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HYPERELASTIC.html
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<html><head><link rel="stylesheet" type="text/css" href="style.css"/></head><body> <H2> <BR> *HYPERELASTIC </H2> <P> Keyword type: model definition, material <P> This option is used to define the hyperelastic properties of a material. There are two optional parameters. The first one defines the model and can take one of the following strings: ARRUDA-BOYCE, MOONEY-RIVLIN, NEO HOOKE, OGDEN, POLYNOMIAL, REDUCED POLYNOMIAL or YEOH. The second parameter N makes sense for the OGDEN, POLYNOMIAL and REDUCED POLYMIAL model only, and determines the order of the strain energy potential. Default is the POLYNOMIAL model with N=1. All constants may be temperature dependent. <P> Let <SPAN CLASS="MATH"><B><IMG WIDTH="18" HEIGHT="35" ALIGN="MIDDLE" BORDER="0" SRC="img2407.png" ALT="$ \bar{I}_1$"></B></SPAN>,<SPAN CLASS="MATH"><B><IMG WIDTH="18" HEIGHT="35" ALIGN="MIDDLE" BORDER="0" SRC="img2408.png" ALT="$ \bar{I}_2$"></B></SPAN> and <SPAN CLASS="MATH"><B><IMG WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0" SRC="img2409.png" ALT="$ J$"></B></SPAN> be defined by: <BR> <DIV ALIGN="CENTER" CLASS="mathdisplay"> <!-- MATH \begin{eqnarray} \bar{I}_1&=&III_C^{-1/3} I_C \\ \bar{I}_2&=&III_C^{-1/3} II_C \\ J&=&III_C^{1/2} \end{eqnarray} --> <TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> <TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG WIDTH="18" HEIGHT="35" ALIGN="MIDDLE" BORDER="0" SRC="img2410.png" ALT="$\displaystyle \bar{I}_1$"></TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG WIDTH="16" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img1168.png" ALT="$\displaystyle =$"></TD> <TD ALIGN="LEFT" NOWRAP WIDTH="50%"><IMG WIDTH="76" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="img2411.png" ALT="$\displaystyle III_C^{-1/3} I_C$"></TD> <TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> (<SPAN CLASS="arabic">711</SPAN>)</TD></TR> <TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG WIDTH="18" HEIGHT="35" ALIGN="MIDDLE" BORDER="0" SRC="img2412.png" ALT="$\displaystyle \bar{I}_2$"></TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG WIDTH="16" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img1168.png" ALT="$\displaystyle =$"></TD> <TD ALIGN="LEFT" NOWRAP WIDTH="50%"><IMG WIDTH="84" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="img2413.png" ALT="$\displaystyle III_C^{-1/3} II_C$"></TD> <TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> (<SPAN CLASS="arabic">712</SPAN>)</TD></TR> <TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2414.png" ALT="$\displaystyle J$"></TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG WIDTH="16" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img1168.png" ALT="$\displaystyle =$"></TD> <TD ALIGN="LEFT" NOWRAP WIDTH="50%"><IMG WIDTH="48" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="img2415.png" ALT="$\displaystyle III_C^{1/2}$"></TD> <TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> (<SPAN CLASS="arabic">713</SPAN>)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"> where <SPAN CLASS="MATH"><B><IMG WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2174.png" ALT="$ I_C$"></B></SPAN>, <SPAN CLASS="MATH"><B><IMG WIDTH="30" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2416.png" ALT="$ II_C$"></B></SPAN> and <SPAN CLASS="MATH"><B><IMG WIDTH="38" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2417.png" ALT="$ III_C$"></B></SPAN> are the invariants of the right Cauchy-Green deformation tensor <!-- MATH $C_{KL}=x_{k,K}x_{k,L}$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="119" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2418.png" ALT="$ C_{KL}=x_{k,K}x_{k,L}$"></B></SPAN>. The tensor <SPAN CLASS="MATH"><B><IMG WIDTH="36" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2419.png" ALT="$ C_{KL}$"></B></SPAN> is linked to the Lagrange strain tensor <SPAN CLASS="MATH"><B><IMG WIDTH="36" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2336.png" ALT="$ E_{KL}$"></B></SPAN> by: <P></P> <DIV ALIGN="CENTER" CLASS="mathdisplay"><!-- MATH \begin{equation} 2E_{KL}=C_{KL}-\delta_{KL} \end{equation} --> <TABLE CLASS="equation" CELLPADDING="0" WIDTH="100%" ALIGN="CENTER"> <TR VALIGN="MIDDLE"> <TD NOWRAP ALIGN="CENTER"><SPAN CLASS="MATH"><IMG WIDTH="146" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2420.png" ALT="$\displaystyle 2E_{KL}=C_{KL}-\delta_{KL}$"></SPAN></TD> <TD NOWRAP CLASS="eqno" WIDTH="10" ALIGN="RIGHT"> (<SPAN CLASS="arabic">714</SPAN>)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> where <SPAN CLASS="MATH"><B><IMG WIDTH="11" HEIGHT="15" ALIGN="BOTTOM" BORDER="0" SRC="img930.png" ALT="$ \delta$"></B></SPAN> is the Kronecker symbol. <P> The Arruda-Boyce strain energy potential takes the form: <BR> <DIV ALIGN="CENTER" CLASS="mathdisplay"> <!-- MATH \begin{eqnarray} U&=&\mu \Bigg\{ \frac{1}{2}(\bar{I}_1-3)+\frac{1}{20\lambda_m^2}(\bar{I}_1^2-9)+\frac{11}{1050\lambda_m^4}(\bar{I}_1^3-27) \nonumber\\&+& \frac{19}{7000\lambda_m^6}(\bar{I}_1^4-81)+\frac{519}{673750\lambda_m^8}(\bar{I}_1^5-243) \Bigg\} \\&+& \frac{1}{D} \left( \frac{J^2-1}{2} - \ln J \right) \nonumber \end{eqnarray} --> <TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> <TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2421.png" ALT="$\displaystyle U$"></TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG WIDTH="16" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img1168.png" ALT="$\displaystyle =$"></TD> <TD ALIGN="LEFT" NOWRAP WIDTH="50%"><IMG WIDTH="347" HEIGHT="63" ALIGN="MIDDLE" BORDER="0" SRC="img2422.png" ALT="$\displaystyle \mu \Bigg\{ \frac{1}{2}(\bar{I}_1-3)+\frac{1}{20\lambda_m^2}(\bar{I}_1^2-9)+\frac{11}{1050\lambda_m^4}(\bar{I}_1^3-27)$"></TD> <TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> </TD></TR> <TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"> </TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG WIDTH="16" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img355.png" ALT="$\displaystyle +$"></TD> <TD ALIGN="LEFT" NOWRAP WIDTH="50%"><IMG WIDTH="300" HEIGHT="63" ALIGN="MIDDLE" BORDER="0" SRC="img2423.png" ALT="$\displaystyle \frac{19}{7000\lambda_m^6}(\bar{I}_1^4-81)+\frac{519}{673750\lambda_m^8}(\bar{I}_1^5-243) \Bigg\}$"></TD> <TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> (<SPAN CLASS="arabic">715</SPAN>)</TD></TR> <TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"> </TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG WIDTH="16" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img355.png" ALT="$\displaystyle +$"></TD> <TD ALIGN="LEFT" NOWRAP WIDTH="50%"><IMG WIDTH="142" HEIGHT="56" ALIGN="MIDDLE" BORDER="0" SRC="img2424.png" ALT="$\displaystyle \frac{1}{D} \left( \frac{J^2-1}{2} - \ln J \right)$"></TD> <TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> </TD></TR> </TABLE></DIV> <BR CLEAR="ALL"> <P> The Mooney-Rivlin strain energy potential takes the form: <P></P> <DIV ALIGN="CENTER" CLASS="mathdisplay"><!-- MATH \begin{equation} U=C_{10}(\bar{I}_1-3)+C_{01}(\bar{I}_2-3)+\frac{1}{D_1}(J-1)^2 \end{equation} --> <TABLE CLASS="equation" CELLPADDING="0" WIDTH="100%" ALIGN="CENTER"> <TR VALIGN="MIDDLE"> <TD NOWRAP ALIGN="CENTER"><SPAN CLASS="MATH"><IMG WIDTH="315" HEIGHT="52" ALIGN="MIDDLE" BORDER="0" SRC="img2425.png" ALT="$\displaystyle U=C_{10}(\bar{I}_1-3)+C_{01}(\bar{I}_2-3)+\frac{1}{D_1}(J-1)^2$"></SPAN></TD> <TD NOWRAP CLASS="eqno" WIDTH="10" ALIGN="RIGHT"> (<SPAN CLASS="arabic">716</SPAN>)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> The Mooney-Rivlin strain energy potential is identical to the polynomial strain energy potential for <SPAN CLASS="MATH"><B><IMG WIDTH="47" HEIGHT="15" ALIGN="BOTTOM" BORDER="0" SRC="img2426.png" ALT="$ N=1$"></B></SPAN>. <P> The Neo-Hooke strain energy potential takes the form: <P></P> <DIV ALIGN="CENTER" CLASS="mathdisplay"><!-- MATH \begin{equation} U=C_{10}(\bar{I}_1-3)+\frac{1}{D_1}(J-1)^2 \end{equation} --> <TABLE CLASS="equation" CELLPADDING="0" WIDTH="100%" ALIGN="CENTER"> <TR VALIGN="MIDDLE"> <TD NOWRAP ALIGN="CENTER"><SPAN CLASS="MATH"><IMG WIDTH="217" HEIGHT="52" ALIGN="MIDDLE" BORDER="0" SRC="img2427.png" ALT="$\displaystyle U=C_{10}(\bar{I}_1-3)+\frac{1}{D_1}(J-1)^2$"></SPAN></TD> <TD NOWRAP CLASS="eqno" WIDTH="10" ALIGN="RIGHT"> (<SPAN CLASS="arabic">717</SPAN>)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> The Neo-Hooke strain energy potential is identical to the reduced polynomial strain energy potential for <SPAN CLASS="MATH"><B><IMG WIDTH="47" HEIGHT="15" ALIGN="BOTTOM" BORDER="0" SRC="img2426.png" ALT="$ N=1$"></B></SPAN>. <P> The polynomial strain energy potential takes the form: <P></P> <DIV ALIGN="CENTER" CLASS="mathdisplay"><!-- MATH \begin{equation} U=\sum_{i+j=1}^{N} C_{ij}(\bar{I}_1-3)^i(\bar{I}_2-3)^j +\sum_{i=1}^{N}\frac{1}{D_i}(J-1)^{2i} \end{equation} --> <TABLE CLASS="equation" CELLPADDING="0" WIDTH="100%" ALIGN="CENTER"> <TR VALIGN="MIDDLE"> <TD NOWRAP ALIGN="CENTER"><SPAN CLASS="MATH"><IMG WIDTH="348" HEIGHT="67" ALIGN="MIDDLE" BORDER="0" SRC="img2428.png" ALT="$\displaystyle U=\sum_{i+j=1}^{N} C_{ij}(\bar{I}_1-3)^i(\bar{I}_2-3)^j +\sum_{i=1}^{N}\frac{1}{D_i}(J-1)^{2i}$"></SPAN></TD> <TD NOWRAP CLASS="eqno" WIDTH="10" ALIGN="RIGHT"> (<SPAN CLASS="arabic">718</SPAN>)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> In CalculiX <SPAN CLASS="MATH"><B><IMG WIDTH="47" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2429.png" ALT="$ N\le 3$"></B></SPAN>. <P> The reduced polynomial strain energy potential takes the form: <P></P> <DIV ALIGN="CENTER" CLASS="mathdisplay"><!-- MATH \begin{equation} U=\sum_{i=1}^{N} C_{i0}(\bar{I}_1-3)^i +\sum_{i=1}^{N}\frac{1}{D_i}(J-1)^{2i} \end{equation} --> <TABLE CLASS="equation" CELLPADDING="0" WIDTH="100%" ALIGN="CENTER"> <TR VALIGN="MIDDLE"> <TD NOWRAP ALIGN="CENTER"><SPAN CLASS="MATH"><IMG WIDTH="275" HEIGHT="67" ALIGN="MIDDLE" BORDER="0" SRC="img2430.png" ALT="$\displaystyle U=\sum_{i=1}^{N} C_{i0}(\bar{I}_1-3)^i +\sum_{i=1}^{N}\frac{1}{D_i}(J-1)^{2i}$"></SPAN></TD> <TD NOWRAP CLASS="eqno" WIDTH="10" ALIGN="RIGHT"> (<SPAN CLASS="arabic">719</SPAN>)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> In CalculiX <SPAN CLASS="MATH"><B><IMG WIDTH="47" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2429.png" ALT="$ N\le 3$"></B></SPAN>. The reduced polynomial strain energy potential can be viewed as a special case of the polynomial strain energy potential <P> The Yeoh strain energy potential is nothing else but the reduced polynomial strain energy potential for <SPAN CLASS="MATH"><B><IMG WIDTH="47" HEIGHT="16" ALIGN="BOTTOM" BORDER="0" SRC="img2431.png" ALT="$ N=3$"></B></SPAN>. <P> Denoting the principal stretches by <SPAN CLASS="MATH"><B><IMG WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2432.png" ALT="$ \lambda_1$"></B></SPAN>, <SPAN CLASS="MATH"><B><IMG WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2433.png" ALT="$ \lambda_2$"></B></SPAN> and <SPAN CLASS="MATH"><B><IMG WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2434.png" ALT="$ \lambda_3$"></B></SPAN> (<!-- MATH $\lambda_1^2$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0" SRC="img2435.png" ALT="$ \lambda_1^2$"></B></SPAN>, <!-- MATH $\lambda_2^2$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0" SRC="img2436.png" ALT="$ \lambda_2^2$"></B></SPAN> and <!-- MATH $\lambda_3^2$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0" SRC="img2437.png" ALT="$ \lambda_3^2$"></B></SPAN> are the eigenvalues of the right Cauchy-Green deformation tensor) and the deviatoric stretches by <!-- MATH $\bar{\lambda}_1$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="20" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="img2438.png" ALT="$ \bar{\lambda}_1$"></B></SPAN>, <!-- MATH $\bar{\lambda}_2$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="20" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="img2439.png" ALT="$ \bar{\lambda}_2$"></B></SPAN> and <!-- MATH $\bar{\lambda}_3$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="20" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="img2440.png" ALT="$ \bar{\lambda}_3$"></B></SPAN>, where <!-- MATH $\bar{\lambda}_i=III_C^{-1/6}\lambda_i$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="109" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="img2441.png" ALT="$ \bar{\lambda}_i=III_C^{-1/6}\lambda_i$"></B></SPAN>, the Ogden strain energy potential takes the form: <P></P> <DIV ALIGN="CENTER" CLASS="mathdisplay"><!-- MATH \begin{equation} U=\sum_{i=1}^{N} \frac{2 \mu_i}{\alpha_i^2}(\bar{\lambda}_1^{\alpha_i}+\bar{\lambda}_2^{\alpha_i}+\bar{\lambda}_3^{\alpha_i}-3)+\sum_{i=1}^{N}\frac{1}{D_i}(J-1)^{2i}. \end{equation} --> <TABLE CLASS="equation" CELLPADDING="0" WIDTH="100%" ALIGN="CENTER"> <TR VALIGN="MIDDLE"> <TD NOWRAP ALIGN="CENTER"><SPAN CLASS="MATH"><IMG WIDTH="372" HEIGHT="67" ALIGN="MIDDLE" BORDER="0" SRC="img2442.png" ALT="$\displaystyle U=\sum_{i=1}^{N} \frac{2 \mu_i}{\alpha_i^2}(\bar{\lambda}_1^{\alp... ...{\alpha_i}+\bar{\lambda}_3^{\alpha_i}-3)+\sum_{i=1}^{N}\frac{1}{D_i}(J-1)^{2i}.$"></SPAN></TD> <TD NOWRAP CLASS="eqno" WIDTH="10" ALIGN="RIGHT"> (<SPAN CLASS="arabic">720</SPAN>)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> <P> The input deck for a hyperelastic material looks as follows: <P><P> <BR> <P> First line: <UL> <LI>*HYPERELASTIC </LI> <LI>Enter parameters and their values, if needed </LI> </UL> <P> Following line for the ARRUDA-BOYCE model: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="13" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img51.png" ALT="$ \mu$"></B></SPAN>. </LI> <LI><!-- MATH $\lambda_{m}$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="25" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2443.png" ALT="$ \lambda_{m}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0" SRC="img369.png" ALT="$ D$"></B></SPAN>. </LI> <LI>Temperature </LI> </UL> Repeat this line if needed to define complete temperature dependence. <P> Following line for the MOONEY-RIVLIN model: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1319.png" ALT="$ C_{10}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2444.png" ALT="$ C_{01}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1320.png" ALT="$ D_{1}$"></B></SPAN>. </LI> <LI>Temperature </LI> </UL> Repeat this line if needed to define complete temperature dependence. <P> Following line for the NEO HOOKE model: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1319.png" ALT="$ C_{10}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1320.png" ALT="$ D_{1}$"></B></SPAN>. </LI> <LI>Temperature. </LI> </UL> Repeat this line if needed to define complete temperature dependence. <P> Following line for the OGDEN model with N=1: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="20" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2445.png" ALT="$ \mu_{1}$"></B></SPAN>. </LI> <LI><!-- MATH $\alpha_{1}$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="21" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2446.png" ALT="$ \alpha_{1}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1320.png" ALT="$ D_{1}$"></B></SPAN>. </LI> <LI>Temperature. </LI> </UL> Repeat this line if needed to define complete temperature dependence. <P> Following line for the OGDEN model with N=2: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="20" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2445.png" ALT="$ \mu_{1}$"></B></SPAN>. </LI> <LI><!-- MATH $\alpha_{1}$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="21" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2446.png" ALT="$ \alpha_{1}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="20" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2447.png" ALT="$ \mu_{2}$"></B></SPAN>. </LI> <LI><!-- MATH $\alpha_{2}$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="21" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2448.png" ALT="$ \alpha_{2}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1320.png" ALT="$ D_{1}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2449.png" ALT="$ D_{2}$"></B></SPAN>. </LI> <LI>Temperature. </LI> </UL> Repeat this line if needed to define complete temperature dependence. <P> Following lines, in a pair, for the OGDEN model with N=3: First line of pair: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="20" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2445.png" ALT="$ \mu_{1}$"></B></SPAN>. </LI> <LI><!-- MATH $\alpha_{1}$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="21" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2446.png" ALT="$ \alpha_{1}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="20" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2447.png" ALT="$ \mu_{2}$"></B></SPAN>. </LI> <LI><!-- MATH $\alpha_{2}$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="21" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2448.png" ALT="$ \alpha_{2}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="20" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2450.png" ALT="$ \mu_{3}$"></B></SPAN>. </LI> <LI><!-- MATH $\alpha_{3}$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="21" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2451.png" ALT="$ \alpha_{3}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1320.png" ALT="$ D_{1}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2449.png" ALT="$ D_{2}$"></B></SPAN>. </LI> </UL> Second line of pair: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2452.png" ALT="$ D_{3}$"></B></SPAN>. </LI> <LI>Temperature. </LI> </UL> Repeat this pair if needed to define complete temperature dependence. <P> Following line for the POLYNOMIAL model with N=1: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1319.png" ALT="$ C_{10}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2444.png" ALT="$ C_{01}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1320.png" ALT="$ D_{1}$"></B></SPAN>. </LI> <LI>Temperature. </LI> </UL> Repeat this line if needed to define complete temperature dependence. <P> Following line for the POLYNOMIAL model with N=2: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1319.png" ALT="$ C_{10}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2444.png" ALT="$ C_{01}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2453.png" ALT="$ C_{20}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2454.png" ALT="$ C_{11}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2455.png" ALT="$ C_{02}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1320.png" ALT="$ D_{1}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2449.png" ALT="$ D_{2}$"></B></SPAN>. </LI> <LI>Temperature. </LI> </UL> Repeat this line if needed to define complete temperature dependence. <P> Following lines, in a pair, for the POLYNOMIAL model with N=3: First line of pair: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1319.png" ALT="$ C_{10}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2444.png" ALT="$ C_{01}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2453.png" ALT="$ C_{20}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2454.png" ALT="$ C_{11}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2455.png" ALT="$ C_{02}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2456.png" ALT="$ C_{30}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2457.png" ALT="$ C_{21}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2458.png" ALT="$ C_{12}$"></B></SPAN>. </LI> </UL> Second line of pair: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2459.png" ALT="$ C_{03}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1320.png" ALT="$ D_{1}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2449.png" ALT="$ D_{2}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2452.png" ALT="$ D_{3}$"></B></SPAN>. </LI> <LI>Temperature. </LI> </UL> Repeat this pair if needed to define complete temperature dependence. <P> Following line for the REDUCED POLYNOMIAL model with N=1: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1319.png" ALT="$ C_{10}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1320.png" ALT="$ D_{1}$"></B></SPAN>. </LI> <LI>Temperature. </LI> </UL> Repeat this line if needed to define complete temperature dependence. <P> Following line for the REDUCED POLYNOMIAL model with N=2: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1319.png" ALT="$ C_{10}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2453.png" ALT="$ C_{20}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1320.png" ALT="$ D_{1}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2449.png" ALT="$ D_{2}$"></B></SPAN>. </LI> <LI>Temperature. </LI> </UL> Repeat this line if needed to define complete temperature dependence. <P> Following line for the REDUCED POLYNOMIAL model with N=3: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1319.png" ALT="$ C_{10}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2453.png" ALT="$ C_{20}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2456.png" ALT="$ C_{30}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1320.png" ALT="$ D_{1}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2449.png" ALT="$ D_{2}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2452.png" ALT="$ D_{3}$"></B></SPAN>. </LI> <LI>Temperature. </LI> </UL> Repeat this line if needed to define complete temperature dependence. <P> Following line for the YEOH model: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1319.png" ALT="$ C_{10}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2453.png" ALT="$ C_{20}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2456.png" ALT="$ C_{30}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1320.png" ALT="$ D_{1}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2449.png" ALT="$ D_{2}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2452.png" ALT="$ D_{3}$"></B></SPAN>. </LI> <LI>Temperature. </LI> </UL> Repeat this line if needed to define complete temperature dependence. <P> <PRE>
Example:
*HYPERELASTIC,OGDEN,N=1
3.488,2.163,0.
</PRE> <P> defines an ogden material with one term: <SPAN CLASS="MATH"><B><IMG WIDTH="20" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2445.png" ALT="$ \mu_{1}$"></B></SPAN> = 3.488, <!-- MATH $\alpha_{1}$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="21" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2446.png" ALT="$ \alpha_{1}$"></B></SPAN> = 2.163, <SPAN CLASS="MATH"><B><IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img1320.png" ALT="$ D_{1}$"></B></SPAN>=0. Since the compressibility coefficient was chosen to be zero, it will be replaced by CalculiX by a small value to ensure some compressibility to guarantee convergence (cfr. page <IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]" SRC="file:/usr/share/latex2html/icons/crossref.png">). <P> <P><P> <BR> Example files: beamnh, beamog. <P> </body></html>