Concept

Fracture energy — where it appears

The work needed to open a unit area of crack in a material, measured in newtons per millimetre. Divided into the modulus over the square of the tensile strength it gives a characteristic length, which separates the sizes at which a member behaves plastically from those at which it behaves as a fracture problem.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

A strength that is a property of the specimen. Nominal strength against size for geometrically similar specimens of one material. On the left the specimen is too small for a crack to run and the strength is a plateau — a plastic limit, and the regime laboratory specimens sit in. On the right a crack releases more energy than it consumes as soon as it starts and the strength falls as the inverse square root of size, which is the regime real structures sit in. The turn happens at D₀ = 120 mm. A 100 mm specimen reads 3.10 N/mm² and a 1500 mm member of the same material carries 1.14: the test overestimates the structure by a factor of 2.71.

The bigger one is the weaker one

Two geometrically similar beams of the same concrete should fail at the same nominal stress, because a strength is supposed to be a material property. They do not. The large one fails at less, and the reason is that a crack releases energy in proportion to a volume and consumes it in proportion to an area.

materials · Size effect $The strength that does not keep pace. Concrete's mean tensile strength against its characteristic compressive strength, with the ratio of the two on the same axis, scaled. The tensile strength goes as f_ck^⅔, so it rises from 1.57 to 5.04 N/mm² over a sixfold rise in the compressive strength, and the ratio between them falls from 11.1% at C20 to 6.0% at C80 — a factor of 1.83. Nothing in a bending or a column calculation ever uses the lower curve, and everything that decides a transition does: when the section cracks, how much minimum steel it needs, how far a bar has to be lapped, what a member without links carries. So a stronger concrete needs more minimum reinforcement, longer laps and a bigger crack-control check, in a member whose ultimate capacity has barely moved. Its real scale is a length: E·G_F/f_ct² is 299 mm here, which is the size at which a member stops behaving plastically and starts behaving like a fracture problem.$

The strength that is never used

Concrete's tensile strength appears in no bending calculation, no column calculation and no shear calculation with links in it. The whole design philosophy is that it cracks and the steel takes over. And it decides where nearly every transition in the subject sits — when a section cracks, how much minimum steel it needs, how far a bar has to be lapped, what a member without links can carry, and how wide a crack opens.

materials · Tensile strength

A crack starts at the corner and runs along the grain. The end of a timber beam 400 mm deep and 140 mm wide, notched on its underside so that 200 mm stands over the support, with the notch's corner 50 mm from the line of the reaction, drawn to scale. A crack starts at the corner and runs along the grain towards mid-span. With a shear strength of 3.5 N/mm² the un-notched end would carry 130.7 kN; Eurocode 5's factor kᵥ of 0.40 on the reduced depth leaves 25.8 kN, 20% of it.

The notch that costs more than its depth

Cut a timber beam's underside away at the support, so that half its depth sits on the bearing, and a stress check on what is left says the end has lost half its shear strength. The beam says otherwise. A crack starts at the notch's corner and runs along the grain, and whether it runs is a question of energy, not stress — so the answer has a size in it. A 400 mm glulam beam notched to half its depth keeps a quarter of its shear resistance, an 800 mm one keeps 18 per cent, and the deeper the beam the less a notch leaves. Sloping the notch buys much of it back, and moving the corner away from the support costs it again.

sections · Shear flow

Named alongside it

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

Size effect Brittle failure Characteristic length Fracture mechanics Anchorage Anisotropy Bond Characteristic strength Concrete strength Crack Crack width Cracking moment

All concepts