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

Problem 1: Part 12

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

The global base vectors are

 

Therefore, the rotation matrix is

 

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

 

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);