Page 95 - Engineered Interfaces in Fiber Reinforced Composites

78 Engineered interfaces in fiber reinforced composites

be constructed from which the differential compliance can be determined. Knowing
the load, P, and the differential, dC/da, the GI, values at any crack length can be
evaluated using Eq. (3.16).
A classic expression for the compliance, C, can be obtained by taking into account
the strain energy due to the bending moment for a perfectly elastic and isotropic
material

2a3
C=- (3.18)
3EI ’
Because I = bh3/12,
8a3
C=- (3.19)
Ebh3 ’

where I is the second moment of area, and h, the half specimen thickness of the DCB
specimen. Combining Eqs. (3.16)-(3.19) yields the strain energy release rate:

P2a2
GI, = - (3.20)
bEI ’
3P6
GI, =- . (3.21)
2ba
Eqs. (3.20) and (3.21) are called the ‘load method’ and ‘displacement method’
(Hashemi et al., 1989), respectively, since only load and displacement records
corresponding to crack lengths are required to evaluate the GI, values, once the
flexural modulus E of the beam is known. These equations also apply to the
WTDCB specimens where the ratio of the crack length to width, a/b, is constant.
As opposed to the simplifying assumption of isotropic materials for Eqs. (3.20)
and (3.2 l), practical composite components are mostly anisotropic and consist of
orthotropic laminates. Further, there are a number of factors that cannot be
properly accounted for in the elastic beam theory as a consequence of the various
aspects of the practical DCB test. These include end rotation and deflection of the
crack tip, effective shortening of the beam from a large deflection of the arms, and a
stiffening effect of the beam due to the presence of the end tabs or hinges bonded to
the specimens. All these factors cause the values of the apparent elastic modulus, E,
calculated from Eq. (3.20) to vary with displacement or crack length. Therefore, a
number of different analytical equations have been proposed in various forms to
ascertain the correction factors in interpreting the experimental data. Among these,
Williams and coworkers (Hashimi et al., 1990a, b; Wang and Williams, 1992) have
presented one of the most rigorous analyses

F 3P6
G- (3.22)
IC - N2b(a + Xh) ’