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: 



The basic points to recall are:






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:  


Formally, the new quantity "critical stress
intensity factor" K_{Ic} would be equal to
(2Y · g)^{½}. This
factor relates to crack growth by giving the energy needed via g, and the energy gained via Y. Let's
forget that now and accept that K_{Ic} somehow takes care
of all energies involved instead of just
the surface energy g and simple strain
energy. Just as Y and g are material "constants", somehow encoding a major property of a given material, so is K_{Ic}. It's just no longer simply the product of the two old parameters but something new that we 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 halfwidth d. 

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


The red line, described by the simple equation with the critical stress intensity factor or fracture toughness K_{Ic} 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. .  
So far I have just renamed things;
there is not really something new. The new inputs come now. They are:




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_{p}; above it 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 changeover from brittlelike 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 halfwidth d_{crit} as shown, using the intersection point between the two curves for fracture only, or plastic deformation only.  
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 halfwidth d_{crit}. You will experience yielding before fracture. That might not be great but still far better than the opposite.  
For cracks smaller than
d_{crit}, 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. 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: 



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. 

Measurements like the one above allow to extract values for K_{Ic} and for the yield stress s_{p}. Of course, we know s_{p} from independent tensile testing experiments already, so we even have a check if everything is as it should be. Here are a few numbers:  


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


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)