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Measuring fracture toughness in biological materials
Author’s Accepted Manuscript Measuring Fracture Toughness in Biological Materials David Taylor
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S1751-6161(17)30295-3 http://dx.doi.org/10.1016/j.jmbbm.2017.07.007 JMBBM2406
To appear in: Journal of the Mechanical Behavior of Biomedical Materials Received date: 26 April 2017 Revised date: 21 June 2017 Accepted date: 4 July 2017 Cite this article as: David Taylor, Measuring Fracture Toughness in Biological Materials, Journal of the Mechanical Behavior of Biomedical Materials, http://dx.doi.org/10.1016/j.jmbbm.2017.07.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Measuring Fracture Toughness in Biological Materials David Taylor1 Trinity Centre for Bioengineering, Department of Mechanical and Manufacturing Engineering, Trinity College Dublin, The University of Dublin, College Green, Dublin 2, Ireland
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Abstract Many biological materials fail by cracking. Examples are bone fractures, contact damage in eggs, splits in bamboo culm and defects in cartilage. The mechanical property that best describes failure by cracking is fracture toughness, which quantifies the ease with which cracks propagate and defines a material’s tolerance for pre-existing cracks and other stress concentrating features. The measurement of fracture toughness presents some challenges, especially for biological materials. To obtain valid results requires care and, in many cases, considerable ingenuity to design an appropriate specimen and test protocol. Common mistakes include incorrect interpretation of the mechanics of loading in unusual specimen designs, and failures occurring at the material’s ultimate tensile strength as a result of specimens or cracks being too small. Interpretation of the resulting toughness data may also present challenges, for example when R-curve behaviour is present. In this article, examples of good and bad practice are described, and some recommendations made.
Keywords: fracture toughness; crack; eggshell; cuticle; soft tissues; cartilage; bone
Introduction
Toughness is an important property for any structural material, but it is a property which is difficult to define and measure. It is a relatively recent invention, first defined in the 1920’s and not introduced into engineering practice until the 1960’s. The need for this parameter becomes clear when we note that there are some materials which have good strength and toughness and yet fail easily. Examples of these materials are glasses and engineering ceramics such as alumina and silicon carbide. Measurements of 1
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yield strength, hardness and elastic (Young’s) modulus for these materials show relatively high values, implying that they can be used in load-bearing applications. However they often break at low applied loads: the key fact here is that they break by cracking. The existence of this different mechanism of failure (cracking rather than yielding) implies the need for a different mechanical property to characterise the material. Another class of materials also illustrates the need for the toughness parameter: some materials have relatively low strength and Young’s modulus but can absorb a large amount of energy before failing, for example some polymers which are used for impact resistance. Though they yield relatively easily they can absorb a lot of energy in post-yield straining. Again we can appreciate that another mechanical property, toughness, is needed to characterise and understand these materials.
However the property called “toughness” is not the same in these two examples. In the first example, the appropriate parameter is the so-called “fracture toughness” which describes the ability of a material to resist the propagation of pre-existing cracks, since it is by this means that glass fails. In the second example the area under the stress/strain curve is more useful because it quantifies the amount of energy that can be absorbed per unit volume of material. Various mechanisms may operate to absorb energy, including multiple cracking, plasticity and viscoelasticity. By contrast, the “fracture toughness parameter” is concerned with the propagation of a single crack. These two parameters are both referred to as the material’s “toughness” in the literature, but they are in fact quite different. Several other toughness parameters have also been defined and used, so care is required when reading the literature to understand exactly what is being measured: a material may perform well according to one measure of toughness but badly according to another.
The present paper will be confined to consideration of the “fracture toughness” parameter. The importance of this parameter becomes clear when we note that most cases of mechanical failure in biological materials occur by the mechanism of cracking. Stiff materials in our bones, teeth and fingernails develop small cracks as a result of normal loading activities (O'Brien et al., 2000): fractures can occur if these cracks grow quickly, as a result of impact for example, or slowly as a result of fatigue failure under repeated loading (Burr, 1997). Soft tissues also fail by cracking: examples are skin damage and cartilage defects, the latter being a major factor in the progression of osteoarthritis (Thambyah et al., 2012). Tooth enamel has an extremely low fracture toughness as a result of its high mineral content (Bajaj and Arola, 2009). Bone achieves a somewhat higher fracture toughness thanks to features in its
microstructure which hinder crack propagation (Nalla et al., 2005a). Other examples of toughness enhancement achieved as a result of microstructure can be found in nature, for example in the shells of molluscs and other aquatic organisms (Barthelat et al., 2007). Toughness may be reduced as a result of aging and disease, and this contributes to the reduced strength of bone observed in debilitating medical conditions such as osteoporosis (Nalla et al., 2004). Better understanding of how natural materials achieve toughness enhancement can lead to improvements in engineering materials through biomimetic approaches.
After an introduction to the definition and measurement of this parameter, the existing literature on its application to biological materials will be reviewed, highlighting the particular challenges which these materials present.
The Definition and Measurement of Fracture Toughness
Fracture toughness is easy to define, but difficulties arise when it comes to the practical measurement of this parameter. The basic concept (Griffith, 1920) defines a parameter Gc which is the energy needed for crack propagation, more specifically the energy per unit area of crack formed. This can be related to the change in the total strain energy in the body, W, with respect to crack area. Assuming a throughthickness crack of length a in a sample of material of constant thickness B, we can define the rate of change of strain energy with crack length as G, thus:
G
1 dW B da
(1)
G can be measured in practical experiments (or in simulated ones using, for example, finite element analysis) by finding the area under the force/displacement curve for a specimen containing a crack and noting how this changes when the crack length is increased (see figure 1).
If this release of strain energy G is large enough to supply the energy needed for crack growth, then the crack will propagate. Depending on the sample and the type of loading, the crack may then grow rapidly and unstably (if G increases monotonically with a) leading to total failure of the specimen, or it may grow in a gradual, stable manner if more applied load or displacement is required to supply the
necessary energy. In either case we can define a critical value, Gc, as the fracture toughness of the material.
Strictly speaking the term Gc is reserved for cases in which the force/displacement behaviour is linear up to failure: so-called “Linear Elastic Fracture Mechanics”. If the behaviour is non-linear, for example due to plasticity, viscoelasticity or microdamage, then the critical energy can still be defined in the same way but it is usually given the term Jc.
This energy-based definition is useful for quantifying and comparing materials, but another term, Kc, is more commonly used in engineering applications because it expresses toughness in terms of the critical stress f required to cause failure by propagation of a crack. In many practical cases the relationship can be expressed as:
K c F f a
(2)
Here F is a factor which depends on the geometry of the body, the geometry and size of the crack, and the type of loading applied. Loading can be either tensile (known as Mode I); in-plane shear (Mode II) or out-of-plane shear (Mode III), as all these types of loading can cause crack propagation. Kc is the critical value of a parameter K known as the stress intensity. Values of K as a function of applied loading and crack length have been determined for many common cases (see for example (Murakami, 1987)). For non-standard cases K can be found using finite element analysis: one approach is to examine the stresses in the vicinity of the crack tip (Janssen et al., 2002). Kc and Gc, both of which are referred to as the “fracture toughness” in published work, are linked by the following relationship:
K c Gc E
(3)
…where E is Young’s modulus (with a slight correction depending on whether the body is loaded in plane stress or plane strain). Several national and international standards exist which define precisely the procedures for experimental measurement of fracture toughness (for example ASTM E1820): figure 2 shows some typical test specimens.
Fracture Toughness Measurement in Stiff Biological Materials
It is convenient to consider two groups of biological materials separately: the hard, stiff materials and the soft tissues. In principle fracture toughness can be measured relatively easily in the hard tissues because their behaviour in the low stress regime (before significant yield or microdamage occurs) approximates to linear elastic and, being relative brittle, they can be induced to fail in this regime by introducing a pre-crack or sharp notch. In practice however, difficulties arise, the most obvious of which is the shape of the test specimen. It is often not possible to obtain enough material to create any of the standard test specimens such as those shown in figure 2, so a certain amount of ingenuity is required to invent a specimen and a test procedure. A good example is the case of bamboo, a material which can be obtained in large quantities but always in the form of a circular tube, known as a culm. Several workers have addressed this problem, and each has devised their own unique test specimen. Figure 2 shows some examples, which can be divided into two types. Some workers retained the tubular shape of the culm and found ways to introduce a crack and to load the tube in such a way as to put the crack under tensile stress, inducing it to propagate (e.g. (Mitch et al., 2010; Taylor et al., 2014). Other workers created specimens similar to the standard ones by machining material from the culm wall (e.g. (Shao et al., 2009; Tan et al., 2011). Difficulties with this latter approach are the fact that the culm wall is curved and that the material is not consistent through thickness: mechanical properties change as one moves from outside to inside.
Examining the published results for the fracture toughness of bamboo (see Table 1) we find enormous differences, with reported Kc values varying by almost three orders of magnitude. Some of the variability in these results can be accounted for by differences in species and in the hydration state of the material, but these effects are relatively minor. The very small value of 0.18MPa√m (Mitch et al., 2010) can be explained by poor understanding of the mechanics involved. These workers used tubular specimens cut directly from bamboo culms. They inserted a split pin into a hole drilled across the tube diameter and caused cracking by applying tensile forces to separate the two halves of the pin (see figure 2). Unfortunately when calculating Kc they used a calibration based on remote loading which is not appropriate in this case. Much higher stresses will arise than expected, partly because the force is being applied locally, via pressure on the walls of the hole, and partly due to through-thickness bending caused by the curvature in the culm walls.
Other workers measured very high values, up to 100MPa√m, which, if true, would mean that bamboo is as tough as steel (Amada and Untao, 2001). The problem which arises here is the anisotropy of the material. Bamboo is extremely anisotropic, being much stronger in the longitudinal direction. Most of the results quoted in Table 1 refer to the Kc value for cracks propagating longitudinally along the culm, but Amada and Untao attempted to measure the toughness in the transverse direction, by introducing a notch (in the form of a cut with a razor blade) into the side of the specimen and applying longitudinal tension. By using cuts of different depth they were able to measure the fracture resistance at different distances from the outer surface. Their Kc values decrease from 100MPa√m near the outside to 20MPa√m near the inside. They also provided data on the tensile strength of small samples taken at different distances from the outer surface, which varied from 400-100MPa. It is evident that, in order to cause failure in the cracked specimens, they had to apply stresses similar to the tensile strength of the material. Under these conditions the specimens will still break at the location where the crack has been introduced, because here the area of the cross section is smaller, but the stress at which failure occurs will be a measure of the tensile strength, not the toughness. Even then it is difficult to understand how values as high as 100MPa√m could have occurred so there may in addition have been some calculation errors.
Something similar has occurred in the work of Low et al, the difference being that they loaded their specimens in bending rather than tension, forcing the failure to begin at the notch root where the bending stresses are highest (Low et al., 2006). Photographs in their paper clearly show that when cracking occurs from the notch the crack grows along the culm, at right angles to the notch direction. Tan et al (2011) used a similar specimen geometry but, realising that crack growth occurred in this direction, they used an appropriate K-solution which had previously been derived for crack growth along a bi-material interface (see figure 2).
The problem by which failure in a toughness test occurs at the material’s ultimate strength, also arises in other materials, especially in soft tissues, so it will be discussed in more detail below. The true value of Kc for longitudinal crack propagation in bamboo appears to be of the order of 1-2MPa√m, based on the consensus of several other papers listed in Table 1. There is certainly more work to be done here to improve on these measurements and to answer important questions such as how this value varies with location in the culm.
Another example which illustrates the difficulty of obtaining an appropriate specimen geometry is the case of eggshell, where one is dealing with a thin curved shell of a material which is so brittle that it is almost impossible to cut a specimen from it without causing damage. There are surprisingly few measurements of Kc for eggshell in the literature, despite the obvious commercial and biological interest in this material and its response to cracks and other defects. Table 2 shows all the available data. Mabe et al conducted tests on whole eggs, applying compression parallel to the egg’s longest axis, using two steel plates (Mabe et al., 2003). This type of test will cause cracks to form and grow from the two contact points, and in principle it should be possible to deduce Kc from the lengths of the cracks produced, though the analysis would be quite complex. Macleod et al used computer modelling to analyse crack growth in eggshell but stopped short of specifying a method for toughness measurement (Macleod et al., 2006). The result obtained for Kc by Mabe et al, of approximately 11MPa√m, is clearly erroneous because there is almost no ceramic material in existence which has a fracture toughness greater than 10MPa√m. The method of analysis is not described in the paper, which only gives an equation that doesn’t contain the crack length as a parameter. Unfortunately this same equation was used more recently by other workers (Xiao et al., 2014) who obtained results of similar magnitude, an example of how incorrect work can propagate in the literature.
Other researchers measured the fracture toughness for eggshell by recording the energy required when using a pair of scissors to cut the material (Gosler et al., 2011). This method has been used successfully for other materials, (Vincent, 1990): by this means one is measuring directly the energy needed to create the crack surfaces, and thus finding Gc. There is a large amount of scatter in the data from Gosler et al, but a typical value, when converted into Kc using equation 3, gives again an impossibly high result of about 20MP√m. It is not immediately obvious why this high value occurred. One practical problem with this type of test is to account for the energy which does not contribute to the creation of new surfaces, such as that lost in friction in the movement of the scissors, but this correction is relatively easy to make by running the test with no specimen or a specimen of known toughness. In this case the problem may be that the scissors will, in addition to creating the cut surface, also cause a lot of damage in the form of microscopic cracking on the sides of the cut, which will consume extra energy.
We took a different approach to the characterisation of eggshell toughness (Taylor et al., 2016), which relies on the fact that if a sphere is axially compressed by a pair of forces L, then the stress field, which is
quite complex near the loading points, takes a very simple form at the equator. Here there is an axial compressive stress c and a circumferential tensile stress t, of equal magnitude, given by:
t
L 2rt
(4)
…where r and t are the radius and thickness of the spherical shell, respectively. We took advantage of this tensile stress by machining a small notch to encourage a pair of cracks to form at a point on the equator and grow up and down towards the poles. We protected the contact point at the poles to prevent cracks from starting there (see figure 3). The K-solution for this case is very simple, the value of F in equation 3 being unity, so Kc can be found from the critical load Lf needed to cause failure by crack propagation from the notch (of total length 2a), as:
Kc
Lf 2rt
a
(5)
In this work we used a machined notch rather than a pre-crack, an approach which is quite common and was used in most of the work on bamboo discussed above. However this must be done with care because there are several ways in which the nature of this notch can affect the result. In our work on eggshell we considered the issue of the root radius of the notch, which is necessarily a finite quantity rather than being zero as assumed for a perfect crack. Of course any real crack will have a finite radius, so the problem is only to find how large this radius can be without affecting the result. In this paper we recorded Kc for several different notch radii and extrapolated back to zero radius (see figure 3): there are also theoretical approaches for predicting the effect of notch radius from first principles (see for example (Taylor, 2007)).
We obtained a value for Kc of 0.3MPa√m for the shells of commercial hens’ eggs (Taylor et al., 2016). This is a credible value which fits in very well with other reported values for related materials. Eggshell is made from calcium carbonate, plus a few percent of organic material. The fracture toughness of pure calcium carbonate mineral in the form of calcite is slightly lower at 0.2MP√m, whilst significantly higher values have been reported for calcite and aragonite in marine organisms, such as mussel shells (Kc = 3, (Fitzer et al., 2015)) and Nacre (Kc = 4.3, (Richter et al., 2011)). These results provide interesting evidence for toughness enhancement in this material system.
Another example involving toughness measurement in stiff, brittle biological materials is our work on insect cuticle (Dirks and Taylor, 2012). We used the tibiae of locusts and other insects. In some cases these limbs approximate quite closely to hollow tubes of circular cross section. Loading these in cantilever bending after introducing a sharp notch in the form of a transverse cut, we were able to take advantage of an existing K-solution of the form of equation 3, for which the appropriate values of F had already been determined (Takahashi, 2002). We found a value for Kc of 4.12MPa√m, which is surprisingly high considering this material’s relatively low stiffness (E = 3.05GPa). As a result, the Gc value is 5.56kJ/m2, which makes insect cuticle one of the toughest materials in Nature. The same approach was later applied to assess the ability of the insect to repair the cut by deposition of extra cuticle (Parle et al., 2016).
When Toughness is Not Constant: the Resistance Curve
An important phenomenon arises in materials which display toughness enhancement: in these materials the measured value of Kc may increase as the crack grows. This phenomenon has been well studied in bone (for example (Nalla et al., 2004; Nalla et al., 2005b)). If a sharp notch is introduced into the material, it is found that a crack will initiate and grow from the notch at a relatively low K value. However, as the crack extends, increasingly large K values are required for continued propagation. This behaviour, when plotted as in figure 4, is known as a resistance curve or R-curve. Normally the critical K value levels out to a constant at large crack extensions, allowing one to define two constants the initiation toughness (at zero crack extension) and the propagation toughness (at large extensions).
The important point to realise here is that this is a real material phenomenon, not an artefact of the test method. In these materials Kc is not constant: it varies depending on the amount of crack growth. The reason is that mechanisms of toughening act at the microstructural scale, therefore some of these mechanisms do not have an effect until the crack has grown an appreciable distance. Examples are crack deflection and the support of the crack faces by unbroken ligaments: in the case of bone these ligaments are osteons (see figure 4), which have a typical diameter of 200m. So as the crack extends
over distances of the order of millimetres, the number of unbroken osteons spanning the crack faces increases, effectively reducing the stress intensity at the crack tip. The R-curve is an important set of data which helps us to understand how toughness is achieved in these kinds of materials. As figure 4 shows, the slope of the R-curve may change, in this case as a result of aging, with important medical consequences.
This is yet another situation in which the nature of the notch or pre-crack has a major effect on the results. The data shown in the figure were obtained by starting with a sharp notch cut into the specimen. If a real crack had been introduced, for example by applying an impact or growing a fatigue crack, then the pre-crack would already be benefitting from the above toughening mechanisms, in which case no R-curve would be measured: the crack would extend only when the “propagation toughness” value of K was applied. Another situation in which the R-curve would not manifest itself is if K rises sufficiently quickly during crack growth. For example consider the results in figure 4 for the bone samples in the 34-41yrs group. The initiation toughness is approximately 2MPa√m. Suppose the initial notch length was 1mm and a stress high enough to cause crack extension is applied. As the crack extends a further 3mm the Kc value will rise to about 3.2MPa, however if the stress remains constant the applied K will also rise in proportion to the square root of the crack length (according to equation 3) and so will now be equal to 4MPa√m. Therefore the crack, once it begins to grow, will keep growing, causing rapid total failure of the specimen, and the Kc value recorded will be the “initiation toughness”. Slow, gradual crack extension will be more likely if the original notch is longer, and also if one applies a fixed displacement rather than a fixed load. Other aspects of the specimen geometry will also come into play. When making any measurements of mechanical properties one should always bear in mind the application to which the results will be put. Traumatic fractures in bone typically occur from very small pre-existing fatigue cracks less than 1mm long, so one could argue that the toughness enhancement seen on the R-curve will not be relevant in this case, but it may be very significant in the gradual development of fatigue cracks leading to stress fractures.
Toughness Control and Strength Control
A problem which was alluded to above in the discussion on bamboo, is examined here in in more detail, and illustrated in figure 5. Suppose tests are carried out to measure Kc, using specimens which contain cracks of different lengths. If the resulting values of the failure stress f, are plotted as a function of
crack length a, using logarithmic scales, then (given equation 3 and assuming constant F) the results should fall on a straight line of slope -0.5. However, if we consider increasingly small values of a, then a critical length ao will be reached where f becomes equal to the material’s ultimate tensile strength, u. For values of a
The values of Kc calculated from tests in which a is too small will be lower than the true value and will depend on a. In these circumstances one is not measuring an inaccurate value of Kc, one is simply not measuring Kc at all. The experiment may appear to be valid, because in most cases the failure will still proceed from the tip of the notch or pre-crack, since the stress will be higher there than elsewhere. Only afterwards when the data are analysed will the error become apparent. International standards for toughness testing contain instructions for various checks which must be carried out to confirm the validity of a Kc value (Zhu and Joyce, 2012). Without going into detail here it should be noted that not only must the crack length be sufficiently large, there are also size limits on other specimen dimensions such as the length of the remaining ligament (equal to the specimen width minus a) and also the specimen thickness (in relation to the establishment of plane strain or plane stress conditions).
Biological materials for which this problem is particularly acute are the soft tissues. These materials often have relatively high ratios of toughness to strength, so ao may be large. Table 3 shows results from the literature for various soft tissues, for which we made estimates of ao (for more details see (Taylor et al., 2012)). It can be seen that in most cases the value of ao is similar to, or larger than, the crack length, implying that the toughness measurements in these papers may be unreliable. For our own tests on porcine muscle we demonstrated by analysing the data that even with cracks several millimetres in length we were always in the regime of strength control. The only exception we could find amongst the soft tissues was the work of Chin-Purcell and Lewis on articular cartilage. They classified their samples into three grades depending on toughness, but in all cases the ao values turn out to be much smaller than the chosen crack length of 0.2mm (Chin-Purcell and Lewis, 1996). This implies that fracture mechanics methods will be useful in understanding the development of crack-like defects in cartilage in vivo.
Another strategy to determine the toughness of soft tissues which can be used to avoid the above problem is to use a tear test. In this test a pre-crack is gradually extended by applying out-of-plane shear forces in an action which is the same as that commonly used to tear a sheet of paper. The approach is useful for thin sheets of flexible material and has been applied to a number of soft biological tissues such as muscle (Purslow, 1985). Values of Gc and Kc can be obtained: strictly speaking the parameter being measured is the shear toughness (usually called the Mode III toughness) rather than the tensile toughness (known as Mode I), though these two parameters are usually closely related in magnitude.
Conclusions
1. Many biological materials fail by cracking, therefore fracture toughness is an important property for characterising their mechanical behaviour.
2. Stiff, brittle materials such as bamboo and eggshell present challenges owing to the difficulty of making a standard test specimen. Some ingenuity is required to generate crack propagation and failure under conditions in which a valid measurement of fracture toughness can be made.
3. Materials which display toughness enhancement via toughening mechanisms at the microstructural level often show R-curve behaviour in which toughness increases with crack extension. Examples are bone and the shells of marine organisms. Care is needed to obtain reliable data and to interpret these data in relation to practical failure cases.
4. A common cause of invalid data is when failure occurs at a nominal stress close to, or equal to, the material’s ultimate tensile strength. Fracture may still occur from crack growth starting at the notch or pre-crack but the measured fracture toughness will be incorrect. The critical length parameter ao is useful in detecting this problem.
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Taylor,D., Kinane,B., Sweeney,C., Sweetnam,D., OÇÖReilly,P., and Duan,K. (2014) The biomechanics of bamboo: investigating the role of the nodes. Wood Science and Technology 49, 345-357. Taylor,D., O'Mara,N., Ryan,E., Takaza,M., and Simms,C. (2012) The fracture toughness of soft tissues. Journal of the Mechanical Behavior of Biomedical Materials 6, 139-147. Taylor,D., Walsh,M., Cullen,A., and OÇÖReilly,P. (2016) The fracture toughness of eggshell. Acta Biomaterialia 37, 21-27. Thambyah,A., Zhang,G., Kim,W., and Broom,N.D. (2012) Impact induced failure of cartilage-on-bone following creep loading: A microstructural and fracture mechanics study. Journal of the Mechanical Behavior of Biomedical Materials 14, 239-247. Vincent,J.F.V. (1990) Structural Biomaterials. Princeton University Press, Princeton, USA. Wang,F., Shao,Z., Wu,Y., and Wu,D. (2013) The toughness contribution of bamboo node to the Mode I interlaminar fracture toughness of bamboo. Wood Sci Tech. 48 1257-1268 Wu,K.S., Van Osdol,W.W., and Dauskardt,R.H. (2006) Mechanical properties of human stratum corneum: Effects of temperature, hydration, and chemical treatment. Biomaterials 27, 785-795. Xiao,J.F., Zhang,Y.N., Wu,S.G., Zhang,H.J., Yue,H.Y., and Qi,G.H. (2014) Manganese supplementation enhances the synthesis of glycosaminoglycan in eggshell membrane: A strategy to improve eggshell quality in laying hens. Poultry Science 93, 380-388. Zhu,X.K. and Joyce,J.A. (2012) Review of fracture toughness (G, K, J, CTOD, CTOA) testing and standardization. Engineering Fracture Mechanics 85, 1-46.
Figure 1: Considering a cracked body of arbitrary shape, subjected to a load L and displacement at the loading point d, the stress intensity K is defined as a function of geometry and loading. The strain energy release rate G can be found as the rate of change of the strain energy W in the body with change in crack area. Critical values Kc and Gc can both be used to define the material’s fracture toughness.
Figure 2: Top row: examples of standard test specimens for fracture toughness measurement: (a) compact tension specimen; (b) single edge notched three point bend specimen. Middle and bottom rows: examples of specialised test specimens created to measure fracture toughness in bamboo: (c) a specimen cut from the curved wall of a culm (Shao et al., 2009); (d) a length of culm loaded across its diameter from one end (Taylor et al., 2014); (e) a length of culm loaded across its diameter from the centre (Mitch et al., 2010); (f) a four point bend specimen with cracks extending longitudinally (Tan et al., 2011).
Figure 3: A series of photographs showing (from left to right) increasing force until failure, applied to an egg with a notch machined at the equator (arrowed). Experimental results showing the effect of notch root radius on measured fracture toughness Kc.
Figure 4: Examples of R-curves for bone (Nalla et al., 2004) from subjects of different ages, showing variation in the initiation and propagation toughness. The photos show examples of uncracked ligaments bridging the cracks (arrowed) measured at different distances from the notch.
Figure 5: The effect of crack length on failure stress. The thick line indicates typical experimental data, showing regimes of strength control, toughness control and mixed control.
Table 1: Values of Fracture Toughness Kc for Bamboo, From Various Sources Kc (MPa√m)
Source
20-100
(Amada and Untao, 2001)
5.5-8.0
(Low et al., 2006)
3.5
(Shioya and Asahina, 2013)
1.7
(Tan et al., 2011)
1.5
(Wang et al., 2013)
1.4
(Taylor et al., 2014)
0.78
(Shao et al., 2010)
0.18
(Mitch et al., 2010)
Table 2: Values of Fracture Toughness Kc for Eggshell, From Various Sources Kc (MPa√m)
Source
11.1
(Mabe et al., 2003)
12.6
(Xiao et al., 2014)
≈20
(Gosler et al., 2011)
0.3
(Taylor et al., 2016)
Table 3: Values of Crack Length and ao for Various Soft Tissues Material
Crack Length a ao (mm) (mm) 0-5 14.1
Source
0-15
31.1
(Purslow, 1985)
0.025 and 0.1
0.11
(Stok and Oloyede, 2007)
0.2
0.0042-0.056
Neocartilage
1.7-2
1.54
TMJ disc
3
1.80
Stratum corneum
0-20
0.051
(Chin-Purcell and Lewis, 1996) (Oyen-Tiesma and Cook, 2001) (Koombua et al., 2006) (Wu et al., 2006)
Porcine muscle Bovine muscle (cooked) Cartilage (superficial layer) Cartilage
(Taylor et al., 2012)
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PII: DOI: Reference:
S1751-6161(17)30295-3 http://dx.doi.org/10.1016/j.jmbbm.2017.07.007 JMBBM2406
To appear in: Journal of the Mechanical Behavior of Biomedical Materials Received date: 26 April 2017 Revised date: 21 June 2017 Accepted date: 4 July 2017 Cite this article as: David Taylor, Measuring Fracture Toughness in Biological Materials, Journal of the Mechanical Behavior of Biomedical Materials, http://dx.doi.org/10.1016/j.jmbbm.2017.07.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Measuring Fracture Toughness in Biological Materials David Taylor1 Trinity Centre for Bioengineering, Department of Mechanical and Manufacturing Engineering, Trinity College Dublin, The University of Dublin, College Green, Dublin 2, Ireland
[email protected]
Abstract Many biological materials fail by cracking. Examples are bone fractures, contact damage in eggs, splits in bamboo culm and defects in cartilage. The mechanical property that best describes failure by cracking is fracture toughness, which quantifies the ease with which cracks propagate and defines a material’s tolerance for pre-existing cracks and other stress concentrating features. The measurement of fracture toughness presents some challenges, especially for biological materials. To obtain valid results requires care and, in many cases, considerable ingenuity to design an appropriate specimen and test protocol. Common mistakes include incorrect interpretation of the mechanics of loading in unusual specimen designs, and failures occurring at the material’s ultimate tensile strength as a result of specimens or cracks being too small. Interpretation of the resulting toughness data may also present challenges, for example when R-curve behaviour is present. In this article, examples of good and bad practice are described, and some recommendations made.
Keywords: fracture toughness; crack; eggshell; cuticle; soft tissues; cartilage; bone
Introduction
Toughness is an important property for any structural material, but it is a property which is difficult to define and measure. It is a relatively recent invention, first defined in the 1920’s and not introduced into engineering practice until the 1960’s. The need for this parameter becomes clear when we note that there are some materials which have good strength and toughness and yet fail easily. Examples of these materials are glasses and engineering ceramics such as alumina and silicon carbide. Measurements of 1
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yield strength, hardness and elastic (Young’s) modulus for these materials show relatively high values, implying that they can be used in load-bearing applications. However they often break at low applied loads: the key fact here is that they break by cracking. The existence of this different mechanism of failure (cracking rather than yielding) implies the need for a different mechanical property to characterise the material. Another class of materials also illustrates the need for the toughness parameter: some materials have relatively low strength and Young’s modulus but can absorb a large amount of energy before failing, for example some polymers which are used for impact resistance. Though they yield relatively easily they can absorb a lot of energy in post-yield straining. Again we can appreciate that another mechanical property, toughness, is needed to characterise and understand these materials.
However the property called “toughness” is not the same in these two examples. In the first example, the appropriate parameter is the so-called “fracture toughness” which describes the ability of a material to resist the propagation of pre-existing cracks, since it is by this means that glass fails. In the second example the area under the stress/strain curve is more useful because it quantifies the amount of energy that can be absorbed per unit volume of material. Various mechanisms may operate to absorb energy, including multiple cracking, plasticity and viscoelasticity. By contrast, the “fracture toughness parameter” is concerned with the propagation of a single crack. These two parameters are both referred to as the material’s “toughness” in the literature, but they are in fact quite different. Several other toughness parameters have also been defined and used, so care is required when reading the literature to understand exactly what is being measured: a material may perform well according to one measure of toughness but badly according to another.
The present paper will be confined to consideration of the “fracture toughness” parameter. The importance of this parameter becomes clear when we note that most cases of mechanical failure in biological materials occur by the mechanism of cracking. Stiff materials in our bones, teeth and fingernails develop small cracks as a result of normal loading activities (O'Brien et al., 2000): fractures can occur if these cracks grow quickly, as a result of impact for example, or slowly as a result of fatigue failure under repeated loading (Burr, 1997). Soft tissues also fail by cracking: examples are skin damage and cartilage defects, the latter being a major factor in the progression of osteoarthritis (Thambyah et al., 2012). Tooth enamel has an extremely low fracture toughness as a result of its high mineral content (Bajaj and Arola, 2009). Bone achieves a somewhat higher fracture toughness thanks to features in its
microstructure which hinder crack propagation (Nalla et al., 2005a). Other examples of toughness enhancement achieved as a result of microstructure can be found in nature, for example in the shells of molluscs and other aquatic organisms (Barthelat et al., 2007). Toughness may be reduced as a result of aging and disease, and this contributes to the reduced strength of bone observed in debilitating medical conditions such as osteoporosis (Nalla et al., 2004). Better understanding of how natural materials achieve toughness enhancement can lead to improvements in engineering materials through biomimetic approaches.
After an introduction to the definition and measurement of this parameter, the existing literature on its application to biological materials will be reviewed, highlighting the particular challenges which these materials present.
The Definition and Measurement of Fracture Toughness
Fracture toughness is easy to define, but difficulties arise when it comes to the practical measurement of this parameter. The basic concept (Griffith, 1920) defines a parameter Gc which is the energy needed for crack propagation, more specifically the energy per unit area of crack formed. This can be related to the change in the total strain energy in the body, W, with respect to crack area. Assuming a throughthickness crack of length a in a sample of material of constant thickness B, we can define the rate of change of strain energy with crack length as G, thus:
G
1 dW B da
(1)
G can be measured in practical experiments (or in simulated ones using, for example, finite element analysis) by finding the area under the force/displacement curve for a specimen containing a crack and noting how this changes when the crack length is increased (see figure 1).
If this release of strain energy G is large enough to supply the energy needed for crack growth, then the crack will propagate. Depending on the sample and the type of loading, the crack may then grow rapidly and unstably (if G increases monotonically with a) leading to total failure of the specimen, or it may grow in a gradual, stable manner if more applied load or displacement is required to supply the
necessary energy. In either case we can define a critical value, Gc, as the fracture toughness of the material.
Strictly speaking the term Gc is reserved for cases in which the force/displacement behaviour is linear up to failure: so-called “Linear Elastic Fracture Mechanics”. If the behaviour is non-linear, for example due to plasticity, viscoelasticity or microdamage, then the critical energy can still be defined in the same way but it is usually given the term Jc.
This energy-based definition is useful for quantifying and comparing materials, but another term, Kc, is more commonly used in engineering applications because it expresses toughness in terms of the critical stress f required to cause failure by propagation of a crack. In many practical cases the relationship can be expressed as:
K c F f a
(2)
Here F is a factor which depends on the geometry of the body, the geometry and size of the crack, and the type of loading applied. Loading can be either tensile (known as Mode I); in-plane shear (Mode II) or out-of-plane shear (Mode III), as all these types of loading can cause crack propagation. Kc is the critical value of a parameter K known as the stress intensity. Values of K as a function of applied loading and crack length have been determined for many common cases (see for example (Murakami, 1987)). For non-standard cases K can be found using finite element analysis: one approach is to examine the stresses in the vicinity of the crack tip (Janssen et al., 2002). Kc and Gc, both of which are referred to as the “fracture toughness” in published work, are linked by the following relationship:
K c Gc E
(3)
…where E is Young’s modulus (with a slight correction depending on whether the body is loaded in plane stress or plane strain). Several national and international standards exist which define precisely the procedures for experimental measurement of fracture toughness (for example ASTM E1820): figure 2 shows some typical test specimens.
Fracture Toughness Measurement in Stiff Biological Materials
It is convenient to consider two groups of biological materials separately: the hard, stiff materials and the soft tissues. In principle fracture toughness can be measured relatively easily in the hard tissues because their behaviour in the low stress regime (before significant yield or microdamage occurs) approximates to linear elastic and, being relative brittle, they can be induced to fail in this regime by introducing a pre-crack or sharp notch. In practice however, difficulties arise, the most obvious of which is the shape of the test specimen. It is often not possible to obtain enough material to create any of the standard test specimens such as those shown in figure 2, so a certain amount of ingenuity is required to invent a specimen and a test procedure. A good example is the case of bamboo, a material which can be obtained in large quantities but always in the form of a circular tube, known as a culm. Several workers have addressed this problem, and each has devised their own unique test specimen. Figure 2 shows some examples, which can be divided into two types. Some workers retained the tubular shape of the culm and found ways to introduce a crack and to load the tube in such a way as to put the crack under tensile stress, inducing it to propagate (e.g. (Mitch et al., 2010; Taylor et al., 2014). Other workers created specimens similar to the standard ones by machining material from the culm wall (e.g. (Shao et al., 2009; Tan et al., 2011). Difficulties with this latter approach are the fact that the culm wall is curved and that the material is not consistent through thickness: mechanical properties change as one moves from outside to inside.
Examining the published results for the fracture toughness of bamboo (see Table 1) we find enormous differences, with reported Kc values varying by almost three orders of magnitude. Some of the variability in these results can be accounted for by differences in species and in the hydration state of the material, but these effects are relatively minor. The very small value of 0.18MPa√m (Mitch et al., 2010) can be explained by poor understanding of the mechanics involved. These workers used tubular specimens cut directly from bamboo culms. They inserted a split pin into a hole drilled across the tube diameter and caused cracking by applying tensile forces to separate the two halves of the pin (see figure 2). Unfortunately when calculating Kc they used a calibration based on remote loading which is not appropriate in this case. Much higher stresses will arise than expected, partly because the force is being applied locally, via pressure on the walls of the hole, and partly due to through-thickness bending caused by the curvature in the culm walls.
Other workers measured very high values, up to 100MPa√m, which, if true, would mean that bamboo is as tough as steel (Amada and Untao, 2001). The problem which arises here is the anisotropy of the material. Bamboo is extremely anisotropic, being much stronger in the longitudinal direction. Most of the results quoted in Table 1 refer to the Kc value for cracks propagating longitudinally along the culm, but Amada and Untao attempted to measure the toughness in the transverse direction, by introducing a notch (in the form of a cut with a razor blade) into the side of the specimen and applying longitudinal tension. By using cuts of different depth they were able to measure the fracture resistance at different distances from the outer surface. Their Kc values decrease from 100MPa√m near the outside to 20MPa√m near the inside. They also provided data on the tensile strength of small samples taken at different distances from the outer surface, which varied from 400-100MPa. It is evident that, in order to cause failure in the cracked specimens, they had to apply stresses similar to the tensile strength of the material. Under these conditions the specimens will still break at the location where the crack has been introduced, because here the area of the cross section is smaller, but the stress at which failure occurs will be a measure of the tensile strength, not the toughness. Even then it is difficult to understand how values as high as 100MPa√m could have occurred so there may in addition have been some calculation errors.
Something similar has occurred in the work of Low et al, the difference being that they loaded their specimens in bending rather than tension, forcing the failure to begin at the notch root where the bending stresses are highest (Low et al., 2006). Photographs in their paper clearly show that when cracking occurs from the notch the crack grows along the culm, at right angles to the notch direction. Tan et al (2011) used a similar specimen geometry but, realising that crack growth occurred in this direction, they used an appropriate K-solution which had previously been derived for crack growth along a bi-material interface (see figure 2).
The problem by which failure in a toughness test occurs at the material’s ultimate strength, also arises in other materials, especially in soft tissues, so it will be discussed in more detail below. The true value of Kc for longitudinal crack propagation in bamboo appears to be of the order of 1-2MPa√m, based on the consensus of several other papers listed in Table 1. There is certainly more work to be done here to improve on these measurements and to answer important questions such as how this value varies with location in the culm.
Another example which illustrates the difficulty of obtaining an appropriate specimen geometry is the case of eggshell, where one is dealing with a thin curved shell of a material which is so brittle that it is almost impossible to cut a specimen from it without causing damage. There are surprisingly few measurements of Kc for eggshell in the literature, despite the obvious commercial and biological interest in this material and its response to cracks and other defects. Table 2 shows all the available data. Mabe et al conducted tests on whole eggs, applying compression parallel to the egg’s longest axis, using two steel plates (Mabe et al., 2003). This type of test will cause cracks to form and grow from the two contact points, and in principle it should be possible to deduce Kc from the lengths of the cracks produced, though the analysis would be quite complex. Macleod et al used computer modelling to analyse crack growth in eggshell but stopped short of specifying a method for toughness measurement (Macleod et al., 2006). The result obtained for Kc by Mabe et al, of approximately 11MPa√m, is clearly erroneous because there is almost no ceramic material in existence which has a fracture toughness greater than 10MPa√m. The method of analysis is not described in the paper, which only gives an equation that doesn’t contain the crack length as a parameter. Unfortunately this same equation was used more recently by other workers (Xiao et al., 2014) who obtained results of similar magnitude, an example of how incorrect work can propagate in the literature.
Other researchers measured the fracture toughness for eggshell by recording the energy required when using a pair of scissors to cut the material (Gosler et al., 2011). This method has been used successfully for other materials, (Vincent, 1990): by this means one is measuring directly the energy needed to create the crack surfaces, and thus finding Gc. There is a large amount of scatter in the data from Gosler et al, but a typical value, when converted into Kc using equation 3, gives again an impossibly high result of about 20MP√m. It is not immediately obvious why this high value occurred. One practical problem with this type of test is to account for the energy which does not contribute to the creation of new surfaces, such as that lost in friction in the movement of the scissors, but this correction is relatively easy to make by running the test with no specimen or a specimen of known toughness. In this case the problem may be that the scissors will, in addition to creating the cut surface, also cause a lot of damage in the form of microscopic cracking on the sides of the cut, which will consume extra energy.
We took a different approach to the characterisation of eggshell toughness (Taylor et al., 2016), which relies on the fact that if a sphere is axially compressed by a pair of forces L, then the stress field, which is
quite complex near the loading points, takes a very simple form at the equator. Here there is an axial compressive stress c and a circumferential tensile stress t, of equal magnitude, given by:
t
L 2rt
(4)
…where r and t are the radius and thickness of the spherical shell, respectively. We took advantage of this tensile stress by machining a small notch to encourage a pair of cracks to form at a point on the equator and grow up and down towards the poles. We protected the contact point at the poles to prevent cracks from starting there (see figure 3). The K-solution for this case is very simple, the value of F in equation 3 being unity, so Kc can be found from the critical load Lf needed to cause failure by crack propagation from the notch (of total length 2a), as:
Kc
Lf 2rt
a
(5)
In this work we used a machined notch rather than a pre-crack, an approach which is quite common and was used in most of the work on bamboo discussed above. However this must be done with care because there are several ways in which the nature of this notch can affect the result. In our work on eggshell we considered the issue of the root radius of the notch, which is necessarily a finite quantity rather than being zero as assumed for a perfect crack. Of course any real crack will have a finite radius, so the problem is only to find how large this radius can be without affecting the result. In this paper we recorded Kc for several different notch radii and extrapolated back to zero radius (see figure 3): there are also theoretical approaches for predicting the effect of notch radius from first principles (see for example (Taylor, 2007)).
We obtained a value for Kc of 0.3MPa√m for the shells of commercial hens’ eggs (Taylor et al., 2016). This is a credible value which fits in very well with other reported values for related materials. Eggshell is made from calcium carbonate, plus a few percent of organic material. The fracture toughness of pure calcium carbonate mineral in the form of calcite is slightly lower at 0.2MP√m, whilst significantly higher values have been reported for calcite and aragonite in marine organisms, such as mussel shells (Kc = 3, (Fitzer et al., 2015)) and Nacre (Kc = 4.3, (Richter et al., 2011)). These results provide interesting evidence for toughness enhancement in this material system.
Another example involving toughness measurement in stiff, brittle biological materials is our work on insect cuticle (Dirks and Taylor, 2012). We used the tibiae of locusts and other insects. In some cases these limbs approximate quite closely to hollow tubes of circular cross section. Loading these in cantilever bending after introducing a sharp notch in the form of a transverse cut, we were able to take advantage of an existing K-solution of the form of equation 3, for which the appropriate values of F had already been determined (Takahashi, 2002). We found a value for Kc of 4.12MPa√m, which is surprisingly high considering this material’s relatively low stiffness (E = 3.05GPa). As a result, the Gc value is 5.56kJ/m2, which makes insect cuticle one of the toughest materials in Nature. The same approach was later applied to assess the ability of the insect to repair the cut by deposition of extra cuticle (Parle et al., 2016).
When Toughness is Not Constant: the Resistance Curve
An important phenomenon arises in materials which display toughness enhancement: in these materials the measured value of Kc may increase as the crack grows. This phenomenon has been well studied in bone (for example (Nalla et al., 2004; Nalla et al., 2005b)). If a sharp notch is introduced into the material, it is found that a crack will initiate and grow from the notch at a relatively low K value. However, as the crack extends, increasingly large K values are required for continued propagation. This behaviour, when plotted as in figure 4, is known as a resistance curve or R-curve. Normally the critical K value levels out to a constant at large crack extensions, allowing one to define two constants the initiation toughness (at zero crack extension) and the propagation toughness (at large extensions).
The important point to realise here is that this is a real material phenomenon, not an artefact of the test method. In these materials Kc is not constant: it varies depending on the amount of crack growth. The reason is that mechanisms of toughening act at the microstructural scale, therefore some of these mechanisms do not have an effect until the crack has grown an appreciable distance. Examples are crack deflection and the support of the crack faces by unbroken ligaments: in the case of bone these ligaments are osteons (see figure 4), which have a typical diameter of 200m. So as the crack extends
over distances of the order of millimetres, the number of unbroken osteons spanning the crack faces increases, effectively reducing the stress intensity at the crack tip. The R-curve is an important set of data which helps us to understand how toughness is achieved in these kinds of materials. As figure 4 shows, the slope of the R-curve may change, in this case as a result of aging, with important medical consequences.
This is yet another situation in which the nature of the notch or pre-crack has a major effect on the results. The data shown in the figure were obtained by starting with a sharp notch cut into the specimen. If a real crack had been introduced, for example by applying an impact or growing a fatigue crack, then the pre-crack would already be benefitting from the above toughening mechanisms, in which case no R-curve would be measured: the crack would extend only when the “propagation toughness” value of K was applied. Another situation in which the R-curve would not manifest itself is if K rises sufficiently quickly during crack growth. For example consider the results in figure 4 for the bone samples in the 34-41yrs group. The initiation toughness is approximately 2MPa√m. Suppose the initial notch length was 1mm and a stress high enough to cause crack extension is applied. As the crack extends a further 3mm the Kc value will rise to about 3.2MPa, however if the stress remains constant the applied K will also rise in proportion to the square root of the crack length (according to equation 3) and so will now be equal to 4MPa√m. Therefore the crack, once it begins to grow, will keep growing, causing rapid total failure of the specimen, and the Kc value recorded will be the “initiation toughness”. Slow, gradual crack extension will be more likely if the original notch is longer, and also if one applies a fixed displacement rather than a fixed load. Other aspects of the specimen geometry will also come into play. When making any measurements of mechanical properties one should always bear in mind the application to which the results will be put. Traumatic fractures in bone typically occur from very small pre-existing fatigue cracks less than 1mm long, so one could argue that the toughness enhancement seen on the R-curve will not be relevant in this case, but it may be very significant in the gradual development of fatigue cracks leading to stress fractures.
Toughness Control and Strength Control
A problem which was alluded to above in the discussion on bamboo, is examined here in in more detail, and illustrated in figure 5. Suppose tests are carried out to measure Kc, using specimens which contain cracks of different lengths. If the resulting values of the failure stress f, are plotted as a function of
crack length a, using logarithmic scales, then (given equation 3 and assuming constant F) the results should fall on a straight line of slope -0.5. However, if we consider increasingly small values of a, then a critical length ao will be reached where f becomes equal to the material’s ultimate tensile strength, u. For values of a
The values of Kc calculated from tests in which a is too small will be lower than the true value and will depend on a. In these circumstances one is not measuring an inaccurate value of Kc, one is simply not measuring Kc at all. The experiment may appear to be valid, because in most cases the failure will still proceed from the tip of the notch or pre-crack, since the stress will be higher there than elsewhere. Only afterwards when the data are analysed will the error become apparent. International standards for toughness testing contain instructions for various checks which must be carried out to confirm the validity of a Kc value (Zhu and Joyce, 2012). Without going into detail here it should be noted that not only must the crack length be sufficiently large, there are also size limits on other specimen dimensions such as the length of the remaining ligament (equal to the specimen width minus a) and also the specimen thickness (in relation to the establishment of plane strain or plane stress conditions).
Biological materials for which this problem is particularly acute are the soft tissues. These materials often have relatively high ratios of toughness to strength, so ao may be large. Table 3 shows results from the literature for various soft tissues, for which we made estimates of ao (for more details see (Taylor et al., 2012)). It can be seen that in most cases the value of ao is similar to, or larger than, the crack length, implying that the toughness measurements in these papers may be unreliable. For our own tests on porcine muscle we demonstrated by analysing the data that even with cracks several millimetres in length we were always in the regime of strength control. The only exception we could find amongst the soft tissues was the work of Chin-Purcell and Lewis on articular cartilage. They classified their samples into three grades depending on toughness, but in all cases the ao values turn out to be much smaller than the chosen crack length of 0.2mm (Chin-Purcell and Lewis, 1996). This implies that fracture mechanics methods will be useful in understanding the development of crack-like defects in cartilage in vivo.
Another strategy to determine the toughness of soft tissues which can be used to avoid the above problem is to use a tear test. In this test a pre-crack is gradually extended by applying out-of-plane shear forces in an action which is the same as that commonly used to tear a sheet of paper. The approach is useful for thin sheets of flexible material and has been applied to a number of soft biological tissues such as muscle (Purslow, 1985). Values of Gc and Kc can be obtained: strictly speaking the parameter being measured is the shear toughness (usually called the Mode III toughness) rather than the tensile toughness (known as Mode I), though these two parameters are usually closely related in magnitude.
Conclusions
1. Many biological materials fail by cracking, therefore fracture toughness is an important property for characterising their mechanical behaviour.
2. Stiff, brittle materials such as bamboo and eggshell present challenges owing to the difficulty of making a standard test specimen. Some ingenuity is required to generate crack propagation and failure under conditions in which a valid measurement of fracture toughness can be made.
3. Materials which display toughness enhancement via toughening mechanisms at the microstructural level often show R-curve behaviour in which toughness increases with crack extension. Examples are bone and the shells of marine organisms. Care is needed to obtain reliable data and to interpret these data in relation to practical failure cases.
4. A common cause of invalid data is when failure occurs at a nominal stress close to, or equal to, the material’s ultimate tensile strength. Fracture may still occur from crack growth starting at the notch or pre-crack but the measured fracture toughness will be incorrect. The critical length parameter ao is useful in detecting this problem.
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Figure 1: Considering a cracked body of arbitrary shape, subjected to a load L and displacement at the loading point d, the stress intensity K is defined as a function of geometry and loading. The strain energy release rate G can be found as the rate of change of the strain energy W in the body with change in crack area. Critical values Kc and Gc can both be used to define the material’s fracture toughness.
Figure 2: Top row: examples of standard test specimens for fracture toughness measurement: (a) compact tension specimen; (b) single edge notched three point bend specimen. Middle and bottom rows: examples of specialised test specimens created to measure fracture toughness in bamboo: (c) a specimen cut from the curved wall of a culm (Shao et al., 2009); (d) a length of culm loaded across its diameter from one end (Taylor et al., 2014); (e) a length of culm loaded across its diameter from the centre (Mitch et al., 2010); (f) a four point bend specimen with cracks extending longitudinally (Tan et al., 2011).
Figure 3: A series of photographs showing (from left to right) increasing force until failure, applied to an egg with a notch machined at the equator (arrowed). Experimental results showing the effect of notch root radius on measured fracture toughness Kc.
Figure 4: Examples of R-curves for bone (Nalla et al., 2004) from subjects of different ages, showing variation in the initiation and propagation toughness. The photos show examples of uncracked ligaments bridging the cracks (arrowed) measured at different distances from the notch.
Figure 5: The effect of crack length on failure stress. The thick line indicates typical experimental data, showing regimes of strength control, toughness control and mixed control.
Table 1: Values of Fracture Toughness Kc for Bamboo, From Various Sources Kc (MPa√m)
Source
20-100
(Amada and Untao, 2001)
5.5-8.0
(Low et al., 2006)
3.5
(Shioya and Asahina, 2013)
1.7
(Tan et al., 2011)
1.5
(Wang et al., 2013)
1.4
(Taylor et al., 2014)
0.78
(Shao et al., 2010)
0.18
(Mitch et al., 2010)
Table 2: Values of Fracture Toughness Kc for Eggshell, From Various Sources Kc (MPa√m)
Source
11.1
(Mabe et al., 2003)
12.6
(Xiao et al., 2014)
≈20
(Gosler et al., 2011)
0.3
(Taylor et al., 2016)
Table 3: Values of Crack Length and ao for Various Soft Tissues Material
Crack Length a ao (mm) (mm) 0-5 14.1
Source
0-15
31.1
(Purslow, 1985)
0.025 and 0.1
0.11
(Stok and Oloyede, 2007)
0.2
0.0042-0.056
Neocartilage
1.7-2
1.54
TMJ disc
3
1.80
Stratum corneum
0-20
0.051
(Chin-Purcell and Lewis, 1996) (Oyen-Tiesma and Cook, 2001) (Koombua et al., 2006) (Wu et al., 2006)
Porcine muscle Bovine muscle (cooked) Cartilage (superficial layer) Cartilage
(Taylor et al., 2012)