Multiscale Modeling of Ligaments and Tendons - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 122

Table 2 Summary of homogenization methods
Homogenization Number of simulations Purpose

Linear Finite number, based on number of

unknowns in linear elasticity tensor

Homogenized linear material
coefficients

Micromechanical Finite number, based on model application Investigate nonlinear behavior

and structure-function
relationships

1st order FE 2

Simulation every time stress and tangent
stiffness are evaluated within macro
model

Nonlinear homogenization for
infinitesimally small
microstructures

2nd order FE 2 Simulation every time stress and tangent

stiffness are evaluated within macro
model

Nonlinear homogenization that
incorporates size effects

The method of homogenization used is dependent on the intended application

*250 lm, which is only 1-2 orders of magnitude less than ligament and tendon
widths, which are *5-30 mm (refer to Sect. 2.3 ). For a comparison of homoge-
nization techniques, refer to Table 2 .

5 Multiscale Modeling of Ligament and Tendon Mechanics

5.1 Introduction

The field of multiscale mechanical modeling, and the multiscale modeling of
ligament and tendon in particular, is in its relative infancy. For example, there
seems to be no clear consensus on the very definition of multiscale modeling, as it
has been applied to a large number of models that vary as to what scales are
included and how the scales are linked. For the purposes of this section, a mul-
tiscale model is defined as a model that addresses two or more physical scales.
This implies that both macroscale and microscale stress and strain are computed
from a single simulation. Furthermore, this requires that there is some form of
linking between scale levels. This linking is based on the appropriate application
of boundary conditions, which may include periodic boundary conditions, pre-
scribed boundary conditions, homogenous boundary conditions and a mixture
thereof. This definition includes micromechanical models in which both a mac-
roscale response and microscale response is described, as well as full nonlinear
homogenizations (e.g., FE 2 methods). It is important to note that this definition
does not include structurally motivated constitutive models, which may utilize the
notion of microstructural features (e.g., uncrimping of collagen fibers [ 70 , 88 ],
fiber recruitment [ 111 ] or fiber families embedded within a ground substance [ 18 ]).
This definition also precludes models that utilize generalized continuums, which
address microscale size effects but do not specifically define microscale stress and
strain [ 225 ]. This section reviews the state of the art in multiscale modeling and
multiscale model validation as they relate to ligament and tendon.