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neohooke_der.tex

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  • Forked from agnumpde / dune-elasticity
    586 commits behind the upstream repository.
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    Oliver Sander authored and sander committed
    [[Imported from SVN: r9296]]
    c0589e8b
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    neohooke_der.tex 3.68 KiB
    \documentclass{article}
    
    \usepackage{amsmath}
    \usepackage{amssymb}
    \usepackage{graphicx}
    \usepackage{natbib}
    \usepackage{psfrag}
    
    \graphicspath{{gfx/}}
    
    \providecommand{\parder}[2]{\frac{\partial #1}{\partial #2}}
    \providecommand{\secparder}[3]{\frac{\partial {#1}^2}{\partial #2 \partial #3}}
    \newcommand{\tr}{\operatorname{tr}}
    
    \begin{document}
    
    \section{The Derivatives of the Energy Functional of a Neo-Hookean Material}
    
    \subsection{Preliminaries: The Derivative of the Determinant of the Deformation  Gradient}
    
    Let
    \begin{equation*}
      J = \det F = \det \nabla (X + u) = \det (I + \nabla u) 
      = \det (I + \frac{\partial u_i}{\partial X^j}).
    \end{equation*}
    The deformation $u$ is a finite element function $u = \sum_{i,j} u_i^j \phi_i^j$.
    
    \subsubsection{In Two Space Dimensions}
    
    \begin{eqnarray*}
      \parder{J}{u_i^j}
      & = & \parder{}{u_i^j}\det (I + \frac{\partial u_i}{\partial X^j}) \\
      & = & \parder{}{u_i^j} \det 
      \begin{bmatrix}
        1 + \sum_k u^0_k \parder{\phi^0_k}{X^0} & \sum_k u^1_k \parder{\phi^1_k}{X^0}  \\
        \sum_k u^0_k \parder{\phi^0_k}{X^1} & 1 + \sum_k u^1_k \parder{\phi^1_k}{X^1}
      \end{bmatrix} \\
      & = & \parder{}{u_i^j}
      [ ( 1 + \sum_k u^0_k \parder{\phi^0_k}{X^0} ) (1 + \sum_k u^1_k \parder{\phi^1_k}{X^1})
      - (\sum_k u^1_k \parder{\phi^1_k}{X^0}) (\sum_k u^0_k \parder{\phi^0_k}{X^1}) ]
    \end{eqnarray*}
    Thus,
    \begin{eqnarray*}
      \parder{J}{u_i^0}
      & = &
      \parder{\phi_i^0}{X^0} \Big( 1 + \sum_k u^1_k \parder{\phi^1_k}{X^1} \Big)
      -
      \Big(\sum_k u^1_k \parder{\phi^1_k}{X^0} \Big) \parder{\phi_i^0}{X^1} \\
      & = &
      \parder{\phi_i^0}{X^0} \Big( 1 + \parder{u^1_k}{X^1} \Big)
      -
      \parder{u^1_k}{X^0} \parder{\phi_i^0}{X^1} \\
      %
      \parder{J}{u_i^1}
      & = &
      \parder{\phi_i^1}{X^1} \Big( 1 + \sum_k u^0_k \parder{\phi^0_k}{X^0} \Big)
      -
      \Big(\sum_k u^0_k \parder{\phi^0_k}{X^1} \Big) \parder{\phi_i^1}{X^0} \\
      & = &
      \parder{\phi_i^1}{X^1} \Big( 1 + \parder{u^0_k}{X^0} \Big)
      -
      \parder{u^0_k}{X^1} \parder{\phi_i^1}{X^0}
    \end{eqnarray*}
    The four second derivatives are
    \begin{eqnarray*}
      \secparder{J}{u_i^0}{u_j^0}
      & = & 0 \\
      %
     \secparder{J}{u_i^0}{u_j^1}
      & = &
      \parder{\phi_i^0}{X^0} \parder{\phi^1_j}{X^1} 
      - \parder{\phi^1_j}{X^0} \parder{\phi^0_i}{X^1}\\
      % 
     \secparder{J}{u_i^1}{u_j^0}
      & = &
      \parder{\phi^0_j}{X^0} \parder{\phi^1_i}{X^1}
      - \parder{\phi_i^1}{X^0} \parder{\phi^0_j}{X^1} \\
      % 
     \secparder{J}{u_i^1}{u_j^1}
      & = & 0
    \end{eqnarray*}
    
    \subsection{The Derivatives of $\tr E$}
    
    \begin{eqnarray*}
      \tr E = \frac 12 \tr (\nabla u + \nabla^T u + \nabla^T u \nabla u)
      = \tr \nabla u + \frac 12 \tr  \nabla^T u \nabla u
    \end{eqnarray*}
    \subsection{First Derivatives of $W$}
    
    \begin{equation*}
      W(u) = \frac{\lambda}{4} ( J^2 -1 ) - (\frac \lambda 2 + \mu) \ln J + \mu \tr E
    \end{equation*}
    
    Thus,
    \begin{eqnarray*}
      \parder{W}{u_i^j}
      & = &
       \frac{\lambda}{4} \parder{}{u_i^j} J^2 
       - (\frac \lambda 2 + \mu) \parder{}{u_i^j} \ln J 
       + \mu \parder{}{u_i^j} \tr E \\
      %
       & = &
       \frac{\lambda J}{2} \parder{J}{u_i^j}
       - (\frac \lambda 2 + \mu) J^{-1} \parder{J}{u_i^j}
       + \mu \tr \parder{}{u_i^j} E
    \end{eqnarray*}
    
    \subsection{Second Derivatives of $W$}
    
    \begin{eqnarray*}
      \secparder{W}{u_i^j}{u_k^l}
      & = &
      \parder{}{u_k^l} \Big[ \frac{\lambda J}{2} \parder{J}{u_i^j}
      - (\frac \lambda 2 + \mu) J^{-1} \parder{J}{u_i^j}
      + \mu \tr \parder{}{u_i^j} E \Big] \\
      % 
      & = & 
      \frac{\lambda}{2} \Bigg[ \parder{J}{u_i^j} \parder{J}{u_k^l} 
      + J \secparder{J}{u_i^j}{u_k^l} \Bigg] \\
      & &
      - (\frac \lambda 2 + \mu) J^{-2} 
      \Bigg[ \secparder{J}{u_i^j}{u_k^l} J - \parder{J}{u_i^j} \parder{J}{u_k^l} \Bigg]
      + \mu \tr \secparder{}{u_i^j}{u_k^l} E
    \end{eqnarray*}
    \end{document}
    
    %%% Local Variables: 
    %%% mode: latex
    %%% TeX-master: t
    %%% End: