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

1. Given the Airy stress function

$F(x,y) = -{P}/{d^3} x^2y^2(3d - 2y)$

a) determine the distribution of all Cartesian stress components in the body, sigma_x , sigma_x , $tau_{xy}$ ;

$F(x,y) = -{P}/{d^3} (3dx^2y^2 - 2x^2y^3)$ = $F(x,y) = {P}/{d^3} (2x^2y^3 -3dx^2y^2)$ .

${partial F}/{partial x}=P/d^3 (4 xy^3 - 6d xy^2)$ ; ${partial F}/{partial y}=P/d^3 (6 x^2y^2 - 6d x^2y)$ .

Cartesian Stress Components

$sigma_x = {partial^2 F}/{partial y^2}$ = $P/{d^3} (12 x^2 y -6 d x^2)$ = ${6P}/{d^3} (2 x^2 y -d x^2)$

$sigma_y = {partial^2 F}/{partial x^2}$ = $P/{d^3} (4 y^3 -6 d y^2)$ = ${2P}/{d^3} (2 y^3 -3 d y^2)$

$tau_{xy} = -{partial^2 F}/{partial x partial y}$ = $-P/{d^3} (12 xy^2 -12d xy)$ = $-{12P}/{d^3} (xy^2 -d xy)$