notations.tex

%%% Packages for notations
\usepackage{amsmath}
\usepackage{upgreek}

%%% Notations not in the table
% Slanted leq and geq 
\let\leq=\leqslant
\let\geq=\geqslant

%%% Spaces
% Real space
\newcommand{\RR}{\mathbb{R}}
% Orthogonal group
\newcommand{\OO}{\mathrm{O}}
% Cyclic group
\newcommand{\ZZ}{\mathbb{Z}}
% Dihedral group
\newcommand{\DD}{\mathbb{D}}

%%% Operators
% Deviatoric part
\newcommand{\dev}[1]{#1'}

%%% Constants
% Zeros
\newcommand{\zero}{\mathbf{0}}
% Identities
\newcommand{\id}[1]{%
\ifnum2=#1\relax
  \mathbf{1}
\else
  \ifnum4=#1\relax
    \mathbf{I}
  \else
    \textcolor{red}{\text{ERROR}}
  \fi
\fi}
% Deviatoric projector
\newcommand{\J}{\mathbf{J}}

%%% Continuous model
% Damage
\newcommand{\dam}{\mathbf{D}}
% Elasticity tensor
\newcommand{\ela}{\mathbf{E}}
% Effective elasticity tensor
\newcommand{\elaeff}{\tilde{\mathbf{E}}}

%%% Harmonic decomposition
% Dilatation tensor
\newcommand{\dil}{\mathbf{d}}
% Harmonic part
\newcommand{\har}{\mathbf{H}}

%%% List of Symbols
\newcommand{\listnotationsname}{List of Symbols}
\newcommand{\listofnotations}{
    \chapter*{\listnotationsname}%
    \label{cha:list_of_symbols}

        \subsection*{Spaces}%
        \label{sec:Spaces}
        \vspace{-1em}
        \begin{longtable}{p{0.1\textwidth} p{0.85\textwidth}}
            \( \RR \)               & Real space                                                                            \\[0.5em]
            \( \ZZ_{n}  \)          & Cyclic group of order $n$                                                             \\[0.5em]
            \( \DD_{n}  \)          & Dihedral group of order $2n$ (group of $n$-gon symmetries)                            \\[0.5em]
            \( \OO(n) \)            & Orthogonal group of dimension $d$                                                     \\[0.5em]
        \end{longtable}
        \vspace{-2em}

        \subsection*{Operators}%
        \label{sec:Operators}
        \vspace{-1em}
        \begin{longtable}{p{0.1\textwidth} p{0.85\textwidth}}
            \( \dev{\bullet} \)     & Deviatoric part of second-order tensor $\bullet$                                      \\[0.5em]
            \( \otimes \)           & Tensor product                                                                        \\[0.5em]
        \end{longtable}
        \vspace{-3em}

        \subsection*{Constants}%
        \label{sec:Constants}
        \vspace{-1em}
        \begin{longtable}{p{0.1\textwidth} p{0.85\textwidth}}
            \( \zero \)             & Null tensor                                                                           \\[0.5em]
            \( \id{2} \)            & Second-order identity tensor                                                          \\[0.5em]
            \( \id{4} \)            & Fourth-order identity tensor                                                          \\[0.5em]
            \( \J \)                & Deviatoric projector (4th-order tensor)                                               \\[0.5em]
        \end{longtable}
        \vspace{-2em}

        \subsection*{Continuous variables}%
        \label{sec:Continuous variables}
        \vspace{-1em}
        \begin{longtable}{p{0.1\textwidth} p{0.85\textwidth}}
            \( \dam \)              & Damage tensor                                                                         \\[0.5em]
            \( \ela \)              & Elasticity tensor                                                                     \\[0.5em]
            \( \ela_0 \)            & Initial elasticity tensor                                                             \\[0.5em]
            \( \elaeff(\dam) \)     & Effective elasticity tensor                                                           \\[0.5em]
        \end{longtable}
        \vspace{-2em}

        \subsection*{Harmonic decomposition}
        \label{sec:hamronic_decomposition}
        \vspace{-1em}
        \begin{longtable}{p{0.1\textwidth} p{0.85\textwidth}}
            \( \mu \)               & Shear modulus                                                                         \\[0.5em]
            \( \kappa \)            & Bulk modulus                                                                          \\[0.5em]
            \( \dil \)              & Dilatation tensor                                                                     \\[0.5em]
            \( \har \)              & Harmonic part                                                                         \\[0.5em]
        \end{longtable}
}