Mechanical testing of cancellous bone (Biomechanics)

write the practical report, which has to be between 1500 and 2500 words. The subject is mechanical testing of cancellous bone. During the practical class, we measured the dimensions of eight bone samples and determined their compressive properties using the materials testing machine. We saw the general pattern of stress versus strain, which looked pretty much similar to that shown in the ASTM standard for testing rigid cellular plastics (i.e. rigid foams). Afterwards, I degreased the specimens by keeping them in alcohol for two weeks, after which I weighed them.

The aim of the experiment was to find out if strength and stiffness depend on bone density, species of animal, and testing condition.

The attached protocol explains how the tests were done, and what data was collected. During the class, you were able to take pictures that show the equipment and the specimens that were tested. The ASTM standard, which I handed out, explains how tests on rigorous porous materials (and cancellous bone counts as a rigorous porous material) ought to be done. We tried to adhere to this as good as possible.

Once I weighed the samples and analysed the results, I realized that I must have made a mistake while calibrating the new load cell, because it appeared that all forces where about a factor two to three too high.

What I have decided therefore is to give you last year’s data. All that data has been put into a single Excel spreadsheet, which you find attached. They consist of Force and Displacement data, collected on 24 samples. Twelve samples were from cow bone, and 12 from pig bone. For each species, 6 experiments were “unconstrained” and 6 were “constrained” (this is explained in the protocol).

In the first worksheet of the file, you will find data on sample height (thickness), diameter and weight. As the protocol explains, after the test the bone cylinders were stored in alcohol for a week to dissolve the marrow, after which they were weighed. The weights are therefore without marrow. If I hadn’t dissolved the marrow, the weights would be too large.

What I would like you to do is work out for each sample a stress-strain graph or at least the stress-strain data. From that, you should be able to obtain the maximum stress (compressive strength; point L in Fig. 1a in the ASTM standard), compressive stiffness (the slope of the stress-strain graph) and failure strain (the strain at compressive failure minus the “zero strain”). The ASTM standard explains how to do all this, in particular Fig. 1a and sections 9.1, 9.2, 9.3, 9.3.2, 9.3.3, 9.4 and 9.4.1.

For the first sample (CanBone1), I have provided formulas in Excel that help you to do all these things. In columns B and C you will find the raw data for displacement and force. Then, in columns D and E I have calculated strain and stress from the thickness and diameter of the specimen (these numbers are also in column H). Next, I made a stress-strain diagram by selecting columns D and E and using the chart wizard. In the chart wizard, choose “xy (scatter)” as your chart type in the first step!

From the graph you can see that by and large all the numbers are positive, although it is a compressive test. In the lectures, I explained that tension is normally positive, and compression negative. However, since all the tests in the sheet are compressive tests, I converted the test results to positive numbers, mainly because the graph looks better that way and matches the ASTM standard.

If you look in the first rows of the data, you will see that there are some negative numbers. However, I just said that I converted all compressive forces and displacements to positive numbers, so can we have negative numbers? The reason is that the signal from the testing machine has a small amount of noise. If you look at the actual numbers in the top rows, you will see that the negative values of the forces and displacements are small – on the order of a few microns and a few Newtons. That is simply the noise level in the system. The load cell we use measures up to 5000 N, so a few Newtons of noise is pretty good.

If you now look at the graph, past the “hump” of maximum stress, you can see that the graph becomes very erratic, as if there is quite a lot of noise. This is however not noise, what it shows is plastic deformation of individual trabeculae. Every time an individual trabecula has undergone sufficient damage, it yields. At that point, the load drops, but the other trabeculae take over. So the load rises again until the next one yields, the load drops, etcetera.

Finally, I worked out the stiffness, compressive strength, the zero strain point, and the failure strain. The stiffness equals the slope of the stress-strain diagram. There is an Excel function called “SLOPE” that does the job. See how I used it! Make sure that you only select stress and strain values from the straight bit of the curve – in my case, this was for stresses between 2 and 8 MPa. The compressive strength is simply the maximum at the first “hump”, for which there is an Excel function “MAX”. We go for the first hump, even though the stress often rises again at higher strains, even to a larger value than that first hump. However, the first hump is the one where the pore structure first starts to fail. It is also the point that we have to take for measuring the compressive strength according to the ASTM testing standard.

To find the zero strain point, I used the idea that Excel can find you SLOPE and INTERCEPT for the best-fit line y=a*x+b, with a the SLOPE and b the INTERRCEPT. The zero strain point is the point where the linear part of the stress-strain curve crosses the horizontal axis (y=0; see ASTM standard). For y=0, the formula for the line becomes 0=a*x+b, or -b=a*x, or -b/a=x, or -INTERCEPT/SLOPE=x. The formula is in column H. The thing left then is to find the failure strain, this is the strain at the point of compressive failure, minus the zero strain. We have found the point of compressive failure using the MAX function, now we need to find the corresponding strain. For that, there is an Excel function LOOKUP. You give it a value, tell in which column to find the value, and from which column you want a corresponding value: see column H cell 7. I have subtracted the zero strain in the same formula.

 
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