For the antiplane strain problem, verify that transformation (14.5.5) will reduce the governing equilibrium equation to relation

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For the antiplane strain problem, verify that transformation (14.5.5) will reduce the governing equilibrium equation to relation (14.5.6). Next show that the separation of variables scheme defined by (14.5.7) will lead to the two equations (14.5.8).

Equation 14.5.5

w (x,y) = W (x, y)/(x, y)

Equation 14.5.6

where uy =  " x' -   1 +; (; (? + u? ) - #x - M) w = q - Mxx Myy) W 0 22    uv2W + Mxx 2 Myy =

Equation 14.5.7

W(x, y)= X(x)Y(y), u(x,y) = op(x)q(y)

Equation 14 .5.8

2 +  + ! ( ) - ( ) ] x = 1 2 X 0 X Xxx 2 1(qy 1  x + [ -  + 1 (4) * - ! () ] x = 0 Y Yyy 49 2 9

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