Characterization of yield surfaces using balanced biaxial tests of cruciform plate specimens

Scripta METALLURGICA et MATERIALIA

Vol. 28, pp. 617-622, 1993 Printed in the U.S.A.

Pergamon Press Ltd. All rights reserved

C H A R A C T E R I Z A T I O N OF Y I E L D S U R F A C E S U S I N G B A L A N C E D B I A X I A L T E S T S OF C R U C I F O R M P L A T E S P E C I M E N S

S. 13. l,in, .I.L. l)i.g, anti I1. M. Zbib l)epartme.t of Mechanical and Materials I:;.gineeri.g Washington Sta, te UMversity, P . I h n a . , WA 9916,1-2920 1~. C. Aifantis l)epartment of Mechanical Engineering and l",ngincering Mecha.nies t\;lichigatl Technological U niversi ty, lh,.ghton, M 1 4993 I (Received October (Revised December

27, 23,

1992) 1992)

Introduction

Characl,eriza,titm of the inelastic material behavior at r,~om temperat.re is .sually acct~mplished thro.gh the evaluation of the yield surface. A quite extensive review cm the earlier w,,rk can be f,,..d in Refs. [I] and [2]. Most of the experimenta.1 study on the evolution of the yield s.rfaee has bee. perfi~rmed with the thin-walled tube specimen under either cotnbivmd tension and torsion, e.g. the work by I~hillips el al. [3], n.nd Michno and Findley [4]; or combi.ed tension and internal pressure, e.g. the w,,rk by ]lecker [5]. Other experimental teeh.iq.es which has been developed for the eharacterizati¢~, of the yield s.rface include the .se c~f the ha.rdness test [6], and t,he biaxial stretching of a cross-shaped plate specimen [7]. While rate-independent plasticity itself is a reasonably well developed s.bject, there seem to be omtinued interests in the use of the plate specimen for cha.racterizing bc~th the inelastic and fracture beha.viors of engineering materials. Perhaps the only and most distinct advantage r~f the plate speeb.en compared to the tube specimen is that the tube specimen can not be .sed for characterizing the materials which are in sheet or plate form. The latest work on the use of plate specimen fi~r characteriziag the inelastic behavior of materiMs is probably the one by Makinde el a/f8], llowever, the test, system described in [8] is act.ally quite similar to that described in Ref. [7] and a commercially available biaxial tension/compression test machine made by SCI1BNCK in Germa.y. f)ther relevant work on the use of pla.te specimen for st.dying the fraet.re behavior under static or fatig.e Ioadings can be found in Refs. [9-12]. Most of the biaxial tension/c~mpressic~n test sysl,erns, s.ch as the SCIIENCK machine and those in Ref. [7, 8], employ two pairs of actuators, one for each I.,adi.g direction, co.lbined with a hydra..lic system and servo controller. Constructing or acquiring n s.ch system is usually qnite costly. In /980, Charvat and Garrett [13] proposed a cost-effective alternative to study the fatigue crack growth behavior of materia,ls under biaxial loadings. The approach was essentially to modify I,hc universa.l test machine to a bia,xial machine by addi.g a horizontal loaxling device. More recently, Ferron and Makinde [14] also designed a spatial pantograph which co.ld be connected to a universal test machine to apply the biaxial loads. Ilowever, as also mentioned in [8], the loading axes of this device are not independent of each other and t,herefore the types of Ioadings which can be applied t,~ the specimen are very limited. The current research is a tentative investigation of using the approach proposed by Cha.rval, and Garrett [13] for fatigue crack growth study to characteri~,e the yield s.rfaee of plate materials. Specimen

The specimen geometry essentially follows that nsed by Brown anti Miller [I l] for the fatigue crack growth

617 0956-716X/93 $6.00 + .00 Copyright (c) 1993 Pergamon Press Ltd.

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test. Ilowever, it) the current stu(ly, we intended to keep the specimen size as small as possible for many obvious reasons such as saving costs on material and machining and testing materials that are limited in size. After a number of adjustments, the final dimension of the specimen is shown in Figure 1. The fo, r slots machined along each edge of the specimen shown in Figure 1 serve two purposes. First, they distribute the applied load along the edge. Second, they unconple the two loading axes by allowing more flexibility for the specimen to deform in two directions. Since this reduces the constraint on the edge, a reasonably uniform stress region can be obtaine(I over the gage section from separate loads applied to each axis. The thickness in the gage section is reduced by two thirds to ensure that maximum stresses occur within this section.

• ~2

'i-z.5

® m

I® c l t

I

i

l® ¢

" ,.,.;!, .

Unit : mm

F I G . 1. Cruciform specimen geometry Although cruciform type of specimen has been used quite extensively, there has been no standard geometry. I. order to ensure that a uniform stress field does exist, a finite element analysis was also performed t() analyze the stress distribution in the specimen shown in Figure I. Since the specimen possesses 3 axes of symmetry, only one eighth is considered. Figure 2a shows the mesh generated using 1,286 three-dimensional solid elements. The analysis is three-dimensional because of the difference between the thicknesses within and o,tside the gage section. Since the loading is applied by pulling pins in the 16 holes d,ring the test, pins are genera.ted in the h~,les in the FEM analysis and nodal forces axe exerted on them. The material constants used in the c¢)ml),ta.tion are listed in TaMe 1. The elastic-plastic model with yon Mises flow rule and linear is(~tropic hardening is asstimed for both specimen and pi,s. TABLE 1 Material Constants f(,r the Specimen and Pins

Specimen Pin

Malerial

Densily (kg/m "~)

Young b Modulu.s (/'a)

ltardenin.q Modul,s (t'a)

Yield Stress (I'o)

Pois.son Ratio

AI 1100 Steel

2700.0 7800.0

7.0× 10 m 2.11 x I())[

6.7× 10~ 5.0x 109

3.44 x 10r 4.6x 10s

0.33 0.25

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The result of the n.merica.1 analysis revea.ls thal, when loading the specimen i . biaxia.I tension, the middle region of tire specimen exhibits only biaxial stretching with negligible shear stresses. The contours of the normal stress are shown in Figure 2b. The fig.re shows that the stress in tile center region of the specimen is uniform over a wide area. The variation in the stress va.lne over this a.rea is less than 0.3%. Moreover, the shear stresses in this area are very .egligible with an order of magnitnde of 3 lower I,han tile normal stresses.

A=0.00x 10r B=6.80x 107 C=!.36510 r D=2-04xl 0r

F I G . 2. (a) Mesh f,,r ,~,,e eighth specime,, and (b) nc, rnlal stress cont,,,rs

E x p e r i m e n t a l A p p a r a t u s and Procedure "File cruciform specimen is loaded biaxially .si~g an lnstron machine and a specia.lly b.ilt horizontal loading frame. The lnstron machine is .see] to apply tile vertical h~ad, and the vertical a.xis of Lhe crucirorm specimen is trea,ted as an ordinary .nia.xial specime.. The horizontal loading system consists of two endplates, fo.r tie bars, a manually driven a.ctua.tor mounted on the left e.dplate, and a load cell connected directly to the actuator (more details about the experimenl, a.l setup are given in [15]). The load is transferred to the specimen through two sets of chains to eliminate any bending due to possible misalignrnents and to distrib.te the loads along the entire edges of the specimen. This helps improve the stress .niformity in the gage seetio,. The whole frame is statically balanced and is suspended on flmr springs. The frame is designed such that it is flexible enough to allow for positioning adjustrnents to eliminate misalignments. The onset of yielding in the specimen is detected by the nse of a strain gage rosette i,stalled at the center of the gage section. The goge used is a WA-06-060W FL 120 rectang.lar rosette made by Measurements G r o . p , Inc. The size of the rosette is 1.52 x 1.52mm with tile three gages overlapped o . the top of each other. The ch¢~ice of a as sm~ll as possible strain rosette is to further tninimize the effects due t¢., any possible variation of the stress and stra.in fields in tile middle section of tile specimen. The rosette meas.res the normal strains in three directions, namely, horizontM direction ( ~ ) , vertical direction (~u), a.nd a. direction 45 degrees from the horizonta.1 axis (c4s). i)uring the test, the shea.r strain in the x-y pla.ne, 7r.u, which is given by 7~.v = 2,4s - ('z + %),

(1)

can be monitored and used I,o cheek the alignment of the test system during the test. The strain and load data are collected and a.na,lyzed usiug an llP3"I97A data. logger and ]1P9127 computer, and plotted automatically on an lip plotter. To determine the yield s.rface, a. constant vertical load (wil,h the va.lne less t h a . the yield load) is first applied to the specimen from the Instron machine. Keeping I,he vertical load ecmstant, the horizontal load is then

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increased slowly using the manual actuator until the material yields. Definition of the onset of yielding will be discussed in the next section. The strain data at yielding, namely, e= and %, a.re then converted to stresses, a~ and ay, using llooke's law. This procedure is repeated for different values of the vertical load, giving a set of data to construct the yield surface. Experimental Results and Discussion Some preliminary experiments on the determination of the yield surface with the current setup have been performed on a commercially p~tre alumimJm, namely, I I00-F. ')'hc specimen shown in Vigure I was machined from a 0.5in thick plate. Typical results, az.cording to the procedure described in the previous section, are shown in Figures 3a and 3b. The figures show the variation in the strains (ez, %, 7xy) as the horizontal load Pr, is increased with the vertical load Py hekt constant; Figure 3a corresponds to P~=0.0 N and Figure 3b correspouds to Pv=2220.0 N. It can be deduced from these figures that 7~u is negligible compared to e= and %, implying

5000

Py=O.O N 3750

7xy

8x

2500

1250 (a) .,,

-250

0

250

500

strain(~G)

750

5000

,,

P,=2220.0 N

++

3750 A

~z~2500 1250

o

-Z50

/

I

0

250

slrain(~8)

II

(b)

I

500

750

F I G . 3. Load-strain curves with (a) P~=0.0 N and (b) Py=2220.0 N

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that a state of biaxial stretching is achieved at the gage section in conl'ormity with tire numerical analysis. Moreover, it is seen from these figures that e~ and c~ increase linearly with an increasing P~ nntil a. slight deviation from linearity is observed towards the top end of the curves. The yield surface was constructed with two different approaches, one with multiple specimens and the other with single specimen. As mentioned in Kef.[l], quite a few definitions have been proposed I,o identify the material yielding for constructing the yield surface. The possible definitions include the proportional lirnit, a small measurable permanent set, the conventional engineering offset of 0.2% strain, point of tangency ,~f stress strain curve with a multiple of elastic slope, and extrapolation methods. In the current work, the definition based on proportional limit was adopted. This definition seems to be essential for the single specimell approach since the amount of plastic strain accumulated during the identifica.tion of each data point for constructing yield surface is very little and the strain hardening effects due to the repetitive use of the specimen can be minimized. Construction of the yield surface with a. single specimen is actually a very common practice. For exa.rnple, see rters. [3,4]. In the multiple specimen approach, six specimens were used to construct tile initial yield surface of AII I00-F. Each specimen was tested under one combination of vertical and horizontal loads, and the proportional limit was identified from each test. The experimental data are shown in Figure 4a as syr,bols. The increment of au betwee. the two neighboring data is about 6.0MPa. The von Mises yield locus for initially isotropic materials is also shown in Figure 4a as the solid line, wlrieh is defined by 2

2

2

where ale is the uniaxial yield stress. The cry i, Equation (2) was taken to be the the averaged value ¢,bl,ained from the aforementioned six tests. In the single specimen approac.h, identification of the proportional limit essentially follows the approa.ch proposed by Phillips and Kasper [16]. After a small number of step loadings, a straight line could Ire traced through the strain readings for an elastic region. If more than two consecutive poinl,s deviate from the elastic line at the same stress level for both rz and ry curves, yielding is assumed to have occurred. For further eonfirtna.ti~n, one more loading s~p may be added. Unloading is followed, and the specimen is tested again with dill'ere,t loading combinations. Figure 4b shows the experimental result based on one specimen. The yon Mises yielct locus is also shown in Figure 4b as the solid line. In addition to the possible error introduced by the inconsistence of the material properties, the possible existence of the shear stress (due to the small, but inevitat~le rnisalignment) may introduce about 0.5% of additional error on the yield stress measurement. By comparing Figure 4b with 4a, although both sets of data seem to indicate that the initial yield surface ~f 1100-F aluminum is very close to tile yon Mises surface, the data from the multiple specimen approach apparently shows more scatter than those obtained with single speci,ven. The inevitalfility of the data. scattering with multiple specimen approa.etn is well known [I 8, 17], It may also be worth ,lenl.ioning that Phillips el al. did q,ite extensive study on the plastic beha.vic~r o1" .I 100-0 alumin,lrb e.g. [18]. Their res.lts indicated that the. initial yield surface of 1J00-0 alurnitmrtJ was also very close to Ihe vcm Mises surl'ace. Conclusions In the current work, a simple anc] cost-effectiw experbnental apparatus for the ewd.at,ion ~f the hi-plane yield surface witln the crnciforrn pla,te specimen has been designed and built. I'reliminary results indicate |,bat this technique can be employed very easily to examine yielding ~d" thi, sheet specir,ens under biaxial stretching conditions. It has also been shown that tile initial yield surface of 1100-F aluminum is very close to the wm Mises yield surface. This system will be used to exarnine the effect of i,..plane anisol, r~py resulting fr~m rolling and prestressing, and will allow for anisotropic yield surface measurer,enl.s.

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YIELD SURFACES

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60

60 "~ 4O

-~40

0.,.

O,,,.

b20

28, No.

J

bZO

0

2O 40 O-x(MPa)

6O

0

ZO 40 Gx(MPa)

6O

~Mises Curve o--Experimental C u r v e FIG. 4. Initial yield surface constructed with (a) six specimens and (b) one specimen Acknowledgement The support of the U.S. Army ll.esearch ()fllce under contra.ct 1)AAL 03-90-G-0151 is gra.tcfully acknowlcdged. References

1. M.J. Michno and W. N. Findley, Int. J. Nonlinea.r Mech., 11, 59 (1976). 2. K. Ikegami, Mechanica.I Beha.vior of Anisotropic Solids: Proceedings of the E,romeeh Colloquium 115, ed. Boehler, J.P., Colloques Inter., p. 201, du (;NIl.S, Pa.ris (1982). 3. A. Phillips, C. S. biu and J. W..lustusson, Acta. Mechanica, 14, 119 (1972). 4. M.J. Michno, Jr. a.nd W. N. Findley, Acta Mechanic,% 18, 163 (1973). 5. S.S. llecker, Metal. TR.ANS. 2, 2077 (1971). 6. D. Lee, F. S. Jaba.ra., and W. A. Backofe,, Trans. 'I'MS-AINIE, 239, 1.476 (1967). 7. E. Shira.tori and K. lkega.mi, J. Mech. Phys. Solids, 16, 373 (1968). 8. A. Makinde, L. Thibodeau and K. W. Neale, Exp. Mech., 32, 138, (1992). 9. I.M. Daniel, Exp. Mech., 22, 188, (1982). 10. D.L. Jones, P. K. Poulose, and II. Liebowitz, Multiaxlal Fatigue, ASTM STP 853, K..I. Miller a.nd M. W. Brown, Eds., p. 413, ASTM, Philadelphia (1985). 11. tl. Kitagawa, R. Yuuki, K. 'Pohgo a.nd M. 'l'anabe, Multia.xial Fatigue, ASTM S'I'P 853, K. J. Miller and M. W. Brown, Eds., p. 164, ASTM, Philadelphia (1985). 12. M.W. Brown and K. J. Miller, Multiaxia.1 Fatig,e, ASTM STP 853, K. J. Miller and M. W. Brown, Eds., p. 135, ASTM, Philadelphia (1985). 13. G. Ferron and A. Makinde, ASTM .1. Testing a.nd F,valuation, 16(3), 253 (1988). 14. I.M. If. Charva.t and G. G. Garrett, ASTM J. Testing and Evaluation, 8(1), 9 (1980). 15. J.L. Ding, S. B. Lin, II. M. Zbib and E. C. Aifantis, in prepa.ration. 16. A. Phillips and R. Kasper, ASME .I. Appl. Mech. 40(4), 891 (1973). 17. M.J. Michno, Jr. and W. N. Findley, ASME J. Eng. Materials k Tech., 14, 724 (1974).

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