√Problem 3c

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.

© Derive the expression of K if a uniform pressure (p) [is] applied throughout the faces of the crack.

The expression for the stress intensity factor KP for a point load P applied at point b is,

K_P = {P sqrt{2}}/sqrt{pi b}.

The force P at a point under uniform pressure p along x is p dx.

By superposition, the stress intensity factor Kp for a pressure along x from the crack tip at x=0 to b will be,

K_p = int{0}{b}{ {p sqrt{2}}/sqrt{pi x} dx}.

= {p sqrt{2}/sqrt{pi}} int{0}{b}{ {1/sqrt{x}} dx } = {p sqrt{2}/sqrt{pi}} 2 sqrt{x} delim{|}{matrix{2}{1}{b 0}}{} = {p sqrt{2}/sqrt{pi}} 2 sqrt{b} = {2 p sqrt{2 b}/sqrt{pi}}

K_p = {{2 p}/pi}sqrt{2 pi b}.

If the pressure is applied throughout the entire crack face, as b goes to infinity Kp also increases indefinitely.


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