10-EFT.WP.Core.Tension v1.0 | Foreword

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Foreword

I. Scope and Objectives

• This volume establishes a unified mechanics for tension, covering line-element tension T_fil(ell,t), interfacial tension sigma_s(x,t), in-body stress sigma_ij(x,t), and wave/transfer quantities such as wave speed c, mechanical impedance Z, reflection R_ref, and transmission T_trans.
• It targets a full pipeline from laboratory measurement to numerical simulation—definition → modeling → discretization → calibration → verification → publication—with reproducibility and cross-volume consistency as first principles.
• Symbols, units, conventions, and the two canonical expressions of arrival time T_arr are aligned with Core.Density, Core.Sea, Core.Threads, and Core.Metrology.

II. Audience and Usage Scenarios

1. Audience: structural/materials/acoustics and signal processing engineers, researchers, platform and foundational library developers.
2. Scenarios:
   • Static solution and verification: infer T_fil or sigma_ij from loads and constraints.
   • Dynamic identification and calibration: estimate T_fil, c, and Z from modes/arrival time T_arr.
   • Interfaces and wetting: infer sigma_s and pressure jumps from curvature and contact angles.
   • Networks and transmission: evaluate joint reflection R_ref and transmission T_trans, and verify power conservation.

III. Core Objects and Dimensions

1. Line element and path: gamma(ell) with arc-length measure d ell, and length L_gamma = ( ∫ 1 d ell ).
2. Tension and stress:
   • T_fil(ell,t) in N; sigma_s(x,t) in N/m; sigma_ij(x,t) in Pa.
   • Line density rho_l = rho_m * A, where rho_m (kg/m^3) and cross-section area A (m^2).
3. Waves and impedance:
   • Uniform string speed c = sqrt( T0 / rho_l ); mechanical impedance Z = rho_l * c.
   • Lossless interface power balance: R_ref + T_trans = 1.
4. Arrival time — two canonical forms (cross-volume unified):
   • Constant-pulled: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
   • Path-wise: T_arr = ( ∫ ( n_eff / c_ref ) d ell )
   • Disparity record: delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |

IV. Relationship to Companion White Papers

• Core.Density: shared conventions for measures/arrival time/uncertainty; this volume references P71-, S72-, Mx-7*, I70-* and aligns with the Density volume’s P91-, S92-.
• Core.Sea: spectral and power-flow metrics are consistent; definitions of S_xx(f), window power U_w, and equivalent noise bandwidth ENBW_Hz are harmonized to avoid double-counting energy.
• Core.Threads: timestamps, retry, and backpressure semantics are shared; the T_arr time base and channel synchronization rules are reused.
• Core.Metrology: units and traceability are consistent; uncertainty propagation and CRLB interfaces are reused.

V. Notation, Units, and Measure Conventions

• Inline symbols use backticks throughout, e.g., T_fil, sigma_s, sigma_ij, u(x,t), c, Z, R_ref, T_trans, n_eff, T_arr.
• Integrals must state measure and domain explicitly: ( ∫ T_fil d ell ), ( ∫ sigma_s dA ), ( ∫ sigma_ij dV ); path-dependent expressions explicitly declare gamma(ell).
• Conflict-name rule: T_fil (N) and T_trans (dimensionless) must never be conflated; n and n_eff are strictly distinguished.
• Unit checks: check_dim(expr) must pass; wave speed c in m/s, impedance Z in kg/s.

VI. Unified Numbering and Citation Discipline

• Postulates P71-*: explicit measures, unit conservation, explicit boundary conditions.
• Minimal equations S72-*: line-element equilibrium, Young–Laplace, string equation, impedance matching, CFL stability, etc.
• Flows Mx-7*: static verification, capillary pressure from curvature, T_arr-based tension back-out, network transmission evaluation, stability and energy audits.
• Bindings I70-*: geometry and sections, static/dynamic solvers, interfacial tension, transmission and calibration, uncertainty interfaces.
• Cross-volume citation uses the fixed format: “See companion white paper Energy Threads Chapter x S/P/M/I…”.

VII. Canonical Assumptions and Applicability

• Small-strain convention: epsilon_ij = (1/2) * ( ∂_i u_j + ∂_j u_i ); for large deformation, configurations and Piola stress maps will be provided.
• Line-element model validity: when A and material parameters vary slowly, T_fil dominates, and shear/bending are secondary.
• Lossless-interface assumption applies only to energy conservation checks; for losses, use Z -> Z + i * Z_loss and make power-flow balance explicit.

VIII. Data, Experiment, and Reproducibility

1. Minimal dataset:
   • Geometry: gamma(ell), L_gamma, coordinate frame and measure metadata.
   • Materials: E, rho_m, A, damping parameters.
   • Observables: u(x,t), event arrival times T_arr, spectra S_xx(f), boundary bc.
2. Metadata binding: bind_to_parameters(ds, params), bind_to_equations(eqn_refs) to ensure one-to-one alignment between derivations and implementations.
3. Reproducibility practice: publish the Mx-7* pipeline parameters, window and ENBW_Hz, and always record delta_form.

IX. Interfaces and Engineering Realization

1. Key interface family (excerpt): tension_static(...), string_wave_solve(...), transmission_coeff(...), calibrate_tension_by_toa(...), fisher_information_tension(...), crlb_tension(...).
2. Quality gates:
   • Dimensional/energy conservation checks pass.
   • Dual-form arrival-time disparity delta_form is recorded and within tolerance.
   • Interface contracts and unit/spectral benchmark cases pass regression.

X. Document Roadmap

• Chapter 1: Tension and geometry foundations (P71-*, measures and units)
• Chapter 2: Line-element equilibrium and constitutive laws (S72-1/2, Mx-71)
• Chapter 3: Membranes and interfacial tension (S72-3/4, Mx-72)
• Chapter 4: Dynamic tension and waves (S72-5/6, Mx-73, T_arr calibration)
• Chapter 5: Joints, branching, and networks (S72-7/8, Mx-74)
• Chapter 6: Transmission, impedance, and throughput (S72-9/10/11, Mx-75)
• Chapter 7: Discretization and numerical stability (S72-12, Mx-76)
• Chapter 8: Normalization and calibration (S72-13/14, Mx-77)
• Chapter 9: Identification, uncertainty, and information bounds (S72-15/16, Mx-78)
• Chapter 10: Engineering implementation and testing (I70-*, Mx-79)