√Problem 3b

ME 5463-001 Fracture Mechanics Exam #1 — David Wagner 2009/10/11 14:16

3. The Westergaard stress function that solves the opening mode problem of a semi-infinite crack subjected to a crack-line force, P, as shown in Figure E3.4 are given by:

Z(z) = {P}/{pi} {1}/{z+b} sqrt{b/z} and Y(z) = 0.

(b) Using the formal definition of the geometric stress intensity factor, derive an expression for K for this combination of geometry and loading.

K = lim{delta+ right 0}{sigma_y |_theta=0} sqrt{2 pi delta}.

The crack tip is at the origin z = 0 + 0i, so let delta=x.

At theta=0, y=0 and z=x, so Z(z) = {P}/{pi} {1}/{z+b} sqrt{b/z} = Z(x + yi) = Z(x) = {P}/{pi} {1}/{x+b} sqrt{b/x}.

The imaginary part of Z(x + 0i) is zero as long as b/x ≥ 0, which it is when approaching the crack tip from the right, as x+ →0.

With Z real under these conditions, Z' is also real, so its imaginary part is zero.

  • Im Z prime(x+) = 0.

sigma_y = Re Z + y(Im Z prime + Im Y prime) =Re Z(x +0i) = {P}/{pi} {1}/{x+b} sqrt{b/x}.

K = lim{delta+ right 0}{sigma_y |_theta=0} sqrt{2 pi delta} = lim{x+ right 0}{{P}/{pi} {1}/{x+b} sqrt{b/x}} sqrt{2 pi x} = {{P}/{pi} {1}/{0+b} sqrt{b}} sqrt{2 pi} = {P sqrt{2 pi b}}/{pi b}

K = {P sqrt{2}}/sqrt{pi b}


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