Using the results of Exercise 13.22, solve the problem of a stress-free spherical cavity in an infinite elastic medium under uniform far-field stress σ^∞_x =σ^∞_y = σ^∞_z = S. Explicitly show that the stress concentration factor for this case is K = 1.5 and compare this value with the corresponding two-dimensional case. Explain why we would expect such a difference between these two concentration factors.

Data from exercise 13.22

Using the basic field equations for spherical coordinates given in Appendix A, formulate the elasticity problem for the spherically symmetric case, where u_R = u(R), u_∅ = u_θ = 0. In particular, show that the governing equilibrium equation with zero body forces becomes:

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2 du 2 du + dR R dR R2 Next solve this equation and show that the general solution can be expressed as C u=CR+ OR = K - R2' 0=08=K+ where C, C2, K, and K are arbitrary constants. 2K2 R3, u=0 K2 R