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ELASTIC.html
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<html><head><link rel="stylesheet" type="text/css" href="style.css"/></head><body> <H2> <BR> *ELASTIC </H2> <P> Keyword type: model definition, material <P> This option is used to define the elastic properties of a material. There is one optional parameter TYPE. Default is TYPE=ISO, other values are TYPE=ORTHO and TYPE=ENGINEERING CONSTANTS for orthotropic materials and TYPE=ANISO for anisotropic materials. All constants may be temperature dependent. For orthotropic and fully anisotropic materials, the coefficients <SPAN CLASS="MATH"><B><IMG WIDTH="53" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2333.png" ALT="$ D_{IJKL}$"></B></SPAN> satisfy the equation: <P> <P></P> <DIV ALIGN="CENTER" CLASS="mathdisplay"><!-- MATH \begin{equation} S_{IJ}=D_{IJKL} E_{KL},\;\;\;\;\;I,J,K,L=1..3 \end{equation} --> <TABLE CLASS="equation" CELLPADDING="0" WIDTH="100%" ALIGN="CENTER"> <TR VALIGN="MIDDLE"> <TD NOWRAP ALIGN="CENTER"><SPAN CLASS="MATH"><IMG WIDTH="269" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2334.png" ALT="$\displaystyle S_{IJ}=D_{IJKL} E_{KL},\;\;\;\;\;I,J,K,L=1..3$"></SPAN></TD> <TD NOWRAP CLASS="eqno" WIDTH="10" ALIGN="RIGHT"> (<SPAN CLASS="arabic">702</SPAN>)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> <P> where <SPAN CLASS="MATH"><B><IMG WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2335.png" ALT="$ S_{IJ}$"></B></SPAN> is the second Piola-Kirchhoff stress and <SPAN CLASS="MATH"><B><IMG WIDTH="36" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2336.png" ALT="$ E_{KL}$"></B></SPAN> is the Lagrange deformation tensor (nine terms on the right hand side for each equation). For linear calculations, these reduce to the generic stress and strain tensors. <P> An isotropic material can be defined as an anisotropic material by defining <!-- MATH $D_{1111}=D_{2222}=D_{3333}=\lambda+2 \mu$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="232" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2337.png" ALT="$ D_{1111}=D_{2222}=D_{3333}=\lambda+2 \mu$"></B></SPAN>, <!-- MATH $D_{1122}=D_{1133}=D_{2233}=\lambda$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="195" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2338.png" ALT="$ D_{1122}=D_{1133}=D_{2233}=\lambda$"></B></SPAN> and <!-- MATH $D_{1212}=D_{1313}=D_{2323}=\mu$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="195" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2339.png" ALT="$ D_{1212}=D_{1313}=D_{2323}=\mu$"></B></SPAN>, where <SPAN CLASS="MATH"><B><IMG WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0" SRC="img44.png" ALT="$ \lambda$"></B></SPAN> and <SPAN CLASS="MATH"><B><IMG WIDTH="13" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img51.png" ALT="$ \mu$"></B></SPAN> are the Lamé constants [20]. <P><P> <BR> <P> First line: <UL> <LI>*ELASTIC </LI> <LI>Enter the TYPE parameter and its value, if needed </LI> </UL> <P> Following line for TYPE=ISO: <UL> <LI>Young's modulus. </LI> <LI>Poisson's ratio. </LI> <LI>Temperature. </LI> </UL> Repeat this line if needed to define complete temperature dependence. <P> <P><P> <BR> Following lines, in a pair, for TYPE=ORTHO: First line of pair: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2340.png" ALT="$ D_{1111}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2341.png" ALT="$ D_{1122}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2342.png" ALT="$ D_{2222}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2343.png" ALT="$ D_{1133}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2344.png" ALT="$ D_{2233}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2345.png" ALT="$ D_{3333}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2346.png" ALT="$ D_{1212}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2347.png" ALT="$ D_{1313}$"></B></SPAN>. </LI> </UL> Second line of pair: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2348.png" ALT="$ D_{2323}$"></B></SPAN>. </LI> <LI>Temperature. </LI> </UL> Repeat this pair if needed to define complete temperature dependence. <P> <P><P> <BR> Following lines, in a pair, for TYPE=ENGINEERING CONSTANTS: First line of pair: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img583.png" ALT="$ E_1$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img585.png" ALT="$ E_2$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2349.png" ALT="$ E_3$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="25" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2350.png" ALT="$ \nu_{12}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="25" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2351.png" ALT="$ \nu_{13}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="25" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img2352.png" ALT="$ \nu_{23}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="30" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2353.png" ALT="$ G_{12}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="30" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2354.png" ALT="$ G_{13}$"></B></SPAN>. </LI> </UL> Second line of pair: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="30" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2355.png" ALT="$ G_{23}$"></B></SPAN>. </LI> <LI>Temperature. </LI> </UL> Repeat this pair if needed to define complete temperature dependence. <P> <P><P> <BR> Following lines, in sets of 3, for TYPE=ANISO: First line of set: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2340.png" ALT="$ D_{1111}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2341.png" ALT="$ D_{1122}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2342.png" ALT="$ D_{2222}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2343.png" ALT="$ D_{1133}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2344.png" ALT="$ D_{2233}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2345.png" ALT="$ D_{3333}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2356.png" ALT="$ D_{1112}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2357.png" ALT="$ D_{2212}$"></B></SPAN>. </LI> </UL> Second line of set: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2358.png" ALT="$ D_{3312}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2346.png" ALT="$ D_{1212}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2359.png" ALT="$ D_{1113}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2360.png" ALT="$ D_{2213}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2361.png" ALT="$ D_{3313}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2362.png" ALT="$ D_{1213}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2347.png" ALT="$ D_{1313}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2363.png" ALT="$ D_{1123}$"></B></SPAN>. </LI> </UL> Third line of set: <UL> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2364.png" ALT="$ D_{2223}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2365.png" ALT="$ D_{3323}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2366.png" ALT="$ D_{1223}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2367.png" ALT="$ D_{1323}$"></B></SPAN>. </LI> <LI><SPAN CLASS="MATH"><B><IMG WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2348.png" ALT="$ D_{2323}$"></B></SPAN>. </LI> <LI>Temperature. </LI> </UL> Repeat this set if needed to define complete temperature dependence. <P> <PRE>
Example:
*ELASTIC,TYPE=ORTHO
500000.,157200.,400000.,157200.,157200.,300000.,126200.,126200.,
126200.,294.
</PRE> <P> defines an orthotropic material for temperature T=294. Since the definition includes values for only one temperature, they are valid for all temperatures. <P> <P><P> <BR> Example files: aniso, beampo1. <P> </body></html>