09 / CHAPTER 07
Balance of angular momentum
Understand why the Cauchy stress tensor is symmetric in the classical case.
ANCHOR EQUATION
Angular momentum
In symbols: σ = σᵀ
In the classical no-couple-stress case, angular momentum balance makes stress symmetric.
Angular momentum makes classical stress symmetric
If a small continuum element had an unbalanced couple, it would spin infinitely fast as its size shrank unless a couple-stress theory were supplied. Classical materials therefore have symmetric Cauchy stress.
Balance moments on a shrinking cube
Surface tractions on opposite faces create moments proportional to shear stresses times face areas and lever arms. In the limit, angular momentum balance requires σ_xy = σ_yx, σ_xz = σ_zx, and σ_yz = σ_zy, so σ = σᵀ. The equality is a consequence of no independent body couples in the classical model.
A measured stress matrix [[100, 12],[8,50]] kPa violates the classical symmetry condition by 4 kPa. Either measurement error, a sign/plane convention error, or couple-stress physics is present.
Symmetry is not a universal law of every continuum
Micropolar fluids, electromagnetic couple stresses, and materials with internal rotations can require nonsymmetric stress plus additional balance laws. State the classical assumption before using σ = σᵀ.
Work it: For a 3D stress matrix, test symmetry and compute the antisymmetric part (σ − σᵀ)/2. Explain what physical model would be needed if it is nonzero.