Nonlinear finite elements/Homework 6/Solutions/Problem 1/Part 13

Problem 1: Part 13

Compute the stress rate at the blue point. This stress rate is expressed in terms of the laminar coordinate system.

The rate constitutive relation of the material is given by

${\cfrac {D}{Dt}}{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{31}\\\sigma _{12}\end{bmatrix}}={\begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\C_{12}&C_{11}&C_{13}&0&0&0\\C_{13}&C_{13}&C_{33}&0&0&0\\0&0&0&C_{44}&0&0\\0&0&0&0&C_{44}&0\\0&0&0&0&0&(C_{11}-C_{12})/2\end{bmatrix}}{\begin{bmatrix}D_{11}\\D_{22}\\D_{33}\\D_{23}\\D_{31}\\D_{12}\end{bmatrix}}~.$

The map between the notations for a composite beam are shown in Figure 14.

Figure 14. Mapping between notations for composite material.

Since the problem is a 2-D one, based on the figure, the reduced constitutive equation is

${\cfrac {D}{Dt}}{\begin{bmatrix}\sigma _{11}\\\sigma _{33}\\\sigma _{31}\end{bmatrix}}={\begin{bmatrix}C_{11}&C_{13}&0\\C_{13}&C_{33}&0\\0&0&C_{44}\end{bmatrix}}{\begin{bmatrix}D_{11}\\D_{33}\\D_{31}\end{bmatrix}}~.$

The laminar $x$-direction maps to the composite $3$-direction and the laminar $y$-directions maps to the composite $1$-direction. Hence the constitutive equation can be written as

${\cfrac {D}{Dt}}{\begin{bmatrix}\sigma _{yy}\\\sigma _{xx}\\\sigma _{xy}\end{bmatrix}}={\begin{bmatrix}C_{11}&C_{13}&0\\C_{13}&C_{33}&0\\0&0&C_{44}\end{bmatrix}}{\begin{bmatrix}D_{yy}\\D_{xx}\\D_{xy}\end{bmatrix}}~.$

Rearranging,

${\cfrac {D}{Dt}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{xy}\end{bmatrix}}={\begin{bmatrix}C_{33}&C_{13}&0\\C_{13}&C_{11}&0\\0&0&C_{44}\end{bmatrix}}{\begin{bmatrix}D_{xx}\\D_{yy}\\D_{xy}\end{bmatrix}}~.$

Plugging in the values, we get

${\cfrac {D}{Dt}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{xy}\end{bmatrix}}={\begin{bmatrix}100&60&0\\60&10&0\\0&0&30\end{bmatrix}}{\begin{bmatrix}2.4837\\0\\-0.5\end{bmatrix}}={\begin{bmatrix}248.37\\149.02\\-15\end{bmatrix}}~.$

The Maple script for this calculation is shown below.

> #
> # Compute stress rate
> #
> # Set up the rate of deformation in Voigt notation
> #
> DLamVoigt := linalg[matrix](3,1,[Dlam[1,1],Dlam[2,2],Dlam[1,2]]);
> #
> # Set up stiffness matrix (Voigt notation)
> #
> CLamVoigt := linalg[matrix](3,3,[C33,C13,0,
>C13,C11,0,
>0,0,C44]);
> #
> # Compute stress rate (Voigt notation)
> #
> DDtSigLamVoigt := evalm(CLamVoigt&*DLamVoigt);

Nonlinear finite elements/Homework 6/Solutions/Problem 1