Computational Methods for Debonding in Composites

4 Analytical and Numerical Investigation of the Length of Cohesive Zone 83
toughness can be obtained from the pure mode fracture toughness of the material
using the expression proposed by Benzeggagh and Kenane [3]:
Gc = GIc +(GIIc − GIc)B η
(4.25)
where η is a mixed-mode interaction parameter. The equivalent interface strength
under mixed-mode loading, τo , can be related to the pure mode interface strengths
using the expression [19]:
(τ o ) 2 =(τ o 3 )2 �
+ (τ o shear )2 − (τ o 3 )2� B η
(4.26)
where τ o 3 and τo shear
respectively.
are the interface strengths under mode I and shear mode loading,
4.4 Generalization of the Length of the Cohesive Zone
for Finite-Sized Geometries
The models used for the estimation of the cohesive length outlined in Sect. 4.2
assume that the crack propagates unstably when the applied energy release rate is
equal to the fracture toughness of the material, Gc. However, depending on the specimen
geometry, unstable crack propagation occurs before the maximum value of the
fracture toughness is attained. The alternative method proposed here to predict the
length of the cohesive zone under mode I loading is based on the relation between
this length and the size effect experienced by the structure.
Schematically representing the R-curve, Fig. 4.2, it is observed that the applied
energy release rate that produces unstable propagation, GIu is equal to the value
where the R-curve and the GI curve are tangent.
The applied energy release rate is a function of the geometry:
GI(a,h)= 1
E ′ σ 2 Nhk(a,h) 2
(4.27)
where σN is a nominal stress [2], h is a geometry-dependent quantity, and k(a,h)
is the shape factor for KI. For a double-cantilever beam (DCB) specimen, taking h
as the thickness of the specimen arm and B the width of the specimen, the applied
energy release rate reads:
GI(a,h)= 12P2a2 (4.28)
E ′ B 2 h 2
By comparing Eqs. (4.27) and (4.28) the nominal stress σN and the shape factor
k(a,h) used in Eq. (4.27) are given as:
σN = P
bh
k(a,h)=2 √ 3 a
h
(4.29)
(4.30)