RECAP of STIFFNESS of a 2D CONSTANT STRAIN TRIANGLE

% K is the stiffness matrix of a 2D constant strain triangle.

% The thickness and material properties are assumed to be constant over the element.

% epsilon = [epsilon.x epsilon.y gamma.xy]' the 3x1 strain vector


% sigma = [sigma.x sigma.y tau.xy]' the 3x1 stress vector


% nu = Poisson's ratio\par

% E = Young's modulus\par

% sigma = D*epsilon  where D 3x3 is a function of E and nu

  D=E*[1-nu nu 0; nu 1-nu 0; 0 0 .5-nu]/((1+nu)*(1-2*nu)) %for PLANE STRAIN
  D=(E/(1-nu*nu))*[1 nu 0; nu 1 0; 0 0 (1-nu)/2]          %for PLANE STRESS

% t is the thickness of element e

% A CST has local nodes at (x1, y1) (x2, y2) (x3, y3) going counter-clockwise

% A CST has area A = 1/2 *det([1 x1 y1; 1 x2 y2; 1 x3 y3])

% A point within a CST has natural coordinates (chi, eta)

% J is the Jacobian of the transformation from (chi, eta) to (x, y)

% J = [ (x1-x3) (y1-y3); (x2-x3) (y2-y3) ]

% each of 3 nodes in a CST has two displacements in the global directions.

% q 6x1 is the vector of nodal displacments, 

% epsilon = B*q where B 3x6 is given by

% B=[(y2-y3)    0    (y3-y1)    0    (y1-y2)    0    ;
        0    (x3-x2)    0    (x1-x3)    0    (x2-x1) ;
     (x3-x2) (y2-y3) (x1-x3) (y3-y1) (x2-x1) (y1-y2) ]/det(J)

% K = t*A*B'*D*B is the 6x6 stiffness matrix for a CST

% Minimization of Potential Energy gives
%	Q.F=inv(K.F)*(F.F-K.EF'*Q.E)
% 	F.E=K.E*Q.E+K.EF*Q.F