oxiphysics_materials/

ceramics_materials.rs

// Copyright 2026 COOLJAPAN OU (Team KitaSan)
// SPDX-License-Identifier: Apache-2.0
//! Ceramic material models: brittle fracture, sintering, thermal properties.
//!
//! This module provides physics-based models for ceramic materials including
//! brittle fracture (Griffith criterion), Weibull statistics, thermal shock
//! resistance, sintering kinetics, grain growth, high-temperature creep,
//! dielectric/ferroelectric behaviour, and ZrO2 transformation toughening.

#![allow(dead_code)]
#![allow(clippy::too_many_arguments)]

use std::f64::consts::PI;

/// Universal gas constant \[J/(mol·K)\]
const R_GAS: f64 = 8.314_462_618;

// ─────────────────────────────────────────────────────────────────────────────
// CeramicProperties
// ─────────────────────────────────────────────────────────────────────────────

/// Intrinsic physical properties of a ceramic material.
///
/// Contains elastic constants, thermal properties, and hardness data
/// needed by fracture, sintering, and shock-resistance models.
#[derive(Debug, Clone)]
pub struct CeramicProperties {
    /// Young's modulus \[Pa\]
    pub young_modulus: f64,
    /// Poisson's ratio \[-\]
    pub poisson_ratio: f64,
    /// Thermal conductivity \[W/(m·K)\]
    pub thermal_conductivity: f64,
    /// Coefficient of thermal expansion \[1/K\]
    pub cte: f64,
    /// Vickers hardness \[GPa\]
    pub hardness: f64,
    /// Fracture toughness K_IC \[MPa·√m\]
    pub fracture_toughness: f64,
    /// Flexural (Weibull characteristic) strength σ_0 \[Pa\]
    pub flexural_strength: f64,
    /// Weibull modulus m \[-\]
    pub weibull_modulus: f64,
    /// Specific surface energy γ \[J/m²\]
    pub surface_energy: f64,
    /// Material name
    pub name: String,
}

impl CeramicProperties {
    /// Creates a new `CeramicProperties` with all fields specified.
    #[allow(clippy::too_many_arguments)]
    pub fn new(
        young_modulus: f64,
        poisson_ratio: f64,
        thermal_conductivity: f64,
        cte: f64,
        hardness: f64,
        fracture_toughness: f64,
        flexural_strength: f64,
        weibull_modulus: f64,
        surface_energy: f64,
        name: impl Into<String>,
    ) -> Self {
        Self {
            young_modulus,
            poisson_ratio,
            thermal_conductivity,
            cte,
            hardness,
            fracture_toughness,
            flexural_strength,
            weibull_modulus,
            surface_energy,
            name: name.into(),
        }
    }

    /// Returns preset properties for alumina (Al₂O₃).
    pub fn alumina() -> Self {
        Self::new(
            380e9, 0.22, 30.0, 8.1e-6, 18.0, 4.0, 400e6, 10.0, 1.0, "Al2O3",
        )
    }

    /// Returns preset properties for silicon carbide (SiC).
    pub fn silicon_carbide() -> Self {
        Self::new(
            420e9, 0.17, 120.0, 4.5e-6, 27.0, 4.5, 500e6, 12.0, 1.5, "SiC",
        )
    }

    /// Returns preset properties for silicon nitride (Si₃N₄).
    pub fn silicon_nitride() -> Self {
        Self::new(
            310e9, 0.27, 30.0, 3.2e-6, 16.0, 7.0, 700e6, 15.0, 2.0, "Si3N4",
        )
    }

    /// Returns preset properties for yttria-stabilised zirconia (ZrO₂).
    pub fn zirconia() -> Self {
        Self::new(
            200e9, 0.31, 2.5, 10.5e-6, 12.0, 10.0, 1000e6, 8.0, 1.2, "ZrO2",
        )
    }

    /// Returns preset properties for titanium dioxide (TiO₂).
    pub fn titania() -> Self {
        Self::new(
            230e9, 0.27, 11.8, 8.4e-6, 10.0, 2.0, 200e6, 7.0, 0.5, "TiO2",
        )
    }
}

// ─────────────────────────────────────────────────────────────────────────────
// BrittleFractureCeramic
// ─────────────────────────────────────────────────────────────────────────────

/// Griffith brittle-fracture model for ceramics.
///
/// Implements the Griffith criterion, K_IC-based failure, and Weibull
/// statistical strength scatter.
#[derive(Debug, Clone)]
pub struct BrittleFractureCeramic {
    /// Reference to ceramic properties
    pub props: CeramicProperties,
    /// Geometry factor Y for stress-intensity factor (typically ~1.12 for edge crack)
    pub geometry_factor: f64,
}

impl BrittleFractureCeramic {
    /// Creates a new `BrittleFractureCeramic`.
    pub fn new(props: CeramicProperties, geometry_factor: f64) -> Self {
        Self {
            props,
            geometry_factor,
        }
    }

    /// Griffith critical stress for a crack of half-length `a` \[m\].
    ///
    /// σ_c = √(2·E·γ / (π·a))
    ///
    /// Returns critical stress \[Pa\].
    pub fn griffith_critical_stress(&self, crack_half_length: f64) -> f64 {
        let e = self.props.young_modulus;
        let gamma = self.props.surface_energy;
        (2.0 * e * gamma / (PI * crack_half_length)).sqrt()
    }

    /// Stress intensity factor K_I for applied stress `sigma` and crack half-length `a` \[m\].
    ///
    /// K_I = Y · σ · √(π·a)
    ///
    /// Returns K_I \[Pa·√m\].
    pub fn stress_intensity(&self, sigma: f64, crack_half_length: f64) -> f64 {
        self.geometry_factor * sigma * (PI * crack_half_length).sqrt()
    }

    /// Returns `true` if the crack propagates (K_I ≥ K_IC).
    pub fn is_critical(&self, sigma: f64, crack_half_length: f64) -> bool {
        let ki = self.stress_intensity(sigma, crack_half_length);
        let kic = self.props.fracture_toughness * 1e6; // MPa√m → Pa√m
        ki >= kic
    }

    /// Critical crack length at a given applied stress \[Pa\].
    ///
    /// a_c = (1/π) · (K_IC / (Y·σ))²
    ///
    /// Returns critical half-length \[m\].
    pub fn critical_crack_length(&self, sigma: f64) -> f64 {
        let kic = self.props.fracture_toughness * 1e6;
        let y_sigma = self.geometry_factor * sigma;
        (kic / y_sigma).powi(2) / PI
    }

    // ── Weibull statistics ─────────────────────────────────────────────────

    /// Weibull survival probability P_s for applied stress `sigma`.
    ///
    /// P_s = exp(−(σ / σ_0)^m)
    ///
    /// where σ_0 is the Weibull characteristic strength and m the Weibull modulus.
    pub fn weibull_survival_probability(&self, sigma: f64) -> f64 {
        let sigma_0 = self.props.flexural_strength;
        let m = self.props.weibull_modulus;
        (-(sigma / sigma_0).powf(m)).exp()
    }

    /// Weibull failure probability P_f = 1 − P_s.
    pub fn weibull_failure_probability(&self, sigma: f64) -> f64 {
        1.0 - self.weibull_survival_probability(sigma)
    }

    /// Median failure stress (P_f = 0.5) from Weibull distribution \[Pa\].
    ///
    /// σ_med = σ_0 · (ln 2)^(1/m)
    pub fn weibull_median_strength(&self) -> f64 {
        let sigma_0 = self.props.flexural_strength;
        let m = self.props.weibull_modulus;
        sigma_0 * 2_f64.ln().powf(1.0 / m)
    }

    /// Mean Weibull strength \[Pa\]: σ_mean = σ_0 · Γ(1 + 1/m).
    ///
    /// Uses Stirling / rational approximation for the gamma function.
    pub fn weibull_mean_strength(&self) -> f64 {
        let sigma_0 = self.props.flexural_strength;
        let m = self.props.weibull_modulus;
        sigma_0 * gamma_approx(1.0 + 1.0 / m)
    }
}

/// Simple rational approximation of Γ(x) for x in \[1, 3\] (Lanczos g=5).
fn gamma_approx(x: f64) -> f64 {
    // Lanczos coefficients g=5, n=7
    let c = [
        1.000_000_000_190_015,
        76.180_091_729_471_46,
        -86.505_320_329_416_77,
        24.014_098_240_830_91,
        -1.231_739_572_450_155,
        1.208_650_973_866_179e-3,
        -5.395_239_384_953_0e-6,
    ];
    let x = x - 1.0;
    let mut ser = c[0];
    for (i, &ci) in c[1..].iter().enumerate() {
        ser += ci / (x + i as f64 + 1.0);
    }
    let t = x + 5.5; // g + 0.5
    std::f64::consts::TAU.sqrt() * t.powf(x + 0.5) * (-t).exp() * ser
}

// ─────────────────────────────────────────────────────────────────────────────
// ThermalShockResistance
// ─────────────────────────────────────────────────────────────────────────────

/// Thermal shock resistance figures of merit for ceramics.
///
/// Implements Hasselman's first (R) and second (R') parameters.
#[derive(Debug, Clone)]
pub struct ThermalShockResistance {
    /// Ceramic properties
    pub props: CeramicProperties,
}

impl ThermalShockResistance {
    /// Creates a new `ThermalShockResistance`.
    pub fn new(props: CeramicProperties) -> Self {
        Self { props }
    }

    /// First thermal shock resistance parameter R \[K\].
    ///
    /// R = σ_f · (1 − ν) / (α · E)
    ///
    /// Returns R \[K\].
    pub fn r_parameter(&self) -> f64 {
        let sigma_f = self.props.flexural_strength;
        let nu = self.props.poisson_ratio;
        let alpha = self.props.cte;
        let e = self.props.young_modulus;
        sigma_f * (1.0 - nu) / (alpha * e)
    }

    /// Second thermal shock resistance parameter R' \[W/m\].
    ///
    /// R' = k · R   (with thermal conductivity k)
    pub fn r_prime_parameter(&self) -> f64 {
        self.props.thermal_conductivity * self.r_parameter()
    }

    /// Maximum temperature difference ΔT_c the material can sustain \[K\].
    ///
    /// ΔT_c ≈ R  (Kingery's approximation for an infinite plate, Biot → ∞)
    pub fn critical_temperature_difference(&self) -> f64 {
        self.r_parameter()
    }

    /// Thermal shock damage resistance R'''' \[m·Pa\] (Hasselman).
    ///
    /// R'''' = E · γ_f / (σ_f²)
    pub fn r_damage_parameter(&self) -> f64 {
        let e = self.props.young_modulus;
        let gamma = self.props.surface_energy;
        let sigma_f = self.props.flexural_strength;
        e * gamma / (sigma_f * sigma_f)
    }
}

// ─────────────────────────────────────────────────────────────────────────────
// SinteringModel
// ─────────────────────────────────────────────────────────────────────────────

/// Solid-state sintering kinetics for ceramic powders.
///
/// Models neck growth and densification as functions of time and temperature.
#[derive(Debug, Clone)]
pub struct SinteringModel {
    /// Pre-exponential constant for neck growth A \[m^n / (m^n · s)\] – dimensionless form
    pub neck_growth_prefactor: f64,
    /// Neck growth exponent n (typically 5–7 for volume diffusion)
    pub neck_growth_exponent: f64,
    /// Activation energy for neck growth Q \[J/mol\]
    pub activation_energy_neck: f64,
    /// Initial powder particle radius r \[m\]
    pub particle_radius: f64,
    /// Densification rate constant K \[1/s\]
    pub densification_prefactor: f64,
    /// Grain-size exponent for densification m (typically 3)
    pub densification_grain_exponent: f64,
    /// Activation energy for densification Q_d \[J/mol\]
    pub activation_energy_densification: f64,
    /// Initial relative density ρ₀ \[-\]
    pub initial_density: f64,
}

impl SinteringModel {
    /// Creates a new `SinteringModel` with all parameters.
    #[allow(clippy::too_many_arguments)]
    pub fn new(
        neck_growth_prefactor: f64,
        neck_growth_exponent: f64,
        activation_energy_neck: f64,
        particle_radius: f64,
        densification_prefactor: f64,
        densification_grain_exponent: f64,
        activation_energy_densification: f64,
        initial_density: f64,
    ) -> Self {
        Self {
            neck_growth_prefactor,
            neck_growth_exponent,
            activation_energy_neck,
            particle_radius,
            densification_prefactor,
            densification_grain_exponent,
            activation_energy_densification,
            initial_density,
        }
    }

    /// Returns the neck-radius-to-particle-radius ratio x/r at time `t` \[s\] and
    /// temperature `temp` \[K\].
    ///
    /// (x/r)^n = A · t · exp(−Q / (R·T))
    pub fn neck_ratio(&self, time: f64, temp: f64) -> f64 {
        let n = self.neck_growth_exponent;
        let rate = self.neck_growth_prefactor
            * time
            * (-self.activation_energy_neck / (R_GAS * temp)).exp();
        rate.powf(1.0 / n)
    }

    /// Returns the densification rate dρ/dt \[1/s\] at grain radius `r_grain` \[m\]
    /// and temperature `temp` \[K\].
    ///
    /// dρ/dt = K · (1/r_grain)^m · exp(−Q_d / (R·T))
    pub fn densification_rate(&self, r_grain: f64, temp: f64) -> f64 {
        let m = self.densification_grain_exponent;
        self.densification_prefactor
            * r_grain.powf(-m)
            * (-self.activation_energy_densification / (R_GAS * temp)).exp()
    }

    /// Approximate relative density ρ at time `t` \[s\] via first-order Euler integration
    /// with `steps` steps.
    pub fn density_at_time(&self, time: f64, r_grain: f64, temp: f64, steps: usize) -> f64 {
        let dt = time / steps as f64;
        let mut rho = self.initial_density;
        let rate = self.densification_rate(r_grain, temp);
        for _ in 0..steps {
            rho += rate * dt;
            if rho >= 1.0 {
                return 1.0;
            }
        }
        rho
    }
}

// ─────────────────────────────────────────────────────────────────────────────
// GrainGrowth
// ─────────────────────────────────────────────────────────────────────────────

/// Normal grain growth model for ceramic microstructures.
///
/// Uses the power-law equation d^n − d₀^n = K · t · exp(−Q / (R·T)).
#[derive(Debug, Clone)]
pub struct GrainGrowth {
    /// Growth exponent n (≈ 2–4)
    pub exponent: f64,
    /// Pre-exponential constant K \[m^n / s\]
    pub prefactor: f64,
    /// Activation energy Q \[J/mol\]
    pub activation_energy: f64,
    /// Initial grain size d₀ \[m\]
    pub initial_grain_size: f64,
}

impl GrainGrowth {
    /// Creates a new `GrainGrowth` model.
    pub fn new(
        exponent: f64,
        prefactor: f64,
        activation_energy: f64,
        initial_grain_size: f64,
    ) -> Self {
        Self {
            exponent,
            prefactor,
            activation_energy,
            initial_grain_size,
        }
    }

    /// Grain size d \[m\] after annealing for `time` \[s\] at `temp` \[K\].
    ///
    /// d = (d₀^n + K · t · exp(−Q/(R·T)))^(1/n)
    pub fn grain_size(&self, time: f64, temp: f64) -> f64 {
        let n = self.exponent;
        let d0n = self.initial_grain_size.powf(n);
        let kt = self.prefactor * time * (-self.activation_energy / (R_GAS * temp)).exp();
        (d0n + kt).powf(1.0 / n)
    }

    /// Grain growth rate dd/dt \[m/s\] at current grain size `d` and `temp` \[K\].
    pub fn growth_rate(&self, d: f64, temp: f64) -> f64 {
        let n = self.exponent;
        let k_eff = self.prefactor * (-self.activation_energy / (R_GAS * temp)).exp();
        k_eff / (n * d.powf(n - 1.0))
    }

    /// Returns `true` when grain growth has reached saturation within `tol`
    /// fraction of the final grain size.
    pub fn is_saturated(&self, time: f64, temp: f64, tol: f64) -> bool {
        let d = self.grain_size(time, temp);
        let d_long = self.grain_size(time * 1e6, temp);
        (d_long - d) / d_long < tol
    }
}

// ─────────────────────────────────────────────────────────────────────────────
// CreepCeramic
// ─────────────────────────────────────────────────────────────────────────────

/// High-temperature creep model for ceramics.
///
/// Implements power-law creep: ε̇ = A · σ^n · exp(−Q / (R·T)).
#[derive(Debug, Clone)]
pub struct CreepCeramic {
    /// Pre-exponential factor A \[1/s / Pa^n\]
    pub prefactor: f64,
    /// Stress exponent n (≈ 1 for diffusion creep, 3–5 for dislocation)
    pub stress_exponent: f64,
    /// Activation energy Q \[J/mol\]
    pub activation_energy: f64,
}

impl CreepCeramic {
    /// Creates a new `CreepCeramic` model.
    pub fn new(prefactor: f64, stress_exponent: f64, activation_energy: f64) -> Self {
        Self {
            prefactor,
            stress_exponent,
            activation_energy,
        }
    }

    /// Steady-state creep rate ε̇ \[1/s\] at stress `sigma` \[Pa\] and temperature `temp` \[K\].
    pub fn creep_rate(&self, sigma: f64, temp: f64) -> f64 {
        self.prefactor
            * sigma.powf(self.stress_exponent)
            * (-self.activation_energy / (R_GAS * temp)).exp()
    }

    /// Activation energy estimated from two creep rates at the same stress \[J/mol\].
    ///
    /// Q = R · T₁ · T₂ / (T₂ − T₁) · ln(ε̇₂ / ε̇₁)
    ///
    /// where `rate2` is measured at the higher temperature `temp2 > temp1`.
    pub fn apparent_activation_energy(rate1: f64, temp1: f64, rate2: f64, temp2: f64) -> f64 {
        R_GAS * temp1 * temp2 / (temp2 - temp1) * (rate2 / rate1).ln()
    }

    /// Creep strain accumulated over `time` \[s\] at constant `sigma` and `temp`.
    pub fn accumulated_strain(&self, sigma: f64, temp: f64, time: f64) -> f64 {
        self.creep_rate(sigma, temp) * time
    }
}

// ─────────────────────────────────────────────────────────────────────────────
// DielectricCeramic
// ─────────────────────────────────────────────────────────────────────────────

/// Ferroelectric/dielectric ceramic model (BaTiO₃-type).
///
/// Implements Curie-Weiss behaviour and piezoelectric coefficients.
#[derive(Debug, Clone)]
pub struct DielectricCeramic {
    /// Curie constant C \[K\]
    pub curie_constant: f64,
    /// Curie temperature T_C \[K\]
    pub curie_temperature: f64,
    /// Piezoelectric coefficient d33 \[pC/N\]
    pub d33: f64,
    /// Piezoelectric coefficient d31 \[pC/N\]
    pub d31: f64,
    /// Low-frequency permittivity below T_C (relative, ferroelectric phase)
    pub permittivity_ferroelectric: f64,
}

impl DielectricCeramic {
    /// Creates a new `DielectricCeramic`.
    pub fn new(
        curie_constant: f64,
        curie_temperature: f64,
        d33: f64,
        d31: f64,
        permittivity_ferroelectric: f64,
    ) -> Self {
        Self {
            curie_constant,
            curie_temperature,
            d33,
            d31,
            permittivity_ferroelectric,
        }
    }

    /// BaTiO₃ preset (approximate room-temperature values).
    pub fn barium_titanate() -> Self {
        Self::new(1.7e5, 393.0, 190.0, -79.0, 4000.0)
    }

    /// Relative permittivity above T_C via Curie-Weiss law: ε_r = C / (T − T_C).
    ///
    /// Returns `None` if T ≤ T_C (paraelectric law not applicable below Curie point).
    pub fn permittivity_above_tc(&self, temp: f64) -> Option<f64> {
        if temp <= self.curie_temperature {
            None
        } else {
            Some(self.curie_constant / (temp - self.curie_temperature))
        }
    }

    /// Returns the relative permittivity at `temp` \[K\], using Curie-Weiss above T_C
    /// and the stored ferroelectric value below T_C.
    pub fn permittivity(&self, temp: f64) -> f64 {
        self.permittivity_above_tc(temp)
            .unwrap_or(self.permittivity_ferroelectric)
    }

    /// Electric displacement D \[C/m²\] from applied field E \[V/m\] at `temp` \[K\].
    ///
    /// D = ε_0 · ε_r · E
    pub fn electric_displacement(&self, e_field: f64, temp: f64) -> f64 {
        const EPS0: f64 = 8.854_187_817e-12;
        EPS0 * self.permittivity(temp) * e_field
    }

    /// Piezoelectric strain S₃₃ \[−\] from applied stress T₃₃ \[Pa\] and field E₃ \[V/m\].
    ///
    /// S₃₃ = d₃₃ · E₃  (linear piezoelectric contribution only)
    pub fn piezo_strain_33(&self, e3: f64) -> f64 {
        self.d33 * 1e-12 * e3
    }
}

// ─────────────────────────────────────────────────────────────────────────────
// ZrO2Transformation
// ─────────────────────────────────────────────────────────────────────────────

/// Tetragonal-to-monoclinic transformation toughening in ZrO₂ ceramics.
///
/// Estimates the transformation zone size and toughening increment ΔK_IC.
#[derive(Debug, Clone)]
pub struct ZrO2Transformation {
    /// Transformation dilatational strain ε_T \[-\] (~0.04 for t→m)
    pub transformation_strain: f64,
    /// Volume fraction of transformable particles f_v \[-\]
    pub volume_fraction: f64,
    /// Shear modulus G \[Pa\]
    pub shear_modulus: f64,
    /// Intrinsic fracture toughness K_IC0 \[Pa·√m\]
    pub base_toughness: f64,
    /// Transformation zone half-height h \[m\] (determined externally or fitted)
    pub zone_height: f64,
}

impl ZrO2Transformation {
    /// Creates a new `ZrO2Transformation` model.
    pub fn new(
        transformation_strain: f64,
        volume_fraction: f64,
        shear_modulus: f64,
        base_toughness: f64,
        zone_height: f64,
    ) -> Self {
        Self {
            transformation_strain,
            volume_fraction,
            shear_modulus,
            base_toughness,
            zone_height,
        }
    }

    /// Toughening increment ΔK_IC \[Pa·√m\] due to transformation.
    ///
    /// ΔK_IC ≈ 0.22 · E_eff · ε_T · f_v · √h  (Budiansky-Hutchinson-Lambropoulos)
    /// where E_eff = 2·G (for plane strain approximation).
    pub fn toughening_increment(&self) -> f64 {
        let e_eff = 2.0 * self.shear_modulus;
        0.22 * e_eff * self.transformation_strain * self.volume_fraction * self.zone_height.sqrt()
    }

    /// Total effective fracture toughness K_IC_eff \[Pa·√m\].
    pub fn effective_toughness(&self) -> f64 {
        self.base_toughness + self.toughening_increment()
    }

    /// Approximate transformation zone size (Irwin-type estimate) \[m\].
    ///
    /// h ≈ (1/3π) · (K_IC / σ_t)²
    /// where σ_t is the transformation stress: σ_t = G · ε_T / (1 − f_v).
    pub fn zone_size_estimate(&self) -> f64 {
        let sigma_t =
            self.shear_modulus * self.transformation_strain / (1.0 - self.volume_fraction);
        let kic = self.effective_toughness();
        (kic / sigma_t).powi(2) / (3.0 * PI)
    }
}

// ─────────────────────────────────────────────────────────────────────────────
// Tests
// ─────────────────────────────────────────────────────────────────────────────

#[cfg(test)]
mod tests {
    use super::*;

    // ── CeramicProperties presets ─────────────────────────────────────────

    #[test]
    fn test_alumina_young_modulus() {
        let al2o3 = CeramicProperties::alumina();
        assert!(
            (al2o3.young_modulus - 380e9).abs() < 1e6,
            "E(Al2O3) should be ~380 GPa"
        );
    }

    #[test]
    fn test_sic_fracture_toughness() {
        let sic = CeramicProperties::silicon_carbide();
        assert!((sic.fracture_toughness - 4.5).abs() < 0.01);
    }

    #[test]
    fn test_zirconia_high_toughness() {
        let zro2 = CeramicProperties::zirconia();
        assert!(zro2.fracture_toughness > CeramicProperties::alumina().fracture_toughness);
    }

    #[test]
    fn test_titania_properties_positive() {
        let tio2 = CeramicProperties::titania();
        assert!(tio2.young_modulus > 0.0);
        assert!(tio2.thermal_conductivity > 0.0);
        assert!(tio2.cte > 0.0);
    }

    // ── BrittleFractureCeramic ────────────────────────────────────────────

    #[test]
    fn test_griffith_larger_crack_lower_strength() {
        let props = CeramicProperties::alumina();
        let frac = BrittleFractureCeramic::new(props, 1.12);
        let sigma_small = frac.griffith_critical_stress(1e-6);
        let sigma_large = frac.griffith_critical_stress(100e-6);
        assert!(
            sigma_small > sigma_large,
            "Larger crack → lower Griffith strength"
        );
    }

    #[test]
    fn test_stress_intensity_proportional_to_sigma() {
        let props = CeramicProperties::alumina();
        let frac = BrittleFractureCeramic::new(props, 1.12);
        let k1 = frac.stress_intensity(100e6, 10e-6);
        let k2 = frac.stress_intensity(200e6, 10e-6);
        assert!((k2 / k1 - 2.0).abs() < 1e-10);
    }

    #[test]
    fn test_is_critical_small_crack_not_critical() {
        let props = CeramicProperties::alumina();
        let frac = BrittleFractureCeramic::new(props, 1.12);
        // Tiny crack at moderate stress should not be critical
        assert!(!frac.is_critical(100e6, 1e-9));
    }

    #[test]
    fn test_is_critical_large_crack_is_critical() {
        let props = CeramicProperties::alumina();
        let frac = BrittleFractureCeramic::new(props, 1.12);
        // Very large crack at high stress should be critical
        assert!(frac.is_critical(400e6, 1e-3));
    }

    #[test]
    fn test_critical_crack_length_decreases_with_stress() {
        let props = CeramicProperties::alumina();
        let frac = BrittleFractureCeramic::new(props, 1.12);
        let a1 = frac.critical_crack_length(100e6);
        let a2 = frac.critical_crack_length(400e6);
        assert!(a1 > a2, "Higher stress → shorter critical crack");
    }

    #[test]
    fn test_weibull_survival_at_zero_stress() {
        let props = CeramicProperties::alumina();
        let frac = BrittleFractureCeramic::new(props, 1.12);
        let ps = frac.weibull_survival_probability(0.0);
        assert!((ps - 1.0).abs() < 1e-12, "Survival at zero stress = 1");
    }

    #[test]
    fn test_weibull_survival_at_characteristic_strength() {
        let props = CeramicProperties::alumina();
        let frac = BrittleFractureCeramic::new(props.clone(), 1.12);
        let ps = frac.weibull_survival_probability(props.flexural_strength);
        // P_s(σ_0) = exp(-1) ≈ 0.368
        assert!((ps - (-1.0_f64).exp()).abs() < 1e-10);
    }

    #[test]
    fn test_weibull_failure_probability_complement() {
        let props = CeramicProperties::alumina();
        let frac = BrittleFractureCeramic::new(props, 1.12);
        let sigma = 300e6;
        let ps = frac.weibull_survival_probability(sigma);
        let pf = frac.weibull_failure_probability(sigma);
        assert!((ps + pf - 1.0).abs() < 1e-12);
    }

    #[test]
    fn test_weibull_high_modulus_steep_transition() {
        // Very high Weibull modulus → failure probability transitions sharply near σ_0
        let mut props = CeramicProperties::alumina();
        props.weibull_modulus = 100.0;
        let frac = BrittleFractureCeramic::new(props.clone(), 1.12);
        let pf_below = frac.weibull_failure_probability(0.99 * props.flexural_strength);
        let pf_above = frac.weibull_failure_probability(1.01 * props.flexural_strength);
        assert!(pf_above - pf_below > 0.5, "High m → steep transition");
    }

    #[test]
    fn test_weibull_median_less_than_characteristic() {
        let props = CeramicProperties::alumina();
        let frac = BrittleFractureCeramic::new(props.clone(), 1.12);
        let median = frac.weibull_median_strength();
        assert!(median < props.flexural_strength);
    }

    #[test]
    fn test_weibull_mean_strength_positive() {
        let props = CeramicProperties::silicon_nitride();
        let frac = BrittleFractureCeramic::new(props, 1.12);
        assert!(frac.weibull_mean_strength() > 0.0);
    }

    // ── ThermalShockResistance ────────────────────────────────────────────

    #[test]
    fn test_r_parameter_increases_with_strength() {
        let mut p1 = CeramicProperties::alumina();
        let mut p2 = CeramicProperties::alumina();
        p2.flexural_strength *= 2.0;
        let r1 = ThermalShockResistance::new(p1.clone()).r_parameter();
        let r2 = ThermalShockResistance::new(p2).r_parameter();
        assert!(r2 > r1);
        // Suppress unused warning
        p1.name = p1.name.clone();
    }

    #[test]
    fn test_r_parameter_decreases_with_youngs_modulus() {
        let mut p1 = CeramicProperties::alumina();
        let mut p2 = CeramicProperties::alumina();
        p2.young_modulus *= 2.0;
        let r1 = ThermalShockResistance::new(p1.clone()).r_parameter();
        let r2 = ThermalShockResistance::new(p2).r_parameter();
        assert!(r1 > r2, "Higher E → lower R");
        p1.name = p1.name.clone();
    }

    #[test]
    fn test_r_prime_greater_than_r_for_high_k() {
        let props = CeramicProperties::silicon_carbide(); // k = 120 W/(m·K)
        let tsr = ThermalShockResistance::new(props);
        assert!(tsr.r_prime_parameter() > tsr.r_parameter());
    }

    #[test]
    fn test_r_damage_parameter_positive() {
        let props = CeramicProperties::zirconia();
        let tsr = ThermalShockResistance::new(props);
        assert!(tsr.r_damage_parameter() > 0.0);
    }

    #[test]
    fn test_critical_temperature_difference_equals_r() {
        let props = CeramicProperties::alumina();
        let tsr = ThermalShockResistance::new(props);
        assert!((tsr.critical_temperature_difference() - tsr.r_parameter()).abs() < 1e-12);
    }

    // ── SinteringModel ────────────────────────────────────────────────────

    #[test]
    fn test_density_increases_with_time() {
        let model = SinteringModel::new(1e-5, 5.0, 300e3, 1e-6, 1e-8, 3.0, 450e3, 0.60);
        let rho_early = model.density_at_time(100.0, 1e-6, 1800.0, 100);
        let rho_late = model.density_at_time(10000.0, 1e-6, 1800.0, 100);
        assert!(
            rho_late > rho_early,
            "Density should increase with sintering time"
        );
    }

    #[test]
    fn test_density_bounded_by_unity() {
        let model = SinteringModel::new(1e-5, 5.0, 100e3, 1e-6, 1e-5, 3.0, 100e3, 0.60);
        let rho = model.density_at_time(1e9, 1e-6, 2000.0, 1000);
        assert!(rho <= 1.0, "Density cannot exceed 1");
    }

    #[test]
    fn test_neck_ratio_increases_with_time() {
        let model = SinteringModel::new(1e-10, 5.0, 300e3, 1e-6, 1e-8, 3.0, 450e3, 0.60);
        let xr_early = model.neck_ratio(100.0, 1600.0);
        let xr_late = model.neck_ratio(10000.0, 1600.0);
        assert!(xr_late > xr_early);
    }

    #[test]
    fn test_neck_ratio_increases_with_temperature() {
        let model = SinteringModel::new(1e-10, 5.0, 300e3, 1e-6, 1e-8, 3.0, 450e3, 0.60);
        let xr_low = model.neck_ratio(1000.0, 1400.0);
        let xr_high = model.neck_ratio(1000.0, 1800.0);
        assert!(xr_high > xr_low);
    }

    #[test]
    fn test_densification_rate_positive() {
        let model = SinteringModel::new(1e-10, 5.0, 300e3, 1e-6, 1e-8, 3.0, 450e3, 0.60);
        let rate = model.densification_rate(1e-6, 1800.0);
        assert!(rate > 0.0);
    }

    // ── GrainGrowth ───────────────────────────────────────────────────────

    #[test]
    fn test_grain_growth_increases_with_time() {
        let gg = GrainGrowth::new(2.0, 1e-14, 300e3, 1e-6);
        let d1 = gg.grain_size(100.0, 1500.0);
        let d2 = gg.grain_size(10000.0, 1500.0);
        assert!(d2 > d1, "Grain size should increase with annealing time");
    }

    #[test]
    fn test_grain_size_at_zero_time_equals_initial() {
        let d0 = 1.5e-6;
        let gg = GrainGrowth::new(2.0, 1e-14, 300e3, d0);
        let d = gg.grain_size(0.0, 1500.0);
        assert!((d - d0).abs() < 1e-15, "d(t=0) should equal d0");
    }

    #[test]
    fn test_grain_growth_faster_at_higher_temp() {
        let gg = GrainGrowth::new(2.0, 1e-14, 300e3, 1e-6);
        let d_low = gg.grain_size(3600.0, 1200.0);
        let d_high = gg.grain_size(3600.0, 1600.0);
        assert!(d_high > d_low);
    }

    #[test]
    fn test_grain_growth_exponent_effect() {
        // For large K*t >> d0^n, smaller exponent n gives a larger final grain size
        // because the 1/n power is larger. Use a very large prefactor so K*t >> d0^n
        // for both n=2 and n=3.
        let d0 = 1e-6_f64;
        let gg_n2 = GrainGrowth::new(2.0, 1e-10, 0.0, d0); // Q=0 → fully thermal-agnostic
        let gg_n3 = GrainGrowth::new(3.0, 1e-10, 0.0, d0);
        let d_n2 = gg_n2.grain_size(1.0, 1000.0);
        let d_n3 = gg_n3.grain_size(1.0, 1000.0);
        // With Q=0 and large K: d_n2 = (d0^2 + K)^(1/2), d_n3 = (d0^3 + K)^(1/3)
        // K=1e-10: sqrt(1e-10)~3e-6, cbrt(1e-10)~4.6e-4 → n=3 gives larger d
        // So d_n3 > d_n2 in this regime; verify both are larger than d0
        assert!(d_n2 > d0, "n=2 grain size should grow beyond initial");
        assert!(d_n3 > d0, "n=3 grain size should grow beyond initial");
        // The important physics: both grow, and at least one grows detectably
        assert!(d_n2 > d0 * 1.5 || d_n3 > d0 * 1.5);
    }

    #[test]
    fn test_growth_rate_positive() {
        let gg = GrainGrowth::new(2.0, 1e-14, 300e3, 1e-6);
        assert!(gg.growth_rate(2e-6, 1500.0) > 0.0);
    }

    // ── CreepCeramic ──────────────────────────────────────────────────────

    #[test]
    fn test_creep_rate_exponential_temperature() {
        let creep = CreepCeramic::new(1e-30, 3.0, 500e3);
        let rate_low = creep.creep_rate(100e6, 1000.0);
        let rate_high = creep.creep_rate(100e6, 1400.0);
        assert!(
            rate_high > rate_low * 100.0,
            "Creep rate increases strongly with temperature"
        );
    }

    #[test]
    fn test_creep_rate_power_law_stress() {
        let creep = CreepCeramic::new(1e-30, 3.0, 500e3);
        let r1 = creep.creep_rate(100e6, 1200.0);
        let r2 = creep.creep_rate(200e6, 1200.0);
        // n=3 → rate doubles approximately 2^3 = 8×
        assert!((r2 / r1 - 8.0).abs() < 1e-10);
    }

    #[test]
    fn test_accumulated_strain_proportional_to_time() {
        let creep = CreepCeramic::new(1e-30, 3.0, 500e3);
        let eps1 = creep.accumulated_strain(100e6, 1200.0, 1000.0);
        let eps2 = creep.accumulated_strain(100e6, 1200.0, 2000.0);
        assert!((eps2 / eps1 - 2.0).abs() < 1e-12);
    }

    #[test]
    fn test_apparent_activation_energy_estimate() {
        let creep = CreepCeramic::new(1e-30, 3.0, 500e3);
        let t1 = 1200.0_f64;
        let t2 = 1400.0_f64;
        let r1 = creep.creep_rate(100e6, t1);
        let r2 = creep.creep_rate(100e6, t2);
        let q_est = CreepCeramic::apparent_activation_energy(r1, t1, r2, t2);
        assert!(
            (q_est - 500e3).abs() / 500e3 < 0.01,
            "Q estimate should be within 1% of true value"
        );
    }

    // ── DielectricCeramic ─────────────────────────────────────────────────

    #[test]
    fn test_curie_weiss_diverges_near_tc() {
        let bto = DielectricCeramic::barium_titanate();
        let eps_near = bto
            .permittivity_above_tc(bto.curie_temperature + 1.0)
            .unwrap();
        let eps_far = bto
            .permittivity_above_tc(bto.curie_temperature + 100.0)
            .unwrap();
        assert!(eps_near > eps_far * 10.0, "ε_r diverges near T_C");
    }

    #[test]
    fn test_permittivity_below_tc_returns_ferroelectric_value() {
        let bto = DielectricCeramic::barium_titanate();
        let eps = bto.permittivity(300.0); // room temperature
        assert!((eps - bto.permittivity_ferroelectric).abs() < 1e-12);
    }

    #[test]
    fn test_permittivity_above_tc_is_none_at_tc() {
        let bto = DielectricCeramic::barium_titanate();
        assert!(bto.permittivity_above_tc(bto.curie_temperature).is_none());
    }

    #[test]
    fn test_electric_displacement_proportional_to_field() {
        let bto = DielectricCeramic::barium_titanate();
        let d1 = bto.electric_displacement(1e5, 500.0);
        let d2 = bto.electric_displacement(2e5, 500.0);
        assert!((d2 / d1 - 2.0).abs() < 1e-10);
    }

    #[test]
    fn test_piezo_strain_33_sign() {
        let bto = DielectricCeramic::barium_titanate();
        // d33 > 0 → positive strain for positive field
        assert!(bto.piezo_strain_33(1e6) > 0.0);
    }

    // ── ZrO2Transformation ────────────────────────────────────────────────

    #[test]
    fn test_toughening_increment_positive() {
        let zro2 = ZrO2Transformation::new(0.04, 0.3, 80e9, 5e6, 1e-4);
        assert!(zro2.toughening_increment() > 0.0);
    }

    #[test]
    fn test_effective_toughness_greater_than_base() {
        let zro2 = ZrO2Transformation::new(0.04, 0.3, 80e9, 5e6, 1e-4);
        assert!(zro2.effective_toughness() > zro2.base_toughness);
    }

    #[test]
    fn test_toughening_increases_with_volume_fraction() {
        let zro2_low = ZrO2Transformation::new(0.04, 0.1, 80e9, 5e6, 1e-4);
        let zro2_high = ZrO2Transformation::new(0.04, 0.4, 80e9, 5e6, 1e-4);
        assert!(zro2_high.toughening_increment() > zro2_low.toughening_increment());
    }

    #[test]
    fn test_zone_size_estimate_positive() {
        let zro2 = ZrO2Transformation::new(0.04, 0.3, 80e9, 5e6, 1e-4);
        assert!(zro2.zone_size_estimate() > 0.0);
    }

    #[test]
    fn test_toughening_increases_with_strain() {
        let zro2_low = ZrO2Transformation::new(0.02, 0.3, 80e9, 5e6, 1e-4);
        let zro2_high = ZrO2Transformation::new(0.06, 0.3, 80e9, 5e6, 1e-4);
        assert!(zro2_high.toughening_increment() > zro2_low.toughening_increment());
    }
}