New experiments for determining yield loci of sheet metal

Journal of

Materials Processing Technology

Journal of Materials ProcessingTechnology60 (I 996) 643-648

ELSEVIER

N e w e x p e r i m e n t s for d e t e r m i n i n g yield loci o f sheet metal W. Mailer a), K. POhiandt b)

=JWieland-Werke AG, Metallwerke, D-89070 Ulm, Germany b)lnstitut far Umformtechnik, Universitat Stuttgart, D- 70174 Stuttgart, Germany

Abstract

Two new methods for determining yield loci of sheet metal are described. In the cross tensile test, by varying longitudinal and transverse stress, the yield locus can be determined in the range of biaxial tensile stress. Starting from a specimen described by KreifSig the test piece geometry was varied by finite element calculations and optimized for obtaining a large zone of homogeneous deformation and high strain before instability occurs. The optimized geometry was verified through photoelastic tests. In the inclined tensile test with suppressed lateral contraction, by varying the angle between the clamps and the direction of movement, states of stress ranging from pure shear to biaxial tensile stress can be obtained. Cross tensile tests were carried out in which the temperature was measured as a function of strain. After a small decrease due to thermoelastic cooling, the beginning of plastic deformation is indicated by the dissipation of deformation heat. By applying this principle, yield loci were determined for various metals. The results were compared with those obtained from uniaxial tensile tests assuming yield criteria by Tresca, v. Mises, Hosford-Backofen and Hill.

Keywords."

yieM loci, cross tensile test, inclined tensile test, thermoelasO'c cooling

1 Introduction

For a computer simulation of technical sheet forming processes, a quantitative description of plastic anisotropy is required. This is given by the yield locus. The yield locus describes all states of stress which enable plastic deformation/1/. R is clearly determined by a curve which applies for biaxial stress, see Fig. 1. In the following text this will be referred to as the yield locus. Various mathematical approximations of the yield locus have been given m the literature. For isotropic materials, according to v. Mises yield occurs if the condition is fulfilled

o,~ =2.[(0, - o,,)~ +(a,, - o,,,)~ + (~,,, -o,) ~]

(l)

Here R is the normal anisotropy. For a more precise description of material behaviour, Hill has proposed different kinds of yield criteria/3/.

# R

O~

(~2 .- R = I

-

-

~

>I

~f

For taking into account plastic anisotropy, the v. Mises criterion may be replaced by the so-called quadratic Hill criterion/2/. If there is no effect of planar anisotropy the quadratic Hill criterion is reduced to the Hosford-Backofen equation:

o,2 = o,~ - 1 2- ~. R- - o , .o,, +o,,2 09244)136/96/$15.00 © 1996 Elsevier ScienceS.A. All rightsreserved PI109244) 136 (96) 023994)

(2)

Fig. 1' Yield loci for various values of normal anisotropy according to Eq. (2).

644

w. Mailer, K POhlandt /Journal of Materials Processing Technology 60 (1996) 643-648

Some special experiments have been applied for the direct determination of yield loci/1/. However, experiments which are easy to carry out give only poor information. Up to now, an accurate determination of yield loci requires an enormous amount of work. This applies to tensile or compressive tests on tubes with superimposed internal pressure or the calculation of yield loci from the crystallographic texture (ODF). Therefore two new methods are presented for determining yield loci by

Fig. 3 shows the distribution of equivalent strain in the specimen at the instant when instability occurs in the vicinity of the notches. The computer simulation was carried out for two materials having different flow curves. Since there is an effect of the material on the maximum strain in the central zone of the specimen a rigorous optimation results into different geometries for different materials.

less experimental work.

2 Fundamentals 2.1 Cross Tensile Test A cross specimen is used for applying a longitudinal and a transverse force simultaneously. By varying the ratio of these forces

12

any point of the yield locus in the region of biaxial stress can be determined.

10 08

2.1.1 Optimum Specimen Geometry

06 KreifAig /4/ described a specimen geometry which had been

04

optimized by stress optical methods for obtaining a large zone of homogeneous deformation. This is required for determining the yield point from temperature measurements (this method has

02

also been applied in this work and is described below). Starting from this geometry, by means of t'mite element calculations a further optimation was carried out for obtaining a high strain before instability occurs. This enables a determination of yield loci of specimens after prestraining up to relatively high strains. The optimum specimen geometry was obtained by varying the parameters Rl; R2; L0 (see Fig.2) while the width B = 50 nun was kept constant.

~ii!~i~i!i!~!i~!~i~!~i!i!~iri!:~i!~!i~i~!~i~i~!!~i!i!i

Fig. 3: Distribution of equivalent strain in the optimized cross tensile test specimen.

As a compromise the following geometry was used for the experiments described below:.

clamped area~

:ii!iiiiiiiiiiiiiiiiiiiiiiii!ili,i.

•

Width

B = 50ram

•

Notch radius

R1 = 7,5 mm

•

Flange radius

R2 = 20 mm

•

Length

L0 = 0

For this geometry the stress distribution was studied by stress optical experiments /5/. In Fig. 4 the lines indicate constant differences of main stresses. A small distance of lines indicates a high gradient of stress. Inside a circle of about 35 mm diameter there is an almost homogeneous distribution of stress while there are high stress gradients in the vicinity of the notches. This confimts the results Fig. 2: Optimized cross tensile test specimen.

of finite element calculations.

W. Mi~ller, K Pahlandt /Journal of Materials Processing Technology 60 (1996) 643-648

645

A iw ,

i

"k

t~

q

Fig. 5: Scheme of the inclined tensile test.

2.2 Inclined Tensile Test Fig. 4: Isochromatic lines of cross tensile specimen (equal forces m both directions).

I n / 6 / a n inclined tensile test was described whereby the test piece consisted of a strip having notches on opposite edges at some distance in axial direction. However, this causes a rather inhomogeneous distribution of strain. Therefore a different kind of

2.1.2 Equivalent Width of Specimen

specimen is proposed for obtaining a large area of homogeneous deformation. This is a prerequisite for the determination of yield

As an approximation, the longitudinal and transverse stresses can be calculated by dividing the applied forces through the geometric (nominal) cross section. However, the stress distribu-

point by temperature measurements. The principle of the test is demonstrated by Fig. 5.

tion through the cross section of the specimen is not exactly

A strip of the material to be tested is clamped in such a way that

homogeneous. For obtaining the true stress distribution in the

the edges of the clamps form an angle against the direction of

zone of biaxial stress, in computer simulations the effect of a transverse stress on the equivalent cross section was taken into account.

movement. This angle is kept constant during the experiment. In all calculations below it is assumed that the width of the free

The problem of deeming the effective or equivalent cross section of the specimen was further complicated by the method applied

contraction is only in thickness direction):

for determining the yield point through temperature measure-

b o >> Lo

area of the specimen is large compared to the "length" so that the lateral contraction in y-direction can be neglected (transverse

ments (see section 3). A rigorous definition of the equivalent cross section of the specimen should enable to determine the local stresses in that zone of the specimen for which the average temperature is determined by measurement. - However, all these effects result into a variation of the equivalent cross section of the order of a few percent. Therefore a good approximation of stresses is obtained by dividing the forces through the nominal cross section B * s.

(3)

In spite of this, in the general case there is no plane strain as can be seen from the deformation tensor (see below). By variation of the angle ct different states of stress can be obtained. In particular the case ct = 0 ° results into a conventional tensile test with suppressed lateral contraction. This corresptrnds to point A in Fig.6.

646

W. Mi~ller, K POhlandt / Journal o f Materials Processing Technology 60 (1996) 643-648

Using Hooke's law the stress tensor can be calculated:

t~IIi I

~

:>J B / ", _

'\

'

1

On

\

/

I i'

II

}

1 o~ = 1-~

E . A w , [(1+ iLt), sin 0~+ (1_ !Lt)] 2-k o

(7)

1 0. = 1-~

E - Aw. [(1+ l.t). sino~_ (1_ i.t)] 2Lo

(8)

,,

!

The direction of the axes of the tensors is defmed by the angle [~ between the directions x and I (see Fig. 7):

t

I ~lx - ~ 4

Fig. 6: Range of the yield locus to be determined by inclined tensile tests.

o~ 2

(9)

Fig. 7 illustrates some distinguished values of angle c~. From Eq. (8) it follows that uniaxial stress is obtained for the angle

,

1-1a l+p

= arcsin~

(10)

For c~ = 90 ° the specimen is deformed by pure shear (point B). It should be noted, however, that an angle of 90 ° can only be approximated because some tensile deformation is required for determining the yield point by temperature measurement. This is

1) o~=g0*

2) IX= o~)~=30"(fOrI.t=1/3)

3)

0~=0" ~tx=/r,/4

indicated by point D which corresponds to a small but finite

~ ~

angle ct. By varying ct the entire range of the yield locus between

I'z

;x~

A and D resp. A' and D' can be obtained, including the points C

II

and C' (urtiaxial state of stress). A prerequisite of this is the specimen being cut out of the sheet in such a way that either the rolling or the transverse direction is parallel to one of the main axes (see below). Hence the inclined tensile test combined with the cross tensile test enables a determination of the entire range

Fig. 7: Orientation of specimen and main axes for different values of the angle.

of the yield locus curve between D and D'. This is sufficient information for many technical applications.

This angle is about 30 ° for ~t = 1/3. For this angle from Eqs. (4)

For a brief theoretical treatment of the experiment, consider a

to (6) it follows for the components of the deformation tensor

small movement of the upper clamp by the distance Aw. In the general case of an angle 0 ° <_ct < 90 ° the elastic deformation

e, (~*) = e,~ (~*) = - p .e~((~* ) =

includes both tensile and shear strain. The supression of lateral contraction in direction y causes an additional reduction of thikkness. By some calculation the deformation tensor can be obtained/7/(for the main axes see Fig, 7 below):

AW (sin0c + 1)

(4)

Ell = Aw , (sino~- 1)

(5)

1 E-Aw 1+11 Lo

IX Aw. sinot 1-11 L o

Lo

(11)

(12)

The section of the yield locus which can be determined through inclined tensile tests is determined from Eqs. (7), (8): -1 < o ~ < gt Oi

em =

Aw

The corresponding stress is obtained from Eq. (7):

O,(~*)-

~" = 2"---~o"

p l+p

(13)

(6) As mentioned before the value -1 can only be approximated

W. Mi~ller, K POhlandt I Journal o f Materials Processing Technology 60 (1996) 643-648

since there must be some increase of volume for obtaining thermoelastic cooling.

647

300

% Z

._.q ¢-

.o 200 =e

3 Experiments

"10

Until now only the cross tensile test has been carried out. Results of inclined tensile tests shall be reported later. The experimental setup for cross tensile tests has been described in ref. /5/. The yield point is determined through measurement of the temperature using an izffrared thermocouple. This method had first been described by SaUat/8/. An elastic tensile deformation, due to the increase of volume, causes a decrease of temperature (Joule Thompson effect). The beginning of plastic deformation is indicated by a sharp increase of temperature due to the dissipation of plastic work, see Fig. 8. The optimum distance of the infrared thcrmocouplc from the specimen was 5 mm for measuring the mean temperature of an area of 10ram diameter (the sensitivity of the thermocouple is limited by an opening angle of 90°). In this area there is homogeneous deformation.

/

P

~> 100

//

._c t/)

.

0 0

3,o z ~

%

Subsequent

E

locus

Z

1,0

(prestraining =.

0

300

J

iiA I M g S i l

yield

ka/I

curve in x

and y:

3%)~

.c:

2,0

0,1

200

Fig. 9: Yield locus of unaUoyed steel from cross tensile tests and from uniaxial tests by assuming various yield criteria.

300 0,2

/

loo

Stress in rolling direction in N/mm 2

•

-

. l i

\

.200.

LL

0

'-0,1

L-

> t-

10o

-0,2

-0,3

w,

0

~

0,2

,

'

~

'

'

0,4

0,6

0,8

1,0

,

~

1,2

,

/

'

1,4

Elongation [%]

0 0

Fig. 8: Temperature vs. elongation for standard tensile test piece.

loo

2~

3oo

Stress in rolling direction in N/mm 2

Fig 10: Yield locus of an aluminium alloy in initial condition and after prestraining. Examples of test results are shown in Figs. 9 and 10. The steel was tested only in initial state whereas the aluminium alloy was tested both in the initial state and after prestraining. Evidently for steel Stl4 05 among the various methods of approximating the yield locus from uaiaxial tensile test data Hill's qua~atic yield criterion is closest to the test results. Fig. l0 illustrates the effects of anisotropic and isotropic strain hardening.

The authors wish to thank Prof. Dr.-Ing. K. Siegert, Institut ~/r Umformtechnik, Universitat Stuttgart, for supporting this work.

648

VE.Mailer, K P~hlandt /Journal o f Materials Processing Technology 60 (1996) 643-648

References

[ 1]

P6hlandt, K.; Hasek, V.: Priffung der Umformeignung von Blechwerkstoffen, in Lange, K.: Umformtechnik. Handbuch far Industrie trod Wissenschaft, Bd. 3: Blechbearbeitung, 2. Aufl., Springer-Verlag 1990.

[2]

Hill, R.: The Mathematical Theory of Plasticity, Oxford University Press, New York 1950.

[3]

Hill, R.: A User-friendly theory of Orthotropic Plasticity in Sheet Metals, Int. J. Mech. Sci. 15 (1993), 19-25.

[4]

K.reiBig, R.: Theoretische trod experimentelle Untersuchungen zur plastischen Anisotropie, Diss. B 1981, Technische Hochschule Karl-Marx-Stadt.

[5]

Mtlller, W.: Beitrag zur Charakterisienmg yon Blechwerkstoffen tinter mehrachsiger Beanspruchung, Diss. Universitat Stuttgart, demnachst.

[6]

Ziebs, J.; et al.: FlieBbedingung und Sparmung-Form~tnderung-Beziehung mit ausgekehlten Flachproben, Z. Metallkde. 66 (1975), 58-66.

[7]

MOiler, W.; POhlandt, K.: Neue Versuche zur Erfassung des richtungsabhangigen Verhaltens von Blechwerkstoffen, Blech Rohre Profile, to be publ.

[81

Sallat, G.: Theoretische und experimentelle Untersuchungen zum FlieBverhalten yon Blechen im zweiachsigen Hauptspanmmgszustand, Diss. TH Karl-Marx-Stadt, 1988.