PRINCIPLES OF THE DRAW-BEND SPRINGBACK TEST

Transcription

1 PRINCIPLES OF THE DRAW-BEND SPRINGBACK TEST DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Jianfeng Wang, B.S., M.S. * * * * * The Ohio State University 2004 Dissertation Committee: Approved by Robert H. Wagoner, Adviser Glenn S. Daehn Peter M. Anderson Adviser Department of Materials Science and Engineering

2 Copyright by Jianfeng Wang 2004

3 ABSTRACT This thesis focuses on the principles of springback for 6022-T4 aluminum sheets, using a special draw-bend test with a range of sheet tensions and tool radii. To model the anisotropic yielding of 6022-T4 sheet, Barlat 91, 96 and 2000 yield functions were implemented into Abaqus/Standard through user material subroutines. A nonlinear kinematic hardening model with multiple back stresses was constructed to closely reproduce the reversed strain hardening behavior of sheet metals. The new material constitutive models are as accurate as the previous work based on two-surface plasticity, but they have simpler mathematical forms and require fewer model parameters. The mechanics of the persistent anticlastic curvature were studied by draw-bend experiments, finite element analysis and an elastic plate-bending theory. The transverse cross-section shape was dictated by a dimensionless parameter, β, which depends on the specimen geometry, tool radius and sheet tension. The rapid decrease of springback angle as the front sheet tension approaches yielding is correlated to a critical values of β, above which the retained anticlastic curvature is small and hence has little impact on the springback angle. However, the anticlastic curvature built up during the forming step persisted after unloading when β is less than 10 15, and thus greatly reduced the final springback. ii

4 In order to quantify the time-dependent springback phenomenon and infer its physical basis, several aluminum alloys were draw-bend tested under conditions promoting the time-dependent response. The time-dependent springback angles are approximately linear with log(time) for times up to a few months, after which the kinetics become slower and saturation is reached in approximately 15 months. Residual stress-driven creep and anelasticity are discussed as the possible sources of the time-dependent springback. For 6022-T4, qualitative agreement was obtained using a crude finite element model, with creep laws derived from constant load creep tests. For the second possibility, novel anelasticity tests following a reverse loading path were performed for 6022-T4 and DQSK steel. Based on the experiments and simulations, it appears that anelasticity is unlikely to play a large role in long-term time-dependent springback of aluminum alloys. iii

5 This is dedicated to my beloved parents. iv

6 ACKNOWLEDGMENTS I would like to express my sincere gratitude to my advisor, Professor Robert H. Wagoner, for providing encouragement and guidance throughout the course of my study. I am also grateful to Professor Peter M. Anderson and Professor Glenn S. Daehn for serving as members of my dissertation committee. I also want to thank Professor David K. Matlock from Colorado School of Mines, for kindly hosting my visit in 2001 and assisting the draw-bend experiment. I gratefully appreciate the help of Dr. Lumin Geng from General Motors Company, for his valuable advice on Abaqus/UMAT programming. I would like to thank Mr. William D. Carden and Mr. Vijay Balakrishnan for providing some of their experimental results for this research. Ms. Christine Putnam is greatly appreciated for her administrative support and assistance on proof-reading various publications. Richard Boger and Lloyd Barnhart are also acknowledged for their kind help on some of the mechanical testings. Finally, I would like to thank my parents for their enormous support and understanding throughout my study. v

7 VITA January 18, Born Zhejiang, P. R. China B.S. Materials Science and Engineering, Shanghai Jiaotong University, P. R. China B.S. Computer Engineering and Application, Shanghai Jiaotong University, P. R. China M.S. Materials Science and Engineering, Shanghai Jiaotong University, P. R. China 1999-present Graduate Research Associate, The Ohio State University. PUBLICATIONS Research Publications Wang, J.F., Wagoner, R.H. and Matlock, D.K., Creep Following Springback. Proceedings of the Plasticity 2003: Dislocations, Plasticity and Metal Forming, ed. A.S. Khan, p Quebec City, Canada. FIELDS OF STUDY Major Field: Materials Science and Engineering vi

8 Studies in: Plasticity and Metal Forming Mechanical Metallurgy Finite Element Method Prof. Robert H. Wagoner Prof. Peter M. Anderson Prof. Glenn S. Daehn Prof. Somnath Ghosh vii

9 TABLE OF CONTENTS Page Abstract Dedication Acknowledgments Vita List of Tables ii iv v vi xi List of Figures xiii Chapters: 1. Introduction Review of Yield Criteria for Sheet Metal Introduction to Plastic Anisotropy Thermodynamic Framework of Elasto-plastic Materials Isotropic Yield Criteria Von Mises and Tresca Yield Criteria Hosford s Non-quadratic Yield Criterion Anisotropic Yield Criteria Hill s family of Anisotropic Yield Criteria Karafillis and Boyce s Yield Criterion Barlat s Family of Anisotropic Yield Criteria Anisotropic Yielding of 6022-T4 Aluminum viii

10 3. Integration of Plastic Rate Equation Introduction to Stress Update Schemes Forward Integration Backward Integration Semi-backward Integration Implementation to Abaqus via UMAT Anisotropic Hardening of 6022-T4 Aluminum Sheet Springback Simulation of Draw-Bend Test with Finite Element Method Introduction Draw-Bend Experiment Materials Draw-Bend Test Experimental Results Effect of Back Force and Tool Radius Effect of Specimen Width Finite Element Results and Discussion Effect of Back Force Effect of Strip Width Choice of Element Shell vs. Solid Conclusions Anticlastic Curvature in Draw-Bend Test Introduction Draw-Bend Experiment Finite Element Models Elastic Theory for Plate Bending Problem Statement and Closed-Form Solution Results for Pure Bending of an Initially Flat Plate ( 1 R x0 = 1 R x0 = 0) Bending of Initially Curved Plate Error Analysis of Elastic Theory Anticlastic Curvature in Draw-Bend Test Effect of Back Force Effect of Specimen Width Application of the Elastic Bending Theory Discussion Conclusions ix

11 6. Time-Dependent Springback Introduction Experimental Materials Draw-Bend Experiments Anelastic Tests Results Static (Time-Independent) Draw-Bend Tests Time-Dependent Draw-Bend Springback Room Temperature Creep Test for 6022-T Creep-Based Springback Simulation Anelastic Deformation after Unloading Discussions Conclusions Conclusions Appendices: A. Numerical Algorithm for Abaqus UMAT B. Draw-Bend Test Data Bibliography x

12 LIST OF TABLES Table Page 2.1 Parameters for Barlat 91, 96 and 2000 yield functions Parameters c i (MPa) and γ i for the mnlk model Chemical composition of 6022-T4 aluminum sheet (in weight percent) Parameters used in finite element model for elastic bending Error introduced by using parabolic function The radii of primary and anticlastic curvature in draw-bend test Chemical composition (in weight pct.) and thickness of aluminum sheets Grain size of aluminum alloys Kinetics of time-dependent springback of 6022-T4 aluminum from measurements at τ 1 = 60s and τ 2 = s (15 months) Long-time kinetics of springback for 6022-T4 aluminum based on remeasurement of samples at τ 1 = 60s and τ 2 = s (7 years) Calculated times to reach fractions (0.2, 0.5 and 0.8) of the saturation strains (δε ) or springback angles (δθ ) B.1 Measured sheet tension, springback angle and the radius of anticlastic curvature for draw-bend tested 6022-T4 sheets (RD) with R/t = 3.5 and W = 50mm xi

13 B.2 Measured sheet tension, springback angle and the radius of anticlastic curvature for draw-bend tested 6022-T4 sheets (RD) with R/t = 14 and W = 50mm B.3 Measured sheet tension, springback angle and the radius of anticlastic curvature for draw-bend tested 6022-T4 sheets (TD) with R/t = 3.5 and W = 50mm B.4 Measured springback angle and the radius of anticlastic curvature for draw-bend tested 6022-T4 sheets (RD) using R/t = 14 tool xii

14 LIST OF FIGURES Figure Page 2.1 Schematics of coordinate systems of a sheet (X-Y -Z) and a tensile sample (x-y-z) cut at an angle θ at the sheet rolling direction Schematics of the yield surface evolution in the stress space: (a) with kinematic and isotropic hardening, (b) with kinematic hardening only, and (c) with isotropic hardening only Evolution of the back stress von Mises and Tresca yield loci compared with experimental data Plane-stress isotropic yield functions Effect of r-value on the yield surface shape Effect of m on stress ratio σ b σ RD Comparison of the calculated and experimental results for 6022-T4: (a) normalized yield stress, and (b) r-value Schematics of return mapping algorithm in stress space Schematics of the return mapping schemes: (a) the backward Euler method, and (b) the tangent cutting plane method Input and output of a user material subroutine Schematics of the Bauschinger test: (a) the fork device, and (b) flow curves after load reversal xiii

15 3.5 Simulated and experimental reverse flow curves after: (a) compression/tension test, and (b) tension/compression test. Markers are experimental data from Balakrishnan (1999), and solid lines are the corresponding simulation results Simulated reverse flow curves using various hardening laws: (a) isotropic hardening and NLK model with single back stress, and (b) mnlk and Geng-Wagoner models Draw-bend experiment: (a) equipment at Colorado School of Mines, and (b) schematics of test procedure and geometry of a deformed sheet after springback Dependence of the springback angle and the anticlastic curvature on the normalized back force: (a) R/t = 3.5 and (b) R/t = Effect of back force on springback for samples with different orientation: (a) θ and (b) 1 R a Dependence of springback angle and anticlastic curvature on specimen width: (a) F b = 0.5 and (b) F b = Comparison of the simulation results with the experimental data: (a) the springback angle θ, and (b) the unloaded anticlastic curvature 1 R a Comparison of the simulated springback angles using (a) the Barlat 96 and 2000 yield functions and (b) the Geng-Wagoner model and the mnlk law Comparison of the simulation and the experimental data for strips with various widths using F b = 0.5: (a) the springback angle, and (b) the anticlastic curvature Comparison of the simulated springback angles using different elements for (a) F b = 0.5 and (b) F b = Simulated springback angles using shell and solid elements (a) An anticlastic surface, and (b) a synclastic surface Schematics of the draw-bend test and an unloaded specimen xiv

16 5.3 Simulated springback angle using various assumptions Rectangular coordinate system for plate bending problems Closed-form solution for: (a) normalized anticlastic deflection, and (b) normalized transverse stress Anticlastic deflection and stress ratio for an elastic material with β = Anticlastic deflection and stress ratio for elastic material with β = Variation of the anticlastic factor φ with β Bending an initially curved plate Maximum anticlastic deflection from finite element simulation Error analysis for using parabolic function Effect of the back force on: (a) springback angle and (b) unloaded anticlastic curvature. Lines are FEM simulation results and markers are experimental data Variation of the maximum anticlastic deflection with the front pulling force in the draw-bend test: (a) loaded and (b) unloaded Moment and normalized section moment of inertia by finite element simulation Springback angle and anticlastic curvature from draw-bend test for (a) F b = 0.5 and (b) F b = Compare simulation and measurement for F b = 0.5 case: (a) springback angle and (b) anticlastic curvature Unbending and springback analysis for draw-bend test Comparison of simulated and analytically predicted maximum anticlastic deflections for draw-bend tested samples after springback: (a) R/t = 3.5 and (b) R/t = xv

17 5.19 Comparison of measured, simulated and analytically predicted anticlastic profiles for draw-bend tested samples: (a) F b = 0.4 and (b) F b = Simulated anticlastic profiles of draw-bend tested samples: (a) for various back forces and (b) various widths at F b = Variation of Searle s parameter β with (a) back force and (b) specimen width Optical micrographs revealing grain structures Uniaxial stress-strain curves for the tested aluminum alloys Draw-bend specimens: (a) schematic geometry before and after springback; (b) tracings at various times following forming for 6022-T4 aluminum Variation of measured time-independent springback angle on: (a) back force; (b) tool radius Schematic of the tension/unloading test: (a) general load path, (b) detail of unloading region Schematic of compression/tension/unloading test: (a) specimen geometry and stabilization fixture; (b) general load path Variation of simulated front force with back force for various R/t ratios Effect of bending radius and back force on residual stress: (a) throughthickness residual stress for various R/t at F b = 0.5; (b) variation of maximum tensile and compressive residual stresses with back force for three R/t ratios Change of springback angle with time after forming: (a) 2008-T4, (b) 5182-O, (c) 6022-T4, (d) 6111-T4. Multiple tests are differentiated by open and closed markers. Slopes shown are in degree/log(s) Measured time-dependent springback angles for6022-t4 from two studies (Carden, 1996, and this work) xvi

18 6.11 Room temperature creep curves for 6022-T4: (a) steady state creep law; (b) primary creep law Room temperature creep behavior for 6022-T4 and DQSK: (a) creep strain; (b) creep rate Abaqus simulation results at three stages: (a) through-thickness stress; (b) Comparison of measured and simulated time-dependent springback Effect of creep law parameters on simulated time-dependent springback: (a) stress exponent N; (b) strength parameter K Anelastic strain after uniaxial tension Anelastic strain after unloading from compression tension test: (a) 6022-T4; (b) DQSK steel Strain paths in draw-bend test: (a) locations of 5 through-thickness integration points (IP); (b) change of strain paths A.1 A first guess for backward Euler algorithm xvii

19 CHAPTER 1 INTRODUCTION Special draw-bend tests are utilized to study both the static and time-dependent springback of high-strength 6022-T4 aluminum alloy sheet. The main results are reported in Chapters 4, 5 and 6, with conclusions following in Chapter 7. In Chapter 2, various yield functions are briefly reviewed, with focus on the Barlat s family of yield criteria that are used for static springback simulation after the draw-bend test for 6022-T4 aluminum sheets. The Barlat 96 and Barlat 2000 yield functions can accurately describe the planar anisotropy, i.e., the angular variation of yield strength and r-value within the sheet plane. Chapter 3 is mainly concerned with the numerical implementation of material constitutive models into finite element code, Abaqus/Standard, through user subroutines. After reviewing the return mapping algorithm with forward, backward and semi-backward integration methods, a nonlinear kinematic hardening model with multiple back stresses (mnlk ) is discussed. The combination of Barlat s anisotropic yield functions and mnlk model ensures accuracy when modeling the anisotropic yielding and non-isotropic hardening behavior of 6022-T4 aluminum sheets. In Chapter 4, the draw-bend device and test procedure are first presented. Aluminum sheets of 50mm wide are draw-bend tested, with the normalized sheet tension 1

20 varying from 0.1 to 1.1 and two tool radii: 3.2mm and 12.7mm. Finite element simulations are also conducted using Barlat s yield functions and anisotropic hardening models (mnlk and a modified hardening law by Geng and Wagoner). The mnlk model parameters are calibrated from reverse flow curves of in-plane tension/compression tests. Draw-bend test results for specimens with widths ranging from 12mm to 50mm are also reported, with 12.7mm tool radius and two selected back forces of 0.5 and 0.9. The role of persistent anticlastic curvature on springback is discussed. A manuscript to be submitted to the International Journal of Plasticity is in preparation. Chapter 5 is dedicated to the study of anticlastic curvature that appeared during the forming step in the draw-bend test, but persisted after unloading for certain test conditions. A closed-form solution, derived from the classical large deformation theory of bending elastic plate, is employed to explain the correlation of the persistent anticlastic curvature and forming conditions. It is realized that the anticlastic surface after springback deviates from a circular shape for small tensions. This transition is determined by the interplay between specimen geometry (W/t) and the curvature of the curled region of unloaded strip (R x ), via a dimensionless parameter β = W 2 R x t. The occurrence of the persistent anticlastic curvature corresponds to a critical β-value of A manuscript of this work has been submitted to the International Journal of Solids and Structures. The time-dependent springback in draw-bend tested aluminum sheets is investigated in Chapter 6. The short-time response is linear on log(time) scale for times up to a few months; after that the kinetics are slower and the time-dependent springback 2

21 angle eventually saturates. Simulation based on residual stress driven creep underestimates the experimental results by a factor of 2, albeit the qualitative agreement. Anelastic strains after unloading from uniaxial tension and compression/tension tests are measured for both 6022-T4 aluminum and two forming grade steels. The fast kinetics of anelasticity suggest that it is not likely the dominant mechanism for longterm response, but it may contribute to the short-time behavior. A manuscript of this work has been submitted for publication in the International Journal of Plasticity. Chapter 7 summarizes the conclusions from Chapters 3 to 6. 3

22 CHAPTER 2 REVIEW OF YIELD CRITERIA FOR SHEET METAL The accuracy of sheet metal forming and springback simulation depends not only on the forming conditions (friction, tool and binder geometry), but also on the choice of the material constitutive models and their numerical implementations into finite element programs. Among these factors, the material constitutive law plays an important role in describing the mechanical behavior of sheet metals, because it is essential to obtain the accurate stress distribution in a formed part in order to correctly predict springback. Two different approaches have been used in sheet metal forming simulations, which are based on either crystal plasticity or phenomenological models. In the polycrystalline plasticity model, a sheet sample is treated as an aggregate of single crystals with preferred orientation distribution (i.e., crystallographic texture). Plastic deformation is then described by the discrete dislocation slips that take place on specific lattice planes and along particular crystallography directions. This method is very powerful, because texture evolution during plastic deformation can be considered. However, it requires a huge amount of computation time to simulate any practical sheet metal forming process [1]. 4

23 In this study, the second approach is employed. Within the framework of continuum plasticity theory, plastic flow occurs when a yield criterion is satisfied. In stress space, the initial yielding can be described by a smooth, continuous surface, F (σ ij ) = 0, which defines the boundary between the elastic and plastic domains. All the points inside the yield surface represent elastic deformation, whereas stress points belonging to the surface are related to plastic states. Compared to the crystal plasticity model, this approach has a simpler mathematical form, and thus it is more feasible for finite element programming. In addition, only a small number of parameters are needed, and they can be easily determined from simple experiments, such as uniaxial tensile tests. Theoretical considerations and experimental evidence have shown that the mathematical form of the yield surface is subject to some restrictions. For example, Bridgman s work [2] demonstrated that the hydrostatic pressure did not cause plasticity for metals, i.e., yielding is pressure independent. As a result, the yield function can be expressed as F (I 2, I 3 ) = 0, where I 2 and I 3 are the second and third invariants of the stress tensor. On the other hand, Drucker s postulate [3] stated that a yield surface must be convex to ensure the uniqueness of plastic strain rate for a given stress state. An associated flow rule was also implied from this postulate for stable (non-softening) materials, so that the direction of the plastic strain rate is normal to the yield surface: ɛ p = λ F σ (2.1) where λ is a plastic multiplier parameter. In other words, a yield surface also acts as a plastic potential from which the plastic flow direction is derived. 5

24 2.1 Introduction to Plastic Anisotropy Sheet metals naturally exhibit mechanical anisotropy because of the preferred grain orientation. The detailed crystallographic texture is determined by the thermomechanical manufacturing history (e.g., hot/cold rolling and annealing). Typically, sheet metals are orthotropic, with mirror symmetry axes aligned with the sheet rolling (RD), transverse (TD) and normal (ND) directions, as shown in Figure 2.1 [4]. y z x σ Y (TD) σ Z(ND) x ψ X(RD) Figure 2.1: Schematics of coordinate systems of a sheet (X-Y -Z) and a tensile sample (x-y-z) cut at an angle θ at the sheet rolling direction. To characterize the plastic anisotropy of the sheet, the Lankford parameter or the plastic strain ratio is generally adopted [5]. The plastic strain ratio, also called r-value, is defined as follows: or more precisely by r ψ = εp 2 ε p, (2.2) 3 r ψ = dεp 2 dε p 3 (2.3) 6

25 where ε p 2 and ε p 3 are true plastic strains in the sample width and thickness directions, respectively. The r-values can be measured by uniaxial tension tests for samples at different angles to the sheet rolling direction, Figure 2.1. In practice, it is quite difficult to precisely measure small changes of the sheet thickness. Since plastic deformation conserves volume, i.e., ε p 1 + ε p 2 + ε p 3 = 0, it is convenient and more accurate to calculate r-value alternatively: r ψ = εp 2 ε p 1 + ε p 2 (2.4) while ε p 1 is the plastic strain in the longitudinal direction. The plastic strain ratio has a profound effect on sheet formability. It is a measure of the resistance to thinning or localized deformation, which usually precedes failure in sheet metal forming. Therefore, high r-value is desirable to achieve good formability. Usually, r-value varies with orientation in the sheet plane, and it is also a function of plastic strain. An averaged r-value is defined as r = r 0 + 2r 45 + r 90 4 (2.5) which closely correlates to the deep drawability of sheet metals [4]. Here, r 0, r 45 and r 90 designate the r-value measured from samples cut at 0, 45 and 90 from the sheet rolling direction. On the other hand, the in-plane variation of the r-value can be evaluated by r = r 0 2r 45 + r 90 2 (2.6) r is a measure of the planar anisotropy, and it relates to the earing profile in deepdrawn products [4]. 7

26 2.2 Thermodynamic Framework of Elasto-plastic Materials In small deformation theory, the total strain is additively decomposed into elastic and plastic parts: ɛ = ɛ e + ɛ p (2.7) For isotropic elasticity, the generalized Hooke s law states that the Cauchy stress tensor σ is proportional to the elastic strain: σ = C el : ɛ e (2.8) where C el is a fourth order isotropic tensor. It can be expressed as: C el = 2GI dev + K1 1 (2.9) The superscript dev denotes the deviatoric part of a tensor. For example, I dev = I (2.10) where I and 1 are the fourth and second identity tensors respectively: (1) ij = δ ij (2.11) (I) ijkl = 1 2 (δ ikδ jl + δ il δ jk ) (2.12) where δ ij is the Kronecker-Delta symbol: δ ij { 1 if i = j 0 otherwise (2.13) The material constants, G and K, are the shear and bulk moduli respectively. They are related to the Young s modulus E and Poisson s ratio ν through G = and K = E. 3(1 2ν) 8 E 1+ν

27 When the elastic limit stress is exceeded, plastic deformation occurs. The onset and continuation of plastic flow is governed by a yield function. In principal stress space, the yield function defines the boundary of an elastic region in which plastic deformation is absent. The stress stays on the yield surface after plasticity occurs. All the possible plastic stress states constitute a hypersurface in the stress space. If the associated flow rule is adopted, this surface also acts as a plastic potential, from which the plastic strain rate is derived. Mathematically, a yield surface can be defined as follows f(σ; α, R) = σ }{{} eq (σ α) [σ 0 + R(q)] = 0 (2.14) }{{} V σ y The motion of the yield surface (kinematic hardening) is described by the translation of its center, represented by the back stress tensor α. The evolutionary law of the back stress will be discussed in the last section of this chapter. The change of yield surface size (isotropic hardening) is indicated by the change of the yield strength, σ y. The initial yield strength (when q = 0) is σ 0, while R represents the isotropic expansion of the yield surface, as graphically shown in Figure 2.2. There are three commonly used isotropic hardening laws in the literature [6]: h iso q linear hardening R(q) = Aq n power law hardening R (1 e mq ) Voce s law with saturation The scalar quantity, q, describes material hardening due to the microstructural change in the course of plastic deformation, such as the increase of dislocation density. Usually, q is the accumulated equivalent plastic strain, and it is defined as q(t) = t 0 ɛ p dτ = Generally q(t) ɛ p (t), except for a monotonic loading path. 9 t ɛp : ɛ p dτ (2.15)

28 σ 3 Subsequent Yield Surface (a) (b) (c) σ y α α σ 0 Initial Yield Surface σ 2 σ 1 Figure 2.2: Schematics of the yield surface evolution in the stress space: (a) with kinematic and isotropic hardening, (b) with kinematic hardening only, and (c) with isotropic hardening only. For an isothermal deformation process, where no heat exchange is involved, the Helmholtz s free energy per unit mass is expressed as [7]: ρψ(ɛ e, θ, q) = 1 2 ɛe : C el : ɛ e + a q 2 θ : θ + R(l)dl (2.16) where, ɛ e, θ and q are strain-like internal variables which are macroscopical measures of an irreversible process, such as plastic deformation; σ, α and R are their associated thermodynamic forces (stress-like variables). ρ is the material s density. The thermodynamic quantities are related through the following equations of state: σ = ρ ψ ɛ e = Cel : (ɛ ɛ p ) (2.17) α = ρ ψ θ R = ρ ψ q 0 = aθ (2.18) = R(q) (2.19) When the temperature field is static during deformation, the Clausius-Duhem inequality reduces to σ : ɛ ρ ψ 0 (2.20) 10

29 where the equality sign only holds for reversible processes. Substituting Equations 2.17, 2.18 and 2.19 into the above inequality, a mechanical dissipation function can be defined as: D mech (σ; α, R) = σ : ɛ p ρ(r q + α : θ) (2.21) Hill s principal of maximum dissipation [8] states that among all the admissible stresses and internal variables, S = {(σ ; α, R ) f(σ ; α, R ) 0}, the exact solution (σ; α, R) maximizes the mechanical dissipation for any given ( ɛ p, q): D mech (σ; α, R) = max {D mech(σ ; α, R )} (2.22) (σ ;α,r ) S Equation 2.22 describes a constrained optimization problem. Using the standard procedure [9], Equation 2.22 is first transformed into a minimization problem by changing the sign of the mechanical dissipation function. Then, a Lagrangian functional is constructed by adding function D mech and the constraint equation f together: L(σ, λ) = D mech + λf(σ; α, R) (2.23) where λ is a Lagrangian multiplier. The exact solution will make the functional L stationary with respect to any variations of σ and λ. Therefore, the associated flow rule and the so-called Kuhn- Tucker conditions are arrived at after optimization: L σ = f ɛp + λ σ = 0 (2.24) f(σ; α, R) 0 (2.25) λ 0 (2.26) λf = 0 (2.27) 11

30 The first equation states that the direction of plastic flow is normal to the yield surface (i.e., the normality rule), while the last two equations establish the loading-unloading criterion. The scalar quantity, λ, is also called the plastic multiplier, and its sign is determined by the plastic yielding and loading-unloading conditions: λ > 0 λ = 0 if f = 0 and f = 0 (2.28) if f = 0 and f < 0, or f < 0 (2.29) One additional conclusion can also be derived from the principle of maximum dissipation, that is, the yield surface must be convex [8]. The evolution laws for the internal variables can be derived from a thermodynamic potential F (σ; α, R) = f(σ; α, R) + b 2a α : α (2.30) Therefore, ɛ p F f = λ = λ σ σ = λa (2.31) F θ = λ α = ɛp b a λα (2.32) F q = λ R = λ (2.33) From the second equation above, and assuming that α = aθ, the evolution of the back stress becomes: α = a ɛ p b λα (2.34) Equation 2.34 is the Armstrong-Frederic type of evolution law [10]. A generalization was made by Chaboche to include multiple back stress tensors in order to better describe cyclic plastic deformations [11, 12], such as ratcheting and cyclic creep [13]. 12

31 Some earlier works did not have a recall term in the evolution equation for the back stress. For example, the following equations were used to describe the change of the back stress [14, 15]: α = { cp dɛ p (Prager) dµ (σ α) (Ziegler) (2.35) where c p is a material constant, and dµ is a parameter that can be determined from the yield condition. The difference between these two models lies in the direction of the back stress. In Prager s model, i.e, the linear kinematic hardening, the yield surface moves in the direction of the plastic strain rate. As pointed out by Ziegler [15], Prager s model does not produce consistent results for 3D and 2D problems. A modified model was proposed to overcome this drawback, so that the yield surface center translates in the radial direction of the yield surface [15]. n β σ β O n dα σ σ α α f n = 0 (1). Prager (2). Ziegler (3). Mroz f n+1 = 0 (2). Ziegle Figure 2.3: Evolution of the back stress Figure 2.2 compares the different evolution laws for the back stress tensor α. Case 1 and 2 correspond to Prager s and Ziegler s rule of back stress evolution respectively. 13

32 For the two-surface plasticity theory [16] and multi-layer theory [17], the evolution of α is determined by the relative position of a bounding surface with respect to the yield surface. As demonstrated by Case 3 in Figure 2.2, the direction of the yield surface motion is parallel to a vector that connects the current stress σ (on the yield surface) and a mapping stress σ β (on the bounding surface). At σ β, the bounding surface has the same outward normal as the yield surface does at σ, Figure 2.2. In this case, f n = 0 and f n+1 = 0 are the yield (loading) and the bounding surfaces, respectively. 2.3 Isotropic Yield Criteria This section briefly summarizes the commonly used isotropic yield functions. Although they cannot describe the plastic anisotropy of sheet metals, they are foundations upon which anisotropic yield criteria are constructed Von Mises and Tresca Yield Criteria The most commonly used yield criterion was proposed by von Mises in 1913 [18]. It stated that plastic deformation began when the elastic distortion energy reached a critical value. Usually, a von Miese equivalent stress is defined as 2σ 2 eq = (σ 11 σ 22 ) 2 + (σ 22 σ 33 ) 2 + (σ 33 σ 11 ) 2 + 6(σ σ σ 2 31) (2.36) Plastic deformation occurs when σ eq exceeds the elastic limit stress. For an isotropic material, σ eq is also the yield stress in a uniaxial tensile test. When expressed in the principal stress space, Equation 2.36 becomes: 2σ 2 eq = (σ 1 σ 2 ) 2 + (σ 2 σ 3 ) 2 + (σ 3 σ 1 ) 2 (2.37) 14

33 In the case of plane-stress (σ 3 = 0), it further reduces to σ 2 eq = σ σ 2 2 σ 1 σ 2 (2.38) An earlier yield criterion was based on the Tresca s maximum shear principle [19]. The Tresca s equivalent stress is σ eq = max { σ 1 σ 2, σ 2 σ 3, σ 3 σ 1 } (2.39) For plane-stress state, Equation 2.39 becomes σ 2 eq = (σ 11 σ 22 ) 2 + 4σ 2 12 (2.40) For many metallic materials, the experimentally measured yield loci fall between the predictions by von Miese and Tresca yield criteria. As shown in Figure 2.4, von Miese yield function can fit experimental data better than Tresca s criterion for many ductile metals, such as aluminum, copper and mild steel [8] Hosford s Non-quadratic Yield Criterion A more general, isotropic yield criterion with a non-quadratic form was proposed by Hosford [20]: 2σeq m = σ 1 σ 2 m + σ 2 σ 3 m + σ 3 σ 1 m (2.41) where 1 m < to ensure convexity of the yield surface. With proper choice of the exponent m, Equation 2.41 can better fit experimentally measured yield loci of sheet metals [21, 22]. It was found that m = 6 and m = 8 are suitable for BCC and FCC polycrystal aggregates [4]. A 2D (plane-stress) version of the Hosford s yield criterion is given by 2σ m eq = σ 1 m + σ 2 m + σ 1 σ 2 m (2.42) 15

34 σ 12 /σ eq Tresca von Mises Mild steel Aluminum Copper 0.2 Hill, σ 11 /σ eq Figure 2.4: von Mises and Tresca yield loci compared with experimental data. The yield locus describe by Equation 2.42 is plotted in the first quadrant of the 2D principal stress space for various m, Figure 2.5. It is clear that the Hosford yield criterion reduces to the Tresca and von Mises yield criteria, when m = and m = 2 respectively. The von Miese yield locus is an ellipse that is circumscribed to the Tresca polygon. When m increases from 2 to, the resulted yield loci are bounded by the von Miese and Tresca yield locus. Larger m tends to increase the curvature of the yield surface at the uniaxial and biaxial stress states, but reduce the curvature near the plain-strain states, Figure Anisotropic Yield Criteria In order to describe orthotropic plasticity of sheet metals, Hill first proposed a quadratic yield function [23]. It has been widely used for sheet forming simulations, 16

35 y m = 2 (von Mises) σ2/σeq m = (Tresca) m = 8 m = 20 x σ 1 /σ eq Figure 2.5: Plane-stress isotropic yield functions. because it can be applied to general stress states, and can be easily implemented into finite element codes. In fact, it is readily available from most commercial finite element codes, such as Abaqus [24] Hill s family of Anisotropic Yield Criteria Hill s anisotropic yield criteria is a generalization of the von Mises criterion [23,25]. The equivalent stress is defined as 2σ 2 eq = F (σ 22 σ 33 ) 2 +G(σ 33 σ 11 ) 2 +H(σ 11 σ 22 ) 2 +2Lσ Mσ Nσ 2 12 (2.43) where F, G, H, L, M and N are material constants. Obviously, Equation 2.43 reduces to the von Mises yield function when all coefficients are set to unity. In case of plane-stress (σ 23 = σ 31 = σ 33 = 0), Equation 2.43 reduces to (G + H)σ Hσ 11 σ 22 + (H + F )σ Nσ 2 12 = 1 (2.44) 17

36 The coefficients G, H and F can be determined from the r-values of the sheet metals measured at different angles to the sheet rolling direction: r 0 = H G, r 90 = H F, r 45 = H F + G 1 2 (2.45) while the determination of N requires shear test. When the principal directions of the stress tensor are coincident with the anisotropic axes, i.e., σ 12 = 0, Equation 2.44 becomes σ 2 1 2r r 0 σ 1 σ 2 + r 0(1 + r 90) r 90 (1 + r 0 ) σ2 2 = σ 2 RD (2.46) where σ 1 and σ 2 are the principal stresses, and σ RD is the yield stress in uniaxial tension for a sample parallel to the sheet rolling direction. Now, only three material properties, i.e., the r-values r 0 and r 90, and the uniaxial tensile stress σ RD, are needed to determine the yield surface. This makes the Hill 48 yield criterion easy and friendly to use in practice. As a special case, normal anisotropy (or, planar isotropy) leads to r 0 = r 90 = r 45 = r, and Equation 2.46 further reduces to σ 2 1 2r 1 + r σ 1σ 2 + σ 2 2 = σ 2 RD (2.47) At the balanced biaxial stress state (i.e., σ 1 = σ 2 = σ b ), the above equation leads to σ b = σ RD 1 + r 2 = r > 1, σ b > σ RD (2.48) Therefore, Equation 2.47 fails to describe the so-called anomalous behavior, which was commonly observed in aluminum alloys [26]. Many aluminum alloys have r- values less than unity, but their biaxial yield strengths are higher than the uniaxial tensile strength. Furthermore, it can be shown that σ 0 r 0 (1 + r 90 ) = σ 90 r 90 (1 + r 0 ) (2.49) 18

37 which always predicts that r 0 > r 90 if σ 0 > σ 90. Therefore, the Hill 48 cannot represent the so-called secondary order anomalous behavior either [27]. Hill 48 yield locus is always elliptic for the plane-stress case, because of its quadratic form. However, r-value can significantly change the shape of the yield locus, Figure The effect of increasing r-value gives larger biaxial tension yield stress, i.e., stronger thinning resistance. Meanwhile, the plane-strain state moves along the dashed line as indicated in Figure It approaches the biaxial stress state as r-value increases. 2 σ 2 /σ eq r = 5 dε 2 = 0 r = 2 r = 1 r = 0.5 r = σ 1 /σ eq Figure 2.6: Effect of r-value on the yield surface shape To remedy the aforementioned drawbacks of the Hill s 48 yield criterion, a more general general yield criterion with a non-quadratic form (Hill 79) was proposed [28]: σeq m = f σ 2 σ 3 m + g σ 3 σ 1 m + h σ 1 σ 2 m + a 2σ 1 σ 2 σ 3 m (2.50) + b 2σ 2 σ 1 σ 3 m + c 2σ 3 σ 1 σ 2 m 19

38 where m is a material constant. For planar anisotropy, four special cases can accommodate the yield anomaly, but only the case IV assures convexity of the yield surface, with a = b = g = f = 0: σ m eq = c σ 1 + σ 2 m + h σ 1 σ 2 m (2.51) The constants c and h can be determined from uniaxial tension test along the sheet rolling direction: h c = 2r + 1 and c + h = 1. Then, Equation 2.51 can be rewritten as 2(1 + r)σ m RD = σ 1 + σ 2 m + (1 + 2r) σ 1 σ 2 m (2.52) Now the ratio of the biaxial stress to the uniaxial stress can be computed: σ b σ RD = m 1 + r (2.53) 2 m 1 As shown in Figure 2.4.1, this new yield criterion is capable of describing the anomalous yield phenomenon, if m < 2. Hosford s non-quadratic isotropic yield function can be extended to incorporate the plastic anisotropy [29]: σ m eq = F σ 1 σ 2 m + G σ 2 σ 3 m + H σ 3 σ 1 m (2.54) The exponent m is related to the crystallographic texture: m = 6 and m = 8 are found to be feasible for BCC and FCC materials respectively. In Hill s model, however, the exponent m is determined from experiments, and thus it is not necessarily an integer. Since Hill 79 criterion can only be used when the principal axes of the stress tensor are aligned with the orthotropic symmetry axes, its application is somewhat limited. A generalization of Hill 1979 yield criterion (case IV) was proposed to include shear terms [30]. Other extensions were also made [25, 31, 32], but they are not discussed 20

39 Stress ratio m=1.5 m=2 m=2.5 σ b =σ RD r-value Figure 2.7: Effect of m on stress ratio σ b σ RD. here. There are other yield criteria appeared in the literature, such as a polynomial yield function with fourth order [33, 34], an empirically constructed yield function based on the Bezier interpolation [35], and a yield criterion that was expressed in polar coordinates [36 38]. A more detailed review of the various yield functions can be found elsewhere [39] Karafillis and Boyce s Yield Criterion A generic anisotropic yield function was proposed by superposing two stress potentials with different weights [40]: φ 1 = S 1 S 2 m + S 2 S 3 m + S 3 S 1 m (2.55) φ 2 = 3 m m 1 ( S 1 m + S 2 m + S 3 m ) (2.56) φ = cφ 1 + (1 c)φ 2 (2.57) 21

40 where c is a weight factor, and S i (i = 1-3) are the principal values of a transformed stress tensor S = L : σ. The fourth order tensor L is determined by the material symmetry and mechanical anisotropy. The tensor L is symmetric: L ijkl = L jikl = L jilk = L klij (2.58) Unlike other works, this yield criterion (K&B 93) is not limited to describe plastic orthotropy. It can be used for materials with lower symmetries, such as monoclinic and trigonal, by properly choosing the the components of L. For orthotropic materials (such as metal sheets), there are only 6 non-zero components in L: (c 2 + c 3 )/3 c 3 /3 c 2 / c 3 /3 (c 3 + c 1 )/3 c 1 / [L] = c 2 /3 c 1 /3 (c 1 + c 2 )/ c c c 6 If c = 0, the K&B 93 yield function reduces to the Barlat 91 yield function, as will be discussed next. The yield locus predicted by the K&B 93 is in very good agreement with experimental data, as well as theoretic calculations based on the Bishop-Hill theory [41] Barlat s Family of Anisotropic Yield Criteria In 3D stress space, the Barlat 91 yield surface is given by [42]: f = σ eq σ y = 0, (2.59) The equivalent stress, σ eq, is calculated from a stress potential: φ = 2σ m eq = S 1 S 2 m + S 2 S 3 m + S 3 S 1 m. (2.60) 22

41 where m is a material constant. Usually m = 6 and m = 8 are used for BCC and FCC metals [42]. S i (i=1,2,3) are the eigenvalues of the so-called transformed deviatoric stress tensor, S: The components of S are [S ] = S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 (2.61) S 11 = c(σ 11 σ 22 ) b(σ 33 σ 11 ) 3 S 22 = a(σ 22 σ 33 ) c(σ 11 σ 22 ) 3 S 33 = b(σ 33 σ 11 ) a(σ 22 σ 33 ) 3 S 12 = hσ 12 S 23 = gσ 13 S 31 = fσ 23 Six material parameters (a, b, c, f, g and h) are needed for the Barlat 91 yield function. Constants a, b and c can be determined from uniaxial tensile tests along the RD, TD and ND directions; while shear tests along these directions are required for f, g and h. For plane-stress case (σ 3 = 0), only a, b, c and h are relevant, and tension in 45 can be used instead of in-plane shear test to determine h. In practice, a biaxial tensile test is utilized instead of compression in the thickness direction for sheet metals, although a few sheet samples can be stacked together for compression test [43]. The Barlat 91 yield function works better than Hill 48 criterion for aluminum sheets. However, some discrepancy was found between experimental data and the model predictions for some aluminum-magnesium sheets [43]. An improved version, the Barlat 94 yield criterion, was proposed [43]: φ = α 1 S 2 S 3 m + α 2 S 3 S 1 m + α 3 S 1 S 2 m (2.62) 23

42 where S i are the principal values of the transformed stress tensor S. The coefficients, α k (k = 1 3), are calculated as follows: α k = α x p 2 1k + α y p 2 2k + α z p 2 3k (2.63) where p ij (i, j=1 3) are the direction cosines between the principal axes of anisotropy and the principal axes of S. A further improvement was achieved by making α k depend on the orientation of the principal axes [43]: α x = α x0 cos 2 (2β 1 ) + α x1 sin 2 (2β 1 ) (2.64) α y = α y0 cos 2 (2β 2 ) + α y1 sin 2 (2β 2 ) (2.65) α z = α z0 cos 2 (2β 3 ) + α z1 sin 2 (2β 3 ) (2.66) where β i (i = 1 3) are the angles between the material symmetry axes and the principal axes of S: { y 1 if S1 S cos β 1 = 3 y 3 if S 1 < S 3 { z 1 if S1 S cos β 2 = 3 z 3 if S 1 < S 3 { x 1 if S1 S cos β 3 = 3 x 3 if S 1 < S 3 Here, {x, y, z} indicate the orthogonal coordinate frame attached to the material, i.e., the symmetry axes of the sheet; while {1, 2, 3} are the principal directions of the transformed stress tensor S. For plane-stress, the non-zero components of the 24

43 transformed stress tensor S are: S 11 = c 1(σ 11 σ 22 ) + c 2 σ 11 3 S 22 = c 1σ 22 + c 3 (σ 11 σ 22 ) 3 S 33 = c 2σ 11 c 1 σ 22 3 S 12 = c 6 σ 12 The coefficients α k now become α x = α x cos 2 (2ψ) + α y sin 2 (2ψ) α y = α x sin 2 (2ψ) + α y cos 2 (2ψ) α z = α z0 cos 2 (2ψ) + α z1 sin 2 (2ψ) where α z0 = 1, and ψ is the angle between the 1-axis and the sheet rolling direction x. In total, 7 parameters are needed in the Barlat 96 yield function: c 1, c 2, c 3, c 6, α x, α y and α z1. These material constants can be determined from three uniaxial tensile tests in the 0, 45 and 90 directions, plus a hydraulic bulging test. One shortcoming of the Barlat 96 yield function is that there is no guarantee of convexity for this yield surface [43]. Therefore, numerical difficulties were encountered in finite element analysis. To remedy this drawback, a new plane-stress yield criterion has been recently formulated (i.e, the Barlat 2000 yield function), using two stress potentials φ 1 and φ 2 [44]: σ m eq = φ 1 + φ 2 2 (2.67) φ 1 = X 1 X 2 m (2.68) φ 2 = 2Y 2 + Y 1 m + 2Y 1 + X 2 m (2.69) 25

44 where X i and Y i (i=1,2) are the principal values of the following transformed stresses: X 11 C 11 C 12 0 s 11 X 22 = C 21 C 22 0 s 22 X C 33 s 12 Y 11 Y 22 Y 12 = C 11 C C 21 C C 33 where s ij (i, j = 1, 2) are the deviatoric stress components in plane-stress. Since φ 1 only depends on X 1 X 2, matrix [C ] has only 3 independent components. Usually, C 12 = C 21 = 0 is imposed [45]. It is proved that this yield function is convex [45]. In addition, its numerical implementation into finite element code is easier than the Barlat 96 function. The 8 material parameters can be determined from the uniaxial tensile test and the balanced biaxial tension test. s 11 s 22 s Anisotropic Yielding of 6022-T4 Aluminum Some of the previously discussed yield functions are used to describe the initial plastic anisotropy for 6022-T4 aluminum sheet. Figure 2.8 compares the predicted yield strengths and r-values with the experimental data for samples at different orientations with respect to the sheet rolling direction. The coefficients of Hill 48 yield function are calibrated from the measured r-values in unaxial tensile tests. As shown in Figure 2.8, the predicted yield stresses cannot match the experimental results. On the other hand, the parameters of the Barlat 91 (2D version) yield function are calculated from the yield stresses, but the predicted r- values do not agree with the measured results. Calculated yield strengths and r-values using the Barlat 96 and 2000 yield functions are in very good agreement with the experimental data, because more parameters are fitted from both the measured yield 26

45 T4 1 Normalized yield stress von Mises Barlat' Barlat'96 Barlat 2000 Hill'48 Exp Angle from RD (degree) (a) r-value von Mises Barlat'91 Barlat'96 Barlat 2000 Hill'48 Exp. (Barlat, 1997) Angle from RD (degree) (b) Figure 2.8: Comparison of the calculated and experimental results for 6022-T4: (a) normalized yield stress, and (b) r-value. strengths and r-value. For the 6022-T4 aluminum sheet, the coefficients of Hill 48, Barlat 91, Barlat 96 and Barlat 2000 yield functions are listed in Table 2.1 [42 44]. 27

46 Hill 48 F G H L M N Barlat 91 a b c f g h Barlat 96 c 1 c 2 c 3 c 4 α x α y α z0 α z Barlat 2000 C 11 C 22 C 33 C 11 C 12 C 21 C 22 C These coefficients are computed from the matrices [L ] and [L ] given in [45]. Table 2.1: Parameters for Barlat 91, 96 and 2000 yield functions. 28

47 CHAPTER 3 INTEGRATION OF PLASTIC RATE EQUATION In this chapter, the numerical implementation of Barlat s anisotropic yield functions and a nonlinear kinematic hardening model with multiple back stress components (denoted as mnlk hereafter) are discussed. The general procedure to integrate the plastic rate equation is first reviewed. Three algorithms, i.e., the forward, backward and semi-backward integration methods are explained. The procedure of implementing material constitutive models into a commercial finite element package ABAQUS/Standard is discussed. To verify the implementation, tension/compression tests are simulated and the simulated flow curves after a strain path reversal are compared with the experimental results for 6022-T4. The proposed algorithm is mathematically simpler than the previously developed Geng-Wagoner hardening law that was based on a two-surface plasticity model [46]. The mnlk model only requires six fitting parameters, which can be derived from the uniaxial compression/tension (C-T) or tension/compression (T-C) tests. 3.1 Introduction to Stress Update Schemes For small strain deformation, rate-independent elasto-plasticity with an associated flow rule, the task of stress update scheme is to solve the the following differential 29

48 equations, using an appropriate integration method: ɛ = ɛ e + ɛ p (3.1) σ = C el : ɛ e (3.2) ɛ p = λa = λ f σ (3.3) V = λh(σ; V ) (3.4) The above governing differential equations are subject to two constrains: f(σ; V ) = σ eq (σ α) σ 0 R(q) = 0 (3.5) λ 0, f 0 and λf 0 (3.6) where V = {α, R} denotes the internal variables collectively, and H specifies the evolution laws of the internal variables. The first constraint states that the yield condition is always satisfied, while the second one is the loading-unloading criterion. The evolution of the internal variables, given by Equation 3.4, can be rewritten in component form, α = λh kin (σ, α) (3.7) R = R(q) (3.8) where the first equation specifies a general kinematic hardening, and the second one is the isotropic hardening. In the stress space, Equations 3.7 and 3.8 describe the translation and expansion of the yield surface. The commonly used isotropic hardening laws are the linear hardening (R = h iso q), the power law hardening (R = R 0 q n ) and the Voce s hardening law with a saturation value (R = R R e mq ). When the associative flow rule is used, such as in this study, the yield surface normal, a, is also the direction of the plastic flow. 30

49 Due to the nonlinear nature of the governing differential equations, an incremental solution procedure is usually required. Let ( ) ( ) n+1 ( ) n denotes an increment over a time interval [t n, t n+1 ]. Then, the above equations can be incrementally solved to obtain { σ n+1, ɛ n+1, ɛ p n+1, V n+1 }, if {σn, ɛ n, ɛ p n, V n } and the total strain increment ɛ are known. For implicit finite element method, the so-called material Jacobian (algorithmic or consistent stiffness matrix, i.e., C alg = δσ δɛ n+1) is also required in order to formulate the element stiffness matrix: N IP ( K e = B T C alg BdV V e w i B T C ep B ) (3.9) V e i where B is the strain-displacement matrix (ɛ = Bu), w i is the weight at the Gaussian integration point, and V e is the element volume. As pointed out by Simo and Taylor [47], the use of consistent stiffness matrix ensured a quadratic rate of convergence, when the Newton-Ralphson method was used at structural level to solve the nodal displacement u from the global equilibrium equation Ku = f ext. Based on how the elasto-plastic rate equations are discretized and integrated, there are three major integration methods, namely the forward, backward and semibackward Euler integration methods, which are summarized in the following sections. 3.2 Forward Integration i=1 The rate equations are first discretized in the time domain, and the current incremental changes in the plastic strain and plastic multiplier are approximated as ɛ p ɛ p n t and λ λ n t, within the current time increment t = t n+1 t n. Notice that the rates at the beginning of the increment are used. The plastic consistency condition can be derived from Equation 3.5 and 3.6: f = f σ : σ + f V V = 0 (3.10) 31

50 where the subscripts σ and V denote the partial derivatives with respect to the stress and the internal variables respectively. Following a standard derivation [48], the plastic multiplier λ is calculated as λ = a : C el : ɛ a : C el : a f V H (3.11) Then, the strain and stress are updated using the forward Euler integration method: ɛ n+1 = ɛ n + ɛ (3.12) ɛ p n+1 = ɛ p n + λa n (3.13) V n+1 = V n + λh (σ n ; V n ) (3.14) σ n+1 = σ n + C el : ( ɛ ɛ p ) (3.15) Finally, the continuum tangent stiffness matrix C ep is obtained by substituting Equation 3.11 back into the above equations: C ep = C el (Cel : a) (C el : a) a : C el : a f V H (3.16) Since the yield condition is not enforced at the end of the current increment, i.e., f n+1 0, the updated stress and internal variables deviate from the exact solutions. Therefore, the forward Euler solution drifts away from the exact one, and an additional step is needed to correct this drift. However, the widely used return mapping algorithm, which is based on the backward Euler integration method, does not suffer from this shortcoming. 3.3 Backward Integration In contrast to the forward method, the incremental quantities are calculated from rates at the end of the current increment in the backward method. That is, ɛ p 32

51 ɛ p n+1 t and λ λ n+1 t. In addition, the yield condition is always enforced at the end of the current increment, so that solution drifting is avoided: ɛ n+1 = ɛ n + ɛ (3.17) ɛ p n+1 = ɛ p n + λa n+1 (3.18) V n+1 = V n + λh n+1 (σ n+1 ; V n+1 ) (3.19) σ n+1 = σ n + C el : ( ɛ ɛ p ) (3.20) f n+1 = σ eq (σ n+1 α n+1 ) σ 0 R(q n+1 ) = 0 (3.21) Since ɛ p n+1, α n+1 and λ n+1 are unknown during the current time increment, an iterative solution procedure is required. According to the return mapping algorithm, which was firstly recognized by Simo and Taylor [47], the current stress is iteratively modified from an elastic trial value: σ n+1 = σ n + C el : ( ɛ ɛ p ) = σ trial n+1 λc el : a (3.22) Notice that the mapping direction is given by the flow vector a, and usually it is not constant during the iteration. The return mapping algorithm consists of two steps, based on the methodology of operator splitting [47]: 1. Elastic Predictor In this step, a trial stress is first calculated, assuming that only elastic deformation happens and all internal variables are frozen, i.e., ɛ p n+1 = 0 and q = 0: σ trial n+1 = σ n + C el : ɛ (3.23) 2. Plastic Corrector If the elastic trial stress lies inside the previous yield surface, then the elastic 33

52 trial stress is readily accepted as the final solution. Otherwise, plasticity occurs during the current increment, because the elastic trial stress exceeds the yield stress, i.e., f trial = σ eq (σ trial n+1 α n ) σ 0 R(q n ) > 0. A plastic correction step is applied to restore the yield condition: σ n+1 = σ trial n+1 σ n+1 = σ trial n+1 λc el : a n+1 (3.24) An iterative solution procedure is required to obtain the stress and internal variables, since the flow direction a n+1 is also to be solved. The return mapping algorithm is graphically shown in Figure 3.1. In the stress space, it can be envisioned as a geometric mapping by which the elastic trial stress is projected onto an updated yield surface. The projecting direction is the current yield surface normal, which is unknown in priori. σ trial n+1 f trial > 0 σ 2 λc el : a n+1 σ n+1 σ n Elastic Region σ 1 f n+1 = 0 Figure 3.1: Schematics of return mapping algorithm in stress space. 34

53 In the plastic correction step, the equations are nonlinear with respect to the incremental plastic multiplier, λ, while the total strain increment is kept constant. These nonlinear equations can be solved numerically by the Newton-Ralphson method. Let χ( λ) = 0 denotes a nonlinear equation of λ. An initial guess is λ (0) = 0. After linearization using the first order Taylor s series expansion, the k th iterative change of λ can be calculated as follows: ( ) (k) dχ χ (k) + δλ (k) = 0 = δλ (k) = χ(k) d λ ( dχ ) (k) (3.25) d λ Then, an update is made for the next iteration, λ (k+1) = λ (k) +δλ (k). The iteration procedure continues until convergence is achieved, i.e., δλ (k+1) T OLn where T OL n is a prescribed tolerance. In the following context, the subscript n+1 will be omitted for quantities that are evaluated at the end of the current increment. After rearrangement, Equations 3.18, 3.19 and 3.21 become δλ (k) Ψ = ɛ p + ɛ p n + λa = 0 (3.26) Υ = V + V n + λh = 0 (3.27) f = 0 (3.28) Both Ψ an Υ are functions of the iterative change of stress, σ (k) and change of the internal variable, V (k). Apply the Newton-Ralphson method: [48]: [ ( Ψ (k) + δλ (k) a (k) + C el 1 : σ (k) + λ (k) a ) (k) σ : σ (k) + ( ) a (k) V V (k)] = 0 Υ (k) + δλ (k) H (k) V (k) + λ (k) [ ( H σ 35 (3.29) ) (k) : σ (k) + ( ) H (k) V V (k)] = 0 (3.30) f (k) + f (k) σ : σ (k) + f (k) V V (k) = 0 (3.31)

54 To solve for the unknowns σ (k), V (k) and δλ (k), the first two equations are written in matrix form after some rearrangements: ] { } 1 σ [M (k) (k) = r (k) δλ (k) ñ (k) (3.32) V (k) where [M] 1 = r = [ C el 1 + λ a σ { Ψ Υ λ H σ }, ñ = λ a V I + λ H V { } a H ] The stress and internal variable increments are then obtained: { } σ (k) = M (k) : r (k) δλ (k) M (k) : ñ (k) (3.33) V (k) Substitute this into Equation (3.32), the k th iterative change of the plastic multiplier is obtained: δλ (k) = f (k) f (k) : M (k) : r (k) f (3.34) (k) : M (k) : r (k) where f = {f σ, f V }. The plastic strain and internal variables are then updated: ɛ p(k+1) = ɛ p(k) C el 1 : σ (k) (3.35) V (k+1) = V (k) + V (k) (3.36) λ (k+1) = λ (k) + δλ (k) (3.37) The complete algorithm is summarized in the following box [48]. 36

55 Return Mapping Algorithm with Backward Euler Integration 1. Initialization: set initial values of plastic strain and internal variables to the last converged values: k = 0: ɛ p(0) = 0, V (0) = V n, λ (0) = 0 and σ (0) = σ trial. 2. Check the yielding condition, evaluate the residuals and check convergence at the k th iteration: f (k) = σ eq (σ (k) α (k) ) σ 0 R(q (k) ) { } (k) r (k) Ψ = Υ If f (k) < T OL f and r (k) < T OL r, convergence is achieved after k- iterations. Otherwise, go to the next step. 3. Calculate the incremental plastic multiplier: [M] 1 = [ C el 1 + λ a σ λ H σ λ a V I + λ H V δλ (k) = f (k) f (k) : M (k) : r (k) f (k) : M (k) : r (k) 4. Compute the incremental stress and internal variables: { } σ (k) = M (k) : r (k) δλ (k) M (k) : ñ (k) V (k) 5. Update the stress and internal variables: ɛ p(k+1) = ɛ p(k) C el 1 : σ (k) V (k+1) = V (k) + V (k) λ (k+1) = λ (k) + δλ (k) σ (k+1) = σ (k) + σ (k+1) k + 1 k, go to 2. ] 37

56 3.4 Semi-backward Integration The backward Euler method is robust and accurate. However, it involves the calculation of the yield surface curvature, i.e., the second derivatives of the yield function with respect to the stress tensor, 2 f σ 2. When a complicated yield criterion is used, these calculations can be quite painful and are possible sources of programming errors. An alternative to the backward Euler method is the tangent cutting plane algorithm [49]. It intends to solve the following differential equations: ɛ n+1 = ɛ n + ɛ (3.38) ɛ p n+1 = ɛ p n + λa n (3.39) V n+1 = V n + λh n (σ n ; V n ) (3.40) σ n+1 = σ n + C el : ( ɛ ɛ p ) (3.41) f n+1 = 0 (3.42) Notice that this algorithm is different from the backward integration method, because it is implicit only in terms of λ. The flow vector a at the beginning of the increment is used. Therefore, this method is often called the semi-backward integration. Figure 3.2 illustrates the difference between the backward and the semi-backward integration methods. An iterative solution procedure can be geometrically envisioned as a projection that brings the elastic trial stress onto an updated yield surface in the stress space. In the backward Euler method, the projection always operates on the initial elastic trial stress, but the direction of return mapping is successively updated. However, in the semi-backward algorithm, each projection uses different starting stresses. For some cases, the tangent cutting plane algorithm may cause stress drift away from the exact solution [50]. 38

57 σ 2 σ n+1 σ trial n+1 σ (1) n+1 σ (k) n+1 σ 2 Cuts σ n+1 σ trial n+1 σ (1) n+1 σ (k) n+1 σ n σ n Elastic Region f n+1 = 0 σ 1 Elastic Region σ 1 f n+1 = 0 (a) (b) Figure 3.2: Schematics of the return mapping schemes: (a) the backward Euler method, and (b) the tangent cutting plane method. 3.5 Implementation to Abaqus via UMAT The commercial FE software Abaqus provides a powerful tool by allowing users to implement their own constitutive models through a user subroutine interface (UMAT). This subroutine is called at every integration point, hence it needs to be accurate, robust and yet computationally efficient [24]. The data passed into a UMAT include the stress, strain and internal valuables from the last converged increment, and the current total strain increment that is computed from the nodal displacement after solving the global equilibrium equation. The outputs of a UMAT are the updated stress and internal variables, and the consistent tangent stiffness matrix for implicit finite element program, Figure

58 {σ n, ǫ p n, α n, R n } ǫ UMAT {σ n+1, ǫ p n+1, α n+1, R n+1 } C alg = δσ δǫ n+1 Figure 3.3: Input and output of a user material subroutine. Springback prediction by FEM relies on the accurate description of the plastic anisotropy of the sheet material. In this study, the anisotropic yielding of a sheet metal is modeled by the Barlat 91, 96 and 2000 yield functions [42 44]. In addition, the anisotropic hardening behavior, following a reversed strain path, is also included in the numerical implementation using the mnlk hardening model. The coefficients in the yield functions are calibrated from the uniaxial yield strengths and the r-values of sheet specimens at 0, 45 and 90 degrees with respect to the sheet rolling direction. The Barlat 96 yield function also requires the balanced biaxial flow stress that can be measured form the bulge test [43]. The anisotropic hardening is modeled by a nonlinear kinematic hardening (NLK ) with an Armstrong-Frederic type of rule for the back stress evolution [10]. The basic NLK hardening model (with one back stress) is available from Abaqus, but it can only be used with simple yield criteria, such as the von Mises and the Hill 48 [24]. Moreover, it cannot fully represent the stress-strain curves observed after a reversed strain path [46]. Therefore, an extended version of the nonlinear kinematic hardening with three back stress components [11] is adopted in this study. 40

59 The evolution of the back stresses is described by the commonly used Armstrong- Frederick [10] law: dα i = c i σ eq (σ α)dq γ i α i dq (3.43) where σ eq and α represent the size and center of the yield surface respectively, c i and γ i are model parameters. The total back stress is the sum of N component back stresses, as suggested by Chaboche to better simulate cyclic plasticity [12]: α = N α i, i = 1, 2,... N. (3.44) i=1 In this study, N = 3 and the model parameters c i and γ i can be fitted from in-plane C-T or T-C tests. The method to obtain these parameters will be discussed in next section. Detailed numerical algorithm of Abaqus UMAT is included in Appendix A. 3.6 Anisotropic Hardening of 6022-T4 Aluminum Sheet The 6 parameters in the mnlk model, i.e., c i and γ i (i = 1-3), can be fitted from the Bauschinger test results. A special device was used to prevent buckling when a sheet sample was subject to compression [51], Figure 3.4(a). The sample was sandwiched between two pairs of sliding forks, on which stabilizing pressure was applied to avoid buckling. Teflon was used between the sheet sample and the forks to reduce friction. For isotropic hardening material, the reverse flow curve is schematically indicated by Curve a in Figure 3.4. When hardening becomes anisotropic, the yield strength in compression is lower than the monotonic flow stress just prior to unloading, and a region of fast strain hardening follows, as demonstrated by Curve b in Figure 3.4. The third characteristic of the Bauschinger effect is the permanent softening. At large strains after the load reversal the reverse flow curve becomes parallel to the monotonic one, with an offset in flow stress, σ ps. 41

60 Pressure σ 1 Stress Monotonic flow curve a b c (a). isotropic hardening σ ps σ 1 Pressure (b) anisotropic hardening without permanent softening (c). with permannet softening Effective strain (a) (b) Figure 3.4: Schematics of the Bauschinger test: (a) the fork device, and (b) flow curves after load reversal. The reversed stress-strain curves from uniaxial tension/compression tests were available for 6022-T4 aluminum with three or four prestrains ranging from 0.01 to 0.08 [52]. A simplex algorithm is used to obtain the optimum values of the six parameters in the mnlk model. The object function is the error defined as follows Error = 1 K σ(k) exp σ (k) FEM 2 (3.45) K k=1 where K is the number of experiment data points, σ exp and σ FEM are the measured and calculated flow stresses after a load reversal respectively. A Fortran program was written to search for the optimal parameters that minimized the object function. The results after optimization are shown in Table 3.1, for both compression/tension and tension/compression tests. σ (k) exp To validate the numerical implementation of the mnlk model, the reverse flow curves generated from the Bauschinger tests of 6022-T4 sheet were simulated by finite 42

61 c 1 c 2 c 3 γ 1 γ 2 γ 3 C-T fit T-C fit Table 3.1: Parameters c i (MPa) and γ i for the mnlk model. element method. One 4-node shell element (Abaqus type S4R) was used in the FE model to simulate the C-T and T-C tests. Figures 3.5(a) and 3.5(b) compare the simulated and measured reverse flow curves after different prestrains, for T-C and C-T tests respectively. As can be seen, the Bauschinger effect is closely reproduced by the mnlk model. The main features of the Bauschinger effect, i.e, the reduced yielding and the fast transient hardening, are clearly captured. For small prestrains ( 0.01), there is little permanent softening, i.e., the reverse flow curve eventually joins the monotonic one, Figure 3.5. The permanent softening is also captured by the mnlk model. In Figure 3.6, simulated reverse flow curves from C-T tests, using different hardening laws, are compared with the experimental data for one test case. Clearly, the isotropic law is incapable of describing neither the fast transient hardening nor the permanent softening behavior after a reversed strain path. The original NLK model, with single back stress, can either reproduce the transient hardening region or the permanent softening, Figure 3.6(a). It cannot capture all three features of the Bauschinger effect. On the other hand, both the Geng-Wagoner model based on two 43

62 True stress (MPa) T4, C-T Monotonic Exp. ε com = ε com = ε com = Effective strain True stress (MPa) T4, T-C 300 Monotonic Exp. ε =0.079 ten ε ten =0.047 ε ten =0.029 ε ten = Effective strain (a) (b) Figure 3.5: Simulated and experimental reverse flow curves after: (a) compression/tension test, and (b) tension/compression test. Markers are experimental data from Balakrishnan (1999), and solid lines are the corresponding simulation results. surface plasticity [46] and the mnlk model can closely simulate the Bauschinger effect. However, the mnlk model is simpler than the Geng-Wagoner model, in terms of the mathematical derivation and numerical implementation. Taking the Bauschinger effect into consideration gives more realistic prediction for draw-bend springback simulation, because the through-thickness stress distribution in forming step can be better represented [46]. If material softens after a load reversal, such as the 6022-T4 used in this study, the bending moment will be overestimated if an isotropic hardening model is used. As a result, the amount of springback is overpredicted [46]. 44

63 Isotropic T4 C-T Monotonic True stress (MPa) NLK (one α ) (c=375, γ=5) NLK (one α ) (c=38000, γ=400) Exp. ε com = Effective strain True stress (MPa) mnlk (α 1,2,3 ) Two surface (Geng 2001) Exp. (Balakrishnan, 1999) ε com = Effective strain (a) (b) Figure 3.6: Simulated reverse flow curves using various hardening laws: (a) isotropic hardening and NLK model with single back stress, and (b) mnlk and Geng-Wagoner models. 45

64 CHAPTER 4 SPRINGBACK SIMULATION OF DRAW-BEND TEST WITH FINITE ELEMENT METHOD Note: Some of the draw-bend experiment data were provided by W.D. Carden. This chapter also utilized the tension-compression test results from V. Balakrishnan. A manuscript to be submitted for publication in the International Journal of Plasticity is in preparation. Abstract Accurate springback prediction is essential for tool design and quality control in sheet forming processes. To understand the springback phenomenon, a series of draw-bend tests were carried out, under carefully controlled laboratory conditions. Aluminum alloy 6022-T4, which has been considered as a potential replacement for forming grade steel to reduce vehicle weight, was tested using a range of tool radii (3.2mm 12.7mm) and normalized sheet tension ( ). Springback angles were measured from the unloaded strips that experienced sequential bending and unbending deformation. It was found that springback decreased with the tool radius and the applied stretching force, with the later dominating. A dramatic drop in springback angle was attributed to the persistent anticlastic curvature in the sheet width direction, when the sheet 46

65 tension approached the yielding force of the material. This secondary curvature significantly increased the section moment of inertia for bending, and thus greatly reduced springback. To understand the mechanics of the anticlastic curvature and its effect on springback, additional tests were conducted for specimens with various width (12mm 50mm). The effect of the anticlastic curvature on springback was clarified. Finite element simulations were carried out, using Barlat s family of anisotropic yield functions (Barlat 91, 96 and 2000) and a non-isotropic hardening model that was developed to reproduce the Bauschinger effect after a load reversal. The effect of the tool radius on the springback was discussed. It is suggested that solid element should be used for small tool radius (R = 3.2mm) where shell assumptions are invalid. 4.1 Introduction Springback denotes the undesired shape change of a formed part after the removal of the forming load. It often causes difficulty in part assembly, and extra work in tool design is needed to compensate for the shape deviation. Therefore, accurate prediction of springback in sheet metal forming operations is essential to reduce the try-out time in tool design, and to improve quality of the stamping parts. Many researchers have established empirical methods [53] for springback prediction and compensation during the past few decades, others have proposed analytical models based on engineering bending theory of beams [54 56]. There are many examples of predicting springback after simple forming operations, such as cylindrical tool bend [57], V-die bend [58 60], U-channel forming [61] and flanging [62]. Although these experiments are not close to the real industrial sheet metal forming processes, scientific understanding was well developed on the 47

66 relationship between springback and material properties and process parameters. It was concluded that springback increased with the increase of material yield strength and the tool radius, but decreased with increasing elastic modulus for an elastic perfectly plastic material after pure bending [55]. When considered strain hardening and plastic anisotropy, it was shown that strain hardening reduced springback [63] while the plastic strain ratio (i.e., r-value) promoted springback [63]. Among all the process variables that affect springback, sheet tension has the most prominent effect in reducing springback [64]. Hence, stretching became an efficient way to minimize springback in many real forming operations. Several shape control methods based on this idea have also been developed, and successfully applied to compensate for the springback in the flange operations. Friction also reduces springback, but in most studied cases it functioned because the sheet tension was increased by friction through a blank holder or a drawbead [65, 66]. Numerous experiments have been conducted on springback behavior of sheet metals under simple test conditions, but there is little information for complicated sheet metal forming operations under controlled test conditions. For automotive industry, this desire becomes even more imperative because of the growing application of high-strength aluminum alloy sheet for structural components, driven by the more stringent requirements of improving fuel consumption and reducing green-house gas emission. The research challenge of springback prediction comes from the fact that aluminum is inferior to steel in formability, and it is prone to springback, primarily because of its lower elastic modulus. In addition, springback prediction becomes more obscured because a formed aluminum sheet part can continue to change shape over time [67 70]. 48

67 While ordinary stretch-bending test is capable of providing direct control of sheet tension, it is unable to represent complicated deformations that are encountered in the practical forming processes in which a sheet metal slides over a rigid tool surface as it is drawn into a die cavity. However, the draw-bend test remedied this shortcoming, and it offered a better opportunity to study the springback behavior [69]. The current draw-bend test was initially designed for friction measurement of coated steel sheets [71, 72]. Although the basic concept is similar to the devices used by others [64, 73, 74], the most important improvement of the current test device is accurate measurement of the deformed shape because a longer drawing distance was adopted [67, 69]. This also makes it possible to observe and examine the side-wall curl [75] in the drawn region of the specimen with an improved accuracy. This sidewall curl presents another form of shape deviation, and it often appeared in U-channel forming and flange operations [62]. A secondary curvature, which is orthogonal to the primary one, was observed in the curled region in the sheet width direction. This curvature has been known as the anticlastic curvature [76], and it has been reported by a few authors in bending flat sheet and composite plate [77]. Although analytical methods are available for pure bending problems, using elastic theory of plate bending, there is a lack of knowledge of the anticlastic curvature in more complex forming processes, such as in the draw-bend test. Some conclusions were attained on how the specimen aspect ratio and bending radius are related to the magnitude of the anticlastic curvatures in four-point bending test [78], but they may not apply to the draw-bend test. In addition to the forming parameters, springback simulation is also sensitive to the choice of the material constitutive models. To accurately calculate the stress/moment 49

68 distribution in the formed part, the mechanical anisotropy of the sheet metals has to be considered. An anisotropic yield function is required to describe the orientationdependent initial yield strength and the plastic strain ratios (r-value) [29]. The modeling of the anisotropic hardening is more complicated, and a few models are available in the literature [10, 16, 79, 80]. In this study, a nonlinear kinematic hardening model was implemented into finite element program Abaqus/Standard via the user subroutine option. This model closely reproduces the characteristics of the Bauschinger effect following a load reversal, yet it has a simpler form compared with the previous work based on a two-surface plasticity theory [46]. The objective of this chapter is to study the springback behavior of 6022-T4 alloy sheets in the draw-bend test under controllable conditions. In particular, the effects of sheet tension, tool radius and anticlastic curvature on springback are clarified. The choice of elements (shell and solid) for springback simulation using finite element method is also discussed for small tool radius (R/t = 3.5). 4.2 Draw-Bend Experiment Series of draw-bend tests were conducted using a special apparatus at Colorado School of Mines that was initially designed to evaluate the friction behavior of coated steel sheets [71, 72]. Two groups of 6022-T4 sheet specimens were prepared for this study. For the first group, rectangular blanks (500mm 50mm 1mm) were sheared with their lengths aligned with either the rolling or transverse direction of the sheet metal. Tooling radii of 3.2 and 12.7mm were used for these tests. Samples from the second group have the same length but various widths ranging from 12mm to 50mm. Only the 12.7mm tool was employed for these strips, with two selected sheet tensions: 50

69 F b = 0.5 and F b = 0.9 (F b is expressed as fractions of the yielding force of the strip in uniaxial tension). After the forming load was released, specimens were removed from the test device, and they were traced onto paper for geometry measurement. In this chapter, only static springback results, measured approximately 60 seconds after unloading, were reported. Time-dependent springback results are presented and analyzed in Chapter Materials To compare with the previous investigation [69], the same aluminum alloy 6022-T4 sheet, provided by the Partnership of Next Generation Vehicles (PNGV), was used in this study. The chemical composition of this alloy is given in Table 4.1 [69]. Si Fe Cu Mn Mg Ti Al Bal. Table 4.1: Chemical composition of 6022-T4 aluminum sheet (in weight percent) Draw-Bend Test The draw-bend test equipment consists of a standard servo-hydraulic mechanical test machine and an attached 90-degree bending frame, Figure 4.1(b). It has two independently controlled hydraulic actuators that are perpendicularly oriented to each other. During the draw-bend test, the upper right actuator was programmed to provide a constant restraining force (i.e, the back force), while the lower one was 51

70 set to move downward at a constant speed. The actual front and back tensions were measured by two load cells that were attached to each hydraulic cylinder. start finish Roller Sample Upper grip F b 1 r R 3 2 R start 4 Lower grip θ Unloaded Loaded 127mm finish springback angle: θ 40mm/s (a) (b) Figure 4.1: Draw-bend experiment: (a) equipment at Colorado School of Mines, and (b) schematics of test procedure and geometry of a deformed sheet after springback Figure 4.1(b) schematically shows the draw-bend test procedure. A pre-bend was first performed by hand to obtain an approximately 90 bent after unloading. Then, a prescribed back force was applied at the left side of the sheet, while the right side was held immobile. During the test, the strip was drawn over a fixed or rotating cylindrical roller for a total travel of 127mm, while the back force was maintained constant. The drawing speed was 40mm/s, with a few exceptions where 10mm/s was used when a small back force (F b < 0.1) was applied. The draw-bend test can closely simulate the realistic sheet metal forming operations, where sequential bending and 52

71 unbending deformation occurs under superposed tension as the strip slides over the tool surface [69]. After the forming operation was completed, the lower grip was opened to allow the sheet to springback freely. The unloaded sample was then traced onto paper for geometry measurement. A typical unloaded strip was schematically shown in Figure 4.1(b), from which four distinguished regions of deformation were identified. In Region 1 and 4, material underwent pure stretching only, and thus they were not interested for springback study. Region 2 is characterized by a relaxed radius, R (originally equals the tool radius R), or the corresponding angle change θ 1. Material in Region 3 has undergone sequential bending and unbending with superposed stretching. The radius of the Region 3 after springback is r, with its corresponding angle change θ 2. r and θ 2 are measures of the so-called side-wall curl which has been reported in channel forming operations [62]. Following an earlier analysis [69], a single parameter, θ, was used to characterize the amount of springback for draw-bend tested specimens, Figure 4.1(b). Since the sheet tension played a dominant role in reducing springback [64, 74], multiple tests were conducted with the normalized back forces ranging from 0.1 to 1.1, at an increment of to improve resolution. Compared to the previous research [69], this study focuses more on the springback behavior in the small R/t range with refined back force intervals. Although an fixed or rotating tool can be used to provide different friction conditions, the effect of friction on springback was not the goal of this study. An industrial drawing lubricant [81] was carefully brushed on both the roller surface and the inner sheet surface to provide medium friction coefficient (approximately 0.15) [69]. 53

72 Previous study discovered a secondary curvature in the sheet width direction [69], which is perpendicular to the primary bending curvature in the curled region of an unloaded strip. This secondary curvature, often called as the anticlastic curvature in the literature [76], was originated from the differential contractions between the top and bottom fibers of a strip in elastic bending, caused by the Poisson s effect [76, 78]. However, the anticlastic curvature persisted after unloading for certain test conditions in the draw-bend test [69,82]. As a result, the springback angle was greatly reduced, because the persistent anticlastic curvature substantially increases of the section moment of inertia for bending. To understand this phenomenon, specimens with various widths (from 12mm to 50mm) were tested at two selected back forces of 0.5 and 0.9 with R/t = 14. Since the anticlastic curvature does not vary significantly within the sidewall curl region, measurements were made at the central position of the curl length, i.e., about 64mm away from the straight leg (Region 4 ). The radius of anticlastic curvature, R a, can be calculated from the following equation, assuming that the transverse cross-section is circular: R a = h 2 + W 2 8 h (4.1) where W is the strip width, and h is the height of the transverse arc. A digital caliper with 0.01mm resolution was used to measure h. 4.3 Experimental Results The static springback results are presented in this section. The effect of back force, tool radius and specimen geometry (width-to-thickness ratio, W/t) are discussed. 54

73 4.3.1 Effect of Back Force and Tool Radius Experiment results were tabulated in Appendix B, see Tables B.1 and B.2 for tool radius of 3.2 and 12.7mm, respectively. The measured springback angle, θ, and the anticlastic curvature, 1 R a, are plotted in Figure 4.2. From the results, it is clear that the stretching force plays a significant role in reducing the springback. In addition to this commonly observed trend, a dramatic drop in the springback angle occurs near F b = This is associated with an abrupt increase of the anticlastic curvature, Figure 4.2(a) and 4.2(b). It is noted that the critical back force, at which the fast decline of springback angle occurs, weakly depends on tool radius. θ (degree) T4 (RD) R/t=3.5 θ Curvature Normalized back force F b /R a (10x -3 mm -1 ) θ (degree) T4 (RD) R/t=14 θ Curvature Normalized back force F b /R a (10x -3 mm -1 ) (a) (b) Figure 4.2: Dependence of the springback angle and the anticlastic curvature on the normalized back force: (a) R/t = 3.5 and (b) R/t = For the draw-bend test, the bending radius has less significant impact on the springback angle than the sheet tension. Figure 4.2 shows that the springback angle 55

74 decreases as the R/t ratio increases. These results are in contrast with simple bending results, but they agree with other investigations [74]. The orientation of strips (RD or TD) also made a noticeable difference, in terms of the variation of the springback angle and the anticlastic curvature with back force. As shown in Figure 4.3(a), the springback angle of the transversely sheared specimens gradually decreases with increasing back force, while the RD samples show a fast decline of springback angle at F b = The anticlastic curvatures are smaller for TD samples, and the variation with F b is less abrupt, Figure 4.3(b). θ (degree) RD TD 6022-T4 R/t= Normalized back force F b (a) Curvature (10-3 mm -1 ) RD TD Normalized back force F b (b) Figure 4.3: Effect of back force on springback for samples with different orientation: (a) θ and (b) 1 R a. 56

75 4.3.2 Effect of Specimen Width In the literature, little work has been done systematically on the effect of the anticlastic curvature on springback in sheet metal forming operations. According to the elementary bending theory of beam, the anticlastic curvature only appears in narrow beams, for which plane-stress state is a good approximation in the width direction [76]. For wide strips, the anticlastic deflection was assumed to nearly disappear or localize in the strip edges [83], so that the deformation is characterized as plane-strain. According to an elastic theory of plate bending [84], the transition from plane-stress state to plane-strain was gradual [85], and the shape of the anticlastic surface was determined by a dimensionless parameter β, which is often called the Searle s parameter [86]: β = W 2 R x t = (W/t)2 R x /t (4.2) where W and t are the strip width and thickness respectively, R x is the radius of the primary (longitudinal) curvature. For draw-bend tested samples, r (Figure 4.1(b)) is used instead of R x in Equation 4.2 when applying the elastic theory to analyze the draw-bend springback. To explore the effect of the anticlastic curvature on springback, additional drawbend tests were conducted for 6022-T4 aluminum sheets with various widths (12 50mm), using R = 12.7mm tool and two back forces: F b = 0.5 and F b = 0.9. Measured springback angle and radius of the anticlastic curvature are reported in Appendix B, Table B.4. As shown in Figure 4.4(a), both the springback angle and the anticlastic curvature decrease as the sheet width increases for large back force (F b = 0.9). At smaller back force (F b = 0.5), the springback angle slightly decreases with the sheet width, followed by an increase for strips wider than 25mm. The 57

76 anticlastic curvature continuously decreases with the specimen width. The maximum change in springback angle from the narrowest (12mm) to the widest strip (50mm) is about 13 and 8 for F b = 0.9 and F b = 0.5, respectively. The maximum standard deviation of θ is about 1.8. The sources of scattering are probably caused by the non-uniform width, hardened strip edges due to shearing and poor alignment of narrow strips during the draw-bend test. θ (degree) T4 (RD) R/t=14 F b =0.5 θ Curvature Width (mm) (a) /R a (x10-3 mm -1 ) θ (degree) θ Curvature 5 2 R/t=14 F =0.9 0 b Width (mm) (b) /R a (x10-3 mm -1 ) Figure 4.4: Dependence of springback angle and anticlastic curvature on specimen width: (a) F b = 0.5 and (b) F b = Finite Element Results and Discussion The static draw-bend tests were simulated using finite element method, for a range of normalized back forces (0.1 F b 1.2) and two R/t-ratios: 3.5 and 14. Different elements were chosen in the 3D analysis: 4-node shell element with reduced integration (S4R), 8-node brick element (C3D8R with reduced integration and 58

77 hybrid C3D8H) and 20-node brick element (C3D20R with reduced integration and C3D20) [24]. For simulations using the shell elements, the sheet strip was discretized into 300 elements of non-uniform size in the longitudinal direction, and 8 element in the width direction. To ensure numerical accuracy, element size is smaller in the contact areas (Regions 2 and 3, Figure 4.1(b)), with one contact node per 4.5 degrees of turn angle [82]. Fifty one through-thickness integration points were used to minimize numerical error [82]. For FE models using the first order solid elements (C3D8H and C3D8R), the finite element mesh consists of 12,000 elements (with 300, 4, and 10 elements in length, width and thickness directions respectively). When higher oder elements (C3D20R and C3D20) were used, only 6 elements were used through the sheet thickness to avoid overwhelming computation cost. For all FE models, mirror symmetry was utilized, and only half of the physical sample was modeled. The Coulomb friction coefficient of 0.15 was used for the lubricated test conditions, while zero friction was assumed if the tool was free to rotate during the draw-bend test [69]. Since springback is determined by the stress/moment distribution in the forming step, accurate springback prediction relies on the proper choice of the material constitutive models. Previous work demonstrated that erroneous simulations were resulted if anisotropic yielding and non-isotropic hardening were neglected or poorly represented [46]. To closely represent the anisotropic yielding of 6022-T4 sheet, Barlat 91 (for solid elements), 96 and 2000 (for shell elements) yield functions [42, 43, 45] were used. A modified anisotropic hardening model [46] and a nonlinear kinematic hardening law [11] were utilized to closely represent the features of the Bauschinger effect after a load reversal. The constitutive models were implemented into a commercial 59

78 finite element package Abaqus/Standard through user material subroutines (UMAT). Details of the Barlat s yield functions and the nonlinear kinematic hardening model can be found in Chapters 2 and 3, while the UMAT algorithms are discussed in Appendix A Effect of Back Force Previously implemented UMAT [46], which utilized the Barlat 96 yield function and a modified anisotropic hardening (the G-W model), was used first to simulate the static springback after the draw-bend test. Compared to the previous study, more back forces are used in the current investigation, in order to reveal the details of the rapid decrease of the springback angle as the sheet tension approaches the yielding, a phenomenon that was reported before [67, 69, 82]. The simulated springback angles and anticlastic curvatures are compared with the experimental results in Figure 4.5(a) and Figure 4.5(b), respectively. The general trend confirms the earlier finding [69], that the sheet tension significantly reduces springback, while the tool radius has less profound effect. For small tensions (F b < ), an increase in the back force by 0.1 causes about 7 degrees of reduction in the springback angle. However, when R/t is increased from 3.5 to 14, θ is decreased by 13 degrees for small back forces, Figure 4.5(a). The correlation between the sudden decrease of θ near F b = and the substantial increase in the anticlastic curvature is in accord with the previous results [69]. The convexity of the Barlat 96 yield function is not proven [43]. To avoid numerical difficulties, the recently proposed Barlat 2000 yield function is implemented in this study [45]. In addition, a nonlinear kinematic hardening model with three back 60

79 θ (degree) R/t=3.5 (Exp.) R.t=3.5 (FEM) R/t=14 (Exp.) R/t=14 (FEM) Barlat'96 yield G-W hardening Normalized back forcre F b (a) Curvature 1/R a (x10-3 mm -1 ) R/t=3.5 (Exp.) R/t=3.5 (FEM) R/t=14 (Exp.) R/t=14 (FEM) Normalized back forcre F b (b) Figure 4.5: Comparison of the simulation results with the experimental data: (a) the springback angle θ, and (b) the unloaded anticlastic curvature 1 R a. stresses (the mnlk model) is employed [11], in which the evolution of the back stress takes the Armstrong-Frederic type of law [10]. This model provides a simpler mathematical form, therefore the numerical implementation is easier than the G-W model [87]. Only 6 parameters are required in the mnlk model, as reported in Table 3.1 of Chapter 3. The detailed procedure used to attain the mnlk model parameters from the Bauschinger test can be found in Chapter 3. As shown in Figure 4.6(a), the use of the Barlat 2000 yield function gives nearly identical simulation results as the Barlat 96, when the G-W hardening model was used for both cases. This is expected because the variation of the yield strength and r-value with orientation are closely reproduced by both the Barlat 96 and 2000 yield functions, as can be seen from Figure 2.8 in Chapter 3. The computation cost by 61

80 80 70 R/t=14, 8x300 S4R G-W hardening mnlk parameters γ 1 =5.4, γ 2 =γ 3 = c 1 /γ 1 =40, c 2 /γ 2 =22, c 3 /γ 3 =15 θ (degree) Exp. 10 Barlat 2000 Barlat' θ (degree) R/t= x300 S4R Exp. Barlat'96 10 mnlk G-W Normalized back force F b Normalized back force F b (a) (b) Figure 4.6: Comparison of the simulated springback angles using (a) the Barlat 96 and 2000 yield functions and (b) the Geng-Wagoner model and the mnlk law. using the Barlat 2000 yield function, however, is increased by approximately 50%, presumably because of two stress potentials are used in formulating the Barlat 2000 yield function (Equation 2.67). The simulated springback angles by the mnlk and G-W models are compared in Figure 4.6(b). The overall agreement is good, although the mnlk model gives less closer match with the experimental data for small back forces. The possible source of error may be that the hardening parameters (c i and γ i, i=1 3) are not optimized, because a crude optimization algorithm was used, see Chapter Effect of Strip Width Finite element simulations were also carried out for draw-bend tests with specimens of various widths (12mm to 50mm), using the Barlat 96 yield function [43] and 62

81 the G-W hardening model [87]. As an ideal plane-stress case, beam element (Abaqus type B21) was also used for one case with R/t = 14 and F b = 0.5. The simulation agrees with the experimental results, that is, θ initially decrease with the strip width, Figure 4.7(a). For narrower samples (W < 25mm), however, an opposite trend was observed and confirmed by finite element simulations. In the limit case of W = 0, the problem can be treated as plane-stress and the springback is maximized. This conclusion is validated by FE simulation, since the 3D results appear to approach the plane-stress result (44.3 by beam element) as the width decreases. The experimental data also supported this conclusion with the smallest width of 12mm. The anticlastic curvatures (measured and simulated) monotonically decrease with strip width, Figure 4.7(b). This is inconsistent with the variation of the springback angle for small back forces, see Figure 4.7(a). Notice that Equation 4.1 was used to calculate the anticlastic curvature, assuming that the cross-section was circular. As will be discussed later in Chapter 5, this assumption is invalid for strips tested at low back forces. Experiments and simulations demonstrate that the anticlastic deflection is concentrated on the sheet edges, while the center area of the specimen is nearly flat. Therefore, Equation 4.1 overestimates the magnitude of the anticlastic curvature for the same arc height h Choice of Element Shell vs. Solid As mentioned in Section 4.3, small tool radii moderately promote the draw-bend springback. Previous experimental results [88] were analyzed by finite element simulations in this chapter. These experiments used a range of R/t (1.8 28) and two back forces (F b = 0.5 and 0.9), with free rotating tool that provided nearly zero friction. 63

82 θ (degree) Exp. FEM (S4R) FEM (B21) R/t=14, F b =0.5 Barlat'96 Yld G-W hardening Width (mm) (a) Curvature (x10-3 mm -1 ) Exp. FEM (S4R) Width (mm) (b) Figure 4.7: Comparison of the simulation and the experimental data for strips with various widths using F b = 0.5: (a) the springback angle, and (b) the anticlastic curvature. θ (degree) T4 F b =0.5 Exp. (Carden, 1996) S4R (Barlat'96) C3D8H (Barlat'91) C3D8R (Barlat'91) C3D20R (Barlat'91) Friction= R/t ratio θ (degree) F b =0.9 0 Exp. (Carden, 1996) S4R (Barlat'96) -20 C3D8H (Barlat'91) C3D8R (Barlat'91) C3D20 (Barlat'91) R/t ratio (a) (b) Figure 4.8: Comparison of the simulated springback angles using different elements for (a) F b = 0.5 and (b) F b =

83 As shown in Figure 4.8, θ initially increases with the tool radius, but decreases when R/t < 3.5 (for F b = 0.5) and R/t < 10.5 (for F b =0.9). Shell element generally works well when the R/t ratio is larger than 5 6 [23, 82], but it fails to predict the decline of the springback angle for small bending radii. This is probably because the assumptions used in the shell formulation, namely the zero through-thickness stress and plane section remaining planar after deformation, are no longer valid. On the other hand, solid elements (linear or quadratic) can qualitatively reproduce the experimental trend. The quadratic elements (C3D20R and C3D20) are in better agreement with the experimental data than the linear elements (C3D8H and C3D8R), presumably because the second order elements are more accurate for bending dominant problems. However, the computation cost is overwhelming for the fully integrated, 20-node solid element (C3D20): about 2 weeks for a typical draw-bend simulation case with 7200 elements (300 in length, 4 in width and 6 through thickness). The use of the C3D20R element (with reduced integration) cuts down the computation time to about 5 days for the same finite element model, because only 8 integration points was used by the C3D20R element instead of 27 by C3D20. In view of the computation efficiency and bending performance, a recently developed, locking-free 8-node brick element appears to be suitable for the draw-bend simulations [89]. Figure 4.9 compares the measured springback angles with the simulation results using both the shell and solid elements for R/t = 3.5. The overall agreement with experiment is satisfactory, even though simulations with the shell element deviate from experiment noticeably near F b = For solid element, improvement is possible if finite element mesh is further refined, or a better 3D yield function [90] is employed to replace the Barlat

84 C3D20 (Barlat'91) S4R (Barlat'96) Exp. θ (degree) R/t= Friction=0.15 G-W hardening Normalized back force F b Figure 4.9: Simulated springback angles using shell and solid elements. 4.5 Conclusions The following conclusions are reached from the draw-bend experiments and finite element simulation for 6022-T4 aluminum sheets: 1. Sheet tension has a dominant role in reducing the static springback of aluminum alloy 6022-T4 after the draw-bend test, while the tool radius has moderate effect in reducing springback. 2. The sudden increase of the springback angle as the normalized back force approaches a critical value is attributed to the occurrence of the persistent anticlastic curvature in the sheet width direction. The critical back force is approximately , and it slightly depends on the tool radius. 3. Sheet orientations (RD or TD) affect the measured springback angle and anticlastic curvature. For transverse strips, the decline of the springback angle 66

85 with sheet tension is gradual, and the corresponding increase of the anticlastic curvature when the normalized back force is near appears to be less abrupt. 4. The Barlat s 2000 yield function and a nonlinear kinematic hardening with multiple back stress components are as accurate as the Barlat 96 yield function and the G-W modified anisotropic hardening that were used in a previous study [46], but they have simpler mathematical forms and therefore are easier for numerical implementation into finite element program. 5. Shell element fails to match the decrease of springback when R/t-ratio is less than 4. Solid element with quadratic interpolation functions is advocated for bending with small radius, but large computational cost is required. Linear solid element, however, is not recommended because of its poor performance in bending applications. 67

86 CHAPTER 5 ANTICLASTIC CURVATURE IN DRAW-BEND TEST Note: A manuscript of this work has been submitted to the International Journal of Solids and Structures for publication. Abstract Draw-bend springback shows a sudden decline as the applied sheet tension approaches the force to yield the strip. This phenomenon coincides with the appearance of persistent anticlastic curvature, which develops during the forming operation and is maintained during unloading under certain test conditions. In order to understand the mechanics of the persistent anticlastic curvature and its dependence on the forming conditions, aluminum sheet strips of widths ranging from 12mm to 50mm were draw-bend tested with various sheet tensions and tool radii. Finite element simulations were also carried out, and the simulated and measured springback angle and anticlastic curvature were compared. Analytical methods based on a large deformation bending theory for elastic plates were employed to understand the occurrence and persistence of the anticlastic curvature. The results showed that the final crosssection shape of a specimen is determined by a dimensionless parameter, which is a function of the sheet width, thickness and radius of the primary curvature in the 68

87 curled region of an unloaded sample. When the normalized sheet tension approaches 1, this parameter rapidly decreases, and significant anticlastic deflection is retained after unloading. The retained anticlastic curvature greatly increases the moment of inertia for bending, and thus reduces springback angle. 5.1 Introduction When a long, flat rectangular sheet of uniform thickness is bent about an axis parallel to one of its edges, say in the x-direction, a transverse curvature is developed in the direction parallel to the bending axis [91]. For elastic deformation, this happens by the differential lateral contraction caused by Poisson s effect. Consequently, the initially flat surface becomes an anticlastic surface, with two orthogonal curvatures in opposite sign, Figure 5.1(a). If the centers of these two curvatures appear on the same side of the surface, the surface is synclastic, Figure 5.1(b). For narrow, initially flat sheets, the ratio between the longitudinal (i.e., primary, x-direction) and the transverse (i.e., secondary, y-direction) curvatures is given by the Poisson s ratio ν, i.e., R y = νr x, according to the fundamental bending theory of beams [76]. The shape of the cross-section of a bent beam or plate depends on a dimensionless parameter, β = W 2 R xt, with W, t and R x being the sheet width, thickness and radius of the primary bending curvature respectively [86]. In the literature, β is also called the Searle s parameter [93]. The anticlastic surface has a constant curvature of ν R x, when β is less than one [92]. In this case, the sheet behaves like a plane-stress beam. However, if β is larger than 20, the anticlastic deflection is mainly confined to the sheet edges, while the sheet central area stays relatively flat [93 95]. Consequently, the deformation state can be characterized as more plate-like, with plane-strain the 69

88 z(x, y) R y z(x, y) R x longitudinal, x (a) transverse, y R x longitudinal, x (b) R y transverse, y Figure 5.1: (a) An anticlastic surface, and (b) a synclastic surface. limiting approximation. The deformation modes maybe interpreted in terms of body forces which tend to suppress the formation of a large circular cross-section [78]. When bending wide sheets to a small radius, the constrained anticlastic curvature causes a biaxial stress state on the tension side of the sheet [96]. The principles of simple elastic plate bending have been extended to bodies of varying thickness [97 100], to the measurement of elastic constants [101, 102], and to specimens plastically deformed in four-point bending tests [78]. In the last case, pertinent to the current work, it was concluded that plasticity affected the magnitude of the anticlastic curvature, but had little effect on springback. There is little literature on anticlastic deflection for more complicated forming processes. Anticlastic displacements up to 1.5 times the sheet thickness have been measured after draw-bending and unloading [69,82], in marked contrast to the simple bending results where theory predicts a maximum deflection of about 10 percent of the sheet thickness. For small sheet tensions, the anticlastic curvature developed in 70

89 the forming step nearly disappeared during unloading, thus having little effect on the final specimen shape (consistent with observations for springback in simple bending). However, as sheet tension was increased to the yield stress of the material, the anticlastic distortion persisted after unloading. This persistent anticlastic distortion increased the moment of inertia of the specimen greatly, and thus reduced springback commensurately [69]. This investigation focuses on the role of anticlastic curvature in springback following draw-bend deformation. The mechanics of the persistent anticlastic curvature is sought, especially its dependence on the forming parameters and the specimen geometry. In order to proceed, the classic theory of bending elastic plate is reviewed. Draw-bend test results are then presented, and they are considered with the aid of the theory and finite element simulations. Discussions and conclusions are then drawn. 5.2 Draw-Bend Experiment The draw-bend test can closely mimic industrial forming processes, where sequential bending and unbending takes place under superposed tension as sheet material is drawn over a rigid tool surface [69]. Unlike other laboratory forming tests, where stretching is usually provided through various locking mechanisms (draw-bead or blank holder), sheet tension can be directly and precisely controlled in the draw-bend test, using a secondary hydraulic cylinder which is programmed to provide constant stretching force during test [71]. In this study, 6022-T4 aluminum sheet were tested using a special draw-bend machine at Colorado School of Mines. Details of this equipment can be found elsewhere [69, 72]. 71

90 F b F b Initial Loaded 1 Unloaded R r 3 2 R mm θ Ẋ = 40 mm/s 127 mm Figure 5.2: Schematics of the draw-bend test and an unloaded specimen. Rectangular specimens were sheared with their lengths parallel to the sheet rolling direction. As shown in Figure 5.2, the draw-bend test procedure is divided into three steps, after the strip was hand-formed around a cylindrical tool to get a 90 degree bent. A prescribed stretching force was first applied to the left end of the strip, while its right end was held immobile. Here, F b (the back force) is the actual sheet tension divided by the yielding force of the specimen in uniaxial tension. Then, the strip was drawn over an unrotating tool by imposing a constant speed of 40mm/s to the right end of the strip, while the back force was kept constant. Standard industrial lubricant [81] was brushed on both the strip and tool surfaces to provide medium friction [69]. After the drawing distance reached 127mm, specimens were unloaded and removed from the test device. Their shapes were traced on paper and then recorded digitally. Traces were first taken one minute after forming and unloading, then repeated at intervals 72

91 up to 15 months for time-dependent springback measurement [70]. The current work only studies the static springback, which is measured approximately one minute after unloading. A typical unloaded specimen is depicted in Figure 5.2, with four deformation regions delineated. Regions 1 and 4 remain straight throughout the test, and thus are not interested for springback study. Regions 2 was in contact with the forming tool before unloading, and its radius of curvature changes from R to R after springback. The important specimen geometry is defined by 3, which has a radius of curvature r after unloading. It is a measure of the so-called sidewall curl that was observed in many sheet-formed parts [103]. Springback is characterized by the angle θ, Figure 5.2. A transverse curvature in the sheet width direction was discovered in draw-bend tested strips [69]. If the cross-section is assumed to be circular, the radius of this anticlastic curvature, R a, is calculated by R a = h 2 + W 2 8 h, (5.1) where h is the arc height of the cross-section [69]. h was measured at the center point of the Region 3 (Figure 5.2) of an unloaded specimen, using a digital caliper with 0.01mm resolution. The cross-section profile was measured by a 0.03mm resolution dial gauge for two draw-bend tested samples. 5.3 Finite Element Models Simple bending of initially flat and curved elastic plates were simulated by finite element to validate the closed-form solution (as will be discussed next). 4-node shell elements with reduced integration (type S4R) were used [24], with 15 integration 73

92 points through sheet thickness (t = 1). adopted in simulations were 65 GPa and 1 3 The elastic modulus and Poisson s ratio respectively. Mirror symmetry was utilized and only one quarter of the plate was modeled, with symmetric boundary conditions applied at the plate edges. Pure bending was attained by applying prescribed rotation to edge nodes. The static springback in draw-bend test was simulated, for a range of back force (0.1 F b 1.2), specimen width (W = 12mm 50mm) and tool radius (3.2mm 12.7mm). Both 2D and 3D analysis were carried out, using plane-stress beam element (Abaqus type B21), and 4-node shell element with reduced integration (S4R), respectively [24]. The sheet strip was modeled by 300 elements of non-uniform size in the longitudinal direction. Smaller elements were used in the contact areas (Regions 2 and 3, Figure 5.2) to ensure numerical accuracy, with one contact node per 4.5 degrees of turn angle [82]. In 3D FE models, only half of the physical strip was modeled because of the mirror symmetry, with 8 elements in the sheet-width direction. Fifty-one integration points were used through the sheet thickness, for both B21 and S4R elements, to minimize numerical error [82]. In order to closely represent the plastic anisotropy of 6022-T4 sheet, the Barlat 96 yield function [43] and a modified anisotropic hardening model [46] were adopted. A friction coefficient of 0.15 was used for lubricated test conditions [69]. Simple bending theory suggests that the draw-bend process is closer to planestrain deformation [83], considering the large width-to-thickness ratio (W/t=55). However, 2D finite element simulations demonstrated that plane-stress (with B21 beam element) results were consistently better than plane-strain (with S4R shell element) [82]. In FE models, plane-strain assumption was enforced by prescribing zero 74

93 T4, R/t=10.5 von Mises yield Isotropic hardening θ (degree) Exp. Plane-stress 10 Plane-strain 3D Normalized back force F b Figure 5.3: Simulated springback angle using various assumptions. lateral displacement to all nodes in the width direction [24]. For the sake of simplicity, only von Mises yield function and isotropic hardening law were used in these simulations. As shown in Figure 5.3, simulations using 2D beam elements are in better agreement with the experiment data than that of plane-strain, Figure 5.3. It is also noted that 2D simulations causes significant error for larger back forces, because neither plane-stress nor plane-strain is able to represent the anticlastic curvature. Full 3D simulations correctly reproduce the fast decline of springback angle when F b > 0.7. This phenomenon is closely related to the persistent anticlastic curvature, which will be explored in detail in the following text. 75

94 5.4 Elastic Theory for Plate Bending Before interpreting the anticlastic curvature during and after draw-bending, the simpler case of pure bending is considered. The classic bending theory of elastic plate is summarized here, while details can be found elsewhere [84, 92]. The results are based on the work of Ashwell [84, 92], which makes use of Marguerre s large deformation theory of plate bending [104] and extends von Kármán s analysis [105]. Unlike the small deformation theory, the membrane stress at the plate middle surface is considered. The general theory is first introduced, with approximations suitable for closedform solution. Then, these results are applied to two cases of interest for the drawbend application: 1) bending of an initially flat sheet, and 2) unbending (straightening) of an initially curved sheet Problem Statement and Closed-Form Solution The problem of bending an initially curved plate is shown schematically in Figure 5.4. A rectangular plate (dimension L, W and t), with initial middle surface shape described by z 0 (x, y) and radii of curvature R x0 and R y0, is subjected to a uniform bending moment m x (per unit width) applied to the plate edges, i.e, at x = ± L 2. Following Ashwell s work [84], the governing differential equation (GDE) for this problem is given as follows: 4 (z z 0 ) = t [ q D t + 2 F 2 z x 2 y + 2 F 2 z 2 y 2 x 2 2 F 2 x y [ ( ) ( ) z 2 2 z 0 F = E x y x y ] 2 z x y ( 2 z 2 z x 2 y 2 z 0 2 z 0 2 x 2 y 2 (5.2) ) ] (5.3) 76

95 n x m x width, W z q(x, y) y x R y n x t length, L m x R x Figure 5.4: Rectangular coordinate system for plate bending problems. where 4 = 4 x x 2 y y 4 is the bi-harmonic differential operator, q(x, y) is the surface pressure acting along the z-axis and D = Et3 12(1 ν 2 ) is the bending flexural rigidity of an elastic plate. F (x, y) is a stress function, from which the tensions per unit length in the plate can be derived: n x = t 2 F y 2, n y = t 2 F x 2, n xy = t 2 F x y (5.4) It is assumed that the plate middle plane has the following shapes before and after deformation [92], respectively: z 0 = z 0 (y) + x2 2R x0, z = z(y) + x2 2R x (5.5) where R x0 and R x are the initial and deformed radii of the primary curvature, respectively. It was further assumed that the undeformed transverse shape at any position of fixed x can be similarly written as z 0 (y) = y2 W 2 (5.6) 2R y0 12R y0 77

96 where R y0 is the initial radius of anticlastic curvature. The unknown to be solved from Equations (5.2) and (5.3) is the anticlastic deflection, z(y). When there is no lateral pressure applied on the plate surface, i.e., q(x, y) = 0, the profiles given by Equations (5.5) and (5.6) satisfy the governing differential Equations (5.2) and (5.3) if D d4 ( z z 0 ) T d2 z dy 4 dy + Et ( z z ) 0 = 0 (5.7) 2 R x R x R x0 ( z E z ) 0 dydy + T x2 = F (5.8) R x R x0 2t where T is a constant. At the plate edges where y = ± W 2 and x = ± L, the boundary 2 conditions are: n y = T (5.9) ( z n x = Et z ) 0 (5.10) R x R x0 [ ] ( 2 ( z z 0 ) 1 m y = D + νd 1 ) (5.11) y 2 y=± R W x R x0 2 ( 1 m x = D 1 ) [ ] 2 ( z z 0 ) + νd (5.12) R x R x0 y 2 x=± L 2 [ ] 3 ( z z 0 ) n yz = D (5.13) y 3 y=± W 2 n xz = 0 (5.14) For an elastic plate with single curvature ( 1 R x0 = 0, 1 R y0 0) subjects to the following boundary conditions at y = ± W 2 : n y = m y = n yz = 0 (5.15) Equations (5.7) and (5.8) reduce to a fourth order, homogeneous ODE, when the radius of primary bending curvature is R x : d 4 z dy 4 + 4γ4 z = 0 (5.16) 78

97 with γ 4 = 3(1 ν2 ) R 2 xt 2. If the initial primary curvature in non-zero, i.e., 1 R x0 order, non-homogeneous ODE is resulted: 0, a forth d 4 z dy + 4γ4 z = 12(1 ν2 ) z 0. (5.17) 4 R x R x0 t When the RHS of Equation (5.17) is small, it can be neglected without causing significant error. The solution of Equation (5.16), i.e., the transverse (anticlastic) deflection, is given as follows: z(y) = cosh γy(c 1 cos γy + C 2 sin γy) + sinh γy(c 3 cos γy + C 4 sin γy). (5.18) Due to the symmetric anticlastic profile, i.e., z(y) is an even function of y, C 2 = C 3 = 0 and the above solution reduces to z(y) = C 1 cos γy cosh γy + C 4 sin γy sinh γy (5.19) The integration constants C 1 and C 4 can be determined from the boundary conditions previously discussed: [ ] d2 z dy 2 [ ] d3 z dy 3 y=± W 2 y=± W 2 = [ 2γ 2 C 4 cos γy cosh γy 2γ 2 C 1 sin γy sinh γy ] = 1 R y0 + ν ν R x0 R x (5.20) = [ 2γ 3 (C 4 C 1 ) cos γy sinh γy 2γ 3 (C 4 + C 1 ) sin γy cosh γy ] = 0. (5.21) Solve the above two equations for constants C 1 and C 4 and substitute them back into Equation (5.19), the anticlastic profile is obtained ( z t = Rx + R ) x ν 1 νr y0 R x0 3(1 ν2 ) [K 1 cosh γy cos γy + K 2 sinh γy sin γy] (5.22) where the constants K 1 and K 2 are defined as follows: ( K1 K 2 ) = 1 sinh γw + sin γw [ ( ) ( ) ( ) ( )] γw γw γw γw sinh cos cosh sin

98 The normal stresses σ 11 and σ 22 are known for for an isotropic elastic plate [78]: σ 11 = ± 6m [ x 1 2ν 2 (K t 2 2 cos γy cosh γy K 1 sin γy sinh γy) ] (5.23) 1 ν ± 2ν 2 (K 3 2 sin γy sinh γy + K 1 cos γy cosh γy) σ 22 = ± 6m x t 2 [ν 2ν(K 2 cos γy cosh γy K 1 sin γy sinh γy)] (5.24) where the plus and minus signs are for tensile and compressive stresses respectively. Notice that the solution is normalized by 6mx, which the longitudinal stress at the t 2 outer fiber of a beam subjected to plane-stress bending Results for Pure Bending of an Initially Flat Plate ( 1 R x0 = 1 R x0 = 0) Equation (5.22) maybe used to visualize the anticlastic curvature for a range of plate widths and primary curvatures. Figure 5.5 shows the variation of the normalized anticlastic deflection and the normalized transverse stress along the plate width direction, for a rectangular plate (W = 50 and t = 1) bent to various curvatures. As shown in Figure 5.5, the anticlastic displacement tends to localize toward the plate edges as β increases, while the plate center remains relatively flat. Correspondingly, more transverse stress is developed in the central area of the plate, but it decays to zero at the plate edges, Figure 5.5. As β increases, the area with biaxial stress state expands, and the transverse stress drops faster near the edges. In the limit of β =, the center of the plate approaches plane-strain, with σ 22 = νσ 11, as illustrated by the thin dotted line in Figure 5.5(b). From the elastic plate theory, it is known that a lateral bending moment, m y = νm x, exists over the flat portion of the deformed plate to maintain a cylindrical surface. However, m y must equal zero at the edges. As pointed out by Fung [106], the near-edge region with localized anticlastic deformation 80

99 Normalized deflection z/t β= β*=13.5 β= β= W=50, t=1 ν=1/3, β=w 2 /R t x Normalized with coordinate y/w (a) Stress ratio σ 22 /σ /3 0.1 β=5 β*=13.5 β=50 β= Normalized width coordinate y/w (b) Figure 5.5: Closed-form solution for: (a) normalized anticlastic deflection, and (b) normalized transverse stress. acts as a boundary layer, through which the later bending moment is built up, from zero at edge, to νm x at the center. The critical value β at which the anticlastic deflection curve starts to change and thus to have an inflection point can be calculated as follows: [ ] d2 z dy 2 y=0 ( ) ( ) γw γw = 0 tan + tanh = 0 (5.25) 2 2 where γw = β 4 3(1 ν 2 ). The first root of the above transcendental equation gives the critical value of β: β = 13.5 when ν = 1 3. The maximum deflection at edges can also be derived. Knowing that sinh(γw ) cosh(γw ) 1 2 eγw as γw, and the constants K 1 and K 2 become ( K1 K 2 ) = e γw 2 [ ( ) ( )] γw γw cos sin 2 2 (5.26) 81

100 Near the plate edges, approximations are also made such that sinh(γy) cosh(γy) 1 2 eγy. Therefore, the anticlastic deflection becomes ( z t = Rx + R ) x ν 1 νr y0 R x0 3(1 ν2 ) e γȳ [cos(γȳ) sin(γȳ)] (5.27) where ȳ = W 2 y is the distance measured from the plate edge toward plate center. For initially flat plate, i.e., 1 R x0 = 1 R y0 = 0, the maximum deflection occurs at the plate edges and it only depends on Poisson s ratio: ( ) z = t max ν 12(1 ν2 ). (5.28) For ν = 1 3, the maximum deflection z max is 10.2% of sheet thickness. β Case W (mm) L (mm) R (mm) Mesh (W L) a b c a b c Table 5.1: Parameters used in finite element model for elastic bending. In order to verify the closed-form solution, a series of elastic finite element simulations were conducted for pure bending of an initially flat sheet (t = 1) with various widths and bending radii, as listed in Table 5.1. The finite element meshes are so chosen that the element aspect ratio is 1. Further refinement showed negligible difference in result, as will be shown later. In order to assess the invariance of the FE results 82

101 for a fixed β, this parameter is rewritten in terms of non-dimensional quantities as follows: β = W 2 Rt = (W/t)2 (R/t) (5.29) The second form reveals the relationship between β, the thickness-normalized specimen width (W/t), and the normalized bending radius (R/t). The simulation results for β = 5 are compared with the analytic solutions in Figure 5.6, using 3 combinations of strip widths and bending radii. The simulated anticlastic profiles agree with the closed-form solutions, for this intermediate case: neither plane-stress nor plane-strain. Normalized deflection z/t W5, R=5 W=10, R=20 W=20, R=80 Analytic t=1, ν=1/3, β= Normalized width coordinate y/w (a) Stress ratio σ 22 /σ /3 W5, R=5 W=10, R=20 W=20, R=80 Analytic t=1, ν =1/3, β= Normalized width coordinate y/w (b) Figure 5.6: Anticlastic deflection and stress ratio for an elastic material with β = 5. For larger β, the analytic solution predicts that the anticlastic deflection is more concentrated toward the sheet edges. This is confirmed by finite element results for β = 100, as shown in Figure 5.7(a). The stress state in the sheet center area is 83

102 close to plane-strain, with stress ratio σ 22 σ 11 approaching the Poisson s constant, see Figure 5.7(b). Mesh refinement shows negligible difference in displacement and stress solution, as illustrated by the selected case for W = 50mm, Figure 5.7. Normalized deflection z/t Theory W=25, R=6.25 W=50, R=25 (12x20) W=50, R=25 (25x40) W=50, R=25 (50x80) W=100, R=100 σ 22 /σ σ /σ =1/ Analytic W=25, W=6.25 W=50, R=25 (12x20) W=50, R=25 (25x40) W=50, R=25 (50x80) W=100, R=100 t=1, ν=1/3, β= Normalized width coordinate y/w (a) t=1, ν =1/3, β= Normalized width coordinate y/w (b) Figure 5.7: Anticlastic deflection and stress ratio for elastic material with β = 100. Conventional knowledge usually distinguishes plane-stress bending from planestrain bending by the ratio of width to thickness [83]. For example, plane-stress bending is assumed for W/t 1 and plane-strain for W/t 1. For these two limiting cases, the primary curvature is proportional to the bending moment M, according to elementary bending theory: { 1 M =, plane-stress EI M(1 ν R 2 ) x, plane-strain EI (5.30) where I is the section moment of inertia. Based on the previous analysis, the transition from plane-stress to plane-strain is not only a function of W/t, if the anticlastic 84

103 deformation is considered. The actual stress state depends not only on W/t, but also on the normalized primary bending radius R x /t (as reflected in the Searle s parameter β). The effect of anticlastic curvature on stress state can be realized as follows. When the anticlastic surface can freely develop (β 1), it has a constant curvature of ˆρ y = ν R x by the elastic Poisson s effect. When considering the width effect as β increases, the anticlastic deflection is suppressed in the central area of the plate. As a result, the transverse curvature varies across width, as can be evaluated from Ashwell s closed-from solution (Equation 5.19): ρ y = d2 z dy 2 = 2ν R x [K 2 cosh γy cos γy K 1 sinh γy sin γy] (5.31) Through Poisson s ratio, the restrained anticlastic curvature will affect the principal one. Now, Equation (5.30) can be generalized to incorporate the change of the principal bending curvature caused by the anticlastic deformation (for the same bending moment), in terms of a dimensionless parameter φ: ρ x = 1 = M R x EI (1 φν2 ) (5.32) Plane-stress and plane-strain are two special cases corresponding to φ = 0 and φ = 1 respectively. Parameter φ has been called anticlastic factor [85], and it depends only on β and the Poisson s constant ν: φ = 1 2 k β [ ] cosh(k β) cos(k β) sinh(k β) + sin(k β) (5.33) where k = 4 3(1 ν 2 ). Figure 5.8 shows that φ is close to zero when β is small, but it saturates for large β. The transition from beam to plate is no longer abrupt. The fast rise of φ correlates to the aforementioned critical value β : φ reaches 0.5 when β = 13.5 for ν =

104 Anticlastic factor φ φ=0.5 β*= Primary curvature ρ = M*(1-φν 2 )/EI x φ=0 (beam) β φ=1 (plate) Figure 5.8: Variation of the anticlastic factor φ with β. Equation (5.31) reduces to d2 z dy 2 = ν R x, as γ 0, i.e., the transverse curvature is constant along the width direction, and it has opposite sign of the primary one. The ratio between these two curvatures is equal to the Poisson s ratio, ν Bending of Initially Curved Plate The foregoing model applies to the initial bending stage of the draw-bend test as an initially flat sheet is drawn over the tool radius. The second stage to be considered is the unbending, or straightening, of the primary curvature as the strip leaves contact with the tooling. Considering the contact constraints with the adjacent tooling, the initial condition of the plate for this stage can be idealized as Rx 0 = R and Ry 0 = (i.e., no anticlastic curvature while in contact with the tool), as shown schematically in Figure 5.9. This starting condition is consistent with FE analysis of the draw-bend operation, which shows essentially no transverse curvature near the tool contact. 86

105 Initial shape y x Thickness t R x0 Length L A A z Section A A z(y) Unbend z 0 (y) = y2 2R y0 y W2 12R y0 m x R x = Straightened shape Width W Figure 5.9: Bending an initially curved plate. The curved plate is straightened by applying a uniform moment m x (per unit width) along the transverse edge, until the primary curvature disappears, Figure 5.9. In order to solve the inhomogeneous ODE (Equation 5.17) in closed-form, the initial plate profile is approximated by parabolic function, as given by Equations (5.5) and (5.6). To calculate the shape of a cross-section A-A, z(y), after unbending, the following approximations are utilized: sinh x sin x x, and cosh x cos x 1 as x 0 (5.34) Then, constants K 1 and K 2 become ( K1 K 2 ) = γw 2 γw 2 γw + γw = ( ) (5.35) 87

106 and the anticlastic deflection reduces to z t = νy2 2R x0 t,, and ( z t ) max = ν 8 W 2 R x0 t = ν 8 β x0 (5.36) According to the foregoing analysis based on the elastic bending theory, straightening an initially curved plate ( 1 R x0 0, 1 R y0 = 0) produces significant anticlastic deflection. The maximum anticlastic displacement is proportional to the Searle s parameter, β x0 = W 2 R x0, and the Poisson s ratio, ν. t Maximum deflection (z/t) max W=5 W=10 W=25 W=50 W=100 Theory Theory: (z/t) max =β x0 ν/8 β x0 =W 2 /R x0 t, t=1, ν=1/3 Elastic material Searle's parameter β x0 (a) Elastic Maximum deflection (z/t) max W=5 W=10 W=25 W=50 W=100 Theory ν=0.5 ν=1/3 Elasto-plastic material Searle's parameter β x0 (b) Elasto-plastic Figure 5.10: Maximum anticlastic deflection from finite element simulation. To validate the closed-form solution, the straightening problem was simulated by finite element method, using both elastic and elasto-plastic material models. For the later case, von Miese yield function and isotropic hardening were used, with a Voce s type of hardening law [6]: σ = e 9.6ε p (Mpa). The FE model is schematically shown in Figure 5.9, where a cylindrical plate is flattened by a uniform moment m x 88

107 at its transverse edge. The cross-sections at x = 0 after unbending are compared with the analytic solution given by Equation (5.36), Figure As will be discussed next, the error caused by the parabolic approximation of a circular cross-section is minimal as x 0. As shown in Figure 5.10(a), elastic simulation results agree with the closed-form solution. The maximum anticlastic deflection, ( ) z, is invariant for t a fixed β, regardless of the specimen widths (W =5mm 100mm) and the initial radii (R x0 =5mm 2000mm). Both simulation and closed-form solution demonstrate that the maximum anticlastic deflection at the plate edges can be a few times the plate thickness, while bending an initially flat plate can only produce z max = 0.102t when ν = 1. This explains why significant anticlastic displacement happens in the forming 3 step of the draw-bend test, where a sheet is straightened as it leaves the tool surface. For an elasto-plastic material, simulation results deviate from the elastic solution when β > 20, Figure 5.10(b). This is because shear stress is required to maintain compatibility between the elastic (ν = 1 ) and plastic (ν = 0.5) regions of an plastically 3 bent plate, while the shear effect is neglected in deriving the closed-form solution [92]. At the same β value, larger deviation exists for narrower plates, because the primary bending radius is smaller and more plastic deformation occurs throughout the plate thickness Error Analysis of Elastic Theory max As previously mentioned, one major assumption was made in order to solve the fourth order ODE of the elastic bending problem. That is, the initial cross-sections in the longitudinal and width directions, and the distorted cross-section in the longitudinal direction, were all approximated by parabola. Or equivalently, curvature is 89

108 x 2 + (z R) 2 = R 2 z R A h C h x = b 2 O b B D z = x2 2R x Figure 5.11: Error analysis for using parabolic function. calculated by ρ = z instead of ρ = z (1+z 2 ) 3 2. As shown schematically in Figure 5.11, the parabolic approximation causes significant error when the chord length, b, is a significant fraction of the radius of a circle, R. The depth of the parabola COD, h, (at x = ± b ), and the depth of the arc AOB, h are calculated by 2 h = b2 (5.37) 2R ( ) 2 b h = R 1 1 (5.38) 2R Apply Taylor s series expansion to the second equation, h = b2 8R + R 8 ( ) 4 ( ) b b + O 2R 2R (5.39) where O( ) indicates the higher order terms. Then, the percentage error introduced by using parabolic function is defined as Err = h h h b, the computed errors are listed in Table 5.2. R %. For various ratios of

109 b R Err (%) Table 5.2: Error introduced by using parabolic function. As can be seen from Table 5.2, the error quickly grows as b R increases. For drawbend test, the unbending starts from an initially circular strip that was wrapped around a tool of radius R, so that b R = 2. Therefore, nearly 100% error is expected if the closed-form solution is applied to a cross-section that is located near the plate edges (x = W, Figure 5.9) Anticlastic Curvature in Draw-Bend Test Two groups of rectangular 6022-T4 aluminum strips, with 0.9mm thickness, were tested in the current work. Specimens from the first group were 50mm wide, and they were tested under various sheet tensions for two tool radii of 12.7mm and 3.2mm. The second group had strip width ranging from 12mm to 50mm, and was tested using 12.7mm tool at two normalized sheet tension: F b = 0.5 and F b = Effect of Back Force As shown in Figure 5.12, the springback angle decreases with the normalized back force, while the anticlastic curvature varies oppositely. As F b approaches , there is a dramatic drop in the springback angle, accompanied by a rapid increase in the anticlastic curvature. The sudden decrease of θ has been attributed to the 91

110 persistent anticlastic curvature, which substantially increases the section moment of inertia when F b 0.7 [69]. θ (degree) R/t=3.5 (Exp.) R.t=3.5 (FEM) R/t=14 (Exp.) R/t=14 (FEM) Barlat'96 yield G-W hardening 8x300 S4R Normalized back forcre F b (a) Curvature 1/R a (x10-3 mm -1 ) Barlat'96 yield G-W hardening 8x300 S4R R/t=3.5 (Exp.) R/t=3.5 (FEM) R/t=14 (Exp.) R/t=14 (FEM) Normalized back forcre F b (b) Figure 5.12: Effect of the back force on: (a) springback angle and (b) unloaded anticlastic curvature. Lines are FEM simulation results and markers are experimental data. Simulated springback angles agree with the experimental data for both bending radii (R/t = 3.5 and R = 14), Figure 5.12(a). It is also noticed that small bending radius causes more springback, which is contrary to the simple bending results. However, it is in accord with other stretch-bending experiments [74]. Both experiment and FEM simulation have shown the decrease of anticlastic curvature when F b , as can be seen from Figure 5.12(b). However, simulation results for small bending radius (R/t = 3.5) shows appreciable deviation from measurement. This is presumably because of the use of shell element which does not 92

111 work well for small radius bending [82]. When R/t < 5, the general shell assumptions, namely zero through-thickness stress and plane section remaining planar after deformation, are no longer valid [8]. Loaded h (mm) Barlat'96 yield G-W hardening t=0.92 mm 1 R/t=14 R/t= Normalized front force F f (a) Unloaded h (mm) R/t=3.5 R/t= Normalized front force F f (b) Figure 5.13: Variation of the maximum anticlastic deflection with the front pulling force in the draw-bend test: (a) loaded and (b) unloaded. The maximum anticlastic deflection (i.e., h) from FE simulation is plotted against the normalized front force (F f ), see Figure Before unloading, the maximum anticlastic deflection decreases with the front force, and it can be as much as 3 times the sheet thickness when F f < 0.5. The remaining depth gradually increases after springback, with a abrupt change as the normalized front force approaches unity, Figure 5.13(b). This indicates that the persistence of anticlastic curvature is determined by the sheet tension. 93

112 The persistent anticlastic curvature has a significant role in reducing springback. As shown in Figure 5.14, the bending moment continuously decreases with the back force, so does the springback angle if there were no sudden change in the moment of inertia of the sheet cross-section. However, the bending rigidity of the strip dramatically increases because of the persistent anticlastic curvature during springback when F f 1.0. The moment of inertia for a circular cross-section, I, can be calculated using standard formulas [107]. As illustrated in Figure 5.14, the normalized moment of inertia, I/I 0, is increased by a factor of about 3 when the front force exceeds yielding. Here, I 0 = W t3 12 is the reference moment of inertia for a flat rectangular cross-section. Because of this sudden increase of bending rigidity, the springback angle is greatly reduced, Figure 5.12(a) R/t=3.5 4 Moment (N*mm) Flat section I 0 = Wt 3 /12 Barlat'96 Yld G-W hardening 8x300 S4R R/t= Normalized back force F b 3 I / I 0 2 Figure 5.14: Moment and normalized section moment of inertia by finite element simulation. 94

113 5.5.2 Effect of Specimen Width The experimental results for springback angle and anticlastic curvature are plotted in Figure 5.15, for specimens with various width from 12mm to 50mm. For both F b = 0.5 and F b = 0.9, the anticlastic curvature decreases with the sample width. However, the springback angle, θ, first decreases with sample width for the case of F b = 0.5, then it increases after a local minimum that happens around W = 25mm, Figure 5.15(a). For F b = 0.9, θ monotonically declines with the specimen width, Figure 5.15(b). θ (degree) θ Curvature (mm -1 ) 2 32 R/t=14, F =0.5 b Width (mm) (a) Curvature (x 10-3 mm -1 ) θ (degree) θ Curvature (mm -1 ) R/t=14, F =0.9 b Width (mm) (b) Curvature (x 10-3 mm -1 ) Figure 5.15: Springback angle and anticlastic curvature from draw-bend test for (a) F b = 0.5 and (b) F b = 0.9. Finite element simulations were carried out for samples tested by F b = 0.5. As shown in Figure 5.16, simulation agrees with the measured springback angle and anticlastic curvature. Result of a 2D simulation using beam elements is also presented in Figure 5.16(a) for comparison. Since plane-stress state is assumed in the sheet 95

114 width direction when beam elements (B21) are used, it gives the highest springback angle. As specimen becomes narrower, θ using 3D shell elements approaches the result predicted by 2D beam elements. θ (degree) Exp. FEM (S4R) FEM (B21) R/t=14, F b = Width (mm) (a) Curvature (x10-3 mm -1 ) R/t=14, F b =0.5 2 Exp. FEM (S4R) Width (mm) (b) Figure 5.16: Compare simulation and measurement for F b = 0.5 case: (a) springback angle and (b) anticlastic curvature Application of the Elastic Bending Theory To apply the closed-form solution, the draw-bend test procedure is divided into three sequential steps: bending, unbending and springback. The first step is trivial, in which the sheet is wrapped around the tool under superposed tension. It is treated as a plane-strain problem, because the strip conforms to the tool surface, and hence transverse displacement can be neglected. 96

115 The second and third steps are much more complicated 3D problems, because anticlastic deformation is present in both steps. In the unbending step, the strip loses its primary curvature ( 1, when it is in contact with the tool) as it slides over R the tool surface. Eventually, it becomes straight in the longitudinal direction, i.e, R (2) x =. Meanwhile, a transverse curvature, 1 R x (2), is developed in the sheet width direction. In the last step, the sample obtains a primary curvature in the side-wall curl region after springback. The radius of this curvature, r, depends on the sheet tension, tool radius and friction condition, as well as the material properties such as the yield surface shape and the strain hardening behavior [69,82]. During springback, the previously developed anticlastic curvature will change from 1 R (2) y to 1 R a. Table 5.3 summarizes the primary and anticlastic curvatures involved in all three steps. Step R x0 R y0 R x R y (1). Bending R (2). Unbending R R y (2) (3). Springback R y (2) r R a Table 5.3: The radii of primary and anticlastic curvature in draw-bend test. The elastic theory of plate bending can be applied for the last step, as schematically shown in Figure The output of the unbending step from FE is used as input for the analysis in the springback step. Then, the predictions by the elastic theory are compared with the FE results; and, for two cases, with experimental data as well. The goal is to explain why the anticlastic curvature persists after springback 97

116 only for small sheet tensions. Apparently, the unbending process involves plasticity, and thus the elastic plate theory will not give satisfactory results. Nonetheless, it helps to understand why the maximum anticlastic deflection before unloading can be as much as 3 times the sheet thickness in the draw-bend test. For the last step, the application of elastic theory is reasonable, since springback is generally an elastic process. R Springback analysis R y0 R x0 m x n x t W L y x z Loaded sample m x n x Figure 5.17: Unbending and springback analysis for draw-bend test. To apply the elastic theory for the springback analysis, three radii of curvature, R x (2), R y (2) and R x (3) from FE simulation, are used as input of the closed-form solution, 98

117 to calculate the anticlastic curvature after springback, R y (3). The predicted maximum anticlastic deflection is then compared with the FE results. As shown in Figure 5.18, the elastic theory prediction agrees qualitatively with the finite element simulations for both tool radii, but it under-estimates the magnitude of the anticlastic deflection. Both FEM and elastic prediction show that the unloaded depth of the anticlastic profile initially increases with the sheet tension, but decreases after the front force exceeds the yielding force of the strip, which corresponds to the occurrence of the persistent anticlastic curvature after springback. It is also noted that the elastic prediction deviates more from the FE results as the normalized front force is larger than unity. On the other hand, the elastic solution is closer to the finite element simulation for larger tool radius. One possible explanation is that springback also involves non-elastic deformation because of the large plastic deformation accumulated before unloading, and the reduction of the flow stress after a reversed strain path [46]. The calculated cross-section profiles are now compared with the FE simulations and the experimental data, for two specimens tested at F b = 0.4 and F b = 0.8 with R/t = 14.0, Figure For small back force, the anticlastic deflection is localized toward the specimen edges, while the cross-section appears to be circular for F b = 0.8. The overall agreement between the elastic prediction, finite element simulation and experimental measurement is fairly satisfactory. However, the elastic solution tends to under-estimate the magnitude of the anticlastic deflection, as previously demonstrated by Figure

118 Unloaded h (mm) Barlat'96 yield G-W hardening R/t=3.5, 8x300 S4R FEM Analytic Normalized front forcre F f (a) Unloaded h (mm) Barlat'96 yield G-W hardening R/t=14, 8x300 S4R FEM Analytic Normalized front forcre F f (b) Figure 5.18: Comparison of simulated and analytically predicted maximum anticlastic deflections for draw-bend tested samples after springback: (a) R/t = 3.5 and (b) R/t = Anticlastic deflection z R/t=14 F b =0.4 Exp FEM Analyitc Noramalized width coordinate y/w (a) Anticlastic deflection z R/t=14 F b = Exp FEM Analytic Noramalized width coordinate y/w (b) Figure 5.19: Comparison of measured, simulated and analytically predicted anticlastic profiles for draw-bend tested samples: (a) F b = 0.4 and (b) F b =

119 5.6 Discussion Based on the previous analysis, the anticlastic deformation in draw-bend tested samples can be characterized by a single dimensionless parameter, β, which combines the influence of the sheet tension, tool radius and sample geometry. As shown in Figure 5.20, the cross-section profiles are plotted for two different groups of drawbend tested specimens. The first group has the same width but the back force is different, Figure 5.20(a); while Figure 5.20(b) shows the effect of the sample width on the cross-section shape. For both cases, it is realized that β uniquley determines the shape of the cross-section. When β < 10 15, all cross-sections are nearly circular. As β increases, the anticlastic deflection tends to localize toward the sheet edges, while the center of sheet is essentially flat. Anticlastic deflection (mm) β=11.92, R/t=14, F =0.7 b β=4.05, R/t=3.5, F b =1.1 β=22.25, R/t=14, F b = Unloaded β=31.98, R/t=3.5, F b =0.2 Width=50mm Normalized width coordinate y/w (a) Anticlastic deflection (mm) R/t=14 F b =0.5 Unloaded β=1.1, W=12.5mm β=0.28, W=6.3mm b=4.32, W=25mm β=19.18, W=50mm β=10.48, W=37mm Normalized width coordinate y/w (b) Figure 5.20: Simulated anticlastic profiles of draw-bend tested samples: (a) for various back forces and (b) various widths at F b =

120 Figure 5.21 summarizes how β varies with the normalized back force and the specimen width. It is worthwhile to notice the similarity between Figure 5.21(a) and Figure 5.12(a). Both the unloaded β (calculated as W 2 ) and the springback angle ( θ) r t decreases with increasing back force. The rapid drop in β happens around F b = , which corresponds to the occurrence of the persistent anticlastic curvature. 40 β=w 2 /(r' t) t=0.92mm 20 β=w 2 /(r' t) R/t=14, F b = t=0.92mm β (unloaded) 20 β (unloaded) R/t=3.5 R/t= Normalized back force F b 1.4 (a) Width (mm) (b) Figure 5.21: Variation of Searle s parameter β with (a) back force and (b) specimen width. 5.7 Conclusions Elastic bending theory and finite element simulation are utilized to investigate the role of forming variables on anticlastic curvature, and springback by draw-bend test. The following conclusions are reached: 102

121 1. Springback steadily decreases as sheet tension increases, with sudden decline of springback angle as the front pulling force approaches yielding. Persistent anticlastic curvature is identified as the cause of this rapid change. Larger tool radius leads to less springback, but in a less important way as sheet tension does. 2. In draw-bend test, anticlastic curvature is developed in the unbending process during forming, and it persists after springback when the applied sheet tension exceeds a critical value. The persistent anticlastic curvature significantly increases the section moment of inertia, and thus dramatically reduces springback. 3. For F b = 0.5 and R/t = 14, the springback angle first decreases, then increases with strip width, but the anticlastic curvature monotonically decreases with specimen width. However, both springback angle and anticlastic curvature decrease with specimen width for F b = The occurrence and persistence of anticlastic curvature in draw-bend test can be explained by elastic bending theory. The cross-section shape after unloading is determined by Searle s parameter, β, which depends on the specimen geometry (W/t) and sheet tension (via the curl radius R x ). The rapid decrease in springback angle at F b = corresponds to a critical β value of 10 15, above which the anticlastic displacement tends to concentrate toward the sheet edges. 5. It is worth noting that the stress state in the lateral direction (plane-stress or plane-strain) cannot be simply identified by the sheet width-to-thickness ratio, 103

122 even for simple bending problems. The radius of primary bending curvature can affect deformation mode too, via Searle s parameter. For draw-bend test, the springback process is closer to plane-stress rather than plane-strain, because of the persistent anticlastic curvature. 104

123 CHAPTER 6 TIME-DEPENDENT SPRINGBACK Note: Some of the experimental data was provided by W.D. Carden. A manuscript of this work has been submitted to the International Journal of Plasticity for publication. Abstract Draw-bend tests, devised to measure springback in previous work, revealed that the specimen shapes for aluminum alloys can continue to change for long periods following forming and unloading. Steels tested under identical conditions showed no time-dependent springback. In order to quantify the effect and infer its basis, four aluminum alloys, 2008-T4, 5182-O, 6022-T4 and 6111-T4, were draw-bend tested under conditions promoting the time-dependent response (small tool radius and low sheet tension). Detailed measurements were made over 15 months following forming, after which the shape changes were difficult to separate from experimental scatter. Earlier tests were re-measured up to 7 years following forming. The shape changes are generally proportional to log(time) up to a few months, after which the kinetics becomes slower. In order to understand the basis of the phenomenon, two models were considered: residual stress-driven creep, and anelastic deformation. In the 105

124 first case, creep properties of 6022-T4 were measured and used to simulate creepbased time-dependent springback. Qualitative agreement was obtained using a crude finite element model. For the second possibility, novel anelasticity tests following reverse-path loading were performed for 6022-T4, aluminum-killed drawing quality steel (AKDQ) and drawing quality special killed steel (DQSK). Based on the experiments and simulations, it appears that anelasticity is unlikely to play a large role in long-term time-dependent springback of aluminum alloys. 6.1 Introduction Springback occurs when sheet metal parts are released from forming tools. If not correctly predicted and compensated for, springback will cause the final part shape to deviate from design specifications and to create assembly problems. For simple forming operations, such as pure and stretching bending [76, 108], U-bending (channel forming) and V-bending [109,110], empirical methods and analytical models have been developed and successfully utilized to predict springback. However, these methods are not suitable for many industrial stamping operations in which complicated deformation paths and evolving contact conditions are present. The draw-bend test realistically simulates the springback situation encountered in many sheet-forming operations, where bending and unbending occur successively, with simultaneous stretching, as material is drawn over a tool surface [69,74, 82]. For most materials and process conditions, the specimen shape is reproducible and static. However, for aluminum alloys draw-bend tested at certain combinations of back force F b (F b is expressed as a normalized quantity by dividing the controlled back force by the force required to yield the sheet in uniaxial tension) and R/t (tool radius/sheet 106

125 thickness), the springback shape was observed to change significantly with time following the forming and unloading steps [67]. This phenomenon had apparently been unreported previously, although stress relaxation of aluminum alloys is better-known, and is closely linked metallurgically [68]. The possibility of time-varying part shapes has significant implications for industries striving to substitute sheet aluminum for sheet steel to reduce mass, while maintaining dimensional tolerances and consistency. The current work is aimed at clarifying and quantifying the basis of time-dependent springback. Process conditions of the draw-bend test (F b and R/t) were chosen to maximize the time-dependent springback and thus to improve resolution. Test results are reported for four aluminum alloys, including both heat-treatable and nonheat-treatable types, for periods of up to 15 months following forming. Preliminary analytical results of simulated time-dependent springback based on creep relaxation of residual stress are compared with experimental measurements for one alloy. The possible mechanisms underlying time-dependent springback are discussed in light of two relevant metallurgical phenomena: creep driven by residual stress [111] and anelastic flow [112]. Both creep and anelasticity are time-dependent deformation processes with slow kinetics at low homologous temperature, but strain still accumulates over a long period of time. Creep usually denotes a slow viscous flow of solid under macroscopically non-zero stress, via atomic diffusion (through lattice or along grain boundary) and dislocation motion (glide or climb). The mechanism of creep depends on composition, microstructure features, temperature and stress [113]. Anelastic flow, on the other hand, is often observed after a path change in stress or strain [114]. In uniaxial tensile test, anelasticity follows unloading and results in a 107

126 hysteresis loop upon re-loading. Anelastic strain is closely related to time-dependent microscopic deformation process [115]. Preliminary analytical results of simulated time-dependent springback based on creep relaxation of residual stress are compared with experimental measurements for one alloy. To identify the possible contribution of anelasticity, tension and tension/compression tests were carried out for 6022-T4, AKDQ (aluminum killed drawingquality) steel and DQSK steel. For 6022-T4, anelastic strain rate and magnitude were compared to their counterparts calculated from measured time-dependent springback angle and to finite element simulation based on creep model. 6.2 Experimental Standard tensile tests, special draw-bend tests, and uniaxial anelastic tests in tension and tension/compression were performed Materials Four aluminum sheet alloys of nominally 1mm thickness were provided by Aluminum Company of America [116]: 2008-T4, 5182-O, 6022-T4 and 6111-T4. The chemical compositions of these alloys are listed in Table 6.1 [117, 118]. Alloy t (mm) Mg Si Cu Mn Fe Al other 2008-T Bal O Bal T Bal. 0.02Ti 6111-T Bal. 0.1Cr Table 6.1: Chemical composition (in weight pct.) and thickness of aluminum sheets. 108

127 The microstructures of as-received materials were examined at the Alcoa Technical Center using optical microscopy at 200 magnification. As shown in Figure 6.1, all the alloys have elongated grains in the longitudinal and long transverse directions (denoted as L and LT, i.e., rolling and transverse directions respectively), but equiaxed grains in the short transverse direction (ST, i.e., thickness direction). Grain counts were taken in all three directions (L, LT and ST), with results reported in Table 6.2. The average grain sizes are estimated based on ASTM Standard E112 [119]: 50µm for 2008-T4 and 6022-T4, 20µm for 5182-O and 56µm for 6111-T4. L LT LT ST 2008 T4 L ST 100 µm 5182 O 6022 T T4 Figure 6.1: Optical micrographs revealing grain structures. 109