Page 77 - 3D Fibre Reinforced Polymer Composites

66 30 Fibre Reififorced Polymer Composites

4.2.2 Representative Volume Element and Effective Properties
A microscopically inhomogeneous composite material can be idealised as a
macroscopically homogeneous continuum when the behaviour of engineering structures
made of the material can be satisfactorily retained. Such idealisation can be realised
over a representative sample of the composite material. Selection of the dimensions of
a representative volume is imperative. The representative volume must be sufficiently
large compared to the scale of the microstructure so that it contains a sufficient number
of individual constituents and microstructural features. It also must be small compared
to the whole structural body so that it is entirely typical of the whole composite
structure on average. For structural scales larger than the representative volume
element, continuum mechanics can be used to reproduce properties of the material as a
whole for structural analysis and design without considering the microstructure of the
material.
For a representative vohmetric element subject to an imposed macroscopically
homogeneous stress or displacement field and no body forces, the average stress and
strain components are defined as:

1
8.. = - JoudV
v,
(4.7)

where 0,. and E,. are the true stresses and strains in the representative volume V or
micro stresses or micro strains, respectively.
When a representative volume element is subject to a prescribed displacement field
on its boundary surface S in the form:

where E; are constant strains, the average strains Eti are identical to the applied

constant strains, i.e., E,. = &:, when there exists perfect interfacial bonding.
When a representative volume element is subject to a homogeneous stress field on
its boundary surface S in the following form:

where 0,: are constant stresses and ni (i=1,2,3) are components of the unit outward
normal vector to the surface of the representative volume, the average stresses are
identical to the applied constant stresses, i.e., qj = 0;. Both conditions in equations
(4.8) and (4.9) are referred to as homogeneous boundary conditions, Le., iso-strain and
iso-stress boundary conditions, respectively. It is worth pointing out that the work done