05.1.2 / Materials

Thermoset cure kinetics

Define calibrated epoxy cure, heat release, and irreversible process shrinkage at the material/ply level before solving the laminate process history. Expand any equation for its physical meaning, variables, units, model connection, and theory source.

State variable and model routing

Degree of cure α is the reacted fraction of the available thermoset chemistry. R15 bounds α to [0,1], prevents reverse reaction, and evaluates kinetics independently at every transport node. Each ply selects its own polymer family and kinetics model in the process definition. Select thermoset cure for a reacting resin or an inert material for a nonreacting layer.

Eq. 05.1.2-01Arrhenius rate constant

k_i(T) = A_i exp[−E_i /(R T)]

Equation detailsExplanation · variables · model connection · reference

Converts each calibrated kinetic prefactor and activation energy into a temperature-dependent reaction-rate constant. Absolute temperature must be used.

Variablesk_irate constant for reaction branch is⁻¹A_ipre-exponential factor for branch is⁻¹E_iactivation energy for branch iJ/molRuniversal gas constant8.314462618 J/(mol·K)Tlocal nodal absolute temperatureK

Model connectionEvaluated from the current thermal iterate before the selected cure law is advanced; it therefore couples cure rate to the transient temperature field.

Theory basisSourour–Kamal cure kinetics

Selectable reaction laws

The model ID chooses one of four rate laws. These laws describe chemistry only after their coefficients have been fit to DSC or equivalent measurements for the actual resin formulation.

Eq. 05.1.2-02Nth-order cure

α̇ = k₁(T)(1−α)ⁿ

Equation detailsExplanation · variables · model connection · reference

Represents a non-autocatalytic reaction whose rate decreases as the unreacted fraction is consumed.

Variablesαdegree of curedimensionless, 0–1α̇instantaneous cure rates⁻¹k₁Arrhenius rate constant from Eq. 05.1.2-01s⁻¹nreaction-order exponentdimensionless

Model connectionThe nth-order cure law. The returned nonnegative rate is passed to the bounded irreversible state update.

Theory basisThermoset kinetic model basis

Eq. 05.1.2-03Kamal–Sourour autocatalytic cure

α̇ = [k₁(T)+k₂(T)α^m](1−α)ⁿ

Equation detailsExplanation · variables · model connection · reference

Adds an autocatalytic branch that grows with degree of cure to the nth-order branch, allowing an isothermal rate peak after the reaction begins.

Variablesk₁, k₂non-autocatalytic and autocatalytic Arrhenius rate constantss⁻¹αdegree of curedimensionless, 0–1mautocatalytic exponentdimensionlessnunreacted-fraction exponentdimensionlessα̇instantaneous cure rates⁻¹

Model connectionThe Kamal–Sourour cure law, also used as the base reaction law before vitrification limitation is applied.

Theory basisSourour–Kamal autocatalytic model

Eq. 05.1.2-04Prout–Tompkins cure

α̇ = k₁(T)α^m(1−α)ⁿ

Equation detailsExplanation · variables · model connection · reference

Uses a single autocatalytic rate constant with reacted- and unreacted-fraction powers. A small positive numerical floor on α prevents a zero-to-a-negative-power singularity.

Variablesk₁Arrhenius rate constants⁻¹αdegree of curedimensionless, 0–1m, nautocatalytic and reaction-order exponentsdimensionlessα̇instantaneous cure rates⁻¹

Model connectionThe Prout–Tompkins cure law. It shares the same bounded state update, heat source, and shrinkage mapping as the other cure laws.

Theory basisNASA composite cure-process formulation

Eq. 05.1.2-05Diffusion-limited Kamal multiplier

f_d = {1 + exp[s_v(α−α_v)]}⁻¹, α̇_limited = f_d α̇_Kamal

Equation detailsExplanation · variables · model connection · reference

Suppresses the chemical Kamal–Sourour rate as cure passes a calibrated vitrification threshold. It is an empirical mobility correction, not an independently predicted glass-transition model.

Variablesf_ddiffusion or vitrification rate multiplierdimensionless, 0–1s_vlogistic vitrification slopedimensionlessα_vdegree of cure at the logistic midpointdimensionlessα̇_Kamalunlimited rate from Eq. 05.1.2-03s⁻¹α̇_limitedmobility-limited cure rates⁻¹

Model connectionThe diffusion-limited cure law. The limited rate, rather than the raw chemical rate, advances α and produces reaction heat.

Theory basisCure-process modeling and vitrification context

Irreversible state integration

Eq. 05.1.2-06Bounded exponential cure update

r_α = 1−αⁿ, αⁿ⁺¹ = min[1, max(αⁿ, αⁿ + r_α{1−exp(−rⁿΔt/r_α)})]

Equation detailsExplanation · variables · model connection · reference

Advances cure over one transport step without consuming more than the remaining unreacted fraction. The outer bounds enforce irreversibility and α≤1.

Variablesαⁿ, αⁿ⁺¹degree of cure at the old and new process timesdimensionlessrⁿselected raw or diffusion-limited kinetic rates⁻¹Δttransport time incrementsr_αavailable unreacted fraction, 1−αⁿdimensionless

Model connectionApplied independently at every active thermoset node inside each staggered thermal/kinetic iteration. The reported rate is (αⁿ⁺¹−αⁿ)/Δt.

Theory basisNumerical cure-process integration context

Reaction heat and chemical shrinkage

Eq. 05.1.2-07Volumetric cure heat source

q_c = ρ H_c α̇

Equation detailsExplanation · variables · model connection · reference

Converts cure progress and mass-specific total heat of reaction into a volumetric source for the one-dimensional heat equation.

Variablesq_cvolumetric cure heat generationW/m³ρlocal effective ply densitykg/m³H_ctotal heat of cure per unit massJ/kgα̇bounded cure rate over the current steps⁻¹

Model connectionIncluded in the thermal right-hand side only when the cure-exotherm option is enabled and reported separately as cure heat generation.

Theory basisNASA composite cure-process heat balance

Eq. 05.1.2-08Local cure-shrinkage free strain

ε^c_k = −s^c_k(α−α₀)

Equation detailsExplanation · variables · model connection · reference

Maps the increase in degree of cure into an anisotropic contraction in the ply material axes. Positive calibrated shrinkage magnitudes therefore produce negative free strain.

Variablesε^c_kcure-shrinkage free strain in material direction kstrains^c_kfull-cure shrinkage magnitude in direction kstrainα₀initial degree of curedimensionlessklocal ply direction 1, 2, or 3index

Model connectionStored at each process node, fitted across each ply, transformed to laminate axes, and optionally included in the process force and moment resultants.

Theory basisNASA cure-induced residual-stress study

Calibration boundary. Pre-exponential factors, activation energies, exponents, total reaction heat, vitrification controls, and shrinkage are resin-system data. R15 defaults to an inert row and does not represent a universal epoxy.

Material inputs and outputs

Input family	Purpose	Primary output
A₁, E₁, A₂, E₂, m, n	Temperature-dependent reaction rate	Cure-rate history
H_c	Total heat of cure per unit mass	Cure-heat history
s^c_1,2,3	Full-process local shrinkage magnitudes	Process-shrinkage history
α₀, α_v, s_v	Initial state and optional mobility limitation	Degree-of-cure history

Theory references

Workbench availability: released models, inputs and compatible study paths. The wider theory library includes reference formulations not available in every Workbench solve.