Equilibrium Equations

Let Ω be a volume occupied by a part of the body in the current configuration, and Γ the boundary of the body. In the Lagrangian formulation, Ω is the volume of space occupied by the material at the current time, which is different from the Eulerian approach where a volume of space through which the material passes is examined. τ is the traction surface on Γ and b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGIbaaaa@39BE@ are the body forces.

Force equilibrium for the volume is then:(1)
Γ τ i d Γ + Ω ρ b i d Ω = Ω ρ v i t d Ω
Material density
The Cauchy true stress matrix at a point of Γ is defined by:(2)
τ i = n j σ j i
Where, n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E9@ is the outward normal to Γ at that point. Using this definition, Equation 1 is written:(3)
Γ n j σ j i d Γ + Ω ρ b i d Ω = Ω ρ v i t d Ω
Gauss' theorem allows the rewrite of the surface integral as a volume integral so that:(4)
Γ n j σ j i d Γ = Ω σ i j x j d Ω
As the volume is arbitrary, the expression can be applied at any point in the body providing the differential equation of translation equilibrium:(5)
σ i j x j + ρ b i = ρ v i t
Use of Gauss' theorem with this equation leads to the result that the true Cauchy stress matrix must be symmetric:(6)
σ = σ T

So that at each point there are only six independent components of stress. As a result, moment equilibrium equations are automatically satisfied, thus only the translational equations of equilibrium need to be considered.