Advanced Methods in Material Forming

Advanced Methods in Material Forming Dorel Banabic Advanced Methods in Material Forming With 264 Figures and 37 Table...

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Advanced Methods in Material Forming

Dorel Banabic

Advanced Methods in Material Forming With 264 Figures and 37 Tables

Prof. Dorel Banabic Technical University Cluj-Napoca Research Centre on Sheet Metal Forming Technology – CERTETA 15 C. Daicoviciu 400020 Cluj-Napoca Romania [email protected]

Library of Congress Control Number: 2007920904 ISBN 978-3-540-69844-9 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Integra Software Services Pvt. Ltd., Pondicherry, India Cover design: eStudio Calamar, Girona, Spain Printed on acid-free paper

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Preface

The 8th International Conference of the European Scientiﬁc Association for Material Forming ESAFORM took place in Cluj-Napoca (Romania) at the end of April 2005 (http://conference.utcluj.ro/esaform2005). The conference was organised by the Research Centre on Sheet Metal Forming Technology CERTETA together with the European Scientiﬁc Association for Material Forming ESAFORM (http://www.esaform.org). The Technical University of Cluj-Napoca and the Romanian Academy were also co-organisers of that scientiﬁc event. The main purpose of the ESAFORM conferences is to bring together specialists from universities, research centres and industry interested in the forming processes of metallic as well as non-metallic materials. The conferences are organised each year by diﬀerent European universities. The previous meetings were organised by the following academic institutions: 1998 – Ecole des Mines de Paris (Sophia Antipolis, France); 1999 – University of Minho (Guimaraes, Portugal); 2000 – University of Stuttgart (Germany); 2001 – University of Liege (Belgium); 2002 – AGH University of Science and Technology – Krakow (Poland); 2003 – University of Salerno (Italy); 2004 – University of Trondheim (Norway). The number of participants in the ESAFORM conferences increased continuously: from about 100 specialists registered in the ﬁrst conference to 280 in the last one. The International Conference ESAFORM 2005 consisted in 24 minisymposia, each of them being coordinated by well-known specialists. The topics of the mini-symposia covered a variety of research domains: Identiﬁcation of models, Numerical strategies, Optimisation, Multi-scale approaches, Constitutive models for metals and polymers, as well as technological problems speciﬁc to Sheet metal forming, Forging, Machining and cutting, Composite forming, etc. Some new mini-symposia included for the ﬁrst time in the ESAFORM conference topics (Anisotropy and formability of materials, Rapid prototyping and rapid tooling, Nano-structuring of metals by severe plastic deformation, Semi-solid processing, etc.) were very successful. A number of 280 specialists from 35 countries (about 80% representing Europe) took part

VI

Preface

in the conference. In the frame of the conference, a number of 6 plenary lectures, 47 keynote papers, 210 session papers and 55 posters were presented. The abstracts (4 pages) were included in the conference proceedings published by the Romanian Academy. Some papers were also published in a special issue of the International Journal of Forming Processes. This volume gathers selected plenary and keynote papers presented in the conference, oﬀering an up-to-date synthesis of the academic and industrial research in the ﬁelds of physical and numerical modelling of materials forming processes. The book is useful for the doctoral fellows, scientists and engineers involved in various domains of materials processing technology. I would like to thank the authors of the papers included in this book, as well as the scientiﬁc reviewers. I am also grateful to the members of the CERTETA centre (especially to Dr. D.S. Comsa) who substantially contributed to the success of the ESAFORM 2005 conference and to the preparation of the book. And last but not least I express my gratitude to Springer publishing house for accepting to publish this book. October 2006

Prof. Dorel Banabic

Contents

Constitutive Modeling for Metals F. Barlat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Material Forming and Dimensioning Problems: Expectations from the Car Industry G. Maeder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Optimization of the Phenomenological Constitutive Models Parameters Using Genetic Algorithms B.M. Chaparro, J.L. Alves, L.F. Menezes and J.V. Fernandes . . . . . . . . . 35 A Metamodel Based Optimisation Algorithm for Metal Forming Processes M.H.A. Bonte, A.H. van den Boogaard and J. Hu´etink . . . . . . . . . . . . . . . 55 Modelling of Permeability and Mechanical Dispersion in a Porous Medium and Comparison with Experimental Measurements F. Loix, V. Thibaut and F. Dupret . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Three-Dimensional Bending of Proﬁles by Stress Superposition S. Chatti, M. Hermes and M. Kleiner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Specimen for a Novel Concept of Biaxial Tension Test – Design and Optimisation W. Hußn¨ atter, M. Merklein and M. Geiger . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Applications of a Recently Proposed Anisotropic Yield Function to Sheet Forming S. Soare, J.W. Yoon, O. Cazacu and F. Barlat . . . . . . . . . . . . . . . . . . . . . . 131

VIII

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Modelling of the Forming Limit Diagrams Using the Finite Element Method L. P˘ ar˘ aianu, D.S. Com¸sa, J.J. Gracio and D. Banabic . . . . . . . . . . . . . . . 151 Recent Advances in Process Design for Sheet and Tube Hydroforming J.C. Gelin, C. Labergere and S. Thibaud . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Evaluating the Press Stiﬀness in Realistic Operating Conditions A. Ghiotti and P.F. Bariani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Fast Material Working: Wire Drawing N.D. Cristescu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 3D-ECAP of Square Aluminium Billets A. Rosochowski, L. Olejnik and M. Richert . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Computer-Aided Tool Path Optimization for Single Point Incremental Sheet Forming M. Bambach, M. Cannamela, M. Azaouzi, G. Hirt and J.L. Batoz . . . . . 233 Study on the Achievable Accuracy in Single Point Incremental Forming J.R. Duﬂou, B. Lauwers and J. Verbert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 On the Finite Element Simulation of Thermal Phenomena in Machining Processes L. Filice, D. Umbrello, F. Micari and L. Settineri . . . . . . . . . . . . . . . . . . . . 263 Numerical Simulation of Wire Coating Pseudoplastic and Viscoplastic Fluids E. Mitsoulis and P. Kotsos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Experimental Study on Behaviour of Woven Composites in Thermo-Stamping Under Nonlinear Temperature Trajectories H.S. Cheng, J. Cao and N. Mahayotsanun . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Thixoforming of Steel: Experiments on Thermal Eﬀects P. Cezard, R. Bigot, V. Favier and M. Robelet . . . . . . . . . . . . . . . . . . . . . . . 309 Study of the Liquid Fraction and Thermophysical Properties of Semi-Solid Steels and Application to the Simulation of Inductive Heating for Thixoforming J. Lecomte-Beckers, A. Rassili, M. Carton, M. Robelet and R. Koeune . . 321

Contents

IX

Simulation of Secondary Operations and Springback – The Implicit Shell Provides a Precise and Rapid Solution ´ Sch¨ E. onbach, G. Glanzer, W. Kubli and M. Selig . . . . . . . . . . . . . . . . . . . . 349 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

Constitutive Modeling for Metals F. Barlat1,2 1

2

Alloy Technology and Materials Research Division, Alcoa Inc., Alcoa Technical Center, 100 Technical Drive, Alcoa Center, PA 15069-0001, USA, [email protected] Center for Mechanical Engineering and Automation, University of Aveiro, Campus Universitario de Santiago, P-3810 Aveiro, Portugal, [email protected]

Summary. This paper reviews aspects of the plastic behavior common in metals and alloys. Macroscopic and microscopic phenomena occurring during plastic deformation are described succinctly. Constitutive models of plasticity at the micro- and macro-scales, suitable for applications to forming, are discussed in a very broad fashion. Approaches to plastic anisotropy are reviewed in a more detailed manner.

Key words: alloy, anisotropy, constitutive model, forming, metal, microstructure, plasticity.

1 Introduction The life of a product follows the typical chronological order: material processing, product manufacturing, product service, failure, disposal or recycling. A product is a functional shape made of one or more materials, which results in a given set of properties and complies to speciﬁcations. The material selection process can be based on the designer experience or on a more analytical and comprehensive approach using databases and optimization methods (see Brechet in Lemaitre, 2001). Today, with advanced computer hardware and software, it is possible to model material processing, product manufacturing, product performance in service, and failure. Material design is somewhat more empirical but new analytical methods are emerging for this purpose (Raabe et al., 2004). Although the ﬁne-tuning of a product manufacturing and performance is empirical, modeling can be an eﬃcient tool to guide and optimize design, to evaluate material attributes, and to predict life time and failure. Moreover, modeling can be used as a research tool for a more fundamental understanding of physical phenomena that can result in the development of improved or new products. In any case, a constitutive model, a mathematical description of a material behavior, is needed. With more than 80000 engineered materials

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Advanced Methods in Material Forming

(see Brechet in Lemaitre, 2001) used in various possible environmental conditions and temperatures, it is impossible to derive universal constitutive models applicable to them all. Restricting the discussion on metals and alloys, even in the context of forming, is still very broad. Therefore, this paper is an attempt to look at plastic deformation for metals and corresponding constitutive models for forming in a very synthetic way. Section 2 describes common aspects of the plastic deformation of metals and alloys and Sect. 3 brieﬂy discusses relevant constitutive models and material parameters which are important for the simulations of forming operations. A slightly more detailed description of plastic anisotropy is provided as an illustration.

2 Plasticity of Metals and Alloys 2.1 Material Processing and Forming From its inception to its ﬁnal realization, a product is subjected to a complex thermo-mechanical path that leads to a geometrical shape and a microstructure. For instance, Table 1 shows the diﬀerent steps necessary to produce a sheet metal part from casting to forming. It involves a sequence of thermal and mechanical operations to which correspond a number of physical phenomena resulting in a material microstructure (Altenpohl, 1998). This table illustrates the complexity of the process. Both geometry and microstructure aﬀect the functionality of the ﬁnal product, its properties and its service performance. The goal of materials scientists and engineers is to develop and improve materials with respect to given properties such as the elastic modulus, strength, ductility, toughness, endurance and corrosion by a careful design of the microstructure. Materials designers are concerned with lattice imperfections, solute content, second-phases, grains and grain boundary structures,

Table 1. Typical ﬂow path for aluminum alloy sheet processing product processing

physical phenomena

microstructure

• •

• •

• •

• • • • •

melting–casting homogenization (heat-treatment) hot rolling cold rolling solution heat-treatment stretching–aging forming

• • •

solidiﬁcation phase transformation (liquid/solid) (solid/solid) plastic deformation recrystallization grain growth

• • • •

crystal structure grain and grain boundary structure crystallographic texture second-phases dislocation structure vacancies and solutes

Constitutive Modeling for Metals

3

crystallographic texture, and the distribution of all these features throughout the material. There are many possible ways to alter a microstructure by manipulating chemical composition and material processing. A given chemical composition itself does not reﬂect the ﬁnal physical properties of a product, which is inﬂuenced by the whole material history after casting. In fact, it is important to note that memory eﬀects can be very persistent. For instance, with reference to Table 1, the homogenization of an ingot at an unsuitable temperature for the particular chemical composition can lead to undesirable eﬀects on its forming and service performances, even after several deformation and recrystallization cycles occurring during the whole process. Process engineers are usually more focused on the shape changes during forming, the appearance of geometrical defects and the existence of a macroscopic residual stress ﬁeld. For instance, in forging, the material must completely ﬁll a die. In hot rolling, a plate must achieve a certain gauge with an acceptable degree of ﬂatness and a minimum amount of residual stresses. In sheet forming, a part is processed successfully if its ﬁnal shape and dimensions fall within the dimensional tolerances after springback. These are only a few examples but in any case, the product must be achieved without fracture. Therefore, ductility is of major importance in forming. 2.2 Macroscopic Observations in Plasticity Aspects of the plastic deformation and ductility of metals and alloys at low and moderate strain rates, subjected to monotonic loading or to a few load cycles, are brieﬂy discussed here. However, it is important to remember that beside plastic deformation, microstructure transformation is the result of temperature changes as well. Moreover, at very high strain rates, dynamic eﬀects lead to additional phenomena that are not discussed here. The stress-strain behavior of metals and alloys at low strain is almost always reversible and linear. The elastic range however, is bounded by the yield limit, the stress above which permanent or inelastic deformations occur. In the plastic range, the ﬂow stress, described by a stress-strain curve, usually increases with the total amount of plastic dissipation or a corresponding measure of accumulated plastic strain, and becomes the new yield stress if the material is unloaded. In general, it is considered that plastic deformation occurs without any volume change and hydrostatic pressure has virtually no inﬂuence on yielding. Experiments conducted at high conﬁnement pressure showed that, though very small, a pressure eﬀect is quantiﬁable and can explain the Strength Differential (SD) eﬀect for high strength steels (Spitzig et al., 1984). The SD eﬀect corresponds to the diﬀerence between tension and compression yield stresses when both tests are conducted independently from an annealed state. Conﬁnement pressure can also signiﬁcantly improve ductility.

4

Advanced Methods in Material Forming

The Bauschinger eﬀect is a common feature in metals and alloys that occurs when a material is deformed up to a given strain, unloaded and loaded in the reverse direction, typically, tension followed by compression. Its signature is that the yield stress after strain reversal is lower than the ﬂow stress before unloading from the ﬁrst deformation step. Bauschinger and SD eﬀects are two diﬀerent phenomena. The ﬂow stress of a material depends on the testing temperature. Moreover, at low absolute temperatures compared to the melting point, time has usually a very small inﬂuence on the ﬂow stress and plasticity in general. However, at higher temperatures, strain rates eﬀects are important. In fact, it has been observed that strain rate and temperature have similar eﬀect on plasticity. Raising the temperature under which an experiment is carried out has a similar eﬀect as decreasing the strain rate. Temperature has another inﬂuence on plasticity. When subjected to a constant stress smaller than the yield limit, a material can deform by creep. A similar phenomenon, called relaxation, corresponds to a decrease in the applied stress when the strain is held constant. Finally, solid state transformations can occur in materials due to an applied stress. These transformations lead to phase changes under stresses that are lower than the yield stress of either phase and can induce plasticity (Taleh et al., 2003). 2.3 Microscopic Observations in Plasticity Commercial metals and alloys used in forming operations are polycrystalline. They are composed of numerous grains, each with a given lattice orientation with respect to macroscopic axes. At low temperature compared to the melting point, metals and alloys deform by dislocation glide or slip and by twinning on given crystallographic planes and directions, which produce microscopic shear deformations. Therefore, the distribution of grain orientations, the crystallographic texture, plays an important role in plasticity. Because of the geometrical nature of slip and twinning deformations, strain incompatibilities arise between grains and produce micro-residual stresses, which are partly responsible for the Bauschinger eﬀect. Slip results in a gradual lattice rotation as deformation proceeds while twinning leads to abrupt changes in lattice orientation. The number of available slip systems determines the nature of the deformation mechanisms. BCC and FCC materials tend to deform by slip because of the large number of available slip systems. However, HCP materials, in which the number of potential slip systems is limited, generally tend to twin as an alternate mechanism to accommodate an imposed deformation. After slip, dislocations accumulate at microstructural obstacles and increase the slip resistance for further deformation, leading to strain hardening with its characteristic stress-strain curve. At higher temperature, more slip systems can be available to accommodate the deformation (Perocheau et al., 2002) but grain boundary sliding, which in

Constitutive Modeling for Metals

5

a sense, is another type of shear, is becoming more predominant. For instance, superplastic forming occurs mainly by grain boundary sliding. In this case, the grain size and shape are important parameters. Atomic diﬀusion is also another mechanism that aﬀects plastic deformation at high temperature and contributes to creep. Commercial materials contain second-phase grains or particles. These phases are present in materials by design in order to control either the microstructure such as the grain size or mechanical properties such as strength. However, some amounts of second-phases are undesired. In any case, the presence of these non-homogeneities alters the material behavior because of their diﬀerences in elastic properties with the matrix as it happens in composite materials, or because of their interactions with dislocations. In both cases, these eﬀects produce incompatibility stresses that contribute to the Bauschinger eﬀect. The mechanisms of failure intrinsic to metals and alloys are plastic ﬂow localization and fracture. Localization tends to occur in the form of shear bands, either micro-bands, which tend to be crystallographic, or macro-bands which are not (Korbel, 1998). Necking in thin sheet occurs in plane strain deformation which, depending on the reference frame the strain is observed, is also a shear deformation mode. In forming, ductile fracture is generally the result of the mechanisms of void nucleation, growth and coalescence. The associated micro-porosity leads to volume changes although the matrix is plastically incompressible, and hydrostatic pressure aﬀects the material behavior. At low temperature compared to the melting point, second-phases are principally the sites of damage. The stress concentration around these phases lead to void nucleation, and growth occurs by plasticity. Coalescence is the result of plastic ﬂow micro-localization of the ligaments between voids. At higher temperature, where creep becomes dominant, cavities nucleate at grain boundaries by various mechanisms including grain sliding and vacancy concentration (Kassner et al., 2003). Generally, the materials subjected to creep and superplastic forming exhibit higher porosity levels than those deformed at lower temperature.

3 Constitutive Modeling 3.1 Modeling Scale In view of the previous section, it is obvious that it is impossible to develop constitutive models for forming applications that can capture all the macroscopic and microscopic phenomena involved in plastic deformation and ductile fracture. Plasticity can be studied at various scales but for forming applications, macroscopic models appear to be more appropriate. Because of the scale diﬀerence between the microstructure and an engineered component, the amount of microscopic material information necessary to store in a forming

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Advanced Methods in Material Forming

simulation would be enormous. It is not possible anyways to track all of the relevant microstructural features in detail. Therefore, it seems more appropriate to integrate them all in a few macroscopic variables. Microscopic models are more suitable as a guide for material design, as a tool for fundamental understanding of plasticity and for inferring suitable formulations at the macroscopic scale. As mentioned above, memory eﬀects in materials can be signiﬁcant and, in principle, integral forms of the constitutive equations should be used to account for this history (Zhou et al., 2003). However, most of the constitutive equations describing plasticity have been developed in diﬀerential forms and have assumed to contain enough of the material history. Constitutive equations have been developed in scalar forms for uniaxial or shear deformation, particularly when coupled to microstructural evolution, and in tensorial forms for multiaxial loading as observed in forming operations. The constitutive equations are often developed in the framework of thermodynamics in order to prevent violation of physical principles. The constitutive laws generally consist of a state equation, sometimes referred to as the kinetic equation, (Krausz et al., 1996) and evolution equations. The state equation describes the relationship between the strain rate ε, ˙ stress σ, temperature T and state variables Xi , which represents the microstructural state of the material. This can be translated, for instance, as σ = f (ε, ˙ T, Xi )

(1)

The evolution equations describe the development of the microstructure through the change of the state variables and can take the form ˙ T, Xj ) X˙ i = f (ε,

(2)

However, the state variables do not need to be connected to a speciﬁc microstructural feature. In this case, (1) and (2) deﬁne a macroscopic model with state (or internal) variables. Since the temperature inﬂuences the kinetics of microscopic mechanisms, it has an eﬀect on plasticity, as discussed above, similar to that of the strain rate. Therefore, in processes involving hot deformation such as rolling, forging or extrusion, these two variables are usually combined in a single quantity, the so-called Zener-Hollomon parameter Z (see Hosford et al., 1983) Z = ε˙ exp (−Q/RT ) (3) thus reducing the number of variables in the constitutive relationships. In (3), R is the gas constant and Q an activation energy, which is determined experimentally for a given material. 3.2 Microscopic Plasticity Modeling Because slip plays a major role in plasticity, it seems important to look at this mechanism in term of both its geometrical eﬀect on anisotropy and its

Constitutive Modeling for Metals

7

eﬀect on strain hardening. Polycrystal description of plasticity has been very successful over the last few decades. This approach is based on the geometrical aspect of plastic deformation, slip and twinning in crystals, and on averaging procedures over a large number of grains. The crystallographic texture is the main input to these models but other parameters, such as the grain shape, can also be included. It is a multiaxial approach and involves tensors. One of the outputs of a polycrystal model is the concept of the yield surface, which generalizes the concept of uniaxial yield stress for a multiaxial stress state. Polycrystal models are also very powerful to describe plastic anisotropy. Because polycrystal models can track the lattice rotation of each individual grain, the material anisotropy is naturally evolving, which makes this approach very attractive. They can be used in multi-scale simulations of forming but they are usually expensive in time and the relevant question is to know if their beneﬁt is worth the cost. Polycrystal modeling aspects including twinning have been treated in a large number of books and publications such as, for instance, Kocks et al. (1978), Gambin (2001), Kalidindi et al. (2001) and Staroselski et al. (2003). Bishop et al. (1951) showed that, for a single crystal obeying the Schmid law, i.e. dislocation glide occurs when the resolved shear stress on a slip system reaches a critical value, the resulting yield surface was convex and its normal was collinear to the strain rate, i.e. that the yield surface is a convex potential. Furthermore, they extended this result for a polycrystal by averaging the behavior of a representative number of grains in an elementary volume without any assumption about the interaction mode between grains or the uniformity of the deformation gradient. Hecker (1976) reviewed a number of multiaxial experiments and did not ﬁnd any signiﬁcant contradiction to these assumptions about normality and convexity. After shearing individual grains, dislocations accumulate in the material, increasing their density which, in turn, leads to strain hardening. The Kocks and Mecking approach (Estrin, 1996) has laid the foundations for many subsequent studies by connecting the dislocation density ρ to the shear ﬂow stress τ using the following state equation √ τ = τo + αμb ρ (4) where τo is the lattice friction stress, μ is the shear elastic modulus, b is the amplitude of the Burgers vector and α is a constant that takes dislocation interactions into account. The dislocation density represents the state of the material and its evolution, which depends on the dislocation production and annihilation rates, and can be represented for instance as (Estrin, 1996) √ dρ dt = k1 + k2 ρ − k3 ρ (5) where ki are coeﬃcients, possibly depending on strain rate and temperature. With this type of approach, it is possible to model time dependent behavior such as creep, and time independent behavior by applying a strain proportional to time. Moreover, parameters describing the microstructure such

8

Advanced Methods in Material Forming

as grain size, second-phase and solute content can be incorporated into the formulation. In this type of approach, two or more state variables can be used such as the forest and mobile dislocation densities (Estrin, 1996). In that case, additional phenomena can be studied such as dynamic strain aging (Kubin et al., 1990; Rizzi et al., 2004) which produce, for instance, serrations in the stress-strain curves. This type of models contributes to the fundamental understanding of plasticity, to microstructure optimization and to the identiﬁcation of relevant parameters that need to be directly or indirectly included in constitutive models. Dislocation models were also used to explain the small pressure eﬀect on plastic deformation. Spitzig and Richmond (1984) showed that in high strength steels the uniaxial yield stress was larger in compression than in tension. They attributed this strength-diﬀerential (SD) eﬀect to the sensitivity of steel to pressure. In fact, these authors conducted experiments under hydrostatic conﬁnement and found a linear dependence between the mean stress (σm = σkk /3) and the eﬀective stress σe (associated to the square root of √ the second invariant of the stress tensor σe = J2 ). To a reasonable approximation, Richmond and Spitzig (1980) used the following yield function to describe their experiments φ = σe + σ {3βσm − 1} = 0

(6)

where σ is the uniaxial ﬂow stress and β is the pressure coeﬃcient. This expression is similar to the yield condition proposed by Drucker et al. (1952) for soils. Spitzig and his co-workers (Spitzig 1979; Spitzig et al. 1976, 1984) conducted experiments for diﬀerent steels and obtained approximately the same pressure coeﬃcient, β = 20 TPa−1 . They also performed experiments on commercial purity aluminum and obtained β = 50 TPa−1 . The volume changes that they observed experimentally were negligible compared to those calculated with the classical ﬂow rule, assuming normality between the yield surface and the strain rate. To explain the SD eﬀect, Jung (1981) proposed a model based on the additional work needed to induce the motion of a dislocation due to the pressure (p) dependence of the elastic shear modulus (μ), expressing β in (6) as β≈

2 dμ 3μ0 dp

(7)

In the previous relationship, μ0 is the shear modulus at atmospheric pressure. This model leads to a pressure coeﬃcient β equal to about 17 and 59 TPa−1 for steel and aluminum, respectively. These values are in good agreement with the experimental values mentioned above. Bulatov et al. (1999) used molecular static calculations with an embedded atom potential to simulate the eﬀect of pressure on dislocation motion. They found that this phenomenon was the result of the interaction of the transient dilatation of moving dislocation with pressure. For aluminum, they computed values for the pressure coeﬃcient

Constitutive Modeling for Metals

9

β of 48 and 63 TPa−1 for screw and mixed dislocations, respectively, which are consistent with the experimental value as well. Practically, for low to medium strength materials and low conﬁnement pressure, this departure from the classical behavior can be neglected. However, this example shows how a parameter calculated using atomistic scale simulations can be transferred to the macroscopic scale. In multi-phase materials, the second phases, whose purpose is usually to increase strength, also contribute to plastic anisotropy. In heat-treatable aluminum alloys, precipitates are intimately linked to texture because they exhibit speciﬁc shapes and crystallographic relationships with the grains. They can inﬂuence anisotropy in a way that depends on their mode of interaction with dislocations. Wilson (1965) showed that binary Al-4%Cu alloys aged with diﬀerent thermal treatments, i.e. containing diﬀerent types of precipitates but the same crystallographic texture, exhibit Bauschinger eﬀects of diﬀerent magnitudes. As noted by Bate et al. (1981), a strong back stress builds up as deformation proceeds in alloys containing nonshearable precipitates. These authors used the results of the elastic inclusion model due to Eshelby (1957) to estimate the values of the back stress. This approach is based on the idea that dislocations accumulate around these particles and produce elastic/plastic strain incompatibilities at the precipitate interface. 3.3 Macroscopic Plasticity Modeling For time-independent plasticity, in a multiaxial stress space, plastic deformation is well described with a yield surface, a ﬂow rule and a hardening law (Barlat et al., 2004). Plastic anisotropy is the result of the distortion of the yield surface shape due to the material microstructural state. In fact, here is an example where the microstructural model, crystal plasticity, guides the development of macroscopic descriptions. The ﬂow rule can be assumed to follow the normality property as discussed by Bishop et al. (1951). Strain hardening can be isotropic or anisotropic. The former corresponds to an expansion of the yield surface without distortion due to an increase of the dislocation density. It is completely deﬁned by a single stress-strain curve. Any other form of hardening, such as kinematic hardening, which corresponds to the translation of the yield surface, is anisotropic. Because stress states are multi-dimensional, it is necessary to describe yielding as a function of the stress tensor. Proper anisotropic plasticity formulations can be obtained if they are developed in the framework of the theory of representation for tensor functions (Boehler, 1978). In this theory, the constitutive equations are expressed such that the material symmetry conditions are automatically veriﬁed. The theorem of representation for a tensor function indicates that the constitutive equation can be expressed as an irreducible form of a set of invariants. Moreover, the theory of representation of tensor function includes the principle of isotropy of space, also called principle of material frame indiﬀerence or objectivity. A model based on this general

10

Advanced Methods in Material Forming

framework was developed for materials exhibiting the orthotropic symmetry such as sheets and plates (Cazacu et al., 2003). A drawback of the general approach is that it is generally diﬃcult to check the convexity condition. However, a subset of this general theory, which is based on linear transformations of the stress tensor, is more suitable for developing convex formulations (Barlat et al., 2005). This approach is detailed below for incompressible materials for which a linear transformation is performed on the stress deviator, s, leading to the transformed stress deviator ˜ s ˜ s = Cs

(8)

C, a fourth order tensor, contains the anisotropy coeﬃcients, accounts for the macroscopic symmetries of the material, and reduces to the identity for s are (Barlat isotropic materials. The associated 1st , 2nd and 3rd invariants of ˜ et al., 2005) sxx + s˜yy + s˜zz ) 3 H1 = (˜ H2 = s˜2yz + s˜2zx + s˜2xy − s˜yy s˜zz − s˜zz s˜xx − s˜xx s˜yy 3 (9) 2 2 2 H3 = 2˜ syz s˜zx s˜xy + s˜xx s˜yy s˜zz − s˜xx s˜yz − s˜yy s˜zx − s˜zz s˜xy 2 where x, y and z are the symmetry axes of the material. Using the variable 3 2 (10) θ = arccos 2H13 + 3H1 H2 + 2H3 2 H12 + H2 the principal values of ˜ s, which are themselves invariant tensorial quantities, are s˜1 =2 H12 + H2 cos θ 3 + H1 (11) s˜2 =2 H12 + H2 cos θ 3 − 2π 3 + H1 s˜3 =2 H12 + H2 cos θ 3 + 2π 3 + H1 In this theory, an anisotropic yield condition is expressed with an isotropic function of s˜1 , s˜2 and s˜3 . It is also possible to use two or more linear transformations, as long as the yield function is isotropic with respect to the 3n (1) (n) variables s˜k , . . . s˜k (for n linear transformations). In sheet forming, plastic anisotropy is an important aspect because it inﬂuences the strain distribution in a part and, consequently, the critical failure spots. In the absence of anisotropic hardening, plastic anisotropy is contained in the shape of the yield surface. For cubic metals, there are usually enough potentially active slip systems to accommodate any shape change. Moreover, compressive and tensile yield strengths are virtually identical. Yielding of such materials is usually represented adequately by an even function of the principal values of the stress deviator sk , such as (Hershey, 1954)

Constitutive Modeling for Metals a

a

a

φ = |˜ s1 − s˜2 | + |˜ s1 − s˜2 | + |˜ s1 − s˜2 | = 2σ a

11

(12)

Here, σ is equated to h (ε) a state variable, which is a function of the dissipated plastic work or the corresponding accumulated plastic strain ε, and correspond to strain hardening. The exponent a is connected to the crystal structure of the material, i.e. 6 for BCC and 8 for FCC (Hosford, 1993). This was established are a result of many polycrystal simulations. Therefore, although this model is macroscopic, it contains some information pertaining to the structure of the material. A recent variation of the yield function described by (12) with two linear transformations leads to a yield surface shown in Fig. 1 (Barlat et al., 2005). The general shape of the yield surface, with regions of high and low curvatures, is consistent with crystal plasticity computations. The anisotropy coeﬃcients are calculated from mechanical tests results or polycrystal property predictions. The level of details of the anisotropic behavior captured by this model, illustrated in Fig. 2, might be excessive for many applications but in certain cases, for instance in the beverage can manufacturing industry, these details are signiﬁcant. For most hexagonal closed packed metals (e.g. Ti, Mg, Zr, etc.), at low temperatures or high strain rates, twinning plays an important role in plastic deformation. The grains cannot accommodate certain shape changes because they lack the necessary deformation systems or because these systems require high activation stresses. Unlike slip, although pressure independent, twinning is sensitive to the sign of the applied stress, which is conducive to a strength diﬀerential (SD) eﬀect. Furthermore, the strong crystallographic 1.5

TD normalized stress

1.0

Yld2004-18p Normalized shear stress contours in 0.05 increments

0.5

0.0

–0.5

–1.0 2090-T3 –1.5 –1.5

–1.0

–0.5 0.0 0.5 RD normalized stress

1.0

1.5

Fig. 1. Plane stress yield surface for a 2090-T3 Al-Li alloy sheet sample represented as contours of constant normalized shear stress σxy /σ

12

Advanced Methods in Material Forming 1.05 Exp. r value Yld2004-18p

Exp. stress Yld2004-18p

1.6

1.2

0.95 0.90

0.8

r value

Normalized stress

1.00

0.85 0.4 0.80 2090-T3 0.75

0

10

20

30 40 50 60 Tensile direction

70

80

0 90

Fig. 2. Uniaxial ﬂow stress and r value (width to thickness strain ratio) directionalities with respect to the angle between tensile and rolling directions as predicted with the yield function Yld2004-18p for a 2090-T3 sheet sample

texture displayed by HCP materials leads to a pronounced anisotropy. To describe the yield asymmetry and anisotropy, Cazacu et al. (2004, 2005) proposed two yield functions, one based on the general theory of tensor representation and the other based on a linear transformation a

a

a

φ = ||s1 | − ks1 | + ||s2 | − ks2 | + ||s3 | − ks3 |

(13)

This formulation, although pressure insensitive, breaks the tension-compression symmetry and, similarly to the yield function in (12), includes anisotropy. For instance, Fig. 3 shows the corresponding yield surfaces at diﬀerent strain levels predicted with this equation and applied to the case of a textured magnesium sheet. The non-isotropic hardening eﬀect is captured by the evolution of the anisotropy coeﬃcients and it corresponds to the rapid changes in texture due to twinning as deformation proceeds. Non-isotropic hardening eﬀects can be described more classically by kinematic hardening and are usually related to the micro-stresses resulting from strain incompatibilities between grains or by the interactions between matrix and second-phases. This type of hardening describes the Bauschinger eﬀect very eﬃciently and can be represented as φ (σ − α) = h (ε)

(14)

σ is the applied stress tensor and α is the back stress tensor, which controls the yield surfacetranslation. Evolution laws for this tensor can take many forms, including the following non-linear expression (Lemaitre et al., 1990) α ˙ =

C (σ − α) ε˙ − γαε˙ σ

(15)

Constitutive Modeling for Metals

13

200

σ yy(MPa)

100

0 1% 5% –100

10% –200 –200

–100

0

100

σ xx(MPa)

200

Fig. 3. Predicted yield locus for textured magnesium sheet. Experimental data (open circle) from Kelley et al. (1968)

where C and γ are the kinematic hardening coeﬃcients. Barlat et al. (1998) used the non-linear kinematic hardening concept to develop a model that was able to explain the inﬂuence of non-shearable precipitates on plastic anisotropy for binary Al-Cu alloys deformed in tension and compression. In this case, the back stress was a function of the volume fraction, shape and habit planes of the precipitates, and of the crystallographic texture. Kinematic hardening can be successfully applied in forming simulations where the loading direction is changed abruptly such as, for instance, the prediction of springback in sheet forming. Other types of anisotropic hardening formulations, which account for the Bauschinger eﬀect, are based on multiple plasticity surfaces. In the case of two surfaces (Dafalias et al., 1975; Hashiguchi, 2005), called loading and yield surfaces, respectively, the loading surface translates into stress space in a direction determined by the applied stresses or strains until it contacts the yield surface. The stress-strain relationships are determined by either surface depending whether contact between the surface is made or not. This concept appears somewhat physical since due to micro-residual stresses, the loading or reloading portion of the stress-strain curve involves the plastic contribution of only the grains that are favorably oriented for slip. A similar concept based on multiple surfaces was also proposed to model anisotropic strain hardening (see Mr´oz in Lemaitre, 2001). In this case, nested surfaces, each with a speciﬁc modulus, translate in stress space. The property of the active surface determines the stress-strain relationships until contact with the new active surface occurs. This type of models is suitable for the description of cyclic plasticity but its formulation might be too complex for forming simulations.

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Advanced Methods in Material Forming

Viscoplasticity describes the time-dependent material behavior when temperature is less than typically half of the absolute melting point. Plastic deformation occurs by the motion of dislocations and the models used for plasticity are still valid. However, it is necessary to include terms containing the strain rate. Therefore, crystal plasticity and yield surface plasticity concepts can be applied in rate-dependent form. In this case, the critical resolved shear stress on the slip systems or the eﬀective stress in the yield function formulations need to be modiﬁed with a viscous term such as m σ = h (ε) ε˙ ε˙0 (16) where ε˙0 is a reference strain rate. Both h (ε) and m depend on the temperature. m is also called strain rate sensitivity parameter. Of course, other viscous terms can be used in constitutive equations. Another approach to viscoplasticity is to assume that there is no yield surface and that any level of stress produces some amount of inelastic deformation, possibly extremely small when the stress is small (Krempl, 1996). In crystal plasticity, this approach is very useful because it eliminates the problem of slip system ambiguity in crystals where more systems than necessary are potentially active. In the continuum approach, the inelastic strain is an increasing function of the diﬀerence between the applied stress and the kinematic stress. Here the kinematic stress is similar to what is called the back stress in the kinematic hardening theories and can be associated to the microresidual stresses that need to be overcome to deform a material plastically. At temperatures that are roughly higher than half of the melting point, diﬀusion and grain sliding mechanisms are more dominant. This is the domain of creep and superplastic deformations. The so-called uniﬁed theories such as that brieﬂy described above for viscoplasticity do not distinguish between the diﬀerent plasticity mechanisms and are therefore able to describe creep as well. For superplastic (Khaleel et al., 2001) and creep (Hoh et al., 2004) forming, the microporosity formed at grain boundaries is a dominant factor, which is necessary to integrate in the constitutive equations of plastic deformation (see Murakami, in Lemaitre 2001). Constitutive models, such as the model proposed by Gurson (1977) and later extended by Tvergaard et al. (in Lemaitre, 2001) σe2 3q2 σm φ (σij , f, σ) = 2 + 2q1 f cosh − 1 + q12 f 2 = 0 (17) 2σe σ contain the porosity as an internal variable. Here, q1 and q2 are constant coeﬃcients and f is the void volume fraction which, along with σ, is a second state variable. The evolution equation for f does not depend on the speciﬁc form of (17). It is the same for all models containing porosity, whether void nucleation takes place or not (Leblond, 2003) f˙ = (1 − f ) ε˙kk

(18)

Constitutive Modeling for Metals

15

F F

Fig. 4. Schematic of microstructural features within a shear localization band

Other aspects of the void growth can be included in this type of formulation, including the presence of hard inclusions in the microvoids or the coalescence of cavities (Siruguet et al., 2005a and b, respectively). To account for porosity or other form of material degradation, another approach consists in using a damage tensor D that modiﬁes the applied stress tensor σ (see Chow in Lemaitre, 2001) −1 σ ˜ = [I − D] σ (19) The resulting eﬀective stress tensor σ ˜ can be used in the classical mechanics formulations and constitutive equations to describe plasticity of damaged materials. Although discussed only brieﬂy in this paper, strain localization and fracture are material intrinsic failure modes that are very important to consider. Shear localization is of particular interest because it can occur in any deformation state, either compressive as in rolling or tensile as in sheet forming. The modeling of shear localization is usually based on the existence of an imperfection in a material (Chien, 2004), which is physically reasonable considering the non-homogeneity of a material microstructure. All the microscopic features described above can contribute to localization through local hardening or softening (Fig. 4). In particular, porosity accounts for additional softening in the shear band. Ductile fracture can be predicted to occur with a Gurson type of constitutive equation by deﬁning a critical porosity level fc above which fracture occurs. In the damage mechanics approach, a critical parameter Dc can be deﬁned for failure as well.

4 Conclusions This paper illustrates the importance of material and process interactions. In principle, modeling of forming and microstructure evolution should be a concurrent process. However, in view of the size of forming simulations and the complexity of materials and physical phenomena occurring during plastic deformation, it seems more eﬃcient to use macroscopic constitutive models

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Advanced Methods in Material Forming

with one or more internal variables to account for the microstructure. Constitutive models at lower scale are, of course, very important for the understanding of the microstructure evolution and to provide a basis for the development of more advanced macroscopic material models.

Acknowledgements The author gratefully acknowledges Professor O. Cazacu, The University of Florida, for fruitful discussions and for providing Fig. 3, and Drs. M.E. Karabin and H. Weiland, Alcoa Technical Center, for their careful reviews of the manuscript.

References Altenpohl D.G., Aluminum: Technology, Applications and Environment, Warrendale, PA, TMS, 1998. Barlat F., Aretz H., Yoon J.W., Karabin M.E., Brem J.C., Dick R.E., “Linear transformation-based anisotropic yield functions”, Int. J. Plasticity, vol. 21, 2005, p. 1009–1039. ˙ Barlat F., Cazacu O., Zyczowski M., Banabic D., Yoon J.W., “Yield surface plasticity and anisotropy”, Continuum Scale Simulation of Engineering Materials – Fundamentals – Microstructures – Process Applications, Raabe D., Roters F., Barlat F., Chen L.-Q., (eds.), Berlin, Wiley-VCH Verlag GmbH, 2004, p. 145–177. Barlat F., Liu J., “Modeling precipitate-induced anisotropy in binary Al-Cu alloys”, Mat. Sci. Eng., vol. A257, 1998, p. 47–61. Bate P., Roberts W.T., Wilson D.V., “The plastic anisotropy of two-phase aluminum alloys – I. Anisotropy in unidirectional deformation”, Acta Metall., vol. 29, 1981, p. 1797–1814. Bishop J.W.F., Hill R., “A theory of the plastic distortion of a polycrystalline aggregate under combined stresses”, Phil. Mag., vol. 42, 1951, p. 414–427. Boehler J.P., Lois de Comportement anisotropes des milieux continus , J. M´ecanique, vol. 17, 1978, p. 153–190. Bulatov V.V., Richmond O., Glazov M.V., “An atomistic dislocation mechanism of pressure-dependent plastic ﬂow in aluminium”, Acta Materialia, vol. 47, 1999, p. 3507–3514. Cazacu O., Barlat F., “A criterion for description of anisotropy and yield diﬀerential eﬀects in pressure-insensitive metals”, Int. J. Plasticity, vol. 20, 2004, p. 2027–2045. Cazacu O., Barlat F., “Application of the theory of representation to describe yielding of anisotropic aluminum alloys”, Int. J. Eng. Sci., vol. 41, 2003, p. 1367–1385. Cazacu O., Plunkett B., Barlat F., “Orthotropic yield criterion for Mg alloy sheets”, Proceedings of the 8th Conference of the European Scientiﬁc Association for Material Forming, Cluj-Napoca, Romania, April 27–29, 2005, Banabic, D., (ed.), Bucharest, The Publishing House of the Romanian Academy, p. 3–10.

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Chien W.Y, Pan J., Tang S.C., “A combined necking and shear localization analysis for aluminum alloys sheets under biaxial stretching conditions”, Int. J. Plasticity, vol. 20, 2004, p. 1953–1981. Dafalias Y.F., Popov E.P., “A model of nonlinearly hardening materials for complex loading”, Acta Mechanica, vol. 21, 1975, p. 173–192. Drucker D.C., Prager W., “Soil mechanics and plastic analysis or limit design”, Quart. Appl. Math., vol. 10, 1952, p. 157–165. Eshelby J.D., “The determination of the elastic ﬁeld of an ellipsoidal inclusion and related problems”, Proc. Roy. Soc. London, vol. A241, 1957, p. 376–396. Estrin Y, “Dislocation density-related constitutive modeling”, Uniﬁed Constitutive Law of Plastic Deformation, Krausz A.S., Krausz K. (eds.), Academic Press, San Diego, CA, 1996, p. 69–106. Gambin W., Plasticity and Texture, Amsterdam, Kluwer Academic Publishers, 2001. Gurson A.L., “Continuum theory of ductile fracture by void nucleation and growth – Part I: Yield criteria and ﬂow rules for porous ductile media”, ASME J. Eng. Materials and Technology, vol. 99, 1977, p. 2–15. Hashiguchi K., “Generalized Plastic Flow Rule”, Int. J. Plasticity, vol. 21, 2005, p. 321–351. Hecker S.S., “Experimental studies of yield phenomena in biaxially loaded metals”, Constitutive Modelling in Viscoplasticity, Stricklin A., Saczalski K.C. (eds.), ASME, New-York, ASME, 1976, p. 1–33. Hershey A.V., “The plasticity of an isotropic aggregate of anisotropic face centred cubic crystals”, J. Appl. Mech., vol. 21, 1954, p. 241–249. Hoh K.C., Lin J., Dean T.A., “Modeling of springback in creep forming thick aluminum sheets”, Int. J. Plasticity, vol. 20, 2004, p. 733–751. Hosford W.F., Caddell R.M., Metal Forming-Mechanics and Metallurgy, Englewood Cliﬀs, NJ, Prentice-Hall, Inc., 1983. Hosford W.F., The Mechanics of Crystals and Polycrystals, Oxford, Science Publications, 1993. Jung J., “A note on the inﬂuence of hydrostatic pressure on dislocations”, Philos. Mag. A, vol. 43, 1981, p. 1057–1061. Kalidindi S.R., “Modeling anisotropic strain hardening and deformation textures in low stacking fault energy fcc metals”, Int. J. Plasticity, vol. 17, 2001, p. 837–860. Kassner M.E., Hayes T.A., “Creep cavitation in metal”, Int. J. Plasticity, vol. 19, 2003, p. 1715–1864. Kelley E.W., Hosford W.F., “Deformation characteristics of textured magnesium”, Trans. TMS-AIME, vol. 242, 1968, p. 654–661. Khaleel M.A., Zbib H.M., Nyberg E.A., “Constitutive modeling of deformation and damage in superplastic materials”, Int. J. Plasticity, vol. 17, 2001, p. 277–296. Kocks U.F., Tom´e C.N., Wenk H.-R., Texture and Anisotropy, Cambridge, University Press, 1998. Korbel A., “Structural and mechanical aspects of homogeneous and nonhomogeneous deformation in solids”, Localization and Fracture Phenomena in Inelastic Solids, P. Perzyna P. (ed.), Wien, Springer-Verlag, 1998, p. 21–98. Krausz A.S., Krausz K., “The constitutive law of deformation kinetics”, Uniﬁed Constitutive Laws of Plastic Deformation, San Diego, CA, Academic Press, (1996), p. 229–279.

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Krempl E., “A small-strain viscoplasticity theory based on overstress“, Uniﬁed Constitutive Laws of Plastic Deformation, Krausz A.S., Krausz K. (eds.), San Diego, CA, Academic Press, (1996), p. 281–318. Kubin L.P., Estrin Y., “Evolution for dislocation densities and the critical conditions for the Portevin-Le Chatelier eﬀect”, Acta Metall. Mater., vol. 38, 1990, p. 697–708. Leblond J.-B., M´ecanique de la rupture fragile et ductile, Paris, Lavoisier, 2003. Lemaitre J. (ed.), Handbook of Materials Behaviour Models, San Diego, Academic Press, San Diego, 2001. Lemaitre J., Chaboche J.-L., Mechanics of Solid Materials, Cambridge, University Press, 1990. Perocheau F., Driver J., “Slip systems rheology of Al-1%Mn crystals deformed by hot plane strain compression”, Int. J. Plasticity, vol. 18, 2002, p. 185–203. Raabe D., Roters F., Barlat F., Chen L.Q. (eds.), Continuum Scale Simulations of Engineering Materials – Fundamentals – Microstructures – Process Applications, Berlin, Wiley-VCH Verlag GmbH, 2004. Richmond O. and Spitzig W.A., “Pressure dependence and dilatancy of plastic ﬂow”, IUTAM Conference, Theoretical and Applied Mechanics, Proc. 15th International Congress of Theoretical and Applied Mechanics, Amsterdam, North-Holland Publishers, 1980, p. 377–386. Rizzi E., H¨ ahner P., “On the Portevin-Le Chatelier eﬀect: Theoretical modeling and numerical results”, Int. J. Plasticity, vol. 20, 2004, p. 121–165. Siruguet K., Leblond J.-B., “Eﬀect of void locking by inclusions upon the plastic behavior of porous ductile solids – I: Theoretical modeling and numerical study of void growth”, Int. J. Plasticity, vol. 20, 2004a, p. 225–254. Siruguet K., Leblond J.-B., “Eﬀect of void locking by inclusions upon the plastic behavior of porous ductile solids – part II: Theoretical modeling and numerical study of void coalescence”, Int. J. Plasticity, vol. 20, 2004b, p. 255–268. Spitzig W.A, “Eﬀect of hydrostatic pressure on plastic ﬂow properties of iron single crystal”, Acta Metall., vol. 27, 1979, p. 523–534. Spitzig W.A, Sober R.J., Richmond O., “The eﬀect of hydrostatic pressure on the deformation behavior of Maraging and HY-80 steels and its implication for plasticity theory”, Metall. Trans., vol. 7A, 1976, p. 1703–1710. Spitzig, W.A., Richmond, O., “The eﬀect of pressure on the ﬂow stress of metals”, Acta Metal., vol. 32, 1984, p. 457–463. Staroselski A., Anand L., “A constitutive model for hcp materials deforming by twinning: Application to magnesium alloy AZ31B”, Int. J. Plasticity, vol. 19, 2003, p. 1843–1864. Taleh L., Sidoroﬀ F., “A micromechanical model of the Greenwood-Johnson mechanism in transformation induced plasticity”, Int. J. Plasticity, vol. 19, 2003, p. 1821–1842. Wilson D.V., “Reversible work-hardening in alloys of cubic metals”, Acta Metall. vol. 13, 1965, 807–814. Zhou Z.-D., Zhao S.-X., Kuang Z.-B., “An integral elasto-plastic constitutive theory”, Int. J. Plasticity, vol. 19, 2003, p. 1377–1400.

Material Forming and Dimensioning Problems: Expectations from the Car Industry G. Maeder Renault s.a.s., Materials Engineering Department, FR TCR LAB 1 36, 1 avenue du Golf – 78288 Guyancourt Cedex, Phone: 33(0)1 76 56 372, Fax: 33(0)1 76 85 03 35, [email protected] Summary. An automotive project is now based on numerical validation to reduce the development time between the project preparation and the start of production. That is particularly true to predict the formability of materials and solve dimensioning problems. But some problems remain and we will give examples of limitations in the case of steel sheet stamping, of tubes or proﬁle bending, of polymer injection. We will also show that the main diﬃculty relates to constitutive equations that require to be more and more representative of the real behaviour of materials.

Key words: automotive project, simulation, stamping, bending, injection.

1 Introduction When choosing a material for designing a part or a set of parts for a particular function, the engineer should consider several factors: the vehicle has to fulﬁl technical, economical and regulation requirements. However, the main factor claimed by the designer relates to the properties of the part once the materials have been formed and/or assembled. The automobile product changes under the inﬂuence of factors that are internal or external to the company, and the time allowed for development shortens with the contribution of representation and virtual design. This is not without consequence on the choice and validation of the material-process couple. In particular, modelling and simulation of forming and assembling processes are becoming compulsory tools for the engineer to accompany development of an automotive project. In this article, we would ﬁrst like to show what an automotive project is today with its major guidelines, its main stages and the evolution factors. We shall then see what are the consequences for the material-process couple with, in particular, the contribution of numerical simulation. Lastly, from a few examples taken from stamping, bending, injection techniques, we shall show

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Advanced Methods in Material Forming

that important progress remains to be accomplished to make the numerical calculation and experimental validation results to coincide.

2 Development of an Automotive Project 2.1 Major Guidelines Simultaneous Engineering: the important change in organization is the passage from a sequential engineering giving way to styling, product design, process design, industrialization, purchasing, suppliers, logistics, after-sales, towards a simultaneous engineering in which all these diﬀerent activities are implemented in parallel. Then, the development process starts from technical requirements to lead to components or physical parts available for assembly, going through functional customer requirements, via the simultaneous product/process design widely making use of simulation, numerical calculation, etc. with diﬀerent steps marked by milestones with well-deﬁned content. Vehicle Breakdown: this development process applies to the elementary functions making up the vehicle. In the case of Renault, there are 24 groups of elementary functions, each placed under the responsibility of a pilot. These functions are, for example, the dashboard, the exhaust line, the axles, the external accessories, etc. The power train is itself considered as a function that is external to the vehicle and has its speciﬁc development logic. Piloting by Customer Requirements: the customer requirements correspond to what a manufacturer can propose to the customer to meet expectations that are more or less formulated: performance/consumption, comfort/behaviour/braking, soundprooﬁng/vibration, reliability/durability. The material-process couple contributes to the achievement of targets of a greater number of customer requirements. 2.2 Main Project Stages The design of a vehicle is divided into two major phases: the upstream phase (project preparation) and the development phase (Fig. 1). –

the upstream phase (project preparation) corresponds to the deﬁnition of the company’s strategy in terms of research of concepts, choice and feasibility of innovations: “State what will be done”.

This upstream phase is divided into two stages. From the “orientations” milestone given by the Product – Planning department, we study several car concepts and diﬀerent technological innovations. Some hypotheses are frozen at the “hypotheses freeze” milestone and the preparatory studies stage leads toward one choice of concept, to the choice of innovations, and to the choice

Material Forming and Dimensioning Problems Project Preparation

Project Development

21

Production

Vehicle Approval to Marketing Entry in Build Try-out Approval Factory Tooling Manufacturing Go Ahead Approval

Styling Freeze 100% Orientations

Hypotheses Freeze

Exploratory Studies

PreContract

Preparatory Studies

Contract

First Design

Detailed Design

Product/ Process Development

Manufacturing

Standard Production

1 2 3 4 5 Digital Mock-up Design Reviews

Fig. 1. Development schedule of a project

of suppliers. The end milestone of this upstream phase is the “precontract milestone”. That is also the ﬁrst milestone of the project development phase. –

the development phase forms the actual vehicle project, with a digital convergence process of the vehicle (design phase) and a physical convergence process leading to the deﬁnition of parts, systems and ﬁnally the vehicle (industrialization phase): “Do what was stated”.

These diﬀerent phases are divided into diﬀerent stages separated by diﬀerent milestones. It is important to point out that, during the “ﬁrst design” and “detailed design” phases, the various milestones scheduled are essentially based on digital validations and design representations (plus a few partial physical test supports). The “contract milestone” determines the Quality-Cost-Delivery objectives of the project. At the “styling freeze” milestone all deﬁnitions corresponding to geometry and interfaces are frozen. The “tooling go ahead” milestone shows the beginning of the physical achievement part of the design. The “product/process development stage includes the manufacturing of the tools. The “vehicle entry in factory” milestone marks the end of the product/process try-out and formalizes the transfer of the engineering responsibility towards the manufacturing responsibility. All Quality-Cost-Delivery objectives are obtained. The “manufacturing approval” milestone marks the beginning of the production, and the “marketing approval” milestone marks the beginning of the standard production stage with the volume asked by the marketing. We must note that the physical achievement of the parts only starts at the “Tooling Go Ahead” milestone and ends at the “Approval to build tryout”. This is possible only by means of highly detailed studies based on the exclusive use of numerical simulation and dimensioning by calculation. And we have to add that this development process imposes the suppliers to integrate it, in the various phases described, for the parts developed outside the company.

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Advanced Methods in Material Forming

2.3 Evolution Factors Evolution of Customer Requirements: the requirements change as a function of market competition, customer expectations and regulations. The consequences of such evolution result in numerous speciﬁcations in which the materials – and therefore the simulation of their behaviour – are representative. For example: – –

– – –

Lightening: processing of new materials, multi-materials, assembly of different materials, etc. Durability: determination of methods (simulation and accelerated tests) to predict the change of properties in materials over time under fatigue solicitations, etc. Safety: taking into account the dynamical properties of materials and assemblies, etc. Acoustics: knowing the acoustic properties of materials after their processing, etc. Perceived Quality: regular clearances and perfectly geometrical ﬁtting between bodywork parts are fundamental elements of perceived quality of a vehicle. These two parameters are directly connected with the ﬁnal geometry of the parts after forming by sheet stamping or polymers injection.

Dimensioning to the Closest Value: a vehicle is now designed more and more in a virtual manner, in order to reduce the development time and the number of actual prototypes that are very expensive. Calculation models are more and more performing with, as a consequence, the necessity to very accurately deﬁne the mechanical, physical and chemical characteristics of materials to introduce them into the calculation models of parts, or in the calculations related to the transformation processes and to the assemblies. This requires that the engineers and the design departments have a good knowledge of the characteristics of the materials, their limitations, and the impact on the durability. This means that no value can be supplied without its conﬁdence interval linked to measurement uncertainties and to the variability of production facilities. Reduction in Vehicle Scheduling: accelerated renewal of models is made with a development schedule that is shorter. For example, the time between “design freeze” (a few weeks before Tooling Go Ahead) and “start of production” milestones has passed from 55 months to 42 months from Laguna I to Laguna II, and from 53 months to 29 months from M´egane I to M´egane II. It is therefore indispensable to freeze the material-process solutions more and more earlier in the development phase, and to reduce the tool definition time to the minimum. For this last point, numerical simulation is primordial. Internationalization: manufacturers are grouping themselves together in view of globalization. Vehicles are manufactured in several countries, in all ﬁve continents where the regulations, customer expectations, competitive markets are diﬀerent. It is also necessary to have a strong local integration: this is very

Material Forming and Dimensioning Problems

23

important because it should often call for local material suppliers, that will have grades that do not necessarily correspond to those in initial speciﬁcations. Adaptation of materials and transformation processes to the local conditions is an important point for internationalization of manufacturers. This adaptation cannot be achieved without numerical simulation. Supplier Relationships: the increase in the diversity of materials and transformation processes associated to them, the increase in the complexity of assemblies between materials lead to outsourcing the transformation processing activities, this outsourcing may be partial or total. In the Renault group, only stamping, painting, assembly, injection of plastic fenders, machining and heat treatment activities remain partially integrated. It is obvious that part suppliers are becoming function or system integrators. They need to master all the numerical simulation tools required in the reduction of development times.

3 Consequences for Materials-Processes 3.1 General Consequences Related to Vehicle Development Outside Project: Innovation Development. Due to shortening of all the development phases of a project, all research or innovation works, – may these be conducted within the manufacturer, within a supplier, or in association with laboratories –, will only be done outside all project framework. Preliminary Project: Introduction of Innovations and Solutions Retained. In the exploratory phase, diﬀerent material-process solutions are studied in parallel. Risk analysis is an important process that comes up against diﬃculties of assessing reliability and durability of the new solution, as well as its ﬁnal cost. In the preparatory phase, the material-process solution is chosen. This solution is validated essentially by modelling and numerical simulation. This means that, as already stated, the characteristics of materials (mechanical, physical data, constitutive equations of static and dynamic behaviour, at various temperatures, etc.) are known to be introduced in the calculation models. Project: Numerical and Experimental Validations. Numerical validation, already initiated in the preparatory phase, is pursued and several milestones correspond to reviews on digital mockups (Digital Mockup Design Review). The design of the parts is ﬁnalized, the materials-processes are speciﬁed, the industrialization scheme is deﬁned. The “product-process production and deﬁnition” is the physical convergence phase that comes after the simulation convergence phase. Any change in the plan of a part, any modiﬁcation in the material-process solution will then have exorbitant consequences on deadline and cost.

24

Advanced Methods in Material Forming

Let us point on, in particular, the time for producing tools that can extend over several months. It should and shall not have any problem at this stage, if the work conducted upstream was of quality. Series: Validation Under Actual Conditions. Follow-up of series-produced vehicles is the best means of validating the reliability and durability of the selected material-process solutions. 3.2 Importance of Numerical Simulation: Case of Stamping All that was stated earlier shows the importance of numerical simulation in the present development of vehicles. Stamping of steel sheets is doubtless the process that opened the way to modelling and numerical simulation. We shall take this example to demonstrate the progress approach. Figure 2 summarizes the progress made from 1985 up to the present day, in the ﬁeld of software applications and their use. However, remember that the correct use of such software applications passes through their “ergonomics” and the training quality of users. Let us give two examples of use in the two main phases of a project. Preliminary Project: simulation enables passage from an initial design in three parts, to a design in a single part (Fig. 3). Project: simulation enables display of the risks of cracks and enables the necessary corrections to be made (Fig. 4). Risks of cracks are indicated by the anows. And the experimental intrinsec charactericstics determined for applying the digital simulation can be also seen on the ﬁgure. Simulation also enables easier anticipation of material changes: substitution from classical steels to very high strength steels or passage to aluminium.

2000 2000 Process Processsurface surfacecreation creation software: software: - -adapted, adapted,robust, robust, - -user 1995 userfriendly, friendly,...... 1995 DieDesigner -> -> DieDesigner Meshing ... ... Meshingsoftware: software: DieMaker, DieMaker ,Simex2000, Simex2000, - -automatic, automatic, - -robust, fast, ... robust, fast, ... 1990 -> 1990 ->Deltamesh Deltamesh Stamping. Industrial Stamping, Industrialsoftware: software: AutoForm, - -validated AutoForm… ,… validated/ /robust robust - -user friendly user friendly -> ->Optris, Optris, Pam PamStamp, Stamp, - compensation of spring back Simex, Simex..., ... - nesting

1985 1985 Research: Research: - -formulas formulas - -finite finiteelements elements - -algorithms algorithms - -......

- ... - - > powerful optimisation software

Fig. 2. Evolution of numerical simulation

2005

Material Forming and Dimensioning Problems

25

Simultaneous engineering Final Design: 1 part

Initial Design: 3 parts

Fig. 3. Integration of simulation in preliminary project phase

Fig. 4. Integration of simulation in project phase

4 Manufacturer’s Expectations in Terms of Forming and Assembling Simulation: Some Examples We pointed out the progresses made by means of numerical simulation. Nevertheless, problems still subsist that are incorrectly or not solved. Their enumeration can serve as a clue towards research work. But, before describing some examples, note the three limitations that are common to all the examples: –

calculation times should be reduced,

26

– –

Advanced Methods in Material Forming

the constitutive equations of materials should be known, boundary conditions are always physically poorly known and numerically not correctly taken into account.

4.1 Limitations of Sheet Stamping Calculations Springback Eﬀect. The sprinkback eﬀect is well known and appear on the Fig. 5 with the exemple of the B section of the part. A positive deformation on one side and a negative deformation on the other side appear compared with the theoretical geometrical form. A method compensating for this springback was developed with Arcelor (Munier, 2004), by designing stamping dies whose die geometry is diﬀerent from that of the ﬁnal part. The ﬁrst applications were encouraging, however, two conditions remained to be fulﬁlled: – –

the numerical accuracy of springback prediction should be whithin 1 mm the calculation time of springback and of the stamping process should be shorter than 1 hour.

Friction Coeﬃcients. In a real stamping operation, friction occurs at the interfaces of sheet-punch, sheet-die, sheet-blankholder, etc. The impact of friction is considerable and the stamping feasibility is directly dependent thereon. The article referenced in (El-Mouatassim, 2002), reviews all the problems related to such friction and proposes a few approaches to reduce Sheet stamping Spring back 3.06

Section A

2.55

2.8

0.151

Section B

Section B 1.3

Section C Y Z

–3.09

–2.15

X

–3.0

Déplacemt norm. –2.1

–1.3

–0.4

0.4

Fig. 5. Example of sheet stamping spring back

3.0 1.3

2.1

Material Forming and Dimensioning Problems

27

them. The present conclusion is that “common sense” remains the best guide. However, sensitivity studies on diﬀerent parameters can help understanding stamping feasibility. Constitutive Equation for TRIP Steels. TRansformation Induced Plasticity steels (TRIP) have a multiphase microstructure constituted by a ferritic matrix, hard bainite and retained austenite. This austenite leads to a martensitic transformation when strain is applied during transformation. Constitutive equations currently used to characterize “classical” materials during forming process don’t take into account this phase transformation. Consequently, forming simulations for TRIP steels are not suﬃciently predictive especially for thickness. The “cross die test” uses a stamping tool which reproduces most of the industrial strain paths. RENAULT uses it to study and classify deep drawability of sheets, roughness, lubricants and coatings (Fig. 6). A square metal sheet is stamped at a given punch depth and blankholder force. A good way to evaluate the robustness of constitutive equations is to compare simulation results with experimental results on cross die test . We use thinning curves to make this comparison on the Fig. 7. The constitutive equations are established by using several mechanical characterization tests (tensile and shearing test,. . . . . . .) to identify quadratic yield criteria associated with isotropic work hardening model – upper curve on the Fig. 7 – or isotropic + kinematic work hardening model (Armstrong– Frederik for example) – bottom curve on the Fig. 7. These results show that we don’t ﬁnd a general rule concerning constitutive equations to have a good prediction of thickness: the accuracy depends on the material and the direction. One way to improve constitutive equations and consequently increase simulation accuracy is to develop a speciﬁc approach for TRIP steel. Renault has been studying diﬀerent ways to do that by developing a mixing law with phase transformation (coupled with quadratic yield criterion) or replacing isotropic work hardening term in Armstrong–Frederick model by a coupled isotropic – kinematical hardening law.

Diagonal direction Rolling direction

Fig. 6. “Cross-die test” geometry

T R IP 800

T R IP 700

T R IP 600

Thickness (mm)

1.10

1.30

1.50

1.70

1.90

2.10 1.90 1.70 1.50 1.30 1.10 0.90

0

1

1.2

1.4

1.6

1.8

2

2.2

0

0.90 –20 0

Thickness (mm)

Thickness (mm)

20

20

20

Experimentation Isotropic work hardening simulation Isotropic + kinematic work hardening simulation

60 80 100 120 140 160 180 200 Curvilinear abscissa (mm)

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.1

1.2

1.3

1.4

1.5

1.6

1.7

0

0

0

50

250

250

100 150 200 Curvilinear abscissa (mm)

250

Experimentation Isotropic work hardening simulation Isotropic + kinematic work hardening simulation

100 150 200 Curvilinear abscissa (mm)

Experimentation Isotropic work hardening simulation Isotropic + kinematic work hardening simulation

100 150 200 Curvilinear abscissa (mm)

50

50

Experimentation Isotropic work hardening simulation Isotropic + kinematic work hardening simulation

Graph of thickness - Diagonal direction

Fig. 7. Comparison experimentations/simulations on thickness (TRIP steels)

40 60 80 100 120 140 160 180 200 Curvilinear abscissa (mm)

Experimentation Isotropic work hardening simulation Isotropic + kinematic work hardening simulation

40 60 80 100 120 140 160 180 200 Curvilinear abscissa (mm)

40

Experimentation Isotropic work hardening simulation Isotropic + kinematic work hardening simulation

Graph of thickness - Rolling direction

Thickness (mm) Thickness (mm) Thickness (mm)

28 Advanced Methods in Material Forming

Material Forming and Dimensioning Problems

29

4.2 Limitations of Bending Calculations for Tubes or Proﬁles Tubes: Numerical simulation is now operational for bending exhaust line tubes. To achieve this, several years were necessary for searching on: – – –

identiﬁcation of constitutive equations speciﬁc to tubes characterization of tube/tool friction factors deﬁnition of robust numerical simulation methods.

However, it is still not possible to do away fully with tests. In fact, we still do not have a feasibility factor that would allows us to know if the numerically optimized solution will generate tube rupture or not. Aluminium Proﬁles: Using aluminium proﬁles space frame structure is an eﬃciency method to lighten vehicle compared to a steel Body-In–White. These proﬁles must be bent to allow shapes compatible for assembling with other parts of the body structure (see Fig. 8). In this study, applied to doorframe proﬁle, we focus on stretch-wrap process with pre-deformation. The main problem of proﬁle bending is to control the geometry distorsion after unloading especially springback. To evaluate springback predictivity of constitutive equations and codes, we made simulations with three diﬀerent softwares (PAMSTAMP2000, PAMSTAMP2G and ABAQUS).These results are compared with experimentations, Fig. 9. Figure 9 shows a diﬀerence between simulations (for the three codes) and experimentations especially for high bending angle and medium predeformation. Material parameters sensitivity study made with Pamstamp2G allowed to consider some ways to increase springback predictivity: –

improve accuracy of mechanical characterisation tests

Fig. 8. Proﬁle bending machine

30

Advanced Methods in Material Forming 12 Sim.,Pam2000,β=10° Sim.,Pam2000,β=20° Sim.,Pam2000,β=30° Sim.,Pam2G,β=10° Sim.,Pam2G,β=20° Sim.,Pam2G,β=30° Exp.,β=10° Exp.,β=20° Exp.,β=30°

10 8 6 4 2 0 0.2

0.3

0.4

0.5

0.6

0.7

0.8

Pre-elongation δF (%)

0.9

1

1.1

Springback gap c(mm)

Springback gap c(mm)

12

Sim.,Pam2000,β=10° Sim.,Pam2000,β=20° Sim.,Pam2000,β=30° Sim.,ABAQUS,β=10° Sim.,ABAQUS,β=20° Sim.,ABAQUS,β=30° Exp.,β=10° Exp.,β=20° Exp.,β=30°

10 8 6 4 2 0 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Pre-elongation δF (%)

Fig. 9. Simulations & experimental results for springback (Aluminium proﬁles)

– – –

develop new process for the constitutive equations identiﬁcation develop new identiﬁcation tests (bending-unbending test) evaluate new yield criteria (like BBC criterion from Prof. BANABIC)

4.3 Limitations of Polymer Injection Simulation Polymer injection is simulated using rheologic calculations. This simulation should enable to predict without testing, the risks inherent to the production of a part, and to modify the drawing, at its design stage, and/or the injection conditions. Thus, it should be possible to avoid reworking of tools and carrying out time-taking debugging operations by correctly clearing all preliminary risks. The current rheology software applications will predict the injection process at these diﬀerent stages: ﬁlling the die, compacting the part including the thermal history of the tools with ﬁlled materials or not, and post-moulding deformation. However, today, a great number of behaviours are still diﬃcult to predict because diﬃcult to model. Part Deformation: Figure 10 shows the case of the wheel cap of Modus, made of polypropylene, ﬁlled with 40% of mineral ﬁllers. A negative deformation of 1.5 mm is obtained on the post-moulded part whereas simulation provides for a positive deformation of 2.45 mm; this induces a contact between the wheel cap and the wheel itself. This diﬀerence can be explained by a calculation of volumetric shrinkage and moulding stresses from mechanical data, without taking into account the experimental shrinkage data measured on plates. Appearance Defects: On a side protection strip of the SCENIC, made of modiﬁed EPDM polypropylene with 15% of mineral ﬁllers, we obtained ﬂow lines and marks of waves on the part. Such appearance defects, to which can be added veins, burns, shrink marks, welding lines, are not quantiﬁed by rheology software applications. Analysis of the results of temperature and pressure provides a tendency only, knowing that

Material Forming and Dimensioning Problems

31

Fig. 10. Comparison simulation/experimentation on deformation after injection

the origin of formation of waves (or tiger stripes) is poorly known (unstable progression of the edge of the die in contact with the wall of the mould?). Pressure Prediction: Figure 11 summarizes the wheel cap of Fig. 9. This ﬁgure compares the experimental and calculated pressure evolution during injection. It can very clearly be seen that the simulated pressure is 30% lower than measured pressure. This overpressure results in inaccuracy of deformation prediction and in the choice of the force to be applied to close the press tooling. The source of the diﬀerence is doubtless the incorrect viscosity value used for the simulation. Shrink Mark Prediction: This concerns the case of PMMA reﬂectors. On these parts, shrink marks can be seen on the edges and on the surface. The volumetric shrinkage is therefore incorrectly predicted by an incorrect integration

Fig. 11. Comparison simulation/experimentation on injection pressure

32

Advanced Methods in Material Forming

of the thickness and by a lack of quantitative results in terms of appearance description. Flow Leaders Prediction: These ﬂow leaders result in deformation due to acceleration of the material in certain areas of the part. This appears in the example of a polycarbonate headlight glass, or also of rear-view mirror shells. However, 2D, 2D1/2 or 3D simulations, even with a local modiﬁcation of the thickness in the radius do not show any deformation. This diﬀerence between experience and simulation is due to the extrapolation of viscosity curves with high and low shear values, as well as the non-integration of the material’s elongational viscosity. Source of Problems: Several applications are available on the market, with meshing as a key point: neutral ﬁbre meshing, dual skin meshing, 3D meshing. Of course, there is a technical know-how for using such applications (and knowledge on their limitations), but the main problem is related to the constitutive equations of materials which are inaccurate. The following can be mentioned as inﬂuencing parameters: – – –

–

–

–

elongational viscosity is very diﬃcult to determine experimentally crystallization can be taken into account but with calculation times multiplied by ten the glass transition temperature (PVT), measured at very low cooling rates (1◦ C/ min) whereas the cooling rates are much higher during injection (200◦ C/ min) the viscosity that is measured over a too small shear strain range and whose values are extrapolated by high and low shear values by means of constitutive equations. the results, calculated from the glass transition temperatures (PVT) obtained from the constitutive equations with approximations, and that use thermo-mechanical data that are far beyond the injection conditions. the orientation of glass ﬁbres (particularly in the thickness) that occurs at the end of ﬂow, at a moment that is diﬃcult to predict by the know-how in simulation today. Furthermore, the applications do not take into account the orientation in thickness nor the anisotropy of the shrinkage.

5 Conclusions The development of an automotive project takes place over a shorter and shorter period, with an increasing and necessary role played by numerical simulation. Numerical simulation for forming is even more important as production of tools requires a time period that is diﬃcult to reduce: the tools should produce parts in accordance with modiﬁcations that are as low as possible. To bring down this calculation time, or to increase the accuracy, it is necessary to have more and more powerful hardware facilities.

Material Forming and Dimensioning Problems

33

But the most important works relate to constitutive equations that require to be more and more representative of the real behaviour of materials. The knowledge of these constitutive equations conditions the use of new materials or the use of assembly of diﬀerent materials. This requires numerous research works that should be developed outside an automotive project, but in universities laboratories, keeping close relationships with the end user (who can be the car manufacturer or the supplier) to limit the sophistication level of results in view of short and medium-term applications.

Acknowledgements The author would like to thank M. El-Mouatassim, R. Vollet, E. Crouan, P. Lory, F. Moussy and E. Vaillant for the examples given in the drafting of this article.

References El-Mouatassim M., Nobile D., Friction and numerical simulation of stamping, Proc. Conf. IBEC 2002. Munier M., Devin J.M., El-Mouatassim M., New Approach for Springback Compensation in Die Design Application for High Strength Steel, Proc. Conf. IDDRG 2004, ed. G. Steinbeck, Sindelﬁngen 2004.

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Optimization of the Phenomenological Constitutive Models Parameters Using Genetic Algorithms B.M. Chaparro1 , J.L. Alves2 , L.F. Menezes3 and J.V. Fernandes3 1

2

3

Departamento de Engenharia e Gest˜ ao Industrial, Escola Superior de Tecnologia de Abrantes, Instituto Polit´ecnico de Tomar, Rua de 17 de Agosto de 1808, 2200-370 Abrantes, Portugal, [email protected] Departamento de Engenharia Mecˆ anica, Departamento de Engenharia Mecˆ anica, Escola de Engenharia da Universidade do Minho, Campus de Azur´em, 4800-058 Guimar˜ aes, Portugal, [email protected] CEMUC – Centro de Engenharia Mecˆ anica da Universidade de Coimbra, Departamento de Engenharia Mecˆ anica, Universidade de Coimbra, Polo II, Pinhal de Marrocos, 3030-201 Coimbra, Portugal, [email protected], [email protected]

Summary. The lack of accuracy of numerical results is still nowadays one of the main drawbacks of sheet metal forming process simulation. One of the main reasons for such a lack of accuracy is the constitutive models used to describe the real material’s mechanical behavior. The most widely used phenomenological constitutive model is based on the classical Hill 1948 yield criterion. In the last decade several new yield criteria have been proposed, with the constraint that a parameter identiﬁcation procedure has not always been clearly set. This study presents a general approach to optimize anisotropic plastic description. A weight-based optimization procedure is presented in order to perform the optimization of several constitutive models based on experimental results. Using this procedure is expected to improve the plastic description with a global optimum solution.

Key words: plasticity, yield function, anisotropy, hardening, genetic algorithms.

1 Introduction The numerical simulation of sheet metal forming processes is particularly sensitive to the numerical modelling of the elastoplastic behaviour of the metallic sheets used in these sorts of processes. It is nowadays accepted that the numerical results of either the thickness reduction or the springback phenomena, among others, are particularly sensitive to the constitutive models

36

Advanced Methods in Material Forming

(Thuiller et al., 2002), (Chaparro et al., 2004). In recent decades, to overcome such diﬃculties, several new yield criteria have been proposed in the literature, in order to improve the description of the plastic behaviour of the metallic sheets (Barlat et al., 1991), (Karaﬁllis et al., 1993), (Cazacu et al., 2001), (Bron et al., 2004), (Cazacu et al., 2004). In fact, enhanced constitutive models including a high number of parameters are needed to improve the accuracy of sheet metal anisotropy description. However, when the number of constitutive parameters increases, the amount and type of experimental tests necessary for their identiﬁcation also increases. In this case, the parameter identiﬁcation procedure for each material becomes highly complex and often the results can be inconsistent. The aim of this study is to present a new identiﬁcation procedure, which allows the material parameters to be identiﬁed automatically based on a minimum error functional approach. To minimize the error functional, a methodology based on a hybrid algorithm is used. The new tool, named DD3MAT, was developed to automatically carry out the identiﬁcation of the full set of material parameters for several enhanced constitutive models, based on an adequate set of experimental results. In the future, this code will be integrated in the main ﬁnite element code DD3IMP, (Menezes et al., 2000), which is dedicated to the numerical simulation of the sheet metal forming processes and has been continuously developed at CEMUC (“Centro de Engenharia Mecˆanica da Universidade de Coimbra”).

2 Plastic Behaviour Description Phenomenological models are usually used to describe the anisotropic behaviour of sheet metals. For each of the so-called yield criteria or yield functions, which describe the yield locus and thus the plastic behaviour, and for each material, it is necessary to determine the ‘right’ set of values of material parameters. In this study the identiﬁcation methodology consists of a best ﬁt procedure between each yield function and the full set of available experimental results (tensile tests, simple shear tests, biaxial tests, etc.). The parameters of the yield criteria proposed by (Hill, 1948), (Barlat et al., 1991), (Karaﬁllis et al., 1993) and one of the two models proposed by (Cazacu et al., 2001) can be identiﬁed by the methodology proposed here. The work-hardening behaviour is described by the Swift law, (Swift et al., 1952). In this study, only the results concerning the isotropic work-hardening are presented. However, the same approach can be easily widened to the identiﬁcation of the kinematic hardening law’s parameters. 2.1 Anisotropic Yield Surfaces A yield function is generally described by the equation, F = K − Y = 0,

Phenomenological Constitutive Models Parameters Optimization

37

where Y is the ﬂow stress, whose evolution is given by a work-hardening law, and K is the equivalent stress computed from the expression of the selected yield criterion. To describe K, and so the anisotropic plastic response of the rolled metallic sheets, several orthotropic yield criteria proposed in the literature were implemented in both DD3IMP and DD3MAT codes, which are brieﬂy reviewed in the next sections. Hill, 1948 The Hill 1948 yield criterion is a generalization of the von Mises’s distortional energy criterion to orthotropy, (Mises, 1913). The yield function is expressed by the quadratic function, (Hill, 1950): 2

2

2

F (σyy − σzz ) + G (σzz − σxx ) + H (σxx − σyy ) + 2Lτyz2 +2M τzx2 +, +2N τxy2 = K 2

(1)

where σxx , σyy , σzz , τxy , τxz and τyz , are the components of the Cauchy stress tensor deﬁned in the orthotropic frame and F , G, H, L, M and N are the Hill coeﬃcients that deﬁne the anisotropy. Barlat, 1991 (YLD91) (Barlat et al., 1991) proposed an extension to orthotropy of the isotropic yield criterion proposed by (Hersey, 1954) and (Hosford, 1972). This criterion, usually referred as YLD91, can be expressed by the following non-quadratic function: m

m

m

|S1 − S2 | + |S1 − S3 | + |S2 − S3 |

= 2K m ,

(2)

where S1 , S2 and S3 are the principal values of the Isotropic Plasticity Equivalent (IPE) stress state, deﬁned by the stress tensor s = L : σ which is obtained from the linear transformation L applied to the Cauchy stress tensor σ. The new IPE stress state can be introduced directly in any of the proposed isotropic yield criteria, since the anisotropy is modelled by the linear transformation L. In case of YLD91 yield criterion, the isotropic yield criterion proposed by (Hosford, 1972) is used. The exponent m characterizes the shape of the yield surface and depends on the material. From (Logan et al., 1980), m is 6 and 8, for BCC and FCC materials, respectively. The material parameters to be identiﬁed are the components of the linear transformation L. Karaﬁllis & Boyce, 1993 (KB93) The yield criterion proposed by (Karaﬁllis et al., 1993) is an extension of the Yld91 yield criterion, using the same linear transformation L. This criterion

38

Advanced Methods in Material Forming

considers two diﬀerent isotropic yield functions, ϕ1 and ϕ2 , and a weighting factor c ∈ [0, 1]: 32k ϕ = (1 − c) ϕ1 + k−1 (3) = 2k 2k +1 2 ϕ1 is the isotropic yield function proposed by (Hosford et al., 1972), 2k

ϕ1 = (S1 − S2 )

2k

+ (S2 − S3 )

2k

+ (S3 − S1 )

,

(4)

where S1 , S2 and S3 are the principal values of the isotropic plasticity equivalent stress tensor and k is a material constant, and ϕ2 is the isotropic yield function given by: 2k 2k 2k (5) ϕ2 = |S1 | + |S2 | + |S3 | . The linear transformation can be expressed by: s = L : σ,

(6)

where s is the deviatoric stress tensor deﬁned in the “isotropic plasticity equivalent” stress space, L is the linear transformation tensor and σ is the Cauchy stress tensor. The linear transformation operator can be expressed as: ⎤ ⎡ −c2 /3 0 0 0 (c2 + c3 ) /3 −c3 /3 ⎢ −c3 /3 (c3 + c1 ) /3 −c1 /3 0 0 0⎥ ⎥ ⎢ ⎢ −c2 /3 −c /3 (c + c ) /3 0 0 0⎥ 1 1 2 ⎥, ⎢ (7) L=⎢ 0 0 0 c4 0 0 ⎥ ⎥ ⎢ ⎣ 0 0 0 0 c5 0 ⎦ 0 0 0 0 0 c6 where c1 , c2 , c3 , c4 , c5 and c6 are the parameters of anisotropy. The unknown parameters for this model are the parameters of the linear transformation and the weighting factor c. k can be a parameter to identify, but (Karaﬁllis et al., 1993), proposed for this parameter a high value. In this study k is considered equal to 15. Adaptation of Drucker for Anisotropy, DL 2001 This criterion, proposed by (Cazacu et al., 2001), is an extension of Drucker’s isotropic yield criterion (Drucker et al., 1949) to orthotropy using the previously given linear transformation L. The extension of Drucker’s yield criterion to orthotropy using the linear transformation approach is given by: 3 2 6 1 2 1 3 K tr s − c tr s = 27 , (8) 2 3 3 where tr (s) is the trace of the stress tensor s obtained after the linear transformation, and c is an additional material parameter. The parameters of the linear transformation and the weighting constant c are the parameters to be identiﬁed.

Phenomenological Constitutive Models Parameters Optimization

39

2.2 Work Hardening For simplicity, only the isotropic work-hardening will be taken into account in this study. For steels, the most commonly used law to describe the isotropic work-hardening is the power law proposed by (Swift et al., 1952), which is most suitable for describing the mechanical behaviour of all materials that do not present saturation behaviour, such as the majority of steels. The Swift law is expressed by: n σ = C (ε0 + εp ) , (9) where C, ε0 e n are material parameters, and εp is the equivalent plastic strain.

3 Material Parameter Identiﬁcation The proposed material parameter identiﬁcation procedure is basically a best ﬁt operation in order to minimize the comparative errors between numerical results, obtained from the constitutive equations (yield criterion and work hardening law), and the available experimental data. A possible way to determine the material’s yield criteria parameters is to adopt some kind of minimization strategy involving the corresponding equation [(1), (2), (3) or (8)] and its derivatives (to evaluate the r-values) for some well deﬁned strain paths. On the other hand, to identify the material’s work-hardening parameters, an analogous strategy can be followed involving (9). If only the isotropic work-hardening is taken into account, the optimization procedure can be carried out independently: ﬁrst the material parameters related to the anisotropic behaviour are identiﬁed and, then, the material parameter identiﬁcation of the isotropic work-hardening law is performed. The number and type of experimental results needed for the above-mentioned parameter identiﬁcation procedure is a function of the used constitutive model. In fact, if the number of experimental values is equal to the number of unknowns, the problem can be solved using a set of equations. However, if the number of experimental values is larger than the number of unknowns, the problem becomes an optimization problem, and several numerical algorithms can be adopted to solve it. In this study a population based (evolutionary-genetic) approach is adopted in order to solve the optimization problem, which is combined with a direct search method. The proposed scheme consists of a hybrid optimizer: a genetic algorithm (Holland, 1975), (Furukawa et al., 1995), (Pal et al., 1996), (Furukawa et al., 2002), (Feng et al., 2004) for the deﬁnition of the population based algorithm, and the Powell algorithm (Powel, 1964) as a direct search method. It must be noted that the best optimization procedure is strongly linked to the characteristics of both the problem and the population. The optimisation problem under study is strongly non-linear and presents several local minima. Even if it is possible to use gradient based algorithms, the

40

Advanced Methods in Material Forming

results depend on the user skills, and additional stochastic mechanisms are needed to lead with the local minima. The proposed algorithm is expensive in terms of CPU time, but it presents the advantage of being user-independent. The choice of a direct search method associated with the genetic algorithm is related to the sub-optimal solution associated with the genetic algorithms.

3.1 Yield Criteria Parameters Identiﬁcation A very eﬀective strategy for the identiﬁcation of yield criteria parameters is the deﬁnition of an error function (objective function), used in the optimization procedure. In this study, the error function is based in the summation of the squares of the errors deﬁned from a comparison between the results obtained from the constitutive equations for a given set of parameters to be identiﬁed and the corresponding experimental values. Such procedure can be considered a generalization of the optimization procedure proposed by (Banabic et al., 2002), (Banabic et al., 2005). The proposed error function also includes some weighting factors which can be diﬀerent for each kind of experimental test (and thus for each kind of strain path). The proposed expression is: 2 2 2 σα στ β σb − 1 + w − 1 + w − 1 2 3 σα exp σb exp στ β exp 2 2 rb rα + w4 − 1 + w − 1 , 5 rα exp rb exp (10)

erroranis =w1

where σα exp and rα exp are the experimental yield stress and r-values, respectively, obtained from the uniaxial tensile tests for a speciﬁc orientation (α) with RD, σb exp is the experimental yield stress obtained from the equibiaxial tensile test, στ β exp is the experimental yield stress under simple shear for a speciﬁc angle (β) with RD, rb exp is the experimental r-value obtained from the equibiaxial tensile tests (or from a compression test normal to the sheet plane) and σα , σb , στ β , rα and rb are the correspondent values but predicted from the constitutive equations. wi (i = 1, .., 5) are weighting factors.

3.2 Work Hardening Parameters Identiﬁcation The identiﬁcation of the work hardening parameters can be performed using several experimental strain paths. In this study, the optimization procedure is carried out using the equivalent values. Such “equivalent values” are computed from the experimental data and from the deﬁnition of the stress states associated with each experimental strain path, using the formulations of the chosen yield criterion, whose parameters must already be identiﬁed. The

Phenomenological Constitutive Models Parameters Optimization

41

stress states associated with each experimental strain path are converted into the equivalent values, which are used in the proposed quadratic error function: 2 nα (i) u σαj 1 errorhard = w1 exp − 1 n (i) j=1 σαj i=1 α 2 nb (k) σbl 1 − 1 nb (k) σ bl exp k=1 l=1 2 nτ (m) w στ βn 1 +w3 −1 , n (m) n=1 στ βn exp m=1 τ

+w2

v

(11)

where u, v and w are the number of experimental values obtained from the uniaxial tensile tests, equibiaxial tensile test and simple shear tests, respectively, nα (i) (for i = 1, .., u) is the number of experimental points for a speciﬁc experimental tensile test, nb (k) (for k = 1, .., v) is the number of experimental points for a speciﬁc experimental equibiaxial tensile test, nτ (m) (for m = 1, .., w) is the number of experimental points for a speciﬁc experimental exp simple shear test, σαj is the equivalent stress value for the point j obtained exp is the equivalent stress for the point l obtained in in the i tensile test, σbl exp is the equivalent stress for the point the k equibiaxial tensile test and στ βn n obtained in the m simple shear test. σαj , σbl and στ βn denote the values computed from the work hardening law for the correspondent experimental point.

4 Optimization Algorithm The material parameter identiﬁcation problem can be considered an inverse optimization problem. This study makes use of a hybrid algorithm for the identiﬁcation of constitutive model parameters. The choice of the hybrid algorithm is based on the properties of the optimization problem. The evolutionary algorithm presents the robustness needed for dealing with local minima, but the solution is a sub-optimal solution. As the genetic algorithm doesn’t involve the formulation of derivatives, a direct search method was chosen to perform the local optimization, which avoids also the computation of the derivatives of the functions involved in the formulation of the problem.

4.1 Evolutionary Algorithm – Genetic Algorithm The Genetic Algorithm (GA) is a selective random search algorithm to achieve a global optimum within a large space of solutions proposed by Holland (Holland, 1975). The genetic algorithm is a population based scheme algorithm

42

Advanced Methods in Material Forming variable a solution (individual)

solutions (population)

1 1 0 0 1

1 0 0 1 0

0 1 0 1 0

0 0 1 1 0

0 0 0 0 1

1 1 0 1 1

1 0 1 0 0

1 0 1 1 1

0 0 0 1 0

0 0 0 1 1

0

1

0

1

1

1

1

0

1

0

1

1

1

position (locus)

(…) 1

0

number of genes

0

1

0

1

0

gene (gene) chromosome (chromosome)

Fig. 1. Biological – genetic algorithm analogies

analogous to natural selection in biological systems. Figure 1 highlights the genetic – biological systems notation analogies. It is possible to see, in Fig. 1, a set of solutions that in biological systems can be considered analogous with the population. Each solution, which in the biological systems is an individual, has a set of genes (genes in biological systems). The position of a given gene is identiﬁed with the locus in genetic and in biological systems. In the numerical application of a genetic algorithm, each solution has a number of genes needed to obtain the solution characteristics, as in biological systems each individual has a number of ﬁxed genes that determine their biological characteristics. From the set of genes of a solution it is possible to calculate a value for the objective function. This value is known as ‘ﬁtness function’ and measures the robustness of the solution. The number of genes needed to keep the solution characteristics is related, in our optimization problem, with the number of variables of the models and the accuracy for each variable, with var maxi −var mini nbitsi = ints logc +1 , (12) pri where nbitsi is the number of bits needed, pri the precision required for the variable i, ints () the superior integer of the argument, c the cardinality of the codiﬁcation, var maxi and var mini the minimum and maximum values for variable i, respectively. For example, if a three variables function is the ﬁtness function (as in case of the Swift model), a solution is obtained from the three variables genes. On the other hand, if for variable a the required accuracy is 0.01, with a ∈ [0, 1], using binary codiﬁcation the expression (12) gives 7 as the minimum number of bits needed to obtain the required accuracy. The used evolutionary algorithm scheme is presented in Fig. 2. For starting the iterative process of the genetic algorithm an initial population is needed. The initial population is usually randomly generated, but it is possible to insert a ‘good’ initial population, if known, in an operation known as a seeding process.

Phenomenological Constitutive Models Parameters Optimization Evolutionary Algorithm

Initial Population Population random generator

Seeding

Cycle Optimization Cycle

Decoding

Evaluation of the fitness function

Selection

Crossover

Mutation

N

Elitism S Test Stop Condition

Fig. 2. Genetic Algorithm

43

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Advanced Methods in Material Forming

In the iterative process, the population is coded using an alphabetic code, and so a decoding procedure is necessary in order to identify its value. In the present case, the value of each variable is not coded, and as alternative the index array of each variable is used. The variable arrays are created before the iterative procedure and have all possible values of each variable. This is done because computationally it is very eﬃcient to decode integer variables. Then, for all the intermediate solutions, the ﬁtness of each solution is calculated using [10] and [11], for the optimization of the yield criterion and work hardening parameters, respectively. The minimization-based technique is applied to the current population in order to select the parents for the deﬁnition of the subsequent population (and next iteration). In this study the selection process is stochastic, the members being chosen for the matting pool using a ﬁtness based roulette. The next generation is obtained combining the solutions in the mating pool. The one point crossover scheme is used. In order to assure genetic diversity, a mutation operator is used. The mutation operator consists of modifying a gene value. This is set to a low probability, less than 5%, in order not to perturb the convergence. It is possible to maintain some members of the population, in order to increase the inﬂuence of the best solution in the overall population, what is called as elitism strategy. If an elitism strategy is used, the number of solutions needed in the mating pool is the population less the number of solutions that go directly to the next generation. The algorithm presented in Fig. 2 is repeated until the end condition is reached. The inﬂuence of each mechanism presented in the convergence and quality of the solution is object of many theoretical and empirical studies (Grefenstette, 1986), (De Jong, 1987), (Goldberg, 1989), (David, 1991), (Koza, 1992), (Back, 1996) beyond the scope of this study. Many of the above studies point out that genetic algorithms works well for a large range of the principal probability parameters. 4.2 Hybrid Algorithm The GA is a sub-optimal solution technique; to ﬁnd the optimal solution an additional search algorithm is needed. The local minimum search algorithm used is based on the Powel algorithm (Powel, 1964). This algorithm basically applies perturbations to the genetic algorithm’s best solutions, using the variable GA arrays, Fig. 3. The best solution of the GA works as starting point for this local minimum optimizer. The variable perturbation is done at a random variable and in a random direction, if the ﬁtness function decreases. The perturbation is done until the ﬁtness function increases. If for the initial random direction the ﬁtness function doesn’t decrease, other direction is tried. This is done until the perturbation doesn’t involve lower ﬁtness functions for all variables.

Phenomenological Constitutive Models Parameters Optimization Hybrid Algorithm

Evolutionary Algorithm

GA

Variable random selection

Sign random selection

Variable perturbation Evaluation of the fitness function FT is better

N

S Change of actualization sign

Variable actualization Evaluation of the fitness function FT is better

N

S

N

FF function unchanged

S

Fig. 3. Hybrid optimization algorithm

45

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Advanced Methods in Material Forming

5 DC06 Steel Parameters Identiﬁcation In order to test the proposed methodology, the material parameter identiﬁcation for 1 mm thick DC06 steel is presented. This steel was characterized in the framework of 3DS European Project. For this identiﬁcation typical values were used for the probably of crossover and mutation, 75% and 5% respectively. For the yield criterion parameter identiﬁcation, the population has 100 solutions and 500 iterations were performed. For the work hardening law parameter identiﬁcation the population has 70 solutions and 120 iterations were made. When experimental data was not available for the identiﬁcation of some speciﬁc parameters, the isotropic values were assumed. 5.1 Yield Parameter Identiﬁcation The yield criterion parameter identiﬁcation was carried out for the constitutive models described in Sect. 2. In order to understand the models’ behaviour, three conditions were used to identify the anisotropy parameters: using only the r-values (Condition A); using the r-values and the initial yield stress (Condition B); and using the r-values and the average of the σα /σ0 values during deformation (0-34% true plastic strain) (Condition C). Table 1 shows the values used for the anisotropic model parameter identiﬁcation. The initial yield stress for the RD direction is 123.6 MPa. Tables 2, 3 and 4 present the parameters found for the three identiﬁcations Conditions, A, B and C, respectively. Figure 4 presents the variation of the r-values in the plane of the sheet. The ﬁgure shows the experimental data points (EXP) and the results obtained with the Hill yield criterion with the parameters identiﬁed using the classical relationship between the Hill parameters and the r0 , r45 and r90 values (HILL48 r0 r45 r90). HILL48 3DS corresponds to the Hill yield criterion results obtained with the parameters delivered in the framework of 3DS European Project, using a gradient based optimisation algorithm (Bouvier et al., 2001). The HILL48, YLD91, KB93 and DL2001 criteria correspond to the results obtained with the parameters identiﬁed with the test procedure of Condition A. Table 1. Experimental data for the DC06 steel (1mm thickness) angle with RD [o ]

r-value

initial yield stress σα /σ0

average of σα /σ0

0 15 30 45 60 75 90

2.530 2.055 1.875 1.840 2.220 2.620 2.720

1.000 1.006 1.005 1.009 1.011 1.027 1.026

1.000 1.016 1.023 1.027 1.017 1.015 0.990

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47

Table 2. Identiﬁed material parameters for the optimization tests Condition A HILL48 YLD91 KB93 DL2001

F 0.249 c1 0.818 c1 0.801 c1 0.772

G 0.301 c2 0.858 c2 0.841 c2 0.824

N 1.286 c3 1.127 c3 1.155 c3 1.201

c6 0.956 c6 0.965 c6 1.077

m 8 c 0.835 c 0.837

K 15

Table 3. Identiﬁed material parameters for the optimization tests Condition B HILL48 YLD91 KB93 DL2001

F 0.256 c1 0.846 c1 0.825 c1 0.777

G 0.305 c2 0.888 c2 0.866 c2 0.829

N 1.358 c3 1.163 c3 1.187 c3 1.209

c6 0.990 c6 0.995 c6 1.084

m 8 c 0.835 c 0.835

K 15

From Fig. 4 is possible to conclude that none of the presented yield criteria parameters could perfectly describe the in plane variation of the r-values. The YLD91 and the KB93 are almost coincident. The curve obtained from the classical relationship between the Hill parameters and the r-values (HILL48 r0 r45 r90) is the worst solution. Figure 5 shows the experimental and numerical yield stress points for the identiﬁcation procedure of Condition A. The results obtained with the DL2001 yield criterion are quite close to the experimental data. On the contrary, all other yield criteria behave badly in the yield stress prediction. The Condition A procedure corresponds to the ‘usual’ set of data used to perform yield parameter identiﬁcation. This procedure is suitable for identifying parameters that will be used in simulations where the accuracy of the ‘strain’ distribution is a major factor as in the prediction of the draw-in, blank dimensions or thickness reduction. However, when the accuracy Table 4. Identiﬁed material yield parameters for the optimization tests Condition C HILL48 YLD91 KB93 DL2001

F 0.256 c1 0.839 c1 0.814 c1 0.771

G 0.304 c2 0.879 c2 0.854 c2 0.822

N 1.343 c3 1.153 c3 1.173 c3 1.201

c6 0.980 c6 0.981 c6 1.073

m 8 c 0.835 c 0.837

K 15

48

Advanced Methods in Material Forming 3.00 2.75 EXP HILL48 YLD91 KB93 DL2001 HILL48 r0 r45 r90 HILL48 3DS

r value

2.50 2.25 2.00 1.75 1.50 0

15

30

45

60

75

90

Angle with RD [º]

Fig. 4. Distribution of r-ratios in the sheet plane for the optimization tests of condition A

of the ﬁnal stress state is mandatory, as for example in springback or wrinkle prediction, the Conditions B or C procedure is preferable once the yield stress experimental data for parameter identiﬁcation is taken into account. The variation of the r-values and the yield stress in the plane of the sheet, as obtained with the sets of parameters delivered with the optimisation procedures Conditions B and C, is very close to the results presented in Figs. 4

1.20

Equiv. Stress [Ya/Y0]

1.15

EXP HILL48 YLD91 KB93 DL2001 HILL48 r0 r45 r90 HILL48 3DS

1.10

1.05

1.00

0.95

0

15

30 60 45 Angle with RD [º]

75

90

Fig. 5. Distribution of yield stress in the sheet plane for the optimization tests of condition A

Phenomenological Constitutive Models Parameters Optimization

49

Table 5. Average errors for each experimental point obtained for all yield parameter identiﬁcation conditions

HILL48 YLD91 KB93 DL2001 HILL48 3DS HILL48 r0 r45 r90

Condition A

Condition B

Condition C

0.189 0.191 0.199 0.188 0.199 0.396

0.304 0.146 0.154 0.108 0.346 0.461

0.236 0.124 0.130 0.123 0.262 0.370

and 5. To emphasize the diﬀerences between the three procedures of optimization, it is preferable to analyse the average error (10) obtained with the various sets of parameters. Table 5 presents the average error obtained with the optimization Conditions A, B and C for all the models. The table shows also the average error obtained with the set of parameters identiﬁed with the classical relationship between the Hill parameters and the r-values (HILL48 r0 r45 r90) and with the parameters delivered in the framework of 3DS European Project (HILL48 3DS). For all identiﬁcation procedures the larger error is less than 0.5% and is obtained with the parameters of the set HILL48 r0 r45 r90. The yield criterion that presents the lowest error is the Drucker corrected to orthotropy (Cazacu et al., 2001). The code DD3MAT delivers a Hill48 set of parameters close to the Hill48 3DS result. It is clear from the above data that the proposed hybrid optimization algorithm behaves quite well in the identiﬁcation of yield criteria parameters. It also, by considering in the optimisation procedure both r-ratio and yield stress experimental data, improves the description of the

Equiv. Stress [MPa]

500 400 300 200 100 0 0.0

Exp 0° Exp 30° Exp 60° Exp 90°

0.1

0.2 Equiv. Strain

0.3

0.4

Exp 15° Exp 45° Exp 75° DD3MAT Hill Swift

Fig. 6. Equivalent stress-strain curves obtained with Hill48 yield criterion, identiﬁed with Condition A, for tensile and shear test at 0, 15, 30, 45, 60, 75 and 90◦ with RD direction

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Advanced Methods in Material Forming Equiv. Stress [MPa]

500 400 300 200 100 0 0.0

0.1

0.2 Equiv. Strain

Exp 0° Exp 30° Exp 60° Exp 90°

0.3

0.4

Exp 15° Exp 45° Exp 75° DD3MAT YLD91 Swift

Fig. 7. Equivalent stress-strain curves obtained with YLD91 yield criterion, identiﬁed with Condition A, for tensile and shear test at 0, 15, 30, 45, 60, 75 and 90◦ with RD direction

material behaviour: the yield stress predictions are clearly improved with low inﬂuence on the r-ratio. 5.2 Work Hardening Parameter Identiﬁcation The work hardening material parameter identiﬁcation made use of all the experimental data available, i.e. seven tensile tests and seven shear tests oriented at 0, 15, 30, 45, 60, 75 and 90◦ with the RD direction. Figures 6, 7, 8 and 9 shows the equivalent stress/strain curves for all the seven directions with RD,

Equiv. Stress [MPa]

500 400 300 200 100 0 0.0

0.1

0.2

0.3

0.4

Equiv. Strain Exp 0° Exp 30° Exp 60° Exp 90°

Exp 15° Exp 45° Exp 75° DD3MAT KB93 Swift

Fig. 8. Equivalent stress-strain curves obtained with KB93 yield criterion, identiﬁed with Condition A, for tensile and shear test at 0, 15, 30, 45, 60, 75 and 90◦ with RD direction

Phenomenological Constitutive Models Parameters Optimization

51

Equiv. Stress [MPa]

500 400 300 200 100 0 0.0

0.1

0.2 Equiv. Strain

Exp 0° Exp 30° Exp 60° Exp 90°

0.3

0.4

Exp 15° Exp 45° Exp 75° DD3MAT DL Swift

Fig. 9. Equivalent stress-strain curves obtained with DL2001 yield criterion, identiﬁed with Condition A, for tensile and shear test at 0, 15, 30, 45, 60, 75 and 90◦ with RD direction

obtained with the HILL48, YLD91, KB93 and DL2001 criteria, respectively. The yield criteria parameters used in this case were identiﬁed with Condition A procedure. The identiﬁed parameters are shown in Tables 6, 7 and 8. In general, the work hardening material parameter identiﬁcation could be performed using just the stress-strain curve obtained in tension for 0◦ with the RD direction, if the anisotropy can be well described with the yield criterion. In the present case, DC06 material, this is not true as can be seen in Table 6. Hardening parameters for the Swift model obtained for all the yield stress models identiﬁed with Condition A model

Y0 [MPa]

C [MPa]

n

error [%]

HILL48 YLD91 KB93 DL2001

123.6 123.6 123.6 123.6

525.6 546.1 551.0 552.1

0.308 0.304 0.303 0.299

0.271 0.163 0.183 0.111

Table 7. Hardening parameters for the Voce model obtained for all the yield stress models identiﬁed with Condition B model

Y0 [MPa]

C [MPa]

n

error [%]

HILL48 YLD91 KB93 DL2001

123.6 123.6 123.6 123.6

530.2 562.5 565.9 555.7

0.306 0.300 0.301 0.299

0.195 0.147 0.169 0.109

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Advanced Methods in Material Forming

Table 8. Hardening parameters for the Voce model obtained for all the yield stress models identiﬁed with Condition C model

Y0 [MPa]

C [MPa]

n

error [%]

HILL48 YLD91 KB93 DL2001

123.6 123.6 123.6 123.6

529.1 558.2 559.2 551.6

0.306 0.301 0.302 0.300

0.209 0.152 0.176 0.109

Figs. 4 to 9. The identiﬁcation can also be done using an isotropic criterion, such as the Von Mises criterion, but this procedure does not consider the anisotropy in stress. In order to overcome these diﬃculties, it is preferable to perform the material parameter identiﬁcation for the work hardening using all the available experimental data and considering the anisotropic criterion identiﬁcation that will be used later in the numerical simulation of the sheet metal forming process.

6 Conclusions This study presents and discusses the material parameter identiﬁcation procedures for the yield criterion and the work hardening law. An automatic tool for parameter identiﬁcation has been developed based on a Hybrid algorithm. The proposed tool was applied to identify the material parameters of deepdrawing DC06 steel. The material parameter identiﬁcation was performed for several yield criteria and using several identiﬁcation conditions. The plastic behaviour description presents lower error if a global optimization is performed instead of the traditional r-value based procedure. The best solution for the work hardening identiﬁcation considers all the available experimental data and the chosen anisotropic criterion identiﬁcation. The proposed identiﬁcation technique, easy to be used, presents eﬃcient and robust performance.

Acknowledgements The author’s are indebted to the Portuguese Foundation for Science and Technology – FCT for ﬁnancial support through the POCTI program.

References Barlat F., Lege D.J., Brem J.C., “A six-component yield function for anisotropic materials”, International Journal of Plasticity, Vol. 7, 1991, p. 693–712. Bron F., Besson J.A., “Yield function for anisotropic materials: Aplication to aluminium alloys”, International Journal of Plasticity, Vol. 20, 2004, p. 937–963.

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Banabic D., Cazacu O., Barlat F., Comsa D.S., Wagner S., Siegert K., “Description of anisotropic behaviour of AA3103-O aluminium alloy using two recent yield criteria”, Proceedings of 6th European Mechanics of Materials Conference, Ed. S. Cescotto, 2002, p. 265–272. Banabic D., Arets H., Comsa D.S., Paraianu L., “An improved analytical description of orthotropy in metallic sheets”, International Journal of Plasticity, Vol. 21, 2005, p. 493–512. Back T., Evolutionary algorithms in theory and in practice: evolutionary strategies evolutionary programming, genetic algorithms, Oxford University Press, 1996. Bouvier S., Teodosiu C., ”Selection and identiﬁcation of elastoplastic models for the materials used in the benchmarks”, Project Digital Die Design (3DS) Progress Report, Laboratoire des Propri´et´es M´ecaniques et Thermodynamiques des Mat´eriaux (LPMTM) – CNRS, University of Paris Nord, France, 2001. Cazacu O., Barlat F., “Generalization of Drucker’s yield criterion to orthotropy”, Mathematics and Physics of Solids, Vol. 6, 2001, p. 613–630. Cazacu O., Barlat F., “A criterion for description of anisotropy yield diﬀerential eﬀects in pressure-insensitive metals”, International Journal of Plasticity, Vol. 20, 2004, p. 2047–2045. Chaparro B.M., Oliveira M.C., Alves J.L., Menezes L.F., “Work hardening models and the numerical simulation of the deep drawing process”, Materials Science Forum, Vols. 455–456, 2004, p. 717–722. Drucker D.C., “Relation of experiments to mathematical theories of plasticity”, Journal of Applied Mechanics, Vol. 16, 1949, p. 349–357. De Jong K. A., “On using genetic algorithm to search program spaces”, Proceeding of 2nd International Conference on Genetic algorithm and Their Applications, Hillsdale, Ed. Lawrence Erlbaum, 1987, p. 210–216. Davis L. D., The handbook of genetic algorithms, Van Nostrand Reinhold, 1991. Feng, X.T., Yang, C., “Coupling recognition of the structure and parameters of non-linear constitutive material models using hybrid evolutionary algorithms”, International Journal for Numerical Methods in Engineering, Vol. 59, 2004, p. 1227–1250. Furukawa T., Yagawa G., “Parameter identiﬁcation of inelastic constitutive equations using an evolutionary algorithm with constitutive individuals”, Proceedings ASME/JSME PVPC Conference, Vol. 305, 1995, p. 437–444. Furukawa T., Sugata T., Yoshimura S., Hoﬀman M., “An automated system for simulation and parameter identiﬁcation of inelastic constitutive models”, Computer Methods in Applied Mechanics and Engineering, Vol. 191, 2002, p. 2235–2260. Grefenstette J.J., “Optimization of control parameters for genetic algorithms”, IEEE Transactions, Vol. SMC-16, 1986, p. 122–128. Goldberg D. E., Genetic algorithms in search, optimization and machine learning, Addison-Wesley, 1989. Hill R., “A theory of the yielding and plastic ﬂow of anisotropic metals”, Proceedings of Mathematical, Physical and Engineering Science, Royal Society London, Vol. A193, 1948, p. 281–297. Hill R., The mathematical theory of plasticity, Oxford, Clarendon Press, 1950. Hersey A.V., “The plasticity of an isotropic aggregate face centered cubic crystals”, Journal of Applied Mechanical Transactions ASME, Vol. 21, 1954, p. 241–249. Hosford W.F., “A generalized isotropic yield criterion”, Journal of Applied Mechanics, Vol. 39, 1972, p. 607–609.

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Holland J.H., Adaptation in neural and artiﬁcial systems, Ann Arbor, University of Michigan Press, 1975. Karaﬁllis A.P., Boyce M.C., “A general anisotropic yield criterion using bounds and a transformation weighting tensor”, Journal of the Mechanics of Physics of Solids, Vol. 41, 1993, p. 1859–1886. Koza J. R., Genetic programming: on the programming of computers by means of natural selection, MIT press, 1992. Logan R.W., Hosford W.F., “Upper-bound anisotropic yield locus calculations assuming (111)-pencil glide”, International Journal of Mechanical Sciences, Vol. 22, 1980, p. 419–430. Menezes L.F., Teodosiu C., “Thee-dimensional numerical simulation of deepdrawing process using solid ﬁnite elements”, Journal of Material Processing Technology, Vol. 97, 2000, p. 100–106. Mises R., “Mechanik der festen korper im plastic-deformablen zustand“, Nachrichten vos der koniglichen gellesschaft des winssenschaften zu Gottingen, Mathematisch-physikalische klasse, 1913, p. 582–592. Pal S., Wathugala G.W., Kundu S., “Calibration of a constitutive model using genetic algorithms”, Computer and Geotechnics, Vol. 19, 1996, p. 325–348. Powel M.J.D., “An eﬃcient method for ﬁnding the minimum of a function of several variables without calculating derivatives”, Computational Journal, Vol. 7, 1964, p. 155–162. Swift H.W., “Plastic instability under plane stress”, Journal of the Mechanics of Physics of Solids, Vol. 1, 1952, p. 1–18. Thuiller S., Manach P.Y., Menezes L.F., Oliveira M.C., “Experimental and numerical study of reverse re-drawing of anisotropic sheet metals”, Journal of Materials Processing Technology, Vol. 125–126, 2002, p. 764–771.

A Metamodel Based Optimisation Algorithm for Metal Forming Processes M.H.A. Bonte, A.H. van den Boogaard and J. Hu´etink University of Twente, Faculty of Engineering Technology, P.O. Box 217, 7500 AE Enschede, The Netherlands, [email protected]; [email protected]; [email protected] Summary. Cost saving and product improvement have always been important goals in the metal forming industry. To achieve these goals, metal forming processes need to be optimised. During the last decades, simulation software based on the Finite Element Method (FEM) has signiﬁcantly contributed to designing feasible processes more easily. More recently, the possibility of coupling FEM to mathematical optimisation algorithms is oﬀering a very promising opportunity to design optimal metal forming processes instead of only feasible ones. However, which optimisation algorithm to use is still not clear. In this paper, an optimisation algorithm based on metamodelling techniques is proposed for optimising metal forming processes. The algorithm incorporates nonlinear FEM simulations which can be very time consuming to execute. As an illustration of its capabilities, the proposed algorithm is applied to optimise the internal pressure and axial feeding load paths of a hydroforming process. The product formed by the optimised process outperforms products produced by other, arbitrarily selected load paths. These results indicate the high potential of the proposed algorithm for optimising metal forming processes using time consuming FEM simulations.

Key words: optimisation, metal forming, ﬁnite element method, metamodelling, hydroforming.

1 Introduction During the last decades, Finite Element (FEM) simulations of metal forming processes have become important tools for designing feasible production processes. In more recent years, several authors recognised the potential of coupling FEM simulations to mathematical optimisation algorithms to design optimal metal forming processes instead of only feasible ones. The basic concept of mathematical optimisation is presented in Fig. 1. Basically, it consists of two major phases: the modelling and the solving of the optimisation problem. The modelling phase consists of:

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1. Selecting a number of design variables the user is allowed to adapt 2. Choosing an objective function, i.e. the optimisation aim 3. Taking into account possible constraints These three items are closely related to each other as depicted in Fig. 1. Both the objective function and the constraints should be quantiﬁed by the design variables. The objective function and constraints are also related to each other in the sense that they are often exchangeable. Consider for example that we would like to make a metal formed product and two relevant properties are the product quality and the costs. Then two approaches can be followed: either the quality is maximised while putting a certain limit on the allowed production costs, or the costs could be minimised while ensuring a certain minimum level of the product quality. In the former case, the quality is clearly the optimisation objective and the costs are constraints, whereas it is just the other way around in the latter case. Next to the modelling phase, mathematical optimisation’s second phase is solving the optimisation problem. This comprises applying an optimisation algorithm to the modelled optimisation problem. The arrows between the modelling and the solving parts in Fig. 1 denote that both phases cannot be seen separately from each other. One should select the right optimisation algorithm for a certain modelled optimisation problem and one should model the optimisation problem cleverly to adjust it to the optimisation algorithm one is planning to apply. If the optimisation model does not match the algorithm, it is likely that the optimisation problem is not solved eﬃciently or cannot be solved at all (Papalambros et al., 2000). This paper focuses on the solving part of optimisation problems in metal forming using time consuming nonlinear FEM simulations. One simulation can easily take hours or even days to execute. It is important to keep this fact in mind when selecting a suitable optimisation algorithm for metal forming processes. One way of optimising metal forming processes is using classical iterative optimisation algorithms (Conjugate gradient, BFGS, etc.), where each function evaluation means running a FEM calculation, see e.g. Kleinermann et al., (2003), Lin et al., (2003) and Naceur et al., (2001). As mentioned above, in case of metal forming these FEM calculations can be extremely time consuming and need to be sequentially evaluated. Furthermore, many classical algorithms require sensitivities, of which the eﬃcient calculation is not straightforward

Fig. 1. The basic concept of mathematical optimisation: modelling and solving

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for FEM simulations. A third diﬃculty concerning iterative algorithms is the risk to be trapped in local optima. Alternatively, several authors have tried to overcome these disadvantages by applying genetic or evolutionary optimisation algorithms, see e.g. Castro et al., (2004), Fourment et al., (2005) and Schenk et al., (2004). Genetic and evolutionary algorithms look promising because of their tendency to ﬁnd the global optimum and the possibility for parallel computing. However, the rather large number of function evaluations that is expected to be necessary using these algorithms is regarded as a serious drawback (Emmerich et al., 2002). Yet another way of optimisation in combination with expensive function evaluations is using approximate optimisation algorithms, of which Response Surface Methodology (RSM) is a well-known representative. RSM is based on ﬁtting a low order polynomial metamodel through response points, which are obtained by running FEM calculations for carefully chosen design variable settings and ﬁnally optimising this metamodel (Myers and Montgomery, 2002). Metamodels are sometimes also referred to as Response Surface models or surrogate models. Allowing for parallel computing and lacking the necessity for sensitivities, RSM is appealing to many authors in the ﬁeld of metal forming, see e.g. Jansson (2002), Jansson et al., (2005) Naceur et al., (2004). Although the practical eﬀectiveness of RSM has been frequently demonstrated, statisticians claim that RSM, being developed for stochastic physical experiments, is theoretically not applicable to deterministic computer experiments such as FEM: running a simulation twice with exactly the same input will generally result in exactly the same answer. They propose the ﬁeld of “Design and Analysis of Computer Experiments” or DACE instead (Sachs et al., 1989a, Sachs et al., 1989b, Santner et al., 2003). DACE is similar to RSM, but interpolates a metamodel through the response points. Allowing for no error at the response points, interpolation better suits the deterministic nature of computer experiments. However, DACE is rarely used for metal forming problems, probably due to its complex statistical nature and the lack of readily available software (Santner et al., 2003). In this paper an optimisation algorithm incorporating both RSM and DACE metamodelling techniques is proposed for metal forming. Section 2 introduces the basic concept of metamodelling and provides a more detailed description of RSM and DACE. The proposed optimisation algorithm is presented in Sect. 3 and the applicability to metal forming is demonstrated in Sect. 4 where it is applied to the optimisation of a hydroforming process. Conclusions are presented in Sect. 5.

2 Metamodelling The principle of metamodelling is presented in Fig. 2 (Kleijnen and Sargent, 2000). The basic idea is to evaluate a certain problem entity, in our case a metal forming process. This problem entity can be modelled by some sort of a simulation model. For metal forming, this simulation model is usually

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Fig. 2. The principle of metamodelling

a nonlinear Finite Element Model (FEM). These nonlinear FEM calculations are very time consuming to evaluate. Therefore, a metamodel or a model from a model (Simpson et al., 2001) is made, which can be quickly evaluated. An accurate metamodel should be valid with respect to both the Finite Element Model and the metal forming process and if it is, it forms a very useful substitute for both the process and the FE model. Kleijnen and Sargent (2000) distinguish four goals that can be served by metamodelling: (i) Understanding the problem entity, (ii) Predicting values of the output or response variable, (iii) Optimisation and (iv) Veriﬁcation and Validation of prior qualitative knowledge on the simulation model with respect to the problem entity. For the optimisation of metal forming processes, the primary interest lies in optimisation as a metamodelling goal. However, the other goals additionally come at no or low computational costs, which is seen as a major advantage of using metamodelling techniques for optimisation purposes. In the next sections, two metamodelling techniques, Response Surface Methodology and Design and Analysis of Computer Experiments or Kriging, are shortly introduced. Prior to ﬁtting the metamodels, a Design Of Experiments (DOE) strategy carefully selects a number of design variable settings for which FEM simulations are being run. These simulations provide a number of response measurements. 2.1 Response Surface Methodology (RSM) Starting with RSM, the response measurements y are presented as the sum of a lower order polynomial metamodel and a random error term ε (Myers and Montgomery, 2002): y = Xβ + ε (1) where X is the design matrix containing the experimental design points and β are the regression coeﬃcients obtained by least squares regression: ˆ = (XT X)−1 XT y β

(2)

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Although (1) seems to be a linear relation of the design variables, the design matrix X can also incorporate terms that are nonlinear with respect to the design variables. Equation 1 should, however, be linear with respect to the regression coeﬃcients (Myers and Montgomery, 2002), which is clearly the case. The metamodel is ˆ y ˆ = Xβ (3) Four possible shapes are commonly applied. They are in ascending complexity: • • • •

linear linear + interaction pure quadratic or elliptic (full) quadratic

A second order RSM metamodel dependent on one design variable is shown in Figure 3(a). The cross marks represent the response measurements. 2.2 Design and Analysis of Computer Experiments (DACE) DACE was proposed by Sacks et al., (1989a), and (1989b) to ﬁt metamodels using deterministic computer experiments. Kriging is used to interpolate between the response measurements. Using Kriging, the random error term ε in (1) is replaced by a Gaussian random function Z(x), which forces the metamodel to exactly go through the measurement points: y ˆ = Xβ + Z(x)

(4)

The ﬁrst part of (4) covers the global trend of the metamodel. The Gaussian random function Z, which accounts for the local deviation of the data from the trend function, has zero mean, variance σ2z and covariance cov (Z(x1 ), Z(x2 )) = σ2z R (x1 − x2 )

(5)

where R is the correlation function and x1 and x2 are two locations, which are determined by the design variable settings at these locations. For the proposed algorithm, a Gaussian exponential correlation function is adopted: R(ϑ, x1 , x2 ) = exp−ϑ(x1 −x2 )

2

Fig. 3. Metamodels based on (a) RSM; (b) DACE

(6)

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As opposed to other possibilities for the correlation function like e.g. cubic splines and ordinary exponential functions, see e.g. Koehler and Owen (1996), Lophaven et al., (2002a), (2002b), and Santner et al., (2003), Gaussian exponential functions are intuitively attractive because they are inﬁnitely diﬀerentiable. Moreover, Gaussian exponential functions are frequently used in literature (Santner et al., 2003) and have been found to give accurate results (Lophaven et al., 2002a). Assume k design variables are present. Then the total correlation function R depends on the k one-dimensional correlation functions Rj as follows (Sacks et al., 1989a): k R (x1 − x2 ) = Rj (x1j − x2j ) (7) j=1

This implies that it is assumed that there is no relation between the diﬀerent dimensions. Adopting the Gaussian correlation function introduced in (6), the total correlation function becomes: R (x1 − x2 ) =

k

exp−ϑj (x1j −x2j )

2

(8)

j=1

Thus, one ϑ is present for each design variable (each dimension). Figure 3(b) presents a Kriging interpolation metamodel, which is ﬁtted through the same response measurements as the RSM metamodel in Fig. 3(a). Note the diﬀerences: the Kriging metamodel’s shape is more complex than the second order polynomial shape of the RSM metamodel. However, the Kriging metamodel interpolates through the response points, whereas the RSM metamodel allows for random error. As was stated in the introduction, it is argued whether allowing for random error is appropriate in case of deterministic computer experiments such as FEM.

3 A Metamodel Based Optimisation Algorithm for Metal Forming The proposed metamodel based optimisation algorithm for the optimisation of metal forming processes using time consuming FEM simulations is presented in Fig. 4. Several steps mentioned in the ﬁgure are explained in the Sects. 1 through 4. Section 5 contains a few words on the implementation of the algorithm. 3.1 Modelling The ﬁrst step is to start with modelling the optimisation problem, i.e. quantifying objective function and constraints and selecting the design variables.

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Fig. 4. A metamodel based optimisation algorithm for metal forming processes

Regarding the constraints, a distinction is made between explicit and implicit constraints. To explain the diﬀerence, running a FEM simulation can be seen as an input-throughput-response model such as the one depicted in Fig. 5. Certain quantities are known beforehand: there is no necessity to run a FEM calculation for evaluating them. The design variables are clear examples of these quantities and there can also be constraints that explicitly depend on the design variables. These constraints are called explicit constraints. In case of metal forming explicit constraints are related to the undeformed product, e.g. constraints on the initial shape of a blank. Quantities that depend on the response require a FEM simulation for evaluating them: they implicitly depend on the design variables. The objective function is generally such an implicit quantity and it is also possible to have implicit constraints. For metal forming, implicit constraints are related to the deformed product, e.g. excessive thinning is not allowed to exceed a speciﬁed limit. It is stressed again that the modelling of an optimisation problem is formally not part of an optimisation algorithm: the algorithm is solely a mean for

Fig. 5. FEM as an input–throughput–response model

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solving the optimisation model. Clever modelling and solving are both crucial for mathematically optimising a problem as was already emphasised in the introduction. 3.2 Design Of Experiments (DOE) When the optimisation problem has been modelled, Fig. 4 shows that the ﬁrst step of the algorithm is to carefully select a number of design sites by a Design Of Experiments (DOE) strategy. A spaceﬁlling Latin Hypercube Design (LHD) is a good and popular DOE strategy for constructing metamodels from deterministic computer experiments (McKay et al., 1979 and Santner et al., 2003) and has been selected for the optimisation of metal forming processes. However, when a metamodel is used for optimisation, it is important that the metamodel gives accurate results in the neighbourhood of the optimum. Often, this optimum will be constrained, i.e. lies on the boundary of the design space. Therefore, an accurate prediction is needed on the boundary, which implies performing measurements on that boundary. An LHD will generally provide design points in the interior of the design space and not on the boundary. To compensate for this lack of points on the boundary, the LHD is combined with a full factorial design, which puts DOE points right in the corners of the design space. This method was also proposed by van Beers and Kleijnen (2004) and Kleijnen and van Beers (2004). Figure 6(a) presents the LHD modiﬁed with a full factorial design for a two dimensional rectangular design space. Unfortunately, the design space will often not be rectangular when explicit constraints are present. In this case, the proposed algorithm will: 1. 2. 3. 4.

check which points of the LHD + full factorial design are non-feasible skip the non-feasible points replace the non-feasible points with new points repeat the above procedure until all points are feasible

Fig. 6. (a) LHD + full factorial design; (b) LHD + full factorial design including explicit constraints

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Replacing the non-feasible points is also done in a spaceﬁlling way by selecting a large number of sets of additional design points. The new set of points is the one for which the minimum point to point distance is maximised. This socalled maximin criterion is used for both the initial DOE and for the case when the user wants to generate additional experimental design points, for example for improving the accuracy of the metamodels. The ﬁnal DOE strategy incorporated in the proposed optimisation algorithm is presented in Fig. 6(b) for two design variables (x1 and x2 ) and two explicit constraints (g1 and g2 ). 3.3 Running the FEM Simulations and Fitting the Metamodels Subsequently, using the settings indicated by the DOE strategy, a number of FEM calculations is run on parallel processors and the response points (objective function and implicit constraint values) are obtained. Following Fig. 4, the next step is to ﬁt for each response seven metamodels: 1. 2. 3. 4. 5.

A linear polynomial using RSM A linear + interaction polynomial using RSM A pure quadratic or elliptic polynomial using RSM A full quadratic polynomial using RSM A Kriging interpolation metamodel with a 0th order polynomial as a trend function 6. A Kriging interpolation metamodel with a 1st order polynomial as a trend function 7. A Kriging interpolation metamodel with a 2nd order polynomial as a trend function

3.4 Validation and Optimisation Metamodel validation based on cross validation (see e.g. Martin and Simpson, 2003) is used to select the best metamodel for the observed response. Using cross validation, one leaves out one, say the ith , of the response measurements and ﬁts the metamodel through the remaining response measurements. The diﬀerence between the real value yi and the value predicted by the metamodel at this location yˆ−i is a measure for the accuracy of the metamodel. One can repeat this procedure for all say n measurement points and calculate the cross validation Root Mean Squared Error (RMSECV ): n 2 (yi − y ˆ−i ) (9) RMSECV = n i=1 As RMSECV approaches 0, the metamodel becomes more and more accurate. Cross validation can also be visualised in a cross validation plot. An example of such a plot is presented in Fig. 7. If the measurements follow the line x = y, the metamodel ﬁts the data well.

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Fig. 7. A cross validation plot

For each response (objective function and implicit constraints) the metamodel outperforming the other six metamodels is selected. These best metamodels for objective function and implicit constraints are added to the explicit constraints in the optimisation model, which is subsequently optimised using a standard Sequential Quadratic Programming (SQP) algorithm, see for example (Haftka and G¨ urdal 1992). In case constraints or Kriging metamodels are present in the ﬁnal optimisation problem, there is a risk of ending up in a local optimum. This problem is overcome by initialising the SQP algorithm at multiple locations. This implies performing many function evaluations, but this is hardly a problem since both RSM and DACE metamodels, being explicit mathematical functions, can be evaluated thousands of times within a second. The DOE points are used as initial locations for the SQP algorithm. The obtained approximate optimum is ﬁnally checked by running one last FEM calculation with the approximated optimal settings of the design variables. The diﬀerence between the approximate objective function value and the real value of the objective function calculated by the last FEM run is a measure for the accuracy of the obtained optimum. If the user is not satisﬁed with this accuracy, the algorithm allows for sequential improvement (e.g. zooming near the optimum) and repeating the procedure presented above until one is satisﬁed with the accuracy. Hence the proposed algorithm incorporates all the advantages of sequential approximate optimisation algorithms.

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3.5 Implementation The optimisation algorithm presented in Fig. 4 and the previous sections was implemented in MATLAB and can be used in combination with any Finite Element code. For the ﬁtting of the DACE/Kriging metamodels, use was made of the MATLAB Kriging toolbox implemented by Lophaven et al., (2002a), (2002b) and Nielsen and Søndergaard (Nielsen).

4 A Metal Forming Application The optimisation algorithm introduced in the previous section is applied to a simple hydroforming process. The product to be hydroformed is presented in Fig. 8(a). Figure 8(b) presents the dimensions. For metal forming, several groups of design variables can be distinguished: 1. Geometrical parameters: (a) Final product geometry (b) Initial workpiece geometry (c) Tool geometry 2. Material parameters 3. Process parameters The group of geometrical parameters is divided further into variables belonging to the ﬁnal, deformed product, e.g. product radii, thicknesses, etc. variables

Fig. 8. (a) a simple hydroformed product; (b) dimensions; (c) typical load paths for hydroforming

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related to the initial, underformed workpiece (blank shape, blank thickness, etc. and variables related to the tool geometry (drawbeads, tool radii and so on). Examples of material parameters are strain hardening coeﬃcients, the initial yield stress or simply several discrete materials in itself. The group of process parameters includes process forces, pressures, tool displacements, friction coeﬃcient, process temperature, etc. For the simple metal forming example considered here, we are interested in optimising the time variation of the internal pressure p and axial feeding u. These are typically process parameters for the hydroforming process. A typical time dependent load path for hydroforming is shown in Fig. 8(c). The velocity of pressure increase α is set to 10 MPa/s during a total hydroforming time of 10 s. Hence, three design variables remain: the time when axial feeding starts t1 , the time when axial feeding stops t2 and the total amount of axial feeding umax . As an optimisation objective, it was chosen to minimise deviations from the initial tube wall thickness. One implicit and one explicit constraint were formulated. The implicit constraint ensures that the ﬁnal product ﬁlls out the die nicely, the explicit constraint makes sure that the time when axial feeding stops is larger than the time when it starts. Convergence problems of the FEM simulations have been encountered when t2 approaches t1 and the amount of axial feeding is high (large umax ). Methods to handle non converged simulations are lacking and is a ﬁeld of open research. An extra explicit constraint has been formulated to overcome the convergence problems. The second explicit constraint makes the ﬁrst explicit constraint redundant, as one can see in the model of the optimisation problem: h − h0 min f (t1 , t2 , umax ) = h0 2 s.t.

gimpl = V ≤ 0 gexpl1 = t1 − t2 ≤ 0 gexpl2 = umax − 9 (t2 − t1 ) ≤ 0 0 s ≤ t1 ≤ 5 s

(10)

3 s ≤ t2 ≤ 10 s 0 mm ≤ umax ≤ 9 mm where h is the ﬁnal wall thickness in the hydroformed product, h0 is the wall thickness of the initial tube and V is the volume between the ﬁnal product and the die. If this volume is larger than zero, there is a gap between the ﬁnal product and the die and the ﬁnal shape of the product is not satisfactory. The 2D FE model of the axisymmetric part is presented in Fig. 9(a). The contact between the product and the die is modelled by contact elements using a penalty formulation. The calculations were performed on 16 parallel processors, which limited the total time to the run time of one calculation,

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Fig. 9. (a) FE model of the initial tube; (b–e) ﬁnal product formed with several arbitrary selected load paths; (f) ﬁnal product formed with optimised load paths

i.e. a couple of minutes for the 2D FE model we are considering. Before the optimisation algorithm is applied, we will ﬁrst arbitrarily select some combinations of the design variables and investigate the eﬀect on the ﬁnal product, the objective function and the implicit constraint. Subsequently, the algorithm is applied and the optimised results are compared to the results obtained with the arbitrarily selected load paths. Figures 9(b) through (e) present the ﬁnal products deformed with the arbitrarily selected load paths. The design variable settings for t1 , t2 and umax and the response values for the objective function f and the implicit constraint gimpl are presented in Table 1. Note that product (a) is the initial undeformed product, which is seen as the product with the perfect wall thickness distribution by the objective function quantiﬁed in (10). For the perfect product, the objective function equals 0. Also note that products (c) and (e) do not satisfy the implicit constraint gimpl , which can also clearly be seen from Figs. 9(c) and (e). The metamodel based optimisation algorithm presented in Sect. 3 is now applied to optimise the wall thickness distribution. The DOE strategy introduced in Sect. 3.2. was applied to generate 16 initial design variable settings

Table 1. Design variable settings and response values Product

t1 (s)

t2 (s)

umax (mm)

f

gimpl

(a) (b) (c) (d) (e) (f)

− 0 0 0 4.8 0

− 0 3 10 6.2 2.5

− 0 9 9 7.7 8.3

0 1.39 0.52 1.42 1.37 0.37

− −0.29 1.79 −0.34 32.64 −0.47

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for which 16 FEM calculations were performed with the in-house code DiekA. Subsequently, the four Response Surfaces and three Kriging metamodels were ﬁtted for both responses (the objective function and the implicit constraint). Based on cross validation, a 0th order Kriging metamodel appeared to be the most accurate metamodel for the objective function. In a similar way, a 1st order Kriging metamodel was identiﬁed to be most accurate for the implicit constraint. Both were included in the optimisation problem, which was subsequently solved using the multistart SQP algorithm described in Sect. 3.4. Several local optima were observed; the global optimum was located at (t1 , t2 , umax ) = (0, 3, 7.9) and the corresponding objective function value was observed to be 0.66. Figure 10(a) shows the approximate optimum located on the metamodel of the objective function. The metamodel is depicted dependent on x2 = t2 and x3 = umax at the constant, optimal level of t1 = 0. Figure 10(b) presents a contour plot of the objective function and the constraints, where one can easily observe that the optimum is constrained by the implicit constraint and the box constraint t2 ≥ 3. To validate the optimum, a FEM calculation was performed with the optimal design variable settings. The actual objective function value was found to be 0.47. The large diﬀerence between the approximate and the actual objective function values at the approximate optimum motivates to sequentially improve the results. Making use of the metamodel visualised in Fig. 10, it was decided to zoom in near the optimum and to relax the box constraint t2 ≥ 3: it was replaced by the new box constraint t2 ≥ 2.5. In total, six batches of each 16 FEM calculations were performed. Each time, the design space was reduced and/or shifted with the aim to ﬁt accurate metamodels in the vicinity of the optimum. The results of all batches are presented in Table 2.

Fig. 10. (a) metamodel of the objective function after batch 1; (b) contour plot of the objective function after batch 1

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Table 2. Optima of the DoC after the 6 batches of 16 FEM calculations each # FEM

t1 (s)

t2 (s)

umax

fopt

factual

16 32 48 64 80 96

0 0 0.8 0 0 0

3.0 1.5 2.5 3.0 3.1 2.5

7.9 7.9 9.0 7.7 8.2 8.3

0.66 0.37 −0.06 0.31 0.37 0.30

0.47 – 0.55 0.50 0.49 0.37

Fig. 11. (a) metamodel of the objective function after batch 6; (b) contour plot of the objective function after batch 6

Figure 11 presents the metamodel of the objective function as well as a contour plot of the objective function after having run the sixth batch of simulations. The obtained optimum (t1 , t2 , umax ) = (0, 2.5, 8.3) is constrained by the box constraint t2 ≥ 2.5. A ﬁnal 97th FEM calculation was performed with the optimised design variable settings. This calculation resulted in a real objective function value of 0.37, which is fairly close to the approximate objective function value of 0.30. Although there is still a small diﬀerence between the approximate and the actual value of the objective function, it was decided to be satisﬁed with the approximate optimum obtained by the metamodel based optimisation algorithm. The optimised settings found by the proposed optimisation algorithm are presented in Table 1 as product (f). The ﬁnal shape of this product is shown in Fig. 9(f). Figure 12 shows the wall thickness throughout the ﬁnal product for all load paths. It can be concluded from the Figs. 9 and 12 and Table 1 that the product deformed with the optimised load paths outperforms the other products formed with arbitrary settings, which demonstrates the good applicability of the proposed algorithm to metal forming.

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Fig. 12. Wall thickness distribution of several hydroformed products

5 Conclusions An optimisation algorithm based on metamodelling techniques is proposed for the optimisation of metal forming using time consuming FEM calculations. It uses both Response Surface Methodology and DACE (or Kriging) as metamodelling techniques. As a Design Of Experiments strategy, a combination of a maximin spaceﬁlling Latin Hypercubes Design with a full factorial design was implemented, which takes into account explicit constraints. Additionally, the algorithm incorporates cross validation as a metamodel validation technique and uses a Sequential Quadratic Programming algorithm for metamodel optimisation. To overcome the problem of ending up in a local optimum, the SQP algorithm is initialised from every DOE point, which is very time eﬃcient since evaluating the metamodels can be done within a fraction of a second. The proposed algorithm allows for sequential improvement of the metamodels to obtain a more accurate optimum. As an example case, the optimisation algorithm was applied to obtain the optimised internal pressure and axial feeding load paths to minimise wall thickness variations in a simple hydroformed product. The product formed with optimised load paths outperforms several products formed with

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arbitrarily chosen load paths, which demonstrates the good applicability of metamodelling techniques to optimise metal forming processes.

Acknowledgements This work is conducted within the framework of project MC1.03162, Optimisation of Forming Processes, which is part of the research programme of the Netherlands Institute for Metals Research (NIMR). The NIMR and its industrial partners are gratefully acknowledged for their support and useful input.

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Lophaven, S., Nielsen, H., Søndergaard, J., “DACE – A MATLAB Kriging Toolbox”, Technical Report IMM-TR-2002-12, Technical University of Denmark – Department of Informatics and Mathematical Modelling, Lyngby, Denmark, 2002. Martin, J., Simpson, T., “A study on the use of Kriging models to approximate deterministic computer models”, in Proceedings of the ASME Design Engineering Technical Conferences DETC, 2003. McKay, M., Beckman, R., Conover, W., “A comparison of three methods for selecting values of input variables in the analysis of output from a computer code”, Technometrics, Vol. 21, 239–245, 1979. Myers, R., Montgomery, D., Response Surface Methodology: Process and Product Optimization Using Designed Experiments, John Wiley and Sons, Inc., New York, USA, 2nd ed., 2002, ISBN 0-471-41255-4. Naceur, H., Guo, Y. Q., Batoz, J. L., Knopf-Lenoir, C., “Optimization of drawbead restraining forces and drawbead design in sheet metal forming process”, International Journal of Mechanical Sciences, Vol. 43(10), 2407–2434, 2001. Naceur, H., Ben-Elechi, S., Knopf-Lenoir, C., Batoz, J., “Response surface methodology for the design of sheet metal forming parameters to control springback eﬀects using the inverse approach”, in Proceedings of NUMIFORM, Columbus OH, USA, 2004. Nielsen, H., “DACE, A MATLAB Kriging toolbox”, http://www.imm.dtu.dk/ hbn/dace/. Papalambros, P. Y., Wilde, D. J., Principles of optimal design, Cambridge University Press, New York, USA, 2000, ISBN 0-521-62727. Sacks, J., Schiller, S., Welch, W., “Design for computer experiments”, Technometrics, Vol. 31, 41–47, 1989. Sacks, J., Welch, W., Mitchell, T., Wynn, H., “Design and analysis of computer experiments”, Statistical Science, Vol. 4, 409–423, 1989. Santner, T., Williams, B., Notz, W., The Design and Analysis of Computer Experiments, Springer-Verlag, New York, USA, 2003, ISBN 0-387-95420-1. Schenk, O., Hillmann, M., “Optimal design of metal forming die surfaces with evolution strategies”, Computers and Structures, Vol. 82, 1695–1705, 2004. Simpson, T., Peplinski, J., Koch, P., Allen, J., “Metamodels for computer-based engineering design: Survey and recommendations”, Engineering with Computers, Vol. 17, 129–150, 2001. van Beers, W., Kleijnen, J., “Kriging interpolation in simulation: A survey”, in R. Ingalls, M. Rossetti, J. Smith, B. Peters, eds., Proceedings of the 2004 Winter Simulation Conference, 2004.

Modelling of Permeability and Mechanical Dispersion in a Porous Medium and Comparison with Experimental Measurements F. Loix1 , V. Thibaut and F. Dupret2 Universit´e Catholique de Louvain Unit´e MEMA – Bˆ atiment Euler, 4, avenue G. Lemaˆıtre, B-1348 Louvain-la-Neuve, Belgium 1 [email protected] 2 [email protected] Summary. A continuous model for the non-isothermal ﬂow in a non-isotropic porous medium is presented and a mechanical dispersion model is proposed. An experimental device is set up in order to measure the permeability and mechanical dispersion components in a typical transversely isotropic cubic block consisting of polymeric needles. Finally, a micro-macro model for the ﬂow of a Newtonian ﬂuid in a porous medium is developed. The micro-model consists of a geometrical network of connectors and junctions. Integration and assembly of the equations provides a macroscopic model which obeys the Darcy or Forchheimer model depending on the dimensionless ﬂow rate. Some peculiarities of the Brinkman model are addressed.

Key words: porous media, darcy model, forchheimer model, brinkman model, mechanical dispersion, micro-macro modelling.

1 Introduction Liquid moulding technologies (Rud et al., 1997) such as the ﬁlling of ﬁber mats represent very attractive forming processes to produce composite parts of the highest quality. The ﬁbers can be of unlimited length, their arrangement and orientation are perfectly controlled, and the principal price to pay when dealing with industrial production is the rather long ﬁlling time needed (this time being much longer than a typical injection moulding cycle for example). Considering the particular cases of Structural Reaction Injection Moulding (SRIM) and Resin Transfer Moulding (RTM), where a reactive mixture activated either by heat or by mixing is injected into the ﬁber mat inside the mould, optimizing these processes requires accurate knowledge of the governing transfer mechanisms during the ﬁlling stage.

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Therefore a relevant model for the ﬂow in a porous medium is needed (Tucker and Dessenberger, 1994). We here focus on the physics of non-isothermal ﬂows in non-isotropic porous media (such as ﬁber mats), possibly with chemical reactions. Therefore, the material constants governing the ﬂow as well as the heat and mass transfer are replaced by appropriate 2nd -, 4th -, or even higher-order tensors (Loix, 2005a), with a signiﬁcant increase of complexity for (i) the experimental and theoretical validation of the governing laws, and (ii) the characterization of the material constants. Two particular eﬀects are present in ﬂows in porous media: (i) permeability, and (ii) mechanical dispersion. First, the medium permeability is characterized by a 2nd -order tensor which plays a key role in momentum transfer. Accordingly and depending on the ﬂow regime, the momentum conservation law can be expressed by means of the Darcy, Forchheimer, or Brinkman model (Lage, 1998 and Tucker, 1994). Secondly, the tortuosity of the porous network causes heat and mass mixing – or mechanical dispersion – which is most often modelled by adding supplementary terms to the molecular thermal and species diﬀusion coeﬃcients. In the work of Mal et al. (Mal et al., 1998 and Mal, 1999), mechanical dispersion is assumed to be governed by a 4th order tensor but this model still needs experimental validation and theoretical justiﬁcation. The ﬁrst objective of this paper (Sect. 2) is to develop a complete continuous model for the ﬂow in a porous medium. The physics of both permeability and mechanical dispersion eﬀects is addressed. Whereas dispersion is wellknown to strongly increase heat and mass diﬀusion, it is besides assumed that this eﬀect could be responsible for the modiﬁed viscosity observed in the Brinkman model, taking into account an increased momentum mixing. Our dispersion model is illustrated by means of typical ﬂows in an anistropic and non-homogeneous medium. The second objective of the paper (Sect. 3) is to describe a set of experiments performed in the anisotropic case (Thibaut, 2004). Three issues are addressed: (i) the conception of a porous medium of controlled symmetry ; (ii) the design of the experimental device ; and (iii) the obtention of permeability or dispersion tensor components from the measurements in order to validate the model. The third goal of this paper (Sect. 4) is to propose a micro-macro model in order to better understand the ﬂow in a non-isotropic porous medium and to draw a link between micro-parameters and macro-properties. It is shown that macroscopic laws can be deduced from a simple microscopic model. For example, Darcy’s model is retrieved from Poiseuille’s assumption at the pore scale. Finally, the relation between the physical properties (permeability, hydrodynamic dispersion) measured in our experiments and the corresponding micro-parameters is investigated.

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2 A Continuous Model for the Flow in a Porous Medium At the macroscopic scale, the liquid and solid phases (resin and ﬁbre mat in SRIM and RTM processes) form a single continuous medium. The physical assumptions of the present paper are: (i) local thermal equilibrium; (ii) rigid solid phase; (iii) no exchange of material between the phases; (iv) constant material properties in both phases; (v) Newtonian ﬂuid rheology; and (vi) no consolidation. Our model has been developed on the basis of the work of Tucker and Dessenberger 1994 and the researches of Mal and Dupret (Mal et al., 1998 and Mal, 1999). Most details have been published in Loix et al., 2002. 2.1 Pressure and Velocity Equations Using the notations of Tucker and Dessenberger (1994), the average mass equation writes as: ∇ · v f = 0, (1) where v f is the Darcy or seepage velocity. Letting Pf f denote the intrinsic average reduced pressure within the liquid and K stand for the permeability ¯ tensor, the average momentum equation can often be expressed by means of the Darcy model: 1 vf = − K · ∇Pf f . (2) ¯ η¯ where η is the ﬂuid viscosity. However, when the ﬂow rate increases, inertia eﬀects can become more important and generate a form drag in addition to the viscous drag. In this case, the Forchheimer model generally provides better results (Lage, 1998), as expressed for an isotropic medium in the form: 0 = −∇Pf f −

η v − Cρ v f v f , K f

(3)

where ρ is the ﬂuid speciﬁc mass, C is a material constant and K is the scalar permeability. When bulk viscosity eﬀects are important with respect to inertia (Tucker and Dessenberger, 1994), the Brinkman model can be used. In the absence of form drag, it writes as: 0 = f ∇Pf f + f ηK−1 · v f + ∇ · ηˆ ∇v f + ∇v f T , (4) ¯ Brinkman term

where ηˆ is the modiﬁed viscosity and f the medium porosity. Although several authors consider that ηˆ = η, some experimental results show that the “ﬁtting” viscosity can in fact diﬀer from the molecular viscosity (cf. Gilver and Altobelli (1994), with ηˆ = 7.5η). Following Loix (2005a), we will here show that momentum mechanical dispersion could be responsible for the observed increased viscosity. In fact, this eﬀect can be expressed by means of a 4th -order

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tensor MD which takes into account the geometry of the porous medium and ¯ ¯ the inﬂuence of velocity. An important observation is that the presence of a divergence term gives to (4) the appropriate mathematical type in such a way that no-slip boundary conditions can be imposed at the medium border. The equation type indeed requires that stress or velocity be imposed along the domain boundary. However, these conditions will generate a ﬂow boundary layer whose thickness is determined by letting in (4) thedrag force beof the same order of magnitude as the Brinkman term, which is O (ˆ η K/f η)1/2 . The latter expression is usually very small since, for common porous materials, the permeability is about 10−5 to 10−8 m2 (Nield and Bejan, 1998) and since ηˆ/η is moderately high (Gilver and Altobelli, 1994). Hence, the inﬂuence of this parameter could be very local in the ﬂow solution, while it could have a higher eﬀect on the heat transfer. 2.2 Temperature and Conversion Degree Equations The energy and species concentration conservation equations write as: ∂T (ρC)f + (ρC)s + (ρC)f v f · ∇T = ∂t ∇ · Ke + KD · ∇T + (ρ)f HR fc (c , T ) , ¯ ¯ and ∂c f + v · ∇c = ∂t f

(5)

De + DD · ∇c + f fc (c , T ), (6) ¯ ¯ where T and c stand for the intrinsic average temperature and concentration, C and ρ denotes heat capacity and speciﬁc mass, Ke and De stand ¯ ¯ for the average molecular thermal and species diﬀusion tensors, HR is the reaction enthalpy, fc denotes the reaction kinetics and the subscripts ‘s’ and ‘f ’ refer to the solid and ﬂuid phases, respectively. The mechanical mixing conduction and diﬀusion tensors KD and DD can ¯ ¯ be modelled by means of a 4th -order dispersion tensor L (Loix et al., 2002, ¯¯ Mal et al., 1998, Mal, 1999): ∇·

1 1 vv (7) K = DD = L : ¯ ¯ . ¯ (ρC)f ¯ D f ¯

v

¯ ¯ Typically, for an isotropic medium, L can be expressed as follows (Loix and ¯ ¯ Dupret, 2003): 1 Lijkl = L⊥ δij δkl + L − L⊥ (δik δjl + δil δjk ) , (8) 2 with 2 non-negative material coeﬃcients L and L⊥ (Rutherford, 1962). On the other hand, a transversely isotropic medium is symmetric for all rotations

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around a given direction, here denoted by l. In this case, the dispersion tensor is characterized by 6 parameters (Loix and Dupret, 2003): Lijkl = L li lj lk ll +L⊥ li lj (δkl − lk ll ) +L⊥ (δij − li lj ) lk ll +L⊥ (δij − li lj ) (δkl − lk ll ) 1 + (L⊥⊥ − L⊥ ) [(δik − li lk ) (δjl − lj ll ) + (δil − li ll ) (δjk − lj lk )] 2 +L [li lk (δjl −lj ll ) + lj lk (δil −li ll )+li ll (δjk −lj lk ) + lj ll (δik − li lk )] . (9) In order to respect the physics of diﬀusion, the tensors K D and DD must be positive deﬁnite and hence (Loix and Dupret, 2003) the coeﬃcients L , L⊥ , L⊥ , L⊥ and L⊥⊥ must be non-negative while L must satisfy the inequality (10) 2 |L | ≤ L⊥ L⊥ + L L⊥⊥ . 2.3 Appication: Flow in a Thin, Non-Homogeneous, Anisotropic Plate Let us here analyze the ﬂow in a thin stratiﬁed medium consisting of a superposition of homogeneous layers (Fig. 1), typically a set of diﬀerently oriented ﬁber mats. The porous plate dimensions are L > l h > 0 along the x, y, and z axis, respectively. We will show that the ﬂow through this kind of medium exhibits mass transfer between the diﬀerent layers. As a particular case, we consider a 2-layered medium with symmetric permeabilities, corresponding to a pair of symmetric mats with respect to the xz plane: ⎤ ⎡ Kxx Kxy 0 (11) [Kij ] = ⎣Kxy Kyy 0 ⎦ for 0 ≤ z ≤ h/2, 0 0 Kzz Medium 1

ez

Upper impermeable border

Pin

ex Pout

ey

h L

l Impermeable lateral borders Medium 2 Lower impermeable border

Fig. 1. Scheme of a thin stratiﬁed porous plate with 2 layers

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⎤ Kxx −Kxy 0 [Kij ] = ⎣−Kxy Kyy 0 ⎦ 0 0 Kzz ⎡

and

for h/2 < z ≤ h.

(12)

It can be shown that, for a transversely isotropic medium, the permeability is characterized by only 2 parameters, K and K⊥ (Rutherford, 1962, Loix and Dupret 2003 and Loix, 2005a). Hence the permeability tensor can be expressed in the alternative form: Kij = K li lj + K⊥ (δij − li lj ).

(13)

To illustrate the competition between dispersion and mass transfer eﬀects, let us consider the ﬂow through a cavity of dimensions 1 m×0.2 m×0.01 m, whose 2 layers are each of 5 mm thickness (Fig. 2). The inlet and outlet pressures are 1.05 bar and 1 bar while the constant resin viscosity is 0.2 kg m−1 s−1 . The lower and upper initial and inlet relative concentrations are c(1) = 0.25 and c(2) = 0.75 (Fig. 3). The porosity is 0.86 for both layers, which are assumed to be transversely isotropic with K⊥ = 10−8 m2 and K = 10−7 m2 while the larger permeability direction l (associated with the ﬁber mat orientation and the symmetry direction of the medium) is parallel to the midsurface of the cavity and oriented in the lower and upper layers at −45◦ and 45◦ with respect to the x-axis: √

(1)

√

(li ) = (li ) = ( √22 , 22√, 0), (2) (li ) = (li ) = ( 22 , − 22 , 0).

(14)

By considering gap-averaged equations, it can be shown that this 2-layered conﬁguration generates a three-dimensional (3D) ﬂow while the pressure ﬁeld will remain one-dimensional (1D). Typical pathlines are depicted in Fig. 2 while the velocity ﬁeld is illustrated in Fig. 4. In order to investigate the

Z

X

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0.1 0.2 0.3 0.4

[m] 0.01

0

0.5 0.6

0

0.1 [m]

[m]

0.7 0.8 0.9 0.2

1

Fig. 2. Pathlines of 2 selected material points through a 2-layered medium

Modelling of Permeability and Mechanical Dispersion (0,0,0.01)

Z

Pin

(0,0,0)

c(2) c(1)

c(2)

Y

X

79

(0,0.2,0) c(1)

Pout

c(2)

c(2)

c(1)

c(1)

Pin

(1,0,0)

Fig. 3. Mesh cross-sections for x = 0.5 m and y = 0.1 m

hydrodynamic dispersion eﬀect and its competition with mass convection, a coeﬃcient L = 0.17 mm has been used to represent the order of magnitude of the tensor Lijkl in (9). This rather low value as compared to the estimates provided by the experimental analysis of Sect. 3 was selected in order to enhance the competition between dispersion and transport eﬀects.

0.01 Z

X

Y

0.1 m/s z = 2.5 mm

.1

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0.1 m/s

z = 7.5 mm

0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

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Fig. 4. Cross-sections of the velocity ﬁeld for x = 0.5 m (top), z = 2.5 mm (middle) and z = 7.5 mm (bottom)

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Advanced Methods in Material Forming 0.01

0 0 0.01

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0 0.01

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0.2 0

0.1

0.2

0.3

0.4

0.5 y = 0.1 m

0.6

0.7

0.8

0.9

1

Fig. 5. Cross-sections of the relative concentration for y = 0.1 m and x = 0.5 m at diﬀerent times from 0 to 20 s (from 0.25 : dark gray, to 0.75 : light gray). All Lijkl coeﬃcients are 0, except L⊥ = 0.17 mm

Modelling of Permeability and Mechanical Dispersion

81 0.01

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0.2 0

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0.6

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Fig. 6. Same as Fig. 5 with L⊥ = 0.017 mm

0.9

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Several simulations have been performed with diﬀerent values of the dispersion tensor Lijkl . Isotropic dispersion was ﬁrst tested with L = L⊥ = L. Subsequently, a transversely isotropic dispersion was investigated by setting to 0 chosen dispersion coeﬃcients (L , L⊥ , L⊥⊥ . . .) (Figs. 5 and 6). From these numerical experiments, it was observed that a signiﬁcant role is here played by L⊥ only. This can be demonstrated by a detailed analysis of our particular problem. In Figs. 5 and 6, it also appears that, when the L⊥ dispersion parameter is increased, the dispersion eﬀect becomes more and more important and ﬁnally dominates mass transport (which is indeed too slow to play any role). In conclusion, complex transfer eﬀects are generated (i) by a jointly nonuniform and non-isotropic permeability, and (ii) by a non-isotropic mechanical dispersion. Experimental validation is needed to assess our mechanical dispersion model.

3 Experimental Analysis 3.1 Principles and Design The focus is here on anisotropic porous media and, more particularly, on transversely isotropic media. Therefore the experimental apparatus must be designed in such a way that the medium orientation can be easily changed with respect to the mean ﬂow direction. Porous Medium Conception The conception of an experimental transversely isotropic porous medium has been led by diﬀerent considerations, viz. (i) that the medium anisotropy must be controlled, (ii) that the relative dimensions of the pores and of the entire medium must be well-chosen, and (iii) that the production of several similar media should be easy. We have chosen to build the medium as an assembly of small elements resulting in a porous cubic block. To get a transversely isotropic medium, the shape of these constitutive elements had to be the same after any rotation around their symmetry direction, keeping in mind that statistical averaging must be taken into account to reach the desired symmetry (in such a way that the orientation of the individual elements can slightly vary around the macroscopic symmetry direction). “Needles” and “discs” answer this objective. For practical reasons, only the medium consisting of needles has been selected in our tests. In order to build a representative macroscopic medium, the length of the “needles” had to be at least one order of magnitude smaller than the entire block. Fibers of 1 mm diameter and 15 mm length and a 10 cm× 10 cm × 10 cm porous block seemed to be a pertinent compromise. With these dimensions

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and shape, one block contains about 50000 to 60000 needle elements and the porous block is easy to design and build. Finally, in order to facilitate the medium assembly process, thermoset polymeric needles were selected. Indeed, with this material, the formation of a coherent medium is automatically obtained by heating and reticulation. Of course care had to be given to almost identically orienting the needles in the appropriate cubic container before starting the reticulation process in the drying oven. Apparatus To produce a ﬂow through a porous medium, the underlying principle consists in imposing a pressure gradient or to replace it by the eﬀect of gravity, thereby resulting in an equivalent loss of head. In our experiment, the ﬂuid is water and the loss of head is induced by a water height diﬀerence between the ﬂow entry and exit whereas the porous block is inserted at a given orientation into a square column, taking care of watertightness near the borders. Moreover, as ﬂows in saturated porous media are to be considered, the exit level is roughly situated at the same height as the middle of the block and a siphon is introduced (Fig. 7). A runoﬀ system is built in order to maintain a constant pressure on the top side of the block. The pressure on the other side is controlled by the height of a barrier in a collector. The latter is designed in such a way that the ﬂuid velocity is suﬃciently low. The ﬂow rate measurement can then be accurate enough. In order to quantify the mixing eﬀect of mechanical dispersion inside the porous medium, the device has been further adapted. First of all, in order to enable the ﬂow of 2 distinctly coloured ﬂuids, watertight separations were Runoff

Fluid Porous medium block Collector

h1

h h2

Draining

Fig. 7. Experimental apparatus – Flow rate resulting from an imposed loss of head

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(a)

(b)

Fig. 8. (a) Siphon system to avoid void formation and to maintain a constant exit pressure; (b) separated zones at the entry and the exit of the porous medium

introduced on the top side of the block (Fig. 8). Mixed ﬂuid is then collected at the bottom of the block in 12 separated zones. A siphon system is still used in order to avoid void formation. 3.2 Measurements and Results Permeability Since the experiments described in Fig. 7 deal with a 1D mean ﬂow, the Darcy equation can be written in macroscopic terms as: V =−

K ΔC , η h

(15)

where V is the macroscopic seepage velocity, η the ﬂuid viscosity, ΔC the loss of head along the block depth (h) and K the permeability in the ﬂow direction. (15) is further rewritten as: Q=−

K ΔC S, η h

(16)

where Q denotes the volumic ﬂow rate across the block cross-section S. Knowing the porous cube dimension, the ﬂuid viscosity, the pressure drop (and hence the loss of head), and the measured ﬂow rate, the permeability component in the mean ﬂow direction can be written as: K=−

Qη h . ΔC S

(17)

Since the permeability tensor for a transversely isotropic medium is characterized by 2 parameters only (Rutherford, 1962 and Loix, 2005a), one for the

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permeability along the symmetry axis (K ) and the second one for the permeability along any perpendicular direction (K⊥ ), these two components can be determined by aligning the needle direction parallel or perpendicular to the mean ﬂow direction. The experiments were repeated several times on 2 macroscopically similar porous blocks (Medium 1 and Medium 2) in order to assess the validity of the measurements, and the entire averaging procedure was repeated twice (noted Exp1 and Exp2). To determine the ﬂow rate, the time for a ﬁxed ﬂuid volume to run across the porous block was measured. Table 1 summarizes the detailed measurements for all handlings. Let us ﬁrst observe that both classes of measurements are consistent. This seems to validate the procedure. Moreover, the running time is clearly higher in the “perpendicular” case than in the “parallel” case. This indicates that the perpendicular permeability component is lower than the parallel component. This assertion is conﬁrmed by the results of Table 2, where we observe that K is from 13% to 14% higher than K⊥ . Even if this diﬀerence is low, it is the same in both cases. In order to validate the Darcy law, this experiment should have been repeated for diﬀerent pressure drops. Nevertheless, it must be noted that our experimental ﬂows are expected to well lie in the linear regime. Indeed, the pore-sized Reynolds number (Tucker and Dessenberger, 1994) is much lower than unity: √ ρV K Rep = ≈ 0.05 < 1. (18) η Table 1. Mean time (s) taken by 1 water liter to ﬂow across the porous medium medium 1 exp 1 exp 2 mean time

medium 2

case

⊥ case

case

⊥ case

29.2284 29.7313 29.4799

33.5436 33.7776 33.6606

28.0563 27.5484 27.8024

32.2741 32.3534 32.3138

Table 2. Permeability components obtained from the measurements medium 1 case viscosity η [Pa.s] ﬂow rate Q[m3 s−1 ] loss of head ΔC [Pa] resulting permeability K[m2 ]

medium 2 ⊥ case

−3

1.12 10 3.4 10−5 1.2753 103 2.96 10−10

case −3

1.14 10 2.97 10−5 1.2753 103 2.65 10−10

⊥ case −3

1.14 10 3.6 10−5 1.2753 103 3.21 10−10

1.14 10−3 3.09 10−5 1.2753 103 2.76 10−10

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Mechanical Dispersion In order to analyse the mixing eﬀect induced by mechanical dispersion, two equivalent but distinguishable ﬂuids are injected at the medium entry. The ﬂuids are made distinct by the adjunction of 99.5 ppm potassium permanganate in one of them (Fig. 8), and the observations are based on measuring the concentration of this substance at the exit of the porous block. Two cases are again studied, with the mean ﬂow perpendicular or parallel to the needle direction, respectively (Figs. 9 (a) and (b)). In a ﬁrst step, we focus on the perpendicular case (a) and examine the equation governing concentration evolution. Choosing the x-axis as aligned with the symmetry direction of the medium, the macroscopic equation for concentration (6) can be written in the steady case and without any source term in the form: ∂C ∂2C ∂2C V = Dx,y 2 + Dy,y 2 , (19) ∂y ∂x ∂y where Dx,y and Dy,y denote the appropriate total diﬀusion components (induced by molecular diﬀusion + dispersion) and V is the macroscopic seepage velocity as aligned with the y-axis. The total diﬀusion tensor is indeed diagonal. The boundary conditions are: ! ! C = C0 , 0 ≤ x < l/2 y = 0, ∂C/∂x = 0, x = 0, and (20) C = 0, l/2 ≤ x < l y = 0, ∂C/∂x = 0, x = l. l

Impermeable walls C0

C0 x

0

y

0 l

y

x

Separated collectors (a)

(b)

Fig. 9. Experimental scheme and axes for the determination of mechanical dispersion: (a) ﬂow perpendicular to the needle direction; (b) ﬂow aligned with the needle direction

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In order to easily draw ﬁrst conclusions from our experiments, we will in addition assume in what follows that the total diﬀusion in the ﬂow direction is negligible in front of the total diﬀusion in the perpendicular direction. This is a quite strong hypothesis that needs conﬁrmation from further work. In this ﬁrst attempt, (19) can be rewritten as: V

∂C ∂2C = Dx,y 2 . ∂y ∂x

(21)

This equation is easily solved by variable separation and Dx,y is ﬁnally obtained by ﬁtting the solution to the measurements. In order to measure the outlet concentrations, the spectrometry technology was used. Samples of colored liquid were collected in 12 separated zones at the porous block exit when the steady-state ﬂow regime was reached. The concentration of each sample was measured while noting its position in the exit zone. Hence, graphs of concentration distribution were ﬁnally drawn. This procedure was again repeated four times for the two blocks in order to validate the measurements (Fig. 10). Since the results are very close together with Medium 1 and Medium 2, this tends to validate the experimental procedure. Fitting the solution to the experiments ﬁnally gives the coeﬃcient Dx,y for both blocks: Dx,y = 1.984 10−5 m2 s−1

and Dx,y = 2.076 10−5 m2 s−1 .

(22)

A similar approach has been applied in the parallel case. Again, the total diﬀusion in the ﬂow direction was assumed to be negligible in front of the total diﬀusion in the perpendicular direction. Hence, with the axes of Fig. 9 (b), the transport-diﬀusion equation V

∂C ∂2C ∂2C = Dx,x 2 + Dy,x 2 , ∂x ∂x ∂y

(23)

∂2C ∂C = Dy,x 2 . ∂y ∂x

(24)

simpliﬁes as V

The related measurements are again consistent for both blocks (Fig. 11) and the ﬁtting of the experimental results to the simpliﬁed solution gives: Dy,x = 6.095 10−6 m2 s−1

and Dy,x = 6.864 10−6 m2 s−1 .

(25)

Analysing the experimental results (Figs. 10 and 11) shows that the “perpendicular” case induces more mixing than the “parallel” case. The diﬀusion components determined in (22) and (25) conﬁrm these observations. However, recalling that these results are based on the assumption of negligible

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Fig. 10. Concentration distribution at the exit of the porous block in 12 separated zones (Thibaut, 2004). The ﬂow is perpendicular to the medium symmetry direction (Fig. 9 (a)). The experiments were performed with the ﬁrst and the second block (left and right ﬁgures)

total diﬀusion in the ﬂow direction, a more accurate approach should consist in removing this assumption and determining the total diﬀusion components by detailed ﬁtting of the calculated results to the measurements. However, it is not clear that the experiments can be accurate enough to provide the required information. Therefore additional experiments might be needed.

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89

Fig. 11. Same as Fig. 10, with the ﬂow parallel to the medium symmetry direction (Fig. 9 (b))

4 A New Micro-Macro Model 4.1 Framework and Justiﬁcation for the Development of a New Model There is a strong lack of models for (possibly reactive) ﬂows in porous media. This issue is however complex since a large number of physical phenomena are involved. First, the law governing the relationship between loss of head and seepage velocity is not always obeying the simple linear Darcy law. In particular, when the shape of the porous network plays a non-negligible role in stress

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generation, additional losses of head must be introduced and the Forchheimer law should be considered (Lage, 1998), with supplementary quadratic terms. Let us recall that this quadratic law is subject to discussion – without complete agreement in the literature – at low ﬂow rates, where a cubic law seems to better agree with experimental results (Firdaouss et al., 1997). In addition, deviations from the Darcy law can mean that inertia eﬀects are non-negligible. In this case, the Brinkman equation (with or without Forchheimer terms) is better suited to model the momentum equations than simple drag forces. However, applying the Brinkman model requires to introduce an additional viscosity, whose presence might be attributed to momentum mixing inside the porous network. The constitutive law governing this additional viscosity remains incompletely known as it was explained in Sect. 2. A second class of phenomena associated with the physics of porous media pertains to the modelling of non-isothermal ﬂows, possibly with chemical reactions. In this case, the tortuosity of the porous network generates an additional heat and mass mixing – the mechanical dispersion – which is assumed in the work of Mal et al. (1998) and Mal (1999) to be governed by a 4th order dispersion tensor. Further complications in the momentum, energy and species conservation equations can result from the porous medium anisotropy. Therefore, the material constants governing the ﬂow as well as the heat and mass transfer must be replaced by appropriate 2nd -, 4th -, or even higher order tensors (Loix, 2005a). In view of these very high modelling diﬃculties, the objective of this section is to develop a micro-macro model and to derive relevant macroscopic constitutive laws in order to accurately understand and predict the phenomena taking place in ﬂows in porous media and in particular in the ﬁlling of ﬁber mats (Loix et al., 2002 and Loix and Dupret 2003). For this purpose, a simple model is developed at the pore microscopic scale by considering a network of “connectors” and “junctions” where all the transfer phenomena are modelled in the simplest way (Loix, 2005a). In other words, the ﬂow complexity is addressed by means of an assembly process at each junction while the ﬂow across each connector is assumed to be governed by simple laws. Hydraulic similarities are used and new variables are introduced. Let us note that a similar approach was developed by Wang et al. (1999). 4.2 Development of a Micro-Macro Porous Medium Model Our microscopic model is inspired from general considerations about ﬂows in porous media. Clearly, a porous medium consists of more or less tortuous pores in which the ﬂuid can ﬂow. While detailed knowledge of the pathlines is impossible and essentially unnecessary, basic analysis indicates that these pathlines can be quite intricate. For instance the ﬂow can separate around some obstacles while some pathlines can meet again after skirting these obstacles.

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91

Let us emphasize that, in order to build a relevant micro-macro model, it will not be necessary to model the full complexity of the real medium at the micro-scale since this complexity will result from the assembly of a very large number of micro-components. Therefore the simplest model is considered at the micro-scale, in order to only capture the principal phenomena involved and satisfy the basic conservation equations. Consequently, our micro-model is based on considering a network of “connectors” and “junctions” (Fig. 12) deﬁned by the following properties: –

A connector is a cylindrical or curved tube of length L and radius R across which the ﬂuid ﬂows from a junction to another junction. The connector volume is assumed to be πR2 L whatever its shape. – A junction is a point of negligible volume where several connectors meet, any ﬂowing direction being possible across each associated connector.

In order to construct the simplest possible micro-scale model, some considerations arising from elementary hydraulics will be used given the similarity between pipe networks and our porous medium network model. First of all, focusing in this study on the steady ﬂow of an incompressible ﬂuid, the basic unknowns will be ﬂow rate and head. The (volumic) ﬂow rate Q[m3 /s] across a connector is deﬁned as " Q= (26) v · n dS, S

where n and S stand for the unit normal and area of any cross-section of the connector, while v denotes the velocity ﬁeld. The head C at a cross-section of a given connector is the averaged sum of dynamic, pressure and elevation contributions: p v2 + z S , C= + (27) 2g ρg where the average of any physical quantity F across S is deﬁned by " 1 F S = F v · n dS, Q S

Ci i

(28)

k Qik

Ck

Fig. 12. Micro-scale model for the ﬂow in a porous medium, as consisting of a network of connectors and junctions

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and where v denotes the velocity norm, g the acceleration of gravity, p the pressure, ρ the ﬂuid speciﬁc mass and z the local elevation. Here, the head is expressed from (27) as an equivalent height [m], while in Sects. 2 and 3, it was expressed as an equivalent pressure [Pa]. The approach presented in this paper provides the simplest way to take mass and momentum conservation laws into account for viscous ﬂuids. It is easy to show that, in the steady incompressible case, the sum of the entering ﬂow rates into any network subset is zero, while the head ﬂux (as calculated as the sum of the products of the entering ﬂow rates by the associated heads) is exactly the power dissipated by friction in this network subset. In order to completely specify the micro-macro model, a connector model, a junction model, and a network model will be successively deﬁned. Connector Model Each connector is a duct across which the loss of head ΔC is governed by the following simple law: 8Lη ΔC = Q, (29) ρgπR4 which can be directly derived from the well-known Poiseuille ﬂow governing laminar loss of head in a cylinder. It is easy to show that a very similar law applies in the case of a curved pipe and this justiﬁes the use of the model provided by (29) in all connectors without loss of generality. Junction Model The simplest possible model consists in assuming that no losses of head take place in the junctions. In fact, Darcy’s law directly derives from this approach at the macro-scale through the linear relationship between loss of head and ﬂow rate provided by (29) in every connector. In order to deﬁne a more complex physics and to represent additional eﬀects such as those of Forchheimer’s or Brinkman’s models, supplementary losses of head must be introduced in the junction model. In a ﬁrst attempt, an entry loss of head from each connector into the associated ending junction is considered, as governed by the following law: ΔC =

gπ 2 R4

Q|Q|,

(30)

where is a dimensionless loss factor which is here, for the sake of simplicity, common to all connectors and which takes its values in the [0, 1] interval. Indeed, in order to avoid model contradictions such as back-ﬂows, we impose that the entry loss of head should not be higher than what we call the disposal head in the connector C d , corresponding to the kinetic part of the total head:

Modelling of Permeability and Mechanical Dispersion

#

2

v v S 2g

C = # d

S

· n dS

v · n dS

=

Q2 . g π 2 R4

93

(31)

The ratio between the entry loss of head, (30), and the connector loss of head, (29), is: ρ|Q| R (32) = Re = Ff Re, 8ηLπ 16L where Re = 2ρ|Q| πRη is a characteristic Reynolds number of the ﬂow in the pores R and Ff = 16L is a shape factor. (32) provides an easy way to interpret the loss of head nonlinearity as associated with pore shape eﬀects (Lage, 1998). It is easy to show that law (30) is suﬃcient to generate Forchheimer’s (but not Brinkman’s) model at the macroscopic level.

Network Model The following notations are adopted: – Cik denotes the connector from junction i to junction k; – Sik is the cross-sectional area of connector Cik ; – Junc(i) stands for the set of junctions linked to junction i by a connector; – Qik stands for the (volumic) ﬂow rate from junction i to junction k; – Ci denotes the head at junction i as calculated after subtraction of all entry losses (therefore, in the present model, this head is unique at each junction). Mass conservation in every junction writes as: Qik = 0,

(33)

k∈Junc(i)

while applying the Poiseuille and entry losses of head provides the following equation in every connector: Ci − Ck =

8Lik η Qik + 2 4 Qik |Qik |. 4 ρgπRik gπ Rik

(34)

The system of (33) and (34) is closed by appropriate boundary conditions, which consist in imposing either the ﬂow rate or the head at each boundary junction. The solution is easily found by applying the Newton-Raphson algorithm. 4.3 Numerical Results Our micro-macro model has been applied to investigate the ﬂow of water (η = 10−3 Pa.s, ρ = 103 kg/m3 ) in the previous experimental 10 cm × 10cm ×

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Advanced Methods in Material Forming Δhx Δhz

=

+ Lx , Rx

Δhy

+

Lxy , Rxy

+ Ly , Ry

+

+

Lxz, Rxz

Lz , Rz

Lyz , Ryz

Lxyz , Rxyz

Fig. 13. Geometry of an elementary cell as a set of connectors and junctions

10cm porous medium. Therefore, a network of equal parallelepipedic cells was designed as shown in Fig. 13, with Δhx = 16 mm, Δhy = Δhz = 2.12 mm, Rx = 0.2285 mm, Ry = Rz = Ryz = 0.25 mm, Rxy = Rxz = 0.135 mm, Lx = 16 mm, Ly = Lz = 2.12 mm, Lyz = 3 mm, Lxy = Lxz = 16.1 mm, while internal connectors were removed (Rxyz = 0 mm). These parameters were selected to approximately represent the geometry and the average topology of the medium to be modelled (Thibaut, 2004). From this micro-macro model, the associated macroscopic relationship between loss of head and seepage velocity has subsequently been determined for various values of (Fig. 14). A least square best ﬁt technique was applied to approximate this relation by means of Darcy’s or Forchheimer’s model. When = 0, the Darcy model is recovered with a permeability Kxx equal to 3.510−10 m2 . This value is very close to the parallel permeability component obtained from the experiments (cf. Table 2). It is important to note that Kxx is strongly linked to the model microscopic parameters. In some cases an analytical relationship between macroscopic and microscopic parameters can even be established. For the linear case ( = 0), with a head gradient along the x axis, the following result is obtained: 4 4 4 Rxy Rxyz Rx4 πΔhx Rxz . (35) +2 +2 +4 Kxx = 8Δhy Δhz Lx Lxy Lxz Lxyz Using similar formulas, the other permeability components are easily determined. However, the agreement with experimental measurements (Table 2) is of much lower quality. When is increased, nonlinear eﬀects play a more and more important role and a kind of Forchheimer’s regime is reached (Fig. 14). However, the Brinkman model cannot be generated by this approach since no momentum mixing eﬀect appears. Moreover, due to the local nonlinearities associated with the assembly process, identifying the exact relationship between the microand macro-scales is more complicated than in the linear case. Hence, this nonlinear regime should be further analysed.

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95

× 104 12

grad P = 6.36E8*vv2Darcy + 3.2E6*vDarcy ε = 0.1

10 ε=0 grad P [N/m3]

8

grad P = 2.85E6*vDarcy

6 ε=1 2 grad P = 6.29E9*vDarcy + 3.3E6*vDarcy

4

2

0

0

0.005

0.01

0.015

0.02 0.025 vDarcy [m/s]

0.03

0.035

0.04

0.045

Fig. 14. Macroscopic relation between head gradient (= pressure gradient if g = 0) and seepage velocity for diﬀerent values of

4.4 Momentum Mixing The ﬂuid arriving from one or more upstream connectors at a junction is redistributed into one or more downstream connectors. Momentum conservation is ensured by the reaction of the solid matrix on the ﬂuid (Fig. 15 (a)) in order to generate a ﬂow obeying the applied model. Assuming now that the additional viscosity of the Brinkman model is generated by momentum mechanical dispersion, it is relevant to introduce directional eﬀects in the new model. The main idea of the law we here investigate relies on the introduction of a ﬂow self-organization such that it minimizes in some way the action of the matrix at the junctions. From Fig. 15 (a) to Fig. 15 (b) for example, the incoming ﬂow has been reduced together with the ﬂow in the vertical connector in order to reduce the matrix reaction. The objective is hence to ﬁnd ﬂow rates Qik that minimize the expression: F= F i · F i, (36) i

where F i is the solid force acting on the ﬂuid at junction i and where the summation is carried out on the set of junctions. In a ﬁrst attempt, we choose

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Advanced Methods in Material Forming C1

Q14

C4

Q42

C2

C1

Q14

α F4

Q42

C4 α

Q43

C2

Q43

F4

C3

C3

(a)

(b)

Fig. 15. Momentum redistribution after a junction and corresponding reaction acting on the ﬂuid: (a) linear model ; (b) modiﬁed model to generate momentum dispersion

for the connector ﬂow the simple previous nonlinear law (34), except that the unknown loss of head parameter ik may now diﬀer from connector to connector. The law then writes as: 2 Q2ik ρg Sik Ci − Ck − sign(Qik ) Qik = (37) 2 ik , g Sik 8Lik η or, after some algebraic manipulations, as: ρg S 2

2 8Likik η (Ci − Ck )

Qik = $ 1+

ρg S 2 4 8Likik η |Ci

−

Ck | g Sik2 ik

.

(38)

+1

In order to complete the formulation, the reaction force F i is developed as a function of Qik : –

The second Newton law applied at junction i provides the relation: " " ∂ (ρv) dV + ρv v · n dS. (39) Fi = Vi ∂t Sik k∈Junc(i)

The 1st term in the right-hand side vanishes because of stationarity while the 2nd term represents the momentum ﬂux due to mass ﬂuxes. Hence, the force is: Fi = − (40) P˙ik , k∈Junc(i)

where P˙ik is the incoming momentum ﬂux across surface Sik . –

Keeping the assumption of a Poiseuille ﬂow in the connector, it can easily be shown that the momentum ﬂux is equal to: ρ Q2ik n . P˙ik = − 3π Sik ik

(41)

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97

Therefore, the complete formulation is ﬁnally given by: & % % Determine ik ∈ [0, 1] minimizing the expression F = i F i · F i , % ⎧ 2 & ρ Qik % F i = k∈Junc(i) 3π ⎪ % ⎪ Sik nik , ⎨ % ρg S 2 2 8L ik (Ci −Ck ) % with ik η % Q = . ik ⎪ 2 % ⎪ ρg S ik ⎩ 1+4 8L ik |C −C | +1 % i k 2 g S ik η

(42)

ik

9 1

× 10−8

8 7 Objective [N²]

0.8

ε

0.6 0.4

6 5 4 3

0.2

2

0

1 0

0.5

1

1.5

2

2.5

0

3

0

0.5

1

1.5

3.5

2.5

3

3.5

2.5

3

3.5

× 10−6 0.04 0.035

3

0.03 2.5 Head [m]

Flow rate [m³/s]

2

α (b)

α (a)

2 1.5

0.025 0.02 0.015 0.01

1 0.5

0.005 0

0.5

1

1.5

2

2.5

3

3.5

0

0

0.5

1

1.5

2

α

α

(c)

(d)

Fig. 16. Same problem as in Fig. 15 for α ∈ [0, π]: (a) values of ij generated by the modiﬁed algorithm. 14 is plotted with ‘+’ symbols, 42 with ‘o’ symbols, and 43 with ‘×’ symbols; (b) objective function F. The continuous curve represents the objective reached by means of the modiﬁed algorithm, while symbols ‘+’ refer to the case 14 = 1; 42 = 0; 43 = 0, and symbols ‘×’ to the case 14 = 1; 42 = 0; 43 = 1; (c) ﬂow rates obtained with the modiﬁed algorithm. Q14 is plotted with ‘+’ symbols, Q42 with ‘o’ symbols, and Q43 with ‘×’ symbols; (d) Head generated at junction 4 by the modiﬁed algorithm. The ‘+’ symbols represent the head generated by the linear model.

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With this formulation, the goal is to let the ﬂow re-arrange itself by a selection of appropriate values of ik in order to increase momentum mixing and by this way to retrieve the Brinkman equation at the macroscale. Unfortunately, at this point, we are not able to conﬁrm that our model reaches this goal. In order to solve the minimum problem (42), a Lagrangian multiplier formulation could have been used. However, the associated system is very complicated to solve. Therefore, the iterative gradient method (which can be very easily implemented) was used. It should be noted that none of these methods ensures to reach the global minimum. Fig. 16 (a) and (b) illustrates the values of ij and of the objective function F as generated by the modiﬁed algorithm for the problem deﬁned in Fig. 15, for diﬀerent angles of the adjacent connector. In each problem, the extremity heads are imposed (C1 = 0.1 m, C2 = C3 = 0 m) and the connector parameters are the same in each branch (R = 0.001 m and L = 0.01 m). It can be observed that the modiﬁed algorithm results in decreasing the incoming ﬂow rate and hence diminishing the incoming momentum ﬂux and the reaction at the junction. The initial objective function F (as calculated from the linear model) is then decreased by a factor 125 ! Fig. 16 (c) and (d) further shows the related ﬂow rates and head.

5 Conclusions and Perspectives This work can be seen as representing a step in the diﬃcult theoretical and experimental analysis of non-isothermal ﬂows in non-isotropic porous media, taking into account the combined eﬀects of permeability and mechanical dispersion. The classical equations of the continuous model governing ﬂows in porous media have been presented. A 4th -order tensor has been introduced to model mechanical dispersion and has been developed in the transversely isotropic case. The ﬂow in a thin, non-homogeneous and transversely isotropic plate has been investigated and the associated transport phenonema have been highlighted. An original experimental set-up has enabled us to perform various permeability and dispersion experiments. The main goals were (i) to demonstrate the possibility of performing experiments on media with controlled anisotropy, (ii) to validate the governing laws for permeability and dispersion in such media, and ﬁnally (iii) to deduce the medium material properties from the measurements. Whereas the analysis is not complete at this stage, it indicates that our method is appropriate for these goals. Typically, we have been able to measure the two permeability components of a transversely isotropic medium consisting of almost aligned polymeric needles. Moreover, assuming that molecular diﬀusion is negligible in front of hydrodynamic dispersion, we have been able to determine some components of the dispersion tensor. Indeed in this case, the ﬁtting of Dx,y and Dy,x to

Modelling of Permeability and Mechanical Dispersion

99

the dispersion model (9) for a transversely isotropic medium gives: L⊥ = 6.5 10−3 m

and L⊥ = 2.15 10−3 m.

(43)

The unique values we have found in the literature are given by Mal et al,. (1998), who propose coeﬃcients 2 orders of magnitude lower than these values. Obviously, the considered medium is diﬀerent. In further work, the permeability measurements should be repeated for different losses of head in order to validate Darcy’s law and to exhibit the onset of Forchheimer’s or Brinkman’s regime. The tensorial form of permeability should also be veriﬁed by performing experiments with the medium symmetry direction oriented at diﬀerent angles with respect to the mean ﬂow direction. Concerning mechanical dispersion, the experimental procedure should be reﬁned in order to determine all the dispersion tensor components. To remove the assumption of negligible total diﬀusion in the ﬂow direction, supplementary experiments (including unsteady experiments) should be carried out and various boundary conditions should be implemented. In order to model the ﬂow in a porous medium and to better understand the related physical eﬀects, a micro-macro model has been developed on the basis of a hydraulic analogy. In a ﬁrst attempt, linear losses of head in the connectors (inspired from the Poiseuille ﬂow in a straight pipe) were introduced while mass conservation provided the way to correctly assemble the equations. It has been shown that these assumptions are suﬃcient to generate Darcy’s ﬂow. Further, the simplest nonlinear loss of head was added at the junction entries in order to generate Forchheimer’s ﬂow. It has been shown that this additional term is appropriate for this goal. Finally, with a view to reproducing the additional viscosity of Brinkman’s model and the (assumed) associated momentum dispersion, a more sophisticated law has been introduced in each connector. However, no deﬁnite conclusions could be drawn as far from this ﬁrst study and hence further work is needed. Numerical experiments have been performed with the parameters adjusted to the needle-made experimental medium. In these experiments, the boundary conditions were ﬁrst imposed in such a way that the parallel permeability component was playing the major role. The calculated value K = 3.5 10−10 m2 was found as very close to the experimental result (Table 2). Similar calculations for the perpendicular component gave K⊥ = 6.710−11 m2 , which is by far smaller than the experimental value. In fact, the ﬁtting required to determine the parameters of the micro-macro model is not obvious and probably not unique. Hence, more tests are simply required to correctly determine the model parameters. In order to validate our hypothesis, which consists in interpreting the additional viscosity of Brinkman’s model as a momentum mechanical mixing eﬀect, the present study should be extended to more reﬁned models and supplementary tests should be carried out. Another important goal will be to deduce relevant relations between the microscopic parameters and the macroscopic physical coeﬃcients.

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The thermal problem has not been considered yet. Nevertheless, since the results obtained are very encouraging, a similar micro-macro method could be used to model the energy equation. However, in order to take into account the inﬂuence of the solid phase, a supplementary network of solid connectors and junctions should be added. Moreover transfer assumptions between the solid and ﬂuid networks should be introduced to take into account the mixing due to microscopic transport. Investigation of the ﬂow in a more complex geometry and, in particular, the ﬁlling of industrial ﬁber mats represents a typical application ﬁeld that could provide possible comparisons between numerical results and experimental measurements.

References Firdaouss M., Guermond J., Quere P. L., “Non-linear corrections to Darcy’s law at low Reynolds numbers”, J. Fluid Mech., vol. 343, 1997, p. 331–350. Gilver R., Altobelli S., “A determination of the eﬀective viscosity for the BrinkmanForchheimer ﬂow model”, J. Fluid Mech., vol. 58, 1994, p. 355–370. Lage J., “Transport phenomena in porous media”, chapter The fundamental theory of ﬂow through permeable media from Darcy to turbulence, p. 1–31, Elsevier Science, 1998. Loix F., Magotte O., Roger H., Dupret F., “A 3D numerical modelling of the RTM and SRIM processes”, 5th Int. ESAFORM Conf. on Material Forming, Krak´ ow, 2002, p. 307–310. Loix F., Dupret F., “Numerical Modelling of Liquid Moulding Processes”, srl N. I. E., Ed., 6th Int. Conf. on Material Forming, Salerno, 2003, p. 835–838. Loix F., “Modelling of transport phenomena in porous”, PhD thesis, UCL, Louvainla-Neuve, December 2005. Loix F., Thibaut V., Dupret F., “Micro-macro modelling of permeability and mechanical dispersion in a porous medium”, vol. 1, Cluj-Napoca, Romania, April 2005, The publishing house of the romanian academy. Mal O., Couniot A., Dupret F., “Non-isothermal simulation of the resin transfer moulding process”, Composites Part A, vol. 29, 1998, p. 189–198. Mal O., “Modelling and Numerical Simulation of Reaction Injection Moulding Processes”, PhD thesis, UCL, Louvain-la-Neuve, 1999. Nield D., Bejan A., Convection in porous media, Springer, 1998. Rudd C. D., Long A. C., Kendall K. N., Mangin C., Liquid Moulding Technologies, Woodhead, 1997. Rutherford A., Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Prentice-Hall, Inc, 1962. Thibaut V., “Etude des ´ecoulements en milieux poreux anisotropes: approche th´eorique, num´erique et exp´erimentale”, Master’s thesis, UCL, 2004. Tucker C., Dessenberger R., “Flow and Rheology in Polymer Composites Manufacturing”, chapter Governing Equations for Flow and Heat Transfer in Stationary Fiber Beds, p. 257–323, Elsevier, 1994. Wang X., Thauvin F., Mohanty K., “Non Darcy ﬂow through anisotropic porous media”, Chemical Engineering Science, vol. 54, 1999, p. 1859–1869.

Three-Dimensional Bending of Proﬁles by Stress Superposition S. Chatti, M. Hermes and M. Kleiner Institute of Forming Technology and Lightweight Construction, University of Dortmund, Baroper Str. 301, 44227 Dortmund, Germany, [email protected], [email protected], [email protected] Summary. The paper shows a number of investigation results about threedimensional-bending of proﬁles. The investigations concentrated on two-dimensional bending using a conventional three-roll-bending machine with a simultaneous deﬂection of the proﬁles towards the third axis by means of a special device. The special device consists of a hydraulic cylinder giving the motion and the required force for bending in the third axis coupled with a guiding tool to guide the proﬁle in the 3Dspace. The most important aspect of these tools is the use of the previous plastic deformation in the 2D-bending zone for an easier bending in the 3D-space through the superposition of lateral forces. By means of this machine set-up both symmetrical and asymmetrical proﬁles can be bent three-dimensionally and the unwanted torsion of asymmetrical proﬁles can be prevented through a compensation moment.

Key words: three-roll-bending, 3D-bending of proﬁles, superposition of stresses.

1 Introduction In recent years, the demand for aluminium and steel proﬁles as important structural elements in traﬃc systems as well as in civil engineering has increased strongly. These proﬁles are used in various ﬁelds of traﬃc engineering in a linear and bent contour (Fig. 1). Bent proﬁles, especially threedimensionally bent proﬁles, provide the design engineer with new degrees of freedom and allow the construction of structures with more advantages regarding e.g. space saving and aerodynamics. Furthermore, the lightweight construction aspect needs to be mentioned by means of which new ways, also through a three-dimensional ﬂexibility in proﬁle shaping, are opened up. For instance, complex parts, so far manufactured by joining diﬀerent parts, can be formed from one piece with improved aerodynamic properties. In addition to their application in automotive engineering, three-dimensionally bent

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Advanced Methods in Material Forming Traffic systems Multipurpose applications

Civil engineering

High precision requirements on straight and curved profiles Lifting and conveying systems

Machines and apparatus construction

Fig. 1. Fields of application of straight and bent proﬁles

proﬁles can be used in several ﬁelds of traﬃc systems like e.g. rail vehicles, utility vehicles, and aerospace. Also in the ﬁelds of architecture and design, for instance in the furniture industry, 3D-structures open up new shaping opportunities. Bent proﬁles as design elements have to meet highest quality demands concerning shape and measurement accuracy. A high bending quality is an important condition for subsequent production steps like assembly and welding, and can therefore reduce the costs of the entire production chain. In some of the above mentioned application ﬁelds, for instance in the automotive industry, high tolerances of curved proﬁles are required which can not be achieved by conventional bending procedures or only at large expenditure. In this case, the direct assembly of proﬁles after bending and their calibration by means of hydroforming before use have to be diﬀerentiated. By this procedure, complex three-dimensionally curved proﬁles can be manufactured in a single manufacturing step including reproducibility of highest quality claims. The quality of the hydroformed workpiece and the reliability of the process is, however, deﬁned by the material and geometrical properties of the initial parts (Chatti 05). For the calibration process certain small shape deviations of the proﬁle contour are sometimes allowed. However, this depends strongly on the complexity of the hydroforming tool contour so that keeping up high accuracy requirements in pre-bending operations represents an important precondition for reaching high process reliability in hydroforming. In case of direct assembly of the proﬁles after bending the quality requirements concerning

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shape and measurement accuracy need to be rated up. This proceeding oﬀers, however, the advantage of economising the relatively expensive process step of hydroforming. Nowadays, three-roll-bending is one of the bending procedures mostly used in industrial manufacturing. With only one set of rolls and by setting only one axis it is possible to achieve diﬀerent radii or a radii distribution along the proﬁle length within the bending plane. In addition to the high ﬂexibility, and thus economy of this procedure, it can achieve high quality standards, too, if suitable measures are used. Some of the measures developed in recent years at the Institute of Forming Technology and Lightweight Construction (IUL), University of Dortmund, aiming at the improvement of two-dimensional-bending of proﬁles are the simulation of three-roll-bending processes by means of semianalytical and ﬁnite-element programmes (Chatti 98, Chatti et al., 01), the conception and establishment of closed-loop process control systems based on algorithmic and neuro-fuzzy controllers (Chatti et al., 03), and the development of adapted methods for the reduction of the cross-sectional deformation during the bending process (Chatti et al., 99). For the three-dimensional-bending of proﬁles a second machine axis is required to deﬂect the proﬁles towards the third direction. At the IUL, a special device is developed for bending proﬁles in the 3D-space, beneﬁting from the previous plastic deformation in the 2D-bending zone of the threeroll-system.

2 State of the Art of the 3D-Bending of Proﬁles Machines for cold forming of proﬁles have already been used since the beginning of the twentieth century. With these machines almost all proﬁle cross-sectional shapes can be formed to diﬀerent workpiece shapes (Kajrup et al., 85). Due to the great number of possible proﬁle cross-sections, bending shapes, demands on the component, and the lot sizes of mass-production, a variety of diﬀerent bending procedures has been used in practice. Some of these procedures are limited in the applicability regarding the proﬁle length, the proﬁle cross-sectional shape, or the proﬁle radius, others are not ﬂexible and involve very high tooling expenses. The procedures of three- and fourroll-bending are universally applicable, whereby three-roll-bending represents the most frequent machine concept in industrial application because of the relatively low machine costs (Chatti et al., 99a, Chatti et al., 03a). Figure 2 shows some bending procedures which are suitable to bend proﬁles threedimensionally. The proﬁle bending procedures can be subdivided into methods of “kinematic shaping” and such of “form-maintaining shaping” (shaping with shape-deﬁning rigid tools) according to the way they are used to form the desired shape. Bending without lateral forces as well as the roll-bending procedures of two-, three-, and four-roll-bending rank among the ﬁrst group.

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Basic stretch bending

Tangential stretch bending

3D-unwind stretch bending

Three-roll-bending

Multi-roll bending

3D-Freeform bending

Profile bending with elastic pad

Laser bending

Rounding during extrusion

Fig. 2. Procedures for 3D-bending or rounding of proﬁles

The second group comprises of the procedures proﬁle bending under presses, stretch bending, and wiper bending. Furthermore, there are more procedure variants of each procedure whose application opportunities are extended by the use of bending tools adapted to the bending task. These are partially procedures which combine bending processes of the two groups (Chatti 98). An exception is laser bending. Laser bending is a new forming technique which allows to form sheet metals and proﬁles by thermal residual stresses instead of external forces (Kraus 97). The kinematic bending processes are more ﬂexible than the formmaintaining processes. The ﬁnal shape of the part is not determined by the shape of the tool, but rather by the relative movements of tool and workpiece. Normally, the ﬁnal shape is produced by a number of successive steps that can be changed easily from workpiece to workpiece, yielding the high ﬂexibility of these processes. Form-maintaining shaping is deﬁned as forming with shape-deﬁning rigid tools that contain the desired workpiece geometry, especially with respect to the curvature, corrected by the springback of the proﬁle during unloading. Due to the ﬁxed geometry of the tool, the geometry of the workpiece is also ﬁxed, leading to high reproducibility and a shorter processing time in many cases (Chatti 98, Vollertsten et al., 99). The following procedures and machine concepts have been developed under the aspect of a purposeful superposition of stresses aiming at the reduction

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of forming forces, proﬁle springback, and/or cross-sectional deformation, or they have an especially high potential for bending proﬁle components three-dimensionally. Stretch bending belongs to the procedure group with form-maintaining shaping and is mainly used in automotive industry for mass production. This procedure allows the manufacturing of three-dimensionally bent proﬁles as well as the insertion of an additional force for inﬂuencing the stress state in the forming zone. The principle of stretch bending, which is subdivided into the procedures tangential stretch bending and unwind stretch bending, is based on the fact that the material is completely plastiﬁed before or during bending by means of a tensile device stretching the proﬁle in longitudinal direction. This stress superposition results in larger strains in the tensile zone and smaller strains in the compression zone of the workpiece during the subsequent bending process, leading to a displacement of the neutral ﬁbre towards the compression zone and to reduced cross-sectional deformation and springback (Sp¨ ath 91, Sp¨ ath 91a, Sprenger 99, Weippert 97). The advantages of high shape and measurement accuracy as well as high reproducibility of stretch bending are faced, however, with the following disadvantages restricting the suitability of this procedure for mass production: – –

Low ﬂexibility (for each proﬁle a new tool is necessary). Very expensive tools and machines (high number of machine axes and complex tool shapes). – Increasing manufacturing costs when bending long proﬁles and big crosssections. A procedure variant for kinematic bending of ﬂat materials and open proﬁles is the roll-bending procedure with superposition of compressive stresses. This procedure variant was carried out within the scope of a project at the IUL (at that time LFU) in the late eighties by Adelhof, Kleiner, and Liewald (Finckenstein et al., 88). In this project, a new bending variant was developed which combines both processes bending and rolling. The high eﬃciency of this bending variant consists in the local plastiﬁcation of the bending cross-section through a longitudinal rolling process, which leads to a substantial decrease of the bending resistance and the bending forces. Figure 3 shows the principles of the roll-bending procedures with a superposition of compressive stresses. A sheet metal or angle proﬁle is rolled between two rolls with adapted shapes and its sheet thickness is reduced in such a way that it reaches a plastiﬁcation of the cross-section between the rolls. A bending device consisting of a bending roll is placed directly behind the rolling stand and moved horizontally or vertically perpendicular to the rolling direction. Following the bending roll motion, the proﬁle is bent to the corresponding direction. The low force at the bending roll shows that the superposition of the compressive stress is mainly responsible for the creation of the proﬁle curvature and not the bending moment.

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Fig. 3. Combined rolling-bending processes

This bending variant showed the following advantages compared to a conventional bending process: – – –

Achievement of higher true strains. Reduction of the cross-sectional deformation. Improvement of the strength properties.

Regarding the 3D-bending, no investigations have been carried out so far using this procedure. In industrial manufacturing three-roll-bending is one of the most commonly used procedures for proﬁle bending because of its high ﬂexibility and its relatively low machine costs. With only one set of rolls and by setting only one axis it is possible to achieve diﬀerent radii or a radii distribution along the proﬁle length. Within a closed-loop process control only one axis has to be controlled as well in order to compensate material and cross-section variations (Chatti et al., 04, Chatti et al., 03). 3D-bending of proﬁles is not directly possible when using a conventional bending machine because the adjustment of the bending rolls leads to a proﬁle bending in the working plane of the machine. However, some standard machines are provided with auxiliary adjustable guiding rolls which allow the bending of screws with a small pitch and hand rails for spiral staircases (through simultaneous rotation of the circular cross-section) or the compensation of twisting moments of asymmetrical proﬁle cross-sections (N.N. 98, N.N. 05). These cases do not concern, however, 3D-bending. To bend stringer proﬁles for airplanes with radii between 5000 mm and 12000 mm, the company Boeing uses four-roll-bending machines of L&F (N.N. 98a). These machines are provided with a multi-axial adjustable exit roll for torsion compensation of strong asymmetrical stringer proﬁles and the production of small three-dimensional curvatures according to the shape of the plane fuselage. Although a superposition of stresses exists here during the 3D-bending the forming forces in the forming zone are not reduced because the third axis is still within the forming zone of the ﬁrst bending plane. Here there is no division between the procedures plastiﬁcation and creation of the curvature in the second plane. Due to the high forces acting on the proﬁle only open proﬁles can be formed, but no proﬁles with undercuts like e.g. double-Tproﬁles or C-proﬁles. Nor can Proﬁles with high true strains (tight radii, high webs) be bent by this procedure. Hollow proﬁles are not intended because

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of the expected high cross-sectional deformation. This procedure seems to be suitable for large radii used in airplane industry. However, it is not usable for complex contours with tight radii or radii distributions as those in proﬁles for automotive industry. To further improve the workpiece guidance during bending, also multi-rollbending machines are used in industry. These machines enable the bending of proﬁle curvatures in opposite direction as well as the production of threedimensionally bent proﬁles. Such 3D-shapes (mainly alternate bending) can only be achieved, however, when bending tubes (circular cross-section). In recent years, also full ﬂexible 3D-freeform bending machines have been developed which are based on the principle of kinematic shaping. The working principle of these machines consists in placing the proﬁle within a guiding device before forming. The proﬁle is afterwards pushed by means of a pusher through a ﬁxed bending tool and through a movable die. The proﬁle bending is produced by a given motion of the die at a simultaneous feed-in motion of the longitudinal proﬁle axis. Due to the kinematic shaping a tool change is not necessary when changing the bending contour for the same proﬁle crosssections, only a new deﬁnition of the relative motion between proﬁle feed-in and ﬁxed as well as movable tool is required (Neugebauer et al., 01). This machine concept is characterised by a restricted proﬁle (tube) length due to the limited length of the pusher. In addition, the forming forces dependence of the proﬁle stiﬀness and the lever arm length is relatively high compared to the kinematic procedure of three-roll-bending. Accordingly, the propulsion technology of the machine is very expensive. In coherence with the high forming forces, cross-sectional deformations and friction problems at the deﬂecting dies are to be taken into consideration. This procedure idea has been realised in diﬀerent machine concepts. Well known examples are the Hexabend (Neugebauer et al., 01), MiiC (N.N. 05a, N.N. 05b), and Nissin (N.N. 05c, N.N. 05d) machines. During the kinematic procedure proﬁle bending with elastic pad the proﬁle is formed between a rigid roll and a polyurethane pad. The roll is pivoted and can be moved vertically. The die is embedded in a rigid steel retainer which can be moved horizontally. The proﬁle is pressed into the pad by the vertical motion of the roll. The pressure on the proﬁle produces the bending moment necessary for forming (Geiger et al., 95, Arnet 99). By using this process it is possible, in principle, to bend proﬁles to spiral shapes and, thus, 3D-contours. The proﬁles have to be placed on the pad with a certain angle to the motion direction of the retainer. A similar forming result could be achieved when the bending roll axis would be arranged not laterally, but inclined to the retainer direction. Here, the high control expenditure which is necessary for the production of reproducible components is problematic. When extending the procedure for 3D-proﬁles the variation of the inﬂuencing parameters will also increase inevitably. Another disadvantage for 2D- and 3D-shapes is the restricted proﬁle length due to the limited pad. Furthermore, a second restriction to some kinds of proﬁles needs to be taken into account

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due to the limited loadability of the polyurethane pad. Only proﬁles with low bending resistance can be bent by this procedure. For the manufacturing of curved proﬁles made of non-metallic materials the rounding during extrusion, which has been developed at the Institute of Forming Technology and Lightweight Construction of the University of Dortmund, competes with the above mentioned bending procedures. In this procedure with kinematic shaping the workpiece is directly rounded during bar extrusion. An external guiding tool applies a lateral force directly after the extrusion die to the leaving proﬁle so that it is deﬂected (Arendes 99, Arendes et al., 99, Arendes et al., 00, Kleiner et al., 00, Klaus 02). The proﬁle forms a circle with a radius depending on the position of the guiding tool in relation to the die. By varying the position of the guiding tool during the extrusion process it is even possible to form shapes with varying radii or combinations of curved and straight shapes. The rounding of the proﬁle is provoked by the change of velocity distribution in the die due to superposed tensile and compression stresses as well as by lateral forces. Because the material in the extrusion die is fully plastic the procedure shows outstanding properties regarding the manufacturing of 2D- and 3D-curved proﬁles. In comparison with bending processes, rounding during extrusion shows some advantages like unreduced proﬁle formability, low cross-sectional deformation, low residual stresses, and, above all, very low proﬁle springback. However, the procedure is limited to certain minimum proﬁle radii which do not cause a collision with the extrusion die opening. Thus, rounding during extrusion is not suitable for the manufacturing of very tight radii. Furthermore, only extrudable alloys (extruded proﬁles) can be formed, which strongly reduces the comparability with bending processes (Becker et al., 03). In these cases, the procedure rounding during extrusion can not replace the bending process. The basics of this procedure and the manufacturing of constant as well as variable two-dimensional curvatures have been already investigated. The production of 3D-curved proﬁles requires, however, the use of a multi-axial guiding tool and is being investigated within the scope of a part project of the Collaborative Research Centre Transregio (SFB/TR10) of the German Research Foundation (DFG).

3 Design of a Suitable Procedure for 3D-Bending The bending procedures mentioned in the state of the art chapter aim at solving the classical problems of proﬁle bending. These are very complex and versatile; however, they can be listed simply in a few points: –

Cross-sectional deformations which can be wrinkles in the compression zone, wall thickness reductions in the tensile zone, reductions of the crosssection height, etc. To this group also belongs the problem of torsion when bending asymmetrical proﬁles.

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–

Inaccurate bending contour which results from an unknown bending behaviour often connected with variations of the proﬁle material. – Technical material changes which arise during the bending process and have a signiﬁcant inﬂuence on the component usability. Material hardening and, for example, too high residual stresses count to this phenomenon. For this problem list diﬀerent solution approaches are available, whereby the procedure stretch bending, roll-bending with superposed compression stress, and rounding during extrusion can be evaluated as very promising. The common ground is the superposition of stress states in bending and rounding respectively. This leads to an additional plastiﬁcation of the material in the bending zone, which is obviously advantageous for the forming process. Rounding during extrusion oﬀers solution approaches for the whole problem spectrum, including the material and springback problems because the material in the die of the extrusion press is fully plastic. Thus, cross-sectional deformations are reduced and the material receives no signiﬁcant hardening eﬀects in forming. The other mentioned procedures try to approximate a plastic state at room temperature. In stretch bending, for example, this state is reached through a superposition of a tensile stress, in the combined rolling-bending procedure through the superposition of a compression stress. This proves the importance of a workpiece pre-plastiﬁcation, whereby the three mentioned procedures also show obvious disadvantages or restrictions (see state of the art) which justify new considerations for the realisation of a new procedure variant. For these reasons it was necessary to carry out pre-investigations in order to develop a new bending procedure for the 3D-bending of proﬁles. It seemed to be reasonable to ﬁnd and systematically investigate new opportunities for the purposeful inﬂuencing of the forming zone in the sense of superposition of stresses. The aim of designing a procedure which exhibits the advantages of kinematic shaping and which also keeps the cross-sectional deformations and springback low is achievable by the combination of a procedure for the purposeful plastiﬁcation of the forming zone with a kinematic procedure for the creation of the desired curvature. Examples for such tendencies are described in the state of the art chapter. This advisement in conjunction with the second demand to develop a concept for the 3D-bending of proﬁles requires a systematic proceeding, taking into account the question: which combinations of bending procedures for the creation of the 3D-contour with the forming procedures according to DIN 8582 (Lange 90) for the plastiﬁcation of the forming zone are reasonable? To create new procedure variants, a proceeding for design systematics recommended in (Pahl et al., 04) has been chosen. Thus, a schema could be established which arranges the procedures according to DIN 8582, belonging to the groups of compression, compression-tensile, tensile, bending, and shear forming in combination with a bending procedure. Due to the claimed

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ﬂexibility in the 3D-eﬃciency of the procedure the parameter of kinematic shaping of the actual bending process was kept constant in this combination strategy. As a result, already known procedures arose from this proceeding, but also new innovative approaches whose technical feasibility had to be evaluated afterwards. From these possible procedure combinations the superposition of the threeroll-bending for plastiﬁcation of the forming zone with the ﬂexible procedure of freeform bending for 3D-bending was selected for the realisation. Because bending represents the simplest forming variant for a proﬁle, the insertion of a bending stress into a favourable plane involving the lowest cross-sectional deformation can be used in order to plasticize the material locally. Afterwards, a second bending procedure can be superposed which produces the desired proﬁle contour. The selection of the three-roll-bending is justiﬁed by the wide industrial use, high ﬂexibility, and low tool and machine costs of this procedure. Another advantage is the fact that the combination of these two procedures can be seen as an extension of a conventional three-roll-bending machine. The kinematic 3D-bending of the proﬁle running through the rolls can be carried out by means of a deﬂecting device inﬂuencing the forming process between the three rolls. The proﬁle curvature, which is normally created by three-roll-bending, could also be irrelevant so that the ﬁrst process only serves the creation of the aforesaid plastic and partly plastic zone respectively. Irrespective of these aspects, the actual curvature is produced kinematically. The cost eﬀectiveness given by this procedure idea should be emphasised. In all other possibilities found higher costs are expected due to more complex technology or higher forces. Figure 4 shows the tool arrangement of the combined procedure which has been realised at the Institute of Forming Technology and Lightweight Construction.

Three-roll arrangement

z

d x Movable deflecting tool

y

Motion direction of the profile

Fig. 4. Superposed three-roll-bending with subsequent proﬁle deﬂection

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4 Machine Set-Up and Working Principle The experimental set-up for this new way of proﬁle bending consists of two essential parts. First, there is a conventional CNC-three-roll-bending-machine available at the IUL used for the bending in the ﬁrst plane. The only change made at this machine is a special drive system to couple and to drive the ﬁrst roll by the middle roll. This is the solution for expected friction problems between proﬁles and rolls. The second important part of the set-up for the realisation of the experiments is a special device that makes it possible to deﬂect the proﬁles towards the third axis directly after the ﬁrst bending process. The force for the bending in this axis is given by a hydraulic cylinder which is charged by a proportional valve controlled by a measuring PC. The path measurement is integrated in the cylinder. The controlling programme is realised with the lab software DIAdem. A hydraulic power unit produces the pressure for the cylinder. For the ﬁrst experimental phase the only moving axis is the hydraulic cylinder (z-axis). The remaining degrees of freedom (x and y-axis) are realised without a drive. They are operated manually and then ﬁxed by clamp-connections. This is satisfactory for the ﬁrst experiments because they can be made in each position by moving only the z-axis. Figure 5 and 6 show the CAD-model of the machine with the designation of the axis and a photo of the machine during an early bending experiment. To reach a high stiﬀness, the machine frame is designed as a welded construction built of great hollow bending beams. This increases the accuracy of the whole machine and the reproducibility of the results. The guidance of the proﬁle after leaving the three-roll-system works with a window ﬁxed on the hydraulic cylinder axis and furnished with Teﬂon-plackets to reduce the friction at the contact zone.

Hydraulic axis

d z

y x

Three-roll-bending machine Machine frame

Fig. 5. Machine set-up

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Fig. 6. Bending device for 3D-proﬁles

5 First Investigation Results The developed procedure and the realised machine set-up allow the investigation of following scenarios: – – – – –

2D-bending of symmetrical proﬁles with variable concave and/or convex curvature, 3D-bending of symmetrical proﬁles with constant or variable curvature, 2D-bending of asymmetrical proﬁles with twisting compensation, 3D-bending of asymmetrical proﬁles with constant or variable curvature, and 2D or 3D-bending of proﬁles with superposition of twisting stresses. This is possible with an additional twisting unit coupled with the hydraulic cylinder.

The objective of the procedure investigations was to prove the existence of certain eﬀects which have the potential for improvements in the area of threedimensional bending of proﬁles. It was important to highlight some tendencies for solutions of classical bending problems which are not solved or only solved with high eﬀorts by many other procedures. First experiments carried out at the IUL include investigations to achieve 2D and 3D-bent symmetrical proﬁles. As symmetrical proﬁles, a hot-rolled

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hollow section (20 × 20 × 2 mm) made of S 235 (EN 10 025), and an extruded proﬁle with the same cross-section made of AlMgSi0.5 were chosen. The advantages oﬀered by these simple and compact proﬁles are a small deformation of the cross-section and low bending forces causing no problems with the stiﬀness of the machine and no bending problems in ﬁrst experiments, thus allowing to concentrate on the eﬀects of the procedure. Due to the high number of degrees of freedom, and therewith the great variety of inﬂuencing factors, a lot of possibilities to set the machine parameters are available. To limit the investigation expenditure, the experiments had to be reduced to some signiﬁcant combinations of inﬂuencing factors, but still allowing to get ﬁrst statements about the combined process. The arrangement of the rolls and the deﬂecting tool is shown schematically in Fig. 7. Also important are here the machine zero points which should be selected in order to enable a comparison between the diﬀerent experimental results. The zero point of the d-axis is deﬁned as the position in which the proﬁle has a force-free contact (no curvature) with all three rolls. The zero point for the x-axis is put on the axis of the second adjustable roll; the distance to the deﬂecting tool represents the axis value. The zero point of the y-axis is situated on the axis of the ﬁxed middle roll. Similar to the d-axis, the

Three-roll arrangement

d=0 y=0

dpos

ycur

x=0

Rollenabstand a

xpos

xcur Deflecting device with hydraulic z-axis y pos

Profile position at d = 0 mm Machine frame

Fig. 7. Schematic representation of the axis system of the machine

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z

5

force-free contact between proﬁle and deﬂecting tool in the z-axis is the position of the zero point. For 3D-bending of proﬁles both the 3-roll-bending machine (d-axis in xyplane) and the hydraulic device (z-axis) have been adjusted and synchronised to each other in order to produce the desired proﬁle contour (Fig. 8). The ﬁrst proﬁle bending in the xy-plane has been superposed with the second one, which is deﬁned by the position of the hydraulic device and the leading window in the x, y, and z-axis. The most extreme position for the y-axis, for example, is the inversion of the contour reached in three-roll-bending. To use and to make evident the advantage of the previous plastic deformation in the three-roll-bending zone for an easier bending in the orthogonal plane, diﬀerent roll adjustment values (d = 0 mm, 13 mm, and 16 mm) have been selected at the middle roll and a superposed force of the hydraulic cylinder with the same adjustment value for the z-axis of 40 mm has been used. In order to obtain a better comparability of the results, the adjustment of the y-axis has been selected in such a way that the bending within the

R 41

9

R 71

y x

Fig. 8. 3D-bending of proﬁles by superposition of bending stresses

x = const. y = const. z = const.

d = 16 mm d = 13 mm d = 0 mm

Radius in mm

2000 1500

d = 13 mm, z = 40 mm, y = –35 mm

1000 500 d = 16 mm, z = 40 mm, y = –35 mm 0

0

50

100 150 Profile length in mm

200

250

Fig. 9. Inﬂuence of previous plastiﬁcation on bending radii

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three-roll-system is neutralised by the deﬂecting tool. This is reached by an adjustment of the y-axis of −35 mm. Thus, only a curvature in xz-plane is created. After an evaluation of the produced contours by means of the coordinate measuring machine PRISMO VAST 5 HTG of the company Zeiss the bending results in the form of radii distributions can be seen in Fig. 9. The radii variations are attributed to surface unevenness, variations of the friction ratio during the process, and variable proﬁle properties. It was found out that the greater the middle roll adjustment causing a higher previous plastic deformation of the proﬁles the greater the proﬁle curvature (smaller proﬁle radius) in the third plane will be. A cause for the curvature increase is the decrease of the proﬁle springback due to the stress superposition. A control of the proﬁle contour and a minimisation of the springback values for all proﬁles would be possible in future developments after more precise investigations of the process parameters.

Ground plate with high stiffness

Force sensor 14

Force in z-axis in kN

12 10 8

d = 0 mm, z = 50 mm

6

d = 13 mm, z = 50 mm d = 16 mm, z = 50 mm

4 2 0

0

5

10

15 Time in s

20

25

30

Fig. 10. Force measurement set-up and inﬂuence of previous plastiﬁcation on bending forces

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In direct correlation with the produced radii it is of interest to make a statement about the forming forces prevailing in the machine axis which are responsible for the bending moment. To investigate the inﬂuence of the previous plastiﬁcation on the bending forces, a device for force measurement consisting of a force sensor put on a stiﬀ plate in the z-axis has been used (Fig. 10). This device allows the direct measurement of the force values in the deﬂecting tool. The investigations have shown that an increase of the d-adjustment causes a decrease of the bending force in z-axis. For comparison reasons two proﬁles have been bent without d-adjustment (without plastiﬁcation, only use of the guiding characteristics of the rolls). In this case, higher force values are shown with the same selected adjustments of the z-axis. The force decrease due to pre-plastiﬁcation should be related to a smaller crosssectional deformation of the proﬁles. These results emphasise the importance of the previous material plastiﬁcation for reaching the advantages described in Sect. 2.

6 Summary For the manufacturing of 3D-curved proﬁles with arbitrary contours, as e.g. signiﬁcant lightweight structure elements, expensive procedures and machines as well as procedures with low ﬂexibility regarding contour changes are used at present. These negative aspects aﬀect the ﬁeld of prototyping and the trend to reduce the number of parts because of an increasing individuality request by the customer. This leads to the demand for a proﬁle bending procedure allowing a free deﬁnable contour which can be manufactured and changed at anytime without high tool costs. In this paper, a new procedure variant for the ﬂexible 3D-bending of proﬁles, based on the aspect of stress superposition, is presented. The procedure variant is based on the superposition of the kinematic three-roll-bending with a procedure of freeform bending. The combination of these procedures creates a design division between a plastiﬁcation of the proﬁle in the forming zone of three-roll-bending and a kinematic production of the bending contour by a subsequently arranged deﬂection device. The aim of the superposition of stresses is the extension of the procedure limits, i.e. the increase of the maximal achievable curvatures at a simultaneous reduction of the forming forces, and thus of the cross-sectional deformations. The eﬀorts aim at the change of the producible shape with high ﬂexibility and without high tool expenditure by means of the kinematic shaping. The feasibility of the procedure and the use of the previous plastic deformation to improve or facilitate the threedimensional-bending of proﬁles have been shown in this paper. To achieve accurate proﬁle contours in the 3D-space which meet the high demands of the industrial practice, a speciﬁc control of the bending device adjustment is necessary.

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References Arendes D., Direkte Fertigung gerundeter Aluminiumproﬁle beim Strangpressen. Dr.Ing. thesis, University of Dortmund, Shaker Verlag Aachen, 1999. Arendes D., Chatti S., Kleiner M., “Forming of Aluminium Extrusions for Structural Elements”, 6th ICTP – International Conference on Technology of Plasticity, Nuremberg, 1999, p. 2337–2342. Arendes D., Kleiner M., Kalz S., Kopp R., “Strangpressen gerundeter Aluminiumproﬁle”, Aluminium, vol. 76 (2000) no. 3, p. 141–148. Arnet H., Proﬁlbiegen mit kinematischer Gestalterzeugung, Dr.-Ing. thesis, University of Erlangen-Nuremberg, Meisenbach Verlag Bamberg, 1999. Becker D., Klaus A., Kleiner M., “Innovative Fertigung von 3D-gekr¨ ummten Strangpressproﬁlen”, ZWF, issue 9, 2003. Chatti S., Optimierung der Fertigungsgenauigkeit beim Proﬁlbiegen, Dr.-Ing. thesis, University of Dortmund, Shaker Verlag Aachen, 1998. Chatti S., Finckenstein E. v., Kleiner M., “Reduktion der Querschnittsverformung beim Biegen von Proﬁlen”, Blech Rohre Proﬁle, vol. 46 (1999) no. 3, p. 38–42. Chatti S., Finckenstein E. v., Kleiner M., “Verfahrenskombination steigert Flexibilit¨ at beim Biegen von Proﬁlen”, Maschinenmarkt, vol. 11 (1999) no. 3, p. 26–30. Chatti S., Klimmek Ch., Kleiner M., “System for Optimisation of Product and Process Design in Proﬁle Bending”, Proceedings of the 4th International ESAFORM Conference on Material Forming, Liege, April 23–25, 2001, vol. 2 (2001), p. 785– 788. Chatti S., Dirksen U., Kleiner M., “Process Control Strategies for Proﬁle Bending”, Proceedings of the 10th SheMet International Conference on Sheet Metal, Ulster, Northern of Ireland, April 14–16, 2003, p. 313–320. Chatti S., Heller B., Ridane N., Kleiner M., “Anpassungsf¨ ahig – Prozesskette zur Herstellung von Leichtbaustrukturen aus Tailor Rolled Blanks”, Maschinenmarkt, vol. 7, 2003, p. 24–27. Chatti S., Dirksen U., Kleiner M., “Optimization of the Design and Manufacturing Process of Bent Proﬁles”, Journal of the Mechanical Behaviour of Materials, vol. 15, no. 6, 2004, ISSN 0334-8938, p. 437–444. Chatti S., Production of Proﬁles for Lightweight Structures, habilitation thesis, University of Franche-Comt´e, France, books on demand, 2005. Finckenstein E. v., Adelhof A., Kleiner M., “Erweiterung der Verfahrensgrenzen beim Hochkantbiegen im Walzrundverfahren”, Blech Rohre Proﬁle, vol. 35 (1988) no. 10, p. 783–787. Geiger M., Arnet H., Engel U., Vollertsen F., “Flexibles Biegen stranggepresster Aluminium-Proﬁle”, Blech Rohre Proﬁle, vol. 42 (1995) no. 1, p. 31–34. Kajrup G., Klein W., “Proﬁlrundbiegen – Anwendung und Entwicklung”, Blech Rohre Proﬁle, vol. 32 (1985) no. 8, p. 419–423. Klaus A., Verbesserung der Fertigungsgenauigkeit und der Prozesssicherheit des Rundens beim Strangpressen, Dr.-Ing. thesis, University of Dortmund, Shaker Verlag Aachen, 2002. Kleiner M., Arendes D., Klaus A., “Verfahren und Vorrichtung zur Ver¨ anderung der Austrittsrichtung des Pressgutes beim Strangpressen” German Patent, 10.02.2000.

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Kraus J., Laserstrahlumformen von Proﬁlen, Dr.-Ing. thesis, University of ErlangenNuremberg, Meisenbach Verlag, 1997. Lange K., Umformtechnik, Handbuch f¨ ur Industrie und Wissenschaft, vol. 3: Blechumformung, Springer, Berlin, 1990. Neugebauer R., Blau P., Drossel W-G., “3D-Freiformbiegen von Proﬁlen”, ZWG, 2001, p. 11–12. N. N., Vorrichtung zum Wendelbiegen, company brochure, Franz Thoman Maschinenbau, 1998. N.N., General Overview CNC Contour Roll Forming Machine, company brochure, October 1998, USA. N. N., www.kaercher-zell.de, Website K¨ archer GmbH, 2005. N.N., www.miic.de. Website of MiiC & Co GmbH, 2005. N.N., Multi-Bender, company brochure, MiiC & Co GmbH, 2005. N.N., Highly improved Functions and Productivity for Tube Bending, company brochure, Nissin Precision Machines Co., Ltd., 2005. N.N., www.nissin-precision.de, Website Nissin Precision machines Co., Ltd., 2005. Pahl G., Beitz W., Feldhusen J., Grote K.-H., Konstruktionslehre, 5th edition., Springer, 2004. Sp¨ ath W., “Gute Chancen – Gebogene Rahmenproﬁle aus Magnesium mit komplizierten Querschnitten automatisch herstellen in Großserien”, Maschinenmarkt, W¨ urzburg, vol. 97 (1991) no. 29, p. 56–61. Sp¨ ath W, “Biegen großer Strangpressproﬁle f¨ ur Aluminiumwagen”, Aluminium, vol. 67 (1991) no. 6, p. 556–557. Sprenger A., Adaptives Streckbiegen von Aluminium-Strangpressproﬁlen, Dr.-Ing. thesis, University of Erlangen-Nuremberg, Meisenbach Verlag Bamberg, 1999. Vollertsten F., Sprenger A., Krause J., Arnet H., “Extrusion Channel and Proﬁle Bending – A Review”, Journal of Materials Processing Technology, vol. 87 (1999) p. 1–27. Weippert R. G., Das adaptive Streckbiegen von Aluminiumhohlproﬁlen, ein Beitrag zum integrierten Technologie-und Innovationsmanagement, VDI Report, 2: Fertigungstechnik, no. 438, Dr.-Ing. thesis, ETH Zurich, 1997.

Specimen for a Novel Concept of Biaxial Tension Test – Design and Optimisation W. Hußn¨ atter, M. Merklein and M. Geiger Chair of Manufacturing Technology, Friedrich-Alexander-University of Erlangen-Nuremberg, 91058 Erlangen, Germany, [email protected], [email protected], [email protected] Summary. Although in times of highest signiﬁcance of process modelling and numerical simulation yield loci get more and more important, these characteristic values do only exist for a few materials primarily at room temperature. In this paper a novel concept of an experimental setup is introduced, with which plastic yielding of sheet metal can be examined also at elevated temperatures. The design of specimen is of great importance for the quality of experimental results, because stress conditions and with it forming behaviour are constituted. Thus, it must be optimized in consideration of stress singularities at corners and for achieving stress states, which are comparable to those of previous yield locus examinations. Most information and details can be obtained from ﬁnite element simulation.

Key words: sheet metal forming, FE-simulation, yield locus.

1 Introduction Numerical simulation by using ﬁnite element method is state of the art not only at scientiﬁc institutes but also in industry. However, the quality of computational results is just as good as the abstract model, i.e. modelling of geometry, kinematics and of course material properties, which are of primary importance in case of mechanical simulation of forming process. Depending on the speciﬁc theory which is used to describe the material behaviour several variables (e.g. tensile strength, anisotropy, yield locus and others) can be taken into account for modelling sheet metal forming. In comparison with parameters that are determined in uniaxial experiments (e.g. uniaxial tensile test) yield locus deﬁnes a starting point of plastiﬁcation as a function of biaxial stress condition (Fig. 1). Since a priori calculation of this material curve is not possible, experimental investigations are necessary. Although determination of yield locus

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Fig. 1. Experiments necessary for determination of yield loci

requires great eﬀorts and no internationally standardized procedure is established, it is mostly used in favour because of its consistent representation of material characteristics according to real sheet metal forming process. Additionally, application of yield loci enables the qualiﬁcation and even quantiﬁcation of hardening behaviour of unknown materials (Grewolls et al., 01; M¨ uller 96).

2 Novel Experimental Setup A novel experimental setup was designed at the Chair of Manufacturing Technology (LFT) to determine yield loci of sheet metals and in particular at elevated temperatures (Geiger et al., 03). A scheme of it is shown in Fig. 2. Since an international standardisation of procedure for determination of yield locus is still missing several diﬀerent techniques have been developed. But they all use identical strategy for specimen’s loading: in plane (Banabic et al., 98; Hoferlin et al., 98; Kuwabara et al., 02). Contrary to them, the setup which is introduced in (Geiger, van der Heyd et al., 05) is based on a punchmotion perpendicular to the sheet. Clamping the sheet causes reaction forces in the specimen arms and consequently leads to an elongation. Assuming that the central region of the sheet over the punch is ideal rigid and therefore undilated and neglecting the radii of die and punch the extension of the specimen within this stretch-forming process (Fig. 3) can be exclusively expressed by geometrical values: 2 h2 + li2 = (li + Δli ) , (1a) and so the elongation can be calculated with

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Fig. 2. Scheme of the novel experimental setup developed at LFT

Δli =

h2 + li2 − li .

The strain is deﬁned as the elongation related to the initial length: + 2 h Δli εi = = + 1 − 1. li li

(1b)

(2)

Equation 2 shows that the strains in a specimen are only depending on each arm length. Thus, a systematic selection of diﬀerent geometries allows variable settings for stress conditions in the specimen. A very important feature of this setup is the possibility of local heating of the specimen. Experimental investigations up to 300◦ C can be realised to achieve material data at elevated temperatures. As heating source a 210 W-diode laser with a special optical characteristics is used, which enables a ring-proﬁle of the beam, where the inner (di ) and the outer diameter (do ) are continuously variable (see Fig. 4).

Fig. 3. Elongation of the specimen due to clamping between die and blank holder

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Fig. 4. Scheme of a proﬁle of the laser beam used for the LFT-setup

3 Specimens’ Geometry Using a completely new experimental set-up means to design a novel and special specimen. Starting with theoretical and analytical investigations on the stress distribution in the origin cruciform geometry the notch eﬀect of the cross is determined. In order to achieve homogenous stress distribution the specimen’s design is optimised based on FE-simulation.

3.1 Analytical Modelling Notch Eﬀect Stress concentration in consequence of notch eﬀects is a common problem for components dimensioning. Standard load cases like the uniaxial tension test are listed in tables and diagrams or can be calculated with a so called shape number (αk ), which deﬁnes the well known relation between the maximum stress (σmax ) and the nominal stress (σn ) in (3) and Fig. 5. σmax = αk · σn

(3)

The shape number depends on geometrical quantities and some special factors, which diﬀer for each load case. As it is shown in (Geiger, Hußn¨ atter et al., 05) it can be calculated to αk = 12.95, i.e. the maximum stress in the corner is about 13 times higher than the nominal stress in the centre of the specimen. Stress Analysis The stress distribution in the sheet can be expressed by analysing the linear elastic stress state. Using a complex variable Green and Zerna were able to state the solution of the basic equations of theory of elasticity in a simpliﬁed way (Green et al., 54). England formulated the stress singularity in the corner

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Fig. 5. Stress singularity in a tensile specimen due to the notch eﬀect

basing on an adequate description for the stress analysis of a wedge (England 71). Applying the boundary conditions of the loaded cruciform specimen the stress along an edge can be given as a logarithmic function of the distance to the notch base (Geiger 2 et al., 05): σx = X + Y log (r)

(4a)

with the two constants

2 2 1+ + log (e) 3π 3π σn ≈ 0.905σn and X= 1 1 2 1+ + log (e) 1− 3π 3π 2 4 − 3π σn ≈ −0.590σn . Y = 1 2 1 1+ 1− + log (e) 3π 3π 2 1−

(4b)

Figure 6 illustrates the standardised stress calculated with the stress analysis in comparison to the simulated stress of a FE-model. Both curves show a similar characteristic: The raise of the nominal stress at the origin (r = 0) is followed by a distinct decrease with growing distance to the corner. Deviations of both curves are caused by simpliﬁcations of modelling the analytical and simulative approach. Recapitulating, the theory of elasticity is very extensive and the results achieved do not ﬁt with the results of FE-simulation. Therefore, it is not an adequate tool for analysing the stress state of biaxial sheet metal tensile testing.

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Fig. 6. Standardised stress for analytical and FE-model

3.2 Numerical Simulation Main Parameters of FE-Simulation Figure 7 shows the real specimen and its model for FE-simulation. In the left picture the specimen is coated with a stochastic pattern necessary for optical strain measurement. Obviously, its geometry is double-symmetric and therefore reduction on a fourth of the model is aspired. The diameter of the outer circle is 230 mm, that of the inner one is 200 mm. The cross-arms have a width of 30 mm, they are arranged perpendicular to each other and so build an angle of 90◦ . This break causes stress concentrations which have to be avoided by developing a special designed geometry for specimen. The,essential data of diﬀerent FE-model types are compared in Table 1, where symbolises the modelling of the whole specimen. Other shapes of geometry are the cruciform specimen (+), where the round area which is clammed during the experiment is omitted, and only a quarter of the specimen (L) because of its double-symmetric property.

Fig. 7. Real cruciform specimen (left) and corresponding FE-model (right)

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Table 1. Real cruciform specimen (left) and corresponding FE-model (right) + + + L element type no. of elements no. of nodes friction no. of increments CPU-time (h)

C3D8I 16012 32017 no 49 2.65

C3D8R 20925 41038 no 55 2.42

C3D8I 14898 30580 no 73 4.40

C3D8R 14898 30580 no 40 0.95

C3D8I 14898 30580 yes 94 4.92

C3D8I 8246 12936 yes 117 1.73

It is a fact, that the lower the invest for computation is the faster a numerical result can be calculated, even though it is more imprecise. Therefore an optimised net of ﬁnite elements must be determined for each work piece that is investigated. Bigger elements can be used in parts of the geometry of marginal stress and strain gradients, whereas smaller elements are necessary in regions of high expected gradients for the exact calculation. In case of the cruciform specimen (Fig. 7), which is drawn by the rolls of the punch two regions are of special interest: On the one hand the central part of the specimen with stress concentration due to the 90◦ -angle between the arms and on the other hand that part of each arm, where contact with the punch occurs. All models are built up by 8-node linear brick elements (C3D8), but for those two diﬀerent integration methods can be used. Equilibrium is aimed at 8 integration points (i.e. one for each node) for type C3D8I, whereas C3D8Relements use the reduced integration, i.e. only one integration point in the centre. Of course this method leads to a decreasing calculation time which is comparable to a reduction of elements’ number. But at the same time the results become inexactly due to averaged values in ﬁelds of large stressgradients. Therefore, it is not acceptable to calculate with reduced integration, so simpliﬁcation of the model itself is the only way to achieve higher computational speed. For further investigation that geometry which represents only a quarter of the specimen is chosen. Since the punch is moved with a velocity of only a few tenth of a millimetre per minute the process is quasi-static and therefore the implicit solver of ABAQUS standard is used. Friction is included as pure rolling and can be calculated with a friction coeﬃcient μR = 0.00625, i.e. 2.5% of μH , the coeﬃcient for dynamic friction. It has no inﬂuence on convergence behaviour and no signiﬁcant eﬀects on calculation time. Improvement of Stress Distribution Biaxial loading of a cruciform specimen let expect stress-concentration where two arms coincide. Bending eﬀects by the force transmission of the punch can

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Fig. 8. FE-simulation of loading an AA 6016

also cause major stresses where the punch gets in contact with the specimen, on the rotating rolls. This fact is conﬁrmed by a FE-simulation (Fig. 8). The calculations have been done for the aluminium alloy AA 6016 which is not only a well known, but also very important material for automotive industry and other ﬁelds of light weight applications. The simulations are based on ﬂow curves which were experimentally determined in tensile tests according to DIN 10002. Moving the punch 5 mm a maximum von Mises stress of 42 N/mm2 occurs at the corner and also the stresses on the rolls (34 N/mm2 ) highly exceed that in the centre of the specimen (12 N/mm2 ). Thus, this origin cruciform shape is not qualiﬁed for the novel setup. As a consequence diﬀerent strategies are applied to inﬂuence the distribution of stresses and strains. Maximum strains should occur in the centre of the specimen and stress concentrations should be reduced to a minimum. Achieving these aims four diﬀerent alternatives are calculated: To defuse the 90◦ -angle each arms are connected with a 4 mm radius in model A. Besides, diﬀerent shapes of kerfs – a circle of 1 mm diameter for model B and a special designed kerf with radii of 2 mm and 15 mm for model C – are integrated to achieve some release in the corner region. Six slits (width 0.5 mm, length 30 mm) in each arm are supposed to control strain distribution in model D, as it is in principle also used in (Kuwabara et al., 02). Figure 9 illustrates the shapes of the diﬀerent models and their eﬀects on stress distribution. Each calculation is based on the same preconditions and material data as the FE-simulation in Fig. 8. Obviously, the radius of model A has no signiﬁcant inﬂuence on the distribution of von Mises stress in the specimen’s centre, but maximum stress in the corner region is explicitly reduced. The circle-kerf in B does not release the corner, even in contrast the maximum stress increases. The variation of the kerf in model C leads to higher strains in the centre region, but stress

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Fig. 9. Diﬀerent geometry shapes and calculated stress distributions

is still to low outside this area. An adequate arrangement of the six slits in each arm – as it is done for model D – allows a precise control of maximum stresses and strains to the centre. Of course stress concentration still appears at the end of slits next to the centre. But these values are – related to the stresses in the centre – the smallest compared to the other models (origin, A–C). In addition, the maximum stresses are localised on a surrounding of 0.5 mm. Table 2 emphasizes the impressions of Fig. 9. Some characteristic values of the models A–D are compared to the origin shape of the cruciform specimen. Although model A is the only alternative which reduces the absolute value of maximum stress in the corner, it is not qualiﬁed due to the high strains on the rolls that are of about 2.5 times higher than the strains in the centre. Also the maximum stresses in model B are not acceptable. Some progress can be seen in the transfer of the kerf from model B to C, but still inadequately. Major problem in this case are the strains on the rolls, which lead to starting plastiﬁcation outside the forming zone.

Table 2. Stresses and strains of diﬀerent specimen shapes

2

σcorner (N/mm ) εcenter (%) σcorner /σcenter εroll /εcenter σcorner /σorigin εcenter /εorigin

origin

A

B

C

D

42 0.24 3.47 1.81 1.00 1.00

37 0.23 3.26 2.56 0.94 0.96

63 0.26 4.97 1.71 1.43 1.06

46 0.30 3.08 1.54 0.88 1.24

59 0.41 2.89 0.97 0.83 1.70

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The slits in model D are the preferred solution. It is the only sample on which the strains in the center are less than on the rolls. This is an important precondition for successful determination of a yield locus diagram. This fact is punctuated by the very high εcenter /εorigin -rate of 1.70. In addition the ratio of stresses in the center and in the corner of the specimen is signiﬁcantly reduced compared to the origin cruciform one of only 83%.

4 Conclusions and Outlook In this paper a novel experimental setup for biaxial tension test is introduced. Therefore, a new approach of specimen is designed and investigated with analytical methods. For further optimisation of the geometrical shape ﬁrst a FE-model has to be created and after that geometrical variations of specimen are analysed. It must be pointed out, that stress singularities resulting from the cruciform shape of the specimen can not be modelled only by the usage of a separately calculated shape number or the very extensive theory of elasticity. Thus FE-simulation promises a great potential for the most realistic modelling of the process and the occurring stresses. Calculating only a quarter of specimen’s geometry has no inﬂuence on simulation results, but leads to a much higher performance of calculation (about 2.85 times) due to the lesser number of ﬁnite elements and nodes. Although the new developed experimental set-up minimises friction eﬀects, some inﬂuence does still remain. This is included in the calculations with a friction coeﬃcient of 6.25 · 10−3 . In order to optimise the specimen’s shape for experimental investigations the origin cruciform specimen is not satisfactory. Thus, several FE-simulations are done to enhance the distributions of stress and strain with the aim to achieve maximum and homogenous strains in the centre and acceptable stresses in the region of the corner, where two arms coincide. An especiallydesigned specimen with six slits in each arm fulﬁls these requirements very good. For further investigations these theoretical results have to be veriﬁed by experiments. An additional improvement of stress ratio in the centre can be achieved by weakening the cross section, either directly by reducing the sheet thickness or indirectly by a local heating of the forming zone.

Acknowledgements The authors would like to gratefully thank the German Research Foundation (DFG) for founding this work. The results are carried out of the research project entitled “Characterisation of yielding of magnesium sheet at elevated temperatures”.

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References Banabic D., M¨ uller W., P¨ ohlandt K., “Experimental Determination of Yield Locus for Sheet Metals”, Proceedings of 1st International ESAFORM Conference, Sophia Antipolis, 17–20 March 1998, Paris, CEMEF, p. 179–182. England A.H., “On stress singularities in linear elasticity”, International Journal of Engineering Science, vol. 9 no. 6, 1971, p. 571–585. Geiger M., Hußn¨ atter W., Kerausch M., Merklein M., Pitz M., Verfahren und Vorrichtung zur Durchf¨ uhrung von Fließortkurven-Tiefungsversuchen an BlechProbek¨ orpern, German patent application DE 103 40 125.3. Germany, 2003. Geiger M., van der Heyd G., Merklein M., Hußn¨ atter W., “Novel Concept of Experimental Setup for Characterisation of Plastic Yielding of Sheet Metal at Elevated Temperatures”, Journal of Advanced Materials Research, vol. 6–8, 2005, p. 657–664. Geiger M., Hußn¨ atter W., Merklein M., “Specimen for a novel concept of the biaxial tension test”, Journal of Materials Processing Technology, vol. 167 no. 2–3, 2005, p. 177–183. Green A.E., Zerna W., Theoretical Elasticity, Oxford, Oxford University Press, 1954. Grewolls G., Kreißig R., “Anisotropic Hardening – Numerical Application of a Cubic Yield Theory and Consideration of Variable r-Values for Sheet Metal”, European Journal of Mechanics – A/Solids, vol. 20 no. 4, 2001, p. 585–599. Hoferlin E., van Bael A., van Houtte P., Steyaert G., De Mar´e C., “Biaxial Tests on Cruciform Specimens for the Validation of Crystallographic Yield Loci”, Journal of Materials Processing Technology, vol. 80–81, 1998, p. 545–550. Kuwabara T., Ikeda S., “Plane-Strain Tension Test of Steel Sheet Using ServoControlled Biaxial Tensile Testing Machine”, Proceedings of 5th International ESAFORM Conference, Krakow, 18–21 April 2002, Krakow, Akapit, p. 499–502. M¨ uller W., Beitrag zur Charakterisierung von Blechwerkstoﬀen unter mehrachsiger Beanspruchung, Berlin, Springer, 1996.

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Applications of a Recently Proposed Anisotropic Yield Function to Sheet Forming S. Soare1 , J.W. Yoon2 , O. Cazacu1 and F. Barlat2 1

2

University of Florida, REEF, 1350 N. Poquito Road, Shalimar, FL 32579, U.S.A., soare@uﬂ.edu, [email protected]ﬂ.edu Material Science Division, Alcoa Technical Center, 100 Technical Drive, Alcoa Center, PA 15069-0001, U.S.A., [email protected], [email protected]

Summary. In this paper the predictive capabilities of a recently proposed yield criterion, CB2001, are assessed. Also, a numerical scheme for identifying the material coeﬃcients is presented. It is shown that although convexity is not a default property of the criterion, it can be achieved numerically. Applications to two sheet forming operations are presented. Using the commercial FE code ABAQUS, simulations of the deep-drawing of a cylindrical cup and springback analysis for unconstrained bending are performed. Two aluminum alloys were considered and modelled with Hill’48 (ABAQUS) and CB2001 (UMAT). The results are also compared with another popular criterion, Yld’96. We conclude that for sheet forming operations were large plastic deformations are involved, accurate ﬁt of the initial plastic anisotropy is a basic condition for successful FE simulations.

Key words: plastic anisotropy, generalized invariants, sheet forming.

1 Introduction Traditionally, metal forming processes have been developed based on expensive experimental trials. In recent years, Finite Element (FE) simulations have been extensively used to reduce the amount of experiments and trial and error involved in the process development. Key for the success of simulations of forming processes is the constitutive model used for description of the plastic behavior. Hill’s (1948) quadratic yield criterion is still the most widely used anisotropic yield criterion. The validity of this yield function has been explored by numerous experiments, the consensus being that it is well suited to speciﬁc metals and textures, especially for steel. Formulations that were particularly intended to improve the description of yielding of materials with a FCC crystal structure, in particular aluminum and its alloys have been subsequently

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proposed. An extensive review of most recent contributions to the description of plastic anisotropy of materials with pronounced texture can be found in Barlat et al., 2003. Within the framework of the theory of representation of tensor functions, Cazacu et al., 2001, 2003, have proposed a general and rigorous method to extend any given yield criterion such as to include any type of material symmetry. Using this method, Cazacu et al., 2001, have proposed an extension of Drucker’s yield criterion (Drucker, 1949) to orthotropy. This yield criterion is deﬁned for fully three-dimensional state of stresses and involves 17 coeﬃcients. Thus, this yield criterion can capture accurately both the yield stress and r-values directionality. The main objective of this paper is to assess the predictive capabilities of this yield criterion. The paper is organized as follows. In Sect. 1 is presented the constitutive framework and the algorithmic aspects related to ﬁnite element discretization. In Sect. 2, Cazacu and Barlat (2001) yield criterion is brieﬂy summarized and examples of application of this criterion to the description of the plastic response of AA 2090-T3, an aluminum alloy with strong anisotropy, and of AA 6111-T4, an alloy with moderate anisotropy, are presented. It is shown that if the material presents strong anisotropy, convexity of the yield function should be added as an additional constraint for the minimization problem associated with the identiﬁcation of the material parameters. Section 3 presents cup drawing simulations for AA 2090-T3. The simulations were done with the commercial ﬁnite element package ABAQUS using a solid model, in order to fully exploit the 3-D formulation of the newly proposed yield function. Predicted results based on Cazacu and Barlat (2001) are further compared with experimental results, simulation results obtained using Hill (1948), and Barlat et al (1996) yield criteria. Finally, in Sect. 4 we present a springback analysis for AA 6111-T4 using Cazacu and Barlat (2001), and Hill (1948) yield criterion, respectively. The results are further compared with data and analysis results reported in Yoon et al., 2002.

2 Constitutive Framework and Stress Update Algorithm In what follows we adopt an updated Lagrangian description, which is adequate for simulating large plastic deformation behavior. First, the initial conﬁguration is taken as reference conﬁguration. Then, the reference conﬁguration is updated to the last converged conﬁguration in a step-by-step procedure. The material particle in its initial conﬁguration is denoted X , whereas its spatial position is denoted by X . Let σ denote the Cauchy stress, and σ˙ denote its Jaumann rate. The deformation gradient is denoted by F and is decomposed as F = ∂u/∂X = RU = VR, with R being the rigid body rotation at the material point, and U and V the right and left stretch tensors. The (spatial) rate of deformation tensor is deﬁned as d = [∂v /∂x +(∂v /∂x )T ]/2, where v is the velocity ﬁeld of the body, and the T superscript means transposed.

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In metal forming applications, the elastic strains are usually much smaller than the plastic strains (two to three orders of magnitude) and this motivates an additive decomposition of the rate of deformation into elastic and plastic parts. Thus, we shall write: d = de + dp However, to account for rigid body rotations all the spatial quantities of interest will be rotated by R with respect to the reference conﬁguration. Thus, we deﬁne Σ = RT σR, D = RT dR D e = RT d e R, D p = RT d p R This rotated tensors are deﬁned on the updated reference conﬁguration and so they are objective. We note here that the time derivative of Σ (see, for example, Simo and Hughes, 1989, or Yoon et al., 1999, for a development based on minimal plastic work deformation paths) is: ˙ = RT σR, ˙ Σ

(1)

˙ is also the rotated Jaumann rate of the Cauchy stress which shows that Σ tensor, and so it is objective. In the space of stress tensors, a yield function in the form f = f (Σ, α) = Σ − K(α) is introduced, satisfying the requirements that it be pressure insensitive, convex, and contain the origin (zero stress) inside its initial level set. The yield function deﬁnes the boundary of the elastic domain: {Σ|Σ < K(α)} If the stress tensor belongs to the elastic domain, the deformation process is entirely reversible (elastic). The equivalent stress Σ = Σ(Σ) will deﬁne the shape of the elastic domain in the stress space (see next section), whereas the hardening function K will deﬁne its size. We further assume an associate ﬂow rule and isotropic hardening. The hardening parameter is identiﬁed by the condition that the plastic work rate associated with a three-dimensional stress-strain state be equivalent with the plastic work rate associated with the one-dimensional experimental hardening curve (usually a uniaxial stress-strain curve): ˙ p = Σ · D p = Σ · γ˙ ∂f = Σ γ˙ ⇒ α˙ = γ˙ W (2) ∂Σ Then the constitutive equations take the form: f := Σ − K(γ) ≤ 0 ˙ = C [D e ] = C [D − D p ] Σ ∂f D p = γ˙ ∂Σ where, above, C is the fourth order tensor of elasticity.

(3) (4) (5)

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The multiplier γ˙ is subject to the restrictions: γ˙

= 0, if f < 0, or if f = 0 and Σ · ∂f /∂Σ ≤ 0 > 0, if f = 0 and Σ · ∂f /∂Σ > 0

(6)

and is determined, as usual, from the consistency condition f˙ = 0. Finally, we notice that the entire set of evolution equations can be recast into the form: ˙ = C ep [D] Σ where the (theoretical) elasto-plastic tangent modulus is given by: ⎧ ⎪ C , if λ˙⎡ = 0;⎤ ⎡ ⎪ ⎪ ⎤ ⎪ ⎪ ∂f ∂f ⎪ ⎨ ⎦ ⎣ ⎦ ⎣ ⊗C C ∂Σ ∂Σ C ep = (7) ⎪ C− ⎡ , if λ˙ > 0. ⎤ ⎪ ⎪ ∂f ∂f ⎪ ⎪ ⎦· ⎪ C⎣ +K ⎩ ∂Σ ∂Σ where ⊗ represents the tensorial product, and · the scalar product. The above system of evolution equations, (3), (4), (5), has to be integrated in order to provide stress updates at material (or integration) points associated with a mesh in a ﬁnite element solution. This is done numerically using a backward Euler integration scheme over a generic time interval [tn , tn+1 ]. If Δt = tn+1 − tn denotes the time increment in a ﬁnite element step, then the strain increment over the step is computed as Δ = D Δ t. Then, after eliminating D p , the integrated form we shall use for stress update is: Σ n+1 − K(γn + Δγ) ≤ 0 ∂f S [ΔΣ] − Δ + Δγ =0 ∂Σ n+1

(8) (9)

where ΔΣ and Δγ are deﬁned by Σn+1 = Σn +ΔΣ, γn+1 = γn +Δγ, and S is the compliance tensor. In addition, it can be proved (Simo and Hughes (1999)) that the numerical counterpart of the loading-unloading conditions, (6), takes the simple form f (Σtrial )

≤ 0 ⇒ Δγ = 0 ⇒ elastic step > 0 ⇒ Δγ > 0 ⇒ plastic step

(10)

where Σtrial := σn + C : Δ. It follows that for an elastic step the solution of system (8), (9), is simply ΔΣ = C [Δ], Δγ = 0, which further implies Σn+1 = Σtrial , and γn+1 = γn . In case of plastic loading, the system (8), (9) becomes . / ∂f S [ΔΣ] − Δ + Δγ ∂Σ R 0 F := := = (11) n+1 r 0 Σ n+1 − K(γn + Δγ) and has to be solved for stress and hardening parameter increments ΔΣ, and Δγ, respectively. It is a 7 × 7 nonlinear system. For quadratic yield functions (like Mises or Hill’48), this system can be reduced to a single

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nonlinear equation (with important consequences in terms of accuracy and eﬃciency). For the non-quadratic case this is not, in general, possible and general approaches for solving nonlinear systems have to be employed. In our case (static analysis) the algorithm at hand is Newton–Raphson. If, say, x := [ΔΣ, Δγ]T , and Δx := [Δ2 Σ, Δ2 γ]T , then the sequence of iterations (x i )i approaching the solution is computed as x

i+1

i

= x + Δx , where

∂F ∂x

i [Δx ] = −F i

The starting point for iterations is taken to be the trial increment : x 0 = [C [Δ], 0]T . With F deﬁned in (11), we have (the subscript n + 1 will be dropped from now on since it is clear that all the computations are performed within the time step [tn , tn+1 ]): 2 ⎤ ⎡ ⎤ ⎡ ∂f ∂ f ∂f −1 ∂F ⎥ ⎢S + Δγ ∂Σ∂Σ ∂Σ ⎥ ⎢Ξ =⎣ ⎦ = ⎣ ∂f ∂Σ ⎦ ∂f ∂x −K −K ∂Σ ∂Σ with Ξ−1 := S + Δγ ∂ 2 f /∂Σ∂Σ . The linear system for increments is then i −1

(Ξ )

2

2

[Δ Σ] + Δ γ

∂f ∂Σ

i

∂f ∂Σ

i = −Ri

· Δ2 Σ − (K ) Δ2 γ = −ri i

From the ﬁrst equation we get Δ2 Σ = −Ξi [Ri + Δ2 γ(∂f /∂Σ)i ], and substituting in the second equation the increment Δ2 γ is computed i ∂f · Ξi [Ri ] ri − ∂Σ Δ 2 γ = i i i ∂f ∂f + (K ) · Ξi ∂Σ ∂Σ where Ri = S [ΔΣi ] − Δ + Δγ i

∂f ∂Σ

i

i

ri = Σ − K(γn + Δγ i ) The stress and hardening parameter can now be updated ΔΣi+1 = ΔΣi + Δ2 Σ, Σi+1 = Σn + ΔΣi+1 Δγ i+1 = Δγ i + Δ2 γ, γ i+1 = γn + Δγ i

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and then a check is performed on the new residuals Ri+1 and ri+1 . If they are under certain tolerances, ||Ri+1 || < T OL, and |ri+1 | < T OL, usually with T OL ≈ 10−7 , then the iteration process stops. For stability and accuracy, the algorithm can be enhanced with a backtracking strategy such as line-searches. That is, instead of accepting the entire increment Δx computed by a Newton–Raphson iteration, only a fraction of it, Δx := μΔx , with μ ∈ (0, 1] is taken as the new increment. The parameter μ is computed from the condition that the magnitude of F reach its minimum along the segment Δx (the direction Δx as computed by Newton–Raphson, is indeed a minimizing direction for the function F · F /2). For a simple and eﬃcient strategy of computing μ we refer the reader to Press et al., 1996. Finally, the algorithmic tangent modulus ∂Σn+1 /∂n+1 of the integrated constitutive law has to be speciﬁed. Starting from dΣ = C [d − dp ] and taking into account that ∂f ∂2f ∂f p = d(Δγ) + Δγ dΣ d = d Δγ ∂Σ ∂Σ ∂Σ∂Σ ∂f dΣ = Ξ d − d(Δγ) ∂Σ Next, from the algorithmic consistency condition

one obtains

df = 0 ⇔

(12)

∂f dΣ − K d(Δγ) = 0 ∂Σ

From (12) we also have

∂f ∂f ∂f dΣ = · Ξ d − d(Δγ) ∂Σ ∂Σ ∂Σ

Combining the last two equations one can solve for d(Δγ): ∂f · Ξ[d] ∂Σ d(Δγ) = ∂f ∂f ] + K · Ξ[ ∂Σ ∂Σ Substituting this expression back into (12) ﬁnally gets the desired formula ⎧ C d, if elastic step ⎨! 0 (Ξ[∂f /∂Σ]) ⊗ (Ξ[∂f /∂Σ]) ∂Σ dΣ = d, if plastic step d = ⎩ Ξ− ∂f /∂Σ · Ξ[∂f /∂Σ] + K ∂ To complete the model, we need to specify the yield potential Σ and the expression of the hardening law. For the simple isotropic hardening considered here, the problem is settled by experimental data from uniaxial tensile tests along the rolling direction (interpolated in some convenient analytic form). The yield potential is discussed in the next section.

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3 Anisotropic Yield Potential In order to describe the anisotropic plastic response of rolled metal sheets, Cazacu and Barlat, 2001, 2003, introduced a general method to extend any isotropic yield function, expressed in terms of the J2 and J3 invariants, into an anisotropic function. The method is based on the theory of representation of tensor functions (see for example Liu, 1982). In Cazacu and Barlat, 2001, this method was used to extend to orthotropy the isotropic yield criterion proposed by Drucker, 1949. In the following, we brieﬂy review the relevant features of the resulting anisotropic yield criterion, that will be referred to as CB2001. Let us denote by S the stress deviator, and deﬁne its invariants as J2 =

1 1 tr(S 2 ), J3 = tr(S 3 ) 2 3

Drucker’s isotropic yield function reads: σ = [27(J23 − cJ32 )]1/6 where c is a material parameter. It can be shown that this isotropic yield surface lies between the surfaces deﬁned by von Mises and Tresca criteria. The method proposed by Cazacu and Barlat, 2001, 2003, to incorporate lower symmetries into the yield function consists in replacing in the expression of the isotropic yield criterion, J2 and J3 by general anisotropic polynomial invariants of the same degree. In Cazacu and Barlat, 2003, are provided expressions for these generalized invariants for orthotropic, transversely isotropic, and cubic symmetries. Here, we present the method for constructing generalized orthotropic invariants. Let (x, y, z) be the coordinate system associated with the orthotropic symmetry group of the material. For a rolled sheet x , y and z represent the rolling, transverse and normal directions, respectively. Thus, the material anisotropy is characterized by the following structural tensors: N 1 = x ⊗ x, N 2 = y ⊗ y, N 3 = z ⊗ z Then, a polynomial P in the stress components is invariant with respect to the symmetry group of the material if (and only if) it is expressible as a polynomial in the quantities: tr(N 1 σ), tr(N 2 σ), tr(N 3 σ), tr(N 1 σ2 ), tr(N 2 σ2 ), tr(N 3 σ2 ), tr(σ3 ) Equivalently, with respect to the (x , y , z ) frame, P is a polynomial in the following quantities: 2 2 2 , σyz , σzx , σxy σyz σzx σx , σy , σz , σxy

A homogeneous polynomial of order two, in the stated quantities, insensitive to pressure, must be of the form:

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J2o =

Advanced Methods in Material Forming

a1 a2 a3 2 2 2 (σx − σy )2 + (σy − σz )2 + (σz − σx )2 + a4 σxy + a5 σyz + a6 σzx 6 6 6

whereas a homogeneous polynomial of order three in the stated quantities, insensitive to pressure, must be of the form: J3o =

1 1 1 (b1 + b2 )σx3 + (b3 + b4 )σy3 + [2(b1 + b4 ) − (b2 + b3 )]σz3 27 27 27 1 1 − (b1 σy + b2 σz )σx2 − (b3 σz + b4 σx )σy2 9 9 1 2 − [(b1 − b2 + b4 )σx + (b1 − b3 + b4 )σy ]σz2 + (b1 + b4 )σx σy σz 9 9 1 1 2 2 − [2b10 σz −b5 σy − (2b10 −b5 )σx ]σxy − [(b6 + b7 )σx − b6 σy − b7 σz ]σyz 3 3 1 2 + 2b11 σxy σyz σzx − [2b9 σy − b8 σz − (2b9 − b8 )σx ]σzx 3

It is worth noting that J2o coincides with Hill’s quadratic yield function. The orthotropic yield function CB2001 is obtained by replacing in the expression of the isotropic Drucker criterion, the invariants of the stress deviator by J2o and J3o , respectively, i.e.: σ = [(J2o )3 − (J3o )2 ]1/6 = k

(13)

For full 3-D stress conditions, the CB2001 yield criterion involves 17 material parameters. These parameters can be determined based on results of shear tests, bulge test, and uniaxial tensile tests corresponding to various orientations of the tensile axis with respect to the rolling direction. Unlike usual yield functions, which require input data from tension tests conducted along three directions in the plane of a sheet, this model can take into account, if available, tensile data for tests conducted at every 15 from the Rolling Direction (RD). For plane stress conditions, CB2001 writes: 3 1 a1 1 6 2 2 2 f = (a1 + a3 )σx − σx σy + (a1 + a2 )σy + a4 σxy 6 3 6 ! 1 1 1 (b1 + b2 )σx3 + (b3 + b4 )σy3 − (b1 σx + b4 σy )σx σy − 27 27 9 02 1 2 = k6 − [(b5 − 2b10 )σx − b5 σy ]σxy 3 Generally, the constant k 6 is eliminated by normalizing the experimental yield stresses with σ0 , the yield stress in the rolling direction. Thus, for 2-D conditions CB2001 involves 9 independent material parameters which have to be determined such that the yield function approaches as close as possible the available data set. More precisely, the material coeﬃcients are to be identiﬁed so that the norm of the diﬀerence between yield function values and data

Applications of Anisotropic Yield Function to Sheet Forming

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points is as small as possible. Then the following distance function is to be minimized with respect to the coeﬃcients ai and bi : N F = wsk [(σθ6 − s6k )2 + wrk (rθ − rk )2 ] + wb(σb6 − s6b )2 k=1

where N is the number of loading directions for which data is available, sk and rk are the corresponding experimental values of yield stress and r–value, and sb is the experimental yield stress for equi-biaxial loading. σθ , rθ and σb are the predicted (theoretical) values with the yield function. The parameters wsk , wrk and wb are weights to be controlled during the identiﬁcation phase allowing for diﬀerentiation among data points. One usually starts with all of them equal to 1.000, and then, in a step-by-step procedure, if some data points need a better ﬁt the corresponding weights are increased. The polynomial criterion thus obtained is not automatically convex, i.e. not every combination of coeﬃcients ai and bi will ensure convexity. To guarantee the convexity of the yield surface, we constrain the minimization process by enforcing the function to be convex at a set of discrete points in the stress space. More precisely, we shall impose that the sections σxy = 0 and σx = 0 of the yield surface (in 2D) be convex sections. From our experience with polynomial functions, this two conditions are enough to guarantee convexity in 2D. Let us take M0 , M1 , . . . , Mnc points on the curve σxy = 0, Fig. 1, corresponding to the angles φ0 , φ1 , . . . , φnc , where φi =

π i, i = 0, 1, . . . , nc nc

Also take M−1 with

π nc 1 Consider the segment [Mi , Mi+2 ] and deﬁne the point Qi+1 = [OMi+1 ] [Mi , Mi+2 ]. Then, impose the constraint φ−1 = −

|OQi+1 | ≤ |OMi+1 |

Fig. 1. Set of points on the σxy = 0 section where convexity constraints are imposed

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In order to get an analytic form of the above relation we note that it is equivalent with the existence of a μ ∈ (0, 1) such that (the points are seen now as vectors applied at the origin O) Qi+1 = μMi+1 On the other hand, we have Qi+1 = λMi + (1 − λ)Mi+2 , for a parameter λ ∈ (0, 1). These two relations are combined to give (in component form) λxi + (1 − λ)xi+2 = μxi+1 λyi + (1 − λ)yi+2 = μyi+1 where the pair (xi , yi ) represents the coordinates of the point Mi . Solving for μ gets xi yi+2 − xi+2 yi μ= xi+1 (yi+2 − yi ) − (xi+2 − xi )yi+1 We clearly have μ > 0 (since Qi+1 = μMi+1 , assuming that an initial guess not so degenerated is available). Then, the constraint to be imposed is μ ≤ 1: c gi+2 := (xi − xi+1 )yi+2 + (xi+1 − xi+2 )yi + (xi+2 − xi )yi+1 ≤ 0 c , for i = −1, 0, . . . , nc − 2. and we obtain nc convexity constraints gi+2 The set of material parameters for which convexity is enforced in this manner (i.e. the above constraints are satisﬁed) is not, in general, a connected domain, but a union of disjoint connected domains. This means that an initial estimate must be situated in the proper set. For example, for AA 2090-T3 alloy, which shows strong anisotropy in the r-values, we used as initial estimate the set of values given in Cazacu and Barlat, 2001. For this set of values, the anisotropy in yielding and r-values was overall well described, but the strain ratio at 45◦ with respect to the rolling direction was under-predicted. Let also note, that since the anisotropic properties of this alloy are strongly anisotropic (in the sense of quadratic norm of the diﬀerence between data), an isotropic initial estimate (e.g. for ak set to 1 bk = 0, which correspond to von Mises yield function raised to the power of six) does not lie in the same connected component of the convexity domain with the best ﬁt for it. A number of 10 to 15 points Mi was enough to enforce convexity everywhere. On the other hand, for anisotropic properties mildly distant from isotropy (as is, for example, AA 6111-T4, exempliﬁed in applications later), the von Mises initial estimate proves to be a good initial guess and convexity is automatically satisﬁed. This identiﬁcation procedure has been coded in Fortran (based on the DN0ONF subroutine from the IMSL library) and used for the applications presented in the next sections. Since in the next sections we will compare the predictions of CB2001 with that of Hill’s quadratic criterion, and that of Yld ’96 (Barlat et al., 1997), in the following we will brieﬂy summarize Yld 96 (more details can be found in

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the cited paper). Thus, let us denote S = L[σ], the image through a linear operator L of the Cauchy stress tensor, and with Si and p i its principal values and directions. Also denote with φi the angles between the symmetry axes (x , y , z ) associated with the material, and the principal directions of the transformed stress tensor S . Then, Yld ’96 criterion is given by the expression: 2σ m = α1 |S2 − S3 |m + α2 |S3 − S1 |m + α3 |S1 − S2 |m

(14)

where, αk are deﬁned as: αk = αx (p k · x )2 + αy (p k · y )2 + αz (p k · z )2 with αx , αy and αz deﬁned by: αx = αx0 cos2 2φ1 + αx1 sin2 2φ1 αy = αy0 cos2 2φ2 + αy1 sin2 2φ2 αz = αz0 cos2 2φ3 + αz3 sin2 2φ3 If the principal values Si are ordered as S1 ≥ S2 ≥ S3 , then the above cosines are computed according to the relations: cos φ1 = y · p 1 , if|S1 | ≥ |S3 |, ory · p 3 , if|S1 | < |S3 | cos φ2 = z · p 1 , if|S1 | ≥ |S3 |, orz · p 3 , if|S1 | < |S3 | cos φ3 = x · p 1 , if|S1 | ≥ |S3 |, orx · p 3 , if|S1 | < |S3 | For orthotropic symmetry, the linear transformation L has to have the form (with the usual tensor to matrix writing convention): ⎤ ⎡ L11 L12 L13 ⎥ ⎢L12 L22 L23 ⎥ ⎢ ⎥ ⎢L13 L23 L33 ⎥ (empty places denote zeros) ⎢ L=⎢ ⎥ L 44 ⎥ ⎢ ⎦ ⎣ L55 L66 and to respect the hydrostatic pressure independence condition it has also to satisfy the relations: L1k + L2k + L3k = 0 The exponent m has a ﬁxed value. For FCC materials, m = 8 is recommended, while for BCC materials a value m = 6 is recommended. Overall, for 3D states of stress the Yld ’96 criterion involves 12 material parameters, whereas for plane stress conditions the number is 9 (taking into account that yield data points are, as usual, normalized with the yield stress value along the rolling direction).

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4 Deep Drawing of a Cylindrical Cup In order to simulate the deep drawing process with a cylindrical punch, we used the commercial ﬁnite element code ABAQUS/Standard in which we implemented the CB2001 model using the User Material (UMAT) option. The geometrical setting and speciﬁc tool dimensions are taken from Yoon et al., 2000, see Fig. 3. The Coulomb coeﬃcient of friction between the blank and the tools was taken equal to 0.1, whereas a force of 22.2 kN was applied on the holder (small enough so that this force does not add additional yielding in the material, but big enough to prevent wrinkling of the blank). The blank sheet to be drawn is made of aluminum alloy AA 2090-T3 for which yield and r-value data was available along seven directions (experimental loading angle θ = 0◦ , 15◦ , 30◦ , 45◦ , 60◦ , 75◦ , 90◦ ). Using the identiﬁcation procedure outlined in the previous section we obtained for the material parameters a1 = 0.803, a2 = 0.66, a3 = 0.27, a4 = 0.59 b1 = 1.71, b2 = −0.022, b3 = −1.18, b4 = −1.7, b5 = −1.003, b10 = −0.023

Fig. 2. Data ﬁt for AA 2090-T3: normalized yield stress and r-value

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As shown in Fig. 2, CB2001 is capable of satisfactorily describing both the anisotropy of the yield stresses and r-values, whereas Hill’s quadratic criterion matches only the three r0 , r45 and r90 values. Yld ’96 gives a better description than Hill ’48, but still the r-value is over-predicted. Figure 3 shows the CB2001 biaxial yield corresponding to constant normalized shear stress σxy /σ0 = 0, 0.025, . . . , 0.8. We remark that the biaxial yield stress σb is also matched accurately (the experimental value is 1.02, whereas the predicted value is 1.018). The stress-strain hardening curve for this alloy is (Yoon et al., 2000): σ = A(B + p )C with A = 646.0 MPa, B = 0.025, and C = 0.227. Due to the orthotropic symmetry of the material, only a quarter of the blank was modelled, and, since the yield criterion oﬀers a 3D characterization of the stress state, we took advantage of this and meshed the blank with three layers of continuum elements

Fig. 3. Sections through yield surface for AA 2090-T3 and geometric setting for the drawing problem

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Fig. 4. Mesh on the blank and ﬁnal drawn cup for AA 2090-T3

(eight-node brick elements with reduced integration). From the available plane stress data, only the 9 coeﬃcients involved in the 2-D form of CB2001 could be identiﬁed. To perform the 3-D simulations, we took: a5 = a6 = 4.0, while the rest of the bi coeﬃcients were set equal to zero (i.e. only the Hill part of the criterion was extended to 3D). Figure 4 shows the mesh used for the blank (1758 elements), and the ﬁnal drawn cup (constructed from the simulated quarter by mirror symmetry).

Fig. 5. Earing prediction for AA 2090-T3 with CB2001, Hill ’48, and comparison with experimental data and Yld 96

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Figure 5 shows the experimental cup heigh proﬁles of the ﬁnal drawn cup (data reported in Yoon et al., 2000), in comparison with the predictions obtained using Hill’s quadratic criterion, YLd’96 and CB2001. It is seen that CB2001 predicts a better proﬁle than Hill’48 and Yld’96. The reason is that Hill’48 overestimates the yield anisotropy of the material, while Yld’96 does not accurately capture the r-value variation.

5 Springback Prediction As another example of application of CB2001 yield criterion, we performed a springback analysis of a AA 6111-T4 blank sheet, 1.0 mm thick. As before, data describing the anisotropic behavior of the rolled sheet was available for yield and r-value in seven directions spanning ninety degrees starting with the rolling direction. For this material the numerical values of the anisotropy coeﬃcients are: a1 = 1.724, a2 = 2.790, a3 = 2.275, a4 = 1.955 b1 = 2.156, b2 = 5.128, b3 = 5.813, b4 = −5.027, b5 = −2.238, b10 = −1.298 Figure 6 shows the variation of the yield stress and r-values as function of the orientation with respect to the rolling direction obtained using CB2001 and Hill(1948) in comparison with data. A good agreement between theoretical and experimental results is obtained with CB2001. Hill ’48, for which the identiﬁcation was based only on the three r-values r0 , r45 , and r90 , gives a good ﬁt for the entire set of r-values but overestimates the yield stresses. In Fig. 7 constant normalized shear stress sections through the CB2001 surface are plotted. The biaxial yield stress computed with CB2001 is 1.003 (experimental value = 1.0). Also, for a 3D simulation we set: a5 = a6 = 2.0, the rest of the bk coeﬃcients being set equal to zero.

Fig. 6. Data ﬁt for AA 6111-T4: normalized yield stress and r-value

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Fig. 7. Sections through yield surface for AA 6111-T4 and geometric setting for the unconstrained bending problem

Figure 7 shows a schematic view of the initial setting used in the ﬁrst step of the analysis of the unconstrained bending of a AA 6111-T4 blank sheet (120 mm length, 30 mm width), with the rolling direction along its length. This ﬁrst step was followed by a second step in which springback was allowed (by removing contact interactions with tools). The die and the punch were modeled as rigid parts. The blank was meshed with 3, 5, 7, and 9 layers of 40(length/2)×20(width) continuum elements (eight-nodes bricks with reduced integration). Thus four simulations were performed corresponding to 3, 5, 7 and 9 integration points across the thickness of the blank. The friction between blank and tools was considered small (0.05). The hardening law for this material was characterized in the form: σ = K(p ) = A − B exp(−Cp ) with A = 429.8 MPa, B = 237.7 MPa, and C = 8.504. The results obtained with CB2001, Hill (1948) and those reported for the same material by Yoon et al., 2002, are given in Table 1. The deﬁnition of the spring back angles is given in Figure 8. The predicted angles in Yoon et al., 2002, were obtained using a hybrid plane stress simulation. The drawing step was simulated with membrane elements after which the data was transferred to a postprocessing step. In this second step bending stresses were reconstructed for shell elements which were then used to simulate the springback of the bent sheet. Signiﬁcant computing time is saved in this way, and it can be seen that the springback angle is correctly predicted. However, both the initial and ﬁnal angle are underestimated. The same translation, but upward, can be remarked

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Table 1. Initial, ﬁnal and springback angles #layers

before springback

after springback

springback angle (diﬀerence)

3 5 7 9 3 5 7 9

72.0◦ 70.0◦ 73.17◦ 73.27◦ 73.92◦ 73.91◦ 73.16◦ 73.26◦ 73.92◦ 73.91◦

60.0◦ 58.0◦ 57.61◦ 60.66◦ 61.41◦ 61.71◦ 57.32◦ 60.43◦ 61.18◦ 61.40◦

12.0◦ 12.0◦ 15.56◦ 12.61◦ 12.51◦ 12.20◦ 15.84◦ 12.83◦ 12.74◦ 12.51◦

experiment Yoon 2002 Hill ’48

CB2001

also in the results from the simulations with continuum elements (Hill’48 and CB2001). This could be explained by the loss of contact between the tip of the punch and blank during the forming stage (when the pressure of the punch is mostly exerted on the ﬂanges of the blank), which can cause a stress reversal in the middle of the blank, suggesting the use of a mixed isotropic-kinematic hardening law to improve the prediction of the initial angle. Although the initial plastic anisotropy is not entirely ﬁt by Hill’48 quadratic potential, it can be seen from the results in Table 1 that both Hill’48 and CB2001 follow the same convergence trend in predicting the spring-back angle (with minor diﬀerences which may be due either to diﬀerent ﬁts for yield strength anisotropy, or constitutive integration scheme). The magnitude of the plastic deformation in this case is small (between 3%−5%). Since both potentials give an accurate ﬁt of the drawing ratio’s anisotropy, one can conclude that for small plastic deformations this data set only, coupled with Hill’48, proves to be suﬃcient for accurate results.

Fig. 8. Measured initial and ﬁnal angles for springback simulation

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6 Conclusions The predictive capabilities of a recently proposed anisotropic yield function, CB2001, were assessed. This yield criterion is deﬁned for fully threedimensional state of stresses and involves 17 coeﬃcients. This yield criterion can capture with accuracy both the yield stress anisotropy and r-values distribution. Yet, this yield function is not automatically convex. It is shown that if the material presents strong anisotropy, convexity of the yield function can be enforced simply by adding an additional constraint in the minimization problem associated with the identiﬁcation of the material parameters. A procedure for the determination of material parameters based on yield stress and r-value data, with convexity numerically imposed at a ﬁnite discrete set of points, was outlined. For materials that don’t exhibit major changes in the shape of their yield surfaces during speciﬁc loading conditions (like the aluminum alloys considered here in forming operations), this proves to be an eﬀective tool. It also allows for 3D simulations and this can be an important advantage over plane stress formulations. The CB2001 yield criterion was applied to the modelling of two aluminum alloys which were then used in two sheet forming operations. First, the plastic anisotropy of the AA 2090-T3 and AA 6111-T4 alloys was characterized in terms of the new yield function and good agreement with the experimental data was obtained. In both cases the classical Hill’s quadratic criterion, based only on r-values, failed to satisfactorily describe the yield stress anisotropy. Then, deep drawing for the AL2090-T3 sheet and unconstrained bending, followed by springback, for the AL6111-T4 sheet were simulated, using both CB2001, Hill’48, and Yld’96 criteria. The ﬁnal proﬁle of the drawn cup was only qualitatively predicted by Hill’48 function, overestimating the height of the proﬁle. In contrast, CB2001, not only described qualitatively well the earing proﬁle, but predicted accurately the height variation. The proﬁle predicted by Yld’96 is between the two mentioned above, and a similar positioning we ﬁnd for its data ﬁt. The three potentials give a sequential approach to the experimental data for plastic anisotropy, clearly suggesting the importance of capturing the initial anisotropy when trying to predict the proﬁle in the deep-drawing problem. The springback analysis revealed that in the range of small plastic deformations the drawing ratio’s anisotropy proves to be enough for accurate results in FE simulations. And when Hill’s quadratic criterion accurately ﬁts this data set, it should be the ﬁrst choice for material modelling. When large plastic deformations are to be expected, as in the case of deep-drawing, accurate ﬁt of the drawing ratio alone is no longer enough and it has to be doubled by an accurate ﬁt of the yield strength also.

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References Barlat F., Maeda Y., Chung K., Yanagawa M., Brem J.C., Hayashida Y., Lege D.J., Matsui K., Murtha S.J., Hattori S., Becker R.C., Makosey S., “Yield function development for aluminum alloy sheet”, J. Mech. Phys. Solids, vol. 45, 1997, p. 1727 Barlat F., Cazacu O., Zyczkowski M., Banabic D., Yoon J.W, Yield surface plasticity and anisotropy in sheet metals, in: Continuum Scale Simulation of Engineering Materials, Fundamentals – Microstructures – Process Applications, Eds: D. Raabe, L.-Q. Chen, Barlat F., Roters F., Publisher: Wiley-VCH Verlag Berlin GmbH, 2003, p. 145–183. Cazacu O., Barlat F., “Generalization of Drucker’s yield criterion to orthotropy”, Mathematics and Mechanics of Solids, vol. 6 no. 6, 2001, p. 613–630. Cazacu O., Barlat F., Application of representation theory to describe yielding of anisotropic aluminum alloys, International Journal of Engineering Science, 41, 12,2003, p. 1367–1385. Drucker D.C., “Relation of experiments to mathematical theories of plasticity”, Journal of Applied Mechanics, 16, 1949, p. 349–357. Liu S.I., “On representation of anisotropic invariants”, International Journal of Engineering Science, no. 20, p. 1099–1109, 1982. Simo J.C., Hughes T.J.R, Computational Inelasticity, Springer Verlag, Berlin, 1999. Press W. H., Teukolsky S. A., Vetterling W.T., Flannery B.P., Numerical Recipes in C, Cambridge University Press, 1996. Yoon J.W., Barlat F., Chung K., Pourboghrat F., Yang D.Y., “Earing predictions based on asymmetric nonquadratic yield function”, International Journal of Plasticity, 16, 2000, p. 1075–1104. Yoon J.W., Pourboghrat F., Chung K., , Yang D.Y., “Springback prediction for sheet metal forming process using a 3D hybrid membrane/shell method”, International Journal of Mechanical Sciences, 44, 2002, p. 2133–2153.

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Modelling of the Forming Limit Diagrams Using the Finite Element Method L. P˘ ar˘ aianu1,2 , D.S. Com¸sa1 , J.J. Gracio2 and D. Banabic1 1

2

TEMA, Aveiro University, Aveiro, Portugal, [email protected], [email protected] CERTETA, Technical University of Cluj Napoca, Cluj-Napoca, Romania, [email protected], [email protected]

Summary. Modeling Forming Limit Diagrams (FLD) using the ﬁnite element method is a quite new approach. The article presents the prediction of both branches of an FLD, based on Hutchinson – Neale model, taking into account the strain rate sensitivity. Abaqus/Standard package has been used to perform the simulations. A numerical algorithm to describe the material behavior has been implemented as a user-subroutine UMAT. The plastic anisotropy of the sheet metal is described by the BBC2003 and Cazacu – Barlat yield criteria. The numerical results have been compared with experimental data for AA3103-0 aluminum alloy. The authors also present the experimental strategy to determine the FLD’s.

Key words: sheet metal forming, forming limit diagrams, ﬁnite element method.

1 Introduction The Forming Limit Diagram (FLD) represents an eﬃcient tool to characterize the formability of sheet metals. The FLD is a curve relating pairs of principal limit strains, which can be obtained at the surface of the sheet metal during a forming process prior to the occurrence of some defects (necking, fracture, etc.). During the last 45 years, the concept of forming limit diagram introduced by Keeler and Backofen (1964) and Goodwin (1968), respectively, has had a remarkable impact in the academic and industrial communities. The importance of the concept consists in the possibility to establish the maximum strains that can occur before necking in a forming process. The concept of FLD has been introduced in order to prevent the material waste and in the same time to allow the reduction of the times and prices related to the development of prototypes. The ﬁrst theoretical FLD models were based on the diﬀuse necking and localized necking theories proposed by Hill (1952) and Swift (1952), respectively. In the seventh decade

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of the previous century, Marciniak and Kuczynski (1967) proposed a theoretical model of strain localization based on the geometrical inhomogeneity already existing in the material. Later on, the Marciniak–Kuczynski model has been extended by Hutchinson and Neale (1978) in order to describe the left branch of the FLD. In 1975, Storen and Rice (1975) proposed the so-called “vertex theory” to model localized necking under biaxial stretching conditions. Recently, Dudzinski and Molinari (1991) proposed “the small perturbation theory” to model the plastic instability. For the last two approaches, the inﬂuence of anisotropy on localized necking cannot be shown. From all of the theories presented above, the most common are Marciniak– Kuczynski and Hutchinson–Neale models. An exhaustive description of the FLD concept may be found in (Banabic et al., 1992). Hora et al. (1996) intended to improve Swift’s criterion by taking into account the experimentally conﬁrmed fact that the onset of necking depends signiﬁcantly on the strain ratio. Diﬀerent strategies to determine the FLD’s were approached in (Butuc et al., 2002, Banabic et al., 2005). The ﬁrst attempt to compute FLD’s using the ﬁnite element method was published by Burford and Wagoner (1989). The FLD predictions are strongly inﬂuence by the shape of the yield loci used in the theoretical model. In 1950, Hill (1948) proposed the ﬁrst yield criterion for anisotropic materials. The mathematical shape of the criterion is a simple quadratic function and the coeﬃcients can be analytically computed. Therefore, it is the most used yield criterion. In 1979 (Hill, 1979), 1990 (Hill, 1990) and 1993 (Hill, 1993) Hill improved his criterion, but the mathematical formulations became more complex and not so easy to use. During the last two decades, many other yield criteria have been proposed aiming to improve the ﬁtting with experimental data. Hosford initiated a new research direction by introducing a yield function based on texture calculations (Hosford, 1972). During the time, the researchers have noticed that using more material parameters in the yield criterion can improve the accuracy of the predictions. Thereby, Barlat extended few times Hosford’s criterion using a linear transformation of the Cauchy stress tensor, transformation proposed by Karanﬁllis and Boyce (1993). In a recent paper, Barlat (2003) introduced a new yield criterion using two linear transformations of the Cauchy stress tensor. The group of yield criteria proposed by Banabic (Banabic et al., 2000) and improved by Paraianu (Paraianu et al., 2003) and later on by Banabic (Banabic et al., 2005) also belongs to “Hosford family”. Using general invariants, Cazacu and Barlat (2001) proposed an extension of Drucker’s (Drucker 1949) yield criterion. Finally, in 2005, Aretz (Aretz et al., 2004) develop a full stress state (3D) yield criterion proposing two formulations taking into account diﬀerent numbers of material parameters. A good synthesis of the yield criteria proposed during the time can be found in (Banabic et al., 2000). The aim of this paper is to predict FLD’s using the ﬁnite element method. For this purpose, two recent yield criteria were chosen (BBC2003 (Banabic

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153

et al., 2005) and Cazacu–Barlat (2001)). In order to evaluate the performances of the yield criteria, the authors compared the theoretical results with experimental data for AA3103-0 aluminum alloy. Moreover, the paper presents practical techniques to determine experimentally FLD’s.

2 Hutchinson–Neale model Figure 1 shows the model proposed by Hutchinson and Neal (1978) to describe the initiation of the necking process due to a geometric imperfection. As one may see, the sheet metal is assumed to have a slightly diﬀerent thickness in some region (“a” has the normal thickness, while “b” is thinner). The ratio of the initial thickness is used to deﬁne a non-homogeneity coeﬃcient: f0 =

tb0 ta0

(1)

In the general case, the imperfection performs a rotation during the straining process. Figure 1 shows the initial position of the groove (deﬁned by the angle Ψ0 ). In the ﬁnite element model, we use Ψ0 = 0 when computing the right branch of the FLD. For the left branch, we assume that the most unfavorable value of Ψ0 corresponds to the plane-strain condition in the (n, t) frame associated to the groove. Using this assumption, we obtain the following formula: 1 (2) Ψ0 = √ 1−ρ where ρ=

Δεa2 Δεa1

(3)

σ1 t a b

t0

b

a

a

σ2 t0

n ψ0 1 2

3

Fig. 1. Hutchinson–Neale model of a geometric imperfection

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is the ratio of the strain increments in region “a”. In the left branch of the FLD, ρ ∈ [−1, 0]. The isotropic hardening behavior is described by a Swift type function: p p m n Y ε˙ = k (a + εp ) b + ε˙ (4) where Y is the yield parameter; k is a material parameter; a is a pre-strain term; b is a pre-strain rate term; n is the hardening exponent and m is the strain rate sensitivity exponent.

3 Description of the Yield Criteria Implemented in the FE Code A yield surface is generally described by an implicit equation having the form: Φ (σ, Yref ) := σ − Yref = 0

(5)

where σ is the equivalent stress and Yref is a yield parameter. 3.1 BBC2003 Yield Criterion The sheet metal is assumed to behave as an orthotropic membrane under plane-stress conditions. The equivalent stress for BBC2003 yield criterion is described by the following mathematical expression (Banabic et al., 2005): 2k 2k (6) σ = [a · (Γ + Ψ) + a · (Γ − Ψ) + (1 − a) · (2Λ)2k ]1 2k where the functions Γ, Ψ and Λ depend on the non-zero components of the stress tensor being deﬁned as follows: ˆ22 σ ˆ11 + M σ 2 + 2 (N σ ˆ11 − P σ ˆ22 ) ˆ12 σ ˆ21 Ψ= + Q2 σ 4 + 2 (Rˆ σ11 − S σ ˆ22 ) + T 2σ Λ= ˆ12 σ ˆ21 4 Γ=

(7)

a and k are material parameters subjected to the constraints 0 ≤ a ≤ 1 and k ∈ N ∗. According to Hosford (1972), the value of the exponent k should be established by taking into account the crystallographic structure of the material: k = 3 for BCC materials; k = 4 for FCC materials.

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The remaining coeﬃcients (M, N, P, Q, R, S and T) are also material parameters. The stress σ ˆij (i, j = 1, 2) components are expressed in an orthonormal basis coincident with the axes of plastic orthotropy: 1 – Rolling Direction (RD); 2 – Transverse Direction (TD); 3 – Normal Direction (ND). One may notice that eqn. [6] is derived from the expression proposed by Barlat and Lian (1989). The shape of the yield surface is deﬁned by a, M , N , P , Q, R, S and T . Their values are evaluated in such a way that the constitutive equations reproduce as well as possible the following characteristics of the material: the uniaxial yield stresses and the r-coeﬃcients associated to the 0◦ , 45◦ and 90◦ directions, as well as the equibiaxial yield stress. One can emphasize that the number of material parameters from (7) is bigger than the number of material characteristics. In order to decrease the number of identiﬁcation equations, we enforce the restriction N = P . The identiﬁcation procedure is based on the Newton–Raphson method. 3.2 Cazacu–Barlat Yield Criterion Assuming that the yielding is not inﬂuenced by the hydrostatic pressure, the connection between yield and stress tensor is expressed through the invariants of the stress deviator: J2 =trS 2 /2 J3 =trS 3 /3

(8)

Based on Drucker’s (1949) isotropic yield criteria, f = J23 − cJ32 = k 2

(9)

6 where c is a constant and k 2 = 27 Yref 3 , Cazacu–Barlat (2001) proposed in 2001 generalized anisotropic expressions of the deviatoric stress invariants: 3 2 f20 = J20 − c J30 = k 2 (10) J30 =

1 1 1 (b1 + b2 ) σx3 + (b3 + b4 ) σy3 + [2 (b1 + b4 ) − b2 − b3 ] σz3 27 27 27 1 1 (11) − (b1 σy + b2 σz ) σx2 − (b3 σz + b4 σx ) σy2 9 9 1 2 − [(b1 − b2 + b4 ) σx + (b1 − b3 + b4 ) σy ] σz2 + (b1 + b4 ) σx σy σy 9 9 σ2 − xz [2b9 σy − b8 σz − (2b9 − b8 ) σx ] 3 2 σxy [2b10 σz − b5 σy − (2b10 − b5 ) σx ] − 3

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Advanced Methods in Material Forming 2 σyz [(b6 + b7 ) σx − b6 σy − b7 σz ] + 2b11 σxy σxz σyz 3 a a2 a3 1 2 2 2 J20 = (σx − σy ) + (σy − σz ) + (σx − σz ) 6 6 6 2 2 2 + a5 σxz + a6 σyz + a4 σxy

−

bk (k = 1, .., 11) and an (n = 1, .., 6) are material parameters. In a two-dimensional stress state, the equivalent stress has the following expression: 3 1 1 6 2 2 2 (a1 + a3 ) σx − a1 σx σy + (a1 + a2 ) σy + 3a4 σxy σ := 2 2 (12) 0 ! (b1 + b2 ) σx3 + (b3 + b4 ) σy3 − 3 (b1 σx + b4 σx ) σx σx 6 = Yref −c 2 −9σxy [(b5 − 2b10 ) σx − b5 σy ] where a1 , a2 , a3 , a4 , b1 , b2 , b3 , b4 , b5 , b10 and c are material parameters. The identiﬁcation procedure, based on Newton–Raphson scheme, calculates the material parameters by enforcing the yield criterion to reproduce the following experimental data: The uniaxial yield stresses associated to the directions deﬁned by 0◦ , 30◦ , 45◦ , 75◦ and 90◦ angles measured from RD – The coeﬃcients of uniaxial plastic anisotropy associated to the directions deﬁned by 0◦ , 30◦ , 45◦ , 75◦ and 90◦ angles measured from RD – The equibiaxial yield stress associated to RD and TD. –

4 Implementation of the BBC2003 and Cazacu–Barlat Yield Criteria in Hutchinson – Neale Model The BBC2003 and Cazacu–Barlat yield criteria have been implemented in the ﬁnite element program Abaqus/Standard as UMAT subroutines. The elastoplastic constitutive model is described by the following expression: {δσ} = [C evp ] {δ (Δε)}

(13)

T

where: {δσ} = [δσ11 , δσ22 , δσ12 ] and {δ (Δε)} = [δ (Δε11 ) , δ (Δε22 ) , 2δ (Δε12 )] are column-vectors containing the perturbations of the stress and strain tensors, respectively. [C evp ] is the consistent elastoplastic tangent modulus calculated using the following matrix formula:

[C

evp

] = [Q] −

Δt H Δt T {g} [Q] + 1 H

T

([Q] {g}) ([Q] {g})

(14)

Forming Limit Diagrams Using Finite Element Method

where:

evp evp evp ⎤ C1122 C1112 C1111 evp evp ⎦ C2222 C2212 [C evp ] = ⎣ evp symm C1212

157

⎡

−1 −1 [Q] = [C e ] + (Δεp ) [M ] {g} =

∂Φ ∂Φ ∂Φ , , 2 ∂σ11 ∂σ11 ∂σ11

(15)

(16)

T

⎤ ∂2Φ ∂2Φ ∂2Φ 2 ⎥ ⎢ ∂σ 2 ∂σ11 ∂σ22 ∂σ11 ∂σ12 11 ⎥ ⎢ 2 2 ⎥ ⎢ ∂ Φ ∂ Φ ⎥ [M ] = ⎢ 2 ⎥ ⎢ 2 ∂σ22 ⎢ ⎥ 2 ∂σ22 ∂σ122 ⎦ ⎣ ∂ Φ ∂ Φ symm 2 + 2 ∂σ12 ∂σ12 ∂σ21

(17)

⎡

⎤ 1υ 0 E ⎢υ 1 0 ⎥ [C e ] = ⎦ ⎣ 1−υ 1 − υ2 00 2 ∂Yref H= p ∂ ε˙

(18)

⎡

Δ¯ εp = ε¯˙p (Δt)

(19)

(20) (21)

Δt is the time increment, ε¯˙p is the equivalent plastic strain rate and Δ¯ εp is the equivalent plastic strain increment. The quantities deﬁned by (16)–(20) are: {g} – column-vector storing the gradient of the yield surface, [M ] – curvature matrix of the yield surface, H – strain rate hardening modulus, [C e ] – classical elastic modulus (depending on the Young’s modulus E and Poisson’s ratio υ). In order to detect the limit deformation, the values of the thickness strain rate corresponding to a pair of elements were stored and compared. One of the elements is placed in region “a”, while the other is placed in region “b” (Fig. 1). The FE simulation ends when the strain rate associated to the second element is seven times greater then the strain rate of the ﬁrst element (Fig. 2). The corresponding point belonging to the FLD is deﬁned by the values of the principal logarithmic strains calculated at the end of the previous time increment for the reference element placed in region “a”. The above procedure has been repeated for seven diﬀerent displacement ratios along the axes 1 and 2 (Fig. 1).

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Advanced Methods in Material Forming ε1a

ε1a*

ε1b*

ε1b

Fig. 2. The dependence between strain rates in regions “a” and “b”

The mesh used in the simulation consisted in 2575 “M3D4” elements (4 node quadrilateral membrane). The mesh associated to region “b” is denser than in region “a”.

5 Application 5.1 Material Characterization The ﬁnite element model has been tested for an AA3103-0 aluminium alloy having the mechanical parameters listed in Table 1. Tables 2 and 3 show the values of the material parameters for BBC2003 and Cazacu–Barlat yield criteria provided by the identiﬁcation procedures. In both cases, Yref is assumed to be the uniaxial yield stress in the rolling direction (Y0 ). The coeﬃcients of the hardening law (4) are also listed in Table 4. The nominal thickness of the sheet metal was 1.2 mm and the value of the imperfection factor used in the calculations was 0.997.

Table 1. Mechanical parameters used in the identiﬁcation procedure σ0 [MPa]

σ30 [MPa]

σ45 [MPa]

σ75 [MPa]

σ90 [MPa]

σb [MPa]

55 r0 0.639

56 r30 0.555

58 r45 0.513

61 r75 0.581

61 r90 0.605

60 Yref [MPa] 55

Table 2. Coeﬃcients of the BBC2003 yield criterion a

M

N

P

Q

R

S

T

K

0.508

0.829

1.055

1.055

1.046

0.964

0.839

0.901

4

Forming Limit Diagrams Using Finite Element Method

159

Table 3. Coeﬃcients of the Cazacu-Barlat yield criterion a1

a2

a3

a4

b1

b2

0.060 b3 –0.706

0.600 b4 0.192

0.872 b5 –0.164

0.268 b10 –0.170

0.174 c 1.400

–1.247

Table 4. Coeﬃcients of the strain hardening law Y [MPa]

a

b

n

m

188

0.0043

0.01

0.226

0.012

5.2 Experimental Techniques Determination of the Limit Strains The limit strains can be determined according to diﬀerent criteria: the occurrence of diﬀuse necking, localized necking or fracture. This paper presents an analysis of the strain localization process and, in connection with this aspect, the method used for establishing the limit strains. For the practical determination, several methods are known: Veerman, Bragard, “double proﬁle”, Kobayashi, Hecker, Zurich Nr.5. The authors preferred Hecker’s method due to the possibilities oﬀered by it. Tests used for the FLD Determination In order to cover the domain of the forming usually met in industry, diﬀerent forming paths of the specimen must be obtained. Such paths should be spanned between the simple traction (ε1 = −2ε2 ) and the equibiaxial stretching (ε1 = ε2 ). They can be obtained by realizing diﬀerent stress states, which in their turn may be obtained by modifying the shape or/and dimensions of the punch (Keeler, Marciniak), the die (Jovignot), the specimen (Brozzo de Luca, Sanz–Grumbach, Nakazima, Hasek, Azrin–Backofen) or the lubricating conditions (Hecker). The Institute of Metal Forming Technology of Stuttgart University (Institut fur Umformtechnik – IFU) performed the following tests to determine the experimental FLD’s: the hydraulic bulge testing for the positive minor strains and tensile testing using two kinds of specimens (with and without notches) for the negative minor strain, respectively. The shapes of the specimens are presented in Fig. 3. Three elliptical dies with the aspect ratios of 1.0, 0.8 and 0.7 were used in order to cover the right side of the FLD (Fig. 3). The major axis of the dies was 100 mm. The specimens were oriented with the major axis perpendicular to the rolling direction, in order to obtain the major strain (ε1 ) oriented along the rolling direction and vice versa. The same algorithm was

Advanced Methods in Material Forming Bulge Test : Geometry 2

1 00

Geometry 1

100 180

80 180 Geometry 4

100

100

Geometry 3

70 180

60 180

Tension Test

40

R2

20

0

Geometry 1

150 250 30

Geometry 2

15

60

120

160

500

Fig. 3. Geometry of the specimens

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161

applied in the case of tensile tests. On this way two FLD’s were obtained for each material: one, for the case of the major strain (ε1 ) oriented in the rolling direction and, the other one, for the case of the major strain (ε1 ) oriented perpendicular to the rolling direction. Specimens The hydraulic bulge tests were performed using circular specimens with 180 mm diameter (Fig. 3). The tensile test was performed with two shapes of the specimens (Fig. 3, geometry 1 and geometry 2, respectively). The geometry of the notched tensile specimen was established using FE simulation, in order to obtain a plane strain in the gauge section. The tensile specimens without notches were obtained by blanking using an appropriate blanking die and the tensile specimens with notches were obtained by shearing and the notches by milling, respectively. A typical set of test specimens is shown in Fig. 3. A circular overlapping grid (5 mm diameter of the circles) was printed on the specimens using an electro-chemical method. Experimental Methodology The bulging tests were made on a hydraulic bulging machine designed and built by IFU. The tensile tests were made on a mechanical tensile testing machine (in the case of tensile specimens without notches) and on a multiaxial testing machine designed and built by IFU (in the case of tensile specimens with notches), respectively. Four and/or ﬁve specimens were used for each test. Testing continued until a fracture was visible in the specimen. In order to measure the circles and/or the axes of the ellipses, a travelling toolmaker’s microscope was used. To deﬁne the FLC, the logarithmic strains ε1 and ε2 from deformed circles (ellipses) both close to, and within, necked or fractured zones were measured. These data points were identiﬁed as good, necked and fractured ellipses and were plotted in the FLD. The necking FLC is a welldeﬁned boundary between good and necked ellipses. Experimental Results The experiments were performed using AA3103-0 aluminum sheets with 1.2 mm thickness. Figure 4 shows the experimental FLD for the AA3103-0 aluminum sheets, in the case when the major strain is oriented along the rolling direction. The diﬀerence between the curves obtained when the major strain is oriented along the rolling direction and perpendicular to the rolling direction, respectively, are not signiﬁcant, because the values of the planar anisotropy coeﬃcients corresponding to RD and TD are very close.

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Advanced Methods in Material Forming 0,8

ε2 = 0 plane strain

0,7

0,6

major strain ε1

0,5

m for

ing

ur it c lim

0,4

rm fo

ε 1=

0,3

ing

f ve (

ure) ract

.

n e( u rv ti c lim

2ε 2

=ε ε1

) i ng eck

2

0,2

0,1

0 –0,4

–0,3

–0,2

–0,1

0 0,1 minor strain ε2

0,2

0,3

0,4

0,5

Fig. 4. Shape of the specimens used in order to determine the Forming Limit Diagrams

5.3 Comparison Between Numerical Results and Experiments Figure 5 shows the yield loci predicted by BBC2003 and Cazacu–Barlat models for AA3103-0. Several experimental points corresponding to diﬀerent biaxial or uniaxial loads were also plotted on the same diagram. One may notice that both yield loci are in a good agreement with experimental data. The yield locus predicted by BBC2003 in equibiaxial stretching is sharper than the one predicted by Cazacu–Barlat. This factor inﬂuences the predictions of FLD’s. Figure 6 presents a comparison between FLD’s computed by the ﬁnite element model based on BBC2003 and Cazacu–Barlat yield criteria. Some experimental points were plotted on the same diagram. The experimental investigation to determine the material parameters for the considered aluminium alloy has been described in previous section. One may see that the right branch of the FLD predicted by Cazacu-Barlat model is more accurate than the one predicted by BBC2003. BBC2003 underestimated the formability of the sheet metal in this region. As concerns the left branch of the FLD, the curves are practically coincident. It seems that the quality of the numerical predictions is strongly inﬂuenced by the shape of

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163

70 60 50

σ_1

40 30 20 Experiments BBC2003 Cazacu-Barlat

10 0 0

10

20

30

40

50

60

70

σ_2

Fig. 5. Yield loci for AA3103-0

0.8 0.7 0.6

eps_1

0.5 0.4 0.3 0.2 0.1

exp-necking exp-good exp-fracture BBC2003 Cazacu-Barlat

0.0 –0.3 –0.2 –0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 eps_2

Fig. 6. Comparison between numerical and experimental FLD’s

the yield surface especially on the right branch of the FLD. A less sharpness eﬀect on the yield locus promotes larger limit strains.

6 Conclusions In order to emphasize the inﬂuence of the yield locus geometry on FLD’s, the authors have implemented two modern yield criteria (BBC2003 and Cazacu– Barlat) in a ﬁnite-element code. The predictions given by these criteria have

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been compared with experimental data. The comparison shows that Cazacu– Barlat formulation is more realistic and in better agreement with experimental data, especially on the right branch of the FLD. But we should notice that Cazacu–Barlat yield criterion needs more experimental data for evaluating the coeﬃcients as compared with BBC2003.

References Aretz H., Barlat F., “General orthotropic yield functions based on linear stress deviator transformations”, CP712, Materials Processing and Design. Modeling, Simulation and Applications, Numiform 2004, p. 147–151. Banabic D., Aretz H., Comsa D.S., Paraianu L., “An improved analytical description of orthotropy in metallic sheets”, International Journal of Plasticity, vol. 21, 2005, p. 493–512. Banabic D., Aretz H., Paraianu L. and Jurco P., “Application of various FLD modelling approaches”, Modelling and Simulation in Materials Science and Engineering, no. 13, 2005, p. 1–11. Banabic D., Bunge H.-J., Pohlandt K. and Tekkaya A. E., “Formability of Metallic Materials”, Banabic D. (ed.), Berlin-Heildelberg, Springer Verlag, 2000. Banabic D., D¨ orr R.I., “Formability of thin sheet metal”, Ed. O.I.D.I.C.M., Bucuresti, 1992, in Romanian. Barlat F., Brem J.C., Yoon J.W., Chung K., Dick R.E., Lege D.J., Pourboghrat F., Choi S.-H., Chu E., “Plane stress yield function for aluminum alloy sheets— part 1: theory”, International Journal of Plasticity, vol. 19, issue 9, 2003, p. 1297–1319. Barlat F., Lian J., “Plastic behavior and stretchability of sheet metals. Part I: A yield function for orthotropic sheets under plane stress conditions”, International Journal of Plasticity, vol. 5, 1989, p. 131–147. Burford D.A., Wagoner R.H., “A more realistic method for predicting limits of metal sheets”, Forming limit diagrams: Concepts, Methods and Applications, Wagoner R.H. (ed.), TMS Publishing House,Warrendale, 1989, p. 167–182. Butuc M.C., Barata da Rocha A., Gracio J. J. and Ferreira Duarte J., “A more general model for forming limit diagrams prediction”, Journal of Materials Processing Technology, vol. 125–126, 2002, p. 213–218. Cazacu O., Barlat F., “Generalization of Drucker’s yield criterion to orthotropy”, Mathematics and Mechanics of Solids, 6, 2001, p. 613–630. Drucker D.C., “Relation of Experiments to Mathematical Theories of Plasticity”, Journal of Applied Mechanics, 16, 1949, p. 349–357. Dudzinski D., Molinari A., “Perturbation analysis of thermoviscoplastic instabilities in biaxial loading”, International Journal of Solids and Structures, vol. 27, 1991, p. 601–628. Goodwin, G.M., “Application of strain analysis to sheet metal forming problems in the press shop”, SAE, paper no. 680093, 1968. Hill R., “On discontinuous plastic states, with special reference to localized necking in thin sheets”, Journal of the Mechanics and Physics of Solids, vol. 1, 1952, p. 19–30

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Hill R., “A theory of the yielding and plastic ﬂow of anisotropic metals”, Proc. Royal Society of London, Series A, 193,no. A 1033, 1948, p. 281–297. Hill R., “Theoretical plasticity of textured aggregates”, Math. Proc. Cambr. Phil. Soc., 85, 1979, p. 179–191. Hill R., “Constitutive modeling of orthotropy plasticity in sheet metal”, Journal of the Mechanics and Physics of Solids, vol. 38, 1990, p. 405–417. Hill R., “A user-friendly theory of orthotropic plasticity in sheet metals”, International Journal of Mechanical Sciences, vol. 35, 1993, p. 19–25. Hora P., Tong L., Reissner J., “A prediction method for ductile sheet metal failure in FE-simulation”, Proceedings of the Numisheet’96 Conference (Dearborn/Michigan), 1996, p. 252–256. Hosford W.F., “A generalized isotropic yield criterion”, Journal of Applied Mechanic, no. 39, 1972, p. 607–609. Hutchinson R.W., Neal K.W., “Sheet necking III.Strain-rate eﬀects”, Mechanics of Sheet Metal Forming, Plenum Press New York – London, 1978, p. 269–285. Karaﬁllis A.P., Boyce, M.C., “A generalized anisotropic yield criterion using bounds and a transformation weighting tensor”, Journal of the Mechanics and Physics of Solids, vol. 41, 1993, p. 1859–1886. Keeler, S.P., Backofen, W.A., “Plastic instability and fracture in sheets stretched over rigid punches”, ASM Trans. Quart 56, 1964, p. 25. Marciniak Z., Kuczynski K., “Limit strains in the processes of stretch forming sheet metal”, International Journal of Mechanical Sciences, vol. 9, 9, 1967, p. 609–612. Paraianu L., Comsa D.S., Cosovici G., Jurco P., Banabic D., “An improvement of the BBC2000 yield criterion”, Proceedings of the ESAFORM 2003 Conference, 2003, p. 215–219. Storen S., Rice J.R., “Localized necking in thin sheets”, Journal of the Mechanics and Physics of Solids, vol. 23, 1975, p. 421–441. Swift H.W., “Plastic instability under plane stress”, Journal of the Mechanics and Physics of Solids, vol. 1, 1952, p. 1–18.

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Recent Advances in Process Design for Sheet and Tube Hydroforming J.C. Gelin, C. Labergere and S. Thibaud FEMTO-ST Institute, Applied Mechanics Laboratory, ENSMM and CNRS, 24 Chemin de l’Epitaphe, 25000 Besan¸con, France, [email protected], [email protected], [email protected] Summary. The paper is concentrated on the last developments related to process design for sheet and tube hydroforming. The paper ﬁrst analysis the ways to properly account ﬂow movements and pressure drops occurring in sheet and tube hydroforming that can interact with sheet or tube deformation during hydroforming described with a ﬂow-structural approach, based on an ALE approach accounting well the structural interactions. Then diﬀerent optimization strategies for process parameters are presented on the basis of cost functions associated to ﬁnal geometry of sheet or tubular components, based on gradient approaches as well as stochastic ones, depending on the number of parameters and on the sensitivity of parameters relatively to the response functions. Finally an integrated design approach based on control of processes is described combining optimization and continuous adjustment of process parameters to get the required parts accounting the machine tool limits and the material ones. Diﬀerent applications are given related to typical components that are used in automotive industry.

Key words: hydroforming, ﬂuid-structure coupling, optimization, process control.

1 Introduction The hydroforming processes for ﬂanges or tubes hydroforming are now increasingly used in automotive industry for manufacturing lightweight components, (Kleiner et al., 2003) and (Merklein et al., 2002). These processes permit to obtain complex structural parts comparatively to classical stamping ones and are used as example in automotive production for standard vehicles or prototype ones, and are often associated to the use of new incoming materials (Vinarcik et al., 2002) and (Alberti et al., 1998). Among diﬀerent hydroforming processes, one classically distinguishes two distinct technologies. In tube hydroforming technology, a tube generally with a circular cross section, is ﬁrst shaped by bending and then clamped in a tooling system, ﬁlled by a ﬂuid and

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then pressurized (Dohmann et al., 1997). It results that the tube takes the inner shape of tools that allow producing complex seamless parts. The ﬂange hydroforming technology consists to take beneﬁt from hydraulic pressure to assist ﬂange deep drawing by replacing the die cavity by a cavity ﬁlled by a hydraulic ﬂuid (Tirosh et al., 1985). In such processes, the pressure in the die cavity decreases the sliding between the ﬂange and the punch and reduces the friction between the ﬂange and the die cavity and also under the blankholder, leading to the increase of Limit Drawing Ratio comparatively to classical deep drawing processes (Gelin et al., 1993). The ﬂange hydroforming process permit to get complex components in a single shape that cannot be produced by classical deep drawing processes. It also permits to get small size details due to the fact that the pressure applies the ﬂange on the dies. In order to perform modeling and simulation of hydroforming processes, it is ﬁrst necessary to properly identify material properties involved in hydroforming. For tube hydroforming, several authors have proposed biaxial experiments where a tube submitted to an inner pressure can be expanded in an open cavity (Manabe et al., 2002; Sokolowski et al., 2000). The results obtained from these experiments show that the hardening exponent has an important role. In order to characterize material behavior through tube expansion tests, some authors have proposed relations issued from analytical modeling (Sokolowski et al., 2000; Asnaﬁ et al. 1999) that permit to get material parameters from inner pressure and axial loads. Another aspect in modeling and simulation of ﬂange or tube hydroforming is associated to the prediction of the limits of formability, i.e. the range of process parameters leading to a component free of defects (necking, fracture, wrinkling and bursting). In that ﬁeld, some authors have attempted to apply classical Forming Limit Diagrams (FLD) to predict necking, bursting and failure (Ko¸c et al., 2002), but the pressure strongly inﬂuences the FLD, so other approaches have been proposed based on ductile fracture mechanics (Lei et al., 2001), whereas other authors proposed to extend the linear stability analysis to 3D complex stress states to describe necking and fracture in tube hydroforming (Lejeune et al., 2003). In order to proceed in ﬂange or tube hydroforming in the process window avoiding failure, bursting and buckling, various authors have proposed strategies to determine the limiting axial loads to apply to pistons or counter pistons in order to prevent risks of buckling and failure (Asnaﬁ et al., 1999). Control strategy based on coupling FE simulation of tube hydroforming and optimization has been proposed by the authors (Gelin et al., 2002; Gelin et al., 2004) that propose a method to incrementally build the inner pressure and axial loads vs. axial feed.

2 Numerical Modeling of Sheet or Tube Deformation In order to fulﬁll requirements of the products quality, stiﬀness constraints or even for aesthetic considerations, accurate description of the hydroforming processes is necessary. In particular, strain contours, thickness variations

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in tubes or ﬂanges and the ﬁnal shape have to be analyzed. However, the hydroforming processes for tubes or ﬂanges are complex processes, the part indeed large displacements, large rotations and strains. For these geometrical non-linearities, it is also necessary to add, for a correct description of the material behavior of sheets or tubes, material non-linearities. In addition, the hydroforming processes generally use tooling systems that introduce other non-linearities as contact and friction. The current tendency in the automobile sector is to reduce the deadlines for designing new vehicles and the impressive technological projection of the data-processing ﬁeld thus involved the use of numerical tools based on the ﬁnite elements method, for the simulation of the hydroforming processes and stamping (Kang et al., 2004). The numerical simulation of hydroforming by the ﬁnite elements method requires the use of ﬁnite elements able to account geometrical and material behavior. The simplest case is the use of 2D ﬁnite elements to deal with the axisymmetric hydroforming problems. However, the shapes of the components that industry requires are often complex and the majority of the parts are obtained starting from ﬂanges or from rather weak tube thicknesses. For taking into account 3D geometries and to model various components, one proposes the use of shells elements (Boisse et al., 1996). This type of elements makes it possible to integrate membrane and bending eﬀects and transverse shearing if necessary. In this context of shells ﬁnite elements, a numerical modeling of the anisotropic elastoplastic behavior has been proposed. The algorithm that is used consists to realize an elastic prediction followed of a possible plastic correction (Boubakar et al., 1995). Concerning the global solution procedure, one concentrated on an approach based on a transient explicit algorithm.

3 Modelling the Fluid/Structure Interactions in Hydroforming 3.1 Fluid-Structure Interactions in Sheet Hydroforming In this paragraph, one focuses on the behavior of the ﬂuid velocity and pressure in the case of ﬂange hydroforming to realize cylindrical or 3D cups. The objective is to set up a model for the simulation of the distribution of ﬂuid pressure between the ﬂange and the die. Moreover, one wishes to determine the variation of the pressure in the closed die where the rise in pressure is obtained through the reduction in volume of ﬂuid caused by the displacement of the punch. In that ﬁeld, pioneer contributions (Tirosh et al., 1985) relate the modeling of the hydroforming of a circular ﬂange and analyze the inﬂuence of pressure and deep drawing ratio on the punch force to be applied, see Fig. 1; they also determined the evolution of the pressure between the ﬂange and the die in using an analytical approach. The present research group (Gelin et al., 1993) has proposed a similar approach to describe the role of the ﬂuid between

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Fig. 1. The hydromechanical deep drawing process with axisymmetric tools

the ﬂange and the die and they have designed an experimental device to measure the pressure of the ﬂuid corresponding to various points of the ﬂange. One describes here a generalized numerical procedure to account the hydrodynamics ﬂuid ﬂow between the sheet metal part and the die, beneath the blankholder. To determine the variation of pressure under the blank holder, one supposes the existence of a hydrodynamic mode (Gelin et al., 1993). To study this hydrodynamic mode, one uses the Navier-Stokes equations in the axisymmetric form that give after integration in the following form, for the radial ﬂuid velocity: 1 ∂p (r) z 2 + C1 z + C2 (1) vr = μ ∂r 2 where vr is the radical ﬂuid velocity, p is the pressure and μ is the ﬂuid viscosity, whereas C1 and C2 are the integration constants determined by the boundary conditions, see Fig. 2. After integration accounting boundary conditions, one obtains the following form for the radial ﬂuid velocity: z h2 ∂p z z − 1 + Vt 1 − (2) vr = 2μ ∂r e e e where Vt is the radial velocity of the blank for the position considered. Pressure Distribution Under the Blank Holder Initially, one determines the ﬂuid ﬂow between the ﬂange and the die. This ﬂow is evaluated for a given position as:

"e Qr = 2πr

vr (z)dz = 2πr 0

−e3 12μ

∂p ∂r

e + Vt 2

(3)

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Blank holder flange Vt

r

Vr

Fluid

e

Die pc

Fig. 2. Representation of the ﬂow between the ﬂange and the die

From equation (3), it is possible to express the radial evolution of the pressure according to the ﬂow in the form: ∂p 12μ Qr e = 3 − + Vt (4) ∂r e 2πr 2 The radial evolution of the ﬂow Qr is obtained by integrating the incompressibility condition on following the thickness of the ﬂuid, one thus has: ⎛ e ⎞ " "e 1 ∂ ⎝ 1 ∂Qr ∂vz ⎠ r vr (z) dz + dz = − Vz = 0 (5) r ∂r ∂z 2πr ∂r 0

0

where Vz is the velocity of the ﬂange following direction z in a given position. The radial evolution of the ﬂow is thus expressed thanks to the relation: ∂Qr = 2πrVz ∂r

(6)

Computation of the Leak-Flow We will determine the ﬂuid ﬂow located between the ﬂange and the die. To simplify the solution, it is considered that the blank holder moves slowly and that one can neglect Vz velocity of the ﬂuid, one thus obtain in this case: Vz ≈ 0 ⇒

∂Qr ≈0 ∂r

(7)

Equation (7) indicates that the ﬂow Qr is practically constant according to the direction of ﬂuid ﬂow.

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punch

pb Limit Position r = rm die

pc

Fig. 3. Discretisation of the ﬂuid ﬂow under the blank-holder

One now search to estimate the radial ﬂow Qr and the pressure variation vs. radial position. Jensen et al. (Jensen et al., 1999) proposed to discretize the ﬂuid ﬂow according to the radial direction in order to solve equation (4). For that, it directly uses the 2D mesh associated to a ﬁnite element simulation. One proposes to employ the same approach with a 3D mesh comprising a symmetry plane. For the continuation of the study, one notes Vir radial velocity, ri the radial position and ei the thickness of ﬁlm ﬂuid corresponding to the node i, see ﬁgure 3. All these geometrical values and velocities are obtained starting from simulation by ﬁnite elements of the ﬂange hydroforming. Calculation is carried out between the position rm and the node nb which is located on the edge external of the ﬂange. The boundary conditions in the case of applied pressure are: ! p (r) = pc ∀r ∈ [0, rm ] (8) pnb = p (rnb ) = 0 where pc is the pressure of the ﬂuid in the cavity. One integrates (4) according to limits (ri ≤ r ≤ ri+1 ) using the following expression: r"i+1

∂p dr = p (ri+1 ) − p (ri ) = ∂r

ri

r"i+1

12μ e3

Qr e + Vt − 2πr 2

dr

(9)

ri

To evaluate the diﬀerence in pressure between the two radial positions ri and ri+1 , one uses a linear approximation of ﬂuid thickness expressed as: e (r) =

ei+1 − ei (r − ri ) + ei ri+1 − ri

et

Vt (r) =

Vti+1 − Vti (r − ri ) + Vti ri+1 − ri

(10)

Equation (9) could be integrated in using a trapeze method that makes it possible to determine, increment by increment, the evolution of the pressure of the ﬂuid between the ﬂange and the die. The evaluation of the leak-ﬂow Qr is obtained by summing the variations of pressure given by (9) on all the node i. By using the boundary conditions (8), one obtains the following relation:

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β α n n b b pnb − pc = − αi Qr + βi i=m

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(11)

i=m

By assuming pnb = 0, one obtains the following formulation: Qr =

pc + β α

(12)

Determination of the Pressure Cavity The pressure in the die cavity is related to leak ﬂow through the compressibility of the ﬂuid in the die cavity expressed as: V1 1 p2 − p1 = ln (13) χ V2 where χ is the compressibility ratio of the ﬂuid in the die cavity. The variation of the volume of ﬂuid is given starting from the volume of deformed sheet and the volume of ﬂuid which escapes from the cavity. One thus obtains the variation of the pressure in the cavity in the form: Vc − Vpi + Vfi 1 i+1 i pc − pc = ln (14) χ Vc − Vpi+1 + Vfi+1 where pic , Vpi and Vfi are respectively the pressure cavity, the volume of sheet and the volume of leakage at the moment ti . The volume of leakage is evaluated thanks to the leak-ﬂow Qr , see (12) so that: Vfi+1 =Vfi + qri+1 · Δti+1 with Vf0 =0 and

Δti+1 = ti+1 − ti

(15)

The hydroforming of a cylindrical cup is chosen as an example. The experiments are related in (Gelin et al., 1993). The tests consist to the deep drawing of a circular ﬂange associated to a pressure cavity generated by the volume variation of an incompressible ﬂuid. The geometry of the tools is given in Fig. 4. The variation of the ﬂuid volume is related to the ﬂange deformation and to the displacement of the punch, the decrease of the volume of the cavity results from the deformation of the sheet metal part. One has established (Gelin et al., 1993) the inﬂuence of the hydroforming parameters (thickness, blank diameter, material, blankholder force, punch velocity) on sheet deformation and on the evolution of the inner die pressure cavity. In Fig. 5, one compares the evolution of the pressure cavity obtained through the numerical procedure above described, with the variation of pressure in the die cavity obtain through experiments. One notices a good agreement between both curves.

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Fig. 4. Geometry of the hydromechanical deep drawing process

Fig. 5. Comparison between simulations and experiments for the die cavity pressure vs. the punch stroke

3.2 Fluid-Structure Interactions in Tube Hydroforming In that part, one analysis the ﬂuid-structure interaction in tube hydroforming where an initially cylindrical tube is pushed in the axial direction by a piston and where the inner pressure inside the tube change the inner shape of the tube, see Fig. 6. The ﬂuid ﬂow in the cavity has to account in order to describe the process as it is carried out in practice. The following developments investigate the way to account ﬂuid-structure interactions when the inner pressure evolves associated to tube hydroforming, i.e. when variation of tube diameter takes place.

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tube

punch

fluid entry

175

punch

fluid movement

deformation of tube surface

Fig. 6. Fluid-Structure interaction in tube hydroforming

Inside the die cavity, the ﬂuid can be considered as a Newtonian viscous ﬂuid and the stress tensor is related to the strain rate tensor as: σ = −pI + 2μD = −pI + 2μ∇S v

(16)

where σ is the Cauchy stress tensor, p is the pressure, v is the ﬂuid velocity, μ is the viscosity and D the strain rate tensor. The balance of momentum and the mass conservation equations are written as: ∂v div (σ) + f = ρ + ∇v.v (17) ∂t div (v) = 0

(18)

where ρ is the ﬂuid density. Equations (17) and (18) are then discretized through the Finite Elements Method, and the well knows P1/P1+ ﬁnite elements have been used. One ﬁrst considers an axisymmetric tube hydroforming problem where the tube is discretized with axisymmetric shell elements avoiding volumetric locking. The ﬂuid in the inner part of the tube is meshed with P1/P1+ elements. In order to ensure a well deﬁned coupling between the ﬂuid in the die cavity and the deformed tube, one has chosen to establish a superposition of the ﬂuid nodes on the inner surface of the tube and structural nodes that are used to calculate the deformation of the tube, see Fig. 7. The boundary conditions are the following: ⎧ v.z = Vp1 on Γv1 ⎧ ⎪ ⎪ ⎨vT .z = Vp1 on Γv1 ⎨ v.z = Vp2 on Γv2 vT .z = Vp2 on Γv2 (19) T v = vT on Γv ⎩ ⎪ ⎪ t = pF .n on ΓvT ⎩ p = pc on Γp where Vp1 and Vp2 stand for the axial velocities at both extremities of the tube and pc is the pressure cavity that one needs to obtain. The ﬂuid is injected in the tube through a small oriﬁce placed on one of the punch (Γp ), Fig. 7.

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Fig. 7. Illustration of the ALE strategy used for ﬂuid-structure interactions in the hydroforming

During tube hydroforming, the tube outer surface moves. The ﬂuid nodes that coincide with this surface have to exhibit such a similar motion, but one have to point out that (17) is expressed in an Eulerian form. So, this equation has been modiﬁed, introducing an Arbitrary Lagrangien Eulerian formulation (ALE) to deal with that problem, expressed as: ∂v mesh + v−v ρ ∇v = div (σ) + f ∂t (20) S σ = −pI + 2μ∇ v where vmesh stands for the velocity of ﬂuid nodes associated to the displacement of the inner part of the tube, and v stands for the ﬂuid velocity along the considered part of the tube, when v = vmesh , the formulation is a Lagrangien one, whereas when vmesh = 0 the formulation is an Eulerian one, see Fig. 7. One consider ss an example, the case consisting to perform the hydroforming of a steel tube with an outer diameter equal to 25.5 mm and a thickness equal 1.6 mm in an axisymmetric cavity with multi-stage cylindrical parts, see Fig. 8.

Fig. 8. Geometrical description of an axisymmetric tube hydroforming problem accounting ﬂuid/structure interactions

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Fig. 9. Fluid volume variation vs. normalized hydroforming processing time

One has represented in Fig. 9 the variation of the oil volume during hydroforming in the inner cavity. It has to be noticed that the ﬂuid volume increase up to 15%, from the initial volume, that corresponds to a signiﬁcant value. One clearly remarks that the ﬂuid volume increases following diﬀerent slopes during processing. At the end of the process, the ﬂuid volume inside the inner die cavity remains approximately constant.

4 Optimisation and Control of the Hydroforming Processes 4.1 Optimization and Control Based on the Pressure The optimization procedures are increasingly used in industry to determine the optimal process parameters. The use of the ﬁnite element method coupled to optimization algorithms is a solution to obtain the optimum parameters. So, the ﬁnite element method is used to evaluate the eﬀect of a set of parameters on the process. The optimization of hydroforming process, in particular, requires the deﬁnition of objective functions representing the quality of the components. These functions are used to quantify shape defects or surface distortions (buckling, wrinkling, and bursting). For example, one can use the objective function expressed as: n hi (p) − h0 2 1 Fobj (p) = (21) n h0 i=1

where h0 is the initial ﬂange or tube thickness hi stands for the thickness at node i and n is the total nodes number involved in the problem. The

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Optimization Method (OM) proposed by the authors (Gelin et al. 2002) consists to combine the simulation of the process and the evaluation of the cost function and then to minimize this cost function with constraints respect in using a SQP algorithm. This method requires computing the gradient of the objective and constraint functions, i.e. to perform sensitivity analyses. Two diﬀerent methods for sensitivity calculations are used to obtain the gradient vector corresponding to the Finite Diﬀerent Method (FDM) and to the Direct Diﬀerentiation Method (DDM). Recently, one has thus developed an optimization module “LSOPTMEF” r as FEM solver. This optimization module includes MPI that uses LS-DYNA library in order to launch several computations in parallel on a LINUX cluster. One also proposes a strategy based on a control algorithm to determine the command laws for hydroforming of parts with better quality. It consists to include a control loop inside the incremental loop used in the ﬁnite element solution procedure. The developed method proposes a coupling between the response surface method and optimal control (Gelin et al., 2004). The employed methodology consists in combining the theory of optimal control with the response surface method (Moving Least Square Approximation). The strategy adopted here consists in building an approximation for the pressure law vs. time. This approximation can be speciﬁed in a simple polynomial form or a more complicated one allowing possibilities to get the form of the law. In a ﬁrst stage, one chose a linear piecewise pressure approximation expressed in the following form: p (t) =

pi+1 − pi pi ti+1 − pi+1 ti t+ ti+1 − ti ti+1 − ti

(22)

In summary, one thus obtains the following optimization diagram Fig. 10. In order to illustrate the possibilities of the diﬀerent approaches, one propose the hydroforming of a tube with an external diameter equal to 49.8 mm, thickness equal to 2 mm and material corresponds to mild steel, in order to evaluate the Optimization Method (OM) and Control Algorithm (CA). One

Optimum parameters Evaluation of the Objective function

Pressure

Pressure curve

p1

p2

p3

p p4 5

SolveurEF 3D PolyForm © LS-DYNA ®

f(p) = 1 N

⎛h − h ⎞ ∑ ⎜⎜ i 0⎟⎟ i =1 ⎝ h0 ⎠ N

2

Optimal solution

time

Modification of parameters (pi) • Quasi-Newton method • Sensibility analysis

Fig. 10. Optimization loop used for process control in hydroforming

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P2

P1

y

z

x

Fig. 11. Position of the tube in the die cavity and the associated boundary conditions

search to determine the evolution of the pressure p(t) and the displacements of the two pistons during hydroforming, see Fig. 11. One uses the objective function given by (21) to measure the quality of the tube. The results of the proposed approach are related in Figs. 12, 13 and 14. 6,5

Value of the objective function

6 DDM for sensitivity analysis FDM for sensitivity analysis

5,5 5 4,5 4 3,5 3 0

2

4 6 Nb of Iterations

8

10

Fig. 12. Comparison of the objective function variation during iterations using DDM and FDM for sensitivity calculations

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70 Pressure curve obtained with control algorithm Pressure curve obtained with optimization method Polynomial (Pressure curve obtained with control algorithm)

Pressure (MPa)

60

50

40

30

20

10

0 0

0,2

0,4 0,6 Loading factor τ

0,8

1

Fig. 13. Comparison of pressure curves vs. loading factor obtained from Control Algorithm (CA) and Optimization Method (OM)

One has to notice from ﬁg. 12 and 13 that both methods (OM) and (AC) converge towards an almost unique solution. One has represented in ﬁg. 14 the thickness variation of the nodes pertaining to the plans P1 and P2 which are located along Z axis (see Fig. 10). Moreover, one observes on Fig. 12 that the objective function converges towards a minimum when one uses the optimization method where the gradient of the objective function is obtained with both sensitivity method (MDF) and (MDD) one. Nevertheless, if CPU times are compared, the Optimization Method (OM) required 49h whereas the Control Algorithm (CA) requires only 14 h. 4.2 Parametric Control for the Hydroforming Process Based on a Volume of Fluid Method In addition to the optimization and control procedure that one have proposed, one search to control the hydroforming process starting from the volume of injected ﬂuid inside the tube. When using a FEM software like LS-DYNA (LS-Dyna, 2003), the internal pressure is often deﬁned as a time dependent function denoted as P(t), but that cause stability problems and even

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2,2

2,1

Thickness (mm)

2

1,9

Thickness of the nodes located in plan 1 obtained with the control algorithm

1,8

Thickness of the nodes located in plan 1 obtained with the optimization method

1,7

Thickness of the nodes located in plan 2 obtained with the control algorithm Thickness of the nodes located in plan 2 obtained with the optimization method

1,6

1,5 0

50

100

150 200 Axial position (mm)

250

300

350

Fig. 14. Thickness variations of the ﬁnal shape obtained with control algorithm and optimization method

continuity ones in certain cases. This aspect can be solved while imposing, not a driving curve for pressure, but the volume of injected ﬂuid according to V(t). The speciﬁc Airbag function available in LS-DYNA (LS-Dyna, 2003) permits to control the volume of injected ﬂuid. It is thus necessary to specify in the program the compressibility modulus K for the ﬂuid (K = 2200 MPa for the water). The pressure is then given accordingly to the ﬂuid volume with the following equation: V0 (t) + L(t) (23) P (t) = K ln V (t) where P(t) is the pressure, V is the volume of ﬂuid in the compressed state, V0 is the volume of ﬂuid in the uncompressed one. L(t) is an added pressure as a function of time. The volume of ﬂuid is calculated from the mass of injected ﬂuid as: M (t) V (t) = (24) ρ where M(t) is the mass of ﬂuid and ρ the density, so:

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"t F(u) du

M(t) = M(0) +

(25)

0

It is thus necessary to determine the mass of ﬂuid contribution necessary for the hydroforming of the tube when one knows the variation of volume ΔV. Accounting that F(u) = ρ.a.u, where a is the slope of the curve, it comes: "t audu ⇒ a =

ΔM = ρΔV = ρ 0

2 (Vf − Vi ) 2ΔV = t2f t2f

(26)

where Vi is the initial volume of ﬂuid and tf is the total processing time. The volume of ﬂuid is deﬁned in the program as being the interior volume of the tube, ranging between the two pistons. As an application of the proposed approach, one search to obtain a liner where the liner the geometry of the die is described in Fig. 15. The geometry of the initial tube corresponds to an external diameter equal 40mm and a thickness equal to 5mm. The material corresponds to an austenitic stainless steel SS304 whose material characteristics are given in Table 1, where the hardening law is expressed with a Hollomon law as σy = Kεn . The discretized problem for the optimization strategy consists in 4 linear parts, the values of pressure or punch strokes are optimized accordingly to 4 values of the normalized loading factor corresponding τ = 0.25, τ = 0.5, τ = 0.75 and τ = 1. So, the optimization problem consists to determine the value of pressure so that the objective function (21) based on the thickness becomes minimum. The SQP method is used in that example due to its ability to account the nonlinear constraints. The optimization tool “LSOPTMEF” developed in our lab uses the parallel computation facilities. The optimization tool allows to get signiﬁcant

Fig. 15. Geometry of the die cavity Table 1. Material parameters corresponding to SS304 steel young’s modulus (MPa)

poisson’s ratio

strength coeﬃcient K (MPa)

hardening exponent n

196000

0.3

1250

0.4

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decreasing of the objective function, see Fig. 16. The evolution of the optimization parameters vs. iterations are piloted in Fig. 17. The optimal pressure resulting from optimization is related in the 18. Then on the basis of the same ﬁnite element model and optimization tool one search to deﬁne the pressure curve from the volume of injected ﬂuid. If one

0,98 0,97 0,96 0,95 0,94 0,93 0,92 0,91 0,9 0

1

2

3

4 iterations

5

6

7

8

Fig. 16. Evolution of the objective function during iterations

140

120

100 Pressure parameters (MPa)

Evolution of the objective function

1 0,99

80

60 pressure value for t = 0.002 pressure value for t = 0.004

40

pressure value for t = 0.006 pressure value for t = 0.008

20

0 0

1

2

3

4 iterations

5

6

Fig. 17. Pressure variation during iterations

7

8

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Fig. 18. Inner volume variation of the tube vs. processing time

imposes the stroke of both pistons at each extremities of the tube, it is easy to estimate the ﬁnal volume of ﬂuid inside the deformed liner Vf as well as the initial volume of ﬂuid in tube Vi , see (23). It is an advantage compared to a control in pressure for which one does get the maximum pressure to apply. In Fig. 17 one shows the pressure variation inside the die cavity when processing, and in Fig. 18 one show the evolution of volume of ﬂuid in the die cavity and the pressure when the mass contribution is controlled. It has to be noted that the evolution of ﬂuid volume follows a regular and quasi-linear increasing curve from t = 0.01. On the other hand, the evolution

140 120

Evolution of the pressure obtained with the optimization procedure

Pressure (MPa)

100 80

Evolution of the pressure obtained with the control volume

60 40 20 0 0

0,002

0,004 0,006 Time (LSDYNA)

0,008

0,01

Fig. 19. Comparison of the pressure curve vs. time obtained with optimization procedure and control volume

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5.433e + 00 5.174e + 00 4.916e + 00 4.657e + 00 4.399e + 00 4.140e + 00 3.882e + 00 3.623e + 00 3.365e + 00 3.106e + 00 2.848e + 00

(a) 5.653e + 00 5.385e + 00 5.117e + 00 4.849e + 00 4.581e + 00 4.313e + 00 4.045e + 00 3.776e + 00 3.508e + 00 3.240e + 00 3.972e + 00

(b)

Fig. 20. (a) thickness distribution obtained with the control volume of ﬂuid; (b) thickness distribution obtained with the optimal pressure curve

of the pressure does not follow the same curve. Indeed, the curve goes up very quickly to reach the elastic limit. Then the curve follows a quasi linear shape and then reaches a maximum corresponding to 120 MPa. The pressure then decreases because the material exceeds the plasticity locus; the contribution

Fig. 21. Strain paths and CLFs associated to liner hydroforming

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of pressure is simply used to increase the inner volume of the part. When the tube and the die are completely in contact, the pressure increases quickly, as one observes at t = 0.018s, see Fig. 19. Now, if one compares the thickness contours on the liner, see Fig. 20, one remark that both approaches (optimization and control in volume of ﬂuid) allow to obtain liners for which the thickness variations remain acceptable. The study has been completed with the analysis of the risks of failure and bursting that could occur during hydroforming. On Fig. 21, one relates the straining paths and the FLDs obtained from the linear stability analysis as related in (Lejeune et al., 2003). It clearly appears no risks of failure or bursting during the hydroforming process.

5 Conclusions In the present paper, diﬀerent approaches and strategies have been investigated for the design and optimal control for sheets and tubes hydroforming processes. One has ﬁrst investigated the role of ﬂuid ﬂow and pressure variation during hydroforming and one has proposed a method to determine the pressure in the die cavity for ﬂange hydroforming. For tube hydroforming, one has clearly investigated the eﬀects of ﬂuid movement and ﬂuid ﬂow in the tube cavity by accounting the ﬂuid velocity associated to the deformation of the inner surface of tube. In terms of optimization and process control, one propose diﬀerent approaches for the determination of the hydroforming pressure and one have also investigated the importance of volume of ﬂuid variation in the inner tube. The resulting algorithms for the control of the hydroforming of metallic liners has been validated.

References Alberti N., Forceliese A., Fratini L., Gabrielli F., Sheet metal forming of titanium blanks using ﬂexible media, Annals of the CIRP 47/1, 217–220 (1998). Asnaﬁ N., Analytical modelling of tube hydroforming, Thin-walled Structures 34, 295–330, (1999). Boisse P., Gelin J.C, Daniel J.L., Computation of thin structures at large strains and large rotations using a simple C0 isoparametric three-node shell element, Computers and Structures 58/2, 249–261 (1996). Boubakar L., Boulmae L., Gelin J-C., Improved ﬁnite element modeling of the stamping of anisotropic elasto-plastic sheet metal parts, Eng.Comput., 13,2-3-4, 143–171 (1996). Braun T., Diot C., Hoglander A., Roca V., An experimental user level of implementation of TCP, rapport de recherche no. 265. Septembre 1995, INRIA. CLEFS CEA, Nouvelles technologies pour l’´energie, CLEFS CEA n◦ 44, (2001). Dohmann F., Hartl C., Tube hydroforming – research and pratical application, J. Mater. Process. Technol. 71, 174–186 (1997).

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Gelin J.C, Delassus P., Modelling and simulation of the Aquadraw Deep Drawing Process, Annals of the CIRP 42/1, 305–308 (1993). Gelin J.C, Laberg`ere C., Application of optimal design and control strategies to the forming of thin walled metallic compoments, J. Mater. Process. Technol. 125–126, 565–572 (2002). Gelin J.C, Laberg`ere C., Application of optimal design and control strategies to the forming of thin walled metallic tubes, Int. J. Forming Processes, Vol.7 n◦ 1–2, 141–158 (2004). Jensen M.R, Olovsson L., Danckert J., Nilsson L., Numerical model for axisymmetrical deep drawing processes, Int. J. Forming Processes, Vol2, n◦ 3–4, 193–210. (1999). Kang B.S, San B.M., Kim J., A comparative study of stamping and hydroforming processes for an automobile fuel tank using FEM, Int J. of Machine Tools & Manufacture, 44, 87–94, (2004). Kleiner M., Geiger M., Klaus A., Manufacturing of light weight components by metal forming, Annals of the CIRP 52/2, 521–542 (2003). Ko¸c M., Atlan T., Prediction of forming limits and parameters in tube hydroforming process, Int J. Mach. Tool Des. Res. 42, 123–138 (2002). Lallouet A., “DP-LOG: un langage logique data-parall`ele”, Actes des 6 journ´ees francophones de programmation logique et programmation par contraintes JFPLC’97, Orl´eans, 26–28 mai 1997, Paris, Herm`es, p. 53–68. Lei L.P, Kang B.S., Kang S.J, Prediction of the forming limit in hydroforming processes using the ﬁnite element method and a ductile fracture criterium, J. Mater. Process. Technol. 113, 673–679 (2001). Lejeune A., Boudeau N., Gelin J.C, Inﬂuence of material and process parameters on bursting during hydroforming processes, J. Mater. Process. Technol., Vol 143–144, 11–17 (2003). Lejeune A., Boudeau N., Gelin J.C, Inﬂuence of material and process parameters on bursting during hydroforming processes, J. Mater. Process. Technol. 143–144, 11–17 (2003). LS-DYNA, keyword user’s manual, PDF ﬁle, version 970, (2003). Manabe K.I, Amino M., Eﬀects of process parameters and material properties on deformation process in tube hydroforming, J. Mater. Process. Technol., Vol 123, 285–291 (2002). Manabe K.I., Amino M., Eﬀects of process parameters and material properties on deformation process in tube hydroforming, J. Mater. Process. Technol. 123, 285–291 (2002). Merklein M., Geiger M., New materials and production technologies for innovative light weight construction, Manufacturing of light weight components by metal forming, J. Mater. Process. Technol. 125–126, 532–536 (2002). Sokolowski T., Gerke K., Ahmetoglu M., Atlan T., Evaluation of tube formabiblity and material characteristics by hydraulic bulge testing of tubes, J. Mater. Process. Technol. 92, 34–40 (2000). Tirosh J, Konvalina P., On the hydroforming deep-drawing process, Int. J. Mech. Sci. 27, 595–607 (1985). Vinarcik E.J., Automotive light metal advances, Part I, Innovative designs and emerging technologies, Light Metal Age 60, 38–41 (2002).

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Evaluating the Press Stiﬀness in Realistic Operating Conditions A. Ghiotti and P.F. Bariani DIMEG – University of Padova, via Venezia 1, 35131 Padova – Italy, [email protected]

Summary. In many forming operations, due to the severe forging loads, the press frame deﬂects elastically and, consequently, upper and lower dies deviate from the nominal relative position. These conditions produce skewed surfaces that, in a too stiﬀ press, can have detrimental eﬀects on the service life of the machine and the tooling as well. This paper presents a new method to evaluate the stiﬀness of presses for forming operations where realistic loading systems are utilised. To this aim, a special testing and calibration apparatus was developed that consists of (i) a loading device capable of generating and at the same time measuring forces and torques in diﬀerent directions and (ii) a special transducer that measures the relative displacement between the ram and the bed of the press. The paper describes the application of the method to the evaluation of the stiﬀness of a screw press.

Key words: forging, presses, stiﬀness.

1 Introduction Improving the geometrical accuracy of forged parts and extending the service life of tooling are two conﬂicting requirements in designing forging operations. The right trade-oﬀ is often the main prerequisite for the technical and economic success of the whole process and must therefore be found, especially when the forged part has a complex and non-symmetric geometry. In this case, the elastic behaviour of the machine tool and particularly its stiﬀness plays an important role. Complex non-symmetric shapes represent still critic applications, for the diﬃculty to well balance the respect of geometric tolerances, the tooling systems preservations and the life of dies. Due to the severe loads that arise during forging operations, the press frame deﬂects elastically and, consequently, upper and lower dies deviate from the nominal relative position. The loss of parallelism of the upset planes produces skewed surfaces and oﬀset of the forged part. Furthermore, in too stiﬀ forging machines this fact can have detrimental eﬀects on the service life of the press and the tooling system as well.

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Guidelines for the design of dies (Ou, 2002) and forging equipment, such as ram guides (Doege, 1990, Jimma, 1992 and Ou et al., 1999) are present in literature, providing indications for the compensations of elastic distortions and temperature eﬀects. However, only recently the attention of researchers has moved on the evaluation of the overall forging system (Doege, 1980, Li and Tong, 1997 and Tomov, 1998). The attempts of dynamic stiﬀness evaluation, by measuring the deformation of forming presses in operating conditions, have developed in two diﬀerent ways: (i) evaluation of stiﬀness by measuring the press frame maximum deformation during upsetting operations (Chodnikiewicz et al., 1991 and Tomov, 1995), and (ii) real-time measure of press deﬂections during combined application of vertical and horizontal force (Chodnikiewicz and Balendra, 2000 and Bariani and Ghiotti, 2002). In the ﬁrst approach, a special loading device generates forces and moments causing linear and angular deﬂections of the press frame that can be measured through deformable markers (Arentoft, 2000). Even if loads on presses are realistic, only the maximum values of deﬂections are evaluated and no information is available on the response of the press to the loading history during forging operations. To provide this typology of information, loads and deﬂections must be monitored in real time as in (ii) (Chodnikiewicz et al., 1994) by overcoming some diﬃculties such as the separation of the eﬀects of vertical and side loads and the very short time available for acquisition (especially for mechanical and screw presses). In this paper an innovative approach to evaluate the press stiﬀness in operating conditions is presented, together with the experimental apparatus and the procedure for hi-speed press testing. The apparatus, called the stiﬀness transducer, is made up of (i) a loading device capable of generating and, at the same time, measuring torques, horizontal and vertical forces and (ii) a noncontact transducer that measures the press frame deﬂections during loading. With this stiﬀness transducer, the response of the machine can be accurately monitored during the forming stroke even in the case of very fast process (e.g. for screw presses and high-speed mechanical presses) and the eﬀects of vertical and horizontal forces and torques can be analyzed separately as well. The stiﬀness transducer and the testing procedure are general enough to be applied to all kinds of presses for bulk and sheet metal forming tasks. In the ﬁrst part of the paper, the ﬂexibility matrix of a press and relevant model are introduced. Then a description of the experimental apparatus and testing procedure is given. Finally, the paper focuses on the results of experiments carried out on a screw press plant.

2 Stiﬀness Modelling In its generalized formulation (Chodnikiewicz et al., 1994), the modelling of ﬂexibility matrix to evaluate the press elasticity includes both linear and nonlinear contributions to the response of a loaded machine (Fig. 1) according to:

Deformation

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191

{f }

Zone 3 Ram Z 1

Zone 2

O X1

{F }

Y1

1

Z O

Zone 1 Press bed

X

Y

Load

Fig. 1. Press deﬂections due to reaction forces during forging

T

{R} = [N LF M ] {A}

(1)

where {R} is the vector of the press Responses, [NLFM] is the Non-Linear Flexibility Matrix and {A} is the vector of the Actions. In the vector of press responses T {R} = {Δx, Δy, Δz, Δα, Δβ, Δγ} (2) the elements Δx, Δy, Δz, and Δα, Δβ, Δγ are the displacements of the ram, respectively along and around the X, Y and Z axes in the OXYZ coordinate system in Fig. 1. These displacements result from both non-linear and linear contribution to the response of the press. In zone 1 of Fig. 1, the displacements are due to the closure of clearances between parts of the press. In zone 2, the response of the press is non linear and the displacements are due to the deformation of surfaces in contact at medium forces. In zone 3, the displacements become again linear and represent the elastic deformation of the whole press frame. In its complete form, the [NLFM] matrix is a 6 × 6 matrix, whose elements λij are the curves that represent the press response acquired during the loading tests. All points in the force-deﬂection curves are logged and used for the description of the press behaviour. In the vector of Actions T {A} = {FX , FY , FZ , MA , MB , MC } (3) the elements Fx , Fy , Fz and MA , MB , MC are, respectively, the forces and the moments that are applied to the press frame during the forming process. To evaluate the elements of the non-linear ﬂexibility matrix in (1), the press must be loaded with a complete set of loading schemes and the consequent deﬂections of the press must be measured as relative displacements between the lower surface of the ram and the bed.

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3 Stiﬀness Evaluation According to the testing operational methodology, the stiﬀness transducer is composed of two devices for the frame loading and a non-contact system equipped with eight eddy current sensors to perform a real-time measure of the relative displacement between the ram and the lower surface of the press. In the development of the loading devices, the main basic requirements that follow were taken into account: (i) the set of force systems are generated through the active movement of the press ram that permits to establish realistic conditions in the testing procedure; (ii) the vertical and horizontal loads are applied in separate tests by using two diﬀerent loading devices. In this way, the limits in evaluating the vertical and horizontal forces contribution to deﬂections, critical when using one measuring system, are solved, on the realistic assumption that the elements Fx , Fy , Fz , MA , MB , MC of the action vector have an independent inﬂuence on the elements Δx, Δy, Δz, Δα, Δβ, Δγ of the vector of responses; and (iii) the real-time measure of relative displacement between the bottom and the top surfaces of the press allows having information of the total loading history during the forging operation. 3.1 The Application of Loads The loading devices can generate and, at the same time, measure torques, horizontal and vertical forces that are applied to a press frame. The prototype of the device to generate the horizontal forces is shown in Fig. 2a. It consists of two wedges: the upper one is ﬁxed to the ram and the lower one to the bed. A maximum horizontal force of 650 kN is generated when the two

Ram

Load cell

Piezoelectric Sensor

Uniform Pressure Area

Calibrated Bolts Wedge

Press bed (a)

(b)

Fig. 2. (a) the device for the application horizontal forces; (b) CAD drawing of the load cell with the embedded piezoelectric sensor

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wedges come into contact during the ram stroke. The small angle of the wedge, which maximizes the ratio between horizontal and vertical forces, makes the eﬀect on measurements caused by the vertical force and the friction at the wedge contact surface negligible. The horizontal force is monitored by a load cell assembled inside the lower edge (see the CAD drawing of the load cell in Fig. 2b), which embeds a piezoelectric sensor for the measurement of load. The load cell is connected to the frame of the lower wedge through four bolts (shown in Fig. 2a) that, due to their calibrated resistance, during the test plastically deform at a predetermined force, protecting the load cell and the press frame from overloading. By positioning the loading device between the ram and the table with diﬀerent orientations, horizontal forces can be generated during the tests with diﬀerent directions. The main components of the loading device to generate vertical forces are shown in Fig. 3a. It is a special die set for upsetting operations that is capable of generating “purely vertical” loading conditions in spite of the tilting of the ram or the deviation from parallelism between the ram and the bed, which is fairly marked in C-frame presses. To this aim, the lower part of the device consists of a stack of three disks that are in contact at two lubricated sliding surfaces (see Fig. 3b). The geometry of sliding surfaces (planar, spherical-concave, spherical-convex, radius) and their location in the stack were optimized by FEM and experimental tests on prototypes as well. The vertical force is monitored by a piezoelectric load cell, embedded in the lowest disk of the stack, whose shape was optimized for a uniform compressionstress ﬁeld. The sensor is suitable for measuring rapidly varying loads, as is required for testing screw- and high-speed mechanical presses. By positioning the device at diﬀerent oﬀ-centre locations on the press bed, torques of diﬀerent magnitude and directions can be applied to the press frame. The vertical loading device, including the load cell, was designed for maximum vertical load of 230 MN.

Ram Ram Specimen Disks stack Load cell

Spherical sliding surface Plane sliding surface

Pressbed bed Press (a)

(b)

Fig. 3. (a) the loading device for the application of vertical force and torques; (b) the sliding surfaces conﬁguration of the stack of disks

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(a)

(b)

Fig. 4. (a) the transducer of the press deﬂections; (b) the measurement of ram tilting

3.2 The Measurement of Deﬂections The deﬂection of the press frame results from the measure of the relative displacement between the ram and the table. The system consists of four supports for the eddy current sensors, ﬁxed to the ram, and the correspondent target surfaces, rigidly linked to the lower surface of upsetting. The use of two sensors for each measurement position, as shown in Fig. 4b, allows determining both the horizontal translations and the relative tilting between the ram and the bottom table by the comparison of the values of the measured distances hs and hl and the evaluation of the angle α. Figure 4a shows the physical equipment during the stiﬀness characterization of a 2300 kN screw press. The supports for sensors are mounted on the press ram while, the target surfaces are ﬁxed at the four corners of the lower table. During the stroke of the ram, which can reasonably be approximated as a rigid die, they measure the variation of the distance from the targets at the four corners of the upsetting surface. The high resolution of the sensors and the elevated sample rate of the acquisition system make it suitable for application in severe dynamic conditions as in mechanical- and in screw-presses. The press deﬂections and the amount of displacements due to misalignments of supports or press guides are evaluated by repeating the tests with and without loading devices.

4 Experiments The stiﬀness transducer was used to test a 2300 kN ﬂywheel screw press. In spite of the low tonnage, this kind of press represents a severe test case for the high ram speed at impact (500 mm/s) and the vibrations of the frame. Figure 5 shows the use of the two loading devices to evaluate the diﬀerent elements of

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Z X Y

Fx

Fy

Fz

MA

MB

Mz

Δx

λ XX

0

0

0

λ XB

0

Δy

0

λ YY

0

λ YA

0

0 0

Δz

0

0

0

0

0

ΔA

0

λ AY

0

λ AA

0

0

ΔB

λ BX

0

0

0

λ BB

0

ΔC

0

0

λ CZ

0

0

λ CC

Fig. 5. The (NLFM) matrix and the use of the loading devices

the [NLFM] matrix for screw presses (Wanheim and Chodnikiewicz, 2002). The horizontal loading device was applied to generate the forces along the X- and Y-axis. When mounted in oﬀ-centre position, the device generates the torque Mz . The vertical loading device was applied in central position to evaluate the eﬀect of the screw mechanism on the rotation Δγ and in oﬀ-centre position in order to generate the tilting Δα and Δβ. In the experiments, a maximum horizontal force of 300 kN (both in the X- and Y-axis directions) and a torque up to 100 kNm (around X-, Y- and Z-axis) were applied. As shown in Fig. 6, the stiﬀness of the press is practically identical in the two horizontal directions. The non-linear part of the response is restricted to a range of forces up to 10kN and the eﬀect of clearances can be detected only when forces are applied in the X-axis direction. 2,5

0,03

λXX

Displacement [mm]

0,02

λYY

1

0,01

0,5 0 –0,5

0

50

100

150

–1

200

250

–0,01

λ AY

–1,5

300

λBX

Tilting [rad]

2 1,5

–0,02

–2 –2,5

–0,03

Load [kN]

Fig. 6. Press deﬂections caused by pure horizontal loads along the X- and Y-axes

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Furthermore, horizontal forces generate the ram tilting, according to the analytical model proposed in (Wanheim and Chodnikiewicz, 2002). Figure 7 shows the response of the press frame to the application of the pure moments MA , MB . The frame deforms diﬀerently under the two loading schemes, showing a higher value of stiﬀness around the Y-axis. No clearance is detected for the application of the pure moments. The twisting of the ram due to the torque MC (see Fig. 8) is very small compared with the rotations Δα and Δβ introduced by the torque MA and MB . No eﬀect of the screw mechanism during pure vertical loading (element λCZ ) was measured. 0,005

2,5 λAA

0,004

0,002 Tilting [rad]

1,5

λBB

λXB

1

λYA

0,001

0,5

0

0

–0,001 0

20

40

60

80

100

–0,5

–0,002

–1

–0,003

–1,5

–0,004

–2

–0,005

Displacement [mm]

0,003

2

–2,5

Load [kNm]

Fig. 7. Press deﬂections caused by pure moments MA and MB (around the X- and Y-axes, respectively)

0,005 0,0045

Tilting [rad]

0,004 0,0035

Z

0,003 0,0025

b

F

λCC

0,002 X

0,0015 0,001

Y

0,0005 0 0

20

40

60

80

100

Load [kNm]

Fig. 8. Press deﬂections caused by pure horizontal loads along the X- and Y-axes

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5 Conclusions A new stiﬀness transducer and testing procedure were developed to evaluate the stiﬀness of forming machines by simulating real process conditions. The main design features of the device are: (i) the set of force systems are generated through the active movement of the press ram that permits to establish realistic conditions in the testing procedure; (ii) the vertical and horizontal loads are applied in separate tests by using two diﬀerent loading devices. In this way, the problem of separating the inﬂuence on press deﬂections of vertical and lateral forces in one measuring system is solved. The behaviour of the press is described by a matrix, whose elements are obtained applying forces and moments to the frame and measuring the correspondent deﬂections. Results are collected in load-deﬂection curves that represent the full response of the machine due to the diﬀerent loading situations. The system was used to evaluate the ﬂexibility matrix of a 2300 kN ﬂywheel screw press. The collected data represent a simple and good description of the press behaviour, useful to predict and optimise the tolerances of the ﬁnal workpiece.

Acknowledgements The work presented in the paper is part of the European Project “IMPRESS” G1RD-CT-2000-02002. The authors are grateful to Vaccari S.p.A support in carrying out the experiments.

References Arentoft M., Eriksen M., Wanheim T., “Determination of six stiﬀnesses for a press”, Journal of Materials Processing Technology, Vol. 105, 2000, p. 246–252. Bariani P., Ghiotti A., “Development of loading devices to evaluate the stiﬀness of forging presses in realistic operating”, Proceedings 6th AITEM International Conference, 2002. Chodnikiewicz K., Tomov, B., Wanheim, T., “WTC Loading Device”, Int. Rep., Inst Manuf. Eng., DTH, 1991. Chodnikiewicz K., Balendra R., Wanheim T., “A new concept for the measurement of press stiﬀness”, Journal of Materials Processing Technology, Vol. 44, 1994, p. 293–299. Chodnikiewicz K., Balendra R., “The Calibration of Metal-forming Presses”, Journal of Material Processing Technology, Vol. 106, 2000, p. 28–33. Doege E., “Static and Dynamic Stiﬀness of Presses and some Eﬀects on the Accuracy of Workpieces”, Annals of CIRP, 1980, 29.1.

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Doege E., Silberbach G., “Inﬂuence of Various Machine Tool Components on Workpiece Quality”, Annals of CIRP, 1990, 39.1. Jimma T., Sekine F., “Eﬀect of Rigidity of Die and Press on Blanking Accuracy of Electronic Machine Parts”, Annals of the CIRP, Vol. 41, 1992, p. 319–322. Li P., Hong G., “On the Stiﬀness of Screw Presses”, Journal of Machines Tools Manufacturing, Vol. 37, 1997, p. 93–100. Ou H., Ferguson W. H., Balendra R., “Assessment of the elastic characteristics of an ‘inﬁnite stiﬀness’ physical modelling press”, Journal of Materials Processing Technology, Vol. 87, 1999, p. 28–36. Ou H., Armstrong C.G., “Die shape compensation in hot forging of titanium aerofoil sections”, Journal of Materials Processing Technology, 2002, p. 347–352. Tomov B., Chodnikiewics K., “A Mechanical Device for Measuring of Displacements and Rotations of a Blanking Forging Press”, Proceedings of International Conference On Advances in Materials and Processing Technologies (AMPT’95), 1995, Dublin. Tomov B., Chodnikiewicz K., “A mechanical device for measuring the displacements and rotation of a blanking or forging press”, Journal of Materials Processing Technology, Vol. 77, 1998, p. 70–72. Wanheim T., Chodnikiewicz K., “An analytical description of press deﬂections and a new set of stiﬀness parameters for presses”, IMPRESS project, Deliverable 6.2, 2002.

Fast Material Working: Wire Drawing N.D. Cristescu Department of Mechanical & Aerospace Engineering, University of Florida, 231 Aerospace Building, PO Box 116250, Gainesville, Florida, 32611-6250, e-mail:cristesc@uﬂ.edu

Summary. The theory of fast material forming starts with the application of the Bingham model to the metal forming. It started with wire drawing, with drawing of tubes either free or with a ﬂoating plug, or with a ﬁxed plug. The extrusion of cylindrical bars at increased temperatures, with ﬂoating glass as a lubricant, was also considered. The theory was done with several mathematical approaches, either direct or with applied.

Key words: wire drawing, tube drawing with ﬂoating plug, drawing of tubes.

1 Introduction The processes of metal extrusion or drawing have been considered within the framework of classical time-independent plasticity theory (see Avitzur 1968). The theory considered here is to describe the inﬂuence of the speed of the process (of the order of 100 m/s) on all the other involved parameters (Cristescu 1975, 1976). We consider mainly wires which are very thin, less than 0.5 mm in diameter. Let us assume that the main mechanical properties of the material can be described with a viscoplastic constitutive equation. Since the elastic part of the strain will be neglected the simplest possible model is a Bingham-type constitutive equation of the form 6 7 σm 1 Dij = 1− √ √ σij (1) 2η 3 IIσ where σm is the mean yield stress which depends on reduction and is an approximation of the isotropic work-hardening law σ = f (ε) of the material, and is the positive part. Thus the model is viscoplastic rigid. A second assumption is that the circular conical die remains rigid during plastic ﬂow and that in the domain where plastic ﬂow takes place (domain II) the process is axi-symmetric. Assuming volume incompressibility the following velocity ﬁeld components in spherical co-ordinate r, θ, ϕ can be obtained (see Fig. 1)

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v = vr = −vf rf2

cosθ r2

,

vθ = vϕ = 0,

(2)

in the domain II deﬁned by r0 ≤ r ≤ rf , 0 ≤ θ ≤ α, 0 ≤ ϕ ≤ 2π. According to (2) the material particles are moving radially towards the apex 0. Since in the domain I (for r > r0 ) the whole rod is moving as a rigid body with the absolute velocity v0 and since in the domain III the rod is moving again as a rigid body with the absolute velocity vf , the spherical surfaces r = r0 and r = rf are discontinuity surfaces for the velocity ﬁeld. Along the surface Γ2 (r = r0 ) the tangential discontinuity of the velocity ﬁeld is obtained from (2) as Δv = −v0 sinθ

along

r = r0 , 0 ≤ θ ≤ α

(3)

while along the surface Γ1 (r = rf ) the tangential discontinuity of the velocity is Δv = −vf sinθ

along

r = rf , 0 ≤ θ ≤ α.

(4)

Another evident formulae is R (5) sinα From (2) the following components of the rate of strain result, in spherical coordinates, v0 R02 = vf Rf2 ,

r=

cosθ cosθ ∂v v = 2vf rf2 3 , Dθθ = = −vf rf2 3 , ∂r r r r 1 sinθ 1 ∂v Drθ = = vf rf2 3 , Drϕ = Dθϕ = 0 2r ∂θ 2 r and obviously Drr + Dθθ + Dϕϕ = 0. Drr =

Fig. 1. Geometry of wire drawing

Dϕϕ =

v , r

(6)

(7)

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Therefore, according to the kinematical velocity ﬁeld chosen, plastic deformation takes place in region II only, while regions I and III remain rigid. Dissipation is calculated in region II, and dissipation due to the velocity discontinuity along Γ1 and Γ2 , while along Γ3 the dissipation is produced by the friction existing there. Γ5 is stress free. The surface Γ6 (x = r0 ) at the entrance and Γ4 (x = r0 cosα) at the exit of the die are used for the computation of the balance of forces. The semi-angle of the die is denoted by α, the drawing stress by σf and a possible back-stress by σb .

2 Basic Equations For the problem under consideration, taking into account the spherical symmetry and (6), the constitutive equation can be written as σm 2 1 1− √ √ (σrr − σθθ ) , Drr = 2η 3 IIσ 3 σm 1 1 1− √ √ (8) Dθθ = (σθθ − σrr ) , 2η 3 IIσ 3 σm 1 1− √ √ σrθ . Drθ = 2η 3 IIσ The second invariant of the stress deviator is IIσ =

1 2 2 . (σrr − σθθ ) + σrθ 3

(9)

Using (6) and (8) the expression for this invariant can be written as 8 vf rf2 8 σm IIσ = √ + 3 η 11cos 2 θ + 1 r 3

(10)

Another group of equations are the equilibrium equations. Since we have σrϕ = σθϕ = 0, σθθ = σϕϕ and since the problem is axially symmetric with respect to ϕ, the set of equilibrium equations reduces to two 1 ∂σrθ 2 (σrr − σθθ ) + σrθ cotgθ ∂σrr + + = 0, ∂r r ∂θ r ∂σrθ 1 ∂σθθ 3σrθ + + = 0. ∂r r ∂θ r

(11)

From (6), (8)2 and (10) we get σrθ = √

sinθ sinθ σ √m + ηvf σf2 3 2 r 11cos θ + 1 3

(12)

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which satisﬁes the requirement that σrθ = 0 for θ = 0. Introducing (12) in (11)2 a diﬀerential equation is obtained, which after integration yields / . $ √ 1 3σm 2 σθθ = √ (13) ln cosθ + + cos θ + C (r) 11 11 where C is a undetermined function depending on r alone. Any one from the two equations (8) taken together with (6), (10) and (13) now yields / . $ √ 3σm 1 + C (r) σrr = √ ln cosθ + cos 2 θ + 11 11 (14) √ cosθ 2 cosθ + 2 3σm √ + 6vf rf η 3 . r 11cos 2 θ + 1 To determine the function C for θ = 0, we obtain 4 1 σm 168 C (r) = − vf rf2 η 3 − √ 3 ln r + C1 r 3 3 12 2

(15)

where C1 is an integration constant, which is determined from the condition r = r0 , θ = 0 we have σrr = σxb . Thus it is found for C1 / . $ √ 14 vf rf2 η σm 3σm 1 − C1 = σxb − √ + √ 4.04145lnr0 − σm . ln 1 + 1 + 11 3 r03 11 3 and for σθθ we get / . $ √ vf rf2 η 4 3σm 1 14 2 − + 3 σθθ = √ ln cosθ + cos θ + 11 3 r3 r0 11 / . $ √ σm 3σm 1 r0 + √ 4.04145ln + σxb − √ − σm ln 1 + 1 + r 11 3 11

(16)

and a similar procedure can be used to determine σrr . This is an approximate value since we have not taken into account the friction. We have to compute now the stress power per unit volume 2σm 8 σij Dij = √ IID + 4ηIID , 3

(17)

where IID =

1 1 2 2 2 2 2 2 Dij Dij = Drr + Dθθ + Dϕϕ + Dθϕ + Dϕr . + Drθ 2 2

(18)

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Using (6) in (17) and (18) we get v2 r4 η 2σm vf rf2 8 2θ + 1 + f f σij Dij = √ 11cos 2 θ + 1 . 11cos 3 6 2r r 3 After integrating over the volume of zone II, and using (5): . 3 / R R 2 R 0 f f 2 2 ˙ = 2πσm vf R ln η 1− W f (α) + πvf f Rf 3 sinα R0 11 × 1 − cosα + 1 − cos 3 α , 3 where

⎧ ⎫ ⎪ ⎪ 2α + 1 + ⎪ ⎪ (cosα) cos ⎪ ⎪ 11 ⎪ ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎨ ⎬ : 9 11 1 1 1 2 −f (α) = √ cos α + 11 − 2 ⎪+ 11 ln cosα + ⎪ 2 3 sin α ⎪ ⎪ ⎪ ⎪ ⎪ : ⎪ 9 ⎪ ⎪ ⎪ ⎪ 12 1 12 ⎩ − ⎭ − ln 1 + 11 11 11

(19)

(20)

The ﬁrst term is the time independent while the second term in (19) corresponds to the time dependent term.

3 Friction Laws We ﬁrst consider a Coulomb friction law τ = μσθθ .

(21)

Here μ is the constant friction coeﬃcient and σθθ is the stress normal to the die surface. Since the velocity along the surface is v = −vf rf2

cosα r2

the rate of the work done by the stress vector on conical die surface Γ3 (r0 ≥ r ≥ rf , θ = α) is ˙Γ W 3

"rf cosα = −μ σθθ vf rf2 2 2πrsinα dr r r0

Introducing here (16) and after some algebra, we get

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. 3 / Rf sinα 4 vf η 1− − 9 Rf R0 ⎤⎫ ⎡√ $ ⎪ 3σm 1 ⎪ −⎥⎪ ln cosα + cos 2 α + ⎪ ⎢ √ 11 ⎪ ⎢ 11 ⎥⎪ ⎪ ⎪ ⎢ ⎥ 2 ⎬ 2 ⎢ ⎥ σm Rf ⎢ R0 14 sinα Rf ⎥ − √ 2.0207 ln + ln − vf η + σxb − ⎥ ⎢ Rf R0 ⎢ 3 R0 $ R0 3 ⎪ ⎥⎪ ⎪ ⎢ √ ⎥⎪ 3σm 1 ⎪ ⎣ ⎦⎪ ⎪ − σm ⎪ − √ ln 1 + 1 + ⎭ 11 11

˙ Γ = −2μπvf Rf2 cotα W 3

(22) Another law in metal plasticity is that the modulus of √shearing stress due to friction is proportional to the yield stress τY = σY 3. For viscoplastic materials this law is generalized as 8 τ = m IIσ (23) where m is a constant friction coeﬃcient and 0 ≤ m ≤ 1. m = 0 means no friction, while m = 1 means adherence on the wall. Using (10), (23) can be written . / vf rf2 8 σm 2 τ = m √ + 3 η 11cos α + 1 . (24) r 3 With this law the rate of work done on the conical die surface is ⎫ ⎧ σ m Rf vf ηsinα √ ⎪ 2 ⎪ ⎪ √ ln + 11cos α + 1 ⎪ ⎪ ⎪ ⎬ ⎨ R 3R 3 0 f cosα / . 2 ˙ Γ = 2πmvf Rf 3 . W 3 Rf ⎪ sinα ⎪ ⎪ ⎪ − 1 × ⎪ ⎪ ⎭ ⎩ R0

(25)

Let us observe that both expressions are variable depending on vf , η and α. The dependency on r is distinct; in the law (21) it decreases with decreasing r, while in the law (23) it increases with increasing r. We assume all the other parameters to stay constants.

4 Drawing Stress We apply now the theorem of powder expanded, extended to the volume V of the viscoplastic domain. We have " " " " σij Dij dV + σi [vi ]dS + σi [vi ]dS = σi vi dS, (26) V

Γ1

Γ2

∂V

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where σi is the component of the stress vector on Γ1 , Γ2 or ∂V while [v] is the jump of the particle velocity at the crossing of the surface Γ1 or of the surface Γ2 . The term on the right-hand side of (26) represents the rate of work done by the stress vector on the surface bounding the volume V . This term can be decomposed in ﬁve parts. On the plane Γ6 where x = r0 we have ˙ Γ = −πR02 v0 σxb , W 6

(27)

˙ Γ = πR2 vf σxf . W f 4

(28)

while on Γ4 where x = rf ,

Along Γ2 the rate of work is zero since this surface is stress free. Along the surfaces where a friction law exists the integral is already computed. Let us consider now the last term from right. The velocity discontinuity is [v] = v0 sinθ. The surface element is dA = r02 sinθ dθ dϕ Thus

"α

" σi [vi ] dA =

2πv0 r02

σi τi sin 2 θ dθ

(29)

0

Γ2

where τi is the vector tangent to Γ2 in the meridian plane. From the global condition of equilibrium of the forces which act on the domain I of the bar we have " 2 ∗ −πσ0 R0 + σ0 dA = 0 Γ2

we determine σ0∗ =

σ0 R02 σ0 = (1 + cosα) 2 2r0 (1 − cosα) 2

Introducing this value in (29) we get "α

" σi [vi ]dA = − (1 + Γ2

or

" Γ2

cosα) πv0 r02 σ0

sin 3 θ dθ 0

2 1 −cosα + cos 3 α + 3 3 πv R2 σ σi [vi ]dA = − 0 0 0 1 − cosα

Similar for the surface Γ4 we get " 1 1 − cos 3 α σi [vi ]dA = πvf rf2 σf (1 + cosα) (1 − cosα) + 3 Γ4

(30)

(31)

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Combining all these formulae, for the second friction law, we have ⎧ ⎡ ⎤ ⎪ 1+ ⎨ 2 σb ⎢ σf sin α 1 3 2⎥ + = ⎣ 2 −cos α+ cos α+ ⎦ 1 2 ⎪ σm σ 32 3 2cosα − cos 2 α − cos 3 α − ⎩ m +2 sin α 3 3 ⎡ ⎤ . / 3 1 − cosα+ R0 Rf 1 2 ηvf ⎣ 11 ⎦ + +2f (α) ln 1− + Rf 3 σm Rf sinα R0 1 − cos 3 α + 3 . 3 / 8 2 ηvf sinα Rf R0 2 √ + √ mctgα ln }+ + 11cos α + 1 1 − Rf σ m Rf R0 3 3 +

2 L σY m 3 Rf σ m

(32)

where

$ √ 11 1 1 −f (α) = √ (cosα) cos 2 α + + 2 11 2 3 sin α $ $ $ 1 12 12 1 1 2 + ln cosα + cos α + − }. − ln 1 + 11 11 11 11 11

and N=

ηvf σ m Rf

(33)

is a “speed eﬀectiveness parameter” expresses the inﬂuence of the speed on the energy dissipated in the domain of viscoplastic deformation. Further the term containing the factor m expresses the energy dissipation due to friction; in this term a component due to the speed inﬂuence is also present. The ﬁnal term in (32) is due to the friction along the die land of length L. In the same equation the dissipation term due to the presence of discontinuity surfaces Γ1 and Γ2 is also involved, but not additively. Since the rates of deformation involved in viscoplastic deformation in region II are very high, the resultant heating was also considered (Cristescu 1980), in order to be able to choose the values of the various constants from the constitutive equation correctly. It was assumed that the heating is adiabatic, that is % " dT %% ρc = σij Dij dV (34) dt %med V

where c is the speciﬁc heat (assumed constant), T is the temperature and dT /dt|med is the mean temperature gradient over the volume V, deﬁned by % " dT %% dT 1 dV . = dt %med V dt V

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On the right-hand side of (34) is the stress power over the volume V , which is given by the above formula. This equation becomes ⎧ . 3 / ⎫ ⎪ ⎪ N R R 2 ⎪ ⎪ 0 f % ⎪ ⎪ 1 − 2f (α) ln + 2 ⎨ ⎬ % πvf Rf σm dT % Rf 3 sinα R0 = % ⎪ ⎪ dt med V ρc ⎪ ⎪ ⎪ × 1 − cosα + 11 1 − cos 3 α ⎪ ⎩ ⎭ 3 This equation was used to estimate the mean rise of temperature due to fast viscoplastic deformation. For steel wire and R0 = 0.5 mm, vf = 1m/s, α = 6◦ , −1 η = 0.34 N mm−2 s, ρc = 3.60 N mm−2 (◦ C) , r% = 20%, an approximate ◦ mean temperature rise of ΔTmed = 84.0 C (28.2◦ C due to plastic deformation and 55.8◦ C due to the speed eﬀect), is obtained, estimating that the particles cross region II in a time interval less than 5×10−4 s. For vf = 50 m/s and η = 0.023 N mm−2 s, the mean temperature rise due to viscolastic deformation is ΔTmed = 121◦ C (29.6◦ C due to plastic deformation and the remaining due to the speed eﬀect). Another estimation of temperature rise was done by integrating numerically the (7.4.34). This is no more done since the temperature rise is less than 150◦ C at the most and thus do not change the constants of the materials signiﬁcantly.

5 Comparison with Experimental Data We have chosen the experimental data by Wistreich 1955 for copper wires with Rf = 1.27 mm and mean yield stress of σY = 349.7 M N/m2 have been drawn at a speed of vf = 33 mm/sec, the friction coeﬃcient is μ = 0.025. The viscosity coeﬃcient for copper was taken from Lindholm and Bessey 1969 η = 2.78 M N/m−2 sec from where N = 0.2069. For these values of the constants, the results shown in Fig. 2 were obtained. For various reductions in area deﬁned by the variation of the drawing stress with semi cone angle α is plotted. The discrepancy with experimental data existing for α relatively big and/or r small is due to the bulging phenomenon. . 2 / Rf r% = 100 1 − R0 Another comparison with experimental data has been done for the total drawing force as measured by Wistreich for semi-cone die angles equal to: 2.29◦ , 8.02◦ and 15.47◦ . These experimental results are shown in Fig. 3 together with the theory. For mild steel Manjoine (see Lindholm and Bessey 1969) and Cristescu 1977 give D ≈ 10−1 − 1s−1 , η ≈ 104 kN m−2 s, for D ≈ 1 − 10 s−1 , η ≈ 1700 kN m−2 s, for D ≈ 10 − 102 s−1 , η ≈ 200 kN m−2 s, and for D ≈ 102 − 103 s−1 , η ≈ 24 kN m−2 s.

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Fig. 2. Variation of drawing stress with semi-cone angle for various reductions in area: comparison between theory and experiment

For higher rates of deformation Campbell 1973 suggests η ≈ 2kN m−2 s and it is varying very little. Let us observe that an increase of vf does not imply an increase on N , since η may decrease signiﬁcantly with the increase of vf . In the example given below, σb = 0 and L = 0, the consideration of these two parameters can be done very easily. In Fig. 4 are given for three reductions, the optimum relative drawing stress for diﬀerent values of N (full lines). From this ﬁgure it can be seen that the drawing stress increases with reduction and become large for very thin wires. A comparison with the experiments was done also by Cristescu 1977 for copper of diameter wires 2R0 = 0.94 mm and steel wires of 2R0 = 1.02 mm, tested slowly. For the die semi-angle α = 8◦ , the length of the land L = 0.5 mm, friction coeﬃcient on the land m = 0.08, friction on the conical surface μ = 0.09, and viscosity of the copper η = 2.94 N mm−2 s. Figure 5 shows some results for α = 8◦ (smaller ﬁgures) and α = 7◦ . That is because α = 8◦ is closer to the optimum die angle. The lines are the theoretical ones. The ﬁgure corresponds to L = 0.5 mm, μ = 0.06, m = 0.08, and η = 2.94 N mm−2 s. Similar results for steel wires.

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Fig. 3. Total drawing force for various reductions and area and three semi-cone angles

6 Other Papers A visio-plastic method, for big deformations, was developed by Dahan and Le Nevez 1983. First some extrusion pieces are obtained and measured. Then by a ﬁnite diﬀerence one is obtaining for a rigid work-hardening and isotropic material, the stress deviator and the strain. Also a Bingham model was used by Ionescu and Vernescu 1988, to describe numerically the wire drawing. The friction conditions are of two types: a Coulomb friction law and a viscoplastic one. The drawing speeds is low, with the viscosity coeﬃcient 2.94 Ns/mm2 . The same theory was presented in a book devoted to “Mathematical Models in Working Metals” by G Camenschi and N. S ¸ andru 2003. The book contains much more than drawing of bars but this theory is presented. In reality the book presents some papers published since 1978. For instance the drawing of bars with asymptotic developments was ﬁrst published in Camenschi et al., 1979 for a friction law of the form (23) and in Camenschi et al., 1983 for a Coulomb friction law. Concerning the drawing theory one is assuming

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Fig. 4. Variation of relative drawing stress and optimum die angle with the speed eﬀectiveness parameter N, for diﬀerent degrees of reduction

Fig. 5. Drawing forces for copper wires

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N < 1, that is slow processes. The solution is obtained for the incompressibil ity condition by introducing the ﬂow function, ψ = ψ (r, θ) = R22 vf ψ 0 r0 , θ and ∂ψ 0 1 1 ∂ψ 0 vr0 = − 02 , vθ0 = 0 r sinθ ∂θ r sinθ ∂r0 0 where the upper index is for dimensionless magnitudes. For asymptotic developments one is writing ψ 0 r0 , θ = ψ00 (θ) + N ψ10 r0 , θ + O N 2 , p0 r0 , θ = p00 r0 , θ + N p01 r0 , θ + O N 2 . and the solution is obtained in dimension quantities with a formula for the stress as well. One is obtaining the drawing force as / . 3 |σf | R1 2 R23 =√ G (α, m) 1 − 3 F (α, m) + N ln σY R1 R2 3N where one have used the friction law (23) and γ=√

m , 1 − m2

"α

sin 2k−1 θdθ , k = 1, 2, 3 2 2 1 − sin θ λ − λ sin θ 1 2 0 √ 3sin2α − 3γcos2α + 2γ 3γ λ1 = √ 2 , sin 2 α 3cosα + 3γsinα Ik =

6γ 2 √ sin α( 3cosα + 2γsinα)2 √ α 2γ + 3sin2α cos 3 2 , F (α, m) = √ α γsin (1 + 2cosα) + 3cosαcos α2 2 m 2 cosαsin 2 α − sinα γsinα − √ cosα I1 3 3 − G (α, m) = γ (1 − cosα) sin2α − √ (1 − cosα) (1 + 2cosα) 3 √ γ 3sin2α + 6γcos2α − 4γ I2 + 4γ 2 I3 √ 3sin2α + 6γsin 2 α . − γ (1 − cosα) sin2α − √ (1 − cosα) (1 + 2cosα) 3 λ2 =

2

The authors have determined also several optimum angles for the drawing, depending on friction, reduction and N. They have also considered a Coulomb friction law in Camenschi et al., 1983.

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The same authors, S ¸ andru and Camenschi 1979 have considered the rolling of a material, by assuming that the rolls can be approximated by two straight lines. They apply an asymptotic series development. The kinetic and dynamic eﬀects, on the upper bound loads in metal forming processes, are due to Tirosh and Kobayashi 1976. It is question of forging, extrusion and piercing. The same kind of approach was applied for rolling of viscoplastic materials (Tirosh et al., 1985). Since the arc of contact area is practically small (α ≈ 5 deg) and the thickness of the sheet is small with respect to the roller diameter, the authors replace the arc of the contact area by a straight line. The strains are σm σrr − σθθ 1 , 1− √ √ Drr = 2 2η 3 IIσ σm 1 1− √ √ Drθ = τrθ . 2η 3 IIσ

and one repeat the theorem of powder expansion and also with the same friction laws. In another paper by Iddan and Tirosh 1996 one renounces at this assumption. The dynamic fracture by spallation in metals was considered by Gilman and Tuler 1970. The stress state associated with plane strain is described, and the eﬀect of the initial conditions and the time dependent conditions of the material on spallation is discussed. Fu and Luo 1995 consider the rigid-viscoplastic FEM to analyze the viscoplastic forming process. The general rigid-viscolastic FEM formulation is given and speciﬁc formulations of isothermal forging processes and superplastics forming are listed also. As an application, the combined extension process of pure lead, which is strain-rate sensitive at room temperature, is analyzed. The simulated results reveal the variation of the forming load with the stroke and its dependence on conditions. On the basis of the metal ﬂow patterns deﬁned by the rigid-viscoplastic ﬁnite-element method, the change of the position of the neutral layer is given and it is found that the occurrence of folding at the ﬂange may be attributed mainly to an abnormal ﬂow pattern. Moreover, the calculated results bring to light the rule of deformation distribution and its dependence on strain rate. Alexandrov and Alexandrova 2000, assuming a rigid viscoplastic material model, show that the velocity ﬁelds adjacent to the surfaces of maximum friction must satisfy sticking conditions. This means that the stress boundary conditions, the maximum friction law, may be replaced by the velocity boundary condition. For planar ﬂows, they show that plastic deformation in the vicinity of maximum friction surfaces is possible. The metal forming processes of aluminum-alloy wheel forging at elevated temperature are analyzed by the ﬁnite element method by Kim et al., 2002.

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A coupled thermo-mechanical model for the analysis of plasic deformation and heat transfer is adapted in the ﬁnite element formulation. The rolling problem is not mentioned in the present book. We would like to mention the paper by Angelov 2004 in which a variational analysis of a rigid-plastic rolling problem was considered. The material is rigid-plastic, strain-rate sensitive and incompressible. A nonlinear friction law is considered.

References Alexandrov S., Alexandrova N., “On the Maximum Friction Law in Viscoplasticity”, Mech. Time-Dependent Materias, vol. 4, 2000, p. 99–104. Angelov T. A., “Variational analysis of a rigid-plastic roling problem”, Int. J. Engng. Sci., vol. 42, 2004, p. 1779–1792. Avitzur B., Metal Forming: Processes and Analysis, McGraw-Hill, New York, 1968. Camenschi G., Cristescu N., S ¸ andru N., “High speed wire drawing”, Archives of Mechanics, vol. 31, 1979, p. 5. Camenschi G., Cristescu N., S ¸ andru N., “Developments of high speed viscoplastic ﬂow through conical converging dies”, Trans. ASME, J. Appl. Mech., vol. 50, no.3, 1983, p. 566–570? Camenschi G., S ¸ andru N., Mathematical Models in Metal Working, Bucharest, 2003. Campbell J. D., “Dynamic Plasticity – Macroscopic and Microscopic Aspects” Materials Science and Engineering, v. 12, no.1, 1973, p. 3–21. Cristescu N., “Plastic ﬂow through conical converging dies, using a viscoplastic constitytive equation”, Int. J. Engn. Sci. vol. 17, 1975, p. 425–433. Cristescu N., “Drawing through conical dies- an analysis compared with experiments”, Int. J. Engn. Sci., vol. 18, 1976, p. 45–49. Cristescu N., “Speed inﬂuence in wire drawing”, Rev, Roum. Sci. Tech. Mec. Appl., vol. 22, no.3, 1977, p. 391–399. Cristescu N., “On the optimum die angle in fast wire drawing”, J. Mech. Working Tech., vol. 3, 1980, p. 275–287. Dahan N., Le Nevez P., M´ethode de calcul des grandes d´eformations plastiques et endommagement dans les pi`eces extrudees., M´emoire et Etudes Scientiﬁques Revue de M´etallurgie, v. 80, no.10, 1983, p. 557–566. Fu M., Luo Z.J., “The simulation of the viscoplastic forming process by the ﬁniteelement Method”, J. Materials Process. Tech., v. 55, 1995, p. 442–447. Gilman J.J., Tuler F.R., “Dynamic Fracture by Spallation in Metals”, Int. J.Fracture Mech., v. 6, no. 2, 1970, p. 169–182. Iddan D., Tirosh J., “Analysis of High-Speed Rolling With Inertia and Rate Eﬀects”. J. Appl. Mech. Trans.ASME. v. 63, no.1, 1996, p. 27–37. Ionescu I.R., Vernescu B., “A numerical method for viscoplastic problem. An application to wire drawing”., Int. J.Engng. Sci., v. 26, no.6, 1988, p. 627– 633. Kim Y. H., Ryou T. K., Choi H. J., Hwang B. B. , “An analysis of the forging processes for 6061 aluminum-alloy wheels”, J. Materials Processing Tech., v. 123, no.2, 2002, p. 270–276. Lindholm U.S., Bessey R.L., “A survey of rate dependent strength properties of metals”, Rept. AFML-TR-69-119., 1969

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Sandru N., Camenschi G., “Viscoplastic ﬂow through inclined planes with applications to the strip drawing”. Lett. Apll. Engng. Sci., v. 17, no.6, 1979, p. 773–784. Tirosh J., Kobayashi S., “Kinetic and Dynamic Eﬀects on the Upper-Bound Loads in Metal-Forming Processes”. J. Appl. Mech. Trans ASME, v. 43, no.2, 1976, p. 314–318. Tirosh J., Iddan D., Pawelshi O., “The Mechanics of High-Speed Rolling of Viscoplastic Materials”. J. Appl. Mech., Trans. ASME, v. 52, 1985, p. 309–318. Wistreich J. G., Proc. Inst.Mech. Engng. London, v. 169, 1955, p. 654–665.

3D-ECAP of Square Aluminium Billets A. Rosochowski1 , L. Olejnik2 and M. Richert3 1

2

3

Design, Manufacture and Engineering Management, University of Strathclyde, 75 Montrose Street, Glasgow G1 1XJ, United Kingdom, [email protected] Institute of Materials Processing, Warsaw University of Technology, Ul. Narbutta 85, 02-524 Warsaw, Poland, [email protected] Faculty of Non-Ferrous Metals, AGH – University of Science and Technology, Aleja Mickiewicza 30, 30-059 Krakow, Poland, [email protected]

Summary. A way of increasing productivity of Equal Channel Angular Pressing (ECAP) by increasing the number of channel turns in the die is being explored. Unlike in other proposals of this type, the channel passages are not in one plane. This leads to a new concept of 3D-ECAP and a possibility of realising the most desirable deformation route BC in the die. The paper explains the above concept in detail and discusses the tool design issues. The laboratory trials of the new process are described and results presented. The structure of commercially pure aluminium 1070 subjected to 3D-ECAP is revealed. Basic mechanical properties are speciﬁed and conclusions formulated.

Key words: ultraﬁne grained metals, severe plastic deformation, equal channel angular pressing.

1 Introduction Compared to common metals, with a grain size of tens or hundreds of μm, UltraFine Grained (UFG) metals, also known as nanocrystalline metals or nanometals, have their grain size reduced to about 0.1–1 μm. This structural change aﬀects many mechanical and physical properties of UFG metals. They have 3–5 times higher yield strength and 2–3 times higher tensile strength. Ductility is usually reduced, but there are some examples of ductility being retained (Wang et al., 2002) or even increased, like in the case of cast magnesium alloys (Yoshimoto et al., 2005) UFG metals excel at very low temperatures giving a unique combination of strength and ductility (Wang et al., 2004). They can also be used at elevated temperatures, despite some tendency to the grain growth. UFG metals can be superplastically formed into products at much higher speed and lower temperature than traditional superplastic metals (Komura et al., 2001).

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Properties of UFG metals destine them for a range of high-tech applications. The ﬁrst practical application of UFG metals was aluminium and copper sputtering targets for physical vapour deposition (Ferrase et al., 2003). The advantages of using UFG sputtering targets include freedom of design, leading to extended life of targets, and more uniform layers of deposited metals. Another application close to implementation is medical implants made of UFG CP titanium (Fokine, 2004). Such implants are characterised by very good biocompatibility and high strength. Other applications will certainly follow, taking advantage of high strength and weight savings so valued in the aerospace and automotive industries. Despite these bright prospects, there has been no substantial commercial manufacturing activity built around UFG metals as yet. As far as the production methods for bulk nanometals are concerned, two main routes are available. The powder metallurgy route, which uses nanopowders, is technically, environmentally and economically problematic. Another route is based on Severe Plastic Deformation (SPD) of metals, which subdivides original coarse grains into much smaller domains (subgrains) by means of various systems of shear bands followed by subgrain rotation (Richert et al., 2001). It enables processing of all kinds of pure metals and alloys using a new class of metal working processes, which do not change the shape of a metal billet (Rosochowski et al., 2004). The most popular batch SPD process is equal channel angular pressing (ECAP), also known as equal channel angular extrusion. It was invented in 1972 in Russia and described in the west in 1981 (Segal et al., 1981). In ECAP (Fig. 1), a square or cylindrical bar is pushed from one passage of a constant proﬁle channel to another passage orientated at an angle of usually 90◦ to the ﬁrst one. Plastic deformation of the material is caused by simple shear in a thin layer at the crossing plane of the channel passages. This shear, together with rotation, converts the white cube in Fig. 1, representing symbolically a small portion of the material, into a parallelepiped. According to (1), the

Shear plane ϕ

ϕ

Fig. 1. Equal channel angular pressing (ECAP)

3D-ECAP of Square Aluminium Billets Route A

Route C

0°

Route BC

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180°

90°

Fig. 2. Three basic options for billet rotation between consecutive passes through ECAP die

equivalent plastic strain, generated by ECAP presented in Fig. 1, is about 1.15 (Segal et al., 1981). √ ε = 2 cot ϕ/ 3 (1) Since the strain required for the desired microstructural changes to occur is usually 4–8, the billet has to be pressed repeatedly through the die. This also gives the opportunity for billet rotation about its axis between each pass of ECAP (Fig. 2). The three fundamental options for this rotation are called A (no rotation), C (180◦ rotation) and BC (90◦ rotation in the same direction). Route BC is especially useful for obtaining a homogeneous microstructure of equiaxed grains separated by high-angle boundaries (Langdon et al., 2000). It is also advantageous in terms of superplastic properties (Komura et al., 2001). There are also other possible combinations of rotations, for example, a 90◦ rotation in the alternating directions, known as route BA . An interesting alternative recently suggested is the rotation pattern: 180◦ , 90◦ and 180◦ . It is referred to as route E and provides the best utilisation of the billet material by minimising so-called end eﬀects (Barber et al., 2004).

2 Three-Dimensional ECAP Manipulation of billets between consecutive passes of ECAP is cumbersome. In order to reduce the number of such manipulations and increase productivity, the number of channel turns in the die can be increased. For example, a two-turn U-shape channel, presented schematically in Fig. 3, is equivalent to route A normally realized outside the die (Segal, 1999). Accordingly, an S-shape channel represents route C (Segal, 1999). What happens to the material inside a multi-turn channel die can be referred to as “in-die rotation” although, physically, it is rather the die which is “rotated” with reference to the billet. The passages of the channels in Fig. 3 are in one plane and hence do not allow a 90◦ rotation, which constitutes route BC .

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Route A

Route C

Fig. 3. Two-turn ECAP for routes A and C

The latest addition to multi-turn ECAP is a process, which allows route BC to be realised inside the die (Rosochowski et al., 2004). This means that, for the ﬁrst time, the ECAP process can be carried out in three dimensions, not just two. This new process has been called three dimensional ECAP (3D-ECAP). As illustrated in Fig. 4a, the paths of the billet material 0-1-2 and 1-2-3 deﬁne the respective planes A and B, which are perpendicular to each other. Such a conﬁguration is equivalent to two classical ECAP operations, as illustrated in Fig. 1, and a 90◦ rotation of the billet between these operations. The number of channel turns is not limited. Figure 5 shows the material path for three-turn, 3D-ECAP obtained by repeating the general idea presented in Fig. 4. The proﬁle of the channel can be either square or round. Figure 6 shows schematically a square channel for a three-turn, 3D-ECAP process. It also shows the shear planes at each turn of the channel. As it is clear from Fig. 4b, these planes are diagonal planes in a cube and, therefore, are orientated at 60◦ (or 120◦ ) to each other. Thus, the channel conﬁguration (a)

(b)

SP 0

C A

120° B 90°

90° 1

3

SP

90° 2

Fig. 4. (a) material path in two-turn, 3D-ECAP; (b) the orientation of shear planes for this case

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Fig. 5. Material path in three-turn, 3D-ECAP

Fig. 6. Square channel for three-turn, 3D-ECAP

presented in Fig. 6 is equivalent to three passes of classical ECAP with 90◦ rotation of the billet between these passes (route BC ).

3 Die Design To realise the channel concept from Fig. 6, a novel die was designed (Fig. 7a). The underlying principle was to avoid any sharp corners along the channel, which could cause stress concentration. This was achieved by splitting the die into four segments. The ﬁrst unsuccessful attempt helped master the design, which eventually led to the segments’ geometry presented in Fig. 7b. The segments were kept together by two tightly ﬁtted pins. Since the pins alone were unlikely to prevent segments from going apart during the forming process, the die was prestressed with two outer rings. This required the outer surface of the die to be slightly conical in order to force it into the rings. The die was made of high speed steel and manufactured using spark erosion. The highest workmanship was necessary to make this multi-segmented prestressed die.

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(b)

Fig. 7. (a) channel conﬁguration in a conical die; (b) the way the die was split into segments

Figure 8 shows the design of a laboratory rig hosting the new die. It had to accommodate out of axis input and output passages of the channel, which was achieved by guiding the punch holder inside the rig and by the rigid rig design. There was a chute in the lower part of the rig, for the removal of processed billets.

Fig. 8. Tool-set design

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4 Experimental Procedure To ﬁt an 8 × 8 mm die channel, the 7.6 × 7.6 × 46 mm Al 1070 billets were machined and wrapped in a 0.1 mm thick Teﬂon tape (Fig. 9). The billets, as well as the die, were lubricated with lanolin. After pressing the ﬁrst billet down the input passage of the channel, the following billets were used to push the ﬁrst one along the channel and eventually out of the die. Figure 10 illustrates this concept for a two-turn, S-shape channel. Since the billets in the 3D-ECAP trials were longer than the two middle passages together, there was always a billet inside the die which extended over three turns. Figure 11 displays such a billet pressed halfway and removed from the die. The billet did not show any ﬂash, which can be attributed to the high quality of die-making and appropriate prestressing. It can also be seen that the material did not ﬁll the channel completely despite a much higher forming force required for the multi-turn ECAP compared to a classical L-shape process. This might be due to the entrapped lubricant. A billet leaving the die is shown in Fig. 12. Its ends are geometrically distorted reﬂecting the mode and history of deformation during 3D-ECAP. Since during the process, billets were in contact inside the die, the end eﬀects (undeformed material inside the billet ends) were reduced, except for the leading end of the ﬁrst billet. This was earlier demonstrated for a two-turn, S-shape ECAP, using FEM simulation (Rosochowski et al., 2002). To achieve a strain larger than 3 × 1.15 = 3.45, the process was carried out two more times, which resulted in a strain of 6.9 after second pass and 10.35 after third pass. The billets were rotated by 90◦ between consecutive passes to conform to in-die rotation taking place during 3D-ECAP. Since the billets leaving the die undergo elastic unloading, they do not ﬁt the input passage of the channel any more. Thus billets for the second and third pass

Fig. 9. Billets for 3D-ECAP trials, machined and wrapped in a Teﬂon tape

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2

1 1

1

2

3

3 2 1

1

2

1

Fig. 10. Sequence of operations for a single pass of the billet through a two-turn, S-shape channel

Fig. 11. Billet pressed halfway

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Fig. 12. Billet after a single pass

were prepared by reducing their lateral dimensions. Their ends were cut oﬀ to provide good contact with the punch and the billet in front. These measures resulted in 12 mm length reduction after ﬁrst pass and 11 mm length reduction after second pass. With the decreasing length, percent reduction for each pass was increasing (Fig. 13). The last percent reduction bar in Fig. 13 refers to the fact that, for some applications, further length reduction by 10 mm may be necessary to remove the skewed ends. Since Al 1070 is a very ductile material, the forming trials were carried out at room temperature. The forming speed was kept low (1 mm/s) to avoid heating up the material, which could lead to grain growth. A 250 kN laboratory press (Fig. 14), equipped with a load cell and a displacement transducer, was used.

Fig. 13. Reduction of billet length after consecutive passes

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Fig. 14. Press used for 3D-ECAP trials

Force [kN]

The process force and displacement were recorded. For the ﬁrst billet, introduced into an empty channel, the force measured during ﬁrst pass was about 10, 30 and 70 kN for consecutive turns. It was rising quicker than strain because of work hardening and increased friction. The subsequent billets were pushed into a channel already ﬁlled with other billets, which changed the forcedisplacement curve into one resembling forward extrusion (Fig. 15). Thus the force reduction was observed during the process due to decreasing friction in the input passage of the channel. Some undulations of this force could be associated with the ends of the billets passing the channel turns. Since the

Pass 1

Pass 2

Pass 3 Stroke [mm]

Fig. 15. Forces recorded during consecutive passes of 3D-ECAP

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billets used for the second and third pass were shorter, the average maximum force was reduced from 71.2 to 68.7 and ﬁnally to 63.5 kN. The maximum punch pressure, resulting from the maximum forming force of 71.2 kN, was 1112.5 MPa.

5 Microstructure The material used for 3D-ECAP experiments was commercial purity aluminium Al 1070. Its chemical composition is given in Table 1. Figure 16 presents the initial grain structure of Coarse Grain (CG) Al 1070, which was revealed by means of the anodizing method of Barker. The average grain size was about 200–300 μm. In order to reveal the grain structure of UFG aluminium, Transmission Electron Microscopy (TEM) was used. The mechanism, which is believed to be responsible for grain reﬁnement, is based on macroscopic as well as microscopic strain localization in the form of shear bands. Figure 17 shows one of those microscopic shear bands visible in the material after ﬁrst pass. The width of this band is about 0.4–0.5 μm. It shows already some fragmentation into smaller domains (cells), probably due to “cutting” it by another system of shear bands. This kind of fragmentation leads to some areas featuring already equiaxed cells (Fig. 18). Whether these cells are subgrains or fully developed grains, can be judged by revealing their misorientation angles. The technique used for this purpose was based on establishing the Kikuchi diﬀraction patterns for all cells under consideration, calculating the orientation of each pattern and determining the angle of misorientation between the neighbouring cells. The cells in Fig. 18 are not fully developed grains because their misorientation angles are below a threshold value of 15◦ . However, with the average size of about 0.7 μm, they are already in the sub-micrometre range. Subsequent passes did not lead to a further noticeable reduction of the average subgrain/grain size. However, the grain boundary misorientation angles continued to evolve from low values after the ﬁrst pass to a mixture of low and high values after the second pass (Fig. 19) and ﬁnally to predominantly higher values after the third pass (Fig. 20). Thus, the evolution of sub-grains into fully developed grains could be observed. Some advanced grains were free of dislocations due to possible recovery and/or recrystallisation. Table 1. Chemical composition (in %) of Al 1070 Al

Fe

Si

Zn

Mg

Mn

Ni

Cu

Cr

99.66

0.216

0.073

0.0104

0.003

0.003

0.003

0.002

0.0015

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Fig. 16. Initial grain structure in Al 1070 Pass 1

Fig. 17. Shear band after ﬁrst pass of 3-turn, 3D-ECAP

6 Mechanical Properties A test widely used for establishing and comparing mechanical properties of various materials is the tensile test. Small billets produced in this research required that tensile specimens were equally small. The main factor inﬂuencing the specimen size was the length-to-diameter ratio, which was kept at the standard level of 5:1. This limited the diameter of specimens to only 2.5 mm (Fig. 21). The specimens were machined by turning.

3D-ECAP of Square Aluminium Billets

Pass 1

Fig. 18. TEM micrograph after ﬁrst pass of 3-turn, 3D-ECAP

Pass 2

Fig. 19. TEM micrograph after two passes of 3-turn, 3D-ECAP

Pass 3

Fig. 20. TEM micrograph after three passes of 3-turn, 3D-ECAP

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Fig. 21. Tensile specimen

Since the gripping ends of the specimens were short, it was decided to use special grips. As illustrated in Fig. 22, the grips were split so that their two halves could be placed around the specimen’s ends and ﬁxed with tubes and pins. The contact surface between the grip and the specimen’s end was conical, which allowed self centring of the specimen. To achieve this eﬀect, the same cone was machined in the specimens in the form of a tangential extension of their radius. Tensile testing was carried out on a 10 kN Zwick machine at room temperature and a speed of 0.5 mm/min. Both the force and the displacement measurements were taken directly from the machine. Three types of the aluminium material were tested: the initial CG material, UFG material after the

Fig. 22. Grip used for tensile testing

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229

ﬁrst pass and UFG material after the second pass. There was not enough material in the billets after third pass to make tensile specimens. Each type of the material was tested three times. Typical force-displacement curves for CG aluminium, and UFG aluminium after the second pass, are shown in Fig. 23. The initial yield strength of the material subjected to two passes has been tripled while its tensile strength doubled. The total elongation has been halved and, judging by a small distance between the yield point and the onset of localization point, the uniform strain is largely reduced. Based on the curves from Fig. 23 and a curve obtained for the material subjected to one pass, the evolution of tensile properties was investigated. Figure 24 displays two strength characteristics and two ductility characteristics as functions of the plastic strain applied to the material during three-turn 3D-ECAP. Both the yield strength R02 (for 0.2% plastic strain) and Rm (UTS) increase in response to the decreasing grain size. Based on the data available, it is impossible to tell exactly how quickly this happens because, in a three-turn process, the ﬁrst non-zero strain value is 3.45. Nevertheless, for the strain larger than 3.45, the rate of strength increase is much slower. Initially, the distance between R02 and Rm decreases, indicating decreasing uniform strain, however, for a larger strain, it increases again. This is conﬁrmed by other measures of ductility which, like area reduction Z, increase after the substantial initial drop or, like percentage (total) elongation A5, decrease only slightly. Thus it can be said that ductility becomes stable for higher strains. It is known from another research on round billets ((Rosochowski et al., 2005), that the strength and ductility trends described 1

Load [kN]

0.8 Initial material

0.6

After 2 passes 0.4

0.2

0 0

1

2 3 4 Displacement [mm]

5

6

Fig. 23. Tensile curves for the initial CG material and for the material subjected to two passes of three-turn 3D-ECAP

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Yield strength R02/tensile strength Rm [MPa]

200

120 100

160

80 120 60

Rm R02 Z A5

80

40

40

20

0

Percentage elongation A5/area reduction Z [%]

230

0 0

2 4 6 Equivalent plastic strain

8

Fig. 24. Evolution of tensile properties of UFG Al 1070

above extend to larger strains, e.g. to a strain of 10.35 resulting from three passes of the same material through a three-turn, 3D-ECAP die.

7 Discussion and Conclusions The number of diﬀerent SPD processes is getting larger every year. Despite the increased choice of SPD processes, the position of ECAP is still very strong. This is due to a very large strain produced during one pass through an L-shape die, simple die design and relatively low forces/tool stresses required for the process. On the other hand, for best results, the process has to be repeated many times, which usually involves error-prone rotations of the billet between consecutive passes. Another problem is friction which, together with a limited punch slenderness, prohibits using billets with the aspect ratio higher than six. In this case, end eﬀects take away a large portion of the usable volume of the billet. ECAP has been tested and perfected over the years mainly due to the original works of its inventor, Vladimir Segal. For example, he invented an ECAP process for plates and movable die walls to reduce friction. Original contributions to ECAP development from other researchers are rare. This research has addressed the issue of ECAP productivity by introducing new process and tool concepts. In particular: –

In order to increase productivity of ECAP, a three-turn version of the process was tested. This was attempted in 3 dimensions with perpendicular channel planes. Thus the advantageous route BC could be realised in-die. – A novel square-channel die design, with a pre-stressed segmented die insert, was proposed for 3D-ECAP. The die performed well despite an increased force caused by the three turns of the billet.

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–

Three passes of the billet through the die produced a large strain of 10.35 and changed the structure of aluminium 1070. In addition to early grain reﬁnement, large misorientation angles were achieved during the second and third pass. – Mechanical properties responded to grain reﬁnement by a 2 to 3-fold increase in strength and reduced ductility. For larger strains, the ductility level stabilised. Increasing the number of channel turns to three reduced the number of passes 3-fold compared to the classical L-shape channel. However, this resulted in more than proportional increase of the process force and, more importantly, tool stresses. For materials harder than pure aluminium, this could be a major limiting factor. Thus, in our opinion, more feasible will be using two-turn channels, which would double productivity and increase the process force “only” three times. Other options are being currently considered to alleviate this problem. For laboratory purposes and in same practical applications, a batch process in the form of classical ECAP or 3D-ECAP could be acceptable. However, the majority of applications would beneﬁt from a continuous SPD process. Currently, a number of such processes, including ECAP-based continuous processes, are being developed.

Acknowledgements The authors would like to thank Scottish Enterprise for the ﬁnancial support under the Proof of Concept Fund initiative.

References Barber R. E., Dudo T., Yasskin P. B., Hartwig K. T., “Product yield of ECAE processed material”, Ultraﬁne Grained Materials III, 2004 TMS Annual Meeting, Charlotte, North Carolina, U.S.A., March 14–18, 2004, p. 667–672. Ferrase S., Alford F., Grabmeier S., D¨ uvel A., Zedlitz R., Strothers S., Evans J., Daniels B., Technology white paper, Honeywell International Inc., 2003. Fokine V. A., “The main directions in applied research and development of SPD nanomaterials in Russia”, Nanomaterials by Severe Plastic Deformation, NanoSPD2, Vienna, Austria, December 9–13, 2002, p. 798–803. Komura S., Furukawa M., Horita Z., Nemoto M., Langdon T. G., “Optimizing the procedure of equal-channel angular pressing for maximum superplasticity”, Materials Science and Engineering A, vol. 297, 2001, p. 111–118. Langdon T. G., Furukawa M., Nemoto M., Horita Z., “Using equal-channel angular pressing for reﬁning grain size”, IOM, vol. 52, no. 4, 2000, p. 30–33. Richert M., St¨ uwe H. P., Richert J., Pippan R., Motz Ch., “Characteristic features of microstructure of AlMg5 deformed to large plastic strain”, Materials Science and Engineering A, vol. 301, 2001, p. 237–243.

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Rosochowski A., Olejnik L., Richert M., “Channel conﬁguration eﬀects in 3D-ECAP”, Nanomaterials by Severe Plastic Deformation, NanoSPD3, Fukuoka, Japan, September 22–26, 2005, p. 179–184. Rosochowski A., Olejnik L., Richert M., “Metal forming technology for producing bulk nanostructured metals”, Journal of Steel and Related Materials – Steel GRIPS, vol. 2, Suppl. Metal Forming 2004, 2004, p. 35–44. Rosochowski A., Olejnik L., “Numerical and physical modelling of plastic deformation in 2-turn equal channel angular extrusion”, Journal of Materials Processing Technology, vol. 125–126, 2002, p. 309–316. Segal V. M., “Equal channel angular extrusion: from macromechanics to structure formation”, Material Science and Engineering A, vol. 271, 1999, p. 322–333. Segal V. M., Reznikov V. I., Drobyshevskiy A. E., Kopylov V. I., “Plastic working of metals by simple shear”, Russian Metallurgy (Metally), vol. 1, 1981, p. 99–105. Wang Y., Ma E., Valiev R. Z., Zhu Y., “Tough nanostructured metals at cryogenic temperatures”, Advanced Materials, vol. 16 no. 4, 2004, p. 328–331. Wang Y., Chen, M., Zhou F., Ma E., “High tensile ductility in a nanostructured metal”, Nature, vol. 419, 2002, p. 912–914. Yoshimoto S., Miyahara Y., Horita Z., Kawamura Y., “Mechanical properties and microstructure of Mg-Zn-Y alloys processed by ECAE”, Nanomaterials by Severe Plastic Deformation, NanoSPD3, Fukuoka, Japan, September 22–26, 2005, p. 769–774.

Computer-Aided Tool Path Optimization for Single Point Incremental Sheet Forming M. Bambach1 , M. Cannamela1 , M. Azaouzi2 , G. Hirt1 and J.L. Batoz2 1

2

Institut f¨ ur Bildsame Formgebung, Intzestr. 10, 52056 Aachen, Germany, [email protected], [email protected], [email protected], GIP-InSIC, 27, Rue d’Hellieule, 88100 Saint-Di´e des Vosges, France, [email protected], [email protected]

Summary. Asymmetric Incremental Sheet Forming (AISF) is a new sheet metal forming process for small batch production and prototyping. In AISF, a blank is shaped by the CNC movements of a simple tool. The standard forming strategies in AISF lead to severe thinning and an inhomogeneous wall thickness distribution. In this paper, several new types of forming strategies are presented that aim at a more homogeneous distribution of material. A forming strategy suitable for computeraided optimization was identiﬁed by ﬁnite element analyses. A “metamodel” was constructed by 162 ﬁnite element calculations in order to test diﬀerent optimization algorithms oﬀ-line for their performance: a Genetic Algorithm (GA), a Particle Swarm Optimization (PSO) algorithm and the simplex search method. The GA was found to be better at detecting the global optimum but lagged behind the PSO in terms of speed.

Key words: incremental sheet forming, forming strategies, ﬁnite element analysis, computer-aided optimization.

1 Introduction Asymmetric Incremental Sheet Forming (AISF) is a sheet metal forming process that uses CNC technology to produce complex sheet metal parts. The conventional forming strategies in AISF are an adaptation of z-level surface machining: the part is split into a series of two-dimensional layers and the plastic deformation is accomplished layer-by-layer through the movements of a simple CNC-controlled forming tool. A layer at constant z-position is formed by an in-plane movement of the tool. On completion of each layer, the tool moves down a small increment along the z-axis and continues to process the subsequent layer until all layers are formed. In order to achieve a good surface quality, the step-down is usually very small, e.g. in the range of 0.2 mm.

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blank holder

blank

forming tool

full positive die

rig

SPIF

TPIF

Fig. 1. Process variants in AISF

Generally, a distinction can be made between “Single PoInt Forming” (SPIF), where the bottom contour of the part is supported by a rig, and “Two-PoInt Forming” (TPIF), where a full or partial positive die supports critical regions of the part (Fig. 1). (Amino et al., 2002) have realized a dedicated forming machine based on TPIF. The conventional forming strategy described above leads to the so-called “sine law” relation between the initial (t0 ) and ﬁnal (t1 ) sheet thickness for a given wall angle α: t1 = t0 sin(90◦ − α) (1) As a consequence, the forming kinematics inherent in AISF entail the following drawbacks: –

The maximum wall angle is limited to 60–70 degrees for the most commonly used sheets of 1.0–1.5 mm thickness. This limitation restricts the potential scope of shapes and applications. – The strong dependence on the feature angle can lead to an inhomogeneous thickness distribution in the ﬁnal part. 1.1 State of the Art for Non-Conventional Forming Strategies in AISF Attempts to mitigate the thinning limit were presented by several authors: (Kitazawa 93) introduced four kinds of multistage strategies for the multistage forming of a dome and compared the distributions of sheet thickness obtained. (Junk et al., 2003) described a multistage strategy to produce a rectangular pyramidal frustum with right wall angles using TPIF. An optimization of the parameter settings for this strategy was performed experimentally by trialand-error, yielding a forming strategy with an initial wall angle of 45 degrees and 15 intermediate stages with a 3 degree increase in wall angle per stage. A drawback being the large forming time, this strategy was further developed by (Hirt et al., 2005) to a strategy combining bending and stretch-deformation. With this strategy, complex industrial parts were realized, again by experimental trial-and-error. (Jeswiet et al., 2001) manufactured a car headlight

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using scaled versions of the original CAD model as intermediate shapes. (Kim et al., 2000) presented an attempt to calculate the optimal intermediate shape in a two-stage forming process. (Giardini et al., 2005) put forward the idea of a two-stage forming process: using a large size tool, a preform was created by punching straight down into the ﬂat sheet. A smaller tool diameter was then used for the ﬁnishing stage. (Kun Daja et al., 2000) show that the strain distribution of an axisymmetric cup can be improved by carefully balancing the action of the tool between areas with a high stiﬀness (edge of the cup) and regions with a low stiﬀness (centre of the cup). An early attempt to use computer-aided optimization in incremental hammering to optimize the part geometry was presented by (Mori et al., 1996). So far, no eﬀorts were made to parameterize forming strategies for AISF and to apply optimization algorithms to the problem of tool path optimization, neither experimentally nor using process models. In this paper, attempts are made to – – –

deﬁne a new class of fully parameterized strategies for SPIF, assess their potential to allow for a homogeneous thickness, single out suitable strategies for computer-aided tool path optimization and – compare the performance of diﬀerent optimization algorithms for the optimization of the sheet thickness distribution in SPIF.

2 New Forming Strategies for SPIF 2.1 Deﬁnition of Strategies In this section, several types of non-conventional forming strategies are presented. The strategies aim at a more homogeneous distribution of wall thickness. They are motivated by the fact that the conventional z-level tool paths in SPIF only cover the side walls while leaving e.g. the bottom of the workpiece undeformed. This type of strategy is shown in the upper part of Fig. 2. In order to also deform the ﬂat bottom of the part of the depicted part, a new “conical” strategy is proposed, where the tool movement starts at the centre of the part and opens up with increasing depth until the desired diameter at

r0

α

d conventional strategy

conical strategy

Fig. 2. Conventional and conical z-movement

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maximum depth is reached. With this strategy, the shape of the side walls is determined indirectly by means of the ﬁnal contour traced by the forming tool. For the in-plane movements, two new strategies are proposed (Fig. 3): –

A “contour” strategy: starting at the centre of the part, the tool moves along self-similar contours of increasing diameter, until the edge of the part at the present depth is reached. The diameter at a certain depth is prescribed by using either the conventional or conical z-movement. This type of strategy uses a radial pitch “dr” to connect subsequent contours. – A “radial strategy”: radial forming consists of radial, star-shaped movements. Each stroke starts at the centre of the part, i.e. after each radial movement, the tool moves back to the centre. The strategy is deﬁned by means of a circumferential pitch “dϕ”. As indicated before, each of the two strategies for the z-movement can be coupled with each of the strategies for the in-plane movement, yielding four types of parameterized strategies. Restricting our attention to conical parts, the forming of a speciﬁc part can be speciﬁed by geometric features (radius of the part at z = 0 mm, cone angle and depth), and the combination of a z-movement (parameter dz) with an in-plane movement (parameter dr or dϕ). The conventional z-level strategy can be recovered by combining the “conventional” z-movement with a contour-type in-plane movement using a single radial pitch per z-level. 2.2 Experimental Results and Discussion The feasibility of the proposed strategies has been tested using a single benchmark geometry and one generic parameter combination for each of the four strategy types. The selected geometry is an axisymmetric cup with a radius r0 of 60 mm, a wall angle of 80 degrees and a depth of 25 mm. The latter value was identiﬁed through test series with increasing forming depth as the depth for which the part starts to break with the conventional forming strategy. As

dϕ dr

contour strategy

radial strategy

Fig. 3. Contour and radial in-plane strategy

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sheet material, DC04 blanks of 1.5 mm thickness were used. The tool diameter was 30 mm. A relatively large pitch of 5 mm per cycle was employed. For the two strategies involving radial punch strokes, a circumferential (angular) pitch of 4◦ was used. The combination of conventional z-movement and inplane “contour” strategy used a radial pitch “dr” of 20 mm. Figure 4 shows a top view on the parts after forming. For evaluation of the thickness distribution, the parts were cut along a radial section, and the thickness was measured every 2 mm using a micrometer gauge (Fig. 5), starting from the centre of the part. The following conclusions can be drawn from the conducted experiments: –

All of the considered strategies enable to form the part in contrast to preliminary tests using the conventional z-level strategy that lead to rupture. – The strategies involving radial strokes allow for a considerable reduction of sheet thickness in the centre of the part. However, the forming time is quite high (∼ 9 min. for the “cone/radial” and ∼ 40 min. for the “conventional/radial” strategy). – The “conventional/contour” strategy provides a thickness distribution comparable to those obtained with the radial movements at a reduced forming time (∼ 2 min.). – Although the “cone/contour” strategy yields a very short manufacturing time of less than 1 min., the sheet thickness could not be reduced in the centre of the part, probably because it lacks the repetitive loading of this region. Furthermore, the part shows a considerable springback and hence the largest deviation to the target geometry among the given strategies.

cone/contour

cone/radial

conv./contour

conv./radial

Fig. 4. Top view on parts manufactured using the four new strategies

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cone/contour cone/radial conv./contour conv./radial

sheet thickness [mm]

1.4 1.3 1.2 1.1 1 0.9 0.8 0.7

0

10

20

30 40 50 60 distance along section [mm]

70

80

Fig. 5. Sheet thickness distributions for the four strategies

–

While the huge pitch values are possibly beneﬁcial for the thickness distribution, a smooth surface would require a dedicated ﬁnishing stage.

3 Finite Element Analysis for the New Strategies The experimental results show that the proposed strategies oﬀer possibilities to take inﬂuence on the sheet thickness distribution of a given part. However, a homogeneous thickness distribution requires a rigorous optimization of the parameter settings, and a parameterization that encompasses the optimal strategy. Given the size of the parameter space, experimental trial-anderror optimization of the tool path should be avoided. In an attempt to assess the feasibility of computer-aided optimization with the new strategies, ﬁnite element analyses were performed for the “cone/contour” strategy, which is the strategy with the shortest tool path (among the investigated). Based on earlier ﬁnite element analyses of the process presented in Bambach et al., the model was realized both in ABAQUS/explicit and ABAQUS/standard. To ensure that the FE calculations are comparable with the experiments, the tool paths that were used for the CNC machine were translated into input ﬁle format for ABAQUS using a dedicated MATLAB routine. A ﬁctitious forming time of 5.31 s was calculated, which corresponds to the duration of the process in reality at full tool speed, i.e. if acceleration and deceleration are neglected. The DC04 sheet was modeled as an elasto-plastic material with isotropic hardening using material data obtained from tensile tests. The results of the FE analyses were tested against experimentally determined values for both sheet thickness and geometry. For evaluation of the ﬁnite element calculations, the sheet thickness and the geometry are compared to experimental data along a radial section in positive x-direction for both the implicit and explicit FE models (Fig. 6). As a measure of the quality of the

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5 experiment explicit FEA implicit FEA

0 z [mm]

–5 –10 –15 –20 –25

0

10

20

30 40 x [mm]

50

60

70

Fig. 6. Comparison of calculated and measured part geometry

calculated geometry, we determine the maximum normal distance dmax and the average normal distance dav between the experimental data and the FE results. Therefore, the normal distance between the two sections (experimental data and FE calculation) is calculated on 141 positions by constructing the normals on the experimental curve and computing the intersections with the FE data curve. The prediction of sheet thickness is judged by means of the maximum deviation dth,max between the experimental data and the FEA. Mesh convergence studies were performed using the experimental data as reference. The blank was meshed with a uniform mesh of 2304 S4R shell elements, which turned out to be a good compromise between accuracy and calculation time. Similar tests were performed for the admissible amount of mass scaling for the explicit FEA. Using the “variable mass scaling” option, scaling to a time step of 10−4 s was found to be a suitable choice. When compared to the more conservative mass scaling to an average time step of 10−5 s the results did not deteriorate considerably, but the calculation time increased from 30 minutes to more than three hours (see Table 1). Since no data are available for friction coeﬃcients in AISF, a sensitivity analysis was performed to assess the role of friction. In ﬁve analyses with varying friction coeﬃcients of 0.0, 0.05, 0.1, 0.2 and 0.5, no considerable inﬂuence on the prediction of geometry and thickness could be found. Based on these results, and given that the sheet was well lubricated in the experiments, the Table 1. Accuracy of explicit and implicit FEA analysis type

average time step [s]

geometry

thickness

dmax [mm]

dav. [mm]

dth,max [%]

calculation time [h]

explicit explicit implicit

10−4 10−5 –

1.82 1.67 1.09

1.19 1.13 0.59

15.62 15.10 11.7

0.48 3.1 7.32

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friciton coeﬃcient was set to zero in the FEA. In the implicit simulation, the mesh size and input data were chosen in accordance to the explicit simulation runs. The results are summarized in Table 1. Although providing very good results as compared to experimental data, the implicit analysis exhibits a relatively large number of increments. The increment size was altered automatically by ABAQUS depending on the success of the iterations. An average of 4.5 iterations had to be performed for each time increment, two of which are so-called “severe discontinuity iterations” which are used to achieve well-deﬁned contact conditions. As a consequence, the analysis time is large even for the small benchmark part and the short tool path, so that an implicit analysis can presently not be used for tool path optimization. A direct comparison between the thickness provided by the explicit and implicit FEA shows that the maximum diﬀerence occurred at the locus where the vertical pitch is performed. This is probably due to the fact that the tool transmits a high kinetic energy during the sudden change from in-plane movement to z-pitch. Plots of the reaction forces obtained by the implicit and explicit calculations corroborate this assumption (Fig. 7): Large deviations between the obtained forces can be found at those points in time where the vertical pitch is performed (see arrows in Fig. 7), while the results are in good agreement throughout the remainder of the calculations.

30 implicit FEA explicit FEA

Fx [kN]

20 10 0 –10 –20

0

0.2

0.4

0.6

0.8

1

0

Fz [kN]

z-level tool path

–10

–20

–30

0

0.2 0.4 0.6 0.8 normalized process time [–]

1

Fig. 7. Comparison between reaction forces obtained by explicit and implicit FEA

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10

Fx [kN]

5 0 –5 –10

0

0.2

0.4

0.6

0.8

1

0

Fz [kN]

spiral tool path

–5

–10

–15

0

0.2 0.4 0.6 0.8 normalized process time [–]

1

Fig. 8. Comparison between reaction forces obtained by explicit and implicit FEA (spiral tool path)

To avoid the inﬂuence of the step-down on the accuracy, additional explicit and implicit analyses were performed using a spiral tool path. For these analyses, the tool forces obtained by the implicit and explicit FEA are in good agreement (see Fig. 8).

4 Optimization Based on the ﬁndings in the foregoing section it can be concluded that the success of computer-aided optimization depends largely on the proper design and para-meterization of the tool path. The ﬁnite element analyses show that both accuracy and short runtime can be provided by the “cone/contour” strategy, if it is re-designed to a spiral tool path. In order to increase the number of parameters, it was decided to focus on a strategy consisting of 3 iterated applications of the spiral “cone/contour” strategy presented above. The strategy was parametrized by the dimensions of its intermediate forms; one depth hi and one diameter di describe one spiral “cone/contour” tool path that produces a conical intermediate form (Fig. 9). The third tool path was ﬁxed in

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Stage 2 h1

Final stage (fixed) h2

H

+ +

d1 d2

D

Fig. 9. Tool paths for the three-stage forming process to be optimized

order to produce the same part geometry for all parameter combinations (D = 120 mm, H = 18 mm), resulting in four parameters per potential solution. The number of possible parameter combinations was decreased by allowing the di to vary only in steps of 10 mm, and restricting the hi to be a multiple of 3 mm. In addition, the following constraints were imposed on the parameters: d1 < d2 < D h1 < h2 < H

(2a) (2b)

With these constraints, 162 parameter combinations were identiﬁed. FEM simulations were run for these combinations to span a solution space. A “metamodell” was built through a linear interpolation between the resulting points, yielding a hypersurface of 5 dimensions. Thus, given an arbitrary point in the solution space the value of the objective function, or “ﬁtness”, can be quickly approximated. This allows for the rapid and repeated evaluation of the diﬀerent methods needed to compare of their performance. The solution space is visualized in Fig. 10 and Fig. 11. In these ﬁgures, the ﬁrst parameter was held constant and slices of the resulting volumetric space taken at even intervals. The color is tied to the value of the function, black is the maximum of approximately 1.035 mm and white is a value of 0 mm. The initial sheet thickness is 1.5 mm as in the analyses before. Two views are presented, showing the slices from two opposing sides. Important to note is how much black appears, and how rapidly the transition to white occurs. The function varies quickly over a small area, but otherwise does not change much. This could mean that 162 values do not adequately describe the topology of the solution space, or the solution space itself could just be ﬂat. Summary tests indicate that the surface does approximate the actual solution space. A surface based on cubic interpolation of the data was also tested, but resulted in unacceptable overestimation of the sheet thickness. The goal of the optimization was to ﬁnd the tool path which distributed the material as evenly as possible throughout the part. As a consequence of volume conservation, the part with the largest possible minimum sheet thickness would have a perfectly homogeneous distribution of thickness. To make any

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d1 = 50 mm

h2

h2

d2

d2

“fitness“ (thickness [mm]) 1.04

h1

h1

0.5

d1 = 60 mm

d1 = 70 mm

0

h2

h2 d2

d2

h1

h1

Fig. 10. Visualization of the solution space, view 1

point thicker would require taking material from another point, reducing the minimum sheet thickness. Thus, the ﬁtness of a given tool path was taken to be the minimum value of thickness in the simulated part resulting from that tool path. Let n t (p) = {ti (p)}i=1 , n = number of nodes, (3) be a set of nodal thicknesses for all nodes in the mesh, where p = (d1 , d2 , h1 , h2 )

(4)

is the design variable. Then, in terms of optimization, a solution for the following problem is sought: maximize min [{ti (p)}] . i

(5)

The objective function here is obtained through a simulation. It cannot be evaluated analytically and so no information about its gradients is available without performing a huge number of simulations. The large solution space,

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d1 = 50 mm

h2

“fitness“ (thickness [mm]) 1.04

h2

0.5 0

d2

d1 = 60 mm

d2

h1

h2

d1 = 70 mm

h1

h2

d2

d2 h1

h1

Fig. 11. Visualization of the solution space, view 2

lack of gradient or other analytical information and use of simulation all suggest that stochastic or search optimization methods would be most appropriate for optimizing tool paths. The work that follows compares a few of these methods in an attempt ﬁnd a suitable algorithm for tool path optimization. Three diﬀerent optimization methods were implemented, all in MATLAB: a Genetic Algorithm (GA), a Particle Swarm Optimizer (PSO), and the simplex method. 4.1 Optimization Algorithms Genetic Algorithm The genetic algorithm is a type of simulated evolution. The parameters of a problem are encoded into an array or “genome” and from these parameters the corresponding ﬁtness value is determined. An individual is a solution represented in this format and placed in a population of other individuals. Based on ﬁtness, individuals are selected for genetic operations that produce new individuals. Two individuals might be combined in some way, called crossover, or one individual might have one or more genes altered in a mutation. This process of creating new solutions continues until the population has converged

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on an optimum or some other termination criterium is met. The algorithm implemented here was written for MATLAB by (Houck et al. 1995). During the course of calibrating the algorithm, it was concluded that maintaining diversity in the gene pool is important to prevent the algorithm from converging prematurely on a local optimum. Although there are many parameters to be considered, the selection mechanism seems to be most important for maintaining diversity. The default selection operator is a normal geometric selection, which often picks just two or three individuals for the crossover and mutation processes. This breeds the poor solutions out in as few as one generation but can result in severe stagnation of the gene pool. A considerable amount of mutation is then required to balance this eﬀect, which is “unnatural” in the sense that natural evolution relies more on the crossover of suﬃciently disparate individuals to maintain diversity. In an eﬀort to maintain diversity while using mutation in moderation, the selection mechanism of the original GA was modiﬁed. Under the new system, the strongest individuals ﬁrst attempt to mate with each other with a certain probability of success, and failing that, attempts are made to mate elite individuals with random ones from the rest of the population. If this, too, fails, two random individuals from the population are crossed over. After all the crossovers have been performed, members of the previous generation are replaced by the oﬀspring at random; “ﬁtness” represents the ﬁtness to reproduce, not to survive. The motivation for this approach is to allow the elite individuals to persist through their oﬀspring, while still allowing for the mixing of elite gene pools with weaker ones to ensure a more thorough exploration of the solution space. As an additional boon to exploration and diversity, a “migration” of randomly generated individuals is brought in to replace a percentage of the population every few generations. Some work has been done in applying GA’s to similar problems, see (Mori et al. 1996) and(Schenk et al. 2004). Particle Swarm Optimization The particle swarm optimizer is another biologially inspired optimization algorithm. It mimics the behavior of ﬂocks of birds or schools of ﬁsh searching for food. A group of particles, the swarm, is inserted into the solution space. Each particle then “ﬂies” through the hyperspace to a new position. Each member of the swarm knows the best solution it has personally found so far and the best solution found by its nearest neighbors, or in some cases, the best solution visited by any member of the swarm. At each iteration, new velocities and positions for each particle are determined from equations of the forms v = c1 v + a c2 (pbest − x) + b c3 (gbest − x) x = x + v

(6) (7)

where x and v are position and velocity, the primed terms are the new values, c1 , c2 , and c3 are positive constants, a and b are random numbers from zero

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to one, pbest is the particle’s own best solution and gbest is the global best solution so far. The optimal values of the constants is not clear, but c1 is typically near 1 and c2 and c3 are close to 2. For this test, the set of parameters was c1 = .729 and c2 = c3 = 1.494 as recommended by (Clerc et al., 2002) and tested by (Trelea et al., 2003). The iterative process continues until convergence or termination. Simplex Method The simplex method diﬀers from the PSO and GA in its approach. Rather than evolving a population of solutions in some way, a simplex is created around an initial guess in the solution space. The simplex is maintained and manipulated iteratively, always seeking to move the weakest vertex of the simplex to a better location. This continues until the simplex has converged to within some speciﬁed tolerance. The MATLAB “Optimization Toolbox” function “fminsearch” was used to test the performance of the simplex method. 4.2 Comparison of Methods When optimizing the hypersurface, the simplex method consistently ran into the nearest local optimum and stayed there, leveling oﬀ after an average of 43 function calls with a standard deviation of 16. In 30 trials, it did not ﬁnd the global optimum once. Only if the initial guess was artiﬁcially set very close to the global optimum did it manage to ﬁnd it. A t-test showed that the average best value found with the simplex method was signiﬁcantly diﬀerent from the worst performing GA, albeit by .01 mm. As a control, the simplex method was then tested against a random search of 43 function calls. The random search outperformed the simplex in this instance, with a t-test supplying weak evidence that this diﬀerence was signiﬁcant. To compare the PSO and the GA, the hypersurface approximating the objective function was optimized many times with each algorithm. One trial consists of a random initial population and 600 function calls. The algorithms were tested with population sizes of 10, 20, and 30. For each trial, the best value found was recorded. Additionally, the number of function calls made before the ﬁtness reached 99.95 % of its steady state value was noted and will be termed “settling time” for this discussion. The threshold was set so high because the diﬀerence from one point in the solution space to the next was so small. As a further consequence of the “ﬂatness” of the solution space, the average value found was the same for both algorithms at all population sizes. The histogram in Fig. 12 shows the diﬀerence in speed between the algorithms at a population size of 20. The x-axis represents the “setting time”, the y-axis gives the frequency, i.e. how many times out of 30 trials the algorithm did settle after a given number of function calls. The trend apparent here persists throughout the data for other population sizes, and t-tests in Table 2 show that the diﬀerence in speed between the PSO and the GA is signiﬁcant at all population sizes.

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PSO, pop = 20 GA, pop = 20

6 5 4 3 2

600

580

550

520

490

460

430

400

370

340

310

280

250

220

190

160

0

130

1 100

Frequency, total count = 30

7

Settling Time (Function Calls)

Fig. 12. Histogram of algorithm settling time in function calls

In addition to the best value found, the “success rate” was deﬁned as the probability that the algorithm will ﬁnd the global optimum. Chi-square tests suggest that for the PSO success rate is rather insensitive to the ratio between population size and number of iterations while for the GA smaller populations allowed more iterations are more successful. In a test where the algorithms of population size 30 were not limited in function calls, the GA and PSO were dead even in terms of success rate and best value found, but the PSO was still nearly twice as fast. 4.3 Conclusions on Optimization For this solution space, the GA was better at ﬁnding the global optimum but lagged behind the PSO in terms of speed. Furthermore, the PSO seemed to be insensitive to the ratio between population size and number of iterations while the GA preferred smaller populations with more iterations. The simplex method was unable to prove itself better than a random search, so it is of little use in this problem. A good portion of the solution space turned out to be fairly “ﬂat”; the 5 dimensional analog of a 3 dimensional plateau. Once Table 2. Statistics summary algorithm alg. pop size

PSO 10

GA 10

PSO 20

GA 20

PSO 30

GA 30

simplex

mean settling 185.7 251.3 242.0 404.7 293.0 416.0 43.1 time std. dev. 121.8 120.4 130.3 130.2 120.2 141.31 16.4 mean best 1.0343 1.0343 1.0341 1.0341 1.0341 1.0341 1.0192 found std. dev. 2.7E-04 3.6E-04 3.7E-04 5.0E-04 4.3E-04 5.0E-04 2.6E-02 sucess rate 0.23 0.40 0.10 0.33 0.17 0.13 0

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the algorithm gets on top of the plateau, there is nowhere else for it to go. This made it diﬃcult to compare the algorithms in terms of the average best value found, but as already noted a comparison of speed and success rate did produce statistically signiﬁcant results. It must be noted, however, that other than population size no other parameters were varied for the PSO and the GA. Given the proper tweaking, it is not inconceivable that either of these algorithms could outperform the other. Tuning these parameters is not trivial, however, and is not the only problem in applying these algorithms to an engineering problem of this nature. The PSO is often praised for its sparing use of parameters, and in fact (Clerc et al. 2002) have written a “parameterless” implementation where the user has only to deﬁne the problem; the PSO then adapts parameters like swarm size and topology as the algorithm progresses. This is in contrast to most GA’s, which have a host of parameters that must be tuned by the user. So far, there is no a priori method for determining what might be a good parameter set for a given problem, and the optimal parameter set varies heavily from problem to problem. Furthermore, engineering applications are bound by practicality. Population sizes must be small and convergence rates high if the objective function is an FEA calculation, as in this case. The average computation time for these calculations on a 3 GHz machine is roughly 1 hour. With the algorithms settling in 200–400 function calls, one optimization would result in 200–400 hours of computation for each optimization, if it could be immediately recognized that the algorithm had settled. This is unacceptable but is already a tall order; tests of PSO’s on common benchmark functions are often allowed to run for hundreds or thousands of iterations. The advantage of a paramterless approach then becomes even more clear, as such computationally expensive procedures cannot be repeated just to tune the parameters. Furthermore, the fastest setting of the PSO took on average 185 function calls to settle, which is more than the 162 used to approximate the solution space in the ﬁrst place. Perhaps a better alternative would be a synthesis of methods, using a PSO or GA to obtain a guess for a more localized search method like the simplex. Still, given the highly parallel nature of the PSO and GA the bottleneck is not processor speed but number of processors. Having even 2 processors would halve the computation time per iteration, and a larger cluster could make the computation time manageable. Another way to circumvent the computa-tion barrier would be to depart from FEA altogether and develop a computationally cheaper model, such as the inverse approach (Batoz et al. 1998). Although being limited compared to FEM, such models can give accurate results for strains and that is all that is needed for an optimization of the thickness distribution.

5 Summary and Outlook In the present paper, parameterized strategies for the optimization of sheet thickness distributions in single point incremental sheet forming are developed. The strategies were found to be a promising alternative to the

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conventional z-level type tool path strategies. Finite element calculations of the strategies were used to identify a strategy that is suitable for computeraided tool path optimization. A “metamodel” was constructed by 162 ﬁnite element calculations. With this representation of the solution space, diﬀerent optimization algorithms were compared: a genetic algorithm (GA), a particle swarm optimizer (PSO), and the simplex search method. Statistically representative evidence can be given that the GA was better at ﬁnding the global optimum but was generally slower than the PSO. Due to the ﬂatness of the solution space near the global optimum, many potentially optimal solutions seem to coexist which caused the simplex method to always get stuck in a local optimum. Future work will largely focus on the parameterization of the tool path, as it determines the topology of the solution space and is therefore crucial to solving the optimization problem. Since full-scale ﬁnite element calculations of the process are time-consuming, simpliﬁed process models are currently being developed in order to allow for a faster evaluation of the outcome of a given forming strategy.

Acknowledgements The authors would like to thank the German Research Foundation (DFG) for the funding received for the ﬁnite element analyses through project HI 790/5-1.

References Amino H., Lu Y., Ozawa S., Fukuda K., Maki T., “Dieless NC Forming of Automotive Service Parts”, Proceedings of the 7th ITCP, Yokohama, 2002, p. 1015–1020. Bambach M.; Ames J.; Azaouzi M.; Campagne L.; Hirt G.; Batoz J.L., “New forming strategies for single point incremental sheet forming: Experimental evaluation and numerical simulation”; Proceedings of the 8th ESAFORM Conference on Material Forming, Cluj-Napoca, 2005, p. 671–674. Batoz J.L., Guo Y.Q., Mercier F., “The Inverse Approach With Simple Triangular Shell Elements for Large Strain Predictions of Sheet Metal Forming Parts,” Engineering Computations, vol. 15 no 7, 1998, p. 864–892. Clerc M., Kennedy J., “The particle swarm: explosion stability and convergence in a multi-dimensional complex space”, IEEE Transactions on Evolutionary Computation, vol. 6 no. 1, 2002, p. 58–73. Giardini C., Ceretti E., Attanasio A., ”Further Experimental Investigations and FEM Model Development in Sheet Incremental Forming”, Advanced Materials Research, vols. 6–8, 2005, p. 501–508. Hirt G.; Ames J.; Bambach M., “A new forming strategy to realise parts designed for deep-drawing by incremental CNC sheet forming”, Steel Research, vol. 76 no. 2/3, 2005, p. 160–166.

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Houck C.R., Joines J.A., Kay M.G., “A genetic algorithm for function optimisation: a MATLAB implementation”. Technical Report NCSU-IE TR 95-09, North Carolina State University. Jeswiet J., Hagan E., “Rapid Proto-typing of a Headlight with Sheet Metal”, Proceedings of the 9th International Conference on Sheet Metal, 2001, pp 165–170. Junk S., Hirt G., Chouvalova I., “Forming Strategies and Tools in Incremental Sheet Forming”; Proceedings of the 10th International Conference on Sheet Metal, 2003, p. 57–64. Kim T.J., Yang D.Y., “Improvement of formability for the incremental sheet metal forming process”, Int. J. Mech. Sci., vol. 42, 2000, p. 1271–1281. Kitazawa K., “Incremental Sheet Metal Stretching-Expanding With CNC Machine Tools”, Proceedings of the 4th ICTP, 1993, p. 1899–1904. Kun Daia Z.R., Wanga, Y.F., “CNC incremental sheet forming of an axially symmetric specimen and the locus of optimization”, Journal of Materials Processing Technology , vol. 102, 2000, p. 164–167. Mori K., Yamamoto M., Osakada K., “Determination of hammering sequence in incremental sheet metal forming using a genetic algorithm”, Journal of Materials Processing Technology, vol. 60, issue 1–4, p. 463–468. Schenk O., Hillmann, M., “Optimal design of metal forming die surfaces with evolution strategies”, Journal of Computers and Structures, vol. 82 issues 20–21, 2004, p. 1695–1705. Trelea I.C., “The particle swarm optimisation algorithm: convergence analysis and parameter selection”, Information Processing Letters, vol. 85, 2003, p. 317–325.

Study on the Achievable Accuracy in Single Point Incremental Forming J.R. Duﬂou, B. Lauwers and J. Verbert Katholieke Universiteit Leuven, Department of Mechanical Engineering, Celestijnenlaan 300B, 3001 Leuven, Belgium, Joost.Duﬂ[email protected], [email protected], [email protected] Summary. Single-Point Incremental Forming (SPIF) is a sheet metal forming technique that is gradually evolving towards industrial applicability. As recent market analysis studies have shown, accuracy is one of the most important limiting factors for the deployment of SPIF in industrial applications. The case studies described in this paper aim to illustrate the state-of-the-art in achievable accuracy for a number of realistic parts having diﬀerent geometric complexity and produced by diﬀerent tool path strategies. A secondary goal of this study is to demonstrate the applicability of SPIF for prototyping or small batch production. The results of the diﬀerent strategies were measured and compared to the geometric speciﬁcations. The achieved accuracy for the respective parts and process strategies are reported.

Key words: single point incremental forming, accuracy, sheet metal, rapid prototyping.

1 Introduction Single Point Incremental Forming (SPIF) is an emerging sheet metal part production technique. In this process, a sheet metal part is formed in a stepwise fashion by a CNC controlled, rotating, spherical tool without the need for a supporting (partial) die. This technique allows a relatively fast and cheap production of prototypes or small series of sheet metal parts (Jeswiet, Micari, et al., 2005)(Park et al., 2003)(Matsubara, 2001). Today the SPIF process is still premature and needs further development before it can be used for industrial applications. The part accuracy and thickness variations, and therefore the resulting part strength, are some of the key concerns. As recent market analysis studies have shown, accuracy is one of the most important limiting factors for the deployment of SPIF in industrial applications (Allwood et al., 2005).

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Distinction should be made between diﬀerent sources of inaccuracy. On the one hand elastic spring back and unwanted plastic deformations may aﬀect the part geometry while still in process. On the other hand, after forming, some parts need to be trimmed to obtain the desired geometry. This trimming process can also introduce inaccuracies due to the residual stresses that exist inside the part after the forming process. While the ﬁrst type of geometric errors is discussed in Sect. 3, deformations due to trimming are touched upon in Sect. 4.

2 Test Setup 2.1 Incremental Forming Set-Up For the experiments described in this paper, a conventional, 3-axis, NCcontrolled milling machine (MAHO 500) was used as incremental forming set-up (Fig. 1). All test parts were made in aluminium 3003-O starting from a ﬂat sheet of 1.2 mm thickness. Al3003-O was chosen because of its good formability and because the forces required to process this material have been well documented (Duﬂou et al., 2005). A special rig was constructed to hold the blank sheets while being formed (Fig. 2). For all strategies, a backing plate was used within

Fig. 1. A CNC controlled 3-axis milling machine equipped with the SPIF rig, while forming a standard cone

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Fig. 2. Exploded view of the SPIF rig

the set-up in order to prevent deformation of the top plane of the part. This backing plate supports the part at the edges where the metal sheet should not deform. A spherical tool (Ø12.7 mm, material: tool steel, Vanadis 23) was used for all applied strategies. The feedrate was set to 2 m/min and the spindle speed was selected such that a rolling movement of the tool over the sheet was obtained. Since it was found that lack of lubrication causes tool failure within a few contours of the tool path, oil was applied to further minimise the friction. 2.2 Measurement Equipment A laser scanner with an accuracy of ±15 μm was used to measure the shape of the manufactured parts. Since the rig was constructed in such a way that it can be removed from the machine without declamping the parts, these could be scanned before declamping. The scanned point cloud was mapped onto the CAD model using the scanned edges of the rig as reference. For each point of the point cloud, the deviation with the CAD model was calculated as the orthogonal distance between the CAD (reference) surface and the point. These deviations were then plotted to obtain a general view of the accuracy of the part. A negative deviation means that the actual part was formed deeper than the CAD model. 2.3 Trimming Facility The device used to trim the parts was an electrical discharge wire-cutting machine (Charmilles Roboﬁl 2000). EDM was chosen because of the limited heat and force that are generated during the cutting process. The parts were trimmed using the same part referencing system as applied for the forming. The cutting location was determined by the original CAD model. Deformations occurring during the incremental forming were thus ignored.

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3 Experimental Results 3.1 Manufactured Parts Two part geometries have been selected. Figure 3 shows the geometry of a scaled version of a solar cooker as used in developing countries (Jeswiet et al., 2005). Important features towards the incremental forming process are the changing slope angle and the sharp edges, which were approximated by the forming tool radius. The second part geometry is a cranial plate, used for reconstructive skull surgery. This is a typical example of a freeform surface (Fig. 3). For each of the geometries, three diﬀerent tool path strategies were tested. 3.2 Solar Cooker Strategy 1: Single Pass Procedure A standard contouring strategy with a constant step-down was applied in this test. This is a tool path strategy normally used for ﬁnishing operations in milling. To keep the scallop height similar in the entire part, a step-down of 0.7 mm was used above rib B (Fig. 4) and 0.3 mm below that level. The overall deviations of the manufactured part compared to the CAD model range from −1.8 mm (overforming) to +5.4 mm (underforming). As can be seen in Fig. 5, there is a lot of spring back of the semi-vertical faces (+1.8 to +2.8 mm). Rib B (Fig. 4) is pulled up when the lower regions of the part were processed. A maximum deviation of +5.4 mm compared to the CAD model was reported in that area. The ribs of the shape were clearly more

Fig. 3. The solar cooker CAD model (left) and the cranial plate CAD model (right)

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Fig. 4. Side section view of the solar cooker (CAD model in dotted line, actual part in full line)

Fig. 5. Top view of the area with positive deformation (underforming)

accurate than the faces; still a deviation of +0.5 to +1.1 mm was measured in that area. As can be seen in Fig. 6, the bottom was formed too deep. Overspinning is resulting in some excess strain, which leads to a secondary bulging eﬀect. Strategy 2: Double Pass, Pre-Shaping A typical problem with the manufacturing of the solar cooker is the deformation of rib B when the lower regions of the cooker are being formed. In order to improve the accuracy of rib B and the semi-vertical faces of the cooker, a two-step strategy was applied. First the sheet was formed to an intermediate pre-shape (see Fig. 7). The geometrical model of the pre-shape is a shallower version of the ﬁnal solar cooker part, where all the edges are blended with a radius of 24 mm. After pre-shaping, the ﬁnal tool path was applied (identical to Strategy 1). Compared to Strategy 1, a signiﬁcant improvement in accuracy was obtained (Fig. 8). The overall deviations of the manufactured part compared to the CAD model range from −2.8 mm to +4.2 mm. The ribs of the shape are more accurate (+0.3 to +0.7 mm), but rib B still shows a deviation of +4.3 mm. The semi-vertical faces are also more accurate with deviations of +0.1 to +2.5 mm. From Fig. 8, it is also clear that the upper part of these faces (the

Fig. 6. Top view of the area with negative deformation (overforming)

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Fig. 7. 3D view of the preshape (left) and the ﬁnal shape (right)

Fig. 8. Top view of the area with more than 1.5 mm of positive deformation (on the left) and the area that was formed more than 1.5 mm too deep (on the right) using Strategy 2

ﬁrst 20 mm) is much more accurate than the lower part (deviations of +0.1 to +0.7 mm) Strategy 3: Double Pass, Reverse Finishing The third tested strategy is a two-step procedure, where Strategy 1 is followed by a ﬁnishing pass. This ﬁnishing pass is the reverse of the Strategy 1 tool path. This approach was chosen to be able to reform rib B and the semivertical faces, which were deformed while forming the lower regions. As can be seen from Fig. 9, the overall accuracy is ranging from −2 mm to +1 mm.

Fig. 9. Top view of the area with more than 0.6 mm of positive deformation (on the left) and the area that was formed more than 0.6 mm too deep (on the right) using Strategy 3

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Fig. 10. Bottom view of the manufactured solar cooker part

Conclusion The results for the tool path strategies described above have been summarized in Table 1. The bottom of the part (Fig. 10) still shows a signiﬁcant deviation, but the reverse strategy did minimize the errors on rib B and most of the deviations on the semi-vertical faces. Another advantage of the reverse ﬁnishing strategy is its ease of use: the tool path generation is straightforward and does not require the calculation of an intermediate shape. 3.3 Cranial Plate Medical implants are typically low batch or one of a kind products, which make them expensive to manufacture using conventional forming techniques. Table 1. Deviations for the solar cooker part (values in mm) strategy (mm)

1

2

3

average absolute deviation avg. neg. dev. avg. pos. dev. minimum maximum sigma (σ)

1.32 −0.58 1.51 −1.78 5.40 1.22

0.96 −0.79 1.07 −2.75 4.20 1.17

0.38 −0.47 0.31 −2.06 0.98 0.51

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SPIF allows for dieless straight-from-CAD manufacturing, making this an ideal production technique for medical applications (Ambrogio et al., 2005). In order to manufacture the cranial plate, the shape of the cranial plate was extended to an approximate hemisphere (Fig. 11). Special care was taken to have a continuous transition between the cranial plate and the extension (0-order continuity). When specifying accuracy requirements, distinction can be made between two geometrical regions, namely the edge of the cranial plate and the central reconstruction area. The accuracy requirements at the edge of the implant are more strict than in the centre of the plate, since the edge has to provide a good ﬁtting overlap with the remaining skull of the patient. The accuracy in the centre of the plate is more of importance for aesthetic reasons, to assure the symmetry of the skull of the patient after the reconstruction procedure. Three diﬀerent shaping strategies have been applied, each with the use of a backing plate. Strategy 1: Single Pass, Contouring The application of a standard contouring tool path provides reasonable results. Maximum deviations of −0.4 to +1.4 mm were measured. As can be seen in Fig. 12, most of the deviations are situated near the edges of the cranial plate, where the hemispherical extension is joined to the actual skull part. This is due to the same eﬀect that deformed rib B in the solar cooker part. When forming the lower regions of the skull part, the upper regions are also deformed since there is a sudden change in wall angle between the skull part and the extension. For the remainder of this paper the focus lies on the deviations of the skull part itself. The deviations of the hemispherical extension were left out since the aim was to increase the accuracy of only the functional part.

Fig. 11. The cranial plate (on the left) and the cranial plate with extension (on the right)

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Fig. 12. Top view of area with more than 0.4 mm spring back marked in gray, as obtained with a Strategy 1 toolpath

Strategy 2: Double Pass, Finishing Spiral Step To improve the accuracy of the edge of the skull part, two operations have been applied. The ﬁrst pass is identical to Strategy 1. The second pass was a spiral tool path applied only on the skull region (Fig. 13). The resulting error at the edge of the skull is almost zero (between 0 and +0.3 mm). A slightly larger error was measured at the centre (maximum error −0.9 mm), but is less critical as described before. Strategy 3: Single Pass, Compensation for Spring Back This strategy is based on the error data obtained in Sect. 3.3.1. The CADmodel was adapted with the following algorithm, to take into account the expected spring back. For each of the points of the scanned point cloud a corresponding point on the CAD model was found. The deviation vectors

Fig. 13. Area that was formed more than 0.4 mm too deep (on the left) and an enlarged version of the used spiral tool path (on the right) used in Strategy 2

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Fig. 14. Top view of the area with more than 0 mm spring back (on the left) and the area that was formed more than 0.4 mm too deep (on the right) when using Strategy 3 Table 2. Deviations for the skull part alone (values in mm) strategy (mm)

1

2

3

average absolute deviation avg. neg. dev. avg. pos. dev. minimum maximum sigma (σ)

0.293 −0.103 0.302 −0.379 1.39 0.199

0.091 −0.345 0.058 −0.909 0.293 0.138

0.109 −0.413 0.082 −0.906 0.523 0.171

going from a CAD point to a scanned point were calculated and multiplied by a factor (0.7 in our case as advised in Bambach et al., 2004). These deviation vectors were subtracted from their corresponding CAD points to obtain a new point cloud. From this point cloud a new CAD model was created. The new tool path was generated on this adapted CAD model. The results are shown in Fig. 14. Conclusion The results for the diﬀerent tool path strategies described above have been summarized in Table 2. As can be concluded, Strategy 2 provided the smallest deviations. Strategy 3 could still be improved by iteratively applying the error compensation algorithm or by choosing a diﬀerent multiplication factor.

4 Trimming Operations Unlike the solar cooker part, the cranial plate has to be removed from its hemispherical extension. Using an EDM wire cutting facility, the cranial plate was trimmed. The inﬂuence of the cutting process was examined by comparing the geometry of the work piece before with the geometry of the part after cutting. Near the edges of the cranial plate, a deviation of +0.3 mm to −0.3 mm was found.

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Fig. 15. Trimmed aluminium cone (left) and trimmed titanium cone (right), both obtained with identical toolpaths

The eﬀect of deformation due to the release of residual stresses during trimming is signiﬁcantly material dependant. As can be witnessed from Fig. 15 for a conical shape, in comparison to the results for aluminium, the obtained ﬁnal shape when processing medical grade titanium substantially deviates from the target geometry.

5 General Conclusion As could be concluded for the results obtained for the two test parts, reprocessing of the workpiece allows to signiﬁcantly improve the part accuracy. Depending on the part topology, diﬀerent strategies provide the most optimal result. The trimming of aluminium parts induces limited additional deformations. When forming other materials like, titanium or stainless steel, special care has to be taken to make sure that parts do not deform excessively due to residual stress release. An annealing step will be required to relieve the stresses inside the part.

References Ambrogio G., De Napoli L., Filice L., Gagliardi F., Muzzupappa M., “Application of Incremental Forming process for high customized medical product manufacturing”, Journal of Materials Processing Technology 162–163 (2005) 156–162 Allwood J., King G.P.F., Duﬂou J.R., “A structured search for applications of the Incremental Sheet Forming process by product segmentation”, Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Eng. Man., vol 219 (2), 2005, p239 Bambach M., Hirt G. and Ames J., “Modeling of optimisation strategies in the incremental CNC sheet metal forming process”, Numiform, 2004, p 1969–1974. Duﬂou J.R., Szekeres A. Vanherck P., “Force Measurements for Single Point Incremental Forming: An experimental Study”, Proceedings of the 11th Internat. Conference on Sheet Metal, Erlangen, 2005 Jeswiet J., Duﬂou J.R., Szekeres A., Levebre P., “Custom Manufacture of a Solar Cooker a case study”, Proceedings of the 11th Internat. Conference on Sheet Metal, Erlangen, 2005

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Jeswiet, J., Micari, F., Hirt, G., Bramley, A., Duﬂou, J., and Allwood, J., “Asymmetric Single Point Incremental Forming of Sheet Metal”, Annals of CIRP 54(2), p 623, 2005 Matsubara S., “A computer numerically controlled dieless incremental forming of a sheet metal”. Proc. of the Inst. of Mech. Engineers, part B, J Eng Manuf, 215, 7, 2001, 959–966. Park J.J., Kim Y.H., “Fundamental studies on the incremental sheet metal forming technique”, J. Mater. Process. Technol. 140 (2003), 447–453.

On the Finite Element Simulation of Thermal Phenomena in Machining Processes L. Filice1 , D. Umbrello1 , F. Micari2 and L. Settineri3 1

2

3

Department of Mechanical Engineering, University of Calabria, 87036 Rende (CS), Italy, {l.ﬁlice, d.umbrello}unical.it Department of Manufacturing, Production and Management Engineering, University of Palermo, 90128 Palermo (PA), Italy, micaridtpm.unipa.it Department of Production Systems and Economics, Polytechnic of Turin, 10129 Torino (TO), Italy, luca.settineripolito.it

Summary. Machining processes are frequently investigated by numerical simulations. Usually 2D analyses are carried out in order to reduce CPU times, considering orthogonal cutting conditions. In this way, the computational time sharply reduces and many process variables may be calculated (i.e. forces, chip morphology, shear angle, contact length). On the other hand, the analysis of thermal aspects involved in machining, for instance the temperature distribution reached in tool, still represents an open problem. Finite element codes are able to simulate a very short process time that is not suﬃcient to reach steady state conditions. Several approaches have been proposed to overcome this problem: in the paper some of them are applied and critically discussed.

Key words: temperature in machining, machining, FEM.

1 Introduction Machining processes have been assumed a growing interest in the last years, even from a purely scientiﬁc point of view, since new possibilities have been made possible by using the numerical simulation (Shaw 05, Kalpakjian 97, Trent et al., 2000, Bothroyd 89, Zorev 66). In fact, if the industry fully requires high performance tools and machines, able to allow high speed and productivity with a limited use of lubricants, which represent a threat for environment, the researchers actively investigate the cutting mechanics laying a foundation on the process understanding, which will result in a better process management and control (T¨ onshoﬀ et al., 2000, Bitans et al., 1965, Oxley et al., 1959). The most appreciate approach is based on the use of numerical codes, which implement the ﬁnite element discretization. The most utilised take into account an updated-Lagrangian formulation, based on the discretization and

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solving of equilibrium equations, and able to manage large deformation processes thanks to eﬃcient remeshing-rezoning algorithms (Filice et al., 1999, Ceretti et al., 1997, Lin et al., 1992). Anyway, despite this high potential, when the above codes are applied to machining processes, some problems appear: the high strain and strain rate imposed by the current cutting speed and the heavy thermal gradients induce several criticisms in the material behaviour modelling, chip formation, temperature evaluation. Furthermore, since a large number of elements is required for obtaining an accurate description of local variables, just a few millisecond may be simulated because the current utilised cutting speed are of the order of hundreds of meters per minute. And this is true also when a strong geometric simpliﬁcation is done, i.e. a 2D plain strain simulation is performed. Thus, a sort of rough balance between what codes can do and what codes cannot do is nowadays possible: –

forces, chip morphology, length of cutting area, wear evolution in simple cases, shear angle in orthogonal cutting conditions may be predicted by using a coupled thermo-mechanical simulation (Ay et al., 1994, Ceretti et al., 2000, Komanduri et al., 1981, Molinari et al., 2002, Altan et al., 2004, Fleischer et al., 2004, Prins 71, Filice et al., 2004, Filice et al., 2003, Umbrello et al., 2004); – wear evolution in complex geometries, correct estimation of thermal gradients and steady state values and eﬀective 3D signiﬁcant numerical simulations are nowadays not exactly possible, and the research on these topics is surely very active (Shirakashi 02, Filice et al., 2005).

In particular, when the thermal phenomena must be taken into account, for instance when the temperature distribution in the tools or other related issues are investigated, another relevant problem arises: currently fully coupled thermo-mechanical FE simulations are not able to follow the machining processes up to steady state conditions, since, in order to reduce the CPU time within reasonable limits, only few milliseconds of the process can be taken into account. Some approaches have been proposed to overcome such drawback (Altan et al., 2004, SFTC-Deform 02, Burte et al., 1990, Yen et al., 2002, Astakhov 99, Yvonnet et al., 2005): most of them are based on a preliminary thermomechanical analysis of the former milliseconds of the machining operation aimed to detect some relevant thermal data (heat ﬂux, contact temperatures on the tool), and on a subsequent pure thermal analysis of the heat transfer phenomenon in the tool up to steady state conditions. Some other researches are aimed, on the contrary, on the determination of the heat ﬂux distribution ﬂowing into the tool through an inverse approach based on the comparison of the numerical and some experimental data, the latter obtained, for instance, by means of embedded thermocouples (Yvonnet et al., 2005, Filice et al., 2004). In the latter case no thermo-mechanical analyses of the machining

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process are necessary and a pure thermal investigation is carried out with relevant advantages in terms of CPU times. In the paper some interesting remarks related with thermal aspects in machining are discussed and the comparison with a set of experimental tests permits to highlight the points of strength and weakness of the current application of the FE techniques in this ﬁeld.

2 Remarks on Thermal Phenomena in Simulation of Orthogonal Cutting In the machining process, as in any other process in which heavy deformations are imposed to the material, relevant quantity of heat is generated. What is more, in machining this phenomenon is also increased due to the friction generated heat. This situation is becoming very critical in the last years, mainly for two diﬀerent reasons: –

the tool material and machine development allow to reach very high cutting speeds, almost impossible up to few years ago. Rotation speeds of the mandrel higher than 18.000 r.p.m. and cutting speeds of several hundreds of m/min are to be considered as common values; – the use of lubricants and coolants is nowadays dissuaded (MQL investigations are increasing) due to the relevant impact on the environment (pollution) and to the heavy inﬂuence on the industrial costs (T¨ onshoﬀ et al., 2000, Altan et al., 2004). How it is well known, the generated heat is partially due to the deformation work on the shear plane (primary shear zone), according to many of the traditional models concerning chip formation theory (Kalpakjian 97, Bitans et al., 1965, Lin et al., 1992). In addition, as above introduced, a secondary shear zone is located on the rake face, along the contact length. In this area the main responsible for the heat generation is friction between the sliding material and the tool (see Fig. 1). At the conventional cutting speeds the largest part of heat is dissipated in the chip and only a low percentage ﬂows towards the tool, as it is well known from the experiments (Kalpakjian 97). According to the above discussed reasons, it is clear that two aspects have to be carefully taken into account as far as heat ﬂow prediction is concerned, namely the evaluation of the heat transfer (ﬁlm) coeﬃcient between the chip and the tool and the friction modelling. The former was experimentally computed by several authors for forging applications but, due to the not clear phenomena which occur in machining (SFTC-Deform 02, Burte et al., 1990), many researchers (Altan et al., 2004, Yen et al., 2002) attempted to tune this constant value with the aim to accelerate the convergence of the ﬁnite

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Fig. 1. Primary and secondary shear zone (Ozel et al., 2004)

element simulations towards steady state conditions, despite the very short cutting time that can be eﬀectively investigated. Friction modelling, on the other hand, requires an accurate set-up. In fact, as it is well known by the theory, along the contact length the shear stress distribution follows a complex law: a constant shear model describes the phenomenon close to the tool tip (sticking zone) while a Coulomb model properly ﬁts in the complementary area of the contact length (sliding zone). Summarising these considerations, it is possible to state that a consistent model able to describe the temperature evolution in a tool-chip system, must be based on the following issues: – –

–

a correct set-up of the thermal properties of the materials (heat capacity and conductivity); a correct set-up of the heat transfer coeﬃcients, towards the environment, by convection, but mostly between the chip and the tool (global ﬁlm coefﬁcient); an accurate model of friction along the contact area.

There is another issue to be taken into account: as mentioned machining simulation can currently investigate only a very short machining time, generally few milliseconds. This time is not suﬃcient to permit that the heat generated in the primary shear zone arrives to the chip-tool interface and aﬀects the temperature distribution in the tool. Therefore the calculated temperatures in the tool depend only on the heat generated by friction in the secondary

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shear zone and the heat transfer coeﬃcient works, more or less, as a partitioning coeﬃcient which determines the heat amounts ﬂowing into the chip and the tool respectively. According to the above considerations it is worth concluding that an eﬀective, scientiﬁcally consistent, numerical analysis of the coupled thermomechanical phenomenon is nowadays not yet possible. Simpler approaches can be developed, based on a proper selection of the heat transfer coeﬃcient or on un-coupled mechanical and thermal analyses: the research here addressed belongs to the latter class. Naturally, a set of experimental tests are necessary in order to measure the cutting temperature which will be used to validate the proposed approach.

3 Acquisition of Temperature in the Tool Cutting experiments were carried out in lathe-turning, using radial feed, as shown in Fig. 2, on an UTITA CNC lathe with a peak power of 30 kW. Orthogonal cutting conditions were simulated, using a tool width of 4 mm vs. a cutting width of 3 mm. Cutting speeds ranging from 100 to 150 m/min, as well as two diﬀerent values of feed rate (0.05 and 0.1 mm/rev) were chosen. In addition, three values of the rake angle were also investigated (0◦ , 10◦ and −10◦ ). The cutting speed was kept constant thanks to the lathe CNC. Two thermocouples with a diameter of 0.5 mm were embedded in the tool applying a force of about 50 N in order to ensure a suﬃcient adhesion (Astakhov 99). Their positions are reported in Fig. 3 and Table 1. The thermocouple positions were measured utilizing a proper optical microscope by observing a posteriori the tool section (along the hole axes) obtained by EDM technology. Temperatures as well as cutting and thrust forces were acquired by an analogical/digital converter with a sampling frequency of 600 Hz. The main issues of the experimental set-up are reported in Table 2. The physical properties both for the tool and the thermocouples are listed in Table 3. Experimental Tool

Fig. 2. The utilized experimental set-up

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Fig. 3. Tool geometry and embedded thermocouples

In particular, for sake of simplicity, a constant value of the tool thermal properties were utilized, namely the average conductivity and average heat capacity for the temperature range occurring in the investigated case according to other researchers (Ozel et al., 2000, Ozel et al., 2004). All the tests were executed in the same environment conditions with almost three repetitions for each of them. The main results of the experimental tests are shown in Table 4. In particular, Fz , Ft and T represent the mean values of the cutting and thrust forces and the measured steady–state temperature, respectively. Table 1. Geometrical position of the embedded thermocouples rake angle

a [mm]

b [mm]

c [mm]

d [mm]

e [mm]

l [mm]

0 +10 −10

0.25 0.22 0.24

0.28 – –

4 4 4

1.82 1.82 1.82

1.01 1.09 1.08

5 5 5

Table 2. Experimental data work material lubricant tool width cutting width cutting insert tool holder rake angle clearance angle thermocouple dynamometer

AISI/SAE 1045, HB 190 absent 4 mm 3 mm Cemented Carbide ISO P20 Microna NOVATEA R151 F 3225 × 20 × 4 γ = 0◦ , −10◦ , 10◦ α = 11◦ Chromel / Alumel (K) quartz three-components dynamometer

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Table 3. Physical properties thermal data of the tool thermal data of the thermocouple

thermal conductivity: 46 W/m/K heat capacity: 2.79∗ 106 J/(m3∗ K) thermal conductivity: 19 W/m/K heat capacity: 3.91∗ 106 J/(m3∗ K)

Table 4. Measured steady state temperatures, cutting and thrust forces Test 1 2 3 4 5

γ ◦

0 0◦ 0◦ −10◦ +10◦

V [m/min]

f [mm/rev]

Fz [N]

Ft [N]

Ta [◦ C]

Tb [◦ C]

100 150 100 100 100

0.1 0.1 0.05 0.1 0.1

745 715 439 740 662

542 432 567 676 478

542 567 432 485 402

480 497 395 – –

The experimental tests conﬁrm several well-assessed literature indications: tool temperature increases and cutting forces decrease at increasing the cutting speed. In addition, a feed rate increasing generates a cutting temperature increasing, even if a lower sensitivity is observed in this case.

4 The Proposed Approach: Global Film Coeﬃcient Tuning As mentioned above, when the thermal aspects must be considered, there is a large contrast between the machining time necessary to reach steady state conditions (usually about 10–20 seconds) and the process time that can be simulated avoiding unacceptable CPU times. Therefore a fully coupled thermo-mechanical analysis is not able to take into account heat diﬀusion into the tool. The machining time that can be eﬀectively simulated is of the order of 10−3 s, according to the current cutting speeds. Figure 4 reports the calculated temperatures corresponding to the thermocouple positions after few milliseconds (short sim) and 0.025sec (long sim, which requires about 20 hours CPU time) when the cutting speed, the feed rate and the rake angle were 100m/min, 0.1 mm/rev and 0◦ , respectively. It is worth pointing out that the calculated data are very far away the experimental ones (according to Fig. 3, Tables 1 and 4: Ta EXP = 542◦ C, Tb EXP = 480◦ C). Since, up to now, it is not possible to extend the simulated time, some alternatives to overcome the described problem have been proposed. All the approaches imply as last step a pure-thermal simulation during which the tool reaches steady state conditions. This type of simulation, in fact, although it is carried out in fully 3D conditions and involves thousands of elements, requires just a few hours, sometimes even minutes. Thus the focus

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TA

TB

Fig. 4. Calculated temperatures

becomes the boundary conditions, namely the temperatures along the contact length. r ﬁnite element code was used as a In this approach, SFTC-Deform-2D component of an inverse procedure able to identify the artiﬁcially modiﬁcation of the global ﬁlm coeﬃcient at the interface between chip and tool. Then, r code was utilized to model a pure thermal simulation to SFTC-Deform-3D implement a short inverse procedure in order to ﬁnd the global ﬁlm coeﬃcient, h, and to verify the results by comparing with the experimental observations. More in detail, as far as plain-strain coupled thermo-mechanical analysis are concerned the workpiece was initially meshed by means of 5000 iso-parametric quadrilateral elements while the tool, modelled as rigid, was meshed into 1000 elements. The material behaviour for AISI 1045 was modelled by using a reliable model proposed by Oxley (Oxley 89); while, as far as friction modelling is concerned, a simple model based on the constant shear hypothesis was implemented (Ceretti et al., 2000), setting m = 0.82. Consequently, a simple inverse approach was used to ﬁnd the global ﬁlm coeﬃcient, h. More in detail, the algorithm was the following: 1. Start the 2D coupled numerical simulations;

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2. Collect the obtained nodal temperatures on the contact length coming from the 2D coupled thermo-mechanical analysis; the simulation is stopped when temperatures reach the steady-state; 3. Apply the obtained nodal temperature of the 2D simulations as boundary conditions for a subsequent 3D thermal analysis (see Fig. 5); 4. Find the predicted steady–state temperature in the tool node corresponding to the thermocouple position (see Fig. 6); 5. Compute the error between the predicted and the experimental steady– state temperature (Table 4); 5.1.If the error is lower than an acceptable value then global interface ﬁlm coeﬃcient, h, is found; 5.2.Else, return to step 1 and increase or decrease global interface heat transfer coeﬃcient, h, according to the obtained error. Of course, on the contrary of the approaches referred to the forming processes, the physical signiﬁcance of the utilized h value is not so clear. It is for sake of simplicity kept constant, according to other researchers (Altan et al., 2004, Fleischer et al, 2004, Altan et al., 2004), but its validity is only measured on the base of the resulting temperature in some track points. As shown in the next Fig. 7, at increasing h steady state conditions are reached quickly, but remarkably diﬀerent temperature histories are calculated. For lower h values, heat dissipated due to friction remains entrapped into the tool and determines a relevant temperature increasing; in turn, for higher h values, heat is transmitted to the chip and it is mostly evacuated. According to the above considerations the proposed inverse approach was applied. It is worth pointing out that a value of h close to 1000 kW/m2 K permitted a satisfactory agreement between the numerical data and the experimental evidences all over the investigated cases, as shown in the next Table 5.

Fig. 5. 3D model and the initial boundary thermal conditions

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Fig. 6. Temperature distributions for h = 1000 kW/m2 K

1200 temperature [°C]

272

h=0 h=100 h=1000 h=10000

1000 800 600 400 200 0 0

0.001

0.002 0.003 0.004 simulation time [sec]

0.005

0.006

Fig. 7. Temperature histories as a function of h [kW/m2 K]

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Table 5. Measured and calculated temperatures at the closest thermocouple position

TEXP TFEM Err%

100 m/min 0.1 mm/rev γ = 0◦

100 m/min 0.05 mm/rev γ = 0◦

150 m/min 0.1 mm/rev γ = 0◦

100 m/min 0.1 mm/rev γ = −10◦

100 m/min 0.1 mm/rev γ = 10◦

542 560 3,3%

432 468 8,3%

567 603 6,3%

485 491 1,0%

402 430 6,9%

5 Discussion of the Results The procedure described in this approach was validated by means of a wide experimental campaign. Despite the satisfactory result, a strategic issue has to be clariﬁed: is it meaningful that a measurable parameter (h) which may be take into account, in a macroscopic point of view, the complex phenomena that occur on the rake face, is used as optimization variable? In other words, although the choice of a large h value permits to determine eﬀective contact temperature values, which, in turn, allows a good prediction of the temperature distribution inside the tool, such procedure appears more a trick than a scientiﬁcally consistent methodology. In fact, more complex phenomena occur on the rake face do the high relative velocity between chip and tool and the high values of temperature and stress. These phenomena should be properly modeled or, at least, the global heat coeﬃcient has to take into account what actually happens in such zone. In addition, other relevant considerations have to be drawn. It is well known that in machining energy is mainly dissipated due to plastic deformation in the primary shear zone and to friction at the chip-tool interface (secondary shear zone). Actually, since the thermo-mechanical analysis is limited to the ﬁrst few milliseconds of the process, the above contact temperatures do not depend on the energy dissipated in the primary shear zone, but only on the energy term due to friction. The last consideration is based on the results of a couple of numerical simulation where, in one case, the percentage of plastic work converted into heat is set equalto to zero. Table 6 reports the temperatures in the contact nodes (i.e. along the contact length) starting form the tool tip: the last column shows that the temperatures are substantially not aﬀected by the mechanical work due to the very short simulation time which does not permit to consider heat diﬀusion. In the Table 6 “m” is the friction factor and “e” the quota of mechanical work converted into heat. Thus, if the friction plays the most relevant role, a very accurate modelling is required. For this reason a model based on the stress distribution, proposed by Zorev (Zorev 66), has been implemented. The model well represents the existence of a sticking zone, close to the tool tip, and a sliding zone up to the end of the contact region.

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Advanced Methods in Material Forming Table 6. Contact nodal temperatures m = 0.82 e = 90% node 1 2 3 4 5 6 7 8

◦

T[ C] 953 984 945 521 308 165 111 58

m = 0.82 e = 0% ◦

T[ C] 834 776 752 689 565 281 148 70

m = 0 e = 90% T[◦ C] 91 94 90 81 73 61 33 23

With the above friction modellization a new set of numerical simulation has been developed, varying the global ﬁlm coeﬃcient “h”. In this way an interesting comparison with the constant shear friction model become possible (Figs. 8, 9 and 10): it is possible to verify that the temperature in the tool are lower respect to the case of constant shear, even if the cutting force are the same.

Fig. 8. Temperature distribution (h = 100 kW/m2 K)

Finite Element Simulation of Thermal Phenomena

Fig. 9. Temperature distribution (h = 1000 kW/m2 K)

Fig. 10. Temperature distribution (h = 10000 kW/m2 K)

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The new conditions result in a completely diﬀerent temperature distribution if a given “h” value is ﬁxed. In particular the temperature in the tool is lower than the chip one, thus the heat ﬂow tends to move through the tool.

6 Conclusion As a conclusion, it can be state that the simulation of orthogonal cutting process is well assessed, up to now, if some variables like the forces and the chip morphology are considered. The thermal analysis, on the other hand, reveals several problems. In the paper it is shown that a possible way to obtain reliable results is the arbitrary modify of the global coeﬃcient “h”. But it is also shown that this value is not physically consistent since just changing the friction model, implementing a more accurate one, its value has to be tuned in order to have a good agreement with the experiments. For instance, to have the same temperature proﬁle in the track point, there is about a factor ten between the global ﬁlm coeﬃcient utilized when the constant shear model or the Zorev one are utilized. In the opinion of the authors the next research step is the development of a reliable model which accurately describes the heat exchange phenomena at the tool-chip interface. This step is strongly required if new consistent extensions of the numerical simulation have to be done (i.e. wear prediction, 3D geometries analysis).

References Altan, T., Yen, Y.C., Sohner, J., Lilly, B., “Estimation of Tool Wear in Orthogonal Cutting using the Finite Element Analysis”, Journal of Materials Processing Technology, Vol. 146, 2004, p. 82–91. Altan, T., Yen, Y.C., Jain, A., “A ﬁnite element analysis of orthogonal machining using diﬀerent tool edge geometries”, Journal of Materials Processing Technology. Vol. 146, 2004, p. 72–81. Astakhov, V.P., Metal Cutting Mechanics, CRC Press, 1999. Ay, H., Yang, W.J., Yang, J.A., “Dynamics of Cutting Tool Temperatures During Cutting Process”, Exp. Heat Transfer, Vol. 7, 1994, p. 203–216. Bitans, K., Brown, R.H., “An Investigation of the Deformation in Orthogonal Cutting”, Int. J. Mach. Tool Des. Res., Vol. 5, 1965, p. 155–165 Boothroyd, G., Fundamentals of Machining and Machine Tools, Marcel Dekker – 2nd ed., 1989. Burte, P.R., Im, Y.T., Altan, T., Semiatin, S.L., “Measurement and Analysis of Heat Transfer and Friction During Hot Forging”, ASME Journal of Engineering for Industry. 1990, p. 332–346. Ceretti, E., Taupin, E., Altan, T., “Simulation of Metal Flow and Fracture Applications in Orthogonal Cutting, Blanking and Cold Extrusion”, Annals of the CIRP, Vol. 46/1, 1997, p. 187–190.

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Ceretti, E., Filice, L., Micari, F., “Basic Aspects and Modeling of Friction in Cutting”, Proceedings of the CIRP International Workshop on Friction and Flow Stress in Cutting and Forming, 2000, p. 73–81. r Deform-User Manual. SFTC-Deform. 2002, Columbus (OH), USA. Filice, L., Micari, F., “Analysis of the relevance of some simulation issues on the eﬀectiveness of orthogonal cutting numerical modelling”, Proceedings. of 2nd CIRP International Workshop on Modeling of Machining Operation, Paris, France, 1999, p. 270–282. Filice, L., Settineri, L., Umbrello, D., Calzavarini, R., Micari, F., “Finite element analysis of the thermo – mechanical behaviour of coated tool in mild steel machining”, Proceeding of the 4th Int. Conference THE coatings in Manufacturing Engineering, Erlangen, Germany, 2004, p. 299–307. Filice, L., Micari, F., Pagnotta, L., Umbrello, D, “Pressure Distribution on the Tool in Cutting: Prediction and Measurement”, International Journal of Forming Processes, Vol. 6, 2003, p. 327–341. Filice, L., Umbrello, D., Micari, F., Settineri, L., ”A simple model for predicting the thermal ﬂow on the tool in orthogonal cutting process”, Proceedings of the 8th CIRP International Workshop on Modeling of Machining Operations, Chemnitz, Germany, 2005. p. 191–197. Fleischer, J., Schmidt, J., Xie, L.J., Schmidt, C., Biesinger, F., “2D Tool Wear Estimation using Finite Element Method”, Proceedings of the 7th CIRP International Workshop on Modeling of Machining Operations, Cluny, France, 2004, p. 82–91. Kalpakjian, S., Manufacturing Processes for Engineering Materials, Addison – Wesley – 3rd ed., 1997. Komanduri, R., Brown, R.H., “On the Mechanics of Chip Segmentation in Machining”, ASME Journal of Engineering for Industry, Vol. 103, 1981, p. 33–51. Lin, Z.C., Lin, S.Y., “A Coupled Finite Element Model of Thermo-Plastic Large Deformation for Orthogonal Cutting”, Journal of Engineering Materials and Technology, Vol. 114, 1992, p. 218–226. Molinari, A., Nouari, M., “Modeling of Tool Wear by Diﬀusion in Metal Cutting”, Wear, Vol. 252, 2002, p. 135–149. Oxley, P.L.B., Palmer W.B., “Mechanics of Orthogonal Machining”. Proc. Inst. Mech. Eng. 1959, p. 623–654. Oxley, P.L.B., “Mechanics of Machining, an Analytical Approach to Assessing Machinability”, Halsted Pr., 1989. Ozel, T., Altan, T., “Determination of workpiece ﬂow stress and friction at the chiptool contact for high-speed cutting”, International Journal of Machine Tools and Manufacture, Vol. 40, 2000, p. 133–152. Ozel, T., Zeren, E., “Determination of work material ﬂow stress and friction for FEA of machining using orthogonal cutting test”, Journal of Material Processing Technology, Vol. 153–154, 2004, 1019–1025. Prins, O.D., “The Inﬂuence of Wear on the Temperature Distribution at the Rake Face”, Annals of the CIRP, Vol. XVIV, 1971, p. 579–584. Shatla, M., Kerk, C., Altan, T., “Process modelling in machining. Part 1: determination of ﬂow stress data”, International Journal of Machine Tools and Manufacture, Vol. 41 2001, p. 1511–1534.

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Shatla, M., Kerk, C., Altan, T., “Process modelling in machining. Part 2: validation and applications of the determined ﬂow stress data”, International Journal of Machine Tools and Manufacture, Vol. 41, 2001, p. 1659–1680. Shaw, M.C., Metal Cutting Principles, Oxford University Press. – 2nd ed., 2005. Shirakashi, T., “Some Diﬃculties on Prediction of Tool Life and Machined Surface Quality Through FEM”, Proceedings of the 5th International ESAFORM Conference on Material Forming, Krakow, Poland, 2002, p. 583–586. T¨ onshoﬀ, H. K., Arendt, C., Ben Amor, R., “Cutting of Hardened Steel”, Annals of the CIRP, Vol. 49/2, 2000, p. 547–566. Trent, E.M., Wright, P. K., Metal Cutting; Butterworth – Heinemann – 4th ed., 2000. Umbrello, D., Hua, J., Shivpuri, R., “Hardness Based Flow stress for Numerical Modeling of Hard Machining AISI 52100 Bearing Steel”, Material Science and Engineering – A, Vol. 374, 2004, p. 90–100. Yen, Y.C., S¨ ohner, J., Weule, J., Schmidt, J., Altan, T., “Estimation of Tool Wear in Orthogonal Cutting using FEM Simulation”, Machining Science Technology, Vol. 6/3, 2002, p. 467–486. Yvonnet, J., Umbrello, D., Chinesta, F., Micari, F., “A Simple Inverse Procedure to Determine Heat Flow on the Tool in Orthogonal Cutting”, In press International Journal of Machine Tools and Manufacture, 2005. Zorev, N.N., Metal Cutting Mechanics, Pergamon Press, 1966.

Numerical Simulation of Wire Coating Pseudoplastic and Viscoplastic Fluids E. Mitsoulis and P. Kotsos School of Mining Engineering and Metallurgy, National Technical University of Athens, Zografou, 157 80, Athens, Greece, [email protected]

Summary. Numerical simulation of the wire-coating process is undertaken for nonNewtonian pseudoplastic and viscoplastic ﬂuids. The Herschel-Bulkley model of viscoplasticity is used, which reduces with appropriate modiﬁcations to the Bingham, power-law and Newtonian models. The analysis is based both on the Lubrication Approximation Theory (LAT), which regards locally a fully developed shear ﬂow, and on a two-dimensional axisymmetric Finite Element Method (FEM). For a given die design the results give distributions of important variables, such as pressure, shear stresses along the die walls and the wire, and the wire tension due to the shearing forces of the ﬂuids on the moving wire. These results are obtained from a full parametric study of the dimensionless power-law index (in the case of pseudoplasticity) and the dimensionless yield stress or Bingham number (in the case of viscoplasticity). Increasing the power-law index or the Bingham number leads to an increase in dimensionless pressure and stresses. In the case of viscoplastic ﬂuids, LAT predicts interesting yielded/unyielded zones, which are however erroneous, as a consequence of using the lubrication approximation. The full 2-D analysis based on FEM shows that such zones exist only after the die exit, where the coating ﬂuid moves on the wire as a rigid body.

Key words: wire coating, pseudoplastic – viscoplastic ﬂuids, die design, simulation.

1 Introduction One of the important forming processes, especially in the plastics industry, is the wire-coating process, in which, a moving bare wire passes through an extruder die head and is coated by a polymer melt supplied under pressure from the extruder (see Fig. 1) (Mitsoulis, 1986a). The process is especially critical for the coating of very thin telephone wires (with diameters of 0.6 mm) for a protective sheathing at high speeds (> 10 m/s). Special-strength polyethylene melts are used (e.g. LDPE). The coating process takes place in minute volumes, and only through numerical simulation can one study the ﬂow behaviour

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Fig. 1. Schematic representation of the wire-coating process with a pressure die (Mitsoulis, 1986a)

inside wire-coating dies (Mitsoulis, 1986b). On the other hand, the industrial design of wire-coating dies is based on empiricism (Haas and Skewis, 1974). It is of importance to study in depth the process and better understand what are the inﬂuencing parameters, which control the coating thickness, the production of a smooth coating, and the increase in production speed without aﬀecting the quality of the coated wire. A numerical simulation approach is therefore necessary in order to improve the eﬀectiveness of such a fast coating process, which consumes a good bulk of polymer resins in the plastics industry. Previous work on the subject has addressed many of these aspects (see review article by Mitsoulis, 1988). More recent work has dealt with issues of viscoelasticity (Mutlu et al., 1998a; 1998b) and slip eﬀects (Ngamaramvaranggul and Webster, 2000). It appears that no work has been done for viscoplastic ﬂuids exhibiting a yield stress. The rheological properties of the coating material play an important role in the ﬂow inside wire-coating dies and inﬂuence the properties of the coated wire. Since many materials used in wire coating are frequently non-Newtonian, exhibiting either pseudoplastic (shear-thinning or -thickening) or viscoplastic (presence of a yield stress) behaviour (see, e.g. Bird et al., 1983), a model that covers both cases should be employed. The Herschel–Bulkley model has the advantage of reducing – with an appropriate choice of parameters – to the Bingham, power-law or Newtonian model. In simple shear ﬂow it takes the form (Bird et al., 1983; see also Fig. 2): n−1

τ = m |γ| ˙ γ˙ = 0,

γ˙ ± τy ,

for |τ | > τy , for |τ | ≤ τy ,

(1a) (1b)

where τ is the shear stress, γ˙ is the shear rate, τy is the yield stress, m is the consistency index, and n is the power-law index. Note that when n = 1 and m = μ (a constant), the Herschel–Bulkley model reduces to the Bingham

Numerical Simulation

Shear Stress, τ

Viscoplastic

0.5 ,n= 1 g H–B n i n= n am, -thin r h a g e n g Sh Bi enin =2 hick t n , r B a H– She

τy yield stress

281

r

Shea

.5 n=0 g P–L, =1 n ian, n o t ng New L, n = 2 keni -thic r P– a e Sh

nin -thin

Pseudoplastic 0

. Shear Rate, γ

Fig. 2. Shear stress vs. shear rate for various pseudoplastic and viscoplastic ﬂuids

model. When τy = 0, the power-law model is recovered, and when τy = 0 and n = 1, the Newtonian model is obtained. It should be noted that in viscoplastic models, when the shear stress τ falls below τy , a solid structure is formed (unyielded). Also, in viscoplasticity, the well-known Bingham number is deﬁned as: n τy H 0 (2) Bn = m Vw where Vw is the wire speed and H0 is a characteristic length, here taken as the wire radius Rw . For purely viscous ﬂuids, Bn = 0. However, at the other extreme of unyielded solids, Bn → ∞. It is, therefore, the purpose of the present work to study the wire-coating process for a given design used in the industry, and provide reference results for pseudoplastic and viscoplastic ﬂuids. A full parametric study will be carried out for the power-law index, n, and the Bingham number, Bn, and the results will be compared between an easy-to-use quasi-one-dimensional analysis based on the Lubrication Approximation Theory (LAT) and a full two-dimensional analysis based on the Finite Element Method (FEM). Engineering quantities of practical importance, such as the pressure drop, the wire tension, and the maximum stresses on the wire and the die wall, will also be given.

2 Mathematical Modelling 2.1 Conservation and Constitutive Equations The wire-coating process for pseudoplastic and viscoplastic ﬂuids, as they pass through a die head under pressure, is described by the general conservation equations of mass and momentum, i.e.

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Advanced Methods in Material Forming

∇·ν =0

(3)

0 = −∇p + ∇ · τ

(4)

where ν = velocity, p = pressure, τ = extra stress. We have assumed that the material is incompressible and ﬂows under creeping conditions (Re ≈ 0), which is a valid approximation in polymer processing for very viscous ﬂuids. The stress tensor τ is related to the rate-of-strain tensor γ˙ by the generalized Newtonian constitutive equation: τ = η(γ) ˙ γ˙

(5)

where η is the apparent viscosity, which depends on |γ|, ˙ the magnitude of the rate-of-strain tensor γ˙ = ∇ν + ∇ν T , given by: |γ| ˙ =

12

1 {γ˙ : γ} ˙ 2

=

1 2 2 2 2 γ˙ zz + γ˙ rr + γ˙ θθ + 2γ˙ rz 2

12

The components of γ˙ in a 2-D axisymmetric domain (r, z, θ) are: ∂vr vr ∂vz ∂vz ∂vr , γ˙ θθ = 2 , γ˙ zr = , γ˙ rr = 2 + γ˙ zz = 2 ∂r ∂z ∂r r ∂z

(6)

(7)

The apparent viscosity η for the Herschel–Bulkley model is given by (Papanastasiou, 1987): ˙ τy [1 − exp (−k |γ|)] n−1 η = m |γ| ˙ , (8) + |γ| ˙ where k is a stress growth exponent (a regularization parameter), which makes the model valid in both yielded and unyielded zones. A large value of k(k > 200 s) mimics well the ideal Herschel–Bulkley model in the limit of very small shear rates. To track down yielded/unyielded zones, we shall employ the criterion that the material ﬂows (yields) only when the magnitude of the extra stress tensor |τ | exceeds the yield stress τy , i.e. $ 1/2 1 1 = = yielded : |τ | = IIτ = {τ : τ } > τy , (9a) 2 2 unyielded :

|τ | ≤τy .

(9b)

2.2 Lubrication Approximation Theory (LAT) The lubrication approximation theory (LAT) considers locally fully developed shear ﬂow between the walls (see Fig. 3, Mitsoulis, 1986a). The conservation of momentum equation then gives dp 1 d = (rτrz ), dz r dr

(10)

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283

Fig. 3. Notation for ﬂow analysis in a wire-coating die according to LAT (Mitsoulis, 1986a)

where τrz = τ is the shear stress in the radial direction. The shear stress is given by the Herschel–Bulkley model in the present work (1). We integrate (10) and apply the boundary conditions as follows: (a) in the annular region, before the ﬂuid meets the wire, u =0 at

r = ri (z)

(11a)

u =0 at

r = r0 (z)

(11b)

(b) in the die region, where ﬂuid and wire travel together, u =Vw

at r = ri = Rw u =0 at r = r0 (z)

(11c) (11d)

Integration of the velocity proﬁle gives the volumetric ﬂow rate Q according to: r" o (z) u(r, z)rdr. (12) Q = 2π ri (z)

In wire coating the following dimensionless variables and parameters are introduced: dP LH0n ∗ cH0n−1 ∗ r z u H0 Q , Ha = , z ∗ = , u∗ = ,ε = ,c = ,Q = n n H0 L Vw L dz mVw mVw Vw H02 (13) where Ha is the Hagen number, L is the length of the die, ε is the aspect ratio of the die, and the rest of the symbols are deﬁned in Fig. 3, with H0 = Rw . After the appropriate manipulations, the following dimensionless momentum equation is obtained: . % ∗ %n−1 / ∗ % du % Bn c∗ du εr∗ % du∗ % + %% ∗ %% + =(Ha) , |τ ∗ | > Bn (14a) ∗ ∗ % ∗% dr dr 2 r dr r∗ =

du∗ = 0. dr∗

|τ ∗ | ≤ Bn

(14b)

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The dimensionless boundary conditions are: (a) in the annular region, before the ﬂuid meets the wire, u∗ (r∗ = ri∗ ) =0 u∗ (r∗ = ro∗ ) =0

(15a) (15b)

(b) in the die region, where ﬂuid and wire travel together, u∗ (r∗ = ri∗ ) =1 u∗ (r∗ = ro∗ ) =0

(15c) (15d)

The conservation of mass becomes: ∗

∗

"ro

Q = 2π

u∗ r∗ dr∗

(16)

ri∗

2.3 Finite Element Method (FEM) – Boundary Conditions The full two-dimensional problem is illustrated in Fig. 4. Due to axisymmetry a two-dimensional analysis is applied. The following boundary conditions are imposed: (a) No-slip conditions along the solid walls. Along the wire boundary, the melt moves with the wire speed Vw . (b) Kinematic boundary condition is imposed on the free surface after the die exit, and its shape is determined by the solution. (c) At entry, the ﬂow rate is given, which corresponds to a desired exit thickness for a given wire speed.

Fig. 4. Notation for ﬂow analysis in a wire-coating die according to FEM (Mitsoulis, 1986b)

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285

3 Method of Solution 3.1 Lubrication Approximation Theory (LAT) The problem unknowns are u∗ , c∗ and the Hagen number (Ha), which includes the unknown pressure gradient dP/dz. For the solution of the equations, we discretize r∗ (ri∗ < r∗ < ro∗ ) in N equally-spaced nodes, having a distance h between them. By using backward ﬁnite diﬀerences for the shear rate, for each node (i) we have: γ˙ ∗ (i) =

du∗ u∗ (i) − u∗ (i − 1) (i) = ∗ dr h

(17)

Thus, the governing (14a) will give the following residuals for each node (i): ⎤ ⎡ % % ∗ % u (i) − u∗ (i − 1) %n−1 ⎥ ⎢ Bn % ⎥ % + %% % ∗ R(i) = ⎢ ∗ % ⎦ ⎣ % u (i) − u (i − 1) % h % % % % h u∗ (i) − u∗ (i − 1) Ha ∗ c∗ × − εr (i) − ∗ (18) h 2 r (i) for i = 2, . . . , N and for |γ| ˙ = 0. This discretization gives N −1 residuals, since we cannot apply it for node 1, for which we apply the boundary condition of no slip (15a) for annular region or (15c) for die region: annular region die region

R(1) = u∗ (1) − 0 R(1) = u∗ (1) − 1

(19a) (19c)

Similarly, on the upper die wall, we apply the boundary condition of no slip (15b) for annular region or (15d) for die region: annular region die region

R(N + 1 ) =u∗ (N ) − 0 ∗

R(N + 1 ) =u (N ) − 0

(19b) (19d)

In the case where the shear rate becomes zero, i.e. in the unyielded zone of plug ﬂow where the velocity is constant, the residuals of the governing (18) are not valid, because |γ| ˙ = 0 and (18) goes to inﬁnity. In this case, the following residual is valid: u∗ (i) − u∗ (i − 1) (20) R(i) = h for i = 2, . . . , N and when |γ| ˙ = 0. The nodes where this equation is valid belong to the unyielded zone, and thus the location of the yield line y0 is found, which separates the yielded from the unyielded zone. The integral for the conservation of mass (16) is calculated using Simpson’s rule for the nodes in the r∗ -direction. Thus, (16) can be written as:

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Advanced Methods in Material Forming ∗

"ro

∗

Q = 2π

u∗ r∗ dr∗ = 2π

N −2 i=1

ri∗

u∗ (i)r(i)+4u∗ (i + 1)r(i + 1)+u∗ (i + 2)r(i + 2) h 3 (21)

Thus, for the conservation of mass we have the following residual: ∗

R(N + 2) = Q∗ − 2π

"ro

u∗ r∗ dr∗ = Q∗ − 2π

N −1 i=1

ri∗

u∗ (i) + u∗ (i + 1) h 2

(22)

Finally, the nonlinear algebraic system of equations has N + 2 equations and N + 2 unknowns (velocity u∗ at N nodes, the constant c∗ , and the parameter Ha). The system is solved with the well-known iterative Newton–Raphson method: X (k+1) = X (k) − J −1 (X (k) )R(X (k) ) (23) where X (k) is the solution at the k-iteration, R(X (k) ) is the residual of the solution at the k-iteration and J(X (k) ) is the Jacobian matrix of the system. Finally, the solution of the equations takes place at each cut in the z ∗ direction, for which there are diﬀerent radii r∗ between the lower wall (wire) and upper (die) wall. Thus, the space in z ∗ is discretized in M nodes, and the problem is solved M times. At every cut, the solution provides the parameter Ha, which includes the pressure gradient dP/dz. The overall pressure drop in the system is given by the following integral using Simpson’s rule: ΔP = L

"1 0

M −2 dP ∗ {dP/dz(j) + 4 dP /dz(j + 1 ) + dP/dz(j + 2 )} ∗ dz = δz dz 3 j=1

(24) The pressure gradient at each node (j) in the z ∗ -direction is calculated from the Hagen number Ha according to: dP mVwn (Ha) = dz LH0n

(25)

The shear stresses at the wall and the wire are calculated a posteriori from the solution of the shear rates there, according to the Herschel–Bulkley model (1). The wire tension is calculated from the stresses on the wire according to the integral: "L F = −2πRw τw dz (26) l

where Rw is the wire radius and τw is the shear stress at the wire. The number of nodes in the z-direction was M = 903 and in the r-direction was N = 301, i.e. the total number of nodes was M ×N = 903×301 = 271, 803!

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287

3.2 Finite Element Method (FEM) The mass and momentum equations are integrated using the Galerkin ﬁnite element method with the velocities vr (or u) and vz (or w) and pressure p as the primary variables (u − w − p formulation) (Mitsoulis et al., 1988). The discretization and details of the FEM are the same as in our previous work. The iterative scheme used is direct substitution (Picard method). The free surface is found in a decoupled manner, by integrating the velocities to construct a streamline (Mitsoulis, 1986b). Convergence is considered satisfactory when changes in the norm-of-the-error and the norm-of-the-residuals are both < 10−3 . The maximum changes in the free surface location were always < 10−5 .

4 Results and Discussion The die design on which all computations are performed is shown in Fig. 5. It corresponds to a good die design (lack of recirculation or excessive stress build-up) used by Du Pont Company for coating thin telephone wires with LDPE polymer melts (Haas and Skewis, 1974; Mitsoulis et al., 1988). The design is designated as GS = 7.9Rw , meaning that it is based on an optimum “gum space” GS of 7.9 wire radii, as found out empirically by Haas and Skewis (1974).

Fig. 5. Die design (GS = 7 .9 Rw ) used by the DuPont company for coating thin telephone wires with LDPE melts (Haas and Skewis, 1974)

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4.1 Pseudoplastic Fluids First the calculations are pursued for pseudoplastic ﬂuids, both shear-thinning (0.1 ≤ n < 1) and shear-thickening (1 < n ≤ 1.3). Newtonian ﬂuids correspond to n = 1. Results are given for the pressure drop, the wire tension, and the shear stresses along the die walls and the wire. Figure 6 shows the dimensionless pressure distribution along the annular and die regions as a function of the power-law index, n. Shear-thinning ﬂuids, representing polymer melts, are those for which 0 < n < 1. As n increases, so does P ∗ in the system. We observe that the pressure distributions go through a maximum with a positive slope in the die region for n > 0.6, which means that for these cases there is adverse recirculation. For n ≤ 0.6, the pressure drop decreases monotonically, there is no recirculation, and thus this die design is appropriate for polymer melts, which usually have n < 0.6. This was also corroborated by the fully 2-D results obtained from the FEM for a Low-Density PolyEthylene (LDPE) melt with n = 0.44 (Mitsoulis et al., 1988, their Fig. 16). The corresponding results for the wire tension distribution are shown in Fig. 7 and have been obtained by integrating the shear stresses along the wire in the die region, starting from the contact point of the ﬂuid with the wire. The wire tension goes through a maximum for all n values. Lower n values give higher F ∗ values. The maximum is also shifted towards the exit as n decreases. These results are important in calculating the extra tension that the wire can sustain on top of the haul-oﬀ tension, so that the wire does not break from the shearing action of the ﬂuid. Figure 8 shows the corresponding shear stress distributions along the wire and the die wall as a function of n. As with P ∗ , higher n values lead to higher

Fig. 6. Pressure distribution in wire coating of pseudoplastic ﬂuids obeying the power-law model according to LAT (Du Pont die design of Fig. 5)

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Fig. 7. Wire tension distribution in wire coating of pseudoplastic ﬂuids obeying the power-law model according to LAT (Du Pont die design of Fig. 5)

Fig. 8. Wall shear stress distribution in wire coating of pseudoplastic ﬂuids obeying the power-law model according to LAT (Du Pont die design of Fig. 5)

τ ∗ values. In all cases, the shear stress at the wire goes through zero and changes sign, thus giving upon integration the maxima of the wire tension in the previous ﬁgures. The results for the shear stresses are important as they are related to the critical shear stresses before the onset of shark-skin or melt fracture that occurs with polymeric materials under high stress conditions (Kazatchkov et al., 2000).

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Fig. 9. Pressure distribution in wire coating of pseudoplastic ﬂuids obeying the power-law model according to LAT and FEM (Du Pont die design of Fig. 5)

It is interesting to see the diﬀerences between the 1-D approach (LAT) and the fully 2-D approach (FEM). A comparison is presented in Fig. 9 for the pressure distribution and select n values. The LAT results are underestimates, as expected, due to not taking into account the radial component of the momentum equation. In particular, for the Newtonian ﬂuid (n = 1), LAT gives an overall pressure drop ΔP ∗ = 41.8, while FEM gives ΔP ∗ = 44.2, a maximum discrepancy of 5.4%. Similar results are found for all n values (7% for n = 0.5 and 10% for n = 0.1). The discrepancies for integrated quantities, such as the wire tension, are even smaller. These diﬀerences are not very big and are acceptable for engineering calculations, meaning that the wire-coating process is essentially a lubrication process, where most of the drastic changes occur in a shear type of ﬂow. 4.2 Viscoplastic Fluids Distributions of Variables Calculations were carried out for the whole range of Bn values, i.e. 0 ≤ Bn < ∞. Sample results are given for the Bingham model [n = 1 in (1)] in Fig. 10 for the pressure, wire tension, and shear stresses. We observe that as Bn increases and the material becomes more solid-like, the pressure drop increases substantially and so do the stresses in absolute values (cf. Bn = 0.1 with Bn = 10). The latter has a big inﬂuence on the wire tension, which gives highly negative values for Bn = 10 (see Fig. 10b). This is obviously an impossible

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Fig. 10. Distributions of variables for diﬀerent Bingham numbers in wire coating of viscoplastic ﬂuids obeying the Bingham model ((1) with n = 1): (a) pressure drop; (b) wire tension; (c) wall shear stresses (Du Pont die design of Fig. 5)

situation for this die design, as it shows that very high forces must be exerted at the haul-oﬀ point to drag the material out of the die. Results with the Herschel–Bulkley model (1) are given in Fig. 11 for the maxima of the pressure drops and the wall shear stresses in the system. As the Bn number increases, so do these quantities, which increase in an exponential manner. The eﬀect of n (eﬀect of pseudoplasticity within viscoplasticity) is only visible for Bn < 10. For these cases the viscous and plastic eﬀects are still competing and none of the two has taken over the other. For Bn > 10, the plastic (yield stress) eﬀects have taken over any viscous eﬀects and the material behaves mostly as a solid plastic. Hence all curves for all n converge to a single one. Yielded/Unyielded Zones Because of the yield stress, viscoplastic ﬂuids have the characteristics of both viscous ﬂuids and plastic solids. A yield line y0 separates the two zones, called

(a)

(b)

Fig. 11. Maximum values for: (a) pressure drop; (b) wall shear stress for diﬀerent Bingham numbers in wire coating of viscoplastic ﬂuids obeying the Herschel–Bulkley model (1)

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yielded and unyielded, respectively. Several cases of viscoplastic ﬂows have been analyzed in the literature and show explicitly these zones, e.g. ﬂow through contractions and extrusion dies (Mitsoulis et al., 1993). Here, a complete qualitative as well as quantitative picture is given, based on a thorough parametric study. With regard to Fig. 12, we observe that for a very low Bn number (Bn = 0.01), there are unyielded zones (shaded black) in the slow ﬂow of the annular region, but the die region is still fully yielded (viscous ﬂow). As Bn increases (Bn = 1), the unyielded zones become much bigger occupying most of the annular region, while the ﬁrst signs of unyielded zones appear in the die region, where the shear stresses pass through zero. A further increase in Bn (Bn = 10) leads to a further increase in unyielded zones, occupying now totally the annular region and a large part of the die region near the die walls, where the ﬂow is slower than next to the moving wire. 4.3 Critique on the Lubrication Approximation The key assumption for obtaining the above results has been the lubrication approximation theory (LAT), which assumes locally fully developed ﬂow and reduces the conservation equations by using only the axial velocity component. This assumption was shown earlier to give good results for the pressure distribution in wire coating of power-law ﬂuids and hence for all integrated quantities resulting from that. It is therefore reasonable to assume that it does also well for viscoplastic ﬂuids, such as the ones used in the present work. However, a closer look at the physics of the viscoplastic problem, and in particular the interesting yielded/unyielded zones shown in Fig. 12, reveals that these cannot be so and that the LAT results for the unyielded zones are contentious. The shaded areas cannot be rigid plugs, since the speeds at entry and exit are diﬀerent, and there is no chance that a constant speed occurs in the central area of the annular or die region. This was borne out by our FEM calculations of the full 2-D problem, which showed the continuing parabolic-like development of the central area velocity proﬁle and the lack of

Fig. 12. Yielded (black)/unyielded zones in wire coating of viscoplastic ﬂuids obeying the Bingham model according to LAT

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Fig. 13. Yielded (black)/unyielded zones in wire coating of viscoplastic ﬂuids obeying the Bingham model according to FEM

any yielded/unyielded zones. Figure 13 shows just that! It is only after the die exit where unyielded zones appear. There a plug velocity proﬁle exists, due to the movement of the coating ﬂuid as a solid plug on the moving wire. An increase of the Bn number brings the solid plug closer to the die exit (towards z = 0). However, the pressure distribution and the ensuing operating variables were within 10% (in the range 0 ≤ Bn ≤ 1000) of the values found by LAT, as was the case for pseudoplastic ﬂuids (see above). It is therefore obvious that the introduction of LAT in lubrication ﬂows with viscoplastic ﬂuids is not valid for such quantities as the yielded/unyielded zones, since it leads to a “paradox”, as pointed out by Lipscomb and Denn (1984). A 2-D analysis is therefore essential in obtaining the correct zones. However, it does not change drastically the other results shown here, especially the pressure distribution and integrated quantities, since wire coating is primarily a lubrication ﬂow. Our FEM 2-D calculations conﬁrmed this.

5 Conclusions Numerical simulations have been undertaken for the wire-coating process of pseudoplastic and viscoplastic ﬂuids in a typical industrial die design. The rheology of the materials has been modelled with the Herschel–Bulkley constitutive equation, which encompasses Bingham, power-law and Newtonian models. First, the lubrication theory was used for each ease and straightforward implementation, and then a full 2-D analysis was undertaken based on the FEM. Parametric studies were undertaken for the power-law index n (pseudoplasticity) and the Bingham number Bn (viscoplasticity). LAT results for the pressure, shear stresses, and wire tension were in good agreement with FEM within a 10% margin, with LAT giving always underestimates. However, LAT for the viscoplastic simulations gave interesting but erroneous results for the yielded/unyielded zones inside the wire-coating die, which were not borne out by the FEM calculations. Real unyielded zones

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exist only outside the die, where the material moves as a rigid body on the coated wire. Thus, although LAT has the advantages of easy-to-program and – implement attributes, it is misleading for viscoplastic ﬂuids as far as unyielded zones are concerned. Also, because of the enhanced speed and memory of today’s personal computers, it is not much faster than the full 2-D FEM analysis. However, LAT is still good for teaching and educational purposes, because of its ease and the good results it gives for engineering variables (pressure, stresses, wire tension). Thus, it can be used for design purposes, in a repetitive manner, using design criteria such as avoidance of recirculation and high stress jumps, which are detrimental to the materials being coated.

Acknowledgements Part of this research is supported by the “HERAKLEITOS” program of the Ministry of Education and Religious Aﬀairs of Greece (#68/0655). The Project is co-funded by the European Social Fund (75%) and National Resources (25%). Another part is supported by a Greek–Slovenia collaborative research project for 2003–2005, ﬁnanced by the General Secretariat of Science and Technology (GGET) of Greece. Financial support from these sources is gratefully acknowledged.

References Bird R.B., Dai G.C., Yarusso, B.J., “The rheology and ﬂow of viscoplastic materials”, Rev. Chem. Eng., vol. 1, 1983, p. 1–70. Haas K.U., Skewis F.H., “The wire coating process: die design and polymer ﬂow characteristics”, ANTEC ’74, Soc. Plast. Eng., vol. 20, 1974, p. 8–12. Kazatchkov I.B., Yip F., Hatzikiriakos S.G., “The eﬀect of boron nitride on the rheology and processing of polyoleﬁnes”, Rheol. Acta, vol. 39, 2000, 583–594. Lipscomb G.G., Denn M.M., “Flow of Bingham ﬂuids in complex geometries”, J. Non-Newtonian Fluid Mech., vol. 14, 1984, p. 337–346. Mitsoulis E., “Fluid ﬂow and heat transfer in wire coating: a review”, Adv. Polym. Technol., vol. 6, 1986a, p. 467–487. Mitsoulis E., “Finite element analysis of wire coating”, Polym. Eng. Sci., vol. 26, 1986b, p. 171–186. Mitsoulis E., Wagner R., Heng F.L., “Numerical simulation of wire-coating lowdensity polyethylene: theory and experiments”, Polym. Eng. Sci., vol. 28, 1988, p. 291–310. Mitsoulis E., Abdali S.S., Markatos N.C., “Flow simulation of Herschel-Bulkley ﬂuids through extrusion dies”, Can. J. Chem. Eng., vol. 71, 1993, p. 147–160. Mutlu I., Townsend P., Webster M.F., “Simulation of cable-coating viscoelastic ﬂows with coupled and decoupled schemes”, J. Non-Newtonian Fluid Mech., vol. 74, 1998a, p. 1–23.

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Mutlu I., Townsend P., Webster M.F., “Computation of viscoelastic cable coating ﬂows”, Int. J. Num. Meth. Fluids, vol. 26, 1998b, p. 697–712. Ngamaramvaranggul V., Webster M.F., “Simulation of coating ﬂows with slip eﬀects”, Int. J. Num. Meth. Fluids, vol. 33, 2000, p. 961–992. Papanastasiou T.C., “Flow of materials with yield”, J. Rheol., vol. 31, 1987, p. 385–404.

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Experimental Study on Behaviour of Woven Composites in Thermo-Stamping Under Nonlinear Temperature Trajectories H.S. Cheng, J. Cao1 and N. Mahayotsanun 1

2145 Sheridan Road, Northwestern University, Dept. of Mech. Engineering, Evanston, IL 60208, USA, [email protected]

Summary. Possessing high speciﬁc strength and stiﬀness, woven composites have received great amount of attention as a potential alternative to sheet metals in aerospace and automobile industries. To successfully simulate the manufacturing process, predict the performance of the end products and provide information to aid design of manufacturing processes, the material model should take consideration of various length scales, dynamic characteristics and material properties under diﬀerent temperatures. Bias extension and tensile tests are among the most important experiments that provide crucial modeling parameters for material characterizations. This paper focuses on experiments under diﬀerent temperature trajectories, as the forming process itself is often conducted under such conditions.

Key words: woven composites, material characterization, bias extension test, thermo-stamping.

1 Introduction Compared to conversional metal materials used in sheet forming processes, woven composites have high speciﬁc strength and stiﬀness, which are signiﬁcant advantages when the part weight is concerned. The aerospace industry is currently the greatest consumer, while the automobile industry is paying considerable amount of attentions to woven composites, with which many panel parts can be fabricated to reduce the weight of vehicles. Thermo-stamping is considered to be an ideal manufacturing process using woven fabric-reinforced thermo-plastic composite sheets. Prior to the forming process, sheets can be heated above the melting temperature, stamped to a shape when they are very easy to deform, and ﬁnally cooled to a rigid solid. The preheating process can take several minutes and the stamping needs only seconds to complete. Common defects observed in sheet metal forming, such as wrinkling and tearing, occur in woven composite thermo-stamping as well.

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To optimise the process, a capable material model involves non-orthogonal behaviour, viscosity and thermo properties is needed. Among various material modelling approaches, the kinematic method is the easiest to apply. First proposed by Mark and Taylor (Mark et al., 1965), it assumed the fabric yarns are inextensible and pin-joined at the cross points. It requires much less computing power than other methods do, but it is not able to provide information about stress distributions due to its pure kinematic nature. Using pin-point assumption, Vu-Khanh, T. and Liu (Vu-Khanh et al., 1995) investigated the ﬁbre redistribution and reorientation during the forming process. A non-orthogonal constitutive model was presented in their work to reﬂect the eﬀect of angle change between ﬁbre yarns on the material properties. The thermal expansion behaviour revolution due to the deformation was predicted by a sub-plies model composing four ﬁctional uni-directional plies. Another approach is the homogenization method presented by Hsiao and Kikuchi (Hsiao et al., 1999), based on the assumptions of instantaneously rigid solid ﬁber suspended in viscous non-Newtonian polymer melts. This approach can model the complete geometrical and mechanical characteristics of the fabric. However it is very expensive computation wise. Conventional ﬁnite element method is eﬀective to simulate the forming process for the woven composites (O’Bradaigh et al., 1991, 1993). Using the strain energy equivalency principle, Zhang and Harding (Zhang et al., 1993) estimated the mechanical properties of a plain weave composite with the aid of the ﬁnite element method. The unit cell modelled with 3D continuum elements only had one direction undulation and the macroscopic mechanical properties were assumed to be orthotropic and linear elastic. The evolution of the unit cell under other deformation modes such as shearing was ignored in their approach. Boisse and others (Boisse et al., 1995, 1997, 2001) developed a constitutive model for a single thread in tension and then built a membrane element based on potential energy minimization to simulate the shaping of thermoplastic glass fabrics. A fully continuum mechanics approach associated with linear shell elements developed by Cao (Cao et al., 2003) will be brieﬂy reviewed later in this paper. A group of international researchers have joined each other in the research of woven composites since the fall of 2001 (http://nwbenchmark. gtwebsolutions.com/index.php), currently focusing on the benchmark tests for the material (Cao et al., 2004). Picture frame and bias-extension tests have been used to study the shear properties of three types of woven composites commingled with continuous E-glass ﬁbres as the reinforcement and PolyPropene (PP) as the matrix – TPEET22XXX (plain weave), TPEET44XXX(twill 2/2 balanced) and TPECU53XXX(twill 2/2 unbalanced). Through their research, bias extension and picture frame tests are both proved to be useful tools to aid and verify the numerical simulations. Since during the industrial forming process, the fabric undergoes signiﬁcant temperature change and high-speed loading (Cao et al., 2004), it is necessary to

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study the material properties under speciﬁc thermo/loading conditions. In the following part of the paper, we will discuss the bias extension and single yarn tensile tests with diﬀerent temperature trajectories and loading speed for TPEET22XXX plain weave material.

2 Materials The material studied in this paper is Plain Weave TPEET22XXX Glass/PP woven composite produced by Saint-Gobain, as shown in Fig. 1. Table 1 listed the fabric parameters.

Fig. 1. Plain weave TPEET22XXX

Table 1. Fabric parameters (Saint-Gobain) ﬁbre diameter, μ m area density, g/m2 yarn linear density, tex thickness, mm yarn count, warp, picks/cm, yarn count, weft , picks/cm, yarn width in the fabric, warp, mm yarn width in the fabric, weft, mm curing temperature, ◦ C

18.5 743 1870 1.2 1.95 1.95 4.27 ± 0.45 4.27 ± 0.45 177

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3 Experimental Setup 3.1 Single Yarn Tensile Tests A Sintech-20/G tensile testing machine was used as the primary testing tool, on which a pair of customized stainless steel grippers were mounted, as shown in Fig. 2. In order to control the temperature, the sample yarn was put inside a thermocouple-monitored oven. The thermocouple was placed within 1cm away from the sample surface to enhance measuring precision. The test sample was directly installed on the grippers. The initial distance between the two grippers is 150 mm. As Fig. 3 demonstrated, the single-yarn tensile test is not sensitive with strain rate under room temperature. The pulling speed varied from 10 mm/min to 200 mm/min, the force-strain curves did not show considerable diﬀerence. Then we ﬁxed the pulling speed at 20 mm/min and changed the temperature trajectories of testing samples. Fig. 4 shows the force-per-yarn vs. strain curves. One group (group 1) was heated to 180◦ C (above the curing temperature) and held for 10 minutes, then air-cooled to the testing target temperature (80◦ C, 100◦ C, 120◦ C, 140◦ C . . .). The other group (group 2) of samples were heated from room temperature to the target temperature directly. As the

Fig. 2. A single-yarn tensile test sample clamped by the customized grippers

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2500 100mm/min a 100mm/min b 200mm/min a 200mm/min b 10mm/min a 10mm/min b

Force/yarn (N)

2000 1500 1000 500 0 0

0.005

0.01

0.015

0.02

0.025

0.03

Strain

Fig. 3. Force/yarn vs. strain, single-yarn, room temperature

graph indicated, temperature trajectories, as well as the target temperature do not aﬀect the tensile stiﬀness of single yarns. In the non-orthogonal model we developed before (Cao et al., 2003), there are two important parameters to be determined by experiments - - - - Young’s modulus E of a single yarn and shear modulus C for the unit cell. The results of tensile experiment eliminated the necessity of modelling E as a function of temperature. 3.2 Bias Extension Tests The bias extension tests share the same tensile machine and grippers with the single yarn tensile test (Fig. 5). Samples were cut into the geometry as shown in Fig. 6. The numbers of yarns across EF, EH, ED, EA, DB, AB, BC and BG are same. The areas of CIGJ and KHLF are clamped by the grippers and 160 80C 120C 160C 180C–160C 180C–120C 180C–80C

140 Force/yarn (N)

120 100

100C 140C 180C 180C–140C 180C–100C

80 60 40 20 0 0

0.005

0.01

0.015 Strain

0.02

0.025

Fig. 4. Force-per-yarn vs. strain, single yarn tensile test, 20 mm/min

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the distance between HF and CG is consistent for a speciﬁc group of tests. Great care was given to make sure that ﬁbre yarns were oriented ±45 degrees to the edges of the grippers. The number of yarns across line EF (as well as other equivalent lines) is 10 or 16. The initial length between HF and SG is kept at 240 mm for 16-yarn samples and 150 mm for 10-yarn samples. Figure 7 shows that higher pulling speed results in stiﬀer responses in bias extension tests. Figure 8 is a schematic drawing of the thermo-stamping system developed at our research partner – Ford Motor Co. The woven composite sample is heated to 200◦ C in an oven indicated at the left hand side (Stage 1 in Fig. 9). After that the temperature is held at 200◦ C for another minute (Stage 2). Then the sample slides to the forming area in the middle to be formed right away. Starting from the point when the sample moves out of the oven, the forming process itself takes less than 2 seconds (Stage 3). The tooling temperature is 80◦ C. However, when the punch stroke is completed, the sample temperature drops to only about 160◦ C and the sample will be held in the tooling for a minute (Stage 4). In this period, the sample temperature reaches

Fig. 5. A bias extension test sample on the customized grippers

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D

H

E

L

F

C

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I

B

A

G

J

Fig. 6. A bias extension test sample

Fig. 7. Tensile force vs. extension, 16-yarn, bias extension

and stays at 80◦ C. There are totally 18 thermocouples attached to the testing fabric. Figure 9 shows their reading during one test. Therefore, it is important to study the material properties under the similar temperature trajectory as in the industrial forming case. Figure 10 shows the tensile force vs. extension curves from the bias extension test when the samples (group 1) were heated above 180◦ C before cooling down to the target temperature. Obviously, the material became stiﬀer as the temperature dropt from the curing point, due to the solidiﬁcation of the matrix. The lower the measuring temperature, the stiﬀer the material becomes. The reason the measured force drops at the end is that the clamping areas on the sample start to fail, because the heat dissipation in those areas is not as good as in other areas. Thus the temperature there is higher than the measuring target temperature. This is a problem we will need to address in future experiments. High temperature environment causes problems in the test such as that the gripper tends to become loose when the temperature is high. Therefore, the grippers have to be tightened during the heating stage to prevent. Sagging is normal when the material reaches curing point.

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Picture frame

Hydraulic Power pack

Infrared Oven

Stamping Press

Load/Unload Station

Fig. 8. CAD rendering of stamping facility at Ford Motor Co.

Figure 11 shows the same type curves for another group (group 2) of samples that were heated to the target temperature directly. The stiﬀness of the material increases with the measuring temperature. This can be explained as that the partially cured matrix will start solidifying when the temperature ﬂuctuates around the target value. The force drop at the end is due to the pull-out of the yarns.

Fig. 9. Temperature trajectories in the thermoforming tests

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Fig. 10. Force vs. extension, 10-yarn, bias extension group 1

Samples at group 1 show much stiﬀer response than their counterparts at group 2 do. For instance, the 180◦ C–80◦ C sample generated over 500 N resistant force at the extension of 10 mm, but the 80◦ C’s resistant force at 10 mm is below 5 N. As pointed out at other reports (Cao et al., 2004), shear deformation is the dominant deformation mechanism at the bias extension test. As presented at (Xue et al., 2003), the constitute equation according to

Fig. 11. Force vs. extension, 10-yarn, bias extension group 2

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the contravariant elastic matrix can be written as below: ⎤ ⎡ ˜1 ˜2 ν˜12 E E ⎧ 11 ⎫ ⎢ 0 ⎥⎧ ⎫ ⎥ ⎨ε˜11 ⎬ ⎨τ˜ ⎬ ⎢ 1 − ν˜12 ν˜21 1 − ν˜12 ν˜21 ⎥ ⎢ ˜2 ˜2 τ˜22 = ⎢ ν˜12 E ⎥ ε˜22 E ⎥⎩ ⎭ ⎩ 12 ⎭ ⎢ 0 τ˜ ⎦ γ˜12 ⎣ 1 − ν˜12 ν˜21 1 − ν˜12 ν˜21 12 ˜ 0 0 G

(1)

˜ ν˜ and G ˜ 12 are dependant on current strains and temperatures. Where E, Our experimental results strongly suggest that shear modulus G need to be strain-rate and temperature history dependant.

4 Conclusions In this paper, we revisited the setup of the tensile and bias extension tests introduced in (Cao et al., 2004). Test results about the eﬀect of tensile speed were presented. Temperature trajectories of the test samples are designed according to the thermocouple measurements taken at the thermoforming tests. We conclude that the single yarn tensile modulus is not temperature trajectory sensitive, but the shear modulus of the fabric is highly dependant on the temperature trajectory. Thus in the material modelling, shear modulus have to be modelled as a function of temperature history of the node. Improving the experiment capability will be part of the future work. We observed that the temperature distribution on the bias extension sample surface is not close to uniform. Areas close to the grippers are always cooler. More tests are needed to study the coupling of tensile and shear modular with various temperature trajectories, as they are considered decoupled at room temperature (Xue et al., 2003).

Acknowledgements The authors appreciate the ﬁnancial support from the Division of Design, Manufacturing and Innovation of National Science Foundation and Ford Motor.

References Boisse P., Cherouat A., Gelin J.C., Sabhi, H., “Experimental study and ﬁnite element simulation of a glass ﬁber fabric shaping process”, Polymer Composites, 16, 83–95. Boisse P., Borr M., Buet K., Cherouat A., “Finite element simulations of textile composite forming including the biaxial fabric behavior”, Composites Part BEngineering, 1997, 28, 453–464.

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Boisse P., Gasser A., Hivet G., “Analyses of fabric tensile behavior: determination of the biaxial tension-strain surfaces and their use in forming simulations”, Composites Part A, 2001, 32, 1395–1414. Cao J, Xue P, Peng XQ, Krishnan N, “An approach in modeling the temperature eﬀect in thermo-stamping of woven composites, Composite Structures 61, 413–420 Composite Structures 61 (2003) 413–420 http://nwbenchmark.gtwebsolutions.com/index.php Cao J, Cheng HS, Yu TX, Zhu B, Tao X.M., Lomov S.V., Stoilova Tz., Verpoest I., Boisse P., Launay J., Hivet G., Liu L., Chen J., de Graaf E.F., Akkerman R., “Benchmark Eﬀort on Material Testing of Woven Composites Fabric”, ESAFORM 2004. Hsiao S.W., Kikuchi N., “Numerical analysis and optimal design of composite thermoforming process”, Computational Methods in Applied Mechanics and Engineering, 1999, 177, 314–318. Mark C, Taylor HM., “The ﬁtting of woven cloth to surfaces”, J Text Inst 1965;47:T477–88. O’Bradaigh CM, Pipes RB., “Finite element analysis of composite sheet-forming process”, Compos Manuf. 1991;2:161–70. O’Bradaigh CM, McGuinness GB, Pipes RB, “Numerical analysis of stress and deformations in composite materials sheet forming: central indentation of a circular sheet”, Compos Manuf 1993; 4: 67–83. Vu-Khanh T., Liu B., “Prediction of ﬁber rearrangements and thermal expansion behavior of deformed woven-fabric laminates”, Composites Science and Technology, 53, 1995, 183–191. Xue P, Peng XQ, Cao J. “A non-orthogonal constitutive model for characterizing woven composite”. Composites Part A—Appl Sci Manuf 2003;34(2):183–93. Zhang Y.C., Harding J., “A numerical micromechanics analysis of the mechanical properties of a plain weave composite”, Computers & Structures, 1990, 36, 839–844.

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Thixoforming of Steel: Experiments on Thermal Eﬀects P. Cezard1,3 , R. Bigot2 , V. Favier1 and M. Robelet3 1

2

3

LPMM – ENSAM, 4, Rue Augustin Fresnel, 57070 Metz Technopˆ ole – France, [email protected] LGIPM – ENSAM, 4, Rue Augustin Fresnel, 57070 Metz Technopˆ ole – France, [email protected] ASCOMETAL CREAS, Avenue France, 57300 Hagondange – France, [email protected]

Summary. This paper presents experimental results on steel thixoforming. The inﬂuence of thermal exchanges with tools and environment on the semi-solid response is analysed. Several rheological experiments such as compression, extrusion or radial ﬁlling test were developed to understand the semi-solid steel behaviour and determine the parameters that have a major inﬂuence on thixoforming. Actually, the temperature of the slug and probably the solid fraction was found a ﬁrst order parameter while the morphology of the solid phase plays a minor role in our experiments.

Key words: thixoforming, high melting point materials, thermal eﬀects.

1 Introduction After more than thirty years of research on semi-solid processes, low melting point materials are now commonly used in industry at the semi-solid state. Thixoforming presents interesting possibilities compared to traditional foundry or forging processes (Fan, 2002, Kapranos et al., 2000 and Suery, 2002). Advantages of thixoforming and rheoforming are classiﬁed in the Table 1. Since several years, diﬀerent authors started to work on thixoforming of high melting point materials (Kapranos et al., 1993, Shimahara and Kopp, 2004, Rassili et al., 2005, Modigell et al., 2005, Omar et al., 2005, Rouﬀ et al., 2000, Kirkwood, 1996). For steel thixoforming, and generally for high melting point materials, the high semi-solid state temperature induced diﬃculties linked to tool materials. Indeed, tools have to keep a good resistance in this ﬁeld of temperature. Moreover they should be magnetic ﬁeld-independant

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Advanced Methods in Material Forming Table 1. Advantages of thixoforming advantages compare to forging

casting

complex and near net shape low forging force bigger life time for tools

mechanical characteristics lower temperature better internal health

(as much as possible) to avoid their heating by induction. Moreover, The high temperature gradient between the slug and air requires to perform the forming step as fast as possible after heating to avoid loss of temperature. These diﬃculties are clearly reduced with low melting point materials. Thermal exchanges between the part and thixoforming tools have a key role on the global behaviour since they change the solid fraction on the part surface and could create a solid skin during forming process (Kapranos et al., 1993). Thus, temperature is in steel thixoforming a ﬁrst order parameter. Shimahara (Shimahara and Kopp, 2004) developed an interesting approach to avoid these problems of thermal exchange and performed isothermal tests by using a furnace which heat tools and slug. Indeed, isothermal tests are the best solution to estimate correctly the inﬂuence of second order parameters such as strain rate or microstructure on the semi-solid behaviour. And for high melting point materials, you could not make considered the test isothermal just because the test is really fast (Kapranos et al., 2001). Rouﬀ (Rouﬀ, 2003) presented the forming possibilities of steel thixoforming compared with conventional forging (Fig. 1) and realized industrial parts such as a steering knuckle using steel thixoforging (Fig. 2). Indeed, the Ascometal Research Centre (CREAS) realised a plate displaying ﬁerce changes of thickness obtained either by forging or by thixoforging using an energy press. After one step so for the same energy value, forging provides just a rough sketch of the part whereas thixoforging provides the ﬁnished part. Moreover, this thixoforged part display a good surface quality and is geometrically and healthy good. As far as traditional hot forging

Fig. 1. Comparison between forging and thixoforging

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Fig. 2. Example of steel thixoforming part (steering knuckle)

(at 1280◦ C) is concerned, for the same force (600 kN), the dies are not closed leading to a partially ﬁnished part (right side on Fig. 1). In the case of thixoforging, the presence of liquid in the semi-solid material ﬁrstly reduces the load level required to deform the material and secondly eases the ﬂow even in very thin parts of the plates. Note that the temperature during thixoforging was about 1435◦ C, which corresponds approximately to 0.4 liquid fraction. Clearly, this experiment permits to appreciate two major advantages of thixoforming upon hot forging concerning the forming of parts displaying complex shape and in particular thin walls, and the reduction of manufacturing steps. More over, the semi-solid slug is still easy handle due to its high solid fraction, and thus the foundry problems linked to transport liquid material are avoided. To go further in the industrial application of steel thixoforging, an industrial part such as a half reduction of an automotive part was thixoforged (Rouﬀ, 2003). The ﬁnished part obtained still in one step is shown in Fig. 2. Here again, the geometry of the part is really good and the internal health is acceptable. These experiments were only tests to prove the possibility of steel thixoforging. Design rules are now required to really beneﬁt from thixoforging capabilities and response for example to the weight reduction requirements of the automotive sector using thin parts. In this paper, the response of semi-solid steel is studied through speciﬁcally developed tests such as axial and radial extrusions experiments. First, after a background introduction, rheological tests developed for the study

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are explained. In the second part, the load-displacement curves obtained for diﬀerent experimental conditions concerning thermal exchanges are analyzed. We ﬁnish this paper with some improvement propositions and conclusions.

2 Experimental Works 2.1 Background The manufacture process of a thixoforged parts presents several steps (Fig. 3.) For each step, several parameters would have an inﬂuence on semi-solid response. Today, the major diﬃculty today is to analyze parameter sensitivity. This paper focuses on the transfer (between the heating position to the forming position) and thixoforming steps, and presents works on thermal eﬀects phenomena. Indeed, for high melting point materials, thermal phenomena become important and their control is essential. They could annihilate the inﬂuence of others parameters. 2.2 Experiments Presentation This part of the paper presents our experimental devices and techniques. Induction Heating Semi solid state is obtained after melting in an induction heater. We generally used 0.6 for solid fraction which corresponds roughly to 1450◦ C for our steel

Fig. 3. Classical planning for a thixoforged part

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grades at thermodynamic equilibrium. It is still diﬃcult to know exactly the solid fraction in the slug when it is. Several authors developed experiments to study this problem (Shimahara and Kopp, 2004, Lecomte-Beckers, 2004) and try to measure the eﬀective solid fraction. They are still far from our experimental conditions of heating. We approximately heat the slug to its ﬁnal temperature in less than 3 minutes by induction heating which give an heating average of 500◦ C/ min. Induction heating cycle was developed by controlling the temperature in the slug with 2 thermocouples as showed in Fig. 4. By successive heating and cooling we arrive to minimize diﬀerences of temperature in the slug. It is worth noticing that a neutral atmosphere (Argon) around the slug during heating is required. It allows to prevent from oxidation that produces a thin skin and consequently could modify the response of the semi-solid slug during forming. Rheological Tests To understand the semi-solid steel behaviour, two diﬀerent rheological tests were performed. Direct Extrusion This experiment consists to reduce the diameter of a cylinder by pushing through an axial ﬁlling die. The cylindrical semi solid slug is 30 mm diameter and 45 mm height. Induction heating is used to obtain the semi-solid state. At the end of the heating step, the ﬁlling die goes down through the induction device (see Fig. 5). This method avoids problems linked to the transfer step from heating to forming. During the experiment, the forming load was measured using a load sensor. The die (piston) is made of ceramic that is resistant and non dependant to magnetic ﬁelds, to support the heating step and the forming step successively. The ram speed of our 6000 kN hydraulic press is here 30 mm/s.

Fig. 4. Thermocouples in the slug during development of heating cycle

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Fig. 5. Principle of our direct extrusion test

Radial Filling Extrusion The radial extrusion test is certainly our more original and ambitious rheological test. The goal of this system is to realise an extrusion where the strain rate is dependant of the ﬁlling thickness. Figure 8 presents a zoom of the active zone. It is possible to change the ﬁlling thickness by adding thickness between upper and lower dies. Usually the ﬁnal sample is not totally extruded so it is necessary to extract it from the lower tool (see Figure 6). 2.3 Experimental Results First, Fig. 7 presents two load-displacement curves for the radial extrusion test developed in this work, obtained in the same conditions except the transfer time (between heating and forming). The load diﬀerence between the two curves came from a transfer time approximately 3 seconds longer, for the highest curve. Then, to point out the key role of thermal exchanges on semi solid steel response, we changed the tool temperature for a same test by using heating collar device. In Fig. 8, for direct extrusion test, two diﬀerent tool temperatures are compared. Figure 9 displays the load-displacement curves for radial extrusion test obtained for three successive experiments so that, because of the thermal

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Fig. 6. Principle of active radial ﬁlling tools

exchanges with “hot” semi-solid slug, the temperature of the tools grows up between each test. To reduce thermal exchanges we realized additional tests with a ceramic based spray on tools. In Fig. 10, the load-displacement curves for two direct extrusion tests, with or without this coating, are presented.

90

80

Load (10 kN)

70

60

50

Optimal transfer + 3 seconds Optimal transfer

40

30 10

15

20

25

30 35 40 Displacement (mm)

45

50

55

60

Fig. 7. Inﬂuence of transfer time. This ﬁgure pointed the importance of the temperature gradient between semisolid slug and environment

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Load (10 kN)

50 Cold dies (20°C) Hot dies (450°C)

40 30 20 10 0 365

370

375 Displacement (mm)

380

385

Fig. 8. Load-Displacement curves for two direct extrusion tests

2.4 Discussion All these results point the major importance of temperature and thermal exchanges for thixoforming process. Table 2 summarize the results presented with their inﬂuence on load in percent. This table shows the major importance of thermal exchanges with dies. Indeed, a modiﬁcation of thermal exchanges, by changing dies temperature, or by reducing exchanges with ceramic layer, could conduct to a 80% increase

100 90

Load (10 kN)

80 70 60 First Test 50

Second Test Third Test

40 30 15

20

25

30

35

40

45

Displacement (mm)

Fig. 9. Load-Displacement curves for three successive tests of radial extrusion

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60

Load (10 kN)

50 40 Without ceramic layer With ceramic layer

30 20 10 0 20

25

30

35 Displacement (mm)

40

45

50

Fig. 10. Load-Displacement curves for two direct extrusion tests with or without ceramic spray layer

of the load. Light change of industrial parameter like transfer time or cadence could also have an inﬂuence on the load up to a 20% increase of the maximum load. Before analysing in details these results, it’s important to make several remarks. The curves presented in this paper are representative of several tests. Numerous campaigns were realized with same parameters to check the repetability of phenomenon and the same major tendencies of thermal eﬀect were observed. After pointing the inﬂuence of thermal exchanges during successive tests, a thermal regulation of tools between each test were realized. Every results presented in this paper (except successive tests results, of course) were obtained by this method to avoid these phenomenon. Figure 7 shows that a small diﬀerence during transfer time could have a strong inﬂuence on semi-solid response. Indeed, for this high temperature, the loss of temperature is huge during the ﬁrst seconds of cooling and naturally inﬂuences the response of semi-solid. The fraction solid grows up especially at the surface of the slug leading to an increase of about 65% of the load required Table 2. Compilation of results ﬁgure and description of test

maximum of the low curve

maximum of the high curve

percent increase

(Fig. (Fig. (Fig. (Fig.

750 kN 750 kN 300 kN 320 kN

800 kN 900 kN 550 kN 550 kN

7% 20% 83% 72%

7) Transfer 9) Successive tests 8) Dies temperature 10) Ceramic layer

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to deform material. By considering this result, the thermal exchanges with tools would be a key point of thixoforming process since the thermal transfer coeﬃcient would be strongly higher. On Fig. 8, when the tool temperature is higher, the global load is lower and its evolution with the displacement is diﬀerent. These results show again the eﬀect of thermal exchanges on the semi-solid response. In the case of the cold tool, the solid fraction, at least at the slug surface, is higher than the one obtained in the case of the “hot” tool leading to an increase of the load. However, it is worth noticing that friction phenomena are also diﬀerent. This “450◦ C” test is far from an isothermal test so it is diﬃcult to study the inﬂuence of other parameters such as the strain rate for example. Figure 9 shows also that the temperature of the tools aﬀects the global forming force and the evolution of this force with the displacement (slope of the curve during the ﬁrst step for example). The fact that the “heating” of the tools by contact with hot slug inﬂuence so much the response of semi solid induced serious questions on how to control all thermal phenomena and moreover, how to estimate thermal coeﬃcients. A thixoforming operation is an extreme forming from a temperature and speed point of view and kinetics problems should be considered. Test presented on Fig. 10 constitute an improvement on thixoforming process. This thin layer of ceramic spray clearly reduced the force required to deform the material. Ceramic has a role of thermal barrier and limits the

600 550 500 450

Cold needle

Force (N)

400 Hot Needle

350 300 250 200 150 100 50 0 0

2

4

6

8 10 12 Indentation Depth (mm)

14

16

18

20

Fig. 11. Load-Displacement curves for two indentation tests with hot or cold needle [14]

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decrease of temperature (and the skin solidiﬁcation). Moreover, this layer could play a lubrication role and improve the ﬂow material and the geometry quality of the ﬁnal part. Since industrially, isothermal thixoforming would be really diﬃcult to manage, a new approach of tool conception by introducing thermal exchanges problems will be necessary. Laying a thermal barrier could be an easy solution as it has just been illustrated. Finally, it is worth noticing that these thermal phenomena, ampliﬁed in the case of high melting point material, are also present for thixoforming of low melting point materials. Indeed, Bigot et al. (2005) presented load displacement curves for an indentation test using cold and hot needle on a Sn-15%Pb semi-solid slug. Figure 11 shows that the lower the tool temperature is the higher the load level. This eﬀect is attributed to the presence of a solid skin at the surface of the semi-solid slug. It proves again that thermal exchanges between the tools, the environment and the semi-solid aﬀect the semi-solid response and should be considered in the understanding of experimental results. 2.5 Conclusion on Experiments These experiments show the necessary work on thermal phenomena to understand and predict the semi-solid materials behaviour especially for high melting point materials. During heating step, during forming step but also during transfer between these two stages, the temperature needs to be controlled. Several others parameters have an inﬂuence on the semi-solid state and its response during forming, but approximation on temperature could be catastrophic to understand physical phenomena which enter into account. Thermal eﬀects have a key role during forming but they also have a critical inﬂuence on ﬁnal mechanical characteristics by changing heat treatment so ﬁnal microstructure of the part. In a scientiﬁc and industrial point of view, the control and identiﬁcation of all this “thermal” area is essential for the development of thixoforming process.

References Bigot R. et al., Characterisation of semi-solid material mechanical behaviour by indentation test, Journal of Materials Processing Technology 160 (2005) 43–53. Fan Z., Semisolid metal processing, Int. Materials reviews 42 (2002) 49–85 Kapranos P. et al., Semi-solid processing of tool steel, Journal de Physique IV, Colloque C7, Suppl´ement au Journal de Physique III, Vol. 3 (1993) 835–840 Kapranos P. et al., Near net shaping by semi-solid processing, Materials and Design 21 (2000) 387–394 Kapranos P. et al., Investigation into rapid compression of semi-solid slugs, Journal of Materials Processing Technology 111 (2001) 31–36. Kirkwood D.H., Semisolid processing of high melting point alloys, 4th Semi Solid Processing Conference, Sheﬃeld, England (1996).

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Lecomte-Beckers J. et al., Characterisation of thermophysical properties of semisolid steels for thixoforming, 8th Semi Solid Processsing conference, Limassol, Cyprus (2004). Modigell M. et al., Rheological properties of semi-solid metallic alloys Proceedings of the 8th esaform conference on material forming, Cluj Napoca, Romania (2005) Omar M.Z. et al., Thixoforming of a high performance HP9/4/30 steel, Materials Science and Engineering A395 (2005) 53–61 Rassili A. et al., Semi-solid forming of steels, Proceedings of the 8th esaform conference on material forming, Cluj Napoca, Romania (2005) Rouﬀ C. et al., Mechanical behavior of semi-solid materials, 6th Semi Solid Processing Conference, Turin, Italy (2000). Rouﬀ C., Contribution ` a la caract´erisation et ` a la mod´elisation du comportement d’un acier ` a l’´etat semi-solide – Application au thixoforgeage, Thesis, Ecole Nationale d’Arts et M´etiers (2003). Shimahara H., and Kopp, R., Investigations of basic data for the semi-solid forging of steels, 8th Semi Solid Processing Conference, Limassol, Cyprus (2004). Su´ery M., Mise en forme des alliages m´etalliques ` a l’´etat semi-solide, M´ecanique et ing´enierie des Mat´eriaux – Hermes Science Publication

Study of the Liquid Fraction and Thermophysical Properties of Semi-Solid Steels and Application to the Simulation of Inductive Heating for Thixoforming J. Lecomte-Beckers1 , A. Rassili2 , M. Carton1 , M. Robelet3 and R. Koeune1 1

2

3

MMS (IMGC, Bˆ at. B52), University of Li`ege, Sart Tilman, 4000 Li`ege, Belgium, {jacqueline.lecomte, marc.carton, r.koeune}@ulg.ac.be Institut Monteﬁore, University of Li`ege, Sart Tilman, 4000 Li`ege, Belgium, [email protected] ASCOMETAL CREAS, BP 750042, 57301 Hagondange Cedex, France, [email protected]

Summary. The thixoforming of steels is so complex that it requires more investigations regarding both the materials and the technical tools dedicated to the elaboration of the process. In this paper we will show the experimental determination of appropriate solidus-liquidus interval on eight diﬀerent steel compositions. This critical parameter was obtained using Diﬀerential Scanning Calorimetry. The paper also presents the results of thermophysical property determination. These parameters are important for the inductive heating phase of a semi-solid forming (SSF) process. Thanks to the simulations of the inductive heating process, the other main results consist on the developments of the heating techniques that are suitable for the achieving of the sine qua none condition to the semi-solid process, which is the uniform temperature distribution in the reheated billet.

Key words: steel, thixoforming, thermophysical properties, liquid fraction, simulation.

1 Introduction Thixoforming – or semi-solid processing – is the shaping of metal components in the semi-solid state. Major challenges for semi-solid processing include broadening the range of alloys that can be successfully thixoformed and developing alloys speciﬁcally for thixoforming. For this to be possible, the alloy must have an appreciable melting range and before forming, the microstructure must consist of solid metal spheroids in a liquid matrix. Characterisation of thermophysical properties of semi-solid steels for thixoforming is useful in two

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ways. First, to study and optimise the behaviour of alloys to be thixoformed, and secondly to obtain parameters to be incorporated in numerical models. A suﬃciently expanded solidus-liquidus interval is required which allows the formation of the desired microstructure under variation of temperature and holding time. As suggested by Meuser and Bleck (2002), the most preferable structure is a globulitic solid phase in a liquid matrix with decreasing viscosity during forming. Aluminium and magnesium alloys are the focus of numerous investigations, but research activities concerning the thixoformability of steel alloys are still in their very beginning. As suggested by Atkinson et al. (2000), for thixoforming the critical parameters must be as follows: – Appropriate solidus-liquidus interval: Pure material and eutectic alloy are not thixoformable for want of a solidiﬁcation interval. In general, the wider the solidiﬁcation interval, the wider the processing window for thixoforming. For multicomponent systems thermodynamic software is available which allows the calculation of the maximum interval, provided basic data is available. – Fraction solid versus temperature: The liquid fraction sensitivity, (df L /dT), deﬁned as the rate of change of the liquid fraction (fL ) with temperature, is a very important parameter for semi-solid forming; it can be obtained experimentally by diﬀerential scanning calorimetry (DSC) and predicted by thermodynamic modelling. This would allow some systematic identiﬁcation of suitable alloying systems. Kazakov (2000) has recently summarised the critical parameters on the DSC curve and the associated fraction liquid versus temperature curve. The critical parameters as suggested by Kazakov are: – The temperature at which the slurry contains 50 % liquid: T1 . – The slope of the curve at fraction liquid fL = 50%: dF/dT(T1 ). To minimize reheating sensitivity this slope should be as ﬂat as possible. – The temperature of the beginning of melting (T0 ). The diﬀerence (T1 −T0 ) determines the kinetics of dendrite spheroidization during reheating. – The slope of the curve in the region where the solidiﬁcation process is complete: dF/dT(Tf ), where Tf is the temperature of end of melting. In Kazakov’s view this should be relatively ﬂat to avoid hot shortness problems. These parameters will be studied on some ferrous alloys in the ﬁrst part of the paper. The second part deals with thermophysical properties. Simulation techniques show great potential to acquire a good understanding of the Semi-Solid Metal (SSM) forming process. These simulations are intended, on one hand, to determine the optimal electrical and geometrical parameters for the inductive heating phase of a SSF process and on the other hand, to obtain an estimation of the forging loads for the thixoforging of an industrial part, e.g. a SKL-ﬂange. The basis of the simulation of the inductive heating process is the solving of the Maxwell’s equations, together with the equation of heat transfer (4). In the third part of the paper, various numerical and experimental results are compared. In particular, a typical time evolution

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of the temperature in the steel part, which is the key parameter in the SSM process, is shown. The basis of the modelling of the forming process is the solving of the deformation equations, taking into account thermal and rate dependence eﬀects. Some important proportionality parameters needed are electrical resistivity, caloriﬁc capacity and thermal conductivity.

2 Experimental Procedures We studied diﬀerent alloys named C38 Asco Modif 1, C38 Asco modif 2, 100 Cr6 Asco modif 1, 50 Mn6 Asco modif 1, 45 Mn5 Asco modif 1 that were modiﬁed for thixoforming properties. As pointed out above, the main critical parameters for thixoforming must be as follow: appropriate solidusliquidus interval and fraction solid versus temperature. These two parameters are obtained from Diﬀerential Scanning Calorimetry (DSC). Secondly the thermophysical properties of the alloys have to be determined. 2.1 Solidus-Liquidus Interval and Fraction Solid Versus Temperature Characterisation The applicability of a material for processing in the semi-solid state is deﬁned by the solidus-liquidus interval and the development of liquid phase in the interesting temperature range. For the evaluation of the solidus and liquidus temperature a Diﬀerential Scanning Calorimetry (DSC) was used. The development of the liquid phase with increasing temperature was calculated using the values from the DSC-measurements. The evaluation of the liquid phase distribution is carried out by the application of a peak partial area integration. The whole area under the enthalpy-area curve is used to determine the melting enthalpy of the material. During DSC measurement, the typical melting peak obtained is shown in Fig. 1. The peak characteristics are: –

The changes of slope, jumps and peaks showing the thermal events (phase transformations, chemical reactions, etc.)

Fig. 1. Melting peak features

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The peak area is the enthalpy variation of the transformation The speciﬁc heat is calculated from the baseline Solidus-liquidus interval: T end of melting-T beginning of melting (Tf −T0 )

We assume that the liquid fraction is proportional to the absorbed energy during the transformation. The sample is heated until total melting. Therefore, the liquid fraction can be calculated considering the peak area of the transformation, as shown in Fig. 2. The characterisation of the melting peak is realised with the following parameters: – –

Total area = 100% of the liquid fraction Beginning and end of melting

The liquid fraction at Ti is determined with the following relation (1): %liquid =

Area(T0 − Ti ) T otalArea

(1)

2.2 Thermophysical Properties Characterization Dilatometry Dilatometry is a technique used to measure the relative dilatation ΔL/L0 of a material submitted to a temperature program (ΔL is the diﬀerence between the length at temperature T and the initial length L0 at room temperature). In our case, the sample holder for powder and pasty sample is required because we carried out the dilatation run tests for high temperatures, where the sample is in liquid state (Fig. 3). Average Expansion Coeﬃcient From the dilatation values obtained and thanks to the relation (2), it was possible to calculate the average expansion coeﬃcient CTE (T1 − T2 ) for the temperature interval (T2 − T1 ).

Fig. 2. Determination of the liquid fraction

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Fig. 3. Sample holder for powdery and pasty samples

ΔL ΔL (T2 ) − (T1 ) L0 L0 CT E(T1 − T2 ) = T2 − T1

(2)

where T2 is the upper temperature limit, T1 is the lower temperature limit and ΔL/L0 is the length change relative to L0 . Density Density ρ(T ) was calculated from the expansion values according to the following relation (3). . 2 3 / ΔL(T ) ΔL(T ) ΔL(T ) (3) +3 + ρ(T ) = ρ0 1 + 3 L0 L0 L0 where ρ0 is the density at reference (mostly ambient) temperature, ΔL(T ) is the expansion of the specimen under investigation and L0 is the specimen length at room temperature. DSC-Cp Determination DSC is a technique in which the diﬀerence in energy input into a substance and a reference material is measured as a function of temperature, while the substance and reference material are subjected to a controlled temperature program. Individual Cp values at diﬀerent temperatures are determined using a sapphire as a standard according to the following (4). Cp =

mStandard DSCSample − DSCBas Cp,Standard mSample DSCStandard − DSCBas

(4)

where Cp is the speciﬁc heat of the sample at temperature T , Cp,Standard is the tabulated speciﬁc heat of the standard at temperature T , mStandard is the mass of the standard, MSample is the mass of the sample, DSCSample is the value of DSC signal at temperature T from the sample curve, DSCStandard is the value of DSC signal at temperature T from the standard curve and DSCBas is the value of DSC signal at temperature T from the baseline.

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Thermal Diﬀusivity – Laser Flash The front face of a cylindrically shaped piece is homogeneously heated by an unfocused laser pulse (Fig. 4). On the rear face of the test piece the temperature increase is measured as a function of time. The mathematical analysis of this temperature/time function allows the determination of the thermal diﬀusivity D(T ) as presented in (Cape and Lehman, 1963) and illustrated in

Fig. 4. Laser pulse on the sample

Fig. 5. Laser ﬂash apparatus schematic

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Fig. 5. As we determined diﬀusivity up until the liquid state, we used a special quartz sample container. Thermal Conductivity Knowledge of thermal diﬀusivity D(T ), density ρ(T ) and speciﬁc heat Cp (T ) allowed the determination of the thermal conductivity χ(T ), calculated according to the Laplace relation (5): χ(T ) = D(T )ρ(T )Cp (T )

(5)

3 Results 3.1 Solidus-Liquidus Interval and Fraction Solid Versus Temperature Characterization Some basic alloys are studied and compared to modiﬁed alloys. The basic alloys are C381 , C801 and 100Cr61 . The modiﬁed alloys are C38 Asco modif 11 and C38 Asco modif 21 , 100 Cr6 Asco modif 11 , 45Mn5 Asco modif 11 and 50 Mn6 Asco modif 11 . All properties are compared to the base alloy C38. The base alloy C38 was used to study the eﬀect of heating rate on DSC curves. The 50 Mn6 Asco modif 1 alloy was used to study the homogeneity of the billet. The results are presented hereafter. Figures 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 and 30 show the DSC signal of the melting peak and Fig. 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 and 31 the corresponding liquid fraction. C38 The DSC signal and the liquid fraction of C38 are shown in Figs. 6 and 7. Diﬀerent heating rates were used (2◦ / min, 10◦ / min and 20◦ / min). The DSC curves show that the DSC signals increase with heating rate but the sensitivity and the peak separation decrease. This fact will probably aﬀect the fast inductive heating, but DSC is limited to 20◦ / min which approaches best the industrial heating rates. All subsequent experiments were therefore conducted with a heating rate of 20◦ / min. The interest of the heating at 2◦ C / min is that it shows the intermediate kink in the liquid fraction versus temperature curve. During melting of C38 we observed three diﬀerent peaks, which are related to the transformation: – – – 1

γ → γ + liquid, peritectic transformation γ + liquid → δ + liquid, and δ + liquid → liquid The composition follows euronorm DIN code. C38 = (carbon 0.38 %), C80 = (carbon 0.80 %), 100 Cr6 = (carbon 1 %, Cr 1.5 %), 50 Mn6 = (carbon 0.5 %, Mn 1.5 %), 45 Mn5 = (carbon 0.45 %, Mn 1 %).

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Fig. 6. DSC signal of C38

Fig. 7. Liquid fraction of C38

Fig. 8. DSC signal of C80

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Fig. 9. Liquid fraction of C80

Fig. 10. Comparison of the DSC signal between C38 and C80

Fig. 11. Comparison of the liquid fraction between C38 and C80

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Fig. 12. DSC signal of 100 Cr6

Fig. 13. Liquid fraction of 100 Cr6

Fig. 14. DSC signal of 100 Cr6 Asco modif 1

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Fig. 15. Liquid fraction of 100 Cr6 Asco modif 1

Fig. 16. Comparison of the DSC signal between C38, 100 Cr6 and 100Cr6 Asco modif 1

Fig. 17. Comparison of the liquid fraction between C38, 100 Cr6 and 100Cr6 Asco modif 1

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Fig. 18. DSC signal of C38 Asco modif 1

Fig. 19. Liquid fraction of C38 Asco modif 1

Fig. 20. DSC signal of C38 Asco modif 2

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Fig. 21. Liquid fraction of C38 Asco modif 2

Fig. 22. Comparison of the DSC signal between C38, C38 Asco modif 1 and C38 Asco modif 2

Fig. 23. Comparison of the liquid fraction between C38, C38 Asco modif 1 and C38 Asco modif 2

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Fig. 24. DSC signal of 45Mn5 Asco modif 1

Fig. 25. Liquid fraction of 45Mn5 Asco modif 1

Fig. 26. Comparison of the DSC signal between C38 and 45Mn5 Asco modif 1

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Fig. 27. Comparison of the liquid fraction between C38 and 45Mn5 Asco modif 1

Fig. 28. DSC signal of 50 Mn6 Asco modif 1

Fig. 29. Liquid fraction of 50 Mn6 Asco modif 1

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Fig. 30. Comparison of the DSC signal between C38 and 50 Mn6 Asco modif 1

Fig. 31. Comparison of the liquid fraction between C38 and 50 Mn6 Asco modif 1

C80, C38 The DSC signal during melting and the liquid fraction of C80 are shown in Figs. 8 and 9. The comparison between C38 and C80 is shown in Figs. 10 and 11. With regard to Kazakov parameters, C80 exhibits better behaviour than C38. The beginning of melting T0 is lower, T1 is lower, the solidiﬁcation interval (Tf − T0 ) is larger (see Table 1), and the slope of the curve dF/dT is ﬂatter at T1 and Tf . 100Cr6, 100 Cr6 Asco modif 1, C38 The DSC signal and the liquid fraction of 100 Cr6 and 100 Cr6 Asco modif 1 are shown in Figs. 12, 13 and 14, 15 respectively. The C38, 100Cr6 and 100Cr6 Asco modif 1 results are shown in Figs. 16 and 17. With regard to Kazakov parameters, 100 Cr6 Asco modif 1 exhibits better behaviour than 100 Cr6 and C38 one. The beginning of melting T0 is lower, T1 is lower, the solidiﬁcation interval (Tf − T0 ) is slightly larger (see Table 1) and the slope of the curve dF/dT is ﬂatter at T1 .

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Table 1. Characteristic temperatures and slopes alloys

T0 (◦ C)

T1 Tf Tf − T0 (◦ C) (◦ C) (◦ C)

slope at T1

slope at Tf

C38 C38 Asco modif 1 45 Mn5 Asco modif 1 50 Mn6 Asco modif 1 (surface) 50 Mn6 Asco modif 1 (half-radius) 50 Mn6 Asco modif 1 (center) C38 Asco modif 2 C80 100 Cr6 100 Cr6 Asco modif 1

1430 1415 1411 1389 1386 1382 1379 1361 1315 1278

1500 1478 1470 1458 1456 1456 1472 1450 1431 1402

0.0200 0.0185 0.0173 0.0136 0.0127 0.0139 0.0145 0.0114 0.0111 0.0097

0.0019 0.0017 0.0020 0.0002 0.0002 0.0002 0.0007 0.0003 0.0004 0.0013

1536 1517 1501 1519 1520 1519 1520 1491 1487 1460

106 102 90 130 134 137 141 130 172 182

C38 Asco Modif 1, C38 Asco Modif 2, C38 The DSC signal and the liquid fraction of C38 Asco modif 1 are shown in Figs. 18 and 19. The DSC signal and the liquid fraction of C38 Asco modif 1 are shown in Figs. 20 and 21. The C38, C38 Asco modif 1 and 2 results are shown in Figs. 22 and 23. With regard to Kazakov parameters, C38 Asco modif 2 exhibits better behaviour than C38 Asco modif 1 and C38 ones. The beginning of melting T0 is lower, T1 is lower, the solidiﬁcation interval (Tf − T0 ) is slightly larger (see Table 1) and the slope of the curve dF/dT is ﬂatter at T1 and Tf . 45Mn5 Asco Modif 1 and C38 The DSC signal during melting and the liquid fraction of C38 Asco modif 1 are shown in Figs. 24 and 25. The comparison between C38 and 50Mn6 Asco modif 1 is shown in Figs. 26 and 27. The conclusions are similar to the other alloys, the behaviour of 45 Mn5 Asco modif 1 is better than C38. 50 Mn6 Asco modif 1 In order to determine the ingot homogeneity, three samples of 50Mn6 Asco modif 1 (named surface, half-radius and centre) were taken and studied. These results are as follows (Figs. 28 and 29). They show that the behaviour of the ingot is rather homogeneous regarding liquid fraction. This is of course important for industrial practice. The comparison between C38 and 50Mn6 Asco modif 1 is shown in Figs. 30 and 31. As regarding Kazakov parameters, the behaviour of 50 Mn6 Asco modif 1 is better than C38 one. The beginning of melting T0 is lower, T1 is

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lower, the solidiﬁcation interval (Tf − T0 ) is slightly larger (see Table 1) and the slope of the curve dF/dT is ﬂatter at T1 and Tf . Alloys Features During Melting Table 1 gives main characteristic temperatures and slopes of the liquid fraction curve during melting at 20◦ / min. Alloys are classiﬁed following decreasing T0 . It is clear that for non alloyed steels the C38 Asco modif 2 gives the best results: T0 and T1 are lower, (Tf − T0 ) is larger, the slopes at T1 and Tf are lower than those of C38. It was used for the simulation of heating. Regarding low alloyed steels, 45 Mn 5 Asco modif 1 is not interesting. On the contrary 100 Cr6, 100 Cr6 Asco modif 1 and 50 Mn 6 Asco modif 1 show good behaviour. 3.2 Thermophysical Properties Characterisation These properties were determined on the alloy C38 Asco Modif 2 and was used for the simulation of heating phase. The results of that simulation are shown in the next part of this study. Dilatometry and CTE The dilatation and CTE results are shown in Figs. 32 and 33. Table 2 gives values of the relative dilatation. Density The Fig. 34 shows the evolution of the density during heating in terms of temperature. The peak observed at 750◦ C is due to the phase transformation austenite-ferrite.

Fig. 32. Evolution of the dilatation during heating

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Fig. 33. Evolution of the CTE during heating Table 2. Relative dilatation during heating temperature (◦ C)

relative dilatation (ΔL/L0 )

100 500 1000 1500

8.39E-04 6.65E-03 1.41E-02 2.45E-02

DSC-Cp Determination The evolution of the speciﬁc heat versus temperature is shown in Fig. 35 and 36. We can see a peak around 750◦ C. This is the phase transformation (austenite-ferrite) already seen from the dilatometry. The dramatic increase about 1400◦ C is due to the sample melting. The speciﬁc heat is directly calculated from the DSC curves. Therefore if a transformation occurs during DSC measurement, it inﬂuences also cp measurement. Fig. 35 gives speciﬁc heat values if we take into account the quantity of heat ΔH released during the transformations, hence the y-axis title “apparent speciﬁc heat”. Ignoring the quantity of heat ΔH, we obtain the speciﬁc heat values shown in Fig. 36.

Fig. 34. Evolution of the density during heating

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Fig. 35. Evolution of the apparent speciﬁc heat during heating

Fig. 36. Evolution of the speciﬁc heat during heating

Thermal Diﬀusivity and Conductivity The behaviour of the diﬀusivity during heating is shown below, Fig. 37. The thermal conductivity is shown in Fig. 38. The behaviour of the thermal conductivity is similar to the thermal diﬀusivity. The values decrease until about 750◦ C. When the phase transformation occurs, the thermal conductivity increases until the beginning of melting. Because of the phase change energy consumption, there is a dramatic diminution during melting, and the more liquid, the smaller the thermal conductivity.

Fig. 37. Evolution of the thermal conductivity during heating

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Fig. 38. Evolution of the thermal diﬀusivity during heating

4 Inductive Heating Simulation Adequate heating cycles are necessary to maintain the globular structure and thus the thixotropic properties of the used steel. In this manner, the simulation of the inductive heating process is of a great help in order to establish the correct heating parameters and by the way to help controlling the heating process. The numerical simulation of the inductive heating consists of the use of adequate softwarebased on the ﬁnite element method for ﬁeld calculations. The software uses the formulations that consider the magneto-thermal coupling involved in the case of the inductive heating process. The software environment used to perform the simulations is called GetDP (Dular et al., 1998). Its main feature is the ability to write the models describing physical phenomena directly in input ﬁles (ASCII Format), in a form similar to their mathematical expression. This enables an eﬃcient evaluation of the eﬀects of changes in modelling decisions, from the straightforward deﬁnition of complex non-linear physical characteristics, to the deﬁnition of the various couplings, solving algorithms and post-processing operations. The steel properties required as an input by the implemented numerical models are of two kinds. First, we need the thermal conductivity and the caloriﬁc capacity experimentally determined earlier in this work. Then, the magnetic permeability and the electrical conductivity shown in Fig. 39 are also necessary. All these properties are temperature dependent, which makes of course, the problem non-linear. The mathematical details of the process were deeply explained in previous works [4]. We will show only some results obtained after the ﬁrst simulations, such as the temperature and the liquid-fraction-distributions in the billet during the re-heating process. We can conclude from the geometrical optimisations that the ideal coil geometry is the parabolic one. For the reheating of small billets like for the SKL ﬂange, a coil of six turns is enough. The electrical parameters are the most important ones and require the use of the exact material properties; they include mainly the heating frequency, power, voltage and current. Figure 40 shows the optimised geometry as well

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Fig. 39. Electrical resistivity

as the temperature colour map of the re-heated billet for the SKL Flange. The model shows a very good temperature distribution within the billet as shown in Fig. 41. The corresponding heating power is given by Fig. 42. In parallel to the simulation of the electrical and geometrical parameters, to deal with a good and uniform temperature distribution, the simulation of the liquid fraction depending on the heating parameters was carried out. The uniform temperature distribution and the use of correct physical properties helped a lot on determining the liquid fraction distribution given by Fig. 43 where we see that with the above heating cycle we can reach 40 % liquid fraction. After setting up the model of the inductive heating simulations, several experiments were realized in order to validate the theoretical assumptions. In order to avoid more expenses for these validations, we decided to use an existing coil of 7 turns even known the coupling is weak. The aim of these experiments is to reproduce by simulations the experimental results and then to adjust the parameters of the model. Figure 44 shows the validation geometry and the temperature colour map in the billet. In order to respect the amount of material used for the SKL Flange, we have reduced the dimensions of the billet. The simulated and measured parameters are in very good agreement if we take into account that for electrical resistivity, we used only a theoretical approach and this is due to the lack of

Fig. 40. Temperature colour map for the SKL Flange billet

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Fig. 41. Temperature distribution in diﬀerent points of the billet

Fig. 42. Simulated heating power in the case of the SKL Flange

equipment that allow the measurement of this parameter. We will later have an electrical resistivity apparatus and the correct parameter will then be introduced in the simulations. Figure 45 gives a comparison between the simulated and measured temperatures in the cold (top) and the hot (centre) part of the billet. The comparison between the simulated and measured temperatures is also shown. A nearly good agreement is to be stressed. The diﬀerence between the results at medium temperatures is due to the energy losses either by radiation or by convection, which is very diﬃcult to handle on one hand and the fact

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Fig. 43. Simulated average liquid fraction distribution in the billet

Fig. 44. Used geometry for the validation tests and the temperature distribution in the billet

that experimental measurements by thermocouples are not so easy in the other hand. Last but not least Fig. 46 represents the used heating power. Note that we have shown in this report only relevant results. Some more validations will be realized in order to improve more and more our simulation model. For the automation of the described process a control system was developed which still needs thermo sensors for the determination of the billet temperature. For the next development step of this system it is planned to dispense these sensors because with a focus on industrial production of thixoforming parts they are not workable. Instead of direct measuring, the billet temperature will be calculated from easy acquirable parameters such as heating power, current etc. by applying a suitable mathematical model. The development of such a model is currently subject of simulation and research work.

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Fig. 45. Temperature validation in the centre and the top of the billet

Fig. 46. Validation experimental heating power

The inductive heating of the steel billets is today controlled by the adjusted heating cycles. Once an appropriate cycle is determined, it is assumed that always the same amount of material is to be re-heated whereas the duration and the amount of liquid fraction as well as the temperature to be reached are identical.

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5 Conclusions Thixoforming of steels is more and more investigated. The outcome of some important development results has required several tests and experiments either numerical or practical. The DSC measurements and corresponding liquid fraction versus temperature were used to study diﬀerent alloys. For non alloyed steels, C38 Asco modif 2 shows better behaviour as regarding T0 , T1 , (Tf − T1 ) and dF/dT (T1 , Tf ) than C38. For low alloyed steels 100 Cr6, 100 Cr6 Asco modif 1 and 50 Mn6 Asco modif 1 show good behaviour. They could be chosen as candidates for thixoforming. Thermophysical properties characterisations are obtained on C38 Asco modif 2 (α, CTE, density, Cp , thermal diﬀusivity and thermal conductivity). As regards to thermal conductivity, a dramatic diminution is observed during melting. These properties were then used for the simulation of heating phase. Thanks to the simulations, a lot of money and eﬀort were saved at diﬀerent levels. Several improvements are still to be made and the thixoforming of steels has not yet revealed all its secrets.

References Atkinson H.V., Kapranos P., Kirkwood D.H., “Alloy Development for Thixoforming”, International Conference on Semi-Solid Processing of Alloys and Composites, Torino, September 27–29 (2000), 445–446; Ed. G.L. Chiarmetta, M. Rosso, Edimet, Italy. Cape J.A., Lehman G.W., “Temperature and Finite Pulse-Time Eﬀects in the Flash Method for Measuring Thermal Diﬀusivity”, Journal of Applied Physics, 37 (1963), 1909. Dular P., Rassili A., Meys B., Henrotte F., Hedia H., Genon A., and Legros W., “A software environment for the treatment of discrete coupled problems and its application to magneto-thermal coupling”, the 4 th International Workshop on “Electric and Magnetic Fields, from numerical models to industrial applications” Marseilles 12–15 May 1998 pp. 547–552; Ed. AIM Association des Ing´ enieurs Diplˆ om´es de l’Institut d’Electricit´e de Mont´eﬁore (Ulg). Kazakov A.A., “Alloy Compositions for Semisolid Forming”, Advanced Materials & Processes, (2000), 31–34. Lecomte-Beckers J., Rassili A., Carton M., and Robelet M., “Characterisation of Thermophysical Properties of Semi-Solid Steels for Thixoforming”, Proceedings of the 8th International Conference on Semi-Solid Processing of Alloys and Composites, Cyprus, September 21–23 (2004); Ed. Worcester Polytechnic Institute, MA. Lecomte-Beckers J., Rassili A., Carton M., and Robelet M., “Study of Liquid Fraction Evolution of Semi-Solid Steels for Thixoforming”, Proceedings of the 8 th ESAFORM Conference on material forming, Cluj-Napoca, Romania, April 27–29 (2005), 1087;Ed. The Publishing House of the Romanian Academy.

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Meuser H., Bleck W., “Determination of Parameters of Steel Alloys in the Semi-Solid State”, Proceedings of the 7th International Conference on Semi-Solid Processing of Alloys and Composites, Tsukuba, Japan (2002), 349; Ed. Yasukata Tsutsui, Manabu Kiuchi, Kiyoshi Ichikawa. Rassili A., Geuzaine Ch., Legros W., Bobadilla M., Cucatto A., Robelet M., Abdelfattah S., Dohmann J., and Hornhardt Ch., “Simulation of adequate Inductive Heating Parameters and The Magneto-Thermal coupling involved in the SSM Processing of Steels”, Proceedings of the 6 th International Conference on Semi-Solid Processing of Alloys and Composites, Torino, September 27–29 (2000); Ed. G.L. Chiarmetta, M. Rosso, Edimet, Italy

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Simulation of Secondary Operations and Springback – The Implicit Shell Provides a Precise and Rapid Solution ´ Sch¨ E. onbach1 , G. Glanzer2 , W. Kubli2 and M. Selig2 1

2

AutoForm Engineering Deutschland GmbH, Oﬃce Pfaﬀenhofen, Ingolst¨ adter Str. 102, D-85276 Pfaﬀenhofen, [email protected] AutoForm Engineering GmbH, Technoparkstrasse 1, CH-8005 Zurich, [email protected], [email protected], [email protected]

Summary. This paper introduces recent advances for the calculation of secondary operations and springback by means of the FEA program AutoForm. A special feature of AutoForm is that the entire forming process including all forming operations, secondary operations and even springback, is calculated with an implicit shell, i.e. with equilibrium control. The advantages of the implicit shell are discussed here. Additionally, the signiﬁcances of a high quality contact algorithm and of ahead reﬁnement are analyzed, for their eﬀect on the accuracy of springback calculations, and various AutoForm solutions are presented. Finally, these advances in sheet metal forming theory are evaluated, by comparison between theoretical results and actual sheet metal parts.

Key words: implicit method, implicit forming simulation, shell element, springback, secondary forming operations.

1 Introduction To meet the needs of the automotive industry regarding environmentallysound vehicles of high quality, reliability and comfort, as well as the need to reduce development costs and lead times, it is common standard to implement forming simulation during development phase. Forming simulation has become fully integrated into product design over the last few years. The ease-of-use as well as the rapidity and accuracy of process simulation are the main reasons for this integration. To further increase the eﬀectiveness of simulation in practice, new problems need to be solved. The current goal regarding forming simulation is to examine the whole process, starting from blank positioning to forming,

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trimming and ﬁnally springback evaluation of the ﬁnished part. The simulation of secondary forming operations and springback, compared to only deep drawing operation, requires a very sophisticated simulation program because these operational steps are physically very “unstable” and complex. However, the current goal is to enhance the simulation software such that, it can be applied to the entire forming process. Furthermore, the simulations need to be expanded from academic examples to practical industrial applications.

2 Characteristics of the Simulation of Secondary Forming Operations and Springback 2.1 State of the Simulation About 10 years ago forming simulation began to be widely applied in practice in the automotive industry. Initially, the main target of the application of simulation systems was to ensure the failure-free production of parts, without requiring physical tools for the try-out. This resulted in signiﬁcant cost reductions and shorter development time for prototyping and process planning. Experience has shown that further cost reductions are possible if forming simulation is applied during product design, and not only for the development of tools. Thus it makes sense to combine the manufacturability analysis of sheet metal parts with product design, and to start the analysis and evaluation process as soon as possible. For this reason, modules have been developed for the AutoForm simulation program which make possible forming simulation already during product design. These modules allow one to evaluate rough initial drafts of the part geometry and therefore of the process capability of serial production, even if the CAD model has not yet been ﬁlleted. In addition to the actual analysis, the software oﬀers a range of additional functions which make the development process easier and more eﬃcient. For example, using AutoForm-OneStep and AutoForm-PartDesigner, the part geometry can be modiﬁed in areas with material failure, and then immediately be evaluated for manufacturability. In special cases, e.g. for outer panels, initial rough tool concepts can be created and analyzed using the modules AutoForm-DieDesigner and AutoFormIncremental (Sch¨ onbach 03, Sch¨ onbach et al., 2003). If the part geometry has been determined to be manufacturable, it is used as the basis for the geometry of the forming tools. As the goal is also to create the forming tools as rapidly and cost-eﬀectively as possible, specialized software systems for process planning should be used. With AutoFormDieDesigner, it is possible to create a fully parameterized tool concept on the basis of the CAD surface data within only a few minutes (Sch¨ onbach et al., 2003). Immediately afterwards, the forming process is simulated and – if needed – the geometry parameters of the tool can be optimized. After deciding on the best concept, modiﬁcations of the process parameters such as

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binder force, lubrication, draw beads or blank outline are easily and quickly done, which can then be validated for eﬀectiveness. In serial toolmaking – in contrast to prototype toolmaking – the whole process is examined, starting from blank positioning to forming and trimming operations until springback of the ﬁnished part. The simulation of secondary forming operations and springback – compared to only deep drawing – requires a very sophisticated simulation program because these operational steps are physically very “unstable” and complex. Therefore, forming simulation today has been limited up to optimization of the drawing operation. However, the current goal is to enhance the simulation software accordingly so that it can be applied to the entire forming process. Furthermore, the simulations must be expanded from academic examples to practical industrial applications. 2.2 Secondary Forming Operations In the simulation of sheet metal forming processes one must distinguish between two diﬀerent deformation modes: basically between stretch-drawing and bending. The ﬁrst deformation mode (stretch-drawing) takes place in the sheet plane and occurs during the drawing of complex parts, with the shape of the part deﬁned (supported) by the tool geometry. The bending enhanced membrane element is suitable for the reproduction of such processes. For secondary forming operations such as restrike, ﬂanging, cam ﬂanging, hemming and also for the springback simulation, the bending deformation mode also applies. In this case, the loading is not restricted to the sheet plane, so the stress distribution over the sheet thickness also becomes important. To reproduce these eﬀects in the simulation software, the bending enhanced membrane element is no longer suﬃcient – a shell element is necessary. For these types of forming operations, the geometry of the part often can not be derived from the tool geometry. Additionally, long tool paths are a common characteristic, although the main part of the forming is locally restricted to a very small range (Fig. 1). Another essential characteristic of secondary forming operations is that often very small radii with respect to the sheet thickness must be formed. This requires greater eﬀorts in terms of adaptive reﬁnement and the contact algorithm. To meet these requirements, it has been necessary to implement a shell element and extended reﬁnement and contact algorithms into the AutoForm system, besides the bending enhanced membrane element. The improved adaptive reﬁnement is realized by an “ahead” reﬁnement, which makes sure that a suﬃcient number of suﬃciently reﬁned elements is always available in areas with a likelihood of signiﬁcant bending and plastiﬁcation, before they are actually needed. The contact algorithm has been improved by extending the previous nodal contact with a combined nodal and elemental contact. In doing so, the boundary conditions are not only applied to the nodes of discrete elements,

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1 2

3

1

Long tool paths

2

Main part of forming is restricted locally to a very small range

3

Very small radii with respect to the sheet thickness must be formed

Fig. 1. Characteristics of secondary forming operations

but continuously all over the mesh. The main advantage of this procedure is that the contact evaluation is more independent of the discretization because separating is also possible in the middle of the element. 2.3 The Implicit Shell Element The implicit method is best suited for the reproduction of slow sheet metal forming processes (which are typical in the automotive industry), where at the end of each increment, static equilibrium control is aimed for. As the process does not need to be accelerated artiﬁcially and the mass does not need to be increased artiﬁcially, (as is the case with the explicit method), no interfering dynamic eﬀects occur which in turn need to be revised by artiﬁcial numerical parameters. With equilibrium control, a very high accuracy level is obtained for the simulation of forming processes, especially for “unstable” bending operations which occur very often in secondary forming operations. As for springback simulation, the accuracy of the operation simulated prior to springback is decisive, with equilibrium control leading to the potential for a new accuracy level in the springback prediction.

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The drawbacks of the implicit method were the computation speed, the high memory requirements and the frequent convergence problems with complex processes. AutoForm successfully solved these issues in the past. The previously implemented bending enhanced membrane element together with a coupled solution (Kubli 96) made it possible to reproduce a large percentage of deep drawing processes, with a high accuracy level and superior computation speed, which made the practical, widespread application of simulation possible even for complex geometries. For secondary forming operations such as ﬂanging, cam ﬂanging or hemming as well as for springback simulation, for which the eﬀects of bending eﬀects and rigid-body motion prevail, the bending enhanced membrane element has reached its limits. For such operations an implicit and fast shell element is required, which combines these advantages: – –

implicit method with equilibrium control, and shell element with the more precise consideration of bending eﬀects.

As a result of these considerations, an implicit shell element with a fully coupled solution has been implemented in the FEA program AutoFormIncremental (Selig 03). The aim was to keep the element formulation as simple as possible while at the same time considering all necessary eﬀects – such as large displacements and bendings – to obtain a high accuracy level. Thus for the shell element formulation – as for the bending enhanced membrane formulation – a triangular element is used. It is a non-conforming element with linear shape functions for all 5 degrees of freedom per node based on the Midlin-Shell theory (Glanzer et al., 2003, Selig 03). The shell element sufﬁces with a single integration point over the area and with several integration points over the thickness. This formulation is compatible with the bending enhanced membrane and allows for switching between the two element types as desired, resulting in further reductions in computing time. The implicit shell solution in AutoForm is a central component for an accurate simulation of secondary forming operations and springback.

3 Reproduction of the Entire Forming Process 3.1 Reproduction of Secondary Forming Operations The strong competition in the automotive industry with respect to development time, costs and quality is creating new demands on part designs and manufacturing processes. The following trends can be seen: – – –

diﬀerent car body parts are joined in composite constructions, a reduction in the number of forming steps is desired, and alternative materials such as high strength steel and high tensile strength steel, aluminum, magnesium, etc. are being used.

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To maintain at a high level or even increase the reliability of the manufacturing process (i.e. production robustness), simulations must accurately reproduce not only the deep drawing process but also all secondary forming operations. A main requirement is that all technical parameters such as the detailed description of the forming process, the precise registration and reproduction of tool movements, sheet thicknesses, sheet dimensions, material properties and tool geometries, must be taken into account and precisely deﬁned in the simulation program. Binder forces, drawbeads and other restraining devices must also be integrated realistically into the simulation. Furthermore, the simulation programs should allow the user to deﬁne these speciﬁcations quickly and easily. The AutoForm UserInterface satisﬁes these requirements. Having integrated all of these parameters and having deﬁned them in the simulation software, the models used in the program must reproduce them as realistically as possible to allow an accurate simulation of all secondary forming operations and of springback. Important considerations include the material model, the previously described implicit shell element, and also numerical parameters such as mesh reﬁnement and contact formulation. AutoForm currently uses Hill and Barlat anisotropic material models. Figure 2 shows an integrated simulation cycle using the example of a door inner panel. The production process consists of four operations. In the ﬁrst operation the blank is positioned in the die and the modiﬁcation of the geometry caused by gravity and by binder closure, respectively, is simulated. The next operation is deep drawing. The geometry of the door inner panel is drawn and 25 mm from the bottom stroke, four relief cuts are made in areas of the window and door storage compartment. Then, the inner area is drawn until the bottom stroke. In the next process step, the outer area of the door frame is formed. To simulate the tool change more precisely, the semi-ﬁnished part is removed from the ﬁrst tool and placed into the second tool. Tipping and positioning of the part in the second tool are automatically done if necessary, which simplify the simulation of the secondary operations. After the forming of the outer frame, several holes are cut into the geometry and the ﬁnal trimming cut is done. Finally, the outer ﬂange and some areas of the inner cuts are ﬂanged and calibrated. This last operation is also carried out after a tool change with automatic positioning of the semi-ﬁnished part and closing of the tool. The drawn part undergoes diﬀerent kinds of deformation modes during the several operations. This has been taken into account to keep the computation time as short as possible, by the application of diﬀerent element types. For the ﬁrst two operations (ﬁrst/ﬁnal drawing of the inner area and forming of the door frame), the stretch-drawing deformation mode prevails. Thus for these process steps the bending enhanced membrane element has been used. For the ﬂanging and calibration of the inner and outer ﬂanges, small radii compared to the sheet thickness are formed, which result in large bending eﬀects. Thus for these operations, the shell element has been used.

Simulation of Secondary Operations and Springback

Blank

355

Op. 4

Δs in %

BEM

Shell

1

–28%

–30%

2

–27%

–29%

3

–25%

–27%

4

–17%

–18%

5

–32%

–31%

6

–26%

–26%

Op. 1

Op. 3

1 Op. 2 3

6

4

2

DC04 (Hill 48) s0= 0.7 mm 1300 x 1100 mm

5

CPU (BEM)

Op.2

2.5 h

Op.4

7.5 h

CPU (Shell)

Op.2

10 h

Op.4

15 h

Fig. 2. Simulation of all secondary forming operations for a door inner panel, calculated on Xeon Processor (3,06 GHz / 512 KB) (Courtesy of Adam Opel AG)

The precondition to allow the switching between the two element types is that the simulation must produce comparable results for operations in which the stretch-drawing deformation mode prevails, independent of the selected element type. In other words for the door inner panel, after the forming of the outer frame, i.e. after the second operation, the same amount of stretching and thickening must exist in the sheet, independent of the selected element type – bending enhanced membrane element or shell element. Only then, can it be assured that the forming simulation provides identical results at the end of the simulation. To verify this, the simulation results determined with the bending enhanced membrane element and the shell element, respectively, are compared to each

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Advanced Methods in Material Forming

other in Fig. 2. In several areas the stretching (Δs in %) in the sheet has been examined (areas 1–6 in the ﬁgure). The examination has been carried out in areas in which the sheet undergoes the most stretching. The ﬁrst three areas are in the inner of the part and are formed in the ﬁrst operation. The next three areas are in the outer frame of the door inner panel and are formed in the second operation. In the inner of the part, the absolute maximum diﬀerence of the thinning amounts to 2%, and to 1% in the door frame. Also, the amount of draw-in matches for the two simulations: For a blank dimension of 1600 × 1400 mm the maximum diﬀerence amounts to 3 mm. Thus for this inner door panel, the bending enhanced membrane element can also be used for the simulation of the ﬁrst two operations without a risk of loss in computation accuracy. By combining the element types, a much shorter computing time is realized without considerably losing accuracy. In the example of the door inner panel, due to use of the membrane enhanced element in the ﬁrst two operations, it was possible to reduce the calculation time after the second operation by 75% and for the full forming process by 50% (see Fig. 2). This means in practice, a signiﬁcantly greater number of simulations can be run instead of just a single validation simulation, e.g. for an optimization or an analysis of robustness. 3.2 Springback Another essential link for the reproduction of the entire forming process is the simulation of springback. Due to springback, the sheet metal part undergoes a deformation such that the part at the end of the forming process does not correspond to the tool geometry. Such geometrical diﬀerences caused by springback mainly occur for two reasons: – –

due to interferences in the residual stress state of the formed parts (by trimming or by cutting holes) after removal of the external forces (removing the formed part from the tool) (Rohleder 02).

To compensate for the springback eﬀects the tool must be reworked, which causes further costs and requires more time. To avoid these additional eﬀorts during die design, the simulation of springback becomes more and more important in the forming simulation. Two kinds of springback computation can be distinguished. First, changes in the geometry caused by springback may occur when changing tools between the operations. Normally, this kind of springback is not measurable in reality, in the semi-ﬁnished part. If major changes in the geometry of the sheet metal part occur, compression in the semi-ﬁnished part may happen in the worst case when the next tool set is closed. Thus it is important to compute this kind of springback, which is called free springback in AutoForm-Incremental, in the simulation. In AutoForm-Incremental, the occurring residual stresses and the resulting changes in the geometry are calculated for the tool change and the simulation is continued with new deformed geometry and the new stress state.

Simulation of Secondary Operations and Springback

357

The necessary boundary conditions for suppression of rigid-body motions of the simulation are determined automatically. Another important aspect for the tool change is that a deformation of the geometry may be caused by the weight of the semi-ﬁnished part. For this reason, AutoForm-Incremental also considers the inﬂuence of gravity on free springback. If the geometry of the semi-ﬁnished part is to be evaluated and analyzed in the simulation after free springback, AutoForm allows for the deﬁnition of support points. The sheet metal part is tipped to the appropriate position based on the support points where it is possible to compare the current geometry to the target geometry. In the second type of springback computation, the springback in the ﬁnished sheet metal part is of interest. After removal of the part from the tool of the last secondary forming operation, springback of the ﬁnished sheet metal part is measured in a measurement ﬁxture. This is done to make sure that the ﬁnal part can be joined to the vehicle body in the appropriate position and with the pre-deﬁned welding points. The measuring ﬁxture mirrors similar boundary conditions as they will later occur in the vehicle. So the part is clamped, and shape deviation and the clamping forces in this clamped position are of interest. For the simulation with AutoForm-Incremental it is possible to deﬁne clamps acting normal to the surface and pilots acting tangential to the surface. Therefore the boundary conditions are not dealt with in the traditional way in Finite Element Analysis by simple elimination of degrees of freedom on nodes – as such boundary conditions move with the material in not ﬁxed directions, which obviously does not correspond to the behavior of clamps or pilots. Instead, AutoForm deals with these boundary conditions as non-linear contact conditions which remain stable in space and are iterated by the equilibrium iteration. The current geometry resulting from the computation can now be compared to a reference geometry, to analyze in detail the amount of springback and consequently the required modiﬁcations of tool and process. As the part is positioned and measured by the measuring instrument in the vehicle coordinate system, AutoForm-Incremental makes it possible to tip the part automatically from the tool coordinate system to the vehicle coordinate system, if needed. Other important aspects for the simulation are the clamping forces. So, the forces which are needed to keep the part in the measuring system in the desired position, are calculated. Later in the BIW (body in white), one has to make sure that the actual clamping forces to join the part to the residual car body, do not exceed the maximum allowed clamping forces of the clamp pliers. Early Determination of Springback The fender (outer panel) in Fig. 3 shows to what extent the simulation of springback in AutoForm has improved after the implicit shell element, the improved adaptive mesh reﬁnement, the improved contact algorithm, new material laws and a number of enhancements concerning the deﬁnition of boundary conditions for springback and springback evaluation have been

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Advanced Methods in Material Forming

Initial Blank A

Drawing A 1. Operation Trimming

2. Operation Trimming Sim: Sim: 2.0 mm Exp: Exp: 2.9 mm Free Springback Computed Normal Displ. Springback in marked areas

Sim: Sim: 10 mm Exp: Exp: 11 mm

Flanging

Cam Flanging

z11 y8 Springback

y7

y6

y9 y10 z12 y2 x1 y3 y5 y4

Sim: Sim: 0.8 mm Exp: Exp: 2.0 mm

Sim: Sim: 11 mm Exp: Exp: 15 mm Computed Normal Displacement Springback occurs in marked areas

AA6016 (Barlat) s0= 1.15 mm 1400 x 600 mm

Fig. 3. Veriﬁcation of the computed springback for a fender (Courtesy of DaimlerChrysler AG)

implemented. The fender is manufactured in four operations (Fig. 3, left side). First the sheet metal part is deep-drawn. After that a hole is cut and the outer boundary is trimmed. Subsequently the inner and outer ﬂanges are ﬂanged and cam ﬂanged. For the simulation of this outer body panel, the new shell element has been used continuously. To reproduce the forming process realistically, it is also important to calculate springback between the forming steps and to integrate these results into the further computations. The part undergoes a change in geometry caused by springback after the trimming and cam ﬂanging. After the drawing operation

Simulation of Secondary Operations and Springback

359

the semi-ﬁnished part is still stabilized by the ﬂange so that it can be assumed that the amount of anticipated springback is so small that it does not have to be considered for the simulation. After trimming, free springback is computed (Fig. 3, left side). After cam ﬂanging, i.e. at the end of the forming process, pre-deﬁned clamps and pilots, by which the boundary conditions have been deﬁned in the measuring ﬁxture, have been considered for the simulation of springback. For the fender in question, 12 clamps have been deﬁned. The normal displacement in the sheet metal part resulting from springback is shown in the lower part of Fig. 3. Signiﬁcant springback is seen above the wheel arch, at the connection of the A pillar and at the headlight (see marking). To verify the simulated springback for the fender in question, on the right side of Fig. 3 the computed geometries with the real geometries after the two forming steps trimming and cam ﬂanging, are shown for the section A above the wheel arch. The computed springback diﬀered from the experimental springback by a maximum of 4 mm (11 mm computed, 15 mm measured) at the end of the forming process (after cam ﬂanging) right above the wheel arch. In remaining areas and after the previous forming step the diﬀerences were lower. The example above shows that the simulation of springback has signiﬁcantly improved due to the above mentioned innovations. Thus the AutoForm program is able to simulate springback qualitatively and quantitatively with a reasonable accuracy for each operation, which makes possible shorter tryout times for the real tool. Compensation of Springback by Means of Simulation To compensate for springback, a stable and precise springback prediction should be calculated during the die design phase, before the tools are produced. To do this, an iterative optimization loop is recommended in which forming simulations take turn with geometry corrections and corrections of other process parameters. The loop is continued until the computed current geometry of the formed sheet metal part after springback, corresponds suﬃciently precisely to the target geometry. The requirement remains that the computational times even for complex geometries should be reasonable, so that several optimization computations can be carried out during the short development times that exist in practice. This is possible for the AutoForm program with its implicit shell formulation. Furthermore, AutoForm recommends the use of AutoForm-DieDesigner for the optimization of springback. With this software, user-deﬁned areas of the tool geometry are automatically modiﬁed until the calculated springback falls under the user-speciﬁed maximum allowable level. The application of AutoForm-DieDesigner ensures that the geometry modiﬁcations are checked to make sure that the corrected geometry is manufacturable, so that for example, no backdrafts are inserted in the draw die and the binder surface remains developable. The aim for the future is to automate this range of functions to facilitate the users’ work.

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4 Conclusion The main goal of the latest version of the AutoForm simulation program is to allow the entire forming process to be reproduced and analyzed in a simulation. The applications of the simulation program have been signiﬁcantly increased by the newly developed implicit shell formulation, the new material laws, the ahead adaptive mesh reﬁnement, and the enhancements in the deﬁnition, calculation and evaluation of springback. The AutoForm program uses the implicit method for both the simulation of secondary forming operations and springback. This makes sense because secondary forming operations are very often bending dominated, so that a static calculation of equilibrium control is essential. The newly developed implicit solver algorithm provides stable calculations in a reasonable computing time. Due to these enhancements it is now possible to also compute – in addition to the drawing operation – all secondary forming operations and springback with reasonable accuracy. Thus, the scope of practical applications for simulations has been enlarged: tool geometries and the process can be analyzed and optimized for all forming operations, and springback simulations for real industrial parts are now possible during process planning and die design.

References Glanzer, G., Selig, M., Steininger, V., Kubli, W., “Umformsimulation mit impliziter Schale – der Weg zur verbesserten R¨ uckfederungsberechnung”, 6. Workshop Simulation in der Umformtechnik, Stuttgart, 20th March 2003 Kubli W., Prozessoptimierte implizite FEM-Formulierung f¨ ur die Umformsimulation großﬂ¨ achiger Blechbauteile, Thesis, ETH Z¨ urich, VDI Verlag vol. 20, no. 204, 1996 Rohleder M.W., Simulation r¨ uckfederungsbedingter Formabweichungen im Produktentstehungsprozess von Blechformteilen, Thesis, Universit¨ at Dortmund, Shaker Verlag 2002 ´ “Zukunftsweisende L¨ Sch¨ onbach, E., osungen in der Umformsimulation und ihr konsequenter Einsatz im Produktentstehungsprozess der Automobilindustrie”, DIF-conference, Wiesbaden, June 2003 Sch¨ onbach, Th., Steininger, V., Kubli, W., “Geschwindigkeitsrausch – Die schnelle Produktentwicklung von Karosserieformteilen mit Simulationstechnik”, Maschinenmarkt, 27/2003, p. 36–39 Selig M., Die implizite Schale in AutoForm, Usermeeting 2003, Barcelona, Spanien, May 2003

Author Index

Alves, J. L., 35 Azaouzi, M., 233 Bambach, M., 233 Banabic, D., 151 Bariani, P. F., 189 Barlat, F., 1, 131 Batoz, J. L., 233 Bigot, R., 309 Bonte, M. H. A., 55 Cannamela, M., 233 Cao, J., 297 Carton, M., 321 Cazacu, O., 131 Cezard, P., 309 Chaparro, B. M., 35 Chatti, S., 101 Cheng, H. S., 297 Com¸sa, D. S., 151 Cristescu, N. D., 199 Duﬂou, J. R., 251 Dupret, F., 73 Favier, V., 309 Fernandes, J. V., 35 Filice, L., 263 Geiger, M., 119 Gelin, J. C., 167 Ghiotti, A., 189 Glanzer, G., 349 Gracio, J. J., 151

Hermes, M., 101 Hirt, G., 233 Hu´etink, J., 55 Hussnaetter, W., 119 Kleiner, M., 101 Koeune, R., 321 Kotsos, P., 279 Kubli, W., 349 Labergere, C., 167 Lauwers, B., 251 Lecomte-Beckers, J., 321 Loix, F., 73 Maeder, G., 19 Mahayotsanun, N., 297 Menezes, L. F., 35 Merklein, M., 119 Micari, F., 263 Mitsoulis, E., 279 Olejnik, L., 215 Paraianu, L., 151 Rassili, A., 321 Richert, M., 215 Robelet, M., 309 Rosochowski, A., 215 ´ 349 Sch¨ onbach, E., Selig, M., 349

362

Author Index

Settineri, L., 263 Soare, S., 131

Umbrello, D., 263 van den Boogaard, A. H., 55 Verbert, J., 251

Thibaud, S., 167 Thibaut, V., 73

Yoon, J. W., 131