SIMULATION OF PLC-EFFECTS IN Al6061/Al 2 O 3 ALLOYS

Transcription

1 SIMULATION OF PLC-EFFECTS IN Al6061/Al 2 O 3 ALLOYS G.V. LASKO 1, Ye.Ye. DERYUGIN 2 and S. SCHMAUDER 1 1 Institute of Material Testing, Material Science and Strength of Materials, University of Stuttgart, Pfaffenwaldring 32, Stuttgart, Germany, Institute of Strength Physics and Materials Science, Siberian Branch of Russian Academy of Sciences; Pr. Akademicheskii 2/1, , Tomsk, Russia Keywords: PLC-Effect, Stress Concentration, Plastic Strain Localization Abstract Interesting and less understood phenomenon of jerky flow - Portevin Le Chatelier effect (PLC) in polycrystals is studied on the mesoscopic level. Apart from kinetic considerations of the phenomenon, the influence of stress concentrations is considered as a driving force of localization phenomena. The model is based on the recently introduced relaxation element method (REM) which maintains an unambiguous connection between the stress drop in local volumes of solids with the plastic deformation in it in combination with the cellular automata method. REM allows to consider local sites of plastic deformation with the gradients of stresses and strains. Modelling of the underlying mesomechanical mechanisms of this phenomenon in polycrystals of Al6061/Al 2 O 3 within the approach mentioned above results in the prediction of the PLC-bands self-organization. The results of the simulations are in very good agreement with observed ones in experiments Introduction Many industrial alloys are typical representatives of materials in which under deformation the different modes of instability of plastic flow can be observed. Up to now the processes of selforganization are less understood. Nevertheless the less investigated processes of selforganization of band structures in metals are of great interest, because the possibility of the prediction of macroscopic properties of materials based on their understanding. From the practical point of view, it is important that PLC effect [1] causes deformation localization and the consecutive formation of the bands of localized shear, as a result of which the initially smooth surface becomes rough. In the connection of application of these alloys in automobile industry, where it is not desirable to have rough surface on the stamped details, the question of the definition of the modes of deformation which select the areas of stable and unstable plastic flow, is very urgent. Usually this interval is defined by testing of macrosamples at different temperatures and with different rates of deformations. However, it is very time consuming and material consuming testing. That is why computer simulation of such an effect is very attractive. It was experimentally proved and theoretically grounded that the evolution of band structures in polycrystalline metals reveals the phenomenon of self-organization. Neither, theory of dislocations, nor the mechanics of solids can explain such a phenomenon adequately. That is why necessary prerequisites have been created for the appearance of new scientific trend - physical mesomechanics [2, 3] filling the gap between two extreme approaches: micro- and macrolevels. From the point of view of physical mesomechanics the development of the band structures in a deformable solid under loading is governed by stress relaxation of stress concentrators on the different scales. The difficulty of the description of the phenomenon of strain localization lies in the fact that it is not possible to formulate a universal physical law of the connection between the plastic deformation and the stresses in the solid because of the 131

2 relaxation nature of the former one. One of the ways to resolve such kind of problems is to apply a new method of description of the stress-strain state of the material under loading - Relaxation Element Method (REM) [4-6]. In the given work the phenomenon of serrated flow is simulated on meso- and macroscale level on the basis of relaxation element method. Computer model The proposed model of the development of the bands of localized plastic deformation operates on the principle of cellular automata [7, 8]. The calculational field is divided into a number of cells in the form of close-packed hexagons, playing the role of elements of structure. Each of such elements of modeled medium possesses the ability to switch the state by discrete jump of plastic deformation, prescribed by a definite relaxation element (RE), which placed at the center of hexagon. RE defines the field of plastic deformation inside the circle region, embracing the hexagon, as a result of stress relaxation in it in the value dσ. According to calculations [4] this field of plastic deformation is characterized by the components p p p ε y = 3Δσ /E, ε x = Δσ /E, ε xy = 0, (2.1) where E is Young s modulus. Coordinate axis 0y is directed along the tensile axis. At that time, around the given site of plastic deformation (1), a non-homogeneous field of internal stresses arises with the components: 2 Δσa σ y 3 1 y x 1 a Δ 2 y = + + Δσ ; 4 2 2r r r r 2 Δσa ; y y x a Δσ = (2.2) x r r r r 2 Δσa yx 2(3 4 ) 12 a + y a y Δσ = 3 + xy r r r Here r 2 =x 2 +y 2, а - is the radius of RE (the dimension of structural element). Together with applied external stress, apparently we will have the field of stresses, characterized by the components: σ y = σ + Δσ y, σ x = σ x и σ xy = σ xy. (2.3) The corresponding distributions of the components Δσ y, Δσ x и Δσ xy are depicted in Fig. 1. The maximum values of Δσ y component without applied external one is equal to Δσ max y = 2Δσ at the boundary of RE and Δ min = -Δσ inside the region of relaxation σ y a b c Fig. 1 The distribution of the components of stress fields σ y (a), σ x (b) and σ xy from the relaxation of σ y - component. In such a manner, the element of the medium can periodically increase its degree of plastic deformation (2.1) and as a stress concentrator (2.2) influence the changing of the stress field in the whole volume of solid. The involvement of structural element into plastic deformation was realized when the shear stress along the direction at an angle of 45 with respect to tensile axis τ=(δσ x Δσ y )/2 + σ/2 attains the critical value τ cr (Tresca criterion) under the external applied 132

3 tensile stress. Interaction of the fields of internal stresses from the different structural elements, undergone plastic deformation, occurs automatically (on supperpositional principle). In the work the working equation has been used with accounting of rigidity of the mashine in the form of the dependency of the changing of external stress dσ in time dt, in which a stress relaxation takes place in the crystallite. 2 M (v0ebdt 3πa Δσ ) dσ =. (2.4) (SE + Ml0) b Here M is the rigidity modulus of the machine, E-is the Young modulus of the specimen, v 0 is the velocity of the free movement of the punch of the machine, a is the radius of the crystallite, while S, l 0 and b is the cross section, length and the width of the working part of the specimen, respectively. Parameter Δσ includes in itself the mechanism of plastic deformation, i.e. the ability of the material into plastic formchanging. The Δσ -value is accepted to be equal to Δσ =2(τ max τ 0 ). This expression has been used when calculating the field of internal stresses, according to equaton (2.3) The time of relaxation dt of the grain involvement into plastic deformation was defined by the value, which correspond to that in the macroband of localized shear of the composite Al+10%Al 2 O 3. σ-ε-diagramm has been constructed by summation of the values dσ, according to expression (2.4). Results The patterns of the formation of the zone of localization is represented in Fig.5. The zone is formed in the form of a separate band of localized deformation over the whole cross-section of the specimen at an angle of 45 with respect to tensile axis. At the beginning the band of plastic deformation with the width of one diameter of the grain is formed. Then the rapid transfer of plastic shears from grain to grain takes place in the vicinity of the front of the given band. As a result the front of the band moves in one grain diameter. The width of the band increases correspondently. In such a manner, the development of the zone of localized plastic deformation on the mechanism of Lüder s band propagation takes place. a b c d Fig. 2: The stages of grain involvement into plastic deformation a) N=50, b) N=150, c) N=300 d) N=700 The examples of the modelled diagrams for the different boundary conditions of loading are represented in Figs

4 Fig. 3: The influence of the rigidity modulus of the machine М on the type of loading diagrams: М, kn/mm = 1.3 x10 2 (1), 1.3 x10 3 (2), 1.3 x10 5 (3), 1.3 x10 8 (4). The influence of the rigidity of the tensile machine on the loading diagrams of a low-carbon steel with E = MPa [12] is depicted in Fig. 3. We observed different types of curves while varying the model parameters: The curve 1 for the «soft» mode of loading (M = 1.3x10 2 3kN/mm) has stair-case type. As the rigidity M of the machine increases from 1.3x10 2 to 1.3x10 8 kn/mm the curve takes more saw-tooth shape. Instead of stairs the amplitude of oscillations of the external stress appears and grows. At all curves the flow plateau is observed, after which the stage of work-hardening follows, caused by an increase in the internal stress field from relaxation elements. The flow plateau is formed on the mechanism of the Lüders band propagation, when the crystallites are involved into plastic deformation, consequently filling up the working part of the specimen (see Fig. 2). Repeated involvement of the grains into plastic deformation starts when practically the whole working part of the specimen is involved into plastic deformation. Further deformation requires essential growth of external applied stress. The rate of loading exerts much influence upon the σ-ε-curves. The less the rate of loading, the more pronounced is the effect of intermittent flow (Fig. 4). The increase in the loading curve results in a decrease in the amplitude of the oscillations of the external stress. Starting from the rate of loading 0,04 mm/s, there exist no oscillations of external load at the curves (curve 5). A further increase in the rate of loading results in disappearing of the yield drop and the flow plateau (Fig. 4). The flow stress increases and the effect of a sharp yield stress disappears (curves 7 and 8 in Fig. 4b). Together with the influence of the rigidity of the testing machine and the rate of loading, defining the boundary conditions of loading, the role of material characteristics itself have also been considered, all other parameters of the model were the same. a b Fig. 4: The influence of the velocity of the punch of the testing machine v 0 on the type of loading diagrams: v 0, mm/s = (1), 0.01 (2), 0.02 (3), 0.03 (4), 0.04 (5), 0.05 (6), 0.08 (7), 0.1 (8). 134

5 Decreasing of the Young s modulus by a factor of 3 from the value 210 GРа for the low-carbon steel to the value 70 GРа for aluminum essentially extends the yield plateau, decreases the effect of strain hardening, and weakly influences the amplitude of the external stress oscillations (Fig. 5) and the average stress at the yield plateau. Shown in Fig. 6 are the σ-ε - curves for different τ cr values, which we connect with the stress of dislocation unpinning. If the dislocations are not pinned by Cottrell clouds (when τ cr = τ 0 ), then the phenomenon of interrupted flow is not observed (lower curve). When τ cr increases, the effect of Portevin-Le Chatelier is enhanced. At the same time the flow stress, the yield plateau and the amplitude of the peaks at the loading diagram are increasing. The performed simulations allowed us to reveal qualitative and quantitative changing curves of loading, depending on the characteristics of the material itself and on the boundary conditions of loading. The obtained characteristics of the changes of the qualitative and quantitative loading diagrams when varying the parameters of the model: rigidity modulus, cross-head velocity, Young s modulus, are in agreement with known experimental findings. [9-12]. E GPa τcr Fig. 5. The influence of Young s modulus on the type of loading diagrams. Fig.6. Influence of the stress of dislocation unpinning on the type of loading diagrams. Conclusion In the work the main results on the simulation by relaxation element method of the processes of localization of plastic deformation, accompanied by the effects of Luders band propagation and PLC-effect are represented. The working formulae for the calculation of the dependency of the increment in the external applied force on rigidity modulus of loading device, boundary conditions of loading and elastic-plastic properties of material is derived. In the course of the simulations the following results have been obtained, which are in good agreement with the known experimental data: Effect of interrupted flow is the consequence of the formation of meso- and macrobands of localized deformation; Increase in the rate of loading suppress the effect of interrupted flow: Saw-tooth type of σ -ε curve is characteristics for the rigid mode of loading (device with the high value of rigidity modulus), in the soft mode of loading, the σ -ε curves have stair-case view. Decrease in Young modulus is accompanied by the attenuation of the effect of strainhardening. Along with it, the following peculiarities of the development of localized plastic deformation have been obtained: 135

6 The formation of a separate macroband for the rigid testing machine occurs under the decreasing in external applied stress. The initial involvement of the grains into plastic deformation occurs on the mechanism of the Lüders band propagation Accumulation of the fields of internal stresses in the volume of polycrystal in the course of repeated grain involvement into plastic deformation results in the effect of strain hardening At the same other conditions decreasing of Young modulus enhances the effect of interrupted flow. The mechanisms of stress relaxation result in arising of PLC-effect, in which the high stress of onset essentially differs from the stress of the following development of strain localization. In particular, interrupted flow can be the consequence of dynamic strain ageing, when the high stress of dislocation unpinning differs from the lower stress of their free movement. The obtained results testify to the high prediction power and advantage of the new developing relaxation element method. Acknowledgements This work was supported by the grant of German Research Foundation (DFG), Grant Schm 746/52-2 and Russian Fund of fundamental investigations, project RFFI a. Financial support is highly appreciated. References 1. Klose, F.B., Ziegenbein A, Weidenmüller, J., Neuhauser H., Hähner, P. (2003). Portevin-Le Chatelier effect in strain and stress controlled tensile test, Comput. Math. Sci Panin, V.E. (2000). Modern problems of physical mesomechanics. In: Int. Conf. MESOMECHA-NICS 2000: Role of Mechanics for Development of Science and Technology, June 13-16, Beijing: Tsinghua University Press, V.1 pp Panin, V.E., Egorushkin, V.E., Panin, A.V. (2006). Physical Mesomechanics of deformed solid as multi-level system. Physical basis of multilevel approach. Phys. Mesomech. V Deryugin, Ye.Ye. (1998). Relaxation element method, Monograph Nauka, Siberian bookpublishing firm SB RAS, Novosibirsk 5. Ye.Ye. Deryugin, Lasko G.V. (1995). Relaxation element method in the problem of mesomechanics and calculations of band structures. In: Panin, V.E. (Ed.), Physical mesomechanics and computer-aided designing of materials. Novosibirsk, Nauka, Siberian book- publishing firm SB RAS, Vol Deryugin,Ye.Ye., Lasko, G.V., Schmauder, S. (1998). Relaxation element method, Comput. Math. Sci 11 (3), Lasko G.V., Deryugin, Ye., Schmauder, S. (2004). Plastic Deformation Development in Polycrystals based on the Cellular Automata and Relaxation Element Method, Lecture Notes in Computer Science, Lasko G.V., Deryugin, Ye., Schmauder, S. (2006). Simulation of the Evolution of Band Structures in Polycrystals on the Basis of Relaxation Element Method and Cellular Automata, Lecture Notes in Computer Science, Zhang, Q., Jiang Z., Jiang H., Chen Z., Wu, X. (2005). On the propagation and pulsation of Portevin-Le Chatelier deformation bands: An experimental study with digital speckle pattern metrology, Int. J. Plasticity

7 10. Krishtal, М.М. (2001). Instability and mesoscopic inhomogenuity of plastic deformation (analytical review). Phys. Mesomech. V. 7 (5) Toyooka, S., Widiastuti, R., Zhang, Q., Kato, H. (2001). Dynamic Observation of Localized Strain Pulsation Generated in the Plastic Deformation Process by Electronic Speckle Pattern Interferometry, Jpn. J. Appl. Phys. V.40 Part 2 No 2A Deryugin, Е.Е., Panin, В.Е., Schmauder, S., Storozhenko, I.V. (2001). Effects of strain localization in composites on the basis of Al with Al 2 O 3, Phys. Mesomech. 4 (3)