Computational Methods for Debonding in Composites

10 Elastoplastic Modeling of Multi-phase Metal Matrix Composite 205
t+∆t S E
t+∆t
S =
(10.43)
1 + 2G∆λ
Also, by writing the rate of change of porosity equation into the incremental form,
we obtain
∆ f = 3(1 − t+∆t f )∆e P m + A∆ēP
(10.44)
where A = A1 + A2. Likewise, for the equivalence of plastic work, we have
σσσ : ∆ e P =(1 − f ) ˆσy∆ēe P
(10.45)
with
t+∆t
ˆσy = t+∆t ˆσy( t ē P + ∆ē P ) (10.46)
After expanding the term σσσ : ∆ eP , we rewrite the equivalence of plastic work at the
end of the time step as
∆λ t+∆t S : t+∆t S + 3 t+∆t σm t+∆t e P m =(1 − t+∆t f ) t+∆t ˆσy∆ē P = 0 (10.47)
Then, defining the functional P as
t+∆t P ≡ ∆λ t+∆t S : t+∆t S + 3 t+∆t σm t+∆t e P m − (1 − t+∆t f ) t+∆t ˆσy∆ē P
(10.48)
Clearly, P = 0 has to be satisfied at the end of all time steps by definition. Likewise,
the yield function has to be equal to zero at the end of all time steps for plastic
loading, that is t+∆t F ( t+∆t S, t+∆t σm, t+∆t ˆσy, t+∆t f )=0 or explicitly,
1t+∆t
S :
2
t+∆t t+∆t ˆσ 2 �
y
S − 2
3
t+∆t f ∗ q1 cosh
� 3q2 t+∆t σm
2 t+∆t ˆσy
�
− 1 − q 2 3( t+∆t f ∗ ) 2
�
= 0
(10.49)
Up to this point, we have presented all the relevant equations for the Gurson-
Tvergaard model. In summary, we have to solve two nonlinear algebraic equations
for every time step.
t+∆t P(∆ē P , ∆e P m )=0
t+∆t F (∆ē P , ∆e P m )=0
where ∆ē P and ∆e P m are the two unknowns (where ∆ē P is the governing parameter).
After solving for ∆ē P and ∆e P m , we can determine ∆ f and then f ∗ .Furthermore,we
can obtain F ′ , ∆λ, S and σm. With these variables determined, one can obtain the
updated local stress for all phases using the TFA equations.
10.4.2 Newton’s Method
For the Gurson-Tvergaard model, there is a system of two nonlinear equations,
namely t+∆t P = 0and t+∆t F = 0. That is, we have two unknowns at the end of