load and displacement are recorded as stress and strain. In other words, for the bone organ as a whole, the deformation is a function of the load applied. Within the bone material, the strain is a function of the stress induced [42]. The International Organization for Standardization (ISO) and the American Society of Testing and Materials (ASTM) develop and maintain standards for testing materials and structures in different industries.
Structural Properties and the Load–Deformation Curve
The load–deformation curve (Figure 3.9) is useful for determining the mechanical properties of whole structures, such as an entire bone, or a bone‐implant construct in fracture repair [35].
The initial curved portion is known as the toe region, where low load invokes relatively large deformation, which reflects the uncrimping of collagen fibres in highly collagenous tissues. The linear portion of the curve is called the elastic region, where the object maintains the capacity to return to its original shape once the load is removed. If loading continues through the elastic region to the yield point, then the structure incurs damage. The plastic region of the curve follows the yield point, wherein the material is no longer capable of returning to its original configuration when the load is removed. In the plastic region, the structure deforms to a greater extent for a given load than in the elastic region. If the load continues to increase, the structure will eventually fail. In a clinical setting, the failure point for bone is the load at which it fractures, but in an experimental setting the failure point for a specific biomechanical test may be defined by the investigator. The ultimate load prior to failure is referred to as the ultimate strength of the material. The failure point of bone typically coincides with peak load, as bones have limited ability to deform plastically. The stiffness of the structure is indicated by the slope of the elastic region of the load–displacement curve. The yield, ultimate and failure strengths of a structure correspond to the yield, ultimate and failure load points on the load–deformation curve. The ultimate and failure strength are usually similar in bone but may be different in other materials. Work to fracture (energy absorbed to failure) of a structure is analogous to the material property of toughness and is represented by the area under the load–displacement curve.
Figure 3.9 Representative load–deformation curve for a whole bone.
Material Properties and the Stress–Strain Curve
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).
The Role of Geometry
Bone geometry markedly influences structural mechanical properties. Axial stiffness, which is the resistance of bone to deformation during loading in tension or compression, is proportional to the cross‐sectional area, while bending and torsional stiffness depend on how the bone material is distributed around the axis of bending or torque.