transverse crack in a laminated composite (bi-material interface) can be found in Ting's

analysis [14-15]. The details of the Ting's methods are described in Appendix A. In this

approach the general solutions of displacement and stresses are derived using strain-

displacement, stress-strain and equilibrium equations in terms of arbitrary constants and

singularities. The singularities are determined when the boundary conditions on the crack

surfaces and continuity conditions at the interface are satisfied.

 The analytic approach is performed to estimate the singularity of a transverse crack

of glass/epoxy and graphite/epoxy (IM7/977-2) composites with stacking sequence [03]T,

[0/90/0]T and [90/0/90]T in Figure 3-1. The crack tip of the transverse crack is placed at

the interface between top and mid layers. The laminate properties of graphite/epoxy and

glass/epoxy are shown in Table 3-1 [26]. The singularity results are compared with the

finite element results. The commercial computer program MATLAB is used to solve the

analytical equations.

Table 3-1. Material properties of glass/epoxy and carbon/epoxy laminates.
 E-glass/epoxy Graphite/epoxy
 E11 (GPa) 38.6 160.9
 E22, E33 (GPa) 8.27 9.62
 G12, G13 (GPa) 4.14 6.32
 G23(GPa) 3.18 4.22
 V12, V13 0.25 0.32
 V23 0.3 0.14
 a'l (10-6 /C) 8.60 -0.512
 U22, U-33 (106 /Co) 22.1 16.3


 Finite element analysis is used not so much to verify the results from the

analytical model but to determine the mesh refinement needed to obtain the proper

singularity at the crack tip. In the finite element analysis a laminated composite beam was

modeled using 8-node solid elements with 20 integration points as shown in Figure 3-2.