HAL Id: jpa-00224817
https://hal.archives-ouvertes.fr/jpa-00224817
Submitted on 1 Jan 1985
HAL
is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire
HAL, estdestinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
SOME EXPERIMENTS OF DETERMINING FLOW STRESS CURVES OF METALS IN SOME
CONTROLLED COURSES OF HIGH STRAIN-RATE WITH A ”SOFT” TESTING-MACHINE
H. Takeyama, Y. Sato, T. Tobe, M. Kato, N. Takatsu
To cite this version:
H. Takeyama, Y. Sato, T. Tobe, M. Kato, N. Takatsu. SOME EXPERIMENTS OF DETERMINING
FLOW STRESS CURVES OF METALS IN SOME CONTROLLED COURSES OF HIGH STRAIN-
RATE WITH A ”SOFT” TESTING-MACHINE. Journal de Physique Colloques, 1985, 46 (C5), pp.C5-
645-C5-650. �10.1051/jphyscol:1985583�. �jpa-00224817�
JOURNAL DE PHYSIQUE
Colloque C5, suppl6ment au n08, Tome 46, aoflt 1985 page C5-645
SOME EXPERIMENTS OF DETERMINING FLOW STRESS CURVES OF METALS I N SOME CONTROLLED COURSES OF H I G H STRAIN-RATE WITH A "SOFT" TESTING-MACHINE
H. ~ a k e ~ a m a i Y . S a t o , T. Tobe, M. Kato and N. Takatsu The Department of Precision Engineering, Tohoku University, Ararnaki-Aza-Aoba, Sendai 980, Japan
R 6 s d
-
Une barre dlHopkinson modifi& pour op6rer B une vitesse de d6formation pr6dgtermini.e, constante ou variable, est dGvelop@ et quelques applications sontmontrks. Des barres d'impact de section non-uniform? dessin6es sur la base de la propagation des ondes 6lastique.s-sont utilisks pour g&&rer les profils d6sires d'hpulsion de mntrainte. I1 est mntri. qu'une barre d'impact ayant la forme d'un tronc de 6 n e perm?t d'obtenir une vitesse de d6formation quasi-constante pendant la d6formation de 1'Bchantillon. Le s y s t h est utilisd pour g6nBrer une vitesse de d6fonnation quasi-constante dans des 6chantillons d 1 6 1 a n c m t diff6rents. La courbe effort-deformation dynamique depuillee des effets de friction entre ll&chantillon et la machine est ainsi obtenue en extra- plant les resultats. Le systime est 6gal-t utilis6 paw 6tudier 1'6coula~~1t des r@taux B vitesse de deformation variable, et quelques exenples sontmontres.Abstract
-
A modified split-Hopkinson-pressure-bar canpression system which operates following a predetermined, constant or variable strain-rate course in specimen is developed and some of its applications are shown. Striker bars of nonunifom cross sections designed on the basis of the elastic-wave-propagation theory are used to generate desired shapes of incident loadinq pulse. It is sham that a striker bar having a shape of frustrum of cone enables constant strain-rate testing throughout the deformation of specimen.
The system is used to generate a nearly constant strain-rate in the specimens with different ratios of diameter to height. The dynamic stress-strain data free £ram the effect of friction between the specimen and tools are then obtained by extrapolating the results. The system is also applied to variable strain-rate tests to investigate the flow characteristics of metals and some examples are demonstrated.
The average strain-rate of workpiece is fairly high in high-energy-rate-forming techniques (the explosive forming, pnemtic-mechanical forming, electric-discharge fonning, etc. ) or high-speed-forming processes ( the wire drawing, rolling of strip, etc.), and the mechanical properties of the mterial have to be determined at high strain-rates. The flaw stress of metal is, h m v e r , a function not only of the instantaneous value of the strain and strain-rate but also of the strain-rate history. For the study of the rate effect, therefore, are required the experiments under definitely controlled strain-rate course.
The cam plastometer is a versatile tester which is capable of deforming a test specimen at an accurately known, predetermined, constant or variable strain-rate.
This type of tester m y be regarded as "hard" in the sense that its components are quite stiff compared with the specimen and its capacity is high enough so that the specimen is loaded at controlled strain-rates. But a part of the cross-head motion is transferred into the elastic strain of the specimen and the tester
,
so it is*Present address : The Dept. of Mech. Eng., Kanagawa Univ., Japan
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985583
C5-646 JOURNAL DE PHYSIQUE
difficult to estimate accurately the workpiece strain especially in the case of a disk-shaped specimen which is comnonly used in dynamic testing. In addition, it is impossible to estimate inertia effects due to the acceleration of moving parts of the tester as the strain-rate is increased. In fact the strain-rates of these testers are restricted to the order of 100 s-i due to factors inherent in their construction/
1/ and they are not generally available.
On the other hand, the split-Hopkinson-pressure-bar (SHPB) compression system is capable of carrying tests at accurately controlled strain-rates of the order of 1000 s';
,
and has been constructed by way of trial in m y laboratories/2,3/. The SHPB system is "soft" in the sense that the deformtion behaviours of specimen aredetermined considering the elastic-strain of the pressure bars or tools. The inertia effect of the bars can be evaluated by taking account of the propagation of the stress wave. However, it has been practically impossible to obtain flow stress curves of specimen following controlled courses of strain-rate since the strain-rate depends on the properties of the material, i.e., its changing resistance to
defonmtion or strain hardening characteristics /4/.
In this study the SHPB system is improved for the prescribed, constant or variable strain-rate testing by modifying the shape of the loading pulse. We pre- estimate the shape from an approximated strain-hardening characteristic for a given course of strain-rate in the specimen and made a striker bar of a varying cross section. The modified system is applied to some tests after controlled courses of strain-rate.
I1
-
STRAIN-RATE CONTROL FOR THE SHPB CCMPFESSION SYSTEM 2.1 Calculation of strain-rate, strain and stressIn this section a general approach to the SHPB cqression test is described.
The arrangement of the pressure bars and specimen is sham in Fig.1. The short compression specimen is sandwiched between the incident and transmitter pressure bars. The loading pulse in the incident pressure bar is initiated by an axial impact from a striker bar which is accelerated to the impact velocity by rubber bands. The instant the striker bar hits against the incident pressure bar, a stress pulse (the incident stress pulse, 6; ) begins to propagate d m the incident bar, passing strain gage A, where its magnitude is picked up. When this incident loading pulse reaches the specimen, a part of the pulse is reflected back toward the impact end, and the remaining part is transmitted through the specimen to the transmitter bar. The magnitude of the reflected stress pulse (CR) is picked up again at gage A, and the pulse transmitted through the specimen to the transmitter bar (the transmitted stress pulse, 6,) is picked up by gage B. An example of experimental records is s h m in Fig. 2. The positive values of
q, GR
and d; stand for compression while the negative for tension.Based on the one-dimensional elastic-we-propagation theory (hereafter referred to as W theory), we can determine the particle velocity, displacement and force at both faces of the specimen as functions of time from three stress-time
characteristics of & , $I and CT. The relations we need are:
velocities:
displacements:
forces :
where the incident and transmitted sides of the specimen are denoted by 1 and 2, respectively, the origin of the time scale t is taken at the instant when the
specimen begins to be carpressed and Eb, Cb and Ab are Young's modulus, wave velocity and cross-sectional area of the pressure bars, respectively.
The average strain-rate, strain and stress in the specimen are given by
where the positive values of stress and strain in the specimen stand for ccmpression and
Ro
and A. the initial height and cross-sectional area of the specimen,respectively.
2.2 Design procedure for tests at prescribed courses of strain-rate
A rectangular stress pulse is produced by the usual method which uses the striker and pressure bars of the same mterial having the same uniform diameter (
refer to Fig.2). Hawever, the strain-rate inthe specimen can not be kept constant by this loading pulse of constant amplitude.
The difference between forces Fl and F2 on b t h faces of the specimen after settling time of the incident pulse, say 20 microseconds, is within 4%in our experiments. The follawing approximate expressions are obtained assuming the equivalence of the forces;
Striker Incident Transmitter bar
Strain
Fig. 1
-
Experirrental arrangement of pressure bars and specimens.. .
Fig. 2
-
Typical pressure bar stresses, S t r a i n trandtted and incident and reflectedpulses. Fig. 3
-
Stress-strain curves at con-stant strain-rates for aluminum.
C5-648 JOURNAL DE PHYSIQUE
With this approximation, the strain-rate
6
and stress 6 in the specimen are proportional to the reflected pulse 6R and transmitted pulseG,
respedively, thusIn order for the specimen to be deformed following the pres-cribd course of strain-rate, an appropriate amplitude variation of reflected-pulse has to be produced. The shape of the reflected pulse is dependent on such parameters as the dimensions and mechanical properties of the specimen and the interface friction between the tools and specimen. For a specific specimen, the shape of loading pulse must, therefore, be selected for getting the shape of reflected pulse which we expect, i.e., the prescribed course of strain-rate in the specimen.
The following procedure is developed by using the approximate expression (7) for testing a specimen following a prescribed strain-rate course.
[l] A relation of flow stress to strain and strain-rate is initially ass& for the prescribed strain-rate course &($I of a given specimen. The quasi-static flow stress vs strain characteristics of the specimen may be used to the first
approximation.
[2] The approximate expression of incident pulse
a=(t)
which provides the specimen with the loading along the given strain-rate course &(t) is predicted in the following way:The reflected pulse &(t) which is proportional to
6 ( t )
is derived first by the u_se of eq.(4'). Secondly, the pre-estimated stress-time characteristics of specimen 6 ( t ) is determined £ram the stress-strain curve assumed inthe step [l] by using &(t)and E ( t ) the latter being the integral of
&t).
The approximate incident pulseat(#
,
which is derived £ram eq.(7), is then- -
dl(t)
=6,(t) - 6,(t),
(7') as the assumed expression for the transmitted pulseaT(t)
is proportional to a())on the basis of eq.(6').[3] The impact velocity and the shape of the striker bar with a varying cross section, which generate the predicted incident pulse
q(t),
are obtained approxhately by using the EWP theory.[4] The striker bar of the varying cross section is machined out. Then it is checked whether the incident pules TI(+) is realized in the incident pressure bar under the velocity pre-estimated in the step [ 3 ] .
[5] By performing the experiment with the striker bar at the impact velocity, we can obtain the strain-rate vs time characteristics of the specimen from eq.(4').
Then it is checked again whether the expected strain-rate course is realized in the specimen.
Results with an increased accuracy are obtained by repeating the above
procedure. In this case, the assumption in the step[l] requires s amodifications ~ mking reference to the results of the step[5].
111 - APPLICATIONS
In order to obtain a pre-estimated shape of the incident pulse, we employ the striker bar of a varying cross section as mentioned above. For the sake of
simplicity in the canputation, the striker bar is approximated by a stepped shaft;
the axial lengths of all segments are assumed to be equal while the cross-sectional area of each segment is not identical. The EWP theory is then used to predict the shape of the stepped shaft. The detailed description of the calculation is found elsewhere /5/.
3.1 Constant strain-rate testing
We begin with an attempt to provide a constant strain-rate course or path for the aluminum (JIS-A1070) specimens of 19.4 mn dia. and 10 mn height, which are annealed for an hour at tanperatme 350'C in an electric furnace. The pressure bars are of stainless steel (JIS-SUS304) finished by centerless grinding. The striker bars are made of stainless steel or brass and are finished by machining.
It is laborious to machine such a stepped shaft consisting of m y sqments.
Therefore, the striker bar having a shape of a frustum of cone is used instead, for the calculated profiles of striker bar resemble it in shape. The experimental, incident pulse obtained with the striker bar agrees approximately with the theoretically pre-estimated pulse/5/.
Fig.3 shows examples of the stress-strain curves obtained in an experiment with aluminum specimens. A brass striker bar is used in these experiments. It can be seen in Fig.3 that the specimens are deformed at almst constant strain-rates in the region of 200-320 s-1.
It is worth noticing that the striker bar designed using the quasi-static relation and without performing the iteration enables constant strain-rate testing within a certain range of strain-rate.
3.2 Flaw stress of disk specimens at high strain-rate
Mechanical properties of sheet metals which are used for high-speed-forming techniques have been usually determined by carrying high-speed tensile-tests.
However, these tests terminate at smll strain levels due to non-uniformity of stress and deformation in the specimen or instabilities associated with necking as the strain-rate is increased. Kolsky was the first to intend determining dynamic mechanical properties of very thin specimens using the SHPB system/6/. It seems difficult to evaluate friction effects exactly in dynamic carpression testing/7/.
But spurious increase in the flow stress due to the friction can be eliminated by applying the procedure of extrapolation/8-ll/.
A series of dynamic canpression tests of OFHC copper specimens 18 mn in diameter is made in which the diameter/height ratio is changed by height variation. The initial diameterheight ratios (Doh.) are 2, 4, 6 and 8. The specimas are vacuum
Current diarneter/height r a t i o
-
I ? k ( l / s ) :
C d
.&-
t h e d e s i r e d s t r a i n - r a t eL W
1-'u courses
V ) L
S t r a i n S t r a i n
( a ) S t r a i n - r a t e ( b ) Stress vs s t r a i n vs strain corresponding t o ( a )
Fig. -4
-
Relation between mean stress Fig. 5-
Stress vs strain and strain- and diameter/height ratio for various rate vs strain curves for OFHC copper values of strain when lubricated with under the controlled course of strain-grease. rate.
C5-650 JOURNAL DE PHYSIQUE
annealed at temperature 600 OC for an hour in an electric furnace. Grease is used as the lubricant.
The strain-rates in the specin-ens varies £ran 1000 s" to 200 s-' in the usual SHPB system which uses a striker bar of uniform cross section. On the other hand, it remains between 650 s-' and 450 s-"n the case of the present modified SHPB system.
Analysis of the experimental results is performed as .follows. First, the mean flaw-stress containing friction effects are graphed against the strain. Secondly, curves are cross-plotted with the mean stress as the ordinate and with the current diameterheight (D/h) ratio as the abscissa for various values of strain. The curves thus obtained for OFHC copper are sham in Fig.4. Extrapolation of a curve back to zero value of D/h in this figure is w e d to be the intrinsic flow stress free from the effect of friction for each strain-rate and strain.
For shorter specimens, the variation of strain-rate during the passage of loading pulse becomes larger in the usual SHPB systems. The Wified SHPB method will give a solution to this problem. For extremely short specimen such as sheet metal, the modified SHPB method ccgnbined with the extrapolation method will be applicable if laminated specimens are used/l2/.
3.3 Tests for changing strain-rate
Dynamic tests with a path of abruptly changing strain-rate in a specimen are needed in order to clarify the effects of strain-rate and its history on the flaw stress curve/l3/. As the strain-rate must be an independent variable in these cases, it needs to be controlled by experimenters.
For instance, the response of metals to an abrupt reduction in the strain-rate can be determined frcm an experiment which enploys the modified SHPB system. An
example of OFHC copper specimens having a D o h 6 ratio of 2 is sham in Fig.5. The initially aimed strain-rate, 800 s-I, is not achieved in the experiment. This may be due to cmitting of the repetitive procedure in the steps [3] and [4] for the
determination of the bar shape mentioned in Section 2.2.
A modified SHPB ccmpression systm is newly developed and is applied successful- ly to some constant strain-rate tests with aluminum and copper at rogn tangerature.
The SHPB systems can be also used to determine accurate flaw stress properties of such materials as plastics and ccmpsite materials as well as those of metals at elevated temperatures.
When the "softtt testing machines are used, a repetition of caplicated procedures is inevitable to obtain the mechanical proppies for a prescribed course of strain- rate. However, the procedures will beccgne to be treatable quickly by ccmputers and their peripheral equipmts which are developing rapidly.
REFERENCES
/1/ Thcanason,P.F., Fqg,B. and Chisholm,A.W.J., Proc. 9th Mach. Tool Des. and Res.
Conf., Birmingham, (1968) 287.
/2/ Campbell, J.D., Materials Science and Engineering, 12 (1973) 3.
/3/ Holzer,A.J., Trans. ASME ser.H, lOl(1979) 231.
/4/ Lindholm,U.S., J. Mech. Phys. Solids, 12(1964) 317.
/5/ Sato,Y. and Takeyam,H., J. Japan Soc. Tech. Plasticity, 24(1983) 744.
/6/ Kolsky,H., Proc. Phys. Soc., B62(1949) 676.
/7/ Holzer,A.J., Ann. CIRP, 29-l(1980) 135.
/8/ Cook,M. and Larke,E.C., J. Inst. Metals, 71(1945) 371.
/9/ Sturgess,C.E.N. and Bramley,A.N., Proc. 11th Mach. Tool Des. and Res. Con£.,
Birmingham, (1970) 803.
/lo/ Sato,Y. and Takeyam,H., J. Japan Soc. Tech. Plasticity, 22(1981) 1023.
/11/ Sato,Y. and Takeyam,H., J. Japan Soc. Tech. Plasticity, 22(1981) 1236.
/12/ E'ukui,S., Kudo,H., Yoshida,K. and Abe,K., Rep. Inst. Sci. Tech., Univ. of Tokyo, 8(1954),135.
/13/ Lipkin,J., Campbe11,J.D. and Swearengen,J.C., J. Mech. Phys. Solids, 26(1978) 251.
