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\title{Computing energy release rates using the Virtual Crack Closure Technique with the Finite Element Method: analytical discussion}

\author{<NAME>$^{1,2,a}$ , <NAME>$^{2,b}$ and <NAME>$^{1,c}$ }

\address{$^{1}$SI2M, IJL, EEIGM, Universit\'e de Lorraine, 6 Rue Bastien Lepage, F-54010 Nancy, France\\$^{2}$Division of Polymer Engineering, Lule\aa\ University of Technology, SE-97187 Lule\aa , Sweden }

{\vspace*{5pt}\address{E-mail: $^{a}$<EMAIL>, $^{b}$<EMAIL>, $^{c}$<EMAIL>}}
%\address{$^{a}$<EMAIL>}, \address{$^{b}$<EMAIL>}, \address{$^{c}$<EMAIL>}

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\begin{abstract}
The effect of debond size and crack tip orientation of mode splitting in the Virtual Crack Closure Technique is analyzed by means of analytical derivations. The total energy release rate is shown to have no direct dependence on the debond angular size, but only an indirect one through the FEM solution of the crack displacement field in the crack tip neighbourhood.
\end{abstract}

%------------------------------------------------------------------------------------------%
%                                   List of acronyms
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\section*{List of acronyms}

\begin{tabular}{ll}
VCCT &  Virtual Crack Closure Technique\\
BEM &  Boundary Element Method\\
FEM &  Finite Element Method\\
\end{tabular}
%\end{table}

%------------------------------------------------------------------------------------------%
%                                   List of symbols
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\section*{List of symbols}

%\begin{table}[!h]
\begin{tabular}{lcl}
$G_{I}$ & $\left[\frac{J}{m^{2}}\right]$  & Mode I energy release rate\\
$G_{II}$ & $\left[\frac{J}{m^{2}}\right]$  & Mode II energy release rate\\
$G_{TOT}$ & $\left[\frac{J}{m^{2}}\right]$  & Total energy release rate\\
$G_{I,r\theta}$ & $\left[\frac{J}{m^{2}}\right]$  & Mode I energy release rate in $r-\theta$ reference frame\\
$G_{II,r\theta}$ & $\left[\frac{J}{m^{2}}\right]$  & Mode II energy release rate in $r-\theta$ reference frame\\
$G_{TOT,r\theta}$ & $\left[\frac{J}{m^{2}}\right]$  & Total energy release rate in $r-\theta$ reference frame\\
$\widetilde{G}_{I,xy}$ & $\left[\frac{J}{m^{2}}\right]$  & Mode I energy release rate of equivalent crack in $x-y$ reference frame\\
$\widetilde{G}_{II,xy}$ & $\left[\frac{J}{m^{2}}\right]$  & Mode II energy release rate of equivalent crack in $x-y$ reference frame\\
$\widetilde{G}_{TOT,xy}$ & $\left[\frac{J}{m^{2}}\right]$  & Total energy release rate of equivalent crack in $x-y$ reference frame\\
$R_{f}$ & $\left[\mu m\right]$  &Fiber radius\\
$a$ & $\left[\mu m\right]$  &Debond size\\
$\Delta a$ & $\left[\mu m\right]$  &Debond increment\\
$\Delta\theta$ & $\left[rad\right]$  & Half debond angular size\\
$\delta$ & $\left[rad\right]$  & Angular size of element at the interface close to the crack tip\\
$u_{x,[A-Z]}$ & $\left[\mu m\right]$  & Displacement along $x$ of a point labeled with a letter in [A-Z]\\
$u_{y,[A-Z]}$ & $\left[\mu m\right]$  & Displacement along $y$ of a point labeled with a letter in [A-Z]\\
$u_{x}$ & $\left[\mu m\right]$  & Displacement along $x$-direction\\
$u_{y}$ & $\left[\mu m\right]$  & Displacement along $y$-direction\\
$u_{r}$ & $\left[\mu m\right]$  & Displacement along $r$-direction\\
$u_{\theta}$ & $\left[\mu m\right]$  & Displacement along $\theta$-direction\\
$F_{x,[A-Z]}$ & $\left[\mu m\right]$  & Force along $x$ at a point labeled with a letter in [A-Z]\\
$F_{y,[A-Z]}$ & $\left[\mu m\right]$  & Force along $y$ at a point labeled with a letter in [A-Z]\\
$F_{x}$ & $\left[\mu m\right]$  & Force along $x$-direction\\
$F_{y}$ & $\left[\mu m\right]$  & Force along $y$-direction\\
$F_{r}$ & $\left[\mu m\right]$  & Force along $r$-direction\\
$F_{\theta}$ & $\left[\mu m\right]$  & Force along $\theta$-direction\\
$\underline{\underline{R}}$ & $\left[-\right]$  & Rotation matrix\\
\end{tabular}
%\end{table}

\clearpage
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%                        FEM formulation with quadrilateral elements
%------------------------------------------------------------------------------------------%
\section{ FEM formulation with quadrilateral elements}

\begin{equation}
\Pi\left(u_{i}\right)=\int_{V}\sigma_{ij}\varepsilon_{ij}dV+\int_{V}\rho\ddot{u}_{i}u_{i}dV-\int_{V}F^{T}_{i}u_{i}dV-\int_{S}f^{T}_{i}u_{i}dS
\end{equation}

\begin{equation}
\Pi\left(u_{i}\right)=\int_{V}\sigma_{ij}\varepsilon_{ij}dV-\int_{V}F^{T}_{i}u_{i}dV-\int_{S}f^{T}_{i}u_{i}dS
\end{equation}

\begin{equation}
\varepsilon_{ij}=\frac{1}{2}\left(u_{i,j}+u_{j,i}\right)
\end{equation}

\begin{equation}
\underline{\varepsilon}\left(x,y\right)=\underline{\underline{\widetilde{B}}}\cdot\underline{u}\left(x,y\right)
\end{equation}

\begin{equation}
\underline{\widetilde{B}}=\begin{bmatrix}
\frac{\partial }{\partial x}&0\\[7.5pt]
0&\frac{\partial }{\partial y}\\[7.5pt]
\frac{\partial }{\partial y}&\frac{\partial }{\partial x}\\
\end{bmatrix}
\end{equation}

%\begin{equation}
%\underline{\varepsilon}\left(x,y\right)=\underline{\underline{\widetilde{B}}}\cdot\underline{u}\left(x,y\right)=\underline{\underline{\widetilde{B}}}\cdot\underline{\underline{N}}\cdot\underline{u}_{N}=\underline{\underline{B}}\cdot\underline{u}_{N}
%\end{equation}

\begin{equation}
\underline{\sigma}=\underline{\underline{D}}\cdot\underline{\varepsilon}
\end{equation}

\begin{equation}
\underline{\sigma}=\begin{bmatrix}
\sigma_{xx}\\\sigma_{yy}\\\tau_{xy}
\end{bmatrix}\qquad\underline{\varepsilon}=\begin{bmatrix}
\varepsilon_{xx}\\\varepsilon_{yy}\\\gamma_{xy}
\end{bmatrix}
\end{equation}

\begin{equation}
\underline{\underline{D}}=\begin{bmatrix}
E_{1}&E_{2}&0\\E_{2}&E_{1}&0\\0&0&G
\end{bmatrix}\qquad\text{with }G=\frac{E}{2\left(1+\nu\right)}\text{ for an isotropic material}
\end{equation}

\begin{equation}
\begin{split}
E_{1}=\frac{E}{1-\nu^{2}}\quad E_{2}=\nu E_{1}\quad&\text{for plane stress}\\
E_{1}=\frac{E\left(1-\nu\right)}{\left(1+\nu\right)\left(1-2\nu\right)}\quad E_{2}=\frac{\nu E_{1}}{1-\nu}\quad&\text{for plane strain}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
\Pi\left(\underline{u}\right)=&\frac{1}{2}\int_{V}\underline{\underline{\varepsilon}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\varepsilon}}dV-\int_{V}\underline{F}^{T}\underline{u}dV-\int_{S}\underline{f}^{T}\underline{u}dS=\\
&=\frac{1}{2}\int_{V}\underline{u}^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\underline{u}dV-\int_{V}\underline{F}^{T}\underline{u}dV-\int_{S}\underline{f}^{T}\underline{u}dS
\end{split}
\end{equation}

\begin{equation}
\delta\Pi\left(\delta\underline{u}\right)=0
\end{equation}

\begin{equation}
\begin{split}
\delta\Pi\left(\delta\underline{u}\right)=&\Pi\left(\underline{u}+\delta\underline{u}\right)-\Pi\left(\underline{u}\right)=\\=&\frac{1}{2}\int_{V}\left(\underline{u}+\underline{\delta u}\right)^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\left(\underline{u}+\underline{\delta u}\right)dV-\int_{V}\underline{F}^{T}\left(\underline{u}+\underline{\delta u}\right)dV-\int_{S}\underline{f}^{T}\left(\underline{u}+\underline{\delta u}\right)dS+\\
-&\frac{1}{2}\int_{V}\underline{u}^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\underline{u}dV+\int_{V}\underline{F}^{T}\underline{u}dV+\int_{S}\underline{f}^{T}\underline{u}dS=\\
=&\cancel{\frac{1}{2}\int_{V}\underline{u}^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\underline{u}dV}+\cancelto{\approx 0}{\frac{1}{2}\int_{V}\underline{\delta u}^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\underline{\delta u}dV}+\\
+&\frac{1}{2}\int_{V}\underline{u}^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\underline{\delta u}dV+\frac{1}{2}\int_{V}\underline{\delta u}^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\underline{u}dV+\\
-&\cancel{\int_{V}\underline{F}^{T}\underline{u}dV}-\int_{V}\underline{F}^{T}\underline{\delta u}dV+\\
-&\cancel{\int_{S}\underline{f}^{T}\underline{u}dS}-\int_{S}\underline{f}^{T}\underline{\delta u}dS+\\
-&\cancel{\frac{1}{2}\int_{V}\underline{u}^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\underline{u}dV}+\cancel{\int_{V}\underline{F}^{T}\underline{u}dV}+\cancel{\int_{S}\underline{f}^{T}\underline{u}dS}=\\
=&\int_{V}\underline{u}^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\underline{\delta u}dV-\int_{V}\underline{F}^{T}\underline{\delta u}dV-\int_{S}\underline{f}^{T}\underline{\delta u}dS
\end{split}
\end{equation}

\begin{equation}
\int_{V}\underline{\delta u}^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\underline{u}dV-\int_{V}\underline{\delta u}^{T}\underline{F}dV-\int_{S}\underline{\delta u}^{T}\underline{f}dS=0
\end{equation}

%\begin{equation}
%\underline{\underline{k}}\cdot\underline{u}_{N}=\underline{f}_{N}
%\end{equation}

\begin{equation}
\underline{u}=\begin{bmatrix}
u_{x}\left(x,y\right)\\u_{y}\left(x,y\right)
\end{bmatrix}\qquad\underline{u}_{N}=\begin{bmatrix}
u_{1,x}\\u_{1,y}\\u_{2,x}\\u_{2,y}\\u_{3,x}\\u_{3,y}\\u_{4,x}\\u_{4,y}\\
\end{bmatrix}\text{ or }\underline{u}_{N}=\begin{bmatrix}
u_{1,x}\\u_{1,y}\\u_{2,x}\\u_{2,y}\\u_{3,x}\\u_{3,y}\\u_{4,x}\\u_{4,y}\\u_{5,x}\\u_{5,y}\\u_{6,x}\\u_{6,y}\\u_{7,x}\\u_{7,y}\\u_{8,x}\\u_{8,y}\\
\end{bmatrix}
\end{equation}

%\qquad\underline{f}_{N}=\begin{bmatrix}
%f_{1,x}\\f_{1,y}\\f_{2,x}\\f_{2,y}\\f_{3,x}\\f_{3,y}\\f_{4,x}\\f_{4,y}\\
%\end{bmatrix}

\begin{equation}
\underline{u}=\underline{\underline{N}}\cdot\underline{u}_{N}
\end{equation}

\begin{equation}
\underline{\underline{N}}=\begin{bmatrix}
N_{1}&0&N_{2}&0&N_{3}&0&N_{4}&0\\
0&N_{1}&0&N_{2}&0&N_{3}&0&N_{4}\\
\end{bmatrix}
\end{equation}

\setcounter{MaxMatrixCols}{20}
\begin{equation}
\underline{\underline{N}}=\begin{bmatrix}
N_{1} & 0     & N_{2} & 0     & N_{3} & 0     & N_{4} & 0     & N_{5} & 0     & N_{6} & 0     & N_{7} & 0     & N_{8} & 0    \\
0     & N_{1} & 0     & N_{2} & 0     & N_{3} & 0     & N_{4} & 0     & N_{5} & 0     & N_{6} & 0     & N_{7} & 0     & N_{8} \\
\end{bmatrix}
\end{equation}


\begin{equation}
\begin{cases}
N_{1}=N_{1}\left(\xi,\eta\right)\\
N_{2}=N_{2}\left(\xi,\eta\right)\\
N_{3}=N_{3}\left(\xi,\eta\right)\\
N_{4}=N_{4}\left(\xi,\eta\right)\\
\end{cases}\quad\text{with}\quad\begin{cases}
\xi=\xi\left(x,y\right)\\
\eta=\eta\left(x,y\right)\\
\end{cases}\text{for isoparametric elements}
\end{equation}

\begin{equation}
\underline{\underline{B}}=\underline{\underline{\widetilde{B}}}\cdot\underline{\underline{N}}=\begin{bmatrix}
\frac{\partial N_{1}}{\partial x}&0&\frac{\partial N_{2}}{\partial x}&0&\frac{\partial N_{3}}{\partial x}&0&\frac{\partial N_{4}}{\partial x}&0\\[7.5pt]
0&\frac{\partial N_{1}}{\partial y}&0&\frac{\partial N_{2}}{\partial y}&0&\frac{\partial N_{3}}{\partial y}&0&\frac{\partial N_{4}}{\partial y}\\[7.5pt]
\frac{\partial N_{1}}{\partial y}&\frac{\partial N_{1}}{\partial x}&\frac{\partial N_{2}}{\partial y}&\frac{\partial N_{2}}{\partial x}&\frac{\partial N_{3}}{\partial y}&\frac{\partial N_{3}}{\partial x}&\frac{\partial N_{4}}{\partial y}&\frac{\partial N_{4}}{\partial x}\\
\end{bmatrix}
\end{equation}

\setcounter{MaxMatrixCols}{20}
\begin{equation}
\begin{split}
\underline{\underline{B}}=&\underline{\underline{\widetilde{B}}}\cdot\underline{\underline{N}}=\\=&\begin{bmatrix}
\frac{\partial N_{1}}{\partial x}&0&\frac{\partial N_{2}}{\partial x}&0&\frac{\partial N_{3}}{\partial x}&0&\frac{\partial N_{4}}{\partial x}&0&\frac{\partial N_{5}}{\partial x}&0&\frac{\partial N_{6}}{\partial x}&0&\frac{\partial N_{7}}{\partial x}&0&\frac{\partial N_{8}}{\partial x}&0\\[7.5pt]
0&\frac{\partial N_{1}}{\partial y}&0&\frac{\partial N_{2}}{\partial y}&0&\frac{\partial N_{3}}{\partial y}&0&\frac{\partial N_{4}}{\partial y}&0&\frac{\partial N_{5}}{\partial y}&0&\frac{\partial N_{6}}{\partial y}&0&\frac{\partial N_{7}}{\partial y}&0&\frac{\partial N_{8}}{\partial y}\\[7.5pt]
\frac{\partial N_{1}}{\partial y}&\frac{\partial N_{1}}{\partial x}&\frac{\partial N_{2}}{\partial y}&\frac{\partial N_{2}}{\partial x}&\frac{\partial N_{3}}{\partial y}&\frac{\partial N_{3}}{\partial x}&\frac{\partial N_{4}}{\partial y}&\frac{\partial N_{4}}{\partial x}&\frac{\partial N_{5}}{\partial y}&\frac{\partial N_{5}}{\partial x}&\frac{\partial N_{6}}{\partial y}&\frac{\partial N_{6}}{\partial x}&\frac{\partial N_{7}}{\partial y}&\frac{\partial N_{7}}{\partial x}&\frac{\partial N_{8}}{\partial y}&\frac{\partial N_{8}}{\partial x}\\
\end{bmatrix}
\end{split}
\end{equation}

\begin{equation}
\delta\underline{u}=\delta\left(\underline{\underline{N}}\cdot\underline{u}_{N}\right)=\underline{\underline{N}}\delta\underline{u}_{N}
\end{equation}

\begin{equation}
\int_{V}\underline{\delta u_{N}}^{T}\underline{\underline{N}}^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\underline{\underline{N}}\cdot\underline{u_{N}}dV-\int_{V}\underline{\delta u_{N}}^{T}\underline{\underline{N}}^{T}\underline{F}dV-\int_{S}\underline{\delta u_{N}}^{T}\underline{\underline{N}}^{T}\underline{f}dS=0
\end{equation}

\begin{equation}
\underline{\delta u_{N}}^{T}\left(\int_{V}\underline{\underline{B}}^{T}\underline{\underline{D}}\cdot\underline{\underline{B}}dV\cdot\underline{u_{N}}-\int_{V}\underline{\underline{N}}^{T}\underline{F}dV-\int_{S}\underline{\underline{N}}^{T}\underline{f}dS\right)=0
\end{equation}

\begin{equation}
\underline{\underline{k}}\cdot\underline{u}_{N}=\underline{F}_{N}\quad\underline{\underline{k}}=\int_{V}\left(\underline{\underline{B}}^{T}\underline{\underline{D}}\cdot\underline{\underline{B}}\right)dV\quad\underline{F}_{N}=\int_{V}\underline{\underline{N}}^{T}\underline{F}dV+\int_{S}\underline{\underline{N}}^{T}\underline{f}dS
\end{equation}

\begin{equation}
\begin{cases}
N_{1}\left(\xi,\eta\right)=\frac{1}{4}\left(1-\xi\right)\left(1-\eta\right)\\
N_{2}\left(\xi,\eta\right)=\frac{1}{4}\left(1+\xi\right)\left(1-\eta\right)\\
N_{3}\left(\xi,\eta\right)=\frac{1}{4}\left(1+\xi\right)\left(1+\eta\right)\\
N_{4}\left(\xi,\eta\right)=\frac{1}{4}\left(1-\xi\right)\left(1+\eta\right)\\
\end{cases}\qquad\begin{cases}
N_{1}\left(\xi,\eta\right)=\frac{1}{4}\left(1-\xi\right)\left(1-\eta\right)\left(-\xi-\eta-1\right)\\
N_{2}\left(\xi,\eta\right)=\frac{1}{4}\left(1+\xi\right)\left(1-\eta\right)\left(\xi-\eta-1\right)\\
N_{3}\left(\xi,\eta\right)=\frac{1}{4}\left(1+\xi\right)\left(1+\eta\right)\left(\xi+\eta-1\right)\\
N_{4}\left(\xi,\eta\right)=\frac{1}{4}\left(1-\xi\right)\left(1+\eta\right)\left(-\xi+\eta-1\right)\\
N_{5}\left(\xi,\eta\right)=\frac{1}{2}\left(1-\xi^{2}\right)\left(1-\eta\right)\\
N_{6}\left(\xi,\eta\right)=\frac{1}{2}\left(1+\xi\right)\left(1-\eta^{2}\right)\\
N_{7}\left(\xi,\eta\right)=\frac{1}{2}\left(1-\xi^{2}\right)\left(1+\eta\right)\\
N_{8}\left(\xi,\eta\right)=\frac{1}{2}\left(1-\xi\right)\left(1-\eta^{2}\right)\\
\end{cases}
\end{equation}

\begin{equation}
\begin{cases}
\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \xi} =-\frac{1}{4}\left(1-\eta\right)\\
\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \xi} = \frac{1}{4}\left(1-\eta\right)\\
\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \xi} = \frac{1}{4}\left(1+\eta\right)\\
\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \xi} =-\frac{1}{4}\left(1+\eta\right)\\
\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \eta}=-\frac{1}{4}\left(1-\xi\right)\\
\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \eta}=-\frac{1}{4}\left(1+\xi\right)\\
\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \eta}= \frac{1}{4}\left(1+\xi\right)\\
\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \eta}= \frac{1}{4}\left(1-\xi\right)\\
\end{cases}
\end{equation}

\begin{equation}
\begin{cases}
\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \xi}=\frac{1}{4}\left(1-\eta\right)\left(2\xi+\eta\right)\\
\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \xi}=\frac{1}{4}\left(1-\eta\right)\left(2\xi-\eta\right)\\
\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \xi}=\frac{1}{4}\left(1+\eta\right)\left(2\xi+\eta\right)\\
\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \xi}=\frac{1}{4}\left(1+\eta\right)\left(2\xi-\eta\right)\\
\frac{\partial N_{5}\left(\xi,\eta\right)}{\partial \xi}=-\xi\left(1-\eta\right)\\
\frac{\partial N_{6}\left(\xi,\eta\right)}{\partial \xi}=\frac{1}{2}\left(1-\eta^{2}\right)\\
\frac{\partial N_{7}\left(\xi,\eta\right)}{\partial \xi}=-\xi\left(1+\eta\right)\\
\frac{\partial N_{8}\left(\xi,\eta\right)}{\partial \xi}=-\frac{1}{2}\left(1-\eta^{2}\right)\\
\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \eta}=\frac{1}{4}\left(1-\xi\right)\left(2\eta+\xi\right)\\
\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \eta}=\frac{1}{4}\left(1+\xi\right)\left(2\eta-\xi\right)\\
\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \eta}=\frac{1}{4}\left(1+\xi\right)\left(2\eta+\xi\right)\\
\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \eta}=\frac{1}{4}\left(1-\xi\right)\left(2\eta-\xi\right)\\
\frac{\partial N_{5}\left(\xi,\eta\right)}{\partial \eta}=-\frac{1}{2}\left(1-\xi^{2}\right)\\
\frac{\partial N_{6}\left(\xi,\eta\right)}{\partial \eta}=-\eta\left(1+\xi\right)\\
\frac{\partial N_{7}\left(\xi,\eta\right)}{\partial \eta}=\frac{1}{2}\left(1-\xi^{2}\right)\\
\frac{\partial N_{8}\left(\xi,\eta\right)}{\partial \eta}=-\eta\left(1-\xi\right)\\
\end{cases}
\end{equation}

\begin{equation}
\underline{p}=\underline{\underline{N}}\cdot\underline{p}_{N}
\end{equation}

\begin{equation}
\underline{p}=\begin{bmatrix}
x\\y
\end{bmatrix}\qquad\underline{p}_{N}=\begin{bmatrix}
x_{1}\\y_{1}\\x_{2}\\y_{2}\\x_{3}\\y_{3}\\x_{4}\\y_{4}\\
\end{bmatrix}\text{ or }\underline{p}_{N}=\begin{bmatrix}
x_{1}\\y_{1}\\x_{2}\\y_{2}\\x_{3}\\y_{3}\\x_{4}\\y_{4}\\x_{5}\\y_{5}\\x_{6}\\y_{6}\\x_{7}\\y_{7}\\x_{8}\\y_{8}\\
\end{bmatrix}
\end{equation}

\begin{equation}
\begin{split}
x = x\left(\xi,\eta\right) = N_{1}\left(\xi,\eta\right)x_{1}+N_{2}\left(\xi,\eta\right)x_{2}+N_{3}\left(\xi,\eta\right)x_{3}+N_{4}\left(\xi,\eta\right)x_{4}\\
y = y\left(\xi,\eta\right) = N_{1}\left(\xi,\eta\right)y_{1}+N_{2}\left(\xi,\eta\right)y_{2}+N_{3}\left(\xi,\eta\right)y_{3}+N_{4}\left(\xi,\eta\right)y_{4}\\
\end{split}
\end{equation}

\begin{equation}
\begin{split}
x = x\left(\xi,\eta\right) =& N_{1}\left(\xi,\eta\right)x_{1}+N_{2}\left(\xi,\eta\right)x_{2}+N_{3}\left(\xi,\eta\right)x_{3}+N_{4}\left(\xi,\eta\right)x_{4}+\\
& +N_{5}\left(\xi,\eta\right)x_{5}+N_{6}\left(\xi,\eta\right)x_{6}+N_{7}\left(\xi,\eta\right)x_{7}+N_{8}\left(\xi,\eta\right)x_{8}\\
y = y\left(\xi,\eta\right) =& N_{1}\left(\xi,\eta\right)y_{1}+N_{2}\left(\xi,\eta\right)y_{2}+N_{3}\left(\xi,\eta\right)y_{3}+N_{4}\left(\xi,\eta\right)y_{4}+\\
& +N_{5}\left(\xi,\eta\right)y_{5}+N_{6}\left(\xi,\eta\right)y_{6}+N_{7}\left(\xi,\eta\right)y_{7}+N_{8}\left(\xi,\eta\right)y_{8}\\
\end{split}
\end{equation}

\begin{equation}
\begin{cases}
\frac{\partial x}{\partial \xi} = \frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \xi}x_{1}+\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \xi}x_{2}+\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \xi}x_{3}+\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \xi}x_{4}\\
\frac{\partial x}{\partial \eta} = \frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \eta}x_{1}+\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \eta}x_{2}+\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \eta}x_{3}+\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \eta}x_{4}\\
\frac{\partial y}{\partial \xi} = \frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \xi}y_{1}+\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \xi}y_{2}+\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \xi}y_{3}+\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \xi}y_{4}\\
\frac{\partial y}{\partial \eta} = \frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \eta}y_{1}+\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \eta}y_{2}+\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \eta}y_{3}+\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \eta}y_{4}\\
\end{cases}
\end{equation}

\begin{equation}
\begin{cases}
\frac{\partial x}{\partial \xi} = & \frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \xi}x_{1}+\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \xi}x_{2}+\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \xi}x_{3}+\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \xi}x_{4}+\\
& +\frac{\partial N_{5}\left(\xi,\eta\right)}{\partial \xi}x_{5}+\frac{\partial N_{6}\left(\xi,\eta\right)}{\partial \xi}x_{6}+\frac{\partial N_{7}\left(\xi,\eta\right)}{\partial \xi}x_{7}+\frac{\partial N_{8}\left(\xi,\eta\right)}{\partial \xi}x_{8}\\
\frac{\partial x}{\partial \eta} = & \frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \eta}x_{1}+\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \eta}x_{2}+\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \eta}x_{3}+\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \eta}x_{4}+\\
& +\frac{\partial N_{5}\left(\xi,\eta\right)}{\partial \eta}x_{5}+\frac{\partial N_{6}\left(\xi,\eta\right)}{\partial \eta}x_{6}+\frac{\partial N_{7}\left(\xi,\eta\right)}{\partial \eta}x_{7}+\frac{\partial N_{8}\left(\xi,\eta\right)}{\partial \eta}x_{8}\\
\frac{\partial y}{\partial \xi} = & \frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \xi}y_{1}+\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \xi}y_{2}+\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \xi}y_{3}+\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \xi}y_{4}+\\
& +\frac{\partial N_{5}\left(\xi,\eta\right)}{\partial \xi}y_{5}+\frac{\partial N_{6}\left(\xi,\eta\right)}{\partial \xi}y_{6}+\frac{\partial N_{7}\left(\xi,\eta\right)}{\partial \xi}y_{7}+\frac{\partial N_{8}\left(\xi,\eta\right)}{\partial \xi}y_{8}\\
\frac{\partial y}{\partial \eta} = & \frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \eta}y_{1}+\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \eta}y_{2}+\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \eta}y_{3}+\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \eta}y_{4}+\\
& +\frac{\partial N_{5}\left(\xi,\eta\right)}{\partial \eta}y_{5}+\frac{\partial N_{6}\left(\xi,\eta\right)}{\partial \eta}y_{6}+\frac{\partial N_{7}\left(\xi,\eta\right)}{\partial \eta}y_{7}+\frac{\partial N_{8}\left(\xi,\eta\right)}{\partial \eta}y_{8}\\
\end{cases}
\end{equation}

\begin{equation}
\begin{cases}
\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial x}=\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \xi}\frac{\partial \xi}{\partial x}+\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \eta}\frac{\partial \eta}{\partial x}\\
\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial x}=\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \xi}\frac{\partial \xi}{\partial x}+\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \eta}\frac{\partial \eta}{\partial x}\\
\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial x}=\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \xi}\frac{\partial \xi}{\partial x}+\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \eta}\frac{\partial \eta}{\partial x}\\
\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial x}=\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \xi}\frac{\partial \xi}{\partial x}+\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \eta}\frac{\partial \eta}{\partial x}\\
\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial y}=\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \xi}\frac{\partial \xi}{\partial y}+\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \eta}\frac{\partial \eta}{\partial y}\\
\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial y}=\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \xi}\frac{\partial \xi}{\partial y}+\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \eta}\frac{\partial \eta}{\partial y}\\
\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial y}=\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \xi}\frac{\partial \xi}{\partial y}+\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \eta}\frac{\partial \eta}{\partial y}\\
\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial y}=\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \xi}\frac{\partial \xi}{\partial y}+\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \eta}\frac{\partial \eta}{\partial y}\\
\end{cases}
\end{equation}

\begin{equation}
\begin{bmatrix}
\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial x}&\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial y}\\
\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial x}&\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial y}\\
\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial x}&\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial y}\\
\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial x}&\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial y}\\
\end{bmatrix}=\begin{bmatrix}
\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \xi}&\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \eta}\\
\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \xi}&\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \eta}\\
\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \xi}&\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \eta}\\
\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \xi}&\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \eta}\\
\end{bmatrix}\begin{bmatrix}
\frac{\partial \xi}{\partial x}&\frac{\partial \xi}{\partial y}\\\frac{\partial \eta}{\partial x}&\frac{\partial \eta}{\partial y}
\end{bmatrix}
\end{equation}

\begin{equation}
\underline{e}_{\xi}=\begin{bmatrix}
\frac{\partial x}{\partial\xi}\\\frac{\partial y}{\partial\xi}
\end{bmatrix}\quad\underline{e}_{\eta}=\begin{bmatrix}
\frac{\partial x}{\partial\eta}\\\frac{\partial y}{\partial\eta}
\end{bmatrix}
\end{equation}

\begin{equation}
\underline{e}_{x}=\begin{bmatrix}
\frac{\partial \xi}{\partial x}\\\frac{\partial \eta}{\partial x}
\end{bmatrix}\quad\underline{e}_{y}=\begin{bmatrix}
\frac{\partial \xi}{\partial y}\\\frac{\partial \eta}{\partial y}
\end{bmatrix}
\end{equation}

\begin{equation}
\underline{\underline{J}}=\begin{bmatrix}
\underline{e}_{\xi}|\underline{e}_{\eta}
\end{bmatrix}=\begin{bmatrix}
\frac{\partial x}{\partial\xi}&\frac{\partial x}{\partial\eta}\\\frac{\partial y}{\partial\xi}&\frac{\partial y}{\partial\eta}
\end{bmatrix}\qquad\underline{\underline{J}}^{-1}=\begin{bmatrix}
\underline{e}_{x}|\underline{e}_{y}
\end{bmatrix}=\begin{bmatrix}
\frac{\partial \xi}{\partial x}&\frac{\partial \xi}{\partial y}\\\frac{\partial \eta}{\partial x}&\frac{\partial \eta}{\partial y}
\end{bmatrix}
\end{equation}

\begin{equation}
\underline{\underline{J}}^{-1}=\begin{bmatrix}
\frac{\partial x}{\partial\xi}&\frac{\partial x}{\partial\eta}\\\frac{\partial y}{\partial\xi}&\frac{\partial y}{\partial\eta}
\end{bmatrix}^{-1}=\frac{1}{\frac{\partial x}{\partial\xi}\frac{\partial y}{\partial\eta}-\frac{\partial y}{\partial\xi}\frac{\partial x}{\partial\eta}}\begin{bmatrix}
\frac{\partial y}{\partial\eta}&-\frac{\partial x}{\partial\eta}\\-\frac{\partial y}{\partial\xi}&\frac{\partial x}{\partial\xi}
\end{bmatrix}
\end{equation}

\begin{equation}
\underline{\underline{g}}=\underline{\underline{J}}^{T}\underline{\underline{J}}=\begin{bmatrix}
\frac{\partial x}{\partial\xi}&\frac{\partial y}{\partial\xi}\\\frac{\partial x}{\partial\eta}&\frac{\partial y}{\partial\eta}
\end{bmatrix}\begin{bmatrix}
\frac{\partial x}{\partial\xi}&\frac{\partial x}{\partial\eta}\\\frac{\partial y}{\partial\xi}&\frac{\partial y}{\partial\eta}
\end{bmatrix}=\begin{bmatrix}
\frac{\partial x}{\partial\xi}\frac{\partial x}{\partial\xi}+\frac{\partial y}{\partial\xi}\frac{\partial y}{\partial\xi}&\frac{\partial x}{\partial\xi}\frac{\partial x}{\partial\eta}+\frac{\partial y}{\partial\xi}\frac{\partial y}{\partial\eta}\\\frac{\partial x}{\partial\eta}\frac{\partial x}{\partial\xi}+\frac{\partial y}{\partial\eta}\frac{\partial y}{\partial\xi}&\frac{\partial x}{\partial\eta}\frac{\partial x}{\partial\eta}+\frac{\partial y}{\partial\eta}\frac{\partial y}{\partial\eta}
\end{bmatrix}
\end{equation}

\begin{equation}
\begin{split}
g=&det\left(\underline{\underline{g}}\right)=\\=&\left(\frac{\partial x}{\partial\xi}\frac{\partial x}{\partial\xi}+\frac{\partial y}{\partial\xi}\frac{\partial y}{\partial\xi}\right)\left(\frac{\partial x}{\partial\eta}\frac{\partial x}{\partial\eta}+\frac{\partial y}{\partial\eta}\frac{\partial y}{\partial\eta}\right)-\left(\frac{\partial x}{\partial\eta}\frac{\partial x}{\partial\xi}+\frac{\partial y}{\partial\eta}\frac{\partial y}{\partial\xi}\right)\left(\frac{\partial x}{\partial\xi}\frac{\partial x}{\partial\eta}+\frac{\partial y}{\partial\xi}\frac{\partial y}{\partial\eta}\right)
\end{split}
\end{equation}

\begin{equation}
dV\left(x,y\right)=\sqrt{g}dV\left(\xi,\eta\right)
\end{equation}

\begin{equation}
\begin{split}
\underline{\underline{k_{e}}}=&\int_{V_{e}\left(x,y\right)}\left(\underline{\underline{B}}^{T}\underline{\underline{D}}\cdot\underline{\underline{B}}\right)dV_{e}\left(x,y\right)=\int_{V_{e}\left(\xi,\eta\right)}\left(\underline{\underline{B}}^{T}\underline{\underline{D}}\cdot\underline{\underline{B}}\right)\sqrt{g}dV_{e}\left(\xi,\eta\right)\\
=&\int_{-1}^{1}\int_{-1}^{1}\left(\underline{\underline{B}}^{T}\underline{\underline{D}}\cdot\underline{\underline{B}}\right)\sqrt{g}d\xi d\eta\approx \sum_{i=1}^{N}\sum_{j=1}^{N}w_{i}w_{j}\left(\underline{\underline{B}}^{T}\left(\xi_{i},\eta_{j}\right)\cdot\underline{\underline{D}}\cdot\underline{\underline{B}}\left(\xi_{i},\eta_{j}\right)\sqrt{g}\right)
\end{split}
\end{equation}

\begin{equation}
k_{e}=\begin{bmatrix}
k_{e|11}&k_{e|12}&k_{e|13}&k_{e|14}&k_{e|15}&k_{e|16}&k_{e|17}&k_{e|18}\\
k_{e|12}&k_{e|22}&k_{e|23}&k_{e|24}&k_{e|25}&k_{e|26}&k_{e|27}&k_{e|28}\\
k_{e|13}&k_{e|23}&k_{e|33}&k_{e|34}&k_{e|35}&k_{e|36}&k_{e|37}&k_{e|38}\\
k_{e|14}&k_{e|24}&k_{e|34}&k_{e|44}&k_{e|45}&k_{e|46}&k_{e|47}&k_{e|48}\\
k_{e|15}&k_{e|25}&k_{e|35}&k_{e|45}&k_{e|55}&k_{e|56}&k_{e|57}&k_{e|58}\\
k_{e|16}&k_{e|26}&k_{e|36}&k_{e|46}&k_{e|56}&k_{e|66}&k_{e|67}&k_{e|68}\\
k_{e|17}&k_{e|27}&k_{e|37}&k_{e|47}&k_{e|57}&k_{e|67}&k_{e|77}&k_{e|78}\\
k_{e|18}&k_{e|28}&k_{e|38}&k_{e|48}&k_{e|58}&k_{e|68}&k_{e|78}&k_{e|88}\\
\end{bmatrix}
\end{equation}

\clearpage
%------------------------------------------------------------------------------------------%
%                        VCCT 1st order
%------------------------------------------------------------------------------------------%

\section{VCCT for first order quadrilateral elements}

\subsection{Definition of crack tip reference frame}

\begin{figure}[!h]
\includegraphics[width=\textwidth]{VCCT-linear.pdf}
\caption{Schematic representation of the discretized crack tip geometry for  $1^{st}$ order quadrilateral elements.}
\end{figure}

\begin{equation}
\underline{\underline{R}}=\begin{bmatrix}
\cos\left(\Delta\theta\right) & \sin\left(\Delta\theta\right) \\
-\sin\left(\Delta\theta\right) & \cos\left(\Delta\theta\right)
\end{bmatrix}\qquad\underline{\underline{R}}^{-1}=\underline{\underline{R}}^{T}=\begin{bmatrix}
\cos\left(\Delta\theta\right) & -\sin\left(\Delta\theta\right) \\
\sin\left(\Delta\theta\right) & \cos\left(\Delta\theta\right)
\end{bmatrix}
\end{equation}

\begin{equation}
\begin{bmatrix}
r \\
\theta
\end{bmatrix}=\underline{\underline{R}}\begin{bmatrix}
x \\
y
\end{bmatrix}\qquad\begin{bmatrix}
x \\
y
\end{bmatrix}=\underline{\underline{R}}^{-1}\begin{bmatrix}
r \\
\theta
\end{bmatrix}
\end{equation}

\subsection{Calculation of displacements and reaction forces}

\begin{equation}
u_{x}=u_{x,M}-u_{x,F}\qquad u_{y}=u_{y,M}-u_{y,F}
\end{equation}

\begin{equation}
u_{r}=\cos\left(\Delta\theta\right) u_{x}+\sin\left(\Delta\theta\right) u_{y}\qquad u_{\theta}=-\sin\left(\Delta\theta\right) u_{x}+\cos\left(\Delta\theta\right) u_{y}
\end{equation}

\begin{equation}
F_{r}=\cos\left(\Delta\theta\right) F_{x,CT}+\sin\left(\Delta\theta\right) F_{y,CT}\qquad F_{\theta}=-\sin\left(\Delta\theta\right) F_{x,CT}+\cos\left(\Delta\theta\right) F_{y,CT}
\end{equation}

\begin{equation}
\begin{cases}
&\left(k_{e,M|11}+k_{e,M|33}\right)u_{x,MCT}+\left(k_{e,M|12}+k_{e,M|34}\right)u_{y,MCT}+\\&+k_{e,M|13}u_{x,M}+k_{e,M|14}u_{y,M}+\left(k_{M|17}+k_{M|35}\right)u_{N,MC|7}+\left(k_{M|18}+k_{M|36}\right)u_{N,MC|8}+\\&+\sum_{i=5}^{6}k_{M|1i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|3i}u_{N,MB|i}+k_{M|31}u_{x,NCOI}+k_{M|32}u_{y,NCOI}=0\\[10pt]
&\left(k_{e,M|21}+k_{e,M|43}\right)u_{x,MCT}+\left(k_{e,M|22}+k_{e,M|44}\right)u_{y,MCT}+\\&+k_{e,M|23}u_{x,M}+k_{e,M|24}u_{y,M}+\left(k_{M|27}+k_{M|45}\right)u_{N,MC|7}+\left(k_{M|28}+k_{M|46}\right)u_{N,MC|8}+\\&+\sum_{i=5}^{6}k_{M|2i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|4i}u_{N,MB|i}+k_{M|41}u_{x,NCOI}+k_{M|42}u_{y,NCOI}=0\\[10pt]
&\left(k_{e,F|77}+k_{e,F|55}\right)u_{x,FCT}+\left(k_{e,F|78}+k_{e,F|56}\right)u_{y,FCT}+\\&+k_{e,F|75}u_{x,F}+k_{e,F|76}u_{y,F}+\left(k_{F|71}+k_{F|53}\right)u_{N,FC|1}+\left(k_{F|72}+k_{F|54}\right)u_{N,FC|2}+\\&+\sum_{i=2}^{3}k_{F|7i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|5i}u_{N,FB|i}+k_{F|57}u_{x,NCOI}+k_{F|58}u_{y,NCOI}=0\\[10pt]
&\left(k_{e,F|87}+k_{e,F|65}\right)u_{x,FCT}+\left(k_{e,F|88}+k_{e,F|66}\right)u_{y,FCT}+\\&+k_{e,F|85}u_{x,F}+k_{e,F|86}u_{y,F}+\left(k_{F|81}+k_{F|63}\right)u_{N,FC|1}+\left(k_{F|82}+k_{F|64}\right)u_{N,FC|2}+\\&+\sum_{i=2}^{3}k_{F|8i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|6i}u_{N,FB|i}+k_{F|67}u_{x,NCOI}+k_{F|68}u_{y,NCOI}=0\\[10pt]
&u_{x,FCT}-u_{x,MCT}-u_{x,DUMMY}=0\\
&u_{y,FCT}-u_{y,MCT}-u_{y,DUMMY}=0\\[10pt]
&u_{x,DUMMY}=0\\
&u_{y,DUMMY}=0\\
\end{cases}
\end{equation}

\begin{equation}
\begin{cases}
&\left(k_{e,M|11}+k_{e,M|33}\right)u_{x,MCT}+\left(k_{e,M|12}+k_{e,M|34}\right)u_{y,MCT}+\\&+k_{e,M|13}u_{x,M}+k_{e,M|14}u_{y,M}+\left(k_{M|17}+k_{M|35}\right)u_{N,MC|7}+\left(k_{M|18}+k_{M|36}\right)u_{N,MC|8}+\\&+\sum_{i=5}^{6}k_{M|1i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|3i}u_{N,MB|i}+k_{M|31}u_{x,NCOI}+k_{M|32}u_{y,NCOI}=0\\[10pt]
&\left(k_{e,M|21}+k_{e,M|43}\right)u_{x,MCT}+\left(k_{e,M|22}+k_{e,M|44}\right)u_{y,MCT}+\\&+k_{e,M|23}u_{x,M}+k_{e,M|24}u_{y,M}+\left(k_{M|27}+k_{M|45}\right)u_{N,MC|7}+\left(k_{M|28}+k_{M|46}\right)u_{N,MC|8}+\\&+\sum_{i=5}^{6}k_{M|2i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|4i}u_{N,MB|i}+k_{M|41}u_{x,NCOI}+k_{M|42}u_{y,NCOI}=0\\[10pt]
&\left(k_{e,F|77}+k_{e,F|55}\right)u_{x,MCT}+\left(k_{e,F|78}+k_{e,F|56}\right)u_{y,MCT}+\\&+k_{e,F|75}u_{x,F}+k_{e,F|76}u_{y,F}+\left(k_{F|71}+k_{F|53}\right)u_{N,FC|1}+\left(k_{F|72}+k_{F|54}\right)u_{N,FC|2}+\\&+\sum_{i=2}^{3}k_{F|7i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|5i}u_{N,FB|i}+k_{F|57}u_{x,NCOI}+k_{F|58}u_{y,NCOI}=0\\[10pt]
&\left(k_{e,F|87}+k_{e,F|65}\right)u_{x,MCT}+\left(k_{e,F|88}+k_{e,F|66}\right)u_{y,MCT}+\\&+k_{e,F|85}u_{x,F}+k_{e,F|86}u_{y,F}+\left(k_{F|81}+k_{F|63}\right)u_{N,FC|1}+\left(k_{F|82}+k_{F|64}\right)u_{N,FC|2}+\\&+\sum_{i=2}^{3}k_{F|8i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|6i}u_{N,FB|i}+k_{F|67}u_{x,NCOI}+k_{F|68}u_{y,NCOI}=0\\[10pt]
&u_{x,FCT}=u_{x,MCT}\\
&u_{y,FCT}=u_{y,MCT}\\[10pt]
&R_{x,DUMMY}=R_{x,FCT}=-R_{x,MCT}=F_{x,CT}\\
&R_{y,DUMMY}=R_{y,FCT}=-R_{y,MCT}=F_{y,CT}\\
\end{cases}
\end{equation}

\begin{equation}
\footnotesize
\begin{cases}
&\left(k_{e,M|11}+k_{e,M|33}+k_{e,F|77}+k_{e,F|55}\right)u_{x,MCT}+\left(k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}\right)u_{y,MCT}+\\&+k_{e,M|13}u_{x,M}+k_{e,M|14}u_{y,M}+k_{e,F|75}u_{x,F}+k_{e,F|76}u_{y,F}+\\&+\left(k_{M|31}+k_{F|57}\right)u_{x,NCOI}+\left(k_{M|32}+k_{F|58}\right)u_{y,NCOI}+\\&+\left(k_{M|17}+k_{M|35}\right)u_{N,MC|7}+\left(k_{M|18}+k_{M|36}\right)u_{N,MC|8}+\left(k_{F|71}+k_{F|53}\right)u_{N,FC|1}+\left(k_{F|72}+k_{F|54}\right)u_{N,FC|2}+\\&+\sum_{i=5}^{6}k_{M|1i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|3i}u_{N,MB|i}+\sum_{i=2}^{3}k_{F|7i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|5i}u_{N,FB|i}=0\\[10pt]
&\left(k_{e,M|21}+k_{e,M|43}+k_{e,F|87}+k_{e,F|65}\right)u_{x,MCT}+\left(k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}\right)u_{y,MCT}+\\&+k_{e,M|23}u_{x,M}+k_{e,M|24}u_{y,M}+k_{e,F|85}u_{x,F}+k_{e,F|86}u_{y,F}+\\&+\left(k_{M|41}+k_{F|67}\right)u_{x,NCOI}+\left(k_{M|42}+k_{F|68}\right)u_{y,NCOI}+\\&+\left(k_{M|27}+k_{M|45}\right)u_{N,MC|7}+\left(k_{M|28}+k_{M|46}\right)u_{N,MC|8}+\left(k_{F|81}+k_{F|63}\right)u_{N,FC|1}+\left(k_{F|82}+k_{F|64}\right)u_{N,FC|2}+\\&+\sum_{i=2}^{3}k_{F|8i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|6i}u_{N,FB|i}+\sum_{i=5}^{6}k_{M|2i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|4i}u_{N,MB|i}=0\\[10pt]
&u_{x,FCT}=u_{x,MCT}\\
&u_{y,FCT}=u_{y,MCT}\\[10pt]
&R_{x,DUMMY}=R_{x,FCT}=-R_{x,MCT}=F_{x,CT}\\
&R_{y,DUMMY}=R_{y,FCT}=-R_{y,MCT}=F_{y,CT}\\
\end{cases}
\end{equation}

\begin{equation}
\scriptsize
\begin{cases}
&u_{y,MCT}=-\frac{k_{e,M|11}+k_{e,M|33}+k_{e,F|77}+k_{e,F|55}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}u_{x,MCT}+\\
&-\frac{k_{e,M|13}u_{x,M}+k_{e,M|14}u_{y,M}+k_{e,F|75}u_{x,F}+k_{e,F|76}u_{y,F}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}+\\
&-\frac{\left(k_{M|31}+k_{F|57}\right)u_{x,NCOI}+\left(k_{M|32}+k_{F|58}\right)u_{y,NCOI}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}+\\
&-\frac{\left(k_{M|17}+k_{M|35}\right)u_{N,MC|7}+\left(k_{M|18}+k_{M|36}\right)u_{N,MC|8}+\left(k_{F|71}+k_{F|53}\right)u_{N,FC|1}+\left(k_{F|72}+k_{F|54}\right)u_{N,FC|2}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}+\\
&-\frac{\sum_{i=5}^{6}k_{M|1i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|3i}u_{N,MB|i}+\sum_{i=2}^{3}k_{F|7i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|5i}u_{N,FB|i}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\\[10pt]

&\left[\left(k_{e,M|21}+k_{e,M|43}+k_{e,F|87}+k_{e,F|65}\right)+\frac{k_{e,M|11}+k_{e,M|33}+k_{e,F|77}+k_{e,F|55}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left(k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}\right)\right]u_{x,MCT}+\\
&+\left(k_{e,M|23}-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}k_{e,M|13}\right)u_{x,M}+\\
&+\left(k_{e,M|24}-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}k_{e,M|14}\right)u_{y,M}+\\
&+\left(k_{e,F|85}-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}k_{e,M|75}\right)u_{x,F}+\\
&+\left(k_{e,F|86}-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}k_{e,M|76}\right)u_{y,F}+\\
&+\left[\left(k_{M|41}+k_{F|67}\right)-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left(k_{M|31}+k_{F|57}\right)\right]u_{x,NCOI}+\\
&+\left[\left(k_{M|42}+k_{F|68}\right)-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left(k_{M|32}+k_{F|58}\right)\right]u_{y,NCOI}+\\
&+\left(k_{M|27}+k_{M|45}\right)u_{N,MC|7}+\left(k_{M|28}+k_{M|46}\right)u_{N,MC|8}+\left(k_{F|81}+k_{F|63}\right)u_{N,FC|1}+\left(k_{F|82}+k_{F|64}\right)u_{N,FC|2}+\\
&-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left[\left(k_{M|17}+k_{M|35}\right)u_{N,MC|7}+\left(k_{M|18}+k_{M|36}\right)u_{N,MC|8}\right]+\\&
-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left[\left(k_{F|71}+k_{F|53}\right)u_{N,FC|1}+\left(k_{F|72}+k_{F|54}\right)u_{N,FC|2}\right]\\
&+\sum_{i=2}^{3}k_{F|8i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|6i}u_{N,FB|i}+\sum_{i=5}^{6}k_{M|2i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|4i}u_{N,MB|i}+\\
&-\frac{\sum_{i=5}^{6}k_{M|1i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|3i}u_{N,MB|i}+\sum_{i=2}^{3}k_{F|7i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|5i}u_{N,FB|i}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}=0\\[10pt]

&u_{x,FCT}=u_{x,MCT}\\
&u_{y,FCT}=u_{y,MCT}\\[10pt]
&R_{x,DUMMY}=R_{x,FCT}=-R_{x,MCT}=F_{x,CT}\\
&R_{y,DUMMY}=R_{y,FCT}=-R_{y,MCT}=F_{y,CT}\\
\end{cases}
\end{equation}

\begin{equation}
\scriptsize
\begin{cases}
&u_{y,MCT}=-\frac{k_{e,M|11}+k_{e,M|33}+k_{e,F|77}+k_{e,F|55}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}u_{x,MCT}+\\
&-\frac{k_{e,M|13}u_{x,M}+k_{e,M|14}u_{y,M}+k_{e,F|75}u_{x,F}+k_{e,F|76}u_{y,F}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}+\\
&-\frac{\left(k_{M|31}+k_{F|57}\right)u_{x,NCOI}+\left(k_{M|32}+k_{F|58}\right)u_{y,NCOI}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}+\\
&-\frac{\left(k_{M|17}+k_{M|35}\right)u_{N,MC|7}+\left(k_{M|18}+k_{M|36}\right)u_{N,MC|8}+\left(k_{F|71}+k_{F|53}\right)u_{N,FC|1}+\left(k_{F|72}+k_{F|54}\right)u_{N,FC|2}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}+\\
&-\frac{\sum_{i=5}^{6}k_{M|1i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|3i}u_{N,MB|i}+\sum_{i=2}^{3}k_{F|7i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|5i}u_{N,FB|i}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\\[10pt]

&\left[\left(k_{e,M|21}+k_{e,M|43}+k_{e,F|87}+k_{e,F|65}\right)+\frac{k_{e,M|11}+k_{e,M|33}+k_{e,F|77}+k_{e,F|55}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left(k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}\right)\right]u_{x,MCT}+\\
&+\left(k_{e,M|23}-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}k_{e,M|13}\right)u_{x}+\\
&+\left(k_{e,M|24}-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}k_{e,M|14}\right)u_{y}+\\
&+\left(k_{e,M|23}+k_{e,F|85}-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left(k_{e,M|13}+k_{e,M|75}\right)\right)\cancelto{\approx 0}{u_{x,F}}+\\
&+\left(k_{e,M|24}+k_{e,F|86}-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left(k_{e,M|14}+k_{e,M|76}\right)\right)\cancelto{\approx 0}{u_{y,F}}+\\
&+\left[\left(k_{M|41}+k_{F|67}\right)-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left(k_{M|31}+k_{F|57}\right)\right]u_{x,NCOI}+\\
&+\left[\left(k_{M|42}+k_{F|68}\right)-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left(k_{M|32}+k_{F|58}\right)\right]u_{y,NCOI}+\\
&+\left(k_{M|27}+k_{M|45}\right)u_{N,MC|7}+\left(k_{M|28}+k_{M|46}\right)u_{N,MC|8}+\left(k_{F|81}+k_{F|63}\right)u_{N,FC|1}+\left(k_{F|82}+k_{F|64}\right)u_{N,FC|2}+\\
&-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left[\left(k_{M|17}+k_{M|35}\right)u_{N,MC|7}+\left(k_{M|18}+k_{M|36}\right)u_{N,MC|8}\right]+\\&
-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left[\left(k_{F|71}+k_{F|53}\right)u_{N,FC|1}+\left(k_{F|72}+k_{F|54}\right)u_{N,FC|2}\right]\\
&+\sum_{i=2}^{3}k_{F|8i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|6i}u_{N,FB|i}+\sum_{i=5}^{6}k_{M|2i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|4i}u_{N,MB|i}+\\
&-\frac{\sum_{i=5}^{6}k_{M|1i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|3i}u_{N,MB|i}+\sum_{i=2}^{3}k_{F|7i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|5i}u_{N,FB|i}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}=0\\[10pt]

&u_{x,FCT}=u_{x,MCT}\\
&u_{y,FCT}=u_{y,MCT}\\[10pt]
F_{x,CT}&=R_{x,FCT}=\\&=\left(k_{e,F|77}+k_{e,F|55}\right)u_{x,FCT}+\left(k_{e,F|78}+k_{e,F|56}\right)u_{y,FCT}+\\
&+k_{e,F|75}\cancelto{\approx 0}{u_{x,F}}+k_{e,F|76}\cancelto{\approx 0}{u_{y,F}}+\\
&+\sum_{i=1}^{4}k_{e,F|7i}u_{N,FC|i}+\sum_{i=1,i\neq\left(5,6\right)}^{8}k_{e,F|5i}u_{N,FB|i}\\
F_{y,CT}&=R_{y,FCT}=\\&=\left(k_{e,F|87}+k_{e,F|65}\right)u_{x,FCT}+\left(k_{e,F|88}+k_{e,F|66}\right)u_{y,FCT}+\\
&+k_{e,F|85}\cancelto{\approx 0}{u_{x,F}}+k_{e,F|86}\cancelto{\approx 0}{u_{y,F}}+\\
&+\sum_{i=1}^{4}k_{e,F|8i}u_{N,FC|i}+\sum_{i=1,i\neq\left(5,6\right)}^{8}k_{e,F|6i}u_{N,FB|i}\\
\end{cases}
\end{equation}

\begin{equation}
\begin{cases}
F_{x,CT}&=\left(k_{e,F|77}+k_{e,F|55}\right)u_{x,MCT}+\left(k_{e,F|78}+k_{e,F|56}\right)u_{y,MCT}+\\
&+\sum_{i=1}^{4}k_{e,F|7i}u_{N,FC|i}+\sum_{i=1,i\neq\left(5,6\right)}^{8}k_{e,F|5i}u_{N,FB|i}\\
F_{y,CT}&=\left(k_{e,F|87}+k_{e,F|65}\right)u_{x,MCT}+\left(k_{e,F|88}+k_{e,F|66}\right)u_{y,MCT}+\\
&+\sum_{i=1}^{4}k_{e,F|8i}u_{N,FC|i}+\sum_{i=1,i\neq\left(5,6\right)}^{8}k_{e,F|6i}u_{N,FB|i}\\
\end{cases}
\end{equation}

\begin{equation}
\begin{cases}
F_{x,CT}&= K_{xx}u_{x}+K_{xy}u_{y}+\\
&+\sum_{i=1}^{4}K_{FC,x|i}u_{N,FC|i}+\sum_{i=1,i\neq\left(3,4,5,6\right)}^{8}K_{FB,x|i}u_{N,FB|i}+\\
&+\sum_{i=5}^{8}K_{FC,x|i}u_{N,MC|i}+\sum_{i=7}^{8}K_{MB,x|i}u_{N,FB|i}\\
F_{y,CT}&= K_{yx}u_{x}+K_{yy}u_{y}+\\
&+\sum_{i=1}^{4}k_{e,F|8i}u_{N,FC|i}+\sum_{i=1,i\neq\left(5,6\right)}^{8}k_{e,F|6i}u_{N,FB|i}\\<commit_msg>commit on 10/082018 at 17:52:18<commit_after>%------------------------------------------------------------------------------------------%
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\usepackage{amsmath}
\usepackage{booktabs}
\usepackage{cite}
\usepackage{float}
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\begin{document}

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\title{Computing energy release rates using the Virtual Crack Closure Technique with the Finite Element Method: analytical discussion}

\author{<NAME>$^{1,2,a}$ , <NAME>$^{2,b}$ and <NAME>$^{1,c}$ }

\address{$^{1}$SI2M, IJL, EEIGM, Universit\'e de Lorraine, 6 Rue Bastien Lepage, F-54010 Nancy, France\\$^{2}$Division of Polymer Engineering, Lule\aa\ University of Technology, SE-97187 Lule\aa , Sweden }

{\vspace*{5pt}\address{E-mail: $^{a}$<EMAIL>, $^{b}$<EMAIL>, $^{c}$<EMAIL>}}
%\address{$^{a}$<EMAIL>}, \address{$^{b}$<EMAIL>}, \address{$^{c}$<EMAIL>}

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%                                       ABSTRACT
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\begin{abstract}
The effect of debond size and crack tip orientation of mode splitting in the Virtual Crack Closure Technique is analyzed by means of analytical derivations. The total energy release rate is shown to have no direct dependence on the debond angular size, but only an indirect one through the FEM solution of the crack displacement field in the crack tip neighbourhood.
\end{abstract}

%------------------------------------------------------------------------------------------%
%                                   List of acronyms
%------------------------------------------------------------------------------------------%

\section*{List of acronyms}

\begin{tabular}{ll}
VCCT &  Virtual Crack Closure Technique\\
BEM &  Boundary Element Method\\
FEM &  Finite Element Method\\
\end{tabular}
%\end{table}

%------------------------------------------------------------------------------------------%
%                                   List of symbols
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\section*{List of symbols}

%\begin{table}[!h]
\begin{tabular}{lcl}
$G_{I}$ & $\left[\frac{J}{m^{2}}\right]$  & Mode I energy release rate\\
$G_{II}$ & $\left[\frac{J}{m^{2}}\right]$  & Mode II energy release rate\\
$G_{TOT}$ & $\left[\frac{J}{m^{2}}\right]$  & Total energy release rate\\
$G_{I,r\theta}$ & $\left[\frac{J}{m^{2}}\right]$  & Mode I energy release rate in $r-\theta$ reference frame\\
$G_{II,r\theta}$ & $\left[\frac{J}{m^{2}}\right]$  & Mode II energy release rate in $r-\theta$ reference frame\\
$G_{TOT,r\theta}$ & $\left[\frac{J}{m^{2}}\right]$  & Total energy release rate in $r-\theta$ reference frame\\
$\widetilde{G}_{I,xy}$ & $\left[\frac{J}{m^{2}}\right]$  & Mode I energy release rate of equivalent crack in $x-y$ reference frame\\
$\widetilde{G}_{II,xy}$ & $\left[\frac{J}{m^{2}}\right]$  & Mode II energy release rate of equivalent crack in $x-y$ reference frame\\
$\widetilde{G}_{TOT,xy}$ & $\left[\frac{J}{m^{2}}\right]$  & Total energy release rate of equivalent crack in $x-y$ reference frame\\
$R_{f}$ & $\left[\mu m\right]$  &Fiber radius\\
$a$ & $\left[\mu m\right]$  &Debond size\\
$\Delta a$ & $\left[\mu m\right]$  &Debond increment\\
$\Delta\theta$ & $\left[rad\right]$  & Half debond angular size\\
$\delta$ & $\left[rad\right]$  & Angular size of element at the interface close to the crack tip\\
$u_{x,[A-Z]}$ & $\left[\mu m\right]$  & Displacement along $x$ of a point labeled with a letter in [A-Z]\\
$u_{y,[A-Z]}$ & $\left[\mu m\right]$  & Displacement along $y$ of a point labeled with a letter in [A-Z]\\
$u_{x}$ & $\left[\mu m\right]$  & Displacement along $x$-direction\\
$u_{y}$ & $\left[\mu m\right]$  & Displacement along $y$-direction\\
$u_{r}$ & $\left[\mu m\right]$  & Displacement along $r$-direction\\
$u_{\theta}$ & $\left[\mu m\right]$  & Displacement along $\theta$-direction\\
$F_{x,[A-Z]}$ & $\left[\mu m\right]$  & Force along $x$ at a point labeled with a letter in [A-Z]\\
$F_{y,[A-Z]}$ & $\left[\mu m\right]$  & Force along $y$ at a point labeled with a letter in [A-Z]\\
$F_{x}$ & $\left[\mu m\right]$  & Force along $x$-direction\\
$F_{y}$ & $\left[\mu m\right]$  & Force along $y$-direction\\
$F_{r}$ & $\left[\mu m\right]$  & Force along $r$-direction\\
$F_{\theta}$ & $\left[\mu m\right]$  & Force along $\theta$-direction\\
$\underline{\underline{R}}$ & $\left[-\right]$  & Rotation matrix\\
\end{tabular}
%\end{table}

\clearpage
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%                        FEM formulation with quadrilateral elements
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\section{ FEM formulation with quadrilateral elements}

\begin{equation}
\Pi\left(u_{i}\right)=\int_{V}\sigma_{ij}\varepsilon_{ij}dV+\int_{V}\rho\ddot{u}_{i}u_{i}dV-\int_{V}F^{T}_{i}u_{i}dV-\int_{S}f^{T}_{i}u_{i}dS
\end{equation}

\begin{equation}
\Pi\left(u_{i}\right)=\int_{V}\sigma_{ij}\varepsilon_{ij}dV-\int_{V}F^{T}_{i}u_{i}dV-\int_{S}f^{T}_{i}u_{i}dS
\end{equation}

\begin{equation}
\varepsilon_{ij}=\frac{1}{2}\left(u_{i,j}+u_{j,i}\right)
\end{equation}

\begin{equation}
\underline{\varepsilon}\left(x,y\right)=\underline{\underline{\widetilde{B}}}\cdot\underline{u}\left(x,y\right)
\end{equation}

\begin{equation}
\underline{\widetilde{B}}=\begin{bmatrix}
\frac{\partial }{\partial x}&0\\[7.5pt]
0&\frac{\partial }{\partial y}\\[7.5pt]
\frac{\partial }{\partial y}&\frac{\partial }{\partial x}\\
\end{bmatrix}
\end{equation}

%\begin{equation}
%\underline{\varepsilon}\left(x,y\right)=\underline{\underline{\widetilde{B}}}\cdot\underline{u}\left(x,y\right)=\underline{\underline{\widetilde{B}}}\cdot\underline{\underline{N}}\cdot\underline{u}_{N}=\underline{\underline{B}}\cdot\underline{u}_{N}
%\end{equation}

\begin{equation}
\underline{\sigma}=\underline{\underline{D}}\cdot\underline{\varepsilon}
\end{equation}

\begin{equation}
\underline{\sigma}=\begin{bmatrix}
\sigma_{xx}\\\sigma_{yy}\\\tau_{xy}
\end{bmatrix}\qquad\underline{\varepsilon}=\begin{bmatrix}
\varepsilon_{xx}\\\varepsilon_{yy}\\\gamma_{xy}
\end{bmatrix}
\end{equation}

\begin{equation}
\underline{\underline{D}}=\begin{bmatrix}
E_{1}&E_{2}&0\\E_{2}&E_{1}&0\\0&0&G
\end{bmatrix}\qquad\text{with }G=\frac{E}{2\left(1+\nu\right)}\text{ for an isotropic material}
\end{equation}

\begin{equation}
\begin{split}
E_{1}=\frac{E}{1-\nu^{2}}\quad E_{2}=\nu E_{1}\quad&\text{for plane stress}\\
E_{1}=\frac{E\left(1-\nu\right)}{\left(1+\nu\right)\left(1-2\nu\right)}\quad E_{2}=\frac{\nu E_{1}}{1-\nu}\quad&\text{for plane strain}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
\Pi\left(\underline{u}\right)=&\frac{1}{2}\int_{V}\underline{\underline{\varepsilon}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\varepsilon}}dV-\int_{V}\underline{F}^{T}\underline{u}dV-\int_{S}\underline{f}^{T}\underline{u}dS=\\
&=\frac{1}{2}\int_{V}\underline{u}^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\underline{u}dV-\int_{V}\underline{F}^{T}\underline{u}dV-\int_{S}\underline{f}^{T}\underline{u}dS
\end{split}
\end{equation}

\begin{equation}
\delta\Pi\left(\delta\underline{u}\right)=0
\end{equation}

\begin{equation}
\begin{split}
\delta\Pi\left(\delta\underline{u}\right)=&\Pi\left(\underline{u}+\delta\underline{u}\right)-\Pi\left(\underline{u}\right)=\\=&\frac{1}{2}\int_{V}\left(\underline{u}+\underline{\delta u}\right)^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\left(\underline{u}+\underline{\delta u}\right)dV-\int_{V}\underline{F}^{T}\left(\underline{u}+\underline{\delta u}\right)dV-\int_{S}\underline{f}^{T}\left(\underline{u}+\underline{\delta u}\right)dS+\\
-&\frac{1}{2}\int_{V}\underline{u}^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\underline{u}dV+\int_{V}\underline{F}^{T}\underline{u}dV+\int_{S}\underline{f}^{T}\underline{u}dS=\\
=&\cancel{\frac{1}{2}\int_{V}\underline{u}^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\underline{u}dV}+\cancelto{\approx 0}{\frac{1}{2}\int_{V}\underline{\delta u}^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\underline{\delta u}dV}+\\
+&\frac{1}{2}\int_{V}\underline{u}^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\underline{\delta u}dV+\frac{1}{2}\int_{V}\underline{\delta u}^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\underline{u}dV+\\
-&\cancel{\int_{V}\underline{F}^{T}\underline{u}dV}-\int_{V}\underline{F}^{T}\underline{\delta u}dV+\\
-&\cancel{\int_{S}\underline{f}^{T}\underline{u}dS}-\int_{S}\underline{f}^{T}\underline{\delta u}dS+\\
-&\cancel{\frac{1}{2}\int_{V}\underline{u}^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\underline{u}dV}+\cancel{\int_{V}\underline{F}^{T}\underline{u}dV}+\cancel{\int_{S}\underline{f}^{T}\underline{u}dS}=\\
=&\int_{V}\underline{u}^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\underline{\delta u}dV-\int_{V}\underline{F}^{T}\underline{\delta u}dV-\int_{S}\underline{f}^{T}\underline{\delta u}dS
\end{split}
\end{equation}

\begin{equation}
\int_{V}\underline{\delta u}^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\underline{u}dV-\int_{V}\underline{\delta u}^{T}\underline{F}dV-\int_{S}\underline{\delta u}^{T}\underline{f}dS=0
\end{equation}

%\begin{equation}
%\underline{\underline{k}}\cdot\underline{u}_{N}=\underline{f}_{N}
%\end{equation}

\begin{equation}
\underline{u}=\begin{bmatrix}
u_{x}\left(x,y\right)\\u_{y}\left(x,y\right)
\end{bmatrix}\qquad\underline{u}_{N}=\begin{bmatrix}
u_{1,x}\\u_{1,y}\\u_{2,x}\\u_{2,y}\\u_{3,x}\\u_{3,y}\\u_{4,x}\\u_{4,y}\\
\end{bmatrix}\text{ or }\underline{u}_{N}=\begin{bmatrix}
u_{1,x}\\u_{1,y}\\u_{2,x}\\u_{2,y}\\u_{3,x}\\u_{3,y}\\u_{4,x}\\u_{4,y}\\u_{5,x}\\u_{5,y}\\u_{6,x}\\u_{6,y}\\u_{7,x}\\u_{7,y}\\u_{8,x}\\u_{8,y}\\
\end{bmatrix}
\end{equation}

%\qquad\underline{f}_{N}=\begin{bmatrix}
%f_{1,x}\\f_{1,y}\\f_{2,x}\\f_{2,y}\\f_{3,x}\\f_{3,y}\\f_{4,x}\\f_{4,y}\\
%\end{bmatrix}

\begin{equation}
\underline{u}=\underline{\underline{N}}\cdot\underline{u}_{N}
\end{equation}

\begin{equation}
\underline{\underline{N}}=\begin{bmatrix}
N_{1}&0&N_{2}&0&N_{3}&0&N_{4}&0\\
0&N_{1}&0&N_{2}&0&N_{3}&0&N_{4}\\
\end{bmatrix}
\end{equation}

\setcounter{MaxMatrixCols}{20}
\begin{equation}
\underline{\underline{N}}=\begin{bmatrix}
N_{1} & 0     & N_{2} & 0     & N_{3} & 0     & N_{4} & 0     & N_{5} & 0     & N_{6} & 0     & N_{7} & 0     & N_{8} & 0    \\
0     & N_{1} & 0     & N_{2} & 0     & N_{3} & 0     & N_{4} & 0     & N_{5} & 0     & N_{6} & 0     & N_{7} & 0     & N_{8} \\
\end{bmatrix}
\end{equation}


\begin{equation}
\begin{cases}
N_{1}=N_{1}\left(\xi,\eta\right)\\
N_{2}=N_{2}\left(\xi,\eta\right)\\
N_{3}=N_{3}\left(\xi,\eta\right)\\
N_{4}=N_{4}\left(\xi,\eta\right)\\
\end{cases}\quad\text{with}\quad\begin{cases}
\xi=\xi\left(x,y\right)\\
\eta=\eta\left(x,y\right)\\
\end{cases}\text{for isoparametric elements}
\end{equation}

\begin{equation}
\underline{\underline{B}}=\underline{\underline{\widetilde{B}}}\cdot\underline{\underline{N}}=\begin{bmatrix}
\frac{\partial N_{1}}{\partial x}&0&\frac{\partial N_{2}}{\partial x}&0&\frac{\partial N_{3}}{\partial x}&0&\frac{\partial N_{4}}{\partial x}&0\\[7.5pt]
0&\frac{\partial N_{1}}{\partial y}&0&\frac{\partial N_{2}}{\partial y}&0&\frac{\partial N_{3}}{\partial y}&0&\frac{\partial N_{4}}{\partial y}\\[7.5pt]
\frac{\partial N_{1}}{\partial y}&\frac{\partial N_{1}}{\partial x}&\frac{\partial N_{2}}{\partial y}&\frac{\partial N_{2}}{\partial x}&\frac{\partial N_{3}}{\partial y}&\frac{\partial N_{3}}{\partial x}&\frac{\partial N_{4}}{\partial y}&\frac{\partial N_{4}}{\partial x}\\
\end{bmatrix}
\end{equation}

\setcounter{MaxMatrixCols}{20}
\begin{equation}
\begin{split}
\underline{\underline{B}}=&\underline{\underline{\widetilde{B}}}\cdot\underline{\underline{N}}=\\=&\begin{bmatrix}
\frac{\partial N_{1}}{\partial x}&0&\frac{\partial N_{2}}{\partial x}&0&\frac{\partial N_{3}}{\partial x}&0&\frac{\partial N_{4}}{\partial x}&0&\frac{\partial N_{5}}{\partial x}&0&\frac{\partial N_{6}}{\partial x}&0&\frac{\partial N_{7}}{\partial x}&0&\frac{\partial N_{8}}{\partial x}&0\\[7.5pt]
0&\frac{\partial N_{1}}{\partial y}&0&\frac{\partial N_{2}}{\partial y}&0&\frac{\partial N_{3}}{\partial y}&0&\frac{\partial N_{4}}{\partial y}&0&\frac{\partial N_{5}}{\partial y}&0&\frac{\partial N_{6}}{\partial y}&0&\frac{\partial N_{7}}{\partial y}&0&\frac{\partial N_{8}}{\partial y}\\[7.5pt]
\frac{\partial N_{1}}{\partial y}&\frac{\partial N_{1}}{\partial x}&\frac{\partial N_{2}}{\partial y}&\frac{\partial N_{2}}{\partial x}&\frac{\partial N_{3}}{\partial y}&\frac{\partial N_{3}}{\partial x}&\frac{\partial N_{4}}{\partial y}&\frac{\partial N_{4}}{\partial x}&\frac{\partial N_{5}}{\partial y}&\frac{\partial N_{5}}{\partial x}&\frac{\partial N_{6}}{\partial y}&\frac{\partial N_{6}}{\partial x}&\frac{\partial N_{7}}{\partial y}&\frac{\partial N_{7}}{\partial x}&\frac{\partial N_{8}}{\partial y}&\frac{\partial N_{8}}{\partial x}\\
\end{bmatrix}
\end{split}
\end{equation}

\begin{equation}
\delta\underline{u}=\delta\left(\underline{\underline{N}}\cdot\underline{u}_{N}\right)=\underline{\underline{N}}\delta\underline{u}_{N}
\end{equation}

\begin{equation}
\int_{V}\underline{\delta u_{N}}^{T}\underline{\underline{N}}^{T}\underline{\underline{\widetilde{B}}}^{T}\underline{\underline{D}}\cdot\underline{\underline{\widetilde{B}}}\cdot\underline{\underline{N}}\cdot\underline{u_{N}}dV-\int_{V}\underline{\delta u_{N}}^{T}\underline{\underline{N}}^{T}\underline{F}dV-\int_{S}\underline{\delta u_{N}}^{T}\underline{\underline{N}}^{T}\underline{f}dS=0
\end{equation}

\begin{equation}
\underline{\delta u_{N}}^{T}\left(\int_{V}\underline{\underline{B}}^{T}\underline{\underline{D}}\cdot\underline{\underline{B}}dV\cdot\underline{u_{N}}-\int_{V}\underline{\underline{N}}^{T}\underline{F}dV-\int_{S}\underline{\underline{N}}^{T}\underline{f}dS\right)=0
\end{equation}

\begin{equation}
\underline{\underline{k}}\cdot\underline{u}_{N}=\underline{F}_{N}\quad\underline{\underline{k}}=\int_{V}\left(\underline{\underline{B}}^{T}\underline{\underline{D}}\cdot\underline{\underline{B}}\right)dV\quad\underline{F}_{N}=\int_{V}\underline{\underline{N}}^{T}\underline{F}dV+\int_{S}\underline{\underline{N}}^{T}\underline{f}dS
\end{equation}

\begin{equation}
\begin{cases}
N_{1}\left(\xi,\eta\right)=\frac{1}{4}\left(1-\xi\right)\left(1-\eta\right)\\
N_{2}\left(\xi,\eta\right)=\frac{1}{4}\left(1+\xi\right)\left(1-\eta\right)\\
N_{3}\left(\xi,\eta\right)=\frac{1}{4}\left(1+\xi\right)\left(1+\eta\right)\\
N_{4}\left(\xi,\eta\right)=\frac{1}{4}\left(1-\xi\right)\left(1+\eta\right)\\
\end{cases}\qquad\begin{cases}
N_{1}\left(\xi,\eta\right)=\frac{1}{4}\left(1-\xi\right)\left(1-\eta\right)\left(-\xi-\eta-1\right)\\
N_{2}\left(\xi,\eta\right)=\frac{1}{4}\left(1+\xi\right)\left(1-\eta\right)\left(\xi-\eta-1\right)\\
N_{3}\left(\xi,\eta\right)=\frac{1}{4}\left(1+\xi\right)\left(1+\eta\right)\left(\xi+\eta-1\right)\\
N_{4}\left(\xi,\eta\right)=\frac{1}{4}\left(1-\xi\right)\left(1+\eta\right)\left(-\xi+\eta-1\right)\\
N_{5}\left(\xi,\eta\right)=\frac{1}{2}\left(1-\xi^{2}\right)\left(1-\eta\right)\\
N_{6}\left(\xi,\eta\right)=\frac{1}{2}\left(1+\xi\right)\left(1-\eta^{2}\right)\\
N_{7}\left(\xi,\eta\right)=\frac{1}{2}\left(1-\xi^{2}\right)\left(1+\eta\right)\\
N_{8}\left(\xi,\eta\right)=\frac{1}{2}\left(1-\xi\right)\left(1-\eta^{2}\right)\\
\end{cases}
\end{equation}

\begin{equation}
\begin{cases}
\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \xi} =-\frac{1}{4}\left(1-\eta\right)\\
\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \xi} = \frac{1}{4}\left(1-\eta\right)\\
\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \xi} = \frac{1}{4}\left(1+\eta\right)\\
\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \xi} =-\frac{1}{4}\left(1+\eta\right)\\
\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \eta}=-\frac{1}{4}\left(1-\xi\right)\\
\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \eta}=-\frac{1}{4}\left(1+\xi\right)\\
\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \eta}= \frac{1}{4}\left(1+\xi\right)\\
\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \eta}= \frac{1}{4}\left(1-\xi\right)\\
\end{cases}
\end{equation}

\begin{equation}
\begin{cases}
\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \xi}=\frac{1}{4}\left(1-\eta\right)\left(2\xi+\eta\right)\\
\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \xi}=\frac{1}{4}\left(1-\eta\right)\left(2\xi-\eta\right)\\
\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \xi}=\frac{1}{4}\left(1+\eta\right)\left(2\xi+\eta\right)\\
\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \xi}=\frac{1}{4}\left(1+\eta\right)\left(2\xi-\eta\right)\\
\frac{\partial N_{5}\left(\xi,\eta\right)}{\partial \xi}=-\xi\left(1-\eta\right)\\
\frac{\partial N_{6}\left(\xi,\eta\right)}{\partial \xi}=\frac{1}{2}\left(1-\eta^{2}\right)\\
\frac{\partial N_{7}\left(\xi,\eta\right)}{\partial \xi}=-\xi\left(1+\eta\right)\\
\frac{\partial N_{8}\left(\xi,\eta\right)}{\partial \xi}=-\frac{1}{2}\left(1-\eta^{2}\right)\\
\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \eta}=\frac{1}{4}\left(1-\xi\right)\left(2\eta+\xi\right)\\
\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \eta}=\frac{1}{4}\left(1+\xi\right)\left(2\eta-\xi\right)\\
\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \eta}=\frac{1}{4}\left(1+\xi\right)\left(2\eta+\xi\right)\\
\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \eta}=\frac{1}{4}\left(1-\xi\right)\left(2\eta-\xi\right)\\
\frac{\partial N_{5}\left(\xi,\eta\right)}{\partial \eta}=-\frac{1}{2}\left(1-\xi^{2}\right)\\
\frac{\partial N_{6}\left(\xi,\eta\right)}{\partial \eta}=-\eta\left(1+\xi\right)\\
\frac{\partial N_{7}\left(\xi,\eta\right)}{\partial \eta}=\frac{1}{2}\left(1-\xi^{2}\right)\\
\frac{\partial N_{8}\left(\xi,\eta\right)}{\partial \eta}=-\eta\left(1-\xi\right)\\
\end{cases}
\end{equation}

\begin{equation}
\underline{p}=\underline{\underline{N}}\cdot\underline{p}_{N}
\end{equation}

\begin{equation}
\underline{p}=\begin{bmatrix}
x\\y
\end{bmatrix}\qquad\underline{p}_{N}=\begin{bmatrix}
x_{1}\\y_{1}\\x_{2}\\y_{2}\\x_{3}\\y_{3}\\x_{4}\\y_{4}\\
\end{bmatrix}\text{ or }\underline{p}_{N}=\begin{bmatrix}
x_{1}\\y_{1}\\x_{2}\\y_{2}\\x_{3}\\y_{3}\\x_{4}\\y_{4}\\x_{5}\\y_{5}\\x_{6}\\y_{6}\\x_{7}\\y_{7}\\x_{8}\\y_{8}\\
\end{bmatrix}
\end{equation}

\begin{equation}
\begin{split}
x = x\left(\xi,\eta\right) = N_{1}\left(\xi,\eta\right)x_{1}+N_{2}\left(\xi,\eta\right)x_{2}+N_{3}\left(\xi,\eta\right)x_{3}+N_{4}\left(\xi,\eta\right)x_{4}\\
y = y\left(\xi,\eta\right) = N_{1}\left(\xi,\eta\right)y_{1}+N_{2}\left(\xi,\eta\right)y_{2}+N_{3}\left(\xi,\eta\right)y_{3}+N_{4}\left(\xi,\eta\right)y_{4}\\
\end{split}
\end{equation}

\begin{equation}
\begin{split}
x = x\left(\xi,\eta\right) =& N_{1}\left(\xi,\eta\right)x_{1}+N_{2}\left(\xi,\eta\right)x_{2}+N_{3}\left(\xi,\eta\right)x_{3}+N_{4}\left(\xi,\eta\right)x_{4}+\\
& +N_{5}\left(\xi,\eta\right)x_{5}+N_{6}\left(\xi,\eta\right)x_{6}+N_{7}\left(\xi,\eta\right)x_{7}+N_{8}\left(\xi,\eta\right)x_{8}\\
y = y\left(\xi,\eta\right) =& N_{1}\left(\xi,\eta\right)y_{1}+N_{2}\left(\xi,\eta\right)y_{2}+N_{3}\left(\xi,\eta\right)y_{3}+N_{4}\left(\xi,\eta\right)y_{4}+\\
& +N_{5}\left(\xi,\eta\right)y_{5}+N_{6}\left(\xi,\eta\right)y_{6}+N_{7}\left(\xi,\eta\right)y_{7}+N_{8}\left(\xi,\eta\right)y_{8}\\
\end{split}
\end{equation}

\begin{equation}
\begin{cases}
\frac{\partial x}{\partial \xi} = \frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \xi}x_{1}+\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \xi}x_{2}+\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \xi}x_{3}+\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \xi}x_{4}\\
\frac{\partial x}{\partial \eta} = \frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \eta}x_{1}+\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \eta}x_{2}+\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \eta}x_{3}+\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \eta}x_{4}\\
\frac{\partial y}{\partial \xi} = \frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \xi}y_{1}+\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \xi}y_{2}+\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \xi}y_{3}+\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \xi}y_{4}\\
\frac{\partial y}{\partial \eta} = \frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \eta}y_{1}+\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \eta}y_{2}+\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \eta}y_{3}+\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \eta}y_{4}\\
\end{cases}
\end{equation}

\begin{equation}
\begin{cases}
\frac{\partial x}{\partial \xi} = & \frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \xi}x_{1}+\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \xi}x_{2}+\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \xi}x_{3}+\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \xi}x_{4}+\\
& +\frac{\partial N_{5}\left(\xi,\eta\right)}{\partial \xi}x_{5}+\frac{\partial N_{6}\left(\xi,\eta\right)}{\partial \xi}x_{6}+\frac{\partial N_{7}\left(\xi,\eta\right)}{\partial \xi}x_{7}+\frac{\partial N_{8}\left(\xi,\eta\right)}{\partial \xi}x_{8}\\
\frac{\partial x}{\partial \eta} = & \frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \eta}x_{1}+\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \eta}x_{2}+\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \eta}x_{3}+\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \eta}x_{4}+\\
& +\frac{\partial N_{5}\left(\xi,\eta\right)}{\partial \eta}x_{5}+\frac{\partial N_{6}\left(\xi,\eta\right)}{\partial \eta}x_{6}+\frac{\partial N_{7}\left(\xi,\eta\right)}{\partial \eta}x_{7}+\frac{\partial N_{8}\left(\xi,\eta\right)}{\partial \eta}x_{8}\\
\frac{\partial y}{\partial \xi} = & \frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \xi}y_{1}+\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \xi}y_{2}+\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \xi}y_{3}+\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \xi}y_{4}+\\
& +\frac{\partial N_{5}\left(\xi,\eta\right)}{\partial \xi}y_{5}+\frac{\partial N_{6}\left(\xi,\eta\right)}{\partial \xi}y_{6}+\frac{\partial N_{7}\left(\xi,\eta\right)}{\partial \xi}y_{7}+\frac{\partial N_{8}\left(\xi,\eta\right)}{\partial \xi}y_{8}\\
\frac{\partial y}{\partial \eta} = & \frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \eta}y_{1}+\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \eta}y_{2}+\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \eta}y_{3}+\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \eta}y_{4}+\\
& +\frac{\partial N_{5}\left(\xi,\eta\right)}{\partial \eta}y_{5}+\frac{\partial N_{6}\left(\xi,\eta\right)}{\partial \eta}y_{6}+\frac{\partial N_{7}\left(\xi,\eta\right)}{\partial \eta}y_{7}+\frac{\partial N_{8}\left(\xi,\eta\right)}{\partial \eta}y_{8}\\
\end{cases}
\end{equation}

\begin{equation}
\begin{cases}
\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial x}=\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \xi}\frac{\partial \xi}{\partial x}+\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \eta}\frac{\partial \eta}{\partial x}\\
\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial x}=\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \xi}\frac{\partial \xi}{\partial x}+\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \eta}\frac{\partial \eta}{\partial x}\\
\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial x}=\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \xi}\frac{\partial \xi}{\partial x}+\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \eta}\frac{\partial \eta}{\partial x}\\
\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial x}=\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \xi}\frac{\partial \xi}{\partial x}+\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \eta}\frac{\partial \eta}{\partial x}\\
\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial y}=\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \xi}\frac{\partial \xi}{\partial y}+\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \eta}\frac{\partial \eta}{\partial y}\\
\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial y}=\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \xi}\frac{\partial \xi}{\partial y}+\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \eta}\frac{\partial \eta}{\partial y}\\
\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial y}=\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \xi}\frac{\partial \xi}{\partial y}+\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \eta}\frac{\partial \eta}{\partial y}\\
\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial y}=\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \xi}\frac{\partial \xi}{\partial y}+\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \eta}\frac{\partial \eta}{\partial y}\\
\end{cases}
\end{equation}

\begin{equation}
\begin{bmatrix}
\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial x}&\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial y}\\
\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial x}&\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial y}\\
\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial x}&\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial y}\\
\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial x}&\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial y}\\
\end{bmatrix}=\begin{bmatrix}
\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \xi}&\frac{\partial N_{1}\left(\xi,\eta\right)}{\partial \eta}\\
\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \xi}&\frac{\partial N_{2}\left(\xi,\eta\right)}{\partial \eta}\\
\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \xi}&\frac{\partial N_{3}\left(\xi,\eta\right)}{\partial \eta}\\
\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \xi}&\frac{\partial N_{4}\left(\xi,\eta\right)}{\partial \eta}\\
\end{bmatrix}\begin{bmatrix}
\frac{\partial \xi}{\partial x}&\frac{\partial \xi}{\partial y}\\\frac{\partial \eta}{\partial x}&\frac{\partial \eta}{\partial y}
\end{bmatrix}
\end{equation}

\begin{equation}
\underline{e}_{\xi}=\begin{bmatrix}
\frac{\partial x}{\partial\xi}\\\frac{\partial y}{\partial\xi}
\end{bmatrix}\quad\underline{e}_{\eta}=\begin{bmatrix}
\frac{\partial x}{\partial\eta}\\\frac{\partial y}{\partial\eta}
\end{bmatrix}
\end{equation}

\begin{equation}
\underline{e}_{x}=\begin{bmatrix}
\frac{\partial \xi}{\partial x}\\\frac{\partial \eta}{\partial x}
\end{bmatrix}\quad\underline{e}_{y}=\begin{bmatrix}
\frac{\partial \xi}{\partial y}\\\frac{\partial \eta}{\partial y}
\end{bmatrix}
\end{equation}

\begin{equation}
\underline{\underline{J}}=\begin{bmatrix}
\underline{e}_{\xi}|\underline{e}_{\eta}
\end{bmatrix}=\begin{bmatrix}
\frac{\partial x}{\partial\xi}&\frac{\partial x}{\partial\eta}\\\frac{\partial y}{\partial\xi}&\frac{\partial y}{\partial\eta}
\end{bmatrix}\qquad\underline{\underline{J}}^{-1}=\begin{bmatrix}
\underline{e}_{x}|\underline{e}_{y}
\end{bmatrix}=\begin{bmatrix}
\frac{\partial \xi}{\partial x}&\frac{\partial \xi}{\partial y}\\\frac{\partial \eta}{\partial x}&\frac{\partial \eta}{\partial y}
\end{bmatrix}
\end{equation}

\begin{equation}
\underline{\underline{J}}^{-1}=\begin{bmatrix}
\frac{\partial x}{\partial\xi}&\frac{\partial x}{\partial\eta}\\\frac{\partial y}{\partial\xi}&\frac{\partial y}{\partial\eta}
\end{bmatrix}^{-1}=\frac{1}{\frac{\partial x}{\partial\xi}\frac{\partial y}{\partial\eta}-\frac{\partial y}{\partial\xi}\frac{\partial x}{\partial\eta}}\begin{bmatrix}
\frac{\partial y}{\partial\eta}&-\frac{\partial x}{\partial\eta}\\-\frac{\partial y}{\partial\xi}&\frac{\partial x}{\partial\xi}
\end{bmatrix}
\end{equation}

\begin{equation}
\underline{\underline{g}}=\underline{\underline{J}}^{T}\underline{\underline{J}}=\begin{bmatrix}
\frac{\partial x}{\partial\xi}&\frac{\partial y}{\partial\xi}\\\frac{\partial x}{\partial\eta}&\frac{\partial y}{\partial\eta}
\end{bmatrix}\begin{bmatrix}
\frac{\partial x}{\partial\xi}&\frac{\partial x}{\partial\eta}\\\frac{\partial y}{\partial\xi}&\frac{\partial y}{\partial\eta}
\end{bmatrix}=\begin{bmatrix}
\frac{\partial x}{\partial\xi}\frac{\partial x}{\partial\xi}+\frac{\partial y}{\partial\xi}\frac{\partial y}{\partial\xi}&\frac{\partial x}{\partial\xi}\frac{\partial x}{\partial\eta}+\frac{\partial y}{\partial\xi}\frac{\partial y}{\partial\eta}\\\frac{\partial x}{\partial\eta}\frac{\partial x}{\partial\xi}+\frac{\partial y}{\partial\eta}\frac{\partial y}{\partial\xi}&\frac{\partial x}{\partial\eta}\frac{\partial x}{\partial\eta}+\frac{\partial y}{\partial\eta}\frac{\partial y}{\partial\eta}
\end{bmatrix}
\end{equation}

\begin{equation}
\begin{split}
g=&det\left(\underline{\underline{g}}\right)=\\=&\left(\frac{\partial x}{\partial\xi}\frac{\partial x}{\partial\xi}+\frac{\partial y}{\partial\xi}\frac{\partial y}{\partial\xi}\right)\left(\frac{\partial x}{\partial\eta}\frac{\partial x}{\partial\eta}+\frac{\partial y}{\partial\eta}\frac{\partial y}{\partial\eta}\right)-\left(\frac{\partial x}{\partial\eta}\frac{\partial x}{\partial\xi}+\frac{\partial y}{\partial\eta}\frac{\partial y}{\partial\xi}\right)\left(\frac{\partial x}{\partial\xi}\frac{\partial x}{\partial\eta}+\frac{\partial y}{\partial\xi}\frac{\partial y}{\partial\eta}\right)
\end{split}
\end{equation}

\begin{equation}
dV\left(x,y\right)=\sqrt{g}dV\left(\xi,\eta\right)
\end{equation}

\begin{equation}
\begin{split}
\underline{\underline{k_{e}}}=&\int_{V_{e}\left(x,y\right)}\left(\underline{\underline{B}}^{T}\underline{\underline{D}}\cdot\underline{\underline{B}}\right)dV_{e}\left(x,y\right)=\int_{V_{e}\left(\xi,\eta\right)}\left(\underline{\underline{B}}^{T}\underline{\underline{D}}\cdot\underline{\underline{B}}\right)\sqrt{g}dV_{e}\left(\xi,\eta\right)\\
=&\int_{-1}^{1}\int_{-1}^{1}\left(\underline{\underline{B}}^{T}\underline{\underline{D}}\cdot\underline{\underline{B}}\right)\sqrt{g}d\xi d\eta\approx \sum_{i=1}^{N}\sum_{j=1}^{N}w_{i}w_{j}\left(\underline{\underline{B}}^{T}\left(\xi_{i},\eta_{j}\right)\cdot\underline{\underline{D}}\cdot\underline{\underline{B}}\left(\xi_{i},\eta_{j}\right)\sqrt{g}\right)
\end{split}
\end{equation}

\begin{equation}
k_{e}=\begin{bmatrix}
k_{e|11}&k_{e|12}&k_{e|13}&k_{e|14}&k_{e|15}&k_{e|16}&k_{e|17}&k_{e|18}\\
k_{e|12}&k_{e|22}&k_{e|23}&k_{e|24}&k_{e|25}&k_{e|26}&k_{e|27}&k_{e|28}\\
k_{e|13}&k_{e|23}&k_{e|33}&k_{e|34}&k_{e|35}&k_{e|36}&k_{e|37}&k_{e|38}\\
k_{e|14}&k_{e|24}&k_{e|34}&k_{e|44}&k_{e|45}&k_{e|46}&k_{e|47}&k_{e|48}\\
k_{e|15}&k_{e|25}&k_{e|35}&k_{e|45}&k_{e|55}&k_{e|56}&k_{e|57}&k_{e|58}\\
k_{e|16}&k_{e|26}&k_{e|36}&k_{e|46}&k_{e|56}&k_{e|66}&k_{e|67}&k_{e|68}\\
k_{e|17}&k_{e|27}&k_{e|37}&k_{e|47}&k_{e|57}&k_{e|67}&k_{e|77}&k_{e|78}\\
k_{e|18}&k_{e|28}&k_{e|38}&k_{e|48}&k_{e|58}&k_{e|68}&k_{e|78}&k_{e|88}\\
\end{bmatrix}
\end{equation}

\clearpage
%------------------------------------------------------------------------------------------%
%                        VCCT 1st order
%------------------------------------------------------------------------------------------%

\section{VCCT for first order quadrilateral elements}

\subsection{Definition of crack tip reference frame}

\begin{figure}[!h]
\includegraphics[width=\textwidth]{VCCT-linear.pdf}
\caption{Schematic representation of the discretized crack tip geometry for  $1^{st}$ order quadrilateral elements.}
\end{figure}

\begin{equation}
\underline{\underline{R}}=\begin{bmatrix}
\cos\left(\Delta\theta\right) & \sin\left(\Delta\theta\right) \\
-\sin\left(\Delta\theta\right) & \cos\left(\Delta\theta\right)
\end{bmatrix}\qquad\underline{\underline{R}}^{-1}=\underline{\underline{R}}^{T}=\begin{bmatrix}
\cos\left(\Delta\theta\right) & -\sin\left(\Delta\theta\right) \\
\sin\left(\Delta\theta\right) & \cos\left(\Delta\theta\right)
\end{bmatrix}
\end{equation}

\begin{equation}
\begin{bmatrix}
r \\
\theta
\end{bmatrix}=\underline{\underline{R}}\begin{bmatrix}
x \\
y
\end{bmatrix}\qquad\begin{bmatrix}
x \\
y
\end{bmatrix}=\underline{\underline{R}}^{-1}\begin{bmatrix}
r \\
\theta
\end{bmatrix}
\end{equation}

\subsection{Calculation of displacements and reaction forces}

\begin{equation}
u_{x}=u_{x,M}-u_{x,F}\qquad u_{y}=u_{y,M}-u_{y,F}
\end{equation}

\begin{equation}
u_{r}=\cos\left(\Delta\theta\right) u_{x}+\sin\left(\Delta\theta\right) u_{y}\qquad u_{\theta}=-\sin\left(\Delta\theta\right) u_{x}+\cos\left(\Delta\theta\right) u_{y}
\end{equation}

\begin{equation}
F_{r}=\cos\left(\Delta\theta\right) F_{x,CT}+\sin\left(\Delta\theta\right) F_{y,CT}\qquad F_{\theta}=-\sin\left(\Delta\theta\right) F_{x,CT}+\cos\left(\Delta\theta\right) F_{y,CT}
\end{equation}

\begin{equation}
\begin{cases}
&\left(k_{e,M|11}+k_{e,M|33}\right)u_{x,MCT}+\left(k_{e,M|12}+k_{e,M|34}\right)u_{y,MCT}+\\&+k_{e,M|13}u_{x,M}+k_{e,M|14}u_{y,M}+\left(k_{M|17}+k_{M|35}\right)u_{N,MC|7}+\left(k_{M|18}+k_{M|36}\right)u_{N,MC|8}+\\&+\sum_{i=5}^{6}k_{M|1i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|3i}u_{N,MB|i}+k_{M|31}u_{x,NCOI}+k_{M|32}u_{y,NCOI}=0\\[10pt]
&\left(k_{e,M|21}+k_{e,M|43}\right)u_{x,MCT}+\left(k_{e,M|22}+k_{e,M|44}\right)u_{y,MCT}+\\&+k_{e,M|23}u_{x,M}+k_{e,M|24}u_{y,M}+\left(k_{M|27}+k_{M|45}\right)u_{N,MC|7}+\left(k_{M|28}+k_{M|46}\right)u_{N,MC|8}+\\&+\sum_{i=5}^{6}k_{M|2i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|4i}u_{N,MB|i}+k_{M|41}u_{x,NCOI}+k_{M|42}u_{y,NCOI}=0\\[10pt]
&\left(k_{e,F|77}+k_{e,F|55}\right)u_{x,FCT}+\left(k_{e,F|78}+k_{e,F|56}\right)u_{y,FCT}+\\&+k_{e,F|75}u_{x,F}+k_{e,F|76}u_{y,F}+\left(k_{F|71}+k_{F|53}\right)u_{N,FC|1}+\left(k_{F|72}+k_{F|54}\right)u_{N,FC|2}+\\&+\sum_{i=2}^{3}k_{F|7i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|5i}u_{N,FB|i}+k_{F|57}u_{x,NCOI}+k_{F|58}u_{y,NCOI}=0\\[10pt]
&\left(k_{e,F|87}+k_{e,F|65}\right)u_{x,FCT}+\left(k_{e,F|88}+k_{e,F|66}\right)u_{y,FCT}+\\&+k_{e,F|85}u_{x,F}+k_{e,F|86}u_{y,F}+\left(k_{F|81}+k_{F|63}\right)u_{N,FC|1}+\left(k_{F|82}+k_{F|64}\right)u_{N,FC|2}+\\&+\sum_{i=2}^{3}k_{F|8i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|6i}u_{N,FB|i}+k_{F|67}u_{x,NCOI}+k_{F|68}u_{y,NCOI}=0\\[10pt]
&u_{x,FCT}-u_{x,MCT}-u_{x,DUMMY}=0\\
&u_{y,FCT}-u_{y,MCT}-u_{y,DUMMY}=0\\[10pt]
&u_{x,DUMMY}=0\\
&u_{y,DUMMY}=0\\
\end{cases}
\end{equation}

\begin{equation}
\begin{cases}
&\left(k_{e,M|11}+k_{e,M|33}\right)u_{x,MCT}+\left(k_{e,M|12}+k_{e,M|34}\right)u_{y,MCT}+\\&+k_{e,M|13}u_{x,M}+k_{e,M|14}u_{y,M}+\left(k_{M|17}+k_{M|35}\right)u_{N,MC|7}+\left(k_{M|18}+k_{M|36}\right)u_{N,MC|8}+\\&+\sum_{i=5}^{6}k_{M|1i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|3i}u_{N,MB|i}+k_{M|31}u_{x,NCOI}+k_{M|32}u_{y,NCOI}=0\\[10pt]
&\left(k_{e,M|21}+k_{e,M|43}\right)u_{x,MCT}+\left(k_{e,M|22}+k_{e,M|44}\right)u_{y,MCT}+\\&+k_{e,M|23}u_{x,M}+k_{e,M|24}u_{y,M}+\left(k_{M|27}+k_{M|45}\right)u_{N,MC|7}+\left(k_{M|28}+k_{M|46}\right)u_{N,MC|8}+\\&+\sum_{i=5}^{6}k_{M|2i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|4i}u_{N,MB|i}+k_{M|41}u_{x,NCOI}+k_{M|42}u_{y,NCOI}=0\\[10pt]
&\left(k_{e,F|77}+k_{e,F|55}\right)u_{x,MCT}+\left(k_{e,F|78}+k_{e,F|56}\right)u_{y,MCT}+\\&+k_{e,F|75}u_{x,F}+k_{e,F|76}u_{y,F}+\left(k_{F|71}+k_{F|53}\right)u_{N,FC|1}+\left(k_{F|72}+k_{F|54}\right)u_{N,FC|2}+\\&+\sum_{i=2}^{3}k_{F|7i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|5i}u_{N,FB|i}+k_{F|57}u_{x,NCOI}+k_{F|58}u_{y,NCOI}=0\\[10pt]
&\left(k_{e,F|87}+k_{e,F|65}\right)u_{x,MCT}+\left(k_{e,F|88}+k_{e,F|66}\right)u_{y,MCT}+\\&+k_{e,F|85}u_{x,F}+k_{e,F|86}u_{y,F}+\left(k_{F|81}+k_{F|63}\right)u_{N,FC|1}+\left(k_{F|82}+k_{F|64}\right)u_{N,FC|2}+\\&+\sum_{i=2}^{3}k_{F|8i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|6i}u_{N,FB|i}+k_{F|67}u_{x,NCOI}+k_{F|68}u_{y,NCOI}=0\\[10pt]
&u_{x,FCT}=u_{x,MCT}\\
&u_{y,FCT}=u_{y,MCT}\\[10pt]
&R_{x,DUMMY}=R_{x,FCT}=-R_{x,MCT}=F_{x,CT}\\
&R_{y,DUMMY}=R_{y,FCT}=-R_{y,MCT}=F_{y,CT}\\
\end{cases}
\end{equation}

\begin{equation}
\footnotesize
\begin{cases}
&\left(k_{e,M|11}+k_{e,M|33}+k_{e,F|77}+k_{e,F|55}\right)u_{x,MCT}+\left(k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}\right)u_{y,MCT}+\\&+k_{e,M|13}u_{x,M}+k_{e,M|14}u_{y,M}+k_{e,F|75}u_{x,F}+k_{e,F|76}u_{y,F}+\\&+\left(k_{M|31}+k_{F|57}\right)u_{x,NCOI}+\left(k_{M|32}+k_{F|58}\right)u_{y,NCOI}+\\&+\left(k_{M|17}+k_{M|35}\right)u_{N,MC|7}+\left(k_{M|18}+k_{M|36}\right)u_{N,MC|8}+\left(k_{F|71}+k_{F|53}\right)u_{N,FC|1}+\left(k_{F|72}+k_{F|54}\right)u_{N,FC|2}+\\&+\sum_{i=5}^{6}k_{M|1i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|3i}u_{N,MB|i}+\sum_{i=2}^{3}k_{F|7i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|5i}u_{N,FB|i}=0\\[10pt]
&\left(k_{e,M|21}+k_{e,M|43}+k_{e,F|87}+k_{e,F|65}\right)u_{x,MCT}+\left(k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}\right)u_{y,MCT}+\\&+k_{e,M|23}u_{x,M}+k_{e,M|24}u_{y,M}+k_{e,F|85}u_{x,F}+k_{e,F|86}u_{y,F}+\\&+\left(k_{M|41}+k_{F|67}\right)u_{x,NCOI}+\left(k_{M|42}+k_{F|68}\right)u_{y,NCOI}+\\&+\left(k_{M|27}+k_{M|45}\right)u_{N,MC|7}+\left(k_{M|28}+k_{M|46}\right)u_{N,MC|8}+\left(k_{F|81}+k_{F|63}\right)u_{N,FC|1}+\left(k_{F|82}+k_{F|64}\right)u_{N,FC|2}+\\&+\sum_{i=2}^{3}k_{F|8i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|6i}u_{N,FB|i}+\sum_{i=5}^{6}k_{M|2i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|4i}u_{N,MB|i}=0\\[10pt]
&u_{x,FCT}=u_{x,MCT}\\
&u_{y,FCT}=u_{y,MCT}\\[10pt]
&R_{x,DUMMY}=R_{x,FCT}=-R_{x,MCT}=F_{x,CT}\\
&R_{y,DUMMY}=R_{y,FCT}=-R_{y,MCT}=F_{y,CT}\\
\end{cases}
\end{equation}

\begin{equation}
\scriptsize
\begin{cases}
&u_{y,MCT}=-\frac{k_{e,M|11}+k_{e,M|33}+k_{e,F|77}+k_{e,F|55}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}u_{x,MCT}+\\
&-\frac{k_{e,M|13}u_{x,M}+k_{e,M|14}u_{y,M}+k_{e,F|75}u_{x,F}+k_{e,F|76}u_{y,F}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}+\\
&-\frac{\left(k_{M|31}+k_{F|57}\right)u_{x,NCOI}+\left(k_{M|32}+k_{F|58}\right)u_{y,NCOI}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}+\\
&-\frac{\left(k_{M|17}+k_{M|35}\right)u_{N,MC|7}+\left(k_{M|18}+k_{M|36}\right)u_{N,MC|8}+\left(k_{F|71}+k_{F|53}\right)u_{N,FC|1}+\left(k_{F|72}+k_{F|54}\right)u_{N,FC|2}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}+\\
&-\frac{\sum_{i=5}^{6}k_{M|1i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|3i}u_{N,MB|i}+\sum_{i=2}^{3}k_{F|7i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|5i}u_{N,FB|i}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\\[10pt]

&\left[\left(k_{e,M|21}+k_{e,M|43}+k_{e,F|87}+k_{e,F|65}\right)+\frac{k_{e,M|11}+k_{e,M|33}+k_{e,F|77}+k_{e,F|55}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left(k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}\right)\right]u_{x,MCT}+\\
&+\left(k_{e,M|23}-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}k_{e,M|13}\right)u_{x,M}+\\
&+\left(k_{e,M|24}-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}k_{e,M|14}\right)u_{y,M}+\\
&+\left(k_{e,F|85}-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}k_{e,M|75}\right)u_{x,F}+\\
&+\left(k_{e,F|86}-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}k_{e,M|76}\right)u_{y,F}+\\
&+\left[\left(k_{M|41}+k_{F|67}\right)-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left(k_{M|31}+k_{F|57}\right)\right]u_{x,NCOI}+\\
&+\left[\left(k_{M|42}+k_{F|68}\right)-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left(k_{M|32}+k_{F|58}\right)\right]u_{y,NCOI}+\\
&+\left(k_{M|27}+k_{M|45}\right)u_{N,MC|7}+\left(k_{M|28}+k_{M|46}\right)u_{N,MC|8}+\left(k_{F|81}+k_{F|63}\right)u_{N,FC|1}+\left(k_{F|82}+k_{F|64}\right)u_{N,FC|2}+\\
&-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left[\left(k_{M|17}+k_{M|35}\right)u_{N,MC|7}+\left(k_{M|18}+k_{M|36}\right)u_{N,MC|8}\right]+\\&
-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left[\left(k_{F|71}+k_{F|53}\right)u_{N,FC|1}+\left(k_{F|72}+k_{F|54}\right)u_{N,FC|2}\right]\\
&+\sum_{i=2}^{3}k_{F|8i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|6i}u_{N,FB|i}+\sum_{i=5}^{6}k_{M|2i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|4i}u_{N,MB|i}+\\
&-\frac{\sum_{i=5}^{6}k_{M|1i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|3i}u_{N,MB|i}+\sum_{i=2}^{3}k_{F|7i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|5i}u_{N,FB|i}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}=0\\[10pt]

&u_{x,FCT}=u_{x,MCT}\\
&u_{y,FCT}=u_{y,MCT}\\[10pt]
&R_{x,DUMMY}=R_{x,FCT}=-R_{x,MCT}=F_{x,CT}\\
&R_{y,DUMMY}=R_{y,FCT}=-R_{y,MCT}=F_{y,CT}\\
\end{cases}
\end{equation}

\begin{equation}
\scriptsize
\begin{cases}
&u_{y,MCT}=-\frac{k_{e,M|11}+k_{e,M|33}+k_{e,F|77}+k_{e,F|55}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}u_{x,MCT}+\\
&-\frac{k_{e,M|13}u_{x,M}+k_{e,M|14}u_{y,M}+k_{e,F|75}u_{x,F}+k_{e,F|76}u_{y,F}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}+\\
&-\frac{\left(k_{M|31}+k_{F|57}\right)u_{x,NCOI}+\left(k_{M|32}+k_{F|58}\right)u_{y,NCOI}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}+\\
&-\frac{\left(k_{M|17}+k_{M|35}\right)u_{N,MC|7}+\left(k_{M|18}+k_{M|36}\right)u_{N,MC|8}+\left(k_{F|71}+k_{F|53}\right)u_{N,FC|1}+\left(k_{F|72}+k_{F|54}\right)u_{N,FC|2}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}+\\
&-\frac{\sum_{i=5}^{6}k_{M|1i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|3i}u_{N,MB|i}+\sum_{i=2}^{3}k_{F|7i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|5i}u_{N,FB|i}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\\[10pt]

&\left[\left(k_{e,M|21}+k_{e,M|43}+k_{e,F|87}+k_{e,F|65}\right)+\frac{k_{e,M|11}+k_{e,M|33}+k_{e,F|77}+k_{e,F|55}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left(k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}\right)\right]u_{x,MCT}+\\
&+\left(k_{e,M|23}-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}k_{e,M|13}\right)u_{x}+\\
&+\left(k_{e,M|24}-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}k_{e,M|14}\right)u_{y}+\\
&+\left(k_{e,M|23}+k_{e,F|85}-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left(k_{e,M|13}+k_{e,M|75}\right)\right)\cancelto{\approx 0}{u_{x,F}}+\\
&+\left(k_{e,M|24}+k_{e,F|86}-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left(k_{e,M|14}+k_{e,M|76}\right)\right)\cancelto{\approx 0}{u_{y,F}}+\\
&+\left[\left(k_{M|41}+k_{F|67}\right)-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left(k_{M|31}+k_{F|57}\right)\right]u_{x,NCOI}+\\
&+\left[\left(k_{M|42}+k_{F|68}\right)-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left(k_{M|32}+k_{F|58}\right)\right]u_{y,NCOI}+\\
&+\left(k_{M|27}+k_{M|45}\right)u_{N,MC|7}+\left(k_{M|28}+k_{M|46}\right)u_{N,MC|8}+\left(k_{F|81}+k_{F|63}\right)u_{N,FC|1}+\left(k_{F|82}+k_{F|64}\right)u_{N,FC|2}+\\
&-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left[\left(k_{M|17}+k_{M|35}\right)u_{N,MC|7}+\left(k_{M|18}+k_{M|36}\right)u_{N,MC|8}\right]+\\&
-\frac{k_{e,M|22}+k_{e,M|44}+k_{e,F|88}+k_{e,F|66}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}\left[\left(k_{F|71}+k_{F|53}\right)u_{N,FC|1}+\left(k_{F|72}+k_{F|54}\right)u_{N,FC|2}\right]\\
&+\sum_{i=2}^{3}k_{F|8i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|6i}u_{N,FB|i}+\sum_{i=5}^{6}k_{M|2i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|4i}u_{N,MB|i}+\\
&-\frac{\sum_{i=5}^{6}k_{M|1i}u_{N,MC|i}+\sum_{i=7}^{8}k_{M|3i}u_{N,MB|i}+\sum_{i=2}^{3}k_{F|7i}u_{N,FC|i}+\sum_{i=1}^{2}k_{F|5i}u_{N,FB|i}}{k_{e,M|12}+k_{e,M|34}+k_{e,F|78}+k_{e,F|56}}=0\\[10pt]

&u_{x,FCT}=u_{x,MCT}\\
&u_{y,FCT}=u_{y,MCT}\\[10pt]
F_{x,CT}&=R_{x,FCT}=\\&=\left(k_{e,F|77}+k_{e,F|55}\right)u_{x,FCT}+\left(k_{e,F|78}+k_{e,F|56}\right)u_{y,FCT}+\\
&+k_{e,F|75}\cancelto{\approx 0}{u_{x,F}}+k_{e,F|76}\cancelto{\approx 0}{u_{y,F}}+\\
&+\sum_{i=1}^{4}k_{e,F|7i}u_{N,FC|i}+\sum_{i=1,i\neq\left(5,6\right)}^{8}k_{e,F|5i}u_{N,FB|i}\\
F_{y,CT}&=R_{y,FCT}=\\&=\left(k_{e,F|87}+k_{e,F|65}\right)u_{x,FCT}+\left(k_{e,F|88}+k_{e,F|66}\right)u_{y,FCT}+\\
&+k_{e,F|85}\cancelto{\approx 0}{u_{x,F}}+k_{e,F|86}\cancelto{\approx 0}{u_{y,F}}+\\
&+\sum_{i=1}^{4}k_{e,F|8i}u_{N,FC|i}+\sum_{i=1,i\neq\left(5,6\right)}^{8}k_{e,F|6i}u_{N,FB|i}\\
\end{cases}
\end{equation}

\begin{equation}
\begin{cases}
F_{x,CT}&=\left(k_{e,F|77}+k_{e,F|55}\right)u_{x,MCT}+\left(k_{e,F|78}+k_{e,F|56}\right)u_{y,MCT}+\\
&+\sum_{i=1}^{4}k_{e,F|7i}u_{N,FC|i}+\sum_{i=1,i\neq\left(5,6\right)}^{8}k_{e,F|5i}u_{N,FB|i}\\
F_{y,CT}&=\left(k_{e,F|87}+k_{e,F|65}\right)u_{x,MCT}+\left(k_{e,F|88}+k_{e,F|66}\right)u_{y,MCT}+\\
&+\sum_{i=1}^{4}k_{e,F|8i}u_{N,FC|i}+\sum_{i=1,i\neq\left(5,6\right)}^{8}k_{e,F|6i}u_{N,FB|i}\\
\end{cases}
\end{equation}

\begin{equation}
\begin{cases}
F_{x,CT}&= K_{xx}u_{x}+K_{xy}u_{y}+\\
&+\sum_{i=1}^{4}K_{FC,x|i}u_{N,FC|i}+\sum_{i=1,i\neq\left(3,4,5,6\right)}^{8}K_{FB,x|i}u_{N,FB|i}+\\
&+\sum_{i=5}^{8}K_{FC,x|i}u_{N,MC|i}+\sum_{i=7}^{8}K_{MB,x|i}u_{N,FB|i}\\
F_{y,CT}&= K_{yx}u_{x}+K_{yy}u_{y}+\\
&+\sum_{i=1}^{4}K_{FC,y|i}u_{N,FC|i}+\sum_{i=1,i\neq\left(3,4,5,6\right)}^{8}K_{FB,y|i}u_{N,FB|i}+\\
&+\sum_{i=5}^{8}K_{FC,y|i}u_{N,MC|i}+\sum_{i=7}^{8}K_{MB,y|i}u_{N,FB|i}\\
{'fraction_non_alphanumeric': 0.3619236017814262, 'fraction_numerical': 0.043287387206313276, 'mean_word_length': 4.241004601652159, 'pattern_counts': {'":': 0, '<': 21, '<?xml version=': 0, '>': 21, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 18, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'dir': 'git-commits-cleaned', 'id': '1922148', 'n_tokens_mistral': 41350, 'n_tokens_neox': 39569, 'n_words': 3002}