11-EFT.WP.Core.DrawingKinetics v1.0 | Chapter 6 Stability Criteria and Instability Modes

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Chapter 6 Stability Criteria and Instability Modes

I. Scope and Objectives

• Provide unified criteria for stability in filament drawing/stretching and the principal instability modes — necking, wave amplification, and winding-coupling instability — to support safe-region design, online monitoring, and control strategies.
• Establish consistent links with Chapter 2 (geometry), Chapter 3 (conservation), and Chapter 4 (constitutive laws), and define measurable margin indicators and thresholds for Mx-15 stability-scan and regression tests.

II. Terminology and Symbols

• Kinematics & geometry: lambda(ell,t), e = ln( lambda ), s = ( d/dt ) ( ln( lambda ) ), v(ell,t), A(ell,t), gamma(ell), d ell.
• Conserved quantities: rho_L = rho * A, J = rho_L * v.
• Tension & constitutive terms: T_fil(ell,t); tangent modulus M_tan = ( dT_fil / de ) |_{s,Theta}; viscous-coefficient family represented by the K_vis set (Chapter 4); elastic coefficients represented by K_el.
• Dimensionless groups: We = s * tau_relax (with tau_relax an effective relaxation time, e.g., tau_relax = ( K_vis / K_el ) under a Maxwell approximation), De = ( tau_relax / T_obs ), Re = ( rho * v * L_ref ) / mu_eff (with mu_eff an effective viscosity and L_ref a reference length).
• Gain & growth rate: for a perturbation delta(•) with normal mode exp( i k ell + g t ), define the growth rate g(k) and amplification G = exp( g * T_obs ).

III. Postulates and Minimal Equations

1. P11-4 (slenderness and small-disturbance postulate)
   Under A / L_ref^2 << 1 and bounded s, a 1D axial approximation is valid; linearization and superposition hold within the operating window.
2. P11-5 (energy acceptability and passivity)
   Over any ts window, constitutive response and boundaries satisfy non-negative dissipation: ( ∫ T_fil * s dt ) >= - E_store_max, consistent with Chapter 5 boundary-power accounting.
3. Linearized minimal equations (S12-6, 1D axial approximation)
   Around a uniform base state X0 = { A0, v0, e0, s0, T0 }, let the perturbation vector be xi = [ delta e, delta v, delta T ]^T. Then
   ( d/dt ) xi = ( A0 + A1 * ( d/dell ) + A2 * ( d^2/dell^2 ) ) xi,
   where coefficient matrices A0, A1, A2 follow from S12-1, S12-2 and Chapter 4 constitutive laws; A2 >= 0 reflects viscous/diffusive regularization (arising from K_vis or surface-energy equivalents).
4. Dispersion relation and long-wave criterion (S12-7)
   • For sinusoidal modes xi ~ exp( i k ell + g t ),
     g(k) = s0 * ( C0 - C2 * ( k * L_ref )^2 ) + O( k^4 ),
     with C0 = ( 1 - ( M_tan / T0 ) ) |_{X0}, and C2 >= 0 set by A2.
   • Long-wave instability (approximate iff): C0 > 0, i.e., M_tan < T0.
5. Generalized Considère criterion (S12-8, necking trigger)
   • At fixed s, Theta_K, if there exists e* such that M_tan( e*, s, Theta_K ) = T_fil( e*, s, Theta_K ), then for e > e* the system enters a necking-susceptible region.
   • Stability margin: Xi_neck = 1 - ( M_tan / T_fil ); Xi_neck > 0 indicates a trend toward instability.
6. Limit draw ratio (S12-9, steady upper bound)
   With DR = lambda_out / lambda_in = exp( e_out - e_in ), and requiring Xi_neck(ell) <= 0 along gamma(ell),
   DR_max = exp( e_limit ), where e_limit = sup { e | M_tan( e, s, Theta_K ) >= T_fil( e, s, Theta_K ) }.

IV. Data Gauges and Manifest

1. Stability indicators (reported versus ts and by window)
   • stab.Xi_neck(ell), stab.g_peak = max_k g(k), stab.k_peak, stab.G_obs = exp( g_peak * T_obs ).
   • group.We, group.De, group.Re, Theta_K.
   • A_reg = C2 (equivalent regularization coefficient), M_tan, T0, s0.
2. Window and measure
   The window length T_obs and reference length L_ref must be explicit; spectral analysis uses S_xx(f) per Chapter 7.
3. Quality gates & compliance
   • gate.stability.neck_max : Xi_neck <= 0
   • gate.stability.gain_max : G_obs <= G_max
   • gate.group.bounds : We, De, Re within declared ranges.

V. Algorithms and Implementation Bindings

1. I10-3 compute_instability_metrics(state) -> dict (reference implementation essentials)
   • Sample the base state: estimate A0, v0, e0, s0, T0 and temperature Theta_K; obtain M_tan from Chapter 4 interfaces.
   • Long-wave test: compute Xi_neck = 1 - ( M_tan / max( T0, eps ) ).
   • Dispersion approximation: form g(k) = s0 * ( C0 - C2 * ( k * L_ref )^2 ), with C0 = Xi_neck, and C2 from K_vis or empirical mapping; find g_peak and k_peak.
   • Dimensionless groups: We = s0 * tau_relax, De = tau_relax / T_obs, Re = ( rho * v0 * L_ref ) / mu_eff.
   • Alerts:
     1. If Xi_neck > 0, flag A_NECKING_RISK.
     2. If g_peak > 0 and G_obs > G_max, flag A_WAVE_AMP.
     3. If Re > Re_max, flag A_INERTIAL_OUT.
   • Output dictionary includes Xi_neck, g_peak, k_peak, We, De, Re, M_tan, T0, s0, C2 and suggested actions.
2. Controller-side guidance (coordinated with I10-1 update_draw_state)
   • If Xi_neck > 0: reduce s or increase the effective K_vis (cooling or raising viscosity), or switch boundary to impedance mode to damp long waves.
   • If g_peak > 0: introduce spectral limiting and S-curve stepping to avoid excitation near k_peak.

VI. Metrology Workflow and Run Graph

• Time-base alignment: ts = alpha + beta * tau_mono (Chapter 8), set T_obs and L_ref.
• Estimate constitutive parameters and tau_relax (continuing from Mx-12), and build lookup tables for M_tan( e, s, Theta_K ).
• For a given operating card (Chapter 5, Mx-10), scan Xi_neck(ell) and g_peak(ell) along gamma(ell).
• Frequency-domain cross-check: use S_xx(f) to extract energy peaks and verify proximity to the band mapped from k_peak.
• Produce the report and threshold comparison against gate.stability.*, and write the manifest for ingestion.

VII. Verification and Test Matrix

1. Minimum required cases
   • Elastic-dominated, low We: must have Xi_neck <= 0, g_peak <= 0.
   • Viscous-dominated, step s: verify C2 > 0 short-wave suppression and downward-opening g(k) versus k.
   • Maxwell-type viscoelasticity: sweep We, record the threshold We* where Xi_neck first crosses from negative to positive.
   • Winding coupling: vary Z_b and omega_w; verify g_peak shifts downward as boundary impedance increases.
   • Thermal drift: vary Theta_K; confirm the effect of tau_relax(Theta_K) on We and the shift of instability thresholds.
2. Boundary and extreme scenarios
   Strong geometric inhomogeneity with rapid thinning of A0; spikes in Re under high acceleration; sensor noise folding and amplification near k_peak.

VIII. Cross-References and Dependencies

• Chapter 2: definitions and measure mapping for e, s, A, v.
• Chapter 3: J = rho_L * v and energy ledgers for self-consistency within the stability window.
• Chapter 4: T_fil constitutive law and computation of M_tan, tau_relax.
• Chapter 5: modulation of dispersion by impedance and winding boundaries.
• Chapter 7: S_xx(f) and the k ↔ f cross-validation.
• Core.Metrology: windowing and uncertainty; Core.Errors: risk scoring and threshold sets; Core.Threads: alert reporting via TS.*.

IX. Risks, Limits, and Open Questions

• The 1D approximation breaks down under strong lateral waves or evolving elliptical sections; extend to transverse-mode coupling in A(ell,t).
• The effective C2 requires empirical or higher-order modeling (surface tension, thermal diffusion, molecular stretch) and must be calibrated for material families.
• Inertial terms at high Re are not explicit in S12-6; extreme high-speed drawing may require second-order momentum and unsteady couplings.
• Non-stationarity in the constitutive law (aging, temperature-sensitive drift) changes the time scales of M_tan and tau_relax; perform online re-estimation.

X. Deliverables and Version Management

1. Artifacts
   • Mx-15 stability-scan scripts and report template (including Xi_neck, g_peak, k_peak, We, De, Re).
   • Baseline maps from constitutive parameters to stability: M_tan( e, s, Theta_K ) and tau_relax(Theta_K).
   • Threshold and alert policy pack: gate.stability.* and linkage rules to TS.bc.*.
2. Version strategy
   Updates to dispersion models or C2 estimation are marked MOD; new instability modes are marked ADD. Compatibility flag compat.stab.v1 is maintained with migration notes in Appendix C.