There exists a variety of approaches to learning and teaching about bone, including the biological-systems, structural, and materials-science perspectives (1). This last consideration interests us most for the present discussion. Composed in large part of solid mineral deposits and organic fibers (for example, see References 2 and 3), bone may be considered as a composite material, similar to artificial fiber-reinforced composites (4) that should therefore follow, in theory, a few fundamental principles of materials science. We focus here on the propensity of many solid materials and composites to feature consistent deformational responses to the application of force at a constant rate. These responses may most readily be measured as changes in the materials’ length. Plots of such deformational responses to force are called stress-strain curves (such as those reported in Reference 5), the primary topic under consideration here.
THE MEANING OF STRESS-STRAIN CURVES
Stress-strain curves for bone (or any other material) display a plot of strain, typically on the x-axis, against stress, on the y-axis (5). Strain may be very simply defined as an “alteration of…dimensions experienced by a solid” (6). Alternatively, strain is a distortion or relative change (7) in a material’s length (∆L); thus, strain is often plotted as a percent change in length from the original length Lo, that is: ∆L/Lo x 100% (5). Stress, an “equilibrating application of force to a body” (6) may simply be conceived of as a force applied over a unit area (F/A), which is also the definition of pressure; alternatively, stress may also be a measured opposing force in response to strain (5, 7). Stress is oftentimes also termed an “applied load” to indicate that stress is the consequence of pressure loads applied to materials. Therefore, stress-strain curves plot the degree of deformation of a material in response to the applied force, or load (5).
THE SHAPE OF STRESS-STRAIN CURVES
Bone, like most materials, will behave according to a more-or-less linear stress-strain relationship under normative conditions (5). This stress/strain ratio—at least at moderate loads and for most materials—typically traces a straight line whose slope is called the Young's modulus, first defined as a material's "strength in resisting flexure of any kind" (8). (The Young's modulus is a term that will be familiar to many students who are preparing for the MCAT®.) This slope is alternatively called the elastic modulus, and is a measure of stiffness (see "Terminology of stress-strain relationships" below). Materials responding to stresses so as to produce a linear stress-strain curve are said to behave in an elastic manner (8,9) and thus can be said to be operating in their elastic region. When operating within its elastic region, a material from which the applied stress is relieved will return to its original length Lo (8,9). For example, heavy metal springs or coils such as those found in box-spring mattress supports, or as part of vehicular shock absorbers, typically operate in the elastic regions of their stress-strain curves, readily returning to their original resting length and shape following deformation due to compression or elongation stress, per Hooke's Law. Hard rubber or plastic balls, such as lacrosse and golf balls, behave similarly: re-assuming their original spherical shape after being bounced hard off of a wall or golf-club head, which temporarily flattens the side of the ball subjected to impact.
Most materials have a yield point, a point at which the stress-strain curve deviates from strict linearity, and where the curve's slope typically becomes noticeably shallower (10). Beyond this point, materials are said to be operating in the plastic region, and will not return to their original length (Lo) upon relief of stress, nor will they even return to the original elastic region (9). At this point, the material under study, having been subjected to plastic strain, is typically permanently deformed from its original shape. Following the example given above, a spring stretched beyond its yield point will not return to its original shape, but will remain permanently deformed, or "stretched out". This propensity of a material to behave differently depending upon its recent state underlies the concept of hysteresis—a (recent-)history dependence of a material's behavior (see below under "Hysteresis, Anisotropy, and Viscoelasticity").
At some degree of stress or strain, a real material will fail—either break or buckle—and the stress-strain curve also breaks off at this point, which is termed the rupture point or failure point (10). A spring pulled too hard can be torn apart, and a golf ball hit too hard can crack at its surface: in both cases, the materials of which these items are made have failed.
TERMINOLOGY OF STRESS-STRAIN RELATIONSHIPS
As described above, a typical curve depicting the stress-strain relationship of a given material (see Figure 1A) will usually feature: a linear elastic region; a yield point where the stress-strain relationship changes (often becoming non-linear), and beyond which the material is now operating in its plastic region; and a failure point where the curve ends, marking where the material under study has failed (either broken or buckled). Several key properties of bone can be determined from such curves. We have found it useful to define most of these terms in pairs of opposites.
Stiffness: The slope (Young's modulus) of the stress-strain curve in the elastic region is a measure of the material's stiffness, or its ability to resist deformation with an applied stress. Materials with a steep linear slope are stiff, and require greater applied force for a given degree of deformation; those with a shallow linear slope are flexible, and deform more readily than stiff materials would with the same applied force (8).
Strength: The y-value at the yield point marks the material's strength: the degree of stress it can withstand before visibly yielding (5)—that is, permanently changing shape. Strength can therefore also be thought of as the highest applied stress after relief from which a material will still be able to return to its original length Lo (8). This stress is marked by the yield point, at the end of the elastic region of the stress-strain curve. Materials with a high y-value at the yield point are strong; those with a low y-value are weak.
Ultimate strength and strain: The ultimate strength of a material may be defined as the highest level of stress it experiences across its stress-strain curve (10). This point may occur anywhere along the curve (often, but not always, at the failure point). The ultimate strain of the material is the highest level of deformation that the material experiences, and is necessarily at the failure point.
Ductility: Materials that have a long plastic region are termed ductile, because they can be stretched and drawn out before the failure point is reached (8). Materials with a short plastic region are termed brittle, and fail (break) with little additional strain beyond the yield point.
Failure strength and failure strain: The failure strength of a material is the value of the level of stress it experiences at failure. The failure strain of the material is the degree of deformation that the material has undergone at the failure point.
Toughness: Toughness is a measure of the energy that can be absorbed before failure (8), and may be determined from the total area under the stress-strain curve (5). Materials may have high, low, or moderate toughness. To have high toughness, materials must typically be both strong and relatively ductile.
It is extremely important to recognize that many of the preceding terms have common colloquial usages. It is essential that students know and understand the technical definitions of these terms, and that the students do not apply to biomechanical problems an understanding of the terminology from the terms' colloquial definitions. Note especially that: a) stiffness and strength are different—stiffness is a measure of the slope of the stress-strain curve, while strength is a value on the y-axis of the curve where the yield point is reached; and that b) strength and toughness—even though these terms' colloquial usages may suggest otherwise—do not refer to the same property (though they are somewhat related, in that very strong materials may also exhibit high toughness; see the definition of "Toughness" above). It also bears noting that materials may have the property of being hard or soft, but these properties cannot be readily determined from a stress-strain curve (11). Hardness is a measure of the ability of a material to resist penetration by a harder object; hardness is measured on a relative hardness scale (e.g., the Mohs hardness scale; see Reference 12). We do not address these terms further in the ensuing discussion.
Once students have learned about the basic principles of stress-strain curves and have applied these principles to bone biomechanics, it is possible to introduce students to the concepts of hysteresis, anisotropy, and viscoelasticity in this physiological context. These concepts will build and extend upon a fundamental understanding of stress-strain curves.
Hysteresis is a dependence of a material's behavior on its history (typically, its recent history); in general, hysteresis emerges as a consequence of the intrinsic properties of a system. This term—hysteresis—for material behavior was first coined by Sir James Alfred Ewing (13,14), with respect to the magnetization of a material in a magnetic field. A magnetic material that starts in an unmagnetized state with no applied magnetic field can be strongly magnetized with the application of a field, and can still retain this magnetism even after the field is removed—that is, returned to zero (15). This maintenance of magnetism in the absence of the field that originally induced it is an excellent example highlighting Ewing's definition of hysteresis in magnetic materials as a "lag" in behavior (14)—in this specific case, a near-permanent lag. The magnetized material remains magnetically polarized long after the removal of the stimulus (the original magnetic field) that induced said polarity in the material: the induced magnetization lingers (or lags); the material retains its magnetic polarity because it was recently in that state, thus displaying hysteresis.
Many solid materials also exhibit hysteresis in their stress-strain curves. With regards to bone, hysteresis may be evident in the different responses of bone to unloading (relief of stress), depending on whether the bone was recently operating in the elastic region or the plastic region of its stress-strain curve (see Figure 1A and discussion below).
Yet, a full representation of the effects of hysteresis is not readily apparent in stress-strain curves of the type displayed in Figure 1A. Such stress-strain curves plot responses only to a single iteration of stress application ending in material failure, thus terminating the experiment on that fragment of bone, and disallowing any further exploration of its properties (hysteresis or otherwise). For a more complete picture of hysteresis in bone stress-strain curves, see (16), which will show that the stress-strain curves for relaxation following relief of stress may trace different paths from those obtained upon initial loading, thus generating a hysteresis loop. Additionally, if the material sample (bone, in this case) is not stressed to failure, multiple hysteresis loops can be produced from sequential iterative cycles of loading and unloading. A stylized representation of hysteresis-loop data adapted from a series of bone hysteresis experiments (16) is presented in Figure 1B. These experiments represent a classic illustration of hysteresis, wherein the material's (bone's) behavior is different based upon whether it was recently subjected to zero stress (then loaded to produce the initial loading curve); or recently subjected to a heavy load at the end of the initial loading curve (then allowed to relax). The result is a differently-shaped stress-strain curve for relaxation as compared with the curve generated from loading.
Nonetheless, the basic concept of hysteresis can still be illustrated and discussed by contrasting the elastic and plastic regions of only a single stress-strain curve (as mentioned above in "The Shape of Stress-Strain Curves," and as shown in Figure 1A). A material that has just recently been stressed within the linear portion (elastic region) of its stress-strain curve will relax back to its original shape or length (Lo) upon relief of stress, as originally noted by Young (8). However, a material that has been stressed beyond the yield point will likely not return to the elastic region. That is, a material that has in its recent past been operating in its plastic region will remain there and not return to its original length upon unloading (9)—thus displaying hysteresis—unless the material is reworked, repaired, or reconstituted. The bone's behavior therefore depends upon its recent state; that is, the bone exhibits hysteresis.
It is worth noting, therefore, that hysteresis may include not only a lag in the effects of a stimulus on a material (as in magnetization), or a general recent-history dependence to a material’s behavior (as in a full hysteresis loop for bone), but also a permanent shift in a material’s response to a given stimulus once the material is stressed to a point past which its material properties have permanently changed (i.e., work hardening—see, for example, Reference 17).
It may be useful to ask students to produce their own examples of hysteresis in either everyday life or in other physiological contexts. A common example that students may be able to offer on their own (or to grasp and relate to if no student independently produces this specific example) is that of a foam mattress. Such mattresses exhibit clear hysteresis, in that whether they are in a full, flat-topped shape or an indented shape depends upon whether someone has recently been lying on top of it. In addition, deformation upon loading is rapid, but a return to the original flattened state takes time—thus, there is a lag in behavior, another illustration of hysteresis. This example closely matches the case of hysteresis in bone. In the realm of physiology, students may remember that voltage-gated sodium channels can occupy two different states at depolarized voltages: open and not inactivated (unblocked), or open and inactivated (blocked). Which state the channel occupies depends upon how long the depolarizing stimulus has been applied. This time-dependent channel behavior, which reflects the channel's most recent history, is also an example of hysteresis.
Early introduction and discussion of this concept of hysteresis in the context of bone permits its ready re-application to later physiology units, for example in aiding students with their understanding of hysteresis in lung compliance curves, which plot volume against pressure over the course of a breathing cycle. For a good example of a clinically-measured lung compliance curve, showing a full hysteresis loop, see Reference 18. Students who have wrestled with the idea of hysteresis in bone behavior should be able to grasp its manifestation in lung pressure-volume loops, where the pressure-volume curve for exhalation traces a different path from the pressure-volume curve for inhalation.
Anisotropy is a directional dependence of a material's behavior: a difference in how a material responds to a stimulus that depends on the direction from which the stimulus is applied. Anisotropy was first studied in relation to wave propagation (e.g., see Reference 19), and again, like hysteresis, was also heavily explored in the context of ferromagnetic materials (e.g., see Reference 20). For clarification, it may be useful to contrast the concept of anisotropy with that of isotropy—the more-common case where a material's properties and behavior are direction-independent. Materials whose structural components are organized in a non-random fashion are likely to exhibit mechanical anisotropy. Bone—whose mineral deposits are organized in the cylindrical osteons, also known as Haversian systems, in the form of lamellae running parallel to the bone's long axis (a structural observation first described in 1689 by Dr. Clopton Havers; see Reference 21)—is one such material.
When a long bone is stressed at different angles of load application, its Young's modulus and the overall stress-strain curve change slope and overall shape (22, 23, 24; also see Figure 2), in accordance with the anisotropic properties of the bone (22; also see Figure 2). These changes are also evident in the repositioning of the yield and failure points (indicated by black and red dots, respectively, in Figure 2). Longitudinal load angles—0o load: direct load application down the long axis of the bone—result in stress-strain curves featuring higher stiffness, strength, and toughness. All of these biomechanical properties—stiffness, strength, and toughness—decrease as the load angle increases towards transverse application: 90o load, applied perpendicular to the long axis of the bone. From a plot of multiple stress-strain curves obtained at different loading angles (see Figure 2), students should be able to observe readily that the biomechanical properties of the long bones change dramatically with direction of loading—a very clear and direct illustration of anisotropy. A recent report (25) has found that these physiological features of bone may be, at least in part, an emergent property of the nanoscale-structure of minerals in the bone extracellular matrix, which appears largely to take the form of aggregates of collagen-fibril-spanning platelets made up of curved mineral nanocrystals.
Again, it can be of value to ask students to produce their own examples of anisotropy in everyday-life or in pure-physiological contexts. A common example of anisotropy that most students should have experienced is that of light penetrating directly into a body of water when the rays enter at perpendicular or near-perpendicular angles, but reflecting off of the water's surface when the light is incident at an oblique angle. Light reflection off of certain beetles' carapaces is also anisotropic, accounting for an observer's seeing different colors of a shell from different angles. For a more clinically-relevant example, see Reference 26 on the anisotropy of ultrasound wave reflection in connective and muscle tissues.
A very simple follow-up question to a discussion of bone anisotropy asks which type of fracture would be more common: a crush (transverse-load) fracture or compression (longitudinal-load) fracture. Students should be able to determine from the family of stress-strain curves depicted in Figure 2 that bone breakage from crushing along the transverse axis would be most common because of the lower strength of the bone in this loading direction.
Viscoelasticity, first explored by such early physical-science pioneers as James Clerk Maxwell (7) is the key concept covered in the Lesson "What do Bone and Silly Putty® Have in Common?: A Lesson on Bone Viscoelasticity." Viscoelastic materials exhibit elasticity, which is a property typically associated with solids (see the preceding discussion of Young's modulus), as well as viscosity, a property more commonly associated with fluids. As discussed above, elastic solid materials (such as steel bars, for example) feature linear, or constant-ratio, stress-strain relationships (such that strain is proportional to the amount of stress, or vice-versa), wherein a material will attain (and maintain) a certain degree of deformation given a specific applied load (27). Such a material will also return to its original length when the load is removed (8,9). Viscous fluids (such as honey), on the other hand, may resist deformation (strain) in response to an applied load (stress), especially if the load is applied at a fast rate; however, viscous fluids can be deformed by slower-rate load application and will typically remain deformed even after the load is removed (27). The combination of these two properties of elasticity and viscosity in a viscoelastic material results in a stress-strain relationship that depends on the rate of strain (9).
Viscoelasticity therefore manifests as differences in a material's mechanical behavior based upon on how fast a deformation it undergoes: different rates of load application result in different responses to that load. This rate-dependent material behavior is sometimes also called a "time-dependence" (9). In short, the property of viscoelasticity will become apparent as changes to the slope and shape of a material's stress-strain curve at different loading rates.
For bone, this property of viscoelasticity can be most easily understood with reference to strain applied longitudinally along the bone's long axis (28). However, it is worth noting that this property applies not only to longitudinal loads inducing compressive or tensile strain parallel to the long bone-osteon axis—"pushing" and "pulling" loads, respectively—but also to torque effects (29). In bone, viscoelasticity is apparent in changes to the shape of the stress-strain curve, which becomes steeper and exhibits a higher yield point, as the strain rate (loading rate) increases (28; also see Figure 3). That is, the faster a long bone is compressed, the greater its stiffness (measured as the slope of the stress-strain curve, or the Young's modulus), and also the greater its strength (represented by a higher-load yield point). The functional consequence of this property is that bone is capable of withstanding short-duration high-impact forces without experiencing plastic deformation. Put plainly, when strained at high rates—at least for very brief (on the time scale of milliseconds) periods—long bones can readily "bounce back" to their original lengths.
However, it is worth nothing that the shorter plastic regions of the stress-strain curves that become apparent under high strain rates carry the consequence that the bone will be less tough than under low strain rates. The functional consequence of this feature of the stress-strain relationship is that the bone is more likely to fail with extensive strain experienced at high rates for extended times. This rate-dependent feature of the stress-strain relationship in bone—that is, its viscoelasticity—also explains why regular running and jumping do not (usually) result in permanent bone deformation (see References 30 and 31), but excessive strain may result in what is termed a "stress fracture," (32,33), or even a complete breakage of the bone.
For the benefit of the curious student who is concerned about underlying mechanisms, it bears mentioning that the viscoelastic properties of bone appear to be dependent in part on the water content of the bone (34), which varies between species (3). Some of these properties may arise either from load-rate-dependent changes in the rate of water flow between compartments along pressure gradients, which can contribute to the stiffening of bone during loading, or from effects of water content on the properties of collagen (see, for example, Reference 35).
These concepts—stress-strain relationships, hysteresis, anisotropy, viscoelasticity—especially when taken all together, may at first seem dauntingly complicated. However, if treated in a systematic way, by focusing on the graphical relationship between the variables of stress (applied load) and strain (deformation), each concept can be tackled in turn to build on students' understanding of the gross biomechanical properties of bone.
These relatively rarely-taught gross-physiology concepts in bone biomechanics can deeply enrich the study of bone anatomy at any course level. The graphical analysis of stress-strain curves is straightforward enough in principle to be taught at an introductory level, where students might have been exposed at most only to algebra and basic physics. In contrast, the inclusion of the concepts of hysteresis, anisotropy, and viscoelasticity as applied to bones can enrich any advanced-level survey of (or even a medical-school-level course in) skeletal anatomy and physiology. The functional context is very intuitive and easily applicable to students' everyday experiences; and the potential for simple and straightforward graphical interpretation can provide useful skill-building exercises. Additionally, the introduction of the concepts of hysteresis and anisotropy in this very-accessible context can help lay the groundwork for re-visiting these concepts in the context of other systems: for example, hysteresis in lung compliance curves (see Reference 18) and anisotropy in the spread of electrical activity in the heart (for an example of an in-depth discussion of these cardiac properties see pp.107-113 of Reference 36).
In the case of viscoelasticity, the use of Silly Putty®, an everyday plaything (see the accompanying Lesson article: "What Do Bone and Silly Putty® Have in Common?: A Lesson on Bone Viscoelasticity"), to display how these properties manifest in real life can provide a very dramatic illustration that students can easily grasp—and quite literally so, if the instructor gives them the putty to handle. We are hopeful that this short primer provides sufficient context and background for instructors at all levels to consider introducing these concepts into their own courses.
We are grateful to our exacting students in PNB 2264 and 2274 for holding us to a high standard of precision and clarity in our language and definitions. We acknowledge the anonymous authors of the Wikipedia page on Young's modulus (Wikipedia https://en.wikipedia.org/wiki/Young%27s_modulus Accessed November 20, 2017) for ideas on considerations to bear in mind regarding potential points of confusion on the relevant mechanics terminology when considered in the light of colloquial usages.
Silly Putty® is a registered trademark of Crayola Properties, Inc.: Silly Putty®. 2012. Crayola LLC, Forks Township, Northampton County, Pennsylvania, USA.
MCAT® is a registered trademark of the Association of American Medical Colleges, Washington, DC, USA.
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