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

From Wikiversity
Jump to navigation Jump to search

Problem 1: Part 15[edit | edit source]

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);