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plotting - How to find the parametric equation of the von Mises yield function?


The von Mises yield function is given by:


Φ(σ1,σ2)=√σ21+σ22−σ1σ2−σy


were σ1 and σ2 are the principal stresses and σy is the yield stress. If Φ(σ1,σ2)=0, σy=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 Φ(σ1,σ2), 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.



Answer



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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