, David B. Burr^{2}, Neil A. Sharkey^{3} and David P. Fyhrie^{4}

(1)

Orthopaedic Surgery, UC Davis Medical Center, Olga, WA, USA

(2)

Department of Anatomy and Cell Biology, Indiana University School of Medicine, Indianapolis, IN, USA

(3)

Kinesiology, Orthopedics and Rehabilitation, Pennsylvania State University, University Park, PA, USA

(4)

Orthopaedic Surgery, Biomedical Engineering, UC Davis Medical Center, Sacramento, CA, USA

Abstract

We have now learned much about the variable structure of the material inside bones. Previously, we learned that the muscle forces acting on the skeleton are quite large relative to the forces exerted on it by the outside world. It is now time to bring these two aspects of skeletal tissue mechanics together: to learn how strong bones are relative to the loads they must bear, and how the mechanical properties of bones depend on their structure.

Keywords

StressStrainStrengthFractureOsteonDensityCancellousCorticalCompositeStrength of bones is dependent upon the material, the microscopic structure and the shape of the whole bone.

A.A. Rauber, 1876, as quoted in Roesler (1987)

## 7.1 Introduction

We have now learned much about the variable structure of the material inside bones. Previously, we learned that the muscle forces acting on the skeleton are quite large relative to the forces exerted on it by the outside world. It is now time to bring these two aspects of skeletal tissue mechanics together: to learn how strong bones are relative to the loads they must bear, and how the mechanical properties of bones depend on their structure.

The mechanical properties of bones are governed by the same principles as those of manmade load-bearing structures. However, the ability of vertebrates to adapt their bone structure to the imposed loading, in terms of both the evolution of a genetic plan for the skeleton and the ability to acutely modify this plan by cellular activity, results in a structure that is highly complex and, when healthy, very effective. The purpose of this chapter is to explain how the various attributes of whole bones combine to determine their monotonic strength and stiffness. (The term monotonic refers to the behavior of a structure loaded steadily to failure, in distinction to repetitive or fatigue loading, which is considered in Chap. 8.)

## 7.2 Fundamentals of Solid Mechanics

To lay a foundation for subsequent discussion of the mechanical properties of bone, this section briefly reviews the mathematical formulation of the mechanical properties of materials and structures. Students trained in engineering may well skip this section; those with no background in this area can find other biologically oriented treatments of this material in an excellent review by Turner and Burr (1993) and books by Currey (2002), Frost (1973), Wainwright et al. (1976) and Cowin (2001).

### 7.2.1 Strength and Stiffness of a Structure

To determine the mechanical properties of a structure—or, more typically, a major part of a structure, such as a beam in a bridge or the shaft of a femur—engineers conduct a load-deformation test. The structure is loaded in a manner appropriate for its actual function—tension, compression, torsion, or bending—and its deformation is recorded as a function of the applied load. Figure 7.1 shows such a test for a femur shaft loaded in compression. Typically, load and deformation are linearly proportional for some time, until the proportional limit is reached; then, the slope of the load-deformation curve is reduced. The point on the load-deformation curve where this begins to happen is known as the yield point because the structure begins to “yield” or plastically deform. If the structure is unloaded after exceeding the yield point it will not return to its original shape, reflective of the plastic deformation that has occurred. Before the yield point the deformation is elastic, meaning that the structure will return to its original shape when the load is removed. If the load continues to increase, an ultimate load is reached; in some materials (e.g., metals), this may not be the point at which a catastrophic failure (or fracture) occurs. The point marking the latter event is known as the failure point. (In cortical bone, the ultimate and failure loads are usually identical.) The strength of the structure may be defined as the load at the yield or failure points, or as the ultimate load, depending on the circumstances. There is no absolute definition of strength, but if it is not otherwise specified, it may be taken as the load at the yield point (Figs. 7.1a and b).

Figure 7.1

A compressive load-deformation test. The peak load point is the largest load carried by the specimen. The slope of the linear portion of the curve (to the yield point) is the stiffness. The area under the curve (orange) is the energy absorbed during the test (a). The yield point is determined using the 95 % secant method (ASTM Standard E-399) one of several approaches that can be used to determine yield (b).

The stiffness or rigidity of the structure is the load required to deform it a given amount. This is the same thing as the slope of the load-deformation curve. The slope is usually measured within the proportional range, where it is constant. Compliance is the reciprocal of stiffness, and is a measure of the ease of deforming the structure. The deformations at the yield and failure points are measures of the extent to which the structure may be deformed.

The work or energy required to yield or fracture the structure is another important mechanical property that is determined by measuring the area under the load-deformation curve out to the yield or failure point.

### 7.2.2 Stress and Strain

The mechanical properties of a structure depend on both its geometry and the properties of the material inside. To determine the properties of the material, it is necessary to “normalize out” the geometric effects. To do this, engineers have defined stress and strain. In its simplest definition, stress is load per unit area, as shown at upper left in Fig. 7.2. The more general definition of static stress is also shown in this figure. If one imagines a small cube of material at a point inside a loaded structure, one can see that each surface of the cube has three loads acting on it: a normal load perpendicular to the surface, and two shear loads parallel to the surface. If these three loads are divided by the area of the cube’s face, they become the three stresses associated with the direction of that face. The cube has six faces, but because there must be equal forces on opposite faces for stability, it turns out that there are 3 × 3 = 9 stress components at each point in a loaded material. The first three of these are normal stress components, and the other six are shear stresses. Some of the shear stresses are equal to one another so that the cube is in moment equilibrium. As a result the total number of independent stress components is ultimately reduced to six: three normal and three shear. We represent these by the symbol σ

_{i }where i = 1 to 6. Stress has more components than a vector (which has only three) and is a second order tensor. Stress is commonly measured in Newtons/m^{2}(N/m^{2}), which is a Pascal (abbreviated Pa). A typical failure stress in bone would be 150 × 10^{6}Pa which is usually abbreviated as 150 MPa (megapascals).Figure 7.2

Simple and tensorial definitions of stress. The cube at right represents a free-body diagram of a small region within the loaded bone segment at left. F represents the normal and shear forces exerted on this region by the stresses in the surrounding bone. When divided by the areas of the cube’s faces (A

_{1}, A_{2}, A_{3}), these forces become the stress tensor components at the cube’s location.Strain is another second order tensor quantity that engineers define to be able to discuss the deformation of a material without regard to its structural geometry. In its simplest definition, strain is the fractional change in dimension of a loaded body; Figure 7.3 illustrates normal and shear strains. One can show that there are, in general, six strain tensor components, three normal and three shear, which describe the deformation of a cube of material at any point in a loaded structure. We represent these by the symbol ε

_{j }, where j = 1 to 6. Strain is a dimensionless ratio, but it is common to measure it in microstrain, or 10^{−6}mm/mm. For example, a strain of 0.02 mm/mm would be the same as 2 % strain or 20,000 microstrain (abbreviated με); this would be a dangerously large strain in a bone.Figure 7.3

Normal (left) and shear (right) strains are defined. A represents the angle through which the side of the shear specimen is rotated. Strains are dimensionless; shear strains may be expressed as radians.

When a structure is loaded in tension, it becomes longer in the load direction, but it usually “waists” or becomes narrower in the transverse directions. (Of course, the opposite phenomenon occurs in a compressed specimen.) The ratio of the transverse strain component to the longitudinal strain component is known as Poisson’s ratio. For geometric reasons, this ratio must be less than 0.5 in ordinary materials. For incompressible materials (e.g., liquids), Poisson’s ratio is 0.5. Typical values for solids, including bone, are about 0.3. (For another biological material, cork, Poisson’s ratio is nearly 0. This makes it easier to push a cork into a wine bottle, but that obviously has nothing to do with why the cork material grows with this unusual property.)

### 7.2.3 Principal Directions

Perhaps it is intuitively clear that the stresses associated with a cube in an arbitrarily loaded structure depend not only on the location of the cube but also on its orientation. In general each face of the cube experiences both normal and shear stresses, and the relative magnitudes of these vary if the cube is placed in different orientations. It happens, however, that it is always possible to find an orientation such that all the shear stresses become zero. These three mutually perpendicular directions are called the principal stress directions. A similar set of principal strain directions exists for each point in a loaded structure. In simple materials, such as steel, these two principal direction sets are identical, but in complex materials like bone they may differ. When examining stress or strain as matrix (or second order tensor) quantities, the principal values are the eigenvalues of the tensor and the principal directions are the eigenvectors. For a second order tensor, there are three eigenvalues and eigenvectors. The word eigen is a German adjective meaning own or self that identifies that the eigenvalues and vectors are intrinsic to the matrix.

### 7.2.4 Strength and Modulus of a Material

A load-deformation curve may be converted to a stress-strain curve by using appropriate formulas to change load to stress and deformation to strain. For example, in a compressed bone shaft, the load is divided by the cross-sectional area of the bone cortex to obtain stress, and the deformation is divided by the original length of the shaft segment to obtain strain. After this is done, all the strength terms previously defined for the structure (yield, ultimate, and failure strengths, along with structural stiffness) can be redefined for the material in terms of stresses (yield, ultimate and failure stress, along with elastic modulus), The area under the stress-strain curve to failure is analogous to the energy to failure of a structure and is called the modulus of toughness for the material.

Stress and strain in a solid are related to one another. In the simplest material the stress is proportional to strain. The proportional law for a simple material was discovered by Robert Hooke and is named in his honor. In its simplest form, Hooke’s law relates one component of stress to its corresponding strain component (e.g., stress and strain in longitudinal tension), and it applies only to the linear portion of the stress-strain curve between the origin and the proportional limit. The slope of this line is the modulus of the material. For normal loading (tension or compression) this is the elastic modulus or Young’s modulus. For shear loading, the modulus is the shear modulus. The normal and shear moduli have the same units as stress, and values for bone are typically of the order of gigapascals (10

^{9}Pa), abbreviated GPa.Box 7.1 Robert Hooke and Thomas Young: The True Theory of Elasticity or Springiness, and A Particular Explication Thereof In Several Subjects In Which It Is To Be Found … CEIINOSSSTTUV

Thus did Robert Hooke (1635–1703) record his law of elasticity as a Latin anagram in 1675. He did this as a result of his well-known quarrel with Isaac Newton, whom Hooke thought had stolen some of his ideas regarding planetary mechanics and other matters. This situation arose from the fact that Hooke held the position of Curator of Experiments in the Royal Society, of which Newton was a contemporaneous member. It was Hooke’s responsibility to present experimental demonstrations at the society’s meetings, and these experiments had often directly demonstrated aspects of what we now know as Newton’s laws, well before Newton published them. Newton, often described as a reclusive and self-absorbed person, refused to acknowledge Hooke’s contributions to these concepts in his Principia until the second edition, after Hooke’s death. In 1678 Hooke presented the solution to his anagram: ut tensio sic vis (as is the deformation, so is the force)—and the theory of springiness that bears his name became public knowledge.

Hooke was a leading scientist of his day. His first published work was the Micrographia in 1665, containing his microscopic studies and introducing the term “cell” to biology. He served as Robert Boyle’s assistant for several years, and invented the air pump used in the experiments leading to Boyle’s law. (Some historians believe that this law also was rightfully Hooke’s.) Subsequently, Hooke pursued many experiments related to respiratory physiology, involving both him and animals as experimental subjects. In the next decade Hooke turned his attention to clock-making. The impetus for this was the need to determine longitude at sea. Because pendulum clocks are inaccurate aboard ship, timekeeping based on the resonant frequency of a spring became of interest. The resulting experiments led Hooke to discover the linear relationship between applied force and extension of a spring. Interestingly, he saw this law as an extension of Boyle’s law, which expresses the “springiness” of air.

Young’s modulus is named for Thomas Young (1773–1829), the physician-physicist who coined the term “modulus of elasticity” and made other important contributions to mechanics. Unlike Hooke, Young had no desire to guard his scientific insights, but his inability to write clearly concealed his ideas about elasticity almost as effectively as Hooke’s anagram: “The modulus of elasticity of any substance is a column of the same substance, capable of producing a pressure upon its base which is to the weight causing a certain degree of compression as the length of the substance is to the diminution of its length.”

Such unintentional obfuscation aside, Young was, like Hooke, a person of enormous breadth of interest and accomplishment. He is perhaps best known for his work in optics, having established both the wave nature of light and, by means of sometimes gruesome and painful experiments on his own eyes, the foundations of physiological optics. In addition to his practice of medicine and his work in physics, Young helped decipher the Rosetta Stone and wrote 61 articles for the newly founded Encyclopaedia Britannica. Much of his scientific work was published anonymously because he was afraid that his medical practice would suffer if the public knew that he was a scientist!

### 7.2.5 Generalized Hooke’s Law: Anisotropy

The Hooke’s law relationship between the stress and strain has a matrix of constant values to inter-relate the six independent components of stress and strain. Using the vector form of the stress and strain each stress component is a linear function of the six strain components:

The matrix notation is written in compressed notation as

where both i and j assume the values 1–6 and j is understood to be summed over these values. The 36 constants of proportionality C

(7.1)

_{ij }are defined as the elastic constants.The vector and matrix notation is a simplification of the tensor notation for the generalized Hooke’s Law where, , is a summation over the stress and strain second order tensor components and the constant components of the fourth order stiffness tensor C. In matrix notation the stress and strain matrices are related to the stress and strain tensor components using Voigt notation as:

The C

_{ij }values are related to the fourth order components of the material tensor. The mapping between C_{ijkl }and C_{ij }depends on how the shear strains and stresses are ordered in the equations just above. The ordering of the shears is arbitrary (there are several ways to order them, e.g. 23-13-12, 12-23-13, 12-13-23…) and you have to make sure that all the parts of your matrix calculations match. In finite elements it is common to use the order 12-23-13 and in this book the order 23-13-12 is used.It turns out that considerations of energy conservation require that the off diagonal terms of the stiffness matrix are equal, e.g., C

The prospect of having 21 constants to determine experimentally is a daunting and essentially impossible task for most biological materials. Fortunately, the number of independent constants needed to relate stress and strain is usually smaller than 21 for most materials because of symmetries in the material microstructure.

_{12}= C_{21}, etc., so that the matrix of C_{ij }constants is symmetric. Taking into account the symmetries, the most general anisotropic material has 21 independent values, and has the form:In isotropic materials (which have no “grain” but behave the same in every direction), there are only two independent elastic constants, and these are related to the elastic modulus, E, and shear modulus, G, which are sometimes known as technical constants. Poisson’s ratio, ν, is considered to be a third technical constant. For isotropic materials, because there are only two elastic constants, there are only two independent technical constants as well, with E = 2G(1 + ν). The stiffness matrix for an isotropic material is:

For materials like bone, which have a grain and thus are anisotropic, the number of elastic constants is between 2 and 21, and the technical constants have different values in different directions. For example, the elastic modulus in tension of a long bone’s diaphysis is greater longitudinally than it is across the shaft.

Most anisotropic materials are not completely so, but have some symmetry to their internal structure. Two common types of limited anisotropy happen to be found in bone, wood, and some other biologic materials. The mechanical properties of orthotropic materials are different in three perpendicular directions (e.g., the longitudinal, radial, and circumferential direction of a long bone). The orthotropic matrix of elastic constants has nine independent values:

The stiffness matrix of an orthotropic material expressed in technical (engineering) constants has very complicated formulae for the constants C

For an orthotropic material the three independent Young’s moduli E

_{ij }. However, the orthotropic compliance matrix, defined by the relationship , is found as the inverse of the stiffness matrix and has a simpler form. In the principal coordinates of an orthotropic material the compliance matrix [S] has the particularly simple form:_{i }, i = 1,2,3 are for the principal directions of the material, there are six values of Poisson’s ratio ν_{ij }and three independent shear moduli G_{ij }for the “i” face in the “j” direction. However, the matrix is symmetric, so that only three of the Poisson’s ratios are independent—in total, therefore, there are nine independent material constants.An interesting property of orthotropic materials loaded along their principal axes and of an isotropic material loaded in any direction is that there is no shear coupling-that is, normal strain does not cause shear stress and vice versa—shear stress does not cause normal strain. This property of orthotropy is related to the theory of how cancellous bone tends to “line up” with the common mechanical loads on a bone (the trajectory theory) (Murray 1936; Fyhrie and Carter 1986; Cowin 1986).

An occasionally useful special case derived from the orthotropic material is where E

_{1}= E_{2}= E_{3}= E, G_{23}= G_{31}= G_{12}= G and all of the “ν_{ij}’s” are equal to ν. The three independent technical constants (E,G,v) of this cubic material property can be useful for materials that have a shear and Young’s modulus that are inconsistent with the isotropic requirement that E = 2G(1 + ν) but for which there is not enough information to define the nine orthotropic constants.Transversely isotropic materials behave similarly in every direction about a single axis of symmetry and have five independent constants. If the axis of symmetry for the material is in the “1” direction, then the compliance matrix (S) has the form:

(Note that if the axis of symmetry is in the “2” or “3” direction that the matrix is different.)

Later in this chapter we show which kinds of bone exhibit these forms of stiffness anisotropy. (Bone is also anisotropic in its strength properties, but there is no straightforward theory covering this behavior.) A much more complete and extremely lucid explanation of anisotropy and elasticity theory may be found in the treatise, Theory of Elasticity of an Anisotropic Elastic Body, by Lekhnitskii (1963).

## 7.3 Determinants of the Strength of a Whole Bone

### 7.3.1 Mechanics

Generally speaking, for a particular mode and rate of loading, the strength (or stiffness) of a structure is a product of (1) its shape and size, and (2) the strength (or stiffness) of the material within. For example, in compression, tension, or shear, the stress in a specimen of cross-sectional area A and bearing a load L is

When this stress reaches the failure stress for the material, the structure will fail. Therefore, by simply rearranging this equation and using the subscript f to represent the failure condition, one obtains an equation for the compressive load that would fail a long bone diaphysis,

where L

where J is the polar moment of inertia (Fig. 7.4). Rearranging this equation, one has, for the failure torque of a circular cylindrical long bone diaphysis loaded in torsion,

where r is now the periosteal radius, and σ

(7.2)

(7.3)

_{f}is the failure load, and σ_{f}is the material strength in the appropriate mode of loading. In this case, the shape and size factor is simply the cross sectional area, A. Analogous equations describe the strength of the diaphysis in other modes of loading. When a circular cylindrical structure is twisted by a torque T, the shear stress varies linearly with distance, r, from the center, and is given by(7.4)

Figure 7.4

Cross-sectional and polar moments of inertia are defined for a bone cross section. The cross section is divided into many small elements of area, dA. The x-axis is assumed to be the neutral axis for bending. Equations are for I and J of a cylindrical tube.

(7.5)

_{sf}is the torsional (or shear) strength of the bone material. The ratio J/r may be called the torsional “shape and size factor” for a circular cylinder.For the hollow circular cylinder approximation of the femur the torsional stiffness is exactly “GJ” with “G” the shear modulus and “J” the polar moment of inertia. For a real bone with a cross-sectional shape other than a circle (or circular tube), however, the torsional rigidity is always smaller because the cross-sections “warp” in the direction of the axis when an axial torque is applied. For any cross-section the equation relating torque (T

_{f }), angle (θ), and length (l) is: where “K” is a factor dependent on shape and size of the cross-section. For a circular (or circular tubular) section like in Fig. 7.4, K is equal to the polar moment of inertia J. For any other cross-section the factor K is smaller than J and can actually be a very small fraction of J for some cross-sections. For an elliptical tube (i.e., an elliptical tube with concentric elliptical inner and outer boundaries) with minor inner and outer radii b_{0}, b and major inner and outer radii a_{0}, a: with q = b_{0}/b = a_{0}/a. The maximum stress is at the end of the minor axis on the outer surface and is: . For this elliptical tube so that J and K differ by less than 5 % if the ratio of the minor to major axis is between 0.8 and one. As a result, for a bone in torsion the mechanical response can often be approximated using a circular cross-section with errors accumulating if the bone isn’t straight or deviates greatly from a circle.When a beam is bent by a moment M, there is tension on one side and compression on the other; the boundary between these regions is called the neutral plane or axis. The tensile and compressive stresses are given by

where I is the cross-sectional moment of inertia and c is the distance from the point in question to the neutral plane. The failure moment is

where σ

(7.6)

(7.7)

_{bf}is the failure stress for bending and c is now the distance to the outermost fibers of the bone. Here, the shape and size factor is I/c.Cross-sectional and polar moments of inertia are defined in Fig. 7.4. Consider that the cross-section of the bone cortex is divided into elements of area Δ A, of which one is shown. J is the sum of the products of each Δ A and the square of its distance from the centroid. I is similarly defined, except that the distance is measured from the neutral axis, which divides the tensile and compressive areas of a bent structure.

Equations of similar form would describe the stiffness of a bone: the primary difference is that a shape and size term is multiplied by the stiffness of the material (the elastic modulus in tension, compression, or bending; the shear modulus in shear or torsion).

### 7.3.2 Examples

Suppose a person’s femoral diaphysis is idealized as a circular cylinder with a periosteal diameter of 3 cm and an endosteal diameter of 1.2 cm. Calculate the diaphyseal failure load for uniaxial compression and the failure moments for bending and torsion.

#### 7.3.2.1 Compression

_{f}= 195 MPa.

Table 7.1

Typical mechanical properties for cortical bone

Property |
Human |
Bovine |
---|---|---|

Elastic modulus, GPa | ||

Longitudinal |
17.4 |
20.4 |

Transverse |
9.6 |
11.7 |

Bending |
14.8 ^{a} |
19.9 ^{b} |

Shear modulus, GPa |
3.51 |
4.14 |

Poisson’s ratio |
0.39 |
0.36 |

Tensile yield stress, MPa | ||

Longitudinal |
115 |
141 |

Transverse |
— |
— |

Compressive yield stress, MPa | ||

Longitudinal |
182 |
196 |

Transverse |
121 |
150 |

Shear yield stress, MPa |
54 |
57 |

Tensile ultimate stress, MPa | ||

Longitudinal |
133 |
156 |

Transverse |
51 |
50 |

Compressive ultimate stress, MPa | ||

Longitudinal |
195 |
237 |

Transverse |
133 |
178 |

Shear ultimate stress, MPa |
69 |
73 |

Bending ultimate stress, MPa |
208.6 ^{a} |
223.8 ^{b} |

Tensile ultimate strain | ||

Longitudinal |
0.0293 |
0.0072 |

Transverse |
0.0324 |
0.0067 |

Compressive ultimate strain | ||

Longitudinal |
0.0220 |
0.0253 |

Transverse |
0.0462 |
0.0517 |

Shear ultimate strain |
0.33 |
0.39 |

Bending ultimate strain |
– |
0.0178 ^{b} |

#### 7.3.2.2 Torsion

#### 7.3.2.3 Bending

Then from Eq. (7.7) one has, letting c = R

If the femur were 42 cm long and supported at its ends, a 5130-N load in the middle would produce this moment. This would be about seven times the body weight of a 70-kg person.

_{p}= 0.015 m,It should be emphasized that the foregoing equations are restricted to fairly simple geometries of the cross section and load application. Bones may frequently depart from these conditions. For example, torsion of a bone having a cross section that departs significantly from axial symmetry, or bending in which the load vector does not pass through the centroid of the cross section, would not accurately follow these descriptions. Even so, it generally remains possible to separate the effects of material properties (which depend on histologic structure) from those of the shape and size of the structure (which depend on anatomy). That is the important point insofar as this discussion is concerned.

Another limitation of this approach is that it assumes that skeletal “shape and size” characteristics include only macroscopic anatomical features, and not microscopic features such as cancellous or osteonal architecture. It is assumed that the mechanical effects of microscopic features can be embodied in the material properties. This strategy, used in continuum mechanics, is consistent with the approach taken when finite element modeling (FEA; see Box 7.3) is used to study skeletal stresses. The gross anatomy of the bone is modeled by the “shape” of the model, and variations in microscopic architecture are satisfied by assigning different material properties to individual regions (elements) within the model. The alternative is to attempt to formulate the shape and size of the microscopic structures, determine their behaviors and then assign properties to the macroscopic structure accordingly Although not yet in widespread use, newer high-resolution imaging techniques combined with multi-scale FEA modeling now provide a platform for this computationally expensive approach The more traditional approach favored here is to simply define material properties on a macroscopic scale, combining the mechanical effects of variations in microscopic architecture with those in microscopic material properties. This approach is also consistent with the common practice of measuring the material properties of bone using specimens small enough to define an anatomical region, but large compared to the size and distribution of microscopic features. For example, in the case of cancellous bone, test specimens are made large enough to contain perhaps 10–50 trabeculae (Brown and Ferguson 1980). For further discussion of this problem, see Fung (1977) and Harrigan et al. (1988). We return to this subject when we discuss the mechanical properties of cancellous bone later in this chapter.

Box 7.2 Mechanical Significance of Bone Fracture Patterns

When a person comes to the hospital with a fracture, the biomechanically knowledgeable physician can often deduce the kind of loading that caused the break from the pattern of the fracture surfaces. To do this, one must keep two things in mind. First, the ultimate stress for bone differs in various modes of loading: it is strongest in compression, weakest in shear, and intermediate in tensile failure (Table 7.1). Secondly, the stress inside a bone depends on orientation as well as position. If the cube in Fig. 7.2 is rotated to some new orientation, the stresses on its surfaces will change. Just as the cube can be rotated to minimize the shear stresses on its surfaces, there is another orientation that will maximize the shear stresses. Because the ultimate stress in compression is much higher than that in shear, when a bone is loaded in compression it tends to fail along the planes which carry the highest shear stresses, not those having the highest compressive stresses. These planes are typically at an angle of 45° to the load direction (Fig. 7.5).

Figure 7.5

Left: In a bent bone, a transverse crack originates on the tensile side (T). On the compressed side (C) the fracture surface often follows a shear plane in one or two directions, producing a wedge-shaped fragment. Right: In a twisted bone, tensile stresses produce a spiral crack that winds around the cortex, and the bone breaks when the crack’s ends are connected by a longitudinal fissure.

When a bone is bent, failure often begins on the tensile side and propagates across the shaft because the tensile stresses on a cross section exceed the strength before the shear stresses on a 45° plane do. When the crack enters the compressed tissue, it again tends to travel along the 45° planes of maximum shear, causing ‘butterfly’ fragments or other types of comminution. When a bone is loaded in torsion the shear stress created by the twisting is across the fibers. In this case, compressive and tensile stresses lie at a 45° angle to the fibers. The tensile stress tends to separate the fibers, producing a fracture surface that spirals around the bone at a 45° angle. The crack then returns to its starting point along a longitudinal shear stress plane. Clinically, each of these fracture patterns is common, indicating that loading of long bones can occur by compression, bending, torsion, or a combination of these.

### 7.3.3 Mechanical Failure of Whole Bones

In considering the mechanical requirements of whole bones, several conflicting demands must be balanced. In normal daily life, the skeleton provides struts (bones) that the muscles can work against to create motion. The stiffer these struts are the more efficient is the action of the muscles. If bones are less stiff, then part of the mechanical output of the muscles is used in deforming the bones, rather than creating useful motion. This wastes metabolic energy. In this context, bones of greater stiffness are desirable. On the other hand, if the individual is in a situation where there is risk of bone fracture from a fall or a blow, less stiff (more compliant) bones would be preferred. This is because, for a given strength, the more compliant structure is able to absorb a greater amount of energy before reaching its failure load (Fig. 7.6). Another consideration is weight: mineralized tissues are twice as dense as other tissues and require considerably more metabolic energy to carry around. The mechanical characteristics of bones, therefore, represent a compromise between the need for stiffness to make muscle actions efficient, the need for compliance to absorb energy and avoid fracture, and the need for minimal skeletal weight. This balance between stiffness and the need for energy absorption to prevent fracture may be one explanation for the often inverse relationship found between stiffness and modulus of toughness in bone (Currey 2004).

Figure 7.6

If two bones have the same strength (failure load), the more compliant one (at right) requires more energy to break it.

Different bones have different functions, and the best solution to the conflicting mechanical demands need not be the same for every bone (Currey et al. 2007). The bones of the inner ear need to be extremely stiff to efficiently transmit sound waves; the fact that they could absorb very little energy before fracture is unimportant because they are protected from extrinsic loads. At the other extreme, the material in the bones of children is much more compliant than those of adults. Currey and Butler (1975) found the femurs of 2- to 8-year-old children to be 68 % as stiff, requiring 45 % more energy to break in bending, than those of 26- to 48-year-old adults. This increased compliance decreases the efficiency of children’s muscle action, but it confers more protection against falling and crashing into things.

The energy required to break a bone is small compared with the energy encountered in daily activities. For example, the energy required to break the shaft of an adult tibia or femur is only about 15 joules (J). The energy released when a 70-kg person falls to the ground from a standing position is about 500 J. The ability to absorb this energy using eccentric muscle contractions and deformations of soft tissues usually prevents fractures from occurring in such a fall. Soft tissue deformation absorbs energy during an accident in two ways. First, loads applied perpendicular to the tissues compress them and propagate stress waves up and down the body. The tissues effective in this way are skin, fat, and muscle (plus fur on animals and clothing on humans). Second, tissues such as fascia, tendons, ligaments, joint capsules, and contracted muscle brace bones against bending by supporting part of the tensile forces. These tissues absorb energy as they are stretched, but they must be nearly as stiff as the bone to be effective. If these energy-absorbing mechanisms are impaired by surprise, restrictions, or incapacity, fracture occurs easily. The propensity of elderly persons to fracture their bones reflects not only the weakness of their bones but also their increased risk of falling and the inability of their muscles, ligaments, and soft tissues to adequately absorb the applied energy, which instead is transmitted to the bones.

Box 7.3 Finite Element Modeling

Finite element analysis, or modeling, is a process by which stresses and strains can be computed in a structure of irregular shape and varying internal properties. The name comes from the process in which the volume of the structure (e.g., the proximal end of a human femur.) is divided into elements, each having a prescribed geometry and a set of mechanical properties (e.g., elastic modulus and Poisson’s ratio). The model also includes boundary conditions that prescribe the forces applied to, or constraints on, portions of the structure. For example, in the case of a human femur, forces may be applied to some of the elements on the articular surface of the head and along the greater trochanter where the abductor muscles act. In addition, the elements at the distal end of the model may be constrained from moving in one or more directions. A computer is used to solve the equations that govern the ways in which the elements deform and the ways that they interact with one another as they deform. This method of analysis allows many problems to be solved that are far too complicated for a “closed form” solution (i.e., a single equation, such as Eq. (7.5), to give a simple example), either because of their irregular geometry or the variability of the material properties inside. Many commercial programs have been developed for finite element analysis, and the method has become very important in orthopaedics because of the irregular shapes of bones, their varying material properties, and the interest in studying the stresses produced when a metal implant is implanted in a bone.

Finite element modeling as a stress analysis tool is widely used in engineering and bioengineering. It is most powerful, however, when the stress analysis is part of an integrated approach that includes a theoretical model of the biological responses of the tissues to loading. Two examples of using finite elements along with empirical biological theories are the works of Hart (1989) and Beaupre and Carter (1992). Recently, some of the principles underlying an improved approach to integrated modeling of the skeleton were systematically presented by Professor Marco Viceconti (2012). The work of Viceconti goes beyond specific applications and puts the use of modeling in skeletal mechanics into the context of the scientific method, including making excellent distinctions between predictive (i.e., empirical) and mechanistic models that embody the theories of biology underlying system response. Mathematical analysis of bones and the skeleton is a small part of the large field of mathematical biology. A rich source and outstanding introduction to mathematical biology can be found in the book of JD Murray (2003).

## 7.4 Material Properties of Cortical Bone

It has been clearly demonstrated that osteonal remodeling of compact bone reduces its flexural fatigue strength and resistance to tension (Carter and Spengler 1978). Compression, shear, and bending strengths are also reduced. This section explores the reasons for this and otherwise examines the mechanical properties of osteonal bone. Osteons should affect the mechanical properties of the cortex in several ways: by the replacement of highly mineralized bone matrix with less calcified material, by increasing cortical porosity, possibly by altering collagen fiber orientation, and by the introduction of cement line interfaces that have different mechanical properties. We first consider the mechanical properties of single osteons, and then consider cortical bone as a composite of numerous such structural units.

### 7.4.1 Properties of Individual Secondary Osteons

In a remarkable series of papers beginning in 1964, Antonio Ascenzi, Ermanno Bonucci, and their colleagues presented a great deal of experimental information about the relationship between structure and mechanical function in individual secondary osteons. Taken together, their papers constitute the major part of our knowledge in this area, and we review them here in some detail.

#### 7.4.1.1 Tensile Properties

In their initial work, Ascenzi and Bonucci (1964) examined the tensile properties of the secondary osteon. Undecalcified longitudinal sections, 20–50 μm thick, were prepared from bovine and human femurs. A sharp needle was used to dissect free the lamina between the Haversian canal and the cement line on one side of an individual osteon. This procedure produced an approximately rectangular segment of osteonal wall about 50 μm wide and 100–300 μm long. Figure 7.7 shows how the osteonal lamellae formed an approximately rectilinear system within the specimens. In their initial experiments, these osteonal segments were simply broken in quasistatic tension, with only the failure load recorded. Later, load-deformation curves were recorded when a microwave extensometer had been developed to measure the extremely small deformations encountered at this scale (Ascenzi and Bonucci 1967).

Figure 7.7

Diagram showing tension test of osteonal wall segment. (a), End view of test slab removed from femur cross section. (b), Relationship of a single osteon’s geometry to the slab in an enlarged end view. (c), Side view of central portion of the test slab. Material has been removed to leave only the lamellae on the right side of the osteon in place. (d), The test slab supported at S and pulled in tension by weight W (Ascenzi and Bonucci 1964, 1967).

The specimens were categorized as either longitudinal (type L) or alternating (type A) in their lamellar structure (c or b, respectively, in Fig. 2.5). Microradiography was used to classify the degree of mineralization as either low or fully calcified. Both dry and wet bone were tested, but only the wet bone data are considered here. Figure 7.8 shows typical curves for secondary osteons with different lamellar orientations and degrees of mineralization. It was found that:

The tensile strength of osteon wall segments was about 114 MPa, a figure that lies in the midrange of data for larger test specimens composed of primary and secondary bone.

Type L osteons had greater tensile strength and lower ultimate deformations than did type A osteons mineralized to the same degree.

The degree of calcification of type A osteons had little effect on the tensile strength of their wall segments. The less mineralized type L osteons were more plastic and deformable than highly mineralized type L osteons, which were extremely stiff. The properties of poorly mineralized type L osteons and all type A osteons (regardless of how well mineralized) were nearly the same (Fig. 7.8).

No differences were found between human and ox osteons, but the authors remarked on the considerable variation in the results for apparently similar osteons. No specific effect of osteocyte lacunae could be mathematically formulated, but examination of the published photographs of failed specimens indicates that lacunae and other irregularities in the lamellar structure may have played an important role in the results. The specimens did not have the reduced central cross section that is preferred in tensile specimens. This factor surely led to stress concentrations at the points where the specimens were gripped, and the published photographs show breakage at these points.

#### 7.4.1.2 Compressive Properties

Next, Ascenzi and Bonucci (1968) studied the compressive properties of single osteons using specimens from a human femur. They had by now developed a technique for cutting cylindrical segments from osteons contained in cross sections 500 μm thick. These minute cylinders were 180–200 μm in diameter and contained the Haversian canal and all but the outermost laminae of the osteon. The specimens were classified as to mineralization and lamellar structure as before, except that now osteons of circumferential structure (type T in Fig. 2.5) were also studied. In this case, they found that type T osteons were strongest and stiffest. Type A osteons were stronger than type L, but there was no difference in their elastic moduli. As before, the more fully mineralized osteons were more elastic, stiff, and brittle, regardless of their type. The specimens uniformly exhibited fracture planes at an angle of 30°–35° to the longitudinal direction, regardless of the lamellar structure or degree of calcification. Thus, the failure mode appeared to be shearing, regardless of the orientation of the collagen fibers.

#### 7.4.1.3 Shear Properties

Ascenzi and Bonucci then examined the shear properties of osteons (1972). A 160-μm-diameter cylindrical punch was used to push out the central lamellae from osteons in sections 300 μm thick. The section was supported from below by a plate containing an 800-μm-diameter circular hole, so that the test region involved the cement line. It should also be noted that the test involved significant bending as well as longitudinal shear (Fig. 7.9). The osteons did not fail at the cement line, but along the margins of the punch. Because there would have been a stress concentration at this boundary, one ought not necessarily accept Ascenzi and Bonucci’s assertion that this proves that the cement line is stronger than the interstitial lamellae, but their results are not consistent with a weak cement line. Type L osteons were the weakest, and types A and T were of equal strength. Again, strength and stiffness increased with calcification in each type. The usefulness of the data from this experiment is limited by the fact that the shear test arrangement departed significantly from “simple shearing”; that is, the lateral distance between the edge of the punch and that of the support plate hole was relatively large. Unless one is willing to make a custom apparatus for each osteon, this condition is a necessary practical compromise.

Figure 7.9

Diagram of relative dimensions of the specimen and test apparatus used in Ascenzi and Bonucci’s (1972) shear experiments. Note that the 300-μm-thick bone section must have been subjected to significant bending loads.

There is another report of a similar experiment. Frost et al. (1961) arranged a simple shear test without regard to individual osteons or cement lines. Using 70-μm-thick sections, a 900-μm-diameter punch, and a “slightly larger” hole in the support plate, they pushed out areas containing several osteons and osteon fragments. The average stress for longitudinal shearing of bone from a human femur was about 74 MPa, considerably above the range reported by Ascenzi and Bonucci.

### 7.4.2 Combined Loading Effects

Ascenzi and Bonnuci’s analyses are theoretically instructive but over simplistic in considering only “pure” loading in compression, tension or shear. In reality, most bone is loaded in bending and incorporates combinations of these types of stresses. Computational analyses under combined loading conditions (Pidaparti and Burr 1992) suggest that collagen fiber orientation has little effect on bending stiffness of bone if collagen fibers are oriented at < 60° with respect to the direction of load (i.e., the longitudinal axis). Thus, bending stiffness is actually compromised to some extent by the presence of only transverse lamellae, or a predominance of Type T osteons. For bending, then, overall mechanical properties are maximized when collagen fibers are oriented at 30° or less to the axis of loading. Torsional stiffness, on the other hand, is maximized when collagen orientation is within 15°–30° of either the longitudinal or transverse axis of the bone. In general and somewhat contrary to the results of Ascenzi and Bonucci, Type A osteons are not well adapted to resist torsional moments.

These analyses show that when loading conditions are unpredictable, as they often are, it is beneficial to the bone for collagen fibers to be oriented nearly longitudinally or at least within 15° of the loading axis. Axial stiffness is reduced by about 40 % as collagen fiber bundles alter orientation from 0° to 90° with respect to the long axis under conditions of bending. This is consistent with the analysis performed by Katz (1981) using the Hashin-Rosen model, and with the measurements by Rho et al. (2001). Although these analyses verified the measurements of Ascenzi under pure loading conditions, we can see that under more realistic conditions the adaptation of collagen fiber orientation is much more complex than the often quoted maxim that “longitudinal collagen fiber bundles are best in tension, and transversely oriented bundles are best in compression.”

## 7.5 Summary

The results of Ascenzi and Bonucci’s experiments are summarized in Table 7.2. Variations in the mechanical properties of osteons with different collagen fiber orientations suggest that they are individually adapted to locally enhance the ability of bone to support a particular type of stress. The lamellae of type L osteons are stronger in tension, but it is relatively easy for their inner lamellae to slip longitudinally relative to the outer lamellae. This observation fits nicely with the frequently reported tendency of osteonal bone tested in longitudinal tension to show “osteon pull-out” (Moyle et al. 1978; Moyle and Bowden 1984). Because type A and type T osteons are weaker in tension, but shear less readily in the longitudinal direction, it is not clear whether macroscopic specimens in which these osteonal types predominate should be weaker or stronger in tension. However, it is clear that in bending, Type L and Type A osteons with fibers oriented less than 60° to the loading axis will maximize the modulus.

Table 7.2

Mechanical properties of different types of osteons

Mode of loading |
Ultimate stress, MPa |
Elastic modulus, GPa |
Ultimate strain, % |
---|---|---|---|

Tension | |||

Type L |
114 ± 17 |
11.7 ± 5.8 |
6.8 ± 2.9 |

Type A |
94 ± 15 |
5.5 ± 2.6 |
10.3 ± 4.0 |

Type T |
– |
– |
– |

Compression | |||

Type L |
110 ± 10 |
6.3 ± 1.8 |
2.5 ± 0.4 |

Type A |
134 ± 9 |
7.4 ± 1.6 |
2.1 ± 0.5 |

Type T |
164 ± 12 |
9.3 ± 1.6 |
1.9 ± 0.3 |

Shear | |||

Type L |
46 ± 7 |
3.3 ± 0.5 ^{a} |
4.9 ± 1.1 ^{b} |

Type A |
55 ± 3 |
4.1 ± 0.4 ^{a} |
4.6 ± 0.6 ^{b} |

Type T |
57 ± 6 |
4.1 ± 0.4 ^{a} |
4.6 ± 0.6 ^{b} |

As mentioned in Chap. 2, it is more likely that collagen fiber bundles are structured in a twisted plywood configuration, rather than completely in a parallel or alternating format between lamellae. From a mechanical standpoint, the twisted plywood configurations should enhance bone’s ability to resist shear stresses developed in torsion and compression loading, because of their multidirectional organization. The twisted plywood configuration would be less than half as resistant as a parallel fiber arrangement to axial tensile forces (Ruggeri et al. 1997), but bone is rarely, if ever, subjected to purely axial tensile forces in-vivo. Moreover, even if it were, osteons with only longitudinally-oriented fibers would be highly susceptible to failure if the bone were loaded in any other orientation because the fibers would allow a preferential pathway for the growth of small fracture planes. Therefore, the twisted plywood organization may be the bone’s adaptive response to withstanding unpredictable loads in many different directions. Bone’s rather complex organization at the microscale provides an elegant example of evolutionary processes and the plasticity of bone at this scale of structure.