Problem 2.37

2.37.  Before answering the question, please show first that the strains are compatible. Also, note that the integral of a partial derivative such as u,_x  contains an arbitrary function of y.

e_x = (s_x - ns_y)/E = c(y² + nx²)/E

e_y = (s_y - ns_x)/E = -c(x² + ny²)/E

g_xy = 0

Check compatibility

e_x,_yy - g_xy,_xy + e_y,_xx  =  2c/E + 0 - 2c/E  =  0,   Yes     

Ee_x = Eu,_x = c(y² + nx²)

Eu = c(xy² + nx³/3) + g(y)

Ee_y = v,_y = -c(x² + ny²)

Ev = -c(x²y + ny³/3 ) + f(x)

g_xy = u,_y + v,_x  = 0;   2cxy + g'(y) - 2cxy + f'(x) = g'(y) + f'(x) = 0

The most general solution is

f'(x) = -g'(y) = arbitrary constant = C

f(x) = Cx + A

g(y) = -Cy + B

Eu = c(xy² + nx³/3)  - Cy + B

Ev = -c(x²y + ny³/3 ) + Cx + A

Interpret A, B, C:

A = arbitrary x-translation

B = arbitrary y-translation

C = arbitrary small rotation about the origin.