is zero with the outward normal $n$. If there is contact ($u_n = g$) the tangential stress $\sigma_t = \sigma\cdot n - \sigma_n n$
vanishes, because we consider a frictionless situation and the normal stress is
negative. The gap $g$ comes with the start configuration of the obstacle and the
-deformable body.
+deformable body. We refer that you have to ensure that the obstacle does not hit
+the boundary of $\Gamma_C$
\section{Derivation of the variational inequality}
Most materials - especially metals - have the property that they show some hardening effects during the forming process.
There are different constitutive laws to describe those material behaviors. The
simplest one is called linear isotropic hardening described by the flow function
-$\mathcal{F}(\tau,\eta) = \vert\tau^D\vert - (\sigma_0 + \gamma\eta)$ where
+$\mathcal{F}(\tau,\eta) = \vert\tau^D\vert - (\sigma_0 + \gamma^{\text{iso}}\eta)$ where
$\eta$ is the norm of the plastic strain $\eta = \vert \varepsilon -
A\sigma\vert$.
It can be considered by establishing an additional term in our primal-mixed formulation:\\
Find a pair $\lbrace(\sigma,\xi),u\rbrace\in \Pi (W\times L^2(\Omega,\mathbb{R}))\times V^+$ with
-$$\left(\sigma,\tau - \sigma\right) - \left(C\varepsilon(u), \tau - \sigma\right) + \gamma\left( \xi, \eta - \xi\right) \geq 0,\quad \forall (\tau,\eta)\in \Pi (W,L^2(\Omega,\mathbb{R}))$$
+$$\left(\sigma,\tau - \sigma\right) - \left(C\varepsilon(u), \tau - \sigma\right) + \gamma^{\text{iso}}\left( \xi, \eta - \xi\right) \geq 0,\quad \forall (\tau,\eta)\in \Pi (W,L^2(\Omega,\mathbb{R}))$$
$$\left(\sigma,\varepsilon(\varphi) - \varepsilon(u)\right) \geq 0,\quad \forall \varphi\in V^+,$$
-with the hardening parameter $\gamma > 0$.\\
+with the hardening parameter $\gamma^{\text{iso}} > 0$.\\
Now we want to derive a primal problem which only depends on the displacement $u$. For that purpose we
set $\eta = \xi$ and eliminate the stress $\sigma$ by applying the projection
theorem (see Grossmann, Roos: Numerical Treatment of Partial Differential
$$\left(P_{\Pi}(C\varepsilon(u)),\varepsilon(\varphi) - \varepsilon(u)\right) \geq 0,\quad \forall \varphi\in V^+,$$
with the projection:
$$P_{\Pi}(\tau):=\begin{cases}
- \tau, & \text{if }\vert\tau^D\vert \leq \sigma_0 + \gamma\xi,\\
- \hat\alpha\dfrac{\tau^D}{\vert\tau^D\vert} + \dfrac{1}{3}tr(\tau), & \text{if }\vert\tau^D\vert > \sigma_0 + \gamma\xi,
+ \tau, & \text{if }\vert\tau^D\vert \leq \sigma_0 + \gamma^{\text{iso}}\xi,\\
+ \hat\alpha\dfrac{\tau^D}{\vert\tau^D\vert} + \dfrac{1}{3}tr(\tau), & \text{if }\vert\tau^D\vert > \sigma_0 + \gamma^{\text{iso}}\xi,
\end{cases}$$
with the radius
-$$\hat\alpha := \sigma_0 + \gamma\xi .$$
+$$\hat\alpha := \sigma_0 + \gamma^{\text{iso}}\xi .$$
With the relation $\xi = \vert\varepsilon(u) - A\sigma\vert$ it is possible to eliminate $\xi$ inside the projection $P_{\Pi}$:\\
$$P_{\Pi}(\tau):=\begin{cases}
\tau, & \text{if }\vert\tau^D\vert \leq \sigma_0,\\
\alpha\dfrac{\tau^D}{\vert\tau^D\vert} + \dfrac{1}{3}tr(\tau), & \text{if }\vert\tau^D\vert > \sigma_0,
\end{cases}$$
-$$\alpha := \sigma_0 + \dfrac{\gamma}{2\mu+\gamma}\left(\vert\tau^D\vert - \sigma_0\right) ,$$
+$$\alpha := \sigma_0 + \dfrac{\gamma^{\text{iso}}}{2\mu+\gamma^{\text{iso}}}\left(\vert\tau^D\vert - \sigma_0\right) ,$$
with a further material parameter $\mu>0$ called shear modulus. We refer that
this only possible for isotropic plasticity.\\
-So what we do is to calculate the stresses by using Hooke's law for linear elastic, isotropic materials
-$$\sigma = C \varepsilon(u) = 2\mu \varepsilon^D(u) + \kappa tr(\varepsilon(u))I = \left[2\mu\left(\mathbb{I} -\dfrac{1}{3} I\otimes I\right) + \kappa I\otimes I\right]\varepsilon(u)$$
+To make things a bit easier from now on we denote
+$$\gamma := \dfrac{\gamma^{\text{iso}}}{2\mu +
+\gamma^{\text{iso}}}\in[0,1)\text{ with }\gamma^{\text{iso}}\in[0,\infty),$$
+$$\beta :=\dfrac{\sigma_0}{\vert\tau^D\vert}.$$ If $\gamma^{\text{iso}}$ tends to zero $\gamma$ tends also to zero. And if $\gamma^{\text{iso}}$ tends to
+infinity $\gamma$ tends to one. This allows us to reformulate our problem as
+follows $$P_{\Pi}(\tau):=\begin{cases}
+ \tau, & \text{if }\vert\tau^D\vert \leq \sigma_0,\\
+ \gamma\tau^D + (1-\gamma)\beta\tau^D
+ + \dfrac{1}{3}tr(\tau), & \text{if }\vert\tau^D\vert >
+ \sigma_0, \end{cases}.$$
+For further details see Suttmeier: On Plasticity with Hardening:
+An Adaptive Finite Element Discretisation, International Mathematical Forum, 5,
+2010, no. 52, 2591-2601.\\
+So what we do is to calculate the stresses
+by using Hooke's law for linear elastic, isotropic materials $$\sigma = C \varepsilon(u) = 2\mu \varepsilon^D(u) + \kappa tr(\varepsilon(u))I = \left[2\mu\left(\mathbb{I} -\dfrac{1}{3} I\otimes I\right) + \kappa I\otimes I\right]\varepsilon(u)$$
with the material parameter $\kappa>0$ (bulk modulus). The variables $I$ and
$\mathbb{I}$ denote the identity tensors of second and forth order. In that
notation $2\mu \varepsilon^D(u)$ is the deviatoric part and $\kappa
$$a'(u^i;\psi,\varphi) =
(I(x)\varepsilon(\psi),\varepsilon(\varphi)),\quad x\in\Omega,$$ $$
I(x) := \begin{cases}
-2\mu\left(\mathbb{I} - \dfrac{1}{3} I\otimes I\right) + \kappa I\otimes I, &
+C_{\mu} + C_{\kappa}, &
\quad \vert \tau^D \vert \leq \sigma_0\\
-\dfrac{\alpha}{\vert\tau^D\vert}2\mu\left(\mathbb{I} - \dfrac{1}{3} I\otimes I
-- \dfrac{\tau^D\otimes\tau^D}{\vert\tau^D\vert}\right) + \kappa I\otimes I,
-&\quad \vert \tau^D \vert > \sigma_0
+\gamma C_{\mu} + (1-\gamma)\beta\left(C_{\mu} -
+2\mu\dfrac{\tau^D\otimes\tau^D}{\vert\tau^D\vert^2}\right) + C_{\kappa}, &\quad
+\vert \tau^D \vert > \sigma_0
\end{cases}
$$
with
-$$\tau^D := C\varepsilon^D(u^i).$$
+$$C_{\mu} := 2\mu\left(\mathbb{I} - \dfrac{1}{3} I\otimes
+I\right)\quad\text{(shear part of the stress strain tensor)},$$
+$$C_{\kappa} := \kappa I\otimes I\quad\text{(bulk part of the stress strain
+tensor)},$$
+$$\tau^D := C\varepsilon^D(u^i).$$
Remark that $a(.;.)$ is not differentiable in the common sense but it is
slantly differentiable like the function for the contact problem and again we refer to
Hintermueller, Ito, Kunisch: The primal-dual active set strategy as a semismooth newton method, SIAM J. OPTIM., 2003, Vol. 13, No. 3, pp. 865-888.
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