Nonlinear finite elements/Homework 6/Solutions/Problem 1/Part 12

Problem 1: Part 12

The global base vectors are

${𝐞}_x=\left[\begin{array}{c}1\\ 0\end{array}\right];{𝐞}_y=\left[\begin{array}{c}0\\ 1\end{array}\right].$

Therefore, the rotation matrix is

${𝐑}_{\text{lam}}=\left[\begin{array}{cc}{𝐞}_x\bullet \widehat{{𝐞}_x}& {𝐞}_x\bullet \widehat{{𝐞}_y}\\ {𝐞}_y\bullet \widehat{{𝐞}_x}& {𝐞}_y\bullet \widehat{{𝐞}_y}\end{array}\right]=\left[\begin{array}{cc}0.7071& 0.7071\\ -0.7071& 0.7071\end{array}\right].$

Therefore, the components of the rate of deformation tensor with respect to the laminar coordinate system are

${𝐃}_{\text{lam}}={𝐑}_{\text{lam}}^T𝐃{𝐑}_{\text{lam}}=\left[\begin{array}{cc}2.4837& -0.5\\ -0.5& 0\end{array}\right].$

The Maple script used to compute the above is shown below.

> #
> # Compute rate of deformation in laminar system
> #
> # Set up global base vectors
> #
> ex := vector([1,0,0]);
> ey := vector([0,1,0]);
> ez := vector([0,0,1]);
> #
> # Set up rotation matrix
> #
> ex_ehatx := dotprod(ex, ehat_x);
> ex_ehaty := dotprod(ex, ehat_y);
> ex_ehatz := dotprod(ex, ehat_z);
> ey_ehatx := dotprod(ey, ehat_x);
> ey_ehaty := dotprod(ey, ehat_y);
> ey_ehatz := dotprod(ey, ehat_z);
> ez_ehatx := dotprod(ez, ehat_x);
> ez_ehaty := dotprod(ez, ehat_y);
> ez_ehatz := dotprod(ez, ehat_z);
> Rlam := linalg[matrix](2,2,[ex_ehatx, ex_ehaty,
> ey_ehatx, ey_ehaty]);
> RlamT := transpose(Rlam);
> #
> # Compute rate of deformation in laminar system
> #
> Dlam := evalm(RlamT&*DefRate&*Rlam);