From 51e6c81c306340231ec8ff67e49a36ce30aa11c1 Mon Sep 17 00:00:00 2001 From: mcbride Date: Wed, 22 Feb 2012 09:59:31 +0000 Subject: [PATCH] step:44 updated introduction git-svn-id: https://svn.dealii.org/trunk@25142 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-44/doc/intro.dox | 63 ++++++++++++++++++++++---- 1 file changed, 54 insertions(+), 9 deletions(-) diff --git a/deal.II/examples/step-44/doc/intro.dox b/deal.II/examples/step-44/doc/intro.dox index 329dc30b0a..155bc210b4 100644 --- a/deal.II/examples/step-44/doc/intro.dox +++ b/deal.II/examples/step-44/doc/intro.dox @@ -351,13 +351,13 @@ The stationarity of the potential follows as + \dfrac{\partial \Pi(\mathbf{\Xi})}{\partial \widetilde{p}} \delta \widetilde{p} + \dfrac{\partial \Pi(\mathbf{\Xi})}{\partial \widetilde{J}} \delta \tilde{J} \\ - &= \int_{\Omega_0} \left\{ - \textrm{grad}\ \delta\mathbf{u} : [ \underbrace{[\widetilde{p} J \mathbf{I}]}_{\equiv \boldsymbol{\tau}_{\textrm{iso}}} - + \boldsymbol{\tau}_{\textrm{vol}}] + &= \int_{\Omega_0} \left[ + \textrm{grad}\ \delta\mathbf{u} : [ \underbrace{[\widetilde{p} J \mathbf{I}]}_{\equiv \boldsymbol{\tau}_{\textrm{vol}}} + + \boldsymbol{\tau}_{\textrm{iso}}] + \delta \widetilde{p}\, [ J(\mathbf{u}) - \widetilde{J}] + \delta \widetilde{J}\left[ \dfrac{\textrm{d} \Psi_{\textrm{vol}}(\widetilde{J})}{\textrm{d} \widetilde{J}} -\widetilde{p}\right] - \right\}~\textrm{d}V + \right]~\textrm{d}V \\ &\quad - \int_{\Omega_0} \delta \mathbf{u} \cdot \mathbf{B}^\text{p}~\textrm{d}V - \int_{\partial \Omega_{0,\boldsymbol{\sigma}}} \delta \mathbf{u} \cdot \mathbf{T}^\text{p}~\textrm{d}A @@ -365,11 +365,21 @@ The stationarity of the potential follows as &=0 \, , @f} for all virtual displacements $\delta \mathbf{u} \in H^1(\Omega)$ subject to the constraint that $\mathbf{u} = \mathbf{0}$ on $\partial \Omega_{\mathbf{u}}$, and all virtual pressures $\delta \widetilde{p} \in L^2(\Omega)$ and virtual dilatations $\delta \widetilde{J} \in L^2(\Omega)$. -One should note that the definitions of the volumetric Cauchy stress and the subsequent tangent differs slightly from the general form given in the section on hyperelastic materials. -This is because the pressure $\widetilde{p}$ is now a primary field. -Note that although the variables are all expressed in terms of spatial quantities, the domain of integration is the reference configuration. -This approach is called a total-Lagrangian formulation. -The approach given in step-18 could be called updated Lagrangian. + +One should note that the definitions of the volumetric Cauchy stress in the three field formulation +$\boldsymbol{\tau}_{\textrm{vol}} \equiv \widetilde{p} J \mathbf{I}$ + and the subsequent volumetric tangent differs slightly from the general form given in the section on hyperelastic materials where +$\boldsymbol{\tau}_{\textrm{vol}} \equiv p J\mathbf{I}$. +This is because the pressure $\widetilde{p}$ is now a primary field as opposed to a constitutively derived quantity. +One needs to carefully distinguish between the primary fields and those obtained from the constitutive relations. + +@note Although the variables are all expressed in terms of spatial quantities, the domain of integration is the initial configuration. +This approach is called a total-Lagrangian formulation . +The approach given in step-18, where the domain of integration is the current configuration, could be called an updated Lagrangian formulation . +These various merits of these two approaches are discussed widely in the literature. +It should be noted however that they are equivalent. + + The Euler-Lagrange equations corresponding to the residual are: @f{align*} &\textrm{div}\ \boldsymbol{\sigma} + \mathbf{b}^\text{p} = \mathbf{0} && \textrm{[equilibrium]} @@ -382,6 +392,21 @@ The first equation is the equilibrium equation in the spatial setting. The second is the constraint that $J(\mathbf{u}) = \widetilde{J}$. The third is the definition of the pressure $\widetilde{p}$. +@note The simplified single-field derivation ($\mathbf{u}$ is the only primary variable) below makes it clear how we transform the limits of integration to the reference domain. +@f{align*} +&= +\int_{\Omega} [-\mathrm{grad}\delta \mathbf{u}:\boldsymbol{\sigma} + \delta \mathbf{u} \cdot\mathbf{b}^\text{p}]~\mathrm{d}v + + \int_{\partial \Omega} \delta \mathbf{u} \cdot \mathbf{t}^\text{p}~\mathrm{d}a \\ +&= +- \int_{\Omega} \mathrm{grad}\delta \mathbf{u}:\boldsymbol{\tau}~\mathrm{d}V ++ \int_{\Omega_0} \delta \mathbf{u} \cdot J\mathbf{b}^\text{p}~\mathrm{d}V + + \int_{\partial \Omega_0} \delta \mathbf{u} \cdot \mathbf{T}^\text{p}~\mathrm{d}A \\ +&= +- \int_{\Omega_0} \mathrm{grad}\delta \mathbf{u}:\boldsymbol{\tau}~\mathrm{d}V ++ \int_{\Omega_0} \delta \mathbf{u} \cdot \mathbf{B}^\text{p}~\mathrm{d}V + + \int_{\partial \Omega_{0,\sigma}} \delta \mathbf{u} \cdot \mathbf{T}^\text{p}~\mathrm{d}A \,. +@f} + We will use the iterative Newton-Raphson method to solve the nonlinear residual equation $R$. For the sake of simplicity we assume dead loading, i.e. the loading does not change due to the deformation. The change in the solution between the known state at $t_{\textrm{n}-1}$ @@ -571,6 +596,26 @@ The procedure to construct the various contributions is as follows: @f] +

The material class

+ +A good object-oriented design of a Material class would facilitate the extension of this tutorial to a wide range of material types. +In this tutorial we simply have one Material class named Material_Compressible_Neo_Hook_Three_Field. +Ideally this class would derive from a class HyperelasticMaterial which would derive from the base class Material. +The three-field nature of the formulation used here also complicates the matter. + +The free energy function for the three field formulation is $\Psi = \Psi_\text{vol}(\widetilde{J}) + \Psi_\text{iso}(\overline{\mathbf{b}})$. +The isochoric part of the Kirchhoff stress ${\boldsymbol{\tau}}_{\text{iso}}(\overline{\mathbf{b}})$ is identical to that obtained using a one-field formulation for a hyperelastic material. +However, the volumetric part of the free energy is now a function of the primary variable $\widetilde{J}$. +Thus, for a three field formulation the constitutive response for the volumetric part of the Kirchhoff stress ${\boldsymbol{\tau}}_{\text{vol}}$ (and the tangent) is not given by the hyperelastic constitutive law as in a one-field formulation. +One can label the term +$\boldsymbol{\tau}_{\textrm{vol}} \equiv \widetilde{p} J \mathbf{I}$ +as the volumetric Kirchhoff stress, but the pressure $\widetilde{p}$ is not derived from the free energy; it is a primary field. + +In order to have a flexible approach, it was decided that the Material_Compressible_Neo_Hook_Three_Field would still be able to calculate and return a volumetric Kirchhoff stress and tangent. +In order to do this, we choose to store the interpolated primary fields $\widetilde{p}$ and $\widetilde{J}$ in the Material_Compressible_Neo_Hook_Three_Field class associated with the quadrature point. +This decision should be revisited at a later stage when the tutorial is extended to account for other materials. + +

Numerical example

The numerical example considered here is a nearly-incompressible block under compression. -- 2.39.5