2. Continuum kinematics for finite strain

Geometrically nonlinear analysis begins with a precise description of how a body moves and deforms. In the small-strain world, we get away with a single linearized strain tensor because displacements and rotations are assumed infinitesimal. Elastomers routinely stretch to 200–500% engineering strain, and metal forming operations involve finite rotations of material fibers. Under these conditions, the linear strain measure is not merely inaccurate — it is fundamentally invalid, because it fails to remain invariant under rigid-body rotation. This chapter builds the kinematic and kinetic framework that all subsequent hyperelastic and finite-strain plasticity formulations rely on.

2.1 Deformation Gradient and the Reference/Current Configuration

We adopt a Lagrangian viewpoint. Let X denote the position of a material point in the undeformed reference configuration Ω₀, and x = φ(X, t) its position in the deformed current configuration Ω. The motion φ is assumed sufficiently smooth. The central kinematic quantity is the deformation gradient:

F = ∂x/∂X = ∂φ/∂X,   F_iJ = ∂x_i/∂X_J

F = I + ∂u/∂X        (u = x − X, displacement)

F is a two-point tensor: it maps reference-configuration line elements to current-configuration line elements, dx = F dX. Its determinant J = det F measures local volume change; for an incompressible elastomer J = 1 exactly, which becomes a constraint rather than a constitutive statement. A deformation is admissible only if J > 0 everywhere — a negative or zero Jacobian signals element inversion, the most common cause of aborted Newton iterations in large-strain FE jobs.

F contains both stretch and rotation. The polar decomposition separates them:

F = R · U = V · R

R : proper orthogonal rotation tensor (RᵀR = I, det R = +1)
U : right stretch tensor (symmetric, reference basis)
V : left stretch tensor (symmetric, current basis)

This decomposition is the reason objective stress rates exist: any constitutive law written directly in terms of the time derivative Ḟ or the Cauchy stress rate σ̇ will be polluted by rigid-body spin, because R evolves even when the material does not stretch.

2.2 Strain Measures: Green-Lagrange and Its Relatives

The change in squared length of a material fiber gives the natural finite-strain measure. With ds² = dx·dx and dS² = dX·dX:

ds² − dS² = 2 dX · E · dX

E = ½ (FᵀF − I) = ½ (C − I)      Green-Lagrange strain
C = FᵀF                          right Cauchy-Green deformation tensor
b = F Fᵀ                         left Cauchy-Green deformation tensor

E is symmetric, vanishes for rigid-body motion (because then F = R and FᵀF = I), and is defined entirely in the reference configuration — a convenient property for hyperelastic stored-energy functions W(C). The eigenvalues of C are the squared principal stretches λ₁², λ₂², λ₃², and E's eigenvalues are ½(λᵢ² − 1). For a uniaxial tension test on an elastomer stretched to λ = 3, the Green-Lagrange strain is E₁₁ = ½(9 − 1) = 4.0, whereas the engineering strain is 2.0 and the logarithmic strain is ln 3 ≈ 1.099. These three numbers describe the same physical state; mixing them in a single constitutive model is a frequent source of order-of-magnitude errors.

Two other measures appear in practice. The Euler-Almansi strain e = ½(I − b⁻¹) lives in the current configuration, and the logarithmic (Hencky) strain ln U is the natural partner for metal plasticity at moderate strains because it additively decomposes under proportional loading. Commercial codes let you request any of these as output, but the stress measure they pair with must match.

2.3 Stress Measures and Their Work-Conjugate Pairs

Three stress tensors dominate finite-strain FE. Each is conjugate to a specific strain rate in the internal virtual work, and confusing them corrupts both the residual and the tangent stiffness.

The Cauchy stress σ is the true physical stress — the traction on a deformed surface. The first Piola-Kirchhoff stress P is generally non-symmetric and is the work conjugate of Ḟ, which makes it convenient for total Lagrangian residuals. The second Piola-Kirchhoff stress S is symmetric and fully referential; it is the natural output of a hyperelastic law S = 2 ∂W/∂C. The internal virtual work identity ties them together:

∫ σ : δd dv = ∫ S : δE dV = ∫ P : δF dV

In a total Lagrangian (TL) formulation, all integrals are taken over Ω₀ and the solver works with S and E. In an updated Lagrangian (UL) formulation, integrals are over Ω and the solver works with σ and the rate of deformation d = sym(∂v/∂x). Both are exact; the choice is one of numerical convenience. Elastomer models are almost always TL because W(C) is naturally referential, while metal-forming codes often prefer UL.

2.4 Objective Stress Rates

Consider a body undergoing pure rigid rotation with constant spin tensor W. The Cauchy stress components in a fixed global frame change even though the material experiences no stress change. Therefore σ̇ is not objective and cannot appear in a rate-form constitutive law. We need a rate that co-rotates with the material. The two standard choices are:

Jaumann rate:   σ̊ = σ̇ − W·σ + σ·W
Truesdell rate: σ̊ = σ̇ − L·σ − σ·Lᵀ + (tr L) σ   (L = Ḟ F⁻¹)

The Jaumann rate uses only the spin W = skew(L), while the Truesdell rate uses the full velocity gradient L and is the one consistent with the Lie derivative of the Kirchhoff stress τ = Jσ. For elastomers, most hyperelastic implementations avoid rate forms entirely by working directly with W(C) and S, sidestepping the objectivity problem. Rate forms matter mainly for finite-strain plasticity and viscoelasticity, where the constitutive update is incremental. A practical warning: Jaumann-rate hypoelastic models exhibit spurious shear oscillation under large simple shear beyond γ ≈ 1, a well-documented artifact that has led many analysts to prefer hyperelastic or Truesdell-based formulations.

2.5 Practical Implications for FE Implementation

When you set up a large-deformation job, three decisions follow directly from this chapter. First, choose the formulation: NLGEOM=ON in Abaqus/Standard activates a TL formulation for hyperelastic materials, while *STEP, NLGEOM with a hypoelastic *ELASTIC definition uses a Jaumann-rate update. Second, verify the Jacobian: request J or VOL output and confirm it stays positive; a negative J at any integration point means the element has inverted and the step must be restarted with a smaller increment. Third, check the stress measure in your post-processing: contour plots of S11 (PK2) and S11 (Cauchy) differ by a factor of J and a rotation, so a 3× stretched elastomer shows PK2 stresses roughly one-third of the Cauchy values.

The kinematic framework established here — F, C, E, the three stress measures, and their conjugate rates — is the common language shared by every subsequent chapter. Hyperelasticity will supply W(C); finite-strain plasticity will supply an evolution law for the plastic part of F; and contact and buckling formulations will reuse the same TL/UL machinery. Getting the kinematics right is not optional bookkeeping; it is the difference between a converged solution and a diverging residual.