Problem 2.1

2.1.  Note that the state of strain in part (b) depends on x, y, and z, and is thus three-dimensional.

Check strain compatibility equations.

a)  e_x = c(x² + y²),   g_xy = 2e_xy  = 2cxy     , e_y = y²,  other strains are zero.

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

b)  e_x = cz(x² + y²),  g_xy = 2e_xy  = 2cxyz,   e_y = y²z,  other strains are zero.

e_x,_yy - g_xy,_xy + e_y,_xx  =  2cz -2cz + 0  =  0

e_y,_zz - g_yz,_yz + e_z,_yy  =  0 + 0 + 0 = 0

e_z,_xx - g_zx,_zx + e_x,_zz  =  0 + 0 + 0 = 0

2e_x,_yz - (-g_yz,_x + g_zx,_y + g_xy,_z),_x   =  4cy - (0 + 0 + 2cxy)_,x  = 4cy - 2cy ≠ 0,  No   unless c = 0

2e_y,_zx- (-g_zx,_y + g_xy,_z + g_yz,_x),_y   =  0 - (0 + 2cxy + 0),_y =  0 if c = 0

2e_z,_xy- (-g_xy,_z + g_yz,_x + g_zx,_y),_z   =  0 - (- 2cxy + 0 + 0),_z = 0