Multiscale Modeling of Ligaments and Tendons - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 117

element (FE) codes often make use of an additive decomposition of the strain
energy into volumetric and deviatoric parts, based on the multiplicative decom-
position of the deformation gradient F [ 108 , 233 , 235 ]. This requires a small
modification of the equations above. The uncoupled strain energy equations are
advantageous for representing these tissues in FE software because they can make
use of element formulations that allow for nearly and fully incompressible material
behavior without element locking [ 233 , 235 ].
4.3 Constitutive Modeling of Viscoelasticity
Ligament and tendon viscoelasticity has most commonly been represented by
quasilinear viscoelastic (QLV) [ 2 , 62 , 74 , 181 , 233 , 235 ] and nonlinear viscoelastic
constitutive models [ 135 , 174 , 179 , 180 ]. The QLV theory postulates that the time
response and the elastic response are independent [ 134 ]. The time response is
described using a relaxation function, while the elastic stress response is typically
described using a hyperelastic constitutive model [ 181 ]. The time dependent stress
is then obtained by convolving the relaxation function with the elastic stress.
According to QLV theory, the relaxation function is implicitly related to the creep
function via a convolution [ 135 ]. Thus, an experimentally measured relaxation
function should predict an experimentally measured creep function. Although a
number of studies have applied the QLV theory successfully to describe the time-
and rate-dependent material behavior of ligaments and tendons [ 62 , 74 , 118 , 215 ,
216 ], several studies have suggested that these materials do not strictly behave as
quasilinear viscoelastic materials [ 59 , 61 , 179 , 217 ]. This has motivated the
development of nonlinear viscoelastic models [ 174 , 179 , 180 ].
The apparent viscoelasticity of ligament and tendon can also be described using
biphasic theory [ 137 , 185 , 236 , 253 ]. Biphasic theory postulates an interaction
between a porous, elastic solid phase and an incompressible fluid phase. Loading
of the biphasic material induces volumetric changes in the elastic phase. This
creates pressure gradients, which drive a time dependent fluid flux through the
porous matrix. Diffusive drag and thus energy dissipation is induced by the local
difference in velocity between the solid and fluid phases. Biphasic materials
exhibit stress relaxation, creep and hysteresis. A necessary component of the field
equations for the biphasic theory is the introduction of additional degrees of
freedom related to the time and spatially varying fluid pressure field (or fluid
velocity), thus making analytical solutions more difficult to compute than for
standard viscoelastic constitutive models. Because of this, quasi-analytic solutions
to biphasic problems have only been obtained for simplified geometries and
loading scenarios [ 13 , 46 ]. These include the confined and unconfined loading of a
cylinder subjected to ramp loading, step loading and harmonic loading [ 13 , 46 ] for
linear material behavior, and for certain nonlinear materials [ 15 , 110 ]. Both flow-
dependent (e.g. biphasic material) and flow-independent mechanisms may be
needed to accurately describe and predict the apparent viscoelasticity of some