Page 80 - 3D Fibre Reinforced Polymer Composites

Micromechanics Models for Mechanical Properties 69

(4.17b)

Noting the definition of concentration matrices in equation (4.16), the following
relationships hold c,[A,] = [I]-c,[A,] and c,[B,] =[I]-c,[B,]. Thus the above
equations can be rewritten as:

(4.18)

By assuming that the strain is uniform throughout the composite, which means that
[Al J=[AzJ=[ZJ, the following simplest equation can be obtained:

[C] = C,[C"'] + C,[C'*'] (4.19)

Similarly, by assuming that stress is uniform throughout the composites, namely
[B1]=[B2]=[fl, we have the following:

[SI = C,[S'"] + c2[S'2' J (4.20)

Equation (4.19) and (4.20) are the Voigt and Reuss approximations, which provide
upper and lower bounds as proved by Hill (Aboudi 1991).
Determination of the concentration matrices in different phases is one of the most
important steps in evaluating the effective overall properties of a composite material.
Mori and Tanaka (1973) presented a method for calculating average internal stress in a
matrix of materials containing misfitting inclusions by using eigenstrains. In the Mori-
Tanaka method, the average strain in the interacting inclusions is approximated by that
of a single inclusion in an infinite matrix subject to the uniform average matrix strain
(Aboudi, 1991), which leads to the following relation:

where superscript 2 indicates the inclusion and superscript 1 corresponds to the matrix,
[T,] is determined from the solution of a single particle imbedded in an infinite matrix
subject to homogeneous displacement boundary conditions defined by the average
matrix strain {F(')}. Substituting the above equation into (4.15) yields a definition of
{E")} in terms of the overall average strains {Z} , and in conjunction with equation
(4.15) leads to the determination of [A2] as follows (Aboudi 1991):

Substituting into equation (4.18) yields the overall stiffness matrix. The overall
compliance matrix can also be obtained similarly.