06/12/2026
Balance of Mass
The balance of mass states that the mass of any material volume remains constant during motion, with no creation or destruction of mass.
In local form, it is expressed as the continuity equation: ∂ρ/∂t + ∇·(ρv) = 0, or in material form as Dρ/Dt + ρ ∇·v = 0.
For incompressible materials, this reduces to the kinematic constraint ∇·v = 0, which must hold at every point in the flow field.
This equation serves as the fundamental link between density and velocity fields and is an essential constraint in all continuum formulations.
Balance of Momentum
The balance of momentum is the continuum version of Newton’s second law, stating that the rate of change of linear momentum equals the sum of surface and body forces.
In differential form, it reads ρ Dv/Dt = ∇·σ + ρ b, known as Cauchy’s equations of motion.
This equation governs the acceleration of every material point and forms the central dynamic field equation in solid and fluid mechanics.
Together with mass conservation and constitutive relations, it closes the initial-boundary-value problem for any continuum body.
Balance of Moment of Momentum
The balance of moment of momentum requires that the rate of change of angular momentum equals the total moment of external forces acting on the body.
In the absence of distributed body couples, this balance enforces local symmetry of the Cauchy stress tensor: σ = σᵀ.
This symmetry is a powerful constraint that reduces the number of independent stress components from nine to six.
When body couples or internal microstructure are present, the balance is modified, allowing asymmetric stress and introducing couple stresses in generalized continuum theories.
Balance of Energy
The balance of energy expresses the first law of thermodynamics locally: the rate of change of kinetic plus internal energy equals the mechanical power of external forces plus the heat supply.
In differential form, it is written as ρ ė = σ : D − ∇·q + ρ r, where D is the rate-of-deformation tensor and q is the heat flux vector.
This equation directly couples mechanical work to internal energy evolution and is essential for all thermo-mechanically coupled problems.
It provides the missing relation needed to close the system once constitutive equations for stress and heat flux are specified.
Balance of Entropy
The balance of entropy is the local expression of the second law of thermodynamics, stating that the rate of entropy production must be non-negative.
In its local form (Clausius–Duhem inequality), it reads ρ η̇ ≥ −∇·(q/θ) + ρ r/θ, where θ is absolute temperature.
This inequality serves as the primary source of thermodynamic restrictions on constitutive equations, ensuring non-negative dissipation.
It is the fundamental tool used to derive thermodynamically consistent material models and to quantify irreversibility in continuum mechanics.
