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.

σtensordiag=(({{σ(Subscript[princ, xx]), 0, 0}, {0, σ(Subscript[princ, yy]), 0}, {0, 0, σ(Subscript[princ, zz])}}));
σtensordiag // 
      MatrixForm

Rotation about z-axis by angle θ

rotmat[θ_] :=(({{Cos[θ], -Sin[θ], 0}, {Sin[θ], Cos[θ], 0}, {0, 0, 1}}));
rotmat[θ] // 
      MatrixForm

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

σrot  =  
        Simplify[
          Transpose[rotmat[θ]] . σtensordiag . 
            rotmat[θ]];
σrot // MatrixForm

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

σrotalt  = 
        Collect[σrot // TrigReduce, {Cos[2  θ], 
            Sin[2 θ]}];
σrotalt // MatrixForm

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

(({{σ(Subscript[lab, xx]), σ(Subscript[lab, xy]), σ(Subscript[lab, xz])}, {σ(Subscript[lab, xy]), σ(Subscript[lab, yy]), σ(Subscript[lab, yz])}, {σ(Subscript[lab, xz]), σ(Subscript[lab, yz]), σ(Subscript[lab, zz])}}))= σrotalt;

\(1. \ σlab\_xx\ \ in\ laboratory\ system\ rotated\ by\ θ\ from\ principal\ axis\ system\)

σ(Subscript[lab, xx])

\(2. \ σlab\_yy\ \ in\ laboratory\ system\ rotated\ by\ θ\ from\ principal\ axis\ system\)

σ(Subscript[lab, yy])

\(3. \ σlab\_xy\ \ in\ laboratory\ system\ rotated\ by\ θ\ from\ principal\ axis\ system\)

σ(Subscript[lab, xy])

\(All\ z - 
      components\ remain\ zero\ except\ the\ original\ diagonal\ term\ \(\(σlab\_zz\)\(:\)\)\)

σ(Subscript[lab, yz])
σ(Subscript[lab, xz])
σ(Subscript[lab, zz])

0

0

σ(Subscript[princ, 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.

\(Notice\ that\ there\ are\ two\ invariants\ of\ the\ general\ \(\(stress\)\(:\)\)\)

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

Simplify[σ(Subscript[lab, xx])  +  σ(Subscript[lab, yy]) ]

And the determinant

Simplify[σ(Subscript[lab, xx])  σ(Subscript[lab, yy])  - (σ(Subscript[lab, xy]))^2]

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 σ(Subscript[princ, xx]) = 10 MPa and all other stress components equal to zero

uniaxial10 =  {σ(Subscript[princ, xx]) -> 10, σ(Subscript[princ, yy]) -> 
          0}

ParametricPlot[{σ(Subscript[lab, xx]), σ(Subscript[lab, xy])} /. 
        uniaxial10
, {θ, 0, π}, 
      AxesLabel -> {"normal stress", "shear stress"}, 
      AspectRatio -> 1, 
      PlotLabel -> " \t \t Mohr Circle for 10 MPa Uniaxial Tension", 
      PlotStyle -> {Thickness[0.01], Hue[1]}]

uniaxialother =  {σ(Subscript[princ, xx]) -> 30, σ(Subscript[princ, yy]) -> 
          10}

\(ParametricPlot[{σlab\_xx, σlab\_xy} /. 
        uniaxialother
, {θ, 0, π}, 
      AxesLabel -> {"normal stress", "shear stress"}, 
      AspectRatio -> 1, PlotRange -> {{0, 40}, {\(-20\), 20}}, 
      PlotLabel -> \*"" \t \t Mohr Circle for σ(Subscript[princ, xx])= 30   σ(Subscript[princ, yy])=10"", 
      PlotStyle -> {Thickness[0.01], Hue[1]}]\)

Comparing this plot with Figure 10-3 in the lecture notes, we see that the maximum and minimum tensile stresses are 10 and 30 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).