\subsection{Cohesive laws}
\label{sec:cohesive-laws}

\subsubsection{Snozzi-Molinari Law\matlabel{ssect:smm:cl:coh-snozzi}}
\begin{figure}[!hbt]
  \centering
  \subfloat[Linear]{\includegraphics[width=0.4\textwidth]{figures/linear_cohesive_law}}
  \qquad
  \subfloat[Bilinear]{\includegraphics[width=0.4\textwidth]{figures/bilinear_cohesive_law}}
  \caption{Irreversible cohesive laws for explicit simulations.}
  \label{fig:smm:coh:linear_cohesive_law}
\end{figure}

\akantu includes the Snozzi-Molinari~\cite{snozzi_cohesive_2013}
linear irreversible cohesive law (see
Figure~\ref{fig:smm:coh:linear_cohesive_law}). It is an extension to
the Camacho-Ortiz~\cite{camacho_computational_1996} cohesive law in
order to make dissipated fracture energy path-dependent. The concept
of free potential energy is dropped and a new independent parameter
$\kappa$ is introduced:
\begin{equation}
  \kappa = \frac{G_\mathrm{c, II}}{G_\mathrm{c, I}}
\end{equation}

where $G_\mathrm{c, I}$ and $G_\mathrm{c, II}$ are the
necessary works of separation per unit area to open completely a
cohesive zone under mode I and mode II, respectively. Their model yields to the
following equation for cohesive tractions $\vec{T}$ in case of crack
opening ${\delta}$:
\begin{equation}
  \label{eq:smm:coh:tractions}
  \vec{T} = \left( \frac{\beta^2}{\kappa} \Delta_\mathrm{t} \vec{t} +
    \Delta_\mathrm{n} \vec{n} \right)
  \frac{\sigma_\mathrm{c}}{\delta}
  \left( 1- \frac{\delta}{\delta_\mathrm{c}} \right)
  = \hat{\vec T}\,
  \frac{\sigma_\mathrm{c}}{\delta}
  \left( 1- \frac{\delta}{\delta_\mathrm{c}} \right)
\end{equation}

where $\sigma_\mathrm{c}$ is the material strength along the fracture,
$\delta_\mathrm{c}$ the critical effective displacement after which
cohesive tractions are zero (complete decohesion), $\Delta_\mathrm{t}$
and $\Delta_\mathrm{n}$ are the tangential and normal components of
the opening displacement vector $\vec{\Delta}$, respectively. The
parameter $\beta$ is a weight that indicates how big the tangential
opening contribution is. The effective opening displacement is:
\begin{equation}
  \delta = \sqrt{\frac{\beta^2}{\kappa^2} \Delta_\mathrm{t}^2 +
    \Delta_\mathrm{n}^2}
\end{equation}
In case of unloading or reloading $\delta < \delta_\mathrm{max}$,
tractions are calculated as:
\begin{align}
  T_\mathrm{n} &= \Delta_\mathrm{n}\,
  \frac{\sigma_\mathrm{c}}{\delta_\mathrm{max}}
  \left( 1- \frac{\delta_\mathrm{max}}{\delta_\mathrm{c}} \right) \\
  T_\mathrm{t} &= \frac{\beta^2}{\kappa}\, \Delta_\mathrm{t}\,
  \frac{\sigma_\mathrm{c}}{\delta_\mathrm{max}}
  \left( 1- \frac{\delta_\mathrm{max}}{\delta_\mathrm{c}} \right)
\end{align}
so that they vary linearly between the origin and the maximum attained
tractions. As shown in Figure~\ref{fig:smm:coh:linear_cohesive_law},
in this law, the dissipated and reversible energies are:
\begin{align}
  E_\mathrm{diss} &= \frac{1}{2} \sigma_\mathrm{c}\, \delta_\mathrm{max}\\[1ex]
  E_\mathrm{rev} &= \frac{1}{2} T\, \delta
\end{align}
Moreover, a damage parameter $D$ can be defined as:
\begin{equation}
  D = \min \left(
    \frac{\delta_\mathrm{max}}{\delta_\mathrm{c}},1 \right)
\end{equation}
which varies from 0 (undamaged condition) and 1 (fully
damaged condition). This variable can only increase because damage is
an irreversible process. A simple penalty contact model has been incorporated
in the cohesive law so that normal tractions can be returned in
case of compression:
\begin{equation}
  T_\mathrm{n} = \alpha \Delta_\mathrm{n} \quad\text{if
    $\Delta_\mathrm{n} < 0$}
\end{equation}
where $\alpha$ is a stiffness parameter that defaults to zero. The
relative contact energy is equivalent to reversible energy but in
compression.

The material name of the linear decreasing cohesive law  is
\code{material\_cohesive\_linear} and its parameters with their
respective default values are:
\begin{itemize}
\item \code{sigma\_c}: 0
\item \code{beta}: 0
\item \code{G\_cI}: 0
\item \code{G\_cII}: 0
\item \code{kappa}: 1
\item \code{penalty}: 0
\end{itemize}
A random number generator can be used to assign a random
$\sigma_\mathrm{c}$ to each facet following a given
distribution (see Section~\ref{sect:smm:CL}).

The bilinear constitutive law works exactly the same way as the linear
one, except for the additional parameter \code{delta\_0} that by
default is zero. Two examples for the extrinsic and intrinsic cohesive
elements and also an example to assign different properties to
intergranular and transgranular cohesive elements can be found in
the folder \code{\examplesdir/cohesive\_element/}.

\subsubsection{Exponential Cohesive Law\matlabel{ssect:smm:cl:coh-exponential}}

Ortiz and Pandolfi proposed this cohesive law in 1999~\cite{ortiz1999}.  The
traction-opening equation for this law is as follows:
\begin{equation}
  \label{eq:exponential_law}
  T = e \sigma_c \frac{\delta}{\delta_c}e^{-\delta/ \delta_c}
\end{equation}
This equation is plotted in Figure~\ref{fig:smm:CL:ECL}. The term
$\partial{\vec{T}}/ \partial{\delta}$ of
equation~\eqref{eq:cohesive_stiffness} after the necessary derivation
can expressed as
\begin{equation}
  \label{eq:tangent_cohesive}
  \frac{\partial{\vec{T}}} {\partial{\delta}} = \hat{\vec{T}} \otimes
  \frac                       {\partial{(T/\delta)}}{\partial{\delta}}
  \frac{\hat{\vec{T}}}{\delta}+ \frac{T}{\delta}  \left[ \beta^2 \mat{I} +
  \left(1-\beta^2\right) \left(\vec{n} \otimes \vec{n}\right)\right]
\end{equation}
where
\begin{equation}
  \frac{\partial{(T/ \delta)}}{\partial{\delta}} = \left\{\begin{array} {l l}
      -e  \frac{\sigma_c}{\delta_c^2  }e^{-\delta  /  \delta_c} &  \quad  if
      \delta \geq \delta_{max}\\
      0 & \quad if \delta < \delta_{max}, \delta_n > 0
    \end{array} \right.
\end{equation}


\begin{figure}[!htb]
  \begin{center}
    \includegraphics[width=0.6\textwidth,keepaspectratio=true]{figures/cohesive_exponential.pdf}
    \caption{Exponential cohesive law}
    \label{fig:smm:CL:ECL}
  \end{center}
\end{figure}


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