2. Linear elastic fracture mechanics (LEFM)

Linear elastic fracture mechanics (LEFM) is the analytical backbone of crack growth simulation. It answers a deceptively simple question: given a crack of known size and orientation in a linearly elastic body, how severe is the stress state at its tip, and when does that severity become sufficient to drive propagation? The framework rests on three pillars — the stress intensity factor K, the asymptotic crack tip stress and displacement fields, and the governing equations that connect them to applied loads and geometry. This chapter develops all three, then shows how they are used in practice.

1. Why a Stress-Based Criterion Fails

A naive approach would compare the peak stress at the crack tip against a material strength. This fails immediately: for a sharp crack in a linear elastic continuum, the elastic solution predicts infinite stress at the tip. The singularity is not a numerical artifact — it is a direct consequence of the geometry. Any criterion based on absolute stress magnitude is therefore meaningless at the tip itself.

The resolution, due to Irwin and Westergaard, is to characterize the intensity of the singularity rather than its peak value. Two cracks with the same singularity strength behave identically near their tips, regardless of the far-field geometry that produced that strength. This single idea makes LEFM both powerful and portable.

2. The Three Loading Modes

Crack tip deformation is decomposed into three independent modes:

• Mode I (opening): crack faces separate symmetrically under tensile stress normal to the crack plane. This is the most common and most damaging mode in engineering practice.
• Mode II (in-plane shear): crack faces slide relative to each other in the crack plane, driven by shear stress.
• Mode III (anti-plane shear): crack faces slide out of plane, driven by transverse shear.

In a general mixed-mode problem, the total field is the superposition of the three. Mode I is treated first because it dominates most fatigue and fracture scenarios, and because mixed-mode criteria are usually expressed as corrections to the Mode I result.

3. The Stress Intensity Factor

The stress intensity factor K quantifies the amplitude of the crack tip singularity. Its general form is:

K = Y · σ · √(π · a)

where σ is the remote applied stress (MPa), a is the crack length (mm or m), and Y is a dimensionless geometry factor that depends on specimen shape, crack location, and loading configuration. The units of K are MPa·√m (or MPa·√mm, depending on the unit system — consistency matters).

The geometry factor Y is where most of the engineering effort lies. For a through-thickness crack in an infinite plate, Y = 1. For a semi-elliptical surface crack in a finite plate, Y depends on crack depth, aspect ratio, plate width, and bending versus membrane loading. Closed-form solutions exist for many standard configurations (Tada, Paris & Irwin; Rooke & Cartwright), and numerical solutions via finite element or boundary element methods are used for everything else.

4. Crack Tip Stress and Displacement Fields

Near the crack tip, the stress field for Mode I takes the asymptotic form:

σ_ij(r, θ) = (K_I / √(2πr)) · f_ij(θ) + higher-order terms

Here r is the radial distance from the tip and θ is the angle measured from the crack plane. The functions f_ij(θ) are universal — they do not depend on specimen geometry. This universality is the heart of LEFM: the spatial distribution of stress near the tip is always the same; only the amplitude K changes.

The leading singular term dominates within a small region called the K-dominant zone. Outside this zone, higher-order terms (notably the T-stress) become significant. LEFM is valid only when the K-dominant zone is large compared to the material's microstructural scale and the plastic zone size.

The displacement field follows a similar structure. For Mode I, the crack opening displacement near the tip is:

u_y(r, θ=π) = (K_I / (2G)) · √(r / (2π)) · (κ + 1)

where G is the shear modulus and κ = 3 − 4ν for plane strain, or (3 − ν)/(1 + ν) for plane stress. This expression is used directly in crack closure and compliance-based crack length measurement techniques.

5. Governing Equations and Validity Limits

The governing equation of LEFM is the relationship between K, the applied stress, and the crack geometry. For Mode I:

K_I = Y · σ · √(π · a)

Fracture occurs when K_I reaches a critical value K_Ic, the plane-strain fracture toughness — a material property measured per ASTM E399. The corresponding failure criterion is:

K_I ≥ K_Ic  →  crack propagation

For fatigue crack growth, the governing equation is the Paris law:

da/dN = C · (ΔK)^m

where ΔK = K_max − K_min is the stress intensity range, and C and m are material constants obtained from da/dN testing per ASTM E647. This equation is the workhorse of damage tolerance analysis and is developed further in later chapters.

6. Worked Example: Through-Crack in a Finite Plate

Consider a 2024-T3 aluminum panel, width W = 200 mm, thickness B = 5 mm, with a central through-crack of total length 2a = 20 mm. The panel carries a remote tensile stress σ = 120 MPa. The yield strength is σ_ys = 345 MPa, and K_Ic = 34 MPa·√m.

Step 1 — Geometry factor. For a central crack in a finite-width plate, the Feddersen correction gives:

Y = √(sec(π·a/W))

With a = 10 mm and W = 200 mm: π·a/W = 0.1571 rad, sec(0.1571) = 1.0125, so Y = 1.0062.

Step 2 — Stress intensity factor.

K_I = 1.0062 × 120 MPa × √(π × 0.010 m)
    = 1.0062 × 120 × 0.1772
    = 21.4 MPa·√m

Step 3 — Validity check. The required minimum dimension is:

2.5 × (K_Ic / σ_ys)² = 2.5 × (34 / 345)² = 0.0243 m = 24.3 mm

The crack length a = 10 mm is less than 24.3 mm, and the thickness B = 5 mm is also less than 24.3 mm. This means the LEFM validity condition is not satisfied — the plastic zone is too large relative to the crack and thickness. The calculated K_I = 21.4 MPa·√m is below K_Ic = 34 MPa·√m, but the result should be treated with caution. In practice, one would either increase the specimen dimensions, use a thicker section, or switch to an EPFM analysis.

This example illustrates a critical engineering point: LEFM gives a number, but that number is only meaningful within its validity envelope. Always check the small-scale yielding condition before trusting a K value.

7. Summary of Key Relationships

The stress intensity factor is the single most important quantity in LEFM. It converts a complex boundary-value problem into a single scalar that can be compared against a material property. The crack tip fields provide the spatial context that justifies this reduction, and the governing equations — K = Yσ√(πa), K ≥ K_Ic, and da/dN = C(ΔK)^m — form the analytical toolkit for both fracture and fatigue analysis. The next step is to apply these tools to real geometries and loading histories, which requires careful attention to geometry factors, validity limits, and unit consistency.