Find the energy release rate from a compliance curve

Measure the compliance, the displacement per unit load, at several crack lengths. The energy release rate equals half the load squared times the slope of compliance with crack length, divided by the thickness. The same formula holds whether the load or the grip displacement is held fixed while the crack advances.

Sandbox Physics · 2026-10-29

A longer crack makes a softer spring

Pull a cracked specimen through its grips and record the load and the grip displacement. In the linear range their ratio is the compliance. A longer crack lets the specimen open more for the same load, so compliance rises with crack length. The stored elastic energy is half the load times the displacement, which is half the load squared times the compliance.

C(a)=\frac{\Delta}{P},\qquad U=\tfrac12P\Delta=\tfrac12P^2C(a)

Two loading routes, one energy release rate

Hold the load fixed and let the crack grow a little. The grips move apart because compliance rose, and the load does work equal to the load times that extra displacement. The stored energy rises by only half of that work. The other half goes to the crack tip.

Now hold the grips fixed instead. No external work is done. The load drops because the specimen became softer, and the stored energy falls. To first order in the small crack advance, the drop equals the amount released under fixed load. Either way, divide by the new crack area to get the energy release rate.

The result needs only the compliance curve. That is why compliance methods are common in fracture tests: they replace the stress field near the tip by a global stiffness measurement.

G=\frac{P^2}{2B}\frac{dC}{da}

Worked example

Work out a double cantilever beam

Split a beam of thickness B and arm height h along its mid-plane to a depth equal to the crack length. Each arm is a cantilever of that length. Simple beam theory, which neglects shear and the rotation at the arm root, gives the total opening per unit load. The compliance grows as the cube of the crack length.

Differentiate and insert into the compliance formula. At fixed load the energy release rate grows as the square of the crack length. Doubling the arm height at the same load cuts it by a factor of eight. Chapter three uses the fixed-grip form of the same result to show why this specimen tears stably.

C=\frac{8a^3}{EBh^3},\qquad G=\frac{P^2}{2B}\cdot\frac{24a^2}{EBh^3}=\frac{12P^2a^2}{EB^2h^3}

How the Lab takes the slope and checks it

The Griffith crack growth Lab solves each specimen once on a finite-element mesh, with the crack plane as a row of nodes that can open. It computes the compliance for every tip position and takes a central difference in crack length. Plates with a central crack are loaded by a uniform stress on the far edge; strips and edge notches are held by rigid grips.

Two checks bound the error. A local estimate uses only the force needed to hold the tip node shut and the opening one element behind it. It agrees with the compliance slope to within 0.32 percent. For a central crack in a plate of finite width, the slope approaches the secant formula from the stress-analysis handbook as the mesh is refined. When the crack spans half the half-width, the ratio goes from 0.976 to 0.987 to 0.994 with 32, 64 and 128 elements across the half-width. The handbook formula is taken as quoted in a 2022 paper by Sargado, Welch and Lüthje.

G_{\rm local}\approx\frac{R_{\rm tip}\,v_{\rm behind}}{\Delta a},\qquad G_{\rm ref}=\frac{\pi\sigma^2a}{E}\left(1-0.025\lambda^2+0.06\lambda^4\right)^2\sec\frac{\pi\lambda}{2},\quad \lambda=\frac{a}{W}

An experiment to think through

Predict, change, explain

Open the gripped-strip preset in the Griffith crack growth Lab. Predict the shape of the energy release rate curve as the crack runs along the strip.
Switch to the plate preset and compare the shape of the same curve.
Explain the flat curve of the strip and the rising curve of the plate with the slope of compliance and the loading route.

Grow a crack and track where the work goes Measure modulus and yield stress

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. Under fixed grips, does the stored elastic energy rise or fall as the crack grows?

Read an explained answer

It falls. The grips do no work, so all of the energy that reaches the tip comes out of the stored elastic energy.

2. In a double cantilever beam at fixed load, the crack length doubles. How does the energy release rate change?

Read an explained answer

It grows as the square of the crack length, so it becomes four times larger.

3. A compliance curve is measured with an error that is the same constant offset at every crack length. Does that offset change the energy release rate?

Read an explained answer

No. A constant offset drops out of the slope. Errors that change with crack length, such as grip slip that grows with load, do enter.

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.