Stress

In [2]:
%%tikz --scale 1.0 --size 500,500 -f png
\draw[line width=2] (0,5) ellipse [x radius=2, y radius=1];
\draw[line width=2, dashed] (2,-5) arc  (0:180:2 and 1);
\draw[line width=2] (2,-5) arc  (0:-180:2 and 1);
\draw[line width=2] (-2.0,5) -- (-2.0,-5);
\draw[line width=2] (2.0,5) -- (2.0,-5);
\draw[line width=6, -latex] (0.0,5) -- (0,8) node[right,text width=2] {\Huge $F$};
\draw[line width=6, -latex] (0.0,-6) -- (0,-8);
\draw[line width=6, dashed] (0.0,-5) -- (0,-6);
\draw[line width=2] (-6,-2) -- (4,-2);
\draw[line width=2] (-4,2) -- (-2,2);
\draw[line width=2, dashed] (-2,2) -- (2,2);
\draw[line width=2] (2,2) -- (6,2);
\draw[line width=2] (4,-2) -- (6,2);
\draw[line width=2] (-6,-2) -- (-4,2);
\draw[line width=2, dashed] (2,0) arc  (0:180:2 and 1);
\draw[line width=2] (2,0) arc  (0:-180:2 and 1);
\fill[line width=2, color=black!10] (2,0) arc  (0:180:2 and 1);
\fill[line width=2, color=black!10] (2,0) arc  (0:-180:2 and 1);
\node at (0,0) {\Huge $A$};
\node at (10,0) {\Huge $\sigma = \frac F A$};

Cauchy tetrahedron

In [3]:
import IPython.display
IPython.display.Image(url='http://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Cauchy_tetrahedron.svg/387px-Cauchy_tetrahedron.svg.png', width=300, embed=True)
Out[3]:

"Cauchy tetrahedron" by Sanpaz - Own work.
Licensed under CC BY-SA 3.0 via Wikimedia Commons.

Stress tensor

${}$

$$ \mathbf{\sigma} = \begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{bmatrix} $$

Visual definition

In [4]:
IPython.display.Image(url='http://upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Components_stress_tensor_cartesian.svg/505px-Components_stress_tensor_cartesian.svg.png', width=300, embed=True)
Out[4]:
  • First subscripted index refers to the index of the unit vector that is normal to the face.
  • Second subscripted index refers to the component of traction vector.