From: mcbride Date: Sat, 11 Feb 2012 20:57:34 +0000 (+0000) Subject: Updated introduction for step-44 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=8cfa6d34be766f11198138a7a396c2e4c6b00938;p=dealii-svn.git Updated introduction for step-44 git-svn-id: https://svn.dealii.org/trunk@25034 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 322cacda9d..1a05c6ce1d 100644 --- a/deal.II/examples/step-44/doc/intro.dox +++ b/deal.II/examples/step-44/doc/intro.dox @@ -8,7 +8,7 @@ Additionally, the three-field formulation employed is valid for quasi-incompress 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 form of the geometrically nonlinear problem was partially considered in step-18: the problem domain evolves with the motion. +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. @@ -33,7 +33,7 @@ The notation adopted here draws heavily on the excellent overview of the theoret basis and numerical algorithms, Computer Methods in Applied Mechanics and Engineering , 85 , 3, - 273--310; + 273-310;
  • C. Miehe (1994), Aspects of the formulation and finite element implementation of large strain isotropic elasticity International Journal for Numerical Methods in Engineering @@ -44,11 +44,11 @@ The notation adopted here draws heavily on the excellent overview of the theoret John Wiley & Sons. - - - - - - -

    Neo-Hookean constitutive model

    - - - - - - - - - - - +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 + && \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 + && \quad {[\textrm{Material}]} \, . +@f} +

    Discretisation of governing equations

    +The three-field formulation used here is effective for quasi-incompressible materials, +that is where $\nu \rightarrow 0.5$ subject to a good choice of the interpolation fields +for $\mathbf{u},~p$ and $\widetilde{J}$. +Typically a choice of $Q_n - P_{n-1} - P_{n-1}$ is made. +A popoular choice is $Q_1 - P_0- P_0$ which is known as the mean dilatation method. +This code can accomodate a $Q_n - P_{n-1} - P_{n-1}$ formulation. +The discontinuous approximation +allows $p$ 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 +$\int_\Omega \Lambda [ \widetilde{J} - 1]~\textrm{d}V$. +Here $\Lambda$ is a Lagrange multiplier to enforce the isochoric constraint. +For further details see Miehe (1994). +We denote the duration of a typical time step as $\varDelta t = t_{\textrm{n}} - t_{\textrm{n}-1}$ +@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 +@f{align*} + \underbrace{\begin{bmatrix} + \mathbf{\mathsf{K}}_{uu} & \mathbf{\mathsf{K}}_{up} & \mathbf{0} + \\ + \mathbf{\mathsf{K}}_{pu} & \mathbf{0} & \mathbf{\mathsf{K}}_{p\widetilde{J}} + \\ + \mathbf{0} & \mathbf{\mathsf{K}}_{\widetilde{J}p} & \mathbf{\mathsf{K}}_{\widetilde{J}\widetilde{J}} + \end{bmatrix}}_{\mathbf{\mathsf{K}}(\mathbf{\Xi}_{\textrm{i}})} + \underbrace{\begin{bmatrix} + \varDelta \mathbf{\mathsf{u}}_{\textrm{i}} \\ + \varDelta \mathbf{\mathsf{p}}_{\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}}) + \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}}) + \end{bmatrix}}_{ \mathbf{\mathsf{F}}(\mathbf{\Xi}_{\textrm{i}}) } +@f} +@f{align*} + \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] + \\ + \varDelta \widetilde{\mathbf{\mathsf{J}}} + & = \mathbf{\mathsf{K}}_{p\widetilde{J}}^{-1} \bigl[ + \mathbf{\mathsf{F}}_{p} + - \mathbf{\mathsf{K}}_{pu} \varDelta \mathbf{\mathsf{u}} + \bigr] + \\ + \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}} + \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} +and thus +@f[ + \underbrace{\bigl[ \mathbf{\mathsf{K}}_{uu} + \overline{\overline{\mathbf{\mathsf{K}}}}~ \bigr] + }_{\mathbf{\mathsf{K}}_{\textrm{con}}}\varDelta \mathbf{\mathsf{u}} + = + \underbrace{ + \Bigl[ + \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}}} +@f] +where +@f[ + \overline{\overline{\mathbf{\mathsf{K}}}} := + \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. +@f[ +\underbrace{\begin{bmatrix} + \mathbf{\mathsf{K}}_{\textrm{con}} & \mathbf{\mathsf{K}}_{up} & \mathbf{0} + \\ + \mathbf{\mathsf{K}}_{pu} & \mathbf{0} & \mathbf{\mathsf{K}}_{p\widetilde{J}}^{-1} + \\ + \mathbf{0} & \mathbf{\mathsf{K}}_{\widetilde{J}p} & \mathbf{\mathsf{K}}_{\widetilde{J}\widetilde{J}} + \end{bmatrix}}_{ {\mathbf{\mathsf{K}}}_{\textrm{store}}} +@f] +

    Numerical example

    +The numerical example considered here is a nearly-incompressible block under compression. +This benchmark problem is taken from +
      +
    1. + S. Reese, P. Wriggers, B.D. Reddy (2000), + A new locking-free brick element technique for large deformation problems in elasticity, + Computers and Structures , + 75 , + 291-304. +