3. Rice location indicator for the Drucker-Prager law

The location indicator of the Rice criterion is defined within the framework of the Drucker-Prager law of behavior. But the definition of a location indicator can be used, more generally, in studies in fracture mechanics, damage mechanics, bifurcation theory, soil mechanics and rock mechanics (and more generally in the context of materials with a softening behavior law).

This indicator defines a state from which the evolution of the mechanical system studied (equations, equilibrium, law of behavior) can lose its uniqueness. In other words, this theory allows:

1. the calculation of the possible initiation state of the location, which is perceived as the limit of validity of classical finite element calculations;

2. the « qualitative » determination of the orientation angles of the location zones.

The location criterion constitutes a limit to the reliability of « classical » finite element calculations.

3.1. The different ways to study location

In the context of studies conducted in soil mechanics, we observed a strong dependence of the numerical solution on finite element discretization. There appears to be a concentration of high values of the cumulative plastic deformations at the level of finite elements and it is noted that this « location zone » changes abruptly with the refinement of the mesh. This phenomenon of location is a source of numerical problems and generates convergence problems in the sense of finite elements.

Localization can be interpreted as an unstable phenomenon, a precursor to a rupture mechanism, characterizing certain types of materials stressed in the inelastic domain. To study instabilities related to location, a distinction is made, on the one hand, between classes of materials with time-dependent behavior and, on the other hand, those that do not depend on time. For materials with time-independent behavior, the approach commonly used is the so-called bifurcation method (it is this method that we are interested in this note). It consists in analyzing the losses in the uniqueness of the problem in speeds. For materials with time-dependent behavior, the uniqueness of the speed problem is often guaranteed and this does not prevent the observation of instabilities during their deformation. For these materials, other approaches must then be used. The most commonly used is the disturbance approach. This approach will not be covered in this note, but for more information consult the notes [bib1], [bib2].

Rudnicki and Rice [bib3] have shown that the study of the location of deformations in rock mechanics is part of the bifurcation theory. This is based on the concept of unstable equilibrium. Rice [bib4] considers that the bifurcation point marks the end of the stable regime. The onset of localization is associated with a rheological instability of the system and this instability corresponds locally to the loss of ellipticity of the equations that govern the continuous incremental equilibrium in speeds. Rice thus proposes a criterion called « bifurcation by location » which makes it possible to detect the state from which, the solution of the mathematical equations that govern the problem at the limits considered and the evolution of the mechanical system studied (equations, equilibrium, law of behavior) lose their uniqueness. This theory allows the calculation of the localization initiation state, which is perceived as the validity limit of classical finite element calculations.

3.2. Theoretical approach

3.2.1. Writing the problem quickly

We consider a structure occupying, at an instant \(t\), the open \(\Omega\) of \({\mathrm{\Re }}^{3}\). The speed problem consists in finding the field of movement speeds \(v\) when the structure is subjected to the speeds of volume forces \({\dot{f}}_{d}\), to the imposed displacement speeds \({v}_{d}\) on a part \({\mathrm{\partial }}_{1}\Omega\) of the border and to the speeds of surface forces \({\dot{F}}_{d}\) on the complementary part \({\partial }_{2}\Omega\).

In the local writing of the problem, the \(v\) travel speed field must therefore verify the problem:

1. \(v\) fairly regular and \(v\mathrm{=}{v}_{d}\) on \({\mathrm{\partial }}_{1}\Omega\)

2. Equations of balance:

\(\text{div}\mathrm{[}\mathrm{L}\mathrm{:}\mathrm{\epsilon }(v)\mathrm{]}+{\dot{\mathrm{f}}}_{d}\mathrm{=}0\) out of \(\Omega\)

\(\mathrm{L}\mathrm{:}\varepsilon (v)\text{.}\mathrm{n}\mathrm{=}{\dot{\mathrm{F}}}_{d}\) out of \({\partial }_{2}\Omega\)

\(\mathrm{n}\) being the outgoing unit normal at \({\mathrm{\partial }}_{2}\Omega\).

• The compatibility conditions (we are limited here to small disturbances):

\(\epsilon (v)=\frac{1}{2}\left[\nabla v+(\nabla v{)}^{T}\right]\)

where the operator \(L\) is generally defined for behavior laws written in incremental form by the relationship:

\(\dot{\sigma }=L(\epsilon ,V):\dot{\epsilon }\)

with:

\(L=\{\begin{array}{ccc}E& & \text{si}F<0\text{ou}F=0\text{et}\frac{b:E:\dot{\epsilon }}{h}\le 0\\ H=& E-\frac{(E:a)\otimes (b:E)}{h}& \text{si}F=0\text{et}\frac{b:E:\dot{\epsilon }}{h}>0\end{array}\)

where \(\mathrm{\sigma }\) is the stress, \(\mathrm{\epsilon }\) the total deformation, \(\mathrm{V}\) a set of internal variables, and \(F\) the plasticity threshold surface. The expressions for \(a,b,E\) and h depend on the formulation of the law of behavior.

3.2.2. Existence and uniqueness results, loss of ellipticity

In this chapter we give some results without demonstrations. However, the reference for these demonstrations is specified.

A sufficient condition for the existence and uniqueness of the previous problem is: \(\dot{\sigma }\mathrm{:}\dot{\epsilon }>0\). This inequality can be interpreted as a definition, in the three-dimensional case, of non-softening. The demonstration is done by Hill [bib5] for standard materials and by Benallal [bib1] for non-standard materials.

The loss of ellipticity corresponds to the moment for which the \(N\text{.}H\text{.}N\) operator becomes singular for a \(N\) direction at a point in the structure. This condition is equivalent to condition: \(\text{det}\left(\mathrm{N}\text{.}\mathrm{H}\text{.}\mathrm{N}\right)=0\). This is the « continuous bifurcation » condition [1] _

in the Rice sense also called acoustic tensor. Rice and Rudnicki [bib3] show that this condition of loss of ellipticity of the local speed problem is a necessary condition for the « continuous or discontinuous » bifurcation [2] _

for the solid. Boundary conditions play no role, only the law of behavior defines the location conditions (location threshold and orientation of the location surface).

Continuous bifurcations thus provide the lower limit of the deformation range over which discontinuous bifurcations can occur.

3.2.3. Analytical resolution for the two-dimensional case.

We pose \(N=({N}_{1},{N}_{2}\mathrm{,0})\) with \({N}_{1}^{2}+{N}_{2}^{2}=1\)

We then have: \(N\text{.}H\text{.}N=\left[\begin{array}{ccc}{A}_{\text{11}}& {A}_{\text{12}}& 0\\ {A}_{\text{21}}& {A}_{\text{22}}& 0\\ 0& 0& C\end{array}\right]\) where Ortiz [bib6] shows that:

\(C={N}_{1}^{2}{H}_{\text{1313}}+{N}_{2}^{2}{H}_{\text{2323}}>0\)

\({A}_{\text{11}}={N}_{1}^{2}{H}_{\text{1111}}+{N}_{1}{N}_{2}({H}_{\text{1112}}+{H}_{\text{1211}})+{N}_{2}^{2}{H}_{\text{1212}}\)

\({A}_{\text{22}}={N}_{1}^{2}{H}_{\text{1212}}+{N}_{1}{N}_{2}({H}_{\text{1222}}+{H}_{\text{2212}})+{N}_{2}^{2}{H}_{\text{2222}}\)

\({A}_{\text{12}}={N}_{1}^{2}{H}_{\text{1112}}+{N}_{1}{N}_{2}({H}_{\text{1122}}+{H}_{\text{1212}})+{N}_{2}^{2}{H}_{\text{1222}}\)

\({A}_{\text{21}}={N}_{1}^{2}{H}_{\text{1211}}+{N}_{1}{N}_{2}({H}_{\text{1212}}+{H}_{\text{2211}})+{N}_{2}^{2}{H}_{\text{2212}}\)

It is therefore sufficient to study the sign of \(\text{det}(A)\) as specified by Doghri [bib7]:

\(\text{det}(A)={a}_{0}{N}_{1}^{4}+{a}_{1}{N}_{1}^{3}{N}_{2}+{a}_{2}{N}_{1}^{2}{N}_{2}^{2}+{a}_{3}{N}_{1}{N}_{2}^{3}+{a}_{4}{N}_{2}^{4}\)

with:

\({a}_{0}={H}_{\text{1111}}{H}_{\text{1212}}-{H}_{\text{1112}}{H}_{\text{1211}}\)

\({a}_{1}={H}_{\text{1111}}({H}_{\text{1222}}+{H}_{\text{2212}})-{H}_{\text{1112}}{H}_{\text{2211}}-{H}_{\text{1122}}{H}_{\text{1211}}\)

\({a}_{2}={H}_{\text{1111}}{H}_{\text{2222}}+{H}_{\text{1112}}{H}_{\text{1222}}+{H}_{\text{1211}}{H}_{\text{2212}}-{H}_{\text{1122}}{H}_{\text{1212}}-{H}_{\text{1122}}{H}_{\text{2211}}-{H}_{\text{1212}}{H}_{\text{2211}}\)

\({a}_{3}={H}_{\text{2222}}({H}_{\text{1112}}+{H}_{\text{1211}})-{H}_{\text{1122}}{H}_{\text{2212}}-{H}_{\text{1222}}{H}_{\text{2211}}\)

\({a}_{4}={H}_{\text{1212}}{H}_{\text{2222}}-{H}_{\text{1222}}{H}_{\text{2212}}\)

We then set \({N}_{1}=\text{cos}\theta\) and \({N}_{2}=\text{sin}\theta\) with \(\theta \in ]\begin{array}{cc}-\frac{\pi }{2};& +\frac{\pi }{2}\end{array}]\). Two cases are then distinguished:

• if \(\theta \text{=+}\frac{\pi }{2}\) then \(\text{det}(A)=0\) if \({a}_{4}=0\);

• if \(\theta \ne +\frac{\pi }{2}\) then \(\text{det}(A)=0\) if \(f(x)={a}_{4}{x}^{4}+{a}_{3}{x}^{3}+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}=0\) with \(x=\text{tan}\theta\).

3.2.4. Root calculation

To solve a polynomial of degree n (like the one defined above, where n=4) it is proposed to use the so-called « Companion Matrix Polynomial » method. The principle of this method consists in looking for the eigenvalues of the matrix (of the Hessenberg type) of order n associated with the polynomial. If we consider the polynomial \(P(x)={x}^{n}+{a}_{n-1}{x}^{n-1}+\text{.}\text{.}\text{.}+{a}_{k}{x}^{k}+\text{.}\text{.}\text{.}+{a}_{1}x+{a}_{0}\). Looking for the roots of this polynomial is the same as looking for the eigenvalues of the matrix:

\(\left[\begin{array}{cccccc}0& 0& 0& 0& 0& -{a}_{0}\\ 1& 0& 0& 0& 0& -{a}_{1}\\ 0& 1& 0& 0& 0& \text{.}\text{.}\text{.}\\ 0& 0& 1& 0& 0& -{a}_{k}\\ 0& 0& 0& 1& 0& \text{.}\text{.}\text{.}\\ 0& 0& 0& 0& 1& -{a}_{n-1}\end{array}\right]\)

This indicator is calculated by option INDL_ELGA from CALC_CHAMP [U4.81.04]. At each integration point, it produces 5 components: the first is the location indicator equal to 0 if \(\mathit{det}(\mathit{N.H.N})>0\) (no location), and equal to 1 otherwise, which corresponds to a possibility of location. The other components provide location directions.

[1]

In a continuous bifurcation, plastic deformation occurs inside and outside the location zone and there is the same law of behavior inside and outside the band.

[2]

In a discontinuous bifurcation, there is a continuity of movement on both sides of the band but we do not have the same behavior. Elastic discharge occurs outside the localization zone, while continuous elasto-plastic loading and deformation occur inside.