From 6642bc80933d26941be90b5ea5f73b16c4c9e03c Mon Sep 17 00:00:00 2001 From: frohne Date: Tue, 19 Feb 2013 00:59:58 +0000 Subject: [PATCH] simplified formulation of a projection with linear hardening before and in section for linearization of the constitutive law git-svn-id: https://svn.dealii.org/trunk@28457 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-42/doc/intro.dox | 47 +++++++++++++++++++------- 1 file changed, 34 insertions(+), 13 deletions(-) diff --git a/deal.II/examples/step-42/doc/intro.dox b/deal.II/examples/step-42/doc/intro.dox index fdcfe129df..509d2e90be 100644 --- a/deal.II/examples/step-42/doc/intro.dox +++ b/deal.II/examples/step-42/doc/intro.dox @@ -54,7 +54,7 @@ row component-by-component and in a pointwise sense. Furthermore we have to distinguish two cases. The continuous and convex function $\mathcal{F}$ denotes the von Mises flow function -@f{gather*}\mathcal{F}(\tau) = \vert\tau^D\vert - \sigma_0¸\quad\text{with}\quad \tau^D +@f{gather*}\mathcal{F}(\tau) = \vert\tau^D\vert - \sigma_0�\quad\text{with}\quad \tau^D = \tau - \dfrac{1}{3}tr(\tau)I,@f} $\sigma_0$ as yield stress and $\vert .\vert$ as the Frobenius norm. If there are no plastic deformations in a particular point - that is $\lambda=0$ - this yields $\vert\sigma^D\vert < @@ -119,14 +119,14 @@ Therein $\varepsilon$ denotes the linearised deformation tensor with $\varepsilo Most materials - especially metals - have the property that they show some hardening effects during the forming process. There are different constitutive laws to describe those material behaviors. The simplest one is called linear isotropic hardening described by the flow function -$\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 -@f{gather*}\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}))@f} +@f{gather*}\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}))@f} @f{gather*}\left(\sigma,\varepsilon(\varphi) - \varepsilon(u)\right) \geq 0,\quad \forall \varphi\in V^+,@f} -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 @@ -140,20 +140,36 @@ Find the displacement $u\in V^+$ with @f{gather*}\left(P_{\Pi}(C\varepsilon(u)),\varepsilon(\varphi) - \varepsilon(u)\right) \geq 0,\quad \forall \varphi\in V^+,@f} with the projection: @f{gather*}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}@f} with the radius -@f{gather*}\hat\alpha := \sigma_0 + \gamma\xi .@f} +@f{gather*}\hat\alpha := \sigma_0 + \gamma{\text{iso}}\xi .@f} With the relation $\xi = \vert\varepsilon(u) - A\sigma\vert$ it is possible to eliminate $\xi$ inside the projection $P_{\Pi}$:\\ @f{gather*}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}@f} -@f{gather*}\alpha := \sigma_0 + \dfrac{\gamma}{2\mu+\gamma}\left(\vert\tau^D\vert - \sigma_0\right) ,@f} +@f{gather*}\alpha := \sigma_0 + \dfrac{\gamma{\text{iso}}}{2\mu+\gamma{\text{iso}}}\left(\vert\tau^D\vert - \sigma_0\right) ,@f} with a further material parameter $\mu>0$ called shear modulus. We refer that this only possible for isotropic plasticity. +To make things a bit easier from now on we denote +@f{gather*}\gamma := \dfrac{\gamma^{\text{iso}}}{2\mu + +\gamma^{\text{iso}}}\in[0,1)\text{ with }\gamma^{\text{iso}}\in[0,\infty),@f} +@f{gather*}\beta :=\dfrac{\sigma_0}{\vert\tau^D\vert}.@f} +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 +@f{gather*}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}.@f} +For further details e.g., 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 @f{gather*}\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)@f} with the material parameter $\kappa>0$ (bulk modulus). The variables $I$ and @@ -195,16 +211,21 @@ In the case of our constitutive law the Fréchet derivative of the semi-linearform $a(.;.)$ at the point $u^i$ is @f{gather*}a'(u^i;\psi,\varphi) = -(I(x)\varepsilon(\psi),\varepsilon(\varphi)),\quad x\in\Omega,@f} @f{gather*} +(I(x)\varepsilon(\psi),\varepsilon(\varphi)),\quad x\in\Omega,@f} +@f{gather*} 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} @f} with +@f{gather*}C_{\mu} := 2\mu\left(\mathbb{I} - \dfrac{1}{3} I\otimes +I\right)\quad\text{(shear part of the stress strain tensor)},@f} +@f{gather*}C_{\kappa} := \kappa I\otimes I\quad\text{(bulk part of the stress strain +tensor)},@f} @f{gather*}\tau^D := C\varepsilon^D(u^i).@f} 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 -- 2.39.5