The effect of strain rate on the post-yield flow of mild steel.pdf

7/27/2019 The effect of strain rate on the post-yield flow of mild steel.pdf

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I. Mech. Phys. Solids,1963, Vol. 11, pp. 49 to 63. Pergamon ress tl Printedn GreatBritain.

THE EFFECT OF STRAINPOST-YIELD FLOW OF

By K. J . MARSH* and J .

RATE ON THEMILD STEEL

D. CAMPBELLDepartment of Engineering Science, University of Oxford

(Received 23rd August, 1962)

SUMMARY

A BRIEF description is given of a rapid-loading hydraulic test machine which was used in anearlier investigation of delayed yielding under constant stress. A modification to the machineis also described, which makes it possible to perform tests at approximately constant strain rate.

Results are given of constant-stress tests on mild steel specimens of four different mean ferritegrain sizes. Some of these results are compared with those obtained in constant strain-rate testson specimens of similar grain size. The experimental data are discussed and it is concluded thatthey may be reasonably represented by an equation of state relating stress, strain and strainrate. For one grain size this equation is depicted by means of a surface shown in isometricprojection.1. INTRODUCTION

IT IS well known that the post-yield plastic flow of iron and mild steel is consider-ably affected by the rate of straining. The flow stress at a given strain has beenfound to increase by a factor of 2 or 3 at strain rates of the order of lo3 set-l(CAMPBELL and DUBY 1956, CAMPBELL and HARDIXG 1961), and smaller increaseshave been observed at lower rates (CLARIS and WOOD 1949, TAYLOR 1957, KRAFFTand SULLIVAN 1959).

Attempts have been made to describe the experimental results by formulatingsemi-empirical relationships between stress and strain rate. Thus CLARK and WOOD(1949) and KRAFFT and SULLIVAN (1959) show an approximately linear relationbetween flow stress and the logarithm of the mean strain rate, while TAYLOR(1957) found a linear relation between flow stress and mean strain rate for a smallrange of low strain rates.

On the basis of dislocation theory, the local plastic strain rate is expected todepend on the applied stress, the dislocation density, and the obstacles to disloca-tion movement which are present in the metal. These obstacles include impurityatoms, grain boundaries, point defects and the dislocation network itself. For analloy of given composition, grain size and heat treatment, the dislocation densityand the obstacles to dislocation movement in slow tests are determined largelyby the strain. However, it has been found (CAMPBELL and DLJBY 1956) that insteel the density of slip bands is greater after rapid straining than after slow strain-ing; it is therefore likely that in general the dislocation sub-structure is a function.

*Nowat NntionalEngineering aboratory,East liilbride, Masgow.49


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.50 K. J. MARSH and J. I). CAMPBELLof the strain history and not merely of the strain. It is also known that dislocations1~0~~ess nertia, and in principle this inertia wil l affect the strain rate: !iowc\ -er,this effect is expected to be negligible except under esplosivc loading.

The local plastic strain rate is given by

where b is the magnitude of the Rurgers \ -e&or. 1 is the length of disloc~ation lii ic.per unit volume, and rj is the mean dislocation velocity.

Neglecting the effects of strain history and dislocation inertia, 7 is dcterminetlby the applied stress CJ nd the existing plastic strain Q,. and 1 is a function of Ed,.Thus we obtain an equation of state of the form

i, = g (0. Ep)..Uternatively, since E ~- E --- n/B, wh(rcmodulus we ma> w&e

c is the total strain ilIl(l E is Yctllngh, . > i, =- g, (a, c).

These equations can be applied on a macroscopic scale provided that thrdistribution of plastic st,rain within the specaimen is uniform. In materials with :Lsharp yield point, however, inhomogeneous plastic flow occurs for a time afterthe initiation of yielding, and the equations arc not, directly, alq~icable during thisperiod. I f it is assumed that yield propagates by the mo\ -ement of a single Liidersband front. the mean plastic strain rate is given 1,)

where x is measured from the end of t,he specimen at which yield is initiated.,rl denotes the position of the yield front at time t. or is the plastic strain at z :_ s,.and I, is the specimen length. Further, the mean plastic strain is given b!.

I t is evident, from tlie abo\ re expressions that i, cannot he obtained as a fiinctioiiof CJ nd cp from measurements of i,*, o and Ed*. It is however possible to pc,rforrnthe inverse calculation if the function g, is assLlnlrtl (Kl1AFb.T 1961).

In an earlier paper (C1~5rr~nr:r,i. nd M.\ i{srl l!M!Z) the ant hors liavc l)rcscntcdthe results of an investigation of the effect of grain size on the dclayctl >.iclding ofmild steel. In that investigation a new hydraiil ic~ rapid-loading niachinc~ was nscctto apply a constant compressive stress, and the strcqs ant1 strain \vcw rcy*c,rded :i4functions of time. The purpose of the present paper is to an:ilb-~ the I lost-yicltlbehaviour in the tests, and to examine to \ vliat cstent tlie rc5nlts arc c*on5istentwith the existence of an equat,ion of state of the kind discussed :tl)o\ -c..

% _kT:~lLAWSA detailed description of the hydraulic rapid-loading machine used to obtain the constant-

stress test results is given elscwirerc (Chwn1~:r.L an(l 21 \null 1902), so that a brief arcount orll~


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The effect of strain rate on the post-yield flow of mild steel 51wil l bc given here, together with a description of the modifications necessary to carry out constantstrain-rate tests.

Fig. 1 shows a sectional diagram of the hydraulic loading machine. Oil is pumped from alarge tank through valves A and B into the reservoir L and the upper cylinder U simultaneously.Controll ing the flow with these valves, the piston P may be kept stationary while any desiredpressure up to 3000 lb/in2 is built up in both the upper cylinder and the reservoir. Valves A and Bare closed when the required pressure has been reached.

Fm. 1. Sectional diagram of hydrauhc rapid-loading machine.

The specimen rests in a recess in the top of the piston rod Q and is just in contact with a coaxialhigh-tensile steel plug forming part of the stress- and strain-measuring head H. Also incorporatedin the head H are three high-tensile steel cantilevers, the ends of which, in the initial position,just touch the conical surface of the piston rod end.

In order to carry out a test, the pressure in the cylinder U is suddenly released by bursting adiaphragm in the special valve D. The piston P moves upwards, compressing the specimenagainst the coaxial plug while the conical surface deflects the cantilevers outwards. The specimenload and deformation are measured from the output of strain gauges fixed to the plug and can-tilevers respectively. After amplification, both outputs are displayed on a double-beam oscillo-scope as simultaneous stress/time and strain/time pulses.

The rise time of the stress/time pulse may be controlled by fitting various contractions inthe damping unit 2. The inherent damping of the system, with no contraction in the. 2 in. dia-meter nominal bore tubing, was sufficient to prevent any appreciable overshoot, the stress pulsehaving a rise time of l-3 m sec. This system was used to obtain the constant-stress test results.A typical oseillograph from a constant-stress test is shown in Fig. 2 (a).

From theoretical consideration of the pressure drop through a contraction in the dampingunit in terms of the strain rate applied to the specimen, it can be shown that, by fitting a suitablecontraction and applying a pressure corresponding to a stress considerably greater than theexpected stress at yield, an approximately constant strain rate should be obtained unti l the speci-men stress becomes appreciably greater than the upper yield stress.


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52 I+2al osrillograph front a danlped dynanlic test is shown in Fig. 2 (1)). In this the i-is


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ITIC:. 2. Osrillograph records of stress and strain. Ipper tract, stress; middle trace, strain;Imvcr trace, hasc lint. (a) Constant-stress trst (1 m ser markers). (b) I)arnped dynamic test

(2.5 m see markers).


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The effect of strain rate on the post-yield flow of mild steel

FIG. 3. Stress, strain and strain rate against time (constant-stress test).

55

ZC

BIG. 4. Stress/strain curves derived from constant-stress tests(mean grain density, 495 grains per mms).


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54 K. J. MARSI~ and J. L). CAMPBELL


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The effect of strain rate on the post-yield flow of mild steel 55

25

Deries

Strain rc0 7+ 5a4I3P 20 IA 0.5 IHe,+,se?n 0.2. 0.1. 0.05L(I 0.020 0.010 0.005x 0.002

FIG. 7. Stress/strain curves derived from constant-stress tests(mean grain density, M grains per mm2).

Some of the experimental results show a considerable scatter about the stress/strain curves drawn, particularly in Figs. 6 and 7. This scatter &s attributed toinaccuracies in the strain-measuring system, to variations in ferri te grain densityfrom specimen to specimen, and to the effects of non-uniform straining. Thestrain-measuring system was not ideally suited to the study of strain-rate effects,the accuracy of strain measurement being of secondary importance in the studyof delay-time phenomena, for which the apparatus was originally designed. Non-uniform wear on the surfaces of the cone and canti lever tips was thought to beresponsible for the difficulty encountered in obtaining reliable calibration curvesfor strain measurement in the later experimental work of Series C-F .

In spite of this scatter, it is considered that the curves drawn in Figs. 4-7provide a reasonable representation of the effect of strain rate on the stress/strainbehaviour, over the range of strain rates from about 0401 to 20 set-1.

The fact that it is possible to draw consistent sets of curves in Figs. 4-7 sllggeststhat the behaviour of the material may be represented by a functional relationshipbetween stress, mean strain and mean strain rate, a.lthough, as pointed out inSection 1, this cannot be considered a fundamental one when the strain distributionis non-uniform.


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r \


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The effect of strain rate on the post-yield flow of mild steel 57be compared directly with the curves of Figs. 4-7. The hydraulic test machinewas therefore modified, as described in Section 2, to permit the performance oftests at approximately constant strain rates.

5: ESPERIMRSTAI, I~KSULTS FROM CONSTAKT STRAIN-RATE TESTSFigs. 8 and 9 show typical sets of curves of stress, strain and strain rate against

time, obtained from damped dynamic tests. In some tests, for instance, that fromwhich the curves of Fig. 9 were obtained, there were two periods of approximatelyconstant strain rate. By reading off values of stress and strain at several intervalsduring a constant strain-rate period, a stress/strain curve could be drawn corrcs-ponding to that value of strain rate and covering the appropriate range of strain.

Sets of such stress/strain curves derived from the results of Test Series E areshown in Figs. 10, 11 and 12. Each set corresponds to a different damping orifice,lower values of strain rate being achieved with smaller diameter orifices. Thedashed curves are those derived from constant-stress Test Series C (Fig. 6).

Similar curves obtained from tests using O-026 and 0.047 in. diameter orifices(Test Series F) are shown in Fig. 13. The dashed curves are those obtained fromTest Series D. (Fig. 7).

Ser~?s EOrifice dl0rn. 0.026 1.

FIG. 10. Stress/strain curves derived from damped dynamic tests(mean grain density, 2033 grains per mm*).


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it.\v

t-_i

i

-1,-I

;,II11 ./

I

-... ..--_.\ --T------- _ ----


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The effect OF strain rate on the post-yield Aow of mild steel 59I t is seen from Figs. IO, I I and 12 that in many cases the agreement with the

curves derived from the constant-stress tests is extremely close. Other resultsare slightly high or low with respect to these curves but there appears to be nosystematic deviation. In general the correlation is good and within the limitsof the accuracy of the strain-me~soring system and the graphical differentiationprocess.

Sertes FOrifre dwxn,, in.

i2

cl 3.6 0 0.50.3.2 Li 0.4302.8 .0.37.2.4 0 0.37II.6 a 0.36t I-6 4 0.300 I.1 s 0.27x I.0 1 0.20* 0.9 SO.173 4 5strmn. %PIG. 13. Stressjstrain CUN~S derived from damped dynamic tests

(mean grain density, a46 grains per mm2).

The results of Series F tests shown in Fig. 13 also show a reasonable correlationwith the curves from the constant-stress tests, although there is a tendency forthe stress values to be low.

6. DISCWSS~OXA number of theoretical studies (SOKOLOVSKY 1048, MALVERN 1951, PLASS

1955, and others) have been made on the assumption that the material obeysan equation of state of the form given in Section 1. Taking the elastic strainrate to be k / E , where E is Youngs modulus, the equation may be written

E d = ci -+ g, (0, IC). (1)An alternative form of the equation is

E ( = ci + g, (a - a,, C) (2)where a, of is the static st~sslstrain relation.


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The effect of strain rate on the post-yield flow of mild steel 61

2.5.3

series cstrm. %

0 0.5. I'00 I.5. 2.0a 2.5I3.00 3.5. 4.0x 4.5+ 5.035 'I / , I I 1 I0005 3.01 0.0: 3.05 0.1 0.2 0.5 I 2 5 IOS+rcln rste. sec~lFIG. 14. Stress against strain rate (logarithmic scale) at constant strain(mean grain density, 2033 grains per mm2).increase gleatly at lower strains as the upper yield point is approached. Similargeneral hehaviour is shown in the curves of Fig. 15. Heplotting these resultsusing (CT u,) as the stress variable showed that no improvement in the correla-tion could be obtained by using (2) instead of (1).

It is concluded from the above discussion that none of the proposed simplified

0 I .o. I.50 2.0A 2.5. 3.0

0.005 0.01 0.02 0.05 0.1 o-2strain rote, WC-

FIG. 15, Stress against strain rate (logarithmic scale) at constant strain(mean grain density, 346 grains per mn?).


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FlC. lfi. lsortretric projeclion of stress, strain, strilin-rate surface(mean grain density, 2033 grains per mm2).


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The effect of strain rate on the post-yield flow of mild steel 63in the present paper, however, the stress was maintained nearly constant duringyielding; it seems probable that under these conditions multiple yield initiationoccurs, so that a condition of sensibly uniform strain is reached sooner. In constantstrain-rate tests, on the other hand, a reduction of stress occurs during yieldingso that multiple yield initiation is less probable. This difference may accountfor some of the discrepancies between the results obtained from the two typesof test.

7. CONCLUSIONSResults of constant-stress compression test,s on nmd steel have been found to

be consistent with the existence of an equation of state defining the plastic strainrate i, as a function of stress CJ nd strain l Further tests in which the strain ratewas maintained approximately constant have confirmed that such an equationis a reasonable representation of the post-yield behaviour of the material. Thedata obtained have been plotted as contour lines in the (a, l and in the (o., e)planes, and also in isometric projection of the three variables.

It has been found that a simpler equation, in which the plastic strain rate istaken to be a function of the difference between the stress and that correspondingto static conditions, does not describe the observed behaviour, except within alimited range of strains in the work-hardening region. The results show thatin this range the overstress does not increase linearly with strain rate; they indicatethat it is more nearly proportional to a fractional power of the strain rate.

REFERENCES19561961

PTOC. Ray. Sac. A 236,24.Response of Metals to H i g h Velocity Deformation,

p. 51. (Interscience, New York and London).Phil . Mag. 7, 933.Proc. Amer. Sac. Test. Mater. 49, 717.Technical Report No. 28, Brown University, Provi-

dence, R.I .

CAMPBELL, J . D. and DuBY, J .CAMPBELL, J . D. and HARDING, J .CAMPBELL, . D. and MARSH, K. J . 1962CLARK , D. S. and WOOD, D. S. 1949COWPER, G. R. and SYMONDS, P. S. 1957KRAFFT, J . M.KRAFF T, J . M. and SULL IVAN,A. M.

MALVERN, L. E. 1951PLASS, H. J ., .J r. 1955SOKOLOvSKY, v. v. 1948TAYLOR, 1). B. (. 1957

196119591961

Report, Serial No. SSC-123, U.S. Naval ResearchLaboratory, Washington, D.C.

Trans. Am.er. Sac. Metals 51, 643.Report, Serial No. SSC-127, U.S. Naval Research

Laboratory, Washington, D.C.J . Appl. Mech. 18, 203.PTOC. 2nd M idwestern Conference ol t Solid

Mechanics, p. 109 (Purdue University).Pri kl. Mat. Mckh., 12, 261.Proc. Conference on Propert ies of Mater ial s at

H igh Rates of Strain, p. 229. I nst. Mech. Eng.London.

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