Mathematical description of stress relaxation of bovine femoral cortical bone

Toshiya Iyo, Naoki Sasaki, Yasuyuki Maki, Mitsuo Nakata

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11 Citations (Scopus)


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 × 105 sec: E(t) = E0{A exp [-(t/τ1)β] + (1 - A)exp(-t/τ2)}, (0 < A, β < 1 and τ1 ≪ τ2), where E0 is the initial value of the relaxation modulus, A is the portion of the first term, τ1 and τ2 are characteristic relaxation times, and β 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 × 105 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 × 104 sec. The same plot for experiments performed for 6 × 105 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/τ1)β] + (1 - A)exp[-(t/τ2)γ]}, (0 < A, β, γ < 1) to describe the stress relaxation of cortical bone. The early stage of the stress relaxation process, which could not be expressed by the first, is well described by the second equation.

Original languageEnglish
Pages (from-to)117-132
Number of pages16
Issue number2
Publication statusPublished - 2006
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Physiology
  • Physiology (medical)


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