## Fracture Mechanics II | |||||||||

Flashback and new Goals | |||||||||

If you have read through the backbone of this chapter, you are now a better person
because you know something about plastic deformation, dislocations, and all the other defects making crystals more colorful
in both meanings of the word. That's good because now I can continue the "Fracture Mechanics" stuff I have started
when you were an ignorant....(insert you job title or whatever else is right). Let's quickly review the essentials of the first Fracture Mechanics module. The figure below illustrates the basic ideas: | |||||||||

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The basic points to recall are:- The rapid growth of pre-existing small "
*microcracks*", characterized by their half-widthand tip radius**d**, leads to fracture in**r***brittle*materials. - On a macroscopic level, the
*fracture criterion*("Griffith criterion") can be derived from an energy consideration. Fracture in uniaxially strained materials occurs at some critical stress**s**if more strain energy is released than surface energy generated. The basic equation for this was (with_{crit}= Young's modulus,*Y***g**= specific surface energy):
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- The criterion is purely
*energetical*and does not take into account the crack tip geometry and the stress concentration there. - The criterion is only applicable to perfectly
*brittle*materials under uniaxial stress with cracks oriented perpendicular to the stress direction.
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In other words: While Griffith's work was seminal in the sense that it produced
major insights into fracture science , it is rather limited and does not allow fracture
engineering. One of the reasons is that even for perfectly brittle materials, the surface
energy g is not really a good parameter. It is not all that well defined for real materials
that have never a clean surface but always a "dirty" one, covered with oxide and God knows what else.We thus need to modify the simple approach, considering in particular ductility and more complex stress situations than just uniaxal deformation. | |||||||||

Fracture and Ductility | |||||||||||||||||||||||||||||

The first thing to do is to rewrite Griffith's criterion to make it more general. We do that by defining two new parameters as follows: | |||||||||||||||||||||||||||||

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Formally, the new quantity "critical stress intensity factor"
would be equal to K_{ Ic}(2 . This factor relates to crack
growth by giving the energy needed via Y · g)^{½}g
, and the energy gained via . Let's forget that now and accept that Y somehow
takes care of K_{ Ic}all energies involved instead of just the surface energy g
and simple strain energy.Just as and Yg are material "constants",
somehow encoding a major property of a given material, so is . It's just no longer simply the
product of the two old parameters but something new that we K_{Ic}measure since we cannot
really calculate it very well.If we know the critical stress intensity factor of some material, we know at what (uniaxial) stress it will fracture if it contains microcracks with half-width .d | |||||||||||||||||||||||||||||

If we now plot the critical stress s versus
crack size _{crit} we get this curve:d | |||||||||||||||||||||||||||||

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The red line, described by the simple equation with the critical stress intensity factor or
fracture toughness separates safe combinations of crack length and stress from combinations
where instantaneous fracture occurs. This is still nothing new but just the Griffith criterion expressed as graph. .K_{Ic} |
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So far I have just renamed things; there is not really something new. The new
inputs come now. They are:- The curve separating save and deadly combinations of stress and crack size (the red curve above) can actually be
*measured*in some special test. That's how we get*numbers*for*K*_{Ic} - So far, everything concerned only
*brittle*materials. But we can take care of*ductility*now. All we have to do is to add a straight line at the specific*yield stress***s**of the material to the figure above_{p}
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The red line, as before, separates separates safe combinations
of crack length and stress from combinations where instantaneous fracture would occur if there
would be no plastic deformation. The mauve line gives the yield stress s; above it _{ p}only plastic deformation will occur.
That means that nowhere in the material can be stresses higher than the yield stress; they would always be released because
dislocations form and move, and thus change the shape of the material in such a way that the stress is reduced.If we now measure
at what kind of combinations of crack width and stress sudden fracture will occur, we get the blue
curve. As one would expect, it is a smooth change-over from brittle-like fracture behavior to ductile behavior, where the
specimen does not fracture under stress, but just "gets longer".Of course, the transitions are not precise and sharp. I tried to do justice to that by using wobbly lines and mixed colors at the boundaries. | |||||||||||||||||||||||||||||

It is convenient to define a critical crack half-width
as shown, using the intersection point between the two curves for fracture only, or plastic deformation only. d_{crit} |
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I guess you see the good news yourself: You don't have to worry anymore about
small microcracks! Sudden fracture does not occur at all
for stresses above the yield stress and cracks smaller than some critical crack half-width
. You will experience d_{crit}yielding before fracture.
That might not be great but still far better than the opposite. | |||||||||||||||||||||||||||||

For cracks smaller than , plastic deformation will
take care of them. Since, as we learned before,
the stress around the crack tip is higher than the global or macroscopic stress, plastic deformation at the crack tip starts
long before the crystal deforms as a whole. This blunts the crack tip and reduces the stress to levels where the crack does
not propagate.d_{crit}This is not a minor effect. The critical crack length might increase to huge values like millimeter! Here is a figure showing this with a real example: | |||||||||||||||||||||||||||||

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This kind of aluminum alloy can live with cracks as long as about 5
mm (!!!) without catastrophical fracturing or fracturing before yielding. In other words: while you can never outsmart the first law of Materials Science, you much prefer that some pressure vessel, for example, starts bulging and sort of getting bigger because it deforms plastically, before it finally cracks and explodes, to sudden fracturing / explosions without any warning. Same thing for that steel girder. Slow bending because it elongates plastically before it finally breaks, while not good, is still much preferable to sudden rupture. And so on. |
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Measurements like the one above allow to extract values for
and for the yield stress K_{Ic}s. Of course, we know _{ p}s
from independent tensile testing experiments already, so we even have a check if everything is as it should be. Here are
a few numbers:_{p} | |||||||||||||||||||||||||||||

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With these numbers we can "somehow" calculate the blue curve and thus
the critical crack length . This means that we now have design criteria; we can start to construct
things that do not break. Fracture d_{crit}engineering starts here. | |||||||||||||||||||||||||||||

Unfortunately, it doesn't end here. There is far more to making safe things that
do not fracture now or ever, provided my kids or my wife don't get to (ab)use it. Topics to consider, for example, are:- Accounting for cases where the specimen dimensions are not far larger than the crack size (always assumed so far). This leads to rather long equations and a lot of tricky approximations.
- Accounting for more complex geometries than just uniaxial stress. This leads to more fracture toughness numbers, labeled,
for example:
and*K*_{IIc}; the "*K*_{IIIc}" so far always referred to simple uniaxial cases._{I} - Considering the precise stresses and stress distributions at the crack tip, and not just the approximations used so
far (that's where the "simple"
comes in). You definitely do not want to know about this.**K** - Calculating
instead of measuring it. Something to avoid at all costs.*K*_{Ic} - Considering rapid deformation, i.e. building up stresses faster than they can be relaxed by dislocation movements. Fun to do experimentally (e.g. in sword fights or by exploding things), but very hard to calculate.
- Considering what could happen fracturewise as time progresses. Keywords are "creep" and "fatigue".
- Going for "safe" design, allowing to make parts that do not fracture even if something goes wrong (your wife uses your knife as a screw driver, your car hits a tree, your satellite is hit by a small meteorite, .... ).
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If you want to know more about this, you need to apply yourself - for a few years. There are no more easy fixes and shortcuts. Sorry. | |||||||||||||||||||||||||||||

Science of Uniaxial Deformation

5.4.2 Dislocation, Plastic Deformation and Hardness

The Cyprus Copper and Bronze Industry

© H. Föll (Iron, Steel and Swords script)