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

Problem 1: Part 15

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Express the stress rate and the modified rate of deformation in the global coordinate system.

The modified laminar rate of deformation is

 

Alternatively, we can write

 

The modified laminar stress rate is

 

Alternatively, we can write

 

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

 

Computing these quantities gives us

 

and

 

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);