— David Wagner 2009/10/25 22:49

√ME 5463-001 Fracture Mechanics, Homework #16

Verify that the equation relating G with K of a single-ended crack under the plane stress condition is the same as that for a central crack.

[Try Eq. 7.7a and K_c for a single ended crack and see 7.78a and b1.]

K_c = 1.12 sigma_c sqrt{pi a} ⇒ sigma_c = K_c/{1.12 sqrt(pi a}} ⇒ sigma_c = K_c/{1.12 sqrt(pi a}}

W_s = 2 a gamma_s

{d/{da}}(2a gamma_s - {{pi a^2 {sigma_c}^2}/{E}}) = 0 = 2 gamma_s - {2 pi a {sigma_c}^2}/E ⇒ 2 gamma_s = {2 pi a {sigma_c}^2}/E ⇒ sigma_c = sqrt{ {gamma_s E}/{pi a} }= K_c/{1.12 sqrt(pi a}} ⇒ {K_c}^2 = {1.25 gamma_s E} = E G_c

The constant is absorbed into G_c by Irwin's assumptions sigma_c= sqrt{{E G_c}/{pi a}}.


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