the neutral axis has the largest effect on resisting bending and torque loads. Two geometric properties, the area moment of inertia and the polar moment of inertia, quantify the contribution of geometry to a particular bone’s resistance to bending and torsion, respectively.
Application of a load at a distance from the centre of a bone induces a bending moment. The bending moment is a product of the magnitude of the force applied and the length of the moment arm about which the force is applied. Moment arm length is the perpendicular distance from the line of action of the force to an axis of rotation. A longer moment arm increases the bending effect of the force applied.
Figure 3.11 Example of a stress–strain curve of a bone sample.
Source: Modified from Lopez [43].
Figure 3.12 Compressive stress–strain behaviour of one compact and two trabecular bone samples demonstrating the influence of apparent density (P) on the material properties of bone. Compact bone is more dense than trabecular bone and exhibits greater stiffness and strength under compressive loading, but tolerates minimal strain before failure. Trabecular bone is porous with a low resistance to compressive stress but is capable of enduring much higher strains than compact bone.
Source: Modified from Keaveny and Hayes [72].
The amount a bone will deform under a bending force is related to the magnitude of the bending moment, the elastic modulus of the bone material and the area moment of inertia (I) about the neutral axis (Figure 3.13). The area moment of inertia is defined as the capacity of a cross‐section to resist bending. For any given bending moment, bone deformation can be reduced by decreasing moment arm length, increasing stiffness of the bone (i.e. larger elastic modulus) or increasing the area moment of inertia.
Figure 3.13 Factors influencing the deformation of bone subjected to a bending force include the magnitude of the bending moment (a product of the force applied (F) and the length of the moment arm), the elastic modulus of the bone material, and the area moment of inertia about a neutral axis.
Factors that affect bone strength and stiffness in torsion are similar to those that operate in bending: the applied torque (force applied to induce a rotation), the length of the bone, the shear modulus and the polar moment of inertia (J) about the torsional axis. The polar moment of inertia is defined as the moment of inertia with respect to an axis perpendicular to the plane of the area. The shear stress created in a bone when loaded under torsion is inversely related to the polar moment of inertia. Thus, in a bone with a high polar moment of inertia, the same torque will result in smaller shear stress than in a bone with a lower polar moment of inertia.
The area moment of inertia and the polar moment of inertia are proportional to the fourth power of the radius for circular cross‐sections (Figure 3.14). For example, if the diaphysis of the bone is considered a hollow cylinder [73–75], small increases in bone diameter will result in exponentially greater bending and torsional strengths. Periosteal callus will contribute substantially more to the bending and torsional stiffness of a bone than endosteal callus, as the new bone material is located further away from the diaphyseal (neutral) axis. Similarly, a small amount of compact bone loss near the marrow cavity may have a relatively small effect on overall bending and torsional rigidity.
Although helpful in fostering a conceptual understanding, assumptions of cylindrical or elliptical geometry underestimate the complexity of bone structure [76]. Experimentally, finite‐element (FE) modelling, wherein geometry and material properties are obtained from quantitative computed tomography (QCT), is used to generate 3D models that more accurately predict the structural response of bones with irregular and variable cross‐sectional characteristics to different loading conditions [77, 78].
Viscoelasticity
Viscoelastic materials exhibit both viscous and elastic characteristics when loaded. Elasticity is the tendency of solid materials to return to their original shape after a deforming force is removed. Viscosity is a measure of a fluid’s resistance to flow (i.e. a viscous fluid will resist motion). Bone is a viscoelastic material because it contains water that can be displaced through the organic matrix. Viscoelasticity refers to the combination of elastic and viscous behaviour where the applied stress results in an instantaneous elastic strain followed by a viscous, time‐dependent strain. In other words, a viscoelastic material will return to its original shape after a deforming force has been removed (i.e. it will show an elastic response) even though it will take time to do so (i.e. it will have a viscous component to this response).
Figure 3.14 Formulae for the area moment of inertia and the polar moment of inertia for a hollow cylindrical cross‐section.
Source: Modified from Morgan and Bouxsein [36].
If a mechanical stress is imposed on a viscoelastic material and held constant, then the resultant strain will increase with time, a phenomenon known as creep (Figure 3.15a). If a constant strain is imposed on a viscoelastic material, then the induced stress will lower with time (stress relaxation) (Figure 3.15b). Viscoelastic materials also display hysteresis, which is the tendency for materials to exhibit different mechanical behaviour based on whether a load is being applied or removed (Figure 3.15c). When a viscoelastic material is loaded and unloaded, the unloading curve is different from the loading curve. The difference between the two curves represents the amount of energy that is dissipated or lost during loading. A key factor in these phenomena is the movement and redistribution of fluid through pores in the viscoelastic biologic tissue [42].
An important characteristic of viscoelastic materials such as bone is strain rate sensitivity, which means that the stress–strain behaviour of the material depends on the rate at which it is loaded (Figure 3.16). Strain rate is the speed or velocity at which a change in dimension (deformation) of a material occurs. The unit quantity for strain rate is inverse time, typically seconds (denoted s−1 or 1/s). As strain rate increases, the stiffness and ultimate strength of the bone increase. The energy absorbing capacity of bone also increases with increasing strain rate until a critical velocity is reached, beyond which this capacity decreases. The critical strain rate is reported to occur at approximately 10−1 to 100/s