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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 and represents a transition in the behaviour of the bone from pseudo‐ductile to brittle [29, 30, 82]. Once the critical velocity is reached, the bone becomes increasingly brittle, resulting in lower strain to failure, lower energy absorbing capacity and reduced fracture toughness [26, 82, 83].

The relationship between gait and strain rate is roughly linear such that the highest strains are experienced at the fastest gaits [84]. Strain rates experienced by horses at walk to canter gaits (2–10 m/s) in vivo are in the range of 1.3–8.3 × 10⁻²/s [81]. At a gallop, horses regularly achieve a velocity of 1.5 × 10⁻¹/s [84], and in the dorsal aspect of the third metacarpal bone of Thoroughbred racehorses strain rates as high as 3 × 10⁻¹/s can be estimated for racing speeds (16–18 m/s) [69, 79]. These are within the range at which bones undergo brittle deformation; however, the relationship between strain rate and catastrophic fracture risk is not straightforward. Whether or not a fracture will occur likely depends on the interaction of strain rate with several other factors including the direction of loading, number of load cycles, magnitude of strain and the presence of pre‐existing fatigue microdamage [80].

Figure 3.15 Behaviour of viscoelastic materials. (a) Creep is an increase in strain under constant stress over time. (b) Stress relaxation is a decrease in stress under constant strain over time. (c) Hysteresis refers to the loss of energy with cyclic loading.

Figure 3.16 The mechanical behaviour of bone is strongly dependent on strain rate. High strain rates lead to higher yield strength and stiffness in cortical bone, but also increased brittleness and reduced fracture toughness. However, strain rates measured on the dorsal metacarpus of Thoroughbred racehorses are intermediate (i.e. between low and high strain rates), ranging from ~10⁻²/s at a walk to ~10⁻¹/s at a gallop [69, 79, 80]; and physiological strains are less than 0.6%. Thus, strain‐rate‐related effects in vivo are likely lower than those observed in ex vivo studies. Very high strain rates (~10³/s) are expected for impact speeds as might occur in a violent collision.

Source: Modified from Davies et al. [81].