The paper establishes exact lower bound on the effective elastic energy of two-dimensional, three-material composite subjected to the homogeneous, anisotropic stress. It is assumed that the materials are mixed with given volume fractions and that one of the phases is degenerated to void, i.e. the effective composite is porous. Explicit formula for the energy bound is obtained using the translation method enhanced with additional inequality expressing certain property of stresses. Sufficient optimality conditions of the energy bound are used to set the requirements which have to be met by the stress fields in each phase of optimal effective material regardless of the complexity of its microstructural geometry. We show that these requirements are fulfilled in a special class of microgeometries, so-called laminates of a rank. Their optimality is elaborated in detail for structures with significant amount of void, also referred to as high-porosity structures. It is shown that geometrical parameters of optimal multi-rank, high-porosity laminates are different in various ranges of volume fractions and anisotropy level of external stress. Non-laminate, three-phase microstructures introduced by other authors and their optimality in high-porosity regions is also discussed by means of the sufficient conditions technique. Conjectures regarding lowporosity regions are presented, but full treatment of this issue is postponed to a separate publication. The corresponding "G-closure problem" of a three-phase isotropic composite is also addressed and exact bounds on effective isotropic properties are explicitly determined in these regions where
This paper deals with the classical problem of material distribution for minimal compliance of twodimensional structures loaded in-plane. The main objective of the research consists in investigating the properties of the exact solution to the minimal compliance problem and incorporating them into an approximate solid-void interpolation model. Consequently, a proposition of a constitutive relation for a porous material arise. The non-smoothness of stress energy functional known from the approach based on homogenization may be thus avoided which is beneficial for the numerical implementation of the scheme. It is next shown that the simplified variant of the proposed formula justifies and generalizes the RAMP (Rational Approximation of Material Properties) interpolation model of Stolpe and Svanberg (Struct Multidisc Optim 22:116-124). In the second part of the paper, explicit formulae for function θ : → [0, 1] describing the distribution of basic isotropic material in the design space ∈ R 2 are derived for various interpolation schemes by the requirement of optimality imposed at each x ∈ . Theoretical considerations are illustrated by a code written in MATLAB for typical optimization problems of a cantilever and MBB beam.
Abstract. Optimization of structural topology, called briefly: topology optimization, is a relatively new branch of structural optimization. Its aim is to create optimal structures, instead of correcting the dimensions or changing the shapes of initial designs. For being able to create the structure, one should have a possibility to handle the members of zero stiffness or admit the material of singular constitutive properties, i.e. void. In the present paper, four fundamental problems of topology optimization are discussed: Michell's structures, two-material layout problem in light of the relaxation by homogenization theory, optimal shape design and the free material design. Their features are disclosed by presenting results for selected problems concerning the same feasible domain, boundary conditions and applied loading. This discussion provides a short introduction into current topics of topology optimization.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.