Home · Governing equations
These are the field identities behind the entries in the library: how deformation is measured, how forces are carried, how mass and momentum are conserved, and how those statements become a discrete system on a mesh.
Motion, stress, and conservation, before any choice of element or solver.
A material line element in the reference placement is mapped to the current placement. Finite-strain measures are built from this tensor. Translation, rotation, stretch, and shear are shown as a homogeneous motion of a grid.
Local volume change. Mass balance in the reference placement is ρ₀ = ρ J. Incompressibility is J = 1, which is the constraint on the principal stretches of rubber.
Strain on the reference configuration. When displacement gradients are small, it reduces to the infinitesimal strain. A pure rotation leaves E at zero and does not leave ε at zero.
The symmetric gradient of displacement. This is the strain used in linear elasticity and in most introductory finite element courses.
A unique rotation R and symmetric stretches U, V. Strain is built from U or from C = U², so a rigid rotation does not count as stretch. Stretch then rotate, or rotate then stretch — the two paths meet.
Finite strain on the current placement, with b = F Fᵀ. Like Green–Lagrange, it vanishes in a rigid motion.
The symmetric part D is stretching; the skew part W is spin. This is the rate that is conjugate to Kirchhoff stress. D, W, and L on a material cross.
Nominal stress: force on a current area, measured per reference area. Work-conjugate to Ḟ. The same force on two areas.
Stress on the reference placement, work-conjugate to Green–Lagrange strain. Hyperelastic laws are usually written as S = ∂W/∂E.
Magnitude of the stress deviator, scaled to match uniaxial tension. J₂ plasticity yields when this reaches the current yield stress. The cylinder in principal-stress space.
Force per current area on a surface with outward normal n. Without couple stresses, σ is symmetric. On a deforming body that traction rides with the current surface. Turn the cut and the same σ produces a different t.
Local mass balance. If the motion is incompressible, this is equivalent to a divergence-free velocity field.
The divergence of stress, together with body force, balances inertia. In a static problem the right-hand side is zero. On a supported body the reactions on the supported area close the global balance.
Linear elasticity in the isotropic case, written with the Lamé parameters. The compact form is σ = ℂ : ε. Uniaxial tension produces lateral strain −ν ε.
The simplest isotropic hyperelastic law. Incompressibility drops the volumetric terms and enforces J = 1 with a pressure. Principal stretches at J = 1.
An objective rate of Cauchy stress. A pure rotation of a constant uniaxial stress changes the laboratory components and leaves ∇σ at zero.
Without couple stresses, Cauchy’s second law is symmetry of σ. It is an algebraic restriction, not a differential one.
The same momentum balance, closed by Newton’s law of viscosity, together with ∇ · v = 0.
The weak form of momentum, and the linear system obtained after interpolation and assembly.
A weak statement of linear momentum. Finite elements replace the trial and test displacements with shape-function expansions. Stationarity of the potential is the same statement when a stored energy exists.
The same shape functions describe geometry and the unknown field. Derivatives in physical space use J = ∂x/∂ξ. Parent square to warped quad.
The strain–displacement matrix B holds the symmetric gradients of the shape functions. Assembly scatters Kᵉ into the global sparse matrix.
After interpolation and assembly, the unknown nodal coefficients satisfy a stiffness system whose right-hand side carries the applied loads.
Constitutive evaluations happen only at the quadrature points. Full versus reduced integration on a bilinear quadrilateral.
A displacement-only space cannot represent ∇ · u = 0 without locking. Pressure is interpolated independently, and the pair must satisfy an inf-sup condition.
Nonlinear problems are reduced to a sequence of linear solves; dynamics add a choice of integrator.
A residual R(u) = 0 is linearized at the current guess. The tangent KT is the Jacobian of that residual. Consistent linearisation is what makes the convergence quadratic.
The linear eigenvalue at which a hinged–hinged column loses uniqueness of the straight solution. Geometric stiffness, not a change in E, is what makes the tangent singular. An imperfect strut.
The load factor λ is an unknown. The extra equation lets the iteration pass a limit point where load control would stall. Riks and Crisfield.
Mass, damping, and a (possibly nonlinear) internal force. Explicit methods lump M; implicit methods form a tangent at each step.
Average acceleration (β = ¼, γ = ½) is unconditionally stable for linear problems. Central difference is the explicit member of the same family.
Gap, pressure, and the requirement that one of them vanish. Friction adds a tangential law and a nonsymmetric tangent.