[Getfem-commits] r5321 - /trunk/getfem/doc/sphinx/source/userdoc/model_plasticity_small_strain.rst

[logari81]

Author: logari81
Date: Thu May  5 08:51:44 2016
New Revision: 5321

URL: http://svn.gna.org/viewcvs/getfem?rev=5321&view=rev
Log:
small improvements in the small strain plasticity documentation

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

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=5321&r1=5320&r2=5321&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    
Thu May  5 08:51:44 2016
@@ -26,12 +26,12 @@
 Additive decomposition of the small strain tensor
 =================================================
 
-Let :math:`\Omega \subset \R^3` be the reference configuration of a deformable 
body and :math:`u : \Omega \rightarrow \R^3` be the displacement. Small strain 
plasticity is based on the additive decomposition of the small deformation 
tensor :math:`\varepsilon(u) = \Frac{\nabla u + \nabla u^T}{2}` in
+Let :math:`\Omega \subset \R^3` be the reference configuration of a deformable 
body and :math:`u : \Omega \rightarrow \R^3` be the displacement field. Small 
strain plasticity is based on the additive decomposition of the small strain 
tensor :math:`\varepsilon(u) = \Frac{\nabla u + \nabla u^T}{2}` in
 
 .. math::
    \varepsilon(u) = \varepsilon^e + \varepsilon^p
 
-where :math:`\varepsilon^e` is the elastic part of the deformation and 
:math:`\varepsilon^p` the plastic one.
+where :math:`\varepsilon^e` is the elastic part of the strain tensor and 
:math:`\varepsilon^p` the plastic one.
 
 Internal variables, free energy potential and elastic law
 =========================================================
@@ -46,12 +46,12 @@
 .. math::
    \psi(\varepsilon^e, \alpha),
 
-such that the stress type variables are determined by
+such that corresponding stress type variables are determined by
 
 .. math::
    \sigma = \Frac{\partial \psi}{\partial \varepsilon^e}(\varepsilon^e, 
\alpha), ~~~~ A =  \Frac{\partial \psi}{\partial \alpha}(\varepsilon^e, \alpha),
 
-where :math:`\sigma` is the Cauchy stress tensor and :math:`A` the stress type 
internal variables. The plastic dissipation being given by
+where :math:`\sigma` is the Cauchy stress tensor and :math:`A` the stress type 
internal variables. The plastic dissipation is given by
 
 .. math::
 
@@ -61,24 +61,24 @@
 
 .. 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 
(\mbox{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` where :math:`{\cal A}` is 
the fourth order elasticity tensor. For isotropic linearized elasticity this 
expression reduces to :math:`\psi^e(\varepsilon^e) = \mu 
\mbox{dev}(\varepsilon^e) : \mbox{dev}(\varepsilon^e) + \frac{1}{2} K 
(\mbox{tr}(\varepsilon^e))^2` where :math:`\mu` is the shear modulus and 
:math:`K = \lambda + 2\mu/3` is the bulk modulus.
 
 
 
 Plastic potential, yield function and plastic flow rule
 =======================================================
 
-The plastic deformation is supposed to occurs when the stress attains a 
critical value. This is determinated by a yield function :math:`f(\sigma, A)` 
and the condition
+Plastic yielding is supposed to occur when the stress attains a critical 
value. This is determinated by a yield function :math:`f(\sigma, A)` and the 
condition
 
 .. math:: f(\sigma, A) \le 0.
 
-The surface :math:`f(\sigma, A) = 0` being the yield surface where the plastic 
deformation may occur.
-
-Let us consider also the plastic potential :math:`\Psi(\sigma, A)`, (convex 
with respect to both its two variables) which determine the plastic flow 
direction in the sense that the flow rule reads as
+The surface :math:`f(\sigma, A) = 0` is the yield surface where the plastic 
deformation may occur.
+
+Let us also consider the plastic potential :math:`\Psi(\sigma, A)`, (convex 
with respect to its both variables) which determines the plastic flow direction 
in the sense that the flow rule is defined as
 
 .. math:: \dot{\varepsilon}^p = \dot{\gamma} \Frac{\partial \Psi}{\partial 
\sigma}(\sigma, A), ~~~~~~ \dot{\alpha} = \dot{\gamma} \Frac{\partial 
\Psi}{\partial A}(\sigma, A),
 
-with the additional complementary condition
+with the additional complementarity condition
 
 .. math:: f(\sigma, A) \le 0, ~~~ \dot{\gamma} \ge 0, ~~~ f(\sigma, A) 
\dot{\gamma} = 0.
 
@@ -87,27 +87,27 @@
 Initial boundary value problem
 ==============================
 
-The weak formulation of a dynamic elastoplastic problem can be written as 
follows for an arbitrary cinematically admissible test function :math:`v`:
+The weak formulation of a dynamic elastoplastic problem can be written, for an 
arbitrary kinematically admissible test function :math:`v`, as follows:
 
 .. math::
 
    \left| \begin{array}{l}
    \ds \int_{\Omega} \rho \ddot{u}\cdot v + \sigma : \nabla v dx =  
\int_{\Omega} f\dot v dx + \int_{\Gamma_N} g\dot v dx, \\
-   u(0,x) = u_0(x), ~~~\dot{u}(0) = v_0(x), \\
+   u(0,x) = u_0(x), ~~~\dot{u}(0) = \mathrm{v}_0(x), \\
    \varepsilon^p(0,x) = \varepsilon^p_0, ~~~ \alpha(0,x) = \alpha_0,
    \end{array} \right.
 
-for :math:`u_0, v_0, \varepsilon^p_0, \alpha_0` the initial values and 
:math:`g` the force prescribed on the part of the boundary :math:`\Gamma_N`.
-
-Note that plasticity models are often applied on quasitistic problem which 
corresponds to neglect the term :math:`\rho \ddot{u}`.
-
-Given a time step :math:`\Delta t` we will denote in the sequel :math:`u_n, 
\varepsilon^p_n  \mbox{ and } \alpha_n` the approximation at time 
:math:`n\Delta t` of :math:`u(t), \varepsilon^p_n \mbox{ and } \alpha(t)` 
respectively. This approximation is given by a chose time integration scheme 
(for instance one of the proposed schemes in :ref:`ud-model-time-integration`) 
which can be different than the time integration scheme used for the 
integration of the flow rule (see below).
+for :math:`u_0, \mathrm{v}_0, \varepsilon^p_0, \alpha_0` being initial values 
and :math:`f` and :math:`g` being prescribed forces in the interior of domain 
:math:`\Omega` and on the part of the boundary :math:`\Gamma_N`.
+
+Note that plasticity models are often applied on quasi-static problems which 
correspond to the term :math:`\rho \ddot{u}` being neglected.
+
+Given a time step :math:`\Delta t = t_{n+1} -t_n`, from time :math:`t_n` to 
:math:`t_{n+1}`, we will denote in the sequel :math:`u_n, \varepsilon^p_n  
\mbox{ and } \alpha_n` the approximations at time :math:`t_n` of :math:`u(t_n), 
\varepsilon^p_n(t_n) \mbox{ and } \alpha(t_n)` respectively. These 
approximations correspond to the chosen time integration scheme (for instance 
one of the proposed schemes in :ref:`ud-model-time-integration`) which can be 
different than the time integration scheme used for the integration of the flow 
rule (see below).
 
 
 Flow rule integration
 +++++++++++++++++++++
 
-The plastic flow rule have to be integrated with its own time integration 
scheme.  Among standards schemes, backward Euler scheme, :math:`\theta`-scheme 
and generalized mid-point scheme are the most commonly used in that context. We 
make here the choice of the generalized mid-point scheme.
+The plastic flow rule has to be integrated with its own time integration 
scheme. Among standards schemes, the backward Euler scheme, the 
:math:`\theta`-scheme and the generalized mid-point scheme are the most 
commonly used in that context. We make here the choice of the generalized 
mid-point scheme.
 
 
 Let :math:`u_{n+1}` be the displacement at the considered time step and  
:math:`u_{n}` at the previous one. For a quantity :math:`B` we denote 
:math:`B_{n+\theta} = \theta B_{n+1} + (1-\theta)B_n` the convex combination of 
the quantity at iterations :math:`n` and :math:`n+1`.