FractureRegime
plain-language theorem explainer
FractureRegime enumerates the five canonical fracture regimes under the J-cost model: elastic, plastic, creep, fatigue, and environmentally assisted. Materials physicists classifying crack growth via the Paris-Erdogan law on the phi-ladder would cite this enumeration when assigning regime-specific exponents. The declaration is an inductive type with five constructors that derives DecidableEq and Fintype instances automatically.
Claim. The fracture regimes form the finite set consisting of the elastic, plastic, creep, fatigue, and environmentally-assisted modes.
background
In the Recognition Science treatment of materials, fracture toughness relates to J-cost on the stress intensity ratio r = K_applied / K_IC. Thresholds are set at r < 1/φ (no propagation), entry into the J(φ) band (sub-critical growth), and r > 1 (fast fracture). The Paris-Erdogan exponent m follows the phi-ladder, with m ≈ 4 for structural metals and m ≈ 2 for ceramics, adjacent classes differing by factor φ ≈ 1.618.
proof idea
The declaration is an inductive definition that introduces five nullary constructors. The deriving clause equips the type with DecidableEq, Repr, BEq, and Fintype instances, enabling automatic decision procedures and cardinality computation.
why it matters
This definition supplies the five regimes required by FractureToughnessCert, which asserts cardinality five and pairs the enumeration with CanonicalCert. It realizes the configDim D = 5 stated in the module for the five canonical material fracture regimes. The structure supports downstream results connecting regime choice to J-cost thresholds and phi-ladder exponents.
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