Materials, component physics, and engineering constraints

Practical stellarator design requires coupled thermal, mechanical, electromagnetic, nuclear, and lifetime models. This note gives explicit model interfaces and theorem targets. Empirical constitutive data remain supplied hypotheses with provenance and applicability envelopes.

Contents

• Material laws and validity envelopes
• Heat loads and cooling
• Mechanical support and stress
• Superconducting magnets and tape strain
• Quench and electrical protection
• Neutron transport, shielding, and breeding
• Damage, erosion, fatigue, and service life
• Power and maintainability
• Theorem backlog

Material laws and validity envelopes

Describe a material by composition, grade, manufacturing history, orientation, joint/interface construction, and evolving state. Possible state variables are temperature T, strain ε, stress σ, irradiation dose d, helium/hydrogen content, cycle count, and accumulated plastic strain.

Source lower/upper envelopes for:

Represent these as explicit law bundles with a validity predicate and interval bounds. Keep deterministic envelopes distinct from a confidence statement about a fitted model. Do not extrapolate unirradiated room-temperature properties into hot irradiated operation.

Knaster–Moeslang–Muroga surveys fusion-material challenges. DEMO divertor work provides concrete component/load examples. Huslage et al. connects REBCO tape mechanics to stellarator optimization. None makes one scalar limit universal for every grade or device.

Heat loads and cooling

Distinguish plasma-facing boundary heat flux q'' in W/m², volumetric neutron/Joule heating Q in W/m³, and total component power in W. Obtain the first from a transport/exhaust certificate or a supplied worst-case envelope, and the second from neutron/electromagnetic models.

For a solid body,

Use tensor k for anisotropic materials. Specify initial temperatures, material interfaces, thermal contact resistance, radiation, cooling, and applied heat boundary conditions. Prove energy balance, positivity/comparison statements, and temperature bounds with the needed regularity.

First exact thermal theorem

Take a steady one-dimensional slab 0≤x≤h with h>0, constant k>0, uniform Q≥0, incoming heat flux q''≥0 at x=h, and convective cooling at x=0 with h_c>0 and coolant temperature T_c. Assume all generated/applied heat exits through the cooled face. The boundary laws are kT'(h)=q'' and kT'(0)=h_c(T(0)-T_c). Solving -kT''=Q gives

Prove the solution, boundary fluxes, monotonicity, and maximum location. With conservative k≥k_min, h_c≥h_c,min and upper input bounds in this constant-coefficient family, certify a temperature margin against a supplied lower allowable-temperature limit. Do not substitute this formula into arbitrary 3-D or variable-coefficient geometry without a comparison theorem.

A synthetic exact-arithmetic test fixture uses the entire parameter box

Monotonicity selects the upper T_c,q'',Q,h and lower k,h_c corner. The maximum is 44002/45 K, less than an illustrative 1000 K limit by 998/45 K. These inputs and limit are invented for a checker/sanity example; they are not sourced material qualification data. The mathematical bound covers the whole stated box, conditional on the slab model.

For a simplified fully conducting interface resistance R_th at the cooling side, add R_th(q''+Qh) under its own boundary law.

For a steady single coolant stream with specific enthalpy h_f(T) satisfying dh_f/dT=c_p(T), no energy storage or other work, and negligible kinetic/potential-energy changes, prove

A network model can give an early flow/temperature certificate. Boiling, critical heat flux, pumping limits, coolant transients, leakage and flow interruption require additional explicit models; a minimum mass-flow assumption cannot simultaneously describe a pump-failure event.

Model an island divertor and detachment as a distinct plasma-edge/exhaust problem. Extract useful measured envelopes from W7-X divertor literature, keeping operating conditions and transfer assumptions.

Mechanical support and stress

For small-strain quasi-static elasticity use

Define supported boundaries and traction interfaces. For a linear elastic submodel set plastic strain to zero explicitly. Require uniform positivity of the elasticity tensor, remove rigid motions with justified constraints, and prove the relevant Korn/trace estimates before existence or a posteriori stress bounds.

An early theorem can analyze a fully axially restrained uniform bar with spatially uniform temperature change ΔT, constant Young modulus E and expansion coefficient α:

Prove this from the bar model; it is not the stress tensor of every heated coil or wall. For an otherwise load-free constant-section bar restrained only at its ends, with spatially varying temperature and constant E,α, the uniform axial stress instead uses the length average: σ=-Eα L⁻¹∫₀ᴸ ΔT(s) ds. A later beam model should distinguish bending, torsion, deflection, and local buckling.

For an isotropic ductile failure proxy define

A feasible pointwise contract is σ_vm ≤ σ_allow(T,d)/SF with declared SF>1 and sourced allowables. This does not automatically cover brittle fracture, REBCO delamination, fatigue, contact, anisotropic failure, or irradiated tungsten fracture. Use separate predicates/constitutive laws for those mechanisms.

Manufacturing and thermal deformations change both clearances and magnetic geometry. Carry a proved displacement/strain bound back into 06-coils-and-design.md.

Superconducting magnets and tape strain

Distinguish operating plasma field from peak field inside the conductor. The superconducting margin requires the latter and its orientation. A useful contract is

throughout the conductor and operating envelope, together with a thermal margin and a valid conductor/current-sharing model. Use specified NbTi, Nb₃Sn, or REBCO data rather than one generic superconductor threshold.

Represent a winding pack as

where r is arclength-parametrized and (d₁,d₂,d₃=r') is a material frame. Prove embedding, positive volume Jacobian, and clearance. For a straight reference fibre with coordinate s, its geometric longitudinal stretch is

The stress-free/reference geometry and fabrication strain must be specified; this geometric stretch alone is not the full constitutive strain of a manufactured tape.

For planar pure bending, the small-strain maximum is proportional to half-width times hard-way curvature, or half-thickness times easy-way curvature. For pure uniform twist of a straight axis at offset radius r_f,

Prove these special cases from the frame formula. Certify the full geometry for combined bend/twist and across every tape/turn, rather than adding approximate strain formulas without an error estimate. Account for winding angle, joints, pack deformation, cooldown, preload, and irreversible degradation. Huslage et al. is a concrete implementation reference.

Quench and electrical protection

Use a specified electric/thermal constitutive law. A simplified superconducting power law is

away from J_cond=0, extended by E(0)=0, with E_c,J_crit,n>0. Here J_cond is vector conductor current density and J_crit the scalar critical density. Its dissipation J_cond·E≥0 is an elementary theorem; the law's empirical applicability is separate.

For a lumped coil with constant L>0,

Prove stored-energy and dump-energy bounds. Couple resistive heat to

with C(T)>0 and validated coefficients. A multi-coil model needs an appropriate positive inductance matrix and mutual-coupling dynamics.

First certify the zero-supply dump case with fixed coefficients. Then include detection delay, hotspot temperature, terminal/inter-turn voltage, current sharing, and thermal diffusion. A normal-operation current margin does not prove survival of a quench; quench protection has its own event and time horizon.

Neutron transport, shielding, and breeding

For D–T fusion, track approximately 17.6 MeV per reaction partitioned into alpha and neutron energy (approximately 3.5 and 14.1 MeV respectively). Use exact model constants or declared input intervals and a reaction-rate envelope from 04.

A linear neutron-transport model for angular flux ψ_n(x,\omega,E,t) is

with explicit incoming boundary conditions and additional neutron-producing reaction terms where needed. Supply microscopic/macroscopic cross-section conventions, isotope densities, and uncertainty from a pinned IAEA FENDL dataset.

Define nuclear heating, dose, and reaction rates as positive integrals of the certified flux against the relevant response kernels. Establish source normalization, centimetre/metre and eV/joule conversions, and any group/mesh error. OpenMC documentation describes simulation/tally conventions; Monte Carlo confidence intervals are statistical statements rather than guaranteed deterministic enclosures.

For a monoenergetic directed beam with constant macroscopic attenuation Σ≥0, no scattering or other sources, and path-length coordinate x (for example normal incidence on a uniform slab),

Prove attenuation and use upper source/lower attenuation bounds. General angular scalar flux need not obey this formula even without scattering; scattering and streaming through ports require a transport estimate.

Define breeding by

assuming positive consumption. Require a specified margin above the consumption and losses relevant to the plant, rather than imposing a universal chosen number. Couple it to a tritium-inventory model with extraction, decay, retention, processing delay, and reserve constraints.

Damage, erosion, fatigue, and service life

Using energy-dependent scalar neutron flux Φ(E,t) and a displacement cross section per atom σ_dpa(E), one damage model is

Do not divide by atomic density again if the cross section already describes displacements per target atom. Specify the displacement model and its conventions; dose is not itself a complete material-degradation law.

Use a nonnegative activation/transmutation inventory ODE N'=A(Φ,T)N+s. Under Metzler/nonnegative-source conditions, prove nonnegative inventories and selected bounds. Track helium/hydrogen production and link it to sourced constitutive envelopes.

For a simplified erosion law,

where Y is sputter yield and n_solid atomic number density. Add redeposition, transients, chemical effects, and mixed-material evolution only through defined extensions. Couple eroded thickness to temperature, strength, and clearance.

For an explicitly assumed cumulative fatigue law such as D_f=Σ_i n_i/N_i, prove the arithmetic implication from a cycle/load envelope to D_f≤D_allow. Its empirical predictive validity is not proved by that arithmetic. Brittle-fracture and creep models require separate load-history/degradation laws.

A service-life theorem should quantify over the stated operation history, source envelopes, temperatures, irradiation state, and uncertainty. Return a conservative time/cycle bound and the active limitation. Use repair/replacement schedules as part of the state evolution where relevant.

Power and maintainability

Define gross electrical output using a specified heat-conversion law and subtract auxiliary heating, cryogenic refrigeration, coolant pumping, current supply, fuel processing, and other recirculating loads. Certify a positive net-power margin only for the coupled model; a plasma triple product or ideal-MHD stability proof is insufficient.

Maintenance requires access volume and collision-free removal paths for blankets, divertor units, coils, and supports, accounting for activation/cooldown constraints where modelled. These geometric constraints may limit design before a physics optimum is reached.

Theorem backlog