ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR STRESS-STRAIN RELATIONS IN PLASTICITY AND RELATED TOPICS TECHNICAL REPORT NO. 2 AN EXPERIMENTAL STUDY OF BIAXIAL STRESS-STRAIN RELATIONS IN PLASTICITY By P. M. NAGHDI J. C. ROWLEY Project 2027 OFFICE OF ORDNANCE RESEARCH, U.S. ARMY CONTRACT DA-20-018-ORD-12099 PROJECT NO. TB 20001(234), DA PROJECT 599-01-004 December, 1953

ACKNOWLEDGMENTS The authors wish to express their appreciation to Messers. C. W. Beadle and G. A. Shifrin, Research Assistants in Engineering Mechanics, for their valuable assistance, both in the improvement of the extensometer and other equipment and in the running of tests. ii

ABSTRACT Experimental results for 10 tubular 24 S-T4 aluminumalloy specimens are reported. The data from these variable-loading path tests under the action of combined tension and torsion are presented and discussed. Included in an appendix is a description of modifications of the experimental equipment given previously. iii

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN AN EXPERIMENTAL STUDY OF BIAXIAL STRESS-STRAIN RELATIONS IN PLASTICITY 1. Introduction Although considerable progress has been made in recent years in the development of initially isotropic theories of plasticity for strainhardening materialsl'2, relatively little is known about initially anisotropic theories of plasticity which account for such phenomenon as the Bauschinger effect3. This is not surprising, since at present too few experimental results are available. To correlate test results with initially isotropic theories of plasticity, some experimenters in the past have made considerable efforts to use test specimens which are reasonably isotropic and uniform. In almost all cases reported, however, it seems to have been difficult to eliminate the nonuniformity and initial anisotropy of the material. Hence, correlation of experimental results with initially isotropic theories of plasticity is only qualitative in nature. Such correlation may, of course, lay the groundwork for further progress in the development of more general theories to account for initial anisotropy, as well as such phenomena as the Bauschinger effect. The present report contains experimental results for 10 tubular specimens subjected to the combined action of tension and torsion with variable loading paths. The tubular specimens, which were made of 24S-T4 aluminum alloy, possessed considerable initial anisotropy*. Also discussed is the significance of the experimental results in the light of the general theory of plasticity. 2. General Background For initially isotropic and plastically incompressible strainhardening materials, two distinct theories of plasticity (flow and deformation) have been employed in recent years. The basic concepts and *See Reference 4. 1

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN consequences of these theories have been discussed by Pragerl'2; while illuminating discussions of the role of the experiments, as well as their correlation with the mathematical theory of plasticity in general, and interpretation of experimental results have been given by Drucker5'6n7 and Prager8. Both of these theories are linear in character; i.e., the linearity of increments of plastic strains in the increments of stress is assumed. It may be mentioned that the slip theory of plasticity9, which has been regarded as a nonlinear theory, has recently been shown by Koiter10o to be included in the class of linear flow theories which are associated with singular yield conditions. While stress-strain relations of the deformation type are valid for radial loading paths (i.e., loading for which all components of stress increase at the same rate), they are invalid for more complex paths of loading. For such paths, stress-strain relations of the flow type must be used. For purposes of clarity, we shal.l discuss briefly the general stress-strain relations of flow and deformation theories and their consequences relevant to the experimental results given in this report. nhe general flow theory which incorporates the history of loading, when expressed in terms of plastic potential f, reads as follows: "H _f 3f Ek for loading, i.e., ~ > 0 ij k k = 0 for unloading or neutral loading, i.e., ak2 - H is a scalar function which may depend on state of stress (aoi), strain (Eij), and/or history of loading. In (1), the dot denotes differentiation with respect to time, and prime and double primes refer to the elastic and plastic components of strain, as it is tacitly assumed that the total strain tensor sij = E'ij + C"ij The loading function (yield condition) f in (1) may be such as to account for various degrees of initial and strain-hardening anisotropy. A few loading functions exhibiting this character have been suggested by Edleman and Druckers. A simple example is f = C. (a ) 2 (2) __________________________ 2 _____________________________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN which accounts for small initial anisotropy. In (2), Cijk~ is a symmetric tensor of quadratic form. For isotropic theories of plasticity, f may be a function of both aij and cij' When f is a function of aij alone, then the flow theory is referred to as the isotropic stress theory. In this connection, it is clear that the simple flow theory C ij - H(J2) Si J2 (3) may be obtained directly from (1), if H is assumed to depend on only J2, and when f = J2, i.e., J3 = 2Si2 Sij s (4) In (3), the stress deviation Sij is given by Sij = ij - S ij; S = ii (5) where bij is the Kronecker delta. The stress-strain relations of the deformation type for initially isotropic strain-hardening materials have been given with complete generality by Prager1l: c ij = P(J2, J3) Sij + Q(J2, J3) tij, (6) where J2 is given by (4) and J3 = 1 Sij Sjk Ski tij = Sik S 2 J2 ij 5kk 3 i

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN It is easily seen that the simple deformation theory C ij = P(J2) Sij (8) is a special case of equation (6). In the experiments on tubular specimens which are to be described, the loading was such that tension alone was followed by torsion (accompanied by various amounts of tension); and the determination of the initial shear modulus Gi when twist began was included. For this reason, it is desirable to examine the predictions of various theories of plasticit appropriate for such loading paths, with reference to the initial shear modulus Gi. As has been pointed out previously*, all isotropic deformation theories predict that the initial shear modulus is given by Gi = Go for loading.1 + 3Go ( -.1 ) Es E (9) =Go for unloading, where Es is the secant modulus with reference to the initiation of twist, and Eo and Go are the elastic moduli in tension and shear. Similarly, all linear isotropic flow theories of plasticity imply that (for all da33/da23 = da/dT) Gi = Go for both loading and unloading. (10) If, however, in equation (1) f is an anisotropic loading function, then during loading Gi will not necessarily be equal to Go. In particular, if f is assumed to be of the form given by (2), then Gi = d-a23 =...... Go. 2 [d~~3 + dc23] 023=0 1 + GO H a2 (C1211) [Clll -- + 21C211] dT See, for instance, Reference 12. 4

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 3. Experimental Results The experiments performed on 10 thin-walled tubular specimens will be described, and their results will be presented in this section. These variable-loading-path tests consist of tension alone, followed by torsion and varying amounts of accompanying tension. The test specimens and their data have been arranged in four groups (designated as A, B, C, D) according to the value of the ratio of tension to torsion at the initiation of the torque. This grouping is indicated in Fig. 1, where, in conformity with engineering notation, az and TQz (or simply a and i) deenote the axial and shearing stresses in the tubular specimen. Also shown in Fig. 1 is the approximate value of the initial slope do/dT for each group. TQz - 1 Initial 2 a d-a Yield / z / Fig. 1 The test data obtained are presented in the form of curves, in Figs. 2 - L1. For each group of specimens, three sets of curves are plotted: axial stress vs. axial strain, showing where twist began; elastic and initial shear moduli, including two elastic runs before and after each experiment (the first run was performed on the virgin tube in each case); and plastic strains occurring after initiation of twist. In each group, the secant modulus was computed to be approximately 7 x 105 psi. In the data computations, the following formulas were used: Axial strain c -=', where 2o is the initial gage length. 5

60 OF TWIST 50 zz.00006,'N/SEC. IN 40 loo 20 IN FIG. 2(a) AXIAL STRESS vs. AXIAL STRAIN TUBE A-I (#27)

121 5 z Q.000.I SEC. YZ21.00013 IN /SEC. x x <n 6 - N o FIRST RUN N 4, 0 SECOND RUN 4 2 Go 3.18 X6 I I I iGi 1.71 X lI06PSI 0i f i O 0 I 2 $ 4 0 1 2 4IN Yez IN x Io. — "Yez IN x 103 FIG. 2(b) ELASTIC AND INITIAL SHEAR MODULI TUBE A-I (*27)

12 12 1 12 I4 00 4' 8" " N o X'z.oooO'1/SEC.ININ d0 4546 47 484 0 4 6 8 0 2 4 6 To 0o FIG. 2(c) PLASTIC STRAINS AND LOADING PATH TUBE A-I (#27) TUBE A -I1 (*27)

60 50... z.00007 /SEC. - IN OTE: a INDICATES BEGINNING OF TWIST 40'o 30 20 Eo= 9.75 X 106PSI 0 2 4 6 8 10 12 14 Z IN X 1X3 FIG. 3(a) AXIAL STRESS vs. AXIAL STRAIN TUBE A-2 (*29)

12 _ 6 10 _ _ _ _5_ _ _ _ 8 ~~~~~44 0 ZOITN/SEC. IN /SEC. x 0 SECOND RUN IlS 6 3~~~~~~ Go=3.36x 10 PSI Gi 1.67 X 106 PSI 0 1 3 4 0 rN~~ ~~ 3 4N ANx 10 —4w N 103. IN wIN FIG. 3(b) ELASTIC AND INITIAL SHEAR MODULI TUBE A-2 (*29)

1212 12 I 0 - tO. 10 Z.000 I LN/SEC. 0 0 0 X o CO 6 _ ____ 6 0. 0 N0 1 0 E*u OO.0005 IN/SE~C.'- IN, Z tl.00005 TN- /SIC 2 _ _ _ d't o1 1 I I _~,..... ~......... 45 46 47 48 49 0 2 4 6 8 0 2 4 6 o' x -. z — I " x o-- PSI X Id3 -~XI0 3 ~ i~~ FIG. 3(c) PLASTIC STRAINS AND LOADING PATH TUBE A-2 (*.29)

60 5C NN 4EZ.00007IN /SEC. NOTE. 1 INDICATES BEGINNING OF TWIST 440'0 - 30 0.0 ro~~~~~~~z E0=9.76 X 106 PSI 10~~~~~ 10 _ _ _ _ 0 2 4 6 8 10 12 14 Ez J X I0o FIG. 4(o) AXIAL STRESS vs. AXIAL STRAIN TUBE A-3 (#31)

12 6 IC ~~~~~~~~~~~~~5 8,?~.00 NYz C.000 IN~/SEC! IN lo IN Ye 2.0001 T4 /SEC.2.0C I-N/E 10 10 IN -x ~~~~~~~~~x w~~ Q. Cl) N Io FRST RUN. Nb 0 SECOND RUN Go c 3.12 X lOp PSI Gi= 1.61 X IOp PSI 0 _ _ _ _ _ _ _ _ 0 I 2 3 4 4 IN 3 IN 3 4Z INXIo — loxi-IN. 10 FIG. 4(b) ELASTIC AND INITIAL SHEAR MODULI TUBE A-3 (#31)

10 C _ _ o Z t.0002 -/SEC.'~~o t) o 0 - - 8 ____ 8rr~i 0 0 0 0. N~~~~~~~~~~~~~~~~~~~~~~~~~~~~I ~F- N ( (0'# ~N oR E449OOOI z 24.a 0 0 2 f/SEC. -i,,, A x dT 0rr... ~,0,. 4748 49 50) " 0 2 4 6 8 0 2 4 6 O'QZ P"I xIO-, o-.. "''"xlo' --- -~'~. x,o' -.. FIG. 4(c) PLASTIC STRAINS AND LOADING PATH TUBE A-3 (#31)

60 50 _ _ _ zI'.O 5 IN/SEC. NOTE: 0 INDICATES BEGINNING IN OF TWIST S. 20 Eo' 9.77 X loro P.SI 10 0 __ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ 0 2 4 6 8 90 12 94 - Z IN jxI(? Psi N FIG. 5(o) AXIAL STRESS vs. AXIAL STRAIN TUBEE A-4 (#34)

12.. 6 3 NIN G 5o FIRST RUN NL 4I —,~~~~TB 2- (#3) o FIG 5(b) ELSTIC RUN I LM 0 T2 3 4 0 A2 3 4 IN IN FIG. 5(b) ELASTIC AND INITIAL SHEAR MODULI TUBE A-4 (.134)

2(.2~ 121 2 1 12 1 12 1121 b 0 ~'4 __ _ 1 _ _ _ _ _ 474849 5051 2 4 6 ~z~I PSI,0_ X; N103_ I C __IIN/ 3 FIG. 5(c) PLASTIC STRAINS AND LOADING TH TUBE A-4 #34) ~41~~~~~~~~~~~~~ FIG. 5(c) PLASTIC STRAINS AND LOADIN.0G01T /SEC L~~~~~UEA4( 4

60 50 Ez.00006/SEC. NOTE: 0 INDICATES BEGINNING OF TWIST1 440'0 x 30 I ) 20~ Eo=9.80 X 106 PSI 10 0 2 4 6 8 10 12 14 IN 3 I Z IN FIG. 6(o) AXIAL STRESS vs. AXIAL STRAIN TUBE A-5 (#35)

12 6 yez.00008 R/SEC. ~,s ~ Iol/SEC. X I:,I I I I Ic 8 4 ~O~ 2 3 4 0 1 2 _ 4 2 I< 0I IN X 1 IN 103 FIG. 6(b) ELASTIC AND INITIAL SHEAR MODULI TUBE A-5 ( 35)

12 12 12 ( 101 10 10 Z'Z.00007LIN/SE~C IC toI 2I ii i i,. 0 ~~~~~~~~~~~~~~~~ 0~~~~~~ ~J, ~ 0 14 4 N Q~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I ~" IN Ez g.00005 INSEC. 2 22 dO' d-.40 dr o' I I 1..... 47 48 49 50 0 2 4 6 8 0 2 4 6 ~, (z PSI X I$ JIN, 103w X s IN -- -A INX FIG. 6(c) PLASTIC STRAINS AND LOADING PATH TUBE A-5 (*35)

50 EZr.00005 /SEC. NOTE: 3 INDICATES BEGI NN ING OF TWIST t 0 gS) 40 to,- o 2 3 4 6 7 8 ~~~~~~x 30~~x10 i-'~ ~ ~ ~ ~~~~~~~ti FIG. 7(o) AXIAL STRESS VS AXIAL STRAIN ~~20~~TUB -I (5) Eomr 9.55 x106 PSi 0 I234 5 678 c IN, 103.. Z IN FIG. 7(a) AXIAL STRESS VS AXIAL STRAIN TUBE B-Il (*51)

12 12 IN~~~~~~~~~~~~~~~~I 2E.00007 - /SEC.~.00O9/SE zga.00 IN IS ~z 00009 IN / SEC. o ~~~~~~~~~~~~60 Co I P O- I St RUN co IL 0-2 d RUN N 4 4 Go03.49 x lop PSI Gi 3.30 106 PSI 2 2 0 2 3 4 0 2 4 IN 03 IN 1'Z IN ez IN FIG. 7(b) ELASTIC AND INITIAL SHEAR MODULI TUBE B-I (*51)

24 0 20 ~~~~~~~~~~~~~~~~~~~~~~~0 0 00 0 0 o 0 2~~~~~0 o o 01'~~~~o 16 0 to~ ~ ~ ~~~ o'I) ~ ~ ~ C 10 0 -12 K~~~~~~~~~~~~~~~~~ U) 0 0. N~~~~~~~~~~~~~~~~~~~~~~ d0 d (T 4 0~~~~~ 0 1_0 20 30_ 40 50 _ _ _ 2 -I_ 3 II&XI03100~~. _AEJ IO3 z..z.N ZIN ro~ ~ ~ ~ ~ ~ ~~~~S x 16 ob ) oI zN Q.~~~~~O FIG. 7(c) PLASTIC STRAINS AND LOADING PATH TUBE B-I ( 51)

50 IN.I,00006 /SEC NOTE: 0 INDIGATES BEGINNING OF TWIST 40 lo 30 20 E9.67 x 10 PS I 0. ~- 2~~~0 I0 0 2 4 6 8 10 12 14 IN. x3 I"Z IN. FIG. 8(a) AXIAL STRESS VS AXIAL STRAIN TUBE C-I (#36)

10 5 I_ _ 0-IN.t RUNI C Go 3.44 x 10E PSI ANDGi A1.61 x 106 Psi 8 4 K K 0 2 3 4 2 0 3 Go=3.44 x 106PSI Gi.6 x 10PSI FIG. 8(b)ELASTIC AND INITIAL SHEAR MODULI TUBE C-I (# 36)

I I I I I II I I T I 0 0 0 0 8 it | o l7~ l l|62.00006 AI j/SEC | I/S dT -f' d~ZI 49 50 0 2 4 6 0 2 4 6 z PSIx IC GZ: XlO - aEjjf Of 0 FIG. 8(c) PLASTIC STRAINS AND LOADING PATH TUBE C-l (#36)

50,,, IN EG~~ "'z-.006 IN /SEC. NOTE: A INDIGATES BEG.NNING OF TWIST 40 30 x 30'~~~a 20 Eo 9.75 x 106 PSI 0 2 4 6 8 10 12 14 IN 303 EZ IN- x10 FIG. 9(a) AXIAL STRESS VS AXIAL STRAIN TUBE C-2 (#50)

10 5 1 21.00001 IN/SEC. Ze IN YezIN 8 _ _ _ _ _ _ 06 3 0-1 RUN 0-2 RUN QI~~~~~~~~~~~~~~~~~~~I N ro N2 G 3.37 X0 PSI 0 23 4 0 2 ~I'I,3 ___ IN 34 _____ ____ 3 BIN FIG. 9(b) ELASTIC AND INITIAL SHEAR MODULI TUBE C-2 (#- 50)

0 0 0 0 0 0 0 0 0 0 ~~~~0~~ to o 0 3~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ U) 0 0 N2'Gz ~ 0001 /SEC. Ez L000006 /SECI o dOd —.025 d't 49 0 2 4 6 0 2 4 6 CT-P SI 10- IN x 103 AE ifN x 1 PSI Z -0 x IN' IN FIG. 9(c) PLASTIC STRAINS AND LOADING PATH TUBE C-2 (#50)

50 IN. NOT. 0 INDICATES BEGINNING Ez.000 SEC. z IN. OF TWIST 40 _ _ _ _'b 30 C) 0 bN 20 E010.02x106 PSI 10~~~~~~~ 0 2 4 6 8 10 12 14 Ez IN. xz03-Bob Z IN. FIG.I0(a) AXIAL STRESS VS AXIAL STRAIN TUBE D-I (#55)

10 2 IN Gz.0006 N/SEC 8 I IN. go, Y --.000I1j SEC'o'b ez IN. 7K 6 U)~~~~~~~- tRNC P, ~~~~~~~~o-,~td RUN HWr~ N ~~o-d-ZRUN 4 2 GO = 3.38 x 10e PSI Gi =1.20 x 10 PSI o I 0 o I 0 2 3 4 0 2 3 4 f IN. $ 13IN. 3 az IN. Gz,.Xo1 —, FIG. 10(b) ELASTIC AND INITIAL SHEAR MODULI TUBE D-I (# 55)

4 _ 0 3; PSI x 103_ IN x 103_ N x 100'K 0 0 0FIG.IO(c) PLASTIC STRAINS AND LOADING PATH TUBE D-I (I55)

50 E,.00005 IN. /SEC NOTE: 0 INDICATES BEGINNING IN. OF TWIST 40'IO 30 a. 20 E0=9.79 x I06PSI 10 0 2 4 6 8 10 12 14 IN. FIG. I1(a) AXIAL STRESS VS AXIAL STRAIN TUBE D-2 (#56)

10 2.00006'/SEC z INN.f r: ~~~~~~.0001 N 10 ~~~~~~~~0 01 St RUN 3-2 nd RUN jI N ~~~~~~~~~~N Go 3.36xIO's PSI Gi 1.09 x 10'sPSI 2 0 2 3 4 I 3 4 iN.X 103 3Y.IS - Ye IN. e I FIG. 11(b) ELASTIC AND INITIAL SHEAR MODULI TUBE D-2(#56)

3 -~~~00 0 0 0 - 9 2 0 5)5253O2 20 soo 0 0 0..d =0 dT ~0 0~~~~ 49 50 5152 53 0 2 4 02 46 FIGL 11(c) PLASTIC STRAINS AND LOADING PATH TUBE D-2 (#56)

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Shearing strain yg = r *' ~0 where g is the angle of twist over the gage length fo ~ Axial stress a = Axial Load 27t (a2-b2) where a and b are the outer and inner radii, respectively. Shearing stress Tg = Go r a = M (o-) (in elastic range) and T = 3 M (in plastic range), Qz 2rt (a3-b3) where the torque a M = 2rt r2 T dr. b In obtaining the last formula, it is assumed that in the plastic range Tgz is essentially uniform across the wall thickness of the tube. A few remarks concerning the significance of the experimental results are in order before their theoretical implications can be treated. The uniformity of the specimens was very good4. All tubes had an 0.075inch nominal wall thickness and 0.75-inch inside diameter, with the exception of Group D-1 and D-2 tubes, which had the same inside diameter but a wall thickness of 0.055 inch***. The anisotropy of the specimens was appreciable, as was previously noted in Reference 4; Fig. 12, showing one half of an axial cross section, is included to emphasize the severity of this condition. The curves in Figs. 2-.1 also illustrate several limitations of the experimental setup. First, a glance at the loading paths as plotted indicates the difficulty that was encountered in maintaining a linear stress path with a machine that is essentially a straining machine. Second, a slight tendency for the shear stress to lag occasionally in the initial *r was assumed equal to the outer radius a of the cross section. ** Ygz is twice the shearing component of the strain tensor. ***See Reference 4 for design requirements on wall thickness.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Fig..12 portions of both elastic and plastic shear stress- shear strain curves (e.g., Fig. 7b) is a reflection of a small residual stickiness of the loading ram in the upper bearing (see Appendix of this report). Finally, the data for tube B-i, Fig. 7 afford an exce.l.lent, evaluation of the small amount of interaction that remains between the axial-extension and angle-of-twist measurements. The majority of the deviations of these plastic strains from zero up until the plastic range is actually reached are probably due to this interaction. A figure of roughly + 10 microinches can be set for the resolution of the strains and must be kept in mind in this connection. 4. Discussion With reference to Figs. 2-11, it is seen that the repeatibility of the experimental results is good and that for Group A specimens this extends over 5 tubes. In this connection, it should be mentioned that the agreement in the values of elastic constants (Figs. 2-6 ) is remarkable in view of the 37

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN For Group A, the shear modulus Gi = (dTgz/dygz)TQZ= 0 is less than the elastic shear modulus G. (Figs. 2-6) by roughly a factor of 2. This result contradicts the predictions of both isotropic linear flow and deformation theories of plasticity and is startling, since normally, on the basis of a simple flow theory (f-J2), one would assume that unloading has taken place. Apparently such a phenomenon has not been reported by previous investigators, e.g., References 13, 14, and 15. While the confirmation of this contradiction must surely be reconsidered, at present it may be attributed to several sources. These include (i) the effect of both initial and strain-hardening anisotropy and (ii) the invalidity of the usually assumed linearity of increments of plastic strain in the increments of stress. We shall consider these possibilities separately. (i) In discussing the influence of anisotropy, we shall accept the validity of the assumption of linearity. Consider the subsequent yield surface f2 for an initially anisotropic strain-hardening material as shown in Fig. 13. This hypothetical loading surface is drawn so that the tangent at A (when ez PATH I (TYPICAL OF GROUP B) PATH I (TYPICAL OF GROUP A) f2 A \ / Fig..13 traveling in a counterclockwise direction) makes an obtuse angle with the axis a. Also shown in Fig. 13 are typical loading paths for Groups A and B specimens.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN As pointed out by Drucker16, when the loading surface has neither corners nor pointed vertices the increment of the plastic strain vctor is normal to the loading surface, Hence, the plastic strain vector d+" at A is along the external normal to f2 and has both axial and shearing components. Clearly then, Gi = (dTQz/dygz)TQ =0 at A (beginning of twist) should not be equal to Go. Furthermore, along the loading path I additional plastic deformation should be produced. It should be noted that had the subsequent yield surface f2 been a blown-up version of a simple isotropic yield surface (such as J2), then at least the initial portion of the loading path I in Fig. 13 would constitute unloading into the elastic domain. It is clear that the experimental plastic strains predicted for Group A specimens are in accord with this evaluation. (ii) If the effects of anisotropy are ignored, then one may attribute the observed behavior of the present experimental results at least partially to the nonlinear dependence of the plastic strain increments on the increments of stress. However, on the basis of experimental evidence7'18, it appears that the assumption of linearity is reasonable, although considerable further experiments must be performed before certainty is achieved on this matter. With reference to Fig..13, if the loading path is as in Group B, then Gi would be expected to have the elastic value Go, considering that unloading has definitely occurred. That this is indeed the case can be seen from Fig. 7, where Gi c Go..he small amount of deviation in these values is not unreasonable, in view of the presence of anisotropy. For Groups C and D, where loading would normally be assumed to occur, the existence of plastic strains after the application of torque is verified. These results are also in accord with the above speculation. Again, the value of Gi for these groups is about 1/2 of Go, contrary to the results predicted by isotropic theories of plasticity. It is the opinion of the present authors that the experimental results given here are explained in the main by the presence of initial and strain-hardening anisotropy. It may, of course, be possible that other factors such as strain rate and time effects, grain size, homogeneity of stress state, and machining of specimens may have influenced the results. EHowever, these effects are felt to be small and only a comprehensive experimental investigation could evaluate their influence. Aside from the initial value of the shear modulus Gi and its consequences discussed above, the general trend of the results agrees favorably with those of other investigatorsl3. Comparisons between experimentally determined plastic strains with those predicted by simple flow and deformation theories are not plotted in Figs. 2-.11 since, in view of the discussion concerning the effects of anisotropy on the yield surface, 759

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN such comparisons appear to be useless. However, it might be of interest to compare the results given here with the predictions of a slip theory of plasticity9. 5. Conclusion An experimental technique has been developed to pursue investigations in plasticity. This method, which permits combined tensiontorsion experiments, has been shown to give good repeatibility. Experimental results for 10 tubular specimens of an aluminum alloy (24S-T4), in which torsion was followed by tension, are presented. These results are discussed in the light of the fundamentals of stressstrain relations in plasticity. It is further concluded that the initial shear modulus Gi is definitely not in agreement with the elastic shear modulus Go as predicted by isotropic flow theories.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN APPENDIX 1. General The accuracy requirements and attendant difficulties peculiar to experimental investigations in plasticity, as opposed to the usual testing procedures in materials testing, have been emphasized recently by Drucker and Stokton'7. Our Technical Report No. 14 outlined briefly how the experimental setup was being developed to include features necessary to meet these rather special requirements. This section records the details of the equipment as it is currently used, and describes developments and modifications that have been made since the issuance of Technical Report No..1. 2. The Extensometer Perhaps the most difficult feature of experimental studies in a state of combined stress is the measurement of the deformations of the specimens. This is particularly true when the state of stress is one of both tension and torsion. In order to determine the average strains with sufficient resolution and precision over a reasonable gage length, a rather special extensometer is required. The main factors under consideration are that the extensometer should preferably have linear characteristics, the strain measurements should be independent, backlash and lag should be negligible, and finally it must lend itself to the coverage of a wide range of deformations, with comparable precision in both elastic as well as plastic regions. The fundamental features of such an instrument were presented in Technical Report No. 1. Referring to Fig. 4 of that report, which shows the extensometer in its original form, several rather basic alterations have been made. The diametrical potentiometers (P2) were found to have unsatisfactory resolution and were therefore removed. Several unsuccessful attempts were made to replace their function by a transducer that could measure circumference or diameter changes. These efforts were thwarted mainly by the difficulty of obtaining a reproducible calibration. 41

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN The angle of twist pick-up arm (A) was replaced by a double cantilever arrangement made up of thin flexure numbers as shown in Figs. 14 and 15. This modification considerably improved the response of the angle of twist measurement and eliminated a tendency to lag under reversal of torque. The axial extension potentiometers (P1) gave a great deal of trouble. Wear of contact areas on the sliders (Cl) and the very short range of motion (relative to the resistance-wire diameter) reduced the repeatibility and resolution to unacceptable values when both twist and extension were being recorded together. Although new contact materials and better resistance-wire material were tried, insufficient improvement was achieved. It finally proved necessary to replace these potentiometers by an extension unit utilizing SR-4 strain gages. This unit has proven through extensive tests to give the desired sensitivity and also the required independence of twist and extension. Figures l14 and.15 show the construction of this unit. Four phosphor bronze spring legs, having two SR-4 type A-7 gages each, are attached to the bottom disc (D). The other ends of these bent strips are fastened to a steel ring which has its top surface ground and polished. This ring serves as a base for three instrument bearings which are fixed to the underside of the top disc. Th'e use of the four spring strip arms makes possible the averaging of any slight tilts that might occur between the two discs.* As evidence of the repeatibility and linearity of the extensometer, typical calibration curves are included as Fig. 16. 3. Tension-Torsion Machine A change in the tension-torsion machine was indicated due to a tendency of the upper load bar to stick in its sleeve bearing. Although an infrequent occurrence, it was quite bothersome. A ball-bearing arrangement substituted in part for the sleeve bearing considerably improved the situation and will possibly allow reversed twist tests to be performed. 4. Recording Oscillograph In direct opposition to static elastic testing, experimental investigations in plasticity require that account be taken at all times of the rates of loading and straining. In this study, al.l variables are recorded continuously on a single oscillograph film. A facsimile of one such film for a variable-loading-path test is shown in Fig. 17. The convenience of having the data completely displayed as a continuous function of time is clearly seen in this typical record. It is, of course, not necessary to know the gage factor of the strain gages, since the unit may be calibrated.

IT~~~~~~~~~~~~~~~~~~~~I - A F aL14WriO x~~~~~~~i'i~~~~Ol1

Desertification~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~::'-:::::-: ~ ~ C S ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~R m,4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:::::::::: LEXURE AR~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:~: Fig. 15::

14 o INCREASING + DECREASING 10..CALIBRATION FOR ANGLE OF TWIST STANDARD ANGLE INCREMENTS OF 4/30 0 I 2 3 4 5 --- DEFLECTION ON FILM -IN. FIG. 16 CALIBRATION OF ANGLE OF TWIST 45

9t1 N' J / iS SECOND TIME MARKER-2 =i I V I /! AXIAL EXTENSI I - k ZERO SUPPRERESCS SION n -- - — NO SUPPR-10 ZEO SUPPRESSI r-7 -4 _Z0 R -: m I 0 _ C ZC o-4 0o rrl m r-~ - o(I (J)

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The use of zero suppression is also indicated in Fig. 17. This method for allowing the record of any variable to be expanded over several widths of the 6-inch film makes possible a considerable increase in resolution. For diagrams of the recording circuits, see Figs. 18 through 22. 47

Co m -4 0 -- -<10 Kft CALIBRATING IO Kfl TORSION RESISTORS GALVANOMETER 900 il 0 ~~~SR-4'TP A-7 SIR( GAGES -F-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~f co 1.3 Kf GALVANOMETER I CALIBRATING 10Kn RESISTORS 0KTESO m FIG.18 LOAD DYNAMOMETER RECORDING CIRCUITS

SENSITIVITY 150Q CALIBRATING RESISTORS 8 SR-4 TYPE A-7 STRAIN GAGES ON 0 sSN lt_ E /EXTENSOMETER Io _- I120N' 2 G)' GALVANOMETER z NOTE: THE TWO 120o FIXED BRIDGE ARMS ARE IN AN OIL BATH FIG 19 CIRCUIT FOR RECORDING AXIAL EXTENSION

15,Q SENSITIVITY 75K 4.7 K 5,O CALIBRATING /~~ ~~ e n o r RESISTORS m 0 o- 6V IOK ~: ZEROING RECORD o — - 3.OV 0- - 1.5V FIG.20 CIRCUIT FOR RECORDING ANGLE OF TWIST (POTENTIOMETER)

or. Nm z ro H~~~~ Z ~~~~~~GALVANOMETER =~~~~~~ 5.6,0 r o561 -4 rn~~~~~~~~~~~L SW. z L CALIBRATING RESISTORS FIG. 21 CIRCUIT FOR RECORDING ANGLE OF TWIST. (BRIDGE)

IK OK, IK (IV~~~~~~~~)~ rub 50K| f?' 509 osoA FIG 22 ZERO SUPPRESSION CIRCUIT

REFERENCES.1. "Recent Developments in the Mathematical Theory of Plasticity", by W. Prager, J. Appl. Phys., 20, 235-241 (1949). 2. "The Stress-Strain Laws of the Mathematical Theory of Plasticity - A Survey of Recent Progress", by W. Prager, J. Appl. Mech., 15, 226-233 (1948). 3. "Some Extensions of Elementary Plasticity Theory", by F. Edelman and D. C. Drucker, J. Franklin Institute, 251, 581 (1951). 4. "Fundamental Experiments in Plasticity - Instrumentation and Preliminary Phase", by P. M. Naghdi and J. C. Rowley, 0. 0. R. Technical Report No. 1, University of Michigan, 1952. 5. "Stress-Strain Relations for Strain Hardening Materials - Discussion and Proposed Experiments", by D. C. Drucker, Proc. First Symp. Appl. Math., 1, 181-187 (1949). 6. "The Relation of Experiments to Mathematical Theories of Plasticity', by D. C. Drucker, J. Appl. Mech., 16, 349 (1949). 7. "The Significance of the Criterion for Additional Plastic Deformation of Metals", by D. C. Drucker, J. Colloid Science, 4, 299 (1949). 8. "On the Interpretation of Combined Torsion and Tension Tests of ThinWall Tubes", by W. Prager, NACA TN1501, 1948. 9. "A Mathematical Theory of Plasticity Based on the Concept of Slip", by S. B. Batdorf and B. Budiansky, NACA TN1871, April 1949. 10. "Stress-Strain Relations, Uniqueness and Variational Theorems for Elastic-Plastic Materials with a Singular Yield Surface", by W. T. Koiter, Qaart. Appl. Math., 11, No. 3, 350-354 (1953)..11. "Strain Hardening Under Combined Stresses", by W. Prager, J.. Appl. Phys., 16, 837-840 (1945). 12. "Preliminary Experiments for Testing Basic Assumptions of Plasticity Theory", by R. W. Peters, N. F. Dow, and S. B.Batdrf, Proc. Soc. Exp. Stress Analys., 7, No. 2, 127-140 (1949). 53

13. "Experimental Studies of Polyaxial Stress-Strain Laws of Plasticity", by B. Budiansky, N. F. Dow, R. W. Peters, and R. P. Shepherd, Proc. First U.S. National Congress Appl. Mech., 503-512 (1951). 14. "An Experimental Investigation of Plastic Stress-Strain Relations", by J. L. M. Morrison and W. M. Shepherd, Proc. Inst. Mech. Eng., 163, 1-9 (1950). 15. "On the Plastic Behavior of Thick Tubes under Combined Torsion and Internal Pressure", by B. Crossland. and R. Hill, J. Mech. and Phys. of Solids, 2, 27-38 (1953). 16. "A More Fundamental Approach to Plastic Stress-Strain Relations", by D. C. Drueker, Proc. First U.S. National Congress Appl. Mech., 487-491 (1951 ). 17. "Instrumentation and Fundamental Experiments in Plasticity"', by D. C. Drucker and F. D. Stokton, to appear in Proc. Soc. Exp. Stress Analysis; also ONR Tech. Rept. No. 68, Brown University, 1952. 18. "Experimental Evidence of Non-Linearity in Plastic Stress-Strain Relations"', by F. D. Stokton, ONR Tech. Rept. No. 88, Brown University, 1953. 54