OOF: Finite Element Analysis of Microstructures

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isotropic

An isotropic element has the same stiffness independent of the loading direction or the plane on which the load is applied.

According to Figure 4.2, there are two independent elastic constants. OOF uses the Young's modulus $E$ and the Poisson's ratio $\nu$ for the two input variables. In the case of plane stress, the $3\times3$ stiffness matrix is:

\begin{displaymath}
\frac{E}{1-\nu^2} \left(
\begin{array}{ccc}
1 & \nu & 0\\
\nu & 1 & 0\\
0 & 0 & \frac{1-\nu}{2}
\end{array}\right).
\end{displaymath}

For plane strain, the stiffness matrix is:

\begin{displaymath}
\frac{E(1-\nu)}{(1+\nu)(1-2\nu)} \left(
\begin{array}{ccc}
1...
...& 1 & 0\\
0 & 0 & \frac{1-2\nu}{2(1-\nu)}
\end{array}\right).
\end{displaymath}

Parameters

young
The Young's modulus $E$ [stress].

poisson
The Poisson's ratio $\nu$ [dimensionless].

alpha
The coefficient of thermal expansion [inverse temperature].

Figure 4.2: Symmetries of the stiffness matrix for Isotropic, Cubic and Hexagonal materials.
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