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The von Mises yield function is given by:

$ \Phi(\sigma_1,\sigma_2)=\sqrt{\sigma_{1}^{2} +\sigma_{2}^{2}-\sigma_{1} \sigma_{2}} - \sigma_y $

were $\sigma_1$ and $\sigma_2$ are the principal stresses and $\sigma_y$ is the yield stress. If $\Phi(\sigma_1, \sigma_2)=0$, $\sigma_y =200$ and using ContourPlot:

contourplot = ContourPlot[Sqrt[sig1^2 + sig2^2 - sig1 sig2] - 200 == 0, {sig1, -300, 300}, {sig2, -300, 300}]

I have:

enter image description here

I need to find the parametric version of $\Phi(\sigma_1,\sigma_2)$, but I'm stuck.

Still now based on this question How to plot a rotated ellipse using ParametricPlot?, I can plot a rotated parametrized ellipse (red and dashed line) obtained from this code:

 a = 300; b = a/2; gamma = Pi/4; pmplot = ParametricPlot[{(a Cos[theta] Cos[gamma] - b Sin[theta] Sin[gamma]), a Cos[theta] Sin[gamma] + b Sin[theta] Cos[gamma]}, {theta, 0 ,2 Pi}, PlotStyle -> {Thick, Red, Dashed}]; Show[contourplot, pmplot]

enter image description here

The problem is to find the values of a and b to fit the parametric equation with the von Mises ellipse.

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ClearAll[a, b, gamma] table = Sqrt[#^2 + #2^2 - # #2] - 200 & @@@ Table[{(a Cos[theta] Cos[gamma] - b Sin[theta] Sin[gamma]), a Cos[theta] Sin[gamma] + b Sin[theta] Cos[gamma]}, {theta, 0, Pi, Pi/4}]; {a, b, gamma} = NArgMin[{Norm @ table, 0 <= gamma <= 2 Pi}, {a, b, gamma}]

{282.843, -163.299, 0.785398}

pmplot = ParametricPlot[{(a Cos[theta] Cos[gamma] - b Sin[theta] Sin[gamma]), a Cos[theta] Sin[gamma] + b Sin[theta] Cos[gamma]}, {theta, 0, 2 Pi}, PlotStyle -> {Thick, Red, Dashed}]; Show[contourplot, pmplot]

enter image description here

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  • $\begingroup$ great solution! Do you know if it's possible to obtain analytically these constants (a,b and gamma)? $\endgroup$ Commented Jun 6, 2018 at 15:13
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    $\begingroup$ @Diogo, for an exact solution, you can use eqns = Thread[ FullSimplify[ Norm[Sqrt[#^2 + #2^2 - # #2] - 200 & @@@ {(a Cos[theta] Cos[gamma] - b Sin[theta] Sin[gamma]), a Cos[theta] Sin[gamma] + b Sin[theta] Cos[gamma]}], Assumptions -> { a > 0, theta == #, 0 <= gamma <= Pi}] & /@ {0, Pi/2, Pi} == 0]; Reduce[Join[eqns, { 0 <= gamma <= 2 Pi}], {a, b, gamma}, Reals] . $\endgroup$ Commented Jun 6, 2018 at 16:04
  • $\begingroup$ thank you. The solutions i get from your code are: a=200sqrt[2], b=200Sqrt[2] and gamma=2 ArcTan[Sqrt[2]/(2 + Sqrt[2])]. Both a and gamma are compatible with your numerical solution, but b appears to be diferent. Do you know why? $\endgroup$ Commented Jun 6, 2018 at 16:16
  • $\begingroup$ Apparently the exact solution is: a=sigmay sqrt(2), b=sigmay sqrt(2/3) and gamma =-2 ArcTan[1 - Sqrt[2]] $\endgroup$ Commented Jun 6, 2018 at 16:26
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Just in case you need a parametric von Mises yield criterion equation for plane stresses.

Von Mises yield criterion:

$\sigma_{1}^{2} - \sigma_{1}\sigma_{2} + \sigma_{2}^{2} = \sigma_{y}^{2}$

were $\sigma_{1}$ and $\sigma_{2}$ are the principal stresses and $\sigma_{y}$ is the yield stress.

Parametric equations for von Mises yield criterion:

$\sigma_{1} = 2\sigma_{y}(\frac{1}{\sqrt{3}}\cos t + \sin t) = 2\sigma_{y}(\cos t - \frac{1}{\sqrt{3}} \sin t) = \frac{4\sigma_{y} \sin{t}}{\sqrt{3}}$

$\sigma_{2} = 2\sigma_{y}(-\frac{1}{\sqrt{3}}\cos t + \sin t) = 2\sigma_{y}(\cos t + \frac{1}{\sqrt{3}} \sin t) = \frac{4\sigma_{y} \sin{(t + \pi/3)}}{\sqrt{3}}$

where $t$ is the parameter, which ranges from $0$ to $2π$ radians.

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