Skip to main content

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


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.



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


Comments

Post a Comment

Popular posts from this blog

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...
Post a Comment
Read more

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...
Post a Comment
Read more

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...
Post a Comment
Read more