xref: /dports/math/deal.ii/dealii-803d21ff957e349b3799cd3ef2c840bc78734305/examples/step-44/doc/intro.dox

<br>

<i>This program was contributed by Jean-Paul Pelteret and Andrew McBride.
<br>
This material is based upon work supported by  the German Science Foundation (Deutsche
Forschungsgemeinschaft, DFG), grant STE 544/39-1,  and the National Research Foundation of South Africa.
</i>

@dealiiTutorialDOI{10.5281/zenodo.439772,https://zenodo.org/badge/DOI/10.5281/zenodo.439772.svg}

<a name="Intro"></a>
<h1>Introduction</h1>

The subject of this tutorial is nonlinear solid mechanics.
Classical single-field approaches (see e.g. step-18) can not correctly describe the response of quasi-incompressible materials.
The response is overly stiff; a phenomenon known as locking.
Locking problems can be circumvented using a variety of alternative strategies.
One such strategy is the  three-field formulation.
It is used here  to model the three-dimensional, 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 is 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 then describe the three-field formulation in detail prior to explaining the structure of the class used to manage the material.
The setup of the example problem is then presented.

@note This tutorial has been developed (and is described in the introduction) for the problem of elasticity in three dimensions.
 While the space dimension could be changed in the main() routine, care needs to be taken.
 Two-dimensional elasticity problems, in general, exist only as idealizations of three-dimensional ones.
 That is, they are either plane strain or plane stress.
 The assumptions that follow either of these choices needs to be consistently imposed.
 For more information see the note in step-8.

<h3>List of references</h3>

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).
A nice overview of issues pertaining to incompressible elasticity (at small strains) is given in Hughes (2000).

<ol>
	<li> J.C. Simo, R.L. Taylor and K.S. Pister (1985),
		Variational and projection methods for the volume constraint in finite deformation elasto-plasticity,
		<em> Computer Methods in Applied Mechanics and Engineering </em>,
		<strong> 51 </strong>, 1-3,
		177-208.
		DOI: <a href="http://doi.org/10.1016/0045-7825(85)90033-7">10.1016/0045-7825(85)90033-7</a>;
	<li> J.C. Simo and R.L. Taylor (1991),
  		Quasi-incompressible finite elasticity in principal stretches. Continuum
			basis and numerical algorithms,
		<em> Computer Methods in Applied Mechanics and Engineering </em>,
		<strong> 85 </strong>, 3,
		273-310.
		DOI: <a href="http://doi.org/10.1016/0045-7825(91)90100-K">10.1016/0045-7825(91)90100-K</a>;
	<li> C. Miehe (1994),
		Aspects of the formulation and finite element implementation of large strain isotropic elasticity
		<em> International Journal for Numerical Methods in Engineering </em>
		<strong> 37 </strong>, 12,
		1981-2004.
		DOI: <a href="http://doi.org/10.1002/nme.1620371202">10.1002/nme.1620371202</a>;
	<li> G.A. Holzapfel (2001),
		Nonlinear Solid Mechanics. A Continuum Approach for Engineering,
		John Wiley & Sons.
		ISBN: 0-471-82304-X;
	<li> T.J.R. Hughes (2000),
		The Finite Element Method: Linear Static and Dynamic Finite Element Analysis,
		Dover.
		ISBN: 978-0486411811
</ol>

An example where this three-field formulation is used in a coupled problem is documented in
<ol>
	<li> J-P. V. Pelteret, D. Davydov, A. McBride, D. K. Vu, and P. Steinmann (2016),
		Computational electro- and magneto-elasticity for quasi-incompressible media immersed in free space,
		<em> International Journal for Numerical Methods in Engineering </em>.
		DOI: <a href="http://doi.org/10.1002/nme.5254">10.1002/nme.5254</a>
</ol>

<h3> Notation </h3>

One can think of fourth-order tensors as linear operators mapping second-order
tensors (matrices) onto themselves in much the same way as matrices map
vectors onto vectors.
There are various fourth-order unit tensors that will be required in the forthcoming presentation.
The fourth-order unit tensors $\mathcal{I}$ and $\overline{\mathcal{I}}$ are defined by
@f[
	\mathbf{A} = \mathcal{I}:\mathbf{A}
		\qquad \text{and} \qquad
	\mathbf{A}^T = \overline{\mathcal{I}}:\mathbf{A} \, .
@f]
Note $\mathcal{I} \neq \overline{\mathcal{I}}^T$.
Furthermore, we define the symmetric and skew-symmetric fourth-order unit tensors by
@f[
	\mathcal{S} \dealcoloneq \dfrac{1}{2}[\mathcal{I} + \overline{\mathcal{I}}]
		\qquad \text{and} \qquad
	\mathcal{W} \dealcoloneq \dfrac{1}{2}[\mathcal{I} - \overline{\mathcal{I}}] \, ,
@f]
such that
@f[
	\dfrac{1}{2}[\mathbf{A} + \mathbf{A}^T] = \mathcal{S}:\mathbf{A}
		\qquad \text{and} \qquad
	\dfrac{1}{2}[\mathbf{A} - \mathbf{A}^T] = \mathcal{W}:\mathbf{A} \, .
@f]
The fourth-order <code>SymmetricTensor</code> returned by identity_tensor() is $\mathcal{S}$.


<h3>Kinematics</h3>

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 nonlinear map between the reference and current configurations, denoted $\boldsymbol{\varphi}$, acts as follows:
@f[
	\mathbf{x} = \boldsymbol{\varphi}(\mathbf{X},t) \, .
@f]
The material description of the displacement of a particle is defined by
@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)
		\dealcoloneq \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
$J(\mathbf{X},t) \dealcoloneq \textrm{det}\ \mathbf{F}(\mathbf{X},t) > 0$
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]

Two important measures of the deformation in terms of the spatial and material coordinates are the left and right Cauchy-Green tensors, respectively,
and denoted $\mathbf{b} \dealcoloneq \mathbf{F}\mathbf{F}^T$ and $\mathbf{C} \dealcoloneq \mathbf{F}^T\mathbf{F}$.
They are both symmetric and positive definite.

The Green-Lagrange strain tensor is defined by
@f[
	\mathbf{E} \dealcoloneq \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]
If the assumption of infinitesimal deformations is made, then the second term
on the right can be neglected, and $\boldsymbol{\varepsilon}$ (the linearised
strain tensor) is the only component of the strain tensor.
This assumption is, looking at the setup of the problem, not valid in step-18,
making the use of the linearized $\boldsymbol{\varepsilon}$ as the strain
measure in that tutorial program 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}
        = (J^{2/3}\mathbf{I})\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)
		\dealcoloneq \dfrac{\partial \mathbf{v}(\mathbf{x},t)}{\partial \mathbf{x}}
		= \textrm{grad}\ \mathbf{v}(\mathbf{x},t) \, ,
@f]
where $\textrm{grad} \{\bullet \}
= \frac{\partial \{ \bullet \} }{ \partial \mathbf{x}}
= \frac{\partial \{ \bullet \} }{ \partial \mathbf{X}}\frac{\partial \mathbf{X} }{ \partial \mathbf{x}}
= \textrm{Grad} \{ \bullet \} \mathbf{F}^{-1}$.


<h3>Kinetics</h3>

Cauchy's stress theorem equates the Cauchy traction $\mathbf{t}$ acting on an infinitesimal surface element in the current configuration $\mathrm{d}a$ 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.
Similarly,  the first Piola-Kirchhoff traction $\mathbf{T}$ which acts on an infinitesimal surface element in the reference configuration $\mathrm{d}A$ 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 Cauchy traction $\mathbf{t}$ and the first Piola-Kirchhoff traction $\mathbf{T}$ are related as
@f[
	\mathbf{t}\mathrm{d}a = \mathbf{T}\mathrm{d}A \, .
@f]
This can be demonstrated using <a href="http://en.wikipedia.org/wiki/Finite_strain_theory">Nanson's formula</a>.

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
$\mathbf{S} = {\mathbf{F}}^{-1} \boldsymbol{\tau} {\mathbf{F}}^{-T}$.


<h3> Push-forward and pull-back operators </h3>

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}} \dealcoloneq \mathbf{F}^{-T} (\bullet)^{\text{cov}} \mathbf{F}^{-1}
	\qquad \text{and} \qquad
	\chi^{-1}_{*}(\bullet)^{\text{cov}} \dealcoloneq \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}} \dealcoloneq \mathbf{F} (\bullet)^{\text{con}} \mathbf{F}^T
	\qquad \text{and} \qquad
	\chi^{-1}_{*}(\bullet)^{\text{con}} \dealcoloneq \mathbf{F}^{-1} (\bullet)^{\text{con}} \mathbf{F}^{-T} \, .
@f]
For example $\boldsymbol{\tau} = \chi_{*}(\mathbf{S})$.


<h3>Hyperelastic materials</h3>

A hyperelastic material response is governed by a Helmholtz free energy function $\Psi = \Psi(\mathbf{F}) = \Psi(\mathbf{C}) = \Psi(\mathbf{b})$ 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[
	\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]

Following the multiplicative decomposition of the deformation gradient, the Helmholtz free energy can be decomposed as
@f[
	\Psi(\mathbf{b}) = \Psi_{\text{vol}}(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*}
	\boldsymbol{\tau}_{\text{vol}} &=
		2 \mathbf{b} \dfrac{\partial \Psi_{\textrm{vol}}(J)}{\partial \mathbf{b}}
		\\
		&= p J\mathbf{I} \, ,
		\\
	\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
$p \dealcoloneq \dfrac{\partial \Psi_{\text{vol}}(J)}{\partial J}$ is the pressure response.
$\mathbb{P}$ is the projection tensor which provides the deviatoric operator in the Eulerian setting.
The fictitious Kirchhoff stress tensor $\overline{\boldsymbol{\tau}}$ is defined by
@f[
	\overline{\boldsymbol{\tau}}
		\dealcoloneq 2 \overline{\mathbf{b}} \dfrac{\partial \Psi_{\textrm{iso}}(\overline{\mathbf{b}})}{\partial \overline{\mathbf{b}}} \, .
@f]


@note The pressure response as defined above differs from the widely-used definition of the
pressure in solid mechanics as
$p = - 1/3 \textrm{tr} \boldsymbol{\sigma} = - 1/3 J^{-1} \textrm{tr} \boldsymbol{\tau}$.
Here $p$ is the hydrostatic pressure.
We make use of the pressure response throughout this tutorial (although we refer to it as the pressure).

<h4> Neo-Hookean materials </h4>

The Helmholtz free energy corresponding to a compressible <a href="http://en.wikipedia.org/wiki/Neo-Hookean_solid">neo-Hookean material</a> 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_{\text{iso}}(\overline{\mathbf{b}})} \, ,
@f]
where $\kappa \dealcoloneq \lambda + 2/3 \mu$ is the bulk modulus ($\lambda$ and $\mu$ are the Lam&eacute; parameters)
and $\overline{I}_1 \dealcoloneq \textrm{tr}\ \overline{\mathbf{b}}$.
The function $\mathcal{G}(J)$ is required to be strictly convex and satisfy the condition $\mathcal{G}(1) = 0$,
among others, see Holzapfel (2001) for further details.
In this work $\mathcal{G} \dealcoloneq \frac{1}{4} [ J^2 - 1 - 2\textrm{ln}J ]$.

Incompressibility imposes the isochoric constraint that $J=1$ for all motions $\boldsymbol{\varphi}$.
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_{\textrm{iso}}(\mathbf{b})} \, ,
@f]
where $ I_1 \dealcoloneq \textrm{tr}\mathbf{b} $.
Thus, the incompressible response is obtained by removing the volumetric component from the compressible free energy and enforcing $J=1$.


<h3>Elasticity tensors</h3>

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}
		= 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
	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 spatial elasticity tensor can be written in the following decoupled form:
@f[
	\mathfrak{c} = \mathfrak{c}_{\text{vol}} + \mathfrak{c}_{\text{iso}} \, ,
@f]
where
@f{align*}
	J \mathfrak{c}_{\text{vol}}
		&= 4 \mathbf{b} \dfrac{\partial^2 \Psi_{\text{vol}}(J)} {\partial \mathbf{b} \partial \mathbf{b}} \mathbf{b}
		\\
		&= J[\widehat{p}\, \mathbf{I} \otimes \mathbf{I} - 2p \mathcal{I}]
			\qquad \text{where} \qquad
		\widehat{p} \dealcoloneq p + \dfrac{\textrm{d} p}{\textrm{d}J} \, ,
		\\
	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}
			+ \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} ] \, ,
@f}
where the fictitious elasticity tensor $\overline{\mathfrak{c}}$ in the spatial description is defined by
@f[
	\overline{\mathfrak{c}}
		= 4 \overline{\mathbf{b}} \dfrac{ \partial^2 \Psi_{\textrm{iso}}(\overline{\mathbf{b}})} {\partial \overline{\mathbf{b}} \partial \overline{\mathbf{b}}} \overline{\mathbf{b}} \, .
@f]

<h3>Principle of stationary potential energy and the three-field formulation</h3>

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.

As mentioned above, we adopt a three-field formulation.
We denote the set of primary unknowns by
$\mathbf{\Xi} \dealcoloneq \{ \mathbf{u}, \widetilde{p}, \widetilde{J} \}$.
The independent kinematic variable $\widetilde{J}$ enters the formulation as a constraint on $J$ enforced by the Lagrange multiplier $\widetilde{p}$ (the pressure, as we shall see).

The three-field variational principle used here is given by
@f[
	\Pi(\mathbf{\Xi}) \dealcoloneq \int_\Omega \bigl[
		\Psi_{\textrm{vol}}(\widetilde{J})
		+ \widetilde{p}\,[J(\mathbf{u}) - \widetilde{J}]
		+ \Psi_{\textrm{iso}}(\overline{\mathbf{b}}(\mathbf{u}))
		\bigr] \textrm{d}v
	+ 	\Pi_{\textrm{ext}} \, ,
@f]
where the external potential is defined by
@f[
	\Pi_{\textrm{ext}}
		= - \int_\Omega \mathbf{b}^\text{p} \cdot \mathbf{u}~\textrm{d}v
			- \int_{\partial \Omega_{\sigma}} \mathbf{t}^\text{p} \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
$\partial \Omega_{\mathbf{u}} \cap \partial \Omega_{\boldsymbol{\sigma}} = \emptyset$.
The prescribed Cauchy traction, denoted $\mathbf{t}^\text{p}$, 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 $\mathbf{b}^\text{p}$.



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 \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{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
			\\
		&\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
			\\
		&=0 \, ,
@f}
for all virtual displacements $\delta \mathbf{u} \in H^1(\Omega)$ subject to the constraint that $\delta \mathbf{u} = \mathbf{0}$ 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 Kirchhoff 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 <em> total-Lagrangian formulation </em>.
The approach given in step-18, where the domain of integration is the current configuration, could be called an <em> updated Lagrangian formulation </em>.
The 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]}
		\\
	&J(\mathbf{u}) = \widetilde{J} 		&& \textrm{[dilatation]}
		\\
	&\widetilde{p} = \dfrac{\textrm{d} \Psi_{\textrm{vol}}(\widetilde{J})}{\textrm{d} \widetilde{J}} && \textrm{[pressure]} \, .
@f}
The first equation is the (quasi-static) 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}\delta \mathbf{u} \cdot [\textrm{div}\ \boldsymbol{\sigma} + \mathbf{b}^\text{p}]~\mathrm{d}v
&=
\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_0} \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 \\
&=
- \int_{\Omega_0} [\mathrm{grad}\delta\mathbf{u}]^{\text{sym}} :\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}
where
$[\mathrm{grad}\delta\mathbf{u}]^{\text{sym}} = 1/2[ \mathrm{grad}\delta\mathbf{u} + [\mathrm{grad}\delta\mathbf{u}]^T] $.

We will use an 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 a quantity between the known state at $t_{\textrm{n}-1}$
and the currently unknown state at $t_{\textrm{n}}$ is denoted
$\varDelta \{ \bullet \} = { \{ \bullet \} }^{\textrm{n}} - { \{ \bullet \} }^{\textrm{n-1}}$.
The value of a quantity at the current iteration $\textrm{i}$ is denoted
${ \{ \bullet \} }^{\textrm{n}}_{\textrm{i}} = { \{ \bullet \} }_{\textrm{i}}$.
The incremental change between iterations $\textrm{i}$ and $\textrm{i}+1$ is denoted
$d \{ \bullet \} \dealcoloneq \{ \bullet \}_{\textrm{i}+1} - \{ \bullet \}_{\textrm{i}}$.

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:
Find $d \mathbf{\Xi}$ such that
@f[
	R(\mathbf{\Xi}_{\mathsf{i}+1}) =
		R(\mathbf{\Xi}_{\mathsf{i}})
		+ D^2_{d \mathbf{\Xi}, \delta \mathbf{\Xi}} \Pi(\mathbf{\Xi_{\mathsf{i}}}) \cdot d \mathbf{\Xi} \equiv 0 \, ,
@f]
then set
$\mathbf{\Xi}_{\textrm{i}+1} = \mathbf{\Xi}_{\textrm{i}}
+ d \mathbf{\Xi}$.
The tangent is given by

@f[
	D^2_{d \mathbf{\Xi}, \delta \mathbf{\Xi}} \Pi( \mathbf{\Xi}_{\mathsf{i}} )
		= D_{d \mathbf{\Xi}} R( \mathbf{\Xi}_{\mathsf{i}}; \delta \mathbf{\Xi})
		=: K(\mathbf{\Xi}_{\mathsf{i}}; d \mathbf{\Xi}, \delta \mathbf{\Xi}) \, .
@f]
Thus,
@f{align*}
 	K(\mathbf{\Xi}_{\mathsf{i}}; d \mathbf{\Xi}, \delta \mathbf{\Xi})
 		&=
 			D_{d \mathbf{u}} R( \mathbf{\Xi}_{\mathsf{i}}; \delta \mathbf{\Xi}) \cdot d \mathbf{u}
 			\\
 				&\quad +
 			 	D_{d \widetilde{p}} R( \mathbf{\Xi}_{\mathsf{i}}; \delta \mathbf{\Xi})  d \widetilde{p}
 			 \\
 			 	&\quad +
 			  D_{d \widetilde{J}} R( \mathbf{\Xi}_{\mathsf{i}}; \delta \mathbf{\Xi})  d \widetilde{J} \, ,
@f}
where
@f{align*}
	D_{d \mathbf{u}} R( \mathbf{\Xi}; \delta \mathbf{\Xi})
 	&=
 	\int_{\Omega_0} \bigl[ \textrm{grad}\ \delta \mathbf{u} :
 			\textrm{grad}\ d \mathbf{u} [\boldsymbol{\tau}_{\textrm{iso}} + \boldsymbol{\tau}_{\textrm{vol}}]
 			+ \textrm{grad}\ \delta \mathbf{u} :[
             \underbrace{[\widetilde{p}J[\mathbf{I}\otimes\mathbf{I} - 2 \mathcal{I}]}_{\equiv J\mathfrak{c}_{\textrm{vol}}} +
             J\mathfrak{c}_{\textrm{iso}}] :\textrm{grad} d \mathbf{u}
 		\bigr]~\textrm{d}V \, ,
 		\\
 	&\quad + \int_{\Omega_0} \delta \widetilde{p} J \mathbf{I} : \textrm{grad}\ d \mathbf{u} ~\textrm{d}V
 	\\
 	D_{d \widetilde{p}} R( \mathbf{\Xi}; \delta \mathbf{\Xi})
 	&=
 	\int_{\Omega_0} \textrm{grad}\ \delta \mathbf{u} : J \mathbf{I} d \widetilde{p} ~\textrm{d}V
 		-  \int_{\Omega_0} \delta \widetilde{J} d \widetilde{p}  ~\textrm{d}V \, ,
 	\\
 	D_{d \widetilde{J}} R( \mathbf{\Xi}; \delta \mathbf{\Xi})
 	&=  -\int_{\Omega_0} \delta \widetilde{p} d \widetilde{J}~\textrm{d}V
 	 + \int_{\Omega_0} \delta \widetilde{J}  \dfrac{\textrm{d}^2 \Psi_{\textrm{vol}}(\widetilde{J})}{\textrm{d} \widetilde{J}\textrm{d}\widetilde{J}} d \widetilde{J} ~\textrm{d}V \, .
@f}

Note that the following terms are termed the geometrical stress and  the material contributions to the tangent matrix:
@f{align*}
& \int_{\Omega_0} \textrm{grad}\ \delta \mathbf{u} :
 			\textrm{grad}\ d \mathbf{u} [\boldsymbol{\tau}_{\textrm{iso}} +  \boldsymbol{\tau}_{\textrm{vol}}]~\textrm{d}V
 			&& \quad {[\textrm{Geometrical stress}]} \, ,
 		\\
& \int_{\Omega_0} \textrm{grad} \delta \mathbf{u} :
 			[J\mathfrak{c}_{\textrm{vol}} + J\mathfrak{c}_{\textrm{iso}}] :\textrm{grad}\ d \mathbf{u}
 		~\textrm{d}V
 		&& \quad {[\textrm{Material}]} \, .
@f}


<h3> Discretization of governing equations </h3>

The three-field formulation used here is effective for quasi-incompressible materials,
that is where $\nu \rightarrow 0.5$ (where $\nu$ is <a
href="http://en.wikipedia.org/wiki/Poisson's_ratio">Poisson's ratio</a>), subject to a good choice of the interpolation fields
for $\mathbf{u},~\widetilde{p}$ and $\widetilde{J}$.
Typically a choice of $Q_n \times DGPM_{n-1} \times DGPM_{n-1}$ is made.
Here $DGPM$ is the FE_DGPMonomial class.
A popular choice is $Q_1 \times DGPM_0 \times DGPM_0$ which is known as the mean dilatation method (see Hughes (2000) for an intuitive discussion).
This code can accommodate a $Q_n \times DGPM_{n-1} \times DGPM_{n-1}$ formulation.
The discontinuous approximation
allows $\widetilde{p}$ and $\widetilde{J}$ to be condensed out
and a classical displacement based method is recovered.

For fully-incompressible materials $\nu = 0.5$ and the three-field formulation will still exhibit
locking behaviour.
This can be overcome by introducing an additional constraint into the free energy of the form
$\int_{\Omega_0} \Lambda [ \widetilde{J} - 1]~\textrm{d}V$.
Here $\Lambda$ is a Lagrange multiplier to enforce the isochoric constraint.
For further details see Miehe (1994).

The linearised problem can be written as
@f[
	\mathbf{\mathsf{K}}( \mathbf{\Xi}_{\textrm{i}}) d\mathbf{\Xi}
	=
	\mathbf{ \mathsf{F}}(\mathbf{\Xi}_{\textrm{i}})
@f]
where
@f{align*}
		\underbrace{\begin{bmatrix}
			\mathbf{\mathsf{K}}_{uu}	&	\mathbf{\mathsf{K}}_{u\widetilde{p}}	& \mathbf{0}
			\\
			\mathbf{\mathsf{K}}_{\widetilde{p}u}	&	\mathbf{0}	&	\mathbf{\mathsf{K}}_{\widetilde{p}\widetilde{J}}
			\\
			\mathbf{0}	& 	\mathbf{\mathsf{K}}_{\widetilde{J}\widetilde{p}}		& \mathbf{\mathsf{K}}_{\widetilde{J}\widetilde{J}}
		\end{bmatrix}}_{\mathbf{\mathsf{K}}(\mathbf{\Xi}_{\textrm{i}})}
		\underbrace{\begin{bmatrix}
			d \mathbf{\mathsf{u}}\\
            d \widetilde{\mathbf{\mathsf{p}}} \\
            d \widetilde{\mathbf{\mathsf{J}}}
		\end{bmatrix}}_{d \mathbf{\Xi}}
        =
        \underbrace{\begin{bmatrix}
			-\mathbf{\mathsf{R}}_{u}(\mathbf{u}_{\textrm{i}}) \\
            -\mathbf{\mathsf{R}}_{\widetilde{p}}(\widetilde{p}_{\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}}_{\widetilde{p}}(\widetilde{p}_{\textrm{i}}) \\
           \mathbf{\mathsf{F}}_{\widetilde{J}}(\widetilde{J}_{\textrm{i}})
		\end{bmatrix}}_{ \mathbf{\mathsf{F}}(\mathbf{\Xi}_{\textrm{i}}) } \, .
@f}

There are no derivatives of the pressure and dilatation (primary) variables present in the formulation.
Thus the discontinuous finite element interpolation of the pressure and dilatation yields a block
diagonal matrix for
$\mathbf{\mathsf{K}}_{\widetilde{p}\widetilde{J}}$,
$\mathbf{\mathsf{K}}_{\widetilde{J}\widetilde{p}}$ and
$\mathbf{\mathsf{K}}_{\widetilde{J}\widetilde{J}}$.
Therefore we can easily express the fields $\widetilde{p}$ and $\widetilde{J}$ on each cell simply
by inverting a local matrix and multiplying it by the local right hand
side. We can then insert the result into the remaining equations and recover
a classical displacement-based method.
In order to condense out the pressure and dilatation contributions at the element level we need the following results:
@f{align*}
		d \widetilde{\mathbf{\mathsf{p}}}
		& = \mathbf{\mathsf{K}}_{\widetilde{J}\widetilde{p}}^{-1} \bigl[
			 \mathbf{\mathsf{F}}_{\widetilde{J}}
			 - \mathbf{\mathsf{K}}_{\widetilde{J}\widetilde{J}} d \widetilde{\mathbf{\mathsf{J}}} \bigr]
			\\
		d \widetilde{\mathbf{\mathsf{J}}}
		& = \mathbf{\mathsf{K}}_{\widetilde{p}\widetilde{J}}^{-1} \bigl[
			\mathbf{\mathsf{F}}_{\widetilde{p}}
			- \mathbf{\mathsf{K}}_{\widetilde{p}u} d \mathbf{\mathsf{u}}
			\bigr]
		\\
		 \Rightarrow d \widetilde{\mathbf{\mathsf{p}}}
		&=  \mathbf{\mathsf{K}}_{\widetilde{J}\widetilde{p}}^{-1} \mathbf{\mathsf{F}}_{\widetilde{J}}
		- \underbrace{\bigl[\mathbf{\mathsf{K}}_{\widetilde{J}\widetilde{p}}^{-1} \mathbf{\mathsf{K}}_{\widetilde{J}\widetilde{J}}
		\mathbf{\mathsf{K}}_{\widetilde{p}\widetilde{J}}^{-1}\bigr]}_{\overline{\mathbf{\mathsf{K}}}}\bigl[ \mathbf{\mathsf{F}}_{\widetilde{p}}
 		- \mathbf{\mathsf{K}}_{\widetilde{p}u} d \mathbf{\mathsf{u}} \bigr]
@f}
and thus
@f[
		\underbrace{\bigl[ \mathbf{\mathsf{K}}_{uu} + \overline{\overline{\mathbf{\mathsf{K}}}}~ \bigr]
		}_{\mathbf{\mathsf{K}}_{\textrm{con}}} d \mathbf{\mathsf{u}}
		=
        \underbrace{
		\Bigl[
		\mathbf{\mathsf{F}}_{u}
			- \mathbf{\mathsf{K}}_{u\widetilde{p}} \bigl[ \mathbf{\mathsf{K}}_{\widetilde{J}\widetilde{p}}^{-1} \mathbf{\mathsf{F}}_{\widetilde{J}}
			- \overline{\mathbf{\mathsf{K}}}\mathbf{\mathsf{F}}_{\widetilde{p}} \bigr]
		\Bigr]}_{\mathbf{\mathsf{F}}_{\textrm{con}}}
@f]
where
@f[
		\overline{\overline{\mathbf{\mathsf{K}}}} \dealcoloneq
			\mathbf{\mathsf{K}}_{u\widetilde{p}} \overline{\mathbf{\mathsf{K}}} \mathbf{\mathsf{K}}_{\widetilde{p}u} \, .
@f]
Note that due to the choice of $\widetilde{p}$ and $\widetilde{J}$ as discontinuous at the element level, all matrices that need to be inverted are defined at the element level.

The procedure to construct the various contributions is as follows:
- Construct $\mathbf{\mathsf{K}}$.
- Form $\mathbf{\mathsf{K}}_{\widetilde{p}\widetilde{J}}^{-1}$ for element and store where $\mathbf{\mathsf{K}}_{\widetilde{p}\widetilde{J}}$ was stored in $\mathbf{\mathsf{K}}$.
- Form $\overline{\overline{\mathbf{\mathsf{K}}}}$ and add to $\mathbf{\mathsf{K}}_{uu}$ to get $\mathbf{\mathsf{K}}_{\textrm{con}}$
- The modified system matrix is called ${\mathbf{\mathsf{K}}}_{\textrm{store}}$.
  That is
  @f[
        \mathbf{\mathsf{K}}_{\textrm{store}}
\dealcoloneq
        \begin{bmatrix}
			\mathbf{\mathsf{K}}_{\textrm{con}}	&	\mathbf{\mathsf{K}}_{u\widetilde{p}}	& \mathbf{0}
			\\
			\mathbf{\mathsf{K}}_{\widetilde{p}u}	&	\mathbf{0}	&	\mathbf{\mathsf{K}}_{\widetilde{p}\widetilde{J}}^{-1}
			\\
			\mathbf{0}	& 	\mathbf{\mathsf{K}}_{\widetilde{J}\widetilde{p}}		& \mathbf{\mathsf{K}}_{\widetilde{J}\widetilde{J}}
		\end{bmatrix} \, .
  @f]


<h3> The material class </h3>

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 Helmholtz 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.


<h3> Numerical example </h3>

The numerical example considered here is a nearly-incompressible block under compression.
This benchmark problem is taken from
- S. Reese, P. Wriggers, B.D. Reddy (2000),
  A new locking-free brick element technique for large deformation problems in elasticity,
  <em> Computers and Structures </em>,
  <strong> 75 </strong>,
  291-304.
  DOI: <a href="http://doi.org/10.1016/S0045-7949(99)00137-6">10.1016/S0045-7949(99)00137-6</a>

 <img src="https://www.dealii.org/images/steps/developer/step-44.setup.png" alt="">

The material is quasi-incompressible neo-Hookean with <a href="http://en.wikipedia.org/wiki/Shear_modulus">shear modulus</a> $\mu = 80.194e6$ and $\nu = 0.4999$.
For such a choice of material properties a conventional single-field $Q_1$ approach would lock.
That is, the response would be overly stiff.
The initial and final configurations are shown in the image above.
Using symmetry, we solve for only one quarter of the geometry (i.e. a cube with dimension $0.001$).
The inner-quarter of the upper surface of the domain is subject to a load of $p_0$.