Fracture Year 1 BaDI John L. Errington MSc

How things break For a structure/material to perform a useful function, it must remain intact (resist fracture) We need to know how materials fracture in order to design adequately We can learn a lot from engineering failures

The study of fracture Historically an empirical science Early 1800s — theory of structural elasticity developed by French mathematicians; ignored by British and American engineers Structures designed according to early theory often fell down

“…when we analyse a situation mathematically, we are really making for ourselves an artificial working model of the thing we want to examine. We hope that this algebraical analogue or model will perform in a way which resembles the real thing sufficiently closely to widen our understanding and enable us to make useful predictions.” J.E. Gordon, Structures, London 1978

“many of the real situations which are apt to arise are so complicated that they cannot be fully represented by one mathematical model. With structures there are often several alternative possible modes of failure. Naturally the structure breaks in whichever of these ways turns out to be the weakest — which is too often the one which nobody had happened to think of, let alone do sums about.” J.E. Gordon, Structures, London 1978

The study of fracture After 1850 — even British and American engineers began to calculate the highest probable tensile stresses in large structures such as bridges Designed for “safety factor” (more descriptively, “factor of ignorance”) of 3-4 Accidents still common; many ascribed to “defective material”

A goad to understanding 1901 — HMS Cobra, a brand-new destroyer, suddenly broke in two and sank in the North Sea in fairly ordinary weather. Why? 1903 — A number of hull strain measurements were made on HMS Wolf, at sea in rough weather. Stresses calculated from the strains were far below the strength of the steel hull (safety factor 5-6).

The first major step toward understanding fracture 1913 — C.E. Inglis, a professor of engineering at Cambridge, published a theoretical paper showing that the approach of the engineers and designers of the day was all wrong Up to that time, designers had plotted the probable distribution of stresses broadly over the whole structure

The first major step toward understanding fracture That approach only works if the material and structure have smooth surfaces and no sudden changes of shape Inglis showed that geometrical irregularities (holes, cracks, sharp corners) may dramatically raise the local stress — over a very small area — higher than the breaking stress of the material, even when the general stress in the surrounding material is low

Why stress is concentrated by a flaw

What is the stress at the tip of the crack? Stress = s Length of crack = L Radius of tip of crack = r Stress at tip of crack:

For round holes (portholes), r = L, so stress around them will be 3s (still within safety factor) For doors/hatchways with sharp corners, r is small while L is large On HMS Wolf, none of the extensometers were placed near hatchways

The sharper the tip of the crack, the higher the stress around it One can decrease the stress if one can blunt the crack by having it end in a round hole

If we could make r sufficiently small, we could get local stresses higher than the fracture strength of any material, not matter how small the crack. So why aren’t structures falling down all over the place?

More on strain energy and fracture

The sixty-four thousand dollar question Whether a structure actually breaks at any particular juncture depends on whether or not it is possible for the strain energy to be converted into fracture energy so as to create a new crack

Tensile fracture depends chiefly on: The price in terms of energy which has to be paid in order to create a new crack The amount of strain energy which is likely to become available to pay this price The size and shape of the worst hole or crack or defect in the structure

Fracture energy When a solid is broken in tension, at least one crack must spread completely across the material, so as to divide it into two parts Therefore, two new surfaces that did not exist before must be created New surfaces

Fracture energy In order to create the new surfaces, all the chemical bonds that previously held them together must have been broken How much energy is required to break chemical bonds? For structural materials, the energy required to break all the bonds in any one plane is about 1 Joule/m 2

Fracture energy 1 Joule/m 2 is really a pathetically small amount of energy For a brittle solid, this is about all of the energy we have to supply to break it (therefore we don’t use brittle materials in tensile applications!)

What makes a material “tough”? Tough materials have a high work of fracture — they require much more energy in order to produce new fracture surface How much more, and why?

Approximate tensile strengths and work of fracture of some solids

What makes a material brittle vs tough? In a brittle solid, the work done during fracture is virtually confined to that needed to break the chemical bonds at, or very near, the new fracture surface (about 1 J/m 2 ) In a tough material, its fine structure is disturbed to a much greater extent (a depth of up to 50 million atoms below the visible fracture surface)

The dislocation mechanism The best understood mechanism for the disruption of fine structure occurs in metals As metals are pulled in tension, they depart from Hookean behavior at moderate stresses, and deform plastically after that point (the material stretches out before it breaks — “necking”)

What is the dislocation mechanism? The layers of atoms slide over one another like a deck of cards This sliding absorbs a great deal of energy

Biological materials and work of fracture Biological materials do not make use of the dislocation mechanism Biological materials and other composites rely on the interfaces between different materials to increase the work of fracture

How do you tell whether something will fracture or not?! A.A. Griffith published a theory in 1920 that explained how we are able to live with material defects and stress concentrations Griffith was the first engineer to approach fracture in terms of energy, rather than stress and strain

Griffith’s theory From an energy point of view, Inglis’s stress concentration is just a mechanism for converting strain energy into fracture energy, just as an electric motor is a mechanism for converting electrical energy into mechanical work Neither will work unless it is continually supplied with the right kind of energy

Example: piece of elastic material, stretched and then clamped at both ends UnstrainedClamped

What is the energy bill to propagate a crack? If the material is 1 unit thick, then Energy debt = WL Where W = work of fracture L = length of crack Energy debt is  L

Where is the energy coming from? Since it is a closed system, the energy has to come from relaxation of strain energy somewhere else in the material The crack gapes a little under stress and relaxes the material immediately behind the crack surfaces

Where is the energy coming from? The two shaded triangular areas give up strain energy Whatever the length of the crack (L), the triangles will keep roughly the same proportions, so their areas will increase as L 2 Strain energy release  L 2

The core of Griffith’s theory The energy debt of a crack increases as L, while its energy credit increases as L 2

Griffith energy release L g = critical Griffith crack length

The consequence Even if the local stress at the crack tip is very high even if it is much higher than the “official” tensile strength of the material the structure is still safe and will not break so long as no crack or other opening is longer than the critical length L g

Calculating L g

The length of a safe crack depends on the ratio of the work of fracture of the material to that of the strain energy Resilient materials (e.g., rubber) therefore tend to have very short L g s (why a balloon goes pop)

Biological/scaling consequences The critical crack length of bone is an absolute, not relative distance (the same for an elephant as for a mouse) The strength and stiffness of bone are the basically the same in all animals The brittleness of bone limits the structural risks that a large animal can accept The larger the animal, the more careful it has to be to avoid putting large stresses on its bones

Composite materials revisited How to design a composite so that it is resistant to fracture: Make the thickness of any of the stiff components much smaller than L g ! In that way they cannot develop a long enough crack for fracture to propagate. Stiff elements