Computational Methods for Debonding in Composites

306 H. Miled et al.
Eqs. 15.28 and 15.30 lead to the Advani and Tucker model [1] which expresses
the effective stiffness tensor as a function of the second and fourth orientation
tensors:
C c � � � �
ijkl p = b1aijkl+ b2 (aijδkl + aklδij)+b3 aikδ jl + ailδ jk + a jkδil + a jlδik
� �
+b4 (δijδkl)+b5 δikδ jl + δilδ jk
(15.31)
An averaging on the compliance tensor gives an expression of the anisotropic
tensor which is similar to Eq. 15.31, with five constants m1,...,m5:
S c ijkl = m1aijkl+
� �
m2 (aijδkl + aklδij)+m3 aikδ jl + ailδ jk + a jkδil + a jlδik
� �
+m4 (δijδkl)+m5 δikδ jl + δilδ jk
(15.32)
In spite of SUD being the inverse of CUD , the effective compliance tensor is not
the inverse of the stiffness tensor given by Eq. 15.31. This represents the weakness
of the two levels approach. Many authors [18, 34], prefer to use the stiffness
tensor to describe the anisotropic properties because of his better agreement with
experiments.
The effective thermal expansion tensor can be also expressed as a function of the
second order orientation tensor [1]
α c = P1a + P21 (15.33)
P1 and P2 are two constants related to the unidirectional thermal properties:
15.4 Results and Discussion
P1 = α1 − α2
P2 = α2
(15.34)
15.4.1 Choice of a Micromechanical Model for the Unidirectional
Properties
In order to compare the three models listed in Sect. 15.3, a composite reinforced
by spherical inclusions is considered (β = 1). The predicted longitudinal Young
modulus is compared to experimental measurements conducted by Simth [38].
The Young’s modulus of the matrix and particles are respectively Em = 3GPa,
νm = 0.4, E f = 76 GPa, and ν f = 0.23. The comparison between predicted and
effective longitudinal Young modulus is done for increasing fiber volume fractions
(Fig. 15.9).
Figure 15.9 shows that Mori-Tanaka model gives the better prediction of the
longitudinal Young modulus for volume fraction going up to 20–30%.