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The figure below shows the micro tetrahedron in an elastic body in equilibrium:

enter image description here

In the textbook, I see the expressions of normal stress($\sigma_{v}$) and shear stress($\tau_{v}$) on the differential plane along the v direction (whose direction vector is {l, m, n}) are as follows:

$$\sigma_ {v} = \sigma_ {1} \times l^2 + \sigma_ {2} \times m^2 + \sigma_ {3} \times n^2 \\ \tau_ {v}^2 = \sigma_ {1}^2 \times l^2 + \sigma_ {2}^2 \times m^2 + \sigma_ {3}^2 \times n^2 - (\sigma_ {1} \times l^2 + \sigma_ {2} \times m^2 + \sigma_ {3} \times n^2)^2 $$

In the above formula, $l^2 + m^2 + n^2 = 1$.

I draw Mohr's circle of l and m according to the above formula, but I get the following results:

f = {{l^2, m^2, 1 - l^2 - m^2}.{σ1, σ2, σ3}, 
    Sqrt[Abs[({l^2, m^2, 
         1 - l^2 - 
          m^2}.{σ1^2, σ2^2, σ3^2}) - ({l^2, m^2, 
         1 - l^2 - 
          m^2}.{σ1, σ2, σ3})^2]]} /. {σ1 ->
      5, σ2 -> 2, σ3 -> -1};
ParametricPlot[Evaluate[f], {l, 0, 1}, {m, 0, 1}, Mesh -> 30, 
 PlotPoints -> 50, AxesOrigin -> {0, 0}]

enter image description here

In fact, the correct shape of the Mohr's circle should be similar to the following pattern:

enter image description here

I want to know how I can draw the correct Mohr's circle about the spatial stress state of direction vector parameters l and m (Pay attention to the difference with three-dimensional Mohr's circle).

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  • 5
    $\begingroup$ Using Sqrt[] guarantees that you'll only see the top half, so you need to do -Sqrt[(* stuff *)] as well. $\endgroup$ Commented Mar 27, 2020 at 12:14
  • 1
    $\begingroup$ @J.M.'stechnicaldifficulties Thank you for your comment. But the circular hole between 2 and 5 still can't be drawn. $\endgroup$
    – user69323
    Commented Mar 27, 2020 at 12:29

1 Answer 1

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I think you should add restriction $l^2+m^2<1$ to draw correctly.

f1 = {{l^2, m^2, 1 - l^2 - m^2}.{σ1, σ2, σ3}, 
    Sqrt[Abs[({l^2, m^2, 1 - l^2 - m^2}.{σ1^2, σ2^2, σ3^2}) - ({l^2, m^2, 1 - l^2 - m^2}.{σ1, σ2, σ3})^2]]} /.{σ1 -> 5, σ2 -> 2, σ3 -> -1};
f2 = {{l^2, m^2, 
      1 - l^2 - m^2}.{σ1, σ2, σ3}, -Sqrt[
      Abs[({l^2, m^2, 1 - l^2 - m^2}.{σ1^2, σ2^2, σ3^2}) - ({l^2, m^2, 1 - l^2 - m^2}.{σ1, σ2, σ3})^2]]} /.{σ1 -> 5, σ2 -> 2, σ3 -> -1};
ParametricPlot[Evaluate[{f1, f2}], {l, 0, 1}, {m, 0, 1}, 
 RegionFunction -> 
  Function[{x, y, l, m}, 
   l^2 + m^2 < 1 && -(6 l^2 + 3 m^2 - 1)^2 + 24 l^2 + 3 m^2 + 1 > 0], 
 Mesh -> 30, PlotPoints -> 50, AxesOrigin -> {0, 0}]

In this way, we can get a graph similar to the textbook.

enter image description here

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