Fracture Mechanics — When Materials Fail

1. Griffith's Energy Criterion

In 1921, Alan Griffith observed that real materials fracture at stresses far below their theoretical strength. His insight: a crack introduces a stress concentration but also releases elastic strain energy. A crack grows when the energy released exceeds the energy needed to create new fracture surfaces.

For a through-crack of half-length a in an infinite plate under remote stress σ, the strain energy release rate is:

The surface energy per unit area is γ (J/m²). The crack propagates when G ≥ G_c, where G_c = 2γ is the critical energy release rate:

This Griffith criterion reveals the key insight: fracture stress scales as 1/√a. Double the crack length → fracture at 71% of the original stress.

2. Stress Intensity Factor K

Griffith's energy approach works globally, but we need a local description of the stress field near a crack tip. Irwin (1957) showed that for any crack geometry, the singular stress field has the form:

where r is the distance from the crack tip, θ is the angle, and f_ij(θ) are dimensionless angular functions. The stress intensity factor K captures everything about the loading and geometry:

Here Y is a dimensionless geometry factor (Y = 1 for an infinite plate, tabulated for other geometries). The singular 1/√r stress field is universal — only its amplitude K changes.

3. Crack Opening Modes

There are three fundamental ways a crack can open, each with its own K:

• Mode I (Opening) — tensile stress normal to the crack plane. Most critical for brittle materials. K_I = σ√(πa).
• Mode II (In-plane Shear) — shear parallel to crack plane, perpendicular to crack front. K_II = τ√(πa).
• Mode III (Out-of-plane Shear / Tearing) — shear parallel to crack front. Common in twisted shafts.

Real cracks are usually mixed-mode. The effective K is written as an equivalent K_eff = √(K_I² + K_II² + K_III²/(1-ν)).

4. Irwin's Plastic Zone

The K-based stress field predicts infinite stress at r = 0 — physically impossible. In ductile materials, yielding limits the stress. Irwin estimated the plastic zone radius by setting σ_yy = σ_yield:

For plane strain (thick specimens), the plastic zone is smaller by a factor of ~3: r_{p,plane strain} = r_p/3. The condition for valid K-based LEFM (linear elastic fracture mechanics) is that the plastic zone is much smaller than other dimensions — specimen size requirements drive standard fracture toughness tests.

5. Fracture Toughness K_Ic

The material property that resists fracture is the critical stress intensity factor K_Ic (plane strain fracture toughness). A crack propagates when:

K_Ic has units of MPa·√m. The relationship to Griffith's G_c is:

6. Fatigue Crack Growth — Paris Law

Under cyclic loading, cracks grow at stress levels well below K_Ic. Paris and Erdogan (1963) found empirically that crack growth per cycle follows a power law:

where ΔK = K_max − K_min = Y · Δσ · √(πa) is the stress intensity range, and C, m are material constants (typically m ≈ 2–4 for metals).

The Paris law gives three regimes on a log(da/dN) vs log(ΔK) plot:

• Region I (threshold) — ΔK < ΔK_th, crack does not grow. Important for infinite life design.
• Region II (Paris regime) — linear log-log relationship, crack grows steadily. Most fatigue life is spent here.
• Region III (fast fracture) — K_max → K_Ic, rapid acceleration to final fracture.

Integrating the Paris law from initial crack size a₀ to critical crack size a_c = (K_Ic/Yσ)²/π gives the number of cycles to failure:

For m ≠ 2, this integral has a closed form:

7. 2D Implementation

A simple 2D fracture simulation can use a spring-lattice model: particles on a grid connected by springs. Each spring breaks when its strain exceeds a threshold ε_c = σ_y/E.

// Spring-lattice fracture (2D, canvas)
const N = 64;
const springs = buildLatticeConnections(particles, N);

function update(dt) {
  // Verlet integration
  for (const p of particles) {
    const acc = computeForces(p, springs) / p.mass;
    p.vel.add(acc.scale(dt));
    p.pos.add(p.vel.scale(dt));
  }

  // Break springs that exceed strain threshold
  for (const s of springs) {
    const currentLen = dist(s.a.pos, s.b.pos);
    const strain = (currentLen - s.restLen) / s.restLen;
    if (Math.abs(strain) > STRAIN_THRESHOLD) {
      s.broken = true;
    }
  }
}

function computeForces(p, springs) {
  const f = new Vec2(0, 0);
  for (const s of springs) {
    if (s.broken) continue;
    const other = s.a === p ? s.b : s.a;
    const delta = other.pos.sub(p.pos);
    const len = delta.length();
    const extension = len - s.restLen;
    f.add(delta.normalized().scale(s.stiffness * extension));
  }
  return f;
}

For more physically accurate fracture, XFEM (Extended FEM) enriches standard finite element shape functions with discontinuous functions across crack surfaces and singular crack-tip functions — avoiding re-meshing as cracks grow.

8. Material Comparison