Sanders’ Energy-Release Rate Integral for Arbitrarily Loaded Shallow Shells and Its Asymptotic Evaluation for a Cracked Cylinder

Sanders’ path-independent, energy-release rate integral I is specialized to an arbitrarily loaded shallow shell containing a stress-free void, assuming linear theory. Two forms of I are given. When the void is a crack, one form reduces to integrals over circles of vanishing radius centered at the tips and is expressible in terms of bending and stretching stressintensity factors. The other form reduces to an integral along the crack. For an elastically isotropic cylindrical shell containing a longitudinal crack and subject to a uniform bending stress at large distances from the crack, the dimensionless form of I depends on Poisson’s ratio v and a dimensionless crack length λ. When λ is small the shell is nearly flat; when λ is large the shell is very thin. An asymptotic formula is obtained for I as λ → ∞. This is done by reducing the boundary-value problem to a coupled set of singular integral equations, scaling, taking the limit as λ → ∞ to obtain inner and outer integral equations, solving the outer equations analytically, and, finally, evaluating I along the crack where the outer solutions dominate. As the evaluation of I does not require an explicit solution of the inner integral equations, an apparently intractible coupled Wiener-Hopf problem is evaded.