2. Mathematical foundations of stress and displacement fields

Geometric deep learning does not replace continuum mechanics; it re-parameterizes it. Before a neural operator can predict a displacement field u(x) or a stress tensor σ(x), the learning problem must be embedded in the same mathematical structure that governs the physics. This section establishes that structure: the equilibrium equations that constrain stress, the compatibility conditions that constrain strain, the constitutive laws that couple the two, and the boundary value problem (BVP) that ties them together. Each of these becomes a design constraint — a loss term, an architectural symmetry, or a hard projection — in the models developed later in the course.

Kinematics: displacement, strain, and the small-strain assumption

Let a body occupy a domain Ω ⊂ ℝ³ with boundary ∂Ω. The displacement field u: Ω → ℝ³ maps each material point to its movement. Under the small-strain assumption, the infinitesimal strain tensor is the symmetric part of the displacement gradient:

ε_ij = ½ (∂u_i/∂x_j + ∂u_j/∂x_i)

The symmetry ε_ij = ε_ji is not an approximation but a definitional consequence of this construction, and it matters for learning: any network that outputs a full 3×3 strain tensor must be constrained (or penalized) to produce symmetric outputs. The six independent components of ε are the natural target of a strain-predicting model, and the antisymmetric part of the displacement gradient — the infinitesimal rotation — carries no strain information at all.

Equilibrium: the balance of linear and angular momentum

Static equilibrium requires that the divergence of the stress tensor balance the body force:

∇·σ + b = 0        (linear momentum balance)
σ = σᵀ             (angular momentum balance)

In index notation, ∂σ_ij/∂x_j + b_i = 0. The symmetry of σ follows from moment balance in the absence of couple stresses. These are three scalar equations in three dimensions, yet the stress tensor has six independent components — the system is underdetermined by three degrees of freedom per point. That deficit is precisely what the constitutive law and compatibility conditions must resolve, and it is the reason a purely equilibrium-based loss is insufficient for training a stress predictor.

Compatibility: strain fields that come from a displacement

Not every symmetric strain field is admissible. If ε is derived from a single-valued displacement field, it must satisfy the Saint-Venant compatibility conditions:

∂²ε_ij/∂x_k∂x_l + ∂²ε_kl/∂x_i∂x_j
  − ∂²ε_ik/∂x_j∂x_l − ∂²ε_jl/∂x_i∂x_k = 0

In three dimensions these reduce to six independent equations. Their practical role in this course is diagnostic: a network trained only on equilibrium can produce stress fields that integrate to a strain field violating compatibility, yielding displacement predictions that are not single-valued (they contain a dislocation-like jump). Compatibility is therefore a candidate hard constraint or a regularizer, and it is the mathematical justification for predicting displacement first and differentiating to obtain strain, rather than predicting strain directly.

Constitutive laws: closing the system

For linear isotropic elasticity, the constitutive relation is Hooke's law in tensor form:

σ_ij = λ ε_kk δ_ij + 2μ ε_ij

where λ and μ are the Lamé parameters, related to Young's modulus E and Poisson's ratio ν by μ = E / (2(1+ν)) and λ = Eν / ((1+ν)(1−2ν)). Substituting into equilibrium and expressing strain through displacement gives the Navier–Cauchy equations:

(λ + μ) ∇(∇·u) + μ ∇²u + b = 0

This is the governing PDE that a physics-informed or operator-learning model must respect. Note the two distinct differential operators — a gradient-of-divergence term and a vector Laplacian — which behave very differently under discretization and under Fourier/spectral representations. For anisotropic or nonlinear materials, the constitutive law becomes a learned or tabulated mapping σ = C(ε), and the equilibrium and compatibility structure remains unchanged. This separation is a key design principle: kinematics and balance are universal; constitutive behavior is material-specific and is where data-driven components belong.

Boundary value problems: the complete specification

A well-posed elasticity BVP specifies the domain, the constitutive parameters, the body force, and boundary conditions on complementary parts of ∂Ω:

• Dirichlet (displacement) conditions: u = ū on Γ_u — prescribed motion, e.g., a clamped face.
• Neumann (traction) conditions: σ·n = t̄ on Γ_t — prescribed surface load, with outward unit normal n.
• Mixed/Robin conditions: elastic foundations or contact springs, coupling u and σ·n.

Well-posedness requires Γ_u ∪ Γ_t = ∂Ω and Γ_u ∩ Γ_t = ∅, plus sufficient Dirichlet constraint to eliminate rigid-body modes (three translations and three rotations in 3D). A model that ignores rigid-body suppression will fit training data yet produce arbitrary constant offsets at inference — a common and easily misdiagnosed failure.

From continuous fields to learnable objects

The figure below summarizes how these four ingredients interlock and where each enters a learning pipeline. The left column shows the physical constraints; the right column shows the corresponding computational role.

A worked example: 1D bar as a sanity check

Before trusting any learned model, verify it on a problem with a closed-form solution. Consider a bar of length L = 1 m, cross-section A = 1×10⁻⁴ m², Young's modulus E = 200 GPa, fixed at x = 0, with a tip load P = 1 kN at x = L and no body force. The BVP reduces to E A u'' = 0 with u(0) = 0 and E A u'(L) = P. The exact solution is linear:

u(x) = P x / (E A) = (1000 · x) / (200e9 · 1e-4) = 5e-5 · x   [m]
σ = P / A = 1e7 Pa = 10 MPa   (uniform)
ε = σ / E = 5e-5

A candidate model should reproduce u(1) = 5×10⁻⁵ m to within a stated tolerance (e.g., 1% relative error) and produce a spatially uniform stress of 10 MPa. If the predicted stress varies along the bar, the equilibrium residual is not being minimized correctly. If u(0) ≠ 0, the Dirichlet term is under-weighted or the rigid-body mode is unconstrained. If the stress is uniform but the tip displacement is off by a constant, suspect a unit inconsistency between E in GPa and lengths in meters — the single most common failure in elasticity pipelines.

What this means for the learning problem

Three practical consequences follow. First, predict displacement, not stress: differentiation is exact and cheap, and it automatically enforces compatibility and strain symmetry. Second, keep the constitutive law modular: for linear elasticity it is a fixed linear map, and for nonlinear or anisotropic materials it becomes a small learned subnetwork whose inputs and outputs are tensors with known symmetry. Third, treat boundary conditions as first-class constraints, not as an afterthought — either through hard enforcement (e.g., distance-function blending for Dirichlet data) or through separately monitored loss terms. These three decisions carry directly into the operator-learning architectures and physics-informed training schemes developed in the following sections.