Table of Contents
√Problem 3a

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.

(a) Derive an exact expression for the Cartesian stress sigma_y valid everywhere throughout the body.

Setup

First, define some coordinate systems.

  • z = x +iy = r_0 e^{i theta_0} = r_0(cos theta_0 + i sin theta_0),
    • cos theta_0 = x/{r_0} = x/{sqrt{x^2+y^2}}, sin theta_0 = y/{r_0} = y/{sqrt{x^2+y^2}};
  • z+b = (x+b) +iy = r_1 e^{i theta_1} = r_1(cos theta_1 + i sin theta_1)
    • cos theta_1 = {x+b}/{r_1} = {x+b}/{sqrt{(x+b)^2+y^2}}, sin theta_1 = y/{r_1} = y/{sqrt{(x+b)^2+y^2}}.

Find Re Z

Z(z) = {P}/{pi} {1}/{z+b} sqrt{b/z} = {P sqrt{b}}/{pi} (z+b)^{-1} z^{-1/2} = {P sqrt{b}}/{pi} {r_1}^{-1} e^{-theta_1 i} {r_0}^{-1/2} e^{{-1/2}theta_0 i} = {P sqrt{b}}/{pi} {1/{r_1 sqrt{r_0}}} (cos{-theta_1} + i sin{-theta_1}) (cos{-1/2}theta_0 + i sin{-1/2}theta_0).

⇒ Re Z(r_0,theta_0,r_1,theta_1) = {P sqrt{b}}/{pi} {1/{r_1 sqrt{r_0}}} (cos{-theta_1} cos{-1/2}theta_0 - sin{-theta_1} sin{-1/2}theta_0 ).

Now, cos(-α) = cos(α) and sin(-α) = -sin(α), so,

Re Z(r_0,theta_0,r_1,theta_1) = {P sqrt{b}}/{pi} {1/{r_1 sqrt{r_0}}} ( cos{theta_1} cos{1/2}theta_0 - sin{theta_1} sin{1/2}theta_0 )

= {P sqrt{b}}/{pi} {1/{r_1 sqrt{r_0}}} ( cos{theta_1} cos{1/2}theta_0 - sin{theta_1} sin{1/2}theta_0 ).

Find y Im Z'

Z'(z) = -{P}/{pi} sqrt{{b}/{z}} {3z+b}/{2z(z+b)^2} = -{P sqrt{b}}/{2 pi} z^{-3/2}(3z+b)(z+b)^{-2}

⇒ Z prime (r_0,theta_0,r_1,theta_1) = -{P sqrt{b}}/{2 pi} {r_0}^{-3/2} e^{-3/2 theta_0 i} (3 r_0 e^{theta_0 i} +b) {r_1}^{-2} e^{-2 theta_1 i}

= -{P sqrt{b}}/{2 pi} (3 {r_0}^{-1/2} e^{-1/2 theta_0 i} +b {r_0}^{-3/2} e^{-3/2 theta_0 i}) {r_1}^{-2} e^{-2 theta_1 i}

= -{P sqrt{b}}/{2 pi} (3 {r_0}^{-1/2} (cos{-1/2 theta_0} + i sin{-1/2 theta_0}) +b {r_0}^{-3/2} (cos{-3/2 theta_0} + i sin{-3/2 theta_0})) {r_1}^{-2} (cos{-2 theta_1} +i sin{-2 theta_1})

= -{P sqrt{b}}/{2 pi}{r_1}^{-2} ( 3 {r_0}^{-1/2} cos{-1/2 theta_0} + i 3 {r_0}^{-1/2} sin{-1/2 theta_0} + b {r_0}^{-3/2} cos{-3/2 theta_0} + i b {r_0}^{-3/2} sin{-3/2 theta_0} ) (cos{-2 theta_1} +i sin{-2 theta_1})

= -{P sqrt{b}}/{2 pi}{r_1}^{-2} ( 3 {r_0}^{-1/2} cos{-1/2 theta_0} cos{-2 theta_1} + i 3 {r_0}^{-1/2} sin{-1/2 theta_0}cos{-2 theta_1} + b {r_0}^{-3/2} cos{-3/2 theta_0} cos{-2 theta_1} + i b {r_0}^{-3/2} sin{-3/2 theta_0} cos{-2 theta_1}

+ i 3 {r_0}^{-1/2} cos{-1/2 theta_0} sin{-2 theta_1} - 3 {r_0}^{-1/2} sin{-1/2 theta_0} sin{-2 theta_1} +i b {r_0}^{-3/2} cos{-3/2 theta_0} sin{-2 theta_1} - b {r_0}^{-3/2} sin{-3/2 theta_0}sin{-2 theta_1})

= -{P sqrt{b}}/{2 pi}{r_1}^{-2} ( 3 {r_0}^{-1/2} cos{-1/2 theta_0} cos{-2 theta_1} + b {r_0}^{-3/2} cos{-3/2 theta_0} cos{-2 theta_1} - 3 {r_0}^{-1/2} sin{-1/2 theta_0} sin{-2 theta_1} - b {r_0}^{-3/2} sin{-3/2 theta_0}sin{-2 theta_1}

+ i 3 {r_0}^{-1/2} cos{-1/2 theta_0} sin{-2 theta_1} + i b {r_0}^{-3/2} cos{-3/2 theta_0} sin{-2 theta_1} + i 3 {r_0}^{-1/2} sin{-1/2 theta_0}cos{-2 theta_1} + i b {r_0}^{-3/2} sin{-3/2 theta_0} cos{-2 theta_1} )

y=r_0 sin theta_0⇒

y Im Z' = -{P sqrt{b}}/{2 pi}{r_1}^{-2}r_0 sin theta_0 (3 {r_0}^{-1/2} cos{-1/2 theta_0} sin{-2 theta_1} + 3 {r_0}^{-1/2} sin{-1/2 theta_0} cos{-2 theta_1} + b {r_0}^{-3/2} cos{-3/2 theta_0} sin{-2 theta_1} + b {r_0}^{-3/2} sin{-3/2 theta_0} cos{-2 theta_1} )

Calculate σ_y

sigma_y = ReZ + y(Im Z prime + Im Y prime)

Im Y prime = 0⇒ sigma_y = ReZ + y Im Z prime=

{P sqrt{b}}/{pi} {1/{r_1 sqrt{r_0}}} ( cos{theta_1} cos{1/2}theta_0 - sin{theta_1} sin{1/2}theta_0 ) + {P sqrt{b}}/{2 pi}{r_1}^{-2}r_0 sin theta_0 (+3 {r_0}^{-1/2} cos{1/2 theta_0} sin{2 theta_1} + 3 {r_0}^{-1/2} sin{1/2 theta_0} cos{2 theta_1} + b {r_0}^{-3/2} cos{3/2 theta_0} sin{2 theta_1} + b {r_0}^{-3/2} sin{3/2 theta_0} cos{2 theta_1} )

= {P sqrt{b}}/{pi r_1}[ {r_0}^{-1/2} ( cos{theta_1} cos{1/2}theta_0 - sin{theta_1} sin{1/2}theta_0 ) + {1/2}{r_1}^{-1}r_0 sin theta_0 (3{r_0}^{-1/2} cos{1/2 theta_0} sin{2 theta_1} + 3 {r_0}^{-1/2} sin{1/2 theta_0} cos{2 theta_1} + b {r_0}^{-3/2} cos{3/2 theta_0} sin{2 theta_1} + b {r_0}^{-3/2} sin{3/2 theta_0} cos{2 theta_1} ) ]


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