Crack Initiation and Crack Damage Stress — Threshold Analysis from Stress–Strain Data

Why the tool gives you several answers

There is no agreed way to pick any of these thresholds, and that is not a gap waiting to be filled — it has been measured. Nicksiar and Martin put six published methods for crack initiation against one set of low-porosity rock tests and found the means agreed to within a few per cent while the scatter *within* each method ran past twenty per cent, with no method statistically preferable to another. A tool that reports one number and calls it the crack initiation stress is therefore hiding the only thing about the measurement that a reader needs to know.

So this page runs every method it can defend from a primary source and prints them together, as a stress and as a percentage of peak, with the citation beside each. Crack closure is picked twice: by the moving-point axial stiffness of Eberhardt and colleagues, and off the crack volumetric strain curve. Crack initiation is picked twice: from the departure of the crack volumetric strain from its elastic plateau, after Martin and Chandler, and by the lateral strain response construction of Nicksiar and Martin, which was designed specifically to remove the user judgement the older strain methods need. Crack damage is the reversal of total volumetric strain. Where they disagree, the disagreement is the result.

The circularity in E and ν, and how it is resolved

Crack volumetric strain is the total volumetric strain minus its elastic part, (1 − 2ν)σ/E — so two of the four thresholds depend on elastic constants, and the elastic constants have to be measured over the linear portion, which is the interval between the other two thresholds. Fitting them over the conventional 40 to 60 per cent of peak instead does not escape the problem, it conceals it: crack initiation sits near 0.42 to 0.47 of peak in uniaxial compression, so the conventional window straddles it in most brittle rocks and reads dilation as elastic response. Eberhardt and colleagues measured exactly this on granite — Poisson's ratio came out 0.31 over the ASTM interval and 0.25 over the crack-closure-to-crack-initiation interval — and showed that a shift of 0.05 in ν moves the picked σci by around forty per cent. It is the largest single sensitivity in the whole analysis.

The way out is that three of the constructions need no elastic constants at all. The volumetric reversal needs only the two strain channels; the axial stiffness pick needs only stress and axial strain; and the lateral strain response needs only the lateral channel and the volumetric reversal. So the tool locates σcd, σcc and a first σci from those, fits E and ν over the interval they bracket, builds the crack volumetric strain, and refines. The fit range actually used is printed in the results table, and you can override it with a fixed range when you are reproducing somebody else's reported modulus.

Noise, and what the figure tells you about it

Every threshold except the peak is a statement about a derivative, and the derivative of a strain-gauge record is mostly noise if you take it naively. Nothing here differentiates raw data: all slopes come from a moving least-squares window whose width is an option and is printed on the figure, both extrema are located by fitting rather than by picking the largest reading, and the two plateau picks refuse to fire on a departure smaller than three standard errors of the window slope. That last rule is why a noisy record produces a *late* crack initiation stress rather than a confident wrong one — and why the shipped sample includes a deliberately worse second specimen next to a clean one, so you can see what degradation looks like before you meet it in your own data.

Common questions

What data do I need?

One row per reading: axial stress in MPa, axial strain, and lateral (radial, circumferential or diametral) strain. Hundreds to a few thousand readings is normal and the traces are decimated for drawing, never for computing. Strain can be in microstrain, percent or dimensionless — the unit is detected from the magnitudes and the figure prints which one it settled on. A `specimen` column lets two or three tests share the table and the figure.

Can I use it without a lateral strain channel?

Partly, and the figure is explicit about the limits. Crack closure by the axial stiffness method and the peak stress both come off the axial record alone. Crack initiation, crack damage and Poisson's ratio all need the specimen's sideways response — volumetric strain is εv = ε₁ + 2ε₃ and there is no way around it — so those cells read as unavailable rather than being estimated.

My lateral strain column is positive. Does that matter?

It matters a great deal, and it is a silent failure rather than a loud one. Rock mechanics takes compression positive, so a specimen bulging sideways has negative lateral strain; a file that records magnitudes instead produces a volumetric strain curve that never reverses, which reads as a rock that never dilated rather than as a data problem. The tool compares the signs of the two channels, negates the lateral one when they agree, and says on the figure that it did. There is an option to state the convention outright.

Which crack initiation method should I quote?

Quote the method as well as the number, whichever you pick — that is the honest answer and it is also what the literature does. If you need a rule of thumb: the crack volumetric strain method is the one most reported historically and the one your reader is most likely to recognise, and the lateral strain response method is the one to prefer when the record is noisy or when you need a repeatable procedure across many specimens, because it involves no judgement about where a curve departs from linear. If the two disagree by more than about ten per cent on a good record, look at the figure before you trust either.

What are the tolerances on the figure, and why are they options?

Both plateau picks are answers to the question "when has this slope reached zero", and a real curve approaches its plateau asymptotically while a real slope has noise on it, so there is no tolerance-free version of that question. The plateau tolerance is stated as a percentage of the curve's own steepest closure or dilation slope, which makes it scale-free, and the stiffness tolerance as a percentage of the plateau stiffness. Both are printed on the figure with the regression window, because a threshold quoted without them is not reproducible.

Where does the sample data come from?

It is synthetic and the page says so. There is no public digitised stress–strain record for the classic tests these methods were written against, and inventing one by hand would be presenting a fabrication as a measurement — which is the exact failure the whole page argues against. Instead the sample is generated from a stated forward model whose four thresholds are inputs rather than results, using elastic constants and a strength taken from published granite tests, and the same generator drives the test suite: thresholds are planted, the picker is run, and the recovery is asserted. Load your own data over it whenever you like.

Which sources are implemented?

Martin & Chandler (1994) for the crack volumetric strain construction and for σcc, σci and σcd read off it; Eberhardt, Stead, Stimpson & Read (1998) for the moving-point axial stiffness crack closure pick; Nicksiar & Martin (2012) for the lateral strain response construction; Nicksiar & Martin (2013) for the σci-to-peak ratios quoted in the notes; and ASTM D7012 §10.3.5 and §10.3.6 for the tangent, secant and straight-line-fit moduli and for Poisson's ratio. All five are cited in the figure footer, so the exported SVG carries its own provenance.

Is this free? Does anything leave my browser?

Free, with no account and no gated options, and nothing is uploaded — the whole analysis runs in this tab. Free exports carry a small watermark, as they do everywhere on this site. A stress–strain record from a client's core is exactly the sort of thing that should not need to pass through anybody else's system to be plotted.