Back from a long silent period... In part 2 of this series, I introduced the variational principle of Hashin and Shtrikman (1962). Then, in part 3, I showed how further assumptions on the reference material led to the energy principle of Hashin and Shtrikman. Both principles are great tools to derive numerical methods for the computation of strain/stress fields in highly heterogeneous materials (where other methods —eg finite elements— would have a hard time doing the same calculation). But the energy principle also allows the derivation of theoretical results, such as rigorous bounds on the macroscopic properties. Why this is so is the topic of today's instalment.
Saturday, December 10, 2011
Monday, October 31, 2011
In the second part of this series, I introduced the variational principle of Hashin and Shtrikman (1962). It should be noted that this principle is valid, regardless of the reference material $\stiffness_0$ (no restriction apply to the elastic moduli of the reference material). In the present instalment, we will see under which circumstances this principle is in fact an energy principle.
Wednesday, October 19, 2011
In part 1 of this series, we introduced the Lippmann-Schwinger equation. Direct discretization of this equation can lead to efficient schemes for the numerical solution of the elementary problem of micromechanics. However, my experience shows that taking first the variational form of this equation, and following standard Garlerkin discretization approaches can lead to much more robust and efficient schemes. In this instalment, we will introduce the so-called variational "principle" of Hashin and Shtrikman (1962),
Tuesday, October 4, 2011
In this series, we will enter the heart of the matter. We will look at non-standard ways of numerically solving the local problem of micromechanics, introducing first the Lippmann-Schwinger equation (in this instalment), then the energy principle of Hashin and Shtrikman (in the next instalment).
Sunday, September 18, 2011
In part 1 of this series, we have seen how the Green operator for strains, $\greeniv_0$, was defined as the operator providing the solution to the problem of elastic equilibrium of a prestressed homogeneous domain. In this instalment, some simple, but useful properties of this operator will be proved.
Sunday, September 11, 2011
Micromechanics aims at computing the macroscopic properties of heterogeneous materials. In other words, we are looking for a stress-strain relationship, and displacements are of but little use to us. We therefore tend to prefer Green functions for strains to Green functions for displacements, which are commonly used in the mathematical theory of elasticity. The Green operator for strain is the essential ingredient for my main research topic, namely polarization techniques.In the first instalment of this series, I'm going to define the Green operator for strains.