Orientation dependence of progressive post-yield behavior of human cortical bone in compression

Abstract

Identifying the underlying mechanisms of energy dissipation during post-yield deformation of bone is critical in understanding bone fragility fractures. However, the orientation-dependence of post-yield properties of bone is still poorly understood. Thus, the objective of this study was to determine the effect of loading direction on the evolution of post-yield behavior of bone using a progressive loading protocol. To do so, cylindrical compressive bone samples were prepared each in the longitudinal, circumferential and radial directions, from the mid-shaft of cadaveric femurs procured from eight middle-aged male donors (51.5 ± 3.3 years old). These specimens were tested in compression in a progressive loading scheme. The results exhibited that the elastic modulus, yield stress, and energy dissipation were significantly greater in the longitudinal direction than in the transverse (circumferential and radial) directions. However, no significant differences were observed in the yield strain as well as in the successive plastic strain with respect to the increasing applied strain among the three orientations. These results suggest that the initiation and progression of plastic strain are independent of loading orientations, thus implying that the underlying mechanism of plastic behavior of bone in compression is similar in all the orientations.

Fig. 1

Specimen preparation and progressive loading protocol for human cortical bone under compression. (a) orientation of bone samples; (b) a custom-made fixture with a thickness of 5 mm for preparing bone specimens with parallel ends; (c) an extensometer of 3 mm gage length attaching to bone specimens; (d) a typical curve of stress vs. strain in the progressive loading with an incremental displacement; (e) a diagram showing how to calculate mechanical properties using the progressive stress–strain curve.

Fig. 2

Relationship between instantaneous modulus (E_i) and instantaneous strain (ε_i) of cortical bone at longitudinal, circumferential and radial directions. Exponential regression was fitted for experimental data with instantaneous strain less than 2% from three orientations: longitudinal direction (E_i = 22.1e^−62.4s_i, R² = 0.83), circumferential direction (E_i = 8.79e^−46.3s_i, R² = 0.45), and radial direction (E_i = 6.44e^−37.5s_i, R² = 0.42).

Fig. 3

Relationships between plastic strain and instantaneous strain of human cortical bone at three different orientations. Linear regression was fitted for experimental data: longitudinal direction (ε_p=0.459ε_i−0.00227, R²=0.95), circumferential direction (ε_p=0.412ε_i−0.00239, R²=0.93), and radial direction (ε_p=0.391ε_i−0.00175, R²=0.84).

Fig. 4

The magnitude of stress relaxation (Δσ₀) as the function of instantaneous strain when bone is loaded in longitudinal, circumferential and radial directions.

Fig. 5

Relationships between the viscoelastic time constant (τ) and instantaneous strain of human cortical bone for longitudinal, circumferential and radial directions.

Fig. 6

Relationships between the asymptotic term of stress relaxation (A) and instantaneous strain of human cortical bone for longitudinal, circumferential and radial directions.

Fig. 7

Plastic strain energy as the function of instantaneous strain when bone is loaded in the longitudinal, circumferential, and radial directions.

Fig. 8

Relationships between the released elastic strain energy and instantaneous strain in the three orientations.

Fig. 9

Hysteresis energy as the function of instantaneous strain when bone is loaded in the longitudinal, circumferential and radial directions.