Mathematical description of stress relaxation of bovine femoral cortical bone.

In 1993 we proposed an empirical formula for describing the relaxation modulus of cortical bone based on the results of stress relaxation experiments performed for 1 x 10(5) sec: [E(t) = E0{A exp[ -(t/tau1)beta] + (1 - A) exp(-t/tau2)}, (0 < A, beta <1 and tau1 << tau2) where E0 is the initial value of the relaxation modulus, A is the portion of the first term, tau1 and tau2 are characteristic relaxation times, and beta is a shape factor [Sasaki et al., J. Biomechanics 26 (1993), 1369]. Although the relaxation properties of bone under various external conditions were described well by the above equation, recent experimental results have indicated some limitations in its application. In order to construct an empirical formula for the relaxation modulus of cortical bone that has a high degree of completeness, stress relaxation experiments were performed for 6 x 10(5) seconds. The second term in the equation was determined as an apparently linear portion in a log E(t) vs t plot at t>1 x 10(4) sec. The same plot for experiments performed for 6 x 10(5) seconds revealed that the linear portion corresponding to the second term was in fact a curve with a large radius of curvature. On the basis of this fact, we proposed a second improved empirical equation E(t) = E0{A exp [ -(t/tau1)beta] + (1 - A) exp[-(t/tau2)gamma]}, (0

Anisotropic viscoelastic properties of cortical bone.

Relaxation Young's modulus of cortical bone was investigated for two different directions with respect to the longitudinal axis of bone (bone axis, BA): the modulus parallel (P) and normal (N) to the BA. The relaxation modulus was analyzed by fitting to the empirical equation previously proposed for cortical bones, i.e., a linear combination of two Kohlraush-Williams-Watts (KWW) functions (Iyo et al., 2003. Biorheology, submitted): E(t)=E0 (A1 exp[-(t/tau1)beta]+(1-A1) exp[-(t/tau2)gamma]), [0 < A1, beta, gamma < 1], where E0 is the initial modulus value E0. Tau1 and tau2(>>tau1) are characteristic times of the relaxation, A1 is the fractional contribution of the fast relaxation (KWW1 process) to the whole relaxation process, and beta and gamma are parameters describing the shape of the relaxation modulus. In both P and N samples, the relaxation modulus was described well by the empirical equation. The KWW1 process of a P sample almost completely coincided with that of an N sample. In the slow process (KWW2 process), there was a difference between the relaxation modulus of a P sample and that of an N sample. The results indicate that the KWW1 process in the empirical equation represents the relaxation in the collagen matrix in bone and that the KWW2 process is related to a higher-order structure of bone that is responsible for the anisotropic mechanical properties of bone.