<?xml version='1.0' encoding='utf-8'?>
<!-- FILE : demiray_model_1981.xml
CREATED : 11th November 2004
LAST MODIFIED : 27th November 2004
AUTHOR : JaeHoon Chung
The Bioengineering Institute
The University of Auckland
MODEL STATUS : This model conforms to the CellML 1.0 Specification released on
10th August 2001, and the 16/01/2002 CellML Metadata 1.0 Specification.
DESCRIPTION : This is a CellML version of the Exponential 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.
CHANGES:
27/11/04 - CML - added some more publication data and altered name to match the model repository template.
--><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="demiray_1981" name="demiray_1981">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
<articleinfo>
<title>Exponential Constitutive Material Law</title>
<author>
<firstname>JaeHoon</firstname>
<surname>Chung</surname>
<affiliation>
<shortaffil>Auckland Bioengineering Institute, The University of Auckland</shortaffil>
</affiliation>
</author>
</articleinfo>
<section id="sec_status">
<title>Model Status</title>
<para>
This model contains no ODEs so cannot be solved in available software (OpenCell or COR). However the model is still valid CellML with full unit consistency. The model is intended to interface with a CMISS model at a larger spatial scale.
</para>
</section>
<sect1 id="sec_structure">
<title>Model Structure</title>
<para>
This exponential material law for isotropic soft biological material was first proposed by H. Demiray in 1981 (see full publication details below). From the stochastic models for the elastic behaviour of biological tissues proposed by Soong and Huang (<emphasis>A Stochastic Model for Biological Tissue Elasticity in Simple Elongation</emphasis>, Journal of Biomechanics, 6, p451-458, 1973) and Lanir (<emphasis>A Structural Theory for the Homogenous Biaxial Stress-Strain Relationships in Flat Collagenous Tissues</emphasis>, Journal of Biomechanics, 12, p423-436, 1979), Demiray observed the form of the strain energy density function to be an exponential function of the first invariant (trace) of the right Cauchy-Green deformation tensor, I1. This argument is further supported by Fung's statement that the strain energy function of soft biological tissues in a one-dimensional case is an exponential function of the stretch ratio (<emphasis>Elasticity of Soft Tissues in Simple Elongation</emphasis>, American Journal of Physiology, 213, p1532-1544, 1967).
</para>
<para>
Demiray assumed that biological materials were elastic, homogenous, isotropic and incompressible. Like the Neo-Hookean material law (a special case of the <ulink url="${HTML_EXMPL_MOONEY_RIVLIN_LAW}">The Mooney-Rivlin Constitutive Material Law, 1951</ulink>, this exponential constitutive law is a function of only the axial Green-Lagrange strain components, and hence it does not penalise shear deformations.
</para>
<para>
The complete original paper reference is cited below:
</para>
<para>
Large deformation analysis of some soft biological tissues, H. Demiray, 1981,
<emphasis>Journal of Biomechanical Engineering</emphasis>
,103, 73-78. (Unfortunately this article doesn't appear to be available as an online version). <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=7278185&dopt=Abstract">PubMed ID: 7278185</ulink>
</para>
</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"/>
</units>
<component cmeta:id="interface" name="interface">
<!-- Variables we expect to be set/controlled externally -->
<variable units="strain" private_interface="out" name="E11"/>
<variable units="strain" private_interface="out" name="E22"/>
<variable units="strain" private_interface="out" name="E33"/>
<variable units="strain" private_interface="out" name="E12"/>
<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-->
<component cmeta:id="equations" name="equations">
<!-- Inputs -->
<variable units="strain" public_interface="in" name="E11"/>
<variable units="strain" public_interface="in" name="E22"/>
<variable units="strain" public_interface="in" name="E33"/>
<variable units="strain" public_interface="in" name="E12"/>
<variable units="strain" public_interface="in" name="E13"/>
<variable units="strain" public_interface="in" name="E23"/>
<variable units="strain" public_interface="in" name="c1"/>
<variable units="strain" public_interface="in" name="c2"/>
<!-- Outputs computed here -->
<variable units="stress" public_interface="out" name="Tdev11"/>
<variable units="stress" public_interface="out" name="Tdev22"/>
<variable units="stress" public_interface="out" name="Tdev33"/>
<variable units="stress" public_interface="out" name="Tdev12"/>
<variable units="stress" public_interface="out" name="Tdev13"/>
<variable units="stress" public_interface="out" name="Tdev23"/>
<!-- Temporary variables -->
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="Tdev11_calculation_eq">
<apply id="Tdev11_calculation">
<eq/>
<ci>Tdev11</ci>
<apply>
<times/>
<cn cellml:units="strain">2.0</cn>
<ci>c1</ci>
<ci>c2</ci>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="strain">2.0</cn>
<ci>c2</ci>
<apply>
<plus/>
<ci>E11</ci>
<ci>E22</ci>
<ci>E33</ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="Tdev22_calculation_eq">
<apply id="Tdev22_calculation">
<eq/>
<ci>Tdev22</ci>
<apply>
<times/>
<cn cellml:units="strain">2.0</cn>
<ci>c1</ci>
<ci>c2</ci>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="strain">2.0</cn>
<ci>c2</ci>
<apply>
<plus/>
<ci>E11</ci>
<ci>E22</ci>
<ci>E33</ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="Tdev33_calculation_eq">
<apply id="Tdev33_calculation">
<eq/>
<ci>Tdev33</ci>
<apply>
<times/>
<cn cellml:units="strain">2.0</cn>
<ci>c1</ci>
<ci>c2</ci>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="strain">2.0</cn>
<ci>c2</ci>
<apply>
<plus/>
<ci>E11</ci>
<ci>E22</ci>
<ci>E33</ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="Tdev12_calculation_eq">
<apply id="Tdev12_calculation">
<eq/>
<ci>Tdev12</ci>
<apply>
<times/>
<cn cellml:units="strain">0.0</cn>
<ci>E12</ci>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="Tdev13_calculation_eq">
<apply id="Tdev13_calculation">
<eq/>
<ci>Tdev13</ci>
<apply>
<times/>
<cn cellml:units="strain">0.0</cn>
<ci>E13</ci>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="Tdev23_calculation_eq">
<apply id="Tdev23_calculation">
<eq/>
<ci>Tdev23</ci>
<apply>
<times/>
<cn cellml:units="strain">0.0</cn>
<ci>E23</ci>
</apply>
</apply>
</math>
</component>
<!--membrane-->
<connection>
<map_components component_2="equations" component_1="interface"/>
<map_variables variable_2="E11" variable_1="E11"/>
<map_variables variable_2="E22" variable_1="E22"/>
<map_variables variable_2="E33" variable_1="E33"/>
<map_variables variable_2="E12" variable_1="E12"/>
<map_variables variable_2="E13" variable_1="E13"/>
<map_variables variable_2="E23" variable_1="E23"/>
<map_variables variable_2="Tdev11" variable_1="Tdev11"/>
<map_variables variable_2="Tdev22" variable_1="Tdev22"/>
<map_variables variable_2="Tdev33" variable_1="Tdev33"/>
<map_variables variable_2="Tdev12" variable_1="Tdev12"/>
<map_variables variable_2="Tdev13" variable_1="Tdev13"/>
<map_variables variable_2="Tdev23" variable_1="Tdev23"/>
<map_variables variable_2="c1" variable_1="c1"/>
<map_variables variable_2="c2" variable_1="c2"/>
</connection>
<group>
<relationship_ref relationship="encapsulation"/>
<component_ref component="interface">
<component_ref component="equations"/>
</component_ref>
</group>
<rdf:RDF>
<rdf:Description rdf:about="rdf:#2279f997-93ad-4147-bf83-b2595b20c2ce">
<bqs:Pubmed_id>7278185</bqs:Pubmed_id>
<bqs:JournalArticle rdf:resource="rdf:#8e311e6a-5729-4ec9-aadf-29319d593d4b"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#cb189d69-1afc-45fb-b1a3-754665b55452">
<rdf:value>
In this simple model we only have one component, which holds the
six equations.
</rdf:value>
</rdf:Description>
<rdf:Description rdf:about="#interface">
<cmeta:comment rdf:resource="rdf:#e97f1071-9167-4be4-9c5e-0af6b1a45ec0"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#66ab762c-4fe0-4c96-a783-f103946d320f">
<vCard:Orgname>The University of Auckland</vCard:Orgname>
<vCard:Orgunit>Auckland Bioengineering Institute</vCard:Orgunit>
</rdf:Description>
<rdf:Description rdf:about="rdf:#16c68bb3-7907-40ce-b3b1-026ac25bfa53">
<dcterms:W3CDTF>1981</dcterms:W3CDTF>
</rdf:Description>
<rdf:Description rdf:about="rdf:#94120a75-121f-475b-b46a-ec452fe2cebd">
<rdf:type rdf:resource="http://www.w3.org/1999/02/22-rdf-syntax-ns#li"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#39f2a85c-4b66-488c-828d-ffb47ee9d53a">
<vCard:Given>H</vCard:Given>
<vCard:Family>Demiray</vCard:Family>
</rdf:Description>
<rdf:Description rdf:about="#demiray_1981">
<dc:title>
The exponential constitutive material law
</dc:title>
<cmeta:comment rdf:resource="rdf:#a23ba7c7-7b0d-4a12-97de-4cea83f3cc45"/>
<bqs:reference rdf:resource="rdf:#2279f997-93ad-4147-bf83-b2595b20c2ce"/>
<bqs:reference rdf:parseType="Resource">
<dc:subject rdf:parseType="Resource">
<bqs:subject_type>keyword</bqs:subject_type>
<rdf:value>
<rdf:Bag>
<rdf:li>biological tissues</rdf:li>
<rdf:li>deformation</rdf:li>
<rdf:li>mechanical constitutive laws</rdf:li>
</rdf:Bag>
</rdf:value>
</dc:subject>
</bqs:reference>
</rdf:Description>
<rdf:Description rdf:about="rdf:#45b78ef8-4668-42df-bd64-7126bd0bbfd7">
<bqs:subject_type>keyword</bqs:subject_type>
<rdf:value rdf:resource="rdf:#94120a75-121f-475b-b46a-ec452fe2cebd"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#0b060431-126d-48f1-aeac-dd961eb220a1">
<vCard:N rdf:resource="rdf:#39f2a85c-4b66-488c-828d-ffb47ee9d53a"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#57a4ad10-0c21-42a0-92eb-4fc23fce8d5f">
<bqs:Person rdf:resource="rdf:#0b060431-126d-48f1-aeac-dd961eb220a1"/>
</rdf:Description>
<rdf:Description rdf:about="#equations">
<cmeta:comment rdf:resource="rdf:#cb189d69-1afc-45fb-b1a3-754665b55452"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#c244a99c-b6b4-4005-aeb7-9cb66111d9dc">
<dc:title>Journal of Biomechanical Engineering</dc:title>
</rdf:Description>
<rdf:Description rdf:about="rdf:#8e311e6a-5729-4ec9-aadf-29319d593d4b">
<dc:creator rdf:resource="rdf:#57a4ad10-0c21-42a0-92eb-4fc23fce8d5f"/>
<dc:title>
Large deformation analysis of some soft biological tissues
</dc:title>
<bqs:volume>103</bqs:volume>
<bqs:first_page>73</bqs:first_page>
<bqs:Journal rdf:resource="rdf:#c244a99c-b6b4-4005-aeb7-9cb66111d9dc"/>
<dcterms:issued rdf:resource="rdf:#16c68bb3-7907-40ce-b3b1-026ac25bfa53"/>
<bqs:last_page>78</bqs:last_page>
</rdf:Description>
<rdf:Description rdf:about="rdf:#d1eca79e-2f52-4d7d-8641-bf6a8880f96d">
<vCard:EMAIL rdf:resource="rdf:#8d6199a0-96d0-4612-be0c-0dc3cef4db2c"/>
<vCard:ORG rdf:resource="rdf:#66ab762c-4fe0-4c96-a783-f103946d320f"/>
<vCard:N rdf:resource="rdf:#18aadb8d-a004-4b13-a525-53c5234639fd"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#e97f1071-9167-4be4-9c5e-0af6b1a45ec0">
<rdf:value>
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:#ede57da6-9618-46bb-93df-008ff352a636">
<dcterms:W3CDTF>2004-11-10</dcterms:W3CDTF>
</rdf:Description>
<rdf:Description rdf:about="rdf:#a23ba7c7-7b0d-4a12-97de-4cea83f3cc45">
<rdf:value>
This is a CellML version of the Exponential 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:#39f4d387-171e-4eee-8b08-447b2d3b0c70">
<rdf:type rdf:resource="http://www.cellml.org/bqs/1.0#reference"/>
<dc:subject rdf:resource="rdf:#45b78ef8-4668-42df-bd64-7126bd0bbfd7"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#18aadb8d-a004-4b13-a525-53c5234639fd">
<vCard:Given>JaeHoon</vCard:Given>
<vCard:Family>Chung</vCard:Family>
</rdf:Description>
<rdf:Description rdf:about="">
<dcterms:created rdf:resource="rdf:#ede57da6-9618-46bb-93df-008ff352a636"/>
<dc:creator rdf:resource="rdf:#d1eca79e-2f52-4d7d-8641-bf6a8880f96d"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#8d6199a0-96d0-4612-be0c-0dc3cef4db2c">
<rdf:type rdf:resource="http://imc.org/vCard/3.0#internet"/>
<rdf:value>jh.chung@auckland.ac.nz</rdf:value>
</rdf:Description>
</rdf:RDF>
</model>
