]> https://gitweb.dealii.org/ - dealii-svn.git/commitdiff
Updated docs
authormcbride <mcbride@0785d39b-7218-0410-832d-ea1e28bc413d>
Wed, 15 Feb 2012 09:28:52 +0000 (09:28 +0000)
committermcbride <mcbride@0785d39b-7218-0410-832d-ea1e28bc413d>
Wed, 15 Feb 2012 09:28:52 +0000 (09:28 +0000)
git-svn-id: https://svn.dealii.org/trunk@25078 0785d39b-7218-0410-832d-ea1e28bc413d

deal.II/examples/step-44/doc/intro.dox

index 1a05c6ce1d06bc9a535e141707f55550c3dafafd..a2f124c1b8e238ac4ee362a10f9fc6d9f92f918c 100644 (file)
@@ -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]
 
-<h3> neo-Hookean materials </h3>
+<h3> Nseo-Hookean materials </h3>
 
 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.
 </ol>          
 
+ @image html "setup.png"
+
 

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