Proeedigs of the 2d WSEAS It. Coferee o Applied ad Theoretial Mehais, Veie, Italy, November 2-22, 26 222 Fuzzy Dyami Charateristi of Corete Material uder Impat Loa GAO SHIQIAO LIU HAIPENG JIN LEI Shool of Mehatroi Egieerig Beijig Istitute of Tehology No.5 Zhogguau South Road, Beijig 8 P. R. CHINA Abstrat:- For differet impat veloity (high, middle ad low), the orete material shows differet harateristis. To desribe suh a harateristi effetively, the oept of fuzzy dyami harateristi of orete material is preseted. Based o this oept, a fuzzy model to predit the peetratio is give ad a total peetratio proe of a projetile agaist semi-iite orete target is aalyzed ad alulated. From the ompariso with the tests, the good agreemet is obtaied. Keywor: orete; fuzzy harateristi; fuzzy model; peetratio Itrodutio For a semi-iite orete target or a thik orete slab, the istataeous peetratio veloity of a projetile will be dereased util to zero beause of the effet of restae applied o the projetile-ose. For differet peetratio veloity, the orete material shows differet harateristis. I geeral, the restae of the orete target applied o the projetile osts of two parts. Oe is from the iertial effet, the other is from the stregth effet of the material. The iertial restae depe maily o the ma of the material (i.e. the dety of material). The stregth restae depe maily o the stregth of the material iludig the dyami stregth. Whe the strikig veloity is high, the great majority of restae is the iertial fore. Whe the strikig veloity is middle, i additio to iertial fore, the majority of restae iludes also the dyami stregth fore. With the derease of the strikig veloity, the stregth fore beomes more major. I the poit of view of military egieerig, the high veloity rages from 8m/s to 5m/s, the middle veloity rages from m/s to 8m/s, ad the low veloity rages from m/s to m/s. I this paper, the strikig veloity is oed i the rage metioed above, the veloity more tha 5m/s is beyod the sope of this study. The orete material is haraterized by brittle ature whih differs from isotropi teaious metal material. It has a great differee betwee the ompreive stregth ad the tele stregth. I additio, ompreibility of orete plays a importat role i the deformatio for high preure ad high strikig veloity beause of the existee of the air voi. It is show from a great deal of theoretial ad experimetal ivestigatios by previous author that, for high strikig veloity, the orete material shows the harateristi whih is the same as ideal ompreible flui; for middle strikig veloity, it shows the harateristi of ideal or strai-hardeig plasti soli; ad for low strikig veloity, it shows the harateristi of geeral elasti-plasti soli. Therefore, for the total proedure of deep peetratio, for differet veloity rages, differet models must be used to predit the total peetratio effetively. I view of the differee i restae betwee
Proeedigs of the 2d WSEAS It. Coferee o Applied ad Theoretial Mehais, Veie, Italy, November 2-22, 26 223 iertial effet ad the stregth effet for differet strikig veloities, a fuzzy model of restae is preseted i this paper. 2 Desriptio of the fuzzy model Durig the peetratio, the restaes of target applied o the projetile are maily from the ormal preure ad osequet slide fritio fore o the surfae of the projetile-ose. The ormal preure osts of the followig four portios. The rst is the stati stregth. The seod is the dyami stregth. The third is the solid iertia. The fourth is the fluid iertia. The geeral expreio of ormal ompreive stre a be writte by sum = () is the stati stregth stre, is the dyami stregth stre, ompreive stre, ad is the solid iertial is the fluid iertial ompreive stre. sta for logial sum. Equatio () a ot be arithmeti sum beause ad ome from differet models. That is, the orete material a ot be meawhile (i the same oditio) odered as both solid ad fluid. To obtai a aurate orret model about ompreive strees, the oept of fuzzy dyami harateristi of orete material is established. By meas of the metho of fuzzy mathematis, we dee a variable as the grade of membership of the orete material. This variable a represet a degree belogig to whih deformatio ad damage (that is, dyami flui damage, dyami soli damage, stati soli damage, et.) the material will subjet to, whe it is peetrated by a projetile. By meas of the grade of membership of the material, the arithmeti form of equatio () a be rewritte as + + + sum = (2),, ad are ompoets of the vetor of the grade of membership. They are grades of membership of stati elasti-plasti solid stregth damage, dyami elasti-plasti solid stregth damage, dyami elasti-plasti solid iertial damage ad dyami fluid iertial damage, respetively. They are satised by the followig ormalized relatioships + = (3a) + = (3b) The grade of membership =,,, } depe maily o the { strikig veloities. For high veloity impat, the orete material a be odered as ideal ompreible fluid [], the grade of membership a be take as, = {.,.,.,.} (4a) For low strikig veloities, the orete material a be odered as geeral elasti-plasti solid. The qua-stati stregth plays a importat role i the restat fore. The grade of membership a be take as = {.,.,.,.} (4b) For middle strikig veloity, the orete material a ot be mply odered as ideal ompreible fluid or elasti-plasti solid. Istead of, it is eeary to oder the orete as a fuzzy material. The grade of membership must be take as =,,, } (5) { the ompoets must satisfy the ormalized relatioship (3a) ad (3b). Their further aalys will be made below. 3 Aalyses of ormal ompreive strees
Proeedigs of the 2d WSEAS It. Coferee o Applied ad Theoretial Mehais, Veie, Italy, November 2-22, 26 224 Muh effort has bee direted at modelig the restae of orete target o the projetile-ose. The empirial formulae ilude Petry formula, ACE formula, NDRC formula, BRL formula, et. [2,3,4]. The aalytial formulae are derived maily by Forrestal, Luk, Brar, et. [5,6,7,8,9,,] based o the dyami avity expao. Their formulae are oderably milar eah other. They all ost of two portios. Oe is the stati restae, the other is the iertial restae of solid. The empirial formulae are based o the dimeoal aalys ad test ivestigatio. The aalytial formulae of dyami avity-expao are based oly o the aumptio of spherial (or ylidrial) avity ad o the aumptio of ostat-veloity expao. I additio, this aalytial theory has ot odered the ma-variatio effet of the projetile aused by respose medium. To deal with suh a problem, for high strikig veloity, Gao [2,3,5,6,7] has preseted a ormal expao theory i whih the boudary of avity is ot oed to spherial surfae ad its propagatio veloity is ot oed to ostat. I this theory, the majority of strees is the hydrostati preure ad the majority of deformatio of the orete targets is the volumetri strai beause the air voi will be rstly gradually ompreed out of orete. Holmquist, Johso ad Cook [4] gave a dyami stregth model for orete subjeted to large strai, high strai rates, ad high preures. ostats that ivolve oly material parameters obtaied from triaxial tests, whih are suggested for orete by Forrestal et al. [8] as B =. ad τ A = Sf S is a empirial ostat obtaied from triaxial tests, ad f is the uoed ompreive stregth.. I equatio (6), the right had de osts of two terms, oe is the stati stregth term ad the other is the iertial term. It is obvious that, i this restat fore model, oly the stati stregth portio ad solid iertial portio have bee odered. Beause the orete material is odered as a elasti-plasti solid i this model, the above two terms will represet the stati stregth stre ad the solid iertial ompreive stre. 3.2 The dyami fluid restat fore model For high strikig veloity, the orete material a be odered as ideal ompreible fluid, i view of whih, Gao [2,3,5,6,7] has preseted a ormal expao model oderig the wave veloity as a variable ad the projetile ma as a variable. The ormal preure o the ose surfae is expreed as * dv = ρ v + ρ l (7) dt 3. The solid restat fore model There are may empirial or aalytial formulae to desribe the axial restat fore o the projetile ose. Oe is the Forrestal s [8] aalytial ad empirial formula whih osequetly developed by Li ad Che [4], whose form is as follows 2 2 = τ A Bρv os ϕ (6) + ϕ is agle betwee axis ad the ormal diretio of projetile ose surfae, v is the istataeous veloity of the projetile, ρ is the dety of orete targets, ( τ A) ad B are ρ is the origial dety of orete material, material, * ρ is the ultimate dety of the orete is the wave veloity of respose medium boudary whih depe o the istataeous ormal veloity of the ose surfae v, ad l is the propagatio distae of respose medium wave. The rst term of equatio (7) is the dyamial fluid restat fore, the seod term is the effet of ma variatio. I this model the stati stregth is
Proeedigs of the 2d WSEAS It. Coferee o Applied ad Theoretial Mehais, Veie, Italy, November 2-22, 26 225 egleted. 3.3 Dyami stregth restat fore model of solid For large strais, large strai rates, ad high preures, the stregth of the orete material will hage relevat to stati state. Holmquist et al. [4] presets a omputatioal ostitutive model for orete subjeted to large strais, high strai rates, ad high preures. I this model, the ormal dyamial stregth restat fore o the ose surfae is expreed as N = f A ( D) + B P * ][ C l ε*] (8) [ + f is the qua-stati ompreive stregth, Bρv 2 os 2 ϕ = (9) * dv = ρ v + ρ l (9d) dt The fuzzy model for sum ormal ompreive stre a be expreed as sum = Sf [ + C + ( ρ v + f [ A ( D) + B P * 2 2ϕ l ε*] + Bρ v os () * dv + ρ dt The fuzzy model for the axial restat fore a be expreed as F e z x sum * S A ) N = () ] D is the damage ( D. ), P * = P / f is e z is the uit vetor i axial diretio of the the ormalized preure ( P is the atual preure), ε* = ε / ε is the dimeole strai rate ( ε is the atual strai rate ad = s ε. is the referee strai rate), ad A, B, N, ad C are the material ostats ( A is the ormalized oheve stregth, B is the ormalized preure hardeig oefiet, N is the preure hardeig expoet, C is the strai rate oefiet). 4 The fuzzy axial restat fore model 4. The axial restat fore Aordig the aalys metioed above, from equatios (6) (7) ad (8), the differet effetive strees a be expreed as = Sf (9a) N = f A ( D) + B P * ][ C l ε*] (9b) [ + projetile ad S A is the urve surfae of projetile ose whih has peetrated the target. 4.2 The aalys of the grade of membership The grade of membership vetor a desribe the harateristi variatio of the orete material with the strikig veloity. Equatio (4a) ad (4b) give us the both extreme ases for the high strikig veloity ad low strikig veloity respetively. For middle strikig veloity (m/s-8m/s), the geeral form is as equatio (5). I order to determie the ompoets of the grade vetor of membership, the liear futio of veloity is hose. They are alled the membership futios whih a be expreed by 8 v = = 7 7 for m / s v 8m / s (2) v = = 7 7 is the istataeous peetratio veloity of the projetile. 5 Calulatios ad ompariso with
Proeedigs of the 2d WSEAS It. Coferee o Applied ad Theoretial Mehais, Veie, Italy, November 2-22, 26 226 experimets By meas of the method, the alulatio ad ompariso with the experimets are made o the deeleratio of peetratio harateristis of ogive-ose projetile ormally peetratig ito thik orete target. We oduted the deeleratio experimets of peetratio with the ogive-ose, steel projetile as show i Fig. Fig.. The projetile used i the experimets The diameter of the projetile 2 a =. 62m. The radius of the ogive ose R =. 77m. The aliber-radius-head (CRH) ψ = 2. 724.The ma of material as elasti-plasti soli), by the model [2] (oderig the orete material as ompreible ideal fluid), ad by the model i this paper. It is obvious that, for the high veloity, the ompreible ideal fluid model is suitable. For low veloity, the Forrestal s solid model should be suitable, but the parameter S must be reasoably determied. I [8], the stati stregth restat fore was too giatly odered that it a ot predit the peak of deeleratio suitably. Every oe of the method metioed above a ot gly be suitable for the total peetratio proe. The fuzzy model i this paper is suitable for the total peetratio proe durig whih the veloity of the projetile varies from the iitial veloity to zero. It is also show that, the alulated results by the fuzzy model i this paper are i good agreemet with the experimet results. the projetile is m = 3. 777kg. The iitial strikig veloity of the projetile is 763 m / s. The dety of orete target is ρ = 3 24kg / m ad the ompreive stregth of it is f 7 2 = 3. N / m. The limit dety of orete target is * 3 ρ = 246kg / m. Fig.3. The deeleratio urves vs time The peetrated orete target i tests is show i Fig.2. Fig.2. The peetrated target The urves of deeleratio vs time of the projetile are show i Fig.3. I Fig.3, there are four urves. Oe is the from experimets. The other three urves are from alulatios repetively by the Forrestal s model [8] (oderig the orete 6 Coluo I terms of the ompariso of the results from the experimets with the results from the alulatio, it is obvious that, i Forrestal s model, the effet of stati stregth was too emphatially odered that the peak of deeleratio is iostet with the reality; i the fluid model, the effet of iertia of the material was too emphatially odered that the depth of peetratio is iostet with the reality of experimets. To overome their defets, the fuzzy model i this paper is suggested for the aalys of total proe of peetratio.
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