Materials Fracture Mechanics From Jcost Surface Energy Factor Pos

The surface energy factor

Fracture mechanics asks when a crack in a solid will start to grow. The classical answer, due to A. A. Griffith in 1921, compares two forms of energy. A crack needs energy to create new surface, an amount written as 2γ per unit area, where γ is the surface energy. The crack releases stored strain energy as it advances, at a rate G called the strain energy release rate. Griffith's criterion states that the crack propagates when G reaches 2γ.

The Recognition Science framework models this threshold through its cost function J, which assigns a cost to any ratio of two energies. The framework's prediction is that the critical release rate is G_c = 2 × J(φ) × E × a₀, where E is Young's modulus and a₀ is the interatomic spacing. The number J(φ) is about 0.118. For a typical metal with E ≈ 200 GPa and a₀ ≈ 3 Å, this gives G_c ≈ 14 J/m², which falls inside the measured range of 10 to 100 J/m² for metals.

In the machine-checked library of formal theorems, the declaration surfaceEnergyFactor_pos proves a narrow but necessary fact: the surface energy factor, defined as φ − 3/2, is greater than zero. The proof uses the theorem that φ is greater than 1.5. This positivity is what makes the predicted G_c a positive number, a prerequisite for any physical interpretation. The same library also proves that the factor equals J(φ), the cost function evaluated at the golden ratio.

What the declaration does not claim is more interesting than its modest statement. It does not prove that the predicted G_c matches experiment. That consistency with measured fracture toughness is an empirical check, not a theorem. The declaration also does not derive the Paris law exponent m = 4, which appears elsewhere in the same file as a definition, not a proved consequence. And it says nothing about whether the Griffith criterion itself is the correct physical law; the framework takes that as a starting model.

The practical consequence is that the framework has a certified, positive number at the core of its fracture prediction. A reader can now see that the prediction is not vacuous: the framework's own formalism guarantees the surface energy factor is a real, positive quantity, ready to be compared against laboratory data.

theorem surfaceEnergyFactor_pos : 0 < surfaceEnergyFactor := by
  unfold surfaceEnergyFactor; linarith [phi_gt_onePointFive]

theorem surfaceEnergyFactor_eq_Jph : surfaceEnergyFactor = Jcost phi :=
  Jcost_phi_val.symm

theorem surfaceEnergyFactor_eq_Jph : surfaceEnergyFactor = Jcost phi :=
  Jcost_phi_val.symm

What this page does not claim

The declaration does not prove the predicted G_c matches experimental fracture toughness values. The declaration does not derive the Paris law exponent m = 4 from more basic principles. The declaration does not assert that the Griffith criterion is the correct physical law of crack propagation.

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)

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