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William Ledoux
04-15-1998, 09:57 AM
Dear Biomch-lers,

I would like to thank everyone who responded to my question last
month concerning the quasi-linear theory. I have found that my data
does not demonstrate the linear reduced relaxation vs. ln time plot that
is a requirement in order to use Fung's QLV. My data was distinctly
bilinear.

****** those who responded:

Michael Sachs
Julie A. Martin
James C. Iatridis
Philippe K. Zysset
Lynne E. Bilston
Barry Myers
Eric Talman
Dianne Pawluk
plus one person who didn't want their name used

****** my original question

Biomch-lers,
I am interested in employing Fung's quasi-linear viscoelastic
theory
to curve
fit stress relaxation data. I am preforming a compression test whereby
I ramp
to a certain displacement and hold while I am measuring the force. My
question concerns the approach employed for the reduced relaxation
response (I
will call this G from here on). Fung's text books (1972 and 1993) and
many
articles in the literature (e.g., Woo, J. Biomech. Eng., 1981, Kwan, J.

Biomech, 1993, and Myers, J. Biomech, 1994) all demonstrate a linear
results
for G. By that, I mean that on a semi log plot of the G, the data are
approximately linear between 2 points refered to as tau1 and tau2. (see
plot
below on the left) These constants are two time constants. Fung
details how
to solve for tau1, tau2 and the third constant (c) by using three
equations (G
at 1 sec, G at infinitely (the end of the experiment) and the slope of
G).
My specific question is what can be done if the data is not
linear? (see
plot below on the right) When I plot G with the log of time I get a
distinctly bi-linear curve. Does anyone have any experience curve
fitting
with bi-linear stress relaxation data such as this? I am considering
the not
using Fung's form of G and instead using a multiple exponential fit.

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tau1 tau2

typical, linear
curve bilinear response

These plots are crude, but they demonstrate my point.

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I have had similar experiences with most tissues that I work with. In
particular, we usually find that G(t) = A(t+1)^B works very well for
bovine pericardium, whic show similar non-linear characteristics. This
is about as much as we have done, although we are planning on doing
more.

Michael Sachs msacks@coeds.eng.miami.edu

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I don't know if this response is too late to be helpful, but I too am
using QLV and just completed a study comparing the linear form of QLV
that you mention and the nonlinear form outlined by Woo in Basic
Orthopedic Biomechanics (Raaven Press, 1991) and J. Biomechanics, v26,
pp.447-452. My results showed differences in time constant parameter
results, and a significant difference in the constant C even with a
small sample size. Since I'm looking at short relaxation times and
don't 'know' G(infinity), I used an iterative technique to fit the
linear equation to data. For the nonlinear model, I used a nonlinear
least squares routine (Levenberg-Marquardt) to fit the data. I would
suggest looking into this full-form nonlinear QLV model, which takes
into account finite strain rate (whereas the linear approximation does
not) if your data is not linear with respect to log time.

Julie A. Martin jmartin@emba.uvm.edu

************************************************** ********************

That the reduced relaxation function is not linear on a semi-log plot
suggests that your material exhibits frequency sensitivity. I prefer
using
a continuous relaxation spectrum rather than discrete spectrum
(multiple exponential fit) because it can describe more behaviors with
relatively few material parameters.

I addressed precisely this issue in a recent paper on the viscoelastic
behavior of the nucleus pulposus of the intervertebral disc by employing

a variable amplitude relaxation spectrum capable of describing frequency

sensitive parameters.

The continuous relaxation spectrum, S(tau), is given as:
S(tau)=c1/tau + c2/tau**2 for tau1