5.1 Introduction: Cracks as the True Failure Feature

Classical strength-of-materials and yield criteria (Chapters 1–2) predict failure when the nominal stress reaches a material limit. This works for ductile components without pre-existing flaws. It fails catastrophically for:

• Brittle materials (high-strength steels, ceramics, cast iron) where cracks initiate at micro-defects.
• Thick sections where plane-strain constraint elevates triaxiality at the crack tip.
• Fatigue cracks that grow sub-critically before final fracture.
• Welded structures with inherent crack-like defects.

Fracture mechanics treats a crack as a geometric singularity and characterizes the severity of the crack-tip stress field by the stress intensity factor $K$. Failure occurs when $K$ reaches a material property called fracture toughness $K_{Ic}$.

Linear Elastic Fracture Mechanics (LEFM) applies when plastic deformation at the crack tip is confined to a small zone compared to crack size and specimen dimensions.

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5.2 Fracture Modes I, II, and III

Irwin classified crack surface displacements into three orthogonal modes:

5.2.1 Mode I — Opening (tension)

Crack faces separate symmetrically perpendicular to the crack plane. This is the most common and dangerous mode in engineering structures (pressure vessels, aircraft skins, bolt holes).

5.2.2 Mode II — In-plane shear (sliding)

Crack faces slide relative to each other in the crack plane. Occurs in mixed-mode problems (gear teeth, adhesive joints under shear).

5.2.3 Mode III — Out-of-plane shear (tearing)

Crack faces slide anti-symmetrically out of the crack plane. Relevant to shafts under torsion, spiral cracks in tubes.

For mixed-mode loading, the total strain energy release rate is:

$$G = \frac{K_I^2}{E'} + \frac{K_{II}^2}{E'} + \frac{K_{III}^2}{2G}$$

where $E' = E$ (plane stress) or $E/(1-\nu^2)$ (plane strain), and $G$ is the shear modulus.

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5.3 Stress Intensity Factor $K$

The stress field near a crack tip in Mode I (for an isotropic linear elastic material) is:

$$\sigma_{ij} = \frac{K_I}{\sqrt{2\pi r}} f_{ij}(\theta) + O(1)$$

where $r$ is the distance from the crack tip, $\theta$ is the angle from the crack plane, and $f_{ij}(\theta)$ are universal angular functions (Williams expansion).

5.3.1 General form

For a crack of depth $a$ in a body under remote stress $\sigma$:

$$\boxed{K_I = Y \sigma \sqrt{\pi a}}$$

where $Y$ is a geometry correction factor (dimensionless) that depends on:

• Crack location (edge, center, surface, embedded).
• Specimen geometry (plate width $W$, thickness $B$).
• Loading configuration (tension, bending, pressurization).

5.3.2 Common $Y$ factors

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5.4 Fracture Toughness $K_{Ic}$

When $K_I$ reaches a critical value $K_{Ic}$, the crack propagates unstably (fast fracture):

$$\boxed{K_I \ge K_{Ic} \Rightarrow \text{fracture}}$$

$K_{Ic}$ is the plane-strain fracture toughness — a material property measured per ASTM E399.

5.4.1 Typical values

5.4.2 Plane stress vs plane strain

• Plane strain ($B$ thick, $K_{Ic}$): maximum constraint, minimum toughness — conservative for thick sections.
• Plane stress ($B$ thin, $K_c$): less constraint, higher apparent toughness, but thickness-dependent.

For design of thick components (pressure vessels, reactor vessels), always use $K_{Ic}$ (plane strain).

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5.5 Plastic Zone Size and LEFM Validity

LEFM assumes the plastic zone at the crack tip is small compared to:

1. Crack length $a$.
2. Ligament size $(W - a)$.
3. Specimen thickness $B$.

5.5.1 Irwin plastic zone estimate (plane stress)

$$\boxed{r_p = \frac{1}{\pi}\left(\frac{K_I}{S_y}\right)^2}$$

5.5.2 Plane strain plastic zone

$$r_p = \frac{1}{3\pi}\left(\frac{K_I}{S_y}\right)^2$$

The plane-strain zone is ~3× smaller than plane stress — thicker sections suppress plasticity at the tip.

5.5.3 ASTM E399 validity requirements

For a valid $K_{Ic}$ measurement:

• $a, B, (W-a) \ge 2.5 (K_{Ic}/S_y)^2$ (plastic zone size criterion).
• Crack length $0.45 \le a/W \le 0.55$.

When LEFM is invalid (large plastic zone), use Elastic-Plastic Fracture Mechanics (EPFM) — Chapter 6 ($J$-integral, CTOD).

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5.6 Energy Release Rate $G$ and Griffith's Criterion

Griffith (1921) showed that crack growth occurs when the energy release rate $G$ exceeds the material's critical energy release rate $G_c$:

$$\boxed{G = \frac{K_I^2}{E'}} \ge G_c$$

5.6.1 Griffith for brittle materials

For ideally brittle materials (glass, ceramics): $G_c = 2\gamma_s$ where $\gamma_s$ is the surface energy per unit area.

5.6.2 Orowan extension for metals

For metals, plastic work at the crack tip dominates: $G_c = 2\gamma_s + G_p \gg 2\gamma_s$.

5.6.3 Relation to $K_{Ic}$

$$K_{Ic} = \sqrt{G_c E'}$$

This connects the energy-based (Griffith) and stress-intensity (Irwin) formulations.

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5.7 Critical Crack Size and Safety Factor

From $K_I = Y\sigma\sqrt{\pi a} = K_{Ic}$:

$$\boxed{a_c = \frac{1}{\pi}\left(\frac{K_{Ic}}{Y\sigma}\right)^2}$$

5.7.1 Safety factor on crack size

$$n_a = \frac{a_c}{a} = \left(\frac{K_{Ic}}{K_I}\right)^2$$

5.7.2 Safety factor on stress

$$n_\sigma = \frac{\sigma_c}{\sigma} = \frac{K_{Ic}}{K_I}$$

5.7.3 Damage-tolerant design philosophy

1. Assume a crack of size $a_i$ exists (from NDT inspection limit).
2. Compute $K_I$ under maximum service load.
3. Verify $K_I < K_{Ic}/n$ (instant fracture check).
4. Grow the crack with Paris law (Ch.6) to estimate remaining life.
5. Retire or inspect before $a$ reaches $a_c$.

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5.8 Mixed-Mode Fracture and Interaction Criteria

Real cracks often experience combined Mode I and Mode II loading. Interaction criteria predict mixed-mode failure:

5.8.1 Maximum circumferential stress ($\sigma_\theta$ criterion)

Crack propagates in the direction where circumferential stress $\sigma_\theta$ is maximum (Mode I locally).

5.8.2 Strain energy density (SED)

Crack propagates in the direction of minimum strain energy density.

5.8.3 Mixed-mode interaction (quadratic)

$$\left(\frac{K_I}{K_{Ic}}\right)^2 + \left(\frac{K_{II}}{K_{IIc}}\right)^2 = 1$$

For many materials, $K_{IIc} \approx K_{Ic}$.

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5.9 Practical LEFM Analysis Procedure

1. Identify the crack: location, orientation, initial size $a_i$ (from inspection or assumed flaw).
2. Determine loading: remote stress $\sigma$ or stress distribution from FEA.
3. Select $Y$: from handbook (Tada, Murakami) or FEA contour integral.
4. Compute $K_I = Y\sigma\sqrt{\pi a}$.
5. Check LEFM validity: $r_p \ll a$, ligament, thickness.
6. Compare to $K_{Ic}$: compute safety factor $n = K_{Ic}/K_I$.
7. Compute $a_c$: maximum allowable crack size.
8. If $K_I$ is sub-critical: proceed to fatigue crack growth analysis (Ch.6).

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5.10 Summary

5.11 Forensic Application: Reading Fracture Surfaces

Fracture surface morphology reveals the failure mechanism and validates LEFM predictions:

5.11.1 Cleavage (brittle) fracture

Flat, crystalline facets with river patterns pointing back to the crack origin. Occurs when $K_I$ exceeds $K_{Ic}$ at low temperature or high strain rate. The origin is often at a micro-defect (inclusion, carbide) at or near the surface.

5.11.2 Ductile fracture (micro-void coalescence)

Dimpled, fibrous surface with shear lips at 45°. Occurs after extensive plastic deformation. $K$ at fracture may exceed $K_{Ic}$ due to tearing resistance ($J_R$ curve, Ch.6).

5.11.3 Fatigue fracture

Smooth beach marks (striations) radiating from the crack origin. The origin location confirms the stress field: subsurface origin → Hertzian contact (Ch.7); surface origin at notch → $K_t$ effect; origin at weld toe → fabrication defect.

5.11.4 Mixed-mode fracture

Herringbone or slanted fracture surface indicates combined Mode I + II. Measure the angle between the crack plane and the fracture surface to estimate $K_{II}/K_I$ ratio.

Design takeaway: Never rely on ultimate tensile strength alone for high-strength alloys, thick sections, or welded structures. Always compute $K_I$, compare to $K_{Ic}$, and integrate crack growth (Ch.6) for damage-tolerant design.