Assessing the effective elastic properties of the tendon-to-bone insertion: a multiscale modeling approach

Abstract

The interphase joining tendon to bone plays the crucial role of integrating soft to hard tissues, by effectively transferring stresses across two tissues displaying a mismatch in mechanical properties of nearly two orders of magnitude. The outstanding mechanical properties of this interphase are attributed to its complex hierarchical structure, especially by means of competing gradients in mineral content and collagen fibers organization at different length scales. The goal of this study is to develop a multiscale model to describe how the tendon-to-bone insertion derives its overall mechanical behavior. To this end, the effective anisotropic stiffness tensor of the interphase is predicted by modeling its elastic response at different scales, spanning from the nanostructural to the mesostructural levels, using continuum micromechanics methods. The results obtained at a lower scale serve as inputs for the modeling at a higher scale. The obtained predictions are in good agreement with stochastic finite element simulations and experimental trends reported in literature. Such model has implication for the design of bioinspired bi-materials that display the functionally graded properties of the tendon-to-bone insertion.

Appendices

Appendix 1: Hill tensor $\mathbb {P}^0$

1.1 Appendix 1.1: Hill tensor for a cylindrical inclusion in a transversely isotropic medium

The nonzero components of the Hill tensor $\mathbb {P}^0$ for a cylindrical inclusion embedded in a transversely isotropic matrix of stiffness $\mathbb {C}^0$ are given according to Suvorov and Dvorak (2002) using Voigt notation, $x_3$ being the axis of rotational symmetry,

$$\begin{aligned} P_{11}^0&=\frac{1}{8}\frac{(C^0_{22}+C^0_{66})+2C^0_{66}}{C^0_{22}C^0_{66}}\,, \end{aligned}$$

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$$\begin{aligned} P_{22}^0&=\frac{1}{8}\frac{(C^0_{22}+3C^0_{66})}{C^0_{22}C^0_{66}}\,, \end{aligned}$$

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$$\begin{aligned} P_{12}^0&=\frac{1}{8}\frac{C^0_{66}-C^0_{22}}{C^0_{22}C^0_{66}}\,, \end{aligned}$$

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$$\begin{aligned} P_{66}^0&=\frac{1}{2}\frac{C^0_{22}+C^0_{66}}{C^0_{22}C^0_{66}}\,, \end{aligned}$$

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$$\begin{aligned} P_{44}^0&=\dfrac{1}{2C^0_{44}}\,, \end{aligned}$$

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$$\begin{aligned} P_{55}^0&=P_{44}^0\,. \end{aligned}$$

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1.2 Appendix 1.2: Hill tensors for a spherical inclusion in an isotropic medium

Considering the case of a spherical inclusion embedded in an isotropic matrix of stiffness $\mathbb {C}^0$ (Suvorov and Dvorak 2002), the components of the Hill tensor $\mathbb {P}^0$ now read as

$$\begin{aligned} P_{11}^0&=\frac{7C^0_{44}+2C^0_{12}}{15C^0_{44}(C^0_{12}+2C^0_{44})}\,, \end{aligned}$$

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$$\begin{aligned} P_{12}^0&=\frac{C^0_{44}+C^0_{12}}{-15C^0_{44}(C^0_{12}+2C^0_{44})}\,, \end{aligned}$$

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$$\begin{aligned} P_{44}^0&=\frac{2(3C^0_{12}+8C^0_{44})}{15C^0_{44}(C^0_{12}+2C^0_{44})}\,, \end{aligned}$$

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$$\begin{aligned} P_{22}^0&=P_{33}^0=P_{11}^0,\;\; P_{55}^0=P_{66}^0=P_{44}^0,\;\; P_{13}^0=P_{23}^0=P_{12}^0\,. \end{aligned}$$

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Appendix 2: Effective stiffness tensor of the hydroxyapatite foam $\mathbb {C}_{\text {Hw}}(x)$

The self-consistent scheme for two interpenetrating (spherical) inclusion phases with a linear elastic and isotropic behavior can be solved according to Hellmich et al. (2004). In such a case, the nonlinear system of equations (3) can be substituted by a system of two nonlinear equations (note that the spatial variable x was omitted here for sake of clarity),

$$\begin{aligned} \dfrac{f_{\text {HA}}(K_{\text {HA}}-K_{\text {Hw}})}{1+\alpha _{\text {Hw}}(K_{\text {HA}}-K_{\text {Hw}})/K_{\text {Hw}}}+\dfrac{(1-f_{\text {HA})}(K_{\text {wp}}-K_{\text {Hw}})}{1+\alpha _{\text {Hw}}(K_{\text {wp}}-K_{\text {Hw}})/K_{\text {Hw}}}=0\,, \end{aligned}$$

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$$\begin{aligned} \dfrac{f_{\text {HA}}(G_{\text {HA}}-G_{\text {Hw}})}{1+\beta _{\text {Hw}}(G_{\text {HA}}-G_{\text {Hw}})/G_{\text {Hw}}}+\dfrac{(1-f_{\text {HA})}(G_{\text {wp}}-G_{\text {Hw}})}{1+\beta _{\text {Hw}}(G_{\text {wp}}-G_{\text {Hw}})/G_{\text {Hw}}}=0\,, \end{aligned}$$

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where the two unknowns $K_{\text {Hw}}$ and $G_{\text {Hw}}$ denote the bulk and shear moduli of the HA foam, respectively. The parameters $\alpha _{\text {Hw}}$ and $\beta _{\text {Hw}}$ are defined as

$$\begin{aligned} \alpha _{\text {Hw}}=\frac{3K_{\text {Hw}}}{3K_{\text {Hw}}+4G_{\text {Hw}}}\,, \;\; \beta _{\text {Hw}}=\frac{6(K_{\text {Hw}}+2G_{\text {Hw}})}{5(3K_{\text {Hw}}+4G_{\text {Hw}})}\,. \end{aligned}$$

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Solving the aforementioned system yields the following components for the stiffness tensor of the hydroxyapatite foam,

$$\begin{aligned}&C^{\text {Hw}}_{11}=K_{\text {Hw}}+\dfrac{4}{3}G_{\text {Hw}}\,, \end{aligned}$$

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$$\begin{aligned}&C^{\text {Hw}}_{12}=K_{\text {Hw}}-\dfrac{2}{3}G_{\text {Hw}}\,, \end{aligned}$$

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$$\begin{aligned}&C^{\text {Hw}}_{44}=G_{\text {Hw}}\,, \end{aligned}$$

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$$\begin{aligned}&C^{\text {Hw}}_{22}=C^{\text {Hw}}_{33}=C^{\text {Hw}}_{11}\,, \;\; C^{\text {Hw}}_{13}=C^{\text {Hw}}_{23}=C^{\text {Hw}}_{12},C^{\text {Hw}}_{55}=C^{\text {Hw}}_{66}=C^{\text {Hw}}_{44}\,. \end{aligned}$$

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