Evolution of anisotropy under plane stress

J. Mech. Phys. Solids, Vol. 45, No. 5, pp. 841-851, 1997 0 1997 ElsevierScienceLtd Printed in Great Britain. All rights reserved

Pergamon

PI1 : S8022-50%(%)08085-3

EVOLUTION

OF ANISOTROPY

0022-5096/97$17.00+0.00

UNDER PLANE STRESS

K. H. KIM Department of Mechanical Engineering, Korea University, 5-l Anamdong, Seongbukku, Seoul, Korea

and J. J. YIN Institute for Advanced Engineering, 541, 5-Ga, Namdaemun-no,

Chung-gu, Seoul, Korea

(Received 30 November 1995 ; in revisedform 12 July 1996)

ABSTRACT Yield loci have been measured for cold rolled steel sheets prestrained by two stage loading. During the first loading, the sheets have been stretched by 3 and 6% tensile strain in the rolling direction. The second loading was at angles to the rolling direction with varying amounts of tensile strain. Then a set of tensile test specimens has been prepared from each of the prestrained sheets. From tensile tests, effects of the twostage prestrains on the subsequent yielding has been investigated. Experiments show that initial orthotropic symmetry is maintained and that the orientations of orthotropy axes change continuously throughout the prestraining process. A simple phenomenological rule for the rotation of orthotropy axes is suggested. 0 1997 Elsevier Science Ltd. All rights reserved. Keywords : A. strain hardening, B. anisotropic material, B. constitutive behavior, C. mechanical testing.

1.

INTRODUCTION

Deformation-induced anisotropy of polycrystalline materials is very important from the practical as well as the theoretical point of view. It is due to microstructural developments during finite deformation such as crystallographic reorientation of single crystals. Since the theory of Bishop and Hill (195 l), extensive efforts have been made to understand the anisotropy of polycrystalline aggregates by mechanisms of slip in single crystals (Budiansky and Wu, 1962 ; Hill, 1965 ; Hutchinson, 1970 ; Asaro and Needleman, 1985 ; Prantil et al., 1993). Such studies provide means to understand evolving texture and suggest a method for producing constitutive models related to microstructural details. However, the theoretical developments are not yet advanced enough to yield practical constitutive equations. Apart from the micromechanical approach, phenomenological investigations based upon experiments have been under progress. Among the macroscopic theories based upon experimental observations, the theory proposed by Hill (1950) has been most successful in its simplicity. Later, more general types of yield function have been 841

K. H. KIM and J. J. YIN

842

proposed to convert a wide range of experimental observations (Bassani, 1977 ; Gotoh, 1977 ; Hill, 1979,1990,1993). Macroscopic theories assume orthotropic symmetry, the axes of which remain unchanged during subsequent loading. For situations where the state of anisotropy is altered by subsequent deformation such as in finite simple shear with respect to the X or Y axis of orthotropy, the theories are not applicable. A recent experimental study (Rim, 1992) showed that orthotropic symmetry is maintained during twisting of cold drawn tubes and the orientations or orthotropy axes are altered. In cases where orthotropic symmetry is maintained during deformation, characterization of the orthotropy axes will be central to the development of macroscopic constitutive equations. Hill (1950) assumed conservation of orthotropic symmetry in the study of length change of tubes during twisting but this assumption has not been substantiated experimentally. Most of the previous experiments (Bailey et al., 1971; Michino et al., 1974; Stout et al., 1983; Eisenberg et al., 1984; Helling et al., 1987) have been designed to study yield loci in stress spaces without reference to possible orthotropic symmetry. In this study experiments are performed to investigate conservation of orthotropic symmetry and evolution of anisotropy under a plane stress condition for cold rolled steel sheets.

2. 2.1.

EXPERIMENT

Experimental Scheme

An experimental method for the study of anisotropy of rolled sheet metals is to perform tensile tests at angles to the rolling direction. With additional compression test data in the thickness direction, a plane stress yield surface in a,+,-~, stress space can be constructed with respect to the known orthotropy axes X and Y (Yin, 1992). Variation of uniaxial yield stress with tensile loading axis orientation can be used to establish orthotropic symmetry. The test method used in the current study is an extension of this scheme. Cold rolled sheets of a low carbon steel widely used in the automotive industry have been selected for the tests. This material has mild initial anisotropy. To enhance the degree of anisotropy, full size sheets have been stretched along the rolling direction by 3 and 6% tensile strains. From the gauge section of the strained full size sheets, medium size tensile specimens shown in Fig. 1 are prepared

-

Fig. 1. Schematic illustration of experimental procedure.

R.D.

1st prsstraln 3%,6X

Evolution of anisotropy

843

Table 1. Combinations offirst and second tensile prestrains First prestrain (%)

Angle 9 30 45” 60” 90” 30” 45” 60” 90”

Second prestrain (%) 1

1

1 -

1 1 1 -

2 2 2 2 2 2 Necking 2

5 5 5 5 5 Necking Necking 5

10 10 10 10 10 Necking Necking 10

15

15

at 30, 45, 60 and 90”. The medium size specimens are then prestrained along their length by 1, 2, 5, 10 and 15% tensile strains. Combinations of the first and second prestrains are shown in Table 1. Final miniature specimens are prepared from each of the gauge sections of the medium size specimens at every 10” to the specimen axis. For each medium size specimen, 18 tensile test data and a set of compression test data are obtained. Figure 1 shows the configurations of full, medium and miniature tensile specimens. To supplement the tensile test data compression tests have been performed. Circular disk type specimens were prepared from the same gauge section of each medium size specimen. From the compression test data, balanced biaxial yield stress has been determined neglecting the effects of pressure on yielding. In tensile and compression tests, yield points have been measured at 0.2% offset strain. Orthotropic symmetry is determined based upon yield stress variation with respect to the tensile axis. In case symmetry is observed, new orientations of the orthotropy axes X and Y are determined for each level of the first and the second prestrains. Figure 2 shows

Fig. 2. Definition of various angles: $, second prestrain angle with respect to the rolling direction; 8, second prestrain angle w.r.t. the X-axis; q$, tensile angle w.r.t. the second prestrain axis; a, tensile angle w.r.t. the X-axis; and 0, X-axis angle w.r.t. the rolling direction.

K. H. KIM and J. J. YIN

844

.

: as received

V

: 3% 1st prestrain : 6% 1st prestrain

l

10

I

0

I

30

I

I

90

60 a

1

120

I

150

I

160

(deg.)

CJx (kgflmd) Fig. 3. Uniaxial yield stress distribution and plane stress yield loci of cold rolled steel sheets for the asreceived state and for the states after the first prestrain of 3 and 6% elongation in the rolling direction. Angle a is as defined in Fig. 2 and 0 = 0 since the X-axis is in the rolling direction.

various angles defined in this study. Once the orientations of the orthotropy axes X and Y are determined, plane stress yield surfaces in a,-q,-t, space can be constructed from the tensile and compressive test data. 2.2.

Experimental Results

Figure 3(a) shows uniaxial yield stress for the as-received state and the states after 3 and 6% tensile elongation along the rolling direction. Solid lines represent the

Evolution of anisotropy

845

quadratic yield function (Hill, 1950) fit to each set of test data. Since the quadratic yield function represents the test data quite well, the plane yield stress loci can be calculated from the curve fit for each state. In Fig. 3(b) the plane stress yield loci are shown in the 0,~~ plane for t,, = 0. The stress components g,, cy and z,~ are defined with respect to the orthotropy axes X and Y along the rolling direction and the transverse direction, respectively. Figure 4 shows the evolution of anisotropy with the progress of second prestrain at angles to the rolling direction for the first prestrain of 3%. In all the three cases of Fig. 4 orthotropic symmetry is maintained throughout the second prestrains. When the axis of the second tensile prestrain is at 30” to the rolling direction as shown in Fig. 4(a) the X-axis tends to rotate towards the loading axis with the progress of the tensile prestrain. When the axis of the second tensile prestrain is at 45” or 60” to the rolling direction the Y-axis rotates towards the loading axis, as shown in Fig. 4(c) and (e). Similar tests have been performed for the first prestrain of 6%. The material was susceptible to necking at strains of 2% or more when the second prestrain axis was at 45” or 60” to the rolling direction. Test data obtained prior to necking are essentially similar to those shown in Fig. 4.

3.

DISCUSSION

Hill (1950) assumed that one of the orthotropy axes is formed in the direction of maximum principal stretch during twisting of a thin walled tube. Kim (1992) observed that during the twisting of cold drawn tubes the orthotropic symmetry is maintained and the orthotropy axes are rotated in the twisting direction. The amount of rotation was coincident with Hill’s assumption in this case. Returning to the current experiment, the X-axis is rotated towards the loading axis when the second tensile prestrain is at 30” to the rolling direction and this seems to be in coincidence with Hill’s assumption, According to Hill’s assumption, when the second prestrain axis is at 45” or 60” to the rolling direction the X-axis has to rotate towards the tensile axis since the X-axis is in the direction of major principal stretch. However, experiment shows that the Y-axis is rotated towards the tensile axis in these cases. Therefore Hill’s assumption is to be confined to monotonic simple shearing (Hill, 1993). To the best of the authors’ knowledge, there seems to be no theory which can explain the orientation changes of orthotropic axes as shown in Fig. 5. Dafalias and Rashid (1989) have studied the rotation of orthotropy axes based upon plastic spin theory. Plastic spin theory has been rigorously developed for the special case of planar double slip of a single crystal without latent hardening (Dafalias, 1993). However, for a typical polycrystal with arbitrary ODF, a sensible definition of plastic spin in an orthotropic representative macro-element seems to be still lacking. In the absence of an appropriate theory, a heuristic approach guided primarily by the experimental observations might be fruitful. With this in mind, we suggest that the rate of orientation change of the orthotropic axes is proportional to the shear strain rate with respect to the principal directions of stress. More specifically, da =(l +C)dEIz

(1)

where 1 and 2 represent the tensile loading direction and the transverse direction for

K. H. KIM and J. J. YIN

846

ly=30”

204

I 0

I 30

I 60 $t

(b)

=r

1 90

I 120

I 150

1 180

(deg.)

yl=30”

10

0

Fig. 4. Uniaxial yield stress distributions and plane stress yield loci for the first prestrain of 3% and th second prestrain of 0, 1,2,5 and 10% applied at e = 30”, 45” and 60”. Solid lines represent Hill’s quadrati yield function fit to each set of test data. Plane stress yield loci have been calculated from the curve f results for I,~ = 0. (a) Uniaxial yield stress distribution for the second prestrain applied at (CI= 30”; (1 plane stress yield loci for the second prestrain applied at $ = 30”; (c) uniaxial yield stress distribution fc the second prestrain applied at $ = 45”; (d) plane stress yield loci for the second prestrain applied i JI = 45”; (e) uniaxial yield stress distribution for the second prestrain applied at JI = 60”; ( f) plane strer yield loci for the second prestrain applied at $ = 60”.

Evolution of anisotropy

847 v =45”

2cc

I 0

I 30

I 120

I 90

60

I 150

I 160

01 (deg.)

I

20 ox

30

40

50

( kgf/mm2)

Fig. 4. Continued.

the second prestraining, fi is the angle between the X-axis direction and the tensile loading direction, as shown in Fig. 2, and C is a constant, the value of which is dependent upon the state of hardening. In (1) it is assumed that simple shear is induced by shear strain dclz within the gauge section of the tensile specimen. Thus the rotation of orthotropy axes can be decomposed into rigid rotation by simple shear and strain induced rotation represented by dqZ and CdqZ, respectively. In actual tensile tests, however, it is difficult to achieve simple shear in the gauge section completely free from the end constraints due to the specimen geometry. Grips have

848

K. H. KIM and .I. J. YIN (e)

4

w=60”

5% 2%

1% 0%

2d

I 0

I 30

I 60

I 90

I 120

I 150

I 180

+t (deg.)

(f) 50 -

y/=60'

0

10-

0

I 10

////I 20 Cx

/ 30

40

I 50

(kgf/mm*)

Fig. 4. Continued.

been designed to allow some degree of rotational freedom and thus to minimize constraint against simple shear during the second prestraining. From Hill’s quadratic yield function and normality flow rule it can be shown that

d&,2 de,, - -

{@+W cos* /3- (f+ 2h) sin* p + (cos* j-sin*

/?)} cos p sin /I g cos* j?+fsin* /I + h - @ +f+ 4h - 2) cos* #Isin’ 1

(2)

(4

50 40

F

_.

,

0

5

~:z’Y~ 10

2nd prestrain (b)

50

-

40

-

30

-

$

20

-

s 0,

IO-

15

lq =

45"

0

experiment

-

eq. (1)

&lI (%)

ye =

60”

0

-

5

experiment eq. (1)

I 15

10

2nd prestrain

20

&ll (%)

2nd prestrain

-50 ’ 0

,

1 20

z+zll (%)

Fig. 5. Rotation of X-axis vs the second tensile prestrain applied at 1(1= 30”, 45” and 60” after the first prestrain of 3%. Experimental values for the angle 0 have been determined from the fit of Hill’s quadratic yield function to each set of the test data shown in Fig. 4. Solid lines represent predictions from (1) for f= 0.3613,9 = 0.3535 and h = 0.4957.

850

K. H. KIM and J. J. YIN

wheref= F/N, g = G/N, and h = H/N (see Hill, 1950). In Fig. 4 a certain variation of yield surface shape is observed during the second tensile prestraining. However for the sake of simplicity we ignore such variation and assume that yield surface shape remains similar. Under this assumption, the values of anisotropy parametersf, g and h can be taken to be constant. From Fig. 2 it is obvious that fi + 8 = $. For a given direction of the second tensile prestrain axis, the angle rc/is constant and thus we have de = - db. Figure 5 shows the predictions from (2) for selected values off, g, h (taken from the as-received state) and some arbitrarily chosen values of C compared with the experimental data.

4.

CONCLUSION

When tensile loading is applied to a cold rolled steel sheet at angles to the rolling direction, the orientations of the orthotropy axes are changed within a few percent of the tensile strain. With the progress of the tensile strain the orthotropy axes are aligned along the tensile loading direction and the transverse direction. Observed rotation of the orthotropy axes cannot be explained within the framework of any existing theory. A simple phenomenological model is suggested in which the rate of rotation of the orthotropy axes is proportional to the shear strain rate with respect to the loading axes. The phenomenological model gives a reasonable agreement with the observations. It is to be investigated further whether conservation of orthotropic symmetry and the rotation of orthotropy axes during loading non-coaxial with the orthotropy axes is a phenomenon generally observed under plane stress conditions.

ACKNOWLEDGEMENTS The authors are grateful to Professor R. Hill for his valuable advice and comments and to Professor D. W. Kim at Seoul National University for his encouragement during the progress of this work. Support from Dr Youngseok Kim of Pohang Steel Company with the test material and from Mr Hyosang Cho of Daewoo Heavy Industry Company with the specimen preparation are greatly appreciated.

REFERENCES Asaro, R. J. and Needleman, A. (1985) Texture development and strain hardening in rate dependent polycrystals. Acta. Metall. 33,923-953. Bailey, J. A., Haas, S. L. and Nawab, K. C. (1971) Anisotropy in plastic torsion. J. Basic Engng, Trans. ASME, Paper No. 71-Met-Y. Bassani, J. L. (1977) Yield characterization of metals with transversely isotropic plastic properties. Znt. J. Mech. Sci. 19,651-660. Bishop, J. F. W. and Hill, R. (1951) A theoretical derivation of the plastic properties of a polycrystalline face-centered metal. Phil. Mag. 42, 1298-1307. Budiansky, B. and Wu, T. T. (1962) Theoretical prediction of plastic strains of polycrystals. Proc. 4th Congr. Appl. Mech. 2,1175-l 185. Dafalias, Y. F. and Rashid, M. M. (1989) The effect of plastic spin effect on anisotropic material behaviour. Znt. J. Plast. 5, 227-246.

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Dafalias, Y. F. (1993) Planar double-slip micromechanical model for polycrystal plasticity. J. Engng Mech. 119, 1260-1284. Eisenberg, M. A. and Yen, C.-F. (1984) The anisotropic deformation of yield surfaces. J. Engng Mat. Tech., Trans. ASME 106,355360. Gotoh, M. (1977) A theory of plastic anisotropy based on a yield function of fourth order (plane stress state)-I. Znt. J. Mech. Sci. 19, 513-520. Helling, D. E., Miller, A. K. and Stout, M. G. (1987) An experimental investigation of the yield loci of 1100-O aluminum, 70 : 30 brass, and an overaged 2024 aluminum alloy after various prestrains. J. Engng Mat. Tech., Trans. ASME 108, 313-320. Hill, R. (1950) The Mathematical Theory of Plasticity. Oxford University Press. Hill, R. (1965) Continuum micro-mechanics of elastoplastic polycrystals. J. Mech. Phys. Solids 13,89-101. Hill, R. (1979) Theoretical plasticity of textured aggregates. Math. Proc. Camb. Phil. Sot. 85, 179-191. Hill, R. (1990) Constitutive modelling of orthotropic plasticity in sheet metals. J. Mech. Phys. Solids 38,4054 17. Hill, R. (1993) A user-friendly theory of orthotropic plasticity in sheet metals. Znt. J. Mech. Sci. 35, 19-25. Hill, R. (1993) Private communication. Hill, R. (1993) Private communication. Hutchinson, J. W. (1970) Elastic-plastic behavior of polycrystalline metals and composites. Proc. R. Sot. Lond. A319,247-272. Kim, K. H. (1992) Evolution of anisotropy during twisting of cold drawn tubes. J. Mech. Phys. Solids 40, 127-l 39. Michino Jr, M. J. and Findley, W. N. (1974) Subsequent yield surfaces for annealed mild steei under dead-weight loading : aging, normality, convexity, corners, Baushinger, and cross effects. J. Engng Mat. Tech., Trans. ASME 96, 51H4. Prantil, V. C., Jenkins, J. T. and Dawson, P. R. (1993) An analysis of texture and plastic spin for planar polycrystals. J. Mech. Phys. Solids 41, 1357-1382. Stout, M. G., Hecker, S. S. and Bourcier, R. (1983) An evaluation of anisotropic effective stress-strain criteria for the biaxial yield and flow of 2024 aluminum tubes. J. Engng Mat. Tech., Trans. ASME 105,242-249. Yin, J. J. (1992) A study on the strain hardening characteristics of anisotropic sheet metals. Ph.D. thesis, Seoul National University, Korea.