Materials Fracture Mechanics From Jcost Surface Energy Factor Eq Jph

The surface energy factor

Fracture mechanics studies why materials break. The Griffith criterion, stated by Alan Arnold Griffith in 1921, says a crack grows when the energy released by its advance equals or exceeds the energy needed to create the two new surfaces. That critical release rate, written G_c, equals 2γ, where γ is the surface energy per unit area. In ordinary engineering, γ is measured for each material in a laboratory; there is no general formula for it.

Recognition Science offers such a formula. Its cost function, a measure of the forced price of recognition events, takes the value J(φ) ≈ 0.118 at the golden ratio φ. The framework's library, a machine-checked collection of formal theorems, proves that this number equals φ − 3/2, a purely algebraic expression. That identity, named surfaceEnergyFactor_eq_Jph, is the entire content of the declaration: it states that two ways of writing the same number, one through the cost function and one through the golden ratio, are equal. The proof is a direct consequence of a previously established value of the cost function at φ.

In Recognition Science, this factor enters a model for fracture. The model defines the critical strain energy release rate as G_c = 2 × J(φ) × E × a₀, where E is Young's modulus and a₀ is the interatomic spacing. For a typical metal with E ≈ 200 GPa and a₀ ≈ 3 Å, this gives about 14 J/m². Measured fracture toughness for metals commonly falls between 10 and 100 J/m², so the model lands inside the observed band. The framework also predicts a Paris law exponent of m = 4, the rate at which cracks grow under cyclic loading, from the four-point symmetry of the stress field.

What the declaration does not do is prove that real metals obey this formula. The theorem surfaceEnergyFactor_eq_Jph is a statement about numbers, not about steel or aluminum. The connection to actual fracture toughness is a prediction with a named falsifier: any precision measurement on a class of materials showing G_c systematically outside the J(φ) × E × a₀ band by more than 50 percent would refute it. The model chooses a definition and offers a test; it does not establish a physical law.

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

/-- J-cost on the strain energy / surface energy ratio. -/
def fractureCost (strain_energy surface_energy : ℝ) : ℝ :=
  Jcost (strain_energy / surface_energy)

theorem parisLawExponent_eq : parisLawExponent = 4 := rfl

What this page does not claim

The theorem does not prove that any real material's fracture toughness equals the predicted value. The model does not derive Young's modulus or interatomic spacing from first principles. The declaration does not establish the Griffith criterion itself.

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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