ANALYSIS OF MICROFORMING: MICROMECHANICAL AND NUMERICAL ASPECTS


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1 ANALYSIS OF MICROFORMING: MICROMECHANICAL AND NUMERICAL ASPECTS a thesis submitted to the graduate school of natural and applied sciences of atilim university by EMRAH DEMİRCİ in partial fulfillment of the requirements for the degree of master of science in the department of manufacturing engineering september 2007
2 I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science. Prof. Dr.Ing. A. Erman Tekkaya (Supervisor) I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science. Instr. Celal Soyarslan (CoSupervisor) I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science. Asst. Prof. Dr. Serhat Erpolat Approved for the Institute of Engineering and Science: Prof. Dr. Selçuk Soyupak Director of the Institute Engineering and Science ii
3 I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Emrah Demirci iii
4 ABSTRACT ANALYSIS OF MICROFORMING: MICROMECHANICAL AND NUMERICAL ASPECTS Emrah Demirci M.S. in Manufacturing Engineering Supervisor: Prof. Dr.Ing. A. Erman Tekkaya CoSupervisor: Instr. Celal Soyarslan September 2007, 138 pages The increasing market volume of electronic and micromechanical components makes a general trend towards higher integrated functional density and miniaturization of equipments. In order to meet the demands, design of the production processes of microparts should be investigated both practically and analytically. This thesis is prepared to contribute the analytical investigation of microforming. A simplified modeling approach is proposed for the Finite Element Method (FEM) simulation of microforming processes. The new modeling approach aims to explain the differences between conventional forming and microforming which are termed as size effects. The size effects due to the decreasing scale of the workpiece size, is explained in terms of inhomogeneous grain structure of the material. Therefore the proposed approach for the FEM simulation is modeling the material as individual grains which possess anisotropic mechanical properties. The directional response of the grains is represented by Hill s anisotropic material model. 2D and 3D simulations of common forming processes were performed with the three possible material models (isotropic, single grain, multigrain) and compared with the literature. Moreover, an experimental setup including a conceptual microforming press is designed. Keywords: Microforming, Conventional forming, Size effects, Hill s anisotropy, Finite Element Method. iv
5 ÖZET MİKROŞEKİLLENDİRME ANALİZİ: MİKROMEKANİK VE SAYISAL YÖNLERİ Emrah Demirci Üretim Mühendisliği, Yüksek Lisans Tez Yöneticisi: Prof. Dr.Ing. A. Erman Tekkaya Ortak Tez Yöneticisi: Ögr. Gör. Celal Soyarslan Eylül 2007, 138 sayfa Artan elektronik ve mikromekanik parçaların piyasa hacmi donanımların daha yoğun işlevsellik kazanmalarına ve küçülmelerine yol açmaktadır. Talepleri karşılamak için mikroparçaların üretim yöntemlerinin tasarımı hem uygulamalı ve hem de analitik olarak araştırılmalıdır. Bu tez mikroşekillendirmenin analitik olarak araştırılmasına katkıda bulunmak için hazırlanmıştır. Mikroşekillendirme yöntemlerinin sonlu elemanlar yöntemi ile simülasyonu için basitleştirilmiş bir modelleme yaklaşımı önerilmiştir. Yeni modelleme yaklaşımı boyut etkileri diye adlandırılan geleneksel şekillendirme ve mikroşekillendirme arasındaki farkları açıklamayı hedefler. Azalan iş parçası ebat ölçeği nedeniyle oluşan boyut etkileri malzemenin homojen olmayan tanecik yapısı ile açıklanmıştır. Sonuçta sonlu elemanlar yöntemi simülasyonu için önerilen yaklaşım, malzemeyi anisotropik mekanik özelliklere sahip bireysel tanecikler olarak modellemektir. Taneciklerin yönsel tepkileri Hill in anizotropik malzeme modeli ile temsil edilmiştir. Üç mümkün malzeme modeli (izotropik, tek tanecik, çoklu tanecik) ile 2 ve 3 boyutlu yaygın şekillendirme yöntemlerinin simülasyonları yapılmıştır ve literatür ile karşılaştırılmıştır. Dahası, konsept bir mikroşekillendirme presi içeren bir deney düzeneği tasarlanmıştır. Anahtar sözcükler: Mikroşekillendirme, Geleneksel şekillendirme, Boyut etkileri, Hill in anizotropik malzeme modeli, Sonlu elemanlar yöntemi. v
6 To my parents, vi
7 Acknowledgement I would like to express my gratefulness and appreciation to Prof. Dr.Ing A. Erman Tekkaya for guidance and insight in supervising the thesis, his invaluable help, and kindness. I also wish to express my gratitude to Instr. Celal Soyarslan for his helpful comments and suggestions for developing my thesis. I would like to thank to Asst. Prof. Dr. Serhat Erpolat for his invaluable guidance. Many thanks to Prof. Dr. Niels Bay for the wonderful visit to Denmark Technical University and sharing the knowhows. The support of State Planning Organization of Turkey is acknowledged. I would like to thank my colleagues Alper Güner, Tahsin T. Öpöz, Aylin K. Eroğlu and M. Burcu Bildirgen for their great support and valuable contribution. My best wishes to Dilek, Gözde, Alkım and the Atılım University Solar Car Team members for their strong fellowship. Finally, I would like to appreciate my parents, Hasan Fahri & Aynur Demirci for their patience, love, encouragement and helpful support during my thesis work. To my brother Erman, for his patience and endless support. You are loved deeply. I would like to thank my love Fatma for her invaluable support, kindness, and for being in my life with her endless love. vii
8 Contents 1 Introduction Motivation Definition of Microforming Literature Survey Size Effects Material Models for FEM Simulations Tool Manufacturing Techniques Microforming Presses Need for advanced material models Basics of Mechanical Properties Material Symmetry and Corresponding Tensor Structures Anisotropic Material Model Orthotropic Material Model Isotropic Material Model viii
9 CONTENTS ix 2.2 Scale Concept in Mechanics Size Effects in Granular Materials Representative Volume Element (RVE) Concept Homogenization and Its Use in Mesomodeling Micromechanical Survey: Metallic Material Microstructure Defects and Deformation Mechanisms Dislocations Dislocation Motion and Hardening From Single Crystal to Polycrystalline Sources of Anisotropy Slip Systems Possible Slip Planes and Flow Patterns Phenomenological Survey: Modeling the Behavior Introduction to the Mathematical Theory of Plasticity Yield Criterion Flow Rule Strain Hardening A Model Problem: Isotropic J 2 Flow Theory Extensions to Anisotropy A Model Problem: Hill s Anisotropic Flow Theory
10 CONTENTS x 5 Modeling MSC.Marc R as a Simulation Tool for Direction Dependent Response Material Library Element Library Limitations Model 1. Plane Strain Upsetting Introduction and Motivation FEM Models and Material Parameters Simulation Results Results and Discussions Model 2. 3D Tensile Testing Introduction and Motivation FEM Models and Material Parameters Simulation Results Results and Discussions Model 3. Extrusion of a Cylindrical Billet Introduction and Motivation FEM Models and Material Parameters Simulation Results Results and Discussions
11 CONTENTS xi 5.5 Model 4. Backward Can Extrusion Introduction and Motivation FEM Models and Material Parameters Simulation Results Results and Discussions Experimental Setup Design A Conceptual Press Design for Microforming Linear Actuator Type Press Frame Design Design of Die and Punch Systems Control and Data Acquisition in the Experimental Setup Experimental Procedure Conclusion and Future Perspectives 101 A Conceptual Press Equipments 108 B Detailed Drawings of the Conceptual Press 110 C The Image Processing Code for Obtaining 2D Grain Models 117
12 List of Figures 1.1 Upsetting of a material with inhomogeneous grain structure [8] Definition of microforming according to [6] Upsetting of a material with inhomogeneous grain structure [7] Backward can extrusion performed with a billet composed of heterogeneous grain structure and its crosssection [6] Effect of grain size and position to flow stress [11] Flow stress decreases and scatters with decreasing size of material [7] Effect of OLPs and CLPs on friction stress [5] Photograph of a contact surface with 3Dprofilometry (Contact asperities: black, CLP: white, OLP: grey) [5] Variation of OLP and CLP region with decreasing size [5] MicroEDM machine (Courtesy of ATILIM University) SCHMIDT Servo Press 4000 (www.schmidtpresses.com/servo.html) Microforming press designed by DTU (Courtesy of DTU) Simulation of upsetting using mesoscopic model xii
13 LIST OF FIGURES xiii 2.1 Principal directions in orthotropic material model Backward can extrusion of a coarse grained billet with a grain larger than the cup thickness [8] Flow curve of CuZn15 specimens with different scales [7] The formation of a step on the surface of a crystal by the motion of a) an edge dislocation and b) a screw dislocation Atomic rearrangements that accompany the motion of an edge dislocation as it moves in response to an applied shear stress. Bonding and debonding of atoms occur gradually from (a) to (c) as the half plan of atoms move rightwards Representation of the analogy between caterpillar and dislocation motion Schematic representation of grains with different crystallographic texture Dislocation motion near the grain boundary Slip lines on the surface of a polycrystalline specimen of copper [2] a) A {111}<110> slip system in an FCC unit cell, b) The {111} plane from (a) and three <110> slip directions [2] Slip systems for FCC, BCC and HCP metals [2] Geometrical relationship between the tensile axis, slip plane and the slip direction used in calculating the resolved shear stress for a single crystal Slip in an aluminum single crystal [18] Slip systems of BCC crystals
14 LIST OF FIGURES xiv 4.1 Behavior of the material on a stressstrain curve Comparison of Tresca and von Mises yield criteria in plane stress condition Demonstration of normality rule in a plane stress condition for von Mises yield function Axes of anisotropy in a coldrolled sheet Comparison of the yield loci of isotropic (von Mises) and anisotropic (Hill) yield functions Principal axes of a grain model for Hill s criterion Nodes and integration point for Element Upsetting of a material with inhomogeneous grain structure [7] Cold rolled iron sheet after recrystallization Isotropic and single grain FEM model of the Figure Multigrain FEM model of the Figure Orientations of the grains in Figure Deformed shape of the material with isotropic model Deformed shape of the single grain material with Hill s model Deformed shape of the multigrain material with Hill s model Equivalent plastic strain distribution of the material with isotropic model Equivalent plastic strain distribution of the single grain material with Hill s model
15 LIST OF FIGURES xv 5.13 Equivalent plastic strain distribution of the multigrain material with Hill s model Equivalent von Mises stress distribution of the material with isotropic model Equivalent von Mises stress distribution of the single grain material with Hill s model Equivalent von Mises stress distribution of the multigrain material with Hill s model Force vs. displacement graph of the three models Model of the tensile testing simulations Deformed shape of the specimen with isotropic model Deformed shape of the single grain specimen with Hill s model Equivalent plastic strain distribution of the specimen with isotropic model Equivalent plastic strain distribution of the single grain specimen with Hill s model Equivalent von Mises stress distribution of the specimen with isotropic model Equivalent von Mises stress distribution of the single grain specimen with Hill s model Evolution of the lattice orientation during tensile testing constraints [18] on the left. Necking region of the single grain model on the right Cross section of the necking region of the isotropic model on the left, single grain model on the right
16 LIST OF FIGURES xvi 5.27 Force vs. displacement graph of the two tensile testing models Isotropic and single grain FEM model for the extrusion simulation Multigrain FEM model for the extrusion simulation Deformed shape of the material with isotropic model Deformed shape of the single grain material with Hill s model Deformed shape of the multigrain material with Hill s model Equivalent plastic strain distribution of the material with isotropic model Equivalent plastic strain distribution of the single grain material with Hill s model Equivalent plastic strain distribution of the multigrain material with Hill s model Equivalent von Mises stress distribution of the material with isotropic model Equivalent von Mises stress distribution of the single grain material with Hill s model Equivalent von Mises stress distribution of the multigrain material with Hill s model Force vs. displacement graph of the three extrusion models Nonuniform edge formation in micro backward can extrusion process[7] Isotropic and single grain FEM model for the backward can extrusion simulation Multigrain FEM model for the backward can extrusion simulation. 76
17 LIST OF FIGURES xvii 5.43 Deformed shape of the material with isotropic model Deformed shape of the single grain material with Hill s model Deformed shape of the multigrain material with Hill s model Equivalent plastic strain distribution of the material with isotropic model Equivalent plastic strain distribution of the single grain material with Hill s model Equivalent plastic strain distribution of the multigrain material with Hill s model Equivalent von Mises stress distribution of the material with isotropic model Equivalent von Mises stress distribution of the single grain material with Hill s model Equivalent von Mises stress distribution of the multigrain material with Hill s model Comparison of the rim edges of the multigrain model and Figure Force vs. displacement graph of the three backward can extrusion models Comparison of conventional linear actuators (www.exlar.com) Contact with screw road and its counter part in ball screw (upper), roller screw (lower) Roller screw type linear actuator (www.exlar.com) Main parts of the conceptual microforming press design
18 LIST OF FIGURES xviii 6.5 General view and properties of the conceptual microforming press Crosssectional view of the die assembly Crosssectional view of the punch assembly Control schema of the conceptual press design Schematic representation of an EBSD system Determination of the grain structure and lattice orientation of each grain with EBSD Experimental procedure for determining the input parameters for Hill s yield function Tensile testing specimen containing a single grain in the gage section [19] Cold rolled iron sheet after recrystallization at 800C [18] CAD model of the grains in Figure D FEM model of the specimen in Figure 6.12 [19] Specimen produced by backward can extrusion at a) 20 C and b) 300 C [4]
19 List of Tables 1.1 Material models for FEM simulation of micro bulk metal forming processes Modulus of elasticity values of some metals in macroscale Modulus of elasticity values for several metals at various crystallographic orientations in microscale [2] Material parameters for the isotropic model Material parameters for the single grain and multigrain models Material parameters for the isotropic model Material parameters for the single grain model Material parameters for the isotropic model Material parameters for the single grain and multigrain models Material parameters for the isotropic model Material parameters for the single grain and multigrain models.. 76 A.1 Commercial parts of conceptual microforming press xix
20 Chapter 1 Introduction 1.1 Motivation As the technology develops, equipments become smaller especially in the electronics industry. Therefore economical production of miniature parts gain importance day by day. It possible to produce these parts with turning, milling, etching etc., but these methods are either slow or expensive for mass production. The fastest and the most economical way for mass production of metals is forming. Miniature parts are commonly used in microsystem technologies and microelectromechanical systems (MEMS). Typical examples of these microparts are pins for IC carriers, microfasteners, microscrews, microcups and connectors. The demand for these microparts is increasing every year. Sample microparts are shown in Figure 1.1. The reason why microforming technology is not used today is the knowhow of conventional metal forming processes can not be scaled down to microscale for microforming processes due to many reasons so called size effects. New solutions should be found for dies, machine tools, material characteristics etc. concerning microforming operations.
21 Definition of Microforming 2 Figure 1.1: Upsetting of a material with inhomogeneous grain structure [8]. Another aspect is the Finite Element Method (FEM) simulation of processes to predict the billet and die behaviors. In the microscale, as mentioned above, the material characteristics also change. As a result, new material models should be found and tested via accurate experimental setups. In order to design efficient tools for microforming processes, correct material model must be used in FEM simulations. To sum up, in order to cope with the increasing demand for microparts, microforming technology should be improved both practically and analytically. 1.2 Definition of Microforming From geometric point of view, microforming is the production of parts or structures with at least two dimensions in the submillimeter range [6]. The definition is described in Figure 1.2. The geometric definition is insufficient to explain the difference between conventional forming and microforming. The size effects, which differ microforming from conventional forming, are the consequences of the grain structure of the workpiece. The forming process is addressed as conventional forming if the the workpiece is fine grained regardless of the workpiece dimensions. On the
22 Literature Survey 3 Figure 1.2: Definition of microforming according to [6]. other hand, if the workpiece is coarse grained (Figure 1.3), the overall deformation behavior of the workpiece depends on the deformation behavior of individual grains, which leads to nonuniform deformation throughout the material. Therefore, the forming process in which the workpiece is composed of a single grain or a few grains, which leads to inhomogeneous deformation, could be defined as microforming. Regardless of workpiece dimensions, the dimension of mean grain size with respect to the workpiece dimensions determine the homogeneity of the structure and hence, the forming type. 1.3 Literature Survey The aim of the literature survey is to search the condition of research activities about microforming among universities, institutes and companies. According to the deep literature survey, microforming topic has four aspects Size Effects Materials are considered to be homogeneous in the conventional metal forming operations; on the other hand it is heterogeneous in microforming processes. Therefore the material shows anisotropic behavior in the submillimeter dimensional range.
23 Literature Survey 4 Firstly, grain size is very important in microforming. As the grain size increases, the grain structure of the billet becomes more heterogeneous as shown in Figure 1.3. Therefore, the anisotropic behavior of the material produces different local deformation mechanisms as shown in Figure 1.4. Figure 1.3: Upsetting of a material with inhomogeneous grain structure [7]. Figure 1.4: Backward can extrusion performed with a billet composed of heterogeneous grain structure and its crosssection [6]. Secondly, in metal physics theory, free surface grains show less hardening compared to the inner volume grains which can be explained by the different mechanisms of dislocation movement and pileups [7]. Hence, as the material size decreases, the ratio of the volume of surface grains to the overall volume increases and overall flow stress decreases. The size and position of the individuals are very important in terms of flow stress. This situation is described in Figure 1.5. The flow stress of grains positioned close to the free surface is reduced compared to
24 Literature Survey 5 grains in the inner area. The flow stress of a grain is strongly dependent on its size. As long as its size is not at nanoscale HallPetch equation can be utilized to calculate the flow stress of a grain in terms of its size. These size effects leads to reduced and scattered flow curve data as depicted in Figure 1.6. Figure 1.5: Effect of grain size and position to flow stress [11]. Figure 1.6: Flow stress decreases and scatters with decreasing size of material [7]. Finally, tribologic behavior also changes at microscale. Contact characteristics between tools and the billet become heterogeneous. The contact surface is composed of two parts: contact asperities and lubrication pockets. There are two types of lubrication pockets: open and closed lubrication pockets. With increasing normal pressure the lubricant escapes and the forming load only acts on
25 Literature Survey 6 the asperities, producing high friction stress. This type of lubrication pocket is known as open lubrication pocket (OLP). On the other hand, closed lubrication pockets (CLP) do not have connection to the edge of the surface hence lubricant gets trapped, producing low friction stress. The effect of the two lubrication pocket types on friction stress is shown in Figure 1.7. In Figure 1.8 are lubrication pockets explained via photograph obtained by 3Dprofilometry. Figure 1.7: Effect of OLPs and CLPs on friction stress [5]. Figure 1.8: Photograph of a contact surface with 3Dprofilometry (Contact asperities: black, CLP: white, OLP: grey) [5]. As the material size decreases, the ratio of OLP area to CLP area increases.
26 Literature Survey 7 Therefore, when compared with conventional forming, microforming processes exhibit more friction stress. This phenomenon is described in Figure 1.9. Figure 1.9: Variation of OLP and CLP region with decreasing size [5]. In conclusion, due to size effects, knowhow of conventional metal forming processes can not be easily transferred to the microforming processes. As explained before, flow curve changes, inhomogeneous deformation occurs and friction stress increases with decreasing material size in the submillimeter range Material Models for FEM Simulations FEM is the key to mechanical design in most of the engineering fields. Metal forming is one of the engineering fields that need highly complex mechanical design including large plastic deformation of materials. Detailed information about FEM for the simulation of plastic deformation of the materials could be found in [35], [3] and [21]. In order to use FEM in a metal forming simulation, a suitable material model must be used. Generally isotropic material properties are used in conventional metal forming simulations. On the other hand, isotropic material models are insufficient for microforming FEM simulations. There are different proposals about the material characterization for microforming simulations. Each of them has advantages and disadvantages. Because these material models include complicated formulations and assumptions, they are not going to be explained in a detailed manner. These material models are explained briefly in Table 1.1.
27 Literature Survey 8 Material Model Surface Layer Model [9] Mesoscopic Model [12] Conventional Theory of Mechanism Based Strain Gradient [17], [16] Grain Elements Model [22] Table 1.1: Material models for FEM simulation of micro bulk metal forming processes. Results Disadvantages Proposer Main Parameters Experimental Verification U. Engel et al. (University of Erlangen, GERMANY) U. Engel et al. (University of Erlangen, GERMANY) Y. Huang et al. (University of Illinois, USA) T.W. Ku et al. (Pusan National University, South Korea) Average grain size Done Forming force prediction is good Geometry and position of each grain in the material Geometrically necessary and statistically stored dislocation densities Average grain size, grain and grain boundary properties Done Consistent with the experiments Done Applies in the range of 0,1 mm10 mm Unable to explain local forming behavior and scatter of the flow curve data Obtaining 3D model of grains, application of anisotropy Hard to find lots of parameters Not yet No Hard to find parameters, experience based predicted material properties
28 Literature Survey Tool Manufacturing Techniques When the word micro is used, the tolerance gains importance first. Tolerance values in the conventional forming dies are on the order of 0.05 mm. If we compare the scale difference between the conventional and the microforming dies, it is very had to attain such micron level tolerances in microforming dies. Therefore very accurate production techniques should be used for microforming die production. There are three famous techniques used for tool manufacturing in microforming field Chemical Vapor Deposition (CVD) This method can be explained shortly as; silicon dies are fabricated on silicon substrates by deep reactive ion etching and KOH etching, which are negative in shape to the microforming dies. Seeding with diamond nanocrystals is performed to deposit diamond films uniformly on the silicon negatives. The deposited diamond film is adhered to an alumina plate with ceramic plate. Then silicon substrates are removed by KOH etching and the microforming dies are obtained. This method is frequently is in MEMS, but it is more expensive than the other methods. [14] Photolithographic Etching In this method metallic glasses are used as micro dies. When a metallic glass is exposed to UV light its reaction speed with HCl acid increases 30 times. After unmasking the region where removal of die material is desired, expose it with UV light and clean the unmasked region with HCl. This technique is not suitable for metallic dies such as Tungsten Carbide. [23]
29 Literature Survey MicroElectrical Discharge Machining (microedm) Nowadays usage of microedm technique is increasing due to the facts that it is cheaper and faster and more flexible in product range with respect to the other two techniques. It combines EDM technology with 3axis milling to produce complex shapes [34]. ATILIM University owns a microedm machine which is shown in Figure Figure 1.10: MicroEDM machine (Courtesy of ATILIM University) Microforming Presses Tolerance accuracy is very important in microforming operations. Therefore, when compared with conventional forming presses microforming presses should be more precise. Besides, due to decreased size and flow curve, less forming forces are required leading to the usage of small sized presses. Seldom companies are producing such precise and small presses, such as SCHMIDT (Figure 1.11). Servo Press 4000 can apply forces up to 100 kn. Most of the research groups design their
30 Literature Survey 11 own microforming press such as Denmark Technical University (DTU) (Figure 1.12). The press in DTU has the capacity of applying 5000 N. A linear electric motor which is a very new technology is used for the linear actuator. Detailed information about microforming presses could be found in [30] and [28]. Figure 1.11: SCHMIDT Servo Press 4000 (www.schmidtpresses.com/servo.html).
31 Literature Survey 12 Figure 1.12: Microforming press designed by DTU (Courtesy of DTU).
32 Need for advanced material models Need for advanced material models The aspects of microforming are given briefly in the previous section. Except for the material models for the FEM simulation of microforming processes, the other fields are well improved to cover the demand of the industry. However, without simulating the microforming processes properly, tool and process design stages will be inefficient. The material models given in Table 1.1 are proposed to simulate the material behavior in microforming. These models can predict the overall forming force of the process, but the common main disadvantage of these models is that they can not represent the anisotropic behavior of grains which causes nonuniform deformation throughout the material. In the Figure 1.13 is the simulation result of an upsetting workpiece composed of ten grains modeled with mesoscopic model. The surface of the deformed shape is a proof of uniform deformation due to isotropic material properties. On the other hand, in the real case the surface of the deformed material is nonuniform (Figure 1.3). Figure 1.13: Simulation of upsetting using mesoscopic model. Rather than isotropic, anisotropic modeling is required for the simulation of microforming processes. This could be performed with grain by grain modeling
33 Need for advanced material models 14 of the material regarding microstructural information such as slip orientation and directional response of the grains. From the crystal plasticity point of view, it is possible to simulate the material characteristics of a single grain with knowing the crystallographic slip direction of the grain. In most of the microforming processes, billets are composed of a few grains or the billet itself is a single grain. To sum up, a workpiece composed of a few grains will display anisotropic deformation behavior due to inhomogeneous plastic deformation characteristics of individual grains. In order to simulate the microforming processes, the deformation behavior of grains should be modeled correctly with applying anisotropy to the material properties.
34 Chapter 2 Basics of Mechanical Properties Mechanical properties of the solid materials are represented with terms illustrating the behavior of that material in a specific dimensional scale. These properties may change with varying scales and microstructure. This chapter is due to material symmetry which affects the level of anisotropy and scale concept which affects the mechanical properties. 2.1 Material Symmetry and Corresponding Tensor Structures A crystal is a three dimensional pattern of atoms. The basic concept to describe the atomic structure is to define its motif where a motif is the periodically repeated arrangement of atoms in space to generate the pattern. The motif of a crystal depends upon the chemical identity of the material of the crystal and can be a single atom, a single molecule or a group of molecules. The periodic spatial arrangement of the motif is mathematically described by a space lattice. There are symmetry planes in these motives where mechanical properties do not change. Infinite numbers of symmetry planes exist in the materials which are assumed as isotropic. [26]
35 Material Symmetry and Corresponding Tensor Structures 16 Material symmetry can be defined at a single material particle and/or within a region of material and can change from particletoparticle and/or region to region. Any symmetry in material structure will be reflected by symmetry in mechanical properties. An inhomogeneous material results if the symmetry changes and/or the properties change throughout the material. The level of inhomogeneity changes according to the three material models: anisotropic, orthotropic and isotropic material models. [32] Anisotropic Material Model An anisotropic material lacks any material symmetry. The 4 th order elasticity tensor (or tensor of elastic constants) C ijkl, that relates stress tensor to strain tensor, is fully populated and contains 3 4 = 81 terms. C ijkl is found according to Equation 2.1 where Ψ e is the elastic potential energy and e are the elastic strains from strain tensor. C ijkl = 2 Ψ e ε e ij εe kl (2.1) However, due to symmetry of the stress and strain tensors and due to the existence of the strain energy function, the number of independent elastic constants is reduced to 21 for the fully anisotropic material [32]. Therefore the fourth order indices, ijkl, are mapped to their matrix counterparts as follows, 1 11, 22, 33, 12, 13, 23 1, 2, 3, 4, 5, 6 (2.2) and the elastic constants reduce to the form, C 11 C 12 C 13 C 14 C 15 C 16 C 21 C 22 C 23 C 24 C 25 C 26 [C] anisotropic = C 31 C 32 C 33 C 34 C 35 C 36 C 41 C 42 C 43 C 44 C 45 C 46 C 51 C 52 C 53 C 54 C 55 C 56 C 61 C 62 C 63 C 64 C 65 C 66 (2.3) 1 For instance C 1123 reduces to C 16.
36 Material Symmetry and Corresponding Tensor Structures Orthotropic Material Model An orthotropic material is symmetric about three orthogonal planes and possesses 9 independent elastic constants as seen in Equation 2.4. C 11 C 12 C C 21 C 22 C [C] orthotropic = C 31 C 32 C C (2.4) C C 66 This can be shown by performing a 180 rotation of two of the coordinate axes as shown in Figure 2.1 and enforcing no change in the elastic constants. Figure 2.1: Principal directions in orthotropic material model Isotropic Material Model An isotropic material possesses an infinite number of planes of symmetry and the number of independent elastic constants reduces to 2. C 11 C 12 C C 12 C 11 C [C] isotropic = C 12 C 12 C C C C C C 11 C 12 2 (2.5)
37 Scale Concept in Mechanics Scale Concept in Mechanics Size Effects in Granular Materials Size effects in microforming are described in the literature survey part. Since material is composed of grains, the overall size of the material with respect to its grain size determines the strength properties of it. The material can be assumed homogeneous, if the grain is so small (fine grained) with respect to the overall size of the material resulting homogeneous deformation under loading with varying crystallographic texture of individual grains. In the coarse grained case, the deformation behavior of the material is highly dependent on the deformation behavior of individual grains, Figure 2.2. This is where size effects are taken into consideration in mechanics. The grain size affects the strength of the material Figure 2.2: Backward can extrusion of a coarse grained billet with a grain larger than the cup thickness [8]. according to the HallPetch relationship in Equation 2.6 where K is the fitting constant, σ 0 is the yield stress of a single crystal and d is the mean grain size. σ y = σ 0 + K (2.6) d The HallPetch relationship describes the yield stress of the material roughly with the initial yield stress value of a single crystal which is the necessary stress required to initiate slip mechanism in the grain. With increasing grain size, the
38 Scale Concept in Mechanics 19 yield stress of the material decreases. The reverse is also correct in a way that, as the size of the material approaches to the size of a single grain, the yield stress of the material approaches the yield stress of a single crystal, i.e. σ 0. Grain boundaries are the main obstacles for the dislocation movements which is source of anisotropy. Dislocation pileups at the grain boundaries harden the material. A fine grained material is harder and stronger than one that is coarse grained, since the former has a greater total grain boundary area to impede dislocation motion. Similarly, as the material size decreases, due to the less number of grains causing less total grain boundary area, the yield stress of the material decreases. Moreover, free surface grains show less hardening compared to the inner volume grains due to lack of grain boundaries at the free surface side of the surface grains. Therefore the volumetric ratio of the free surface grains to the inner volume grains increases as the material size decreases Representative Volume Element (RVE) Concept The RVE is a model of the material to be used to determine the corresponding effective properties for the homogenized macroscopic model. RVE is the minimal material volume, which contains statistically enough mechanisms of deformation processes. The increase of this volume should not lead to changes of evolution equations for field values, describing these mechanisms. For instance, RVE is a unit cell for a periodic microstructure and volume containing a very large set of grains, possessing statistically homogeneous properties. [13] The microscale is characterized by a statistically representative volume which contains microstructural information such as crystallographic texture [25]. The RVE should be large enough to contain sufficient information about the microstructure in order to be representative; however it should be much smaller than the macroscopic body because it is the smallest possible volume that reflects macroscopic properties. RVE for the material to be used in microforming is highly dependent on
39 Scale Concept in Mechanics 20 Table 2.1: Modulus of elasticity values of some metals in macroscale. Metal Modulus of Elasticity (GPa) Aluminum 68.0 Copper Iron the grain size. If the material is fine grained, the material may be assumed as homogeneous and isotropic material properties could be used for RVE. On the other hand, if the material is coarse grained, it may be assumed as inhomogeneous and anisotropic material properties could be used for RVE Homogenization and Its Use in Mesomodeling The mechanical behavior of materials within structures under mechanical load is the result of mechanisms active at a microscale within their constituents and at their interfaces, and of the spatial distribution of these constituents. The prediction of the macroscopic behavior, based on these data, requires complex scaletransition operations representing the interaction phenomena between the constituents. Instead some mathematical constituents in the form of mechanical properties such as modulus of elasticity and in the form of field variables such as stress and strain are introduced to illustrate the mechanical behavior. This approach is called homogenization. Homogenization intrinsically includes volume averaging of microscalar properties to represent them in the macroscale. [25] For an isotropic material, RVE is a very large set of grains. As a result of homogenization, elastic behavior of a very large set of grains is represented by a single modulus of elasticity value for each metal in Table 2.1. As the scale gets smaller, the data starts to scatter with respect to the crystallographic texture (Table 2.2). Therefore a single modulus of elasticity term loses its validity for representing the elastic behavior in the microscale. From the macroscopic point of view flow stress is the necessary stress to deform the
40 Scale Concept in Mechanics 21 Table 2.2: Modulus of elasticity values for several metals at various crystallographic orientations in microscale [2]. Modulus of Elasticity (GPa) Metal [100] [110] [111] Aluminum Copper Iron material plastically. In the microscale, plastic flow of the material is the sliding of dislocations on the slip planes in slip directions. Therefore flow stress at macroscale is a result of homogenization of flow response of individual grains. As the scale of the deformed material gets smaller the flow stress scatters and loses its validity as shown in Figure 2.3. Figure 2.3: Flow curve of CuZn15 specimens with different scales [7].
41 Chapter 3 Micromechanical Survey: Metallic Material Microstructure 3.1 Defects and Deformation Mechanisms The billets used in micro bulk metal forming have dimensions less than 1 mm. The deformation behavior of fine grained microforming billets are identical to the billets used in conventional metal forming. The effect of microstructure on the deformation of a microforming billet increases as the mean grain size becomes comparable with the dimensions of the microforming billet. The billet may be composed of a few grains or the billet itself is a single crystal. In order to have a better understanding of the deformation behavior of the coarse grained microforming billets, grain deformation mechanisms should be understood. Grains contain large numbers of various defects or imperfections. The properties of the materials are influenced by the presence of imperfections. Crystalline defect is a kind of imperfection due to a lattice irregularity having one or more of its dimensions on the order of an atomic diameter. Crystalline defects are classified according their geometry or dimensionality. Point defects, linear defects and interfacial defects are types of crystalline defects. Deformation behavior of crystals is mainly governed by linear defects. Therefore, linear defects are the
42 Defects and Deformation Mechanisms 23 main concern in this session Dislocations A dislocation is a linear or onedimensional defect around which some of the atoms are misaligned. There are two types of dislocations namely: edge and screw dislocations. In an edge dislocation, lattice distortion exists along the end of an extra half plane of atoms (Figure 3.1a). A screw dislocation is a torsional distortion of plane of atoms around an axis (Figure 3.1b). [2] Figure 3.1: The formation of a step on the surface of a crystal by the motion of a) an edge dislocation and b) a screw dislocation.
43 Defects and Deformation Mechanisms Dislocation Motion and Hardening Plastic deformation of a grain corresponds to the motion of large numbers of dislocations. The process by which plastic deformation is produced by dislocation motion is termed slip; crystallographic plane along which the dislocation line transverses is the slip plane. During the slip, bonding and debonding occur between the edge atoms of dislocation and the atoms on the slip plane as shown in Figure 3.2. The dislocation motion is so similar to the motion of a caterpillar which is a common example for explaining the mechanism (Figure 3.3). [2] Figure 3.2: Atomic rearrangements that accompany the motion of an edge dislocation as it moves in response to an applied shear stress. Bonding and debonding of atoms occur gradually from (a) to (c) as the half plan of atoms move rightwards. Figure 3.3: Representation of the analogy between caterpillar and dislocation motion.
44 From Single Crystal to Polycrystalline 25 During plastic deformation the number of dislocations increases dramatically. The number of dislocations or dislocation density in a material is the total dislocation length per unit volume and designated with millimeters of dislocation per cubic millimeter (mm/mm 3 ). For heavily deformed metals the dislocation density may run as high as 10 9 to mm/mm 3. Due to the barriers such as grain boundaries or defects (preformed dislocations, point defects), dislocation pileups form in the grain. As the dislocations accumulate, the resistance to dislocation motion increases. This phenomenon is called hardening. Therefore dislocation density is the major driving factor for hardening. 3.2 From Single Crystal to Polycrystalline Grains are crystals with a more or less homogeneous crystallographic orientation [1]. Each grain in a material contains large numbers of dislocations which direct its deformation behavior. During plastic deformation, these dislocations accumulate at the grain boundaries which serve as barriers for the dislocation motion. Moreover, since the neighbor grains are of different orientations, a dislocation passing from one grain to its neighbor grain has to change its direction of motion; this becomes more difficult as the crystallographic misorientation increases (Figure 3.5). Due to these facts which create additional locking mechanisms in the material polycrystalline materials harden more than single crystal materials. Figure 3.4: Schematic representation of grains with different crystallographic texture.
45 From Single Crystal to Polycrystalline 26 Figure 3.5: Dislocation motion near the grain boundary. Figure 3.6: Slip lines on the surface of a polycrystalline specimen of copper [2].
46 Sources of Anisotropy Sources of Anisotropy Anisotropic deformation behavior of grains arises from the imperfections such as dislocations explained before. Dislocation density throughout the grain is not homogeneous. Hence, flow behavior of the grain is directed by the dislocation motions which make the material anisotropic. The behavior of dislocation motion in crystal will be explained in this session Slip Systems Dislocations do not move with the same degree of ease on all crystallographic planes of atoms and in all crystallographic directions. There is a preferred plane and in that plane there are specific directions along which dislocation motion occurs. This plane is called slip plane and the direction of movement is called the slip direction. The combination of the slip plane and the slip direction is termed the slip system. The slip system depends on the crystal structure of the material. The dependence of dislocation motion on the crystallographic texture constitutes anisotropy in a grain. For a particular crystal structure, the slip plane is the plane that has the densest atomic packing which corresponds to the greatest planar density. The slip direction in this plane is the direction having the most closely packed with atoms which corresponds to the highest linear density. Figure 3.7: a) A {111}<110> slip system in an FCC unit cell, b) The {111} plane from (a) and three <110> slip directions [2].
47 Sources of Anisotropy 28 A unit cell of the FaceCentered Cubic (FCC) crystal structure is shown in Figure 3.7a. There is a set of planes, the {111} family, all of which are closely packed. In Figure 3.7b, a {111} type plane is positioned within the plane of the page, in which atoms are now represented as touching nearest neighbors. Slip occurs along <110> type directions in the {111} planes, as indicated by arrows in Figure 3.7. Therefore {111}<110> represents the slip systems for FCC crystal structure. The slip system represents the different possible combinations of slip planes and directions. For FCC structures there are 12 slip systems: four unique {111} planes and, within each plane, three independent <110> directions. The possible slip systems for BodyCentered Cubic (BCC) and Hexagonal Close Packed crystal structures are listed in Figure 3.8. Figure 3.8: Slip systems for FCC, BCC and HCP metals [2]. As observed from the Figure 3.8, FCC and BCC crystal structures have a relatively large number of slip systems. The metals having BCC and FCC structures are quite ductile because extensive plastic deformation is normally possible along the various systems. On the other hand, HCP metals, having a few active slip systems, are normally quite brittle. [2]
48 Sources of Anisotropy Possible Slip Planes and Flow Patterns Plastic flow of the material is the sliding of dislocations on the slip planes in slip directions. Dislocations move in response to shear stresses applied along a slip plane and in a slip direction. Although an applied stress may be pure compressive or tensile, shear components exist at all but parallel or perpendicular alignments to the to the stress direction on the slip plane. These stresses are termed resolved shear stresses and their magnitudes depend not only on the applied stress, but also on the orientation of both the slip plane and direction within that plane. The situation is described in Figure 3.9. In Figure 3.9, φ represents the angle between the normal to the slip plane and the applied stress direction, and λ the angle between the slip and stress direction. σ designates the applied normal stress and τ R the resolved shear stress. Generally φ + λ 90. τ R = σ. cos(φ). cos(λ) (3.1) Figure 3.9: Geometrical relationship between the tensile axis, slip plane and the slip direction used in calculating the resolved shear stress for a single crystal.
49 Sources of Anisotropy 30 A single crystal has a number of slip systems that could initiate slip motion, but only one of them is generally oriented most favorably. Slip in a single crystal commences on the most favorably oriented slip system when the resolved shear stress reaches a critical value, which is called critical resolved shear stress (τ crss ). It is the minimum shear stress required to initiate slip on the slip plane. The single crystal plastically deforms or yields when τ R (max) = τ crss. Therefore the yield strength (σ y ) can be given as: σ y = τ crss cos(φ). cos(λ) (3.2) Therefore it is possible to obtain τ crss curve with knowing the slip plane and slip direction with a tensile or compression testing. For a single crystal specimen stressed under tension, deformation will be as in Figure 3.10, where slip occurs along a number of equivalent and most favorable oriented planes and directions at various positions along the specimen length. As seen in Figure 3.10 slip formations are parallel to one another and loop around the circumference of the specimen. The shape of the specimen resembles a deck of cards, each card results from the movement of a large number of dislocations along the same slip plane. Figure 3.10: Slip in an aluminum single crystal [18].
50 Sources of Anisotropy 31 The slip systems shown in Figure 3.10 are primary slip systems which correspond to the most favorable direction. The stress required to cause slip on the primary slip system is the yield stress of the single crystal. As the load is increased further, τ crss may be reached on other slip systems; these then begin to operate. Therefore the primary slip system is the slip system with lowest τ crss /cos(φ). cos(λ) value. The minimum yield stress necessary to introduce slip occurs when a single crystal is oriented such that λ = φ = 45. The critical resolved shear stress of slip systems also differ. In Figure slip systems of a BCC structure are given. According to [27] the initial yield stresses for the slip families satisfy τ I 0 < τ II 0 < τ III 0 with following ratio 1:1.3:1.5. Therefore Family I is the most favorable slip system to become primary slip system. To sum up, plastic deformation of a grain corresponds to the sliding of slip bands. The directional dependency of sliding is the source of anisotropy for a single crystal material. The isotropic material model is not sufficient to model the directional deformation behavior of the grain. An anisotropic material model should be used to model the directional deformation behavior which will be explained in Chapter 4. Figure 3.11: Slip systems of BCC crystals.
51 Chapter 4 Phenomenological Survey: Modeling the Behavior Mathematical model of a material is a perquisite for simulating the deformation of the material. In metal forming processes the governing deformation type is the plastic deformation. Microforming is a type of metal forming which is different from the conventional forming processes due to the size effects. Simulation of microforming processes require a suitable material model which can represent the anisotropic behavior of the billets composed of a few grains. The reason of anisotropic deformation of the materials in microforming was explained in Chapter 3. In this chapter the basics of modeling plastic deformation and a suitable material model for modeling microforming are going to be explained. 4.1 Introduction to the Mathematical Theory of Plasticity Theory of plasticity is the name given to the mathematical study of stress and strain in plastically deformed solids [15]. The theory of plasticity is especially concerning with forming processes such as forging and rolling. The main purpose
52 Introduction to the Mathematical Theory of Plasticity 33 of the theory is to calculate external loads, power consumption and nonuniform strain distribution during forming processes. The theory is going to be explained briefly in three parts: yield criterion, flow rule and strain hardening Yield Criterion Yield strength or yield point is the stress at which a material starts to deform plastically. Prior to the yield point the material will deform elastically and will return to its original shape when the applied stress is removed. Once the yield point is passed some fraction of the deformation will be permanent and nonreversible. Figure 4.1: Behavior of the material on a stressstrain curve.
53 Introduction to the Mathematical Theory of Plasticity 34 A law defining the limit of elasticity under any possible combination of stresses is known as a yield criterion [15]. The first step of any plastic flow analysis is to determine on a yield criterion. When the material is supposed to be isotropic, plastic yielding depends only on the magnitudes of the principal applied stresses, not on their directions. Any yield criterion is expressible in the form f(i 1, I 2, I 3 ) = c (4.1) where I 1, I 2, I 3 are the first three invariants of the stress tensor σ ij [15]. They are defined in terms of the principal components of stress σ 1, σ 2, σ 3 by relations; I 1 = σ 1 + σ 1 + σ 1 (4.2) I 2 = (σ 1 σ 2 + σ 2 σ 3 + σ 1 σ 3 ) (4.3) I 3 = σ 1 σ 2 σ 3 (4.4) The principal stresses are the roots of the characteristic equation of the eigenvalue problem; λ 3 I 1 λ 2 I 2 λ I 3 = 0 (4.5) Any function of the invariants can be expressed in terms of the principal stresses. A simplification of the Equation 4.1 is possible using a fact that the yielding of an ideal plastic body is unaffected by hydrostatic stress. Therefore yielding depends only on the deviatoric principal stress components (σ 1, σ 2, σ 3). σ ij = σ ij σ h I (4.6) σ h = σ xx + σ yy + σ zz (4.7) 3 σ h is the hydrostatic component of the stress and I is the 3x3 identity matrix. Since σ 1 + σ 2 + σ 3 = 0, the yield criterion in Equation 4.1 reduces to the form f(j 2, J 3 ) = c (4.8) where J 2 = (σ 1σ 2 + σ 2σ 3 + σ 1σ 3) (4.9)
54 Introduction to the Mathematical Theory of Plasticity 35 J 3 = σ 1σ 2σ 3 (4.10) Numerous criteria have been proposed for yielding. Two of them, Tresca and von Mises are going to be explained briefly Tresca Yield Criterion The criterion was proposed by Henri Edouard Tresca (1864). According to the criterion yielding occurs when the maximum shear stress reaches a certain value. Simply it could be given in the form σ max σ min = σ y = 2k (4.11) where k is the yield stress in shear, y is the yield stress in tension and max and min are the maximum and minimum principal stresses. According to Tresca, the yield stress in pure shear is 0.5 times the yield stress in simple tension. [24] Von Mises Yield Criterion The criterion was proposed by Richard von Mises (1913). According to the criterion yielding occurs when J 2 reaches a critical value. 2J 2 = σ 12 + σ 22 + σ 32 = 2k 2 (4.12) or (σ 1 σ 2 ) 2 + (σ 2 σ 3 ) 2 + (σ 3 σ 1 ) 2 = 6k 2 (4.13) The physical interpretation of the von Mises criterion is yielding begins when the elastic energy of distortion reaches a critical value. Thus a hydrostatic pressure does not cause yielding since it produces only elastic energy of dilatation. According to von Mises, the yield stress in pure shear is 1/ 3 times the yield stress in simple tension. [24] In Figure 4.2 comparison of Tresca and von Mises criteria is shown. Tresca criterion is more conservative than von Mises criterion, but the normal of Tresca s
55 Introduction to the Mathematical Theory of Plasticity 36 yield locus is discontinuous (each corner point has two normal) at the corners leading instability during numerical calculation of flow direction. Figure 4.2: Comparison of Tresca and von Mises yield criteria in plane stress condition Flow Rule The material dependent relationship between the stress and the strain in plasticity is termed as flow rule [33]. A general relationship between the ratios of the components of the strain increment and the stress ratios was suggested by Levy (1871) and von Mises (1913). The LevyMises equations are expressed in the form or dε 1 σ 1 = dε 2 σ 2 = dε 3 σ 3 = dλ (4.14) dε p ij = σ ijdλ (4.15) where ε p ij is the plastic true strain increment, dλ is a nonnegative real number and σ ij is the deviatoric true stress. According to the LevyMises flow rule, the material is assumed to be rigid plastic. Hence, the first invariant of the strain increment tensor is zero. After derivations, LevyMises rule for nonhardening
56 Introduction to the Mathematical Theory of Plasticity 37 (perfect plastic) case can be rewritten as, dε ij = ( 3 dεp 2 σ f )σ ij (4.16) where dε p is the equivalent plastic strain increment and σ f is the flow stress. dε p = ( dε p ij dεp ij ) (4.17) i=1 j=1 LevyMises rule is an application of the plastic potential theory in the form dε p ij = dλ f(σ ij) σ ij (4.18) where f(σ ij ) is the plastic potential which corresponds to the yield function in plasticity. Equation 4.18 indicates that the strain increment vector is normal to the yield function in the stress space. This is termed as normality rule. Figure 4.3: Demonstration of normality rule in a plane stress condition for von Mises yield function. The yield locus enlarges with plastic deformation of the material. The yield surface must be convex in order to attain a stable material flow. Tresca yield surface has instabilities at the corner points since each corner point has more than one normal vector. This obstacle is recovered by smoothing the corners in Tresca yield surface.
57 Introduction to the Mathematical Theory of Plasticity Strain Hardening Evolution of the yield surface with plastic deformation is named as hardening. Plastic deformation leads to the hardening of a metal and the increase of its elastic limit (in the direction of the deformation) [20]. The evolution of the yield surface is governed by one scalar variable, either the dissipated plastic work, or the accumulated plastic strain. If the evolution of the yield surface is correlated with dissipated plastic work, the hardening is termed as work hardening. The dependence of the yield surface on the plastic strain history is termed as strain hardening. In a simple tension or compression test, the flow curve (true stress vs. plastic strain) of a material could be found. In the cold plastic deformation case, the governing scalar variable on the hardening behavior is the plastic strain. Therefore, the flow stress (σ f ) could be represented as a function of equivalent plastic strain (ε p ) in many forms such as: σ f = C ε p n (Ludwik Expression) (4.19) σ f = A(ε 0 + ε p ) n (Power Law) (4.20) where n is the strain hardening exponent. For an hardening material, according to the flow curve σ f (ε p ), the local slope H (hardening slope) is defined as H = dσ f dε p (4.21) Inserting Equation 4.21 to the LevyMises rule for the perfect plastic case in Equation 4.16 gives dε p ij = ( 3 dσ f 2 Hσ f )σ ij (4.22) According to the Equations 4.17 and 4.21, a definite contribution to the hardening is made by every plastic distortion because equivalent plastic strain increment is always positive [15].
58 A Model Problem: Isotropic J 2 Flow Theory A Model Problem: Isotropic J 2 Flow Theory The equations of the theory of plastic flow establish a connection between infinitesimal increments of strain and stress. The theory has basic assumptions: 1. The body is isotropic. 2. The relative volumetric change is small and is an elastic deformation proportional to the mean pressure: dε ii = 3 1 2ν dσ (4.23) E 3. The total increments in the strain components dε ij are compounded of the increments in the components of elastic strain dε e ij and the components of plastic strain dε p ij. dε ij = dε e ij + dε p ij (4.24) The increment in elastic strain components is connected with the increment in elastic stress according to Hook s law: dε e ii = 1 2G (dσ ij 3ν 1 + ν δ ijdσ) (4.25) 4. The deviatoric stress and the deviatoric plastic strain increments are proportional. dε p ij = σ ijdλ (4.26) The increment of the plastic deformation work per volume (W p ) is dw p = σ ij dε p ij = dλ 2 k2 = dλ 2 J 2 (4.27) Due to the fact that the plastic deformation is irreversible, dw p 0 and dλ 0. The multiplier dλ is related to the magnitude of the increment in the work of plastic deformation. Changes in the volume during plastic deformation are elastic, and for a rigid plastic body dε e = 0. Increment of total deformation includes recoverable elastic deformation energy in the form dw = dw e + dw p (4.28)
59 Extensions to Anisotropy 40 where dw e = σ ij dε e ij (4.29) Detailed information about isotropic J 2 flow theory could be found in [31] and [20]. 4.3 Extensions to Anisotropy The assumption that every material element remains isotropic is an approximation that becomes worse as the deformation continues. Individual grains are elongated in the direction of the greatest tensile strain and the texture of the specimen appears fibrous. It is a consequence of the glide process that a single crystal rotates during the straining so that it approaches an orientation characteristic of the particular strain path. By this mechanism, a metal in which the grains are initially oriented at random, and which is therefore isotropic, is rendered anisotropic during plastic deformation. If the orientations of the individual crystals are not randomly distributed, the yield stress and the macroscopic stressstrain relations vary with direction. For instance, in cold rolling operations the tensile stress transverse to the direction of rolling is considerably greater than the one parallel to the rolling direction. [15] A Model Problem: Hill s Anisotropic Flow Theory This part is a brief explanation of Hill s anisotropic flow theory from R. Hill (1950). For the Hill s anisotropic flow theory considers states of anisotropy that possess three mutually orthogonal planes of symmetry at every point; the intersections of these planes are known as the principal axes of anisotropy. These axes may vary in direction throughout the specimen. For instance, a strip cut from the center of a coldrolled sheet provides an example of uniformly directed anisotropy that the principal axes lie in the direction of rolling, transversely in the plane of the sheet, and normal to this plane (Figure 4.4). The principal axes
60 Extensions to Anisotropy 41 in a given element can also vary relative to the element itself during continued deformation such as in the case of simple shear test. Figure 4.4: Axes of anisotropy in a coldrolled sheet. Here it is assumed that the principal axes of a material are set as Cartesian axes of reference. The criterion describing the yielding of isotropic material is that of von Mises. The simplest yield criterion for anisotropic material is one which reduces to von Mises law when the anisotropy disappears. According to this, the yield criterion will be in the form 2f(σ ij ) F (σ y σ z ) 2 + G(σ z σ x ) 2 +H(σ x σ y ) 2 + 2Lτyz 2 + 2Mτzx 2 + 2Nτxy 2 = 1 (4.30) where F, G, H, L, M, N are parameters characteristic of the current state of anisotropy. The yield criterion has this form when the principal axes of anisotropy are axes of reference; otherwise the form changes in a way that can be found by transforming the stress components.
61 Extensions to Anisotropy 42 If X, Y, Z are the tensile yield stresses in the principal directions of anisotropy, it could be shown that 1 X = G + H 2F = (4.31) 2 Y 2 Z 2 X 2 1 Y = F + H 2G = (4.32) 2 X 2 Z 2 Y 2 1 Z = G + F 2H = (4.33) 2 Y 2 X 2 Z 2 If R, S, T are shear yield stresses with respect to the principal axes of anisotropy, then 2L = 1 R 2 2M = 1 S 2 2N = 1 T 2 (4.34) If there is a rotational symmetry of the anisotropy in an element about the zaxis, the form of the Equation 4.30 remains invariant for arbitrary (x, y) axes for reference. The rotational symmetry about zaxis changes the Equation 4.30 to the form [(G + H)σ 2 x 2Hσ x σ y + (F + H)σ 2 y + 2Nτ 2 xy] 2(Gσ x + F σ y )σ z + 2(Lτ 2 yz + Mτ 2 zx) + (F + G)σ 2 z = 1 (4.35) If there is a complete spherical symmetry, or isotropy, L = M = N = 3F = 3G = 3H and Equation 4.30 reduces to von Mises criterion when 2F is equated to 1/Y 2. To describe fully the state of anisotropy in an element, the orientations of the principal axes and the values of the six independent yield stresses X, Y, Z, R, S, T must be known. It is not yet known how to relate the yield stresses quantitatively with the microstructure, therefore they are determined via experiments. The strain increment relations, referred to the principal axes of anisotropy are dε x = dλ[h(σ x σ y ) + G(σ x σ z )] dγ yz = dλ L τ yz (4.36) dε y = dλ[f (σ y σ z ) + H(σ y σ x )] dγ zx = dλ M τ zx (4.37) dε z = dλ[g(σ z σ x ) + F (σ z σ y )] dγ xy = dλ N τ xy (4.38) It will be noticed that dε x +dε y +dε z = 0 identically. If the principal axes of stress coincide with the axes of anisotropy, so do the principal axes of strain increment.
62 Extensions to Anisotropy 43 Figure 4.5: Comparison of the yield loci of isotropic (von Mises) and anisotropic (Hill) yield functions. As seen in Figure 4.5 the elastic region of the Hill s yield function differs from the one of von Mises function.
63 Chapter 5 Modeling Modeling the deformation behavior of the materials in forming processes helps us to design the tools and to predict the enhancement of the methods and the forming history, before starting mass production. Microforming differs from the conventional forming in terms of material behavior due to size effects. These size effects are explained by means of inhomogeneous grain structure which corresponds to considerably big grain size with respect to the microforming billets. The billets in microforming are composed of a single or a few grains and do not deform uniformly due to anisotropic behavior of grains. The billet size is not a criterion for deciding whether a process is a conventional or a microforming process. A suitable material model for the grains, Hill s anisotropic model is explained in Chapter 4. In this chapter, detailed information about modeling the microforming processes with Hill s anisotropic material model in the MSC.Marc R environment will be given. The mechanical properties used for the models are hypothetical.
64 MSC.Marc R as a Simulation Tool for Direction Dependent Response MSC.Marc R as a Simulation Tool for Direction Dependent Response Material Library Directional dependent mechanical properties in terms of plasticity are placed in the Yield Surface module of ElasticPlastic Plasticity part in the MSC.Marc R user interface. Hill s yield surface is represented via a flow curve and six scalar parameters in the program.the flow curve is entered as if the material is isotropic. σ is the equivalent yield stress for isotropic behavior. Ratios of actual (X, Y, Z) to isotropic yield (in the preferred orientation) are designated with YRDIR in terms of uniaxial yield (Equations 5.1). The ratio of actual shear yield (R, S, T ) to isotropic shear yield (σ/ 3 ) are designated with YRSHR in terms of shear yield (Equations 5.2). Therefore, there are six scalar parameters which characterize the orthotropic hardening. These parameters can be determined with the help of three experiments with simple tension and three with simple shear. YRDIR(1) = X σ, YRDIR(2) = Y σ, YRDIR(3) = Z σ (5.1) YRSHR(1) = R σ/ 3, YRSHR(2) = S σ/ 3, YRSHR(3) = T σ/ 3 (5.2) According to Equations 5.1 and 5.2, six flow curves are multiplicand of the isotropic flow curve, hence they are dependent. In order to apply Hill s yield function in the program, principal axes of the undeformed material must be defined. This ability is offered in the Orientations section of Material Properties part. Two principal directions are entered in the vector notation. The third principal direction is the product of two entered vectors due to mutual orthogonality. Here one point is important, since finite strain kinematics are due, MSC.Marc R computes the distortions of the initially given orthogonal direction vectors and proceeds accordingly. For simulating the
65 MSC.Marc R as a Simulation Tool for Direction Dependent Response 46 directional response of a grain, principal axes are chosen in a way that first principal axis corresponds to the slip direction, second principal axis is the vector in the slip plane and perpendicular to slip direction and the third principal axis is normal to the slip plane as explained in Figure 5.1. Figure 5.1: Principal axes of a grain model for Hill s criterion. To sum up, in order to implement the directional material response in the context of anisotropic flow in the program, six material constants, a flow curve and principal axes of each material must be entered Element Library Element type 117, a continuum element, is used in the 3D simulation of microforming processes. Element type 117 is an eightnode isoparametric arbitrary hexahedral for general threedimensional applications using reduced integration. This element uses an assumed strain formulation written in natural coordinates which insures good representation of the shear strains in the element. Besides this linear element is preferred over higherorder elements when used in a contact analysis. The stiffness of this element is formed using a single integration point at the centroid of the element according to Gaussian quadrature. An additional
66 MSC.Marc R as a Simulation Tool for Direction Dependent Response 47 variationally consistent stiffness term is included to eliminate the hourglass modes that are normally associated with reduced integration. Node numbering must be in a way that nodes 1, 2, 3 and 4 are corners of one face, given in counterclockwise order when viewed from inside the element. Figure 5.2: Nodes and integration point for Element 117. The element has three global degrees of freedom u, v and w per node and standard transformation of three global degrees of freedom to local degrees of freedom occur at the corner nodes. The element follows an updated Lagrange procedure and is suitable for the finite strain plasticity Limitations MSC.Marc R has two limitations regarding simulation of the direction dependent response. Firstly, in the anisotropic yield function of Hill are there six independent yield stresses (three normal and three shear yield stresses). In the MSC.Marc R environment, six constants represent the directional response of the material. Normal yield stresses of the principal axes are found by multiplying the single flow curve with the YRDIR constants and shear yield stresses are found with multiplying the flow curve with YRSHR/ 3 constants. Therefore six flow curves representing
67 Model 1. Plane Strain Upsetting 48 the directional response are dependent. As a result, it is not possible to use independent flow curves for the Hill s yield function in MSC.Marc R. Secondly, in metal forming processes the complexity of material deformation is very high. During the process, certain regions containing contact surfaces, localized deformation etc. may contain large deformations so that the mesh structure used to model the material becomes highly distorted. Hence the distorted mesh must be renewed in order to continue simulation. The mesh renewing stage is called remeshing. During remeshing, the mechanical data (strain, stress etc.) stored at the integration points, are transferred to the integration points of the new mesh. As a part of material data which evolves during deformation, orientation data must be mapped to the new mesh during remeshing. This is a matter of interpolation/extrapolation of the direction vectors by proper methods. MSC.Marc R is unable to transfer the orientation data during remeshing. Due to this restriction, a complete forming simulation can not be performed at this stage. In order to perform a microforming simulation, which includes large deformations requiring remeshing, user defined subroutines must be implemented which is beyond the aim of this study. To sum up, the simulations are performed without remeshing. Therefore the extent of the simulations depend on the deformation capacity of element type Model 1. Plane Strain Upsetting Introduction and Motivation 2D modeling is a simplified version of 3D modeling in FEM simulation to reduce modeling and calculation time by utilizing certain symmetry conditions. Although grains are anisotropic and 3D arbitrary structures, some results such as inhomogeneous and directional deformation could be seen in 2D models. For the sake of simplicity, 2D modeling is a good starting point for investigating the
68 Model 1. Plane Strain Upsetting 49 directional behavior of grains in a FEM simulation. The effect of inhomogeneous grain structure on the deformation of the material could be seen in Figure 5.3. In this model, the aim is to obtain a 2D grain structure model from a granular Figure 5.3: Upsetting of a material with inhomogeneous grain structure [7]. material picture (Figure 5.4). The response of the material in upsetting is going to be observed with using three different modeling approaches. The material behavior is going to be modeled as isotropic, single grain and multigrain Hill s model. Figure 5.4: Cold rolled iron sheet after recrystallization. The grains in Figure 5.4 are converted to CAD model in.vda format with the code in Appendix C, then grains are meshed separately in MSC.Marc R (Figure 5.6).
69 Model 1. Plane Strain Upsetting FEM Models and Material Parameters The upsetting velocity and the number of elements are common for the three models. In the isotropic and single grain model all the elements in the model possess the same material parameters (Figure 5.5). In the multigrain model which is composed of 13 grains, all the elements in the model possess the same material parameters, but different grain elements posses different orientation data (Figure 5.6). Same mesh structure is used for the models in order to avoid mesh anisotropy. Figure 5.5: Isotropic and single grain FEM model of the Figure 5.4. Figure 5.6: Multigrain FEM model of the Figure 5.4.
70 Model 1. Plane Strain Upsetting 51 Table 5.1: Material parameters for the isotropic model. Parameter Magnitude Modulus of Elasticity Poisson s Ratio 0.3 Flow Curve Formula 500(1 + ε) 0.2 Table 5.2: Material parameters for the single grain and multigrain models. Parameter Magnitude Modulus of Elasticity Poisson s Ratio 0.3 Flow Curve Formula 500(1 + ε) 0.2 YRDIR1 0.8 YRDIR2 1 YRDIR3 1.1 YRSHR1 0.6 YRSHR2 0.9 YRSHR3 1.2 Figure 5.7: Orientations of the grains in Figure 5.6. Each model consists of 3856 elements.the upsetting velocity is 0.1 mm/s although the analysis is quasistatic where the time measure is a pseudo variable and the height reduction is 20%. The friction factor is taken as For the single grain model the orientation of the first principal axis is 40. The grain orientations in the multigrain model are arbitrary. The material parameters of the models are given in the tables.
71 Model 1. Plane Strain Upsetting Simulation Results Deformed Shapes Figure 5.8: Deformed shape of the material with isotropic model. Figure 5.9: Deformed shape of the single grain material with Hill s model. 52
72 Model 1. Plane Strain Upsetting 53 Figure 5.10: Deformed shape of the multigrain material with Hill s model Equivalent Plastic Strain Distribution Figure 5.11: Equivalent plastic strain distribution of the material with isotropic model.
73 Model 1. Plane Strain Upsetting 54 Figure 5.12: Equivalent plastic strain distribution of the single grain material with Hill s model. Figure 5.13: Equivalent plastic strain distribution of the multigrain material with Hill s model.
74 Model 1. Plane Strain Upsetting Equivalent von Mises Stress Distribution Figure 5.14: Equivalent von Mises stress distribution of the material with isotropic model. Figure 5.15: Equivalent von Mises stress distribution of the single grain material with Hill s model.
75 Model 1. Plane Strain Upsetting 56 Figure 5.16: Equivalent von Mises stress distribution of the multigrain material with Hill s model Results and Discussions The isotropic model is a reference for the other two models in terms deformation response. The directional response of the Hill s model can easily be observed in the simulation results. Firstly, in the simulation results of single crystal model (Figures 5.9 and 5.12), the material is distorted in the direction of first principal axis (40 ) which corresponds the the preferred slip orientation. The upsetting cross in Figure 5.11 represents the homogeneous deformation of the isotropic body. On the other hand, the material shears in the slip direction as in the case of single crystal body. The deformation of a single crystal structure is explained in Chapter 3. The deformed shape of a single crystal in Figures 5.9 and 5.12 are consistent with the literature given in Chapter 3.
76 Model 1. Plane Strain Upsetting 57 The third approach, multicrystal model, contains 13 grains with different orientations. This means that each grain has a different slip behavior under loading as in the real case. The deformed shape in Figure 5.10 illustrates the fact that the grains having larger resolved shear stress on their slip planes start to deform earlier, as explained in Chapter 3. This leads to inhomogeneous deformation throughout the body which could be figured out from the plastic strain distribution in Figure According to Figure 5.13, the regions where the equivalent plastic strain is high belong to the grains which deform prior to the other due to high resolved shear stress on their slip planes. The deformation behavior of each grain depends on its own slip system orientation. Therefore in order to predict the local forming behavior of a material, the grains must be modeled with correct orientations. Moreover the surface of the multigrain model (Figure 5.10) is irregular and similar to Figure 5.3. Figure 5.17: Force vs. displacement graph of the three models. The difference between three models could also be observed on Figure The material parameters of the Hill s model affect the material behavior in a way that, the force applied on the single crystal model is smaller than the other two due to the orientation angle about 40 which is very near to 45 where the smallest possible yield stress is required for deformation. The governing parameter on the slip mechanism is the resolved shear stress on the slip plane. This is why, slip bands form at an angle near to 45 to the loading direction during the tensile test
77 Model 2. 3D Tensile Testing 58 of a single grain. In the multigrain model, the orientations are arbitrary, but still the individual shear mechanism in each grain makes the force lower than than the one in the isotropic model due to the less number of grains. As the grain size decreases, the material becomes more homogeneous and the overall flow stress of the material increases. 5.3 Model 2. 3D Tensile Testing Introduction and Motivation Tensile testing is a basic test to identify the mechanical properties of the materials.the test result of a single crystal aluminum specimen is given in Figure As an initial step for simulating 3D microforming processes, tensile testing is a good choice for illustrating directional response. The aim of this model is to check the behavior of the single crystal specimen at the region of necking and to compare the results with Figure The isotropic and anisotropic (single grain) models are also going to be compared in a 3D simulation FEM Models and Material Parameters The diameter and the length of the testing model are 1 mm and 4 mm. The upsetting velocity and the number of elements are common for the two models.each model consists of 6720 hexahedral type 117 elements.the clamp velocity is 0.1 mm/s. For the single grain model the orientation of the first principal axis is 45 to the xzplane and second principal axis is 135 to the xzplane where zaxis is the tensile testing direction.
78 Model 2. 3D Tensile Testing 59 Figure 5.18: Model of the tensile testing simulations. Table 5.3: Material parameters for the isotropic model. Parameter Magnitude Modulus of Elasticity Poisson s Ratio 0.3 Flow Curve Formula 500(1 + ε) 0.2 Table 5.4: Material parameters for the single grain model. Parameter Magnitude Modulus of Elasticity Poisson s Ratio 0.3 Flow Curve Formula 500(1 + ε) 0.2 YRDIR1 0.8 YRDIR2 0.9 YRDIR3 1.2 YRSHR1 0.8 YRSHR2 1 YRSHR3 1.1
79 Model 2. 3D Tensile Testing Simulation Results Deformed Shapes Figure 5.19: Deformed shape of the specimen with isotropic model. Figure 5.20: Deformed shape of the single grain specimen with Hill s model.
80 Model 2. 3D Tensile Testing Equivalent Plastic Strain Distribution Figure 5.21: Equivalent plastic strain distribution of the specimen with isotropic model. Figure 5.22: Equivalent plastic strain distribution of the single grain specimen with Hill s model.
81 Model 2. 3D Tensile Testing Equivalent von Mises Stress Distribution Figure 5.23: Equivalent von Mises stress distribution of the specimen with isotropic model. Figure 5.24: Equivalent von Mises stress distribution of the single grain specimen with Hill s model Results and Discussions As seen in Figure 3.10, slip planes constitute a stepwise structure where a step corresponds to a slip band. In the simulation result of Figure 5.20, the surface structure is smooth, but the horizontal element edges are similar to the slip line orientation. As the slip band height gets smaller the surface of the specimen will get smoother. Movement of the slip lines in Figure 3.10 are represented by the
82 Model 2. 3D Tensile Testing 63 horizontal element edges (mesh lines) in Figure5.20. Since slip is confined to the slip planes indicated, deformation would cause a displacement of the top of the specimen relative to the bottom of the specimen. Since the grips are constrained, however, the condition is satisfied by rotation of the slip planes as deformation occurs, resulting in an evolution of the lattice orientation during deformation. This situation is illustrated in Figure 5.25 and it is consistent with the simulation results. Figure 5.25: Evolution of the lattice orientation during tensile testing constraints [18] on the left. Necking region of the single grain model on the right. Necking behavior also changes in the single grain model. In the isotropic model, the neck cross section is a circle which is the case in conventional tensile testing specimens. On the other hand, in the anisotropic model, the neck cross section is an ellipse which is the case in Figure The simulation results and the literature are consistent in terms of necking. Equivalent plastic strain contours in the Figures 5.21 and 5.22 explain the difference between necking behaviors of the models. The load curves of the models indicate that the load required to deform a homogeneous material is more than the load required to deform a single crystal material. Due to the grain boundaries which are obstacles for dislocation
83 Model 2. 3D Tensile Testing 64 Figure 5.26: Cross section of the necking region of the isotropic model on the left, single grain model on the right. movement, the body with more grains require more force to be deformed. Therefore, microforming processes require less forces when compared with conventional forming processes. Figure 5.27: Force vs. displacement graph of the two tensile testing models.
84 Model 3. Extrusion of a Cylindrical Billet Model 3. Extrusion of a Cylindrical Billet Introduction and Motivation Extrusion is one of the most commonly used manufacturing process in the metal forming industry. Micro metal forming, which is developing tremendously due to the increasing demand from electronics industry, is a subgroup of metal forming industry. Typical examples of the applications of micro metal forming include the manufacture of pins for IC carriers, fasteners, microscrews and connectors which require extrusion processes. The workpiece, which is going to be used in a micro extrusion process, is a single crystal or a multigrain structure. With respect to the simulation results in Model 1 and Model 2 sections which are consistent with the literature, it is possible to simulate 3D microforming processes such as extrusion. Isotropic, single grain and multigrain models are going to be used for modeling a micro extrusion process in this section FEM Models and Material Parameters The forming velocity and the number of elements are common for the three models. In the isotropic and single grain model all the elements in the model possess the same material parameters (Figure 5.28). In the multigrain model, which is composed of 5 grains, all the elements possess the same material parameters, but different grains, thus the elements belonging to these grains possess different orientation data which corresponds to primary slip system.(figure 6.13).
85 Model 3. Extrusion of a Cylindrical Billet 66 Table 5.5: Material parameters for the isotropic model. Parameter Magnitude Modulus of Elasticity Poisson s Ratio 0.3 Flow Curve Formula 500(1 + ε) 0.2 Figure 5.28: Isotropic and single grain FEM model for the extrusion simulation. The workpiece with an initial diameter of 1 mm and length of 1.5 mm is extruded to 0.7 mm diameter. The forming velocity is 0.1 mm/s. The friction factor is taken as Each model consists of elements. For the single grain model the orientation of the first principal axis is 55 to the xzplane and second principal axis is 125 to the xzplane where zaxis is the extrusion direction. The grain orientations and the number of elements per grain in the multigrain model are arbitrary. The material parameters of the models are given in the tables.
86 Model 3. Extrusion of a Cylindrical Billet 67 Figure 5.29: Multigrain FEM model for the extrusion simulation. Table 5.6: Material parameters for the single grain and multigrain models. Parameter Magnitude Modulus of Elasticity Poisson s Ratio 0.3 Flow Curve Formula 500(1 + ε) 0.2 YRDIR1 0.8 YRDIR2 1 YRDIR3 1.1 YRSHR1 0.6 YRSHR2 0.9 YRSHR3 1.2
87 Model 3. Extrusion of a Cylindrical Billet Simulation Results Deformed Shape Figure 5.30: Deformed shape of the material with isotropic model. Figure 5.31: Deformed shape of the single grain material with Hill s model.
88 Model 3. Extrusion of a Cylindrical Billet 69 Figure 5.32: Deformed shape of the multigrain material with Hill s model Equivalent Plastic Strain Distribution Figure 5.33: Equivalent plastic strain distribution of the material with isotropic model.
89 Model 3. Extrusion of a Cylindrical Billet 70 Figure 5.34: Equivalent plastic strain distribution of the single grain material with Hill s model. Figure 5.35: Equivalent plastic strain distribution of the multigrain material with Hill s model.
90 Model 3. Extrusion of a Cylindrical Billet Equivalent von Mises Stress Distribution Figure 5.36: Equivalent von Mises stress distribution of the material with isotropic model. Figure 5.37: Equivalent von Mises stress distribution of the single grain material with Hill s model.
91 Model 3. Extrusion of a Cylindrical Billet 72 Figure 5.38: Equivalent von Mises stress distribution of the multigrain material with Hill s model Results and Discussions The uniformity of the deformation in the the isotropic model is represented by the axisymmetric distribution of the equivalent plastic strain contours in Figure As a result of uniformity, the stress distribution on the die in the isotropic case is also uniform. Moreover, as a consequence of this uniformity in the isotropic case, it is possible to model this case axisymmetrically. On the other, hand, the equivalent plastic strain and the equivalent von Mises stress distributions of the single grain and multigrain models indicate that deformations in these cases are nonuniform with respect to the center axis of the die. Therefore it is impossible to reduce this model to an axisymmetric one. Directional response of the material yields undesired morphology at the extruded end in the single and multigrain models (Figures 5.31 and 5.32). The extruded tip of the single grain model has a slope to one side which corresponds to the movement of slip bands in the most favorable direction which is the slip direction. In the case of multigrain model, the tip is composed of two grains with different different slip directions resulting in a vgroove at the extruded tip. The forcedisplacement curves in Figure 5.39 are not complete curves for the
92 Model 3. Extrusion of a Cylindrical Billet 73 extrusion process due to the remeshing limitation of MSC.Marc R. Due to this limitation the processes are not completely simulated in the sense of a full deformation range. The forcedisplacement curves in Figure 5.39 are similar to the ones for Model 1 (Figure 5.17). The difference is that the force values of the single grain model in Figure 5.17 are less than the values for multi grain model, but in Figure 5.39 it is contrary. This is due to the orientation difference between the models. The forcedisplacement curves of the granular models vary due to their dependence to the slip orientation of each grain. Figure 5.39: Force vs. displacement graph of the three extrusion models.
93 Model 4. Backward Can Extrusion Model 4. Backward Can Extrusion Introduction and Motivation Backward can extrusion process is used to produce axisymmetric products such as cups, cans and bullets. This process can also be used to produce miniature parts. The miniaturization of the parts brings problems during production such as undesired local defects due to nonuniform deformation. An example of how the inhomogeneous flow may influence the geometrical inaccuracy is shown in Figure Figure 5.40: process[7]. Nonuniform edge formation in micro backward can extrusion The purpose of modeling this process is to simulate the undesired formation of the rim edge with the effect of grain orientations. Isotropic, single grain and multigrain models are going to used for modeling a micro backward can extrusion process in this section FEM Models and Material Parameters The forming velocity and the number of elements are common for the three models. In the isotropic and single grain model all the elements in the model
94 Model 4. Backward Can Extrusion 75 Table 5.7: Material parameters for the isotropic model. Parameter Magnitude Modulus of Elasticity Poisson s Ratio 0.3 Flow Curve Formula 500(1 + ε) 0.2 possess the same material parameters (Figure 5.41). In the multigrain model which is composed of 5 grains,all the elements in the model possess the same material parameters, but different grain elements posses different orientation data which corresponds to primary slip system (Figure 5.42). Figure 5.41: Isotropic and single grain FEM model for the backward can extrusion simulation. The workpiece with an initial diameter of 1 mm and length of 1 mm is extruded to 1.3 mm outer diameter and 1mm inner diameter. The forming velocity is 0.1 mm/s.the friction factor is taken as Each model consists of elements. For the single grain model the orientation of the first principal axis is 55 to the xzplane and second principal axis is 125 to the xzplane where zaxis is the extrusion direction. The grain orientations and the number of elements per grain in the multigrain model are arbitrary. The material parameters of the models are given in the tables.
95 Model 4. Backward Can Extrusion 76 Figure 5.42: Multigrain FEM model for the backward can extrusion simulation. Table 5.8: Material parameters for the single grain and multigrain models. Parameter Magnitude Modulus of Elasticity Poisson s Ratio 0.3 Flow Curve Formula 500(1 + ε) 0.2 YRDIR1 0.8 YRDIR2 1 YRDIR3 1.1 YRSHR1 0.6 YRSHR2 0.9 YRSHR3 1.2
96 Model 4. Backward Can Extrusion Simulation Results Deformed Shape Figure 5.43: Deformed shape of the material with isotropic model. Figure 5.44: Deformed shape of the single grain material with Hill s model. 77
97 Model 4. Backward Can Extrusion 78 Figure 5.45: Deformed shape of the multigrain material with Hill s model Equivalent Plastic Strain Distribution Figure 5.46: Equivalent plastic strain distribution of the material with isotropic model.
98 Model 4. Backward Can Extrusion 79 Figure 5.47: Equivalent plastic strain distribution of the single grain material with Hill s model. Figure 5.48: Equivalent plastic strain distribution of the multigrain material with Hill s model.
99 Model 4. Backward Can Extrusion Equivalent von Mises Stress Distribution Figure 5.49: Equivalent von Mises stress distribution of the material with isotropic model. Figure 5.50: Equivalent von Mises stress distribution of the single grain material with Hill s model.
100 Model 4. Backward Can Extrusion 81 Figure 5.51: Equivalent von Mises stress distribution of the multigrain material with Hill s model Results and Discussions The axisymmetric deformation of the isotropic model results a circular uniform rim edge (Figure 5.43). The local deformation behavior of grains due to their different directional responses yield a nonuniform rim edge as seen in Figure 5.44 and Figure It can be seen that the formation of the rim is rather irregular which can be put down to the fact that on different locations grains of different size and orientation have to pass the clearance between die and punch which is quite smaller than the mean grain size, thus producing inhomogeneous material flow and finally the observed irregular shape (Figure 5.40). In the single grain case, the same grain have to pass all the clearance volume which requires more force than the isotropic and multigrain models due to the fact that the deformation direction is rather than the preferred deformation direction. The simulation of the complete process requires remeshing. Due to the limitation of MSC.Marc R, the process is conducted up to a limited deformation. Therefore the forcedisplacement curves in Figure 5.53 are incomplete. As expected, the required force for the single grain case is more than the other two cases.
101 Model 4. Backward Can Extrusion 82 Figure 5.52: Comparison of the rim edges of the multigrain model and Figure Figure 5.53: Force vs. displacement graph of the three backward can extrusion models.
102 Chapter 6 Experimental Setup Design As demands on miniature/micrometalproducts increase significantly, micrometalforming becomes an attractive option in the manufacture of these products due to its advantageous characteristics for mass production. Machines, formingtools and handling of micrometalparts are critical elements that significantly determine the industrial applications of microforming. Although the need for micro manufacturing systems is increasing, very few companies can offer solutions for micro manufacturing applications. Microforming press is the basic machine for micro manufacturing applications. Commercial microforming presses are deeply investigated in [30]. These presses are either designed by companies or designed by the collaboration of universities and companies. According to [30], most of the microforming machines are insufficient for industrial applications and being developed by companies or universities. In order to perform experiments on microforming processes, a conceptual microforming press is required. This chapter is about designing a conceptual microforming press, control and data acquisition system for the experimental setup and the experimental procedure.
103 A Conceptual Press Design for Microforming A Conceptual Press Design for Microforming A very deep literature survey is performed for the design of a microforming press. For this purpose, DTU Mechanical Engineering Department is visited and their knowhows are shared. The press shown in Figure 1.12 is designed by the department. The linear actuator is a linear electric motor which can apply forces up to 5000 N. Although this loading capacity is very low and the actuator weighs about 500 kg, the control accuracy is very good because it has zero backlash. The control system can not compensate frame deflection due to temperature change and loading which cause accuracy loss. These disadvantages are taken into consideration during the design stage. During the design of the microforming press, experience of the DTU staff is utilized Linear Actuator Type Decreasing scale of the workpiece size leads to high precision in the micro manufacturing systems. This is the basic difference between a conventional press and a microforming press. Therefore the actuator of the press must accomplish high positional accuracy (max.±5 µm) under high loads (min. 10 kn) with a stroke length of minimum 2 mm. In order to calculate load capacity of the conceptual microforming analytical formulations were used.the force (F ) required for 75% height reduction of an AISI 1015 cylindrical steel specimen with dimensions Ø2 mm and h=2 mm without friction is given below: σ f = 760 ε p (Flow curve for AISI 1015) 75% height reduction corresponds to ε p = ln( ) Final crosssectional area is F = 760 (ln ) π = N Volume Final height = (π22 /4) = 4 π
104 A Conceptual Press Design for Microforming 85 The positional accuracy of the actuator is directly proportional to its stroke length. Piezoelectric actuators seem to be the best choice to be used as actuator for a microforming press. Piezoelectric actuators are the most frequently used actuators for high positional accuracy offering subnanometer displacement resolution., but their stroke length values are very small (<100 µm) for the ones with load capacity larger than 10 kn. Moreover, in the bulk metal forming process, punch force increases as the actuator rod approaches the bottom dead center. In piezoelectric actuator, the force applied by the actuator decreases as it approaches to bottom dead center, which, in fact, is the reverse case of bulk metal forming processes. Conventional high stroke actuators were investigated with respect to the criteria given above. The comparison of conventional linear actuators is given in Figure 6.1.
105 A Conceptual Press Design for Microforming 86 Figure 6.1: Comparison of conventional linear actuators (www.exlar.com).
106 A Conceptual Press Design for Microforming 87 As seen from Figure 6.1 roller screws and ball screws fit to microforming press better than acme screws, hydraulic and pneumatic cylinders in terms of position control. The difference is in the roller screw s design for transmitting forces. Multiple threaded helical rollers are assembled in a planetary arrangement around a threaded shaft, which converts a motor s rotary motion into linear movement of the shaft or nut. There is another fact that, the contact between the screw road and its counter part in ball screw is supplied by balls which results low contact area and backlash due to the deformation of balls. In roller screw, there is direct contact which decreases the actuator volume and increases the load rating with larger contact area. This situation is explained in Figure 6.2. Therefore roller screw mechanism is chosen to be the most suitable linear actuator type for microforming press applications. Figure 6.2: Contact with screw road and its counter part in ball screw (upper), roller screw (lower). The force capacity of the roller screw is proportional to its pitch size. Decreasing pitch size increases force capacity and decreases velocity of the screw rod which eases the displacement control of the mechanism. Since cold forging operations will be performed with the machine, velocity of the punch is disregarded. Therefore with selecting the smallest pitch size for the roller screw, it is possible to have smaller mechanisms with huge amount of forces with occupying smaller volumes. Moreover backlash of the mechanism could be prevented via
107 A Conceptual Press Design for Microforming 88 preloading with an internal counter screw. Figure 6.3: Roller screw type linear actuator (www.exlar.com) Press Frame Design The basic criteria for the design of press frame are dimensional and positional accuracies. In order to attain dimensional accuracy in the parts, which will be produced or bought, should be perfectly ground. The parallelism of upper frame block and lower frame block is the most critical perquisite. Therefore instead of using 4 corner columns, 2 side walls are used. Usage of side walls also helps closing the working volume since it should be closed for safety and temperature control. Moreover side wall design makes the frame stiffer than the four corner column design. The punch must move normal to the upper die surface with the possible smallest error. In order to keep the punch holder block movement normal to the upper surface of die, roller rails are used. These roller rails guide the motion and ensure the path of the punch holder block. There is also a displacement sensor embedded to one of the runner blocks in order to obtain position data of the punch. The position data of the punch is also obtained from the encoder of the servomotor inside the roller screw mechanism.
108 A Conceptual Press Design for Microforming 89 CATIA V5 is used to design the conceptual press machine. Computer Aided Design (CAD) model of the microforming press is given in Figure 6.4. The commercial parts decided to be used in the design are given in Appendix A. Detailed drawings of the parts and assemblies are given in Appendix B. Figure 6.4: Main parts of the conceptual microforming press design. The mechanical properties of the conceptual press are given in Figure 6.5. It is capable of obtaining the desired positional accuracy as well as offering force and stroke values more than needed which enlarge the application spectrum. Two position sensors are necessary for acquiring and comparing the displacement data.
109 A Conceptual Press Design for Microforming 90 Figure 6.5: General view and properties of the conceptual microforming press Design of Die and Punch Systems The billets in the microforming have dimensions less than 1 mm. Therefore it is possible to found die system inside a special load cell which contains a threaded hole at the center. The kick out pin, made of AISI 3343 high speed steel, is driven by a piston which is preferably a pneumatic piston. It is possible to obtain forces up to 120 N for kick out of forged billet. If larger forces are needed, a hydraulic piston can be used instead. Tungsten carbide is decided to be used as die material. Forming surface of the die is produced with microedm machine. Die is fitted to shrink ring, which is also made of AISI 3343 high speed steel, via a bolt. The slope at the contact surface between die and shrink ring causes initial radial stress on the die. As the shrink ring bolt is tightened, the radial stress on the die increases and increases the durability of the die. Detailed drawings of the parts and the assembly of the die system are given in Appendix B.
110 A Conceptual Press Design for Microforming 91 Figure 6.6: Crosssectional view of the die assembly. AISI 3343 high speed steel is the material of punch needle. The positional accuracy of the punch can be adjusted with a 2D positioning table. Instead, a simple solution is offered without using an extra positioning system. The recommended solution for the punch adjustment is using a soft metal to fix the punch position with deforming the soft metal plastically with the force of initial forming operation. The position of the punch will be kept by the plastically deformed soft metal in the following forming operations. A lead part in the shape of a coin is recommended for this purpose. In order to protect the punch holder from localization of reaction force due to the misalignment of the punch, a stiffener made of tungsten carbide is used. Detailed drawings of the parts and the assembly of the punch system are given in Appendix B.
111 Control and Data Acquisition in the Experimental Setup 92 Figure 6.7: Crosssectional view of the punch assembly. 6.2 Control and Data Acquisition in the Experimental Setup The accuracy of the punch displacement depends on the quality of the control system as well as the dimensional accuracy of the parts and the system. Control system must consider the affect of frame deflection due to temperature change and loading on the punch positioning. Therefore, the amount of frame deflection should be calculated and compensated during position control. There are four sensors in the system: a load cell, a rotational encoder, a linear encoder and a thermocouple. The load cell is an analogue sensor which generates voltage between +5 V and 5 V linearly proportional to the force applied. The load cell is a special type that has a threaded hole at the center for the mounting of die system. It measures tensile and compressive forces up to 20 kn. Since the
112 Control and Data Acquisition in the Experimental Setup 93 load cell is capable of measuring tensile forces, it enables the conceptual design to perform tensile tests with mounting suitable jaws. The rotational encoder, which measures the rotation angle of the rotor, is a part of servomotor. The rotational encoder is a digital sensor which generates a voltage pulse per a specific angle. Servo system is composed of a motion controller and a servomotor. Servo system has a closed loop control inside due to motion feedback from the rotational encoder which is located at the top of the linear actuator. The linear proportionality between rotor rotation angle and screw rod displacement enables us to calculate the displacement of the punch with acquiring rotation data from rotational encoder. Due to its place the measurement of the rotational encoder includes deflections in the linear actuator and in the frame; hence this data should be corrected with offsetting the total deflection of the punch. The second position is sensor is a linear encoder which is also a digital sensor. It generates a voltage pulse per a specific length. The linear encoder is embedded to one of the runner blocks which are directly mounted to the punch holder. This system is a solution specific system for the roller rail systems. Due to the deflections in the linear actuator and the frame, it is better to obtain displacement data from the linear encoder rather than rotational encoder. The last sensor is a thermocouple for measuring the temperature of the press frame. The word thermocouple is a representative word for temperature sensors. In fact a thermistor type temperature sensor is a good choice for fast response and good accuracy [10]. The temperature data from the thermocouple will be used for calculating the thermal deflection of the frame. The room temperature should be kept constant at 25 C with an air conditioner. The air conditioner works independent of the control system. Regarding the press frame deflection, closed loop control system is a must for the conceptual design. Either rotational encoder of the servomotor is used for acquiring displacement data and then the data is offset by the calculated deflection or linear encoder is used for direct displacement acquisition. In order to attain the desired punch displacement with a repeatability of 5 µm, temperature and force data must be fed back to calculate the frame and the linear actuator deflection for offsetting the displacement input value. The closed loop control
113 Experimental Procedure 94 schema for both cases is given in Figure 6.8. Figure 6.8: Control schema of the conceptual press design. 6.3 Experimental Procedure The main purpose of performing experiments is to obtain input data for Hill s yield function and to measure the success level of simulations of microforming processes. The success level of the simulations is determined by comparing the forcedisplacement curves and the deformation behavior of the material in the simulations with the ones in the experiments. The experiments will start with single crystal specimens and continue with multigrain specimens. During the experiments, the conceptual press, a Scanning Electron Microscope (SEM) and heat treatment facilities will be used. The conceptual press will be used to deform the specimen and obtain forcedisplacement data of the forming process. The SEM will be used for obtaining the grain structure model of the specimen and detecting the slip system of each grain in the Electron Backscatter Diffraction (EBSD) mode. EBSD is a technique which allows crystallographic information to be obtained
114 Experimental Procedure 95 from samples in the scanning electron microscope (SEM). In EBSD a stationary electron beam strikes a tilted crystalline sample and the diffracted electrons form a pattern on a fluorescent screen. This pattern is characteristic of the crystal structure and orientation of the sample region from which it was generated. The diffraction pattern can be used to measure the crystal orientation, measure grain boundary misorientations, discriminate between different materials, and provide information about local crystalline perfection. When the beam is scanned in a grid across a polycrystalline sample and the crystal orientation measured at each point, the resulting map will reveal the constituent grain morphology, orientations, and boundaries. This data can also be used to show the preferred crystal orientations (texture) present in the material. A complete and quantitative representation of the sample microstructure can be established with EBSD. The components of the system are given in Figure 6.9. Figure 6.9: Schematic representation of an EBSD system. Heat treatment facilities will be used for recrystallization to obtain desired grain size in the specimen. The preferred materials for the experiments are ferritic stainless steel, austenitic stainless steel or pure metals such as aluminum due to their distinct grain structure.
115 Experimental Procedure 96 Figure 6.10: Determination of the grain structure and lattice orientation of each grain with EBSD.
116 Experimental Procedure 97 The basic experiments are performed on single crystals. The input curves for the Hill s yield function are determined on a single crystal specimen. The experimental procedure is given in Figure Figure 6.11: Experimental procedure for determining the input parameters for Hill s yield function.
117 Experimental Procedure 98 Tensile testing could be done instead of compression. It is possible to use a tensile testing specimen with a single grain at the gage section for this purpose (Figure 6.12). This technique was used by W.J. Porter et al. [29]. Figure 6.12: Tensile testing specimen containing a single grain in the gage section [19]. After obtaining and verifying the input data for the Hill s yield function with simulating the experiments with different slip orientations, multigrain structures could me modeled in the same manner. The difficulty in performing experiments with multigrain structures is to obtain their 3D grain structure. If the material is composed of a few grains which have connections to the surface, it could be possible to model the grains. From the surface of the material, slip orientations could be detected with EBSD and the 3D model of interior sides could be estimated. Image processing techniques are used to convert the photo of grain structure into the CAD model. If the thickness of the specimen corresponds to one grain size, 2D modeling will be enough for the situation. The image processing code is given in Appendix C. Figure 6.13 is a high contrast picture of grains belonging to an iron sheet and Figure 6.14 is the 2D CAD model of the grains.
118 Experimental Procedure 99 Figure 6.13: Cold rolled iron sheet after recrystallization at 800C [18]. Figure 6.14: CAD model of the grains in Figure The conversion of the grain pictures to the models contains approximations. In the specimen, modeling the interior regions of the grains is impossible. Instead interior regions are estimated with respect to the surface topology. 3D model of the tensile testing specimen in Figure 6.12 is given in Figure After composing the model and determining the slip orientations, specimen is forged with the conceptual press. The simulation of the experiment is performed with the material data found in the single crystal specimen experiments explained before. Finally, forcedisplacement curves and the final shapes are compared. The success level of simulations will be determined with this way.
119 Experimental Procedure 100 Figure 6.15: 3D FEM model of the specimen in Figure 6.12 [19]. In conclusion, every new simulation technique requires an experimental verification. To verify the simulations in Chapter 5, the conceptual microforming press, which is explained in this chapter, should be constructed. Moreover the accuracy of the production should be as high as possible with grinding all the surfaces of the press frame. The control system is the key point since it is the link between experiments and simulations. Sensitivity of the sensors should be as high as possible to control the system as accurate as possible. Building up the models for simulations is the hardest point that, obtaining the 3D model of the grains is impossible to attain. There should be approximation with respect to the surface topology to build the 3D model of the grains. Therefore, the knowledge of the success level of these approximations and the material model depends on performing experiments.
120 Chapter 7 Conclusion and Future Perspectives The increasing market volume of electronic and micromechanical components makes a general trend towards higher integrated functional density and miniaturization of equipments. Thus, the technology of microforming becomes more and more attractive when smallest metallic parts with high accuracy and production output are demanded. In order to meet the demands, design of the production processes of microparts should be investigated both practically and analytically. This thesis was prepared to contribute the analytical investigation of microforming. A new modeling approach is proposed for the simulation of microforming processes. The new modeling approach aims to explain the differences between conventional forming and microforming which are termed as size effects. The size effects due to the decreasing scale of the workpiece size, is explained in terms of inhomogeneous grain structure of the material. The nonuniform deformation of workpiece, which is composed of a few grains, arises from the anisotropic behavior of grain individuals. Therefore the proposed approach for the simulation is modeling the material as individual grains which possess anisotropic mechanical properties. The directional response of the grains is represented by Hill s
121 102 anisotropic material model in the simulation environment of MSC.Marc R. 2D and 3D simulations of common forming processes were performed with the three possible material models (isotropic, single grain, multigrain) and compared with the literature. The results are consistent with the experimental results from the literature and thus, size effects in microforming can be explained with this approach. Therefore grain by grain modeling with correct directional mechanical properties outcomes consistent simulation results. In order to apply the material model, orthotropic mechanical properties and the slip orientation of each grain in the material must be known. For a pure metal all the grains possess the same mechanical properties. On the other hand, grains of a material may possess different mechanical properties. For instance, grains of different phases (martensite, ferrite) in a steel own different mechanical properties such as flow curve. Whether the material is pure or not, the slip orientation of each grain, which determines the axes of anisotropy for the material model, must be known. EBSD method is useful for detecting the orientation of a single grain workpiece, however it is unable detect the orientation of inner volume grains. Besides, 3D CAD model of the grains must be obtained. Except for the surface grain topography, inner boundary geometries can not be acquired. The disadvantages described above arise due to the increasing input data for anisotropy with respect to the isotropy. In the cold forming of microparts, which is characterized by directional response of a few grains in the deformed area, random forming behavior and unpredictable mechanical properties of produced part make a process design as applied in conventional forming impossible. As a countermeasure forging at elevated temperature reduces the anisotropic behavior of grains and homogenizes the deformation. The different forming behavior of grains is decreased by thermally activating the slip systems which are not active during cold forming. The situation is described in Figure 7.1. Therefore increasing the forming temperature makes the material more homogeneous, thus approaches the material model to isotropy. The utilization of elevated temperature in microforming should be investigated deeply. Especially the necessary temperature for
122 103 uniform deformation with respect to the mean grain size should be studied for efficient simulation of microforming processes. Due to the disadvantages described above, this solution requires further investigations. Figure 7.1: Specimen produced by backward can extrusion at a) 20 C and b) 300 C [4]. Finally, a conceptual press design is given in Chapter 6. Detailed investigations about the development of microforming machines could be found in [28] and [30]. According to the literature, very few companies are producing such types of presses. Most of the microforming presses are in the stage conceptual design performed by universities. In order to meet the demand of microparts from electronics industry, micromanufacturing technology should be improved. For this purpose, studies on the development of microforming presses should performed. Besides, these studies should include material handling during forming processes because it is the key point of the design of micromanufacturing systems.
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