Derive the strain functions

Question

Response to the following:

1. Compute the extensional [A], coupling [B], and bending stiffness [D] matrices of a bimetallic (copper and aluminum) strip with each lamina 2 mm thick. For aluminum E = 71 GP a, α = 23(10^-6 )/ ?C. For copper E = 119 GP a, α = 17(10^-6 )/ ?C.

Assume the Poisson's ratio is 0.3 for both materials.

2. 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)

where A, B, C, Q, and R are constants.

(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.