scholarly journals Explicit effective elasticity tensors of two-phase periodic composites with spherical or ellipsoidal inclusions

2016 ◽  
Vol 94-95 ◽  
pp. 100-111 ◽  
Author(s):  
Quy-Dong To ◽  
Guy Bonnet ◽  
Duc-Hieu Hoang
Keyword(s):  
Two Phase ◽  
Periodic Composites ◽  
Effective Elasticity ◽  
Elasticity Tensors

Which Elasticity Tensors are Realizable?

1995 ◽  
Vol 117 (4) ◽  
pp. 483-493 ◽  
Author(s):  
Graeme W. Milton ◽  
Andrej V. Cherkaev
Keyword(s):  
Building Blocks ◽  
Elasticity Tensor ◽  
Isotropic Phase ◽  
Order Tensor ◽  
Two Phase ◽  
Effective Elasticity ◽  
Elasticity Tensors ◽  
Two Phases ◽  
The Given ◽  
Fourth Order Tensor

It is shown that any given positive definite fourth order tensor satisfying the usual symmetries of elasticity tensors can be realized as the effective elasticity tensor of a two-phase composite comprised of a sufficiently compliant isotropic phase and a sufficiently rigid isotropic phase configured in an suitable microstructure. The building blocks for constructing this composite are what we call extremal materials. These are composites of the two phases which are extremely stiff to a set of arbitrary given stresses and, at the same time, are extremely compliant to any orthogonal stress. An appropriately chosen subset of the extremal materials are layered together to form the composite with elasticity tensor matching the given tensor.

VARIATIONAL METHODS, SIZE EFFECTS AND EXTREMAL MICROGEOMETRIES FOR ELASTIC COMPOSITES WITH IMPERFECT INTERFACE

1995 ◽  
Vol 05 (08) ◽  
pp. 1139-1173 ◽  
Author(s):  
ROBERT LIPTON ◽  
BOGDAN VERNESCU
Keyword(s):  
Size Effects ◽  
Variational Principles ◽  
Volume Ratio ◽  
Imperfect Interface ◽  
Relative Importance ◽  
Two Phase ◽  
Effective Elasticity ◽  
And Variational Principles ◽  
Elastic Composites ◽  
Compliance Mismatch

We introduce new bounds and variational principles for the effective elasticity of anisotropic two-phase composites with imperfect bonding conditions between phases. The monotonicity of the bounds in the geometric parameters is used to predict new size effect phenomena for monodisperse and polydisperse suspensions of spheres. For isotropic elastic spheres in a more compliant isotropic matrix we exhibit critical radii for which the stress state, external to the spheres, is unaffected by their presence. Physically all size effects presented here are due to the increase in surface to volume ratio, as the sizes of the inclusions decrease. The scale at which these effects occur is determined by the parameters [Formula: see text] and [Formula: see text]. These parameters measure the relative importance of interfacial compliance and phase compliance mismatch.

Effective Elasticity Tensors in Context of Random Errors

2015 ◽  
Vol 121 (1) ◽  
pp. 55-67 ◽  
Author(s):  
Tomasz Danek ◽  
Mikhail Kochetov ◽  
Michael A. Slawinski
Keyword(s):  
Random Errors ◽  
Effective Elasticity ◽  
Elasticity Tensors

scholarly journals On Choosing Effective Elasticity Tensors Using a Monte-Carlo Method

2015 ◽  
Vol 63 (1) ◽  
pp. 45-61 ◽  
Author(s):  
Tomasz Danek ◽  
Michael A. Slawinski
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Effective Elasticity ◽  
Elasticity Tensors

On the effective elasticity of a two-dimensional homogenised incompressible elastic composite

1988 ◽  
Vol 110 (1-2) ◽  
pp. 45-61 ◽  
Author(s):  
Robert Lipton
Keyword(s):  
Two Dimensions ◽  
Two Dimensional ◽  
Elastic Materials ◽  
Elastic Composite ◽  
Effective Elasticity ◽  
Elasticity Tensors

SynopsisThe set of effective elasticity tensors for all two-dimensional mixtures of two isotropic incompressible elastic materials taken in prescribed proportion is described. In two dimensions the effective tensors are completely characterised by bounds on their eigenvalues.

scholarly journals Closed-form solutions for the effective conductivity of two-phase periodic composites with spherical inclusions

2013 ◽  
Vol 469 (2151) ◽  
pp. 20120339 ◽  
Author(s):  
Q. D. To ◽  
G. Bonnet ◽  
V. T. To
Keyword(s):  
Closed Form ◽  
Effective Conductivity ◽  
Approximate Solutions ◽  
Static Structure Factor ◽  
Static Structure ◽  
Two Phase ◽  
Closed Form Solutions ◽  
Periodic Composites ◽  
Spherical Inclusions ◽  
Volume Limit

In this paper, we use approximate solutions of Nemat-Nasser et al. to estimate the effective conductivity of two-phase periodic composites with non-overlapping spherical inclusions. Systems with different inclusion distributions are considered: cubic lattice distributions (simple cubic, body-centred cubic and face-centred cubic) and random distributions. The effective conductivities of the former are obtained in closed form and compared with exact solutions from the fast Fourier transform-based methods. For systems containing randomly distributed spherical inclusions, the solutions are shown to be directly related to the static structure factor, and we obtain its analytical expression in the infinite-volume limit.

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