A strain rate dependent model with decreasing Young’s Modulus for cortical human bone

<jats:title>Abstract</jats:title> <jats:p>In the existing literature, some studies have observed an increase in the elastic modulus of human cortical bone with strain rate, which has been described as a consequence of the viscoelastic properties of the bone. However, these results contradict the findings of other studies, in which an independence or decrease of the elastic modulus with strain rate is observed, which could be explained by other non-viscoelastic mechanisms. This research studies the dynamic behavior of human cortical bone specimens and investigates their mechanical properties . A full and objective strain rate dependent model is proposed and used to describe the experimental results obtained from uniaxial tensile tests of twenty-one human rib cortical bone specimens from twelve male post mortem human subjects (average age of 68.5 ± 12.3 years). In addition, a general discussion of some families of viscoelastic models is given and the caution with which they should be used when dealing with complex materials such as bone. The main experimental finding is that in the range of strain rate analyzed (<jats:inline-formula> <jats:tex-math> <?CDATA $\dot{\varepsilon }=0.10-0.60$?> </jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:mi>ε</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̇</mml:mo> </mml:mrow> </mml:mover> <mml:mo>=</mml:mo> <mml:mn>0.10</mml:mn> <mml:mo>−</mml:mo> <mml:mn>0.60</mml:mn> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="bpexacd458ieqn1.gif" xlink:type="simple" /> </jats:inline-formula>), there is a significant decrease in Young’s modulus (<jats:italic>E</jats:italic> ≈ 18 GPa for <jats:inline-formula> <jats:tex-math> <?CDATA $\dot{\varepsilon }=0.10\ {{\rm{s}}}^{-1}$?> </jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:mi>ε</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̇</mml:mo> </mml:mrow> </mml:mover> <mml:mo>=</mml:mo> <mml:mn>0.10</mml:mn> <mml:mspace width="0.33em" /> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">s</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="bpexacd458ieqn2.gif" xlink:type="simple" /> </jats:inline-formula> and <jats:italic>E</jats:italic> ≈ 8 GPa for <jats:inline-formula> <jats:tex-math> <?CDATA $\dot{\varepsilon }=0.50\ {{\rm{s}}}^{-1}$?> </jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:mi>ε</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̇</mml:mo> </mml:mrow> </mml:mover> <mml:mo>=</mml:mo> <mml:mn>0.50</mml:mn> <mml:mspace width="0.33em" /> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">s</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="bpexacd458ieqn3.gif" xlink:type="simple" /> </jats:inline-formula>), which is not of viscoelastic origin. Moreover, the most frequently used viscoelastic models analyzed in this study predict how the elastic modulus should not vary markedly with strain rate for small strains. In fact, the observed behavior seems related to the findings of other researchers who observed that the microcraking damage depends on the strain rate in the same sense found in our work. This allows us to interpret the qualitative results as a consequence of the microcracking that takes place within the cortical bone, and not related to viscoelastic effects.</jats:p>