The simulation of large deformations for elasto-plastic materials remains indispensable for forming and the calculation of structures. This is a problem that is always on the agenda due to its importance in applications, where industrial operations are limited by the appearance of deformation localization phenomena and plastic instabilities

In this paper, the focus is set on metallic materials. Within their grains many micro shear bands appear which eventually grow into macro shear bands, provoking failure. A distinctive phenomenon is that neither the micro bands or, all the more, the macro ones are crystallographic. They are not due to the coarsening of the sliding on slip planes. Now, the appearance of shear bands is often related to the geometrical features of the specimen. This is the case in

The appearance of shear bands is always related to the fact that the applied stresses do not have the same principal axes as the specimen. This happens in

The proposed approach is validated by an application in the case of the channel-die compression test. Thus, the phenomenological elastoplastic behaviour with isotropic strain hardening has been numerically integrated and coupled with a criterion for predicting instabilities based on Rice’s criterion. This criterion has the advantage to be three-dimensional and to apply in the elastoplastic case. Contrarily to other criteria used in early works, it is not limited in its field of application either by its two-dimensional character or by its limitation to the rigid plastic behaviour

Rudnicki et al.

in

In this work we consider an orthotropic material of three orthotropic axes noted

where:

In the case of small elastic deformations

So we can rewrite the relationship:

In this isoclinic configuration, the tensors should be represented by Lagrangian quantities^{T} such that:

For that we will adopt here the revolving referential formulation

where W is the spin of the global rotations. Writing the tensor w defines the choice of the rotating frame. Two cases will be considered in this work:

- corotational Jaumann derivative such as:

- spin W based on Green Naghdi’s derivative.

where the tensor

In this work the orientation Q will be defined by means of three angles of Bunge

The rotation rate is written:

We have adopted a three-dimensional kinematics writing which allows us to define a rotating material frame of reference whose deformation gradient and velocity gradient tensors are described in the form of an upper triangular(tr) matrix. The original idea of a material rotation frame was due to Mandel ^{TM}, which uses a Lagrangian formulation, the UMAT subroutine allows to program customized laws of behavior in such a rotational kinematic framework

where (

The strain velocity gradient tensor L is then defined in the base

For a gradient of deformation F defined by:

we have the following writing in the deformed configuration:

To configure the distorted configuration, we just use the orthonormal coordinate system _{i} be:

The transformation gradient

The introduction of the rotating reference system allows us to define the tensors

Where Q is the matrix defined in (8) by the three Bunge angles

To describe the law of behavior, the criterion used is Hill’s

where:

and

so that:

with Swift's law of hardening:

where the coefficients

The formalism of the generalized standard material allows us to write.

These relations are supplemented by the elastic law which is written in the orthotropic reference system by:

if

then

and equation (23) becomes:

and

or:

In the Eulerian configuration this is written:

with:

Writing the decomposition of the strain rates:

allows to write:

This can be written in the form:

with:

The objective derivative is written:

or:

which becomes:

with:

and

Two cases can be distinguished:

i) Corotational referential (Jaumann where w = 0, in which case the law is written:

and:

Using the first Piola Kirchhoff tensor, we can write the incremental law in the form:

with:

Let us now consider the channel-die test in which the height of the sample changes from h to h_{0}. We note

We easily relate the quantities γ_{2} and γ_{3} to the angles

Since F is a triangular tensor, the velocity gradient

The stress tensor is such that the faces perpendicular to axis 1 are free, i.e. T_{11} = 0. We also assume the absence of friction T_{12} = T_{13} = 0. Hence:

In this study we limit ourselves to the shear band phenomenon, for which the instability is characterized in terms of bifurcation of the equilibrium path _{c} which is the moment of the passage from a homogeneous state to a non-homogeneous one and which corresponds to a discontinuity in the velocity gradient

The onset of shear bands has been linked to a number of material factors. For metallic alloys, a review has been proposed by Pineau et al. _{c} the material is stable and the field

where П is the first Piola-Kirchhoff tensor.

Indeed from:

we obtain:

Hence, Rice's condition of instability is:

with:

This section presents two analyses: first the analysis of constraints based on the responses of stresses and shear strains as a function of the shortening deformation, then the analysis of instability followed by a discussion of the results.

The numerical resolution of the system of differential equations is done using the 4^{th} order Runge Kutta method. The programming is written under the Matlab environment and the prediction of instability is done by a test of Rice’s criterion.

In this section, we focus on behavioural change and the effectiveness of the proposed criterion, in order to analyse and discuss the numerical results obtained and to simulate the anisotropic elastoplastic behaviour independent of rotation speed.

We illustrated the results in the case of a power type hardening law defined by:

To demonstrate the feasibility of the computation of the mechanical behaviour, Hill’s coefficients had to be specified. The example chosen in the literature was a mild steel sheet studied by Habracken et al.

To show the influence of anisotropy on the precocity of shear banding, the axes of orthotropy of the material must not coincide with those of the channel-die. Now, they are determined by three Bunge angles φ_{1}, φ and φ_{2}. As noted before, φ_{1} was set to 30 °. There was no point in varying simultaneously the two other angles. So φ was set at 90° and φ_{2} ranged from 0 to 180°.

The material parameters

As pointed out above, the use of an incremental constitutive law implies the choice of an objective derivative, for example Jaumann’s and Green-Naghdi’s. Both have been tested in the present work.

In

_{2 }and χ_{3} as a function of shortening Ɛ. They show a slight increase in the angles for the tangential deformation values in the Green Naghdi case and almost no change in the Jaumann case.

In this section, we present an analysis of the results obtained when integrating Rice’s criterion with anisotropic elastoplastic phenomenological behaviour and isotropic strain-hardening in the case of a planar channel-die compression test. The analytical approach proposed above led to the results below. Other authors have integrated only the forming limit diagrams flow limit curves (FLC) in deformations by applying Rice’s criterion to the elasto-plastic strain hardening in uniaxial and plane tension

Our approach consists in stopping when the determining function (58) passes through a minimum or else changes its sign. We trace the evolution of the stress and the force under control of the change of orientations. We start with a predictive calculation of instability. If Rice’s condition of bifurcation is fulfilled, we pass to a new increment of the orientation by a jump of 15°.

The results of the numerical simulations can be found in

_{2} gives the direction of the applied compression (vertical in the channel-die) in the (1,2) plane of the sheet, that is, between the rolling and transverse directions. For most applications, and especially deep-drawing, this is the key factor. There are definitively directions of this plane in which shear bands are more likely to appear.

Orientation φ2 (°) |
60 ° |
75 ° |
90 ° |
105 ° |
120 ° |
135 ° |
150 ° |
180 ° |

Ɛ |
0,3230 |
0,2478 |
0,3777 |
0,3742 |
0,5723 |
1,2931 |
1,0137 |
0,6928 |

σ33/E |
0,000728 |
0,000737 |
0,000866 |
0,000847 |
0,000877 |
0,001031 |
0,001129 |
0,001070 |

σ22/E |
0,001210 |
0,001315 |
0,001624 |
0,001537 |
0,001476 |
0,001528 |
0,001410 |
0,001186 |

The analysis of two initial orientations (φ_{1} and φ are fixed and φ_{2} varied between 0 ° and 180 ° with a step of 5 °) studied and schematized in _{2} on the appearance of plastic instabilities. The studies on single crystals confirm this evolution as well as the effect of microtextures.

The present paper shows how an orthotropic material submitted to channel-die compression can be modelled in the framework of large strains and how it is affected by shear banding. Several points came out when doing so.

Large strain formulation involves the use of numerous tensors but the calculations are greatly simplified by the introduction of a material rotation frame in which the deformation tensor takes the form of an upper triangular matrix.

Rice’s criterion, which necessitates the large strain formulation, predicts efficiently shear banding as a bifurcation in the deformation path. It is linked only to considerations of continuum mechanics, regardless of material factors often presented as the source of these instabilities. In fact, they only accommodate it according to the specific microstructure of the considered solid.

Along with numerous previous works, the paper shows the sensitivity of shear banding to anisotropy. Its precocity relies heavily on the position of the applied stress with respect to the axes of the material. Here the latter was only mildly orthotropic, with a plastic behaviour represented by Hill’s quadratic criterion. Nevertheless, their onset varied from ε = 0.25 to ε = 1.29 in a rotation around an axis.

The next step in the research will be to introduce this formulation in a finite element code using a Lagrangian formulation, so as to retain the advantage of the formalism of large strains.