[Getfem-commits] r5293 - in /trunk/getfem/doc/sphinx/source/userdoc: gasm_high.rst model_plasticity_small_strain.rst

[Yves . Renard]

Author: renard
Date: Wed Apr 13 09:26:14 2016
New Revision: 5293

URL: http://svn.gna.org/viewcvs/getfem?rev=5293&view=rev
Log:
work in progress

Modified:
    trunk/getfem/doc/sphinx/source/userdoc/gasm_high.rst
    trunk/getfem/doc/sphinx/source/userdoc/model_plasticity_small_strain.rst

Modified: trunk/getfem/doc/sphinx/source/userdoc/gasm_high.rst
URL: 
http://svn.gna.org/viewcvs/getfem/trunk/getfem/doc/sphinx/source/userdoc/gasm_high.rst?rev=5293&r1=5292&r2=5293&view=diff
==============================================================================
--- trunk/getfem/doc/sphinx/source/userdoc/gasm_high.rst        (original)
+++ trunk/getfem/doc/sphinx/source/userdoc/gasm_high.rst        Wed Apr 13 
09:26:14 2016
@@ -567,7 +567,7 @@
 
   - ``Norm_sqr(v)`` for ``v`` a vector or a matrix gives the square of the 
euclidean norm of a vector or of the |Frobenius| norm of a matrix. For a vector 
this is equivalent to ``v.v`` and for a matrix to ``m:m``.
 
-  - ``Normalized(v)`` for ``v`` a vector or a matrix gives ``v`` divided by 
its euclidean (for vectors) or |Frobenius| (for matrices) norm. In order to 
avoid problems when ``v`` is small, it is implemented as ``Normalized_reg(v, 
1E-25)``.
+  - ``Normalized(v)`` for ``v`` a vector or a matrix gives ``v`` divided by 
its euclidean (for vectors) or |Frobenius| (for matrices) norm. In order to 
avoid problems when ``v`` is close to 0, it is implemented as 
``Normalized_reg(v, 1E-25)``.
 
   - ``Normalized_reg(v, eps)`` for ``v`` a vector or a matrix gives a 
regularized version of ``Normalized(v)`` : ``v/sqrt(|v|*|v|+eps*eps)``.
 

Modified: 
trunk/getfem/doc/sphinx/source/userdoc/model_plasticity_small_strain.rst
URL: 
http://svn.gna.org/viewcvs/getfem/trunk/getfem/doc/sphinx/source/userdoc/model_plasticity_small_strain.rst?rev=5293&r1=5292&r2=5293&view=diff
==============================================================================
--- trunk/getfem/doc/sphinx/source/userdoc/model_plasticity_small_strain.rst    
(original)
+++ trunk/getfem/doc/sphinx/source/userdoc/model_plasticity_small_strain.rst    
Wed Apr 13 09:26:14 2016
@@ -61,7 +61,7 @@
 
 .. math:: \psi(\varepsilon^e, \alpha) = \psi^e(\varepsilon^e) + \psi^p(\alpha).
 
-In the case of linearized elasticity, one has :math:`\psi^e(\varepsilon^e) = 
\frac{1}{2} ({\cal A}\varepsilon^e) :\varepsilon^e` for :math:`{\cal A}` the 
fourth order elasticity tensor en more precisely :math:`\psi^e(\varepsilon^e) = 
\mu \mbox{dev}(\varepsilon^e) : \mbox{dev}(\varepsilon^e) + \frac{1}{2} K 
(tr(\varepsilon^e))^2` for isotropic linearized elasticity for :math:`\mu, K = 
\lambda + 2\mu/3` the shear and bulk modulus, respectively.
+In the case of linearized elasticity, one has :math:`\psi^e(\varepsilon^e) = 
\frac{1}{2} ({\cal A}\varepsilon^e) :\varepsilon^e` for :math:`{\cal A}` the 
fourth order elasticity tensor en more precisely :math:`\psi^e(\varepsilon^e) = 
\mu \mbox{dev}(\varepsilon^e) : \mbox{dev}(\varepsilon^e) + \frac{1}{2} K 
(\mbox{tr}(\varepsilon^e))^2` for isotropic linearized elasticity for 
:math:`\mu, K = \lambda + 2\mu/3` the shear and bulk modulus, respectively.
 
 
 
@@ -161,35 +161,86 @@
 
 The complementarity equation :eq:`flowrule3` is then prescribed with the use 
of a well chosen complementarity function, as in [HA-WO2009]_ for :math:`r > 0` 
such as:
 
-.. math:: \ds \int_{\Omega} ((-f(\sigma_{n+\theta}, A_{n+\theta}) - r\Delta 
\xi)_+ + f(\sigma_{n+\theta}, A_{n+\theta}) ) \lambda dx = 0,   \forall \lambda
-
-
-pb : need of :math:`A_{n+\theta}` ... Would be more simple in 
:math:`f(\sigma_{n+1}, A_{n+1})`
-
-and ... the following should be more simple :
-
 .. math:: \ds \int_{\Omega} (\Delta \xi - (\Delta \xi + r f(\sigma_{n+\theta}, 
A_{n+\theta}))_+) \lambda dx = 0,   \forall \lambda
+
+or
+
+.. math:: \ds \int_{\Omega} (f(\sigma_{n+\theta} + (-f(\sigma_{n+\theta}, 
A_{n+\theta}) - \Delta \xi/r)_+ , A_{n+\theta}) ) \lambda dx = 0,   \forall 
\lambda
+
+
+pb : need of :math:`A_{n+\theta}` 
+
+
 
 Plane strain approximation
 ++++++++++++++++++++++++++
 
-To be described
+A plane strain approximation is a 2D problem which corresponds to the 
deformation of a long cylindrical object where the strain in  the length 
direction (assumed to be along the :math:`z` axis) is considered small compared 
to the ones in the other directions and is neglected. It result in a plane 
strain tensor of the form
+
+.. math:: \varepsilon(u) = \left(\hspace{-0.5em}\begin{array}{ccc} 
\varepsilon_{1,1} & \varepsilon_{1,2} & 0 \\ \varepsilon_{1,2} & 
\varepsilon_{2,2} & 0 \\ 0 & 0 & 0 \end{array}\hspace{-0.5em}\right).
+
+We denote
+
+.. math:: \bar{\varepsilon}(u) =  \left(\hspace{-0.5em}\begin{array}{cc} 
\varepsilon_{1,1} & \varepsilon_{1,2} \\ \varepsilon_{1,2} & \varepsilon_{2,2} 
\end{array}\hspace{-0.5em}\right)
+
+the non neglected components of the strain tensor. The adaptation to the plane 
strain approximation to plastic model is most of the time an  easy task.
+
+An isotropic linearized elastic response reads
+
+.. math:: \bar{\sigma} = \lambda \mbox{tr}(\bar{\varepsilon}^e) I + 
2\mu\bar{\varepsilon}^e,
+
+and
+
+.. math:: \sigma_{33} = \lambda \mbox{tr}(\bar{\varepsilon}^e) = 
\Frac{\lambda}{2(\lambda+\mu)}.\mbox{tr}(\bar{\sigma})
+
+thus 
+
+.. math:: \mbox{tr}(\sigma) = \Frac{3\lambda+2\mu}{2(\lambda+\mu))} 
\mbox{tr}(\bar{\sigma}),
+
+.. math:: \mbox{Dev}(\sigma) = \sigma - \Frac{3\lambda+2\mu}{6(\lambda+\mu))} 
\mbox{tr}(\bar{\sigma}) I.
+
+
+
+
 
 Plane stress approximation
 ++++++++++++++++++++++++++
 
-To be described
+The plane stress approximation describe generally the 2D membrane deformation 
of a thin plate. It consist in prescribing the stress tensor to have only 
in-plane nonzero components, i.e.
+
+.. math:: \sigma = \left(\hspace{-0.5em}\begin{array}{ccc} \sigma_{1,1} & 
\sigma_{1,2} & 0 \\ \sigma_{1,2} & \sigma_{2,2} & 0 \\ 0 & 0 & 0 
\end{array}\hspace{-0.5em}\right).
+
+We will still denote 
+
+.. math:: \bar{\sigma} =  \left(\hspace{-0.5em}\begin{array}{cc} \sigma_{1,1} 
& \sigma_{1,2} \\ \sigma_{1,2} & \sigma_{2,2} \end{array}\hspace{-0.5em}\right)
+
+the in-plane components of the stress tensor. For elastoplasticity, it 
consists generally to apply the 2D plastic flow rule, prescribing the out-plane 
components of the stress tensor to be zero with the additionnal variables 
:math:`\varepsilon^e_1`, :math:`\varepsilon^e_2`, :math:`\varepsilon^e_3` being 
unknown (see for instance [SO-PE-OW2008]_).
+
+For an isotropic linearized elastic response, one has :math:`\sigma = \lambda 
\mbox{tr}(\varepsilon^e) + 2\mu\varepsilon^e` such that
+
+.. math:: \varepsilon^e = \left(\hspace{-0.5em}\begin{array}{ccc} 
\varepsilon^e_{1,1} & \varepsilon^e_{1,2} & 0 \\ \varepsilon^e_{1,2} & 
\varepsilon^e_{2,2} & 0 \\ 0 & 0 & \varepsilon^e_{3,3} 
\end{array}\hspace{-0.5em}\right).
+
+with
+
+.. math:: \varepsilon^e_{3,3} = 
-\Frac{\lambda}{\lambda+2\mu}(\varepsilon^e_{1,1} + \varepsilon^e_{2,2})
+
+so that
+
+.. math:: \bar{\sigma} = \lambda^* \mbox{tr}(\bar{\varepsilon}^e) + 
2\mu\bar{\varepsilon}^e ~~~\mbox{ with } \lambda^* = 
\Frac{2\mu\lambda}{\lambda+2\mu}
+   :label: plane_stress_iso
+
+Moreover
+
+.. math:: \|\mbox{Dev}(\sigma)\| = \left(\|\bar{\sigma}\|^2 - 
\Frac{1}{3}(\mbox{tr}(\bar{\sigma}))^2\right)^{1/2}.
 
 Some classical laws
 +++++++++++++++++++
 
-Von Mises vs Tresca threshold ...
-
 
 Tresca : :math:`\rho(\sigma) \le \sigma_y` where :math:`\rho(\sigma)` spectral 
radius of the Cauchy stress tensor and :math:`\sigma_y` the uniaxial yield 
stress (which may depend on some hardening internal variables.
 
 Von Mises :  :math:`\|\mbox{Dev}(\sigma)\| \le \sqrt{\frac{2}{3}}\sigma_y` 
where
-:math:`\mbox{Dev}(\sigma) = \sigma - \frac{1}{3}\mbox{Tr}(\sigma)I` the 
deviatoric part of :math:`\sigma` and :math:`\|\sigma\| = \sqrt{\sigma:\sigma}`.
+:math:`\mbox{Dev}(\sigma) = \sigma - \frac{1}{3}\mbox{tr}(\sigma)I` the 
deviatoric part of :math:`\sigma` and :math:`\|\sigma\| = \sqrt{\sigma:\sigma}`.
 
 
 Perfect isotropic associated elastoplasticity with Von-Mises criterion 
(Prandl-Reuss model)
@@ -204,7 +255,7 @@
 
 The generalized mid-point scheme for the integration of the plastic flow rule 
reads:
 
-.. math:: \varepsilon^p_{n+\theta} - \varepsilon^p_{n} = \theta 
\alpha(\sigma_{n+\theta}, A_{n+\theta}) \Delta \xi 
\sqrt{\frac{3}{2}}\Frac{\mbox{Dev}(\sigma_{n+\theta})}{\|\mbox{Dev}(\sigma_{n+\theta})\|}.
+.. math:: \varepsilon^p_{n+\theta} - \varepsilon^p_{n} = \theta 
\alpha(\sigma_{n+\theta}, A_{n+\theta}) \Delta \xi 
\Frac{\mbox{Dev}(\sigma_{n+\theta})}{\|\mbox{Dev}(\sigma_{n+\theta})\|}.
 
 Choosing the factor :math:`\alpha(\sigma_{n+\theta}) = 
\|\mbox{Dev}(\sigma_{n+\theta})\|` and still with :math:`\Delta \xi = 
\Frac{\Delta \gamma}{\alpha(\sigma_{n+\theta})}` this gives the equation
 
@@ -216,9 +267,33 @@
 
 which is a linear expression with respect to :math:`u_{n+1}` (but not with 
respect to :math:`\Delta \xi`).
 
-
-
-
+**Plane strain approximation**
+
+The plane strain approximation has the same expression replacing the 3D strain 
tensors by the in-plane ones :math:`\bar{\varepsilon}^p` and  
:math:`\bar{\varepsilon}(u_{n+\theta})`.
+
+.. math:: \bar{{\mathscr E}}^p(\bar{u}_{n+\theta}, \theta \Delta \xi, 
\bar{\varepsilon}^p_{n}) = \Frac{1}{1+2\mu\theta\Delta 
\xi}(\bar{\varepsilon}^p_{n} + 2\mu\theta\Delta \xi 
\mbox{Dev}^*(\bar{\varepsilon}(\bar{u}_{n+\theta}))),
+
+where :math:`\mbox{Dev}^*(\bar{\varepsilon}) = \bar{\varepsilon} - 
\Frac{\mbox{tr}(\bar{\varepsilon})}{3} \bar{I}` is still the 3D deviator.
+
+Moreover, for the yield condition, 
+
+.. math:: \mbox{Dev}(\sigma) = 2\mu\mbox{Dev}(\varepsilon(u) - \varepsilon^p) 
= 2\mu\left(\varepsilon(u) - \varepsilon^p - 
\Frac{\mbox{tr}(\bar{\varepsilon}(u)) - \mbox{tr}(\bar{\varepsilon}^p)}{3} 
I\right)
+
+.. math:: \begin{array}{rcl} \|\mbox{Dev}(\sigma)\| &=& 
2\mu\sqrt{\left\|\bar{\varepsilon}(u) - \bar{\varepsilon}^p - 
\Frac{\mbox{tr}(\bar{\varepsilon}(u)) - \mbox{tr}(\bar{\varepsilon}^p)}{3} 
\bar{I}\right\|^2 + \Frac{(\mbox{tr}(\bar{\varepsilon}(u)) - 
\mbox{tr}(\bar{\varepsilon}^p))^2}{9}} \\ &=& \sqrt{\left\|\bar{\sigma} - 
\Frac{3\lambda+2\mu}{6(\lambda+\mu)}\mbox{tr}(\bar{\sigma})\bar{I} \right\|^2 + 
\Frac{\mu^2}{9(\lambda+\mu)^2}\mbox{tr}(\bar{\sigma})^2 } \end{array}
+
+**Plane stress approximation**
+
+For plane stress approximation, we use :eq:`plane_stress_iso` which gives
+
+.. math::  \bar{\varepsilon}^p_{n+\theta} - \bar{\varepsilon}^p_{n} = \theta 
\Delta \xi \mbox{Dev}^*(\bar{\sigma}_{n+\theta}) =  \theta \Delta \xi 
\mbox{Dev}^*(\lambda^*\mbox{tr}(\bar{\varepsilon}^e_{n+\theta})\bar{I} + 2\mu 
\bar{\varepsilon}^e_{n+\theta}) = \theta \Delta 
\xi\left(\Frac{\lambda^*-2\mu}{3}\mbox{tr}(\bar{\varepsilon}^e_{n+\theta})\bar{I}
 + 2\mu\bar{\varepsilon}^e_{n+\theta}\right)
+
+thus with :math:`\beta = \Frac{\lambda^*-2\mu}{3}` one has
+
+.. math::  (1+2\mu\theta \Delta \xi)\bar{\varepsilon}^p_{n+\theta} + 
\beta\theta \Delta \xi \mbox{tr}(\bar{\varepsilon}^p_{n+\theta})\bar{I} = 
\bar{\varepsilon}^p_{n} + \theta \Delta 
\xi\left(\beta\mbox{tr}(\bar{\varepsilon}(u_{n+\theta}))\bar{I} + 
2\mu\bar{\varepsilon}(u_{n+\theta})\right)
+
+By inverting this relation we find for :math:`A = \bar{\varepsilon}^p_{n} + 
\theta \Delta \xi\left(\beta\mbox{tr}(\bar{\varepsilon}(u_{n+\theta}))\bar{I} + 
2\mu\bar{\varepsilon}(u_{n+\theta})\right)`
+
+.. math::  \bar{{\mathscr E}}^p(\bar{u}_{n+\theta}, \theta \Delta \xi, 
\bar{\varepsilon}^p_{n}) = \Frac{1}{1+2\mu\theta\Delta \xi} A - \left( 
\Frac{\beta\theta\Delta \xi}{(1+2\mu\theta\Delta 
\xi)(1+(2\mu+2\beta)\theta\Delta \xi)} \right) \mbox{tr}(A)\bar{I}
 
 
 Isotropic elastoplasticity with linear isotropic and kinematic hardening and 
Von-Mises criterion
@@ -249,6 +324,11 @@
 
 Note that the isotropic hardening modulus do not intervene in :math:`{\mathscr 
E}^p(u_{n+\theta}, \theta \Delta \xi, \varepsilon^p_{n})` but only in 
:math:`f(\sigma, A)`.
 
+**Plane strain approximation**
+
+
+The plane strain approximation has the same expression replacing the 3D strain 
tensors by the in-plane ones :math:`\bar{\varepsilon}^p` and  
:math:`\bar{\varepsilon}(u_{n+\theta})`.
+
 Souza-Auricchio elastoplasticity law (for shape memory alloys)
 ==============================================================
 
@@ -297,6 +377,11 @@
 
 stupid ? 
 
+**Plane strain approximation**
+
+
+The plane strain approximation has the same expression replacing the 3D strain 
tensors by the in-plane ones :math:`\bar{\varepsilon}^p` and  
:math:`\bar{\varepsilon}(u_{n+\theta})`.
+
 
 
 Some classical modelizations
@@ -310,6 +395,19 @@
 Elasto-plasticity bricks 
 +++++++++++++++++++++++++
 
+
+
+Generic brick
+=============
+
+The generic brick add the following terms:
+
+.. math:: \int_{\Omega} \sigma_{n+1} : \nabla v dx +  \int_{\Omega} .. math:: 
\ds \int_{\Omega} (\Delta \xi - (\Delta \xi + r f(\sigma_{n+\theta}, 
A_{n+\theta}))_+) \lambda dx = 0,   \forall \lambda
+
+
+Other bricks
+============
+
 to be done: ::
 
       getfem::add_elastoplasticity_brick