Stress concentration

Why does a scratch far too small to weigh anything cut a steel plate's strength so sharply?

Take a steel plate ten millimetres thick and draw a scalpel across it, leaving a scratch a tenth of a millimetre deep. You have removed perhaps a thousandth of the cross-section, too little to weigh. Yet under a fluctuating load that plate may now crack at the scratch, at a stress the unscratched plate carries indefinitely.

The natural first thought is that strength must be proportional to the material left — which predicts a penalty of one part in a thousand. So either that thought is wrong, or something other than the missing metal is doing the damage.

Think about how load crosses a section. Every bit of force applied at one end must arrive at the other, travelling through whatever material is available. Picture the lines of force as traffic in lanes: put an obstruction in the middle and the traffic does not vanish, it crowds into the lanes beside it. The question is not how much road was lost but how tightly the traffic is squeezed going past — which already tells us the answer depends on the obstruction's shape, not its area.

Inglis worked this out in 1913 for an elliptical hole in a large plate under tension. Let a be the half-length of the ellipse measured across the load, and rho the radius at its tip. The stress at that tip comes out as the remote stress multiplied by

Kt = 1 + 2 times the square root of (a divided by rho).

Read that carefully, because it settles the question. The ratio inside the root is a shape ratio: how deep the flaw is compared with how sharply it ends. Nothing about the plate's width or the removed area appears at all.

Test it on a case we can check. A circular hole is an ellipse with equal axes, so its tip radius equals its half-width and the ratio is one: Kt is 3. A round hole triples the stress at its edge no matter how small it is — a pinhole is as bad as a coin-sized hole, which lost area cannot begin to explain.

Now the scratch. Suppose it is one millimetre deep and, being cut by something sharp, ends in a tip radius of a micrometre. The ratio is a thousand, its square root about thirty-two, so Kt is about sixty-five. The stress at the bottom of that scratch is sixty-five times the stress around it. That is the answer, and it comes out of the geometry alone.

Let us falsify the rival account rather than merely dismissing it. If lost cross-section were the mechanism, making a flaw bigger would always make things worse. But the standard repair for a crack in a plate is to drill a round hole at the crack tip: that removes more material and makes the part stronger, because it replaces a near-zero tip radius with the drill's and drops Kt from enormous to about three. No area-based account can produce that result.

Three refinements, where the honest engineering lives.

First, Kt is an elastic number, and steel is not merely elastic. Load the plate statically and the metal at the notch tip yields, blunting it and shedding load to its neighbours. That is why a ductile plate with a hole in it barely loses static strength: what governs is yielding across the remaining section, not the peak. The notch bites under fatigue, in brittle materials, at low temperatures, and wherever the metal cannot flow.

Second, the notch's severity in fatigue is not the full Kt. The measured strength reduction, conventionally Kf, is smaller by an amount depending on material and notch size, because the peak stress occupies a tiny volume and a crack must grow out of it into material far less stressed. Design practice therefore uses tested notch-sensitivity data rather than the geometric factor.

Third, the reconciliation with the sibling explanations here. Push the tip radius toward zero — a real crack rather than a machined notch — and Inglis gives infinite stress. That is not a discovery but a breakdown: elastic stress at a crack tip is not a usable quantity. Griffith's response was to abandon peak stress for an energy balance, the account behind the fibre-strength explanation in this collection, which yields strength falling as the inverse square root of flaw depth. The division of labour is clean: Kt describes flaws with a measurable tip radius, and the energy criterion takes over once the tip is atomically sharp. Tempered glass sidesteps both, holding its flaws shut under residual compression — it does not reduce Kt, it reduces the stress Kt multiplies.

Now the numbers, softest first. The value 3 for a circular hole is exact given Inglis's assumptions: a truly infinite plate under uniform remote tension. In a plate of finite width it differs with the ratio of hole to width, which is why handbooks print curves rather than a constant — I will not quote a figure. The value 65 is arithmetic on dimensions I chose for illustration, not a measurement of any real scratch. Which exposes the weakest link: the tip radius of an actual scratch is not measurable on a real component, so Kt for a genuine surface defect is not computable. It tells you which direction to move in, not a number for a stress report.

The load-bearing claim is the shape ratio itself: the multiplier is set by the flaw's depth relative to its tip radius, not by the material it removes — which is why polishing, radiusing a corner and stop-drilling a crack all work.

• Why stress gradient explains a small notch being less damaging than its Kt suggests.
• How the stress intensity factor replaces Kt once the tip radius goes to zero.

Every term the collection defines is gathered in the glossary.