diff --git a/doc/manual/manual-constitutive-laws.tex b/doc/manual/manual-constitutive-laws.tex new file mode 100644 index 000000000..6c1b701a2 --- /dev/null +++ b/doc/manual/manual-constitutive-laws.tex @@ -0,0 +1,324 @@ +\subsection{Neo-Hookean}\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, Lamé'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{Small-Deformation Plasticity}\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{Visco-Elasticity} + +% 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\epsilon(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}\epsilon(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{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} +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} +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. + + +\subsection{Summary}\index{Material!List} + +The list of all the materials available in Akantu is summarized in Tables \ref{tab:smm:cl:summary:list} as well as the keyword required for each material and the assosiated material properties. + +\begin{table}[h!] + \begin{center} + \begin{tabular}[c]{ m{3.5cm} | l | c | p{3.5cm} } + Material & Keyword & Parameter & Description \\ + \hline + %%%%%%%%%%%%%%%%% + Linear elastic isotropic & \code{elastic} & - & Table \ref{tab:smm:cl:summary:base}\\ + \hline + %%%%%%%%%%%%%%%%%% + Linear elastic orthotropic & \code{elastic\_orthotropic} & \code{Cij} & Tangent matrix coefficients (i,j = 1,2, ... voigt size ) \\ + \hline + %%%%%%%%%%%%%%%%%% + Linear elastic anisotropic & \code{elastic\_anisotropic} & \code{n1} & Direction of main material axis \\ + & & \code{n2} & Direction of secondary material axis \\ + & & \code{n3} & Direction of tertiery material axis \\ + & & \code{Cij} & Tangent matrix coefficients (i,j = 1,2, ... voigt size ) \\ + & & \code{alpha} & Proportion of viscous stress\\ + \hline + %%%%%%%%%%%%%%%%%% + Neohookean (Finite-strain) \cite{Belytschko:2000} & \code{neohookean} & - & Table \ref{tab:smm:cl:summary:base}\\ + \hline + %%%%%%%%%%%%%%%%%% + Standard linear solid \cite{simo92} & \multirow{2}{*}{\code{sls\_deviatoric}} & - & Table \ref{tab:smm:cl:summary:base}\\ + & & \code{Eta} & Viscosity\\ + & & \code{Ev} & Stiffness of the viscous element \\ + \hline + %%%%%%%%%%%%%%%%%% + Elasto-plastic linear isotropic hardening & \code{plastic\_linear\_isotropic\_hardening} & - & Table \ref{tab:smm:cl:summary:base}\\ + & & \code{h} & Hardening modulus\\ + & & \code{sigma\_y} & Yielding stress\\ + \hline + %%%%%%%%%%%%%%%%%% + Visco-plastic & \code{visco\_plastic} & - & Table \ref{tab:smm:cl:summary:base}\\ + & & \code{rate} & Rate sensitivity component\\ + & & \code{edot0} & Reference strain rate\\ + & & \code{ts} & Time \\ + \hline + \end{tabular} + \end{center} + \caption{List of material properties with their corresponding keywords and material parameters.} + \label{tab:smm:cl:summary:list} +\end{table} + +\vspace{0.5cm} + +In addition to the properties presented in Table \ref{tab:smm:cl:summary:list}, every material also has the parameter ''\code{rho}`` which corresponds to the density. The properties listed in Table \ref{tab:smm:cl:summary:base} correspond to the parameters required to describe a linear elastic isotropic material, however those parameters are also commun to most of the materials available as previously described in Table \ref{tab:smm:cl:summary:list}. + +\begin{table}[h!] + \begin{center} + \begin{tabular}[c]{ l | p{6.5cm} }\label{tab:smm:cl:summary:base} + Parameter & Description \\ + \hline + %%%%%%%%%%%%%%%%% + \code{rho} & Density\\ + \code{E} & Young's modulus\\ + \code{nu} & Poisson's ratio\\ + \code{Plane\_Stress} & Plane stress simplification (only 2D problems)\\ + \hline + \end{tabular} + \end{center} + \caption{List of material parameters shared my most materials} + \label{tab:smm:cl:summary:base} +\end{table} + +\IfFileExists{manual-extra_materials.tex}{\input{manual-extra_materials}}{} + +\IfFileExists{manual-cohesive_laws.tex}{\input{manual-cohesive_laws}}{} + + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "manual" +%%% End: