]> https://gitweb.dealii.org/ - dealii-svn.git/commitdiff
changed definition of vol free energy to match standard
authormcbride <mcbride@0785d39b-7218-0410-832d-ea1e28bc413d>
Thu, 16 Feb 2012 14:05:32 +0000 (14:05 +0000)
committermcbride <mcbride@0785d39b-7218-0410-832d-ea1e28bc413d>
Thu, 16 Feb 2012 14:05:32 +0000 (14:05 +0000)
git-svn-id: https://svn.dealii.org/trunk@25099 0785d39b-7218-0410-832d-ea1e28bc413d

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

index ad84d560b9585fd240ea0094abef5ade46f92846..572474a1082a04ba67542ec5258d24b3c9266d72 100644 (file)
@@ -112,7 +112,7 @@ It is also symmetric and positive definite.
 The Green-Lagrange strain tensor is defined by
 @f[
        \mathbf{E}:= \frac{1}{2}[\mathbf{C} - \mathbf{I} ]
-               = \underbrace{\frac{1}{2}[\textrm{Grad}^T\ \mathbf{U} + \textrm{Grad}\ mathbf{U}]}_{\boldsymbol{\varepsilon}}
+               = \underbrace{\frac{1}{2}[\textrm{Grad}^T \mathbf{U} +  \textrm{Grad}\mathbf{U}]}_{\boldsymbol{\varepsilon}}
                        + \frac{1}{2}[\textrm{Grad}^T\ \mathbf{U}][\textrm{Grad}\ \mathbf{U}] \, ,
 @f]
 If the assumption of infinitesimal deformations is valid, then the second term
@@ -242,7 +242,7 @@ The Helmholtz free energy corresponding to a compressible <a href="http://en.wik
 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 $\mathcal{G}:=\bigl[ \frac{1}{2} [{J}^{2} - 1 ] - \textrm{ln}( {J}) ] \bigr]$.
+In this work $\mathcal{G}:=\frac{1}{4} [ J^2 - 1 - 2\textrm{ln}J ]$.
 
 Incompressibility imposes the isochoric constraint that $J=1$ for all motions $\mathbf{\varphi}$.
 The Helmholtz free energy corresponding to an incompressible neo-Hookean material is given by
index bffb02b48951c507404298c0999690b177980ba8..506d58121c651b45f0218f23eab925d422b7674b 100644 (file)
@@ -583,7 +583,7 @@ private:
 // That is $\overline{I}_1 :=\textrm{tr}(\overline{\mathbf{b}})$.
 // In this example the SEF that governs the volumetric
 // response is defined as
-// $ \Psi_{\text{vol}}(\widetilde{J})  = \kappa \bigl[ \frac{1}{2} [ \widetilde{J}^{2} - 1 ] - \textrm{ln}( \widetilde{J}) ] \bigr]  $
+// $ \Psi_{\text{vol}}(\widetilde{J})  = \kappa \frac{1}{4} [ \widetilde{J}^2 - 1 - 2\textrm{ln}\widetilde{J} ]$.
 // where $\kappa:= \lambda + 2/3 \mu$ is the <a href="http://en.wikipedia.org/wiki/Bulk_modulus">bulk modulus</a> and
 // $\lambda$ is <a href="http://en.wikipedia.org/wiki/Lam%C3%A9_parameters">Lame's first parameter</a>.
 template<int dim>
@@ -645,7 +645,7 @@ public:
        // Derivative of the volumetric free energy wrt $\widetilde{J}$
        // return $\frac{\partial \Psi_{\text{vol}}(\widetilde{J})}{\partial \widetilde{J}}$
        double get_dPsi_vol_dJ(void) const {
-               return kappa * (J_tilde - 1.0 / J_tilde);
+               return (kappa / 2.0) * (J_tilde - 1.0 / J_tilde);
        }
 
        // Second derivative of the volumetric free energy wrt $\widetilde{J}$
@@ -654,7 +654,7 @@ public:
        // We calculate
        // $\frac{\partial^2 \Psi_{\textrm{vol}}(\widetilde{J})}{\partial \widetilde{J} \partial \widetilde{J}}$
        double get_d2Psi_vol_dJ2(void) const {
-               return kappa * (1.0 + 1.0 / (J_tilde * J_tilde));
+               return ( (kappa / 2.0) * (1.0 + 1.0 / (J_tilde * J_tilde)));
        }
 
        // Return various data that we choose to store with the material

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