# Extraction of the effect of microstructure on extreme elastic properties

### Project 1. Laminar elastic composites with crystallographic symmetry

(Francfort and Murat, 1986) derive an explicit formula for the effective elasticity tensor of a multiply layered composite from two isotropic elastic materials in prescribed proportion. Together with R. James and A. Lutoborsky we investigate multiply layered composites with crystalographic symmetry. It is shown that these formulae can be represented as a group average over the crystallographic group (James, Lipton, and Lutoborsky, 1990). The special case of cubically symmetric composites made by multiple layering is considered. The extremal property of laminar composites (Avellaneda, 1987) is used to obtain new optimal bounds on the effective shear modulii for elastic composites with cubic symmetry.

This work has appeared in R.D. James, R. Lipton, and A. Lutoborsky. Laminar elastic composites with crystallographic symmetry SIAM Journal on Applied Mathematics. 50 (1990) 683--702.
This work is supported by NSF Grant DMS-8718881

### Project 2. Optimal bounds on the effective elastic tensors for orthotropic composites

We consider the totality of orthotropic composites made from two isotropic linear elastic components in fixed proportion. The elastic properties of orthotropic composites are characterized by nine independent moduli. We provide bounds for six of these, namely the three Youngs moduli and three in-plane shear moduli. The bounds are optimal and correlate the six moduli.

This work appeared in R. Lipton. Proceedings of the Royal Society of London A 444 (1994) 399--410.
This work is supported by NSF Grant DMS-9205158.

### Project 3. Optimal bounds for the effective energy of two phase elastic incompressible composites

In this work with Robert Kohn we derive optimal upper and lower bounds on the elastic energy for composites made from two isotropic incompressible materials. These bounds are expressed in terms of the volume fraction of the two materials the average applied deviatoric strain, and the shear modulii of the two materials. The energy bounds are shown to be obtainable by two-phase laminar elastic configurations of the two materials.

This work appeared in R.V. Kohn and R. Lipton. Optimal bounds for the effective energy of a mixture of isotropic, incompressible, elastic materials. Archive for Rational Mechanics and Analysis. 102 (1988) 331--350.

### Project 4. The G-closure set for two-phase incompressible elastic composites in two dimensions

In this work we derive the G-closure set for all mixtures of two isotropic incompressible elastic constituent materials in fixed proportion. Physically the G-closure is the set of all effective elasticity tensors obtained by mixing two incompressible elastic materials in prescribed proportion.

This work appeared in R. Lipton. On the effective elasticity of a two-dimensional homogenized incompressible elastic composite. Proceedings of the Royal Society of Edinburgh. 110A (1988) 45--61.

## References

1. M. Avellaneda, Optimal Bounds and Microgeometries for Elastic Two-Phase Composites. SIAM J. Appl. Math. 47 (1987) 1216--1228
2. G. Francfort and M. Murat. Homogenization and optimal bounds in linear elasticity. Arch. Rational Mech. Anal. 94 (1986) 307--334.
3. R.D. James, R. Lipton, and A. Lutoborsky. Laminar elastic composites with crystallographic symmetry SIAM Journal on Applied Mathematics. 50 (1990) 683--702.
4. R.V. Kohn and R. Lipton. Optimal bounds for the effective energy of a mixture of isotropic, incompressible, elastic materials. Archive for Rational Mechanics and Analysis. 102 (1988) 331--350.
5. R. Lipton. On the effective elasticity of a two-dimensional homogenized incompressible elastic composite. Proceedings of the Royal Society of Edinburgh. 110A (1988) 45--61.