Local indicators ================== The purpose of these indicators is to identify the areas of the structure where, at a particular moment, either a discharge or a loss of radiality of the stress field occurs. They are produced by post-processing a static or dynamic calculation, 2D or 3D, using an elastic law of behavior or not. They take the form of scalar fields whose analysis can be performed by plotting their isovalues by a graphics post-processor. .. _RefNumPara__34685017: Discharge indicators ----------------------- Local total discharge indicator ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This indicator measures at point :math:`M` and between the instant :math:`t` and :math:`t+\Delta t`, the relative variation in the constraint norm in the Von Misès sense. It is formally written: :math:`{I}_{1}=\frac{\parallel \sigma (M,t+\Delta t)\parallel -\parallel \sigma (M,t)\parallel }{\parallel \sigma (M,t+\Delta t)\parallel }`. This quantity is negative in the event of a local dump at point M. Standard :math:`\parallel \sigma (M,t)\parallel` can be written in four different ways depending on the choice of the modeler: 1. :math:`\parallel \sigma (M,t)\parallel =\sqrt{\frac{3}{2}{\sigma }^{D}\text{.}{\sigma }^{D}}`, where :math:`{\sigma }^{D}` is the deviatory part of the stress tensor (this standard is useful in plasticity with isotropic work hardening). 2. :math:`\parallel \sigma (M,t)\parallel =\sqrt{\frac{3}{2}\sigma \text{.}\sigma }`, where we consider the totality of the stress tensor in order to detect decreases in hydrostatic pressure, for example. 3. :math:`\parallel \sigma (M,t)\parallel =\sqrt{\frac{3}{2}({\sigma }^{D}-X)\text{.}({\sigma }^{D}-X)}`, with :math:`X` the return stress tensor in the case of an elastoplastic law with kinematic work hardening. 4. :math:`\parallel \sigma (M,t)\parallel =\sqrt{\frac{3}{2}(\sigma -X)\text{.}(\sigma -X)}` Local elastoplastic discharge indicator ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This indicator makes it possible to know if the discharge remains elastic or if there would be a risk of plasticization if pure kinematic work hardening was used. This is an "extreme" indicator, considering that, for metals, isotropic and kinematic workings are both present. Option DERA_ELGA therefore calculates (in addition to the components DCHA_V and DCHA_T) the components: i_decha= IND_DCHA: * IND_DCHA =0 initial value without constraints. * IND_DCHA =1 if elastic load * IND_DCHA =2 if plastic load * IND_DCHA =-1 if lawful elastic discharge (regardless of the type of work hardening) * IND_DCHA =-2 if abusive discharge (we would have laminated with kinematic work hardening). VAL_DCHA: indicates the proportion of output of the criterion (see later). The operation is as follows: for laws VMIS_ISOT * only at each integration point, at each point of integration, at each time :math:`t`, based on the stress tensor :math:`\sigma (t)`, the cumulative equivalent plastic deformation :math:`p(t)`, and the isotropic work hardening curve :math:`R(p(t))`, * Initialization: IND_DCHA =0, VAL_DCHA =0. * as long as :math:`p(t)=0`, IND_DCHA =1 (elasticity), * if IND_DCHA =-2 (abusive discharge criterion met), we no longer do anything * if :math:`\Delta p(t)>0`: * if we are in charge, so if the angle between the stress increase tensor and the total stress tensor is "small":, :math:`\frac{\tilde{\sigma }(t-\Delta t)\cdot \Delta \tilde{\sigma }}{∣∣\tilde{\sigma }(t-\Delta t)∣∣∣∣\Delta \tilde{\sigma }∣∣}>0` IND_DCHA =2; (points between A and B in the figure). * if you are in a landfill, IND_DCHA =-2 (rare case: this means that in a short time, you cross the charging surface in a direction away from that of the previous step) * if :math:`\Delta p(t)=0`: * if IND_DCHA >-1, we calculate the tensor :math:`X` (center of the elasticity domain if we were using pure kinematics, so if the initial load surface, represented by a circle in the deviating plane, had been translated to the current point) by: :math:`X=\sigma (t)\frac{R(p)-R(0)}{R(p)}` * if IND_DCHA =-1, we use the :math:`X` tensor already calculated. * we calculate the "kinematic" criterion :math:`{(\sigma (t)-X)}_{\mathrm{eq}}\le R(0)` * If this criterion is met, then the discharge would also be elastic with kinematic work hardening, so it is "legal": IND_DCHA =-1 (any point between B and E in FIG. 2). To apply the criterion to the next moment, we store :math:`X` (6 components) * otherwise, IND_DCHA =-2 and keep this value (because the rest of the calculation would be modified if the work hardening were kinematic). (points between E and F, or between C and C' in the following figure). :math:`\text{VAL\_DCHA}=\frac{∥\sigma -X∥}{{R}_{0}}` Figure 2.1.2-a: elastoplastic discharge indicator .. image:: images/10000000000001CB000001CDE1C030889D837FE7.png :width: 3.6744in :height: 3.052in .. _RefImage_10000000000001CB000001CDE1C030889D837FE7.png: * **Comments**: * The disadvantage of this method is that it is not systematic: the user must "think" of calling option DERA_ELGA. * The criterion obtained is relatively severe: it assumes that pure isotropic work hardening is replaced by pure kinematic work hardening. One possibility of evolution would be to define a threshold surface radius greater than :math:`{R}_{0}`. Radiality loss indicators --------------------------------- Local load radiality indicator ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This indicator measures at point :math:`M` and between the instant :math:`t` and :math:`t+\Delta t`, the variation in the direction of the constraints. It is written: :math:`{I}_{2}=1-\frac{\mid \sigma (M,t)\text{.}\Delta \sigma \mid }{\parallel \sigma (M,t)\parallel \parallel \Delta \sigma \parallel }`, where the dot product "." is associated with one of the previous four standards. This quantity is zero when the radiality is maintained during the time increment. This criterion can also be interpreted as the quantity :math:`1-\mathrm{cos}(\theta )`, where :math:`\theta` is the angle between :math:`\sigma` and :math:`\Delta \sigma`. This true-value indicator varies between 0 for radial loads and 1. It is useful in particular for determining the validity of an elasto-plastic solution in fracture mechanics, Error indicator due to time discretization ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ It provides a measure of error :math:`\eta` due to time discretization, directly related to the rotation of the normal to the load surface. We calculate the angle between :math:`{n}^{\text{-}}`, the normal to the plasticity criterion at the start of the time step (instant t-), and :math:`{n}^{\text{+}}`, the normal to the plasticity criterion calculated at the end of the time step as follows: :math:`{I}_{\eta }=\frac{1}{2}∣∣\Delta n∣∣=\frac{1}{2}∣∣{n}^{\text{+}}-{n}^{\text{-}}∣∣=∣\mathrm{sin}(\frac{\alpha }{2})∣`. This indicator is directly related to the norm of variation from normal to convex in plasticity (this can be easily generalized to any normal-flowing elastoplastic law), and it can also be interpreted as the sine of half the normal two angle.This provides a measure of the error (see [:ref:`4 <4>`]). This criterion is operational for Von Mises elastoplastic behaviors under isotropic, linear and mixed kinematics work hardening. It can be used to control the automatic subdivision of the time step.