Multiscale Modeling of Ligaments and Tendons - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 123

5.2 Micromechanical Modeling of Ligament and Tendon
We define a micromechanical model as a 2D or 3D model that specifies a
microscale geometry (e.g. a unit cell), applies macroscale boundary conditions
(e.g. simulated tensile loading combined with periodic boundary conditions) and
solves the governing equations over the simulation domain. Such a model yields
both a microscale response (e.g. within the unit cell) and a macroscale response
(e.g. reaction force of a unit cell subjected to tensile deformation). Since even
simple 2D geometries do not generally have tractable solutions, these models
almost exclusively rely on computational methods, most commonly the FE
method. Models within the literature are primarily 2D and focused on equilibrium
elasticity, although a biphasic model and 3D models have been proposed [ 137 ,
187 ]. Boundary conditions include fixed boundary conditions (e.g. zero dis-
placement on a model edge), prescribed boundary conditions (e.g. a prescribed
load or displacement on a model edge), periodic boundary conditions and a
combination of these. Both linear and nonlinear micromechanical models have
been proposed for biological tissues. Linear models (ubiquitous in the study of
trabecular bone, e.g. [ 165 , 172 , 176 , 197 , 223 ]) generally seek to obtain homog-
enized coefficients of the linear elastic stiffness tensor. Due to the nonlinear nature
of ligament and tendon, most micromechanical models for this application are
nonlinear and seek to explore nonlinear behavior as the primary goal, although
some studies have reported homogenized linear coefficients as well (e.g. [ 187 ].).
The utility of microscale models is found in a number of ways. Some models are
used to study certain structure function relationships (e.g. how certain microscale
structures affect macroscale behavior) [ 84 , 187 ]. Other models are used to study
microscale damage mechanisms [ 205 ], and still others investigate microscale
mechanotransduction [ 137 ].
Micromechanical models have been used to examine the structure-function
relationship between fibril shape, fibril aspect ratio, the stiffness of fibrils and the
inter-fibril matrix [ 83 , 84 ]. In these studies, a 2D plane strain model was used to
examine force transfer between adjacent collagen fibrils via an inter-fiber matrix
[ 84 ]. A unit cell was created that consisted of a discretized fibril embedded within
a matrix material (Fig. 8 ). The fibrils were given cylindrical or parabloidal (i.e.
tapered) endpoints [ 83 ]. The unit cell was subjected to homogenous boundary
conditions in which a displacement was applied to the sides of the model. The
aspect ratio of the fiber and applied load were varied parametrically and their
influence on the fibril stress, inter-fiber force transfer and strain was examined.
Simulations revealed that fiber strain displayed a dependency on the end shape of
the fibril, on the fibril aspect ratio and the ratio of the fibril stiffness to the matrix
stiffness. The effect of tapered fibril ends was to decrease stress within fibers. The
effect of increasing the stiffness of the inter fibril matrix was to increase load
sharing between fibril and the matrix, which yielded decreased fibril strains. By
utilizing a unit cell approach, this study was able to study the influence of struc-
ture-function relationships that would be difficult if not impossible to investigate