write a general program which computes the constitutive matrix,

 

(1 ) write a general program which computes the constitutive matrix,
Aij= (A11,A12, A16, A22, A26, A66)
Bij= (B11, B12, B16, B22, B26, B66)
Dij= (D11, D12, D16, D22, D26, D66)

Base on input parameter,
Given N= total number of layers
Given fiber orientations (theta1,theta2, theta3, …,thetaN-1, ThetaN)
Given E1/E2= 10,
given poisson12 = 0.25 Therefore poisson21 = poison 12 (E2/E1)

(1a) we all know for each layer, we have
Q matrix for the kth layer = (Q11,Q12, Q16, Q22,Q26,Q66)
We define Q non-dim = (1/(E2)) (Q11,Q12,Q16,Q22,Q26,Q66)

For example, use equation 2.66, Q11 /E2 = (E1/E2)/(1-poisson12*poisson 21)
Computer Q non-dim
The above Q is assume we are referring to theta= 0

We now specialize the above to arbitrary value of the fiber angle, see equation 2.85, page 77
Q11bar (1/E2) = (1/E2) [ Q11 cos4 + 2(Q12+2Q66) sin2cos2 + Q22 sin4]

A11 non-dim = A11 /(E2t) A12 non-dim = A12 / (E2t) A16 non-dim= A16/ (E2t)
A22 non-dim = A22 /(E2t) A26 non-dim = A26 / (E2t) A66 non-dim= A66/ (E2t)

 

Then use equation 4.24 , page 198, to compute
A11/(E2t) = (1/E2t) [ summation from k=1 to k=N for Q11bar(function of theta) . (zk-z k-1)]
A12/(E2t) = (1/E2t) [ summation from k=1 to k=N for Q12bar(function of theta) . (zk-z k-1)]

A16/(E2t) = (1/E2t) [ summation from k=1 to k=N for Q16bar(function of theta) . (zk-z k-1)]

A22/(E2t) = (1/E2t) [ summation from k=1 to k=N for Q22bar(function of theta) . (zk-z k-1)]

A26/(E2t) = (1/E2t) [ summation from k=1 to k=N for Q26bar(function of theta) . (zk-z k-1)]

A66/(E2t) = (1/E2t) [ summation from k=1 to k=N for Q66bar(function of theta) . (zk-z k-1)]

Remember t is total thickness of the entire laminate
heis the thickness of each individual layer, therefore, (N) ( he) = t

Similarly, compute Bij non-dim, and computer Dij non-dim where

B11 non-dim = B11 /(E2t2) B12 non-dim = B12 / (E2t2) B16 non-dim= B16/ (E2t2)
B22 non-dim = B22 /(E2t2) B26 non-dim = B26 / (E2t2) B66 non-dim= B66/ (E2t2)

 

D11 non-dim = D11 /(E2t3) D12 non-dim = B12 / (E2t3) D16 non-dim= D16/ (E2t3)
D22 non-dim = D22 /(E2t3) D26 non-dim = D26 / (E2t3) D66 non-dim= D66/ (E2t3)

 

2. do the specific example of layups

2a (0,0,0,0,90,90,90,90,0,0,0,0,90,90,90,90)
Question, is the above the same as (0,90,0,90) if so, why
(90,90,90,90,0,0,0,0,90,90,90,90,0,0,0,0)
Question, is the above the same as (90,0,90,0) if so, why

2b (0,45,90,-45,0,45,90,-45)symm this means total of 16 layers

2c for case of total of only four layers, computer by hand, and also by computer
And make sure your hand calculations agrees with your computer
calculations for Aijnon, Bijnon, Dij non

(22, -17,-43, -36) total of four layers
(32,-45,-23,11) total of four layers
(0,0,0,0) total of four layer

2d do the special case of isotropic orthotropic layers
(0,0,0,0) but using E1/E2= 1.0

 

3. read chapter 16,William D. Callister book “Materials Science and Engineering”
Do problems, 16.2, 16.6, 16.9, 16.10