From: bangerth Date: Wed, 15 Feb 2012 17:45:07 +0000 (+0000) Subject: Since all png files are copied into the same directory, they need to have a unique... X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=18e967fabd6e183299feb76d7af1bb9380436cb1;p=dealii-svn.git Since all png files are copied into the same directory, they need to have a unique prefix, for which we typically choose 'step-xx.'. Do so here. Also fix a typo. git-svn-id: https://svn.dealii.org/trunk@25087 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-44/doc/intro.dox b/deal.II/examples/step-44/doc/intro.dox index a2f124c1b8..24a2733694 100644 --- a/deal.II/examples/step-44/doc/intro.dox +++ b/deal.II/examples/step-44/doc/intro.dox @@ -1,29 +1,29 @@

Introduction

-The subject of this tutorial is nonlinear solid mechanics. -A three-field formulation is used to model the fully-nonlinear (geometrical and material) response of an isotropic continuum body. -The material response is approximated as hyperelastic. +The subject of this tutorial is nonlinear solid mechanics. +A three-field formulation is used to model the fully-nonlinear (geometrical and material) response of an isotropic continuum body. +The material response is approximated as hyperelastic. Additionally, the three-field formulation employed is valid for quasi-incompressible as well as compressible materials. -The objective of this presentation is to provide a basis for using deal.II for problems in nonlinear solid mechanics. -The linear problem was addressed in step-8. -A non-standard, hypoelastic-type form of the geometrically nonlinear problem was partially considered in step-18: a rate form of the linearised constitutive relations are used and the problem domain evolves with the motion. -Important concepts surrounding the nonlinear kinematics are absent in the theory and implementation. -Step-18 does, however, describe many of the key concepts to implement elasticity within the framework of deal.II. +The objective of this presentation is to provide a basis for using deal.II for problems in nonlinear solid mechanics. +The linear problem was addressed in step-8. +A non-standard, hypoelastic-type form of the geometrically nonlinear problem was partially considered in step-18: a rate form of the linearised constitutive relations are used and the problem domain evolves with the motion. +Important concepts surrounding the nonlinear kinematics are absent in the theory and implementation. +Step-18 does, however, describe many of the key concepts to implement elasticity within the framework of deal.II. -We begin with a crash-course in nonlinear kinematics. -For the sake of simplicity, we restrict our attention to the quasi-static problem. -Thereafter, various key stress measures are introduced and the constitutive model described. +We begin with a crash-course in nonlinear kinematics. +For the sake of simplicity, we restrict our attention to the quasi-static problem. +Thereafter, various key stress measures are introduced and the constitutive model described.

List of references

-The three-field formulation implemented here was pioneered by Simo et al (1985) and is known as the mixed Jacobian-pressure formulation. -Important related contributions include those by Simo and Taylor (1991), and Miehe (1994). -The notation adopted here draws heavily on the excellent overview of the theoretical aspects of nonlinear solid mechanics by Holzapfel (2001). +The three-field formulation implemented here was pioneered by Simo et al (1985) and is known as the mixed Jacobian-pressure formulation. +Important related contributions include those by Simo and Taylor (1991), and Miehe (1994). +The notation adopted here draws heavily on the excellent overview of the theoretical aspects of nonlinear solid mechanics by Holzapfel (2001).
    -
  1. J.C. Simo, R.L. Taylor and K.S. Pister (1985), +
  2. J.C. Simo, R.L. Taylor and K.S. Pister (1985), Variational and projection methods for the volume constraint in finite deformation elasto-plasticity, Computer Methods in Applied Mechanics and Engineering , 51 , 1-3, @@ -34,12 +34,12 @@ The notation adopted here draws heavily on the excellent overview of the theoret Computer Methods in Applied Mechanics and Engineering , 85 , 3, 273-310; -
  3. C. Miehe (1994), +
  4. C. Miehe (1994), Aspects of the formulation and finite element implementation of large strain isotropic elasticity International Journal for Numerical Methods in Engineering - 37 , 12, + 37 , 12, 1981-2004; -
  5. G.A. Holzapfel (2001), +
  6. G.A. Holzapfel (2001), Nonlinear Solid Mechanics. A Continuum Approach for Engineering, John Wiley & Sons.
@@ -47,14 +47,14 @@ The notation adopted here draws heavily on the excellent overview of the theoret

Notation

-There are various fourth-order unit tensors. +There are various fourth-order unit tensors. The fourth-order unit tensors $\mathcal{I}$ and $\overline{\mathcal{I}}$ are defined by @f[ - \mathbf{A} = \mathcal{I}:\mathbf{A} + \mathbf{A} = \mathcal{I}:\mathbf{A} \qquad \text{and} \qquad - \mathbf{A}^T = \overline{\mathcal{I}}:\mathbf{A} \, . + \mathbf{A}^T = \overline{\mathcal{I}}:\mathbf{A} \, . @f] -Note $\mathcal{I} \neq \overline{\mathcal{I}}^T$. +Note $\mathcal{I} \neq \overline{\mathcal{I}}^T$. Furthermore, we define the symmetric and skew-symmetric fourth-order unit tensors by @f[ \mathcal{S} := \dfrac{1}{2}[\mathcal{I} + \overline{\mathcal{I}}] @@ -67,80 +67,80 @@ such that \qquad \text{and} \qquad \dfrac{1}{2}[\mathbf{A} - \mathbf{A}^T] = \mathcal{W}:\mathbf{A} \, . @f] -The fourth-order SymmetricTensor returned by identity_tensor is $\mathcal{S}$. +The fourth-order SymmetricTensor returned by identity_tensor is $\mathcal{S}$.

Kinematics

-Let the time domain be denoted $\mathbb{T} = [0,T_{\textrm{end}}]$, where $t \in \mathbb{T}$ and $T_{\textrm{end}}$ is the total problem duration. -Consider a continuum body that occupies the reference configuration $\Omega_0$ at time $t=0$. +Let the time domain be denoted $\mathbb{T} = [0,T_{\textrm{end}}]$, where $t \in \mathbb{T}$ and $T_{\textrm{end}}$ is the total problem duration. +Consider a continuum body that occupies the reference configuration $\Omega_0$ at time $t=0$. Particles in the reference configuration are identified by the position vector $\mathbf{X}$. -The configuration of the body at a later time $t>0$ is termed the current configuration, denoted $\Omega$, with particles identified by the vector $\mathbf{x}$. +The configuration of the body at a later time $t>0$ is termed the current configuration, denoted $\Omega$, with particles identified by the vector $\mathbf{x}$. The nonlinear map between the reference and current configurations, denoted $\mathbf{\varphi}$, acts as follows: @f[ \mathbf{x} = \boldsymbol{\varphi}(\mathbf{X},t) \, . @f] -The material description of the displacement of a particle is defined by +The material description of the displacement of a particle is defined by @f[ \mathbf{U}(\mathbf{X},t) = \mathbf{x}(\mathbf{X},t) - \mathbf{X} \, . @f] The deformation gradient $\mathbf{F}$ is defined as the material gradient of the motion: @f[ - \mathbf{F}(\mathbf{X},t) + \mathbf{F}(\mathbf{X},t) := \dfrac{\partial \boldsymbol{\varphi}(\mathbf{X},t)}{\partial \mathbf{X}} = \textrm{Grad}\mathbf{x}(\mathbf{X},t) = \mathbf{I} + \textrm{Grad}\mathbf{U} \, . @f] -The determinant of the of the deformation gradient -$J(\mathbf{X},t):= \textrm{det} \mathbf{F}(\mathbf{X},t) > 0$ -maps corresponding volume elements in the reference and current configurations, denoted -$\textrm{d}V$ and $\textrm{d}v$, -respectively, as +The determinant of the of the deformation gradient +$J(\mathbf{X},t):= \textrm{det} \mathbf{F}(\mathbf{X},t) > 0$ +maps corresponding volume elements in the reference and current configurations, denoted +$\textrm{d}V$ and $\textrm{d}v$, +respectively, as @f[ \textrm{d}v = J(\mathbf{X},t) \textrm{d}V \, . @f] An important measure of the deformation in terms of the spatial coordinates is the left Cauchy-Green tensor $\mathbf{b} := \mathbf{F}\mathbf{F}^T$. -The left Cauchy-Green tensor is symmetric and positive definite. -Similarly, the (material) right Cauchy-Green tensor is defined by $\mathbf{C} := \mathbf{F}^T\mathbf{F}$. -It is also symmetric and positive definite. +The left Cauchy-Green tensor is symmetric and positive definite. +Similarly, the (material) right Cauchy-Green tensor is defined by $\mathbf{C} := \mathbf{F}^T\mathbf{F}$. +It is also symmetric and positive definite. -The Green-Lagrange strain tensor is defined by +The Green-Lagrange strain tensor is defined by @f[ - \mathbf{E}:= \frac{1}{2}[\mathbf{C} - \mathbf{I} ] + \mathbf{E}:= \frac{1}{2}[\mathbf{C} - \mathbf{I} ] = \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] -where $\boldsymbol{\varepsilon}$ is the linearised strain tensor used when the assumption of infinitesimal deformations is valid. -Note, the use of $\boldsymbol{\varepsilon}$ as the strain measure in step-18 is questionable. +where $\boldsymbol{\varepsilon}$ is the linearised strain tensor used when the assumption of infinitesimal deformations is valid. +Note, the use of $\boldsymbol{\varepsilon}$ as the strain measure in step-18 is questionable. In order to handle the different response that materials exhibit when subjected to bulk and shear type deformations we consider the following decomposition of the deformation gradient $\mathbf{F}$ and the left Cauchy-Green tensor $\mathbf{b}$ into volume-changing (volumetric) and volume-preserving (isochoric) parts: @f[ - \mathbf{F} - = (J^{1/3}\mathbf{I})\overline{\mathbf{F}} - \qquad \text{and} \qquad - \mathbf{b} - = \overline{\mathbf{F}}\overline{\mathbf{F}}^T - = (J^{2/3}\mathbf{I})\overline{\mathbf{b}} \, . + \mathbf{F} + = (J^{1/3}\mathbf{I})\overline{\mathbf{F}} + \qquad \text{and} \qquad + \mathbf{b} + = \overline{\mathbf{F}}\overline{\mathbf{F}}^T + = (J^{2/3}\mathbf{I})\overline{\mathbf{b}} \, . @f] -Clearly, $\textrm{det} \mathbf{F} = \textrm{det} (J^{1/3}\mathbf{I}) = J$. +Clearly, $\textrm{det} \mathbf{F} = \textrm{det} (J^{1/3}\mathbf{I}) = J$. The spatial velocity field is denoted $\mathbf{v}(\mathbf{x},t)$. The derivative of the spatial velocity field with respect to the spatial coordinates gives the spatial velocity gradient $\mathbf{l}(\mathbf{x},t)$, that is @f[ - \mathbf{l}(\mathbf{x},t) - := \dfrac{\mathbf{v}(\mathbf{x},t)}{\mathbf{x}} + \mathbf{l}(\mathbf{x},t) + := \dfrac{\mathbf{v}(\mathbf{x},t)}{\mathbf{x}} = \textrm{grad}\mathbf{v}(\mathbf{x},t) \, , @f] -where $\textrm{grad}(\bullet):= \textrm{Grad}(\bullet) \mathbf{F}^{-1}$. +where $\textrm{grad}(\bullet):= \textrm{Grad}(\bullet) \mathbf{F}^{-1}$.

Kinetics

Cauchy's stress theorem equates the Cauchy traction $\mathbf{t}$ acting on an infinitesimal surface element in the current configuration to the product of the Cauchy stress tensor $\boldsymbol{\sigma}$ (a spatial quantity) and the outward unit normal to the surface $\mathbf{n}$ as @f[ \mathbf{t}(\mathbf{x},t, \mathbf{n}) = \boldsymbol{\sigma}\mathbf{n} \, . @f] -The Cauchy stress is symmetric. +The Cauchy stress is symmetric. Similarly, the first Piola-Kirchhoff traction $\mathbf{T}$ acts on an infinitesimal surface element in the reference configuration is the product of the first Piola-Kirchhoff stress tensor $\mathbf{P}$ (a two-point tensor) and the outward unit normal to the surface $\mathbf{N}$ as @f[ \mathbf{T}(\mathbf{X},t, \mathbf{N}) = \mathbf{P}\mathbf{N} \, . @@ -149,19 +149,19 @@ The first Piola-Kirchhoff stress tensor is related to the Cauchy stress as @f[ \mathbf{P} = J \boldsymbol{\sigma}\mathbf{F}^{-T} \, . @f] -Further important stress measures are the (spatial) Kirchhoff stress $\boldsymbol{\tau} = J \boldsymbol{\sigma}$ -and the (referential) second Piola-Kirchhoff stress +Further important stress measures are the (spatial) Kirchhoff stress $\boldsymbol{\tau} = J \boldsymbol{\sigma}$ +and the (referential) second Piola-Kirchhoff stress $\mathbf{S} = {\mathbf{F}}^{-1} \boldsymbol{\tau} {\mathbf{F}}^{-T}$.

Push-forward and pull-back operators

Push-forward and pull-back operators allow one to transform various measures between the material and spatial settings. -The stress measures used here are contravariant, while the strain measures are covariant. +The stress measures used here are contravariant, while the strain measures are covariant. The push-forward and-pull back operations for second-order covariant tensors $(\bullet)^{\text{cov}}$ are respectively given by: @f[ - \chi_{*}(\bullet)^{\text{cov}}:= \mathbf{F}^{-T} (\bullet)^{\text{cov}} \mathbf{F}^{-1} + \chi_{*}(\bullet)^{\text{cov}}:= \mathbf{F}^{-T} (\bullet)^{\text{cov}} \mathbf{F}^{-1} \qquad \text{and} \qquad \chi^{-1}_{*}(\bullet)^{\text{cov}}:= \mathbf{F}^{T} (\bullet)^{\text{cov}} \mathbf{F} \, . @f] @@ -177,16 +177,16 @@ For example $\boldsymbol{\tau} = \chi_{*}(\mathbf{S})$.

Hyperelastic materials

-A hyperelastic material response is governed by a Helmholtz free energy function $\Psi$ which serves as a potential for the stress. +A hyperelastic material response is governed by a Helmholtz free energy function $\Psi$ which serves as a potential for the stress. For example, if the Helmholtz free energy depends on the right Cauchy-Green tensor $\mathbf{C}$ then the isotropic hyperelastic response is @f[ - \mathbf{S} + \mathbf{S} = 2 \dfrac{\partial \Psi(\mathbf{C})}{\partial \mathbf{C}} \, . @f] -If the Helmholtz free energy depends on the left Cauchy-Green tensor $\mathbf{b}$ then the isotropic hyperelastic response is +If the Helmholtz free energy depends on the left Cauchy-Green tensor $\mathbf{b}$ then the isotropic hyperelastic response is @f[ - \boldsymbol{\tau} - = 2 \dfrac{\partial \Psi(\mathbf{b})}{\partial \mathbf{b}} \mathbf{b} + \boldsymbol{\tau} + = 2 \dfrac{\partial \Psi(\mathbf{b})}{\partial \mathbf{b}} \mathbf{b} = 2 \mathbf{b} \dfrac{\partial \Psi(\mathbf{b})}{\partial \mathbf{b}} \, . @f] @@ -196,26 +196,26 @@ Following the multiplicative decomposition of the deformation gradient, the Helm @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*} - \boldsymbol{\tau}_{\text{vol}} &= - 2 \mathbf{b} \dfrac{\partial \Psi_{\textrm{vol}}(J)}{\partial \mathbf{b}} + \boldsymbol{\tau}_{\text{vol}} &= + 2 \mathbf{b} \dfrac{\partial \Psi_{\textrm{vol}}(J)}{\partial \mathbf{b}} + \\ + &= p J\mathbf{I} \, , \\ - &= p J\mathbf{I} \, , + \boldsymbol{\tau}_{\text{iso}} &= + 2 \mathbf{b} \dfrac{\partial \Psi_{\textrm{iso}} (\overline{\mathbf{b}})}{\partial \mathbf{b}} \\ - \boldsymbol{\tau}_{\text{iso}} &= - 2 \mathbf{b} \dfrac{\partial \Psi_{\textrm{iso}} (\overline{\mathbf{b}})}{\partial \mathbf{b}} - \\ &= \underbrace{( \mathcal{I} - \dfrac{1}{3} \mathbf{I} \otimes \mathbf{I})}_{\mathbb{P}} : \overline{\boldsymbol{\tau}} \, , @f} -where -$p = - 1/3 \textrm{tr} \boldsymbol{\sigma} = - 1/3 J^{-1} \textrm{tr} \boldsymbol{\tau}$ -is the hydrostatic pressure and $\mathbb{P}$ is the projection tensor and provides the deviatoric operator in the Eulerian setting. +where +$p = - 1/3 \textrm{tr} \boldsymbol{\sigma} = - 1/3 J^{-1} \textrm{tr} \boldsymbol{\tau}$ +is the hydrostatic pressure and $\mathbb{P}$ is the projection tensor and provides the deviatoric operator in the Eulerian setting. The fictitious Cauchy stress tensor $\overline{\boldsymbol{\tau}}$ is defined by @f[ - \overline{\boldsymbol{\tau}} - := 2 \overline{\mathbf{b}} \dfrac{\partial \Psi_{\textrm{iso}}(\overline{\mathbf{b}})}{\partial \overline{\mathbf{b}}} \, . + \overline{\boldsymbol{\tau}} + := 2 \overline{\mathbf{b}} \dfrac{\partial \Psi_{\textrm{iso}}(\overline{\mathbf{b}})}{\partial \overline{\mathbf{b}}} \, . @f] -

Nseo-Hookean materials

+

Neo-Hookean materials

The Helmholtz free energy corresponding to a compressible neo-Hookean material is given by @f[ @@ -240,22 +240,22 @@ Thus, the incompressible response of obtained by removing the volumetric compone

Elasticity tensors

-We will use a Newton-Raphson strategy to solve the nonlinear boundary value problem. -Thus, we will need to linearise the constitutive relations. +We will use a Newton-Raphson strategy to solve the nonlinear boundary value problem. +Thus, we will need to linearise the constitutive relations. The fourth-order elasticity tensor in the material description is defined by @f[ - \mathfrak{C} + \mathfrak{C} = 2\dfrac{\partial \mathbf{S}(\mathbf{C})}{\partial \mathbf{C}} = 4\dfrac{\partial^2 \Psi(\mathbf{C})}{\partial \mathbf{C} \partial \mathbf{C}} \, . @f] The fourth-order elasticity tensor in the spatial description $\mathfrak{c}$ is obtained from the push-forward of $\mathfrak{C}$ as @f[ \mathfrak{c} = J^{-1} \chi_{*}(\mathfrak{C}) - \qquad \text{and thus} \qquad + \qquad \text{and thus} \qquad J\mathfrak{c} = 4 \mathbf{b} \dfrac{\partial^2 \Psi(\mathbf{b})} {\partial \mathbf{b} \partial \mathbf{b}} \mathbf{b} \, . @f] -The fourth-order elasticity tensors (for hyperelastic materials) possess both major and minor symmetries. +The fourth-order elasticity tensors (for hyperelastic materials) possess both major and minor symmetries. The fourth-order spatial elasticity tensor can be written in the following decoupled form: @f[ @@ -263,17 +263,17 @@ The fourth-order spatial elasticity tensor can be written in the following decou @f] where @f{align*} - J \mathfrak{c}_{\text{vol}} + J \mathfrak{c}_{\text{vol}} &= 4 \mathbf{b} \dfrac{\partial^2 \Psi_{\text{vol}}(J)} {\partial \mathbf{b} \partial \mathbf{b}} \mathbf{b} \\ &= J(\widetilde{p} \mathbf{I} \otimes \mathbf{I} - 2p \mathcal{I}) - \qquad \text{where} \qquad + \qquad \text{where} \qquad \widetilde{p} := p + \dfrac{\textrm{d} p}{\textrm{d}J} \, , - \\ - J \mathfrak{c}_{\text{iso}} + \\ + J \mathfrak{c}_{\text{iso}} &= 4 \mathbf{b} \dfrac{\partial^2 \Psi_{\text{iso}}(\overline{\mathbf{b}})} {\partial \mathbf{b} \partial \mathbf{b}} \mathbf{b} \\ - &= \mathbb{P} : \mathfrak{\overline{c}} : \mathbb{P} + &= \mathbb{P} : \mathfrak{\overline{c}} : \mathbb{P} + \dfrac{2}{3}(\overline{\boldsymbol{\tau}}:\mathbf{I})\mathbb{P} - \dfrac{2}{3}( \mathbf{I}\otimes\boldsymbol{\tau}_{\text{iso}} + \boldsymbol{\tau}_{\text{iso}} \otimes \mathbf{I} ) \, , @@ -286,12 +286,12 @@ where the fictitious elasticity tensor $\overline{\mathfrak{c}}$ in the spatial

Principle of stationary potential energy

-The total potential energy of the system $\Pi$ is the sum of the internal and external potential energies, denoted $\Pi_{\textrm{int}}$ and $\Pi_{\textrm{ext}}$, respectively. -We wish to find the equilibrium configuration by minimising the potential energy. +The total potential energy of the system $\Pi$ is the sum of the internal and external potential energies, denoted $\Pi_{\textrm{int}}$ and $\Pi_{\textrm{ext}}$, respectively. +We wish to find the equilibrium configuration by minimising the potential energy. -We denote the set of primary unknowns by -$\mathbf{\Xi}:= \{ \mathbf{u}, p, \widetilde{J} \}$. -The independent kinematic variable $\widetilde{J}$ enters the formulation as a constraint on $J$ enforced by the Lagrange multiplier $p$ (the pressure). +We denote the set of primary unknowns by +$\mathbf{\Xi}:= \{ \mathbf{u}, p, \widetilde{J} \}$. +The independent kinematic variable $\widetilde{J}$ enters the formulation as a constraint on $J$ enforced by the Lagrange multiplier $p$ (the pressure). The three-field variational principle used here is given by @f[ @@ -299,38 +299,38 @@ The three-field variational principle used here is given by \Psi_{\textrm{vol}}(\widetilde{J}) + p[J(\mathbf{u}) - \widetilde{J}] + \Psi_{\textrm{iso}}(\overline{\mathbf{b}}(\mathbf{u})) - \bigr] \textrm{d}v + \bigr] \textrm{d}v + \Pi_{\textrm{ext}} \, . @f] where the external potential is defined by @f[ - \Pi_{\textrm{ext}} + \Pi_{\textrm{ext}} = - \int_\Omega \mathbf{b} \cdot \mathbf{u}~\textrm{d}v - \int_{\partial \Omega_{\sigma}} \overline{\mathbf{t}} \cdot \mathbf{u}~\textrm{d}a \, . @f] -The boundary of the current configuration $\partial \Omega$ is composed into two parts as +The boundary of the current configuration $\partial \Omega$ is composed into two parts as $\partial \Omega = \partial \Omega_{\mathbf{u}} \cup \partial \Omega_{\sigma}$, -where -$\partial \Omega_{\mathbf{u}} \cap \partial \Omega_{\boldsymbol{\sigma}} = \emptyset$. -The prescribed Cauchy traction, denoted $\overline{\mathbf{t}}$, is applied to $ \partial \Omega_{\boldsymbol{\sigma}}$ while the motion is prescribed on the remaining portion of the boundary $\partial \Omega_{\mathbf{u}}$. -The body force per unit current volume is denoted $\mathbf{b}$. +where +$\partial \Omega_{\mathbf{u}} \cap \partial \Omega_{\boldsymbol{\sigma}} = \emptyset$. +The prescribed Cauchy traction, denoted $\overline{\mathbf{t}}$, is applied to $ \partial \Omega_{\boldsymbol{\sigma}}$ while the motion is prescribed on the remaining portion of the boundary $\partial \Omega_{\mathbf{u}}$. +The body force per unit current volume is denoted $\mathbf{b}$. The stationarity of the potential follows as @f{align*} - R(\mathbf\Xi;\delta \mathbf{\Xi}) + R(\mathbf\Xi;\delta \mathbf{\Xi}) &= D_{\delta \mathbf{\Xi}}\Pi(\mathbf{\Xi}) \\ &= \dfrac{\partial \Pi(\mathbf{\Xi})}{\partial \mathbf{u}} \cdot \delta \mathbf{u} + \dfrac{\partial \Pi(\mathbf{\Xi})}{\partial p} \delta p - + \dfrac{\partial \Pi(\mathbf{\Xi})}{\partial \widetilde{J}} \delta \tilde{J} + + \dfrac{\partial \Pi(\mathbf{\Xi})}{\partial \widetilde{J}} \delta \tilde{J} \\ &= \int_{\Omega_0} \bigl[ - \textrm{grad}\delta\mathbf{u} : [ \boldsymbol{\tau}_{\textrm{iso}} + \boldsymbol{\tau}_{\textrm{vol}}] + \textrm{grad}\delta\mathbf{u} : [ \boldsymbol{\tau}_{\textrm{iso}} + \boldsymbol{\tau}_{\textrm{vol}}] + \delta p [ J(\mathbf{u}) - \widetilde{J}] + \delta \widetilde{J}[ \dfrac{\textrm{d} \Psi_{\textrm{vol}}(\widetilde{J})}{\textrm{d} \widetilde{J}} - p] - \bigr]~\textrm{d}V + \bigr]~\textrm{d}V \\ &\quad - \int_{\Omega_0} \delta \mathbf{u} \cdot \mathbf{b}~\textrm{d}v - \int_{\partial \Omega_{0,\boldsymbol{\sigma}}} \mathbf{u} \cdot \overline{\mathbf{t}}~\textrm{d}a @@ -338,89 +338,89 @@ 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 $\delta p \in L^2(\Omega)$ and virtual dilatations $\delta \widetilde{J} \in L^2(\Omega)$. -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. +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. The Euler-Lagrange equations corresponding to the residual are: @f{align*} - &\textrm{div} \boldsymbol{\sigma} + \mathbf{b} = \mathbf{0} && \textrm{[equilibrium]} + &\textrm{div} \boldsymbol{\sigma} + \mathbf{b} = \mathbf{0} && \textrm{[equilibrium]} \\ - &J(\mathbf{u}) = \widetilde{J} && \textrm{[dilatation]} + &J(\mathbf{u}) = \widetilde{J} && \textrm{[dilatation]} \\ &p = \dfrac{\textrm{d} \Psi_{\textrm{vol}}(\widetilde{J})}{\textrm{d} \widetilde{J}} && \textrm{[pressure]} \, . @f} 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 $p$. +The second is the constraint that $J(\mathbf{u}) = \widetilde{J}$. +The third is the definition of the pressure $p$. -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. +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 $t_{\textrm{n}-1}$ -and the currently unknown state at $t_{\textrm{n}}$ is denoted $\varDelta \mathbf{\Xi}^{\textrm{n}} = \mathbf{\Xi}^{\textrm{n}} - \mathbf{\Xi}^{\textrm{n}-1}$. -The incremental change between iterations $\textrm{i}$ and $\textrm{i}+1$ is denoted -$\varDelta \mathbf{\Xi}^{\textrm{n}}_{\textrm{i}} := +and the currently unknown state at $t_{\textrm{n}}$ is denoted $\varDelta \mathbf{\Xi}^{\textrm{n}} = \mathbf{\Xi}^{\textrm{n}} - \mathbf{\Xi}^{\textrm{n}-1}$. +The incremental change between iterations $\textrm{i}$ and $\textrm{i}+1$ is denoted +$\varDelta \mathbf{\Xi}^{\textrm{n}}_{\textrm{i}} := \varDelta \mathbf{\Xi}_{\textrm{i}} = \mathbf{\Xi}_{\textrm{i}+1} - \mathbf{\Xi}_{\textrm{i}}$. -Assume that the state of the system is known for some iteration $\textrm{i}$. -The linearised approximation to nonlinear governing equations to be solved using the Newton-Raphson method is: +Assume that the state of the system is known for some iteration $\textrm{i}$. +The linearised approximation to nonlinear governing equations to be solved using the Newton-Raphson method is: Find $\varDelta \mathbf{\Xi}_{\textrm{i}}$ such that @f[ - R(\mathbf{\Xi}_{\mathsf{i}+1}) = - R(\mathbf{\Xi}_{\mathsf{i}}) + R(\mathbf{\Xi}_{\mathsf{i}+1}) = + R(\mathbf{\Xi}_{\mathsf{i}}) + D^2_{\varDelta \mathbf{\Xi}_{\textrm{i}}, \delta \mathbf{\Xi}} \Pi(\mathbf{\Xi_{\mathsf{i}}}) \cdot \varDelta \mathbf{\Xi}_{\textrm{i}} \equiv 0 \, , @f] -then set +then set $\mathbf{\Xi}_{\textrm{i}+1} = \mathbf{\Xi}_{\textrm{i}} -+\varDelta \mathbf{\Xi}_{\textrm{i}}$. -The tangent is given by ++\varDelta \mathbf{\Xi}_{\textrm{i}}$. +The tangent is given by @f[ - D^2_{\varDelta \mathbf{\Xi}, \delta \mathbf{\Xi}} \Pi( \mathbf{\Xi}^{\mathsf{(i)}} ) + D^2_{\varDelta \mathbf{\Xi}, \delta \mathbf{\Xi}} \Pi( \mathbf{\Xi}^{\mathsf{(i)}} ) = D_{\varDelta \mathbf{\Xi}} R( \mathbf{\Xi}^{(\mathsf{i})}; \delta \mathbf{\Xi}) =: K(\mathbf{\Xi}^{(\mathsf{i})}; \varDelta \mathbf{\Xi}, \delta \mathbf{\Xi}) \, . @f] -Thus, +Thus, @f{align*} K(\mathbf{\Xi}^{(\mathsf{i})}; \varDelta \mathbf{\Xi}, \delta \mathbf{\Xi}) - &= + &= D_{\varDelta \mathbf{u}} R( \mathbf{\Xi}^{(\mathsf{i})}; \delta \mathbf{\Xi}) \cdot \varDelta \mathbf{u} \\ &\quad + D_{\varDelta p} R( \mathbf{\Xi}^{(\mathsf{i})}; \delta \mathbf{\Xi}) \varDelta p - \\ + \\ &\quad + D_{\varDelta \widetilde{J}} R( \mathbf{\Xi}^{(\mathsf{i})}; \delta \mathbf{\Xi}) \varDelta \widetilde{J} \, , @f} -where +where @f{align*} D_{\varDelta \mathbf{u}} R( \mathbf{\Xi}; \delta \mathbf{\Xi}) &= - \int_\Omega \bigl[ \textrm{grad} \delta \mathbf{u} : + \int_\Omega \bigl[ \textrm{grad} \delta \mathbf{u} : \textrm{grad} \varDelta \mathbf{u} [\boldsymbol{\tau}_{\textrm{iso}} + \boldsymbol{\tau}_{\textrm{vol}}] - + \textrm{grad} \delta \mathbf{u} :[J\mathfrak{c}_{\textrm{vol}} + J\mathfrak{c}_{\textrm{iso}}] :\textrm{grad} \varDelta \mathbf{u} + + \textrm{grad} \delta \mathbf{u} :[J\mathfrak{c}_{\textrm{vol}} + J\mathfrak{c}_{\textrm{iso}}] :\textrm{grad} \varDelta \mathbf{u} \bigr]~\textrm{d}V \, , \\ - &\quad + \int_\Omega \delta p J \mathbf{I} : \textrm{grad} \varDelta \mathbf{u} ~\textrm{d}V + &\quad + \int_\Omega \delta p J \mathbf{I} : \textrm{grad} \varDelta \mathbf{u} ~\textrm{d}V \\ D_{\varDelta p} R( \mathbf{\Xi}; \delta \mathbf{\Xi}) &= - \int_\Omega \textrm{grad} \delta \mathbf{u} : J \mathbf{I} \varDelta p ~\textrm{d}V + \int_\Omega \textrm{grad} \delta \mathbf{u} : J \mathbf{I} \varDelta p ~\textrm{d}V - \int_\Omega \delta \widetilde{J} \varDelta p ~\textrm{d}V \, , \\ - D_{\varDelta \widetilde{J}} R( \mathbf{\Xi}; \delta \mathbf{\Xi}) + D_{\varDelta \widetilde{J}} R( \mathbf{\Xi}; \delta \mathbf{\Xi}) &= -\int_\Omega \delta p \varDelta \widetilde{J}~\textrm{d}V + \int_\Omega \delta \widetilde{J} \dfrac{\textrm{d}^2 \Psi_{\textrm{vol}}(\widetilde{J})}{\textrm{d} \widetilde{J}\textrm{d}\widetilde{J}} \varDelta \widetilde{J} ~\textrm{d}V @f} -Note that the following terms are termed the geometrical stress and the material contributions to the tangent matrix: +Note that the following terms are termed the geometrical stress and the material contributions to the tangent matrix: @f{align*} -& \int_\Omega \textrm{grad} \delta \mathbf{u} : - \textrm{grad} \varDelta \mathbf{u} [\boldsymbol{\tau}_{\textrm{iso}} + \boldsymbol{\tau}_{\textrm{vol}}]~\textrm{d}V +& \int_\Omega \textrm{grad} \delta \mathbf{u} : + \textrm{grad} \varDelta \mathbf{u} [\boldsymbol{\tau}_{\textrm{iso}} + \boldsymbol{\tau}_{\textrm{vol}}]~\textrm{d}V && \quad {[\textrm{Geometrical stress}]} \, , - \\ -& \int_\Omega \textrm{grad} \delta \mathbf{u} : - [J\mathfrak{c}_{\textrm{vol}} + J\mathfrak{c}_{\textrm{iso}}] :\textrm{grad} \varDelta \mathbf{u} - ~\textrm{d}V + \\ +& \int_\Omega \textrm{grad} \delta \mathbf{u} : + [J\mathfrak{c}_{\textrm{vol}} + J\mathfrak{c}_{\textrm{iso}}] :\textrm{grad} \varDelta \mathbf{u} + ~\textrm{d}V && \quad {[\textrm{Material}]} \, . @f} @@ -451,10 +451,10 @@ We denote the duration of a typical time step as $\varDelta t = t_{\textrm{n}} - @f[ \mathbf{\mathsf{K}}( \mathbf{\Xi}_{\textrm{i}}^{\textrm{n}})\mathsf{d}\mathbf{\Xi}_{\textrm{i}}^{\textrm{n}} - = + = \mathbf{ \mathsf{F}}(\mathbf{\Xi}_{\textrm{i}}^{\textrm{n}}) @f] -such that +such that @f{align*} \underbrace{\begin{bmatrix} \mathbf{\mathsf{K}}_{uu} & \mathbf{\mathsf{K}}_{up} & \mathbf{0} @@ -466,27 +466,27 @@ such that \underbrace{\begin{bmatrix} \varDelta \mathbf{\mathsf{u}}_{\textrm{i}} \\ \varDelta \mathbf{\mathsf{p}}_{\textrm{i}} \\ - \varDelta \widetilde{\mathbf{\mathsf{J}}}_{\textrm{i}} + \varDelta \widetilde{\mathbf{\mathsf{J}}}_{\textrm{i}} \end{bmatrix}}_{\varDelta \mathbf{\Xi}_{\textrm{i}}} = \underbrace{\begin{bmatrix} -\mathbf{\mathsf{R}}_{u}(\mathbf{u}_{\textrm{i}}) \\ -\mathbf{\mathsf{R}}_{p}(p_{\textrm{i}}) \\ - -\mathbf{\mathsf{R}}_{\widetilde{J}}(\widetilde{J}_{\textrm{i}}) + -\mathbf{\mathsf{R}}_{\widetilde{J}}(\widetilde{J}_{\textrm{i}}) \end{bmatrix}}_{ -\mathbf{\mathsf{R}}(\mathbf{\Xi}_{\textrm{i}}) } = \underbrace{\begin{bmatrix} \mathbf{\mathsf{F}}_{u}(\mathbf{u}_{\textrm{i}}) \\ \mathbf{\mathsf{F}}_{p}(p_{\textrm{i}}) \\ - \mathbf{\mathsf{F}}_{\widetilde{J}}(\widetilde{J}_{\textrm{i}}) + \mathbf{\mathsf{F}}_{\widetilde{J}}(\widetilde{J}_{\textrm{i}}) \end{bmatrix}}_{ \mathbf{\mathsf{F}}(\mathbf{\Xi}_{\textrm{i}}) } @f} @f{align*} - \varDelta \mathbf{\mathsf{p}} + \varDelta \mathbf{\mathsf{p}} & = \mathbf{\mathsf{K}}_{\widetilde{J}p}^{-1} \bigl[ - \mathbf{\mathsf{F}}_{\widetilde{J}} - - \mathbf{\mathsf{K}}_{\widetilde{J}\widetilde{J}} \varDelta \widetilde{\mathbf{\mathsf{J}}} \bigr] + \mathbf{\mathsf{F}}_{\widetilde{J}} + - \mathbf{\mathsf{K}}_{\widetilde{J}\widetilde{J}} \varDelta \widetilde{\mathbf{\mathsf{J}}} \bigr] \\ \varDelta \widetilde{\mathbf{\mathsf{J}}} & = \mathbf{\mathsf{K}}_{p\widetilde{J}}^{-1} \bigl[ @@ -494,9 +494,9 @@ such that - \mathbf{\mathsf{K}}_{pu} \varDelta \mathbf{\mathsf{u}} \bigr] \\ - \Rightarrow \varDelta \mathbf{\mathsf{p}} + \Rightarrow \varDelta \mathbf{\mathsf{p}} &= \mathbf{\mathsf{K}}_{\widetilde{J}p}^{-1} \mathbf{\mathsf{F}}_{\widetilde{J}} - - \underbrace{\bigl[\mathbf{\mathsf{K}}_{\widetilde{J}p}^{-1} \mathbf{\mathsf{K}}_{\widetilde{J}\widetilde{J}} + - \underbrace{\bigl[\mathbf{\mathsf{K}}_{\widetilde{J}p}^{-1} \mathbf{\mathsf{K}}_{\widetilde{J}\widetilde{J}} \mathbf{\mathsf{K}}_{p\widetilde{J}}^{-1}\bigr]}_{\overline{\mathbf{\mathsf{K}}}}\bigl[ \mathbf{\mathsf{F}}_{p} - \mathbf{\mathsf{K}}_{pu} \varDelta \mathbf{\mathsf{u}} \bigr] @f} @@ -507,7 +507,7 @@ and thus = \underbrace{ \Bigl[ - \mathbf{\mathsf{F}}_{u} + \mathbf{\mathsf{F}}_{u} - \mathbf{\mathsf{K}}_{up} \bigl[ \mathbf{\mathsf{K}}_{\widetilde{J}p}^{-1} \mathbf{\mathsf{F}}_{\widetilde{J}} - \overline{\mathbf{\mathsf{K}}}\mathbf{\mathsf{F}}_{p} \bigr] \Bigr]}_{\mathbf{\mathsf{F}}_{\textrm{con}}} @@ -518,7 +518,7 @@ where \mathbf{\mathsf{K}}_{up} \overline{\mathbf{\mathsf{K}}} \mathbf{\mathsf{K}}_{pu} \, . @f] -Note that due to the choice of $p$ and $\widetilde{J}$ as discontinuous at the element level, all matrices that need to be inverted are defined at the element level. +Note that due to the choice of $p$ and $\widetilde{J}$ as discontinuous at the element level, all matrices that need to be inverted are defined at the element level. @f[ \underbrace{\begin{bmatrix} @@ -532,17 +532,17 @@ Note that due to the choice of $p$ and $\widetilde{J}$ as discontinuous at the e

Numerical example

-The numerical example considered here is a nearly-incompressible block under compression. -This benchmark problem is taken from +The numerical example considered here is a nearly-incompressible block under compression. +This benchmark problem is taken from
    -
  1. +
  2. S. Reese, P. Wriggers, B.D. Reddy (2000), - A new locking-free brick element technique for large deformation problems in elasticity, + A new locking-free brick element technique for large deformation problems in elasticity, Computers and Structures , 75 , 291-304. -
+ - @image html "setup.png" + @image html "step-44.setup.png" diff --git a/deal.II/examples/step-44/doc/results.dox b/deal.II/examples/step-44/doc/results.dox index 829c0626c6..dd5dfe7dd7 100644 --- a/deal.II/examples/step-44/doc/results.dox +++ b/deal.II/examples/step-44/doc/results.dox @@ -4,13 +4,13 @@
- @image html "Q1P0_Ref_2.png" + @image html "step-44.Q1P0_Ref_2.png"

Two refinement levels for a tri-linear displacement formulation.

- @image html "Q2P1_Ref_1.png" + @image html "step-44.Q2P1_Ref_1.png"

One refinement level for a tri-quadratic displacement formulation.

@@ -19,13 +19,13 @@
- @image html "Q1P0_Ref_4.png" + @image html "step-44.Q1P0_Ref_4.png"

Four refinement levels for a tri-linear displacement formulation.

- @image html "Q2P1_Ref_3.png" + @image html "step-44.Q2P1_Ref_3.png"

Three refinement levels for a tri-quadratic displacement formulation.

diff --git a/deal.II/examples/step-44/doc/Q1P0_Ref_2.png b/deal.II/examples/step-44/doc/step-44.Q1P0_Ref_2.png similarity index 100% rename from deal.II/examples/step-44/doc/Q1P0_Ref_2.png rename to deal.II/examples/step-44/doc/step-44.Q1P0_Ref_2.png diff --git a/deal.II/examples/step-44/doc/Q1P0_Ref_4.png b/deal.II/examples/step-44/doc/step-44.Q1P0_Ref_4.png similarity index 100% rename from deal.II/examples/step-44/doc/Q1P0_Ref_4.png rename to deal.II/examples/step-44/doc/step-44.Q1P0_Ref_4.png diff --git a/deal.II/examples/step-44/doc/Q2P1_Ref_1.png b/deal.II/examples/step-44/doc/step-44.Q2P1_Ref_1.png similarity index 100% rename from deal.II/examples/step-44/doc/Q2P1_Ref_1.png rename to deal.II/examples/step-44/doc/step-44.Q2P1_Ref_1.png diff --git a/deal.II/examples/step-44/doc/Q2P1_Ref_3.png b/deal.II/examples/step-44/doc/step-44.Q2P1_Ref_3.png similarity index 100% rename from deal.II/examples/step-44/doc/Q2P1_Ref_3.png rename to deal.II/examples/step-44/doc/step-44.Q2P1_Ref_3.png diff --git a/deal.II/examples/step-44/doc/setup.png b/deal.II/examples/step-44/doc/step-44.setup.png similarity index 100% rename from deal.II/examples/step-44/doc/setup.png rename to deal.II/examples/step-44/doc/step-44.setup.png