Derive the middle-surface strains and curvatures

Question

Solve the following problem:

Given the following displacements and rotation functions

u_o = A cos(αx) sin (βy)

v_o = B sin(αx) cos (βy)

w  = C sin(αx) sin (βy)

Φ_x = Q cos(αx) sin (βy)

Φ_y = R sin(αx) cos (βy)

(a) Derive the middle-surface strains and curvatures.

(b) Derive the strain functions as a function of x, y, and z.

(c) For A = B = C = 0.001, Q = R = 2, α = β = π, x = y = 1/2, t = 0.002, evaluate and plot the strain as a function of the thickness coordinate z.