J Mater Sci (2012) 47:4514–4519 DOI 10.1007/s10853-012-6296-3
Modelling of equal channel angular pressing using a mesh-free method T. Fagan • R. Das • V. Lemiale • Y. Estrin
Received: 4 November 2011 / Accepted: 23 January 2012 / Published online: 29 February 2012 Ó Springer Science+Business Media, LLC 2012
Abstract Severe plastic deformation (SPD) processes are widely recognised as efficient techniques to produce bulk ultrafine-grained materials. As a complement to experiments, computational modelling is extensively used to understand the deformation mechanisms of grain refinement induced by large strain loading conditions. Although considerable research has been undertaken in the modelling of SPD processes, most of the studies have been accomplished using mesh-based methods, such as the finite element method (FEM). Mesh-based methods have inherent difficulties in modelling high-deformation processes because of the distortions in the mesh and the resultant inaccuracies and instabilities. As an alternative, a meshfree method called smoothed particle hydrodynamics (SPH) is used. The effectiveness of this technique is highlighted for modelling of one of the most popular SPD techniques, equal channel angular pressing. A benchmark between SPH and FE calculation is performed. Furthermore, a number of simulations under different processing
T. Fagan V. Lemiale Division of Mathematics, Informatics and Statistics, CSIRO, Clayton, Australia T. Fagan Department of Mechanical and Aerospace Engineering, Monash University, Clayton, Australia R. Das (&) Department of Mechanical Engineering and Centre for Advanced Composite Materials, University of Auckland, Auckland, New Zealand e-mail:
[email protected] Y. Estrin Department of Materials Engineering, Centre for Advanced Hybrid Materials, Monash University, Clayton, Australia
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conditions are compared to existing literature data. A satisfactory agreement is found, which indicates that SPD processes can be approached by mesh-free methods, such as SPH.
Introduction Severe plastic deformation (SPD) processes use special procedures of metal forming to impose very high strains on materials leading to exceptional grain refinement whilst simultaneously maintaining the original specimen or workpiece dimensions. New and unusual properties, such as enhanced strength, significant increase of fatigue limit and improved superplastic forming capability, have been demonstrated for a wide range of different metals and alloys processed by SPD [1–6]. Ultrafine-grained (UFG) materials are defined as polycrystals having an average grain size less than 1 lm. Furthermore, bulk UFG materials have fairly homogenous and reasonably equiaxed microstructures, with a large proportion of the grain boundaries having high angles of misorientation. The presence of a large fraction of high-angle grain boundaries is one of the main factors responsible for the exceptional properties of SPD processed materials [6]. Significant research has been directed towards an improved understanding of the deformation mechanisms leading to grain refinement down to sub-micron range [7–10]. In this context, numerical modelling is a potent approach to help optimising processing techniques [11]. However, since large plastic deformations are induced by SPD processes, conventional finite element-based methods (FEM) formulated in a Lagrangian framework are usually not appropriate due to severe distortion of the FE mesh. Indeed, it is well known that the integrity of elements has profound effects on
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the quality and accuracy of the FE solutions [12], and thus other methods not restricted by mesh distortion are potentially useful to model these problems. Smoothed particle hydrodynamics (SPH) is a particlebased numerical method which is used to obtain solutions to systems of partial differential equations. The problem geometry is discretised into ‘particles’ that represent specific material volumes. The method is based on the use of local interpolation from the surrounding discrete particles to construct continuous field approximations that can be used in the discretisation of field equations. For details of SPH fundamentals, see references [13–15]. SPH does not use any underlying grid or mesh structure to represent the problem geometry and so avoids the difficulties associated with conventional mesh-based methods in maintaining the integrity and quality of the mesh under large strains. SPH has been traditionally used for modelling fluid flows [14–16]. In recent years, there has been a growing interest in applying SPH to a wide variety of solid mechanics problems [17–20]. A critical attribute of Lagrangian methods lies in their ability to track the material history through an entire simulation. In SPH, this can be achieved for particles. SPH can predict the thermal and mechanical history of a workpiece without the need for data mapping between evolving (discretised) geometric configurations. This provides significant capability to track the variables, such as cumulative plastic strain, damage or evolving microstructural variables. Therefore, SPH appears to be conceptually well suited to simulate large strain problems, including SPD processes. By applying the SPH method to equal channel angular pressing (ECAP) (one of the most studied and established SPD processes), we show that it is indeed an effective alternative to FEM. After a brief overview of the SPH method specifically in the context of elastoplastic deformation, comparison of SPH with FEM is first performed. It is shown that under the conditions considered, both methods produce similar results. Then the ECAP process is applied to a perfectly plastic material and a material exhibiting strain hardening. Considerable differences in the deformation patterns are predicted, and the observed trend is consistent with existing data. Finally, we investigate the effect of the outer channel angle, a parameter known to play an important role in ECAP. Again, the results compare favourably to previously reported data from the literature. The applicability of SPH to simulations of large strain problems, such as SPD processes, is therefore established.
SPH formulation with elastoplastic linear isotropic hardening In this section, a brief overview of the numerical framework is provided, the emphasis being on the implementation of
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elastoplasticity in the SPH formulation. More details on SPH can be found in the references [13–15]. The SPH discretisation produces a set of coupled ordinary differential equations that are integrated using an improved second-order Euler predictor–corrector method. Firstly, the equations are solved to determine the current deviatoric stress and local density. The density is then used to compute the pressure. The constraints of the deviatoric stress by the plasticity model are then evaluated. The final deviatoric stress and the pressure, along with the current particle velocity, are then used to calculate the acceleration of the particles. Finally, the equation of motion for each particle is integrated along with the kinematic equations to give updated particle positions. The radial return plasticity model by Wilkins [21] is used to model the elastoplastic deformation. A trial deviatoric stress Str is calculated assuming an initial elastic response. Upon yielding, an increment of plastic strain can be estimated as: Dep ¼
rvm ry 3G þ H
ð1Þ
where rvm is the equivalent von Mises stress, ry is the current yield stress, G is the shear modulus and H is the strain hardening coefficient considered to be constant for simplicity. The plastic strain is then updated as: ep ¼ ep þ Dep
ð2Þ
The current yield stress is given by: ry ¼ r0y þ H ep
ð3Þ
where r0y is the initial yield stress. Therefore, the yield stress increment Dry at each time step is calculated as: Dry ¼ H Dep
ð4Þ
Finally, the deviatoric stress S at the end of a time step is given by: S ¼ rs Str where rs is the radial scale factor given by: ry rs ¼ rvm
ð5Þ
ð6Þ
Modelling approach and problem configuration Figure 1 shows the setup for the numerical model. The commercial package MSC.Marc was used to model the experiment with the FE method. The elements were approximately 0.5 mm2 in size, which represented a total of 2,975 elements in the work-piece chosen. The analysis assumed plane strain conditions. For the SPH simulations, an in-house program was used. An initial particle separation of 0.25 mm was implemented.
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The work-piece was modelled as an elastoplastic solid. The ECAP die channel and the piston were rigid solids. The contact between the work-piece and the tools was assumed to be frictionless. A work-piece with a height of 59.75 mm and crosssectional dimensions of 12.5 9 12.5 mm2 was considered. The material used was a copper alloy with density of 8,960 kg/m3, Young’s modulus of 124 GPa and Poisson’s ratio of 0.33. The initial yield stress was assumed to be 400 MPa. Comparison of the SPH method with the FE method
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edge effects than the FEM one by the contact boundary conditions used. Figure 3 shows the end of the work-piece for both FEM and SPH to highlight the similarity in the final deformed shape and the plastic strain distribution predicted by the two techniques. The height of the tip is approximately 2.6 mm in the FEM solution and 3 mm in the SPH solution, a 15% difference. This agreement was deemed satisfactory as there are a number of numerical factors that may influence the final results. In particular, in an earlier paper [25], it was shown that the numerical resolution (e.g. the mesh size in finite element calculations) had a noticeable effect on the final billet shape. The observed geometrical gaps between the work-piece and the tools tended to converge to a particular value with a finer resolution. However, this increase in accuracy implied a significantly longer computing time. Therefore, following the conclusions of this earlier work, a compromise was adopted in the present paper to avoid unnecessarily long simulations. In summary, overall this first SPH–FEM comparison has shown that both methods give comparable results, demonstrating the relevance of SPH in the modelling of SPD processes.
Previously, results of FE analysis have been compared to ECAP experiments reported in several studies [22–24], and it has been established that FE solutions can be considered reliable in the modelling of ECAP. To verify SPH as a valid modelling tool for ECAP, it was therefore first benchmarked against FEM on a single ECAP pass simulation. In these simulations, the outer channel angle, W, was 45°. Figure 2 shows the equivalent (von Mises) plastic strain field calculated by both the methods. A good qualitative agreement is observed both in terms of the predicted deformed shape and with regard to the plastic strain distribution. It should be noted that the FE solution field appears smoother because the result is extrapolated from the Gauss points and plotted at the nodes, whereas the SPH solution is directly plotted on the particles with no further extrapolation. The level of strain experienced by the work-piece towards the lower edge and the upper edge towards the piston (indicated by the arrows in Fig. 2b) is where there are noticeable quantitative differences. A higher level of plastic strain is predicted by SPH in these areas. This may be attributed to the differences between the two methods in the way contact between the work-piece and the tooling is treated. The SPH solution seems to be more affected by
Fig. 2 Equivalent plastic strain field: FEM (left) and SPH (right)
Fig. 1 Geometry used for the ECAP simulations
Fig. 3 Close-up view of the front end of the metal work-piece: FEM (left) and SPH (right)
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Effect of process parameters on material deformation using SPH To further demonstrate that SPH is a potent alternative to FEM, a number of ECAP simulations were conducted using SPH under different conditions. First, the effect of strain hardening was investigated. Next, the effect of the variation of the outer channel angle W was studied. Variation in strain hardening In this study, a linear isotropic hardening was assumed; refer to the section describing the elastoplastic algorithm implemented in SPH. The yield stress is given by Eq. 3, which only involves a single parameter: the constant strain hardening coefficient, H. Two different cases were considered, namely an ideally plastic material with zero H and a material exhibiting a non-zero strain hardening. Figure 4 shows the equivalent plastic strain field for the two cases, the left and right columns being the cases for perfectly plastic and strain hardening materials, respectively. In Fig. 4a, the initial deformation has begun with a band of high-plastic strains near the bend of the channel. In contrast, for the strain hardening material, the front of the billet has become curved and a corner gap has occurred (Fig. 4b). The final plastic strain distribution is more homogeneous for the perfectly plastic material (Fig. 4c). For the strain hardening case, a higher level of plastic strain is predicted in the upper section of the work-piece, with a reduced level through the middle and lower sections. For this case, the corner gap has also increased in size for the final deformed geometry (Fig. 4d). Both the deformation patterns including the formation of the corner gap and the Piston Displacement (%)
H = 0 GPa = 0°
plastic strain distribution are consistent with earlier results obtained under similar conditions [26]. The final shapes of the work-piece are also distinct, with a nearly flat, unchanged surface shown in Fig. 4c, as opposed to a rotation and exposure of the lower face as observed in Fig. 4d. Similar patterns were found by Leo et al. [27] under comparable conditions. Figure 5 highlights the effective corner angle, Weffective, of *56°, which represents the angle of curvature, or arc curvature, of the work-piece. Segal [28], and subsequently Iwahashi et al. [29] and Alkorta and Gil Sevillano [30], proposed equations for calculating strain after any number of passes as a function of the inner and outer channel angles. For materials where a corner gap is produced, Kim et al. suggested that the value of W used in these equations be changed from the die corner angle to the arc curvature of the work-piece, thus resulting in a more accurate representation of the final strain [26]. Table 1 shows the average equivalent plastic strain for the region ABCD, indicated in Fig. 1. According to the SPH prediction, the presence of strain hardening results in an average reduction in plastic strain of about 15% as compared to the theoretical prediction [26]. The average strain predicted by SPH is about 20% lower than the value predicted by Iwahashi et al. [29] and Alkorta and Gill Sevillano [30] for the perfectly plastic material, and, respectively, 17 and 15% lower for the strain hardening material. Variation of the outer channel angle Next we study the deformation and plastic strain distribution variation as a function of the outer channel angle, W.
H = 1.86 GPa = 0°
(a)
(b)
45
Fig. 5 Definition of the effective outer angle as proposed by Kim et al. [26]
(c)
(d)
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Table 1 Comparison of average equivalent plastic strain across a region of the work-piece after one ECAP pass with the literature [29, 30] H (GPa)
Fig. 4 Equivalent plastic strain field for different values of the strain hardening coefficient, H
Average equivalent plastic strain SPH
Iwahashi et al. [29]
Alkorta and Gil Sevillano [30]
0
0.917
1.15
1.15
1.86
0.780
0.943
0.917
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In these simulations, a perfectly plastic material was considered. The resulting deformation shown in Fig. 6 demonstrates the effect of the channel angle on the strain field and the leading face of the work-piece. The 0° channel angle completely fills the corner and no gap is produced (Fig. 6a). The 45° channel angle seems to initially introduce a corner gap, as shown in Fig. 6b. However, as the work-piece deformation progresses, the corner gap diminishes. Figure 6d shows that the 45° channel produces a strain reduction in the lower section and a moderate, fairly homogenous strain field in the middle and upper sections of the work-piece. There is also an angular point in the case of the 45° angle channel. The overall strain for the 45° angle channel is lower compared with the 0° angle channel. In the latter case, a high degree of homogeneity of strain distribution throughout the work-piece and little deformation of the leading face was found (Fig. 6c). This compares well with the results of Balasundar et al. [31], where a reduction in the equivalent plastic strain was found with an increase in the corner angle, W. Figure 7 shows the change in the internal deformation patterns for different magnitudes of the outer channel angle. This result suggests that the inclusion of an outer channel angle increases the shear angle with respect to the flow direction, resulting in reduced overall deformation in the work-piece. The angle h1 predicted by the SPH simulation for the W = 0° case is in agreement with the analytically derived shear angle of 26.6° provided by Zhu et al. [10].
Piston
= 0°
Figure 8 shows the force on the piston as a function of its displacement. It is seen that the non-zero angle leads to a reduction in the overall force required to press the workpiece, which is consistent with previous FEM analysis [32]. The reason for a dip observed in the upper curve (for W = 0°) may be due to a ‘‘trapping’’ and brief accumulation of material in the lower left corner, allowing the remaining metal to flow over the top (see Fig. 9). In other numerical experiments, this has been termed ‘dead metal zone’ [33]; it tends to alter the flow in a similar manner as a non-zero outer channel angle. The clearing of the accumulated metal could result in the piston force returning to its mean value. For the case of W = 45°, the force required for ECAP was found to be on average 27% lower than that in the case of W = 0°.
Fig. 7 Simulation of shear angle for a W = 0° and b W = 45°
= 45°
Displacement (%)
(a)
(b)
41
Fig. 8 Force-displacement curve for the piston for different values of the outer channel angle, W
(c)
(d)
100
Fig. 6 Plastic deformation behaviour for different outer channel angles
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Fig. 9 Accumulation of metal in the channel corner, referred to as ‘dead metal zone’
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Conclusions Smoothed particle hydrodynamics has been proposed as an alternative to FEM in the simulation of metal working processes involving large strains. To demonstrate its suitability for modelling such processes, ECAP was chosen as a representative SPD process for benchmarking, since the characteristics of this process are well established with considerable literature available. The initial comparison between the SPH and FEM solutions based on a simplified constitutive model has demonstrated that both methods predict similar results, especially in terms of the final deformed shape of the workpiece and the plastic strain distributions. Next the effect of strain hardening was analysed by considering a perfectly plastic material and a strain hardening material with a constant hardening coefficient. In accord with previous reports in literature, it was found that strain hardening changes the flow pattern and the level of plastic strain quite appreciably. The effect of the outer channel angle was also investigated. It was found that a sharp corner angle configuration induces larger plastic strains and requires a larger pressing force. The final shape of the work-piece also exhibits significantly less distortion of the end part. All these observations are consistent and agree well with the literature. From this work, it is concluded that SPH can be used as an alternative to FEM in the simulation of SPD processes, the main advantage being the elimination of numerical problems associated with mesh distortion. This will provide a significant advantage in the modelling of SPD processes in which materials undergo extremely large deformations. In this study, a simplified elastoplastic constitutive model, which does not account for microstructure evolution of the material, was used. Future work will focus on the development of material models that do account for the evolution of microstructure with straining. Utilising the strengths of SPH, a new class of numerical models will be developed. Acknowledgements We would like to acknowledge the funding through CSIRO, as well as the Mechanical and Aerospace Engineering and Materials Engineering Departments of Monash University.
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