25 research outputs found
Three-phase plane composites of minimal elastic stress energy: High-porosity structures
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
low-porosity 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 the
stress energy bound is optimal.Comment: Added section 4.3 and figures 9-11. Minor editorial changes for the
improvement of clarit
A Note on Optimal Design of Multiphase Elastic Structures
The paper describes the first exact results in optimal design of three-phase
elastic structures. Two isotropic materials, the "strong" and the "weak" one,
are laid out with void in a given two-dimensional domain so that the compliance
plus weight of a structure is minimized. As in the classical two-phase problem,
the optimal layout of three phases is also determined on two levels: macro- and
microscopic. On the macrolevel, the design domain is divided into several
subdomains. Some are filled with pure phases, and others with their mixtures
(composites). The main aim of the paper is to discuss the non-uniqueness of the
optimal macroscopic multiphase distribution. This phenomenon does not occur in
the two-phase problem, and in the three-phase design it arises only when the
moduli of material isotropy of "strong" and "weak" phases are in certain
relation.Comment: 8 pages, 4 figure
Dynamics and stationary configurations of heterogeneous foams
We consider the variational foam model, where the goal is to minimize the
total surface area of a collection of bubbles subject to the constraint that
the volume of each bubble is prescribed. We apply sharp interface methods to
develop an efficient computational method for this problem. In addition to
simulating time dynamics, we also report on stationary states of this flow for
<22 bubbles in two dimensions and <18 bubbles in three dimensions. For small
numbers of bubbles, we recover known analytical results, which we briefly
discuss. In two dimensions, we also recover the previous numerical results of
Cox et. al. (2003), computed using other methods. Particular attention is given
to locally optimal foam configurations and heterogeneous foams, where the
volumes of the bubbles are not equal. Configurational transitions are reported
for the quasi-stationary flow where the volume of one of the bubbles is varied
and, for each volume, the stationary state is computed. The results from these
numerical experiments are described and accompanied by many figures and videos.Comment: 19 pages, 11 figure