This document contains two problems involving the analysis of strains and curvatures in layered composite materials. Problem 1 involves calculating the extensional and bending matrices of a bimetallic strip made of copper and aluminum layers. Problem 2 gives displacement and strain functions to derive middle surface strains and curvatures, and express strains as a function of thickness coordinate z, plotting the results.

1. Problem 6.1
Compute the extensional coupling and bending matrices of a bimetallic(copper
and aluminum) strip with each layer 2 mm thick. For aluminum,
. For copper, . Assume Poisson's ratio is
0.3 for both materials.
Solution
Here the materials are metals (homogeneous)
Also, there is no temperature change.
Give the aluminum the subscript "1" and the copper the subscript "2".
We have 2 layers each have
The metal layers will be treated as composite laminates.
For aluminum layer:
For copper layer:

2. Problem 6.2
Given the following displacements and relation functions,
Where A, B, C, Q, and R are constants.
a) Derive the middle surface strains and curvatures.

3. b) Derive the strain functions as a function of x, y, and z.
c) For evaluate
and plot the strains as a function of the thickness coordinate z.
Solution
a) Strains and curvatures of middle plane:
b) From requirement no. (a) the expressions of are the
requirements.
c) I took the limits of z from –t/2 to +t/2 (over the cross section thickness)
Results are shown below

5. Appendix
MATLAB code for plotting strains vs Z of requirement (C) f problem
6.2
clear;clc;close all
A=0.001; B=A; C=A;
Q=2; R=Q;
alfa=pi; beta=alfa;
x=0.5; y=x;
t=0.002;
z=linspace(-t/2,t/2,100);
eps_xx=-A*alfa*sin(alfa*x)*sin(beta*y)-z*(-
Q*alfa*sin(alfa*x)*sin(beta*y));
eps_yy=-beta*B*sin(alfa*x)*sin(beta*y)-z*(-
beta*R*sin(alfa*x)*sin(beta*y));
eps_xy=0.5*(beta*A*cos(alfa*x)*cos(beta*y)+alfa*B*cos(alfa*x)*cos(bet
a*y))-
z*(beta*Q*cos(alfa*x)*cos(beta*y)+alfa*R*cos(alfa*x)*cos(beta*y));
figure
plot(eps_xx,z)
xlabel('epsilon_x_x')
ylabel('z')
figure
plot(eps_yy,z)
xlabel('epsilon_y_y')
ylabel('z')
figure
plot(eps_xy,z)
xlabel('epsilon_x_y')
ylabel('z')