From 45423dddcc66f65e3e5778618bfe25ee882477d8 Mon Sep 17 00:00:00 2001 From: mcbride Date: Sat, 14 Jan 2012 21:52:28 +0000 Subject: [PATCH] Updated introduction for step-44 git-svn-id: https://svn.dealii.org/trunk@24901 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-44/doc/intro.dox | 299 ++++++++++++++++++++++++- 1 file changed, 298 insertions(+), 1 deletion(-) diff --git a/deal.II/examples/step-44/doc/intro.dox b/deal.II/examples/step-44/doc/intro.dox index 4b98a14bda..191c57005c 100644 --- a/deal.II/examples/step-44/doc/intro.dox +++ b/deal.II/examples/step-44/doc/intro.dox @@ -1,4 +1,301 @@

Introduction

-

List of references

+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 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. +The geometrically nonlinear problem was partially considered in step-18: 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. + +

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 see Holzapfel (2001), among numerous others. + +
    +
  1. 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, + 177-208; +
  2. J.C. Simo and R.L. Taylor (1991), + Quasi-incompressible finite elasticity in principal stretches. Continuum + basis and numerical algorithms, + Computer Methods in Applied Mechanics and Engineering , + 85 , 3, + 273--310; +
  3. 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, + 1981-2004; +
  4. G.A. Holzapfel (2001), + Nonlinear Solid Mechanics. A Continuum Approach for Engineering, + John Wiley & Sons. +
+ +

Notation

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

Kinematics

+ +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 nonlinear map between the reference and current configurations, denoted $\mathbf{\varphi}$, is given by +@f[ + \mathbf{x} = \mathbf{\varphi}(\mathbf{X},t) \, . +@f] +The material description of the displacement of a particle is defined by +$\mathbf{U}(\mathbf{X},t) = \mathbf{x}(\mathbf{X},t) - \mathbf{X}$. + +The deformation gradient $\mathbf{F}$ is defined as the material gradient of the motion: +@f[ + \mathbf{F}(\mathbf{X},t)} + := \dfrac{\partial \mathbf{\varphi}(\mathbf{X},t)}{\partial \mathbf{X}} + = \text{Grad}\mathbf{x}(\mathbf{X},t) \, . +@f] +The determination 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. + +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}} \, . +@f] +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}} + = \textrm{grad}\mathbf{v}(\mathbf{x},t) \, . +@f] + +

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 $\mathbf{\sigma}$ (a spatial quantity) and the outward unit normal to the surface $\mathbf{n}$ as +@f[ + \mathbf{t}(\mathbf{x},t, \mathbf{n}) = \mathbf{\sigma}\mathbf{n} \, . +@f] +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} \, . +@f] +The first Piola-Kirchhoff stress tensor is related to the Cauchy stress as +@f[ + \mathbf{P} = J \mathbf{\sigma}\mathbf{F}^{-T} \, . +@f] +Further important stress measures are the (spatial) Kirchhoff stress $\mathbf{\tau} = J \mathbf{\sigma}$ +and the (referential) second Piola-Kirchhoff stress +$\mathbf{S} = \mathbf{F}^{-1}\mathbf{\tau}\mathbf{F}^{-T}$. + + +

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 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} + \qquad \text{and} \qquad + \chi^{-1}_{*}(\bullet)^{\text{cov}}:= \mathbf{F}^{T} (\bullet)^{\text{cov}} \mathbf{F} \, . +@f] + +The push-forward and pull back operations for second-order contravariant tensors $(\bullet)^{\text{con}}$ are respectively given by: +@f[ + \chi_{*}(\bullet)^{\text{con}}:= \mathbf{F} (\bullet)^{\text{con}} \mathbf{F}^T + \qquad \text{and} \qquad + \chi^{-1}_{*}(\bullet)^{\text{con}}:= \mathbf{F}^{-1} (\bullet)^{\text{con}} \mathbf{F}^{-T} \, . +@f] +For example $\mathbf{\tau} = \chi_{*}(\mathbf{S})$ + + +

Hyperelastic materials

+ +A hyperelastic material response is governed by a Helmholtz free energy function $\Psi$ which serves as a potential for the stress. +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} + = 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 +@f[ + \mathbf{\tau} + = 2 \dfrac{\partial \Psi(\mathbf{b})}{\partial \mathbf{b}} \mathbf{b} + = 2 \mathbf{b} \dfrac{\partial \Psi(\mathbf{b})}{\partial \mathbf{b}} \, . +@f] + +Following the multiplicative decomposition of the deformation gradient, the Helmholtz free energy can be decomposed as $\Psi(\mathbf{b}) = \Psi(\mathbf{J})_{\text{vol}} + \Psi(\wideline{\mathbf{b}})_{\text{iso}}$. +Similarly, the Kirchhoff stress can be decomposed into volumetric and isochoric parts as $\mathbf{\tau} = \mathbf{\tau}_{\text{vol}} + \mathbf{\tau}_{\text{iso}}$ where: +@f{align*} + \mathbf{\tau}_{\text{vol}} &= + 2 \mathbf{b} \dfrac{\partial \Psi(\mathbf{J})}{\partial \mathbf{b}} + \\ + &= p \mathbf{I} \, , + \\ + \mathbf{\tau}_{\text{iso}} &= + 2 \mathbf{b} \dfrac{\partial \Psi(\overline{\mathbf{b}})}{\partial \mathbf{b}} + \\ + &= \underbrace{( \mathbb{I} - \dfrac{1}{3} \mathbf{I} \otimes \mathbf{I})}_{\mathbb{P}} : \overline{\mathbf{\tau}} +@f} +where $p = - 1/3 \textrm{tr} \mathbf{\sigma}$ 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 $\overline{\mathbf{\sigma}}$ is defined by +@f[ + \overline{\mathbf{\sigma}} + := 2 \overline{\mathbf{b}} \dfrac{\partial \Psi(\overline{\mathbf{b}})}{\partial \overline{\mathbf{b}}} \, . +@f] + + +

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. + +The fourth-order elasticity tensor in the material description is defined by +@f[ + \mathscr{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 $\mathscr{c}$ is obtained from the push-forward of $\mathscr{C}$ as +@f[ + \mathscr{c} = J^{-1} \chi_{*}(\mathscr{C}) + \qquad \text{and thus} \qquad + J\matscr{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 spatial elasticity tensor can be written in the following decoupled form: +@f[ + \mathscr{c} = \mathscr{c}_{\text{vol}} + \mathscr{c}_{\text{iso}} +@f] +where +@f{align*} + J \mathscr{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 \matscr{I}) + \qquad \text{where} \qquad + \widetilde{p} := p + \dfrac{\textrm{d} p}{\textrm{d}J} \, , + \\ + J \mathscr{c}_{\text{vol}} + &= 4 \mathbf{b} \dfrac{\partial^2 \Psi_{\text{iso}}(\overline{\mathbf{b}})} {\partial \mathbf{b} \partial \mathbf{b}} \mathbf{b} + \\ + &= \mathbb{P} : \mathscr{\overline{c}} : \mathbb{P} + + \dfrac{2}{3}(\overline{\mathbf{\tau}}:\mathbf{I})\mathbb{P} + - \dfrac{2}{3}( \mathbf{I}\otimes\mathbf{\tau}_{\text{iso}} + + \mathbf{\tau}_{\text{iso}} \otimes \mathbf{I} ) \, , +@f} +where the fictitious elasticity tensor $\overline{\mathscr{c}}$ in the spatial description is defined by +@f[ + \overline{\mathscr{c}} + &= 4 \overline{\mathbf{b}} \dfrac{ \partial^2 \Psi_{\text{iso}}(\overline{\mathbf{b}})} {\partial \overline{\mathbf{b}} \partial \overline{\mathbf{b}}} \overline{\mathbf{b}} \, . +@f] + +

Principle of stationary potential energy

+ +The total potential energy of the system $\Pi$ is the sum of the internal and external potential energies, denoted $\Pi_{\text{int}}$ and $\Pi_{\text{ext}}$, respectively. +We wish to find the equilibrium configuration by minimising the potential energy. + +We denote the set of primary unknowns by +$\mathbf{\Xi}:= \{ \mathbf{u}, p, \widetilde{J} ; \Lambda \}$. +The independent kinematic variable $\widetilde{J}$ enters the formulation as a constraint on $J$ enforced by the Lagrange multiplier $p$ (the pressure). +The Lagrange multiplier $\Lambda$ is to enforce the incompressibility constraint $J=1$. + +The three-field variational principle used here is given by +@f[ + \Pi(\mathbf{\Xi}) := \int_\Omega \lbracket[ + \Psi_{\text{vol}}(\widetilde{J}) + + p(J(\mathbf{u}) - \widetilde{J}) + + \Psi_{\text{iso}}(\mathbf{b}(\mathbf{u})) + + \Lambda(\widetilde{J}-1) + \rbracket] \textrm{d}v + + \Pi_{\text{ext}} \, . +@f] +where the external potential is defined by +@f[ + \Pi_{\text{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 +$\partial \Omega = \partial \Omega_{\mathbf{u}} \cup \partial \Omega_{\sigma}$, +where +$\partial \Omega_{\mathbf{u}} \cap \partial \Omega_{\sigma} = \emptyset$. +The prescribed Cauchy traction, denoted $\overline{\mathbf{t}}$, is applied to $ \partial \Omega_{\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}) + &= 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} + \\ + &= \int_{\Omega_0} \lbracket[ + \textrm{grad}\delta\mathbf{u} : [ \mathbf{\tau}_{\text{iso}} + \mathbf{\tau}_{\text{vol}}] + + \delta p [ J(\mathbf{u}) - \widetilde{J}] + + \delta \widetilde{J}[ \dfrac{\textrm{d} \Psi_{\text{vol}}(\widetilde{J})}{\textrm{d} \widetilde{J}} - p + \Lambda] + \rbracket]~\textrm{d}V + \\ + &\quad - \int_{\Omega_0} \delta \mathbf{u} \cdot \mathbf{b}~\textrm{d}v + + \int_{\partial \Omega_{0~\sigma}} \mathbf{u} \cdot \overline{\mathbf{t}}~\textrm{d}a + \\ + &=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 an updated-Lagrangian formulation. -- 2.39.5