Stress Strain, Principle Axes, and Mohr's Circle

General Stress State  in Principle 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

General Two Dimensional Stress State coordinate system by rotating the Principle sytem by θ around z-axis

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

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

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

Set general two-dimensional state equal to the rotated principle 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;

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

σ(Subscript[lab, xx])

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

σ(Subscript[lab, yy])

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

σ(Subscript[lab, xy])

All z - components vanish except the original diagonal

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

σ(Subscript[lab, zz])

Equations 1, 2, and 3 are the equations that express a circle (known as Mohr's circle of stress) (see accompanying class notes)

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]