Transactions on Engineering Sciences vol 6, 1994 WIT Press, ISSN

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1 Finite element modelling of strain localisation in shear bands during a cutting process J. Pierry, X.C. Wang Department MSM, University of Liege, 6, quai Banning, 4000 Liege, Belgium ABSTRACT Several simulations of the cutting process were carried out on two types of materials: a metal and a rock. For that purpose, two adequate laws implemented in the large strain finite element code LAGAMINE were used. These laws allow strain localisation in shear bands. The beginning of this localisation is detected by different means: by a bifurcation criterion, in the sense of the Rice analysis (1) or by a scalar based on the evolution of the strains during the cutting process. The loading is realised by the displacement of the cutter, step by step. After each step, the criterion are computed, indicating a progression of the area where localised strains could be produced. Classical finite element meshes lock the progression of shear bands. Therefore, remeshing techniques are developed and applied so as to observe the chips formation into the material. These procedures are based on classical remeshing techniques or on new ones, like the spectral method proposed by Belytschko (2,3) or the band dedicated interface element meshing. INTRODUCTION Cutting processes are widely used in various industry fields such as metallurgy and well boring for example. Unfortunately, optimising these processes is usually completed through field observation or expensive laboratory tests. Finite element simulations of the cutting process can help understanding the behaviour of a material against which a tool is pressed. Hence it can become a helpful support to improve cutting efficiency, e.g. to design new cutters.

2 522 Localized Damage DETECTION OF STRAIN LOCALISATION AND RHEOLOGY We are concerned with failure mechanisms that progressively lead to the formation of one or several chips in the material. We need to know where and under which conditions failure takes place. It is well known that breaking mostly occurs by strain concentration in a thin area. This phenomenon is called shear band localisation. The constitutive laws describing the material rheology must allow this localisation and its detection. Moreover, during the cutting process, the material undergoes large deformations. It is thus imperative that the finite element code allows the occurrence of such deformations. The finite element code LAGAMINE that has been developed for several years at the University of Liege at MSM department meets these demands. Several laws suitably describing various material rheologies have been successfully implemented in this code. We have used two of them to describe two different types of materials: a metal on the one hand, that is ruled by the Gurson elastoplastic law and a sandstone on the other hand, described thanks to an hypoplastic non-linear incremental law, called CLoE (4). Bifurcation criterion The detection of the strain localisation can be realised by the computation of the so-called bifurcation criterion in the sense of the Rice analysis. Results are shown on figure 1 for steel cutting and on figures 2 and 3 for rock cutting. Two double-headed arrows are drawn where the criterion is fulfilled. Their respective slopes shows the corresponding direction of the potential shear band. As shown in figure 1, for a metal, the band is quasi-linear, its slope is more or less 45, the other direction, which is approximately orthogonal to the other, doesn't seem to have any physical meaning. In a rock (figures 2 and 3), the area is much more widespread but the two directions seem to have their own physical meaning: an example of possible band shape has been added on each figure in bold line. The first one is similar to that of a slipping surface in a slope. The second band is a straight line that develops upwards to the left. As we mentioned, these lines are only an example of possible bands. Due to the extension of the bifurcation area, their shape and position remain very 'drawer-dependent*. Refining the mesh doesn't make the area any thinner. It is thus essential that we would find another criterion to define the shear bands with satisfactory precision.

3 Localized Damage 523 Figure 1: Bifurcation arrows in steel cutting tool displacement 0.5 mm Figure 2 Figure 3 Bifurcation arrows in rock cutting Tool displacement 0.2 mm Scalar band indicator Another criterion has been tested in order to detect more precisely the occurrence of a shear band into the material. It is an expression of the evolution of the deviatoric component of strain during a time step from A tob: A-B Examples are shown in figures 4 and 5 in rock cutting for two different meshes. Similar indicators that could be more reliable and easy to compute are tested so as to better capture the bands.

4 524 Localized Damage Figure 4: Scalar indicator in a coarse mesh Figure 5: Scalar indicator in a fine mesh POST-BIFURCATION ANALYSIS AND REMESHING TECHNIQUES Once the area where failure is likely to occur is determined, the point is to allow the effective development of the bands into the material. Classical finite element meshes lack that possibility and it is thus impossible to analyse the post-bifurcational behaviour, i.e. the formation of chips, unless the potential displacements field is enriched. This can be obtained through a change in the mesh, involving various procedures globally called remeshing operation. Various remeshing techniques can be considered: Classical remeshing It is based on successive refinements of the mesh: the tool is moved forward until the localisation criterion is fulfilled in a predetermined rate of elements. The simulation is interrupted and the remeshing process begins. A new mesh that is thicker in the interesting area is produced. The simulation can either then be restarted from the beginning using this new mesh or resumed. In the latter, data from the old mesh must be appropriately transferred to the new one, so that the force balance resulting from the preceding simulation is kept or at least not too much disturbed. Unless this last step in the remeshing procedure is well cared for, it will lead to numerical divergence when the simulation is carried out further on. This method is used in modelling metal forming processes (5). Fitting it to the cutting peculiarities is far from obvious. Several problems must be worked out. For example, the refinement must be much more dense and focused on the area where rupture is expected. It should also be oriented, i.e. anisotropic, so as to see the shear band actually develop.

5 Localized Damage 525 Sectrl A second strain field, that is able to represent strains occurring in the shear band is superimposed to the classicalfiniteelement's one. This new field is only present where strain localisation is expected. If the band propagation is to be progressive, several spectral bands will be considered. This method has been implemented in the LAGAMINE code by WANG X.C.(3). An example is shown on figure 6: one spectral band is superimposed on thefiniteelements mesh where the bifurcation arrows showed a shear band could develop (see figure 1). The simulation is then carried out over again. A comparison of the strainfieldobtained with and without the spectral band in a cross section A-B is shown on figure 7. It can be seen that strains are much higher in the band region when a spectral band is superimposed. These results prove the efficiency of the method to account for shear bands in a material. 0,4 Finite elements + spectral band 0,1 Finite elements A OjO\ 0,0 02 0,4 0,6 0,8 If) Figure 6: Superimposition of a Figure 7: Comparison of strains spectral band on the predicted shear with and without a spectral band band area Remeshing using interface elements The meshes obtained thanks to the preceding methods scarcely remain valid for a long time since some of their elements keep undergoing large distortions. They demand several remeshing procedures that interrupt the simulation and lead to meshes made up of an increasing number of elements. In order to keep up with the code's limitations and to overcome the difficulties arising from the use of these techniques, we have developed a new and original method, so-called remeshing with interposition of interface elements that is described hereafter. A slipping surface is introduced in the mesh where the localisation criterion is fulfilled. Specific interface elements are stuck to both sides of the discretised crack, so that the geomaterialfiniteelements located on either side of the line can deform independently.

6 526 Localized Damage They are only related to one another by the interface law that is a contact law such as the Coulomb's dry friction contact law where the unique parameter is the friction angle. To model the rock's interface behaviour more adequately, we also adapted the CLoE law to the particular cinematic of interface elements. This new law can deal with the shear band's behaviour specificities and can represent its rheology provided the law's parameters correspond to the behaviour of the material within the band and not to that of the rock. Simulations with approximate parameters showed a smoothing of the crack's influence on the global rock's behaviour(6). An example is shown hereafter for rock cutting. A new mesh involving two cracks was inferred from the general bifurcation shape in the 'unremeshed' model as shown in figures 2 and 3. The new mesh is shown infigure8, the interface law is Coulomb with a low friction angle so that the facing portions of rock can slip rather easily. We obtain a rock in which two chips have fully taken shape. Again, the simulation is restarted from the beginning. The deformed configuration obtained after a displacement of 0.34 mm of the tool is shown in figure 9. Figure 8: Mesh with two complete chips of rock Figure 9: Deformed configuration of the mesh involving two chips CONCLUDING REMARKS The few results presented in this paper demonstrate the efficiency of finite element simulation of the cutting process. We proposed two different criterion to find out the area where bands are likely to appear and their directions. Two original methods have been developed to take these bands into account in the model. We aim to use several methods in combination: for example, we could first refine the mesh until the shear band is localised with enough accuracy and interface elements could then be progressively inserted to watch the effective development of the crack into the material, i.e. the building up of a chip. This procedure should of course ideally be automatic and progressive so that the simulations can be resumed rather than restarted.

7 REFERENCES Localized Damage RICE, J.R: The localisation of plastic deformation. Theoretical and applied mechanics, W. KOITER ed., 1976, North-Holland. 2. BELYTSCHKO, T. and FISH, J: Spectral superposition on finite elements for shear banding problems. Proc. of the 5th. Int. Symp. on Numerical Methods in Engineering, 1989, vol 1, pp WANG, X.C: Modelisation numdrique des problemees avec localisation de la deformation en bandes de cisaillement, 1993, these de doctorat en Sciences Appliqudes, Universitd de Liege, Belgique. 4. CHARLIER, R., CHAMBON, R., DESRUES, J. and HAMMAD, W. - Shear band bifurcation in soil modelling : a rate type constitutive law for explicit localisation analysis. Third Int. Conf. on Constitutive laws for Engng. materials, Tucson, Ed Desai and Krempl, ASME Press, HABRAKEN, A. M. and CESCOTTO, S. - An automatic remeshing technique for finite element simulation of forming processes. Intl. Jl. for Num. Meth. in Eng., Vol 30, pp , PIERRY, J: Moddlisation par elements finis de Faction mdcanique d'un outil a diamant polycristallin, 1992, mdmoire de maitrise en Sciences Appliqudes, University de Liege, Belgique.