Stress Strain, Principal Axes, and Mohr's Circle

General Stress State  in Principal Axes  System

This is a general state, we will rotate about the z-axis and compare the result to a general two-dimensional stress state.

$\left(\begin{array}{lll}{\mathrm{\sigma princ}}_{\mathrm{xx}}& 0& 0\\ 0& {\mathrm{\sigma princ}}_{\mathrm{yy}}& 0\\ 0& 0& {\mathrm{\sigma princ}}_{\mathrm{zz}}\end{array}\right)$

Rotation about z-axis by angle θ

$\left(\begin{array}{lll}\mathrm{Cos}\left[\theta \right]& -\mathrm{Sin}\left[\theta \right]& 0\\ \mathrm{Sin}\left[\theta \right]& \mathrm{Cos}\left[\theta \right]& 0\\ 0& 0& 1\end{array}\right)$

Transformation to general two-dimensional stress state coordinate system by rotating the principal system by θ around z-axis

$\left(\begin{array}{lll}{\mathrm{Cos}\left[\theta \right]}^2{\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{Sin}\left[\theta \right]}^2{\mathrm{\sigma princ}}_{\mathrm{yy}}& \mathrm{Cos}\left[\theta \right]\mathrm{Sin}\left[\theta \right]\left(-{\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)& 0\\ \mathrm{Cos}\left[\theta \right]\mathrm{Sin}\left[\theta \right]\left(-{\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)& {\mathrm{Sin}\left[\theta \right]}^2{\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{Cos}\left[\theta \right]}^2{\mathrm{\sigma princ}}_{\mathrm{yy}}& 0\\ 0& 0& {\mathrm{\sigma princ}}_{\mathrm{zz}}\end{array}\right)$

$\left(\begin{array}{ccc}{\mathrm{Cos}\left[\theta \right]}^2{\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{Sin}\left[\theta \right]}^2{\mathrm{\sigma princ}}_{\mathrm{yy}}& \mathrm{Cos}\left[\theta \right]\mathrm{Sin}\left[\theta \right]\left(-{\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)& 0\\ \mathrm{Cos}\left[\theta \right]\mathrm{Sin}\left[\theta \right]\left(-{\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)& {\mathrm{Sin}\left[\theta \right]}^2{\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{Cos}\left[\theta \right]}^2{\mathrm{\sigma princ}}_{\mathrm{yy}}& 0\\ 0& 0& {\mathrm{\sigma princ}}_{\mathrm{zz}}\end{array}\right)$

Writing the same equation in a slightly different way...

$\left(\begin{array}{lll}\frac{1}{2}\mathrm{Cos}\left[2\theta \right]\left({\mathrm{\sigma princ}}_{\mathrm{xx}}-{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)+\frac{1}{2}\left({\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)& \frac{1}{2}\mathrm{Sin}\left[2\theta \right]\left(-{\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)& 0\\ \frac{1}{2}\mathrm{Sin}\left[2\theta \right]\left(-{\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)& \frac{1}{2}\mathrm{Cos}\left[2\theta \right]\left(-{\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)+\frac{1}{2}\left({\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)& 0\\ 0& 0& {\mathrm{\sigma princ}}_{\mathrm{zz}}\end{array}\right)$

$\left(\begin{array}{ccc}\frac{1}{2}\mathrm{Cos}\left[2\theta \right]\left({\mathrm{\sigma princ}}_{\mathrm{xx}}-{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)+\frac{1}{2}\left({\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)& \frac{1}{2}\mathrm{Sin}\left[2\theta \right]\left(-{\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)& 0\\ \frac{1}{2}\mathrm{Sin}\left[2\theta \right]\left(-{\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)& \frac{1}{2}\mathrm{Cos}\left[2\theta \right]\left(-{\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)+\frac{1}{2}\left({\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)& 0\\ 0& 0& {\mathrm{\sigma princ}}_{\mathrm{zz}}\end{array}\right)$

Naming the coefficients of the two-dimensional state in the rotated principal system

$\left(\begin{array}{ccc}{\mathrm{\sigma lab}}_{\mathrm{xx}}& {\mathrm{\sigma lab}}_{\mathrm{xy}}& {\mathrm{\sigma lab}}_{\mathrm{xz}}\\ {\mathrm{\sigma lab}}_{\mathrm{xy}}& {\mathrm{\sigma lab}}_{\mathrm{yy}}& {\mathrm{\sigma lab}}_{\mathrm{yz}}\\ {\mathrm{\sigma lab}}_{\mathrm{xz}}& {\mathrm{\sigma lab}}_{\mathrm{yz}}& {\mathrm{\sigma lab}}_{\mathrm{zz}}\end{array}\right)=\mathrm{\sigma rotalt};$

$1.{\mathrm{\sigma lab}}_{\mathrm{xx}}\mathrm{in}\mathrm{laboratory}\mathrm{system}\mathrm{rotated}\mathrm{by}\theta \mathrm{from}\mathrm{principal}\mathrm{axis}\mathrm{system}$

${\mathrm{\sigma lab}}_{\mathrm{xx}}$

$\frac{1}{2}\mathrm{Cos}\left[2\theta \right]\left({\mathrm{\sigma princ}}_{\mathrm{xx}}-{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)+\frac{1}{2}\left({\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)$

$\frac{1}{2}\mathrm{Cos}\left[2\theta \right]\left({\mathrm{\sigma princ}}_{\mathrm{xx}}-{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)+\frac{1}{2}\left({\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)$

$2.{\mathrm{\sigma lab}}_{\mathrm{yy}}\mathrm{in}\mathrm{laboratory}\mathrm{system}\mathrm{rotated}\mathrm{by}\theta \mathrm{from}\mathrm{principal}\mathrm{axis}\mathrm{system}$

${\mathrm{\sigma lab}}_{\mathrm{yy}}$

$\frac{1}{2}\mathrm{Cos}\left[2\theta \right]\left(-{\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)+\frac{1}{2}\left({\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)$

$\frac{1}{2}\mathrm{Cos}\left[2\theta \right]\left(-{\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)+\frac{1}{2}\left({\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)$

$3.{\mathrm{\sigma lab}}_{\mathrm{xy}}\mathrm{in}\mathrm{laboratory}\mathrm{system}\mathrm{rotated}\mathrm{by}\theta \mathrm{from}\mathrm{principal}\mathrm{axis}\mathrm{system}$

${\mathrm{\sigma lab}}_{\mathrm{xy}}$

$\frac{1}{2}\mathrm{Sin}\left[2\theta \right]\left(-{\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)$

$\frac{1}{2}\mathrm{Sin}\left[2\theta \right]\left(-{\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}\right)$

$\mathrm{All}z-\mathrm{components}\mathrm{remain}\mathrm{zero}\mathrm{except}\mathrm{the}\mathrm{original}\mathrm{diagonal}\mathrm{term}{\mathrm{\sigma lab}}_{\mathrm{zz}}:$

$0$

$0$

$0$

$0$

${\mathrm{\sigma princ}}_{\mathrm{zz}}$

${\mathrm{\sigma princ}}_{\mathrm{zz}}$

Equations 1, 2, and 3 are the equations that express a circle known as Mohr's circle of stress (see accompanying class notes). The equations show the way in which the stress tensor components in a two-dimensional state of stress (a "biaxial" stress state) vary with orientation of the coordinate system in which the stresses are described.

$\mathrm{Notice}\mathrm{that}\mathrm{there}\mathrm{are}\mathrm{two}\mathrm{invariants}\mathrm{of}\mathrm{the}\mathrm{general}\mathrm{stress}:$

The trace (or twice the offset of Mohr's Circle):

$\mathrm{Simplify}\left[{\mathrm{\sigma lab}}_{\mathrm{xx}}+{\mathrm{\sigma lab}}_{\mathrm{yy}}\right]$

${\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}$

${\mathrm{\sigma princ}}_{\mathrm{xx}}+{\mathrm{\sigma princ}}_{\mathrm{yy}}$

And the determinant

$\mathrm{Simplify}\left[{\mathrm{\sigma lab}}_{\mathrm{xx}}{\mathrm{\sigma lab}}_{\mathrm{yy}}-\left({\mathrm{\sigma lab}}_{\mathrm{xy}}\right)^2\right]$

${\mathrm{\sigma princ}}_{\mathrm{xx}}{\mathrm{\sigma princ}}_{\mathrm{yy}}$

${\mathrm{\sigma princ}}_{\mathrm{xx}}{\mathrm{\sigma princ}}_{\mathrm{yy}}$

These last two results are precisely the trace and determinant of the x and y terms in the original diagonal form of the stress state, thus illustrating the invariance of these quantities under rotation of coordinate rotations.

Example of Mohr's circle for two-dimensional body in uniaxial tension with ${\mathrm{\sigma princ}}_{\mathrm{xx}}$ = 10 MPa and all other stress components equal to zero

$\mathrm{ParametricPlot}\left[\left\{{\mathrm{\sigma lab}}_{\mathrm{xx}},{\mathrm{\sigma lab}}_{\mathrm{xy}}\right\},\left\{\theta ,0,\pi \right\},\mathrm{AxesLabel}\to \left\{''normal\; stress'',''shear\; stress''\right\},\mathrm{AspectRatio}\to 1,\mathrm{PlotLabel}\to ''\backslash t\; \backslash t\; Mohr\; Circle\; for\; 10\; MPa\; Uniaxial\; Tension'',\mathrm{PlotStyle}\to \left\{\mathrm{Thickness}\left[0.01\right],\mathrm{Hue}\left[1\right]\right\}\right]$

$⁃\mathrm{Graphics}⁃$

$⁃\mathrm{Graphics}⁃$

Comparing this plot with Figure 10-3 in the lecture notes, we see that the maximum and minimum tensile stresses are 10 and 0 MPa (from intercepts with x axis), as expected, and the maximum shear stress is ±5 MPa and it is experienced on a plane oriented at 2θ = 90° or θ = 45° to the tensile axis (remember that angles on Mohr's circle plots are twice the angle in the body).

Created by Mathematica  (October 5, 2005)