Computational Methods for Debonding in Composites

4 Analytical and Numerical Investigation of the Length of Cohesive Zone 81
Therefore, the length of the cohesive zone under mode I loading for isotropic
materials can be written as a function of the fracture toughness of the material GIc:
l ∞ cz = M E′ GIc
(τ o ) 2
(4.8)
For orthotropic materials under mode I or mode II loading, the relation between
the energy release rate and the stress intensity factor can be written as [9]:
GI = K 2 I
� � 1
a11a22 2
2
GII = K 2 a11
II √
2
��a22 � 1
2
a11
��a22 � 1
2
a11
+ 2a12 + a66
2a11
+ 2a12 + a66
2a11
� 1 2
� 1 2
(4.9)
(4.10)
where a11,a22,a12 and a66 are the components of the compliance matrix.
Therefore, the length of the cohesive zone for orthotropic materials under pure
mode I or mode II loading can be written as:
l ∞ Icz = MI
l ∞ E
IIcz = MII
′ IIGIIc �
τo shear
where E ′ I and E′ II are obtained from Eqs. (4.9) and (4.10) as:
E ′ I =
E ′ II =
� �
a11a22
− 1
2
2
� a11
√2
��a22 � 1
2
a11
� �
−1 �a22
� 1
2
a11
E ′ IGIc � � (4.11)
τo 3
� (4.12)
+ 2a12 + a66
2a11
+ 2a12 + a66
2a11
�− 1 2
�− 1 2
(4.13)
(4.14)
Under plane stress, a11 = 1
E11 , a22 = 1
E22 , a12 = − ν12
E22 and a66 = 1 . Therefore,
G12
Eqs. (4.13) and (4.14) can be written for plane stress problems as:
E ′ I =
E ′ II = E22
Q
� �
E22
Q
� � 1
E11
2
E22
(4.15)
(4.16)