Int. J. Mechatronics and Manufacturing Systems, Vol. 4, No. 5, 2011 441 Inverse method for flow stress parameters identification of tube bulge hydroforming considering anisotropy T. Zribi, A. Khalfallah* and H. Belhadjsalah Laboratoire de Génie Mécanique, Ecole Nationale d Ingénieurs de Monastir, Av, Ibn El Jazzar, 5019, Monastir, Tunisia E-mail: temim_zribi@yahoo.fr E-mail: khalfallah_a@yahoo.fr E-mail: hedi.belhadjsalah@enim.rnu.tn *Corresponding author Abstract: The objective of this paper is to determine the behaviour parameters of hydroformed tubular materials of annealed 25CrMo4 steel seamless tubes. An inverse identification procedure has been developed to determine the flow stress parameters from experimental free bulge and simple tensile tests. Two inverse identification strategies are proposed. In the first one, the tubular material behaviour is assumed isotropic, and then the parameters of the effective stress-strain relationship are derived from experimental bulge height versus the forming pressure. In the second one, the planar anisotropy of Hill 48 yield criterion is considered. Firstly, the simple tensile test is used to fit the isotropic strain hardening law. Secondly, the anisotropic parameter is identified by inverse technique from the experimental bulge test. The obtained results are compared with experimental measurements to validate the proposed strategies. It is proven that inverse identification methods could be a better alternative to analytical methods for flow stress parameters determination. Moreover, the anisotropy parameter has shown a large effect on the hydroformed tube responses, particularly on the wall thickness. Consequently, this material anisotropy should be considered within the flow stress identification of tube bulge tests. Keywords: tube hydroforming; inverse method; parameter identification; anisotropy; flow stress. Reference to this paper should be made as follows: Zribi, T., Khalfallah, A. and Belhadjsalah, H. (2011) Inverse method for flow stress parameters identification of tube bulge hydroforming considering anisotropy, Int. J. Mechatronics and Manufacturing Systems, Vol. 4, No. 5, pp.441 453. Biographical notes: T. Zribi is a PhD student. He started his researches in 2007. He works on inverse identification methods of flow stress parameters of tubes in hydroforming process. His PhD thesis advisor and co-advisor are the co-authors of this paper. Actually, he is a temporary attached Teacher of Mechanical Engineering at the High Institute of Applied Sciences and Technology in Sousse, Tunisia. A. Khalfallah is an Assistant Professor of Mechanical Engineering at the High Institute of Applied Sciences and Technology in Sousse, Tunisia. He received his PhD in Applied Mechanics from the University of ElManar-Tunisia in 2004. His research interests focus on inverse parameter identification of Copyright 2011 Inderscience Enterprises Ltd.
442 T. Zribi et al. constitutive models describing sheet metal forming and hydroforming processes. He is a permanent researcher at the Mechanical Engineering Laboratory at the University of Monastir (ENIM), Tunisia. H. Belhadjsalah is a Professor of Applied Mathematics at the University of Monastir in the National Engineering School of Moanstir, Tunisia. He received his PhD in Numerical Analysis at the University of Paris 11, France in 1990. His main research interests are varied including the advanced engineering mathematics, numerical simulation (FEM), non-linear mechanical structure and material behaviour in forming process and industrial systems. This paper is a revised and expanded version of a paper entitled Inverse method for flow stress parameters identification of tube bulge hydroforming considering anisotropy presented at the 7th EUROMECH Solid Mechanics Conference, Lisbon, Portugal, 7 11 September 2009. 1 Introduction Tube hydroforming (THF) is a forming process that becomes interesting in the last years. The main advantages of the THF over conventional forming processes are weight reduction of components, enhancing part rigidity and improving dimensional tolerances. The application of this process is widely spread to manufacture components such as camshafts, radiator frames, engine cradles, roof rails, seat frames, etc., in the automotive aircraft and aerospace industries (Dohmann and Hartl, 1994, 1996). The success of forming results are influenced by several forming factors, such as loading paths, part and tooling design, material properties, tribology, etc., (Ahmed and Hashmi, 1997; Koç et al., 2001). Among the forming parameters, the material characteristics have a significant influence on the formability of the final products. They have an impact on the internal pressure, bulge height, wall thickness distribution and bulge profile. The well-evaluated material properties improve accuracy in simulation results and then a good prediction of the forming process. Since unconstrained THF is a biaxial stress state, the free bulge test is a well-established experiment to determine the flow stress relationship. Several studies concerning the evaluation of tubular material flow stress have been reported in the literature (Koç and Altan, 2001; Strano and Altan, 2004). In some methods, the free bulge region is assumed as circular or elliptical profiles, in order to easily establish analytical models. These assumptions could have an impact on simulation results and therefore on the final products. FE simulations represent powerful tools to predict process parameters by reducing expensive experimentations and trial and error design process. Furthermore, material properties are necessary to conduct FE analysis. Usually, mechanical properties of the tubes are very often determined from tensile test of sheet metal used for rolling the tube or from samples cut from already rolled tubes. Another alternative is used by conducting tensile tests directly on the whole tube (in the case of seamless or extruded tubes). It is known that the maximum effective strain reached during tensile tests is lower than which obtained after THF process. Moreover, during THF process stress state is biaxial or triaxial while in tensile test is uniaxial (Strano and Altan, 2004; Lianfa and Cheng, 2007). For all these reasons, testing methods based on multiaxial stress state are needed to improve accuracy in the determination of
Inverse method for flow stress parameters identification of tube bulge 443 tubular material properties. Sokolowski et al. (2000) have proposed an analytical method to evaluate tube formability and material characteristics based on hydraulic bulge testing. Strano and Altan (2004) developed an inverse energy approach to determine strain-stress relationship of tubular materials for hydroforming applications. Hwang et al. (2007) have proposed an analytical method to predict flow stress law of tubular materials by assuming the profile of free bulge region as an elliptical surface. These works among others aimed at the determination of flow stress parameters are based on the assumption of isotropic material behaviour. In fact, manufacturing operations of tubes such as rolling, welding, extrusion, etc., are necessary anisotropic. In this paper, an inverse identification method is proposed to improve accuracy in the prediction of material properties of a tube formed by hydraulic bulge pressure. A model including planar anisotropy and isotropic hardening law are considered. The experimental data were taken from a published work of Bortot et al. (2008) from University of Brescia, Italy. Experiments are conducted on annealed 25CrMo4 seamless tubes with 40 mm external diameter and 2 mm initial wall thickness. The validation of the developed method is based on the comparison between the experimental thickness, the meridian radius of curvature at the pole with those obtained by the proposed methodology to assess the accuracy of this method. This comparison is also performed between the Bortot et al. (2008) work and the proposed method. 2 Inverse parameter identification method In this section, an inverse identification procedure has been developed to determine flow stress from experimental results of free THF test. This method is based on a finite element simulation coupled with an optimisation method and a set of experimental tests. The experimental database is consisted of accessible material responses by experiments. In this paper the experimental data represent the recorded bulge height at the pole versus the internal forming pressure. Other experimental measurements as thickness and meridian radius of curvature were kept for validation purposes. The inverse procedure acts iteratively in a manner that minimises the gap between the simulation results (FE calculus) and the corresponding experimental responses. The aim of this method is to find the best material parameters for which numerical results match with experimental data. It is based on the minimisation of a cost function [equation (1)] which defines a least-square approximation between predicted curves calculated with FE code and experimental ones. Identified parameters are adjusted automatically until the error between the numerical results and the corresponding experiment is less than a predefined infinitesimal value δ, called tolerance. N 2 exp 1 num Ri R i f( P) = δ exp N i= 1 R (1) i In equation (1), P represents the set of the material parameters, N the number of experimental points, R exp i and R num i are the experimental and the corresponding simulated response, respectively.
444 T. Zribi et al. 2.1 Optimisation method The inverse identification procedure is based on an optimisation method. Several optimisation methods such as conjugate gradient, Newton method, etc., are based on the evaluation of the sensitivity of the cost function. But there exists also direct optimisation methods which operate by direct evaluation of the cost function. Among these last methods, the Nelder Mead simplex method (Nelder and Mead, 1965) is chosen in this optimisation procedure. A simplex-based method constructs an evolving pattern of n + 1 points in R n that are viewed as the vertices of a simplex. Starting from initial simplex, a new simplex is formed at each iteration by geometrical transformation of its vertices. The vertices are evaluated by reflection, expansion, contraction or shrinkage transformations until the simplex size becomes smaller than a termination criterion (Wright, 1995). This minimised method ensures a good estimation of the parameters and it converges toward global solutions. Moreover, it can be easily linked to any finite element code as a result of its modular conception. Figure 1 shows a flowchart of the applied inverse method. 2.2 Finite element model The finite element analysis of a free hydraulic bulging tube is conducted using a commercial FE code ABAQUS. A general-purpose shell element was chosen (S4R) for the spatial discretezation of the tube. The general-purpose shells are capable of deformation due to transverse shear stress. The (S4R) element is a four-node element. Each node has six degrees of freedom and uses an independent bilinear interpolation function. The (S4R) element is a reduced integration element. Reduced integration element reduces the amount of CPU time necessary for analysis of the model, which is advisable for iterative processes using FE calculation such as inverse identification, and it typically provides accurate results. But if a hourglassing can occur due to the reduced number of integration points in the (S4R) element, an hourglass stabilisation control feature is built into the element. Five integration points were used through the thickness of the element and the tube is meshed with 3,800 shell elements. This number of elements is shown to be sufficient to ensure a precise solution of the numerical simulation. The Figure 2 shows the geometry and the finite element mesh used to simulate numerically the free THF. It is composed of a die which is assumed to be a rigid body and a tube with elastoplastic behaviour. The die entrance radius (R d ) is 6 mm, the external tube diameter (D 0 ) is 40 mm, the tube thickness (t 0 ) is 2 mm and the length of tube (L) and its bulge area (w) are 210 mm and 40 mm, respectively. During the simulation, the die is assumed to be a rigid body, and the tube behaviour is elastoplastic with an isotropic strain hardening behaviour. The tube ends are fixed during THF test; therefore, displacements at the ends of the tube were kept at zero. The clearance between dies and tube was set to zero. Figure 3 presents the geometric model of the tube.
Inverse method for flow stress parameters identification of tube bulge 445 Figure 1 Flow chart of the inverse procedure to identify material parameters Figure 2 Geometrical FE model for free bulge tube simulation (see online version for colours)
446 T. Zribi et al. Figure 3 Geometrical model In the finite element simulation, the internal pressure was defined as a time dependent curve. It was increased linearly according to different time increments similarly the experimental procedure; it reached the pressure of 70 MPa in the total simulation time. Figure 4 shows the stress state of an element at the pole dome while the bulging tests. It is a biaxial stress state under the stresses σ ϕ and σ θ in the meridian and circumferential directions. During the bulge tests of thin walled-tube, the stress in the thickness direction σ t, is considerably small and it can be neglected. The radii r ϕ and r θ represent the meridian radius of curvature and the hoop radius in the circumferential direction. Figure 4 Definition of the stress state at the pole of free hydroformed tube
Inverse method for flow stress parameters identification of tube bulge 447 3 Identification strategies 3.1 Isotropic case In this section, an identification strategy for material parameters of tube bulge test is presented. This methodology is based on the inverse identification method by considering isotropic elastoplastic behaviour. Therefore, the Swift isotropic hardening law: ( 0 p ) n σ = K ε + ε (2) and the von Mises yield criterion are assumed. The purpose of this identification strategy is to find the hardening modulus K, the strain hardening coefficient n and the pre-strain ε 0 from the experimental curve of the free tube bulge test. The experimental results are obtained from the measurement of the bulge height versus the internal pressure at the pole. The experimental data were taken from a published work of Bortot et al. (2008) from university of Brescia, Italy. The experiments were conducted on annealed 25CrMo4 seamless tubes. The tube dimensions were obviously taken the same as they are used in the numerical simulation. The tube bulge test was carried out at room temperature and with fixed tube edges. The initial parameters and the identified ones are presented in Table 1. The identification procedure has been stopped after its convergence toward the best material parameters that minimise the gap between the experimental and simulated data. Table 1 Initial and identified parameters in the isotropic case Parameters Initial values Identified values K(MPa) 1,000 842.39 n 0.3 0.17 ε 0 0.001 0.0013 Figure 5 Evaluation of the cost function with iteration (see online version for colours)
448 T. Zribi et al. Figure 5 shows the evolution of the identification error versus the number of iterations. It can be seen that the cost function vanishes after about 60 iterations and the identification procedure converges toward a global minimum, despite the large number of iterations. Figure 6 shows the experimental data and the numerical curve corresponding to the identified parameters after running the inverse identification procedure. It is shown that the identified parameters represent a very good agreement of the experimental points. It is worth to note that the curve of the bulge height versus the internal pressure at the pole of Bortot et al. (2008) work (Figure 6) was obtained using an analytical model and FEM simulation for a validation purpose of their approach. Comparing the presented results and those of Bortot et al., it is clear that the inverse identification methodology can make a better prediction of the tubular material properties. Moreover, in the described approach, no assumptions on the geometry (spherical or elliptical) of the bulged area are considered. The validation of the proposed approach is conducted on the comparison between the experimental points of the thickness and the meridian radius of curvature at the pole versus the bulge height. Even though the tubular material parameter of the flow stress are different, in Figure 7, it is shown that the prediction of the thickness based on inverse parameter identification of the flow stress are similar to the thickness predicted by the Bortot et al. model. It is found that the error between experimental and simulated responses is in the range of 4%. Concerning the meridian radius of curvature, it is shown in Figure 8 that the prediction results by the proposed strategy are different from experimental points. Since Bortot et al. have used the measured radius of curvature as input data in their model identification thus, it is expected that the results are too closer to the experimental responses. Figure 9 shows the obtained stress-strain relationship in the isotropic case. It is different from the flow stress that was determined by Bortot et al. methodology, since the methodologies were different. Figure 6 Comparison between experimental and numeric bulge height (see online version for colours)
Inverse method for flow stress parameters identification of tube bulge 449 Figure 7 Comparison between experimental and simulated thickness at the pole with final identified parameters (see online version for colours) Figure 8 Comparison between experimental and numerical radius of curvature at the pole with final identified parameters (see online version for colours)
450 T. Zribi et al. Figure 9 Comparison between flow stress curves obtained by different identification strategies (see online version for colours) 3.2 Anisotropic case In this section, the material anisotropy is taken into account and it will be identified from experimental data using the proposed inverse identification method. The aim of this task is to find a better prediction of the thickness, the meridian radius of curvature and to identify accurately the flow stress which considers real material properties. In this case, the tubular material is considered as anisotropic with an isotropic strain hardening law. The anisotropic plastic behaviour is described by the Hill (1948) 48 yield criterion in its particular form of planar anisotropy and in the case of plane stress state. σ = ( ( r ))( 2 2) + ( r ) 3 1+ 1/ σφ + σθ 2σφσ θ 2 1 2/ This anisotropy is defined by a unique coefficient denoted r. Where σ ϕ, σ θ are the principal stresses in the meridian and circumferential directions, respectively. If r = 1, the effective stress can be reduced to von Mises equivalent stress. The Hollomon law is used to describe the isotropic strain hardening evolution for the best fitting of the stress-strain relation for the uniaxial tensile test. It is worth to mention that at the beginning, the Swift work hardening law was used to fit the experimental stress-strain curve for the uniaxial tensile test, but the pre-strain coefficient ε 0 was found to be neglectable. In fact, no experimental strain data at initial yielding is available. It can be seen from Figure 9 which plots the experimental tensile test curve. As one can see the first point is about 3% of plastic deformation which is far beyond strain at initial yielding. (3)
Inverse method for flow stress parameters identification of tube bulge 451 n ( εp ). σ = K (4) This identification strategy is performed in two steps: In the first step, the parameters of the strain hardening law (K and n) are obtained from the experimental tensile test by a least square method for a best fitting of the experimental points with the Hollomon equation [equation (4)]. This equation seems to be suitable for fitting the equivalent strain-stress relationship in the simple tensile test (Figure 9). In the second step, the Hollomon parameters K and n identified from the simple tensile test will be used as input data into the finite element code. The planar anisotropic coefficient r is identified using inverse procedure by minimising the gap between the experimental bulge height at the pole versus internal pressure data and the corresponding simulated response. The initial parameters K and n are arbitrarily chosen to start the first step optimisation of Hollomon s parameters by least square method. The second step of the identification procedure of the anisotropic coefficient starts by an initial isotropic coefficient r = 1. The obtained parameters for this identification strategy are reported in the Table 2. Table 2 Initial and identified parameters in the anisotropic case Parameters Initial values Identified values K(MPa) 1,000 1,020.3 n 0.3 0.31 r 1.0 2.26 Figure 6 represents the final bulge height versus internal pressure corresponding to the final identified parameters. It shows a disagreement with experimental data and with the curve obtained in the isotropic case. The error is relatively high comparing with the first case. Even though, the observed discrepancy between the obtained curve in the anisotropic and the target curve (experimental data), the validation task is performed by comparing the experimental thickness and the meridian radius of curvature and those predicted by this approach. Figure 7 shows the predicted thickness at the pole versus the bulge height. It is clear that the anisotropic coefficient influences scientifically the thickness of the tube. Taking into account of the anisotropy influence on material behaviour in this identification strategy, the meridian radius of curvature is better predicted than which is obtained in the isotropic case (Figure 8). The flow stress curve predicted by this strategy is plotted on the Figure 9. 4 Flow stress relationship This section summarises the final results obtained by the proposed identification strategies in the aim to determine the stress-strain relationship. Figure 9 displays the flow stress curves obtained by the proposed approaches. The parameters of Swift strain hardening law are determined using the least square method for best fitting the flow stress curves plotted in Figure 9. The parameters of the stress-strain relations are summarised in Table 3. It is observed a larger difference among the methodology adopted for the evaluation of the stress-strain relationship.
452 T. Zribi et al. Table 3 Flow stress parameters obtained by the proposed approaches Identification strategy K εo n Isotropic case (bulge test) h = f(p) 795.51 0.001 0.15 Anisotropic case 1,212 0.003 0.32 Bortot et al. 1,490 0.193 0.77 5 Conclusions In this study, we have investigated the determination of flow stress parameters for tubular materials using free bulge tube test by inverse identification procedure and we have taken into account the material anisotropy effects. Our purpose is a contribution for obtaining realistic and general characteristics of tubular material based on real physical properties of materials. Mainly, two identification strategies of the flow stress curves parameters are proposed. In the first strategy, an isotropic tubular material behaviour was assumed. The experimental bulge height at the pole versus the applied internal pressure was used to determine the flow stress parameters by inverse approach. In the second strategy, an assumption of anisotropic material was adopted. Then the uniaxial tensile test and the bulge test were used to identify completely the flow stress by taking into account the anisotropy effect. The validation of the identified stress-strain relationship parameters obtained by the proposed strategies was performed to assess the ability of these approaches for a better description of the tubular material behaviour. It is shown that anisotropy affects significantly the tubular material, since it influences on the bulge height, the meridian radius of curvature and mainly on the tube thickness. In near future work, an inverse identification of flow stress parameters of tubular materials will be based simultaneously on experimental bulge height and thickness evolution versus internal pressure and minimising a multiobjectif cost function. References Ahmed, M. and Hashmi, M.S.J. (1997) Estimation of machine parameters for hydraulic bulge forming of tubular components, Journal of Materials Processing and Technology, Vol. 64, Nos.1/3, pp.9 23. Bortot, P., Ceretti, E. and Giardini, C. (2008) The determination of flow stress of tubular material for hydroforming applications, Journal of Materials Processing and Technology, Vol. 203, Nos. 1 3, pp.381 388. Dohmann, F. and Hartl, Ch. (1994) Liquid-bulge-forming as a flexible production method, Journal of Materials Processing and Technology, Vol. 45, Nos. 1/4, pp.377 382. Dohmann, F. and Hartl, Ch. (1996) Hydroforming a method to manufacture lightweight parts, Journal of Materials Processing and Technology, Vol. 60, Nos. 1/4, pp.669 676. Hill, R. (1948) A theory of the yielding flow of anisotropic metals, Proceedings of the Royal Society A, Vol. 193, pp.281 297. Hwang, Y.M, Lin, Y.K. and Altan, T. (2007) Evaluation of tubular materials by a hydraulic bulge test, International Journal of Machine Tools and Manufacturing, Vol. 47, No. 2, pp.343 351. Koç, M. and Altan, T. (2001) An overall review of the tube hydroforming (THF) technology, Journal of Materials Processing and Technology, Vol. 108, No. 3, pp.384 393.
Inverse method for flow stress parameters identification of tube bulge 453 Koç, M., Aue-u-lan, Y. and Altan, T. (2001) On the characteristics of tubular materials for hydroforming experimentation and analysis, International Journal of Machine. Tools and Manufacturing, Vol. 41, No. 5, pp.761 772. Lianfa, Y. and Cheng, G. (2007) Determination of stress-strain relationship of tubular material with hydraulic bulge test, Thin-Walled Structures, Vol. 46, No. 2, pp.147 154. Nelder, J.A. and Mead, R. (1965) A simplex method for function minimization, Computer Journal, Vol. 7, No. 4, pp.308 313. Sokolowski, T., Gerke, K., Ahmetoglu, M. and Altan, T. (2000) Evaluation of tube formability and material characteristics: hydraulic bulge testing of tubes, Journal of Materials Processing and Technology, Vol. 98, No. 1, pp.34 40. Strano, M. and Altan, T. (2004) An inverse energy approach to determine the flow stress of tubular materials for hydroforming applications, Journal of Materials Processing and Technology, Vol. 146, No. 1, pp.92 96. Wright, M.H. (1995) Direct search methods: once scorned, now respectable, in Griffiths, D.F. and Watson, G.A. (Eds.): Numerical Analysis, pp.191 208, Addison Wesley Longman, Harlow, UK.