Stress near a crack tip and the stress intensity factor

Close to a sharp crack tip, every stress component grows as one over the square root of the distance to the tip, with one common amplitude: the stress intensity factor. Its square divided by the effective modulus equals the energy release rate. The stress picture and the energy picture therefore give the same growth criterion.

Sandbox Physics · 2026-10-29

One amplitude sets the stress near the tip

Put the origin at the tip of a crack that is opened by tension across its faces. At a small distance from the tip, the stress components all have the same form: an amplitude, divided by the square root of the distance, times a fixed function of the angle. The angular functions are universal. The geometry and the loads enter only through the amplitude.

That amplitude is the stress intensity factor. It has units of stress times the square root of length, so it must contain a length from the specimen, usually the crack length. For a centre crack in a large plate it is the remote stress times the square root of pi times the half-length. In a plate of finite width, a correction factor depending on the ratio of half-length to half-width multiplies it.

\sigma_{yy}(r,\theta=0)=\frac{K_I}{\sqrt{2\pi r}},\qquad K_I=\sigma\sqrt{\pi a}\,F\!\left(\frac{a}{W}\right),\qquad F=\left(1-0.025\lambda^2+0.06\lambda^4\right)\sqrt{\sec\frac{\pi\lambda}{2}}

Stress intensity and energy release rate are one criterion

The near-tip field carries the energy that reaches the tip, so the energy release rate can be computed from the amplitude alone. The result is the square of the stress intensity factor divided by an effective modulus. In plane stress, for thin sheets, that modulus is Young’s modulus. In plane strain, for thick bodies, it is larger by one over one minus Poisson’s ratio squared.

For the centre crack this reproduces chapter one: the square of the stress times pi times the half-length, over the modulus. A critical stress intensity factor and a fracture energy are two ways of stating the same material property, for a crack that grows straight ahead under opening.

G=\frac{K_I^2}{E'},\qquad K_{Ic}=\sqrt{E'G_c},\qquad E'_{\rm stress}=E,\qquad E'_{\rm strain}=\frac{E}{1-\nu^2}

Worked example

Read the stress intensity factor from photoelastic fringes

A photoelastic sheet between crossed circular polarizers shows dark fringes where the principal stress difference is a whole multiple of a material fringe value divided by the thickness. For the near-tip field, the principal stress difference equals the stress intensity factor times the sine of the angle, over the square root of two pi times the distance. Each fringe is therefore a closed loop whose radius is largest straight above the tip.

Measure the largest radius of a loop of known order and solve for the stress intensity factor. The loop radius grows as the square of the stress intensity factor, so doubling the load makes each fringe loop four times larger. In practice a uniform stress parallel to the crack, absent from this one-term field, tilts and stretches the loops; careful measurements fit additional terms. The one-term reading is a first estimate.

\sigma_1-\sigma_2=\frac{K_I\,|\sin\theta|}{\sqrt{2\pi r}}=\frac{N f_\sigma}{t}\quad\Rightarrow\quad K_I=\frac{N f_\sigma}{t}\sqrt{2\pi r_{\max}}

How small the broken region has to be

The square-root field cannot hold all the way to the tip. In a metal the material yields where the field exceeds its yield stress. A first estimate of the size of that region sets the opening stress ahead of the tip, from the first formula of this chapter, equal to the yield stress and solves for the distance. It ignores the load that the yielded material passes on to its surroundings, so it only gives the order of magnitude. Bouchbinder, Fineberg and Marder list two requirements: linear elasticity must hold away from the tip, and all nonlinear processes must stay in a region that is small compared with the crack and the specimen.

If the region is not small, the energy release rate still exists as an energy flow, but a single critical value no longer predicts failure. That situation returns in chapter six, where a strength length comparable to the notch spreads the onset of cracking over an interval of grip openings.

r_p\approx\frac{1}{2\pi}\left(\frac{K_I}{\sigma_Y}\right)^2

An experiment to think through

Predict, change, explain

In the photoelastic stress viewer, choose the plate with a hole. Predict where the fringes crowd most densely around the hole edge.
Raise the stress level and count how the fringes near the hole multiply. Then open the stress–strain tester and read the yield stress of the plastic specimen.
Explain why a hole raises the stress by a finite factor while a sharp crack makes it grow without limit, and use the yield stress to estimate how close to the tip the one-term field still applies.

See stress concentrate around a hole Measure modulus and yield stress Grow a crack and track where the work goes

The fracture Labs run simplified two-dimensional models in scaled units. Each Lab states its model, units and limits in its notes and can export the record behind its readouts.

Try explaining it yourself

1. A thick plate in plane strain and a thin sheet in plane stress have the same stress intensity factor. Which has the larger energy release rate, if Poisson’s ratio is 0.3?

Read an explained answer

The thin sheet. In plane strain the effective modulus is larger by one over 0.91, so the energy release rate is 0.91 times that of the sheet.

2. The load on a photoelastic specimen doubles. How does the largest radius of the third-order fringe loop change?

Read an explained answer

It grows four times, because the loop radius scales as the square of the stress intensity factor.

3. A centre crack has half-length equal to half of the plate half-width. By how much does the finite-width correction raise the stress intensity factor?

Read an explained answer

The ratio is one half. The polynomial factor is about 0.9975 and the square root of the secant of pi over four is about 1.189, so the factor is about 1.19.

F\!\left(\tfrac12\right)=\left(1-0.025\cdot\tfrac14+0.06\cdot\tfrac1{16}\right)\sqrt{\sec\tfrac{\pi}{4}}\approx0.9975\times1.189\approx1.19

Sources to read alongside

Bouchbinder, Fineberg & Marder, Dynamics of Simple Cracks, arXiv:0911.0173; Annual Review of Condensed Matter Physics 1, 371–395 (2010)
Section 1, equations (1) to (3), defines the Griffith point of a rigidly gripped strip and the surface energy per unit area. Section 2 states the energy balance, the inverse-square-root stress near the tip, the relation between energy release rate and stress intensity, the principle of local symmetry and the assumptions of linear elastic fracture mechanics. It also reports the micro-branching onset near 0.4 of the Rayleigh speed in PMMA and glass. Section 3 discusses small scales at the tip, including cohesive zones and phase-field models. The course quotes these statements; it does not use the review’s data.
Sargado, Welch & Lüthje, Approximate formulas for the energy release rate of a crack perpendicular to a material interface, arXiv:2210.06315v1 (2022)
Equations (1) to (3) restate the energy released by a centre crack in a large plate and its energy release rate, with the effective modulus for plane stress and plane strain. Equation (29) quotes the Tada, Paris and Irwin secant correction for a centre-cracked strip. The course uses it as the finite-width reference for the centre-crack benchmark. The 1973 handbook itself was not consulted.
Tanné, Li, Bourdin, Marigo & Maurini, Crack nucleation in variational phase-field models of brittle fracture, Journal of the Mechanics and Physics of Solids 110, 80–99 (2018); preprint hal-01568702
Section 2 gives the AT2 energy (equation 2 with the AT2 choice), the characteristic length and the relation between critical stress intensity factor and fracture energy (equation 5), and the uniform-bar strength (Table 1). Section 3.1, equations (7) to (11), gives the singular field at a V-notch; when the notch closes into a crack, its amplitude is the mode-I stress intensity factor. Section 4 treats the regularization length as a material length through size effects. Benchmark B4 checks the browser model’s uniform-bar strength against Table 1.