materials-fracture-mechanics

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.

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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

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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))

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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.

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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

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Output Format

When using this skill, structure the answer as:

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

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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