Structural Engineering & Statics
where the load goes
Concept

Plastic zone — where it appears

The region of yielded material ahead of a crack tip, of a size proportional to the square of the stress intensity divided by the yield stress. When the load comes off, the surrounding elastic material closes on it and leaves it in compression, which is why an overload slows the growth that follows it rather than hastening it.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

What a crack makes of a notch, against how sharp the notch is. The fatigue notch factor against the root radius, at a fixed elastic factor of 3 in a 430 MPa steel. K_t is a property of the shape and does not move along this axis at all — the dashed line — while the factor fatigue actually feels climbs toward it from below. At a 1 mm root the answer is 2.40, which is 30 per cent of the notch relieved; at 0.1 mm it is 1.38, and a notch that concentrates by 3 elastically is barely felt. Nothing has changed about the stress field: what has changed is that the peak is confined to a smaller volume than the material's own process size, so the crack starts against an average rather than against a maximum.

The notch a crack does not feel in full

The elastic concentration factor is a property of shape and knows nothing about size, which is what makes it so useful and so misleading. A fatigue crack starts against an average over a volume the material owns, so two notches with the same factor and different radii have different fatigue strengths — and the stronger the steel, the less of that relief it gets.

materials · Stress concentration The same cycles, and two different lives. Two sequences made of exactly the same cycles — blocks of 6.00M at 40 N/mm² alternating with blocks of 0.22M at 90, on a detail starting with a 0.50 mm flaw. Taking the small cycles first, the crack reaches its critical 87.7 mm after 12.36M cycles; taking the large ones first, after 6.36M. Miner's sum at the moment of failure is 0.99 for the first and 0.68 for the second, so a rule that predicts failure at a sum of one is right to within a per cent about the first and 48 per cent unconservative about the second. The mechanism is on the axes: ΔK rises with the crack, so a large block met late finds a longer crack and does more with it.

The record played backwards

Miner's rule adds damage, and a sum has no order. A crack does, twice over: a large block met late finds a longer crack and does more with it, and an overload leaves a plastic zone that slows everything after it. The same cycles rearranged fail at 12.4 million or at 6.4, and a Miner sum that is right to one per cent about the first is out by half about the second. One cycle in four million can add fifty-four per cent to a life.

materials · Fatigue Between the two lines, a curve below both. The stress at which a plate of grade 355 steel of toughness 100 MPa√m, with an edge-crack factor of 1.12 fails, against the length of the crack in it on a logarithmic scale. Dashed: the fracture-mechanics line, the toughness over Y√(πa), and the yield stress; the usual check takes whichever is lower. Solid: Dugdale's strip-yield model, in which a thin strip ahead of the crack yields and the two mechanisms act together. The two dashed lines cross at the transition length, 20.1 mm, and there the strip-yield stress is 288 N/mm², 0.81 of both: the lower of the two simple answers overstates the plate by 23 per cent. At 20 mm the strip-yield stress is 289 N/mm² against 355 N/mm². Far from the crossing on either side the curve joins the line it approaches.

The crack between the two checks

A cracked plate is checked twice: once for fracture, as if the steel could not yield, and once for yielding, as if the crack could not grow, and whichever answer is lower is taken. Each check is right far from the other. Where they cross — at a crack of twenty millimetres in an ordinary structural steel — the plate fails at four fifths of both, because a strip of yielded steel ahead of the crack is by then two and a half times as long as the crack itself, and neither check has a place for it.

materials · Fracture One steel, one crack, three plates. The strip-yield failure stress of plates with an edge crack, at −20 °C in a grade 355 steel whose master-curve reference temperature is −38 °C, against the crack length on a logarithmic scale, for plates 12, 25 and 60 mm thick, each with the toughness of its own crack front at the 5 per cent level: 88, 77 and 66 MPa√m. At a 20 mm crack the 12 mm plate fails at 267 N/mm², the 25 mm plate at 242 and the 60 mm plate at 214. Dashed, the simple check with the 25 mm toughness, the lower of fracture and yield: 274 N/mm² at the same crack, for every thickness.

The toughness that belongs to the plate

A steel's cleavage toughness is measured on a specimen 25 mm thick, and a crack through a plate starts at the weakest spot its front passes through — so a crack through a 60 mm flange samples more weak spots than the test did, and is less tough, and one through a 12 mm web samples fewer. Put each plate's own toughness into the strip-yield assessment and the simple check, which overstates a cracked plate by 23 per cent at the specimen's thickness, overstates a 60 mm flange by 36 per cent and a 100 mm plate by 45, while a thin web pays much of the strip-yield debt back.

materials · Fracture

Named alongside it

The objects these essays reach for when they reach for this one.

Size effectBrittle fractureCritical crackFatigueFractureFracture toughnessStress intensityCrack growthCrack initiationDamageDetail categoryDuctility

All concepts