate Models and Optimal Design
Abstract
We consider the optimal design of linearly elastic, solid plates, that is, we seek the stiffest plate that can be made of a given amount of material. For large values of the ratio between the maximum allowable thickness and the minimum allowable thickness the stiffest plate cannot be obtained within the class of plates with slowly varying thickness; basically, this is caused by the cubic dependence of the rigidity tensor on the thickness function.
An extended class of plate models that allow for fields of stiffeners is described and it is shown how effective rigidity tensors can be obtained by homogenization or by a smear-out method based on continuity considerations. A computational optimization has been carried out and the numerical results indicate that use of this generalized plate model regularizes the optimization problem.
Martin P. Bendsøe
The Effective Dielectric Coefficient of a Composite Medium: Rigorous Bounds from Analytic Properties
Abstract
The analytic properties of the bulk effective dielectric coefficient εe of a composite medium, viewed as a function of the component coefficients ε1,ε2,…, are reviewed in Section I, and are then used to discuss rigorous bounds for (real and complex) εe when various types of partial information are available about the medium. General methods are described in Section II for constructing optimal and rigorous bounds for both real and complex εe in two-component composites. A method for constructing rigorous bounds for εe in composites made of more than two components is described in Section III. Optimum bounds are found for real εe. For complex εe the optimization problem has not been fully resolved. Nevertheless, bounds for complex εe are obtained that are better than some other bounds that have recently been derived in Refs. 6 and 7.
David J. Bergman
Variational Bounds on Darcy’s Constant
Abstract
Prager’s variational method of obtaining upper bounds on the fluid permeability (Darcy’s constant) for slow flow through porous media is reexamined. By exploiting the freedom one has in choosing the trial stress distributions, several new results are derived. One result is a phase interchange relation for permeability; when the fluid-phase and particle-phase are interchanged for a fixed geometry, we find an upper bound on a linear combination of the complementary permeabilities. Another result is a proof of the monotone properties of the bounds. The optimal two-point bounds from this class of variational principles are evaluated numerically and compared to exact results of low density expansions for assemblages of spheres.
James G. Berryman
Micromodeling of Void Growth and Collapse
Abstract
Use of the hollow sphere model to describe the response of porous materials and powders under tensile or compressive stress is described. Topics include pore pressure effects, powder compaction equations, rate effects, load maxima, and deviatoric effects. A recent hybrid empirical-microstructural model for creep compaction of heated metal powders (HIP) is described in some detail.
M. M. Carroll
On Bounding the Effective Conductivity of Anisotropic Composites
Abstract
There has recently been a renewal of interest in bounding the effective moduli of composite materials. Several factors are responsible, including attention to applications in structural optimization [1,18,20,28]. The developments of the past several years include new ways of applying old variational principles such as those of Hashin and Shtrikman. They also include two entirely new methods for proving bounds: one based on compensated compactness, and the other on explicit representation formulas.
Robert V. Kohn, Graeme W. Milton
Thin Plates with Rapidly Varying Thickness, and their Relation to Structural Optimization
Abstract
There is a close relationship between problems of structural optimization and the analysis of media with microstructure. The optimal design of variable thickness plates is a case in point: for certain problems, plates with “stiffeners” formed by rapid thickness variation can be stronger per unit volume than any traditional, uniform or slowly varying plates. To resolve such a design problem one must introduce a “generalized plate model,” representing the overall effect of a microstructure of stiffeners on the behavior of the plate.
Robert V. Kohn, Michael Vogelius
Modelling the Properties of Composites by Laminates
Abstract
Laminate materials of the type introduced by Schulgasser [1] and Bruggeman [66,67] are studied and the extent to which they can simulate the transport properties of other composites is explored. Laminates with chirality, with especially high or low field concentrations, or which attain various bounds are constructed. The Hashin-Shtrikman bounds on the shear modulus are demonstrated to be optimal, being attained by a hierarchical laminate material. While the conductivity function of two-component composites can be simulated by laminates, an example suggests this does not extend to five-component composites. Attention is drawn to the connection between conductivity functions, Stieltjes functions, and bounds.
G. W. Milton
Waves in Bubbly Liquids
Abstract
Consider a gas-bubble liquid mixture with β the gas volume fraction p the pressure and ρ the density of the mixture. Let ceff be the effective sound speed of the mixture and τ = ρ−1 the specific volume. We have that
$$ {\text{c}}_{{{\text{eff}}}}^{2} = \frac{{{\text{dp}}}}{{{\text{d}}\rho }} = \frac{1}{{{\text{d}}\rho {\text{/dp}}}} = \frac{1}{{{\text{d}}{{\tau }^{{ - 1}}}{\text{/dp}}}} = \frac{1}{{\kappa \rho }} $$
(1.1)
where the compressibility κ is defined by
$$ \kappa {\text{ = }}\frac{{ - 1}}{\tau }{\mkern 1mu} \frac{{{\text{d}}\tau }}{{{\text{dp}}}} $$
(1.2)
i.e. the change of volume with respect to pressure. Now let us assume that density and compressibility of the mixture are simply the averages over the two component values
$$ \rho {\text{ = }}{{\rho }_{{\text{g}}}}\beta {\text{ + }}{{\rho }_{\ell }}({\text{1 - }}\beta ),{\mkern 1mu} \kappa {\text{ = }}{{\kappa }_{{\text{g}}}}\beta {\text{ + }}({\text{1 - }}\beta ){{\kappa }_{\ell }} $$
(1.3)
where subscripts denote liquid or gas. The density of the gas is typically 1000 times smaller than that of the liquid while the compressibility of the liquid is negligible. Combining (1.1) and (1.3) with this simplification gives the formula
$$ {\text{c}}_{{{\text{eff}}}}^{2} = \frac{{\kappa _{{\text{g}}}^{{ - 1}}}}{{{{\rho }_{\ell }}\beta ({\text{1 - }}\beta )}} $$
(1.4)
If now p = const. ργ for the gas with γ the ratio of specific heats, we have κ g −1 =γp and hence
$$ {\text{c}}_{{{\text{eff}}}}^{2} = \frac{{\gamma \rho }}{{{{\rho }_{\ell }}\beta (1 - \beta )}} $$
(1.5)
R. Caflisch, M. Miksis, G. Papanicolaou, L. Ting
Some Examples of Crinkles
Abstract
Many physical problems can be phrased as the problem of minimizing some energy functional E[f] over a given class of admissible functions f. It can happen that there is a minimizing sequence fn that approaches a limit f̄ but f̄ does not minimize E, either because f̄ is not in the admissible class or because E is not lower semicontinuous. In the examples that I discuss here, this happens because the derivatives <Inline>#</Inline> are highly discontinuous and do not approach f̄’ in the limit. I call such sequences crinkles, and call the limiting function f̄ the carrier of the crinkle. Young [1] has written a book on the subject; he calls such sequences generalized curves. In control theory the same sort of thing is also called a chattering state.
A. C. Pipkin
Microstructures and Physical Properties of Composites
Abstract
A question that has often been raised is: “How accurate is the effective medium approximation?” The question is significant in view of the fact that different effective medium theories, derived with the same goal of describing a “random” composite, can produce drastically different predictions. In the first part of this paper I illustrate with several examples that different versions of effective medium theories are actually associated with different underlying microstructures. This fact explains a major part of the discrepancies in the predictions of various effective medium theories. The recognition of the role of microstructure naturally raises to the forefront the need for a general and precise method for incorporating structural information in the calculation of electric and elastic properties of composites. The second half of the paper addresses part of this problem by presenting a first-principle approach to the calculation of effective elastic moduli for arbitrary periodic composites. By using Fourier coefficients of the periodic system as structural inputs, the new method offers the advantage of circumventing the need for explicit boundary-conditions matching across material interfaces. As a result, it can handle complex unit cell geometries just as easily as simple cell geometries.
Ping Sheng
Remarks on Homogenization
Abstract
Homogenization is concerned with the relations between microscopic and macroscopic scales but different mathematical problems can be associated to this general question: one of them is to give a probabilistic framework where microscopic quantitites are functions depending on a parameter ω lying in a probability space and macroscopic quantities are expectations of them [this may be the subject of another workshop], another one is to consider asymptotic expansions where one considers functions u(x,x/ε), where u(x,y) is periodic in y, which are called microscopic values, the macroscopic quantities being obtained by averaging in y.
Luc Tartar
Variational Estimates for the Overall Response of an Inhomogeneous Nonlinear Dielectric
Abstract
For any problem that can be formulated as a “minimum energy” principle, a procedure is given for generating sets of upper and lower bounds for the energy. It makes use of “comparison bodies” whose energy functions may be easier to handle than those in the given problem. No structure for the energy functions is assumed in the formal development but useful results are most likely to follow when they are convex. When applied to linear field equations, the procedure yields the Hashin-Shtrikman variational principle, and so can be regarded as its generalization to nonlinear problems.
The procedure is applied explictly to a boundary value problem for an inhomogeneous, nonlinear dielectric. Then, a slight extension which describes randomly inhomogeneous media is applied, to develop bounds for the overall energy of a nonlinear composite, which reduce to the Hashin-Shtrikman bounds in the linear limit. Sample results are shown for a simple two-phase composite.
J. R. Willis
Backmatter
Metadata
Title
Homogenization and Effective Moduli of Materials and Media
Editors
J. L. Ericksen
David Kinderlehrer
Robert Kohn
J.-L. Lions
Copyright Year
1986
Publisher
Springer New York
Electronic ISBN
978-1-4613-8646-9
Print ISBN
978-1-4613-8648-3
DOI
https://doi.org/10.1007/978-1-4613-8646-9