FEM simulation of the forming of textured aluminum sheets

Materials Science and Engineering A256 (1998) 51 – 59

FEM simulation of the forming of textured aluminum sheets JianGuo Hu a,*, J.J. Jonas a, Takashi Ishikawa b a

b

Department of Metallurgical Engineering, McGill Uni6ersity, Montreal H3A 2B2, Canada Department of Materials Processing, School of Engineering, Nagoya Uni6ersity, Nagoya 464 -8603, Japan Received 17 April 1998; received in revised form 13 July 1998

Abstract A texture-based fourth order strain-rate potential was used directly in an elastoplastic finite element code (ABAQUS) to model sheet metal forming. The method is based on the Taylor model of crystal plasticity and therefore takes the presence of texture into consideration. The deep drawing of cold-rolled and annealed aluminum sheet was simulated using this code in conjunction with a specially developed UMAT subroutine. The full geometry of deep drawing, including the disposition of the tools and the effect of friction, was taken into account in the simulations and both shell and brick elements were employed. The influences of element type, as well as of friction and blank holder force, are discussed. The thickness variations induced by deep drawing are also described. Reasonable agreement was obtained between the predicted and measured earing profiles for an annealed aluminum sheet. The steps to be followed to bring about further improvements in the accuracy of the simulations are also discussed. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Finite element method; Texture; Anisotropy; Strain-rate potential; Shell elements; Deep drawing; Aluminum sheet

1. Introduction The polycrystalline materials used in engineering exhibit significant plastic anisotropy that can be attributed principally to the presence of crystallographic texture. Such plastic anisotropy has pronounced effects on subsequent forming processes. Numerous attempts have been made in recent years to derive constitutive relations that describe the initial and induced anisotropy of the material, which are particularly useful when finite element simulations of sheet metal forming are to be performed. Generally speaking, three types of approaches can be used to characterize the anisotropy of sheet metals. The most common approach is to represent the anisotropy of the material by a phenomenological yield criterion and to determine the associated anisotropy coefficients by means of simple mechanical tests. * Corresponding author. Present address: Department of Materials Processing, School of Engineering, Nagoya University, Nagoya 4648603 Japan. Tel.: + 81 52 7893255; fax: + 81 52 7893574; e-mail: [email protected]

Several anisotropic yield criteria of this type have been employed for FEM simulations; for example, Hill’s quadratic criterion [1,2] and Gotoh’s fourth order criterion [3–5]. Although other yield criteria [6–8] have also been proposed, most of the latter have not been tested in elasto-plastic FEM simulations. One problem with these criteria is that they are unable to describe some rather common behaviors in steels and aluminum alloys [9]. The second approach is to combine a polycrystal plasticity model (such as the Taylor model) with a finite element code [10–12]. This method can take the initial crystallographic texture as well as texture evolution during deformation explicitly into account. The polycrystalline aggregate is modeled as a collection of a few hundred grains, with the assumption that the strain-rate is uniform in each aggregate. However, these simulations are rather lengthy and are generally conducted on supercomputers. Recently, Becker et al. [13], Qiu et al. [14], and Anand, and Balasubramanian [15,16] have all shown that the results of finite element simulations using Taylor-type models are not completely satisfactory, essentially because the strain-rate homogeneity requirement is too stringent. Furthermore, the required

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J. Hu et al. / Materials Science and Engineering A256 (1998) 51–59

calculation times are so long that it is difficult to analyze in detail the influences of all the process and material parameters involved in realistic forming processes. The third approach combines some of the advantages of the first and second methods described above and involves fitting the constants of a simple analytical function to data produced by plasticity calculations, rather than by experimentation. The plasticity calculations and the regression fitting exercise can in turn be applied either to: (1) the yield surface or; (2) the plastic work rate. In the former case, the analytical function is fitted to the yield surface in the five components of stress. This was first proposed by Barlat and Lian [17], extended to include all six stress components by Barlat et al. [18], and then applied to finite element simulations of sheet metal forming by Chung and Shah [19] and Hayashida et al. [20]. The latter case involves a function of the strain-rate rather than the stress components, and the analytical function is fitted to the inverse potential (not the yield surface). Van Houtte et al. [21,22], Barlat et al. [23] and Arminjon and Bacroix [9,24] have all proposed functions of the strain-rate that provide the components of the stresses by differentiation with respect to strain-rate. The fourth order strain-rate potential of Arminjon et al. [9] and Arminjon and Bacroix [24] has been incorporated directly into FEM simulations by Bacroix and Gilormini [25]. An advantage of this technique is that regression analysis of the inverse potential function is easier than that of the yield surface and can be introduced directly into FE codes [25]. Only two experimental tests, initial measurement of the texture and a uniaxial tension test, are needed to specify the anisotropic strain-rate potential. Recently, Zhou et al. [26] extended the two-dimensional user material (UMAT) subroutine developed for the commercial finite element code ABAQUS into one suitable for three-dimensional FE simulations with brick elements, although they were unable to simulate the complete drawing of a cup because of numerical divergences in their calculations. The purpose of this paper is to overcome some insufficiencies of the previous work [25,26]. First, with regard to the treatment of Bacroix and Gilormini [25], their frictionless and bottomless simulations are developed into the realistic deep drawing of a textured aluminum sheet. Then, on the basis of these improvements and the fact that a completely drawn cup can be formed, the influence of several important parameters on earing behavior during deep drawing was investigated and the thickness variations that develop were determined. In this way it is shown that parameters such as the coefficient of friction and the blank

holder force have significant effects on the plastic anisotropy that is induced during deep drawing.

2. Methodology

2.1. Fourth order plastic strain-rate potential The fourth-order strain-rate potential used in the present work has the following form [9,22,25]: 22

c(E: P)= % aK K=1

XK (E: P)

E: P 3

(1)

where [E: P]T = [o; P11,o; P22,o; P12,o; P13,o; P23], X1 = (o; P11)4

X2 = (o; P22)4

X5 = (o; P12)4

X6=(o; P11)3o; P22

X8=(o; P11)2(o; P11)2

X3 = (o; P23)4

X4=(o; P13)4

X7=(o; P22)3o; P11

X9=(o; P11)2(o; P23)2

X10=(o; P11)2(o; P13)2

X11=(o; P11)2(o; P12)2

X12=(o; P22)2(o; P23)2

X13=(o; P22)2(o; P13)2

X14=(o; P22)2(o; P12)2

X15=(o; P23)2(o; P13)2

X16=(o; P23)2(o; P12)2

X17=(o; P13)2(o; P12)2

X18=o; P11o; P22(o; P23)2

X19=o; P11o; P22(o; P13)2

X20=o; P11o; P22(o; P12)2 X21=o; P11o; P23o; P13o; P12

X22=o; P22o; P23o; P13o; P12

and

E: P 1

= 2{(o; P11)2 + (o; P22)2 + o; P11o; P22 + (o; P12)2 + (o; P13)2 +(o; P23)2}2 (2) Therefore, five independent components of the plastic strain-rate tensor o; P, (o; P11o; P22o; P12o; P23o; P13), exist on the basis of the incompressibility of plastic deformation. Here, the strain-rate potential is a homogeneous function of degree one with respect to positive multipliers and orthotropic symmetry is assumed, so that the potential is only suitable for textures with this type of symmetry. Nevertheless, for the plane stress state that applies when shell elements are employed, the potential in Eq. (1) is composed of only nine non-zero XK terms, as follows: X1=(o; P11)4

X2 = (o; P22)4

X5=(o; P12)4

X6=(o; P11)3o; P22

X3 = 0

X7 = (o; P22)3o; P11

X8 = (o; P11)2(o; P22)2

X9 = 0

X10 = 0

X11=(o; P11)2(o; P12)2

X12=0

X13 = 0

X15=0

X16 = 0

P 2 22

P 2 12

X14=(o; ) (o; ) X17 = 0

X18 = 0

X20=o; o; (o; ) P P 11 22

P 2 12

X19 = 0 X21=0

X4 = 0

X22 = 0

J. Hu et al. / Materials Science and Engineering A256 (1998) 51–59

and Eq. (2) can be substituted by 1

E: P = 2{(o; P11)2 +(o; P22)2 +o; P11o; P22 +(o; P12)2}2

(3)

The aK coefficients in Eq. (1) characterize the plastic anisotropy that results from a particular crystallographic texture. They are calculated from ‘adjustment’ coefficients (b mn l )K and the series expansion coefficients C mn of the orientation distribution function of the texl ture [27]: mn aK =% % %C mn i (b i )K.

(4)

i m n

The (b mn i )K coefficients do not depend on the texture, so they can be calculated in advance using the following least-squares procedure [9]: : P) −(b mn : P)]2 =min. %% % % [M mn i (E i )KXK (E

(5)

o; P i m n

: P) is obtained by integrating the product Here M mn i (E of the Taylor factor M(g,E: P) and the generalized har: P) over all of orientation space monic function Gmn i (E (g): : P)= M mn i (E

&

M(g,E: P)Gmn i (g)dg.

(6)

g

Thus the (b mn i )K coefficients only depend on the Taylor factor and the XK (E: P) terms that define the form of the strain-rate potential. The full constraint model [28] is used here. Since the Taylor factors are affected by the grain interaction conditions, the fitting procedure is based on a crystallographic description of the plastic behavior. It should be stressed that, once the XK (E: P) functions have been chosen, the fitting for (b mn i )K is only performed once if texture evolution is not taken into account. Thus the strain rate potential can be readily fitted to initial textures possessing orthorhombic symmetry.

2.2. Strain-rate and stress tensor The total strain-rate is first split into an elastic and a plastic part: E: =E: e + E: P

(7)

53

positive multipliers, the stress tensor can be obtained from the potential with the aid of the normality rule. For plastically incompressible materials, the deviatoric stress components, Sij, can be derived from: S=tc



n

22 (c(E: P) XK (E: P) = tc % aK( /(E: P P (E:

E: P 3 K=1

where S= (S1, S2, S3)= (S11 − S33, S22 − S33, 2S12) for the shell elements, and tc is the reference stress associated with the corresponding yield function f(S)=tc, which is also a homogeneous function of degree one with respect to positive multipliers. In the full constraint theory of crystal plasticity, the reference stress tc is actually the resolved shear stress for slip. It reflects the extent of strain hardening that occurs during deformation. The value of tc can be determined from the flow stress measured in a tensile test performed along the X1 axis or the rolling direction [25]: tc = tc (g)

hc =

(tc (g

(11)

For a given total strain increment Do, defined as the integral of E: in the rotating frame, the implicit equations relating to the strain increment and deviatoric stress were integrated numerically to obtain the corresponding stress components using the backward Euler method. When the unknowns Do P11, Do P22, and Do P12 were solved, the Newton–Raphson method was employed. ABAQUS uses the displacement-based finite element method. As required by the ABAQUS code, the UMAT subroutine is composed of three parts; these deal with: (1) stress and hardness state variable updating; (2) the elastic-to-plastic transition; and (3) calculation of the consistent tangent modulus known as the Jacobian matrix in ABAQUS. The detailed procedures for the numerical calculation are described in Ref. [25] for a state of plane stress and in Ref. [26] for 3-dimensional calculations.

3. FEM simulations of deep drawing

s; =2mde6(E: e)+ lTr(E: e)I

3.1. Materials

where s; is the conventional Jaumann rate of the Cauchy stress in a global, fixed frame, m and l are the two elastic Lame´ constants (m is the shear modulus), and I is an identity tensor. For the plastic part, as the strain-rate potential in Eq. (1) is homogeneous of degree one with respect to

(10)

The corresponding hardening rate hc can then be calculated from the tensile curve using

For the elastic part, the rate form of Hooke’s isotropic law is used: (8)

(9)

An aluminum sheet with a thickness of t=1.0mm was employed to simulate deep drawing. The aluminum sheet was made by normal cold rolling (NCR) to 90% reduction and annealing at 350°C for 1 h. The manufacturing process and measurement of the initial texture

J. Hu et al. / Materials Science and Engineering A256 (1998) 51–59

54

Fig. 1. ODF for the present 90% cold-rolled and annealed (at 350°C) aluminum sheet.

Fig. 2. Yield loci in strain-rate space obtained from the fourth-order plastic strain-rate potential.

were described in detail elsewhere [29]. The series expansion coefficients of the orientation distribution function (ODF) were calculated by using the McGill ‘texture menu’ [30]. The ODF for the 90% cold-rolled and annealed sheet is illustrated in Fig. 1. As shown in this figure, the sheet contains both relatively strong recrystallization and retained rolling texture components. The former consists of the cube {001}Ž100 and the latter the Cu {112}Ž111, S3 {123}Ž634 and Bs {110}Ž112 ideal orientations. This mixture is typical of the cold-rolled and recrystallized textures that appear in annealed aluminum sheets. The anisotropy of the sheets was expressed by the fourth-order strain-rate potential of Eq. (1). With the aid of the relations given in Eq. (4), the ak coefficients of the potential were determined from the ‘adjustment’ (b mn i )K coefficients calculated using the full constraint theory and the C mn i coefficients of the measured texture data. Table 1 summarizes the values of the aK coefficients of the potentials for the present textured material. In order to analyze the influence of the shear strainrate component on the in-plane plastic anisotropy, the yield loci obtained from the fourth-order plastic strainrate potential were calculated. The results are shown in Fig. 2 plotted in strain-rate space. As indicated in Fig.

2, these loci exhibit significant changes in shape when the shear strain-rate component o; 12 is increased. Even though the yield surfaces do not rotate, their ellipticity (the ratio of major axis to minor axis) is reduced with increasing shear strain-rate. Thus an increase in shear strain-rate leads to a reduction in the in-plane anisotropy of the sheet. Nevertheless, when the shear strain rate is small (B o; 12 5 0.1), the effect is not very significant.

3.2. Deep drawing simulations The deep drawing simulations were carried out using the ABAQUS code in conjunction with the specially developed UMAT subroutine. The following power-law relation linking the flow stress in shear (t) to the total shear strain (g) on all the slip systems was employed to describe the hardening behavior of the sheet:



t= t0



h0g +1 t0n

n

(12)

Here t0, h0, and n are the critical resolved shear stress at yielding, initial hardening rate and strain hardening index, respectively. From the stress-strain curves measured in the tensile tests performed along the RD of the

Table 1 Values of the aK coefficients of the plastic strain-rate potential for the present aluminum sheet K

aK

K

aK

K

aK

K

aK

K

aK

1 2 3 4 5

9.3223 9.2284 10.5440 10.5084 10.4676

6 7 8 9 10

18.7522 18.2132 27.9077 19.5291 19.1121

11 12 13 14 15

18.7426 19.3361 18.2758 18.4168 18.5113

16 12 18 19 20

18.6509 18.6210 20.1776 18.3563 18.0528

21 22

1.3397 0.2647

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For the section on the effect of element type, the C3D8R type of element was also employed, but only a single layer of thickness was taken into account. C3D8R in ABAQUS terminology refers to 8-node linear brick elements with reduced integration and hourglass control. The initial mesh and the total number of elements for the blank were the same as in the S4R case, but there were twice as many nodes as in the former case. The Coulomb friction coefficients between the blank and the tools (punch, blank holder and die) were prescribed as 0.01–0.25. Rigid surface elements were selected for the interfaces between the blank and the tools. Forces of 650–2700 N were adopted for the blank holder ‘pressure’. A punch stroke of 35 mm was considered to lead to the formation of a complete cup.

3.3. Deep drawing tests

Fig. 3. Geometry of the tools used in the deep drawing simulations. (The radii and all dimensions are in mm.).

sheet, the initial hardening parameters were determined as t0 =2.5 MPa, h0 =2000 MPa, and n = 20. The geometry of the tools used in the simulations is indicated in Fig. 3. Because of orthotropic symmetry, only a quarter of the blank was employed for the simulations. The S4R type of ABAQUS element was adopted for the initial blank mesh. S4R in ABAQUS terminology means a 4-node doubly curved thin or thick shell element with reduced integration, hourglass control and finite membrane strains. The total numbers of nodes and elements were 133 and 112, respectively, for a quarter of the blank. The initial mesh is shown in Fig. 4. The 1, 2, and 3 axes were aligned with the rolling, transverse and normal directions of the sheet, respectively.

Fig. 4. The initial mesh of a quarter of the blank.

Circular blanks with a thickness of 1.00 mm and a diameter of 78.0 mm were used for the deep drawing tests, together with a flat-bottomed punch of diameter 40.0 mm and a die of diameter 42.5 mm. The punch and die profile radii were 8.0 mm and 6.0 mm, respectively. The same tool and sheet dimensions were employed in the simulations. After deep drawing, the ear heights of the drawn cups were measured. The earing percentages were then calculated using the following equation: e(i)=

Dhi hi − hmin = hmin hmin

(13)

4. Results

4.1. Predicted thickness contours during deep drawing The thickness variations predicted by the simulations along the circumferential and radial directions are illustrated in Fig. 5. In this case, shell elements were employed, together with a friction coefficient of m=0.10 and a blank holder force (BHF) of 650N. As shown in Fig. 5, there is a continuous decrease in thickness along the bottom of the cup and near the punch radius from the beginning of deep drawing to the end of the stroke. However, there is an equivalent increase in thickness in the outer half of the blank in the initial stages until the punch stroke reaches 50%, see Fig. 5 (a). As drawing proceeds, the region containing the increase in thickness begins to move towards the outer rim, until, when the complete cup has formed, the thickness increase is restricted to the edge of the cup. Furthermore, as shown in Fig. 5 (d), the sheet displays four-ears which are located at 0 and 90° with respect to RD. These predictions are in general good agreement with the experimental results [31]. Further details regarding the

56

J. Hu et al. / Materials Science and Engineering A256 (1998) 51–59

Fig. 5. Contours of thickness during deep drawing (m =0.05 and BHF =650N). (a) Punch stroke = 50% (b) Punch stroke = 65% (c) Punch stroke =85% (d) Punch stroke= 100%

earing profiles are provided in Fig. 9, which is presented in Section 4.4. The variations in maximum thickness during deep drawing are summarized in Fig. 6. From the figure, it can be seen that the maxima increase more rapidly in the final stages of drawing than in the beginning; whereas the minimum thickness decreases more significantly in the beginning stages of drawing and remains almost unchanged in the final stages.

4.2. Effect of friction on the earing profile The influence of friction between the tools and the sheet on the earing profiles of a drawn cup is presented in Fig. 7. It is evident that low and high friction coefficients have different effects on the ear profiles. The earing percentages increase when the coefficient is increased from m =0.01 to m =0.15, see Fig. 7 (a). However, the earing percentages decrease with an increase in

coefficient from m= 0.15 to m= 0.25, see Fig. 7 (b). This phenomenon can be attributed to the increases in punch load and shear stress that accompany increasing friction. In the case of low coefficients, more friction increases the punch load and promotes the development of plastic anisotropy, together with more sheet thinning and thickening. In the case of high coefficients, more friction leads to an increase in the shear strain-rate o; 12 which, as indicated in Fig. 2, reduces the anisotropy. In the latter case, the increase in shear strain-rate is the dominant factor; whereas in the former case, or when the coefficient is less than 0.15, the increase in punch load is more important.

4.3. Effect of blank holder force on the earing profiles Drawing simulations were performed employing various blank holder forces (together with a friction coeffi-

J. Hu et al. / Materials Science and Engineering A256 (1998) 51–59

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Fig. 6. Evolution of maximum and minimum thickness during deep drawing.

cient of m = 0.05. The influence of blank holder force is shown in Fig. 8. It is evident that greater blank holder forces lead to more accentuated earing profiles. This is because higher blank holder forces are associated with higher punch loads; this in turn is similar to the case of low friction coefficients discussed above. Even though complete cups cannot be drawn under conditions of high friction because of the occurrence of fracture, on the basis of the results concerning the influence of friction, it can be concluded that increasing the blank holder force under conditions of high friction will be not as significant as in the case of low friction coefficients.

4.4. Effect of element type on the earing profiles The earing profiles predicted using for shell and brick elements are compared with experimental data in Fig. 9. It can be seen that the shell elements are more sensitive to plastic anisotropy than the brick elements. This may be because two shear stresses or strain-rate components along the thickness directions are neglected in the shell elements. By comparing the simulations and the experimental data, it can be seen that the shell element predictions overestimate the experimental trends, whereas the brick element ones underestimate the extent of earing. The present results also suggest that it will be necessary to use several layers of elements through thickness and to take texture evolution during deep drawing into account if more accurate predictions of earing behavior are to be obtained.

5. Discussion It was demonstrated above that the fourth order strain-rate potential proposed by Arminjon and Bacroix [9,24] can be employed to simulate the deep drawing of textured aluminum sheets. Such calculations can include evaluating the effects of a series of forming parameters (such as friction and blank holder force). Reasonable results were obtained, even when shell elements were used and only the initial texture was taken into consideration. As reported in detail by Hu et al. [32], the adoption of shell elements involves less computation time and describes plastic anisotropy in the rolling plane more sensitively than when brick elements are used. These advantages indicate that shell elements could well be employed in the analysis of realistic deep drawing processes, even when the texture evolution that takes place during drawing is taken into account. Savoie et al. [33] measured the textures formed in cups during drawing and found that they differed considerably from the initial textures. For example, it is apparent that the initial cube (recrystallization) component disappears and only orientations close to the P {011}Ž111 are relatively stable, as long as the strain path is close to plane strain during drawing. These trends in texture evolution could increase or decrease the ear heights. In practice, due to the change in strain path from rolling (plane strain) to deep drawing, the orthotropic strain-rate potentials lose their symmetry. Concurrently, the texture evolves away from its initial ortho-

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J. Hu et al. / Materials Science and Engineering A256 (1998) 51–59

Fig. 7. Effect of friction on earing profiles (BHF = 650N). (a) Low friction coefficients (b) High friction coefficients

rhombic character towards a cylindrical one. As mentioned by Zhou et al. [26], strain-rate potentials of triclinic symmetry have to be employed under these conditions. Moreover, the errors introduced by using orthotropic potentials as first approximations have yet to be evaluated. The present analysis indicates that these errors may not be very large. The influence of through-thickness texture gradients was not taken into consideration in this investigation. This could also have contributed to the discrepancy between the present predictions and the experimental measurements. In order to improve the computational accuracy, it will therefore be necessary to overcome the above-men-

tioned deficiencies. Such improvements will be discussed in detail elsewhere.

Fig. 8. Effect of blank holder force (BHF) on the earing profiles (m = 0.05).

Fig. 9. Effect of element type on the earing profiles (m= 0.05 and BHF= 650N).

6. Conclusions A fourth order strain-rate potential was incorporated into a commercial ABAQUS code and used to simulate the deep drawing of textured aluminum sheets. Reasonable agreement was obtained between the predicted and measured earing profiles pertaining to a cold-rolled and annealed sheet. The following conclusions can be drawn from this work:

J. Hu et al. / Materials Science and Engineering A256 (1998) 51–59

(1) In the early stages of drawing, there is generalized thickening in the blank; in the final stages, this thickening is concentrated near the edge of the cup. By contrast, there is a monotonous trend to thinning along the bottom and near the punch radius. (2) When the coefficient of friction is low, increases in friction promote ear development; conversely, when the friction is high, increases in friction suppress ear development. (3) Increasing the blank holder force enables the extent of earing to increase. However, this trend is less marked when the coefficient of friction is high. (4) Shell element predictions overestimate the extent of earing; whereas brick element predictions underestimate earing.

Acknowledgements The authors are indebted to the Natural Sciences and Engineering Research Council of Canada (NSERC) and to the Kingston Research and Development Center (KRDC) of Alcan International Limited for financial support of the part of this research carried out at McGill University, Montreal, Canada. They are also grateful to Dr Y. Zhou (Suralform Aluminum International Limited, Canada), to Drs J. Savoie and S. MacEwen (KRDC, Alcan International Limited, Canada), to Professor K. Ikeda (Tohoku University, Japan) as well as to Associate Professor N. Yukawa (Nagoya University, Japan) for their helpful discussions and encouragement.

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