Progressive post-yield behavior of human cortical bone in compression for middle-aged and elderly groups

Abstract

In this study, a progressive loading regimen (load-dwell-unloading-dwell-reloading) was applied on bone samples to examine the compressive post-yield response of bone at increasing strain levels. Cortical bone specimens from human tibiae of two age groups (middle-aged group: 53+/-2 years, 4 females and 4 males, elderly group: 83+/-6 years, 4 females and 4 males) were loaded in compression using the progressive loading scheme. Modulus degradation, plastic deformation, viscous response, and energy dissipation of bone during post-yield deformation were assessed. Although initial modulus was not significantly different between the two age groups, the degradation of modulus with the applied strain in the elderly group was faster than in the middle-aged group. The modulus loss (or microdamage accumulation) of bone occurred prior to plastic deformation. Plastic strain had a similar linear relationship with the applied strain for both middle-aged and the elderly group although middle-aged bone yielded at a greater strain. The viscoelastic time constant changed similarly with increasing strain for the two groups, whereas a higher magnitude of stress relaxation was observed in the middle-aged group. Energy dissipation was investigated through three pathways: elastic release strain energy, hysteresis energy, and plastic strain energy. The middle-aged group had significantly greater capacity of energy dissipation than the elderly group in all three pathways. The information obtained may provide important insights in age-related effects on bone fragility.

Fig. 1

Schematic representation of specimen preparation and mechanical testing.

Fig. 2

Illustration of methods for determining the mechanical properties of bone in each load–dwell–unload–dwell cycle during the progressive loading test.

Fig. 3

Methods for determination of the yield strain (ε_y) and viscoelastic time constant (τ). (a) Determination of yield strain (ε_y) based on a linear regression of the linear portion of the instantaneous strain (ε_i) and the plastic strain (ε_p) curve by making ε_p = Aε_i−B = 0 (i.e., ε_y = B/A); (b) determination of viscoelastic time constant (τ) based on an exponential equation fitting of the stress relaxation vs. time curve.

Fig. 4

Comparison between the stress–strain curve of monotonic loading test and the envelop of the stress–strain curve of the progressive cyclic loading test on bone specimens from the same donor.

Fig. 5

Photographs of two typical failure patterns observed among the compressed specimens: oblique fracture and cone-shaped fracture.

Fig. 6

Relationship between the instantaneous modulus and applied strain and the exponential regression fitting of the curves for middle-aged (E_i = 18.8e^{−52.7(ε_i−0.0012)}, R² = 0.96) and elderly (E_i = 19.0e^{−64.3(ε_i−0.002)}, R² = 0.95) groups. The value of m is greater for elderly bone (m = 64.3) than middle-aged bone (m = 52.7), suggesting a faster modulus loss for elderly bone.

Fig. 7

Relationship between the plastic strain (ε_p) and the applied strain (ε_i) and the linear regression fitting of the linear part of the curves for both middle-aged (ε_p = 0.60 ε_i−0.0052, and R² = 0.98) and elderly groups (ε_p = 0.61 ε_i−0.0046, and R² = 0.95). The yield strain (ε_y) is determined at the intersection of the regression line with the horizontal axis.

Fig. 8

Relationship between the viscoelastic time constant (τ) and the applied strain (ε_i) for both middle-aged and elderly groups.

Fig. 9

Relationship between the stress relaxation (Δσ₀) and the applied strain (ε_i) for both middle-aged and elderly groups.

Fig. 10

Relationship between plastic strain energy dissipation (Upl) and applied strain (ε_i) and the linear regression fitting of the curves for both middle-aged (R² = 0.98) and elderly groups (R² = 0.97).

Fig. 11

Relationship between elastic release strain energy dissipation (Uer) and the applied strain (ε_i) for both middle-aged and elderly groups. Linear regression was applied in the region with strain more than 1%.

Fig. 12

Relationship between hysteresis energy dissipation (Uhy) and the applied strain (ε_i) for both middle-aged and elderly groups. Linear regression was applied in the region with strain more than 1%.

Fig. 13

Relationship between the plastic strain (ε_p) and modulus loss (= 1−E_i/E₀) for both middle-aged and elderly groups.

Fig. 14

Fluorescence image of a damaged zone of bone in compression stained in basic fuchsin. Cross-hatched microcracks oriented in the directions of the maximum shear stress are observed.