Thixoforming : Semi-solid Metal Processing

222j 7 A Physical and Micromechanical Model for Semi-solid Behaviour
7.2
Basic Concepts of Micromechanics and Homogenization
Most engineering materials are heterogeneous in nature. They generally consist of
different constituents or phases, which are distinguishable at specific scales. Each
constituent may show different physical properties (e.g. elastic moduli, thermal
expansion, yield strength) and/or material orientations. However, in many engineering
applications, a structure component may contain numerous such constituents
such that it is impractical or even impossible to account for each of them for
engineering design and analysis. When studying the properties of a real material,
we need to define the microscopic length scale at which the properties of interest are
directly relevant to determine the overall or effective property of the material from
which the component is made. The term overall properties means the properties
averaged over a certain volume of the heterogeneous material. For such overall
properties to be meaningful, the average taken over by an arbitrary volume element
comparable to the relevant scale must be the same as for the heterogeneous material
sample underconsideration. Heterogeneousmaterialsthatmeet thisrequirement are
said to be macroscopically homogeneous. In addition, if in a heterogeneous material
the microscopic length parameter d and the macroscopic length parameter D can be
identified for a length scale of interest such as d/D 1, then the heterogeneous
materialismicroscopicallyhomogeneousatthelengthscaleD.Avolumeelementwith
characteristic dimension D is called a representative volume element (RVE) because
the overall properties on any RVE would be the same. In other words, the overall
properties of each RVE represent the overall properties of the heterogeneous material.
Modelling based on micromechanics and homogenization techniques aims at
relating microstructure features and local behaviour defined at a relevant length scale
to the overall properties. It is based on the solution of Eshelby, who determined the
stress and strain tensors within an inclusion embedded in a homogeneous matrix [8].
Details of the scale transition methodology and micromechanics were given by, for
example, Kr€oner [9, 10], Mura [11] and Zaoui [12]. To sum up, this approach has three
steps:
1. Definition of the representative volume element: The size of the RVE must be such that
it includes a very large number of heterogeneities and also be statistically
homogeneous and representative of the local continuum properties, so that
appropriate averaging schemes over these domains give rise to the same mechanical
properties. These properties correspond to the overall or effective
mechanical properties. Because it is impossible to have knowledge of every detail
of the microstructure, only relevant statistical (we are not interested in periodic
microstructure) information on the microstructure is incorporated in the definition
of RVE. These average features allow the identification of certain mechanical
phases that dictate the overall behaviour.
2. Definition of concentration equations: Local strains are different from the overall
strain due to the inhomogeneities within the material. Concentration equations
aim to relate local and overall variables such as strain, strain rate or stress fields.