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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 — including the momentum equations in spatial, integral, and referential form — before any choice of element or solver. See also Basic Equations in the library.
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. Referential form: ρ₀ = ρ J. If the motion is incompressible, this is equivalent to ∇ · v = 0.
Cauchy’s first law: divergence of stress and body force balance inertia. Static problems set a = 0. On a supported body the reactions on the support close the global balance.
Global statement on any part of the body. The divergence theorem plus localization recovers ∇ · σ + ρ b = ρ a. Essential and natural boundary data partition the surface into support and loaded faces.
The same physics on the undeformed placement, with P = J σ F⁻ᵀ. Total Lagrangian finite elements discretize this form; Updated Lagrangian rewrites it on the current mesh. Basic equations.
Without couple stresses, Cauchy’s second law is symmetry of σ. Equivalently P Fᵀ = F Pᵀ. It is an algebraic restriction, not a differential one.
First law: internal energy rate equals stress power, heat efflux (−∇ · q), and supply r. Fourier’s law q = −k ∇θ closes conduction in the usual thermal problems.
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.
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.
Elliptic profile on a patch of half-width a. Peak at the centre, zero at the edges. Increase the approach and watch the patch grow. For a discrete ContactFEA mesh of two cantilevers, see the beam contact figure.
A sharp crack in a linear solid is singular. A cohesive zone is not. A phase field smears the surface into a band of width ℓ. The contour integral extracts the energy release in either case.
Leading term at a sharp crack. Stress is unbounded as r → 0; energy release is finite. Modes I, II, and III each have their own K. The ligament stress has slope −½. A finite plate with a center crack is in the center-crack figure.
Energy released per unit area of crack advance. E′ = E in plane stress, E/(1−ν²) in plane strain. Fracture when G reaches Gc, or K reaches Kc.
Rice’s path-independent integral. Any path from the lower face to the upper that encloses the tip gives the same J. Change the contour and the number does not move.
Area under the law is the fracture energy; ℓch is the process-zone size. The tip stress is ft, not infinite. Change the law. A displacement-controlled DCB path is in the DCB cohesive figure.
Ambrosio–Tortorelli regularization of a surface. The 1-D profile is d = exp(−|n|/ℓ). Change ℓ and watch the band.
Degraded tensile energy plus toughness times crack density. As ℓ → 0 the second term recovers Gc × Area(Γ). g(d) = (1−d)² + κ.
Direction of the next increment under mixed-mode I–II. Equivalently θc = 2 arctan[(K_I − √(K_I² + 8 K_II²))/(4 K_II)].
On a cohesive crack in a saturated solid. Pressure is continuous across the faces; its normal gradient is not, because the fracture exchanges fluid with the intact medium. Process zone.
The same contour around a rigid-line tip is the configurational force on that tip. It vanishes at θn = arctan(1/√ν). Fibre neutrality is that statement for a thin elastic fibre.
A mixture of skeleton and fluid. Drainage turns total stress into effective stress; in more than one dimension that coupling is not a heat equation.
What the skeleton constitutive law sees. α = 1 − K/Ks. Terzaghi’s principle is the saturated soil limit α = 1. Phases and Bishop’s unsaturated split.
Relative discharge through the pores. Unsaturated flow replaces k by k(Sw). Gravity is a body force on both constituents.
Mass balance of the fluid. Undrained response is M → ∞ with no flow. That limit is incompressible, and equal-order u–p elements lock.
Excess pressure in a column whose total vertical stress is known. Isochrones and U(Tv) follow. Advance the time factor.
Triangular initial pressure in a fill; an accreting layer thickens at rate m. Finite-strain consolidation lets k and mv depend on void ratio. Self-weight isochrones.
In two or three dimensions the total stress redistributes. Uncoupled Terzaghi is monotonic; Biot coupling overshoots. Cryer’s sphere.
A unidirectional ply is transversely isotropic. A laminate of such plies is integrated through the thickness by classical lamination theory.
Longitudinal modulus of a unidirectional ply. The transverse modulus is closer to a series estimate, or to Halpin–Tsai. Five independent constants remain for transverse isotropy.
Plane-stress stiffness of the ply in material axes. The remaining entries of the 3×3 matrix Q follow from E₂, G₁₂, and the Poisson ratios. Rotate the fibres and this matrix is transformed to Q̄(θ).
Uniaxial stiffness along x for a ply whose fibres sit at θ to x, with c = cos θ and s = sin θ. At 45° the shear coupling is largest.
Force and moment resultants on the midplane. A is membrane stiffness, D bending, B the coupling that a symmetric stack removes. The ABD matrix is the section law a shell element actually uses.
Matrix, fibre, and interface. The slip is w = u_f − u_m. The fibre is a line that does not share nodes with the matrix mesh. An RVE of random fibres is the geometry; ERS and its enrichments are the discrete model.
Uniaxial tension along x, fibres in the xy-plane. The fibre is unstrained at θn(ν); a reduced model then returns Exc = Em. Turn the fibres and the composite moduli follow.
A finite element analysis is a sequence of stages. A GPU pays only in those that are data-parallel and that own wall time.
Overall speedup when a fraction p of the wall time runs s times faster. Meshing cannot save a factorisation, and a faster constitutive kernel cannot save a PCIe copy. Click the stages.
Stress and tangent at Gauss points are independent per element. That is the first kernel worth moving, especially in plasticity. Assembly is the first place those threads collide.
The Krylov iteration is a GPU job if A stays on the device. A small sparse factorisation often should not move. Explicit dynamics skip A entirely: the internal force is the kernel.