Does Compression Grow Cracks?

Stress intensity conventions

At a stress ratio of R = −1, the choice between full-range and clipped ΔK is worth a factor of 2ⁿ in life. It is usually the largest single lever in a spectrum fatigue deck, and the conservatism runs the opposite direction from what most people assume.

A random vibration crack growth analysis comes back short. Somewhere in the input deck sits a decision most analysts make once, early, and never revisit: what stress intensity range do you hand to the Paris law when the load cycle swings negative?

At a stress ratio of R = −1, the cycle spends half its time in compression. Two conventions are in common use, and they differ by a factor of two in the driving force. At a Paris exponent of three, a factor of two in the range is a factor of eight in the growth rate. That single choice will swamp nearly everything else in the analysis.

This post works through the physics, the arithmetic, and — the part that trips people up — which direction the conservatism actually runs.

The two conventions

Write the stress intensity range in the usual way:

with

where $Y$ is the geometry factor, $\sigma$ the remote stress, and $a$ the crack length.

Convention 1 — count the full range. At R = −1, $\sigma_{min} = -\sigma_{max}$, so $K_{min} = -K_{max}$ and

Convention 2 — clip the compression. Set the negative excursion to zero. The cycle becomes R = 0 and

Same physical load history. Two answers a factor of two apart. Everything below is about which one to use and why.

The physical argument for clipping

The case for discarding the compressive half is straightforward and mostly correct: a crack with its faces in contact cannot transmit tensile stress across the tip. Once the remote load goes compressive, the faces close, load transfers through contact rather than around the tip, and the stress intensity stops accumulating. No crack tip singularity, no driving force, no growth.

If that were the whole story, clipping would be simply correct and there would be nothing to discuss.

It isn’t the whole story, because cracks do not open and close at zero load.

Elber’s discovery and the opening stress

In 1970, Elber observed that fatigue cracks in aluminum remained closed for part of the tensile portion of the cycle. The plastic wake left behind by the advancing tip — material stretched at the tip, then unloaded into the wake — props the faces apart. The crack does not open until the applied stress exceeds an opening stress $\sigma_{op}$ that is generally above zero.

The consequence is that only part of the nominal range is effective:

with $K_{op}$ corresponding to $\sigma_{op}$. Elber’s ratio is

and the Paris law is written against the effective range rather than the nominal one:

Newman later gave a widely used analytical form for $\sigma_{op}/\sigma_{max}$ as a function of R, the constraint factor $\alpha$ (roughly 1 for plane stress, 3 for plane strain), and the ratio $\sigma_{max}/\sigma_0$ where $\sigma_0$ is the flow stress. This is the closure model embedded in NASGRO’s crack growth equation, and it is the reason the code asks for those inputs.

The key point for our purposes: closure is a middle position. Full-range $\Delta K$ credits nothing to closure. Clipping credits exactly the compressive half. Real closure typically credits somewhat more than clipping, because $\sigma_{op}$ is usually positive.

Working the arithmetic

Take a center-cracked plate under remote tension, $Y \approx 1$, in an aluminum alloy with representative Paris constants. Let

• $C = 1 \times 10^{-9}$ (in/cycle, ksi·in^1/2 units)
• $n = 3.0$
• $\sigma_{max} = 20$ ksi, fully reversed
• $a_i = 0.05$ in, $a_c = 0.50$ in

Separate and integrate the Paris law:

For n = 3 the integral evaluates in closed form:

Full range, $\Delta\sigma = 40$ ksi:

Clipped, $\Delta\sigma = 20$ ksi:

The ratio is exactly $2^n = 8$, as it must be — the geometry integral is identical and only the range scales. The convention alone is worth a factor of eight in life.

For n = 3.5, a common value for aluminum in the mid-Paris regime, the factor is $2^{3.5} = 11.3$. For steels with n near 3, it stays close to 8. For some titanium alloys with n approaching 4, it reaches 16.

This is why the convention deserves more scrutiny than it usually gets. It is not a second-order modeling detail. In most spectrum fatigue analyses it is the single largest lever in the deck.

Where the clipping assumption breaks down

Clipping is defensible in many situations. It is not universally safe, and the failure modes are worth knowing because they cluster in exactly the structural details where fatigue cracks actually initiate.

Compressive overloads re-sharpen the tip

A large compressive excursion flattens the plastic wake and drives $\sigma_{op}$ down, sometimes below zero. Subsequent cycles then see a larger effective range than the steady-state closure model predicts. This effect runs opposite in sign to the clipping assumption: compression, having been assumed harmless, has made the following cycles more damaging.

Thin sections and low constraint

Newman’s constraint factor $\alpha$ approaches 1 under plane stress. Closure develops less, $\sigma_{op}$ drops, and the crack stays open further into the compressive excursion than a clipped model credits. Thin-walled structure is the usual home for random vibration problems.

Tensile residual stresses

This is the important one for joints. A tensile residual field — from cold working gone wrong, from welding, from machining, from interference-fit fasteners, from bearing at a lug hole — superimposes on the remote load. The local stress at the crack tip can remain tensile while the remote load is compressive. The crack never closes, and the full range is driving growth even though the nominal cycle looks half-harmless.

Bolted joints, lug holes, and fastener rows are where remote-load reasoning is least reliable. They are also, not coincidentally, where the cracks are.

Short cracks

Closure develops with the plastic wake, which by definition does not exist yet at a freshly initiated crack. Short cracks grow faster than long-crack data predicts, partly for this reason. If a meaningful fraction of your life is spent at small crack sizes, the closure credit you are assuming is not yet earned.

The sequence problem

Suppose you decide to do this properly and model closure explicitly rather than clipping. NASGRO will let you. This is where spectrum fatigue gets difficult, and it is worth understanding why before committing.

Closure is history-dependent. The opening stress $\sigma_{op}$ is not a property of the current cycle. It is set by the plastic wake, which is set by the largest prior overload and how recently it occurred. Turning on closure means the analysis is no longer a function of the cycle content of the spectrum — it is a function of the cycle order.

Three consequences follow immediately.

First, rainflow output is order-scrambled by construction. Rainflow counting extracts closed hysteresis loops from a stress history and returns a table of ranges and means. The extraction deliberately discards sequence. Feeding that table to a sequence-dependent damage model requires you to invent an ordering, and there is no unique correct choice. Low-high, high-low, and random orderings can differ substantially in predicted life — and the spread widens as the spectrum’s dynamic range increases.

Second, you have signed up for retardation modeling. Once the plastic wake is being tracked, an overload followed by smaller cycles produces retardation: the enlarged plastic zone elevates $\sigma_{op}$ and the following cycles grow the crack more slowly than their range alone would suggest. This is a real and often large effect. Modeling it means selecting Willenborg, Wheeler, or a strip-yield model, and each carries fitting parameters — the Wheeler exponent, the Willenborg shutoff ratio — that need justification against test data you may not have.

Third, block spectra stop being coherent. A two-block construction — a high-level block followed by a low-level block — is a perfectly reasonable idealization for a cycle-content damage model. Under a sequence-dependent model it is something else entirely: all the retardation lands at one interface, and reversing the block order changes the answer. The idealization that made the spectrum tractable is precisely what the closure model is sensitive to.

None of this makes closure modeling wrong. It makes it a substantial commitment, and it explains why many experienced analysts decline it in spectrum work and reach for clipping instead. That is a defensible position, arrived at for good reasons.

But notice what has happened. Clipping was adopted to avoid the difficulty of the sequence problem. Its correctness as a physical assumption was not what recommended it. Those are different justifications, and it is worth being honest about which one is operating.

Which way does the conservatism run?

Here is the part that gets stated backwards more often than any other point in this subject.

Set the three positions side by side:

• Full range — most conservative. Requires no justification. If you pass with it, you are done and no reviewer will ask a second question.
• Clipped — a relaxation relative to full range, worth a factor of $2^n$. Physically better motivated, but the motivation has to be argued, and the argument must address residual stress at the critical location.
• Explicit closure — a further relaxation. Most physically accurate, most defensible in principle, and the hardest to execute because of everything in the previous section.

The practical consequence: do not lead with the compression convention when an analysis comes back short.

Work the levers that cost nothing first. Allowances the specification already grants. Bookkeeping errors in exposure duration or cycle counts. Fidelity improvements that happen to run favorably — using an actual rainflow R distribution rather than forcing every cycle to R = −1, for instance, which is a better representation of the physics and reduces damage as a side effect rather than as its purpose.

Only after those are exhausted should the compression convention come into play, and then with a written basis that addresses the residual stress state at the location that governs.

There is a reason for the ordering beyond mere caution. A relaxation you spent and did not need is a liability. If the analysis clears the requirement with clipping applied, and a reviewer later asks what the answer looks like on full range, you want to be able to say we checked, and we pass either way. If you cannot say that, you have converted a technical result into a defense of a modeling assumption — and you will be defending it under conditions less favorable than the ones in which you chose it.

The question to ask your deck

Most of the confusion in practice comes from not knowing which convention is actually running. Before debating what it should be, establish what it is.

1. How is $\Delta K$ computed for a negative-R cycle? Full peak-to-valley, or clipped to the tensile portion?
2. Is closure enabled? If so, which model, and what constraint factor?
3. What convention was the material data fit under? This is the one that silently ruins otherwise careful work. If the $da/dN$ curve was regressed from R = −1 test data reduced on a full-range basis, then applying clipped $\Delta K$ against that curve credits the closure benefit twice — once in the data reduction, once in the analysis. NASGRO material files vary in this respect. Check before changing anything.
4. If closure is on, what ordering was assumed, and how sensitive is the answer to it?

Question 3 deserves particular emphasis. It is entirely possible to make a change that is correct in isolation and wrong in combination with the material data, and the error is invisible in the output.