35-EFT.WP.EDX.OrientedTension v1.0 | Chapter 4 Axioms & Minimal Equations of Oriented Tension (P/S)

Chapter 4 Axioms & Minimal Equations of Oriented Tension (P/S)

I. Abstract & Scope
This chapter establishes the axiom system and the minimal set of equations for oriented tension. With the orientation energy density W_orient as the core, and under symmetry, zero-baseline & positive-definiteness, and frame-objectivity, we derive the minimal constitutive form mapping W_orient to the oriented tension tensor T_fil_ij, together with the minimal dynamical closure for the order tensor. All symbols use English notation in backticks; SI units apply. No ToA terms appear here.

II. Dependencies & References

• Orientation geometry & distributions: Chapter 3 S80-1/2 (f_orient normalization; Q_ij definition).
• Symbols & dimensions: Chapter 2 (P80-1/4/5/6).
• Metrology & inversion: Chapter 5 (M80-1…4).
• Coupling & transport: Chapter 6 (S80-5/6).
• Energy accounting & power partition: Chapter 7 (S80-7/8).

III. Normative Anchors (added in this chapter, P80-/S80-)

• P80-2 (Orientation-Symmetry Axiom): apolar materials remain invariant under n_hat → -n_hat; polarity must be modeled explicitly in polar materials.
• P80-3 (EDX-Accounting Axiom): orientation energy and power terms close within a control volume; W_orient participates in the conservation and dissipation split of Chapter 7.
• P80-9 (Zero-Baseline & Positivity Axiom): in the isotropic base state Q_ij=0, W_orient=0, and for admissible perturbations W_orient ≥ 0.
• P80-10 (Frame-Objectivity Axiom): W_orient and the derived T_fil_ij are objective (invariant) under rigid-body rotations.
• S80-3 (Energy-Based Constitutive Form of Oriented Tension):
  T_fil_ij = ( ∂W_orient / ∂(∂_i u_j) ) + T^{(Q)}_{ij},
  where T^{(Q)}_{ij} collects the non-elastic/orientation-mapped part via Q_ij (detailed in the text).
• S80-4 (Minimal Dynamics of the Order Tensor):
  ∂_t Q_ij + u_k ∂_k Q_ij − Ω_{ik} Q_{kj} − Q_{ik} Ω_{kj} = − ( Q_ij − Q_ij^eq ) / tau_relax + D_Q ∇^2 Q_ij + S_ij,
  with Ω_{ij} = ( ∂_i u_j − ∂_j u_i ) / 2. S_ij is an orientation source (e.g., field-induced alignment), and Q_ij^eq is the local equilibrium order.

IV. Body Structure

I. Background & Problem Statement

• From an energetic perspective, oriented tension is obtained from variations of W_orient(Q_ij, ∇Q_ij, …), and coexists with elastic contributions depending on the deformation gradient.
• The minimal equations must satisfy: symmetry (P80-2), energy-closure (P80-3), zero-baseline and positivity (P80-9), and objectivity (P80-10); and provide identifiable parametrizations for Chapter 5 metrology and Chapters 6–7 coupling/energy equations.

II. Key Equations & Derivations (S-series)

1. S80-3 (W_orient → T_fil_ij)
   • Elastic-like contribution (if W_orient contains ∂_i u_j):
     T^{(el)}_{ij} = ∂W_orient / ∂(∂_i u_j) (unit Pa).
   • Orientation-mapped contribution (minimal linear closure):
     T^{(Q)}_{ij} = Λ_{ijkl} Q_{kl},
     where Λ_{ijkl} satisfies material symmetries and objectivity (unit Pa). Nonlinear closures may be written as T^{(Q)}_{ij} = ∑_m c_m 𝓘^{(m)}_{ij}(Q) with invariant-based tensor bases 𝓘^{(m)}.
   • Total oriented tension: T_fil_ij = T^{(el)}_{ij} + T^{(Q)}_{ij}.
2. S80-4 (Dynamics of Q_ij)
   • Advection & rotation: ∂_t Q_ij + u_k ∂_k Q_ij − Ω_{ik} Q_{kj} − Q_{ik} Ω_{kj}.
   • Relaxation & diffusion: − ( Q_ij − Q_ij^eq ) / tau_relax + D_Q ∇^2 Q_ij.
   • Sources: S_ij built from invariants or external fields (E_vec, B_vec, u_vec), preserving symmetry and tracelessness.
   • Energetic compatibility: the quadratic approximation W_orient ≈ (1/2) A Q_ij Q_ij + (1/2) K ∂_k Q_ij ∂_k Q_ij (A≥0, K≥0) is consistent with the positivity of tau_relax and D_Q.

III. Methods & Flows (M-series)

• M80-16 (Axiom-Consistency Audit): input candidate W_orient and Λ_{ijkl}, automatically check P80-2/3/9/10 and dimensions; output a consistency report and a minimal counterexample set.
• M80-17 (Zero-Field Linearization & Identifiability): linearize S80-3/4 near Q_ij≈0; derive identifiability conditions and experiment-design matrices for Λ_{ijkl}, A, K, tau_relax, D_Q.
• M80-18 (Constitutive Parameter Estimation): with Chapter 5 metrology, minimize L = L_T + L_Q + L_reg to fit {Λ_{ijkl}, A, K, tau_relax, D_Q}; output {posterior, evidence} with uncertainties.

IV. Cross-References within/beyond this Volume

• Chapter 3: geometric–statistical inputs S80-1/2 (Q_ij, f_orient).
• Chapter 5: metrology/inversion pipelines for Q_ij and T_fil_ij (M80-1…4).
• Chapter 6: coupling terms and transport coefficients use T_fil_ij and Q_ij from this chapter (S80-5/6).
• Chapter 7: inclusion of W_orient and power terms in energy accounting (S80-7/8).

V. Validation, Criteria & Counterexamples

1. Positive criteria:
   • W_orient(Q_ij, ∇Q_ij) satisfies zero-baseline & positivity; objectivity passes.
   • T_fil_ij from S80-3 matches measured tension in units/dimensions and symmetry.
   • S80-4 is stable in linear regimes and identifiable on experimental timescales (posterior convergence and improved evidence).
2. Negative criteria:
   • W_orient yields negative energy or violates n_hat → -n_hat invariance.
   • Λ_{ijkl} breaks material symmetry or objectivity.
   • Removing orientation terms (e.g., Λ_{ijkl}→0 or A→0) does not degrade fit quality (mechanism falsified).
3. Contrasts:
   • Evidence comparisons among {elastic-only, orientation-only, elastic+orientation}.
   • {isotropic Λ, anisotropic Λ} discrimination via T_fil_ij patterns.
   • {with K, without K} impacts on interfacial/gradient-sensitive observables.

VI. Deliverables & Figure List

1. Deliverables:
   • ConstitutiveCard.json (W_orient structure, Λ_{ijkl}, A,K,tau_relax,D_Q with units/dimensions).
   • Identifiability.md (linearization and design matrices).
   • EvidenceReport.md (positive/negative criteria, evidence ratios, counterexample summary).
2. Figures/Tables (suggested):
   • Tab. 4-1 Minimal parametrizations of W_orient with units.
   • Tab. 4-2 Symmetries and independent counts of Λ_{ijkl}.
   • Fig. 4-1 Stability spectrum and parameter sensitivities of S80-4.
   • Tab. 4-3 Evidence & error comparisons for {elastic-only, orientation-only, combined}.