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bilinear relaxation with qlv

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  • bilinear relaxation with qlv

    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.

    |
    * | *
    |
    * | *
    |
    * | *
    |
    * | *
    |
    * | *
    |
    * |
    * * * *
    |
    * |
    |________________
    |____________________
    tau1 tau2

    typical, linear
    curve bilinear response


    These plots are crude, but they demonstrate my point.


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

    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

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

    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
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