Computational Methods for Debonding in Composites

7 Interaction Between Intraply and Interply Failure in Laminates 155
For the evaluation of the traction, the equivalent displacement jump is calculated
via
�
δb = 〈δn〉 2 + δ 2 r
(7.62)
where subscript b refers to the current mixed mode and δr is defined as the total shear
displacement jump: δ 2 r = δ 2 s + δ 2
t . Normal relative displacements only contribute
when positive, hence the use of the MacAuley operator, which is defined as 〈x〉 =
(x + |x|)/2.
The mixed mode ratio β and the related variable B are computed according to
β =
B =
δr
δr + 〈δn〉
β 2
1 + 2β 2 − 2β
(7.63)
(7.64)
The onset criterion and the propagation criterion for the current mixed mode ratio
are
δ 0 b =
�
(δ 0 n ) 2 �
+ (δ 0 r ) 2 − (δ 0 n ) 2�
Bη (7.65)
and
δ f
b =
δ 0 n δ f
�
n + δ 0 r δ f
r − δ 0 n δ f
�
n Bη δ 0 (7.66)
b
respectively, where η is a material parameter that is to be obtained from experimental
data and is related to mode interaction.
With this, the bilinear constitutive law for the current mixed mode ratio is complete
and the damage variable can be updated. Upon monotonic loading, the damage
variable, d, is calculated with
f �
δb δb − δ
d = 0 �
� b
δb δ f
b − δ 0 � , d ∈ [0,1] (7.67)
b
To obtain the correct loading-unloading-reloading behavior, the following condition
is added
d˙ ≥ 0 (7.68)
which demands storage of d as history variable at each integration point, similar to
the description by Jiang et al. [12], which is different from the description by Turon
et al. [25], where the maximum total displacement jump, δb, is stored as history
variable and substituted into Eq. (7.67).