Variational Methods in Continuum Mechanics

Variational Principles as the Language of Mechanics

Continuum mechanics is the theory of deformable bodies: beams, plates, fluids, rubbers. Its equations (of equilibrium, motion) are derived from variational principles. This is not merely a mathematical convenience: the variational formulation directly generates the finite element method—the main tool in engineering computations. Every time an engineer launches ANSYS or COMSOL, they are solving a variational problem.

The Principle of Virtual Displacements

Statement: a body Ω is in equilibrium under the action of body forces f and surface loads t on a part ∂Ω_t.

Principle of virtual displacements: a body is in equilibrium if and only if the total virtual work is zero for any admissible virtual displacement δu:

δW = ∫_Ω σᵢⱼ δεᵢⱼ dV − ∫Ω fᵢ δuᵢ dV − ∫{∂Ω_t} tᵢ δuᵢ dS = 0

Term explanations:

• σᵢⱼ — stress tensor (6 independent components for symmetric σ)
• δεᵢⱼ = (δuᵢ,ⱼ + δuⱼ,ᵢ)/2 — virtual strain (symmetrized gradient)
• fᵢ — components of body forces (for example, gravity, fi = ρgᵢ)
• tᵢ = σᵢⱼnⱼ — components of surface loads

This is the "weak" formulation of the equilibrium equations. The "strong" form follows from integration by parts: ∂ⱼσᵢⱼ + fᵢ = 0 in Ω, σᵢⱼnⱼ = tᵢ on ∂Ω_t.

Principle of Minimum Potential Energy

For elastic bodies (non-dissipative):

Total potential energy: Π[u] = ∫_Ω W(ε(u)) dV − ∫Ω f·u dV − ∫{∂Ω_t} t·u dS

where W(ε) is the density of elastic potential energy (for Hookean material: W = λ(trε)²/2 + μ tr(ε²)).

Principle: among all admissible displacement fields (satisfying kinematic boundary conditions), the equilibrium field minimizes Π.

The Euler-Lagrange equation for Π: ∂ⱼσᵢⱼ + fᵢ = 0—the equations of equilibrium!

Thus, the equilibrium equations of mechanics are first-order conditions for the minimum of potential energy. Variational calculus "derives" the laws of mechanics.

FEM: Ritz Method with Finite Elements

The finite element method (FEM) is a numerical realization of the Ritz method for variational problems in mechanics.

Idea: partition Ω into finite elements Ωₑ (triangles, tetrahedra). Within each element, approximate u by polynomial shape functions Nᵢ(x): u ≈ Σᵢ uᵢ Nᵢ(x), where uᵢ are nodal displacements.

Substitute into Π: Π[u] = Π(u₁,...,uN)—a function of N real numbers.

∂Π/∂uᵢ = 0 → KU = F (a system of linear equations)

where K = ∫_Ω BᵀCB dV is the stiffness matrix (B is the strain matrix, C is the elasticity tensor), F is the load vector.

Dimension: N = 3 × (number of nodes). For an aircraft model—millions of degrees of freedom. ANSYS solves such systems within hours.

Mixed Principles

The classical Lagrange principle is “single-field” (only u). Mixed principles introduce multiple fields.

Hellinger–Reissner principle (two fields, u and σ):

Π_HR[u, σ] = ∫_Ω [σ:ε(u) − W*(σ)] dV − ∫Ω f·u dV − ∫{∂Ω_t} t·u dS

where W*(σ) is the conjugate energy (Legendre transform of W(ε)).

The Euler-Lagrange equations yield simultaneously the compatibility equations for strains and equilibrium equations. Convenient when σ and u are approximated independently.

Castigliano's principle: for structures with known forces—the displacement at the point of force application equals the derivative of the complementary energy with respect to that force. Simple for trusses and beams.

Full Analysis: Bending of an Euler–Bernoulli Beam

Problem: a beam of length L, clamped at x=0 (fixed: u(0)=u'(0)=0), the free end loaded by a force P at x=L. Find the deflected shape of the beam.

Functional: Π[u] = ∫₀ᴸ (EI/2)(u'')² dx − P·u(L)

where EI is the beam's stiffness.

Euler-Lagrange equation: ∂Π/∂(δu) = 0 → EI u'''' = 0 on (0, L).

Boundary conditions: u(0) = u'(0) = 0 (clamping), EI u''(L) = 0, EI u'''(L) = P.

Solution: u'''' = 0 → u = ax³ + bx² + cx + d. From BC: d = 0, c = 0, 6aL + 2b = 0, 6aEI = P.

a = P/(6EI), b = −PL/(2EI). Deflection: u(x) = Px²(3L−x)/(6EI).

Deflection at the end: u(L) = PL³/(3EI)—the standard formula of structural mechanics!

Nonlinear Elasticity and Hyperelasticity

For large deformations (rubber, biological tissues) the linear Hooke's law fails.

Elastic energy: W = W(F), where F = I + ∇u—the deformation gradient. The problem: min ∫_Ω W(F) dV − ∫ f·u dV.

Euler-Lagrange equation: Div P + f = 0, P = ∂W/∂F—Piola-Kirchhoff stress.

Models: neo-Hookean (W = μ(I₁−3)/2), Mooney–Rivlin (W = C₁(I₁−3) + C₂(I₂−3)). Used in biomechanics (heart, blood vessel, skin modeling), rubber products, soft robotics.

Principle of Minimum Potential Energy

In statics of a solid deformable body, the fundamental principle is: the equilibrium configuration minimizes the total potential energy Π = U_internal − W_external, where U_internal is the elastic energy of deformation, W_external is the work of external forces. This is a variational problem: minimize Π[u] over the displacement field u(x) subject to boundary conditions. The Euler–Lagrange equation for this functional is the equilibrium equation of the elastic body.

Euler's and Navier–Stokes Equations for Fluids

The motion of an ideal (inviscid) fluid is governed by the Euler equations: ∂v/∂t + (v·∇)v = −∇p/ρ. They are derived from Hamilton's principle with Lagrangian L = (ρ/2)|v|². For viscous fluids—Navier–Stokes equations with an additional term νΔv. The existence and smoothness of Navier–Stokes solutions is one of the Millennium Prize Problems ($1$ million from the Clay Institute).

The Finite Element Method (FEM)

FEM is a practical tool for solving variational problems in continuum mechanics:

1. Partition the body into finite elements (triangles, tetrahedra, hexahedra)
2. Approximate the displacement field in each element with a linear/quadratic basis function
3. Reduce the variational problem to a system of linear equations KU = F (K is the stiffness matrix, F the external forces vector)
4. Solve and post-process (compute stresses, strains)

Industrial FEM packages: ANSYS, ABAQUS, COMSOL, NASTRAN. Modern extensions: isogeometric analysis (NURBS basis), meshfree methods (SPH, MPM for plastic deformations).

Principle of Virtual Work

An alternative formulation: at equilibrium, the work of all external and internal forces on any virtual displacement is zero. This gives the "weak formulation" of the variational problem—FEM's foundation.

Applications

• Civil engineering: analysis of bridges, skyscrapers, dams (Burj Khalifa, Akashi Kaikyo Bridge)
• Aviation: analysis of load-bearing aircraft structures (Boeing 787, Airbus A350)
• Automotive engineering: crash tests via numerical modeling
• Biomechanics: modeling of bones, implants, blood flow
• Geophysics: modeling tectonics, earthquakes