10-EFT.WP.Core.Tension v1.0 | Chapter 1 — Tension and Geometric Foundations

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Chapter 1 — Tension and Geometric Foundations

I. Concepts and Domains

• Objects and carriers: line elements (filament/rod), membranes/interfaces (surface), and solids (body). The associated field variables are T_fil(ell,t) (N), sigma_s(x,t) (N/m), and sigma_ij(x,t) (Pa).
• Path and parameterization: gamma(ell) denotes an arc-length parameterized curve with ell ∈ [0, L_gamma], where L_gamma = ( ∫ 1 d ell ).
• Tangent and normal: t_hat(ell) = d gamma / d ell / | d gamma / d ell |; n_hat is a unit normal defined by the chosen cross-section or boundary.
• Line density and materials: material density rho_m (kg/m^3), area A(ell) (m^2), and line density rho_l(ell) = rho_m(ell) * A(ell) (kg/m). Displacement field u(x,t).

II. Geometry and Measures (curve/surface/volume)

• Line measure: line integrals always state d ell, e.g., ( ∫_Gamma T_fil d ell ), and specify the path Gamma = { gamma(ell) }.
• Surface measure: across a membrane/interface use dA, e.g., ( ∫_S sigma_s dA ), where S is the parameterized surface domain.
• Volume measure: inside a body use dV, e.g., ( ∫_V sigma_ij dV ); declare the base measure mu when needed.
• Change of variables: if a general curve parameter s is used, supply the Jacobian | d ell / ds |, e.g., ( ∫ T_fil( ell(s) ) * | d ell / ds | ds ).
• Postulate P71-1 (measure explicit): every integral ∫ must name its domain and differential (d ell/dA/dV); any path-dependent quantity must explicitly declare gamma(ell).

III. Units and Dimensional Consistency

1. Baseline dimensions: [T_fil]=N, [sigma_s]=N/m, [sigma_ij]=Pa, [rho_l]=kg/m, [c]=m/s, [Z]=kg/s.
2. Conservation rule: every derivation and implementation must pass check_dim(expr) prior to publication; never mix dimensionless ratios (e.g., T_trans) with dimensional tension T_fil.
3. Typical checks:
   • ( ∫ rho_l d ell ) has units of kg;
   • ( ∫ T_fil * ∂_ell u d ell ) has units of J.
4. Postulate P71-2 (unit/dim conservation): each term in a derivation must retain consistent dimensions; any cross-domain mapping must document how unit(x) and dim(x) transform.

IV. Carriers (line/surface/volume) and Internal Force Mapping

• Axial resultant (from body stress to line tension):
  With section normal aligned to the axis n_hat = t_hat, the sectional resultant is N = ( ∫_A sigma • n_hat dA ). Define the line tension as T_fil = N • t_hat.
• Membrane tension and edge ring:
  Edge resultant: F_edge = ( ∮_∂S sigma_s * t_edge ds ), with t_edge the boundary tangent.
• Dimensional reduction (body→surface→line) must state cross-section/contact geometry, normal orientation, and differentials to avoid mixing directions and measures.

V. Boundaries, Tractions, and Condition Types

• Dirichlet (displacement): prescribe u = u_bar (m) at endpoints or on boundaries.
• Neumann (traction/force): prescribe end force T_end (N) at line ends; prescribe line force density tau_edge (N/m) on membrane edges.
• Robin (mixed): alpha * u + beta * t = g, with units chosen so the equation is homogeneous.
• Postulate P71-3 (boundary explicitness): boundary type, orientation, units, and geometric support must be stated together; traction-type boundaries must include measure and domain for integration.

VI. Tangential Equilibrium in 1D (Minimal Equation)

• S72-1 (tangential balance): ∂_ell T_fil(ell,t) + q_tan(ell,t) = 0, where q_tan is the distributed load along t_hat (N/m).
• Projected body force: with body force density b(x,t) (N/m^3), its linewise projection satisfies q_tan = ( ∫_A b • t_hat dA ).
• Note: the form applies to axial force balance under small deformation; for geometric nonlinearity, use incremental or total forms in Chapter 2.

VII. Baseline Quantities from Geometry to Dynamics

1. Line density: rho_l(ell) = rho_m(ell) * A(ell).
2. Uniform string wave speed: c = sqrt( T0 / rho_l ) (T0 constant tension).
3. Mechanical impedance (traveling wave): Z = rho_l * c.
4. Arrival-time anchors (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: delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |

VIII. Canonical, Publication-Ready Snippets

• Line mass on an interval [a,b]: m_[a,b] = ( ∫_a^b rho_l(ell) d ell )
• Axial strain energy: U_axial = ( ∫_0^{L_gamma} ( T_fil(ell) * ( ∂_ell u(ell) ) / 2 ) d ell )
• Line work input: W_input = ( ∫_0^{L_gamma} T_fil(ell) * ∂_ell u(ell) d ell )
• Stress-to-tension (restated): T_fil(ell,t) = ( ( ∫_{A(ell)} sigma(x,t) • n_hat dA ) • t_hat )

IX. Cross-References and Consistency Anchors

• Mass/density symbols align with EFT.WP.Core.Density v1.0: rho_m, rho_l, n_eff, T_arr (see that volume’s Appendix A and Chapter 9).
• Spectral and window metrics align with Core.Sea: S_xx(f), U_w, ENBW_Hz.
• Thread-time baseline and arrival-time sampling semantics align with Core.Threads Chapter 3.
• Fixed cross-volume citation: See companion white paper Energy Threads Chapter x S/P/M/I…; key anchors here are P71-* and S72-1.

X. Publication Checklist (Mx-70 — Geometry–Measure–Unit Audit)

• gamma(ell), L_gamma, differentials (d ell/dA/dV), and integration domains are explicit.
• All formulas pass check_dim(expr); units of T_fil, sigma_s, sigma_ij are correct.
• End and boundary conditions are classified as Dirichlet/Neumann/Robin with physical meaning and units.
• If arrival time is used for calibration or identification, both forms and delta_form are reported.
• When reducing from body stress to line tension, section normal, dimensional reduction integrals, and directional projections are stated.

Terminology and Numbering Recap (for reuse by later chapters)

• P71-1: measure explicit; P71-2: unit/dim conservation; P71-3: boundary explicit.
• S72-1: ∂_ell T_fil + q_tan = 0 (tangential equilibrium).
• Core definitions: gamma(ell), L_gamma, t_hat, n_hat, rho_l = rho_m * A, T_fil, sigma_s, sigma_ij, c = sqrt( T0 / rho_l ), Z = rho_l * c, dual-form T_arr and delta_form.