Materials Fracture Mechanics From Jcost Fracture Cost At Threshold

The threshold theorem

Fracture mechanics studies when a crack in a solid starts to propagate. The classical Griffith criterion states that a crack grows when the strain energy release rate, the energy available per unit area of new crack surface, reaches twice the surface energy per unit area. The Recognition Science framework, which derives physical laws from a forced cost function, encodes this threshold in a formal definition. Its cost, a measure of the price of a recognition event, is applied to the ratio of strain energy to surface energy.

The declaration fractureCost_at_threshold is a proved theorem in the framework's machine-checked library of formal theorems. It states that when the strain energy and the surface energy are equal, the fracture cost is exactly zero. In the language of the framework, this is the point of indifference, the precise balance where the crack neither advances nor retreats. The theorem also proves the cost is never negative for positive energies, meaning the cost function is well-behaved at all physical energy ratios.

The framework goes further with a specific numerical prediction. It defines a surface energy factor, equal to the golden ratio minus one and a half, which evaluates to about 0.118. The predicted critical strain energy release rate is twice this factor times the Young's modulus times the interatomic spacing. For typical metals, this gives a value near 14 joules per square meter, which sits inside the 10 to 100 joules per square meter range measured for real materials. The framework also predicts the Paris law exponent, which describes how fast a crack grows per loading cycle, is exactly 4, a value derived from the three spatial dimensions.

What the theorem does not claim is that these numerical agreements are proven. The threshold theorem is a statement about the cost function reaching zero at a balanced ratio, and that is fully machine-checked. The prediction that this balance corresponds to real material fracture, with the specific value of 14 joules per square meter, is a hypothesis with a named falsifier: any precision measurement on a class of materials showing the critical release rate systematically outside the predicted band by more than 50 percent would refute it. The framework's library proves the mathematics of the cost function, not the physics of every material.

theorem fractureCost_at_threshold (e : ℝ) (h : e ≠ 0) :
    fractureCost e e = 0 := by
  unfold fractureCost; rw [div_self h]; exact Jcost_unit0

theorem fractureCost_at_threshold (e : ℝ) (h : e ≠ 0) :
    fractureCost e e = 0 := by
  unfold fractureCost; rw [div_self h]; exact Jcost_unit0

theorem fractureCost_nonneg (s sur : ℝ) (hs : 0 < s) (hsur : 0 < sur) :
    0 ≤ fractureCost s sur := by
  unfold fractureCost; exact Jcost_nonneg (div_pos hs hsur)

/-- RS surface energy factor: J(φ) ≈ 0.118. -/
def surfaceEnergyFactor : ℝ := phi - 3 / 2

/-- Paris law exponent: m = 4 = configDim + 1. -/
def parisLawExponent : ℕ := 4

What this page does not claim

The numerical agreement with metal fracture toughness is not a theorem, only a hypothesis with a named falsifier. The threshold theorem does not claim that real cracks always propagate at the exact predicted energy release rate. The framework does not claim to derive the Paris law exponent from first principles without the prior assumption of three spatial dimensions.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Materials/FractureMechanicsFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: