Computational Methods for Debonding in Composites

10 Elastoplastic Modeling of Multi-phase Metal Matrix Composite 219
10.6 Closing Remarks
In this paper, the combined TFA-GPM approach is implemented to predict the
overall elastoplastic behavior of n-phase fiber-reinforced composites. The localization
rule is based on the TFA method while the prediction of the mechanical
concentration factors is estimated by the EMT theory. The yield criterion of the
matrix phase is extended from the von-Mises yield criterion to the Gurson-Tvergaard
yield criterion so as to take into account the effects of void growth under plastic
deformation. The results show that the Gurson-Tvergaard yield criterion does provide
a better approximation in the prediction of overall elastoplastic behavior of
different types of 2-phase fiber-reinforced composites. Furthermore, simulation of a
4-phase fiber-reinforced composite is also provided, this certainly demonstrates the
power of the existing model. More importantly, a rigorous mathematical analysis of
the possible ranges of ∆eP m and ∆ēP has been provided. This certainly reduces the
computational effort for guessing the possible values of ∆eP m and ∆ēP . In Sect. 10.5,
the evaluation and verification of the proposed TFA-GPM model under different
cases have been accomplished. This assures the validity of the proposed model and
it provides a level of confidence for other applications.
In summary, the two significant contributions in this research work include:
1. The Gurson-Tvergaard yield criterion gives better results compared to the von-
Mises yield criterion since the effect of void growth has been considered in the
analysis.
2. A rigorous mathematical analysis was performed to obtain the necessary conditions
within the ranges for the change of the mean plastic strain ∆eP m and the
change of the effective plastic strain ∆ēP . This certainly provides a better start
for the iteration of the algorithm.
Appendix – The Four Partial Derivatives
∂P
=
)
∂(∆e P m
∂F
∂(∆e P m ) = − 2GSE : S E
+ 2q1 ˆσ 2 y
3
S E : S E
(1 + 2G∆λ) 2
�
1 − 4G∆λ
�
∂(∆λ)
1 + 2G∆λ ∂(∆e P m )
+3(σ E m − 2cm∆e P m )+ ˆσy∆ē
P ∂(∆ f )
∂(∆e P m)
(10.70)
(1 + 2G∆λ) 3
∂(∆λ)
∂(∆e P m ) − 2q23 ˆσ 2 y f ∗
∂ f
3
∗
∂(∆e P m )
�
∂ f ∗
∂(∆e P � �
3q2σm
cosh −
m ) 2 ˆσy
3q2cm f ∗ � ��
3q2σm
sinh
(10.71)
2 ˆσy 2 ˆσy