Bones provide a structural framework to attach muscles and organs, enable movement through the attachment of muscles, provide physical protection of organs (such as the skull for the brain and the rib cage for the lungs), store minerals (calcium and phosphorus) and some fats (in the yellow marrow), and produce red blood cells (in the red marrow). The stiff nature of bone clearly enables it to form a semirigid framework, enable motion (because how could muscles do their job with flexible bones?), and provide organ protection. We will see it also means that large bones can serve these functions and still be hollow and filled with the soft marrow. There are long bones, as in the arms and legs; short cube-like bones; flat bones, as in the skull and ribs; and irregularly shaped bones, as in the pelvis and vertebrae.

Bone is a complex composite material, with living and nonliving matter. The living matter includes the cells osteoblasts and osteoclasts, which, respectively, make new bone and resorb (erode) existing bone, and osteocytes, which are former osteoblasts buried in bone they have made. Bone experiences net growth during childhood (with osteoblasts outperforming osteoclasts), steady state during most of adulthood (with the effects of osteoblasts and osteoclasts balancing each other), and net decrease in older age (with osteoclasts outperforming osteoblasts), leading to osteoporosis [71]. Excluding water , the nonliving matter of bone is 40% by weight (60% by volume) collagen and 60% by weight (40% by volume) calcium hydroxyappatite (). The  nm rod or plate crystals with hexagonal symmetry of the ceramic-like calcium hydroxyappatite are bound by the elastomer-like collagen. The inorganic ceramic component gives compact bone its large strength (a large elastic constant Y) and a large ultimate compressive stress (UCS). The collagen component makes bone much more flexible than a ceramic and much more stable under tension and bending. If you let a turkey leg sit for 24 h in 1 M HCl it becomes very flexible because the ceramic crystals have been dissolved and all that remains is a collagen structure [10]. About 1% of the organic component is proteoglycans (mucopolysaccharides). About 25% of the volume of bone is water, % of which is bound to the collagen. Spongy (or trabecular) bone has voids with lateral dimensions of 50–500 m.

Figure 4.3a shows the structure of a typical long bone, such as the femur. It has a long tubular shaft, the diaphysis (die-a’-phi-sus), which is a relatively thin shell of compact, cortical, or dense bone for strength. We will see later in this chapter that this type of hollow design maintains much of the strength of the corresponding solid structure, but with much less weight. At either end, the shaft broadens to form the epiphyses (e-pi-fi-sees’), where there is an overlayer of articular cartilage for lubrication and inside the bone, beneath the compact bone, is trabecular, cancellous, or spongy bone, which is a porous mesh of trabeculae (tra-bic’-you-lee) that can absorb shock. This porous bone is also found in the bones in the spinal column, where it provides some structural support and absorbs shock. Figure 4.3b shows that the layer of cortical bone in the shaft is thick and it becomes relatively very thin at the proximal end, where it surrounds the trabecular bone. There is bone marrow in the hollow shaft, the diaphysis. In short and irregular bones, spongy bone is encircled by a thin layer of compact bone, while in flat bones it is sandwiched by it.

Ligaments and tendons are dense connective tissue with a dense network of fibers, with few cells and little ground substance. Ligaments are tough bands of fibrous connective tissue. They are 55–65% water and 35–45% dry matter, which consists of 70–80% collagen (mostly type I), 10–15% elastin, and a small amount, 1–3%, of proteoglycans. The collagen (Fig. 4.4) gives ligaments their high tensile strength. The collagen helices assemble into microfibrils (4 nm in diameter), which assemble into subfrils (20 nm in diameter), which assemble into fibrils (50–500 nm in diameter), and then into collagen fibers (100–300 m in diameter) with fibroblast cells that synthesize the collagen.

The dry weight of tendons is 75–85% collagen (95% type I and 5% type III or V), elastin, and 1–2% proteoglycans. The structural hierarchy (Fig. 4.5) is like that of ligaments except they are arranged into packets called fascicles. Also, the bundles of collagen fibers are more parallel in tendons than in ligaments, as seen in Fig. 4.6.

In contrast, the dry matter of skin is 56–70% collagen (mostly type I), 5–10% elastin, and 2–4% proteoglycans.

In each of these soft materials, the collagen gives it tensile strength, while the elastin gives it elasticity, which is more important in ligaments than in tendons.

There are three types of cartilage: Hyaline (high’-uh-lun) cartilage, the most common in adults, is found in the ventral ends of ribs and covering the joint surfaces of bones. Elastic cartilage is more flexible, and is found in the external ear and eustachian tubes. Fibrocartilage occurs in the intervertebral disks. Cartilage that lines the bones in synovial joints (1–6 mm thick) is also called articular cartilage; it serves as a self-renewing, well-lubricated load bearing surface with wear prevention. It is most often hyaline cartilage, except in joints, such as the knee (the menisci), which contain fibrocartilaginous disks.

Articular cartilage is not meant to serve as a shock absorber to cushion forces or slow joint rotation [45], because it is so thin that it can absorb very little energy even though it is less stiff than cortical bone. In fact, it absorbs much less energy than muscles resisting joint rotation (eccentric contractions, Chap. 5) or the bones on either side of the joint (see Problem 4.12). About 30% of cartilage by mass is a solid matrix of collagen (40–70% of the dry mass, type II collagen and several other types: V, VI, IX, X, and XI) and proteoglycan (15–40% of the dry mass) and 70% is water and inorganic salts (as seen the structure in Fig. 4.7). Chondrocyte cells that manufacture the cartilage organic material comprise less than 5–10% of the volume. Cartilage is viscoelastic because it is a very flexible, porous material (50 Å voids) with voids that are filled with water. The water dissipates energy as it flows through the voids under compression.

In tension the collagen of the solid phase carries most of the load, while in compression both the solid and liquid phases carry the load. The viscoelasticity of cartilage is controlled by the exudation of the fluid through the pores in this biphasic material that is responsible for the lubrication of synovial joints.

In the harmonic regime, elastic materials are modeled as perfect springs obeying Hooke’s Law. This is usually expressed as
where F is the force felt by an object attached to a spring, with spring constant k, when the spring is extended a distance x. When the spring is extended a distance x, say to the right, the attached body feels a restoring force kx to the left (Fig. 4.8).

In examining such Hookean materials we will need to alter this viewpoint a bit. There is a length of spring or material for which there is no restoring force. We will call this equilibrium length . In (4.1), x is implicitly the deviation from this equilibrium length, the deformation . For reasons that will become clear soon, we prefer to refer x to this equilibrium length and so
Also, in studying problems with springs, we usually examine the effect of the spring forces on other masses. Here we are concerned with the effect of other forces on materials modeled as springs. Therefore we consider the force applied to the spring-like object, , which is the negative of the above force F felt by the object attached to the spring, and is
We have also changed notation so that length of the material is L and its relaxed length is .

When the material is relaxed. When there is a positive (Figs. 4.8b and 4.9a), the material is under tension and (Figs. 4.8c and 4.9b), the material is under compression and .

Equation (4.3) represents the extensive properties of the material. While this is very important, we first want to examine the intensive properties of the material. If the object has a cross-sectional area A and length L, we can rewrite (4.3) as
Each fraction represents an intensive parameter. The applied force/area, , is called the stress . The fractional increase in length, , is called the strain (or the engineering strain) . is the elongation. The normalized spring constant, , is called either Young’s modulus or the elastic modulus and is represented by Y (or E). This modulus is a fundamental intensive property of the material. Consequently,
This linear constitutive relationship describing this material is valid only for small strains. It is usually valid for , but the range of validity really depends on the type of material. We have ignored any change in cross-sectional area. There is usually a change in A with a change in L (see below), which we will usually ignore here.

As seen in Fig. 4.8, tensile stress means 0$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_4_Chapter_IEq30.gif”>. A compressive stress means and leads to a compressive strain . For such elastic materials in the proportional (or harmonic or Hookean) regime the stress–strain relation is linear, as is seen in Fig. 4.1. The units of stress and modulus Y are both those of force/area, such as N/m ( Pa) or the more convenient unit of N/mm ( MPa); we will usually use these last two equivalent units. Strain, , is unitless. Remember from Table 2.​6 that  psi.

In addition to these linear relations between stress and strain, there are other types of deformations (Fig. 4.10). Figure 4.11a shows the geometry of shear deformations with force F and shear stress . ( is not torque here.) The response is the shear strain, and for small deviations . The shear stress and strain are related by
where G is the shear modulus. This shear deformation is related to the torsion of the top of a cylinder, with the bottom fixed, as seen in Fig. 4.11b, where the torsion T is related to the deformation angle .

Let us consider the deformation of a cylinder with the long axis along the z-axis. We have already called the axial strain response in the z direction , but because stresses lead to strain deformations in different directions, we could be more specific (for the moment) and call it . We assumed earlier that the cross-sectional area of such a cylinder does not change under tension or compression, but it does to a certain extent. We will call the fractional strains in these lateral x and y directions—the lateral or transverse strains— and , respectively. (In more advanced discussions, these three x , y, and z components of strain are really referred to as , , and .) For the linear deformation described above, symmetry implies that . For a given material there is a relationship between these longitudinal and lateral strains provided by Poisson’s ratio
For isotropic materials, the range of possible is , although materials with negative are not found in nature. For anisotropic materials, such as many materials in the body, can exceed 0.5. For metals and many engineering materials , but it tends to be higher for biological materials. For bone, ranges from 0.21 to 0.62 [47]. For tissues like those in the brain, .

After this deformation the new volume is the old one , when each strain has magnitude (Fig. 4.12). Using Poisson’s ratio, the new volume is the old volume and the fractional change in the volume is . For example, with this fractional volume change is . If , there is no volume change even with the change in shape. Similarly, after deformation the new cross-sectional area is the old one , and the fractional change in area is .

Like Y and G, is an intensive property of the material. For isotropic materials they are interrelated by
For example, if , the shear and elastic moduli are related by .

How much do our bones shorten under compression? We will assume that the relation is valid until the stress reaches its maximum just before fracture occurs, which is called the ultimate compressive stress (UCS) and which is 170 MPa for compact bone; this is a good approximation for this calculation. Then and the bone shortens by
and fractionally by
How much does the femur shorten when you stand on one foot? With no stress the femur is  mm long. The body weight of 700 N (70 kg) is distributed over the femur cross-sectional area , so  MPa. The femur shortens by only )) . This corresponds to a strain of . In units of microstrain (), this is 100 microstrain (or ).

The maximum stress in compression is the UCS. What is the strain at the UCS (assuming linear behavior)? At the breaking limit the bone shortens by , which corresponds to a fractional shortening of . The femur shortens by (()) or 0.5 cm. This is a fractional decrease of , and a microstrain of 10,000 .

There are several essentially equivalent ways to determine the potential energy stored in elastic materials. From (4.1), in a spring the potential energy (PE) is
Changing to coordinates relative to the equilibrium position and changing the length to L gives
Because , , and , we see that . Therefore
where the volume . Because , this can be expressed as
The potential energy per unit volume is an intensive quantity.

This is equivalent to integrating
using or
where the last integral equals the area under the curve in Fig. 4.13 and can be obtained by replacing by and integrating to get .

How can we design the best elastic storage medium for the body, such as would be desired for tendons? We would want (1) to store the maximum amount of potential energy for a given applied force and (2) the medium to withstand as large a as possible.

(1) The stored energy is
so we would want to maximize the length , minimize the cross-sectional area A, and minimize Y.

(2) However, to withstand a large we need to keep below the threshold for damage (which for tension is called the ultimate tensile stress, UTS), so we have a limit for how small we could make A to keep UTS. Also, there is a limit to how much the element can be lengthened for a large , given its motion requirements, such as that for a tendon. Because , there are limits on how large both and can be. This sets a limit on the length and, because , a limit on how small Y can be.

There is a tradeoff in the optimal values of , A, and Y set by these two criteria. We want long and thin tendons with a small Y, but there are limits. In this design problem we also have to recognize that the medium, such as a tendon, is not perfectly harmonic or even elastic; all materials are really viscoelastic.

Let us return to the example of running in Fig. 3.​34 [2]. The force on the Achilles tendon is 4,700 N. With a cross-sectional area of , we see that  MPa. Given the maximum stress for tendons, the UTS, is  MPa, during running the stress in these tendons is not far from the damage threshold. It is not surprising that the Achilles tendons of athletes occasionally snap, either partially or totally.

Using the stress–strain relation shown in Fig. 4.14, this stress leads to a strain of . The length of the Achilles tendon is  mm, so this strain corresponds to the tendon lengthening by 15 mm and

This is exactly the amount of energy we stated was being stored in the Achilles tendon during every step of a run.

How much energy is stored in the bones during this step? Let us examine the largest bone, the femur. We will use  mm and , and so . Also . The upward normal force in Fig. 3.​34 is 6,400 N, which we will assume is transmitted all the way to the femur. The stress is and

If the same enegry is stored in the tibia and fibula, then at most 3–4 J is stored in these long bones, which is a very small fraction of the 100 J kinetic energy lost per step.

Elastic energy recovery from tendons and ligaments may be important in motion, which means the energy storage is mostly elastic, it can be recovered in phase (which often means fast enough) to assist the motion, and failures due to stresses at high loading values and repetitive actions are not attained [66]. Longer tendons, such as Achilles tendons in the lower limb ( mm long), can stretch more and tend to store more energy than shorter tendons (why?), but recoil and release the energy slower. These shorter tendons include the shoulder internal rotator muscle tendons used in throwing ( mm) and the patellar tendon ( mm). While the shorter tendons in the shoulder store less energy per tendon, there are many in parallel so much energy can still be stored in them and in shoulder ligaments, and with faster release.

Many body materials cannot be modeled as Hookean springs, even for small stresses. This is typically true for collagenous tissues, such as tendons, skin , mesentery (which are the folds attaching the intestines to the dorsal abdomen), the sclera , cartilage, and resting skeletal muscle. Experimentally, for resting muscles and the materials shown in Fig. 4.21 it is found that for larger strains
This is very different than that for materials such as bone for which, with a that is independent of stress. Instead they are characterized by the slope of the stress-strain curve, d/d, which is called the tangent modulus of elasticity , which is a function of stress or strain, as in (4.22).

Equation (4.22) can be integrated after bringing the and terms to opposite sides of the equation

where is a constant. Exponentiating both sides gives
where . Because , we see that and so
This is illustrated in the passive curves in Fig. 5.​21. (See Appendix C for more information about this method of solution.)

Sometimes the Lagrangian strain is defined, where L is the length and is the length with no stress. Equation (4.26) becomes
with .

At larger strains this exponential form may not work well. In the neo-Hookean regime, finite strain, , is defined, which for small deformations approaches the small-strain approximation, . Soft, neo-Hookean materials tend to follow a linear relationship between stress and (or E), and not a linear or exponential relationship between stress and (or ). This neo-Hookean regime and other more general ways to define strain are described in Problems 4.20–4.22.

Such non-Hookean stress–strain curves are typical for materials with fibers. As mentioned earlier, there are large strains for small stresses where the tangled fibers are being aligned (in this toe regime), but much larger stresses are required to achieve much higher strains where the already-aligned fibers are being stretched (Fig. 4.24). Try this by stretching yarn. The fibers begin to tear at the UTS, corresponding to the load seen in Fig. 4.25.

For tendons (Fig. 4.14) and ligaments (Fig. 4.25) the tangent modulus of elasticity increases with strain at low strain (in the toe region) and then becomes constant (in the linear region) until it ruptures (and so, before rupture it is not well described by (4.22)). For tendons, the linear regime begins when they are stretched % (range from 1.5–4%) or stressed to  MPa (5–30 MPa), and in the linear regime the tangent modulus is  GPa (0.6–1.7 GPa) [80].

We will now return to the deformation of Hookean materials, like bone. We will revisit the properties of these non-Hookean materials in the discussions of viscoelasticity and muscles.