ON THE USE OF HOMOGENEOUS POLYNOMIALS TO DEVELOP ANISOTROPIC YIELD FUNCTIONS WITH APPLICATIONS TO SHEET FORMING

Transcription

1 ON THE USE OF HOMOGENEOUS POLYNOMIALS TO DEVELOP ANISOTROPIC YIELD FUNCTIONS WITH APPLICATIONS TO SHEET FORMING By STEFAN C. SOARE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA

2 Copyright 2007 by STEFAN C. SOARE 2

3 ACKNOWLEDGMENTS I thank my committee members for their kind patience and understanding. I thank Dr Yoon from Alcoa for his constant encouragements. There were many difficult moments I could not have overcome without his support. I thank my colleagues Mike Nixon, Brian Plunkett and Joel Stewart for their friendship, collaboration and understanding. 3

4 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTER 1 INTRODUCTION Yield Surface Identification of Material Properties A Review of Some Orthotropic Yield Functions Hill s Quadratic and Other Attempts Gotoh s Fourth Order Polynomial A Sixth Order Polynomial: CB A More Direct Approach: BBC The Use of Linear Transformations to Model Anisotropy Outline of the Dissertation FOURTH, SIXTH AND EIGHTH ORDER HOMOGENEOUS POLYNOMIALS Convexity Fourth Order Polynomial Criterion for Plane Stress States Bounds on Coefficients and Data Identification Procedure for Poly Applications to the Modeling of Two Aluminum Alloys and of a Mild Steel Sixth and Eighth Order Polynomials (Plane Stress) Extensions to 3D Stress States Applications to Deep Drawing The Numisheet 93 Square Cup Test Prediction of Earing in Cylindrical Cups Discussion and Further Examples A MORE GENERAL APPROACH: CRITERIA WITH TENSION-COMPRESSION SYMMETRY/ASYMMETRY A General Formulation Modeling of Plastic Properties with Strength Differential Effect Implementation Details

5 4 APPLICATIONS TO THE PREDICTION OF FORMING LIMIT DIAGRAMS Marciniak and Kuczynski Model Integration of the M-K Equations Computing the Stress in Zone A Computing the Strains in Zone B Applications to Two Aluminum Alloys AA5182-O: A First Experimental Data Set AA5182-O: A Second Experimental Data Set AA3104-H APPENDIX A SOME ALGEBRAIC DETAILS REGARDING THE CONVEXITY OF POLY4 142 B POLY4 INITIAL GUESS FOR POLY C RETURN MAPPING ALGORITHM D THE JACOBIAN FOR THE ALGEBRAIC SYSTEM OF ZONE B REFERENCES BIOGRAPHICAL SKETCH

6 Table LIST OF TABLES page 2-1 Experimental data for AA2090-T3, AA2008-T4 and NUM 93 mild steel Poly4 coefficients for AA2090-T3, AA2008-T4 and NUM 93 mild steel Poly6 coefficients for isotropic Mises, AA2090-T3, Mat 1, and Mat Poly8 coefficients for isotropic Mises and AA2090-T3. For both we have a 1 = Draw-in predictions of Poly4 FE simulation and experimental data Poly8 coefficients for the two materials Mat 1 and Mat Poly7 coefficients for the two materials Mat 1 and Mat Experimental data for AA5182-O, (B), Banabic et. al (2005b), AA5182-O, (W), Wu et al. (2003), and AA3104-H19. Note that for all three cases only data at 0 o, 45 o and 90 o is experimental. The data in between was generated with Poly4. The biaxial yield strength and r-value data (where available) is experimental Poly4 coefficients for AA5182-O(B), AA5182-O(W), and AA3104-H Poly6 coefficients for AA5182-O(B), AA5182-O(W1), AA5182-O(W2), AA H19(1) and AA3104-H19(2) Poly8 coefficients for AA5182-O(W1), AA5182-O(W2), AA3104-H19(1) and AA H19(2). For all we have a 1 =

7 Figure LIST OF FIGURES page 1-1 Geometrical setting of the uniaxial tests for which directional yield strength and r-value are measured/predicted Poly4 material characterization of AA2090-T3 using the identification procedure in Gotoh (1977). Directional yield strength and r-value Projection on the biaxial plane of the Poly4 yield surface for AA2090-T3, and the Poly4 biaxial yield curve for the same material. Although the biaxial yield curve is convex, the yield surface is not Yield curves in the biaxial plane according to Hosford s isotropic criterion Projections on the biaxial plane of the Poly4 yield surface for AA2090-T3 and AA2008-T Poly4 material characterization of AA2090-T3. Directional yield strength and r-value; comparison with Yld96 is also shown Poly4 material characterization of AA2008-T4. Directional yield strength and r-value; comparison with Yld96 is also shown Poly4 material characterization of the NUM 93 mild steel. Directional yield strength and r-value Projections on the biaxial plane of the Poly6 and Poly8 yield surfaces for AA T Poly6 and Poly8 material characterization of AA2090-T3. Directional yield strength and r-value; comparison with Yld2004 is also shown Tool dimensions for the square cup problem Profile of the drawn square cup, at 40 mm punch stroke, and definitions of draw-in (dimensions in mm) Thickness strain variation along the OB diagonal as predicted by FE simulations with Poly4 and Von Mises, and as measured at 15 mm punch stroke Thickness strain variation along the OB diagonal as predicted by FE simulations with Poly4 and Von Mises, and as measured at 40 mm punch stroke Geometrical setting and tool dimensions for the deep drawing with cylindrical punch simulation Typical mesh used on the blank

8 2-13 Fully drawn cup (final configuration). Only a quarter of the blank was simulated, the full cup being constructed by symmetry considerations Profiles of AA2090-T3 drawn cups simulated with Poly4, 2D and 3D. Yld96 simulation is also shown, data after Yoon et al. (2006) Profiles of AA2090-T3 drawn cups simulated with Poly6-3D and Poly8-3D. The Yld2004 simulation is also shown, with data after Yoon et al. (2006) Evolution of the profile of the cup during the drawing of the AA2090-T3 blank, simulated with Poly Projections on the biaxial plane of the Poly6 yield surfaces for Mat 1 and Mat Poly6 material characterization of Mat 1 and Mat 2. Directional yield strength and r-value Profiles of fictitious material MAT 1 cup as predicted by Poly6 (FEM), and by analytical formula (2 78) proposed by Yoon et al. (2006). For comparison, the experimental profile of AA2090-T3 is also shown Profiles of fictitious material MAT 2 cup as predicted by Poly6 (FEM), and by analytical formula (2 78) proposed by Yoon et al. (2006). For comparison, the experimental profile of AA2090-T3 is also shown Poly87 material characterization of Mat 1. Directional yield strength in tension and compression, and r-value in tension and compression Projection on the biaxial plane of Mat 1 yield surface Poly87 material characterization of Mat 2. Directional yield strength in tension and compression and r-value in tension and compression Projection on the biaxial plane of Mat 2 yield surface Profiles of the two fictitious materials MAT 1 and Mat 2 cups as predicted by the Poly87 simulation. For comparison, the experimental profile of AA2090-T3 is also shown Evolution of the profile of the Mat 2 cup during drawing process (simulated with Poly87) Geometrical setting for M-K analysis Positioning on the hardening curve of the two zones of the sheet. Deformation instability (necking) is triggered when zone B approaches the flat portion of the hardening curve Rotation of the groove during the stretching of the sheet

9 4-4 Poly4 and Poly6 material characterization of AA5182-O, with data from Banabic et. al (2005b). Directional yield strength and r-value Projections on the biaxial plane of the Poly4 and Poly6 yield surfaces for AA O(B) The Poly4 and Poly6 biaxial yield curves (σ xy = 0) for AA5182-O(B) The evolution of the ratio in 4 3 during plane strain loading Forming limit diagram for the AA5182-O(B) alloy Poly6 and Poly8 material characterization of AA5182-O, with data from Wu et al. (2003). Directional yield strength and r-value Projections on the biaxial plane of the Poly6 and Poly8 yield surfaces for AA O(W) The evolution of the ratio in 4 3 during plane strain loading for AA5182-O(W) Poly6 and Poly8 material characterization of AA5182-O, with data from Wu et al. (2003) and optimization toward the inscribed anisotropic hexagon. Directional yield strength and r-value Projections on the biaxial plane of the Poly6 and Poly8 yield surfaces for AA O(W): optimization toward the inscribed anisotropic hexagon The Poly6 and Poly8 biaxial yield curves (σ xy = 0) for AA5182-O(W); with Yld96 biaxial points as input The Poly6 and Poly8 biaxial yield curves for AA5182-O(W); the Poly6 and Poly8 biaxial yield curves are optimized toward the inscribed anisotropic hexagon Forming limit diagram for the AA5182-O alloy described in Wu et al. (2003); Yld96 points in the biaxial plane used as input Forming limit diagram for the AA5182-O(W). The Poly6 and Poly8 biaxial yield curves are optimized toward the inscribed anisotropic hexagon Poly6 and Poly8 material characterization of AA3104-H19. Directional yield strength and r-value. Biaxial data included in optimization Projections on the biaxial plane of the Poly6 and Poly8 yield surfaces for AA H The Poly6 and Poly8 biaxial yield curves (σ xy = 0) for AA3104-H19; biaxial points from Wu et al. (2003) were used as input for optimization The Poly6 and Poly8 biaxial yield curves for AA3104-H19; optimized towards the anisotropic inscribed hexagon

10 4-22 Forming limit diagram for AA3104-H19. Comparison with Yld96 also shown Forming limit diagram for AA3104-H19. Biaxial yield curves optimized toward the anisotropic inscribed hexagon Poly6 and Poly8 material characterization of AA3104-H19. Directional yield strength and r-value. Optimized toward the inscribed hexagon Projections on the biaxial plane of the Poly6 and Poly8 yield surfaces for AA H19. Optimized toward the inscribed hexagon The evolution of the ratio in 4 3 during plane strain loading for AA3104-H

11 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ON THE USE OF HOMOGENEOUS POLYNOMIALS TO DEVELOP ANISOTROPIC YIELD FUNCTIONS WITH APPLICATIONS TO SHEET FORMING Chair: Oana Cazacu Major: Mechanical Engineering By STEFAN C. SOARE August 2007 Within the framework of phenomenological plasticity, this work proposes a new look at polynomial anisotropic yield functions. Their range of applications has been severely limited in the past by convexity issues. A simple constrained optimization scheme is proposed here to solve this problem. It is shown that homogeneous polynomials of fourth, sixth and eighth order can be powerful tools for modeling almost any type of variation in directional properties of plastic flow. The plane stress formulations, suitable for sheet forming applications, are considered first. Then a simple method for generating minimalist 3D extensions, based mostly on the plane stress criterion, is presented. The polynomial approach is then employed to develop a more general method of designing yield surfaces. The new method allows for yield surfaces without a center of symmetry, a property useful for modeling metals with strength differential, that is, metals with different yielding properties in tension and compression. Several applications to deep drawing and the prediction of forming limit diagrams are considered. Particular attention is paid to the prediction of earing in cylindrical cups, as a method of validation of the yield function. The prediction of the forming limit curve in biaxial stretching is also an important test of the capabilities of a yield function, the most important factor being the correct description of the biaxial yield curve. It is shown that polynomial yield functions can be successfully used for both earing and FLD prediction. 11

12 CHAPTER 1 INTRODUCTION Traditionally, metal forming processes have been developed based on expensive experimental trials. In recent years, finite element (FE) simulations have been extensively used to reduce the amount of experiments and trial and error involved in the process development. Key for the success of simulations of forming processes is the constitutive model used for the description of the plastic behavior. Within the theoretical framework of associative plasticity this reduces to specifying the yield surface and the hardening law. The hardening law describes the way the yield surface evolves in the stress space during the deformation process. This evolution can be isotropic, which means the yield surface expands the same amount in all directions preserving the initial shape, or it can be anisotropic, meaning that simultaneously with its expansion the yield surface also changes its shape. Depending on the type of loading, anisotropic hardening can also be responsible for changes in the structural properties/symmetries of the material. Here we focus only on the effects of the initial anisotropy on the deformation process and therefore the hardening of the yield surface will be assumed isotropic, leaving only the yield surface itself as the main topic of this text. Not long ago, W.F. Hosford synthesized the qualities of a good anisotropic yield function as follows, Hosford (1993), pp. 139: To be useful, an anisotropic yield criterion should describe material behavior with reasonable accuracy and also involve only a small number of material parameters that can be easily evaluated by simple tests (e.g., tension tests). Even today, in the modern era of the computer, this opinion seems to be strongly rooted in the metal forming community, with a special emphasis on the small number of material parameters. However, as recent investigations in the numerical simulation of plastic deformation have shown, for example the prediction of limit strains during sheet stretching, (e.g. Wu et al., 2003), or the prediction of earing in cylindrical cups (e.g. Yoon et al., 2004, 2006), reasonable accuracy must leave more and more room to the 12

13 very accurate description of the initial plastic anisotropy of the sheet. In practical terms this means that more involved formulations of yield criteria, with an increased number of material parameters, have to gain acceptance. This chapter starts with a brief review of the concept of yield surface for orthotropic sheet metals and its identification input based on data from mechanical tests (directional yield strength and r-value). Then past and current state of the art in the development of yield criteria are presented. It concludes with an outline of the rest of the text. 1.1 Yield Surface The yield function is a nonnegative real function defined on the direct product between the stress space and the space of hardening parameters R 9 R k (σ, ξ) f(σ, ξ) R + For each fixed ξ, its level surfaces, f ξ (σ) = 0, bound the elastic domains and characterize the directions of the associate (normal) plastic flow. A one-to-one relationship between strain increments and stress is guaranteed if these level surfaces are convex. This is the case if the yield function itself is a convex function. This convexity requirement has also been motivated by physical considerations in Bishop and Hill (1951) where the principle of maximum dissipation is proposed for the selection of the direction of plastic flow: given a plastic strain increment dɛ p, the actual state of stress σ is the one that maximizes the plastic dissipation: (σ σ ) dɛ p 0, ( )σ R 9 with f(σ ) 0 (1 1) It can be easily shown that if the above inequality is satisfied, then the yield surface must be convex and the direction of plastic flow must be along the normal to the yield surface (commonly known as normality rule, or associative flow rule) dɛ p = dλ f σ (1 2) 13

14 with dλ being the proportionality factor. Thus the yield surface not only bounds the elastic domain, but also defines the direction of plastic flow. The hardening parameters ξ describe the evolution in the stress space of the yield surface during the deformation process. Since only isotropic hardening will be considered here, the set of parameters ξ is reduced to a single scalar, denoted from now on with ɛ p, timing the history of the deformation and the yield function can be rewritten in the more familiar form f(σ, ξ) = σ(σ) H(ɛ p ) (1 3) with σ : R 9 R + being a convex function, usually called the equivalent stress, and with H : R + R + a positive nondecreasing function. For anisotropic yield surfaces the hardening parameter is in general defined as the normalized equivalent plastic work. This leads us to the second important assumption about the yield function. The yield function is assumed to be first order positive homogeneous with respect to the stress tensor: σ(ασ) = ασ(σ), ( )σ R 9, ( )α R + (1 4) Using Euler s lemma on first order homogeneous functions and the normality rule, we then have the following equivalence relation for the rate of plastic work dw p := σ dɛ p = σdλ (1 5) which prompts for the following natural definition of the hardening parameter dɛ p = dλ = dɛ p = dw p dw p σ = ɛp = σ (1 6) The yield function must also be pressure independent: f(σ + pi) = f(σ), ( )σ R 9, ( )p R (1 7) with p being the pressure, and I the second order identity tensor. This condition is based on experimental facts. It has been observed that superimposing relatively high 14

15 confinement pressures on any stress state does not affect the yielding of the material. Also, in associative plasticity pressure-independence guarantees that the plastic deformation is isochoric (which must be true for single crystals, where plastic flow is due to dislocation motion, thus generating isochoric deformation fields). Indeed, differentiating in 1 7 with respect to p and using 1 2 we obtain ( ) f f I = 0 tr = 0 tr(dɛ p ) = 0. σ σ the tr operator indicating the trace of a second order tensor. Crystal structure, dislocation activity, orientation of the grains (texture) are the main generators of anisotropy in metals, Hull and Bacon (2001), Hosford (1993), Kocks et al. (2000), Gambin (2000). They all come into play during sheet fabrication by rolling. At macroscopic level, in a first approximation, the rolled sheet can then be considered as having orthotropic symmetry, with the rolling direction (RD), the transverse direction (TD), and the normal direction (ND) as the three orthogonal intersection lines of the three orthogonal planes of symmetry. In what follows, these three directions will be taken as the x, y and z axes, respectively, of an orthogonal coordinate system associated with the sheet (also referred to as material frame or axes). With respect to this coordinate system the matrix of the components of the Cauchy stress tensor will be denoted as usual σ x σ xy σ xz [σ] = σ yx σ y σ yz σ zx σ zy σ z with σ yx = σ xy, σ zx = σ xz, and σ zy = σ yz. The orthotropic symmetry group of the material is then generated by the 180 o rotation about the x-axis, call it Q 1, and about the y-axis, call it Q 2. The necessary and sufficient condition condition for a yield function f to be orthotropic, i.e., be invariant with respect to the symmetry group, is f(q T i σq i ) = f(σ), i = 1, 2, ( )σ R 9 (1 8) 15

16 where the right superscript T denotes transposition. Note that this condition is written in invariant form, that is, it is not tied to any particular coordinate system. A general representation of the yield function f as a scalar function of a list of invariants of combinations of the stress tensor σ and the two rotations Q i can then be obtained from the above equation based on what is commonly known as the theory of representation of anisotropic tensorial functions, Wang (1970), Liu (1982), Boehler (1987). While this approach might offer valuable insight in problems involving the constitutive modeling of complicated sets of anisotropic tensorial quantities, for our problem it is better to follow a more direct, and simpler, approach based on the direct representation of the yield function as a function of the components of the stress tensor with respect to the symmetry axes. At a first look this might seem restrictive, since it ties the form of the yield function to one particular coordinate system. However, in most practical applications, and in particular in FE implementations, one deals only with this component form of the yield function, and therefore having the above invariance condition expressed directly in component form with respect to the material frame is much more useful. Thus, with respect to the symmetry axes we write the yield function in the form f = f(σ x, σ y σ z, σ xy, σ xz, σ yz ), σ := [σ x, σ y σ z, σ xy, σ xz, σ yz ] T R 6 (1 9) Then, with respect to the symmetry axes, the 180 o rotation Q 1 about the x-axis has the matrix of components [Q 1 ] = (1 10) and from equation 1 8 we obtain f(σ x, σ y, σ z, σ xy, σ xz, σ yz ) = f(σ x, σ y, σ z, σ xy, σ xz, σ yz ), ( )σ R 6 (1 11) 16

17 In other words, the plane (xz) is plane of material symmetry for the yield potential f, if f does not depend on the sign of the shear components σ xy and σ xz. A similar reasoning repeats for the reflection about the plane (yz) to obtain f(σ x, σ y, σ z, σ xy, σ xz, σ yz ) = f(σ x, σ y, σ z, σ xy, σ xz, σ yz ), ( )σ R 6 (1 12) Thus, the yield function f in equation 1 9 is orthotropic if and only if it satisfies equations 1 11 and Remark. In particular, for plane stress conditions, say parallel with the (xy)-plane, a yield function f is orthotropic if and only if it satisfies f(σ x, σ y, σ z, σ xy ) = f(σ x, σ y, σ z, σ xy ), ( )σ R 3 (1 13) Remark. It is worth noting here that lower symmetries than orthotropic, like monoclinic, cannot be modeled with plane stress criteria. Lower symmetries mean dependence on the sign of some of the out-of-plane shear components, and thus information concerning yielding under full stress states is required. Another general property of a yield function concerns its symmetry with respect to the reversal of the loading direction. In general, plastic deformation is caused mainly by the motion of dislocations within the crystal lattice. This deformation mode leaves the crystal lattice unchanged and therefore the yielding of the crystal is unaffected by the sign of the applied stress. At macroscopic level this means that the yield function has a center of symmetry: there exists σ 0 R 9 such that f(σ 0 σ) = f(σ 0 + σ), ( )σ R 9 (1 14) Since in this work kinematic hardening (meant to model the Bauschinger effect by translating the yield surface in the stress space) of the material is not considered in our constitutive model, the particular case when the yield function is symmetric with respect 17

18 to the origin of the stress space suffices for our purposes f( σ) = f(σ), ( )σ R 9 (1 15) For example, for a uniaxial loading the above equation states that the material yields at the same stress in compression as in tension. It should be noted, however, that plastic deformation of a crystal can be also caused by twinning (folding of the crystal lattice about a mirror plane) a deformation mechanism favored especially in hexagonal close-packed (HCP) metals (magnesium, titanium), although it can also occur in face centred-cubic (FCC) (aluminum, copper) and body centred-cubic (BCC) (iron) metals, Hosford (1993). This deformation mechanism is strongly dependent on the direction of the applied stress and at macroscopic level it manifests through yield surfaces that do have a center of symmetry. This effect is referred to in the literature as strength differential, SD. Asymmetric yield surfaces will be discussed in Chapter 3 of this text. For the moment we limit our discussion to symmetric yield surfaces only. 1.2 Identification of Material Properties Convexity, pressure-independence, independence on the sign of shear and tension - compression symmetry are all general properties required for an orthotropic yield function to have. Further detail on the shape of the yield function is acquired for each class of metallic sheets by studying its in-plane yielding properties. The commonly used procedure in industrial practice is mechanical testing. More precisely, directional properties of the sheet can be experimentally characterized by a set of uniaxial tests. Samples are cut from the sheet at several angles θ from the rolling direction, and then tested to determine their directional yield strength σ θ (in tension), see Figure 1-1. The stress state in the sample (parallel with the θ-direction of loading) has with respect to the symmetry axes the components 18

19 [σ] = σ θ cos 2 θ cos θ sin θ 0 cos θ sin θ sin 2 θ (1 16) The first order positive homogeneity of the yield function makes possible an explicit formula for its predicted directional yield strength: σ θ σ(cos 2 θ, sin 2 θ, cos θ sin θ, 0, 0) = H(0) = σ 0 (1 17) Note. We will often refer to the equivalent stress itself, σ, as being the yield function and denote it with f, or F. It will be clear from context wether we refer to the norm of the stress, that is, to the expression of the equivalent stress, or to the full yield surface. Thus, with this new convention, the above formula will be often encountered in this text in the form σ θ := σ θ σ 0 = [F (cos 2 θ, sin 2 θ, cos θ sin θ, 0, 0)] 1 (1 18) Another indicator of anisotropic properties measurable during uniaxial tests is the Lankford value, or r-value. It is defined as the ratio, Hosford and Caddell (1983), pp. 265, r = ɛ w = ln(w/w 0) ɛ t ln(t/t 0 ) (1 19) where ɛ w denotes the strain in the width (TD) direction, ɛ t denotes the strain in the thickness (ND) direction, and w, w 0 denote the current and initial width, t and t 0 the current and initial thickness. From a physical point of view, this ratio measures how drawable the sheet is (its resistance to thinning). Note that for an isotropic material (with smooth yield surface) the directional yield strength and r-value are constant. In general, the r-value does not vary with strain, Hosford and Caddell (1983), and therefore instead of the above formula, the ratio of the strain increments can be taken as its definition r θ = dɛ w dɛ t (1 20) 19

20 where, as for the measurement of the directional yield stress, θ is the angle of the test sample with respect to the rolling direction. Here w is the width of the sample along the direction perpendicular to the axis of the sample (parallel to the direction of loading). Neglecting the elastic part of the strain increment and using the normality rule 1 2, equation 1 20 takes the form r θ = [ ( f f sin 2θ cos 2 θ + f )] / ( f sin 2 θ + f ) σ xy σ y σ x σ x σ y (1 21) in which the components of the stress tensor are given by For example, for the uniaxial test along the rolling direction we have θ = 0 and then r 0 = f/ σ y f/ σ x + f/ σ x = dɛp y dɛ p x = f/ σ y f/ σ x = r r 0 = /r 0 (1 22) The relevance of this formula is better understood if we consider the deep drawing process, see Figure 2-11 for a schematic of it, where a blank sheet is pressed with a punch into the cavity of a die. The flange area of the blank, while still between the die and holder is mainly in a state of uniaxial compression near the rim. Let us now focus on an infinitesimal material element of the blank with its axis parallel with the RD direction, but positioned at the rim on the transverse direction of the sheet. This element is in a state of uniaxial compression and assuming tension/compression symmetry we see from 1 22 that a higher r 0 value implies a bigger y-component of the plastic increment, that is bigger increase of the height of the final cup. The reasoning is similar for any element along the rim. Thus in formula 1 19, or in 1 20, the width direction in the deep drawing process is essentially the radial direction along which the cup wall increases (due to the compression loading, the strain along the width is positive). Therefore high r-values are in general associated with sheet metals that can be drawn into deep cups, whereas low r-values indicate smaller depths of deep-drawing. From a geometrical point of view, knowledge of the directional yield strength implies knowledge about only one curve on the yield surface. Knowing the r-value adds 20

21 information about the gradient of the yield surface along this curve. The standard practice is to have θ = 0 o, 45 o, 90 o, but for some applications some metals, e.g. the capricious aluminum alloy AA2090-T3, may require a denser set of locations, for example θ = 0 o, 15 o, 30 o, 45 o, 60 o, 75 o, 90 o. The biaxial yield curve (the intersection of the yield surface with the (σ x, σ y )-plane) bounds stress states which are routinely encountered during many of the forming processes (any process that involves stretching), and therefore it requires a more precise characterization. For this purpose the biaxial value of the yield strength, σ b, is measured by loading a cruciform sample with the stress field σ x = σ y = σ b, σ xy = 0 (equi-biaxial loading). This indicates on the biaxial yield curve the location of the point σ x = σ y. Additional detail on the shape of the biaxial yield curve is obtained by measuring for the same equi-biaxial state of loading the biaxial r-value, r b, defined by r b = dɛp y dɛ p x = f/ σ y f/ σ x (1 23) This ratio specifies the inclination of the outward normal at the curve, at the equi-biaxial stress point. 1.3 A Review of Some Orthotropic Yield Functions The two primordial yield functions are the isotropic functions of Tresca (1864) and von Mises (1913). According to Tresca s criterion, the material yields when the maximum shear stress reaches a certain value. Thus, if the principal values of the stress tensor are denoted by σ 1, σ 2 and σ 3, then Tresca s yield function has the expression f(σ) = Max{ σ 1 σ 2, σ 2 σ 3, σ 3 σ 1 } (1 24) The von Mises s criterion states that yielding occurs when the (Euclidean) norm of the stress deviator reaches a certain value f(σ) = 3 2 σ tr(σ) 3 I (1 25) 21

22 or, in the more familiar component form f(σ) = { 1 [ (σx σ y ) 2 + (σ y σ z ) 2 + (σ z σ x ) 2] + 3 ( σxy 2 + σxz 2 + σ 2 2 yz) } 1/2 (1 26) As later research into the plasticity of metals showed, e.g., Hershey (1954), Hosford (1972), neither Tresca s yield function nor von Mises s holds the absolute truth: FCC structured lattices have in general yield surfaces with shapes closer to Tresca s hexagon (enclosing it), while BCC structured lattices have in general yield surfaces closer to von Mises s ellipsoid (inscribed in it). That is, most of the experimental data falls between the two yield functions. Nevertheless, as we shall see, these two yield functions have been the starting point for much of the significant later work on anisotropic yield functions Hill s Quadratic and Other Attempts Historically the first anisotropic yield function, Hill s quadratic criterion, Hill (1948), is still today one of the most popular anisotropic yield functions. It is a direct extension of von Mises s isotropic criterion to include orthotropic symmetry. With respect to the symmetry axes: f(σ) = [ F (σ y σ z ) 2 + G(σ z σ x ) 2 + H(σ x σ y ) 2 + 2Lσ 2 yz + 2Mσ 2 zx + 2Nσ 2 xy] 1/2 (1 27) Its simple analytic expression allows for simple convexity conditions, and also for easy coefficient identification (through explicit formulas). The yield function is real and convex if and only if its coefficients satisfy the inequalities, Hill (1990), F + G > 0, F G + GH + HF > 0, N > 0, M > 0, L > 0. (1 28) The function is well suited for metals like steel (BCC), which have in general mild variations of their directional properties, and also have ellipsoidal yield surfaces. However, since only four material coefficients are available for in-plane properties, it cannot accurately predict both the directional variation of the yield strength and of the r-value. Its identification usually takes as input only one of the two data sets and has large error 22

23 in the prediction of the other. As noticed by its author himself, Hill (1950), pp. 330, when applied to earing prediction for circular deep-drawn cups (see the next chapter), the quadratic criterion can predict at most four ears, the typical number for most BCC and FCC structured lattices. Still, some FCC alloys, like brass and some grades of aluminum alloys, produce cups with six or even eight ears (see next chapter). In spite of the mentioned limitations, the simplicity of its formulation and the ease with which its coefficients can be identified, rendered Hill s quadratic a status of universality within the metal forming industry, being adopted for any analysis that needed the description of anisotropic plastic properties. However, experimental evidence that the quadratic criterion was not a universal criterion continued to gather, and around 1970 s it was also observed that some grades of aluminum had an equi-biaxial yield strength ratio σ b := σ b /σ 0 greater than one, but in the same time had an average r-value less than one, Woodthorpe et al. (1970), Mellor and Parmar (1978). Since the quadratic criterion could not model this type of anisotropy, strangely enough, it was labeled at the time as anomalous behavior (of course, there is nothing anomalous in the behavior of aluminum). Recognizing the limitations of his quadratic criterion, actually quite early, Hill (1950), p 330, suggests the use of a general homogeneous polynomial in the form P (2D) n (σ) = i+j+2k=n a ijk σ i xσ j yσ 2k xy (1 29) as yield function for plane stress states. The integers i, j, k are all nonnegative. Note. For brevity, we will often write the yield function as n-th order homogeneous. The [ ] 1/n. actual yield function above is, however, P (2D) n Compared with 1 27, formula 1 29 might be considered at most heuristic, but Bourne and Hill (1950) explore the capabilities of the third order plane stress orthotropic polynomial (n = 3 in the above formula): f(σ) = a 1 σ 3 x + a 2 σ 2 xσ y + a 3 σ x σ 2 y + a 4 σ 3 y + (a 5 σ x + a 6 σ y )σ 2 xy 1/3 (1 30) 23

24 the absolute value being needed due to the odd order of the polynomial. Besides this abusive use of formula 1 29 (the homogeneity degree n should be an even integer), little improvement over the quadratic criterion is reported, the third order polynomial still not being able of a reasonable simultaneous description of yield and r-value variation (although it has a superior number of material parameters, six, over the plane stress quadratic criterion). To deal with the so called anomalous behavior, Hill proposed two more non-quadratic criteria. In Hill (1979) it is proposed that yielding occurs for f(σ) = [p σ y σ z m + q σ z σ x m + r σ x σ y m +a 2σ x σ y σ z m + b 2σ y σ z σ x m + c 2σ z σ x σ y m ] 1/m (1 31) with m, p, q, r, a, b, and c material parameters (the function is similar with the one proposed by Hosford (1979)). Notable, in the above formula there is no dependence on the shear components of the stress tensor, that is, the criterion is applicable only when the axes of loading coincide with the symmetry axes. And in Hill (1990) the following plane stress criterion is proposed f(σ) = { σ x + σ y m + k m [(σ x σ y ) 2 + 4σ 2 xy] m/2 +(σ 2 x + σ 2 y + 4σ 2 xy) m/2 1 [ 2a(σ 2 x σ 2 y) + b(σ x σ y ) 2 ] } 1/m (1 32) with m, k, a, b material parameters. Although both 1 31 and 1 32 improve on the prediction of the biaxial yield strength, they bring little or no progress in the prediction of the directional properties (1 31 cannot even be applied for a simple off axis test) Gotoh s Fourth Order Polynomial Gotoh (1977) was the first to explore formula 1 29 for n = 4. He thus considered the fourth order polynomial 24

25 f(σ) = a 1 σ 4 x + a 2 σ 3 xσ y + a 3 σ 2 xσ 2 y + a 4 σ x σ 3 y + a 5 σ 4 y +(a 6 σ 2 x + a 7 σ x σ y + a 8 σ 2 y)σ 2 xy + a 9 σ 4 xy (1 33) This was the first yield function that could simultaneously describe both yield strength and r-value directional properties. It was also the first yield function capable of modeling the so called anomalous behavior of aluminum. Using equations 1 18 and 1 21, Gotoh obtained a nine by nine system of linear equations which lead him to a set of explicit formulas for the identification of the coefficients a i above. The basic input data for identification is then σ 0, σ 45, σ 90, r 0, r 45, r 90, and σ b. Two more data points being needed, Gotoh suggested the use of σ θ and r θ with θ either o, or o. This approach, however, may sometimes output unsatisfactory predictions for both directional yield strength and r-value. Taking for example the case of aluminum alloy AA2090-T3, Gotoh s identification procedure generates the results in Figures 1-2 and 1-3. We used θ = (45 o + 90 o )/2, and as corresponding data the averages (σ 60 + σ 75 )/2 and (r 60 + r 75 )/2. In Figure 1-2 we note that the Poly4 predictions feature additional maxima or minima of the directional properties (around 15 o ) over the experimental data trend. And although the biaxial yield curve is convex, the yield surface is not convex, Figure 1-3. As the degree of the polynomial increases in formula 1 29, this type of behavior, with undesired local oscillations in the prediction of the directional properties and lack of convexity, becomes more and more the rule, rather than the exception, when the coefficients are identified without any regard for the positivity and convexity of the yield function A Sixth Order Polynomial: CB2001 Any anisotropic (orthotropic, in our case) yield function can be reduced through appropriate conditions on its parameters to an isotropic criterion. This reduction, however, cannot not be trivial. It must coincide with some meaningful isotropic criterion, like von Mises, or Tresca, or any other generally accepted isotropic description of plastic 25

26 properties. The question is then, how could an isotropic yield function be generalized to an anisotropic one? Any answer to this question would then provide a method for generating anisotropic yield criteria. Cazacu and Barlat (2001) proposed a rigorous method to extend any isotropic yield function expressed in terms of J 2 and J 3 to orthotropy. The authors start from the well known fact that any isotropic yield function can be represented in the form f(σ) = g(j 1 (σ), J 2 (σ), J 3 (σ)) (1 34) where J i (σ) are the three invariants of the stress tensor, J 1 (σ) = tr(σ), J 2 (σ) = tr(σ 2 ), tr(σ 3 ). In particular, for pressure independent materials the above form reduces to f(σ) = g(j 2 (σ ), J 3 (σ )) (1 35) with σ the deviatoric stress. The authors then use the list of anisotropic polynomial invariants of the stress tensor for the orthotropic case, e.g., Liu (1982), to generate the following anisotropic, pressure independent, polynomial generalizations of J 2 and J 3 to orthotropic symmetry J o 2 = a 1 6 (σ x σ y ) 2 + a 2 6 (σ y σ z ) 2 + a 3 6 (σ z σ x ) 2 + a 4 σ 2 xy + a 5 σ 2 yz + a 6 σ 2 zx (1 36) J3 o = 1 27 (b 1 + b 2 )σx (b 3 + b 4 )σy [2(b 1 + b 4 ) (b 2 + b 3 )]σz (b 1σ y + b 2 σ z )σx (b 3σ z + b 4 σ x )σy [(b 1 b 2 + b 4 )σ x + (b 1 b 3 + b 4 )σ y ]σ 2 z (b 1 + b 4 )σ x σ y σ z 1 3 [2b 10σ z b 5 σ y (2b 10 b 5 )σ x ]σ 2 xy 1 3 [(b 6 + b 7 )σ x b 6 σ y b 7 σ z ]σ 2 yz 1 3 [2b 9σ y b 8 σ z (2b 9 b 8 )σ x ]σ 2 zx + 2b 11 σ xy σ yz σ zx (1 37) The orthotropic criterion is then obtained by simply substituting the above J o 2 and J o 3 invariants into the original expression of the isotropic criterion f(σ) = g(j o 2, J o 3) (1 38) 26

27 In the cited paper this method is then applied to obtain an orthotropic version of Drucker s isotropic criterion. As mentioned before, most metals with random texture have their yielding points between Tresca s and Mises s yield surfaces. To better represent their yielding, Drucker (1949) proposed the following isotropic criterion f(σ) = (J 2 ) 3 c(j 3 ) 2 (1 39) with c a material parameter. Note, however, that Drucker s criterion is still an extension of von Mises s quadratic and it does not include as particular case Tresca s criterion for which a representation in terms of J 2 and J 3 is much more complicated, see for example Malvern (1969). The Cazacu and Barlat (2001) orthotropic extension of this criterion, CB2001, is then f(σ) = (J o 2) 3 c(j o 3) 2 (1 40) Thus, the method suggested in the cited paper does generate anisotropic yield criteria from isotropic ones. However, its outcome should be further analyzed on a case-by-case basis. For example, for CB2001 some comments are in order. First, it should be remarked that the constant c is no longer needed in the anisotropic version of the criterion. Indeed, without loss of generality it can be multiplied into the b k coefficients of J3. o Second, the CB2001 criterion is a quite particular form of a sixth order polynomial. Thus, its plane stress restriction reduces to formula 1 29 with n = 6. Third, the coefficients of the sixth order polynomial in the form of 1 29 depend through complicated nonlinear formulas on the coefficients a k and b k of J2 o and J3. o This nonlinearity further restricts the range of applications of CB2001 by placing an unnecessary burden on the identification process and on the positivity and convexity conditions (like most of the anisotropic polynomial criteria, CB2001 is not by default real-valued and convex). This is confirmed in Soare et al. (2007a) where the modeling capabilities of this criterion are explored. When applied to the deep drawing of AA2090-T3 the CB2001 criterion could predict only four ears. However, 27

28 as it will be shown later, the same problem is easily solved with six ears prediction by a general sixth order polynomial, see also Yoon et al. (2006) A More Direct Approach: BBC2003 Banabic et al. (2000), and later Banabic et al. (2005a), proposed the following simple algebraic form, close to a polynomial form, as plane stress yield function: f(σ) = a [(Γ + Ψ) m + (Γ Ψ) m ] + (1 a)(2λ) m (1 41) where Γ = Lσ x + Mσ y 2 (1 42) Ψ = [ (Nσx P σ y 2 ) 2 + Q 2 σ 2 xy] 1/2 (1 43) Λ = [ (Rσx Sσ y 2 ) 2 + T 2 σ 2 xy] 1/2 (1 44) The homogeneity degree m is considered fixed and, inspired by Hosford s theory (see next section), associated with the crystal class. Thus m = 8 for FCC metals, and m = 6 for BCC metals. The BBC2003 yield function, as called by its authors, then has 9 material parameters (the same as the fourth order polynomial of Gotoh). An application of this criterion to the modeling of AA2090-T3 is also attempted in Banabic et al. (2005a). The results are similar to the CB2001 modeling of this alloy, Soare et al. (2007a), CB2001 having 10 plane stress material parameters. Therefore it cannot predict six ears for the corresponding drawn cup. However, BBC2003 enjoys over CB2001 two essential properties which endow it with much more practical value: it is always real-valued and convex (provided 0 a 1) The Use of Linear Transformations to Model Anisotropy A scalar function having a vector as argument is isotropic if and only if it depends only on the magnitude (norm) of the vector. That is, if it is a radial function. To make then an analogy with vectors, isotropic yield functions depend only on the magnitude of 28

29 the stress tensor, not on its orientation with respect to a particular coordinate system. The analogy is complete if by magnitude we understand the magnitude of the traction vector on a certain surface. Three such traction vectors are needed to completely characterize the stress tensor. If these vectors are the principal directions of the stress tensor, the corresponding magnitudes are the principal stresses σ i and any isotropic yield function has the form f(σ) = g(σ 1, σ 2, σ 3 ) (1 45) with g a scalar symmetric function: g(σ i, σ j, σ k ) = g(σ 1, σ 2, σ 3 ), for any permutation (i, j, k) of (1, 2, 3). A famous theorem of Davis (1957) then says that if the function g is convex, then the function f is convex too (note that g is a scalar function of three arguments, whereas f has as argument a matrix, or a tensor). Anisotropy then means that besides the magnitudes σ i, the orientation of the stress tensor with respect to the symmetry axes must also be incorporated into the function g. The simplest method to achieve this is to make a linear transformation on the stress tensor (its deviator, more precisely) s = Lσ (1 46) At each material particle, the representation of the (fourth order Euclidean) tensor L is inevitably tied to a fixed triad (or frame) of vectors associated with the corresponding particle, in particular they may represent the axes of symmetry, and then the transformed tensor s incorporates now information about its orientation with respect to this triad. The anisotropic criterion is then f(σ) = g(s 1, s 2, s 3 ) (1 47) where now s i are the principal stresses of the transformed stress tensor s. This is the essence of the method based on linear transformations. One of the benefits of this method is that it generates convex functions (if a linear transformation is performed on the arguments of a convex function, the new function is also convex). In practice, it is used 29

30 with several generalizations depending on the number of material parameters involved, on the complexity of the experimental data involved. Hosford (1972) proposed an isotropic yield criterion based on the principal values s i of the stress deviator σ in the form f(σ) = 1 2 ( s 1 s 2 a + s 2 s 3 a + s 3 s 1 a ) (1 48) Depending on the value of the exponent a [1, ), the yield surface generated by this function can span the entire domain between Tresca (m {1, }) and von Mises (m {2, 4}) surfaces (it can actually describe also yield surfaces that inscribe von Mises s ellipsoid, but these cases have no physical meaning in our case). In Hosford (1972) it was remarked that the exponent a can be used as a parameter to fit experimental data and theoretical predictions for different crystal structures. Later work, Hosford (1979), Logan and Hosford (1980), showed that the exponent a = 8 was in best agreement with the results predicted by the Taylor-Bishop-Hill (TBH) plasticity theory of a polycrystal, Taylor (1938), Bishop and Hill (1951), for FCC structured lattices, whereas a = 6 best fitted the predictions of the TBH theory for BCC structured lattices, see Figure 1-4. The function 1 48 is the isotropic function g in 1 47 generally used as the starting point for anisotropic theories. Barlat et al. (1991) proposed the anisotropic yield function 1 47, called Yld91, with L 11 L 12 L L 12 L 22 L L 13 L 23 L [L] =, L L L 66 30

31 the components of the linear transformation with respect to the symmetry axes (with the usual tensor to matrix writing convention), and with g the Hosford s criterion Not all the components are independent. If L in 1 46 had σ as argument, instead of σ, then to satisfy the hydrostatic pressure independence condition the components L ik have to satisfy the relations : L 1k + L 2k + L 3k = 0 Thus Yld91 has six material parameters for general 3D stress states. It incorporates Hill s quadratic when the exponent a = 2, the advantage of Yld91 s formulation being the variable exponent a which allows Yld96 yield surfaces shapes other than ellipsoids. The reduced number of material parameters, however, limits the modeling capabilities of the directional properties. Karafilis and Boyce (1993) used the following isotropic extension of Hosford s criterion g(s 1, s 2, s 3 ) = 1 2 [(1 c)φ 1(s 1, s 2, s 3 ) + cφ 2 (s 1, s 2, s 3 )] (1 49) where φ 1 is given by 1 48, and φ 2 (s 1, s 2, s 3 ) = 3 a 2 a ( s 1 a + s 2 a + s 3 a ) (1 50) They motivate their choice by the need for more general isotropic yield surfaces, the additional function φ 2 being capable of modeling yield surfaces outside the quadratic of Mises. This is quite irrelevant for both FCC and BCC type of lattices. Their resulting anisotropic yield function, although incorporates Yld91, does not bring any significant improvement over Yld91. However, their idea to combine two different yield functions has been a source of inspiration for the creation of several other yield functions, e.g., Barlat et al. (2003), Bron and Besson (2004), Banabic et al. (2005a). Barlat et al. (1997) propose the following extension of Yld91, known as Yld96: f(σ) = 1 2 (α 1 s 2 s 3 a + α 2 s 3 s 1 a + α 3 s 1 s 2 a ) (1 51) 31