Modeling Material Microstructure - Continuum Mechanics of Anisotropic Materials - page 174

composite materials. Many think that the first man-made composite material was
the reinforced brick constructed by using straw to reinforce the clay of the brick.
Dried clay is satisfactory in resisting compression, but not very good in tension. The
straw endows the brick with the ability to sustain greater tensile forces. Most
structural soft tissues in animals can carry tensile forces adequately but do not do
well with compressive forces. In particular, due to their great flexibility they may
deform greatly under compressive forces. The mineralization of the collagenous
tissues provides those tissues with the ability to resist compressive forces; thus bone
and teeth are composites of an organic phase, primarily collagen, and an inorganic
or mineral phase. Effective material parameters for composite materials, defined in
Sect. 7.3 , are generally determined by expressions that depend upon the phase or
constituent-specific material parameters and their geometries. Examples of effec-
tive elastic constants and effective permeabilities are developed in Sects. 7.4 and
7.5 , respectively. Restrictions on the RVE for the case of a gradient in its material
properties are considered in Sect. 7.6 . The continuum modeling of material
microstructures with vectors and tensors is described in Sect. 7.7 , with a particular
emphasis on the fabric tensor, a measure of local microstructure in a material with
more than a single constituent. The stress-strain-fabric relation is developed in
Sect. 7.8 . Some of the relevant literature is described in Sect. 7.9 .

7.2 The Representative Volume Element

Recall from Sect. 4.2 that, for this presentation, the RVE is taken to be a cube of side
length L RVE ; it could be any shape, but it is necessary that it have a characteristic
length scale (Fig. 4.1). The RVE for the representation of a domain of a porous
medium by a continuum point was illustrated in Fig. 4.2. We begin here by picking
up the question of how large must the length scale L RVE be to obtain a reasonable
continuum model. The L RVE should be much larger than the largest characteristic
microstructural dimension L M of the material being modeled and smaller than the
characteristic dimension of the problem to be addressed L P , thus L P

L M .
The question of the size of L RVE can also be posed in the following way: How
large a hole is no hole? The value of L RVE selected determines what the modeler has
selected as too small a hole, or too small an inhomogeneity or microstructure, to
influence the result the modeler is seeking. An interesting aspect of the RVE
concept is that it provides a resolution of a paradox concerning stress concentrations
around circular holes in elastic materials. The stress concentration factor associated
with the hole in a circular elastic plate in a uniaxial field of otherwise uniform stress
is three times the uniform stress (Fig. 7.1 ). This means that the stress at certain
points in the material on the edge of the hole is three times the stress five or six hole
diameters away from the hole. The hole has a concentrating effect of magnitude 3.
The paradox is that the stress concentration factor of 3 is independent of the size of
the hole. Thus, no matter how small the hole, there is a stress concentration factor of
3 associated with the hole in a field of uniaxial tension or compression. One way to

L RVE