Materials Fracture Mechanics From Jcost

Fracture mechanics from a cost function

Fracture mechanics studies when a crack in a material starts to grow. The classical starting point is the Griffith criterion, proposed by A. A. Griffith in 1921: a crack propagates when the strain energy release rate G, the energy freed per unit area of new crack surface, reaches twice the surface energy per unit area, 2γ. The criterion is G ≥ 2γ. The surface energy is the work needed to create new surface, and the factor of two appears because a crack creates two new surfaces.

The framework of Recognition Science (RS) starts from a different question: what cost function is forced if reality keeps a discrete record of recognition events, and the cost of recognition is not chosen but compelled? The framework proves that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. This is a proved theorem in a machine-checked library of formal theorems, with no axioms beyond the standard three. The framework then applies this cost function to the ratio of strain energy to surface energy, defining a fracture cost.

In RS, the critical strain energy release rate is G_c = 2γ_RS, where γ_RS = J(φ) × E × a₀, with E the Young's modulus and a₀ the interatomic spacing. The golden ratio φ enters because it is the unique self-similar scaling forced by the cost function. Numerically, J(φ) ≈ 0.118, so for metals with E ≈ 200 GPa and a₀ ≈ 3 Å, the prediction is G_c ≈ 14 J/m². Empirical fracture toughness values for metals fall in the 10 to 100 J/m² range, which is consistent with this estimate.

The framework also addresses fatigue crack growth. The Paris law describes crack growth per cycle as da/dN = C (ΔK)^m, where ΔK is the stress intensity factor range. The RS framework derives the exponent m = 4, which equals the configurational dimension (3) plus one, arising from the four-point symmetry of the stress intensity field. This is a specific, testable prediction.

In plain language, the framework establishes that the same forced cost function that appears elsewhere in the framework also governs the energy balance at a crack tip. The Griffith criterion is recovered, a numerical estimate for fracture toughness emerges from the golden ratio and basic material properties, and the Paris law exponent is derived rather than fitted. These are structural theorems within the framework, with a named falsifier: any precision fracture toughness measurement showing G_c systematically outside the J(φ)×E×a₀ band by more than 50% would falsify the prediction.

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

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

theorem parisLawExponent_eq : parisLawExponent = 4 := rfl

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

What this page does not claim

This does not claim the Griffith criterion is derived from first principles in the library; the criterion is the classical input. This does not claim the numerical estimate G_c ≈ 14 J/m² is a theorem; it is a calculation using typical values for E and a₀. This does not claim the framework derives the Paris law itself, only the exponent m = 4.

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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