International Journal of Material Phyic. ISSN 97-39X Volume 5, Numer 1 (1), pp. 15- International Reearch Pulication Houe http://www.irphoue.com Dimenion Effect on Dynamic Stre Equilirium in SHPB Tet Department of Phyic, Faculty of Science, Univerity of Barah-Iraq E-mail: njarrih@yahoo.com Atract The tate of dynamic equilirium of tre wave in a pecimen under tet y SHPB i not automatically achieved due to the effect of loading rate and the pecimen dimenion. Therefore, a guideline for proper SHPB experiment and analytical invetigation have een conducted to examine the proce of wave equilirium in different material pecimen. Compreive experiment on different material with SHPB were conducted to determine the effect of length and diameter a well a the denity of the material on the time required for the tre equilirium to e reached within the pecimen. The loading rate i kept in all meaurement a the loading pule more like the actual loading tre pule in SHPB to have a etter idea of the real repone of the material under a contant train rate. Although, the actual incident pule haped i not very uitale for a parametric tudy of the thickne effect. Keyword: SHPB, tre equilirium, multiple reflection, dimenion effect. 1. Introduction The plit Hopkinon preure ar (SHPB) technique (Fig. (1)) i a well-etalihed method ued for the determination of high train rate propertie of material. In the SHPB tet, the ample (a mall olid cylinder of the teted material) i andwiched etween two long-high trength teel ar. The ample can e compreed y a tre pule generated y impacting the end of one of the teel ar (incident ar) uing a
1 teel projectile. The projectile ha the ame diameter a the teel ar, and a length that i appropriate for providing uitale loading pule duration. The tre pule in the ar are recorded y train gauge placed in equiditant from the ample. The tre-train propertie can e otained from the amount of the tre pule reflected and tranmitted y the ample, auming that tre equilirium exit throughout the ample. When the tre wave travel along the incident ar in a poitive direction, it hit the firt interface etween the incident ar and the ample. The difference in impedance etween the ar and the ample make the wave partially reflect ack in the negative direction a a reflected pule R, while the ret of the wave pae through the firt interface a a tranmitted pule (from the firt interface) at time zero T. The T travel toward the econd interface etween the ample and the tranmitter ar and again due to the impedance mimatch, part of T will reflect ack toward the firt face and the ret will tranmit into the tranmitter ar. The time required for the T pule to reach the econd face of the ample, or in other word, the time required for the pule to travel etween the two face of the ample i called the travere time tt, which defined a: tt c where i the ample length and c i the wave peed in the ample. Therefore, after one period of travere time, the reflection occur at the econd face of the ample creating R1 and T1. Thi proce continue, o that multiple reflection occur within the ample and a ucceion of reflected wave ecome "trapped" inide the ample propagating ack and forth etween the two interface [1]. Theoretically, reflected wave thu "trapped" in thi manner will undergo an infinite numer of reflection etween the interface; however, at each reflection the intenity of the reflected tre will decreae ince a portion of the wave i tranmitted each time. Eventually, the trapped wave will have decayed to negligile amplitude. The effect of multiple reflection within the ample i to caue a diperion of the incident wave. Thu, if the incident wave ha a harp rie time efore reaching a contant maximum tre, the tranmitted wave will have a le harp rie time. Thee multiple reflection caue a non-uniform tre ditriution that may lead to inaccurate etimate of the initial tre/train propertie of the ample []. The theory of SHPB analyi i aed on the equation T I R. Thi equation i only true if the force and therefore the tree are equal on oth ide of the ample. Thi equilirium condition will not arie immediately a the tre wave i incident on a SHPB ample, ut occur after everal reflection have taken place inide the ample.
Dimenion Effect on Dynamic Stre Equilirium in SHPB Tet 17 Fig. 1: Schematic diagram for SHPB ytem.. Theory Wave propagation ehaviour for elatic ar i well-undertood and mathematically predictale [8], and from the elementary wave theory, the wave equation can e hown: u 1 u (1) x C t o Where C E / i the fundamental longitudinal wave velocity, and u i the diplacement. From the wave equation, the tre in the ar can e hown to e C v, where i the denity of the ar and v i the velocity of the particle of the ar ujected to the pule. Conider an incident elatic wave of compreive tre I moving to the right a in Fig. (), through the ar 1 of cro-ectional area A 1. Thi wave i partially reflected and partially tranmitted at the urface of dicontinuity AB, where another ar of cro-ectional area A i perfectly attached to 1. If A were zero, the wave would e reflected completely, whilt if 1 and were of identical area and material, then the incident wave would e totally tranmitted. However, ince 1 and have different area and are of different material, then at AB the incident wave mut e reflected and tranmitted. V 1 1 I A 1, Z 1 R A T A, Z B Fig. : Interface etween two ar.
18 The tre wave tranmitted through i T, and reflected ack through 1 i R. Where the initial tre in 1 i I Z11 at the plane AB the following condition are atified: i. the force acting on the plane AB acting from 1 and are equal at all time, and, ii. the particle velocity in plane AB, in the material for 1 and are equal. iii. According to (i) we have, auming I, R and T are taken to e compreive, then A1 ( I R ) A T, () I, and R are aociated with wave travelling in oppoite direction, therefore, (ii) give I R T (3) where denote particle peed and ucript I, R, and T refer to incident, reflection, and tranmiion. In general the tre () i related to denity (), ound peed (c), and particle peed () y:. c Z The tranmitted and reflected tree can e derived to e; A 1Z T I T I () A1 Z1 AZ The reflected tre i otained a: A Z A1 Z1 R I R I (5) A1 Z1 AZ Where T and R are the tranmiion and reflection coefficient repectively. 3. Analyi and Computation The principle of SHPB technique are well-documented []. The theory of SHPB how that the nominal train, train rate., and nominal tre are given y the following equation: c t Rdt (). (7) c R A ET (8) A
Dimenion Effect on Dynamic Stre Equilirium in SHPB Tet 19 Incident ar Sample Tranmitter ar A Z B1 A Z B A Z l Fig. 3: Preure ar and ample. Conider an elatic tre wave incident on the firt interface reflection from thi interface occur at time t= a in Fig. (3), where: (B1). The where, and A are the length and the cro-ectional area of the ample, while c, A, and E are the wave peed, cro-ectional area and Young' modulu for the ar repectively. The aove equation have een derived auming that tre equilirium exit in the ample. The tre at the incident ar/pecimen interface ( B1 ) i ( I R )A B1 A And the tre at pecimen/tranmitter ( B ) ar interface i T A B A Where I, R and T are the incident, reflected and tranmitted tre in the ar repectively. Auming incompreile platicity, then Ao o AS S (wher e A o i the original cro-ectional area of the pecimen and A S i the intantaneou cro- ectional area of the pecimen, and A i the cro-ectional area of the ar. firt
I = incident tre, = reflected tre at B1, R T1= tranmitted tre at the interface B at time t= / c which i called the travere time (tt), and, RS1= reflected tre at B at time tt. If the incident tre wave ha a finite duration, then the tre I may e time dependent. At t = 1 travere time, the tre i tranmitted into the ample. It i important to note at thi tage that if I i compreive (+ve), then according to equation () T1 will alo e compreive (+ve), while from equation (5) R may e compreive (+ve) or tenile (-ve) depending on the mechanical impedance Z of the ample and it croectional area A. Equation and 5 can e re-written for the SHPB a; T A Z I A Z A Z (8) R A Z A Z I (9) A Z A Z a: and the tranmiion and reflection coefficient can e written at the interface B1 A Z T 1 = A Z A Z (1) A Z R 1 = A Z A Z A Z (11) At the interface B the reflection will occur inide the ample, o the coefficient i denoted a R and the tranmiion coefficient a T (where the tre wave ha tranmitted partially into the tranmitter ar). R A Z A Z R 1 A Z A Z (1) A Z T (13) A Z A Z
Dimenion Effect on Dynamic Stre Equilirium in SHPB Tet 1 For a compreive incident tre, the tranmitted tre will alway e compreive; while the reflected tre can e tenile or compreive. Uually A Z A Z making R 1 negative and R poitive []. The uild-up of the reflected and the tranmitted pule in the preure ar caued y the multiple reflection etween the interface, and the uild up of the tranmitted and reflected tre pule in the SHPB ample can e equated a; At travere time TS T1 I and R R1 I, and at 1 travere time T1 T TS T1T I, RS1 R RS R1T 1 I and R1 R1 I1. Where at the econd travere time the tre are; T T1T I1, RS R RS1 T1 I R1 T1 I1 T1 I and R R1 I R1T 1T I and o on. After N of travere time the tranmitted and reflected tree equal ; TN R1 T ( N ) T1T I ( N 1), for N and RN R1 ( IN T ( N 1) ), for N 1 repectively. In tandard SHPB theory, the tranmitted tre T i proportional to the actual tre of the ample. Thi cannot e correct unle tre equilirium ha een achieved. Stre equilirium occur when the equation I R T i atified. So, the equilirium condition can e achieved when the ratio 1 i atified. I T R. Reult and Dicuion Undertanding the way the SHPB ytem operate help to make ome prediction aout the nature of the computed reult, uch a: 1) The tranmitted pule alway ha the ame ign a the incident pule a can e een from equation (8) while from equation (9), the reflected pule doe not alway have the ame ign, ut rather depend on the value of the cro-ectional area and the impedance of the ample compared with thoe of the ar. If A Z A Z, the reflected pule will e of oppoite ign to the incident pule. ) After the tre pule pae through the ample the multiple reflection inide the ample take a long time to decay, hence equilirium no longer exit etween the ar and the ample until the time tend to infinity. So the equilirium ratio T /( I R ) will ocillate with a period of tt -the time required for the pule to return to the interface. The aove pule are hown in Fig. () for HDPE, Nylatron, CFC and Aluminium ample with length/diameter ratio of /8. It can eaily e noticed that the more dene material and higher ound peed (higher impedance ample), the horter time required for the equilirium to e achieved. 3) Becaue of the time required for the pule to propagate through the ample, the tranmitted pule tart one travere time (tt) after the reflected pule. Thi delay
can e compenated experimentally y appropriate poitioning of the train gauge, ut would e inconvenient for ample of different thicknee. ) In general, the maller the cro-ectional area of the ample, the greater the reflected pule and the maller the tranmitted pule and longer time i required for the equilirium a hown in Fig. (5). 5) The trend for all the computed reult i that the normalied tranmitted pule tend to 1 and the reflected pule vanihe to zero [1]. The time thi proce take depend on the cro-ectional area and the mechanical impedance of the ample and ar. ) The reult in Fig. () how that when the length of the ample increae the time required for the equilirium increae a well. while when the diameter increae the time decreae. 7) From the previou reearche, the ideal length to diameter ratio of the ample i half. Thi ratio wa found to avoid the arrelling effect when the length i ig and to avoid the friction effect at thin ample. 8) Due to Poion ratio, the comination of L/D variation have een examined and hown in Fig. (5). Thi mean when the length i changed the diameter ha to e changed a well to keep the ame L/D ratio with ame value. 1..8 Equlirium Normalzed tre.... -. -. -. Stre wave Tranmitted Reflected -.8-1. NLT 8 1 1 ar. time Fig. : Incident, tranmitted, Reflected wave and the equilirium for nylatron ample with L/D ratio /8.
Dimenion Effect on Dynamic Stre Equilirium in SHPB Tet 3 1 1 1 HDPE Diameter (mm) 8 CFC Nylatron Duralumin 8 1 Time eq (u) Fig. 5: Diameter veru time for complete equilirium for HDPE, Nylatron, caron Firegla, and Aluminium. 1 8 Sample Length (mm) CFC Nylatron HDPE 8 1 Time eq (u) Fig. : Length-time for HDPE, Naylatron and CFC.
Y =13.38-.3599 X HDPE 1 1 8 Length (mm) Diameter (mm) Y =1.559+.1898 X length diameter 8 1 1 1 1 18 Time eq (u) Fig. 7: Length and Diameter veru time for HDPE. 1 Y =11.9378-.8 X NLT 1 8 Length(mm) Y =1.93+.3811 X Diameter (mm) diameter length 1 3 5 7 Time eq (u) Fig. 8: Length and Diameter veru time for nylatron.
Dimenion Effect on Dynamic Stre Equilirium in SHPB Tet 5 CFC 1 Y =13.5-.339 X 1 1 Length(mm) 8 Diameter(mm) Y =1.781+.1885 X 8 1 1 1 1 18 Time eq (u) Fig. 9: Length and Diameter for CFC. 5. Concluion The dynamic tre equilirium in material under SHPB teting ha een invetigated uing the multiple reflection of the tre pule inide the pecimen within the elatic limit of the material. It wa illutrated that the equilirium condition, which i one of the fundamental requirement in material dynamic property teting, i not atified automatically when a SHPB i ued to determine the dynamic repone of the material under tet. To enure tudying the effect of dimenion on the dynamic equilirium, the loading rate mut e examined and kept the ame [7]. Alo, very high loading rate may caue localized failure in the pecimen near the front face when impacted y the tre pule from the incident ar, therefore in the preent work the loading rate, which related to the incident pule i kept the ame in all meaurement. A reduction in the pecimen thickne may lead to achieving early dynamic tre equilirium epecially in oft material. However, the thickne cannot e reduced indefinitely, and the friction effect will e more pronounced in thin pecimen. The large difference etween the initial front and ack-urface tree due to the large thickne caue a evere non-equilirium in the pecimen. It i thu, neceary to quantitatively undertand the effect of the pecimen dimenion on the dynamic tre equilirium in order to properly deign SHPB experiment, o valid reult for the teted material can e otained.
In an experiment, the peak tre of the pecimen at a certain contant train rate i part of the experimental goal and i not variale if the experiment i properly deigned; therefore, it i important to decreae the amplitude of the initial tre in the fronturface tre pule to facilitate early equilirium in the pecimen. Thu, in thi work, the incident pule generated with a uitale rie time to achieve the equilirium in a horter time. Reference [1] Al-Maliky N, (1997), PhD thei, Loughorough Univerity, UK [] Parry D, and, Al-Maliky N (199, Journal de Phyique IV,Colloque C8, upplement au III, Volume, Septemre 199. [3] Song B and Chen W.,, Experimental Mechanic, Vol. (3), p3. [] Kolky H, 193, Stre wave in olid, Dover Pu. INC. [5] Jonon W, Impact trength of material, 197, Edward Arnold Pu. Limited. [] - Song B., and Chen W. Dynamic tre equiliration in Split Hopkinon Preure Bar Tet on Soft Material. Experimental Mechanic, Vol (3) 1, 3-31 [7] Chen W, Lu F., Frew D J, and Foretal M J Dynamic Compreion Teting of oft Material, Tran. ASME, J. Appl. Mech., 9,, 1-3 [8] Marai S T, Tait R B, Cloete T J and Nurick G N Material teting at high train rate uing the plit Hopkinon preure ar. Latin American Journal of Solid and Structure, Vol 1,, 319-339.