3. Behaviour That’s Not Elastic

1. Contrast plastic deformation with elastic deformation
2. Provide at least one atomic level definition of plastic deformation and one macroscopic definition
3. Explain why the term "plastic" is used to refer to permanent deformation
4. Sketch generalized curves for the stress-strain behaviour of metals, ceramics, and polymers
5. Summarize the need to test ceramics in bending, rather than in a tensile test
6. Describe the general theory behind tempered glass, whether thermally or chemically                     
7. Illustrate the final stress distribution through the thickness of a sheet of tempered glass                    
8. Explain how residual stresses increase the strength and improve the safety of tempered glass
9. Given all of the others, calculate any one of the following for a 3-point bend test on a rectangular sample: sample height, sample width, span between lower supports, load, peak stress in sample
10. Justify the statement that a scientific model need only be as good as it needs to be for the present discussion
11. Using two examples from the fields of materials science or solid state chemistry, demonstrate the usefulness and the limitations of a scientific model

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Exciting! Now we can start to think about what would happen to the atoms when we apply enough load to move beyond the elastic region. Let’s call this type of deformation permanent deformation since the sample dimensions will have changed if we load beyond the elastic region, even if the load is removed. If the dimensions change permanently and the sample is made up of atoms we can once again consider our simple hard sphere model and state that during permanent deformation the atoms must move to new positions, even when the load is released. 

This is illustrated in Figure 1 for the sample before loading, and in Figure 2 while the load is applied. Atoms have moved to new positions and the sample dimensions have changed. Even when the load is removed, although the elastic strain is reversed (the atoms move closer to one another) the atoms remain in their new positions, as shown in Figure 3).

The obvious name for this type of deformation is permanent deformation but there is another name that is very commonly used - plastic deformation.

Okay, I’m being intentionally a little provocative with my title here, but the usage of the term plastic to refer to permanent deformation does cause confusion for many students first learning this topic since the term plastic is often misused in general usage.  We often say, "plastic" when we really mean, "polymer."  Of course, polymer sounds a little more technical and unfamiliar to many people, so I can understand the familiar ease of the term plastic.  However, plastic actually refers to the permanence of the deformation and derives from the Greek plastikos meaning to sculpt or shape.  Although polymers are very important and useful in plastic surgery it is the permanence of the changes to anatomical features, whether cosmetic or reconstructive that gives the medical specialization its name.

Previously we saw that most metals have an initial linear elastic region. But what happens to the stress-strain behaviour if we load beyond this elastic region?

Figure 4 shows the generalized stress-strain behaviour for a typical metal. Plastic deformation begins close to the end of the linear elastic region, al- though it is a little tricky to determine exactly where plastic deformation begins. We’ll explore this in more detail a little later.

You've got a lot of important new concepts and terminology under your belt by now.  For example, you know that stress is force supported by an area and that strain is elongation distributed over a length.  You know that most materials have a linear elastic relationship between stress and strain for relatively lower loads and that the constant of proportionality relating stress to strain is the Young's modulus.  But now we have a challenging concept to tackle and that is to gain an intuitive understanding of Young's modulus.  This can actually be surprisingly difficult and it is not uncommon to find students in the final year of their degrees who still have some confusion as to what Young's modulus really represents.  Sometimes it is helpful to draw from our everyday life to help gain this understanding.  For example, if you bend a wooden stick and it snaps you gain an understanding of the fracture strength of that wood.  On the other hand, how can bending a piece of wood help you to understand the Young's modulus?  Part of the problem is that there is no single clear term in common usage in the English language that describes the Young's modulus.  Sometimes we'll refer to the elasticity or how bendy something is, or even how elastic something is, but unfortunately, none of these terms are specific an unambiguous.  Elasticity or elastic could refer to a low (or even a high) Young's modulus, or it could refer to the quality of a material being able to support high values of elastic strain.  Bendy is certainly no better as it doesn't provide any clue as to high or low values of a property and also makes no reference to whether we are referring to elastic or plastic deformation.  So, at this point the word stiffness often comes along and offers up its pleasant familiarity and colloquial liberty and lulls us into thinking that it is the word we've been looking for to describe Young's modulus.  Certainly it does provide some useful and specific information.  High stiffness is universally understood to mean that something does not deflect or elongate much under load and low stiffness would mean the opposite.  In fact, it goes further because I think it is safe to say that most people would agree the term stiffness only describes deformation that is recoverable, or not permanent.  Confusion with this term comes from your new membership in the elite group of people being trained as an engineer!  Engineers (particularly civil and mechanical engineers) have taken the commonly used word stiffness and attributed it some additional meaning.  As an engineer you will know that the deflection of a beam under a given load depends not only on the material property, the Young's modulus, but also the shape of the beam.  For instance, an I-beam will deflect less than the same amount of material in a solid rectangular cross-section.  This is where the so-called second moment of area comes into play (and you will likely learn about in a mechanics class).  So, you see, the stiffness, as used by engineers includes not only the material property, but also the shape and you need to be careful with this usage when communicating with other engineers.

So, Figure 4 has shown us what plastic deformation looks like in a typical metallic sample, but what might it look like for the other material classes? It seems obvious that it should be possible to cause plastic deformation in polymers, since the common name for them is "plastics." (Of course, by this point you know better than to be so cavalier with your terminology as to call a polymer a plastic. Perish the thought.) But are all polymers capable of plastic deformation? What about ceramics? Great questions, all of them, but let’s slow down a little and first look at the generalized stress-strain behaviour for typical polymers and ceramics. 

How convenient, Figure 5 shows these very curves along with the curve for a typical metal all plotted on the same axes.  You’ll notice a few important features of the curves. First off, the Young’s modulus of ceramics is higher than that of metals which is in turn higher than the "Young’s Modulus" of polymers.

 You’ll also notice in Figure 5 that the strength seems to follow the same trend with ceramics being strongest and polymers the weakest. This is generally true, but it is worth mentioning now that this is only when ceramics are loaded in compression. In tension ceramics are really weak. You may think that the ceramic curve appears to be entirely linear elastic until fracture. You are correct. This is because ceramics are brittle. That is, they don’t undergo plastic deformation, at least, not around room temperature. The polymer curve also has an interesting appearance in that it dips down and then comes back up a little. That can happen with a lot of polymers. The reason for this we’ll explore a little later.

Why did I put Young's modulus in quotes just then?  Good question.  Although we commonly do use the term Young's modulus when referring to the elastic behaviour of polymers, some purists will argue that this is misleading or even inaccurate since the elastic response of polymers depends on many different bond types including various primary bonds and secondary bonds; more to come on these later.  As you'll recall from the previous notes, the Young's modulus is related to the behaviour of a single bond type.  For this reason, you will sometimes see the term elastic modulus used instead of Young's modulus when referring to polymers and composite materials.

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Say you decide that it is a good choice to include a ceramic component in a design. How do you validate the strength provided by the suppliers? For that matter, how do you determine the strength of a ceramic in the first place? Well, since ceramics are so strong and brittle it is very difficult to machine  them into the nice tensile coupons that we saw earlier and are used for metals and polymers. The high strength causes your cutting tools to become dull very quickly and what’s more, the ceramic itself is likely to shatter during the machining operation. Finally, and this is a little more subtle, even if we did manage to make a nice ceramic tensile sample, it is very challenging to align a ceramic sample exactly along the loading axis. You see, metals and polymers are relatively ductile, that is, they have high plastic strain to fracture.

For this reason, even if we put a metal or polymer sample into a tensile testing machine at a slight angle, once the test starts the sample will deform and become aligned with the tensile loading axis, as show in Figure 6. A ceramic will fracture at a very low value of strain while the sample is still experiencing significant shear stresses. A valid tensile test should expose the sample to only a uniaxial tensile load. For these reasons we often test ceramics in bending. The beam geometry is quite simple and relatively easy to prepare on a water cooled diamond saw, for example. A common form of bend test is the three-point bend test and is illustrated in Figure 7.

The lower surface of a beam loaded as shown will be in tension, while the top surface will be in compression. This means that the stress state on the top and bottom surfaces actually have opposite sign. To have opposite sign, the stress therefore must have passed through zero stress. This is called the neutral axis and you’ll learn more about this in your mechanics course.  Anyway, since the stress varies through the thickness we need to think about where on (or in) the beam we are interested in knowing the stress.  Since ceramics perform poorly in tension, the beam will fracture first on the lower surface, specifically near the middle, underneath where the load is applied. The equation that determines the stress in the middle of the lower surface is

If you were in lecture, you likely saw me standing on a thin piece of tempered glass and you may have even seen it deforming elastically.  The glass that I stood on was conventionally thermally tempered glass.  This is quite an interesting material.  When the glass is at a high temperature the surfaces are rapidly cooled.  The microstructure of glass is actually quite complicated and there are 3-dimensional networks of bonds within the glass.  At elevated temperatures the molecules are further apart, however when the surfaces are rapidly cooled some of this excess volume from the higher temperature is locked into the glass.  The central portion of the glass however continues to cool more slowly and the molecules are better able to organize and achieve a smaller final volume.  This means that the final piece of tempered glass, once it has cooled, contains residual stresses.  The surfaces are being pushed together as the center tries to contract.  The surfaces are therefore in residual compression, while the center is in residual tension.    Because of this, when I stand on the tempered glass beam and you see the beam bend the lower surface is still under a net compressive load.  Since ceramics are strong in compression, the glass does not break and I don't get hurt.  This is why tempered glass is stronger than conventional glass.  It also explains why tempered glass breaks into small pieces if it does fracture.  Ever seen a broken car window?  The pieces of glass are really small.  Ever broken a screen on your phone?  Lots of small pieces, right?  This is because the residual strain energy in the glass attempt to convert to surface energy in the form of new surfaces during fracture.  How do you make a lot of surface energy?  Lots of small pieces.  (As a side note: your phone screen is chemically tempered, but the same principle with residual compressive stresses at the surfaces applies.)

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Okay, this may seem obvious but I think it is worth mentioning: in science we strive to understand the world around us and in so doing come up with new ways to improve the world.  In order for our human brains to attempt to understand the universe and the things in it we come up with scientific models.  These scientific models are like analogies that allow us to simplify really complicated things and to predict behaviour.  For example, one of the early models of the atom is the Bohr model, which is a planetary model.  That is, if we are using the Bohr model we can think of the electrons as if they were little planets orbiting around the sun.  It’s familiar and that is comforting to humans.  What’s more?  It even predicts some behaviour of the atom accurately, specifically energy absorbed and released by electrons when they move from one orbital to another.  Of course, the Bohr model has some limitations and so along the way, as scientists came across data that the Bohr model did not predict, they released improvements, like firmware updates, to the Bohr model so that it still worked and then eventually when the updates just weren’t cutting it, a new model was released.  A little like an iPhone, but I digress.  The point here is that a model does not need to be complete or accurate in every way.  If it was, we wouldn’t need a model.  A model only needs to be as good as it needs to be for the purpose that you are using it.  More on this later.

So, we’ve learned that plastic, or permanent, deformation involves atoms moving to new equilibrium positions. These new equilibrium positions will still be at the same equilibrium interatomic spacing, but the atoms will be in new positions. Ceramics don’t undergo plastic deformation at room temperature, which is another way of saying that they are brittle. Ceramics are hard to test in tension so the next best thing is a bend test.

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