On the Calculation of Energy Release Rates for Cracked Laminates with Residual Stresses

Prior methods for calculating energy release rate in cracked laminates were extended to account for heterogeneous laminates and residual stresses. The method is to partition the crack tip stresses into local bending moments and normal forces. A general equation is then given for the total energy release rate in terms of the crack-tip moments and forces and the temperature difference experienced by the laminate. The analysis method is illustrated by several example test geometries. The examples were verified by comparison to numerical calculations. The residual stress term in the total energy release rate equation was found to be essentially exact in all example calculations. Introduction Schapery and Davidson,1 Hutchinson and Suo,2 and Williams3 derived general methods for calculating energy release rates in a variety of laminate specimens from the crack-tip values for bending moments, normal forces, and shear forces.3 References [1] and [2] considered heterogeneous beams. Although Ref. [3] considered only homogeneous beams, it was noted that it is trivial to extend it to include any heterogeneous elastic arrangement of the layers. When heterogeneity is present including differential thermal expansion properties and the structure is subjected to a change in temperature, however, there will be residual stresses that are not included in prior analyses and may contribute to energy release rate. This paper extends the prior methods to a general method for calculating the role of residual stresses in fracture of cracked laminates subjected to a uniform change in temperature. Energy Release Rate with Residual Stresses Figure 1 shows a general laminated or multilayered structure having a crack with moments (M (0) 1 , M (0) 2 , and M (0) 3 ) applied to each arm and axial forces (N1, N2, and N3) on each arm resolved into point loads applied at arbitrary locations (n1, n2, and n3). By force and moment balance, the resultants local to the small crack tip region of length δL are: N3 = N1 +N2 (1) M1 = M (0) 1 +N1 ( h1 2 − n1 ) (2) M2 = M (0) 2 +N2 ( h2 2 − n2 ) (3) M3 = M1 +M2 −N1(h2 − n3 + n1) +N2(n3 − n2) (4) The “layers” indicate the possibility of multiple layers within each arm of the general composite beam. Shear forces may be included (as discussed in Ref. [3]), but are not included here because prior results on laminate residual stress effects suggest that shear corrections are not needed to analyze residual stress energy