Uniaxial and Multiaxial Fatigue Life Prediction of the Trabecular Bone Based on Physiological Loading: A Comparative Study

Abstract

Fatigue assessment of the trabecular bone has been developed to give a better understanding of bone properties. While most fatigue studies are relying on uniaxial compressive load as the method of assessment, in various cases details are missing, or the uniaxial results are not very realistic. In this paper, the effect of three different load histories from physiological loading applied on the trabecular bone were studied in order to predict the first failure surface and the fatigue lifetime. The fatigue behaviour of the trabecular bone under uniaxial load was compared to that of multiaxial load using a finite element simulation. The plastic strain was found localized at the trabecular structure under multiaxial load. On average, applying multiaxial loads reduced more than five times the fatigue life of the trabecular bone. The results provide evidence that multiaxial loading is dominated in the low cycle fatigue in contrast to the uniaxial one. Both bone volume fraction and structural model index were best predictors of failure (p < 0.05) in fatigue for both types of loading, whilst uniaxial loading has indicated better values in most cases.

Appendix

Implementation of Gait Loading

A set of polynomial functions derived from Matlab relative to gait loading during normal walking were used to formulate the time dependent behaviour of trabecular bone under fatigue analysis.

$$\begin{aligned} F_{x} = - 1E6\left( { - 0.3434t^{7} + 1.1756t^{6} - 1.5667t^{5} + 1.0239t^{4} - 0.3369t^{3} + 0.0490t^{2} - 0.0013t + 0.0002} \right) \hfill \\ F_{y} = - 1E5\left( { - 1.1068t^{7} + 3.8818t^{6} - 4.8999t^{5} + \, 2.4244t^{4} - \, 0.0797t^{3} {-} \, 0.2734t^{2} + \, 0.0542t - 0.0010} \right) \hfill \\ F_{z} = - 1E5 \, \left( { - \, 2.9006t^{7} + \, 7.0557t^{6} {-} \, 3.5732t^{5} {-} \, 3.5934t^{4} + \, 4.4087t^{3} {-} \, 1.6199t^{2} + \, 0.2244t \, + \, 0.0048} \right) \hfill \\ \end{aligned}$$

The initial gait loading reduce to the trabecular loading as the following;

Normal force, F _z  = −1E5 (−2.1755t ⁷ + 5.2918t ⁶ − 2.6799t ⁵ − 2.6951t ⁴ + 3.3065t ³ − 1.2149t ² + 0.1683t + 0.0036)

Torsional moment, T = 1E3 (1.1352t ⁷ − 3.9095t ⁶ + 5.1417t ⁵ − 3.1659t ⁴ + 0.8622t ³ − 0.0542t ² − 0.0168t − 0.0003)

Thus, T _xy  = 1E3 (0.8514t ⁷ − 2.9321t ⁶ + 3.8563t ⁵ − 2.3744t ⁴ + 0.6467t ³ − 0.0407t ² − 0.0126t − 0.0002)

Influencing Parameter in Fatigue Life Prediction

Figure 9 shows increasing cycles to failure with function of fatigue strength coefficient while other parameters are constant. This figure also shows a small deviation (lower than 5%) of different Q value and predetermined fatigue strength coefficient in all cases. Q values determines the number of evaluation points used in the search for the critical plane and thus it controls the computational time. A smooth transition in the results indicates that the specified search resolution for the critical plane is sufficient to correctly capture the fatigue response. A mesh convergence also has been done on Fig. 10 to estimate accurate fatigue life in low cycle fatigue and effective plastic strain and this sensitivity analysis is consistent under static initial modulus and fatigue properties. Details on the definition of fatigue parameter are shown in Fig. 11. Fatigue strength exponent, b and fatigue strength coefficient, σ′_f normally cover the entire range of high cycle fatigue that can be explained in logarithmic increase of fatigue cycles. Both increase of these two parameter will increase fatigue failure prediction. In contrast to high cycle fatigue, effect of fatigue ductility coefficient, ε′_f on low cycle fatigue prediction is insignificant to plastic strain. However the prediction of number of cycle to failure is affected. Increasing value of fatigue ductility coefficient could result in higher prediction of fatigue cycle in which the same effect can be obtained by reducing the value of fatigue ductility exponent, c. In this study, consistent initial fatigue parameter and coefficients derived from the bovine bone and consider valid in all type of analysis regarding to fatigue life prediction.

Figure 9

Effect of search resolution, Q, on critical plane setting evaluation in fatigue analysis. The accuracy of the algorithm is determined by the spacing of the points, which can be selected by the critical point evaluation as the search resolution setting Q = (N + 1).

Figure 10

Mesh convergence analysis performed to obtain the optimum number of elements in prediction of cycle to failure and effective plastic strain.

Figure 11

Typical fatigue life curve under combination Basquin and Coffin–Manson.