<?xml version='1.0' encoding='utf-8'?>
<!-- FILE : mooney_rivlin_law_1951.xml
CREATED : 28th December 2003
LAST MODIFIED : 18th February 2004
AUTHOR : Catherine Lloyd
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 Mooney-Rivlin constitutive material law, defining the relation between the eight 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="rivlin_saunders_1951_version01" name="rivlin_saunders_1951_version01">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
<articleinfo>
<title>The Mooney-Rivlin Constitutive Material Law</title>
<author>
<firstname>Vijayaraghavan</firstname>
<surname>Rajagopal</surname>
<affiliation>
<shortaffil>Bioengineering Institute, University of Auckland</shortaffil>
</affiliation>
</author>
</articleinfo>
<section id="sec_status">
<title>Model Status</title>
<para>
This is the original unchecked version of the model imported from the previous
CellML model repository, 24-Jan-2006.
</para>
</section>
<sect1 id="sec_structure">
<title>Model Structure</title>
<para>
This is an example of how CellML can be used to describe a material law which models the passive, mechanical behaviour of a material. The material law is to calculate the stress induced in a body when a certain amount of deformation (strain) is imposed on it. The calculated stresses are used to solve a system of equations that govern the mechanics of large deformations. The example presented here is the Mooney-Rivlin constitutive material law, which defines the relationship between eight independent strain components and the stress components.
</para>
<para>
The Mooney-Rivlin equation was developed by Rivlin and Saunders to describe the deformation of highly elastic bodies which are incompressible (volume is conserved during deformation) and isotropic (the material has the same mechanical properties in all directions at a material point). They formulated the material law as a strain energy function in terms of the first and second principal invariants (quantitative information on the deformation) of the deformation. The formulation is called a strain energy function as the energy is conserved during deformation of these materials under constant temperature. A typical load-extension ratio curve for rubber in simple extension is shown in the figure below.
</para>
<para>
The model was implemented in a manner that could be used for peforming finite element model simulations on the CMISS software program developed at the Bioengineering Institute, University of Auckland. Examples of how the CellML model is used can be seen <ulink url="http://cmiss.bioeng.auckland.ac.nz/development/examples/5/5g/index.html">here</ulink>.
</para>
<para>
The complete original paper reference is cited below:
</para>
<para>
<ulink url="http://80-www.jstor.org.ezproxy.auckland.ac.nz/browse/00804614/ap000767/00a00000?config=jstor&frame=noframe&userID=82d8bf54@auckland.ac.nz/018dd5533b005010c9029&dpi=3">Large Elastic Deformations of Isotropic Materials VII. Experiments on the Deformation Of Rubber</ulink>, R. S. Rivlin and D. W. Saunders, 1951, <ulink url="http://80-www.jstor.org.ezproxy.auckland.ac.nz/browse/00804614?config=jstor&frame=noframe&userID=82d8bf54@auckland.ac.nz/018dd5533b005010c9029&dpi=3">
<emphasis>Philosophical Transactions of the Royal Society of London, series A</emphasis>
</ulink>, 243, 251-288. (Full text (HTML) and <ulink url="http://80-www.jstor.org.ezproxy.auckland.ac.nz/cgi-bin/jstor/printpage/00804614/ap000767/00a00000/0?backurl=/cgi-bin/jstor/listjournal/00804614/ap000767/00a00000/0%3fconfig%3djstor%26frame%3dnoframe%26userID%3d82d8bf54@auckland.ac.nz/018dd5533b005010c9029%26dpi%3d3&config=jstor&backcontext=citation&userID=82d8bf54@auckland.ac.nz/018dd5533b005010c9029&dpi=3&action=download">PDF</ulink> versions of the article are available on the <emphasis>Philosophical Transactions of the Royal Society of London, series A</emphasis> website.)
</para>
<informalfigure float="0" id="fig_mooney_rivlin_diagram">
<mediaobject>
<imageobject>
<objectinfo>
<title>mooney_rivlin</title>
</objectinfo>
<imagedata fileref="rivlin_1951.png"/>
</imageobject>
</mediaobject>
<caption>A typical load-extension ration curve for rubber in simple extension (Rivlin, R.S. and Saunders, D.W. 1951)</caption>
</informalfigure>
</sect1>
</article>
</documentation>
<!-- Global units -->
<units name="strain">
<unit units="dimensionless"/>
</units>
<units name="stress">
<unit units="dimensionless"/>
</units>
<units name="pole">
<unit units="dimensionless"/>
</units>
<units name="curvature">
<unit units="dimensionless"/>
</units>
<units name="scale">
<unit units="dimensionless"/>
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<!-- Variables we expect to be set/controlled externally -->
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<variable units="strain" private_interface="out" name="E13"/>
<variable units="strain" private_interface="out" name="E23"/>
<variable units="strain" private_interface="out" name="c1" initial_value="0"/>
<variable units="strain" private_interface="out" name="c2" initial_value="0"/>
<!-- Variables we want to make available externally -->
<variable units="stress" public_interface="out" private_interface="in" name="Tdev11"/>
<variable units="stress" public_interface="out" private_interface="in" name="Tdev22"/>
<variable units="stress" public_interface="out" private_interface="in" name="Tdev33"/>
<variable units="stress" public_interface="out" private_interface="in" name="Tdev12"/>
<variable units="stress" public_interface="out" private_interface="in" name="Tdev13"/>
<variable units="stress" public_interface="out" private_interface="in" name="Tdev23"/>
</component> <!--interface-->
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<!-- Inputs -->
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<!-- Outputs computed here -->
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<apply>
<plus/>
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<times/>
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<ci>c1</ci>
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<apply>
<times/>
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<apply>
<plus/>
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<apply>
<times/>
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<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="Tdev12_calculation_eq">
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<apply>
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<apply>
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<ci>E12</ci>
<ci>c2</ci>
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<apply id="Tdev13_calculation">
<eq/>
<ci>Tdev13</ci>
<apply>
<minus/>
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<ci>E13</ci>
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<apply>
<minus/>
<apply>
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<ci>E23</ci>
<ci>c2</ci>
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<!--membrane-->
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<dcterms:modified rdf:resource="rdf:#e6978095-e882-4769-bc97-69bdb70a872e"/>
<rdf:value>
Added metadata to the model.
</rdf:value>
<rdf:type rdf:resource="http://www.cellml.org/metadata/1.0#modification"/>
<cmeta:modifier rdf:resource="rdf:#d111e65e-da44-4a83-9d82-f9de634a9ef7"/>
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<rdf:Description rdf:about="rdf:#1e90367e-3c18-4c4c-add6-021998bebc4f">
<rdf:value>
This is a CellML version of the Mooney-Rivlin constitutive material law,
defining the relation between the eight 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>
<rdf:Description rdf:about="rdf:#13bb3ab2-6ddf-43b7-a163-5d1992bb635e">
<vCard:ORG rdf:resource="rdf:#c2cab628-694c-4830-841c-a6297e9f99fb"/>
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<rdf:Description rdf:about="rdf:#c7563801-3254-48b0-bf3a-30bf5c7661f0">
<bqs:subject_type>keyword</bqs:subject_type>
<rdf:value>
<rdf:Bag>
<rdf:li>rubber</rdf:li>
<rdf:li>mooney-rivlin</rdf:li>
<rdf:li>isotropic</rdf:li>
<rdf:li>elastic</rdf:li>
<rdf:li>mechanical constitutive laws</rdf:li>
<rdf:li>constitutive material law</rdf:li>
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<vCard:Given>R</vCard:Given>
<vCard:Family>Rivlin</vCard:Family>
<vCard:Other>S</vCard:Other>
</rdf:Description>
<rdf:Description rdf:about="rdf:#4f30a002-ed44-4bdd-8dd1-bbadba0a1c7c">
<rdf:value>
In this simple model we only have one component, which holds the
six equations.
</rdf:value>
</rdf:Description>
<rdf:Description rdf:about="rdf:#38c1f9c5-4c86-44ef-83d1-6a8887f21fbd">
<vCard:Given>D</vCard:Given>
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<vCard:Other>W</vCard:Other>
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</rdf:Description>
<rdf:Description rdf:about="rdf:#7a7fadce-c8d7-4ad7-8b89-d0e1821efe82">
<dcterms:W3CDTF>2003-11-28</dcterms:W3CDTF>
</rdf:Description>
<rdf:Description rdf:about="rdf:#e912d78c-2223-4140-9ba6-330643019915">
<vCard:Given>David</vCard:Given>
<vCard:Family>Nickerson</vCard:Family>
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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>
</rdf:Description>
<rdf:Description rdf:about="rdf:#fc6d71c5-b1fb-4667-8dfb-0bdec158197c">
<dcterms:W3CDTF>1951-04-24</dcterms:W3CDTF>
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<dc:title>Philosophical Transactions of the Royal Society of London Series A</dc:title>
</rdf:Description>
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<cmeta:comment rdf:resource="rdf:#4f30a002-ed44-4bdd-8dd1-bbadba0a1c7c"/>
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<dcterms:W3CDTF>2004-02-18</dcterms:W3CDTF>
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<dc:title>
Large Elastic Deformations of Isotropic Materials. VII. Experiments on the Deformation of Rubber
</dc:title>
<bqs:volume>243</bqs:volume>
<bqs:first_page>251</bqs:first_page>
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<rdf:value>d.nickerson@auckland.ac.nz</rdf:value>
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<vCard:Orgname>The University of Auckland</vCard:Orgname>
<vCard:Orgunit>The Bioengineering Institute</vCard:Orgunit>
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<dc:title>
Mooney-Rivlin constitutive material law
</dc:title>
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<vCard:Given>Catherine</vCard:Given>
<vCard:Family>Lloyd</vCard:Family>
<vCard:Other>M</vCard:Other>
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</model>
