Stress-Strain relations#
Poisson’s ratio \(\nu\) relates strains with stresses in perpendicular directions.
The complete stress strain relation in 3D is defined as:
\[\begin{split}\begin{bmatrix}
\epsilon_{xx} \\
\epsilon_{yy} \\
\epsilon_{zz} \\
2\epsilon_{xy} \\
2\epsilon_{yz} \\
2\epsilon_{zx}
\end{bmatrix} =
\frac{1}{E}
\begin{bmatrix}
1 & -\nu & -\nu & 0 & 0 & 0 \\
-\nu & 1 & -\nu & 0 & 0 & 0 \\
-\nu & -\nu & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 2(1+\nu) & 0 & 0 \\
0 & 0 & 0 & 0 & 2(1+\nu) & 0 \\
0 & 0 & 0 & 0 & 0 & 2(1+\nu) \\
\end{bmatrix}
\begin{bmatrix}
\sigma_{xx} \\
\sigma_{yy} \\
\sigma_{zz} \\
\sigma_{xy} \\
\sigma_{yz} \\
\sigma_{zx}
\end{bmatrix}\end{split}\]
This is treated in chapter 4 of the lecture notes Introduction to Continuum Mechanics (Hartsuijker and Welleman, 2007).
Instructions from lecture#
This topic is presented in Dutch in a lecture available from 0:15:50 to 1:24:00 here.
Exercises#
Problems 1-3 in chapter 5 of the lecture notes Introduction to Continuum Mechanics (Hartsuijker and Welleman, 2007). Instead of applying the circle of Mohr, apply the transformulation formulas. Answers are available in chapter 5 of the lecture notes.