Multiscale Elastic Models of Collagen Bio-structures: From Cross-Linked Molecules to Soft Tissues

Abstract

Mechanics of collagen bio-structures at different scales (nano, micro, and macro) is addressed, aiming to describe multiscale mechanisms affecting the constitutive response of soft collagen-rich tissues. Single-scale elastic models of collagen molecules, fibrils, and crimped fibers are presented and integrated by means of consistent inter-scale relationships and homogenization arguments. In this way, a unique modeling framework based on a structural multiscale approach is obtained, which allows to analyze the macroscale mechanical behavior of soft collagenous tissues. It accounts for the dominant mechanisms at lower scales without introducing phenomenological descriptions. Comparisons between numerical results obtained via present model and the available experimental data in the case of tendons and aortic walls prove present multiscale approach to be effective in capturing the deep link between histology and mechanics, opening to the possibility of developing patient-specific diagnostic and clinical tools.

Appendix

The incremental elastic equilibrium solution for the multi-layered aortic cylinder, comprising $N$ identical media lamellar units (MLUs) and loaded with a uniform internal pressure increment $\dot{p}$, is herein briefly reported. Reference is made to the linearly elastic solution proposed in [67], considering the problem as an axisymmetric generalized plane strain problem, characterized by a given non-vanishing constant direct strain increment $\dot{\varepsilon}_o$ along the cylinder axis $z$. Accordingly, for the $k$th MLU, the incremental components of the displacement field in a cylindrical system of coordinates result in

$$ \dot{u}_{\rho}^k= A_1^k\rho^{\alpha_k} +A_2^k\rho^{-\alpha_k} + \dot{\varepsilon}_o a_1^k \rho + B^k a_{2}^k \rho^2, $$

(36)

$$ \dot{u}_{\varphi}^k=B^k \rho z, $$

(37)

$$ \dot{u}_{z}^k =\dot{\varepsilon}_o z. $$

(38)

The non-trivial increments of strain components are

$$ \dot \varepsilon_\rho^k = \frac{\partial \dot u_\rho^k}{\partial \rho},\qquad \dot \varepsilon_\varphi^k =\frac{\dot u_\rho^k}{\rho},\qquad \dot \gamma_{z\varphi}^k =B^k\rho, $$

(39)

and the components of stress increments are

$$ \dot{\sigma}_{\rho}^k=A_1^k\beta_{\rho 1}^k\rho^{\alpha_k-1} +A_2^k\beta_{\rho 2}^k\rho^{-\alpha_k-1} + \dot{\varepsilon}_o \beta_{\rho 3}^k + B^k \beta_{\rho 4}^k \rho, $$

(40)

$$ \dot{\sigma}_{\varphi}^k=A_1^k\beta_{\varphi 1}^k\rho^{\alpha_k-1} +A_2^k\beta_{\varphi 2}^k\rho^{-\alpha_k-1} + \dot{\varepsilon}_o \beta_{\varphi 3}^k + B^k \beta_{\varphi 4}^k \rho,\\ $$

(41)

$$ \dot{\sigma}_{z}^k=A_1^k\beta_{z 1}^k\rho^{\alpha_k-1} +A_2^k\beta_{z 2}^k\rho^{-\alpha_k-1} + \dot{\varepsilon}_o \beta_{z 3}^k + B^k \beta_{z 4}^k \rho, $$

(42)

$$ \dot{\tau}_{z\varphi}^k=A_1^k \beta_{\gamma 1}^k\rho^{\alpha_k-1} +A_2^k \beta_{\gamma 2}^k\rho^{-\alpha_k-1} + \dot{\varepsilon}_o \beta_{\gamma 3}^k + B^k \beta_{\gamma 4}^k \rho, $$

(43)

where

$$ \beta_{q 1}^k = \bar{C}^k_{g2}+\alpha_k\bar{C}^k_{g3}, \qquad \beta_{q 2}^k = \bar{C}^k_{g2}-\alpha_k\bar{C}^k_{g3}, \\ $$

(44)

$$ \beta_{q 3}^k = \bar{C}^k_{g1}+(\bar{C}^k_{g2}+\bar{C}^k_{g3})a_1^k, \qquad \beta_{q 4}^k = \bar{C}^k_{g6}+(\bar{C}^k_{g2}+2\bar{C}^k_{g3})a_2^k, $$

(45)

with the index $q=z,\varphi,\rho,\gamma$ corresponding to the $g$-index values $g=1,2,3,6$, respectively, and with

$$ \alpha_k^2=\bar{C}^k_{22}/\bar{C}^k_{33}, $$

(46)

$$ a_1^k = (\bar{C}^k_{12}-\bar{C}^k_{13}) / [\bar{C}^k_{33}(1-\alpha_k^2)], \qquad a_2^k=(\bar{C}^k_{26}-2\bar{C}^k_{36})/[\bar{C}^k_{33}(4-\alpha_k^2)], $$

(47)

$\bar{C}_{ij}^k$ denoting the $(i,j)$ component of the stiffness matrix ${\mathbb{\bar C}}^k$ for the $k$th MLU.

It is worth pointing out that during the deformation path both strain and stress components, and thereby also the strain-dependent MTU material properties, are $z$- and $\varphi$-independent.

Assuming the MLUs as perfectly bonded layers, the $3N$ unknown constants $A_1^k,\,A_2^k,\,B^k$ (with $k=1...N$) are solved by imposing the following $3(N-1)$ continuity conditions at the MLU’s interfaces:

$$ \dot{u}_\rho^k(r_k) = \dot{u}_\rho^{k+1}(r_k), \qquad \dot{u}_\varphi^k(r_k) = \dot{u}_\varphi^{k+1}(r_k), \qquad \dot{\sigma}_r^k(r_k) = \dot{\sigma}_r^{k+1}(r_k), $$

(48)

and the three equilibrium incremental relationships:

$$ \dot{\sigma}_r(r_i)=-\dot{p}, \qquad \dot{\sigma}_r(r_i+h_a)=0, \qquad 2\pi\sum_{k=1}^{N} \int_{r_{k-1}}^{r_k}\rho^2 \dot{\tau}_{z\varphi}^k d\rho = 0, $$

(49)

the latter prescribing the average torque related to the tangential stress increment $\dot{\tau}_{z\varphi}$ to be zero.