Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. II: The vigdergauz microstructure

Abstract

For modeling coherent phase transformations, and for applications to structural optimization, it is of interest to identify microstructures with minimal energy or maximal stiffness. S. Vigdergauz has shown the existence of a particularly simple microstructure with extremal elastic behavior, in the context of two-phase composites made from * This work was done while Y. G. was a student at the Courant Institute. † The work of R. V. K. was partially supported by ARO contract DAAL 03-92-G-0011 and NSF grants DMS-9404376 and DMS-9402763. 1 isotropic components in two space dimensions. This "Vigdergauz microstructure" consists of a periodic array of appropriately shaped inclusions. We provide an alternative discussion of this microstructure and its properties. Our treatment includes an explicit formula for the shape of the inclusion, and an analysis of various limits. We also discuss the significance of this microstructure (i) for minimizing the maximum stress in a composite, and (ii) as a large volume fraction analog of Michell trusses in the theory of structural optimization.

On hierarchical structures and reiterated homogenization

“…non-iterated) geometries found by Vigdergauz [46,47,48,49,50] which are optimal for the e ective bulk modulus. A very readable treatment of his work can be found in [14]. There is also a new class of optimal iterated structures found recently by Sigmund [43].…”

On the comparison of material interpolation schemes and optimal composite properties in plane shape optimization

“…In the latter case of an auxetic material, values of parameters in the RAMP and SIMP Table 1 Requirements on the parameters in GRAMP, RAMP and SIMP interpolation schemes imposed by the conditions of optimality (U app ≥ U opt ) or material isotropy (U app ≥ U HS ) Constitutive properties predicted by Hashin-Shtrikman approach are realizable on certain isotropic microstructures like 3rd rank sequential laminates (Francfort and Murat 1986) or coated circles (Hashin 1962;Grabovsky and Kohn 1995a). On the other hand, composites of minimal compliance can be arranged as orthotropic 2nd rank orthogonal sequential laminates or, if det N > 0, in a form of the Vigdergauz microstructures (Vigdergauz 1989;Grabovsky and Kohn 1995b) or 4th rank sequential laminates (Allaire and Aubry 1999).…”

Shape optimization in an elastic plate under remote shear: from single to interacting holes

“…Particularly, the equistress isolated hole is simply an ellipse [Cherepanov 1974] with eccentricity δ 0 elongated along the far field eigendirection of the maximum |P 0 |, |Q 0 |. Some specific arrangements of the optimal interacting holes are found in [Cherepanov 1974;Vigdergauz 1976;Grabovsky and Kohn 1995;Vigdergauz 1996]. Commonly, the equistress shapes are smooth with no angular points.…”