with $\sigma_0$ as yield stress. If there is no plastic deformation - that is $\lambda=0$ - this yields $\vert\sigma^D\vert < \sigma_0$
and otherwise if $\lambda > 0$ it follows that $\vert\sigma^D\vert = \sigma_0$. That means if the stress is smaller as the yield stress
there are only elastic deformations. Therein the Index $D$ denotes the deviator part of the stress $\sigma$ which
-is dedined as
+is defined as
$$\sigma^D = \sigma - \dfrac{1}{3}tr(\sigma).$$
It describes the hydrostatic part of the stress tensor in contrast to the volumetric part. For metal the hydrostatic
stress composes the main indicator for the plastic deformation.\\
exerts on the boundary $\Gamma_C$ of the body.\\
\noindent
-The existence and uniqueness of $(u,\lambda)\in V\times K$ of this saddle point
-problem has been stated in Glowinski, Lions and Tr\'{e}moli\`{e}res: Numerical
+The existence and uniqueness of the analytical solution $(u,\lambda)\in V\times
+K$ of this saddle point problem has been stated in Glowinski, Lions and Tr\'{e}moli\`{e}res: Numerical
Analysis of Variational Inequalities, North-Holland, 1981.\\
\noindent
Now we describe an algorithm that combines the Newton-method, which we use for
the nonlinear constitutive law, with the semismooth Newton method for the contact. It
-works as follows:
+sums up the results of the sections before and works as follows:
\begin{itemize}
\item[(0)] Initialize $\mathcal{A}_k$ and $\mathcal{F}_k$, such that $\mathcal{S} = \mathcal{A}_k \cup \mathcal{F}_k$ and $\mathcal{A}_k \cap \mathcal{F}_k = \emptyset$ and set $k = 1$.
\item[(1)] Assemble the Newton matrix $a'(U^k;\varphi_i,\varphi_j)$ and the right-hand-side $F(U^k)$.
F\left(U^{k+1}\right) \vert < \delta$ then stop, else set $k=k+1$ and go to
step (1).
\end{itemize}
+\noindent
+Compare to step-41 step (1) is added but it should be clear from the sections
+above that we only linearize the problem. In step (2) we have to solve a linear
+system of equations again. And now the solution has to fulfill two stopping criterias. $\mathcal{A}_{k+1} = \mathcal{A}_k$ makes sure that the contact zones
+are iterated and the second ensures an accurate enough residual which means that
+the plastic zones are also iterated.
\section{Implementation}