10-EFT.WP.Core.Tension v1.0 | Chapter 2 — Line-Element Equilibrium and Constitutive Laws

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Chapter 2 — Line-Element Equilibrium and Constitutive Laws

I. Scope and Baseline Assumptions

• Object: slender line elements (rod/filament) with dominant axial response; bending and shear are not primary and, where needed, are approximated via equivalent axial loading.
• Geometry: path gamma(ell), arc-length ell ∈ [0, L_gamma], unit tangent t_hat(ell), differential d ell.
• Materials: cross-section A(ell), modulus E(ell), material density rho_m(ell), line density rho_l(ell) = rho_m(ell) * A(ell).
• Loads: distributed load’s tangential component q_tan(ell) (N/m); end force T_end (N); body force density b(x) (N/m^3) projects to the line via q_tan = ( ∫_A b • t_hat dA ).
• Equilibrium recalled from Chapter 1: S72-1 : ∂_ell T_fil(ell,t) + q_tan(ell,t) = 0.

II. Axial Small-Strain Constitutive Law and Strain

• Axial displacement u(ell) with small-strain definition epsilon(ell) = ∂_ell u(ell).
• One-dimensional Hooke relation:
  S72-2 : T_fil(ell) = E(ell) * A(ell) * epsilon(ell)
• Thermal and pre-tension corrections (optional):
  Linear thermal strain epsilon_T = alpha(ell) * ΔT(ell), equivalent T_fil = E A ( epsilon - epsilon_T ) + T_pre.
• Kelvin–Voigt viscoelastic extension (optional):
  T_fil = A * ( E * epsilon + eta * ∂_t epsilon ), with viscosity eta (Pa·s).

III. Static Solution Workflow (Mx-71 — Solve & Verify)

• Declare gamma(ell), d ell, properties E(ell), A(ell), and loading q_tan(ell).
• From S72-1, obtain the internal force field:
  T_fil(ell) = C0 - ( ∫_0^ell q_tan(s) ds ), with constant C0 fixed by boundary conditions or T_end.
• From S72-2, compute strain:
  epsilon(ell) = T_fil(ell) / ( E(ell) * A(ell) ).
• Displacement field:
  u(ell) = C1 + ( ∫_0^ell [ T_fil(s) / ( E(s) * A(s) ) ] ds ), with C1 set by displacement-type boundary conditions.
• Unit audit: [T_fil]=N, [epsilon]=1, [u]=m; enforce check_dim(expr).
• Conservation and equivalence check: end work versus strain energy ( ∫ T_fil * ∂_ell u d ell ).

IV. Boundary Conditions and Closure

• Dirichlet: u(0)=u0 or u(L_gamma)=uL.
• Neumann: T_fil(0)=T0 or T_fil(L_gamma)=TL.
• Robin: alpha * u + beta * T_fil = g (along t_hat), with compatible units across alpha, beta, g.
• End-balance and geometric supports must conform to P71-3 (boundary explicitness).

V. Variable Sections and Piecewise Materials

1. The formulas remain unchanged; only the integrands vary in space:
   • epsilon(ell) = T_fil(ell) / ( E(ell) * A(ell) )
   • u(ell) = C1 + ( ∫_0^ell [ T_fil(s) / ( E(s) * A(s) ) ] ds )
2. Piecewise constant: partition [0,L_gamma] into {[ell_k, ell_{k+1}]}; integrate on each segment with E_k, A_k, enforcing continuity of displacement and force at interfaces.

VI. Catenary and Self-Weight Effects (Axial-Dominant Approximation)

• Line weight w(ell) = rho_l(ell) * g (N/m), vertical axis y, and a near-constant horizontal tension component H.
• Classical solution (uniform w, geometric nonlinearity beyond small deflection):
  y(x) = y0 + a * cosh( ( x - x0 ) / a ), with a = H / w.
• Total tension magnitude:
  |T_fil(x)| = sqrt( H^2 + ( w * s(x) )^2 ), where s(x) is horizontal arc measure; in practice, design on H and check epsilon_max = |T_max| / ( E A ).

VII. Equivalent Loads for Body Force, Temperature, and Prestress

• Body force to line load: q_tan = ( ∫_A b • t_hat dA ); with gravity b = [0,0,-rho_m g], q_tan depends on the angle between t_hat and gravity.
• Temperature field: epsilon_T = alpha * ΔT; equivalent internal force correction ΔT_fil = - E A epsilon_T.
• Prestress/pre-tension: T_pre acts as a boundary constraint for C0 or as an initial internal force field.

VIII. Error and Uncertainty Propagation (Aligned with Core.Density Ch. 10)

• Parameter uncertainties in E, A, rho_l, q_tan propagate via Delta method to T_fil and u:
  cov( u(ell) ) ≈ J_u * cov(theta) * J_u^T, with theta = [E, A, …], J_u = ∂u/∂theta.
• Fisher information and CRLB for tension identification follow S72-15/16 (detailed in Chapter 9).

IX. Coupling to Arrival-Time Calibration (Cross-Volume Consistency)

• If wave speed is used to back-out equivalent constant tension T0 (see Chapter 4), use
  c = L_gamma / T_arr and T0 = rho_l * c^2.
• Always record the two-form disparity:
  delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |.
• Calibration passes only if the static solution and T0 inferred from T_arr agree within tolerance.

X. Engineering Bindings (I70-2 — Summary)

• tension_static(path, loads, bc, law="hooke") -> TensionFieldRef: solver implementing S72-1 and S72-2; supports piecewise E, A and thermal corrections.
• stress_to_tension(sigma, A, orientation) -> float: dimensional reduction from body/surface stress to line tension; orientation must define t_hat and n_hat.
• Metadata outputs: differentials and units; check_dim(expr) passed; boundary conditions and constants C0, C1 recorded with an uncertainty budget.

XI. Publication Checklist (Mx-71-CHK)

• gamma(ell), d ell, E(ell), A(ell), rho_l(ell), q_tan(ell), and boundary conditions are explicit.
• Provide closed-form or numerical expressions for T_fil(ell), epsilon(ell), u(ell), with constants identified.
• Verify energy consistency ( ∫ T_fil * ∂_ell u d ell ) versus end work.
• For temperature/body-force/pre-tension, provide equivalent formulations and unit audit.
• If arrival time is used in calibration, record delta_form per Section IX and compare against the static result.

Terms & Numbering Recap (anchors for reuse)

• Postulates: P71-1 (measure explicit), P71-2 (unit/dim conservation), P71-3 (boundary explicit).
• Minimal equations: S72-1 (tangential balance), S72-2 (Hooke line tension).
• Related identities: rho_l = rho_m * A, q_tan = ( ∫_A b • t_hat dA ), u(ell) = ( ∫ T_fil / (E A) d ell ), c = L_gamma / T_arr, T0 = rho_l * c^2, delta_form.