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

## Problem 1: Part 12

Rotate the rate of deformation so that its components are with respect to the laminar coordinate system.

The global base vectors are

${\displaystyle \mathbf {e} _{x}={\begin{bmatrix}1\\0\end{bmatrix}};~\qquad \mathbf {e} _{y}={\begin{bmatrix}0\\1\end{bmatrix}}~.}$

Therefore, the rotation matrix is

${\displaystyle \mathbf {R} _{\text{lam}}={\begin{bmatrix}\mathbf {e} _{x}\bullet {\widehat {\mathbf {e} _{x}}}&\mathbf {e} _{x}\bullet {\widehat {\mathbf {e} _{y}}}\\\mathbf {e} _{y}\bullet {\widehat {\mathbf {e} _{x}}}&\mathbf {e} _{y}\bullet {\widehat {\mathbf {e} _{y}}}\end{bmatrix}}={\begin{bmatrix}0.7071&0.7071\\-0.7071&0.7071\end{bmatrix}}~.}$

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

${\displaystyle \mathbf {D} _{\text{lam}}=\mathbf {R} _{\text{lam}}^{T}\mathbf {D} \mathbf {R} _{\text{lam}}={\begin{bmatrix}2.4837&-0.5\\-0.5&0\end{bmatrix}}~.}$

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);