Tendons and Ligaments: Current State and Future Directions - Multiscale Modeling in Biomechanics and Mechanobiology - page 180

measured deflection of the A-bands of muscle fibers. Although not a direct valida-
tion of the 2D strains within the unit cell, this validation provides evidence of the
accuracy of the FE simulations. By creating micromechanical FE simulations driven
by macroscopic loading, this study was able to utilize modeling as a means for inves-
tigating microscale strain concentrations, something that would have been difficult
using experimental methods alone.
The aforementioned studies utilized 2D simulations. In our own research, we have
used 3D micromechanical FE models to study structure-function relationships in
tendon and ligament tissue [ 133 ]. The aim of this research was to examine how fibril
organization contributes to the elastic volumetric response. The volumetric response
is quantified using the Poisson's ratio in linear theory and the Poisson's function
in nonlinear theory. Experimentally observed Poisson's ratios range from 1.0 to 3.0
for tendon and ligament [ 129 , 139 ], yet the structural underpinnings for these large
values are not known. It was hypothesized that a planar, crimped arrangement of
fibrils would not account for these large Poisson's ratios, while a helical organization
of fibrils would.
To test this hypothesis, 3D unit cells were created that explicitly modeled collagen
fibrils embedded within a matrix material (Fig. 8.11 , top). The fibrils were given
crimped, helical, and combined crimped with a superhelical organization (Fig. 8.11 ,
top). The models were given periodic boundary conditions and subjected to simulated
tensile loading in the fiber direction, which yielded a homogenized macroscale stress-
strain response and a homogenized Poisson's function. For a subset of models, tensile
strains of 8 % were applied and the nonlinear stress-strain response and the Poisson's
function were obtained (Fig. 8.10 , bottom). For all other models, small strains (0.5 %)
were applied and homogenized Poisson's ratios were obtained.
Models with planar crimp (both with and without a helical twist) could generate
the classic nonlinear response, but only models with a helical twist could gener-
ate large Poisson's ratios (Fig. 8.11 , bottom). This suggests that helical twisting of
fibrils (which has been observed histologically [ 247 , 248 ]) may contribute to the
large experimentally measured Poisson's ratios. A parametric study which varied
crimp angle, helical twist, the number of fibrils, and the stiffness of the fibrils and
matrix suggested that the large Poisson's ratios were predicted across a range of
physiologically relevant values for the these parameters. This study highlights the
utility of homogenized micromechanical models in testing structure function based
hypothesis that are otherwise difficult to address. Furthermore, it demonstrates the
use of 3D until cells with a nonrectangular cross section for the use of nonlinear
homogenization.
In the previously discussed studies, boundary conditions on the micromechanical
model were applied a priori to the microscale models. In a recent study, micromechan-
ical models were combined with a macroscale simulation to solve the localization
problem [ 249 ]. In a localization problem, a macroscale deformation (generally
computed from a continuum-based FE simulation) is applied to an RVE, which
is then solved in order to obtain the microscale stress and strain [ 211 ]. In this
study, an analytically based homogenization was used as the constitutive model
for a macroscopic FE simulation. Briefly, the analytical homogenization modeled