Solve a system of symbolic equations

Hi all,

I am discovering symbolic calculation in Matlab. I have to solve a system of equations that I don't want to solve by hand... :D

It is a mechanical problem based on the classical laminate theory for composite materials. The aim is to exploit data from tensile coupons tests.

% Stack-up of the coupon 
% [orientation (in °), thickness (in mm)]
plies = sym([-60 1; 60 1; 60 1; -60 1]);
plies(:,1) = plies(:,1).*pi/180;

syms E1 E2 G12 nu12 nu21;                           % Mechanical properties of the ply
syms Q11 Q12 Q22 Q66;                               % Stiffness matrix in the orthogonal frame
syms Q11L Q12L Q22L Q66L;                           % Stiffness matric in the laminate frame
syms A11 A12 A22 A66 D11 D12 D16 D22 D26 D66;       % Laminate ABD matrix

nu21 = nu12 * E2/E1;

Q11 = E1/(1-nu12*nu21);
Q12 = nu21*E1/(1-nu12*nu21);
Q22 = E2/(1-nu12*nu21);
Q66 = G12; 

Qply = [    Q11     Q12     0;
            Q12     Q22     0;
            0       0       Q66];

ATemp = zeros(3,3);
% Generates Aij of the laminate ABD matrix
for i=1:size(plis,1)

      ALam = ATemp + frameRotation(Qply, plies(i,1)).*plies(i,2);
      ATemp = ALam;

end
clear ATemp

subexpr(ALam)

% Test hypothesis:
% 1st tensile test in the direction 1
%  - Tensile load NL1
%  - 2 deformation gages dir 1 et 2 : j11 et j21
% 2nd tensile test in the direction 2
%  - Tensile load NL2
%  - 2 deformation gages dir 1 et 2 : j12 et j22

% System 1 :
% { NL1 = A11 j11 + A12 j21
% {   0 = A12 j11 + A22 j21
% System 2 :
% { NL2 = A22 j12 + A12 j22
% {   0 = A12 j12 + A11 j22
% ==> 4 equations / 4 unknowns (E1, E2, nu12, G12)

NL1 = (1000);
NL2 = (500);
j11 = (0.000001);
j21 = (0.0000001);
j12 = (0.0000002);
j22 = (0.000002);

solu = solve(ALam(1,1)*j11 + ALam(1,2)*j21 - NL1, ...
             ALam(1,2)*j11 + ALam(2,2)*j21, ...
             ALam(2,2)*j12 + ALam(1,2)*j22 - NL2, ...
             ALam(1,2)*j12 + ALam(1,1)*j22, ...
             E1, E2, G12, nu12)

And the frameRotation function :

function Qlam = frameRotation(Qply, theta)
% This function computes the ply stiffness matrix Q in the laminate frame
% Qlam = Qply(theta)

Q11 = Qply(1,1);
Q12 = Qply(1,2);
Q22 = Qply(2,2);
Q66 = Qply(3,3);

t = theta;

Q11L = Q11*cos(t)^4 + Q22*sin(t)^4 + 2*(Q12+Q66)*sin(t)^2*cos(t)^2;
Q12L = (cos(t)^4+sin(t)^4)*Q12 + (Q11+Q22-2*Q66)*sin(t)^2*cos(t)^2;
Q16L = (Q11-Q12-Q66)*sqrt(2)*sin(t)*cos(t)^3 + (Q12-Q22+Q66)*sqrt(2)*cos(t)*sin(t)^3;
Q22L = Q22*cos(t)^4 + Q11*sin(t)^4 + 2*(Q12+Q66)*sin(t)^2*cos(t)^2;
Q26L = (Q11-Q12-Q66)*sqrt(2)*sin(t)^3*cos(t) + (Q12-Q22+Q66)*sqrt(2)*cos(t)^3*sin(t);
Q66L = (cos(t)^4+sin(t)^4)*Q66 + (Q11+Q22-2*Q12-Q66)*2*sin(t)^2*cos(t)^2;

Qlam = [    Q11L    Q12L    Q16L;
            Q12L    Q22L    Q26L;
            Q16L    Q26L    Q66L];

The problem is that I don't understand the results that the solve function gives me :

>> [solu.E1 solu.E2 solu.G12 solu.nu12]

ans =

[ (961*z1)/169, z1,  z, -31/13]
[            0, z1,  z,      0]
[            0,  0, z1,      z]

What are these z and z1?? Do you understand where is the problem?

Thank you very much!

Max

Answer 1

[Walter Roberson]

MuPAD has found that the set of solutions is most easily expressed by introducing some temporary variables, z and z1, instead of fully writing out the same subexpression each time.

Imagine for example that one was solving a quadratic with real-valued solutions. Instead of writing out some long expression on both parts, with the second solution just being the negative of the first, MuPAD might introduce a temporary variable z to represent the long expression, and then say that the results are [z, -z] where z is ....

Unfortunately the interface between MuPAD and MATLAB fails to return back what the temporary expression is in some cases, in some releases.

Consider firing up the mupad notebook; see http://www.mathworks.com/help/symbolic/mupad.html

There are some alternatives to firing up the MuPAD notebook, but they mostly involve giving a sequence of MuPAD commands into an evalin() expression. For example, telling MuPAD to write the results to a file.