Biomedical Engineering Reference
In-Depth Information
4.4 Concluding Remarks
We conclude with thoughts on selecting a viscoelastic model that is appropriate for
a specific material and application.
The simplest approach by far is to use a linear viscoelastic model, with the
assumption of small strain. Nearly every viscoelastic material can be modeled
using linear viscoelasticity over a range of conditions, and the first assessment to
make is whether this range of conditions encompasses those in which the material
will be studied. Geometric linearity becomes an increasingly poor assumption with
increasing strain. As a rule of thumb, it is a poor approximation for strain levels
above 10-15 % in cases in which a constitutive model is to be applied to con-
ditions that differ substantially from those used to fit the constitutive law. In the
example used in this chapter, strain levels exceeded this range substantially, but
accounting for geometric nonlinearity was not needed for comparing amongst
identical uniaxial stretches applied to a single specimen. However, if these data
were to be applied to predict the response of a tissue to a large strain, multiaxial
stress state, an appropriate pullback to the reference configuration would need to
be applied to the fitted models prior to generalizing to a 3D constitutive law. For
this, we refer the reader to the texts of Gurtin et al. [ 72 ] and of Bower [ 73 ].
The validity of an assumption of material viscoelasticity can be assessed in a
number of ways. Qualitatively, a good indication from a ramp-loading test that a
nonlinear model is required is a force-displacement that is concave-up: all linear
viscoelastic models yield predictions that are linear or concave-down. If such
curvature is significant over strain rates and amplitudes of interest, a nonlinear
viscoelastic model must be used.
If a nonlinear viscoelastic model is required, quasi-linear viscoelasticity is a
great place to start. The Fung QLV model is often fully adequate if a single
reduced relaxation function is appropriate for all levels of tissue stretch. An
example of how to check this is the set of tests shown in Fig. 1 . If, as in Fig. 1 , the
reduced relaxation function varies with the degree that a tissue is stretched, a
different framework is required.
The Fung QLV model is easily extended through the Generalized Fung QLV
model described in this chapter to allow for reduced relaxation functions that vary
with strain. However, all of the tedious computation associated with the Fung QLV
model is compounded when fitting the Generalized Fung QLV model. The
Adaptive QLV model is far simpler than either of these models to fit, and has an
especially simple form for many standard ramp-and-hold tests.
The principle by which nonlinearity is incorporated into the Adaptive QLV
model differs fundamentally from that of the Fung and Generalized Fung QLV
models. The consequence of this is that the choice of constitutive model cannot be
based upon the simplicity of the Adaptive QLV model alone. Instead, tests must be
performed to assess the suitability of model predictions. The incremental ramp-
and-hold protocol always provides data for such assessment, and in the case of the
reconstituted collagen specimens discussed in this chapter the Generalized Fung
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