Materials physics — the microscopic structural and defect-physics foundations of macroscopic materials behaviour. Distinguished from materials science (broader, engineering-adjacent) by its focus on reductive physical mechanism rather…
materials-physics
Defect concentration: c = exp(-E_f/k_B·T) (Arrhenius equilibrium)
The equilibrium vacancy concentration in a crystal at temperature T follows an Arrhenius law c = exp(-E_f/k_B·T), where E_f is the vacancy…
Hall-Petch strengthening: σ_y = σ_0 + k_HP·d^(-1/2)
Hall (1951) and Petch (1953) independently established the inverse-square-root grain-size dependence of yield strength in polycrystalline…
Nabarro-Herring creep: ε̇ ∝ σ·D·Ω/(k_B·T·d²)
Nabarro (1948) and Herring (1950) derived the bulk-diffusion creep law: under applied stress σ at high homologous temperature (T/T_melt ≳…
Defect Arrhenius: c(E_f=0)=1; c(E_f=k_B·T)=exp(-1)
Sympy-exact witness of the Arrhenius defect-concentration limiting cases. Setup: c = exp(-E_f/(T·k_B)). Identity 1 (zero formation…
Hall-Petch limit: σ_y(d=1)=σ_0+k_HP; σ_y(d→∞)=σ_0
Sympy-exact witness of the Hall-Petch limiting behaviour. Setup: σ_y = σ_0 + k_HP/√d. Identity 1 (d=1 reference-grain-size anchor):…
Nabarro-Herring grain ratio: ε̇(2d)/ε̇(d) = 1/4
Sympy-exact witness of the grain-size-quadratic scaling of Nabarro-Herring creep. Setup: ε̇_NH ∝ σ·D·Ω/(T·k_B·d²) — inverse-square in…