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Material properties are determined using a standardized bone specimen and the results of tests are represented graphically on a stress–strain curve. The stress–strain curve is analogous to a load–deformation curve for bone structural properties, with the distinction of being normalized to load distribution and specimen geometry.

Stress ( σ ) is the force (F) divided by the area (A) of the surface that the force acts on (Figure 3.10a). Forces directed perpendicular to a planar surface are called normal forces, and forces directed parallel to a planar surface are called shear forces. When a force acts perpendicular (normal) to the surface of an object, it exerts a normal stress. When a force acts parallel to the surface of an object, it exerts a shear stress. The units of stress are force over area, and the most common unit is the pascal (Pa), which is equal to 1 N over 1 m² (N/m²). The pascal is a very small unit, therefore physiological stresses are more commonly expressed in megapascals (MPa) (1 MPa is equal to 1 000 000 Pa).

Strain (ε) is a change in dimension that develops within a material in response to stress, divided by the original dimension (Figure 3.10b). Strain may be normal (i.e. a change in length or width) or shear (i.e. a change in shape). Normal strain refers to the length (or width) of a structure divided by its original length (or width) and is therefore dimensionless but commonly measured in units of microstrain (με), so that a strain of 0.01 (1%) would be 10 000 microstrain. For reference, maximum strains in the third metacarpal bones of Thoroughbred racehorses galloping at racing speeds of 16 m/s have been measured in the range of 3250–5670 με (0.3–0.6%) [69]. Shear strain is the amount of angular deformation from a right angle lying in the plane of interest in a sample (Figure 3.10c). Shear strain is expressed in radians ( γ ) or degrees (1 rad = 57.3°).

Figure 3.10 Diagrammatic representations of stresses and strains. (a) A force directed perpendicular to a surface (i.e. a normal force) is described as a compressive (F1) or tensile (F2) force depending on its direction. A force acting parallel to a surface (F3) is called a shear force. Stress (σ) is defined as force (F) divided by the cross‐sectional area (A) of the surface to which it is applied (σ = F/A). (b) Strain (ε) is defined as a change in dimension divided by the original dimension (ε = ΔL/L). (c) Shear strain is the amount of angular deformation (a) of a right angle lying in the plane of interest in a material, which is expressed in radians (γ).

Source: Modified from Morgan and Bouxsein [36].

Change in one dimension is accompanied by a change in a perpendicular dimension. The relative amount of change in perpendicular dimensions is represented by Poisson’s ratio. For example, in a tensile test, lengthening of a structure is accompanied by a narrowing of the width. The quotient of strains in longitudinal and transverse directions is called Poisson’s ratio ( ν ), defined as ν = −(ΔW/W)/(ΔL/L). It is a measure of how loading in the longitudinal direction (axially) affects the structure transversely (laterally). Typically, axial tension results in transverse contraction, while axial compression results in transverse bulging. Poisson’s ratio for bone typically has values between 0.2 and 0.5 (average: 0.3) [70].

As in the load–deformation curve, once the yield point is exceeded, increased applied stress results in permanent deformation of the material. Permanent deformation occurs in the plastic region of the curve, which extends from the yield point to the failure point. Ductility is a measure of the ability of a material to deform plastically prior to failure, and brittleness is the opposite of ductility. The total area under the stress–strain curve (Figure 3.11) is a measure of the energy absorbed to failure or toughness.

Stress–strain curves demonstrate that compact and trabecular bone have significantly different material properties influenced by porosity (or apparent density). Compact bone has higher apparent density than trabecular bone and withstands high compressive stress but will fail at strains exceeding 2% [35, 71]. Trabecular bone is porous and can therefore absorb a significant amount of energy and tolerate up to 30% strain prior to failure [35, 47]. The strength and stiffness of trabecular bone vary with apparent density but are generally less than that of compact bone (Figure 3.12).