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      <title>MATS1</title>
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      <div class="title_refTopic" id="xps10_pagetitle">MATS1 </div>
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      <h4 class="refClassification">Material Stress Dependence </h4>
      <p class="para_refTopic">Specifies stress-dependent material properties for use in applications involving nonlinear materials.  This entry is used if a MAT1, MAT2, or MAT9 entry is specified with the same MID in a nonlinear solution sequence (SOLs 106 and 129). </p>
      <div class="title_refCategory">Format: </div>
      <div class="refCategoryContent">
         <table border="1" width="100%" align="center">
            <colgroup span="1">
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                  <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">MATS1 </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">TID </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">TYPE </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">H </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">YF </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">HR </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">LIMIT1 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">LIMIT2 </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">Example: </div>
      <div class="refCategoryContent">
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               <col span="1" width="10.77*">
               <col span="1" width="9.23*">
               <col span="1" width="10*">
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               <col span="1" width="10*">
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                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">MATS1 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">17 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">28 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">PLASTIC </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">0.0 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">1 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">1 </p>
                  </td>
                  <td align="center" colspan="1" rowspan="1">
                     <p class="para_td">2.+4 </p>
                  </td>
                  <td align="center" 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>
            </tbody>
         </table><br></div>
      <div class="title_refCategory">Fields:</div>
      <div class="refCategoryContent">
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                  <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 MAT1, MAT2, or MAT9 entry.  (Integer &gt; 0) </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">TID </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Identification number of a TABLES1 or TABLEST entry.  If H is given, then this field must be blank.  See Remark 3. (Integer &ge; 0 or blank) </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">TYPE </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Type of material nonlinearity.  See Remarks.  (Character:  &ldquo;NLELAST&rdquo; for nonlinear elastic or &ldquo;PLASTIC&rdquo; for elastoplastic.) </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">H </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Work hardening slope (slope of stress versus plastic strain) in units of stress.  For elastic-perfectly plastic cases, H = 0.0.  For more than a single slope in the plastic range, the stress-strain data must be supplied on a TABLES1 entry referenced by TID, and this field must be blank.  See Remark 2. (Real) </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">YF </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Yield function criterion, selected by one of the following values (Integer): </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">1  =  von Mises (Default) </p>
                     <p class="para_td">2  =  Tresca </p>
                     <p class="para_td">3  =  Mohr-Coulomb </p>
                     <p class="para_td">4  =  Drucker-Prager </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">HR </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Hardening Rule, selected by one of the following values (Integer): </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">&nbsp; </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">1  =  Isotropic (Default) </p>
                     <p class="para_td">2  =  Kinematic </p>
                     <p class="para_td">3 = Combined isotropic and kinematic hardening </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">LIMIT1 </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Initial yield point.  See <a class="links" href="javascript:void(0)" onclick="top.openFile('QRG/17topic_13.html#ref81064');return(false);">Table 16-2</a>.  (Real) 
                     </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">LIMIT2 </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Internal friction angle, measured in degrees, for the Mohr-Coulomb and Drucker-Prager yield criteria.  See <a class="links" href="javascript:void(0)" onclick="top.openFile('QRG/17topic_13.html#ref81064');return(false);">Table 16-2</a>.  (0.0 &le; Real &lt; 45.0) 
                     </p>
                  </td>
               </tr>
            </tbody>
         </table><br><a name="ref81064"></a><div class="title_tableWrapper">Table 16-2. Yield Functions Versus LIMIT1 and LIMIT2 </div>
         <table border="1" cellpadding="8" width="100%" align="center">
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               <col span="1" width="35.93*">
               <col span="1" width="33.41*">
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               <tr>
                  <th align="center" colspan="1" rowspan="1">
                     <p class="para_th"><b class="uiTerm">Yield Function (YF) </b></p>
                  </th>
                  <th align="center" colspan="1" rowspan="1">
                     <p class="para_th"><b class="uiTerm">LIMIT1 </b></p>
                  </th>
                  <th align="center" colspan="1" rowspan="1">
                     <p class="para_th"><b class="uiTerm">LIMIT2 </b></p>
                  </th>
               </tr>
            </thead>
            <tbody>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">von Mises (1)or Tresca (2) </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Initial Yield Stress in Tension, <em>Y</em><sub class="subscript">1</sub></p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Not used </p>
                  </td>
               </tr>
               <tr>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Mohr-Coulomb (3)or Drucker-Prager (4) </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">2*Cohesion, <em>2c</em> (in units of stress) 
                     </p>
                  </td>
                  <td align="left" colspan="1" rowspan="1" valign="top">
                     <p class="para_td">Angle of Internal Friction &phi; (in degrees) </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">If TYPE = &ldquo;NLELAST&rdquo;, then MID may refer to a MAT1 entry only.  Also, the stress-strain data given in the TABLES1 entry will be used to determine the stress for a given value of strain.  The values H, YF, HR, LIMIT1, and LIMIT2 will not be used in this case. </p>
               <p class="para_item">Thermoelastic analysis with temperature-dependent material properties is available for linear and nonlinear elastic isotropic materials (TYPE = &ldquo;NLELAST&rdquo;) and linear elastic anisotropic materials.  Four options of constitutive relations exist.  The relations appear in <a class="links" href="javascript:void(0)" onclick="top.openFile('QRG/17topic_13.html#ref83952');return(false);">Table 16-3</a> along with the required Bulk Data entries. 
               </p><a name="ref83952"></a><div class="title_tableWrapper">Table 16-3. Constitutive Relations and Required Material Property Entries </div>
               <table border="1" cellpadding="8" width="100%" align="center">
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                     <col span="1" width="52*">
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                  <thead>
                     <tr>
                        <th align="center" colspan="1" rowspan="1">
                           <p class="para_th"><b class="uiTerm">Constitutive Relation </b></p>
                        </th>
                        <th align="center" colspan="1" rowspan="1">
                           <p class="para_th"><b class="uiTerm">Required Bulk Data Entries </b></p>
                        </th>
                     </tr>
                  </thead>
                  <tbody>
                     <tr>
                        <td align="left" colspan="1" rowspan="1" valign="top">
                           <p class="para_td">{&sigma;}=[<em>G</em><sub class="subscript">e</sub>(<em>T</em>)] {ϵ} 
                           </p>
                        </td>
                        <td align="left" colspan="1" rowspan="1" valign="top">
                           <p class="para_td">MATi and MATTi where i=1, 2, or 9 </p>
                        </td>
                     </tr>
                     <tr>
                        <td align="left" colspan="1" rowspan="1" valign="top">
                           <p align="center"><img align="bottom" src="graphics/bulkmno-equ24.gif"></p>
                        </td>
                        <td align="left" colspan="1" rowspan="1" valign="top">
                           <p class="para_td">MAT1, MATT1, MATS1, and TABLES1 </p>
                        </td>
                     </tr>
                     <tr>
                        <td align="left" colspan="1" rowspan="1" valign="top">
                           <p align="center"><img align="bottom" src="graphics/bulkmno-equ25.gif"></p>
                        </td>
                        <td align="left" colspan="1" rowspan="1" valign="top">
                           <p class="para_td">MAT1, MATS1, TABLEST, and TABLES1 </p>
                        </td>
                     </tr>
                     <tr>
                        <td align="left" colspan="1" rowspan="1" valign="top">
                           <p align="center"><img align="bottom" src="graphics/bulkmno-equ26.gif"></p>
                        </td>
                        <td align="left" colspan="1" rowspan="1" valign="top">
                           <p class="para_td">MAT1, MATT1, MATS1, TABLEST, and TABLES1 </p>
                        </td>
                     </tr>
                  </tbody>
               </table><br><p class="para_item">In <a class="links" href="javascript:void(0)" onclick="top.openFile('QRG/17topic_13.html#ref83952');return(false);">Table 16-3</a> {&sigma;} and {ϵ} are the stress and strain vectors, [<em>G<sub class="subscript">e</sub></em>] the elasticity matrix, <img align="bottom" src="graphics/bulkmno-ufig04.gif" border="0"> the effective elasticity modulus, and <em>E</em> the reference elasticity modulus. 
               </p>
            </li>
            <li>
               <p class="para_item">If TYPE = &ldquo;PLASTIC&rdquo;, either the table identification TID or the work hardening slope H may be specified, but not both.  If the TID is omitted, the work hardening slope H must be specified unless the material is perfectly plastic.  The plasticity modulus (<em>H</em>) is related to the tangential modulus (<em>E</em><em><sub class="subscript">T</sub></em>) by 
               </p>
               <p align="center"><img align="bottom" src="graphics/bulkmno-equ27.gif"></p>
               <p class="para_item">where <em>E</em> is the elastic modulus and <em>E<sub class="subscript">T</sub></em>=<em>dY/d</em>ϵ is the slope of the uniaxial stress-strain curve in the plastic region.  See <a class="links" href="javascript:void(0)" onclick="top.openFile('QRG/17topic_13.html#ref62897');return(false);">Figure 16-</a>. 
               </p><a name="ref62897"></a><div class="figure">
                  <p align="center"><img align="bottom" src="graphics/bulkmno-fig03.gif"></p>
               </div>
               <div class="title_figure">Figure. Stress-Strain Curve Definition When H Is Specified in Field 5 </div>
            </li>
            <li>
               <p class="para_item">If TID is given, TABLES1 entries (Xi, Yi) of stress-strain data (ϵ<sub class="subscript"><em>k</em></sub>, <em>Y<sub class="subscript">k</sub></em> must conform to the following rules (see <a class="links" href="javascript:void(0)" onclick="top.openFile('QRG/17topic_13.html#ref28777');return(false);">Figure 16-</a>): 
               </p>
               <ul>
                  <li>
                     <p class="para_item">If TYPE = &ldquo;PLASTIC&rdquo;, the true stress-logarithmic strain curve must be defined in the first quadrant.  The first point must be at the origin (X1 = 0, Y2 = 0) and the second point (X2, Y2) must be at the initial yield point (<em>Y</em><sub class="subscript">1</sub> or 2<em>c</em>) specified on the MATS1 entry.  The slope of the line joining the origin to the yield stress must be equal to the value of <em>E</em>.  Also, TID may not reference a TABLEST entry. 
                     </p>
                  </li>
                  <li>
                     <p class="para_item">If TYPE = &ldquo;NLELAST&rdquo;, the full stress-strain curve (-∞ &lt; <em>x</em> &lt; ∞) may be defined in the first and the third quadrant to accommodate different uniaxial compression data.  If the curve is defined only in the first quadrant, then the curve must start at the origin (X1 = 0.0, Y = 0.0) and the compression properties will be assumed identical to tension properties. 
                     </p><a name="ref28777"></a><div class="figure">
                        <p align="left"><img align="bottom" src="graphics/bulkmno-fig04.gif"></p>
                     </div>
                     <div class="title_figure">Figure. Stress-Strain Curve Definition When TID Is Specified in Field 3 </div>
                  </li>
               </ul>
            </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">MID is restricted to the identification number of a MAT1 entry. </p>
            </li>
            <li>
               <p class="para_item">LIMIT2 is ignored. LIMIT1 is only used if TID is blank and H is specified, i.e., for a bilinear plastic material. </p>
            </li>
            <li>
               <p class="para_item">Only YF = 1 (von Mises yield criterion) is supported. </p>
            </li>
            <li>
               <p class="para_item">If HR = 3, a mixed hardening factor of 0.5 is used. HR = 3 may only be used with a bilinear plastic material, i.e. TID must be blank. </p>
            </li>
            <li>
               <p class="para_item">TID can reference a TABLEST entry for TYPE = &ldquo;PLASTIC&rdquo; but not for TYPE = &ldquo;NLELAST&rdquo;. </p>
            </li>
            <li>
               <p class="para_item">TYPE = &ldquo;NLELAST&rdquo; can only be used with the CROD element. The nonlinear elastic material does not support temperature-dependent properties. Hence, <a class="links" href="javascript:void(0)" onclick="top.openFile('QRG/17topic_13.html#ref83952');return(false);">Table 16-3</a> does not apply. 
               </p>
            </li>
            <li>
               <p class="para_item">If TYPE = &ldquo;NLELAST&rdquo;, the full stress-strain curve (tension and compression) must be specified. </p>
            </li>
            <li>
               <p class="para_item">If the slope of the line joining the origin and second point in TABLES1 (for TYPE = &ldquo;PLASTIC&rdquo; and given TID) is not equal to the value of E in MAT1 entry, the strain value at the second point will be adjusted accordingly. </p>
            </li>
            <li>
               <p class="para_item">If TYPE = &ldquo;PLASTIC&rdquo;, all tangent modulii E<sub class="subscript">Ti</sub> must satisfy the following: 
               </p>
               <ul>
                  <li>
                     <p class="para_item">HR = 1:         0 &le; E<sub class="subscript">Ti</sub> &lt; E 
                     </p>
                  </li>
                  <li>
                     <p class="para_item">HR = 2:         0.0001 * E &lt; E<sub class="subscript">Ti</sub> &lt; E 
                     </p>
                  </li>
                  <li>
                     <p class="para_item">HR = 3:         0.0001 * E &lt; E<sub class="subscript">T</sub> &lt; E 
                     </p>
                  </li>
               </ul>
            </li>
         </ol>
      </div>
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