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An investigation of the dynamic behavior of an annealed low carbon steel by means of stress pulse amplification
Int. J, Mech. 8ci. P e r g a m o n Press Ltd.
1967. Vol. 9, pp. 415-431. P r i n t e d in Great Britain
AN I N V E S T I G A T I O N OF T H E DYNAMIC B E H A V I O R OF AN A N N E A L E D LOW CARBON STEEL B Y MEANS OF STRESS P U L S E AMPLIFICATION N. P. SUH College of Engineering, University of South Carolina, Columbia, S.C.
(Received 6 February 1967 and in revised form 10 March 1967) S u m m a r y - - T h e dynamic behavior of an annealed low carbon steel, A I S I 1020, was investigated by a new method based on stress pulse amplification. The steel had a dynamic yield stress of 70,300 psi at the strain-rate of 123 sec -1. The delay yielding time was 9 × 10 -5 sec. When it is tested at the stress level of 103,000 psi and the strain-rate of 180 sec -1, the delay yielding time was about 3.4 × 10 -5 sec. The experimental results indicate t h a t the upper and the lower yield points observed at low strain-rates do not exist at high strain-rates, and t h a t the effects of intermittent loadings are additive. There also appears to be the possibility t h a t the obstacle to yielding m a y be different at different stress levels. The experimental method consisted of amplifying an elastic, compressive stress pulse by propagating it through a truncated cone machined at one end of a bar and observing the change in the shape of the wave reflected from the conical end, which was against a rigid wall. NOTATION
A,B C
amplitudes of stress waves, constants velocity of wave propagation, (E/p)t Do diameter of the largest section of the cone DT diameter of the smallest section of the cone E Young's modulus $ function for the wave propagating toward the apex g function for the wave propagating away from the apex k wave number = 21r/~ a constant r radial co-ordinate measured from the apex of the cone R. radius at the cone-bar discontinuity t time U
Ct+r
V
displacement of particles along the axial direction
E,,/E Ar
A and P o" Gb O"c
time difference between the wave reflected at the lateral surface and the wave reflected at the end of conical section strain-rate wave length mass density stress magnitude of the incident stress wave in the bar magnitude of the stress wave reflected at the lateral surface measured in the bar 415
416
N.P. SvH
Subscripts i incident wave r reflected wave p plastic region t transmitted wave 1. I N T R O D U C T I O N THE EFFECTS of strain-rate and temperature on the yield stress and flow stress of metals have been investigated b y many investigators. 1-~5 It has been shown that the yield stress increases with strain-rate and decreases with increase in temperature. I t has also been observed that some metals such as annealed low carbon steel exhibit delay yielding phenomena, which depend on stress level and temperature. ~3-25 Although many of the works that have been published are of the highest caliber, various averaging techniques were used in computing the stress and strain-rate. This paper presents a method for determining the yield stress of metals at various stress levels and strain-rates, which seems to minimize this limitation because of the extremely thin plastic region at yield. Basically, the method consists of generating an elastic wave, amplifying it in a converging section such as a conical end of a rod, and observing the reflected wave. A significant change is observed in the reflected wave when the amplitude of the stress reaches the yield stress; this is caused by the reflection of part of the wave at the elasto-plaatic boundary which is established in the cone. It will be shown that this method can also measure the delay yielding time very accurately. This method was applied to annealed low carbon steel, AISI 1020, to determine its yield stress under the dynamic loading conditions as well as its delay yielding time. The work done in the past m a y be generally divided into two groups. One technique consists of investigating the resulting plastic deformation of a specimen, then comparing it with the initial energy input, to compute the average stress. The other technique, which is more currently used, makes use of stress wave propagation. MacGregor and Fisher S,a demonstrated the parallel effects of strain-rate and temperature. The strain-rate was determined b y measuring the change in cross-sectional area of the necked region of a tensile specimen during loading. Because a slow tensile-testing machine was used, their experiments were limited to low strain-rates from 5 × 10 -5 sec -1 to 500 × 10-5 sec -1. Manjoine 4 did similar experiments for strain-rates from 10 -e to 103 see -1 ; high strain-rates were obtained b y loading a tensile specimen axially b y letting a flywheel hit one end of it. This technique, however, permitted only the measurement of the average strain -rate, since the movement of both ends of the tensile specimen was measured. Campbell 5, s attached strain gages to his specimen and assumed that the specimen reached the upper yield point when plastic deformation extended 2 cm from the loaded end. Taylor ~ and Davies s investigated the difference in the yield stress under dynamic loading and under static loading b y a ball indentation technique. They determined the minimum load required to indent a steel surface with a steel ball statically and dynamically. They found that the yield stress under the dynamic
Dynamic behavior of an annealed low carbon steel
417
loading conditions is about twice the static yield stress for materials with low static yield stresses. The differencebetween the dynamic and static yield stresses was not as great for metals with high static yield stresses. Recently a substantial amount of work has been done by Goldsmith et al. ~11 on static and dynamic indentation, to determine more exact indentation-force relations and the flow of the displaced metal. Mok and DuffyTM analyzed the indentation process of a hard ball impacting on various specimens. However, Mahtab et al. 13 suggested, based on their results from experiments and dimensional analysis, that their dynamic tip flattening of a conical projectile is a better method of determining the dynamic behavior of metals than the indentation method. Taylor and Qninney~ also investigated the problem by shooting lead projectiles against an anvil connected to two specimens. As the anvil moved, it stretched both of these specimens equally. The yield stress obtained by this method was found to be much higher than the yield stress observed at quasistatic loading. Taylor 7 and Whiffin14 fired fiat, mild steel projectiles against a rigid wall at speeds ranging from 810 to 2120 ft/sec. By assuming that the elastic-plastic boundary propagates away from the impacted zone with a constant velocity at a constant yield stress, they computed the yield stress from the final shape of the specimen. Their results showed that the yield stress under the dynamic loading conditions was substantially higher than the static yield stress. During World War II, yon Karman 15 and Taylorle derived independently the theoretical relations for plastic wave propagation. Although the theory itself does not take into account the strain-rate effect, the experiments performed by Duwez and Clark 17 with long wires, to verify the theory, yielded the result that the yield stress of mild steel under dynamic loading conditions was 2.9 times the static yield stress. Kolsky TM investigated the dynamic stress-strain relations for copper and lead by modifying the Hopkinson bar. The specimens were sandwiched between steel bars and the incident and transmitted stress waves were analyzed to obtain the average yield stress. A similar technique was later used by Krafft, SulUvan and Tipper 19. Recently, Hauser et al.2°, 91performed similar experiments by sandwiching a specimen considerably smaller in diameter than the wavegnide. Wood and Clark 2s.z5 first investigated the time required for the initiation of plastic deformation in annealed low carbon steel subjected to rapidly applied tensile stress. Hendrickson and Wood 24 advanced a dislocation model to correlate the experimental data, assuming that a sufficient number of dislocation loops must pile up at an obstacle to overcome the theoretical shear stress of the metal. The dislocation loop generation was assumed to be dependent upon thermal activation. As far as the stress pulse propagation in a conical body is concerned, no exact mathematical solution has been found because of the complexity of the problem involved. An approximate one-dimensional analysis made by Landon and Qninney22 is valid only for very small conicity. Through various experiments performed using three different waveguides of various shapes, it was
418
N . P . SvH
f o u n d t h a t a s m a l l p o r t i o n o f t h e s t r e s s p u l s e i n c i d e n t t o t h e c o n i c a l s e c t i o n is a l w a y s r e f l e c t e d a t i ts l a t e r a l s u r f a c e . 2~ T h e r e f o r e , i n a n a l y z i n g t h e w a v e p r o p a g a t i o n in a c o n i c a l b o d y b a s e d o n t h e L a n d o n - Q u i n n e y m o d e l , t h e w a v e reflected at the lateral surface m u s t be considered. 2. G E N E R A L
DESCRIPTION
OF THE
TECHNIQUE
A qualitative discussion of the technique will be given in this section in order to facilitate the quantitative reasoning of the next section. Fig. 1 is a schematic representation of the experimental arrangement. An elastic pulse is generated in the bar by striking it with the hammer, which swings like a pendulum. At the opposite end of the bar is a truncated conical section whose end is against a rigid wall. The arrangement is similar to that of the classical Hopkinson bar. The pulse propagates down the bar without much change in shape, if its wavelength is long compared to the radius of the bar. 27 After the pulse enters the conical section, its amplitude is gradually amplified and a part of the pulse is reflected as a tension pulse at the lateral surface of the cone. When the compression pulse reaches the end of the truncated cone, the pulse is reflected as a compression wave because the bar is against a rigid wall. The returning wave, measured by the strain gages, will then consist of a small tension wave superimposed on a compression wave, provided the dynamic yield stress of the bar is more than twice the amplitude of the incident stress wave. STATIC
v
SR-O
i ORCE
MEASURING CIRCUIT TRIGGER
FIa. 1. Schematic representation of the experimental arrangement. The stress wave amplitude can be amplified to varying degrees by truncating the cone at different places. As the stress wave is amplified b y successively greater ratios, the sum of the incident compression wave and the returning compression wave exceeds the dynamic yield stress of the bar at some point. I f yielding occurs without any delay, this first occurs near the peak of the wave front as the m a x i m u m stress is first reached at this point; otherwise, the yielding m a y commence a little after the stress amplitude reaches the peak value because of the delayed yielding phenomenon, za-25 At any rate, the plastic zone is established near the cone-rigid wall boundary because the stress wave amplification is greatest at this point. As soon as the bar deforms plastically, the mechanical impedance of the plastically deformed zone changes. Then a part of the incident compression wave which has just reached the elastic-plastic boundary is reflected back as a tension wave, because the mechanical impedance of the plastically deformed zone is less than that of the undeformed zone. The returning wave measured by the strain gages is thus the sum of the tension wave reflected at the lateral surface of the cone, the tension wave reflected at the elasto-plastie boundary, and the compression wave reflected at the cone-rigid wall boundary. The
D y n a m i c b e h a v i o r o f a n a n n e a l e d low c a r b o n steel
419
p o r t i o n of t h e w a v e w h i c h is t r a n s m i t t e d t o t h e p l a s t i c z o n e t r a v e l s s e v e r a l t i m e s across t h e p l a s t i c r e g i o n b e f o r e i t is c o m p l e t e l y t r a n s m i t t e d i n t o t h e m a i n p o r t i o n o f t h e b a r , b u t t h e p l a s t i c z o n e is so s m a l l t h a t t h e t i m e lag is negligible. T h e d y n a m i c y i e l d s t r e s s c a n b e d e t e r m i n e d f r o m m e a s u r e m e n t s of t h e i n c i d e n t a n d t h e reflected w a v e s , b y g r a d u a l l y c h a n g i n g t h e a m p l i f i c a t i o n r a t i o so t h a t t h e w a v e reflected f r o m t h e l a t e r a l s u r f a c e c a n b e isolated. T h e s t r a i n - r a t e c a n b e f o u n d a c c u r a t e l y f r o m t h e slope o f t h e s t r a i n - t i m e c u r v e of t h e i n c i d e n t w a v e . T h e s e p o i n t s will b e f u r t h e r d i s c u s s e d in d e t a i l in t h e following section. 3. T H E O R E T I C A L
ANALYSIS
Since t h e r e is n o e x a c t m a t h e m a t i c a l t h e o r y for s t r e s s w a v e p r o p a g a t i o n in a conical b o d y , t h e o n e - d i m e n s i o n a l t h e o r y o f w a v e p r o p a g a t i o n in a conical m e d i u m will b e c o n s i d e r e d t o o b t a i n a t t e n u a t i o n ratio. 2~ T h e a n a l y t i c a l r e s u l t s will b e l a t e r m o d i f i e d t o i n c l u d e t h e w a v e reflected a t t h e l a t e r a l surface a n d t o find t h e c o r r e c t a m p l i f i c a t i o n r a t i o w h e n t h e conical a n g l e is large. T h e a s s u m p t i o n s u s e d in t h e following d e r i v a t i o n a r e t h a t t h e p a r t i c l e m o t i o n a t a n y s p h e r e o f r a d i u s r is p a r a l l e l t o t h e axis, t h a t t h e cone is of s m a l l solid angle, t h a t t h e w a v e l e n g t h is large c o m p a r e d t o t h e d i a m e t e r of t h e cone in t h e r e g i o n o f t h e c o n e w h e r e t h e w a v e is p r o p a g a t e d , t h a t s t r e s s is u n i f o r m o n s p h e r i c a l surfaces, a n d t h a t t h e l a t e r a l a c c e l e r a t i o n is negligible. U n d e r t h e s e a s s u m p t i o n s , t h e w a v e e q u a t i o n for a conical m e d i u m m a y b e w r i t t e n as Os (rv) 1 ~2 (rv) ~r ~ = C~ ~t ~
(1)
T h e s o l u t i o n for e q u a t i o n (1) m a y b e w r i t t e n as v = 1 [ I ( c t + r) + g ( C t - r)] r
(2)
w h e r e t h e f u n c t i o n g is for t h e w a v e t r a v e l l i n g a w a y f r o m t h e a p e x a n d f is for t h e w a v e a p p r o a c h i n g t h e a p e x . I f we c o n s i d e r o n l y t h e w a v e a p p r o a c h i n g t h e a p e x , e q u a t i o n (2) m a y b e w r i t t e n as v = ~l(Ct+r)
(3)
T
The strain then becomes: ~v Or
1 f'(u) --~ f(u) r
where u = Ct+r,
~u Or
~u
I t s h o u l d b e n o t e d t h a t u r e p r e s e n t s a c h a r a c t e r i s t i c of e q u a t i o n (1). T h e s t r e s s c a n b e related to strain by: o =
Or
(4)
S u p p o s e we h a v e a w a v e g e n e r a t e d a t r = R 0, w h e r e R 0 r e p r e s e n t s t h e c o n e - b a r d i s c o n t i n u i t y , of t h e f o r m :
w h e r e A is a c o n s t a n t c o r r e s p o n d i n g t o w a v e l e n g t h . T h e w a v e is a s s u m e d t o s t a r t a t t -~ - R o / C , so t h a t w h e n t h e f r o n t o f t h e w a v e r e a c h e s t h e a p e x , t -- 0. S u b s t i t u t i n g e q u a t i o n (5) i n t o e q u a t i o n (4) a n d s o l v i n g for t h e f u n c t i o n f ( C t + r ) , the stress at the front of t h e w a v e t = - r / C a n d r -- R b e c o m e s : o =
- A ~R0
(6)
420
N . P . SUH
T h u s , we h a v e s h o w n t h a t t h e a m p l i t u d e of t h e s t r e s s a t t h e w a v e f r o n t is amplified so t h a t it is i n v e r s e l y p r o p o r t i o n a l to t h e d i s t a n c e f r o m t h e a p e x , or i n v e r s e l y p r o p o r t i o n a l t o t h e r a d i u s ratio. T h e a m p l i f i c a t i o n r a t i o g i v e n b y e q u a t i o n (6) is v a l i d only for e x t r e m e l y s m a l l c o n i c i t y b e c a u s e of t h e a s s u m p t i o n s u s e d in d e r i v i n g t h e e q u a t i o n . As soon as t h e conical a n g l e is m a d e finite, t h e a m p l i f i c a t i o n r a t i o will n o longer b e a f u n c t i o n of t h e first p o w e r of t h e r a d i u s r a t i o . As a n e x t r e m e e x a m p l e , s u p p o s e t h a t t h e conical e n d is r e p l a c e d b y a series of s m a l l e r s e c t i o n s as s h o w n in Fig. 2. I n t h i s case, tile w a v e i n c i d e n t a t each step will be
J
FIC. 2. R e p l a c e m e n t of a cone w i t h a cascade of s h o r t cylinders.
(A) COMPRES$1ON
WAVE
NCIDENTFRoNT~ WAVE
\
\
TENSION WAVEREFLECTED/
RFACE } AT THELATERAL //
WAV FRONT/y~ E % / v
(c)
NETRE(TA)~N(:)AVE - FIO. 3. S u p e r p o s i t i o n of w a v e s t o e x p l a i n t h e e x p e r i m e n t a l r e s u l t s w h e n t h e r e is n o p l a s t i c d e f o r m a t i o n . reflected as well as t r a n s m i t t e d . I t c a n b e easily s h o w n t h a t t h e a m p l i t u d e o f t h e t r a n s m i t t e d w a v e will b e a m p l i f i e d as t h e s q u a r e of t h e r a d i u s r a t i o r a t h e r t h a n as t h e first p o w e r of t h e r a d i u s r a t i o . T h e a m p l i f i c a t i o n r a t i o for a conical b o d y w i t h a finite c o n i c i t y will t h e n b e t h e n t b p o w e r of t h e r a d i u s ratio, w h e r e n is a p o s i t i v e c o n s t a n t b e t w e e n i a n d 2. T h e r e f o r e , e q u a t i o n (6) s h o u l d b e m o d i f i e d as a = -A
(7)
E q u a t i o n (7) does n o t t a k e i n t o a c c o u n t t h e p o r t i o n of p u l s e reflected a t t h e l a t e r a l s u r f a c e o f t h e cone. I t will b e a s s u m e d t h a t t h e p o r t i o n o f t h e pulse w h i c h r e a c h e s t h e s m a l l e s t s e c t i o n o f t h e c o n e w i t h o u t b e i n g reflected a t t h e l a t e r a l surface is a m p l i f i e d in a c c o r d a n c e w i t h e q u a t i o n (7). I n o r d e r t o find t h e yield s t r e s s let Fig. 3(A) r e p r e s e n t t h e w a v e i n c i d e n t t o t h e conical s e c t i o n , as m e a s u r e d b y t h e s t r a i n gages s h o w n in Fig. 1. I f t h e b a r h a d a fiat end, t h e r e t u r n i n g w a v e w o u l d b e n e a r l y t h e s a m e as Fig. 3(A). T h e solid c u r v e of Fig. 3(B) is o b t a i n e d b y a s s u m i n g t h a t a fixed f r a c t i o n of t h e i n c i d e n t w a v e
Dynamic behavior of an annealed low carbon steel
421
is returned a t the lateral surface of the cone. The curve is plotted AT ahead of Fig. 3(A) because the reflection starts to take place right after the pulse reaches the discontinuity between the conical section and the bar, Fig. 3(C) is obtained b y adding Figs. 3(A) and 3(B). The shape of the wave in the bar reflected at the truncated cone in contact with the rigid wall should look like Fig. 3 (C), provided t h a t the amplitude of the incident stress wave is less t h a n one-half the dynamic yield stress. Fig. 4 illustrates the case when the amplitude of the stress wave reaches the dynamic yield stress. Fig. 4(A) is the incident wave, which would be returned without a change in shape if the bar had a fiat end. Fig. 4(B) is similar to Fig. 3(B), which represents the portion of the wave reflected at the lateral surface of the conical end. Fig. 4(C) represents the wave reflected at the elasto-plastie boundary when there is delay yielding. Fig. 4(D) represents the sum of the waves which returns as a reflected wave to the strain gages of Fig. 1. I t should be mentioned that the peak of the wave on the left m a y not exist if the particular metal being tested does not have a n y delayed yielding.
A-
-tI-
(B)
---~REFLECTED TENSION WAVE W^ ,~- \AT THE LATERAL SURFACE/
I
/ _/t DELAY YIELDING TiME
"l
(c) REFLECTED TENSIO~ WAVE AT THE ELASTO-{ PLASTIC BOUNDARY| (D)
~ET REFLECTEDWAVE A Fio. 4. Superposition of waves to explain the experimental results when there is plastic deformation. I t should be emphasized t h a t Figs. 4(A), 4(B) a n d 4(D) are measurable ones and thus, in practice, Fig. 4(C) has to be deduced from these measured figures. "Between Fig. 4(D) and Fig. 3(C) there is a large difference in shape which takes place over a small difference in amplification ratio. The dynamic yield stress can be computed from Fig. 4 as:
(8) ~b is the amplitude of the incident stress wave shown in Fig. 4(A). The ratio (Doll)r)" is the amplification ratio based on equation (7). ~c is subtracted from a~ since a, is the portion of the pulse reflected from the lateral surface of the cone. The factor 2 is to account for the reflected wave as well as the incident wave. The negative sign in front of ~0 accounts for the tensile n a t u r e of ac, while the negative sign in front of the right-hand side indicates that the yield stress is obtained under compression. ~c can be experimentally obtained for a n y DT b y generating a stress pulse of low amplitude so that the plastic region is not 28
422
N . P . SuH
generated. Once t h e p e r c e n t a g e of the incident w a v e reflected at the lateral surface is known, ac can be c o m p u t e d for all stress levels. The strain-rate is equal to the m a x i m u m strain divided by the t i m e t a k e n for the stress w a v e to reach its m a x i m u m from zero. Thus, =
- e(-~--c) lD0]" 1 E ..... '\DTT! " ~
(9)
where E is the d y n a m i c Y o u n g ' s m o d u l u s and At is t h e t i m e t a k e n to reach the peak. I t should be n o t e d t h a t At can be changed a r b i t r a r i l y either b y using a bar w i t h v a r y i n g cross-sectional area or b y g e n e r a t i n g stress w a v e s in a different m a n n e r . DT at yielding can be d e t e r m i n e d where there is a drastic change in t h e r e t u r n i n g w a v e shape b e t w e e n two n e a r l y equal values of DT. A n o t h e r c o n v e n i e n t m e t h o d of d e t e r m i n i n g the presence of a n y plastic d e f o r m a t i o n was to m a k e successive hits w i t h the s a m e conical end of the s a m e bar a n d measure t h e reflected w a v e s to see if there was a n y change in the a m p l i t u d e s of the reflected w a v e s o b t a i n e d from the first two successive impacts. I t should be m e n t i o n e d t h a t this m e t h o d of d e t e r m i n i n g t h e d y n a m i c yield stress should be applied w i t h c a u t i o n w h e n the m e t a l to be tested possesses t h e delay yielding p h e n o m e non. I f t h e delay t i m e of t h e m e t a l to be tested is longer t h a n the d u r a t i o n of the stress pulse, the reflected w a v e will n o t show a n y effect of plastic deformation, a l t h o u g h it would h a v e u n d e r g o n e plastic d e f o r m a t i o n had t h e pulse lasted for a longer period of time. Since, in t h e e x p e r i m e n t s described here, only two successive i m p a c t s were m a d e to detect a n y change in t h e successive reflected waves, the d y n a m i c yield stress d e t e r m i n e d in this m a n n e r is valid only for a stress pulse shorter t h a n t h a t used in these experiments. H o w e v e r , if the m e t a l does not possess the delay yielding p h e n o m e n o n , t h e n t h e d y n a m i c stress determ i n e d b y this m e t h o d is the true incipient yield stress regardless of the d u r a t i o n of loading. The a m o u n t of t h e incident w a v e reflected at the elasto-plastic b o u n d a r y depends on the impedances of the plastically d e f o r m e d and elastic regions. The specific i m p e d a n c e is g i v e n b y (see A p p e n d i x for details): Specific i m p e d a n c e -- [~-~ p}
(10)
where ~a/~e is the stiffness or m o d u l u s a n d is equal to the Y o u n g ' s modulus in the elastic region, a n d p is mass density. T h e v a l u e of ~a/Oein the plastic region is always leas t h a n the Y o u n g ' s modulus. W h e n ~a/ae in the plastic region is zero, t h e w a v e is c o m p l e t e l y reflected as a tension w a v e at the elasto-plastic b o u n d a r y and w h e n Oa/~e is equal to the Y o n n g ' s modulus, all t h e w a v e is t r a n s m i t t e d w i t h no r e f l e c t i o n . I f Oa/~ is n e g a t i v e such as during the t r a n s i t i o n f r o m t h e u p p e r yield p o i n t to the low yield point, the expressions for the velocity of the plastic w a v e p r o p a g a t i o n and the i m p e d a n c e become imaginary, which is n o t a t t a i n a b l e , and thus, equal to zero. Therefore, in this case, m o s t of the incident w a v e m u s t be reflected as a tension w a v e at the elasto-plastic b o u n d a r y . ~s 4. E X P E R I M E N T S The e x p e r i m e n t a l a r r a n g e m e n t was as shown s c h e m a t i c a l l y in Fig. 1. T h e bars, which were m a d e in t h r e e sections, were h u n g on wires. The longer, s t a t i o n a r y section consisted of two bars in series: t h e bar with the strain gages was m a d e of cold rolled A I S I 1018 steel a n d t h e specimen b a r w i t h t h e conical end was m a d e of annealed A I S I - 1 0 2 0 steel. This was done because t h e annealed steel bar b e c a m e easily m a g n e t i z e d u p o n r e p e a t e d impacts, and, as a consequence, t h e strain gages act as a source of e l e c t r o m o t i v e force during t h e passage of t h e stress pulse, eatming erroneous m e a s u r e m e n t s . The entire s t a t i o n a r y section w i t h t h e conical end was p u s h e d against a ground, hardened, 3 × 5 in. flat plate of tool steel w i t h a static force of a b o u t 10 lb, b y m e a n s of two springs. This plate was in t u r n m o u n t e d on a larger steel cylinder a p p r o x i m a t e l y 6 in. dis. and 14 in. long. All c o n t a c t surfaces were lapped fiat a n d s m o o t h w i t h 5 micron a l u m i n u m oxide p o w d e r to ensure t h e t o t a l trausmission of w a v e s across the interface. One end of t h e s t a t i o n a r y bar w i t h t h e strain gages was m a c h i n e d r o u n d to ensure t h a t t h e i m p a c t s b y the shorter bar, which will be called the h a m m e r , be m a d e a t its center. T h e h a m m e r was m a d e of annealed A I S I 1020 steel a n d was 15 in. long for m o s t of the
Dynamic behavior of an annealed low carbon steel
423
experiments. The h a m m e r was raised to a predetermined height to generate a stress pulse of a given a m p l i t u d e . The impacting ends of the h a m m e r and the bar were heat treated to prevent plastic deformation. The full conical angle was 30 ° in all experiments. The strain pulses were measured by ~ in. long foil strain gages (FAP-12-12, BLH} and a dual-beam oscilloscope (Tektronix 502 A}. The strain gages were mounted opposite each other in order to cancel the effect of bending. Pictures of the strain traces were taken by Polaroid cameras mounted on the oscilloscope. The experiments were conducted by first making a short cone, the end of which was lapped fiat perpendicular to its axis. I n successive experiments the conical section was made longer and longer to obtain greater stress amplification. After a new conical section was machined, at least two successive impacts were made to compare the shapes and amplitudes of the reflected waves. In the absence of any plastic deformation, the reflected waves were identical: A part of the conical end, at least ¼ in. long, was cut off after each experiment in order to eliminate any work-hardened portion of the cone. By gradually changing the diameter of the smallest section of the truncated cone (DT), the point at which plastic yielding began to occur was found. When the h a m m e r rebounded after the stress pulse returned to it, it was prevented from making the second impact, so the specimen was loaded only once per impact. Many of the experimental results obtained were checked by repetition. The static yield stress was determined by means of a typical tensile test. The specimens were made from the same rod t h a t was used to make the dynamic measurements. Some of the experiments were repeated and it was found t h a t they could be reproduced exactly. The strain-rate was of the order of 3 × 10 -~ sec -1. The upper static yield stress was determined to be 40,500 psi and the lower yield stress was determined to be 37,500 psi. The particular steel specimen used in these experiments was chemically analyzed by the manufacturer. The chemical composition is as follows: Carbon Manganese Phosphorus Sulphur Silicon
0.19 per cent 0.50 per cent 0.01 per cent 0.028 per cent 0.04 per cent
Copper Nickel Chromium Aluminum Tin
0.04 per cent 0.03 per cent 0.02 per cent 0.01 per cent 0.008 per cent
This paper is based on three different types of experiments performed with annealed low carbon steel. The first set of experiments was performed to verify the validity of the theoretical model and prediction. Stress pulses of different amplitudes were generated by changing the height of the hammer, and the size of the conical section was changed until yielding occurred at a given stress level. Another set of experiments was performed to investigate the dynamic behavior of the metal at stresses higher than the incipient yield stress. The third set of experiments was done to determine the influence of the pulse length, by changing the length of the hammer from 15 to 30 in. 5. E X P E R I M E N T A L
RESULTS
AND DISCUSSION
(a) Verification of the theoretical model Figs. 6-11 show the experimental results. These figures were prepared by superimposing m a n y experimental results such as t h a t shown in Fig. 5, When D r was equal to or greater than 0.340 in., the shapes and the amplitudes of the reflected waves from successive impacts were the same, indicating t h a t there was no plastic deformation (Fig. 6). The similarity between Fig. 6 and Fig. 3(C) should be noted ; both are cases where the amplitude of the stress does not exceed the dynamic yield stress of the metal. I n Fig. 6 the difference between the horizontal levels of the incident wave and the reflected wave is due to the tension wave reflected at the lateral surface of the cone. I n order to further verify the theoretical model qualitatively, a case involving plastic deformation will be illustrated here. This is most drastically illustrated by the first loading curve shown in Fig. 11, which is for DT = 0.250in. I n this case, there was plastic deformation as shown by the successive changes in the reflected waves under repeated loading. The similarity between this curve and Fig. 4(D) should be noted.
424
N. P. SUH _
L.
l 1 !
I n c i d e n t and reflected Dr : 0 " 3 4 0 in
]
;
Reflected waves P r = 0 . 2 5 0 in
Ist
loading
3rd
loading
FIG. 5. A t y p i c a l e x p e r i m e n t a l record.
INCIDENT
REFLECTED
WAVE
DT . . . . . . . . . . . . . . . . . . . . . . . .
WAVE
0.340"
AMPLITUDE OF STRESS . . . . . . IN BAR (INCIDENT WAVE)
1 0 , 2 0 0 psi
F r o . 6. E x p e r i m e n t a l r e s u l t s w h e n DT = 0.340 in.
f
INCIDENT
WAVE
REFLECTED
WAVE
,., 2nd . . . . . . 3rd . . . . . . 4th --x--x-
DT ...........................
0.330"
AMPLITUDE OF STRESS . . . . . . 10,200 psi IN BAR (INCIDENT WAVE) STRAIN
RATE . . . . . . . . . . . . . . .
FIG. 7. E x p e r i m e n t a l r e s u l t s w h e n
123 sec?l
Dr = 0.330 in.
wove.,
Dynamic behavior of an annealed low carbon steel LOADINGS iNCIDENT
I st 2nd . . . . . .
WAVE
REFLECTED
DT . . . . . . . . . . . . . . . . . . . . . . . . . . . . AMPLITUDE
OF
INCIDENT
STRESS
WAVE
0.330" WAVE
IN
BAR--- IO~-~OOpsl
FxG. 8. Experimental result~ when D r = 0.330 for a longer pulse length (30 in. hammer).
INCIDENT
WAVE
S
REFLECTED
WAVE
2rid . . . . 6th --x--~-
DT . . . . . . . . . . . . . . . . . . . . . . . .
0.320"
AMPLITUDE OF STRESS . . . . . . 10,200psi IN BAR (INCIDENT WAVE) F I G . 9. E x p e r i m e n t a l results when D r = 0.320 in.
INCIDENT
REFLECTED
WAVE
j l
I st -
-
2nd . . . . . . . 3rd - - x - - x 4th --A--,
OT . . . . . . . . . . . . . . . . . . . . . . . . . AMPLITUDE
OF
INCIDENT
WAVE
LOADINGS
STRESS
0300"
WAVE
FxG. 10. Experimental results when
IN
BAR . . . . . . 10,200 psi
DT =
0.300 in.
425
426
N . P . SuH
The amplification ratio will be e v a l u a t e d q u a n t i t a t i v e l y . As shown in Table 1, a large n u m b e r of e x p e r i m e n t s were p e r f o r m e d at three different heights of the h a m m e r to d e t e r m i n e D T at which yielding t o o k place. Using these data, the yield stress and the INCIDENT
WAVE
LOADtNGS
REFLECTED
2nel . . . . . . WAVE
WAVE
/<.;~'- ~ k ~
3~d
17th
O T ............................ AMPLITUDE MAXIMUM
OF
0.250"
STRESS
STRAIN
.......
10.200psi
R A T E . . . . . . 180 sec -~
FIG. 11. E x p e r i m e n t a l results w h e n D T = 0"250 in. c o n s t a n t n of e q u a t i o n (7) were d e t e r m i n e d b y s u b s t i t u t i n g the d a t a into e q u a t i o n (7). F o r t h r e e s e p a r a t e sets of data, three sets of t h e following e q u a t i o n can be o b t a i n e d : (o'b - oc), {DT,~ n (ab-oc)~ = ~D-~%!
(iI)
where the subscripts I a n d 2 refer to different sets of data. T h e values of n t h u s d e t e r m i n e d are all w i t h i n 3.4 per c e n t of 1.55. T h e r e m a r k a b l y close v a l u e s of n indicate t h a t the relation, e q u a t i o n (8), is a sound one to use. Table 1 also shows the error estimation. T h e p e r c e n t a g e of the w a v e reflected a t the lateral surface is found to be n e a r l y the same b e t w e e n D r --- 0.370 in. and D T = 0.250 in. T h e reason it does n o t change linearly w i t h DT, as one m i g h t expect, is due to t h e multirefleetion of t h e w a v e s t a k i n g place in t h e cone. Because of t h e large angle b e t w e e n t h e w a v e front a n d t h e lateral surface, even t h e p a r t of the w a v e which is reflected a t t h e lateral surface p r o p a g a t e s forward until it is reflected a few m o r e t i m e s a t o t h e r p a r t s of the lateral s u r f a c e . " TABLE 1. EXPERIMENTAL DATA AND RESULTS Wave
Stress a m p l i t u d e reflected H a m m e r of t h e incident DT at ~ at the height w a v e in t h e b a r yielding lateral (in.) (psi) (in.) surface (%) Exp. 1 Exp. 2 Exp. 3 Max. error Max. dev. f r o m the average value
12 24 36
7,240 10,200 12,500
0.265 0.33 0.375
1.1%
_+ 2% _+2 %
_+1.8%
n
Yield stress c o m p u t e d by e q u a t i o n (8)
8 8 8
1.58 1.58 1.50
70,380 70,380 70,380
+_5
+ 4%
( E v a l u a t e d a t n = 1.55)
+1.5
± 3.4%
_+4.0%
Dynamic behavior of an annealed low carbon steel
427
The delay yielding time at this stress level, 70,380 psi, was found to be 9 x 10-6 sec. The strain-rate was 123 sec -1, which was obtained by considering the middle 80 per cent of the wave front. When this value was compared with the results obtained by Hendrickson a n d Wood 24, it was found t h a t it was almost the same as their results. Although the steel used in the present experiments had 0.02 per cent more carbon and 0.11 per cent more manganese t h a n that used by Hendrickson and Wood, it had 0.007 per cent less phosphorus and 0.012 per cent less sulphur. Therefore, the fact that both of these results agree so closely is not too surprising. On the other hand, the fact that even the data shown by these investigators sometimes deviated from one to another as much as by one order of magnitude in the delay time, such a close agreement between the two results, which were obtained by entirely different methods, must be partially a coincidence. I t should be mentioned at this point t h a t all the experimental data presented here are from a single rod specimen about 20 in. long. The experiments performed with other specimens of the same material cut from different rods showed a delay yielding period as long as 3 x 10-4 sec. E v e n this value was within the experimental scatter of Hendrickson and Wood's work. The foregoing quantitative and qualitative evaluation of the experimental data indicates that the theoretical model advanced in this paper is a reasonable one. (b) Behavior of the steel beyond the incipient yield point Fig. 7 presents the experimental results obtained at D r = 0"330 in. The reflected wave from the first loading is almost identical to the reflected wave shown in Fig. 6 which was for D r = 0.340 in. However, the reflected wave from the second impact has a much lower amplitude, indicating that there was plastic deformation at the conical end at DT = 0.330 in., whereas when D r = 0.340 in., both of the reflected waves were identical. The first reflected wave of Fig; 7 did not show a n y plastic deformation, being identical to the reflected wave of Fig. 6, because the delay time for yielding was nearly the same as the duration of the pulse. This was substantiated b y the experiments performed using the longer hammer (30 in.) as shown in Fig. 8. Fig. 8 shows that after an elapse of 9 x 10-6 see., the amplitude of the reflected wave suddenly decreased to a lower value. This lower amplitude corresponds to the second reflected wave shown in Fig. 7. The decrease in amplitude is due to the tension wave reflected at the elasto-plastic boundary, which is due to the smaller specific impedance of the plastically deformed region, as discussed earlier. I t should be mentioned here that the transition in the amplitude after a given delay period was sometimes more gradual t h a n the results shown in Fig. 8. I t is interesting to note that although the third reflected wave (Fig. 7) has nearly the same amplitude as the second reflected wave, the amplitude of the fourth reflected wave is somewhat lower t h a n the others. This indicates that the specific impedance in the plastic region decreased further due to softening of the metal. One of the most important results of these experiments is t h a t the dynamic stress-strain relation at and near the incipient yield point m a y be quite different from that of the static case. Under dynamic loading them seem to be no upper and lower yield points, which exist at low strain-rates. I f there were upper and lower yield points, the impedance would have been zero during the transition from the upper to the lower yield point, since ~a/~e is negative, provided the theoretical argument given in the earlier section on the impedance of the plastically deformed zone, equation (10), is correct. Therefore, the incident wave would have been reflected predominantly as a tension wave. However, the experimental results show only a small contribution b y the tension wave reflected at the elasto-plastic boundary. Therefore, the change observed in Figs. 7 and 8 is probably caused b y the decrease in the slope of the strew-strain curve at the yield point. On the other hand, it is quite feasible t h a t our understanding of the plastic wave propagation during the transition from the upper yield point to the low yield point m a y need further improvement. The question whether the effect of several loadings is additive will be considered here. The results shown in Fig. 7 indicate t h a t although the first loading was purely elastic, i.e. no macroscopic plastic deformation, the effect of the first loading was present when the specimen was loaded the second time. Also, the experiments with a longer hammer (Fig. 8) indicate t h a t the behavior of metal was dependent upon the total loading time
428
N . P . SUH
regardless of the particular pulse length used. The amplitude of the latter half of the first loading curve of Fig. 8 is the same as the second loading curve of Fig. 7. I t should be pointed out here t h a t in any one set of experiments the successive tests were made within an interval of less than 3 min. ttendrickson and Wood 24 derived the theoretical expression for delay time based on the assumption t h a t the delay time is the time required to generate enough dislocation loops at the F r a n k - R e a d source to overcome the ideal shear strength of the metal at a dislocation pile-up. The generation of dislocation loops was assumed to be controlled by the thermally activated release of the source dislocation from its "atmosphere". Therefore, the fact t h a t the effect of the loading is additive implies that the dislocations which were released from the " a t m o s p h e r e " must not return to their original position after unloading, if the proposed model is a correct one. This was observed in copper crystals by Young. 29 He observed t h a t the dislocations which moved under a low stress did not return to their original positions upon unloading, while the dislocations t h a t moved under a high stress did partially return toward their original positions upon unloading.
WAVE EXPECTED-~ i IF DELAYING MECHANISM [~
IS ALWAYS THE
;A! ~~
// ~ \ fACTUAL WAVE
Fro. 12. Figure illustrating the difference in the reflected wave for different delay yielding mechanisms. The experiments done with DT = 0.320 in. (Fig. 9) are interesting in t h a t during the second loading the metal continuously softened. The rate of softening is faster at high stresses. Upon the sixth loading, the reflected wave shows work hardening, as indicated by the increase in the amplitude of the reflected wave. The experiments done with DT ---- 0.300 in. (Fig. 10) further illustrate the rapid rate of softening at high stresses; the first reflected wave of Fig. 10 is nearly the same as the second reflected wave of Fig. 9. Fig. 11 presents the results obtained when the metal was loaded to an extremely high stress, 103,000 psi, at a strain-rate of 180 sec -1. These values are obtained from equations (8) and (9) when D r = 0.250 in. The sharp drop in amplitude of the initial reflected wave and the delay time of 3.4 × 10 -5 see should be noted. The experiments performed using a stress pulse with small amplitude to stay within the elastic limit indicate t h a t about 8 per cent of the incident wave is reflected at the lateral surface. One of the interesting results of these experiments is indicated by the amplitude of the first reflected wave at the wave front (Fig. 11). The delay time shown here is not as meaningful as t h a t shown by Fig. 8, as the stress level was not constant for an appreciable period of time. I t is probable t h a t the delay time, which decreases at increasing stress levels, is shorter in this test than the rise time of the applied stress wave. I n this case, yielding would occur before the m a x i m u m stress is reached, and the test is one of constant strain-rate. However, ff the delay time is not shorter than the rise time of the applied stress wave, ff the same delay yielding mechanism is at work at all stress levels, and ff only the rate of dislocation loop generation changes as a function of stress amplitude, as assumed in the analysis of Hendrickson and Wood 24, then the amplitude of the wave front must be t h a t shown by the dotted line instead of the results obtained (Fig. 12). I f this is the case, the solid line of Fig. 12 indicates t h a t at this level the pinning mechanism which was responsible for the delayed yielding at a low stress leveI had already been overcome without any delay, when the applied stress was increased to a higher value. This can be done if the stress required to overcome the obstacle is not always equal to the theoretical ideal shear strength of the metal, thus requiring different numbers of dislocation pile-ups for shear motion to continue. Based on his experimental results, Young aa postulated a similar mechanism for pure, single-crystal copper. F u r t h e r experiments with stress pulses
Dynamic behavior of an annealed low carbon steel
429
of short rise time need to be performed to resolve this question. Indeed, such experimental results may be useful in checking the model of yielding advanced by Hahn sl. Another interesting fact from Fig. 11 is that the metal work hardens from the second loading onwards, without the initial softening of the metal as observed when the applied stress was low. Furthermore, as indicated by the third and fourth curves, a continuous hardening and softening takes place even after the metal work hardens, as indicated by the top horizontal level of the reflected waves. All the loadings shown in Fig. 11 were done within 24 hr at room temperature, so it is unlikely that there was any aging effect. This could be due to the partial return of dislocations to their original positions upon unloading, as Young observed in copper at high stress levels, z° 6. C O N C L U S I O N The m e t h o d used here provides a useful m e a n s o f investigating the d y n a m i c b e h a v i o r o f metals. The incipient yield stress of annealed low c a r b o n steel with 0.19 p e r cent c a r b o n is f o u n d to be 70,380psi with the d e l a y time of 9 x 10 -5 sec. As t h e a m p l i t u d e o f stress increases, t h e delay time becomes shorter in a c c o r d a n c e w i t h published results. B a s e d on the experimental results presented in this paper, it can be concluded t h a t the effect of m e c h a n i c a l loading is additive, a n d t h a t t h e u p p e r a n d lower yield points m a y n o t exist a t high strain-rates. There also appears to be the possibility t h a t the e x a c t cause for d e l a y yielding m a y be different a t different stress levels. Acknowledgment--This work was done at the University of South Carolina under the sponsorship of the National Science Foundation, Grant GK-899, for which the author is grateful. The author wishes to thank Mr. R. S. Lee and Mr. C. R. Rogers for their help in the preparation of this paper. The helpful comments of Prof. Milton C. Shaw of the Carnegie Institute of Technology are gratefully acknowledged. Mr. W. J. Dockcry of the Atlantic Steel Company kindly furnished the chemical analysis of the steel.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. ll. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
B. HOPKINSON, Prec. R. Soc. A74, 498 (1905). C. W. MAGGREGORand J. C. FISHER, J. appl. Mech. 12, A217 (1954). C. W. MAcGR~.OORand J. C. FISHEtt, J. appl. Mech. 13, A l l (1946). M. J. MANJOI~r~,J . appl. Mech. l l , A211 (1944). J. D. CAMPBELL,J. Mech. Phys. Solids l, 113 (1953). J. D. CAMPBELL,Acta Met. 1, 706 (1953). G. I. TAYLOR,J . lnst. civil Engr 26, 486 (1946). R. M. DAVrES, Prec. R. Soc. A197, 416 (1949). W. GOLDSMITHand P. T. LYMAN,J . appl. Mech. 27, 717 (1960). W. GOLDSMrrHand C. H. YEW, Proceedings of the 4th U.S. National Congress of Applied Mechanics, Vol. 1, p. 177, ASME, New York {1963). C. H. YEW and W. GOLDSMITH,J . appl. Mech. 64 APM-38 (1964). C. H. Moor and J. DU~FY, Int. J . mech. Sci. 7, 355 (1965). F. U. MAHTAB,W. JOHNSON and R. A. C. SLATER,Int. J . mech. Sci. 7, 685 (1965). A. C. WHIFFI~, Prec. R. Soc. A194, 1038 (1948}. T. voN KARMAN,National Defense Res. Counc. Rep. A-29 (1942). G. I. TAYLOR,British Official Report No. RC 329 (1942}. P. E. DuwEz and D. S. CLARK, Prec. am. ~oc. Test. Mater. 47, 502 (1947). H. KOLSKY, Prec. phye. See. London 65, 677 (1949). J. M. K . R ~ , A. M. SULLIVA~and C. F. TIPPEtt, Prec. R. Soc. A221, 114 (1953). F. E. HAUS~.Rand C. A. WrS,rER, Report to Convair, Series 133, No. 4, University of California, Berkeley (1960). F. E. HAUS~.R, J. A. Sr~MONS and J. E. Dott~, Report to Convair, Series 133, No. 3, University of California, Berkeley (1960).
430
N . P . SUH
J. W. LANDON and H. QUINNEY, Prec. R. Soc. AI03, 622 (1923). D. S. CLARK, Trans. Am. Soc. Metals 46, 34 (]954). J. A. I-IENDRICKSO~and D. S. WOOD, Trans. Am. Soc. Metals 50, 498 (1958). D. S. Wood and D. S. C L ~ x , Trans. Am. Soc. Metals 43, 571 (1951). N. P. SUH and W. F. STOKEY,to be published. R. M. DAVIES, Phil. Trans. A 2 0 0 , 375 (1948). A. H. CO~rRELL, Prec. Conf. Properties (~ Materials at High Rates of Strain, London, p. 1 Inst. Mech. Engrs (1957). 29. F. W. YouNo JR., J. appl. Phys. 32, 1815 (1961). 30. F. W. YOUNG Jm, J. appl. Phys. 33, 963 (1962). 31. G. T. HAHN, Acta Met. 10, 727 (1962). 22. 23. 24. 25. 26. 27. 28.
APPENDIX
Wave reflection at the elaeto-plastic boundary I n this analysis all the assumptions made in deriving equation (1) of the text are assumed valid. I n addition, the material in the plastic state will be assumed to be linearly
=,; zo:
/
PLASTICZONE
r i
X =
FIG. 13. Cone with the elastic-plastic boundary. work-hardening ideal plastic metal. With the co-ordinate system defined as shown in Fig. 13, the displacement due to the incident wave l/~ may be expressed as:
V~ =
A~ sinb(Ct+x) ro+X
(A.I)
where A~ is the amplitude of the incident wave at a point where (re+x) is unity, kC = w = angular frequency, and r 0 is the distance from the apex of the cone to the elasto-plastie boundary. The displacement due to the reflected wave V~and the displacement due to the transmitred wave V~m a y be represented b y :
V~= Ar s i n k ( C t _ x ) + _ B , cosk(Ct-x) ro+X ro+X A, sink~(C~t+x)+ B ~ cosk~(C~t+x) Vt = ro+x ro+x
(A.2)
where the subscript p denotes the plastic region. The boundary conditions at x = 0 are V~+ ~ = V,, (A.3) A c c o r d i n g to the strain.rate independent theory of plastic wave propagation, zs, 16 the velocity of wave propagation in the plastic region is given b y
c~
= ~
0 = ~,/0
(A.4)
Dynamic behavior of an annealed low carbon steel
431
Combining equations (A. 1), (A.2) and (A.3), the reflected and the t r a n s m i t t e d amplitudes are found to be 2r o ko(1 - a) B, = B, = res kt(l + f l ) s + (1 - ~ ) 2 A,
2roS k'(~ + 1) At = r~ ks(1 +fl)~+ (1 - a ) ~A~ rot k2(1 -fl~) - (1 - a ) z A where E~ E ak, (E~ O)t f~ = -Z- = ( ~ p ) ,
(A.5)
1967. Vol. 9, pp. 415-431. P r i n t e d in Great Britain
AN I N V E S T I G A T I O N OF T H E DYNAMIC B E H A V I O R OF AN A N N E A L E D LOW CARBON STEEL B Y MEANS OF STRESS P U L S E AMPLIFICATION N. P. SUH College of Engineering, University of South Carolina, Columbia, S.C.
(Received 6 February 1967 and in revised form 10 March 1967) S u m m a r y - - T h e dynamic behavior of an annealed low carbon steel, A I S I 1020, was investigated by a new method based on stress pulse amplification. The steel had a dynamic yield stress of 70,300 psi at the strain-rate of 123 sec -1. The delay yielding time was 9 × 10 -5 sec. When it is tested at the stress level of 103,000 psi and the strain-rate of 180 sec -1, the delay yielding time was about 3.4 × 10 -5 sec. The experimental results indicate t h a t the upper and the lower yield points observed at low strain-rates do not exist at high strain-rates, and t h a t the effects of intermittent loadings are additive. There also appears to be the possibility t h a t the obstacle to yielding m a y be different at different stress levels. The experimental method consisted of amplifying an elastic, compressive stress pulse by propagating it through a truncated cone machined at one end of a bar and observing the change in the shape of the wave reflected from the conical end, which was against a rigid wall. NOTATION
A,B C
amplitudes of stress waves, constants velocity of wave propagation, (E/p)t Do diameter of the largest section of the cone DT diameter of the smallest section of the cone E Young's modulus $ function for the wave propagating toward the apex g function for the wave propagating away from the apex k wave number = 21r/~ a constant r radial co-ordinate measured from the apex of the cone R. radius at the cone-bar discontinuity t time U
Ct+r
V
displacement of particles along the axial direction
E,,/E Ar
A and P o" Gb O"c
time difference between the wave reflected at the lateral surface and the wave reflected at the end of conical section strain-rate wave length mass density stress magnitude of the incident stress wave in the bar magnitude of the stress wave reflected at the lateral surface measured in the bar 415
416
N.P. SvH
Subscripts i incident wave r reflected wave p plastic region t transmitted wave 1. I N T R O D U C T I O N THE EFFECTS of strain-rate and temperature on the yield stress and flow stress of metals have been investigated b y many investigators. 1-~5 It has been shown that the yield stress increases with strain-rate and decreases with increase in temperature. I t has also been observed that some metals such as annealed low carbon steel exhibit delay yielding phenomena, which depend on stress level and temperature. ~3-25 Although many of the works that have been published are of the highest caliber, various averaging techniques were used in computing the stress and strain-rate. This paper presents a method for determining the yield stress of metals at various stress levels and strain-rates, which seems to minimize this limitation because of the extremely thin plastic region at yield. Basically, the method consists of generating an elastic wave, amplifying it in a converging section such as a conical end of a rod, and observing the reflected wave. A significant change is observed in the reflected wave when the amplitude of the stress reaches the yield stress; this is caused by the reflection of part of the wave at the elasto-plaatic boundary which is established in the cone. It will be shown that this method can also measure the delay yielding time very accurately. This method was applied to annealed low carbon steel, AISI 1020, to determine its yield stress under the dynamic loading conditions as well as its delay yielding time. The work done in the past m a y be generally divided into two groups. One technique consists of investigating the resulting plastic deformation of a specimen, then comparing it with the initial energy input, to compute the average stress. The other technique, which is more currently used, makes use of stress wave propagation. MacGregor and Fisher S,a demonstrated the parallel effects of strain-rate and temperature. The strain-rate was determined b y measuring the change in cross-sectional area of the necked region of a tensile specimen during loading. Because a slow tensile-testing machine was used, their experiments were limited to low strain-rates from 5 × 10 -5 sec -1 to 500 × 10-5 sec -1. Manjoine 4 did similar experiments for strain-rates from 10 -e to 103 see -1 ; high strain-rates were obtained b y loading a tensile specimen axially b y letting a flywheel hit one end of it. This technique, however, permitted only the measurement of the average strain -rate, since the movement of both ends of the tensile specimen was measured. Campbell 5, s attached strain gages to his specimen and assumed that the specimen reached the upper yield point when plastic deformation extended 2 cm from the loaded end. Taylor ~ and Davies s investigated the difference in the yield stress under dynamic loading and under static loading b y a ball indentation technique. They determined the minimum load required to indent a steel surface with a steel ball statically and dynamically. They found that the yield stress under the dynamic
Dynamic behavior of an annealed low carbon steel
417
loading conditions is about twice the static yield stress for materials with low static yield stresses. The differencebetween the dynamic and static yield stresses was not as great for metals with high static yield stresses. Recently a substantial amount of work has been done by Goldsmith et al. ~11 on static and dynamic indentation, to determine more exact indentation-force relations and the flow of the displaced metal. Mok and DuffyTM analyzed the indentation process of a hard ball impacting on various specimens. However, Mahtab et al. 13 suggested, based on their results from experiments and dimensional analysis, that their dynamic tip flattening of a conical projectile is a better method of determining the dynamic behavior of metals than the indentation method. Taylor and Qninney~ also investigated the problem by shooting lead projectiles against an anvil connected to two specimens. As the anvil moved, it stretched both of these specimens equally. The yield stress obtained by this method was found to be much higher than the yield stress observed at quasistatic loading. Taylor 7 and Whiffin14 fired fiat, mild steel projectiles against a rigid wall at speeds ranging from 810 to 2120 ft/sec. By assuming that the elastic-plastic boundary propagates away from the impacted zone with a constant velocity at a constant yield stress, they computed the yield stress from the final shape of the specimen. Their results showed that the yield stress under the dynamic loading conditions was substantially higher than the static yield stress. During World War II, yon Karman 15 and Taylorle derived independently the theoretical relations for plastic wave propagation. Although the theory itself does not take into account the strain-rate effect, the experiments performed by Duwez and Clark 17 with long wires, to verify the theory, yielded the result that the yield stress of mild steel under dynamic loading conditions was 2.9 times the static yield stress. Kolsky TM investigated the dynamic stress-strain relations for copper and lead by modifying the Hopkinson bar. The specimens were sandwiched between steel bars and the incident and transmitted stress waves were analyzed to obtain the average yield stress. A similar technique was later used by Krafft, SulUvan and Tipper 19. Recently, Hauser et al.2°, 91performed similar experiments by sandwiching a specimen considerably smaller in diameter than the wavegnide. Wood and Clark 2s.z5 first investigated the time required for the initiation of plastic deformation in annealed low carbon steel subjected to rapidly applied tensile stress. Hendrickson and Wood 24 advanced a dislocation model to correlate the experimental data, assuming that a sufficient number of dislocation loops must pile up at an obstacle to overcome the theoretical shear stress of the metal. The dislocation loop generation was assumed to be dependent upon thermal activation. As far as the stress pulse propagation in a conical body is concerned, no exact mathematical solution has been found because of the complexity of the problem involved. An approximate one-dimensional analysis made by Landon and Qninney22 is valid only for very small conicity. Through various experiments performed using three different waveguides of various shapes, it was
418
N . P . SvH
f o u n d t h a t a s m a l l p o r t i o n o f t h e s t r e s s p u l s e i n c i d e n t t o t h e c o n i c a l s e c t i o n is a l w a y s r e f l e c t e d a t i ts l a t e r a l s u r f a c e . 2~ T h e r e f o r e , i n a n a l y z i n g t h e w a v e p r o p a g a t i o n in a c o n i c a l b o d y b a s e d o n t h e L a n d o n - Q u i n n e y m o d e l , t h e w a v e reflected at the lateral surface m u s t be considered. 2. G E N E R A L
DESCRIPTION
OF THE
TECHNIQUE
A qualitative discussion of the technique will be given in this section in order to facilitate the quantitative reasoning of the next section. Fig. 1 is a schematic representation of the experimental arrangement. An elastic pulse is generated in the bar by striking it with the hammer, which swings like a pendulum. At the opposite end of the bar is a truncated conical section whose end is against a rigid wall. The arrangement is similar to that of the classical Hopkinson bar. The pulse propagates down the bar without much change in shape, if its wavelength is long compared to the radius of the bar. 27 After the pulse enters the conical section, its amplitude is gradually amplified and a part of the pulse is reflected as a tension pulse at the lateral surface of the cone. When the compression pulse reaches the end of the truncated cone, the pulse is reflected as a compression wave because the bar is against a rigid wall. The returning wave, measured by the strain gages, will then consist of a small tension wave superimposed on a compression wave, provided the dynamic yield stress of the bar is more than twice the amplitude of the incident stress wave. STATIC
v
SR-O
i ORCE
MEASURING CIRCUIT TRIGGER
FIa. 1. Schematic representation of the experimental arrangement. The stress wave amplitude can be amplified to varying degrees by truncating the cone at different places. As the stress wave is amplified b y successively greater ratios, the sum of the incident compression wave and the returning compression wave exceeds the dynamic yield stress of the bar at some point. I f yielding occurs without any delay, this first occurs near the peak of the wave front as the m a x i m u m stress is first reached at this point; otherwise, the yielding m a y commence a little after the stress amplitude reaches the peak value because of the delayed yielding phenomenon, za-25 At any rate, the plastic zone is established near the cone-rigid wall boundary because the stress wave amplification is greatest at this point. As soon as the bar deforms plastically, the mechanical impedance of the plastically deformed zone changes. Then a part of the incident compression wave which has just reached the elastic-plastic boundary is reflected back as a tension wave, because the mechanical impedance of the plastically deformed zone is less than that of the undeformed zone. The returning wave measured by the strain gages is thus the sum of the tension wave reflected at the lateral surface of the cone, the tension wave reflected at the elasto-plastie boundary, and the compression wave reflected at the cone-rigid wall boundary. The
D y n a m i c b e h a v i o r o f a n a n n e a l e d low c a r b o n steel
419
p o r t i o n of t h e w a v e w h i c h is t r a n s m i t t e d t o t h e p l a s t i c z o n e t r a v e l s s e v e r a l t i m e s across t h e p l a s t i c r e g i o n b e f o r e i t is c o m p l e t e l y t r a n s m i t t e d i n t o t h e m a i n p o r t i o n o f t h e b a r , b u t t h e p l a s t i c z o n e is so s m a l l t h a t t h e t i m e lag is negligible. T h e d y n a m i c y i e l d s t r e s s c a n b e d e t e r m i n e d f r o m m e a s u r e m e n t s of t h e i n c i d e n t a n d t h e reflected w a v e s , b y g r a d u a l l y c h a n g i n g t h e a m p l i f i c a t i o n r a t i o so t h a t t h e w a v e reflected f r o m t h e l a t e r a l s u r f a c e c a n b e isolated. T h e s t r a i n - r a t e c a n b e f o u n d a c c u r a t e l y f r o m t h e slope o f t h e s t r a i n - t i m e c u r v e of t h e i n c i d e n t w a v e . T h e s e p o i n t s will b e f u r t h e r d i s c u s s e d in d e t a i l in t h e following section. 3. T H E O R E T I C A L
ANALYSIS
Since t h e r e is n o e x a c t m a t h e m a t i c a l t h e o r y for s t r e s s w a v e p r o p a g a t i o n in a conical b o d y , t h e o n e - d i m e n s i o n a l t h e o r y o f w a v e p r o p a g a t i o n in a conical m e d i u m will b e c o n s i d e r e d t o o b t a i n a t t e n u a t i o n ratio. 2~ T h e a n a l y t i c a l r e s u l t s will b e l a t e r m o d i f i e d t o i n c l u d e t h e w a v e reflected a t t h e l a t e r a l surface a n d t o find t h e c o r r e c t a m p l i f i c a t i o n r a t i o w h e n t h e conical a n g l e is large. T h e a s s u m p t i o n s u s e d in t h e following d e r i v a t i o n a r e t h a t t h e p a r t i c l e m o t i o n a t a n y s p h e r e o f r a d i u s r is p a r a l l e l t o t h e axis, t h a t t h e cone is of s m a l l solid angle, t h a t t h e w a v e l e n g t h is large c o m p a r e d t o t h e d i a m e t e r of t h e cone in t h e r e g i o n o f t h e c o n e w h e r e t h e w a v e is p r o p a g a t e d , t h a t s t r e s s is u n i f o r m o n s p h e r i c a l surfaces, a n d t h a t t h e l a t e r a l a c c e l e r a t i o n is negligible. U n d e r t h e s e a s s u m p t i o n s , t h e w a v e e q u a t i o n for a conical m e d i u m m a y b e w r i t t e n as Os (rv) 1 ~2 (rv) ~r ~ = C~ ~t ~
(1)
T h e s o l u t i o n for e q u a t i o n (1) m a y b e w r i t t e n as v = 1 [ I ( c t + r) + g ( C t - r)] r
(2)
w h e r e t h e f u n c t i o n g is for t h e w a v e t r a v e l l i n g a w a y f r o m t h e a p e x a n d f is for t h e w a v e a p p r o a c h i n g t h e a p e x . I f we c o n s i d e r o n l y t h e w a v e a p p r o a c h i n g t h e a p e x , e q u a t i o n (2) m a y b e w r i t t e n as v = ~l(Ct+r)
(3)
T
The strain then becomes: ~v Or
1 f'(u) --~ f(u) r
where u = Ct+r,
~u Or
~u
I t s h o u l d b e n o t e d t h a t u r e p r e s e n t s a c h a r a c t e r i s t i c of e q u a t i o n (1). T h e s t r e s s c a n b e related to strain by: o =
Or
(4)
S u p p o s e we h a v e a w a v e g e n e r a t e d a t r = R 0, w h e r e R 0 r e p r e s e n t s t h e c o n e - b a r d i s c o n t i n u i t y , of t h e f o r m :
w h e r e A is a c o n s t a n t c o r r e s p o n d i n g t o w a v e l e n g t h . T h e w a v e is a s s u m e d t o s t a r t a t t -~ - R o / C , so t h a t w h e n t h e f r o n t o f t h e w a v e r e a c h e s t h e a p e x , t -- 0. S u b s t i t u t i n g e q u a t i o n (5) i n t o e q u a t i o n (4) a n d s o l v i n g for t h e f u n c t i o n f ( C t + r ) , the stress at the front of t h e w a v e t = - r / C a n d r -- R b e c o m e s : o =
- A ~R0
(6)
420
N . P . SUH
T h u s , we h a v e s h o w n t h a t t h e a m p l i t u d e of t h e s t r e s s a t t h e w a v e f r o n t is amplified so t h a t it is i n v e r s e l y p r o p o r t i o n a l to t h e d i s t a n c e f r o m t h e a p e x , or i n v e r s e l y p r o p o r t i o n a l t o t h e r a d i u s ratio. T h e a m p l i f i c a t i o n r a t i o g i v e n b y e q u a t i o n (6) is v a l i d only for e x t r e m e l y s m a l l c o n i c i t y b e c a u s e of t h e a s s u m p t i o n s u s e d in d e r i v i n g t h e e q u a t i o n . As soon as t h e conical a n g l e is m a d e finite, t h e a m p l i f i c a t i o n r a t i o will n o longer b e a f u n c t i o n of t h e first p o w e r of t h e r a d i u s r a t i o . As a n e x t r e m e e x a m p l e , s u p p o s e t h a t t h e conical e n d is r e p l a c e d b y a series of s m a l l e r s e c t i o n s as s h o w n in Fig. 2. I n t h i s case, tile w a v e i n c i d e n t a t each step will be
J
FIC. 2. R e p l a c e m e n t of a cone w i t h a cascade of s h o r t cylinders.
(A) COMPRES$1ON
WAVE
NCIDENTFRoNT~ WAVE
\
\
TENSION WAVEREFLECTED/
RFACE } AT THELATERAL //
WAV FRONT/y~ E % / v
(c)
NETRE(TA)~N(:)AVE - FIO. 3. S u p e r p o s i t i o n of w a v e s t o e x p l a i n t h e e x p e r i m e n t a l r e s u l t s w h e n t h e r e is n o p l a s t i c d e f o r m a t i o n . reflected as well as t r a n s m i t t e d . I t c a n b e easily s h o w n t h a t t h e a m p l i t u d e o f t h e t r a n s m i t t e d w a v e will b e a m p l i f i e d as t h e s q u a r e of t h e r a d i u s r a t i o r a t h e r t h a n as t h e first p o w e r of t h e r a d i u s r a t i o . T h e a m p l i f i c a t i o n r a t i o for a conical b o d y w i t h a finite c o n i c i t y will t h e n b e t h e n t b p o w e r of t h e r a d i u s ratio, w h e r e n is a p o s i t i v e c o n s t a n t b e t w e e n i a n d 2. T h e r e f o r e , e q u a t i o n (6) s h o u l d b e m o d i f i e d as a = -A
(7)
E q u a t i o n (7) does n o t t a k e i n t o a c c o u n t t h e p o r t i o n of p u l s e reflected a t t h e l a t e r a l s u r f a c e o f t h e cone. I t will b e a s s u m e d t h a t t h e p o r t i o n o f t h e pulse w h i c h r e a c h e s t h e s m a l l e s t s e c t i o n o f t h e c o n e w i t h o u t b e i n g reflected a t t h e l a t e r a l surface is a m p l i f i e d in a c c o r d a n c e w i t h e q u a t i o n (7). I n o r d e r t o find t h e yield s t r e s s let Fig. 3(A) r e p r e s e n t t h e w a v e i n c i d e n t t o t h e conical s e c t i o n , as m e a s u r e d b y t h e s t r a i n gages s h o w n in Fig. 1. I f t h e b a r h a d a fiat end, t h e r e t u r n i n g w a v e w o u l d b e n e a r l y t h e s a m e as Fig. 3(A). T h e solid c u r v e of Fig. 3(B) is o b t a i n e d b y a s s u m i n g t h a t a fixed f r a c t i o n of t h e i n c i d e n t w a v e
Dynamic behavior of an annealed low carbon steel
421
is returned a t the lateral surface of the cone. The curve is plotted AT ahead of Fig. 3(A) because the reflection starts to take place right after the pulse reaches the discontinuity between the conical section and the bar, Fig. 3(C) is obtained b y adding Figs. 3(A) and 3(B). The shape of the wave in the bar reflected at the truncated cone in contact with the rigid wall should look like Fig. 3 (C), provided t h a t the amplitude of the incident stress wave is less t h a n one-half the dynamic yield stress. Fig. 4 illustrates the case when the amplitude of the stress wave reaches the dynamic yield stress. Fig. 4(A) is the incident wave, which would be returned without a change in shape if the bar had a fiat end. Fig. 4(B) is similar to Fig. 3(B), which represents the portion of the wave reflected at the lateral surface of the conical end. Fig. 4(C) represents the wave reflected at the elasto-plastie boundary when there is delay yielding. Fig. 4(D) represents the sum of the waves which returns as a reflected wave to the strain gages of Fig. 1. I t should be mentioned that the peak of the wave on the left m a y not exist if the particular metal being tested does not have a n y delayed yielding.
A-
-tI-
(B)
---~REFLECTED TENSION WAVE W^ ,~- \AT THE LATERAL SURFACE/
I
/ _/t DELAY YIELDING TiME
"l
(c) REFLECTED TENSIO~ WAVE AT THE ELASTO-{ PLASTIC BOUNDARY| (D)
~ET REFLECTEDWAVE A Fio. 4. Superposition of waves to explain the experimental results when there is plastic deformation. I t should be emphasized t h a t Figs. 4(A), 4(B) a n d 4(D) are measurable ones and thus, in practice, Fig. 4(C) has to be deduced from these measured figures. "Between Fig. 4(D) and Fig. 3(C) there is a large difference in shape which takes place over a small difference in amplification ratio. The dynamic yield stress can be computed from Fig. 4 as:
(8) ~b is the amplitude of the incident stress wave shown in Fig. 4(A). The ratio (Doll)r)" is the amplification ratio based on equation (7). ~c is subtracted from a~ since a, is the portion of the pulse reflected from the lateral surface of the cone. The factor 2 is to account for the reflected wave as well as the incident wave. The negative sign in front of ~0 accounts for the tensile n a t u r e of ac, while the negative sign in front of the right-hand side indicates that the yield stress is obtained under compression. ~c can be experimentally obtained for a n y DT b y generating a stress pulse of low amplitude so that the plastic region is not 28
422
N . P . SuH
generated. Once t h e p e r c e n t a g e of the incident w a v e reflected at the lateral surface is known, ac can be c o m p u t e d for all stress levels. The strain-rate is equal to the m a x i m u m strain divided by the t i m e t a k e n for the stress w a v e to reach its m a x i m u m from zero. Thus, =
- e(-~--c) lD0]" 1 E ..... '\DTT! " ~
(9)
where E is the d y n a m i c Y o u n g ' s m o d u l u s and At is t h e t i m e t a k e n to reach the peak. I t should be n o t e d t h a t At can be changed a r b i t r a r i l y either b y using a bar w i t h v a r y i n g cross-sectional area or b y g e n e r a t i n g stress w a v e s in a different m a n n e r . DT at yielding can be d e t e r m i n e d where there is a drastic change in t h e r e t u r n i n g w a v e shape b e t w e e n two n e a r l y equal values of DT. A n o t h e r c o n v e n i e n t m e t h o d of d e t e r m i n i n g the presence of a n y plastic d e f o r m a t i o n was to m a k e successive hits w i t h the s a m e conical end of the s a m e bar a n d measure t h e reflected w a v e s to see if there was a n y change in the a m p l i t u d e s of the reflected w a v e s o b t a i n e d from the first two successive impacts. I t should be m e n t i o n e d t h a t this m e t h o d of d e t e r m i n i n g t h e d y n a m i c yield stress should be applied w i t h c a u t i o n w h e n the m e t a l to be tested possesses t h e delay yielding p h e n o m e non. I f t h e delay t i m e of t h e m e t a l to be tested is longer t h a n the d u r a t i o n of the stress pulse, the reflected w a v e will n o t show a n y effect of plastic deformation, a l t h o u g h it would h a v e u n d e r g o n e plastic d e f o r m a t i o n had t h e pulse lasted for a longer period of time. Since, in t h e e x p e r i m e n t s described here, only two successive i m p a c t s were m a d e to detect a n y change in t h e successive reflected waves, the d y n a m i c yield stress d e t e r m i n e d in this m a n n e r is valid only for a stress pulse shorter t h a n t h a t used in these experiments. H o w e v e r , if the m e t a l does not possess the delay yielding p h e n o m e n o n , t h e n t h e d y n a m i c stress determ i n e d b y this m e t h o d is the true incipient yield stress regardless of the d u r a t i o n of loading. The a m o u n t of t h e incident w a v e reflected at the elasto-plastic b o u n d a r y depends on the impedances of the plastically d e f o r m e d and elastic regions. The specific i m p e d a n c e is g i v e n b y (see A p p e n d i x for details): Specific i m p e d a n c e -- [~-~ p}
(10)
where ~a/~e is the stiffness or m o d u l u s a n d is equal to the Y o u n g ' s modulus in the elastic region, a n d p is mass density. T h e v a l u e of ~a/Oein the plastic region is always leas t h a n the Y o u n g ' s modulus. W h e n ~a/ae in the plastic region is zero, t h e w a v e is c o m p l e t e l y reflected as a tension w a v e at the elasto-plastic b o u n d a r y and w h e n Oa/~e is equal to the Y o n n g ' s modulus, all t h e w a v e is t r a n s m i t t e d w i t h no r e f l e c t i o n . I f Oa/~ is n e g a t i v e such as during the t r a n s i t i o n f r o m t h e u p p e r yield p o i n t to the low yield point, the expressions for the velocity of the plastic w a v e p r o p a g a t i o n and the i m p e d a n c e become imaginary, which is n o t a t t a i n a b l e , and thus, equal to zero. Therefore, in this case, m o s t of the incident w a v e m u s t be reflected as a tension w a v e at the elasto-plastic b o u n d a r y . ~s 4. E X P E R I M E N T S The e x p e r i m e n t a l a r r a n g e m e n t was as shown s c h e m a t i c a l l y in Fig. 1. T h e bars, which were m a d e in t h r e e sections, were h u n g on wires. The longer, s t a t i o n a r y section consisted of two bars in series: t h e bar with the strain gages was m a d e of cold rolled A I S I 1018 steel a n d t h e specimen b a r w i t h t h e conical end was m a d e of annealed A I S I - 1 0 2 0 steel. This was done because t h e annealed steel bar b e c a m e easily m a g n e t i z e d u p o n r e p e a t e d impacts, and, as a consequence, t h e strain gages act as a source of e l e c t r o m o t i v e force during t h e passage of t h e stress pulse, eatming erroneous m e a s u r e m e n t s . The entire s t a t i o n a r y section w i t h t h e conical end was p u s h e d against a ground, hardened, 3 × 5 in. flat plate of tool steel w i t h a static force of a b o u t 10 lb, b y m e a n s of two springs. This plate was in t u r n m o u n t e d on a larger steel cylinder a p p r o x i m a t e l y 6 in. dis. and 14 in. long. All c o n t a c t surfaces were lapped fiat a n d s m o o t h w i t h 5 micron a l u m i n u m oxide p o w d e r to ensure t h e t o t a l trausmission of w a v e s across the interface. One end of t h e s t a t i o n a r y bar w i t h t h e strain gages was m a c h i n e d r o u n d to ensure t h a t t h e i m p a c t s b y the shorter bar, which will be called the h a m m e r , be m a d e a t its center. T h e h a m m e r was m a d e of annealed A I S I 1020 steel a n d was 15 in. long for m o s t of the
Dynamic behavior of an annealed low carbon steel
423
experiments. The h a m m e r was raised to a predetermined height to generate a stress pulse of a given a m p l i t u d e . The impacting ends of the h a m m e r and the bar were heat treated to prevent plastic deformation. The full conical angle was 30 ° in all experiments. The strain pulses were measured by ~ in. long foil strain gages (FAP-12-12, BLH} and a dual-beam oscilloscope (Tektronix 502 A}. The strain gages were mounted opposite each other in order to cancel the effect of bending. Pictures of the strain traces were taken by Polaroid cameras mounted on the oscilloscope. The experiments were conducted by first making a short cone, the end of which was lapped fiat perpendicular to its axis. I n successive experiments the conical section was made longer and longer to obtain greater stress amplification. After a new conical section was machined, at least two successive impacts were made to compare the shapes and amplitudes of the reflected waves. In the absence of any plastic deformation, the reflected waves were identical: A part of the conical end, at least ¼ in. long, was cut off after each experiment in order to eliminate any work-hardened portion of the cone. By gradually changing the diameter of the smallest section of the truncated cone (DT), the point at which plastic yielding began to occur was found. When the h a m m e r rebounded after the stress pulse returned to it, it was prevented from making the second impact, so the specimen was loaded only once per impact. Many of the experimental results obtained were checked by repetition. The static yield stress was determined by means of a typical tensile test. The specimens were made from the same rod t h a t was used to make the dynamic measurements. Some of the experiments were repeated and it was found t h a t they could be reproduced exactly. The strain-rate was of the order of 3 × 10 -~ sec -1. The upper static yield stress was determined to be 40,500 psi and the lower yield stress was determined to be 37,500 psi. The particular steel specimen used in these experiments was chemically analyzed by the manufacturer. The chemical composition is as follows: Carbon Manganese Phosphorus Sulphur Silicon
0.19 per cent 0.50 per cent 0.01 per cent 0.028 per cent 0.04 per cent
Copper Nickel Chromium Aluminum Tin
0.04 per cent 0.03 per cent 0.02 per cent 0.01 per cent 0.008 per cent
This paper is based on three different types of experiments performed with annealed low carbon steel. The first set of experiments was performed to verify the validity of the theoretical model and prediction. Stress pulses of different amplitudes were generated by changing the height of the hammer, and the size of the conical section was changed until yielding occurred at a given stress level. Another set of experiments was performed to investigate the dynamic behavior of the metal at stresses higher than the incipient yield stress. The third set of experiments was done to determine the influence of the pulse length, by changing the length of the hammer from 15 to 30 in. 5. E X P E R I M E N T A L
RESULTS
AND DISCUSSION
(a) Verification of the theoretical model Figs. 6-11 show the experimental results. These figures were prepared by superimposing m a n y experimental results such as t h a t shown in Fig. 5, When D r was equal to or greater than 0.340 in., the shapes and the amplitudes of the reflected waves from successive impacts were the same, indicating t h a t there was no plastic deformation (Fig. 6). The similarity between Fig. 6 and Fig. 3(C) should be noted ; both are cases where the amplitude of the stress does not exceed the dynamic yield stress of the metal. I n Fig. 6 the difference between the horizontal levels of the incident wave and the reflected wave is due to the tension wave reflected at the lateral surface of the cone. I n order to further verify the theoretical model qualitatively, a case involving plastic deformation will be illustrated here. This is most drastically illustrated by the first loading curve shown in Fig. 11, which is for DT = 0.250in. I n this case, there was plastic deformation as shown by the successive changes in the reflected waves under repeated loading. The similarity between this curve and Fig. 4(D) should be noted.
424
N. P. SUH _
L.
l 1 !
I n c i d e n t and reflected Dr : 0 " 3 4 0 in
]
;
Reflected waves P r = 0 . 2 5 0 in
Ist
loading
3rd
loading
FIG. 5. A t y p i c a l e x p e r i m e n t a l record.
INCIDENT
REFLECTED
WAVE
DT . . . . . . . . . . . . . . . . . . . . . . . .
WAVE
0.340"
AMPLITUDE OF STRESS . . . . . . IN BAR (INCIDENT WAVE)
1 0 , 2 0 0 psi
F r o . 6. E x p e r i m e n t a l r e s u l t s w h e n DT = 0.340 in.
f
INCIDENT
WAVE
REFLECTED
WAVE
,., 2nd . . . . . . 3rd . . . . . . 4th --x--x-
DT ...........................
0.330"
AMPLITUDE OF STRESS . . . . . . 10,200 psi IN BAR (INCIDENT WAVE) STRAIN
RATE . . . . . . . . . . . . . . .
FIG. 7. E x p e r i m e n t a l r e s u l t s w h e n
123 sec?l
Dr = 0.330 in.
wove.,
Dynamic behavior of an annealed low carbon steel LOADINGS iNCIDENT
I st 2nd . . . . . .
WAVE
REFLECTED
DT . . . . . . . . . . . . . . . . . . . . . . . . . . . . AMPLITUDE
OF
INCIDENT
STRESS
WAVE
0.330" WAVE
IN
BAR--- IO~-~OOpsl
FxG. 8. Experimental result~ when D r = 0.330 for a longer pulse length (30 in. hammer).
INCIDENT
WAVE
S
REFLECTED
WAVE
2rid . . . . 6th --x--~-
DT . . . . . . . . . . . . . . . . . . . . . . . .
0.320"
AMPLITUDE OF STRESS . . . . . . 10,200psi IN BAR (INCIDENT WAVE) F I G . 9. E x p e r i m e n t a l results when D r = 0.320 in.
INCIDENT
REFLECTED
WAVE
j l
I st -
-
2nd . . . . . . . 3rd - - x - - x 4th --A--,
OT . . . . . . . . . . . . . . . . . . . . . . . . . AMPLITUDE
OF
INCIDENT
WAVE
LOADINGS
STRESS
0300"
WAVE
FxG. 10. Experimental results when
IN
BAR . . . . . . 10,200 psi
DT =
0.300 in.
425
426
N . P . SuH
The amplification ratio will be e v a l u a t e d q u a n t i t a t i v e l y . As shown in Table 1, a large n u m b e r of e x p e r i m e n t s were p e r f o r m e d at three different heights of the h a m m e r to d e t e r m i n e D T at which yielding t o o k place. Using these data, the yield stress and the INCIDENT
WAVE
LOADtNGS
REFLECTED
2nel . . . . . . WAVE
WAVE
/<.;~'- ~ k ~
3~d
17th
O T ............................ AMPLITUDE MAXIMUM
OF
0.250"
STRESS
STRAIN
.......
10.200psi
R A T E . . . . . . 180 sec -~
FIG. 11. E x p e r i m e n t a l results w h e n D T = 0"250 in. c o n s t a n t n of e q u a t i o n (7) were d e t e r m i n e d b y s u b s t i t u t i n g the d a t a into e q u a t i o n (7). F o r t h r e e s e p a r a t e sets of data, three sets of t h e following e q u a t i o n can be o b t a i n e d : (o'b - oc), {DT,~ n (ab-oc)~ = ~D-~%!
(iI)
where the subscripts I a n d 2 refer to different sets of data. T h e values of n t h u s d e t e r m i n e d are all w i t h i n 3.4 per c e n t of 1.55. T h e r e m a r k a b l y close v a l u e s of n indicate t h a t the relation, e q u a t i o n (8), is a sound one to use. Table 1 also shows the error estimation. T h e p e r c e n t a g e of the w a v e reflected a t the lateral surface is found to be n e a r l y the same b e t w e e n D r --- 0.370 in. and D T = 0.250 in. T h e reason it does n o t change linearly w i t h DT, as one m i g h t expect, is due to t h e multirefleetion of t h e w a v e s t a k i n g place in t h e cone. Because of t h e large angle b e t w e e n t h e w a v e front a n d t h e lateral surface, even t h e p a r t of the w a v e which is reflected a t t h e lateral surface p r o p a g a t e s forward until it is reflected a few m o r e t i m e s a t o t h e r p a r t s of the lateral s u r f a c e . " TABLE 1. EXPERIMENTAL DATA AND RESULTS Wave
Stress a m p l i t u d e reflected H a m m e r of t h e incident DT at ~ at the height w a v e in t h e b a r yielding lateral (in.) (psi) (in.) surface (%) Exp. 1 Exp. 2 Exp. 3 Max. error Max. dev. f r o m the average value
12 24 36
7,240 10,200 12,500
0.265 0.33 0.375
1.1%
_+ 2% _+2 %
_+1.8%
n
Yield stress c o m p u t e d by e q u a t i o n (8)
8 8 8
1.58 1.58 1.50
70,380 70,380 70,380
+_5
+ 4%
( E v a l u a t e d a t n = 1.55)
+1.5
± 3.4%
_+4.0%
Dynamic behavior of an annealed low carbon steel
427
The delay yielding time at this stress level, 70,380 psi, was found to be 9 x 10-6 sec. The strain-rate was 123 sec -1, which was obtained by considering the middle 80 per cent of the wave front. When this value was compared with the results obtained by Hendrickson a n d Wood 24, it was found t h a t it was almost the same as their results. Although the steel used in the present experiments had 0.02 per cent more carbon and 0.11 per cent more manganese t h a n that used by Hendrickson and Wood, it had 0.007 per cent less phosphorus and 0.012 per cent less sulphur. Therefore, the fact that both of these results agree so closely is not too surprising. On the other hand, the fact that even the data shown by these investigators sometimes deviated from one to another as much as by one order of magnitude in the delay time, such a close agreement between the two results, which were obtained by entirely different methods, must be partially a coincidence. I t should be mentioned at this point t h a t all the experimental data presented here are from a single rod specimen about 20 in. long. The experiments performed with other specimens of the same material cut from different rods showed a delay yielding period as long as 3 x 10-4 sec. E v e n this value was within the experimental scatter of Hendrickson and Wood's work. The foregoing quantitative and qualitative evaluation of the experimental data indicates that the theoretical model advanced in this paper is a reasonable one. (b) Behavior of the steel beyond the incipient yield point Fig. 7 presents the experimental results obtained at D r = 0"330 in. The reflected wave from the first loading is almost identical to the reflected wave shown in Fig. 6 which was for D r = 0.340 in. However, the reflected wave from the second impact has a much lower amplitude, indicating that there was plastic deformation at the conical end at DT = 0.330 in., whereas when D r = 0.340 in., both of the reflected waves were identical. The first reflected wave of Fig; 7 did not show a n y plastic deformation, being identical to the reflected wave of Fig. 6, because the delay time for yielding was nearly the same as the duration of the pulse. This was substantiated b y the experiments performed using the longer hammer (30 in.) as shown in Fig. 8. Fig. 8 shows that after an elapse of 9 x 10-6 see., the amplitude of the reflected wave suddenly decreased to a lower value. This lower amplitude corresponds to the second reflected wave shown in Fig. 7. The decrease in amplitude is due to the tension wave reflected at the elasto-plastic boundary, which is due to the smaller specific impedance of the plastically deformed region, as discussed earlier. I t should be mentioned here that the transition in the amplitude after a given delay period was sometimes more gradual t h a n the results shown in Fig. 8. I t is interesting to note that although the third reflected wave (Fig. 7) has nearly the same amplitude as the second reflected wave, the amplitude of the fourth reflected wave is somewhat lower t h a n the others. This indicates that the specific impedance in the plastic region decreased further due to softening of the metal. One of the most important results of these experiments is t h a t the dynamic stress-strain relation at and near the incipient yield point m a y be quite different from that of the static case. Under dynamic loading them seem to be no upper and lower yield points, which exist at low strain-rates. I f there were upper and lower yield points, the impedance would have been zero during the transition from the upper to the lower yield point, since ~a/~e is negative, provided the theoretical argument given in the earlier section on the impedance of the plastically deformed zone, equation (10), is correct. Therefore, the incident wave would have been reflected predominantly as a tension wave. However, the experimental results show only a small contribution b y the tension wave reflected at the elasto-plastic boundary. Therefore, the change observed in Figs. 7 and 8 is probably caused b y the decrease in the slope of the strew-strain curve at the yield point. On the other hand, it is quite feasible t h a t our understanding of the plastic wave propagation during the transition from the upper yield point to the low yield point m a y need further improvement. The question whether the effect of several loadings is additive will be considered here. The results shown in Fig. 7 indicate t h a t although the first loading was purely elastic, i.e. no macroscopic plastic deformation, the effect of the first loading was present when the specimen was loaded the second time. Also, the experiments with a longer hammer (Fig. 8) indicate t h a t the behavior of metal was dependent upon the total loading time
428
N . P . SUH
regardless of the particular pulse length used. The amplitude of the latter half of the first loading curve of Fig. 8 is the same as the second loading curve of Fig. 7. I t should be pointed out here t h a t in any one set of experiments the successive tests were made within an interval of less than 3 min. ttendrickson and Wood 24 derived the theoretical expression for delay time based on the assumption t h a t the delay time is the time required to generate enough dislocation loops at the F r a n k - R e a d source to overcome the ideal shear strength of the metal at a dislocation pile-up. The generation of dislocation loops was assumed to be controlled by the thermally activated release of the source dislocation from its "atmosphere". Therefore, the fact t h a t the effect of the loading is additive implies that the dislocations which were released from the " a t m o s p h e r e " must not return to their original position after unloading, if the proposed model is a correct one. This was observed in copper crystals by Young. 29 He observed t h a t the dislocations which moved under a low stress did not return to their original positions upon unloading, while the dislocations t h a t moved under a high stress did partially return toward their original positions upon unloading.
WAVE EXPECTED-~ i IF DELAYING MECHANISM [~
IS ALWAYS THE
;A! ~~
// ~ \ fACTUAL WAVE
Fro. 12. Figure illustrating the difference in the reflected wave for different delay yielding mechanisms. The experiments done with DT = 0.320 in. (Fig. 9) are interesting in t h a t during the second loading the metal continuously softened. The rate of softening is faster at high stresses. Upon the sixth loading, the reflected wave shows work hardening, as indicated by the increase in the amplitude of the reflected wave. The experiments done with DT ---- 0.300 in. (Fig. 10) further illustrate the rapid rate of softening at high stresses; the first reflected wave of Fig. 10 is nearly the same as the second reflected wave of Fig. 9. Fig. 11 presents the results obtained when the metal was loaded to an extremely high stress, 103,000 psi, at a strain-rate of 180 sec -1. These values are obtained from equations (8) and (9) when D r = 0.250 in. The sharp drop in amplitude of the initial reflected wave and the delay time of 3.4 × 10 -5 see should be noted. The experiments performed using a stress pulse with small amplitude to stay within the elastic limit indicate t h a t about 8 per cent of the incident wave is reflected at the lateral surface. One of the interesting results of these experiments is indicated by the amplitude of the first reflected wave at the wave front (Fig. 11). The delay time shown here is not as meaningful as t h a t shown by Fig. 8, as the stress level was not constant for an appreciable period of time. I t is probable t h a t the delay time, which decreases at increasing stress levels, is shorter in this test than the rise time of the applied stress wave. I n this case, yielding would occur before the m a x i m u m stress is reached, and the test is one of constant strain-rate. However, ff the delay time is not shorter than the rise time of the applied stress wave, ff the same delay yielding mechanism is at work at all stress levels, and ff only the rate of dislocation loop generation changes as a function of stress amplitude, as assumed in the analysis of Hendrickson and Wood 24, then the amplitude of the wave front must be t h a t shown by the dotted line instead of the results obtained (Fig. 12). I f this is the case, the solid line of Fig. 12 indicates t h a t at this level the pinning mechanism which was responsible for the delayed yielding at a low stress leveI had already been overcome without any delay, when the applied stress was increased to a higher value. This can be done if the stress required to overcome the obstacle is not always equal to the theoretical ideal shear strength of the metal, thus requiring different numbers of dislocation pile-ups for shear motion to continue. Based on his experimental results, Young aa postulated a similar mechanism for pure, single-crystal copper. F u r t h e r experiments with stress pulses
Dynamic behavior of an annealed low carbon steel
429
of short rise time need to be performed to resolve this question. Indeed, such experimental results may be useful in checking the model of yielding advanced by Hahn sl. Another interesting fact from Fig. 11 is that the metal work hardens from the second loading onwards, without the initial softening of the metal as observed when the applied stress was low. Furthermore, as indicated by the third and fourth curves, a continuous hardening and softening takes place even after the metal work hardens, as indicated by the top horizontal level of the reflected waves. All the loadings shown in Fig. 11 were done within 24 hr at room temperature, so it is unlikely that there was any aging effect. This could be due to the partial return of dislocations to their original positions upon unloading, as Young observed in copper at high stress levels, z° 6. C O N C L U S I O N The m e t h o d used here provides a useful m e a n s o f investigating the d y n a m i c b e h a v i o r o f metals. The incipient yield stress of annealed low c a r b o n steel with 0.19 p e r cent c a r b o n is f o u n d to be 70,380psi with the d e l a y time of 9 x 10 -5 sec. As t h e a m p l i t u d e o f stress increases, t h e delay time becomes shorter in a c c o r d a n c e w i t h published results. B a s e d on the experimental results presented in this paper, it can be concluded t h a t the effect of m e c h a n i c a l loading is additive, a n d t h a t t h e u p p e r a n d lower yield points m a y n o t exist a t high strain-rates. There also appears to be the possibility t h a t the e x a c t cause for d e l a y yielding m a y be different a t different stress levels. Acknowledgment--This work was done at the University of South Carolina under the sponsorship of the National Science Foundation, Grant GK-899, for which the author is grateful. The author wishes to thank Mr. R. S. Lee and Mr. C. R. Rogers for their help in the preparation of this paper. The helpful comments of Prof. Milton C. Shaw of the Carnegie Institute of Technology are gratefully acknowledged. Mr. W. J. Dockcry of the Atlantic Steel Company kindly furnished the chemical analysis of the steel.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. ll. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
B. HOPKINSON, Prec. R. Soc. A74, 498 (1905). C. W. MAGGREGORand J. C. FISHER, J. appl. Mech. 12, A217 (1954). C. W. MAcGR~.OORand J. C. FISHEtt, J. appl. Mech. 13, A l l (1946). M. J. MANJOI~r~,J . appl. Mech. l l , A211 (1944). J. D. CAMPBELL,J. Mech. Phys. Solids l, 113 (1953). J. D. CAMPBELL,Acta Met. 1, 706 (1953). G. I. TAYLOR,J . lnst. civil Engr 26, 486 (1946). R. M. DAVrES, Prec. R. Soc. A197, 416 (1949). W. GOLDSMITHand P. T. LYMAN,J . appl. Mech. 27, 717 (1960). W. GOLDSMrrHand C. H. YEW, Proceedings of the 4th U.S. National Congress of Applied Mechanics, Vol. 1, p. 177, ASME, New York {1963). C. H. YEW and W. GOLDSMITH,J . appl. Mech. 64 APM-38 (1964). C. H. Moor and J. DU~FY, Int. J . mech. Sci. 7, 355 (1965). F. U. MAHTAB,W. JOHNSON and R. A. C. SLATER,Int. J . mech. Sci. 7, 685 (1965). A. C. WHIFFI~, Prec. R. Soc. A194, 1038 (1948}. T. voN KARMAN,National Defense Res. Counc. Rep. A-29 (1942). G. I. TAYLOR,British Official Report No. RC 329 (1942}. P. E. DuwEz and D. S. CLARK, Prec. am. ~oc. Test. Mater. 47, 502 (1947). H. KOLSKY, Prec. phye. See. London 65, 677 (1949). J. M. K . R ~ , A. M. SULLIVA~and C. F. TIPPEtt, Prec. R. Soc. A221, 114 (1953). F. E. HAUS~.Rand C. A. WrS,rER, Report to Convair, Series 133, No. 4, University of California, Berkeley (1960). F. E. HAUS~.R, J. A. Sr~MONS and J. E. Dott~, Report to Convair, Series 133, No. 3, University of California, Berkeley (1960).
430
N . P . SUH
J. W. LANDON and H. QUINNEY, Prec. R. Soc. AI03, 622 (1923). D. S. CLARK, Trans. Am. Soc. Metals 46, 34 (]954). J. A. I-IENDRICKSO~and D. S. WOOD, Trans. Am. Soc. Metals 50, 498 (1958). D. S. Wood and D. S. C L ~ x , Trans. Am. Soc. Metals 43, 571 (1951). N. P. SUH and W. F. STOKEY,to be published. R. M. DAVIES, Phil. Trans. A 2 0 0 , 375 (1948). A. H. CO~rRELL, Prec. Conf. Properties (~ Materials at High Rates of Strain, London, p. 1 Inst. Mech. Engrs (1957). 29. F. W. YouNo JR., J. appl. Phys. 32, 1815 (1961). 30. F. W. YOUNG Jm, J. appl. Phys. 33, 963 (1962). 31. G. T. HAHN, Acta Met. 10, 727 (1962). 22. 23. 24. 25. 26. 27. 28.
APPENDIX
Wave reflection at the elaeto-plastic boundary I n this analysis all the assumptions made in deriving equation (1) of the text are assumed valid. I n addition, the material in the plastic state will be assumed to be linearly
=,; zo:
/
PLASTICZONE
r i
X =
FIG. 13. Cone with the elastic-plastic boundary. work-hardening ideal plastic metal. With the co-ordinate system defined as shown in Fig. 13, the displacement due to the incident wave l/~ may be expressed as:
V~ =
A~ sinb(Ct+x) ro+X
(A.I)
where A~ is the amplitude of the incident wave at a point where (re+x) is unity, kC = w = angular frequency, and r 0 is the distance from the apex of the cone to the elasto-plastie boundary. The displacement due to the reflected wave V~and the displacement due to the transmitred wave V~m a y be represented b y :
V~= Ar s i n k ( C t _ x ) + _ B , cosk(Ct-x) ro+X ro+X A, sink~(C~t+x)+ B ~ cosk~(C~t+x) Vt = ro+x ro+x
(A.2)
where the subscript p denotes the plastic region. The boundary conditions at x = 0 are V~+ ~ = V,, (A.3) A c c o r d i n g to the strain.rate independent theory of plastic wave propagation, zs, 16 the velocity of wave propagation in the plastic region is given b y
c~
= ~
0 = ~,/0
(A.4)
Dynamic behavior of an annealed low carbon steel
431
Combining equations (A. 1), (A.2) and (A.3), the reflected and the t r a n s m i t t e d amplitudes are found to be 2r o ko(1 - a) B, = B, = res kt(l + f l ) s + (1 - ~ ) 2 A,
2roS k'(~ + 1) At = r~ ks(1 +fl)~+ (1 - a ) ~A~ rot k2(1 -fl~) - (1 - a ) z A where E~ E ak, (E~ O)t f~ = -Z- = ( ~ p ) ,
(A.5)