Materials And Fracture Mechanics

Tutor and diagnostics skill for how materials fail under load. It complements the repo's strength-of-materials skills by focusing on defects, crack growth, and fatigue rather than just nominal stress.

Important: Use this skill to screen failure plausibility and teach the concepts. Real integrity decisions still require metallurgy, NDE, and service history.

Module 1 - Stress, Strain, And Elastic Screening

def engineering_stress(force_lbf, area_in2):
    """Nominal stress in psi."""
    if area_in2 <= 0:
        return None
    return force_lbf / area_in2

def engineering_strain(delta_length_in, original_length_in):
    """Engineering strain."""
    if original_length_in <= 0:
        return None
    return delta_length_in / original_length_in

def hooke_law_stress(E_psi, strain):
    """Elastic stress from Hooke's law."""
    return E_psi * strain

Module 2 - Stress Intensity And Critical Crack Size

def stress_intensity_ksi_sqrtin(stress_ksi, crack_length_in, geometry_factor=1.12):
    """
    Mode-I stress intensity factor:
    K_I = Y * sigma * sqrt(pi * a)
    """
    import math
    if crack_length_in <= 0:
        return None
    return geometry_factor * stress_ksi * math.sqrt(math.pi * crack_length_in)

def critical_crack_size_in(KIC_ksi_sqrtin, stress_ksi, geometry_factor=1.12):
    """
    Solve for crack size at fast fracture: K_I = K_IC
    """
    import math
    if stress_ksi <= 0 or geometry_factor <= 0:
        return None
    return (KIC_ksi_sqrtin / (geometry_factor * stress_ksi))**2 / math.pi

def critical_stress_ksi(KIC_ksi_sqrtin, crack_length_in, geometry_factor=1.12):
    """
    Critical far-field stress for a given crack size.
    """
    import math
    if crack_length_in <= 0 or geometry_factor <= 0:
        return None
    return KIC_ksi_sqrtin / (geometry_factor * math.sqrt(math.pi * crack_length_in))

Module 3 - Fatigue Screening

def goodman_utilization(mean_stress_ksi, alternating_stress_ksi,
                        ultimate_strength_ksi, endurance_limit_ksi):
    """
    Goodman relation utilization.
    Values > 1 indicate fatigue failure risk.
    """
    if ultimate_strength_ksi <= 0 or endurance_limit_ksi <= 0:
        return None
    return (mean_stress_ksi / ultimate_strength_ksi +
            alternating_stress_ksi / endurance_limit_ksi)

def paris_cycles(ai_in, af_in, delta_sigma_ksi, C, m, geometry_factor=1.12,
                 steps=1000):
    """
    Numerically integrate Paris law:
    da/dN = C * (Delta K)^m
    """
    import math
    if ai_in <= 0 or af_in <= ai_in or steps <= 0:
        return None
    da = (af_in - ai_in) / steps
    cycles = 0.0
    a = ai_in
    for _ in range(steps):
        delta_k = geometry_factor * delta_sigma_ksi * math.sqrt(math.pi * a)
        dadn = C * delta_k**m
        if dadn <= 0:
            return None
        cycles += da / dadn
        a += da
    return cycles

Use Paris law only in the stable crack-growth regime. It does not cover crack initiation or final unstable fracture.

Module 4 - Failure Logic

Observation	Likely concern
High nominal stress but no flaw evidence	Strength / yielding problem
Modest stress with known defect	Fracture toughness problem
Repeated cyclic loading	Fatigue crack growth problem
Low temperature or sour service	Reduced toughness / embrittlement concern
Corrosion pits at stress concentrators	Pit-to-crack transition risk

Output Format

When using this skill, structure the answer as:

Loading mode and material basis
Whether the problem is strength-, toughness-, or fatigue-controlled
Equation setup and units
Crack or life estimate
What material data are still missing

Learning Resources

Resource	Notes
Callister and Rethwisch	Broad materials foundation
Anderson, Fracture Mechanics	Standard fracture mechanics text
Dowling, Mechanical Behavior of Materials	Strong fatigue coverage