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      <title>MATHP</title>
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      <div class="title_refTopic" id="xps10_pagetitle">MATHP </div>
      <hr noshade="true">
      <h4 class="refClassification">Hyperelastic Material Properties </h4>
      <p class="para_refTopic">Specifies material properties for use in fully nonlinear (i.e., large strain and large rotation) hyperelastic analysis of rubber-like materials (elastomers). </p>
      <div class="title_refCategory">Format: </div>
      <div class="refCategoryContent">
         <table border="1" width="100%" align="center">
            <colgroup span="1">
               <col span="1" width="10*">
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               <col span="1" width="10*">
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               <col span="1" width="10*">
            </colgroup>
            <thead>
               <tr>
                  <th align="center" colspan="1" rowspan="1">
                     <p class="para_th">1 </p>
                  </th>
                  <th align="center" colspan="1" rowspan="1">
                     <p class="para_th">2 </p>
                  </th>
                  <th align="center" colspan="1" rowspan="1">
                     <p class="para_th">3 </p>
                  </th>
                  <th align="center" colspan="1" rowspan="1">
                     <p class="para_th">4 </p>
                  </th>
                  <th align="center" colspan="1" rowspan="1">
                     <p class="para_th">5 </p>
                  </th>
                  <th align="center" colspan="1" rowspan="1">
                     <p class="para_th">6 </p>
                  </th>
                  <th align="center" colspan="1" rowspan="1">
                     <p class="para_th">7 </p>
                  </th>
                  <th align="center" colspan="1" rowspan="1">
                     <p class="para_th">8 </p>
                  </th>
                  <th align="center" colspan="1" rowspan="1">
                     <p class="para_th">9 </p>
                  </th>
                  <th align="center" colspan="1" rowspan="1">
                     <p class="para_th">10 </p>
                  </th>
               </tr>
            </thead>
            <tbody>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">MATHP </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">MID </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">A10 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">A01 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">D1 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">RHO </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">AV </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">TREF </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">GE </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" bgcolor="gray" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">NA </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">ND </p>
                  </td>
                  <td align="center" bgcolor="gray" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" bgcolor="gray" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" bgcolor="gray" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" bgcolor="gray" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" bgcolor="gray" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">A20 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">A11 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">A02 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">D2 </p>
                  </td>
                  <td align="center" bgcolor="gray" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" bgcolor="gray" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" bgcolor="gray" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" bgcolor="gray" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">A30 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">A21 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">A12 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">A03 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">D3 </p>
                  </td>
                  <td align="center" bgcolor="gray" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" bgcolor="gray" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" bgcolor="gray" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">A40 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">A31 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">A22 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">A13 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">A04 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">D4 </p>
                  </td>
                  <td align="center" bgcolor="gray" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" bgcolor="gray" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">A50 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">A41 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">A32 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">A23 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">A14 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">A05 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">D5 </p>
                  </td>
                  <td align="center" bgcolor="gray" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">TAB1 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">TAB2 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">TAB3 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">TAB4 </p>
                  </td>
                  <td align="center" bgcolor="gray" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" bgcolor="gray" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" bgcolor="gray" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">TABD </p>
                  </td>
                  <td align="center" bgcolor="gray" colspan="1" rowspan="1">
                     <p class="para_td">&nbsp; </p>
                  </td>
               </tr>
            </tbody>
         </table><br></div>
      <div class="title_refCategory">Fields:</div>
      <div class="refCategoryContent">
         <table cellpadding="9" width="100%" align="center">
            <colgroup span="1">
               <col span="1" width="16*">
               <col span="1" width="84*">
            </colgroup>
            <thead>
               <tr>
                  <th align="left" colspan="1" rowspan="1">
                     <p class="para_th"><b class="uiTerm">Field </b></p>
                  </th>
                  <th align="left" colspan="1" rowspan="1">
                     <p class="para_th"><b class="uiTerm">Contents </b></p>
                  </th>
               </tr>
            </thead>
            <tbody>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">MID </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Identification number of a MATHP entry.  (Integer &gt; 0; No Default) </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Aij </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Material constants related to distortional deformation.  (Real; Default = 0.0) </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Di </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Material constants related to volumetric deformation.  [Real &ge; 0; Default for D1 is 10<sup class="superscript">3</sup> · (A10+A01); Default for D2 through D5 is 0.0.] 
                     </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">RHO </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Mass density in original configuration.  (Real; Default = 0.0) </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">AV </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Coefficient of volumetric thermal expansion.  (Real; Default = 0.0) </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">TREF </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Reference temperature.  See &ldquo;<b class="uiTerm">MAT1 </b>&rdquo;.  (Real; Default = 0.0) 
                     </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">GE </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Structural damping element  coefficient.  (Real; Default = 0.0) </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">NA </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Order of the distortional strain energy polynomial function.  (0 &lt; Integer &le; 5; Default = 1) </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">ND </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Order of the volumetric strain energy polynomial function.  (0 &lt; Integer &le; 5; Default = 1) </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">TAB1 </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Table identification number of a TABLES1 entry that contains simple tension/compression data to be used in the estimation of the material constants Aij.  xi values in the TABLES1 entry must be stretch ratios <em>l/l</em><sub class="subscript">0</sub> and yi values must be values of the engineering stress <em>F/A</em><sub class="subscript">0</sub>. Stresses are negative for compression and positive for tension. If this convention is not followed the job may fail to converge or issue a traceback. (Integer &gt; 0 or blank) 
                     </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">TAB2 </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Table identification number of a TABLES1 entry that contains equibiaxial tension data to be used in the estimation of the material constants Aij.  xi values in the TABLES1 entry must be stretch ratios <em>l/l</em><sub class="subscript">0</sub>.  yi values must be values of the engineering stress <em>F/A</em><sub class="subscript">0</sub>.  <em>l</em> is the current length, F is the current force, <em>A</em><sub class="subscript">0</sub> is the cross-sectional area.  In the case of pressure of a spherical membrane, the engineering stress is given by <em>Pr</em><sub class="subscript">0</sub>&lambda;<sup class="superscript">2</sup>/2<em>t</em><sub class="subscript">0</sub> where <em>P</em>=current value of the pressure and <em>r</em><sub class="subscript">0</sub>, <em>t</em><sub class="subscript">0</sub>=initial radius and thickness.  (Integer &gt; 0 or blank) 
                     </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">TAB3 </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Table identification number of a TABLES1 entry that contains simple shear data to be used in the estimation of the material constants Aij.  xi values in the TABLES1 entry must be values of the shear tangent &gamma;  and yi values must be values of the engineering shear stress <em>F/A</em><sub class="subscript">0</sub> ·.  (Integer &gt; 0 or blank) 
                     </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">TAB4 </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Table identification number of a TABLES1 entry that contains pure shear data to be used in the estimation of the material constants Aij.  xi and yi values in the TABLES1 entry must be stretch ratios &lambda;<sub class="subscript">1</sub>=<em>l/l</em><sub class="subscript">0</sub> and values of the nominal stress <em>F/A</em><sub class="subscript">0</sub>.<em>l</em>  is the current length, F is the current force, <em>l</em><sub class="subscript">0</sub> and <em>A</em><sub class="subscript">0</sub> are the initial length and cross-sectional area, respectively in the 1-direction.  (Integer &gt; 0 or blank) 
                     </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">TABD </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Table identification number of a TABLES1 entry that contains pure volumetric compression data to be used in the estimation of the material constants Di.  xi values in the TABLES1 entry must be values of the volume ratio <em>J</em>=&lambda;<sup class="superscript">3</sup> where &lambda;=<em>l/l</em><sub class="subscript">0</sub> is the stretch ratio in all three directions; yi values must be values of the pressure, assumed positive in compression.  (Integer &gt; 0 or blank) 
                     </p>
                  </td>
               </tr>
            </tbody>
         </table><br></div>
      <div class="title_refCategory">Remarks: </div>
      <div class="refCategoryContent">
         <ol type="1" start="1">
            <li>
               <p class="para_item">The generalized Mooney-Rivlin strain energy function may be expressed as follows: </p>
               <p align="center"><img align="bottom" src="graphics/bulkmno-equ23.gif"></p>
               <p class="para_item">where <img align="bottom" src="graphics/bulkmno-ufig02.gif" border="0"> and <img align="bottom" src="graphics/bulkmno-ufig03.gif" border="0"> are the first and second distortional strain invariants, respectively; <em>J</em>=det <em>F</em> is the determinant of the deformation gradient; and 2D1=<em>K</em> and 2(A10 + A01)=<em>G</em> at small strains, in which <em>K</em> is the bulk modulus and <em>G</em> is the shear modulus.  The model reduces to a Mooney-Rivlin material if NA=1 and to a Neo-Hookean material if NA = 1 and A01 = 0.0.  See Remark 2. For Neo-Hookean or Mooney-Rivlin materials no continuation entry is required.  T is the current temperature and T<sub class="subscript">0</sub> is the initial temperature. 
               </p>
            </li>
            <li>
               <p class="para_item">Conventional Mooney-Rivlin and Neo-Hookean materials are fully incompressible.  Full incompressibility is not presently available but may be simulated with a large enough value of D1.  A value of D1 higher than 10<sup class="superscript">3</sup> · (A10 + A01) is, however, not recommended. 
               </p>
            </li>
            <li>
               <p class="para_item">Aij and Di are obtained from least squares fitting of experimental data.  One or more of four experiments (TAB1 to TAB4) may be used to obtain Aij.  Di may be obtained from pure volumetric compression data (TABD).  If all TAB1 through TAB4 are blank, Aij must be specified by the user.  Parameter estimation, specified through any of the TABLES1 entries, supersedes the manual input of the parameters. </p>
            </li>
            <li>
               <p class="para_item">IF ND=1 and a nonzero value of D1 is provided or is obtained from experimental data in TABD, then the parameter estimation of the material constants Aij takes compressibility into account in the cases of simple tension/compression, equibiaxial tension, and general biaxial deformation.  Otherwise, full incompressibility is assumed in estimating the material constants. </p>
            </li>
            <li>
               <p class="para_item">See <a class="links" href="javascript:void(0)" onclick="top.popUpPage('QRG/linkInter_target_not_resolved.html#bas_nonlinear:ref90420:-aux,-home','linkInter','menubar=yes,status=yes,resizable=yes,toolbar=yes,scrollbars=yes,location=no,directories=no');return(false);">&ldquo;Hyperelastic Material&rdquo; </a>in the <em>NX Nastran Basic Nonlinear Analysis User&rsquo;s Guide</em> for further details. 
               </p>
            </li>
         </ol>
      </div>
      <div class="title_refCategory">Remarks related to SOLs 601 and 701: </div>
      <div class="refCategoryContent">
         <ol type="1" start="1">
            <li>
               <p class="para_item">Only A10, A01, D1, A20, A11, A02, A30, A21, A12, A03, and RHO parameters are considered for Mooney-Rivlin material model, i.e., NA&le;3, ND=1. TABi and TABD may be used to obtain these parameters. </p>
            </li>
         </ol>
      </div>
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