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.
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
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
\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:
\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
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\\