2. Continuum mechanics foundations

Linear finite element analysis rests on a quiet but powerful assumption: displacements are small enough that the difference between the undeformed and deformed configuration can be ignored. Once a structure undergoes large rotations, finite strains, or buckling, that assumption collapses. The remedy is not a better element or a finer mesh — it is a consistent continuum mechanics framework in which kinematics, strain, and stress are all defined with respect to a clearly stated reference configuration. This chapter establishes that framework: the deformation gradient, the strain measures derived from it, the stress tensors that pair with each strain measure, and the objectivity (frame-indifference) requirement that any valid constitutive law must satisfy.

Reference and Current Configurations

Let a body occupy the reference configuration Ω₀ at time t = 0. A material point is identified by its position vector X in Ω₀. After deformation, the same point occupies position x in the current configuration Ω. The motion is the mapping x = φ(X, t), and the displacement is u = x − X. Every quantity in nonlinear analysis is tagged as either material (referential, uppercase indices, defined on Ω₀) or spatial (current, lowercase indices, defined on Ω). Mixing the two is the single most common source of sign and transformation errors in nonlinear codes.

The Deformation Gradient

The deformation gradient is the tangent map of the motion:

F = ∂x/∂X = I + ∂u/∂X,   F_iJ = δ_iJ + ∂u_i/∂X_J

F is a two-point tensor: it maps material line elements to spatial line elements, dx = F dX. Its determinant J = det F measures local volume change; J > 0 is required for a physically admissible deformation (no material interpenetration). The polar decomposition F = R U = V R separates rigid rotation R (orthogonal, det = +1) from stretch U (right Cauchy–Green stretch, material) and V (left stretch, spatial). This decomposition is the conceptual key to objectivity: rigid-body rotation lives entirely in R, while deformation lives in U or V.

Strain Measures

Because F contains rotation, it cannot serve directly as a strain measure. The standard remedy is to form the right Cauchy–Green tensor C = FᵀF, which is purely material and rotation-free. From C we obtain the Green–Lagrange strain:

E = ½(C − I) = ½(FᵀF − I)

E is the work-conjugate material strain: it vanishes for any rigid-body motion and reduces to the small-strain tensor ε when displacements and displacement gradients are small. On the spatial side, the left Cauchy–Green tensor b = F Fᵀ and the Almansi strain e = ½(I − b⁻¹) play the analogous role. For hyperelasticity, the strain-energy density is written as a function of C (or its invariants I₁, I₂, I₃), which automatically guarantees objectivity.

Stress Tensors and Their Pairings

Three stress tensors appear routinely in nonlinear formulations, and each pairs with a specific strain measure through the virtual work density:

The internal virtual work can be written equivalently in any pairing, e.g. ∫ σ : δd dv = ∫ S : δE dV, where d = sym(Ḟ F⁻¹) is the spatial rate-of-deformation tensor. S is symmetric and is the natural output of a material model formulated on the reference configuration; P is generally non-symmetric and is what appears in the weak form when integrating over Ω₀. A total Lagrangian formulation uses (S, E); an updated Lagrangian formulation uses (σ, d) with the configuration updated each increment.

Objectivity (Frame Indifference)

A constitutive law is objective if it is unaffected by a superimposed rigid-body motion. Under a change of observer x* = Q(t) x + c(t), with Q orthogonal, the deformation gradient transforms as F* = Q F. Objectivity requires:

• Material strain measures are invariant: C* = C, E* = E.
• Spatial stress rotates: σ* = Q σ Qᵀ.
• Second Piola–Kirchhoff stress is invariant: S* = S.

This is why hyperelastic models are written as W(C) rather than W(F), and why hypoelastic rate equations of the form σ̇ = C : d are not objective unless an objective stress rate (Jaumann, Green–Naghdi, Truesdell) is used. The Jaumann rate, for example, subtracts the spin W = skew(ḞF⁻¹) from the material time derivative of σ. Choosing the wrong rate produces spurious stress under large rotation — a classic failure mode in shell and beam formulations.

Why This Matters in Practice

Consider a rubber-like cantilever modeled with a Neo-Hookean hyperelastic material, loaded by a tip moment that rotates the free end by 90°. A small-strain formulation predicts a stress distribution based on the linearized strain, but the actual deformation involves finite rotation with negligible stretch. If the constitutive routine is written in terms of F without objectivity enforcement, the rigid rotation generates spurious stress. The correct approach: compute C = FᵀF, evaluate W(C), derive S = 2 ∂W/∂C, then push forward to σ = J⁻¹ F S Fᵀ. The resulting stress field should be nearly zero in the rotated but unstretched region — a direct numerical verification of objectivity.

Practical Checklist for Implementation

1. Declare the reference configuration explicitly and store X for every integration point.
2. Compute F from the displacement gradient: F = I + ∂u/∂X.
3. Verify J = det F > 0 at every quadrature point; flag or cut the step if violated.
4. Form C = FᵀF and evaluate the material model in terms of C or its invariants.
5. Return S (symmetric, material) and the material tangent ∂S/∂E for the element residual and stiffness.
6. Push forward to σ only for post-processing or spatial formulations; never feed σ back into a material model written on Ω₀.
7. Run a rigid-rotation patch test: apply a pure rotation, confirm zero strain energy and zero stress.

These seven steps form the backbone of any total Lagrangian nonlinear FE implementation. The next chapter builds on this kinematic foundation by introducing the weak form and its linearization, where the material and geometric tangent operators derived here become the building blocks of the Newton–Raphson iteration.