Crystallization consolidation and interface bonding

The optional WB5 thermoplastic models are history-driven calculations attached to a transient thermal case. Solve the temperature history first, then evaluate crystallization at each saved thermal node, void consolidation at a selected depth, and contact and healing at a selected interface. A model must be explicitly linked to the case; a material name or an unlinked model record does not activate it.

These models currently use one-way postprocessing. Their evolving states do not change the thermal solution, layer thickness used by that solution, laminate stiffness or residual stress. In particular, the PEEK branch described here does not return latent heat or crystallization shrinkage to the solver. The separate coupled crystallization formulation in the material-state chapter describes a different execution path. Do not assume that its feedback terms are active when using the optional PEEK history model.

The workflow is: material and boundary data → transient temperature and selected pressure histories → crystallization, void or interface model → state history and interpretation. Void content measures gas in a cell; intimate contact measures touching interface area; healing describes molecular interdiffusion on that area. None of these quantities can be substituted for another.

The parallel and series Velisaris–Seferis records represent two crystallization contributions, each advanced with a nonisothermal Nakamura clock. For branch i, let αᵢ be its relative conversion, nᵢ its Avrami exponent and Kᵢ(T) its temperature-dependent rate in inverse seconds. The accumulated clock preserves thermal history rather than recalculating conversion from the current temperature alone.

The initial clock reproduces the supplied initial conversion. A branch initially at complete conversion remains saturated until the full-melting reset. The distinction between K and the isothermal Avrami coefficient k is important: K = k^1/n, so K has units s⁻¹ while k has units s⁻ⁿ. Entering k directly in the K table changes both units and kinetics.

The contribution weight w lies between zero and one. The series combination is zero when a branch with nonzero weight has zero conversion; a branch of zero weight is excluded without division by zero. X is the absolute matrix crystallinity on the basis supplied with X_max, not the crystallinity of the fiber-filled composite. The interface reports matrix crystallinity, relative crystallinity, and the primary and secondary conversions as percentages.

Five editable temperature points define each K table. Rates are interpolated linearly, with zero endpoint values and no growth outside the specified window. The algorithm splits a saved temperature segment at each table knot and at the full-melting threshold. It then integrates the piecewise-linear interpolation exactly. At or above the melting threshold both clocks are reset to zero, allowing recrystallization on cooling. This instantaneous complete-melting rule is a modeling extension; partial melting, reorganization and melting-rate kinetics are not resolved.

Use DSC measurements from the actual polymer grade and relevant thermal histories to fit the rate tables, exponents, contribution weight and limiting crystallinity. The published parallel exponents 2.5 and 1.5 provide a literature example, but the seeded WB5 rate values, weight, limiting fraction and melting threshold are illustrative inputs rather than a fitted Seferis dataset. The series exponents also require calibration. Both records currently apply the same matrix kinetics at every thermal node; they do not assign different polymers to different layers. The formulation and implementation must be distinguished from a legacy file named crystallization.m whose coefficients describe degradation rather than crystallization.

The void model is a uniform periodic dry-fiber-bed reduction of the model of Simacek and colleagues. A cell contains incompressible fiber and resin together with a gas region. Transverse resin flow and the response of the fiber bed govern the change in cell thickness. It is a local representative cell, not a resolved distribution of interacting voids.

Let q be gas volume divided by the initial cell volume, φ₀ the initial void fraction, c = 1−φ₀, h₀ the initial cell thickness and V_f0 the initial fiber fraction. Constituent conservation gives the current thickness h, fiber fraction V_f and porosity φ directly from q.

Use compression-positive applied absolute pressure P, absolute gas pressure p_g, resin viscosity η, permeability K, fixed void spacing L and effective bed viscosity η_b. Integrating the symmetric Darcy pressure profile across the cell gives a mean resin pressure p_g−ηL²(ḣ/h)/(12K). Adding the compressive bed stress σ_b gives the balance below.

All pressures in this balance are in pascals, viscosities in Pa·s, permeability in m² and spacing in metres; q̇ is in s⁻¹. Applied pressure exceeding the gas and bed resistance reduces q. The reverse imbalance can produce rebound while flow remains active. This implementation derives its balance from the pressure field rather than copying the printed force and volume equations literally; it retains the mean gas-pressure term, a compression-consistent flow sign and dimensionally consistent resistance.

Temperature T is in kelvin in this expression, with a separately specified fitted offset C_η. A flow activation temperature freezes the geometry below the threshold. It is not a crystallization or cure calculation. The example carbon/PEEK coefficients and geometry require calibration before use with another tape. The bed modulus and effective bed viscosity can be zero, as in the source example.

Sealed gas conserves pV/T and cannot disappear at finite pressure. Vented gas remains at ambient pressure; evacuated gas provides no gas resistance. A closed vented or evacuated cell does not nucleate a new void on unloading. The reported equivalent radius is obtained from the gas cross-sectional area qh₀L and is a two-dimensional area descriptor, not a spherical bubble radius.

A backward-Euler update with bisection and step doubling controls the scalar integration error. Thermal knots, flow-threshold crossings and pressure events are resolved explicitly. Check constant-coefficient compaction, gas equilibrium, rebound, constituent conservation and maximum packing before interpreting a process run. Gas diffusion, capillarity, void migration, a void-size distribution and two-way thickness or stiffness feedback are outside this reduction.

Intimate contact is the fraction of an interface whose opposing surfaces touch closely enough for molecular bonding to become possible. The WB5 contact record uses a lumped asperity squeeze-flow law. It does not resolve a fractal hierarchy of individual asperities. Pressure and effective contact-flow viscosity determine an accumulated exposure J, and a roughness coefficient R_c converts that exposure into contact fraction D_c.

J is dimensionless when pressure is in pascals, viscosity in Pa·s and time in seconds. The fifth-power initial term treats an existing contact fraction D₀ as equivalent prior exposure. It deliberately differs from adding D₀ to the zero-state fifth-root solution, which would not preserve the same constitutive state during continued loading. New contact is monotonic; reopening and contact damage are not included.

The contact viscosity uses η_c = A exp[B/(T+C)], evaluated only above the contact activation temperature. The illustrative fit A = 132.95 Pa·s, B = 2969 K and C = −273.15 K reproduces the legacy Celsius-denominator expression while T is supplied in kelvin. This empirical law must not be mistaken for an activation-energy Arrhenius law. Contact-flow viscosity is a separate parameterization from the viscosity used by the void Darcy-flow calculation.

For contact, gauge pressure is already net compression. Absolute pressure is converted by subtracting ambient pressure once, and negative net compression is clipped to zero. Global pressure excludes roller additions; an upper or lower total includes the global profile and the selected face contribution once. The two opposing pressures are not added. Unloading stops new pressure-driven contact but does not remove contact already formed.

After surfaces touch, compatible thermoplastic chains need time at a sufficiently high temperature to interdiffuse across the interface. Healing therefore depends on both thermal history and the time at which each portion of the interface first made contact. A fully hot interface with no contact cannot develop a bond in this model.

The exposure integral advances only above the healing activation temperature. Welding time is in seconds, temperatures in kelvin, activation energy in J/mol and R = 8.314462618 J/(mol·K). The AS4/PEEK example uses 0.11 s at 400°C, activation energy 57300 J/mol and a 338°C activation threshold. These are example inputs, not universal values for all PEEK tapes or polymer pairs.

A new contact patch is born at time τ and begins with zero healing. The dimensionless h₀ in this section is the initial healing of the area already in contact, not the cell thickness h₀ used in the void model. Keeping these states separate prevents a late-forming patch from receiving the thermal exposure that occurred before contact.

The area integral follows Butler-type contact-age coupling with nonisothermal healing. It is not approximated by multiplying the final contact fraction by the healing potential of an interface assumed to touch from the start. The reported healing potential is useful for separating thermal opportunity from contact delay, but coupled bonding is the quantity that includes both histories. A strength is reported only when a fully healed strength is supplied; the scaling is not a qualified design allowable.

The numerical method integrates exposure with midpoint step doubling, limits the new contact area and healing exposure per step, and assigns patch birth exposures using two-point quadrature in contact area. Mature patches are accumulated as fully bonded area. Refine the maximum step, contact increment and tolerance together when checking convergence. Cooling pauses the active mechanisms, while reheating resumes their accumulated histories. Healing may continue after unloading if the interface remains hot and in contact, as assumed here. Debonding, degradation, crystallization-dependent mobility, incompatible polymers and feedback to conductivity or mechanical properties are excluded.

Use the thermal case's PEEK crystallization model link for either crystallization record, the Void consolidation model link for the cell calculation, and the contact-only or coupled-bonding links for the interface calculation. The coupled model references its contact record. If both interface links are populated, they must identify the same contact model so depth, pressure and integration assumptions cannot diverge.

Void and bond histories use a selected normalized depth, with zero at the lower surface and one at the upper surface. They are single-depth histories, not measured or predicted distributions through the complete thickness. Crystallization instead follows each saved thermal node. For a moving process with positive constant speed, distance follows x = vt; this is a time-to-position mapping, not a new spatial transport solve.

First refine the saved thermal output so heating peaks, cooling intervals and roller events are adequately resolved. Then refine each postprocessor's internal integration. An accurate integration of interpolated temperature cannot recover a peak that was never saved. Use isothermal crystallization, fifth-root contact, quarter-power healing and a delayed-contact comparison as limiting checks. Record the model IDs, coefficients, pressure convention, depth, numerical settings and parameter sources with the resulting histories. Experimental validation requires DSC, rheology, void measurements or bond tests appropriate to the quantity being predicted.

1. Velisaris, C. N. and Seferis, J. C. (1986). Crystallization kinetics of polyetheretherketone (PEEK) matrices. Polymer Engineering and Science, 26, 1574–1581. doi:10.1002/pen.760262208.
2. Nakamura and colleagues (1972). Some aspects of nonisothermal crystallization of polymers I. Journal of Applied Polymer Science, 16, 1077–1091. doi:10.1002/app.1972.070160503.
3. Kelly, C., Hay, J., Turner, R. and Jenkins, M. (2020). The effect of a secondary process on the analysis of isothermal crystallisation kinetics by differential scanning calorimetry. Polymers, 12, 19. doi:10.3390/polym12010019. Used for the branch-combination relations, not PEEK calibration.
4. Simacek, P., Advani, S. G., Gruber, M. and Jensen, B. (2013). A non-local void filling model to describe its dynamics during processing thermoplastic composites. Composites Part A, 46, 154–165. doi:10.1016/j.compositesa.2012.10.015.
5. Tierney, J. and Gillespie, J. W. (2006). Modeling of in situ strength development for the thermoplastic composite tow placement process. Journal of Composite Materials, 40, 1487–1506. doi:10.1177/0021998306060162.
6. Yang, F. and Pitchumani, R. (2002). Healing of thermoplastic polymers at an interface under nonisothermal conditions. Macromolecules. doi:10.1021/ma010858o.
7. Butler, C. A., McCullough, R. L., Pitchumani, R. and Gillespie, J. W. (1998). An analysis of mechanisms governing fusion bonding of thermoplastic composites. Journal of Thermoplastic Composite Materials. doi:10.1177/089270579801100404.