Biomedical Engineering Reference
In-Depth Information
biological soft tissues [ 109 , 110 ]. Poroviscoelastic formulations have been pro-
posed that utilize a viscoelastic continuum model within the solid phase. These
approaches have found utility in the field of cartilage mechanics [ 109 , 110 , 154 ].
4.4 Computational Modeling
Analytical solutions to the equations of motion for the mechanics of ligaments and
tendons can only be obtained for simplified geometries and loading scenarios (e.g.,
uniaxial tension-compression). For complex geometries and loading patterns such
as simulation of the mechanics of a ligament within an intact joint, the geometry
and governing equations must be discretized and solved numerically [ 233 , 235 ].
The FE method is by far the most commonly used numerical method in the field of
biosolid mechanics. Commercial and freely available software packages support
preprocessing, solution and postprocessing the nonlinear FE problems. Many
studies in the literature have used FE methods for the simulation of ligament and
tendon mechanics (e.g., [ 64 , 65 , 75 , 235 ]). In addition to elastic problems, the FE
method can also be used to solve viscoelastic problems and biphasic problems. In
the past, addressing these types of problems was more difficult due to the lack of a
FE framework specifically designed for biological applications. To address this
issue, our lab developed FEBio, a nonlinear implicit finite element framework
designed
specifically
for
analysis
in
computational
solid
biomechanics
( www.febio.org )[ 148 ].
4.5 Homogenization
Although continuum based constitutive models are useful for describing macro-
scopic behavior, they do not address the mechanical behavior that occurs at lower
length scales and are not always useful for the study of structure-function rela-
tionships between the microscale and the macroscale. Because of the multiscale
structure of ligaments and tendons, it is sometimes desirable to use models that can
simultaneously describe both macroscale and microscale behavior. This is the goal
of multiscale modeling in mechanics, and homogenization is part of the foundation
of multiscale modeling. Homogenization is the process of obtaining a macroscopic
stress-strain response from a material with a known heterogeneous microstructure
[ 78 , 122 , 211 ]. It is based on the concept of a representative volume element
(RVE), which can be considered representative of the continuum [ 82 , 91 , 120 ]
(Fig. 6 ). An RVE must be large enough to be statistically representative of the
material microstructure, but it must still satisfy the continuum assumption that its
dimensions are much smaller than the macroscale dimension [ 82 ]. For the case of a
perfectly periodic microstructure (e.g., a lattice of spherical particles), the RVE
reduces to a unit cell [ 172 ]. In a homogenization, the RVE is subjected to the
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