The Isotropic and Cubic Material Designs. Recovery of the Underlying Microstructures Appearing in the Least Compliant Continuum Bodies

Abstract

The paper discusses the problem of manufacturability of the minimum compliance designs of the structural elements made of two kinds of inhomogeneous materials: the isotropic and cubic. In both the cases the unit cost of the design is assumed as equal to the trace of the Hooke tensor. The Isotropic Material Design (IMD) delivers the optimal distribution of the bulk and shear moduli within the design domain. The Cubic Material Design (CMD) leads to the optimal material orientation and optimal distribution of the invariant moduli in the body made of the material of cubic symmetry. The present paper proves that the varying underlying microstructures (i.e., the representative volume elements (RVE) constructed of one or two isotropic materials) corresponding to the optimal designs constructed by IMD and CMD methods can be recovered by matching the values of the optimal moduli with the values of the effective moduli of the RVE computed by the theory of homogenization. The CMD method leads to a larger set of results, i.e., the set of pairs of optimal moduli. Moreover, special attention is focused on proper recovery of the microstructures in the auxetic sub-domains of the optimal designs.

Auxeticity of Concentric Auxetic-Conventional Foam Rods with High Modulus Interface Adhesive

“…It has also been found that novel properties, distinct from conventional or auxetic materials, arise when a composite structure is made from both conventional and auxetic materials. In addition to a recent comprehensive survey of auxetic materials and structures [ 37 ], the reader is also referred to a recent focus issue on “auxetics and other systems of anomalous characteristics” [ 38 , 39 , 40 , 41 , 42 ], as well as papers in this special issue [ 43 , 44 , 45 , 46 , 47 , 48 ].…”

An Explicit Construction of the Underlying Laminated Microstructure of the Least Compliant Elastic Bodies

“…Another combinatorial method of choosing the optimal microstructures as the starting point for an inverse homogenization procedure was presented in ref. . The recovery and approximation of the Pareto optimal microstructures appear independently in the case of vector optimization, e.g., in problem of minimizing the compliances of the elastic body for the multiple loading conditions, see ref.…”

Recovery of the Auxetic Microstructures Appearing in the Least Compliant Continuum Two‐Dimensional Bodies

“…In these articles, the reconstruction of microstructures in subdomains where the auxetic properties [ 6–13 ] of the optimal, nonhomogeneous isotropic or cubic elastic material appear has been undertaken. In the present article, being an extension of the previous work, [ 14 ] the homogenization method based on parametric description of the shape of the domain filled by the given isotropic material inside the representative volume elements (RVEs), instead of the combinatorial search among the admissible microstructures, is implemented. Within the 2D setting discussed, the local cubic symmetry of the material reduces to the local symmetry of a square.…”