From 1b2ebedf4bf587b2a22dd7841299129320e5d584 Mon Sep 17 00:00:00 2001 From: mcbride Date: Wed, 15 Feb 2012 09:28:52 +0000 Subject: [PATCH] Updated docs git-svn-id: https://svn.dealii.org/trunk@25078 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-44/doc/intro.dox | 14 ++++++++------ 1 file changed, 8 insertions(+), 6 deletions(-) diff --git a/deal.II/examples/step-44/doc/intro.dox b/deal.II/examples/step-44/doc/intro.dox index 1a05c6ce1d..a2f124c1b8 100644 --- a/deal.II/examples/step-44/doc/intro.dox +++ b/deal.II/examples/step-44/doc/intro.dox @@ -192,7 +192,7 @@ If the Helmholtz free energy depends on the left Cauchy-Green tensor $\mathbf{b} Following the multiplicative decomposition of the deformation gradient, the Helmholtz free energy can be decomposed as @f[ - \Psi(\mathbf{b}) = \Psi(\mathbf{J})_{\text{vol}} + \Psi(\overline{\mathbf{b}})_{\text{iso}} \, . + \Psi(\mathbf{b}) = \Psi_{\text{vol}}(\mathbf{J}) + \Psi_{\text{iso}}(\overline{\mathbf{b}}) \, . @f] Similarly, the Kirchhoff stress can be decomposed into volumetric and isochoric parts as $\boldsymbol{\tau} = \boldsymbol{\tau}_{\text{vol}} + \boldsymbol{\tau}_{\text{iso}}$ where: @f{align*} @@ -215,24 +215,24 @@ The fictitious Cauchy stress tensor $\overline{\boldsymbol{\tau}}$ is defined by := 2 \overline{\mathbf{b}} \dfrac{\partial \Psi_{\textrm{iso}}(\overline{\mathbf{b}})}{\partial \overline{\mathbf{b}}} \, . @f] -

neo-Hookean materials

+

Nseo-Hookean materials

The Helmholtz free energy corresponding to a compressible neo-Hookean material is given by @f[ \Psi \equiv \underbrace{\kappa [ \mathcal{G}(J) ] }_{\Psi_{\textrm{vol}}(J)} - + \underbrace{\bigl[c_1 [ \overline{I}_1 - 3] \bigr]}_{\Psi(\overline{\mathbf{b}})_{\text{iso}}} \, , + + \underbrace{\bigl[c_1 [ \overline{I}_1 - 3] \bigr]}_{\Psi_{\text{iso}}(\overline{\mathbf{b}})} \, , @f] where $\kappa := \lambda + 2/3 \mu$ is the bulk modulus and $\overline{I}_1 := \textrm{tr}\overline{\mathbf{b}}$. The function $\mathcal{G}(J)$ is required to be strictly convex and satisfy the condition $\mathcal{G}(1) = 0$. -In this work $\mathcal{G}:=\bigl[ \frac{1}{2} [ \widetilde{J}^{2} - 1 ] - \textrm{ln}( \widetilde{J}) ] \bigr]$. +In this work $\mathcal{G}:=\bigl[ \frac{1}{2} [{J}^{2} - 1 ] - \textrm{ln}( {J}) ] \bigr]$. Incompressibility imposes the iscohoric consraint that $J=1$ for all motions $\mathbf{\varphi}$. The Helmholtz free energy corresponding to an incompressible neo-Hookean material is given by @f[ \Psi \equiv - \underbrace{\bigl[ c_1 [ I_1 - 3] \bigr] }_{\Psi(\mathbf{b})_{\textrm{iso}}} \, , + \underbrace{\bigl[ c_1 [ I_1 - 3] \bigr] }_{\Psi_{\textrm{iso}}(\mathbf{b})} \, , @f] $ I_1 := \textrm{tr}\mathbf{b} $. Thus, the incompressible response of obtained by removing the volumetric component from the compressible free energy. @@ -298,7 +298,7 @@ The three-field variational principle used here is given by \Pi(\mathbf{\Xi}) := \int_\Omega \bigl[ \Psi_{\textrm{vol}}(\widetilde{J}) + p[J(\mathbf{u}) - \widetilde{J}] - + \Psi_{\textrm{iso}}(\mathbf{b}(\mathbf{u})) + + \Psi_{\textrm{iso}}(\overline{\mathbf{b}}(\mathbf{u})) \bigr] \textrm{d}v + \Pi_{\textrm{ext}} \, . @f] @@ -543,4 +543,6 @@ This benchmark problem is taken from 291-304. + @image html "setup.png" + -- 2.39.5