3. Global indicators#

These indicators are intended to detect whether, during the history of the structure and up to the current moment \(t\), and for an area of the structure chosen by the modeler, there has been a loss of proportionality of the load (these indicators therefore leave a trace of history unlike local indicators which are instantaneous). They can only be used in the context of elastoplastic behavior with isotropic work hardening (in 2D or 3D).

3.1. Indicator on plasticity parameters#

This quantity makes it possible, in the case of Von Mises plasticity with isotropic work hardening, on the one hand to know (on average over a zone \({\Omega }_{S}\) in the \(\Omega\) domain) whether the stresses and the plastic deformations have the same directions and if the plastic threshold is reached at the current moment, and on the other hand whether the plastic deformation has changed direction during the course of history. This quantity is written as:

\({I}_{3}=\frac{1}{{\Omega }_{S}}{\int }_{{\Omega }_{S}}(1-\frac{\mid \sigma \text{.}{\varepsilon }^{p}\mid }{({\sigma }_{Y}+R(p))p})d\Omega\),

where \({\sigma }_{Y}\) is the initial plastic threshold, \(R\) the extension of the load surface due to work hardening and \(p\) the cumulative plastic deformation. The dot product is associated with the norm in the Von Misès sense. This indicator is standardized and has a value between zero and one. It is void if the load has maintained its proportionality at each point of \({\Omega }_{S}\) throughout the past story.

Note 1:

The indicator is not affected if, during the story, there were elastic discharges and then reloads without a change in the direction of the stresses when one returns to the threshold (cf. [Figure 3.1-a]).

_images/Object_24.svg

Figure 3.1-a : Load path between \({t}_{1}\) and \({t}_{2}\) in the constraint deviatoric plane

Note 2:

In the formulation of this indicator, three ingredients related to plasticity are involved:

  • the difference between the direction of the stresses and the current plastic deformations ( \(\sigma \text{.}\varepsilon p\ne \parallel \sigma \parallel \parallel {\varepsilon }^{p}\parallel\) ),

  • the position of the constraints in relation to the current threshold ( \(\parallel \sigma \parallel \le ({\sigma }_{Y}+R(p))\) ),

  • the difference between the current plastic deformation standard and the cumulative plastic deformation ( \(\parallel {\varepsilon }^{p}\parallel \ne p\) ) .

A loss of proportionality could have occurred throughout history without the indicator detecting it through the first two ingredients (i.e., one can have coincidence of the directions of the stresses and the plastic deformations at the end of loading and being on the plastic threshold). On the other hand, we will have, \(\parallel {\varepsilon }^{p}\parallel \ne p\) and the indicator will necessarily be greater than zero, so the user will be warned of the loss of proportionality.

Note 3:

If the indicator necessarily detects a loss of proportionality in an area, in practice it is necessary for the latter to contain enough material points with non-proportional loading. In fact, if one chooses a very large area with few points concerned, the normalization of the indicator carried out with the division by the volume of the zone implies a certain « crushing » towards zero of the value of the quantity. Typically, for a structure containing a defect that is a source of non-proportionality, it is in the interest of choosing an integration area \({\Omega }_{S}\) surrounding the defect with a low neighborhood in order to obtain a significant value of the indicator.

3.2. Energy indicator#

This indicator has the same function as the previous one, but is based on energy density. It is written:

\({I}_{4}=\frac{1}{{\Omega }_{S}}{\int }_{{\Omega }_{S}}(1-\frac{\psi }{\omega })d\Omega\),

where \(\omega\) is the deformation energy density defined by: \(\omega (t)={\int }_{0}^{t}\sigma \text{.}\dot{\varepsilon }d\tau\), and \(\psi\) is the elastic energy density associated with the tensile curve if we consider the nonlinear elastic material. More precisely, this quantity is written as:

\(\begin{array}{}\psi (\varepsilon (t))=\frac{1}{2}{\text{Ktr}}^{2}(\varepsilon )+\frac{2\mu }{3}{\parallel \varepsilon \parallel }^{2},\text{si}\parallel \sigma \parallel <({\sigma }_{Y}+R(p)),\\ \psi (\varepsilon (t))=\frac{1}{2}{\text{Ktr}}^{2}(\varepsilon )+\frac{{R}^{2}(P)}{6\mu }+{\int }_{0}^{P}R(q)\text{dq},\text{si}\parallel \sigma \parallel =({\sigma }_{Y}+R(p)),\end{array}\)

with \(K\) the compressibility module, \(\mu\) the Lamé shear coefficient, the Lamé shear coefficient, \(R\) the threshold of the traction curve associated with the norm of plastic deformation \(P=\parallel {\varepsilon }^{p}\parallel\) (this one may be different from the true plastic threshold, because \(P\ne p\) if the loading is not proportional). This indicator is also normalized between 0 and 1. It is zero for a loading that has always kept its \((\psi \ne \omega )\) proportional character.