3. Crash-specific PDE formulations

A physics-informed neural network for crash response prediction is only as trustworthy as the governing equations it is asked to satisfy. Crash mechanics is not a single equation but a coupled system: momentum balance governs motion, strain–displacement relations define kinematics, and constitutive laws close the system by linking stress to deformation history. In a PINN, each of these is converted into a residual that is evaluated at collocation points and driven toward zero by the loss function. This chapter develops the strong and residual forms of each component for a transient, finite-strain, elastoplastic crash problem, and shows how to assemble them into a training-ready residual set.

Reference configuration and sign conventions

All formulations below use a Lagrangian description with the undeformed configuration Ω₀ as reference. Let X be the material coordinate, u(X, t) the displacement, and F = ∂u/∂X + I the deformation gradient. The first Piola–Kirchhoff stress P is used as the primary stress measure because it is work-conjugate to F and yields a clean momentum residual in the reference frame. Tensile stress and positive displacement are positive; compression is negative. This convention is held consistently in every residual and in the accompanying figure.

Momentum balance

The strong form of linear momentum balance in the reference configuration, neglecting body forces for a crash event dominated by inertia and contact, is

ρ₀ ∂²u/∂t² − ∇_X · P = 0   in Ω₀ × (0, T]

with initial conditions u(X,0) = u₀ and ∂u/∂t(X,0) = v₀, and traction boundary conditions P·N = t̄ on Γ_t. For a PINN, the momentum residual at a collocation point (X_i, t_i) is

r_mom = ρ₀ ü(X_i, t_i) − ∇_X · P(X_i, t_i)

where both the second time derivative and the divergence of stress are obtained by automatic differentiation of the network outputs. The density ρ₀ is a known constant (for steel, 7850 kg/m³; for aluminium, 2700 kg/m³), so the residual is a pure function of network derivatives. A common practical pitfall is that second-order time derivatives are noisy when the temporal collocation spacing is coarse relative to the impact pulse; sampling t_i densely during the first 2–5 ms of the event, where acceleration peaks, materially improves residual conditioning.

Strain–displacement relations

Crash problems are finite-strain, so the small-strain tensor is inadequate. The Green–Lagrange strain

E = ½ (FᵀF − I) = ½ (∇_X u + (∇_X u)ᵀ + (∇_X u)ᵀ ∇_X u)

is the natural kinematic companion to P. The quadratic term is not optional: in a frontal crash, engineering strains in folding zones routinely exceed 10–20%, and dropping the quadratic term introduces errors of the same order. The kinematic residual is therefore not a separate equation to be penalized but a definitional constraint enforced by construction — the network predicts u, and F, E, and their derivatives are computed from it. The only residual worth adding is a compatibility check when displacement is predicted through a separate potential or when multiple sub-networks share a domain:

r_kin = E − ½ (FᵀF − I)   (≈ 0 by construction; used for consistency audits)

Constitutive laws: elastoplasticity with strain-rate sensitivity

The constitutive residual is where crash PINNs most often fail, because plasticity is history-dependent and path-dependent, while a vanilla PINN evaluates residuals pointwise in time. The standard J₂ flow theory with isotropic hardening and rate sensitivity is written as follows. The trial stress is

P_trial = P_n + C : (F − F_n)

with C the fourth-order elastic tangent. The yield function is

f = σ_eq(P_trial) − σ_y(ε̄ᵖ, ε̄̇ᵖ) ≤ 0

where σ_eq is the von Mises equivalent stress and the hardening law includes rate dependence, for example the Cowper–Symonds form

σ_y = σ_y0 (1 + (ε̄̇ᵖ / D)^(1/q)) + H ε̄ᵖ

with σ_y0 the quasi-static yield stress, H the hardening modulus, and D, q material rate parameters. For a typical automotive steel, representative values are σ_y0 ≈ 350 MPa, H ≈ 1.2 GPa, D ≈ 40 s⁻¹, q ≈ 5; these are illustrative and must be replaced by coupon-test data for any real prediction. The plastic multiplier follows from the consistency condition ḟ = 0, and the flow rule is

Ḟᵖ = λ̇ ∂f/∂P,   ε̄̇ᵖ = λ̇,   λ̇ ≥ 0,  λ̇ f = 0

In residual form for training, the constitutive block becomes three residuals evaluated at each collocation point:

r_yield  = max(0, f) / σ_y0                      (yield violation, normalized)
r_flow   = ‖Ḟᵖ − λ̇ ∂f/∂P‖ / ‖Ḟᵖ‖ + ε₀           (flow-rule consistency)
r_hard   = ε̄̇ᵖ − λ̇                                (hardening evolution)

The normalization by σ_y0 and by the plastic rate magnitude is essential: without it, the yield residual (units of stress, ~10⁸ Pa) dominates the momentum residual (units of N/m³) by many orders of magnitude, and the optimizer ignores kinematics entirely. The small constant ε₀ prevents division by zero in the elastic regime.

Assembling the training residual set

The total loss is a weighted sum of the mean-squared residuals over collocation, initial, and boundary sets:

L = w_mom · mean(r_mom²) + w_yield · mean(r_yield²)
  + w_flow · mean(r_flow²) + w_hard · mean(r_hard²)
  + w_ic · mean(r_ic²) + w_bc · mean(r_bc²)

A workable starting point for a steel crash box under 10 m/s impact is w_mom = 1, w_yield = 10, w_flow = 1, w_hard = 1, w_ic = w_bc = 100, with all residuals non-dimensionalized as above. The initial and boundary weights are high because those constraints are cheap to satisfy and anchor the solution. During training, monitor each residual term separately; a yield residual that plateaus while momentum falls indicates the network has learned a mostly elastic response and is not activating plasticity — increase collocation density in the folding region or raise w_yield.

What to verify before trusting a trained model

Residual magnitudes alone do not prove correctness. Three checks are worth running on any trained crash PINN. First, a patch test: impose a uniform uniaxial strain on a small domain and confirm the network reproduces the analytic elastic stress to within 1%. Second, a yield-onset check: ramp the load until σ_eq reaches σ_y0 and verify that the plastic multiplier remains zero below yield and activates smoothly above it — a discontinuous jump signals an over-weighted yield residual. Third, a rate sensitivity sweep: run the same geometry at 1 m/s and 10 m/s and confirm the predicted peak force rises in accordance with the Cowper–Symonds parameters; if the two curves coincide, the rate term has been effectively disabled by normalization or weighting. These checks isolate constitutive errors from kinematic or momentum errors and are far cheaper than a full validation against explicit finite element results.