Biological Three-Point Bending & Fracture Toughness Testing

Just as with other mechanical tests, three-point bending (3PB) and fracture toughness tests provide useful information about a material. 3PB and fracture toughness are both compressive tests. Their setups are very similar, but fracture toughness testing is essentially a variation of 3PB with additional steps. Both tests tell us different information about how materials act, such as strength, elasticity, or toughness. This information can be useful for material selection, such as for an effective heart valve or hip replacement.

In biological materials, such as bone, this information can help us understand the consequences of medications, treatments, or lifestyles. I use both of these methods regularly to understand how aging and fatty diets impact bone health on a mechanical level. I will use bone as an example throughout both methods.

An example of the general set-up for three-point bending or fracture toughness testing for mouse and rat long bones. Where the bones span between two points that hold the sample up. Then a third point loads the bone from above between the other two points. Credit to the author.

There are standards for mechanical tests for each material class. Biological materials often use these standards as a starting point. For example, many standards for mechanical tests outline ideal sample dimensions. Sometimes we can machine samples to these dimensions, but sometimes we have to work with what we have.

For example, a procedure for mechanical testing on tissues and biomaterials from Biomomentum states that the ideal ratio of sample width to length should be more than 16. However, if a mouse bone was 16 times as long as it was wide, mice would be terrifying.

Biological materials do not always fit neatly into the categories defined by these standards. Therefore, it is always important to know the material and the assumptions of the tests and calculations you are working with.

A useful resource for various tissue and biomaterials mechanical testing information is Biomomentum.

While using standard-inspired setups can make our lives easier, it can limit the strength of the conclusions we can draw about biological material. Better methods are available for certain measures we get from 3PB or fracture toughness, but they can require more time and money. If you have no idea what a material/tissue might be doing mechanically, it may not be worth starting with the more costly option. 3PB is a relatively fast, cheap, and easy-to-perform compression bending method, and the results can hint if we should do more involved testing.

For both of these tests on biological materials, we need accurate geometric information. This information is often obtained from non-invasive methods, such as computed tomography (CT scans). This helps us calculate more accurate values.

General Assumptions for Both Tests

• A general rule of thumb is that any biological material being tested by either of these methods will likely not meet all of the assumptions. This includes sample size, cross-sectional shape, and orientation. Meaning that the results are estimates.

• For a simple stress-strain test the load is applied relatively slowly and uniformly over a cross-section.

• The geometry of the sample is not changing over time as the load is applied.

• The geometry of the test fixture matters! We assume this is consistent between all our samples of a study. For example, the span between supports is in the calculations. If comparing between studies, make sure this is consistent or normalize the values.

• Simple 3PB generally assumes that material is homogenous and fibers are not interacting.

• All load is normal to the surface so the center of the sample is experiencing max stress and strain.

• There are sign conventions that tension stresses are positive (+) and compression stresses are negative (-). Thus, for compression tests like three-point bending and fracture toughness, your initial force-displacement and stress-strain curves may appear negative if you do not flip the signs.

Three-Point Bending (3PB)

Three-point bending is a fairly straightforward test with minimal sample prep. It can tell us mechanical and material properties, such as how stiff or how strong a material is with existing defects. For biological materials, 3PB is very useful for telling us about how they act as they are in the body, flaws and all. For small samples, it can be considered a structural mechanical test because it can potentially test a whole organ, like whole bones.

The load-displacement and stress-strain curves generated by 3PB on biological samples may be extremely messy. Despite that, the general points of interest for the curves are still identifiable. If they are not, you may need to evaluate the validity of that sample's test (e.g. evidence of rolling on the supports, building vibrations, etc.). Measures we get from 3PB:

A summary of the measures calculated from three-point bending data.

Notes on 3PB Measurement Definitions and Usage

• Stiffness and elastic modulus are often confused with each other in biological materials, but they are different concepts. Elastic modulus is a material property that describes how resistant a material is to elastic deformation. A material with a higher elastic modulus can absorb more load and return to its original shape when the load is removed. This property is independent of the geometry of the specimen, while stiffness is not. However, stiffness is a more accessible concept to a general audience.

• Strength is often used interchangeably with ultimate strength. Ultimate strength is the maximum amount of force a material can withstand before it fails. However, there is also yield strength (yield stress), which is the force at which a material begins to deform permanently (plastic deformation).

• Toughness is the ability of a material to absorb energy before it fails.

• You may find strength and toughness used interchangeably, especially in biological publications with material testing, but they are not equivalent. This is a misconception that extends into the medical field as well due to the overlaps in the measures in old testing methods. For example, strength is often still considered a key measure of bone health but sometimes it may seem to cover both strength and toughness concepts. However, it is important to differentiate between the two because you can have a weaker bone that is tougher. This means a bone cannot handle a big load but it can absorb energy to resist fracture. Kids are a good example of this. They can seem indestructible and one part of that is that their bones are incredibly tough and can deform quite a lot before breaking. But since their bones are still developing, their bones are ultimately weaker than adults.

• You will notice that there is a toughness measure in the 3PB test. This is an acceptable measure to report, however, it often has a large variation in biological samples with the basic 3PB method. The large variation often makes it uninformative. However, toughness, PYD, and work-to-fracture are helpful measures to help determine if fracture toughness testing would be beneficial.

Drawbacks of 3PB

• Existing defects and their distribution in the sample can significantly impact the test, and are often unknown.

• Sensitive to the geometry of the sample (size, length, etc.).

• Sensitive to loading rate. Review your tissue's literature to find a range of values. E.g. for mouse bones, our lab uses 5 mm/min for 3PB.

• There is a large potential for variation in biological samples. Natural variation plus the sources of variation in 3PB methods and calculations add up. In bone, 30-50% variation of measures is considered normal.

Fracture Toughness Testing

Fracture toughness testing has a 3PB step, but the process is more involved than that. These steps help reduce some of the sources of variation so that the toughness measures have less than 20% error in biological samples.

Toughness is a property of a material that allows it to be strong and ductile, which makes it hard to break. Fracture toughness tests tell us about how a material resists crack growth. Another way to think about it is that it tells us how much energy the material can absorb before a crack begins to grow stably and then unstably to fracture.

With fracture toughness testing, we take advantage of the fact that stress concentrates on sharp defects by introducing a massive defect (a notch) to the material for 3PB. The introduced defect is so large and sharp in comparison to other defects in the material, that we can assume any existing defects have a negligible impact. This allows us to remove the influence of unknown defects in a sample and measure the toughness of the material itself.

Measures from fracture toughness testing:

A summary of the points used in fracture toughness calculations on a force-displacement curve and what they represent.

Notes & Additional Steps of Fracture Toughness Testing

• We assume that the notch is perpendicular to the loading punch of the 3PB setup. Since biological samples are not perfect, it may be difficult to notch consistently every time. Practice your method before using your samples. For example for bone, we aim for a notch reaching about ~1/3 of the thickness that is parallel to the anterior-lateral axis.

• Using a small Dremel saw followed by a feather blade with polishing media to sharpen the notch can help immensely with getting consistent notches and results.

• 3PB data for fracture toughness testing is analyzed the same as normal 3PB data. The other values, like elastic modulus, do not mean much in this case because of the massive defect influencing the mechanical properties.

• After 3PB is complete, the fracture surface (cross-section) is imaged. For bone, we use scanning electron microscopes to see the notch in great detail. Ideally, the growth of the crack is monitored in real time, but this requires a specialized setup.

• The half-notch angle along with the geometric measures necessary for fracture toughness calculations can be measured from the cross-sections using software. For example, we analyze our cross-sectional image with custom MATLAB code.

• It is recommended to add additional quality assurance steps to ensure your results are valid. If your samples fail your quality assurance tests, it can mean your notch was not large enough, too large, or was not loaded directly under the loading punch. Failure of quality assurance tests often explains results that are significantly different. For example in bone, the notch must reach the marrow cavity but be less than halfway through the bone. The half-notch angle must be less than 110 degrees (half-notch < 110 degrees). The notch must be sufficiently parallel to the lateral-medial axis (valid if < 30-degree difference between sides). We visually inspect the cross-section images and use ImageJ to measure angles on them.

• Kc toughness calculations require plugging in the 3PB and geometric data. The equations are generally the same for the tissue but the angle and load values in the equation will change depending on which Kc value you are calculating.

• A sanity check is that Kc, initiation is always less than Kc, max. If this is not true, there may be an error in calculation or testing. We often do not use Kc, instability as it can be difficult to tell the true failure point without close monitoring.

• Be aware that fracture toughness equations used with biological materials may assume a cross-sectional shape for simplicity. For example, long bones for small rodents are treated as thick-walled pipes with circular cross-sections. The values are estimates.

An example of what a valid notch looks like in a mouse bone. The colored outlines and red dots along the notch edge show the result of the MATLAB code, which analyzes the image for geometric information. The degrees noted on either side of the notch are an example of the quality assurance test we perform to ensure our samples have valid, parallel notches.

Drawbacks of Fracture Toughness Testing

• Fracture toughness testing is sensitive to the notch geometry. This is often done with a feather blade or saw, and can be highly user-dependent. Our lab uses a saw with a highly controlled stage, which improves our repeatability of notch geometry but quality assurance tests are still necessary to identify invalid notches.

• Fracture toughness is an intensive process with notching, mechanical testing, sample preparation for imaging, image analysis, and calculations. Our lab uses this method with careful consideration of the questions we want to ask.

• By introducing the large defect in the sample, the measures you normally get from 3PB are not valid. So you need an additional sample to collect that information by normal 3PB.

References

Silva MJ. Bone Mechanical Testing by Three-Point Bending. [PDF available online] 2016.

https://www.med.upenn.edu/pcmd/3pt-4pt-bending.html

Three-point bending flexural test [Youtube Video]

Ritchie RO, Koester KJ, Ionova S, Yao W, Lane NE, Ager JW 3rd. Measurement of the toughness of bone: a tutorial with special reference to small animal studies. Bone [Internet]. 2008 Nov;43(5):798–812. Available from: http://dx.doi.org/10.1016/j.bone.2008.04.027

Turner CH, Burr DB. Basic biomechanical measurements of bone: a tutorial. Bone [Internet]. 1993 Jul;14(4):595–608. Available from: http://dx.doi.org/10.1016/8756-3282(93)90081-k