Computational Methods for Debonding in Composites

7 Interaction Between Intraply and Interply Failure in Laminates 149
Fig. 7.5 Update of ∆κ in through linear interpolation
The first estimate is obtained assuming
h(κn, ˙κn)=h(κn−1, ˙κn−1) (7.32)
From Eq. (7.31), it follows that if ∆κ in is higher than the correct value of ∆κ,
∆κ out is lower than that value, and vice versa. So if we set
∆κ in
2
= ∆κout
1
it is secured that the correct value of ∆κ lies between ∆κin 1
the two conditions
∆κ in
2
h(∆κ in
2 ) ≥ hmin
(7.33)
and ∆κin
2 .Andifweadd
≥ 0 (7.34)
(7.35)
the search for the true ∆κ must converge when the next estimate for ∆κ in
k is each
time computed from linear interpolation (see Fig. 7.5), except when h(∆κ in
2 )=hmin
and h(∆κ out
2 ) < hmin. In that case the material point has failed and h is fixed at hmin
and the stress obtained with h(∆κ in )=hmin is the correct stress.
7.2.3 Consistent Linearization
For proper convergence of the model, it is of great importance that a consistent
tangent is used. The derivation of the consistent tangent is given below.
We start with expanding the constitutive law 7.12 around a small variation:
The expression for δm is:
δg = 0 ⇔ δσ = Dδε− Dmδλ − ∆λ Dδm (7.36)
δm = ∂m
∂σ
∂m
δσ+ δ∆κ (7.37)
∂κ