Book contents
- Frontmatter
- Contents
- List of figures
- Preface
- 1 Introduction
- 2 Some equations of interest and numerical approaches to solving them
- 3 Duality transformations in two-dimensional media
- 4 Translations and equivalent media
- 5 Some microstructure-independent exact relations
- 6 Exact relations for coupled equations
- 7 Assemblages of spheres, ellipsoids, and other neutral inclusions
- 8 Tricks for generating other exactly solvable microgeometries
- 9 Laminate materials
- 10 Approximations and asymptotic formulas
- 11 Wave propagation in the quasistatic limit
- 12 Reformulating the problem of finding effective tensors
- 13 Variational principles and inequalities
- 14 Series expansions for the fields and effective tensors
- 15 Correlation functions and how they enter series expansions†
- 16 Other perturbation solutions
- 17 The general theory of exact relations and links between effective tensors
- 18 Analytic properties
- 19 Y-tensors
- 20 Y-tensors and effective tensors in electrical circuits†
- 21 Bounds on the properties of composites
- 22 Classical variational principle bounds
- 23 Bounds from the Hashin-Shtrikman variational inequalities
- 24 Bounds using the compensated compactness or translation method
- 25 Choosing the translations and finding microgeometries that attain the bounds†
- 26 Bounds incorporating three-point correlation functions†
- 27 Bounds using the analytic method
- 28 Fractional linear transformations as a tool for generating bounds†
- 29 The field equation recursion method†
- 30 Properties of the G-closure and extremal families of composites
- 31 The bounding of effective moduli as a quasiconvexification problem
- Author index
- Subject index
13 - Variational principles and inequalities
- Frontmatter
- Contents
- List of figures
- Preface
- 1 Introduction
- 2 Some equations of interest and numerical approaches to solving them
- 3 Duality transformations in two-dimensional media
- 4 Translations and equivalent media
- 5 Some microstructure-independent exact relations
- 6 Exact relations for coupled equations
- 7 Assemblages of spheres, ellipsoids, and other neutral inclusions
- 8 Tricks for generating other exactly solvable microgeometries
- 9 Laminate materials
- 10 Approximations and asymptotic formulas
- 11 Wave propagation in the quasistatic limit
- 12 Reformulating the problem of finding effective tensors
- 13 Variational principles and inequalities
- 14 Series expansions for the fields and effective tensors
- 15 Correlation functions and how they enter series expansions†
- 16 Other perturbation solutions
- 17 The general theory of exact relations and links between effective tensors
- 18 Analytic properties
- 19 Y-tensors
- 20 Y-tensors and effective tensors in electrical circuits†
- 21 Bounds on the properties of composites
- 22 Classical variational principle bounds
- 23 Bounds from the Hashin-Shtrikman variational inequalities
- 24 Bounds using the compensated compactness or translation method
- 25 Choosing the translations and finding microgeometries that attain the bounds†
- 26 Bounds incorporating three-point correlation functions†
- 27 Bounds using the analytic method
- 28 Fractional linear transformations as a tool for generating bounds†
- 29 The field equation recursion method†
- 30 Properties of the G-closure and extremal families of composites
- 31 The bounding of effective moduli as a quasiconvexification problem
- Author index
- Subject index
Summary
Variational principles have long been known in the context of both conductivity type problems and elasticity problems. Their application to composites was initiated by Hill (1952), who used them to show that the Voigt (1889, 1910) and Reuss (1929) estimates of the elastic moduli of polycrystals were in fact bounds. A new type of variational principle was discovered by Hashin and Shtrikman (1962a, 1962b, 1963), which become famous because it lead to optimal bounds on the conductivity, bulk, and shear moduli of isotropic composites of two isotropic phases. Hill (1963b) gave rigorous proof of their variational principles, showing how they could be derived from the classical variational principles. Subsequently, Hashin (1967) generalized the variational principles to inhomogeneous elastic bodies (not just composites) subject to body forces and mixed boundary conditions. Cherkaev and Gibiansky (1994) extended all of these variational principles to media with complex moduli. Talbot and Willis (1985) extended the Hashin and Shtrikman variational principles to nonlinear media. Other variational inequalities for nonlinear media, based on comparisons with linear inhomogeneous media, were obtained by Ponte Castañeda (1991). In a development that falls outside the range of this book, Smyshlyaev and Fleck (1994, 1996) extended the Hashin and Shtrikman variational principles to elastic composites, where the elastic energy depends not only on the strain, but also on the strain gradient.
Classical variational principles and inequalities
We have seen how to manipulate equations into a form where the tensor entering the constitutive law is self-adjoint and positive-definite.
- Type
- Chapter
- Information
- The Theory of Composites , pp. 271 - 290Publisher: Cambridge University PressPrint publication year: 2002