Computational Methods for Debonding in Composites

146 F.P. van der Meer and L.J. Sluys
The derivatives of f and g with respect to ∆κ and σ read:
with
∂ f
∂∆κ
1 ∂h
= − (σ · Pσ + σ · p) (7.16)
h ∂∆κ
∂ f
= Pσ + p (7.17)
∂σ
∂g
∂m
= Dm∂∆λ + ∆λ D
∂∆κ ∂∆κ ∂∆κ
∂g
∂σ
= I + Dm⊗ ∂∆λ
∂σ + ∆λ D∂m
∂σ
(7.18)
(7.19)
∂h V
= H +
∂∆κ ∆t
(7.20)
∂m
= P
∂σ
(7.21)
∂m
∂∆κ
∂∆λ
∂σ
1 ∂h
= − (2Pσ + p) (7.22)
h ∂∆κ
∂∆λ ∂m
=
∂m ∂σ
∂∆λ
∂∆κ =
1
√ +
m · Qm ∂∆λ ∂m
·
∂m ∂∆κ
∂∆λ
∂m
= − ∆κQm
(m · Qm) 3/2
(7.23)
(7.24)
(7.25)
The return mapping algorithm is presented in Fig. 7.2, in which B is a matrix
containing the partial derivatives of f and g with respect to ∆κ and σ
⎡ ⎤
∂ f ∂ f
⎢
B =
∂∆κ ∂σ⎥
⎣ ⎦ (7.26)
∂g ∂g
∂∆κ ∂σ
and D con is the consistent tangent (for derivation, see Sect. 7.2.3).
In the algorithm, ∆κ is initialized at a nonzero value. For this purpose the value
for ˙κ is stored for each integration point, additional to the state variable κ. Inorder
to improve the stability of the return mapping algorithm, the Newton Raphson loop
that solves for ∆κ and σ is reformulated in a form analogous to Heun’s method.
When the strength approaches zero, this algorithm may fail, due to the singularity
of f for h = 0. When the first algorithm fails, another return mapping scheme is
entered. In this algorithm, there is a double iteration loop. The outer loop searches
for the correct value of ∆κ, while the inner loop solves the plasticity equations for