Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 132 (2015) 274 – 281
The Manufacturing Engineering Society International Conference, MESIC 2015
Application of the Upper Bound Theorem to indentation processes with tilted punch by means of Modular Model C. Bermudoa,*, F. Martína, L. Sevillaa a
Department of Manufacturing Engineering, University of Malaga. C/Dr. Ortiz Ramos, s/n E-29071 Malaga. SPAIN.
Abstract Nowadays, due to the new deformation processes that are being developed like the Incremental Forming Process (IFP) or the Localized-Incremental Forging Process (LIFP), indentation processes are regaining importance again. In this paper, in order to cover the largest possible number of cases in the study of indentation a tilted punch is considered. Thus, using combinations of flat and tilted punches makes possible the analysis of complex geometries. Present study shows the analysis of the indentation process by the Upper Bound Theorem (UBT) when it is carried out with a tilted punch. A modular model is developed. This modular model is compound by three modules, each with two Triangular Rigid Zones (TRZ), adapting better to the configuration of the punch this way. © by by Elsevier Ltd.Ltd. This is an open access article under the CC BY-NC-ND license © 2015 2016The TheAuthors. Authors.Published Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-reviewunder underresponsibility responsibility Scientific Committee of MESIC Peer-review of of thethe Scientific Committee of MESIC 2015 2015. Keywords: Indentation; Upper Bound Theorem; Triangular Rigid Zone; Tilted punch; Modular distribution.
1. Introduction On the one hand, until today, the indentation processes have been considered as secondary processes due to the deformations obtained. This deformation, often, are localized, small and, especially in the hardening tests, have a superficial effect [1, 2]. However, nowadays, with the recent industry development, indentation is gaining importance. There are several indentation based processes, like the Localized-Incremental Forging Process (LIFP) or the Multiple Indentation Process (MIP), both classified as Bulk Metal Forming Processes, that apply lower forces to
* Corresponding author. Tel.: +34 951952427. E-mail address:
[email protected]
1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of MESIC 2015
doi:10.1016/j.proeng.2015.12.495
C. Bermudo et al. / Procedia Engineering 132 (2015) 274 – 281
obtain the same deformation similarly to other current processes, do not require complex dies and are very flexible in their implementation with CNC machines [3]. On the other hand, the Upper Bound Theorem (UBT) by means of modules of Triangular Rigid Zones (TRZ), developed by F. Martín in 2009 [4] and based on studies carried out by Kudo [5-7] in the 60s, is a method that allows to obtain the necessary deformation effort to achieved the estimated shape in the work-piece deformed, depending of the processes used. This technique is characterized by its easy application, after the mathematical model is generated. Just introducing known parameters of the process, as the measurements of the dies or the depth required, the model will provide the effort that has to be applied to reach such deformation. Within the present research, the indentation processes are study from the UBT perspective. A new modular model has been developed adapted to these kind of processes. In this paper, an indentation case carried out with a tilted punch is analyzed (Fig. 1). After the mathematical formulation acquired for the tilted case, different geometries can be study combining the tilted shape with the previous case of flat punch [8, 9] is possible, thus making feasible the resolution of indentation cases with geometrically more complex punches. Punch V
Work-piece
Fig 1. Indentation with tilted punch
Also, the tilted option is an up to date case study [10, 11], due to its application in the manufacturing industry [12, 13]. So, in the research field treated so far, include a proper study of the indentation process with tilted punch is essential 2. Methodology To accomplish a specific plastic deformation, the exact value of the minimum energy required to obtain such deformation is not necessary. The UBT provides the energy value that guarantees the deformation of the work-piece. Despite the UBT does not provide the exact amount of energy required, it does provide the minimum value at which it is ensured the work-piece will deform plastically. In addition, due to its simplicity, this method is especially applicable under plane strain conditions, compared with other methods. The inherent complexity of the plasticity theory over the years [2] has conditioned the development of these analytical methods. Conversely, the kinematic-geometrical application of the UBT by means modules of TRZ has made possible an analytical implementation with limited complexity. Although the result obtained by this new model is approximately, it provides a significant qualitative approach based on a simple resolution. The TRZ is the kinematic-geometrical option that allows reaching precise solutions of the main factors involved in an indentation process. This TRZ option is applied, initially, in suppositions of plane strain. The TRZ model assumes that the stresses and strains that cause deformation in the material only take place in the delimiting planes of each zone. Along these planes is where the discontinuity speeds occur (Fig. 2). The other points inside the considered TRZ will move at the same speed and in the same direction. TRZ2 TRZ1
TRZ3
Fig. 2. TRZ consideration
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From a theoretical point of view and under general conditions, the UBT expresses that: [14] The general expression of this work, in plane strain supposition, can be expressed as:
䌿T ·v ·dS i
Sv
i
v
> @
˺䌿 k· v* ·dS*D 䌿 Ti ·v*i·dSF SD
(1)
SF
Being: Ti: External strengths applied on the work-piece; vi: Real velocities field Sv: Surfaces where external loads are applied; SD*: Discontinuity Surfaces vi*: Kinematically admissible virtual velocities field SF: External surfaces exposed to external surface stresses In order to cope an indentation with tilted punch, a simple inclination is posed. The punch is vertically symmetrical and only one type of inclination is presented. So, a punch with just one slope angle and, as a result, just one peak, is considered (Fig. 1). The implementation takes place applying the optimal model developed [15] shown in Fig. 3. The optimal model developed consists of three modules of two TRZ each. Two of these modules are placed under the punch and the third one is positioned on the outside. This third module represents the material that is not in direct contact with the die but also is under deformation due to the pressure of the material below the punch that is flowing outside the punch laterally.
V
m θ2 θ3
h2 h1
θ1
A b
C
B b
h3
b’
Fig. 3. Optimal modular model implemented to a tilted case of indentation
There is a problem that arises when the first instants of the penetration are examined. At these deformation process first instants, there is not a full contact of the punch with the work-piece (Fig. 4), as in Fig. 3.
Fig. 4. Indentation first instants
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Therefore, an intermediate study is necessary for the phase where the punch is entering the work-piece and the base contact of the punch is not completed. Thanks to the modular consideration implemented, divide the initial phase in full contact stages is possible, enlarging the model whenever the punch increase its penetration depth. Consequently, several infinite thickness problems are analyzed until a full contact of the punch is presented (Fig. 5).
Fig. 5. Modular model adaptation
The main difficulty that entails the study of the inclined punch is to specify a model to follow based on the optimal model considered in previous cases with flat punches. For this reason, the following solutions have been considered. Option 1 A partial contact between the punch and the work-piece is assumed. Maintaining the optimum angle justified in a flat punch case (48.2°) for θ1 [15] is possible to deduce θ2 and θ3 as well as b', h1, h2, and h3, being θ2 and θ3 equal in this case. The problem presented in this configuration is that as the punch slope angle increases, h1 decreases, beginning to reach negative values from angle values of 42°. So this option is excluded because of the restrictions that presents in terms of the punch configuration. Option 2 This configuration keeps h3 as the optimum height found for the flat punch case, maintaining the 48.2° inclination only for the outer angle, θ3. θ1 and θ2, like h1 and h2, can be deduced with the other variables. Nevertheless, this approach also shows negative values for h1 for 30° of slope or more. So it is equally discarded. Option 3 In this option, the optimum height for h1 and the optimum angle for θ1 and θ3 remains. Also, the module bases are not placed to be half of the contact, so the modules can vary depending on the punch penetration. This configuration presents inconsistencies in the base length of the second module, since the base is reduced as it increases the inclination of the punch, presenting negative values from angles of 20°. This option is either not applicable. Option 4 To maintain the optimum angle for punch flat geometry is not possible since some of the related variables present negative values. Accordingly, continue with the optimum height for h1 is determined to guarantee that the first module will cover enough material and omit the loss possibility of any module, as seen in previous options. In addition, the modules under the direct influence of the punch cover again half of contact surface of the punch acting. The following account deduces angle θ2 and θ3. If θ3=θ2 and the exterior module base is equal to the previous modules bases, the outer base is b'=b. The final p/2k values for the finite consideration overcome the infinite consideration values when these are at shape factors lower than the unit, even when working with reasonable low angles in the punch. So it cannot be considered as valid results. However, considering θ3=θ1, the option 4 does not show this problem on the final results, regardless of the inclination of the punch. Therefore, this option is considered for the final setting of the tilted punch analysis.
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3. Results To verify the results obtained in the application of the new modular configuration of the UBT to indentations processes with tilted punches, a comparison with the Finite Element Method (FEM) based program, DEFORM 2D, has been made. Providing the analysis with DEFORM 2D, the results for the punch slopes study show that there are only differences in the efforts values until a complete filling of the punch is achieved (Fig. 6 and Fig. 8), i.e., until a complete contact of the tilted area with the work-piece under deformation (Fig. 3). At the same time, the results obtained with the UBT by means of TRZ modules present differences for variations of the slope angle. These differences persist through the rest of the penetration (Fig. 8). 4
Force[kN]
3
2
1 5˚̊ 5º
10˚̊ 10º
20º 20˚̊
45˚̊ 45
0 0
5
10 15 Penetration depth [mm]
20
25
Fig. 6. Different slope results
4
Force [kN]
3
2
1 5º 5˚̊
10º 10˚̊
20˚̊ 20º
45˚̊ 45
0
0
1 2 Penetration depth [mm] Fig. 7. Initial deformation detail
3
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So, except for the stage where the surface of the tilted punch is not in total contact, the results are similar, coming to the conclusion that to apply the mathematical solution for tilted punch is only necessary in the early stages of the process, until the base of the die has penetrated completely the work-piece. This result behavior and treatment decision in the application of the UBT for tilted punches resembles the adopted behavior in the SLF implementation, where is understood that, once the filling of the punch is fulfilled, this area becomes part of the punch, by which this can be interpreted as a flat punch and the area of the work-piece suffering more distortion is located directly under that area [16]. 5
Force [kN]
4
3
2
1 10º 5˚̊
20º 10˚̊
30º 20˚̊
45º 45˚̊
0 0
1
2 3 4 Penetration depth [mm]
5
6
Fig. 8. UBT results for different inclinations
Thus, make a comparison with the results obtained with FEM is now possible. In Fig. 9 and Fig. 10 can be observed that, using both methods, the new analytical method of the UBT and the numerical method used, the results are similar and follow the same pattern. 4
Force [kN]
3
2
1 20º̊ DEF. 2D 20
20º 20 ̊ UBT
0 0
5
10
15
Penetration depth [mm]
Fig. 9. UBT and FEM comparative for 20°
20
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4
Force [kN]
3
2
1
45 ̊ DEF. 2D
45 ̊ UBT 45º 0 0
5
10
15
20
Penetration depth [mm]
Fig. 10. UBT and FEM comparative for 45°
Therefore, in cases where the punch used for the indentation process is tilted, the introduction of the tilted punch model is only necessary at the early stages of the deformation process. Once the p/2k values for the tilted case exceed the p/2k values obtained in the implementation of the modular model for flat punches, the tilted case model is replaced by the flat case model and the analysis continue with the initial model in its flat configuration. 4. Conclusions For the analysis of an indentation process with a tilted punch, several options have been considered, proving that only with option 4, where h1 is maintained as the optimal height determined for an indentation with flat punch and θ3=θ2, knowing that when this happens, b'=b. Therefore, an analytical solution, applying the new modular UBT consideration is presented, for indentation cases with tilted punches. Also, in this paper, for the correct validation of the analytical model developed, a comparison with FEM is made. We found that, only in the early stages of the deformation process, the results obtained for different inclinations differed. After the surface of the punch was in total contact with the work-piece, the results had no significant differences, concluding that, once the surface of the punch is totally full, this area of deformed material of the workpiece becomes part of the punch. Now, this new area joined to the punch can be interpreted as a flat punch as a whole and so, the area of the work-piece suffering more distortion is located directly under this new area. As a result, the tilted indentation becomes an indentation with a flat punch. The comparison with the results obtained with FEM show that the solution achieved with the application of the new modular configuration of the UBT for indentation with a tilted punch are acute due to the similarity in the evolution of both analysis. The results presented with this study, proves, once more, the suitability of the UBT in the analysis of an indentation process. Thereby, a method that simplifies the existing methods in the needed efforts study in the deformation carried out with an indentation process is presented. On the one hand, from the mathematical perspective, the application of the new modular configuration of the UBT involves a less complicated implementation compared to other methods, such as the FEM. On the other hand, from the analytical perspective, the modular consideration presents a simpler final equation, simplifying the mathematical resolution of the problem.
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5. Acknowledgements The authors want to thank the University of Malaga – Andalucía Tech, International Campus of Excellence, and the FPU program of the Ministry of Education, Culture and Sport of Spain for its contribution on this paper 6. References [1] C. Bernal, A. M. Camacho, J. M. Arenas, and E. M. Rubio. Analytical procedure for geometrical evaluation of flat surfaces formed by multiple indentation processes. Applied Mechanics and Materials, 217-219 (2012), pp. 2351-2356. [2] J. Chakrabarty, Theory of plasticity, Elsevier Science, Oxford (U.K.), 2006. [3] A. M. Camacho, M. M. Marín, E. M. Rubio, and M. A. Sebastian. Application of different simulation strategies for the analysis of multistroke localised-incremental forming operations. Materials Science Forum, 2013 (2012), pp. 19-24. [4] F. Martín, Desarrollo, integración y optimización en el estudio del proceso de forja mediante el teorema del límite superior a través del modelo de bloques rígidos triangulares, Ingeniería de Fabricación, Universidad de Málaga, Málaga, 2009. [5] H. Kudo. An upper-bound approach to plane-strain forging and extrusion-i. International Journal of Mechanical Sciences, 1 (1960), pp. 57-83. [6] H. Kudo. An upper-bound approach to plane-strain forging and extrusion-ii. International Journal of Mechanical Sciences, 1 (1960), pp. 229252. [7] H. Kudo. An upper-bound approach to plane-strain forging and extrusion-iii. International Journal of Mechanical Sciences, 1 (1960), pp. 366368. [8] C. Bermudo, F. Martín, and L. Sevilla. Analysis and selection of the modular block distribution in indentation process by the upper bound theorem. Procedia Engineering, 63 (2013), pp. 388-396. [9] C. Bermudo, F. Martín, and L. Sevilla. Optimización del modelo modular en procesos de indentación mediante el teorema del límite superior. Anales de Ingeniería Mecánica, 19 (2014), p. 97. [10] I. G. Goryacheva, H. Murthy, and T. N. Farris. Contact problem with partial slip for the inclined punch with rounded edges. International Journal of Fatigue, 24 (2002), pp. 1191-1201. [11] A. Sackfield, D. Dini, and D. A. Hills. The finite and semi-infinite tilted, flat but rounded punch. International Journal of Solids and Structures, 42 (2005), pp. 4988-5009. [12] A. Babaei, G. Faraji, M. M. Mashhadi, and M. Hamdi. Repetitive forging (rf) using inclined punches as a new bulk severe plastic deformation method. Materials Science and Engineering: A, 558 (2012), pp. 150-157. [13] L. Tajul, T. Maeno, and K.-i. Mori. Successive forging of long plate having inclined cross-section. Procedia Engineering, 81 (2014), pp. 2361-2366. [14] W. Johnson and P. P. B. Mellor, Engineering plasticity, Ellis Horwood Limited, Oxford (U.K.), 1983. [15] C. Bermudo, F. Martín, and L. Sevilla. Selection of the optimal distribution for the upper bound theorem in indentation processes. Materials Science Forum, 797 (2014), pp. 117-122. [16] W. Prager and P. G. Hodge, Theory of perfectly plastic solids, John Wiley & Sons, Michigan (U.S.A.), 1951.
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