One of the most curious things about materials is not how strong they are but rather how weak they are! It is possible to calculate how strong a material should be by finding the strength of a single inter-atomic or inter-molecular bond and multiplying by the number of bonds in the cross sectional area. Unfortunately this gives a strength that is at least ten times that of the best samples we can normally make.
The graph below is a detail of the stress – separation diagram for a typical metal. The curve crosses the x-axis (zero stress) at r/r0 =1, i.e. the standard separation of the atoms/molecules. If an attempt is made to separate the particles then a negative (attractive) force results that attempts to restore them to their original position. The theoretical maximum stress occurs at 25 % strain.
The theoretical maximum stress is achieved at point D. The line BD is approximately half that BC. Using the triangle ABC show that the theoretical maximum stress is equal to E/8 where E is the Young’s Modulus for the metal.
Piano wire, also called music wire, is one of the strongest steels readily available. It is a high carbon steel that has been extensively work hardened. Its Young’s Modulus is around 84 GPa and has an ultimate tensile strength of approximately 2400 MPa. Compare this to the theoretical maximum stress. Repeat your comparison for structural steel (Young’s Modulus 200 GPa, ultimate tensile strength 400 MPa).
So what is it that makes materials so weak? The answer varies depending on what material you are looking at but for ductile, crystalline materials i.e. metals the answer is dislocations. We tend to think of crystals being perfectly aligned arrays of regularly spaced atoms. This is far from the truth. For a start most crystalline materials are in fact polycrystalline, that is made up from many small crystals that have grown independently and then locked together. Even within these individual crystal grains there are numerous faults in the pattern, called dislocations.
Clicking on the above link will take you to a sheet that will allow you to make simple models of these dislocations and help you to understand how they behave. Save the atom sheet jpeg on your computer.
Print say ten copies of the atom sheet at the end of this worksheet and stack them together. The registration crosses at the corners will help you line them up. Since the sheet is only 20 × 28 atoms in size, the stack is ten atoms thick and a real atom is approximately 0.2 nm in diameter it follows that your model crystal grain is only 4 nm × 5.6 nm × 2 nm. Still small compared to the real thing in most instances; grains vary from a few nm across (10-9 m) up to several millimetres. Typically though they are in the order of micrometres across (10-6 m). A typical slice through a real grain therefore might involve several hundred million atoms and on the scale of your model, be hundreds of metres across. Fix that idea in your mind; a few million sheets of paper, each several times larger than an Olympic stadium, covered in atom models, each just a few millimetres in diameter.
Now if you want to permanently distort your model crystal grain you need to take one sheet and slide it, and all those above it exactly one atom spacing over the sheets below; quite easy for you’re A4 model, next to impossible for the 100 m plus model. Remember that all the atoms need to move at the same time and so each and every bond in the plane that is being moved needs to be stretched simultaneously.
In diagram a the grain is undisturbed. A shear force is applied in b and the planes begin to move. If the force were removed the atoms would return to their original positions; the distortion is elastic. In c the force is far greater and so the resulting distortion is also greater. In d the distortion is so great that the bonds have broken and been remade with the adjacent atoms. The atoms would not now return to their original positions and so the distortion is plastic. The atoms rapidly drop into their new equilibrium positions in e.
The huge force required to switch all the bonds at once explains why the theoretical strength is so large.
An edge dislocation occurs when an incomplete plane of atoms is positioned between two normal planes. You can make a simple model by cutting one of your sheets in half along a line between the atoms. Now instead of moving a whole sheet at once, only a single edge needs to be moved. The diagram shows a free edge dislocation (the dark, half-plane). When a shear force is applied the complete plane to its left can split by breaking the bonds of a single line of atoms maybe a few thousand long. This is far easier to achieve than a whole sheet at a time.
The dislocation can move gradually through the adjacent planes. You can see this in the following diagram. Only a small force is needed to make the adjacent planes split and slide through the dislocation. This movement of a dislocation through a crystal grain is called slip.
When you look at the bonds between the two planes either side of the half plane but beyond the dislocation you should notice that they are stretched. Either side of the half plane the bonds are squashed. This means that there are tensions and compressions present and so there are permanent stresses in the crystal lattice.
The dislocation itself can slip gradually, one atom at a time. You can reproduce this with your model. Place the half sheet of atoms on the top of the stack and then place a single whole sheet above that. With a pair of scissors cut in to the top sheet to the depth of a few atoms. Using sticky tape attach one edge of the cut sheet to the half sheet below it. If you cut a little deeper then you can attach a little more of the top sheet to the lower. Eventually the entire sheet has been cut in half and taped to the sheet below. The dislocation however still exists as the new half sheet and the process can begin again.
The movement need not even start at the edge of the dislocation and so a single dislocation can actually be slipping at several points at the same time.
A screw dislocation is a little harder to visualise but is still quite easy to construct. Take each of your atom sheets and along a line of atoms cut in to the centre of the sheet. Stack two sheets together; the cuts should line up. Now tape the right hand side of top sheet to the left hand side of the bottom sheet. Place another sheet on top and repeat the process until you have enough sheets to make a clear model. The screw dislocation could also be described as a spiral stair case since you could ‘walk’ your finger from the top to the bottom of the stack without ever leaving the surface.
Other, more complex, compound dislocations are also possible.