4. Crack tip plasticity

Linear Elastic Fracture Mechanics (LEFM) treats the material ahead of a crack as a perfectly elastic continuum, so the stress at the crack tip is predicted to rise as 1/√r and become infinite as r → 0. No real metal behaves this way. Above the yield strength, deformation becomes plastic, stresses are redistributed, and the singular elastic field is truncated. This chapter quantifies that truncation with two classical models — Irwin's and Dugdale's — and then establishes the conditions under which LEFM remains a valid engineering tool.

4.1 Why the Elastic Singularity Must Be Truncated

For a Mode I crack, the near-tip stress normal to the crack plane is

σ_yy(r) = K_I / √(2πr)          (LEFM, r → 0)

Setting σ_yy = σ_ys (the uniaxial yield strength) gives a first, purely formal estimate of the plastic zone extent along the crack plane:

r_y = (1 / 2π) · (K_I / σ_ys)²

This "first-order" estimate is too small, because it ignores the fact that yielding redistributes load to the surrounding elastic material. Two corrections dominate engineering practice.

4.2 Irwin's Model: Effective Crack Length and the Factor of Two

Irwin's insight is that the plastic zone behaves like an extension of the crack. The material inside the plastic zone cannot carry the singular stress, so the elastic field is pushed outward. Irwin models this by replacing the physical crack length a with an effective crack length:

a_eff = a + r_y
K_I,eff = Y · σ · √(π · a_eff)

where Y is the geometry factor. Enforcing self-consistency — the plastic zone size must be computed from the effective stress intensity — yields the Irwin plastic zone radius:

Plane stress:  r_p = (1 / 2π) · (K_I / σ_ys)²
Plane strain:  r_p = (1 / 6π) · (K_I / σ_ys)²

The plane-strain value is roughly one-third of the plane-stress value because the through-thickness constraint raises the effective yield resistance to approximately √3 · σ_ys. This is why thin sheets and thick plates of the same alloy exhibit very different apparent toughness.

4.3 Dugdale's Model: The Strip-Yield Cohesive Zone

Dugdale (and Barenblatt) proposed a different idealisation for thin sheets: a narrow strip of yielded material of length ρ ahead of the crack tip, carrying a constant closure stress equal to σ_ys. The strip is treated as a cohesive zone that shields the crack tip. The resulting plastic zone length is

ρ = (π / 8) · (K_I / σ_ys)²

Comparing the two models at the same K_I:

The Dugdale estimate is about 2.5× the Irwin plane-stress value because the strip geometry concentrates the yielded material in a single plane rather than distributing it over a circular region. Neither model is "correct"; each is a consistent idealisation matched to a different constraint state.

4.4 The LEFM Validity Envelope

LEFM is acceptable only when the plastic zone is small relative to the crack length, the ligament, and the specimen thickness. The standard screening criteria are:

• Crack length: a ≥ 2.5 (K_IC / σ_ys)² (ASTM E399).
• Ligament: (W − a) ≥ 2.5 (K_IC / σ_ys)².
• Thickness: B ≥ 2.5 (K_IC / σ_ys)² to ensure plane-strain conditions.

When these are violated, the measured toughness is no longer a material constant — it becomes thickness- and geometry-dependent, and the analyst must move to elastic-plastic fracture mechanics (EPFM) with J-integral or CTOD parameters.

4.5 Visualising the Plastic Zone and Its Consequences

The figure below shows the crack-tip region with the Irwin circular plastic zone, the Dugdale strip-yield zone, and the effective crack tip position. It also indicates how the stress distribution is truncated at σ_ys and redistributed into the surrounding elastic field.

4.6 Practical Screening Workflow

Before trusting a K-based calculation, run this check:

1. Compute K_I from the applied stress, crack length, and geometry factor.
2. Estimate r_p using the Irwin plane-strain formula (conservative for thick sections).
3. Verify a, W − a, and B each exceed 2.5 (K_IC / σ_ys)².
4. If any criterion fails, switch to EPFM: compute J or CTOD, or use a cohesive-zone model.
5. For thin sheets, use the Dugdale strip-yield estimate to size the plastic zone and check net-section yielding.

4.7 Limitations of LEFM — A Consolidated View

The Irwin and Dugdale models are corrections within LEFM, not replacements for it. Their limitations define the boundary of the theory:

• Small-scale yielding (SSY) required. The plastic zone must be embedded in the elastic singularity field. When r_p approaches a, the K-dominance is lost.
• No stable crack growth prediction. LEFM gives initiation, not the rising R-curve behaviour of ductile metals.
• No unloading or cyclic plasticity. Reverse yielding under fatigue is outside the monotonic assumptions of both models.
• Constraint dependence. Plane stress and plane strain give different r_p and different apparent toughness; a single K_IC is not geometry-independent unless the thickness criterion is met.
• No large-deformation or necking effects. Thin ligaments that neck before fracture require finite-strain treatment.

In practice, the Irwin correction is applied iteratively: compute K, update a_eff, recompute K, and repeat until convergence. For most structural alloys under SSY, two or three iterations suffice. When convergence is slow or the correction exceeds 10%, the problem has left the LEFM domain and an EPFM analysis is mandatory.

The next chapter introduces the J-integral, which extends fracture mechanics into the elastic-plastic regime where the Irwin and Dugdale corrections are no longer sufficient.