<?xml version='1.0' encoding='utf-8'?>
<!-- FILE : Polyconvex_law_2005.cml
CREATED : 24th January 2007
LAST MODIFIED : 7th February 2007
AUTHOR : Jesse Ashton
Bioengineering Institute
The University of Auckland
MODEL STATUS : This model conforms to the CellML 1.0 Specification released on
10th August 2001, and the 16/1/02 CellML Metadata 1.0 Specification.
DESCRIPTION : This file contains a CellML description of the Polyconvex Anisotropic (Orthotropic) constitutive material law (anisotropic strain-energy function for soft collagenous tissues), defining the relation between the nine independent strain components and the stress components.
CHANGES:
18/02/04 - CML - Completed the Metadata.
--><model xmlns="http://www.cellml.org/cellml/1.0#" xmlns:cmeta="http://www.cellml.org/metadata/1.0#" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:bqs="http://www.cellml.org/bqs/1.0#" xmlns:cellml="http://www.cellml.org/cellml/1.0#" xmlns:dcterms="http://purl.org/dc/terms/" xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#" cmeta:id="PoleZero" name="itskov_2006">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
<articleinfo>
<title>Exponential Polyconvex Anisotropic Strain-Energy Function</title>
<author>
<firstname>Jesse</firstname>
<surname>Ashton</surname>
<affiliation>
<shortaffil>Auckland Bioengineering Institute, The University of Auckland</shortaffil>
</affiliation>
</author>
</articleinfo>
<section id="sec_status">
<title>Model Status</title>
<para>
This is a CellML version of the Polyconvex constitutive material law for Orthotropic, Incompressible materials, defining the relation between the nine independent strain components and the stress components. It is assumed that the strain components will be controlled externally by the application using this CellML model.
</para>
<para>
Please note that this CellML model has been implemented in such a manner that it could be used for peforming finite element model simulations on the CMISS software program developed at the Auckland Bioengineering Institute, The University of Auckland. For additional information on implementation of cellML files in CMISS, please refer to the following <ulink url="http://www.bioeng.auckland.ac.nz/people/nickerso/cmiss/help.html">Link</ulink>.
</para>
</section>
<sect1 id="sec_structure">
<title>Model Structure</title>
<para>
ABSTRACT: Polyconvexity of a strain-energy function is a very important mathematical condition, especially in the context of a boundary-value problem. In the present paper, we propose an exponential polyconvex anisotropic strain-energy function. It is given by a series with an arbitrary number of terms and associated material constants. Each term of this series a priori satisfies the condition of the energy- and stress-free natural state so that no additional restrictions have to be imposed. Due to the exponential form, the proposed hyperelastic model is suitable for soft biological tissues. Thus, a good agreement with experimental data on different types of tissues is achieved.</para>
<para>
The original paper reference is cited below:
</para>
<para>
A Polyconvex anisotropic strain-energy function for soft collagenous tissues, M. Itskov, A.E. Ehret and D. Mavrilas, 2006.
<emphasis>Biomechanics and Modeling in Mechanobiology</emphasis>, 5(1), 17-26. <ulink url="http://www.ncbi.nlm.nih.gov/entrez/16362195">PubMed ID: 16362195</ulink> </para>
</sect1>
</article>
</documentation>
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<units name="stress">
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<variable units="stress" public_interface="out" private_interface="in" name="Tdev22"/>
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<!-- Outputs computed here -->
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<times/>
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<minus/>
<cn cellml:units="dimensionless">4</cn>
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<minus/>
<cn cellml:units="dimensionless">2</cn>
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<minus/>
<cn cellml:units="dimensionless">4</cn>
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<minus/>
<cn cellml:units="dimensionless">2</cn>
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<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>E21</ci>
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<power/>
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<power/>
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<cn cellml:units="dimensionless">2</cn>
</apply>
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<times/>
<apply>
<power/>
<ci>detc</ci>
<cn cellml:units="dimensionless">2</cn>
</apply>
<ci>x11denom</ci>
<ci>x21denom</ci>
</apply>
</apply>
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<times/>
<ci>mu2</ci>
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<plus/>
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<divide/>
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<exp/>
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<times/>
<apply>
<minus/>
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</apply>
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<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless">1</cn>
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<exp/>
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<apply>
<minus/>
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</apply>
<ci>b2</ci>
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<plus/>
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<divide/>
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<power/>
<ci>k1</ci>
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</apply>
<ci>x12denom</ci>
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<divide/>
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<times/>
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<plus/>
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<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless">4</cn>
</apply>
<ci>E33</ci>
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<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless">2</cn>
</apply>
<ci>E21</ci>
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<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>E23</ci>
<ci>E31</ci>
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<plus/>
<apply>
<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless">4</cn>
</apply>
<ci>E33</ci>
<ci>E12</ci>
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<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless">2</cn>
</apply>
<ci>E12</ci>
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<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>E13</ci>
<ci>E32</ci>
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<power/>
<ci>x12</ci>
<cn cellml:units="dimensionless">2</cn>
</apply>
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<times/>
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<power/>
<ci>detc</ci>
<cn cellml:units="dimensionless">2</cn>
</apply>
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<ci>x22denom</ci>
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<times/>
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<minus/>
<cn cellml:units="dimensionless">4</cn>
</apply>
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<times/>
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<cn cellml:units="dimensionless">2</cn>
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<ci>E13</ci>
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<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>E12</ci>
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<minus/>
<cn cellml:units="dimensionless">4</cn>
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<times/>
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<minus/>
<cn cellml:units="dimensionless">2</cn>
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<ci>E31</ci>
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<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>E21</ci>
<ci>E32</ci>
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<power/>
<ci>x12</ci>
<cn cellml:units="dimensionless">2</cn>
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<power/>
<ci>x22</ci>
<cn cellml:units="dimensionless">2</cn>
</apply>
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<times/>
<apply>
<power/>
<ci>detc</ci>
<cn cellml:units="dimensionless">2</cn>
</apply>
<ci>x12denom</ci>
<ci>x22denom</ci>
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<eq/>
<ci>Tdev12</ci>
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<times/>
<cn cellml:units="dimensionless">0.5</cn>
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<plus/>
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<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless">1</cn>
</apply>
<ci>mu1</ci>
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<exp/>
<apply>
<times/>
<apply>
<minus/>
<ci>k1tilde</ci>
<cn cellml:units="dimensionless">1</cn>
</apply>
<ci>b1</ci>
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<times/>
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<times/>
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<minus/>
<cn cellml:units="dimensionless">4</cn>
</apply>
<ci>E22</ci>
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<minus/>
<cn cellml:units="dimensionless">2</cn>
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<ci>E13</ci>
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<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>E12</ci>
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<ci>E12</ci>
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<times/>
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<minus/>
<cn cellml:units="dimensionless">4</cn>
</apply>
<ci>E11</ci>
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<times/>
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<minus/>
<cn cellml:units="dimensionless">2</cn>
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<ci>E32</ci>
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<power/>
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<power/>
<ci>x11</ci>
<cn cellml:units="dimensionless">2</cn>
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<times/>
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<power/>
<ci>detc</ci>
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<ci>x11denom</ci>
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<cn cellml:units="dimensionless">4</cn>
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<times/>
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<minus/>
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<ci>E12</ci>
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<times/>
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<power/>
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<times/>
<ci>detc</ci>
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<times/>
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<plus/>
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<times/>
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<minus/>
<cn cellml:units="dimensionless">4</cn>
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<ci>E33</ci>
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<times/>
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<minus/>
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<ci>E12</ci>
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<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>E13</ci>
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<times/>
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<minus/>
<cn cellml:units="dimensionless">1</cn>
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<ci>mu2</ci>
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<exp/>
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<minus/>
<ci>k2tilde</ci>
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<times/>
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<minus/>
<cn cellml:units="dimensionless">4</cn>
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<ci>E22</ci>
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<times/>
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<minus/>
<cn cellml:units="dimensionless">2</cn>
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<ci>E13</ci>
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<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>E12</ci>
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<plus/>
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<cn cellml:units="dimensionless">4</cn>
<ci>E12</ci>
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<times/>
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<minus/>
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<ci>E11</ci>
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<times/>
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<minus/>
<cn cellml:units="dimensionless">2</cn>
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<ci>E32</ci>
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<power/>
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<power/>
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<times/>
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<power/>
<ci>detc</ci>
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<plus/>
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<times/>
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<minus/>
<cn cellml:units="dimensionless">4</cn>
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<ci>E33</ci>
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<times/>
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<minus/>
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<ci>E12</ci>
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<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>E13</ci>
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<power/>
<ci>x12</ci>
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<times/>
<ci>detc</ci>
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<minus/>
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<minus/>
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<times/>
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<plus/>
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<times/>
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<minus/>
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<exp/>
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<minus/>
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<power/>
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<power/>
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<cn cellml:units="dimensionless">2</cn>
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<cn cellml:units="dimensionless">2</cn>
</apply>
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<ci>x21denom</ci>
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<cn cellml:units="dimensionless">4</cn>
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<cn cellml:units="dimensionless">2</cn>
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<ci>E21</ci>
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<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>E23</ci>
<ci>E31</ci>
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<power/>
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<cn cellml:units="dimensionless">2</cn>
</apply>
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<times/>
<ci>detc</ci>
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<ci>x21denom</ci>
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<divide/>
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<times/>
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<plus/>
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<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless">4</cn>
</apply>
<ci>E33</ci>
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<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless">2</cn>
</apply>
<ci>E21</ci>
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<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>E23</ci>
<ci>E31</ci>
</apply>
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<apply>
<minus/>
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<minus/>
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<cn cellml:units="dimensionless">1</cn>
</apply>
<ci>b2</ci>
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<plus/>
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<divide/>
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<apply>
<plus/>
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<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>E13</ci>
<ci>E21</ci>
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<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless">2</cn>
</apply>
<ci>E23</ci>
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<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless">4</cn>
</apply>
<ci>E11</ci>
<ci>E23</ci>
</apply>
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<plus/>
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<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>E21</ci>
<ci>E32</ci>
</apply>
<apply>
<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless">4</cn>
</apply>
<ci>E22</ci>
<ci>E31</ci>
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<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless">2</cn>
</apply>
<ci>E31</ci>
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<apply>
<power/>
<ci>x12</ci>
<cn cellml:units="dimensionless">2</cn>
</apply>
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<power/>
<ci>x22</ci>
<cn cellml:units="dimensionless">2</cn>
</apply>
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<times/>
<apply>
<power/>
<ci>detc</ci>
<cn cellml:units="dimensionless">2</cn>
</apply>
<ci>x12denom</ci>
<ci>x22denom</ci>
</apply>
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<divide/>
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<times/>
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<plus/>
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<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless">4</cn>
</apply>
<ci>E33</ci>
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<apply>
<minus/>
<cn cellml:units="dimensionless">2</cn>
</apply>
<ci>E21</ci>
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<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>E23</ci>
<ci>E31</ci>
</apply>
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<power/>
<ci>x12</ci>
<cn cellml:units="dimensionless">2</cn>
</apply>
<ci>k2</ci>
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<times/>
<ci>detc</ci>
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<ci>x22denom</ci>
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<divide/>
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<times/>
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<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless">4</cn>
</apply>
<ci>E33</ci>
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<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless">2</cn>
</apply>
<ci>E21</ci>
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<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>E23</ci>
<ci>E31</ci>
</apply>
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<ci>k1</ci>
</apply>
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<times/>
<ci>detc</ci>
<ci>x12denom</ci>
</apply>
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<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="Tdev22_calculation_eq">
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<eq/>
<ci>Tdev22</ci>
<apply>
<times/>
<cn cellml:units="dimensionless">0.5</cn>
<apply>
<plus/>
<apply>
<times/>
<ci>mu1</ci>
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<plus/>
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<divide/>
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<times/>
<apply>
<exp/>
<apply>
<times/>
<apply>
<minus/>
<ci>i1tilde</ci>
<cn cellml:units="dimensionless">1</cn>
</apply>
<ci>a1</ci>
</apply>
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<power/>
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<cn cellml:units="dimensionless">2</cn>
</apply>
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<times/>
<ci>x11denom</ci>
<ci>x21denom</ci>
</apply>
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<minus/>
<cn cellml:units="dimensionless">1</cn>
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<exp/>
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<minus/>
<ci>k1tilde</ci>
<cn cellml:units="dimensionless">1</cn>
</apply>
<ci>b1</ci>
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<power/>
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</apply>
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<power/>
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<cn cellml:units="dimensionless">2</cn>
</apply>
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<times/>
<ci>x11denom</ci>
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</apply>
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<divide/>
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<times/>
<apply>
<plus/>
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<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless">4</cn>
</apply>
<ci>E11</ci>
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<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless">2</cn>
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<ci>E23</ci>
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<cn cellml:units="dimensionless">4</cn>
<ci>E31</ci>
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<minus/>
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<ci>E32</ci>
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<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>E12</ci>
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<power/>
<ci>x21</ci>
<cn cellml:units="dimensionless">2</cn>
</apply>
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<times/>
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<power/>
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<cn cellml:units="dimensionless">2</cn>
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<ci>x11denom</ci>
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<times/>
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<plus/>
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<times/>
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<minus/>
<cn cellml:units="dimensionless">4</cn>
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<ci>E33</ci>
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<minus/>
<cn cellml:units="dimensionless">2</cn>
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<ci>E21</ci>
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<times/>
<cn cellml:units="dimensionless">4</cn>
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<cn cellml:units="dimensionless">4</cn>
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<ci>E12</ci>
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<cn cellml:units="dimensionless">4</cn>
<ci>E13</ci>
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<power/>
<ci>detc</ci>
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<ci>x11denom</ci>
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<times/>
<ci>mu2</ci>
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<plus/>
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<divide/>
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<times/>
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<exp/>
<apply>
<times/>
<apply>
<minus/>
<ci>i2tilde</ci>
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<power/>
<ci>x12</ci>
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<times/>
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<minus/>
<cn cellml:units="dimensionless">1</cn>
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<exp/>
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<minus/>
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<times/>
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<ci>k2</ci>
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<power/>
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<times/>
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<divide/>
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<plus/>
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<times/>
<cn cellml:units="dimensionless">4</cn>
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<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless">2</cn>
</apply>
<ci>E23</ci>
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<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless">4</cn>
</apply>
<ci>E11</ci>
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<plus/>
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<times/>
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<minus/>
<cn cellml:units="dimensionless">4</cn>
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<ci>E11</ci>
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<times/>
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<minus/>
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<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>E12</ci>
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<power/>
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<times/>
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<power/>
<ci>detc</ci>
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<times/>
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<plus/>
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<times/>
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<minus/>
<cn cellml:units="dimensionless">4</cn>
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<minus/>
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<minus/>
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<ci>E12</ci>
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<times/>
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<plus/>
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<times/>
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<minus/>
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<exp/>
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<times/>
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<minus/>
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<divide/>
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<times/>
<cn cellml:units="dimensionless">4</cn>
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<minus/>
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<minus/>
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<times/>
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<minus/>
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<cn cellml:units="dimensionless">4</cn>
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<minus/>
<cn cellml:units="dimensionless">4</cn>
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<cn cellml:units="dimensionless">4</cn>
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<power/>
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<ci>E21</ci>
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<times/>
<cn cellml:units="dimensionless">4</cn>
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<times/>
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<minus/>
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<power/>
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<times/>
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<minus/>
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<ci>E12</ci>
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<exp/>
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<minus/>
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<apply>
<minus/>
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<apply>
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<ci>x22denom</ci>
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<divide/>
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<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="dimensionless">4</cn>
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<apply>
<minus/>
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<ci>E31</ci>
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<apply>
<power/>
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</apply>
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<apply>
<times/>
<apply>
<power/>
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</apply>
<ci>x12denom</ci>
<ci>x22denom</ci>
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<divide/>
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<times/>
<apply>
<plus/>
<apply>
<times/>
<apply>
<minus/>
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<ci>E22</ci>
<ci>E13</ci>
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<apply>
<times/>
<apply>
<minus/>
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</apply>
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<apply>
<times/>
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<ci>E23</ci>
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<apply>
<plus/>
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<apply>
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<ci>E32</ci>
</apply>
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<apply>
<times/>
<apply>
<power/>
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</apply>
<ci>x12denom</ci>
</apply>
</apply>
</apply>
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<vCard:Given>A</vCard:Given>
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<dcterms:W3CDTF>2007-01-24</dcterms:W3CDTF>
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<vCard:FN>Vignesh Kumar</vCard:FN>
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<rdf:value>
In this simple model we only have one component, which holds the
six equations.
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We'll use this component as the "interface" to the model, all
other components are hidden via encapsulation in this component.
</rdf:value>
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<dc:title>A Polyconvex Anisotropic Strain-Energy Function for Soft Collagenous Tissues</dc:title>
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<vCard:Given>Jesse</vCard:Given>
<vCard:Family>Ashton</vCard:Family>
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<dc:creator rdf:resource="rdf:#828dd465-dd45-4e3e-9073-7b068911eb37"/>
<rdf:value>This CellML file deals with the introduction of an exponential polyconvex anisotropic strain energy function, for use with modeling applications in regards to soft biological tissues.</rdf:value>
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<vCard:Given>D</vCard:Given>
<vCard:Family>Mavrilas</vCard:Family>
</rdf:Description>
<rdf:Description rdf:about="rdf:#fcdc7d35-3582-4c8c-bd7a-832c9036776e">
<vCard:Orgname>The University of Auckland</vCard:Orgname>
<vCard:Orgunit>Auckland Bioengineering Institute</vCard:Orgunit>
</rdf:Description>
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<bqs:subject_type>keyword</bqs:subject_type>
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<rdf:Bag>
<rdf:li>polyconvex</rdf:li>
<rdf:li>soft tissue</rdf:li>
<rdf:li>constitutive material law</rdf:li>
<rdf:li>mechanical constitutive laws</rdf:li>
<rdf:li>anisotropic strain-energy</rdf:li>
</rdf:Bag>
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<dc:creator rdf:resource="rdf:#c0bbcf35-7d1c-462c-92de-356e84a984c7"/>
<rdf:value>This is a CellML version of the Polyconvex constitutive material law for Orthotropic, Incompressible materials, defining the relation between the nine independent strain components and the stress components. It is assumed that the strain components will be controlled externally by the application using this CellML model.</rdf:value>
</rdf:Description>
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<vCard:Orgname>Auckland Bioengineering Institute</vCard:Orgname>
<vCard:Orgunit>The University of Auckland</vCard:Orgunit>
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<dc:title>Biomechanics and modelling in mechanobiology</dc:title>
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<rdf:Description rdf:about="rdf:#79498ae9-887c-4058-9b94-2d9991c22ec6">
<vCard:Given>Jesse</vCard:Given>
<vCard:Family>Ashton</vCard:Family>
<vCard:Other/>
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<dcterms:W3CDTF>2007-01-24T00:00:00+13:00</dcterms:W3CDTF>
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<vCard:Given>M</vCard:Given>
<vCard:Family>Itskov</vCard:Family>
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<vCard:FN>Vignesh Kumar</vCard:FN>
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