manual-constitutive-laws.tex
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manual-constitutive-laws.tex
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			\section{Constitutive Laws \label{sect:smm:CL}}\index{Material}
			In order to compute an element's response to deformation, one needs to
			use an appropriate constitutive relationship. The constitutive law is
			used to compute the element's stresses from the element's strains.

			In the finite-element discretization, the constitutive formulation is
			applied to every quadrature point of each element. When the implicit
			formulation is used, the tangent matrix has to be computed.

			The chosen materials for the simulation have to be specified in the
			mesh file or, as an alternative, they can be assigned using the
			\code{element\_material} vector. For every material assigned to the
			problem one has to specify the material characteristics (constitutive
			behavior and material properties) in a text file (\eg material.dat) as
			follows:
			\begin{cpp}
			material %\emph{constitutive\_law}% %\emph{<optional flavor>}% [
			name = $value$
			rho = $value$
			...
			]
			\end{cpp}
			\index{Constitutive\_laws} where \emph{constitutive\_law} is the adopted
			constitutive law, followed by the material properties listed one by line in the
			bracket (\eg \code{name} and density \code{rho}). Some constitutive laws can
			also have an \emph{optional flavor}. For example a non-local constitutive law
			can be flavored by a weight function. The file needs to be loaded in \akantu
			using the \code{initialize} method of \akantu (as shown in
			Section~\ref{sec:writing_main})
			\begin{cpp}
			initialize("material.dat", argc, argv);
			\end{cpp}
			% or, alternatively, the \code{initFull} method.
			% \begin{cpp}
			% model.initFull("material.dat");
			% \end{cpp}

			In order to conveniently store values at each quadrature in a material
			point \akantu provides a special data structure, the
			\code{InternalField}. The internal fields are inheriting from the
			\code{ElementTypeMapArray}. Furthermore, it provides several functions for
			initialization, auto-resizing and auto removal of quadrature points.

			Sometimes it is also desired to generate random distributions of
			internal parameters. An example might be the critical stress at which the
			material fails. To generate such a field, in the material input file,
			a random quantity needs be added to the base value:
			\begin{cpp}
			sigma_c = $base$
			sigma_c = $base$ uniform [$min$, $max$]
			sigma_c = $base$ weibull [$\lambda$, $m$]
			\end{cpp}

			All parameters are real numbers. For the uniform distribution, minimum
			and maximum values have to be specified.
			Random parameters are defined as a $base$ value to which we add a random number
			that follows the chosen distribution.

			The
			\href{http://en.wikipedia.org/wiki/Uniform\_distribution\_(continuous)}{\emph{Uniform}}
			distribution is gives a random values between in $[min, max)$

			The \href{http://en.wikipedia.org/wiki/Weibull\_distribution}{\emph{Weibull}}
			distribution is characterized by the following cumulative distribution function:
			\begin{equation}
			F(x) = 1- e^{-\left({x/\lambda}\right)^m}
			\end{equation}
			which depends on $m$ and $\lambda$, which are the shape parameter and the scale
			parameter.

			The following sections describe the constitutive models implemented in
			\akantu. In Appendix~\ref{app:material-parameters} a summary of the
			parameters for all materials of \akantu is provided.


			\subsection{Elasticity}\index{Material!Elastic}

			The elastic law is a commonly used constitutive relationship that can be used
			for a wide range of engineering materials (\eg metals, concrete, rock, wood,
			glass, rubber, etc.) provided that the strains remain small (\ie small
			deformation and stress lower than yield strength).

			The elastic laws are often expressed as $\mat{\sigma} =
			\mat{C}:\mat{\varepsilon}$ with where $\mat{\sigma}$ is the Cauchy stress tensor,
			$\mat{\varepsilon}$ represents the infinitesimal strain tensor and $\mat{C}$ is the
			elastic modulus tensor.

			\subsubsection{Linear isotropic\matlabel{ssect:smm:linear-elastic-isotropic}}

			The linear isotropic elastic behavior is described by Hooke's law, which states
			that the stress is linearly proportional to the applied strain (material behaves
			like an ideal spring), as illustrated in Figure~\ref{fig:smm:cl:elastic}.
			\begin{figure}[!htb]
			\begin{center}

			\subfloat[]{
			\begin{tikzpicture}
			\draw[thick,latex-latex] (0,5) node[left] {$\sigma$} \|- (5,0) node (x) [right, below] {$\varepsilon$};
			\draw[thin] (1.5,1.5) -- (2.5,1.5) -- (2.5,2.5) node [midway, right] {E};
			\draw[very thick,color=red] (0,0) -- (4,4);
			\draw[very thick,latex-latex,color=red] (1,1) -- (3,3);
			\end{tikzpicture}
			\label{fig:smm:cl:elastic:stress_strain} }
			\hspace{0.05\textwidth} \subfloat[]{
			\raisebox{0.125\textwidth}{\includegraphics[width=0.25\textwidth,keepaspectratio=true]{figures/hooke_law.pdf}}
			\label{fig:smm:cl:elastic:hooke} }
			\caption{(a) Stress-strain curve for elastic material and (b)
			schematic representation of Hooke's law, denoted as a spring.}
			\label{fig:smm:cl:elastic}
			\end{center}
			\end{figure}
			The equation that relates the strains to the
			displacements is: % First the strain is computed (at every gauss
			point) from the displacements as follows:
			\begin{equation}
			\label{eqn:smm:strain_inf}
			\mat{\varepsilon} =
			\frac{1}{2} \left[ \nabla_0 \vec{u}+\nabla_0 \vec{u}^T \right]
			\end{equation}
			where $\mat{\varepsilon}$ represents the infinitesimal strain tensor,
			$\nabla_{0}\vec{u}$ the displacement gradient
			tensor according to the initial configuration. The constitutive equation
			for isotropic homogeneous media can be expressed as:
			\begin{equation}
			\label{eqn:smm:material:constitutive_elastic}
			\mat{\sigma } =\lambda\mathrm{tr}(\mat{\varepsilon})\mat{I}+2 \mu\mat{\varepsilon}
			\end{equation}
			where $\mat{\sigma}$ is the Cauchy stress tensor
			($\lambda$ and $\mu$ are the the first and second Lame's
			coefficients).

			In Voigt notation this correspond to
			\begin{align}
			\left[\begin{array}{c}
			\sigma_{11}\\
			\sigma_{22}\\
			\sigma_{33}\\
			\sigma_{23}\\
			\sigma_{13}\\
			\sigma_{12}\\
			\end{array}\right]
			&= \frac{E}{(1+\nu)(1-2\nu)}\left[
			\begin{array}{cccccc}
			1-\nu & \nu & \nu & 0 & 0 & 0\\
			\nu & 1-\nu & \nu & 0 & 0 & 0\\
			\nu & \nu & 1-\nu & 0 & 0 & 0\\
			0 & 0 & 0 & \frac{1-2\nu}{2} & 0 & 0 \\
			0 & 0 & 0 & 0 & \frac{1-2\nu}{2} & 0 \\
			0 & 0 & 0 & 0 & 0 & \frac{1-2\nu}{2} \\
			\end{array}\right]
			\left[\begin{array}{c}
			\varepsilon_{11}\\
			\varepsilon_{22}\\
			\varepsilon_{33}\\
			2\varepsilon_{23}\\
			2\varepsilon_{13}\\
			2\varepsilon_{12}\\
			\end{array}\right]
			\end{align}

			\subsubsection{Linear anisotropic\matlabel{ssect:smm:linear-elastic-anisotropic}}
			This formulation is not sufficient to represent all elastic material
			behavior. Some materials have characteristic orientation that have to be taken
			into account. To represent this anisotropy a more general stress-strain law has
			to be used. For this we define the elastic modulus tensor as follow:

			\begin{align}
			\left[\begin{array}{c}
			\sigma_{11}\\
			\sigma_{22}\\
			\sigma_{33}\\
			\sigma_{23}\\
			\sigma_{13}\\
			\sigma_{12}\\
			\end{array}\right]
			&= \left[
			\begin{array}{cccccc}
			c_{11} & c_{12} & c_{13} & c_{14} & c_{15} & c_{16}\\
			c_{21} & c_{22} & c_{23} & c_{24} & c_{25} & c_{26}\\
			c_{31} & c_{32} & c_{33} & c_{34} & c_{35} & c_{36}\\
			c_{41} & c_{42} & c_{43} & c_{44} & c_{45} & c_{46}\\
			c_{51} & c_{52} & c_{53} & c_{54} & c_{55} & c_{56}\\
			c_{61} & c_{62} & c_{63} & c_{64} & c_{65} & c_{66}\\
			\end{array}\right]
			\left[\begin{array}{c}
			\varepsilon_{11}\\
			\varepsilon_{22}\\
			\varepsilon_{33}\\
			2\varepsilon_{23}\\
			2\varepsilon_{13}\\
			2\varepsilon_{12}\\
			\end{array}\right]
			\end{align}

			\begin{figure}[h]
			\centering
			\begin{tikzpicture}
			\draw[thick,latex-latex] (90:3) node[left] {$\vec{e_2}$} \|- (0:3) node [right, below] {$\vec{e_1}$};
			\draw[ultra thick,latex-latex] (150:3) node[left] {$\vec{n_2}$} -- (0,0) -- (20:3) node [right] {$\vec{n_1}$};
			\end{tikzpicture}
			\caption{Material basis}
			\end{figure}

			To simplify the writing of input files the \mat{C} tensor is expressed in the
			material basis. And this basis as to be given too. This basis $\Omega_{\st{mat}}
			= \{\vec{n_1}, \vec{n_2}, \vec{n_3}\}$ is used to define the rotation $R_{ij} =
			\vec{n_j} . \vec{e_i}$. And $\mat{C}$ can be rotated in the global basis $\Omega
			= \{\vec{e_1}, \vec{e_2}, \vec{e_3}\}$ as follow:


			\begin{align}
			\mat{C}_{\Omega} &= \mat{R}_1 \mat{C}_{\Omega_{\st{mat}}} \mat{R}_2\\
			\mat{R}_1 &= \left[
			\begin{array}{cccccc}
			R_{11} R_{11} & R_{12} R_{12} & R_{13} R_{13} & R_{12} R_{13} & R_{11} R_{13} & R_{11} R_{12}\\
			R_{21} R_{21} & R_{22} R_{22} & R_{23} R_{23} & R_{22} R_{23} & R_{21} R_{23} & R_{21} R_{22}\\
			R_{31} R_{31} & R_{32} R_{32} & R_{33} R_{33} & R_{32} R_{33} & R_{31} R_{33} & R_{31} R_{32}\\
			R_{21} R_{31} & R_{22} R_{32} & R_{23} R_{33} & R_{22} R_{33} & R_{21} R_{33} & R_{21} R_{32}\\
			R_{11} R_{31} & R_{12} R_{32} & R_{13} R_{33} & R_{12} R_{33} & R_{11} R_{33} & R_{11} R_{32}\\
			R_{11} R_{21} & R_{12} R_{22} & R_{13} R_{23} & R_{12} R_{23} & R_{11} R_{23} & R_{11} R_{22}\\
			\end{array}\right]\\
			\mat{R}_2 &= \left[
			\begin{array}{cccccc}
			R_{11} R_{11} & R_{21} R_{21} & R_{31} R_{31} & R_{21} R_{31} & R_{11} R_{31} & R_{11} R_{21}\\
			R_{12} R_{12} & R_{22} R_{22} & R_{32} R_{32} & R_{22} R_{32} & R_{12} R_{32} & R_{12} R_{22}\\
			R_{13} R_{13} & R_{23} R_{23} & R_{33} R_{33} & R_{23} R_{33} & R_{13} R_{33} & R_{13} R_{23}\\
			R_{12} R_{13} & R_{22} R_{23} & R_{32} R_{33} & R_{22} R_{33} & R_{12} R_{33} & R_{12} R_{23}\\
			R_{11} R_{13} & R_{21} R_{23} & R_{31} R_{33} & R_{21} R_{33} & R_{11} R_{33} & R_{11} R_{23}\\
			R_{11} R_{12} & R_{21} R_{22} & R_{31} R_{32} & R_{21} R_{32} & R_{11} R_{32} & R_{11} R_{22}\\
			\end{array}\right]\\
			\end{align}

			\subsubsection{Linear orthotropic\matlabel{ssect:smm:linear-elastic-orthotropic}}

			A particular case of anisotropy is when the material basis is orthogonal in which case the elastic modulus tensor can be simplified and rewritten in terms of 9 independents material parameters.

			\begin{align}
			\left[\begin{array}{c}
			\sigma_{11}\\
			\sigma_{22}\\
			\sigma_{33}\\
			\sigma_{23}\\
			\sigma_{13}\\
			\sigma_{12}\\
			\end{array}\right]
			&= \left[
			\begin{array}{cccccc}
			c_{11} & c_{12} & c_{13} & 0 & 0 & 0 \\
			& c_{22} & c_{23} & 0 & 0 & 0 \\
			& & c_{33} & 0 & 0 & 0 \\
			& & & c_{44} & 0 & 0 \\
			& \multicolumn{2}{l}{\text{sym.}} & & c_{55} & 0 \\
			& & & & & c_{66}\\
			\end{array}\right]
			\left[\begin{array}{c}
			\varepsilon_{11}\\
			\varepsilon_{22}\\
			\varepsilon_{33}\\
			2\varepsilon_{23}\\
			2\varepsilon_{13}\\
			2\varepsilon_{12}\\
			\end{array}\right]
			\end{align}

			\begin{align}
			c_{11} &= E_1 (1 - \nu_{23}\nu_{32})\Gamma \qquad c_{22} = E_2 (1 - \nu_{13}\nu_{31})\Gamma \qquad c_{33} = E_3 (1 - \nu_{12}\nu_{21})\Gamma\\
			c_{12} &= E_1 (\nu_{21} - \nu_{31}\nu_{23})\Gamma = E_2 (\nu_{12} - \nu_{32}\nu_{13})\Gamma\\
			c_{13} &= E_1 (\nu_{31} - \nu_{21}\nu_{32})\Gamma = E_2 (\nu_{13} - \nu_{21}\nu_{23})\Gamma\\
			c_{23} &= E_2 (\nu_{32} - \nu_{12}\nu_{31})\Gamma = E_3 (\nu_{23} - \nu_{21}\nu_{13})\Gamma\\
			c_{44} &= \mu_{23} \qquad c_{55} = \mu_{13} \qquad c_{66} = \mu_{12} \\
			\Gamma &= \frac{1}{1 - \nu_{12} \nu_{21} - \nu_{13} \nu_{31} - \nu_{32} \nu_{23} - 2 \nu_{21} \nu_{32} \nu_{13}}
			\end{align}

			The poison ratios follow the rule $\nu_{ij} = \nu_{ji} E_i / E_j$.

			\subsection{Neo-Hookean\matlabel{ssect:smm:cl:neohookean}}\index{Material!Neohookean}
			The hyperelastic Neo-Hookean constitutive law results from an
			extension of the linear elastic relationship (Hooke's Law) for large
			deformation. Thus, the model predicts nonlinear stress-strain behavior
			for bodies undergoing large deformations.

			\begin{figure}[!htb]
			\begin{center}
			\includegraphics[width=0.4\textwidth,keepaspectratio=true]{figures/stress_strain_neo.pdf}
			\caption{Neo-hookean Stress-strain curve.}
			\label{fig:smm:cl:neo_hookean}
			\end{center}
			\end{figure}

			As illustrated in Figure~\ref{fig:smm:cl:neo_hookean}, the behavior is initially
			linear and the mechanical behavior is very close to the corresponding linear
			elastic material. This constitutive relationship, which accounts for compressibility,
			is a modified version of the one proposed by Ronald Rivlin \cite{Belytschko:2000}.

			The strain energy stored in the material is given by:
			\begin{equation}\label{eqn:smm:constitutive:neohookean_potential}
			\Psi(\mat{C}) = \frac{1}{2}\lambda_0\left(\ln J\right)^2-\mu_0\ln J+\frac{1}{2}
			\mu_0\left(\mathrm{tr}(\mat{C})-3\right)
			\end{equation}
			\noindent where $\lambda_0$ and $\mu_0$ are, respectively, Lam\'e's first parameter
			and the shear modulus at the initial configuration. $J$ is the jacobian of the deformation
			gradient ($\mat{F}=\nabla_{\!\!\vec{X}}\vec{x}$): $J=\text{det}(\mat{F})$. Finally $\mat{C}$ is the right Cauchy-Green
			deformation tensor.

			Since this kind of material is used for large deformation problems, a
			finite deformation framework should be used. Therefore, the Cauchy
			stress ($\mat{\sigma}$) should be computed through the second
			Piola-Kirchhoff stress tensor $\mat{S}$:

			\begin{equation}
			\mat{\sigma } = \frac{1}{J}\mat{F}\mat{S}\mat{F}^T
			\end{equation}

			Finally the second Piola-Kirchhoff stress tensor is given by:

			\begin{equation}
			\mat{S} = 2\frac{\partial\Psi}{\partial\mat{C}} = \lambda_0\ln J
			\mat{C}^{-1}+\mu_0\left(\mat{I}-\mat{C}^{-1}\right)
			\end{equation}

			The parameters to indicate in the material file are the same
			as those for the elastic case: \code{E} (Young's modulus), \code{nu} (Poisson's
			ratio).


			\subsection{Visco-Elasticity\matlabel{ssect:smm:cl:sls}}
			% Standard Solid rheological model, see [] J.C. Simo, T.J.R. Hughes,
			% "Computational Inelasticity", Springer (1998), see Sections 10.2 and 10.3
			Visco-elasticity is characterized by strain rate dependent
			behavior. Moreover, when such a material undergoes a deformation it
			dissipates energy. This dissipation results in a hysteresis loop in
			the stress-strain curve at every loading cycle (see
			Figure~\ref{fig:smm:cl:visco-elastic:hyst}). In principle, it can be
			applied to many materials, since all materials exhibit a visco-elastic
			behavior if subjected to particular conditions (such as high
			temperatures).
			\begin{figure}[!htb]
			\begin{center}

			\subfloat[]{
			\includegraphics[width=0.4\textwidth,keepaspectratio=true]{figures/stress_strain_visco.pdf}
			\label{fig:smm:cl:visco-elastic:hyst}
			}
			\hspace{0.05\textwidth}
			\subfloat[]{
			\raisebox{0.025\textwidth}{\includegraphics[width=0.3\textwidth,keepaspectratio=true]{figures/visco_elastic_law.pdf}}
			\label{fig:smm:cl:visco-elastic:model}
			}
			\caption{(a) Characteristic stress-strain behavior of a visco-elastic material with hysteresis loop and (b) schematic representation of the standard rheological linear solid visco-elastic model.}
			\label{fig:smm:cl:visco-elastic}
			\end{center}
			\end{figure}
			The standard rheological linear solid model (see Sections 10.2 and 10.3
			of~\cite{simo92}) has been implemented in \akantu. This model results from the
			combination of a spring mounted in parallel with a spring and a dashpot
			connected in series, as illustrated in
			Figure~\ref{fig:smm:cl:visco-elastic:model}. The advantage of this model is that
			it allows to account for creep or stress relaxation. The equation that relates
			the stress to the strain is (in 1D):
			\begin{equation}
			\frac{d\varepsilon(t)}{dt} = \left ( E + E_V \right ) ^ {-1} \cdot \left [ \frac{d\sigma(t)}{dt} + \frac{E_V}{\eta}\sigma(t) - \frac{EE_V}{\eta}\varepsilon(t) \right ]
			\end{equation}
			where $\eta$ is the viscosity. The equilibrium condition is unique and
			is attained in the limit, as $t \to \infty $. At this stage, the
			response is elastic and depends on the Young's modulus $E$. The
			mandatory parameters for the material file are the following:
			\code{rho} (density), \code{E} (Young's modulus), \code{nu} (Poisson's
			ratio), \code{Plane\_Stress} (if set to zero plane strain, otherwise
			plane stress), \code{eta} (dashpot viscosity) and \code{Ev} (stiffness
			of the viscous element).

			Note that the current standard linear solid model is applied only on the deviatoric part of the strain tensor. The spheric part of the strain tensor affects the stress tensor like an linear elastic material.

			\subsection{Small-Deformation Plasticity\matlabel{ssect:smm:cl:plastic}}\index{Material!Small-deformation Plasticity}
			The small-deformation plasticity is a simple plasticity material
			formulation which accounts for the additive decomposition of strain
			into elastic and plastic strain components. This formulation is
			applicable to infinitesimal deformation where the additive
			decomposition of the strain is a valid approximation. In this
			formulation, plastic strain is a shearing process where hydrostatic
			stress has no contribution to plasticity and consequently plasticity
			does not lead to volume change. Figure~\ref{fig:smm:cl:Lin-strain-hard}
			shows the linear strain hardening elasto-plastic behavior according to
			the additive decomposition of strain into the elastic and plastic
			parts in infinitesimal deformation as
			\begin{align}
			\mat{\varepsilon} &= \mat{\varepsilon}^e +\mat{\varepsilon}^p\\
			{\mat{\sigma}} &= 2G(\mat{\varepsilon}^e) + \lambda \mathrm{tr}(\mat{\varepsilon}^e)\mat{I}
			\end{align}

			\begin{figure}[htp]
			\centering
			{\includegraphics[scale=0.4, clip]{figures/isotropic_hardening_plasticity.pdf}}
			\caption{
			Stress-strain curve for the small-deformation plasticity with linear isotropic hardening.
			}
			\label{fig:smm:cl:Lin-strain-hard}
			\end{figure}

			\noindent In this class, the von Mises yield criterion is used. In the von Mises yield criterion, the yield is independent of the hydrostatic stress. Other yielding criteria such as Tresca and Gurson can be easily implemented in this class as well.

			In the von Mises yield criterion, the hydrostatic stresses have no effect on the plasticity and consequently the yielding occurs when a critical elastic shear energy is achieved.
			\begin{equation} \label{eqn:smm:constitutive:von Mises}
			f = \sigma_{\st{eff}} - \sigma_y = \left(\frac{3}{2} {\mat{\sigma}}^{\st{tr}} : {\mat{\sigma}}^{\st{tr}}\right)^\frac{1}{2}-\sigma_y (\mat{\varepsilon}^p)
			\end{equation}
			\begin{equation} \label{eqn:smm:constitutive:yielding}
			f < 0 \quad \textrm{Elastic deformation,} \qquad f = 0 \quad \textrm{Plastic deformation}
			\end{equation}
			where $\sigma_y$ is the yield strength of the material which can be function of plastic strain in case of hardening type of materials and ${\mat{\sigma}}^{\st{tr}}$ is the deviatoric part of stress given by
			\begin{equation} \label{eqn:smm:constitutive:deviatoric stress}
			{\mat{\sigma}}^{\st{tr}}=\mat{\sigma} - \frac{1}{3} \mathrm{tr}(\mat{\sigma}) \mat {I}
			\end{equation}

			After yielding $(f = 0)$, the normality hypothesis of plasticity determines the direction of plastic flow which is normal to the tangent to the yielding surface at the load point. Then, the tensorial form of the plastic constitutive equation using the von Mises yielding criterion (see equation 4.34) may be written as
			\begin{equation} \label{eqn:smm:constitutive:plastic contitutive equation}
			\Delta {\mat{\varepsilon}}^p = \Delta p \frac {\partial{f}}{\partial{\mat \sigma}}=\frac{3}{2} \Delta p \frac{{\mat{\sigma}}^{\st{tr}}}{\sigma_{\st{eff}}}
			\end{equation}

			In these expressions, the direction of the plastic strain increment (or equivalently, plastic strain rate) is given by $\frac{{\mat{\sigma}}^{\st{tr}}}{\sigma_{\st{eff}}}$ while the magnitude is defined by the plastic multiplier $\Delta p$. This can be obtained using the \emph{consistency condition} which impose the requirement for the load point to remain on the yielding surface in the plastic regime.

			Here, we summarize the implementation procedures for the
			small-deformation plasticity with linear isotropic hardening:
			\begin{enumerate}
			\item Compute the trial stress:
			\begin{equation}
			{\mat{\sigma}}^{\st{tr}} = {\mat{\sigma}}_t + 2G\Delta \mat{\varepsilon} + \lambda \mathrm{tr}(\Delta \mat{\varepsilon})\mat{I}
			\end{equation}
			\item Check the Yielding criteria:
			\begin{equation}
			f = (\frac{3}{2} {\mat{\sigma}}^{\st{tr}} : {\mat{\sigma}}^{\st{tr}})^{1/2}-\sigma_y (\mat{\varepsilon}^p)
			\end{equation}
			\item Compute the Plastic multiplier:
			\begin{align}
			d \Delta p &= \frac{\sigma^{tr}_{eff} - 3G \Delta P^{(k)}- \sigma_y^{(k)}}{3G + h}\\
			\Delta p^{(k+1)} &= \Delta p^{(k)}+ d\Delta p\\
			\sigma_y^{(k+1)} &= (\sigma_y)_t+ h\Delta p
			\end{align}
			\item Compute the plastic strain increment:
			\begin{equation}
			\Delta {\mat{\varepsilon}}^p = \frac{3}{2} \Delta p \frac{{\mat{\sigma}}^{\st{tr}}}{\sigma_{\st{eff}}}
			\end{equation}
			\item Compute the stress increment:
			\begin{equation}
			{\Delta \mat{\sigma}} = 2G(\Delta \mat{\varepsilon}-\Delta \mat{\varepsilon}^p) + \lambda \mathrm{tr}(\Delta \mat{\varepsilon}-\Delta \mat{\varepsilon}^p)\mat{I}
			\end{equation}
			\item Update the variables:
			\begin{align}
			{\mat{\varepsilon^p}} &= {\mat{\varepsilon}}^p_t+{\Delta {\mat{\varepsilon}}^p}\\
			{\mat{\sigma}} &= {\mat{\sigma}}_t+{\Delta \mat{\sigma}}
			\end{align}
			\end{enumerate}

			We use an implicit integration technique called \emph{the radial
			return method} to obtain the plastic multiplier. This method has the
			advantage of being unconditionally stable, however, the accuracy
			remains dependent on the step size. The plastic parameters to indicate
			in the material file are: \code{$\sigma_y$} (Yield stress) and
			\code{h} (Hardening modulus). In addition, the elastic parameters need
			to be defined as previously mentioned: \code{E} (Young's modulus),
			\code{nu} (Poisson's ratio).

			\subsection{Damage}

			In the simplified case of a linear elastic and brittle material, isotropic
			damage can be represented by a scalar variable $d$, which varies from $0$ to $1$
			for no damage to fully broken material respectively. The stress-strain
			relationship then becomes:
			\begin{equation*}
			\mat{\sigma} = (1-d)\, \mat{C}:\mat{\varepsilon}
			\end{equation*}

			where $\mat{\sigma}$, $\mat{\varepsilon}$ are the Cauchy stress and strain
			tensors, and $\mat{C}$ is the elastic stiffness tensor. This formulation relies
			on the definition of an evolution law for the damage variable. In \akantu, many
			possibilities exist and they are listed below.

			\subsubsection{Marigo\matlabel{ssect:smm:cl:damage-marigo}}

			This damage evolution law is energy based as defined by Marigo \cite{marigo81a,
			lemaitre96a}. It is an isotropic damage law.
			\begin{align}
			Y &= \frac{1}{2}\mat{\varepsilon}:\mat{C}:\mat{\varepsilon}\\
			F &= Y - Y_d - S d\\
			d &= \left\{
			\begin{array}{l l}
			\mathrm{min}\left(\frac{Y-Y_d}{S},\;1\right) & \mathrm{if}\; F > 0\\
			\mathrm{unchanged} & \mathrm{otherwise}
			\end{array}
			\right.
			\end{align}
			In this formulation, $Y$ is the strain energy release rate, $Y_d$ the
			rupture criterion and $S$ the damage energy. The non-local version of
			this damage evolution law is constructed by averaging the energy $Y$.

			\subsubsection{Mazars\matlabel{ssect:smm:cl:damage-mazars}}
			This law introduced by Mazars \cite{mazars84a} is a behavioral model to
			represent damage evolution in concrete. The governing variable in this damage
			law is the equivalent strain $\varepsilon_{\st{eq}} =
			\sqrt{<\mat{\varepsilon}>_+:<\mat{\varepsilon}>_+}$, with $<.>_+$ the positive
			part of the tensor.
			The damage the is defined as:
			\begin{align}
			D &= \alpha_t^\beta D_t + (1-\alpha_t)^\beta D_c\\
			D_t &= 1 - \frac{\kappa_0 (1- A_t)}{\varepsilon_{\st{eq}}} - A_t \exp^{-B_t(\varepsilon_{\st{eq}}-\kappa_0)}\\
			D_c &= 1 - \frac{\kappa_0 (1- A_c)}{\varepsilon_{\st{eq}}} - A_c
			\exp^{-B_c(\varepsilon_{\st{eq}}-\kappa_0)}\\
			\alpha_t &= \frac{\sum_{i=1}^3<\varepsilon_i>_+\varepsilon_{\st{nd}\;i}}{\varepsilon_{\st{eq}}^2}
			\end{align}
			With $\kappa_0$ the damage threshold, $A_t$ and $B_t$ the damage parameter in
			traction, $A_c$ and $B_c$ the damage parameter in compression, $\beta$ is the
			shear parameter. $\alpha_t$ is the coupling parameter between traction and
			compression, the $\varepsilon_i$ are the eigenstrain and the
			$\varepsilon_{\st{nd}\;i}$ are the eigenvalues of the strain if the material
			were undamaged.

			The coefficients $A$ and $B$ are the post-peak asymptotic
			value and the decay shape parameters.

			\IfFileExists{manual-constitutive-laws-non_local.tex}{\input{manual-constitutive-laws-non_local.tex}}{}

			\IfFileExists{manual-extra_materials.tex}{\input{manual-extra_materials}}{}

			\IfFileExists{manual-cohesive_laws.tex}{\input{manual-cohesive_laws}}{}


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