Nonlinear finite elements/Homework 6/Solutions/Problem 1/Part 15

Problem 1: Part 15

The modified laminar rate of deformation is

$\left[\begin{array}{c}{D}_{xx}\\ D_{yy}\\ D_{xy}\end{array}\right]=\left[\begin{array}{c}2.4837\\ -14.902\\ -0.5\end{array}\right]$

Alternatively, we can write

${𝐃}_{\text{lam}}=\left[\begin{array}{cc}2.4837& -0.5\\ -0.5& -14.902\end{array}\right].$

The modified laminar stress rate is

$\frac{D}{Dt}\left[\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\sigma }_{xy}\end{array}\right]=\left[\begin{array}{ccc}100& 60& 0\\ 60& 10& 0\\ 0& 0& 30\end{array}\right]\left[\begin{array}{c}2.4837\\ -14.902\\ -0.5\end{array}\right]=\left[\begin{array}{c}-645.76\\ 0\\ -15\end{array}\right].$

Alternatively, we can write

$\frac{D}{Dt}({\mathit{\sigma }}_{\text{lam}})=\left[\begin{array}{cc}-645.76& -15\\ -15& 0\end{array}\right].$

To get the global stress rate and rate of deformation, we have to rotate the components to the global basis using

$\frac{D}{Dt}(\mathit{\sigma })={𝐑}_{\text{lam}}\frac{D}{Dt}({\mathit{\sigma }}_{\text{lam}}){𝐑}_{\text{lam}}^T;𝐃={𝐑}_{\text{lam}}{𝐃}_{\text{lam}}{𝐑}_{\text{lam}}^T.$

Computing these quantities gives us

$\frac{D}{Dt}(\mathit{\sigma })=\left[\begin{array}{cc}-337.88& 322.88\\ 322.88& -307.88\end{array}\right].$

and

$𝐃=\left[\begin{array}{cc}-6.7092& -8.6929\\ -8.6829& -5.7092\end{array}\right].$

The Maple code for the above computations is given below.

> #
> # Apply plane stress condition
> #
> Dxx := DLamVoigt[1,1];
> Dyy := -C13*Dxx/C11;
> #
> # Updated laminar rate of deformation
> #
> DLamVoigtUpd := linalg[matrix](3,1,[Dlam[1,1], Dyy, Dlam[1,2]]);
> #
> # Updated laminar stress
> #
> DDtSigLamVoigtUpd := evalm(CLamVoigt&*DLamVoigtUpd);
> #
> # Rotate back to global basis
> #
> PlaneStressSig := array(1..2,1..2,symmetric):
> PlaneStressSig[1,1] := DDtSigLamVoigtUpd[1,1]:
> PlaneStressSig[2,2] := DDtSigLamVoigtUpd[2,1]:
> PlaneStressSig[1,2] := DDtSigLamVoigtUpd[3,1]:
> evalm(PlaneStressSig);
> GlobalPlaneStressSig := evalm(Rlam&*PlaneStressSig&*RlamT);
> PlaneStressDlam := array(1..2,1..2,symmetric):
> PlaneStressDlam[1,1] := DLamVoigtUpd[1,1]:
> PlaneStressDlam[2,2] := DLamVoigtUpd[2,1]:
> PlaneStressDlam[1,2] := DLamVoigtUpd[3,1]:
> evalm(PlaneStressDlam);
> GlobalPlaneStressDlam := evalm(Rlam&*PlaneStressDlam&*RlamT);