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HYPERFOAM.html
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<html><head><link rel="stylesheet" type="text/css" href="style.css"/></head><body> <H2> <BR> *HYPERFOAM </H2> <P> Keyword type: model definition, material <P> This option is used to define a hyperfoam material. There is one optional parameters, N. N determines the order of the strain energy potential. Default is N=1. All constants may be temperature dependent. <P> The hyperfoam 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} \left[ \lambda_1^{\alpha_i}+\lambda_2^{\alpha_i}+\lambda_3^{\alpha_i}-3+\frac{1}{\beta_i}(J^{-\alpha_i \beta_i}-1) \right] \end{equation} --> <TABLE CLASS="equation" CELLPADDING="0" WIDTH="100%" ALIGN="CENTER"> <TR VALIGN="MIDDLE"> <TD NOWRAP ALIGN="CENTER"><SPAN CLASS="MATH"><IMG WIDTH="370" HEIGHT="67" ALIGN="MIDDLE" BORDER="0" SRC="img2460.png" ALT="$\displaystyle U=\sum_{i=1}^{N} \frac{2 \mu_i}{\alpha_i^2} \left[ \lambda_1^{\al... ...ha_i}+\lambda_3^{\alpha_i}-3+\frac{1}{\beta_i}(J^{-\alpha_i \beta_i}-1) \right]$"></SPAN></TD> <TD NOWRAP CLASS="eqno" WIDTH="10" ALIGN="RIGHT"> (<SPAN CLASS="arabic">721</SPAN>)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> where <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$"></B></SPAN> are the principal stretches. The parameters <SPAN CLASS="MATH"><B><IMG WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2461.png" ALT="$ \beta_i$"></B></SPAN> are related to the Poisson coefficients <SPAN CLASS="MATH"><B><IMG WIDTH="17" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2462.png" ALT="$ \nu_i$"></B></SPAN> by: <P> <P></P> <DIV ALIGN="CENTER" CLASS="mathdisplay"><!-- MATH \begin{equation} \beta_i=\frac{ \nu_i}{1 -2 \nu_i} \end{equation} --> <TABLE CLASS="equation" CELLPADDING="0" WIDTH="100%" ALIGN="CENTER"> <TR VALIGN="MIDDLE"> <TD NOWRAP ALIGN="CENTER"><SPAN CLASS="MATH"><IMG WIDTH="92" HEIGHT="44" ALIGN="MIDDLE" BORDER="0" SRC="img2463.png" ALT="$\displaystyle \beta_i=\frac{ \nu_i}{1 -2 \nu_i}$"></SPAN></TD> <TD NOWRAP CLASS="eqno" WIDTH="10" ALIGN="RIGHT"> (<SPAN CLASS="arabic">722</SPAN>)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> <P> and <P> <P></P> <DIV ALIGN="CENTER" CLASS="mathdisplay"><!-- MATH \begin{equation} \nu_i= \frac{ \beta_i}{1+2 \beta_i}. \end{equation} --> <TABLE CLASS="equation" CELLPADDING="0" WIDTH="100%" ALIGN="CENTER"> <TR VALIGN="MIDDLE"> <TD NOWRAP ALIGN="CENTER"><SPAN CLASS="MATH"><IMG WIDTH="96" HEIGHT="51" ALIGN="MIDDLE" BORDER="0" SRC="img2464.png" ALT="$\displaystyle \nu_i= \frac{ \beta_i}{1+2 \beta_i}.$"></SPAN></TD> <TD NOWRAP CLASS="eqno" WIDTH="10" ALIGN="RIGHT"> (<SPAN CLASS="arabic">723</SPAN>)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> <P> <P><P> <BR> <P> First line: <UL> <LI>*HYPERFOAM </LI> <LI>Enter parameters and their values, if needed </LI> </UL> <P> Following line for N=1: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="20" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2445.png" ALT="$ \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}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2465.png" ALT="$ \nu_{1}$"></B></SPAN>. </LI> <LI>Temperature. </LI> </UL> Repeat this line if needed to define complete temperature dependence. <P> Following line for N=2: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="20" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2445.png" ALT="$ \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}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2465.png" ALT="$ \nu_{1}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2466.png" ALT="$ \nu_{2}$"></B></SPAN>. </LI> <LI>Temperature. </LI> </UL> Repeat this line if needed to define complete temperature dependence. <P> Following lines, in a pair, for N=3: First line of pair: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="20" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2445.png" ALT="$ \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}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2465.png" ALT="$ \nu_{1}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2466.png" ALT="$ \nu_{2}$"></B></SPAN>. </LI> </UL> Second line of pair: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2467.png" ALT="$ \nu_{3}$"></B></SPAN>. </LI> <LI>Temperature. </LI> </UL> Repeat this pair if needed to define complete temperature dependence. <P> <PRE>
Example:
*HYPERFOAM,N=2
0.164861,8.88413,2.302e-5,-4.81798,0.,0.
</PRE> <P> defines a hyperfoam material with two terms in the series. <P> <P><P> <BR> Example files: beamhf. <P> </body></html>