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The general form for a three-dimensional stress field is given by
where the diagonal terms represent tensile or compressive stresses and the off-diagonal terms represent shear stresses. A stress field (in MPa) is given by
To solve for the principal stresses, it is necessary to construct the following matrix (again in MPa):
σ1, σ2, and σ3 can be solved from the equation
σ3 - Iσ2 + IIσ - III = 0
where
I = σxx + σyy + σzz
II = σxxσyy + σxxσzz + σyyσzz - σ2xy - σ2xz - σ2yz
III = σxxσyyσzz - σxxσ2 yz - σyyσ2 xz - σzzσ2xy + 2σxy σxz σyz
I, II, and III are known as the stress invariants. Find σ1, σ2, and σ3 using a root-finding technique. |
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