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 $b$ are the body forces.

Force equilibrium for the volume is then:(1)

\int_{Γ}^{} τ_{i} d Γ + \int_{Ω} ρ b_{i} d Ω = \int_{Ω} ρ \frac{\partial v_{i}}{\partial t} d Ω

Where,

$ρ$: 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

is the outward normal to

Γ

at that point. Using this definition, Equation 1 is written:(3)

\int_{Γ} n_{j} σ_{j i} d Γ + \int_{Ω} ρ b_{i} d Ω = \int_{Ω} ρ \frac{\partial v_{i}}{\partial t} d Ω

Gauss' theorem allows the rewrite of the surface integral as a volume integral so that:(4)

\int_{Γ} n_{j} σ_{j i} d Γ = \int_{Ω} \frac{\partial σ_{i j}}{\partial 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)

\frac{\partial σ_{i j}}{\partial x_{j}} + ρ b_{i} = ρ \frac{\partial v_{i}}{\partial 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.