RECENT DEVELOPMENTS OF SPH IN MODELING EXPLOSION AND IMPACT PROBLEMS

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1 Recent developments of SPH n modelng exploson and mpact problems III Internatonal Conference on Partcle-based Methods Fundamentals and Applcatons PARTICLES 2013 M. Bschoff, E. Oñate, D.R.J. Owen, E. Ramm & P. Wrggers (Eds) RECENT DEVELOPMENTS OF SPH IN MODELING EXPLOSION AND IMPACT PROBLEMS M. B. LIU¹, D. L. FENG¹ AND Z. M. GUO 2,3 ¹ Insttute of Mechancs Chnese Academy of Scences Beng , Chna e-mal: lumoubn@mech.ac.cn, fengdanle@mech.ac.cn 2 Laborator de Càlcul Numèrc (LaCàN) Unverstat Poltecnca de Catalunya (UPC) C. Jord Grona 1-3, Campus Nord, Barcelona, Span e-mal: guozhmng@lve.cn 3 North Unversty of Chna Xueyuan Road Tayuan, Shanx, Chna e-mal:guozhmng@lve.cn Key words: Smoothed Partcle Hydrodynamcs (SPH), Exploson, Impact, Shaped Charge, Explosve Weldng Abstract. Exploson and mpact problems are generally characterzed by the presence of shock waves, ntense localzed materals response and ntensve loadngs. Most of the wave propagaton hydro-codes for such problems use tradtonal grd based methods such as fnte dfference methods (FDM) and fnte element methods (FEM). Though many successful achevements have been made usng these methods, some numercal dffcultes stll exst. These numercal dffcultes generally arse from large deformatons, large nhomogenetes, and movng nterfaces, free or movable boundares. Smoothed partcle hydrodynamcs (SPH) s a Lagrangan, meshfree partcle method, and has been wdely appled to dfferent areas n engneerng and scence. SPH method has been ntensvely used for smulatng hgh stran hydrodynamcs wth materal strength, due to ts specal features of meshfree, Lagrangan and partcle nature. In ths paper, some recent developments of the SPH n modellng exploson and mpact problems wll be ntroduced. A modfed scheme for approxmatng kernel gradent (kernel gradent correcton, or KGC) has been used n the SPH smulaton to acheve better accuracy and stablty. The modfed SPH method s used to smulate a number of problems ncludng 1D TNT detonaton, lnear shaped charge and explosvely drven weldng. The effectveness of the modfed SPH method has been demonstrated by comparatve studes of the SPH results wth data from other resources. 1 INTRODUCTION Exploson and mpact problems are generally characterzed by the presence of shock waves, 428

2 M. B. Lu, D. L. Feng and Z. M. Guo. ntense localzed materals response and ntensve loadngs. A typcal hgh explosve (HE) exploson conssts of the detonaton process through the HE and the later expanson process of the gaseous products to the surroundng medum [1]. For mpact problems wth dfferent mpactng speeds, solds under extreme stuatons behave lke fluds. Two typcal stuatons are hgh velocty mpact (HVI) and penetraton. In HVI, the knetc energy of the system domnates and forces the sold materal to deform extremely and the materal actually flows. In the events of penetraton, the materals can even be broken nto peces that fly, n addton to the extremely large deformaton [2]. Recently more and more analyses of exploson and mpact problems are based on numercal smulatons wth the advancement of the computer hardware and computatonal technques. Most of the applcatons are generally grd-based numercal methods such as the fnte element methods (FEM) or fnte dfference methods (FDM). Though many successful achevements have been made for these methods n modelng exploson and mpact problems, some numercal dffcultes stll exst. These numercal dffcultes generally arse from large deformatons, large nhomogenetes, and movng nterfaces, movable boundares when smulatng exploson and mpact problems ncludng HE detonaton and exploson of explosve gas, deformaton of sold materals, and nteracton of fluds and solds. Smoothed partcle hydrodynamcs (SPH) method [3, 4] s a Lagrangan, meshfree partcle method. In SPH, partcles are used to represent the state of a system and these partcles can freely move accordng to nternal partcle nteractons and external forces. Therefore t can naturally obtan hstory of flud/sold moton, and can easly track materal deformatons, free surfaces and movng nterfaces. Durng the last decades, SPH has been appled to modelng hgh explosve detonaton and exploson, and hydrodynamcs wth materal strength such as mpact and penetratons [1, 5-8]. However, most of the exstng works on SPH modelng of exploson and mpact problems are based on conventonal SPH method, whch s beleved to have poor performances especally n modelng problems wth hghly dsordered partcles [5]. They usually lack quanttatve and even qualtatve comparsons wth expermental results, and also lack valdaton and verfcaton n energy conservaton. In ths paper, we shall present a modfed SPH model and the modfed SPH model wll be appled to a number of exploson and mpact problems to demonstrate ts effectveness. 2 SPH METHODOLOGY 2.1 Basc concepts of SPH In conventonal SPH method, there are bascally two steps n obtanng an SPH formulaton, kernel and partcle approxmatons. The kernel approxmaton s to represent a functon and ts dervatves n contnuous form as ntegral representaton usng the smoothng functon and ts dervatves. In the partcle approxmaton, the computatonal doman s dscretzed wth a set of partcles. A feld functon and ts dervatve can then be wrtten n the followng forms [9] < m f( x ) >= f( x ) W( x x, h) N = 1 ρ (1) 429

3 M. B. Lu, D. L. Feng and Z. M. Guo. < m f( x ) >= f( x ) W N = 1 ρ (2) where < f ( x ) > s the approxmated value of partcle ; f ( x ) s the value of f ( x ) assocated wth partcle ; x and x are the postons of correspondng partcles; m and ρ denote mass and densty respectvely; h s the smooth length; N s the number of the partcles n the support doman; W s the smoothng functon representng a weghted contrbuton of partcle to partcle. The smoothng functon should satsfy some basc requrements, such as normalzaton condton, compact supportness, and Delta functon behavor [4, 10]. 2.2 SPH equatons of moton For hydrodynamcs of fluds and solds wth materal strength, the followng governng equatons of contnuum mechancs apply Dρ v = ρ Dt x Dv 1 σ = Dt ρ x De σ v = Dt ρ x Dx = v Dt where the scalar densty ρ, and nternal energy e, the velocty component v, and the total stress tensor σ are the dependent varables. The spatal coordnates x and tme t are the ndependent varables. The summaton n equaton (3) s taken over repeated ndces, whle the total tme dervatves are taken n the movng Lagrangan frame. The total stress tensor σ n equaton (3) s made up of two parts, one part of sotropc pressure p and the other part of shear stress S. The hydrodynamc pressure s computed from an equaton of state (EOS). For explosve gas, as the sotropc pressure s much larger than components of vscous shear stress, the vscous shear stress can be neglected. For sold materals, the shear stress can be computed from the consttutve equatons of correspondng materals. Therefore usng abovementoned SPH approxmatons, the followng SPH equatons of moton can be obtaned N d ρ m W = ρ ( v v ) dt = 1 ρ x N d σ σ W v = m + + Π 2 2 dt = 1 ρ ρ x de 1 P P W 1 ( v v ) N = m + + Π S ε + H dt 2 = 1 ρ ρ x ρ dx = v dt where ε s the stran rate tensor, Π and H stand for the component of the devator stress (3) (4) 430

4 M. B. Lu, D. L. Feng and Z. M. Guo. tensor, the artfcal vscosty and the artfcal heat separately [4]. 2.3 Kernel gradent correcton The conventonal SPH method has been hndered wth low accuracy and ts accuracy s also closely related to the dstrbuton of partcles, selecton of smoothng functon and the support doman. Though dfferent approaches have been proposed to mprove the partcle nconsstency and hence the SPH approxmaton accuracy, these approaches are usually not preferred for hydrodynamc smulatons because the reconstructed smoothng functon can be partally negatve, non-symmetrc, and not monotoncally decreasng. Exploson and mpact nvolve fast expanson of explosve gas, rapd deformaton and even lquefacton of sold materals, and quck damage on target materals. These lead to hghly dsordered partcle dstrbuton, whch can serously nfluence computatonal accuracy of SPH approxmatons. Hence an SPH approxmaton scheme, whch s of hgher order accuracy and s nsenstve to dsordered partcle dstrbuton, s necessary. In ths paper, the kernel gradent n SPH approxmatons s mproved wth a kernel gradent correcton (KGC) technque [11]. In the KGC technque, a modfed or corrected kernel gradent s obtaned by multplyng the orgnal kernel gradent wth a local reversble matrx L( r ), whch s obtaned from Taylor seres expanson method. In two-dmensonal C spaces, the new kernel gradent of the smoothng functon W can be obtaned as follows C W = L( r ) W (5) L ( ) 1 W W x y x x = V W W x y y y r (6) where x = x x, y = y y. It s found that for general cases wth rregular partcle dstrbuton, varable smoothng length, and/or truncated boundary areas, the SPH partcle approxmaton scheme wth kernel gradent correcton s of second order accuracy. 3 EUATIONS OF STATE AND CONSTITUTIVE MODELING 3.1 Equatons of state For deal explosve, the standard Jones-Wlkns-Lee (JWL) equaton [12] of state can be employed. The pressure of the explosve gas s gven by R1 R2 η ωη η 1 2 ωη p = A(1 ) e + B(1 ) e + ωηρ0e (7) R R where η s the rato of the densty of the explosve gas to the ntal densty of the orgnal explosve. e s the nternal energy of the hgh explosve per unt mass. A, B, R 1, R 2 and ω are coeffcents obtaned by fttng the expermental data. E s the ntal nternal energy of the hgh explosve per unt mass. Values of the correspondng coeffcents can be found n [4]. 431

5 M. B. Lu, D. L. Feng and Z. M. Guo. For non-deal explosve, the JWL++ equaton of state can be used to model the detonaton process wth chemcal reacton, whle the pressure conssts of the contrbutons from both reacted and non-reacted explosve. The Tllotson equaton [13] s employed to descrbe pressure-volume-energy behavor of metals under hgh temperature, pressure and stran rate as follows b 2 p = a+ ηρ e+ Aµ + Bµ 1 0 ω0 b p = a+ ηρ e+ Aµ 2 0 ω0 p = 3 2 ( p4 p2)( e es ) p + ( e e ) bηρ e p aηρ e Aµ e e 0 x 4= 0 + ( + ) ω0 ρ η =, µ = η 1, ω = 1 + ρ s e0η where a, b, A, B,,, e 0, e s and e are the parameters determned by the materal, and p 1 to p4 phase, the vapor and lqud mxture, and the vapor phase [13]. s are the pressure of four dfferent phases of materal, e.g., the sold phase, the lqud e 2 x (8) 3.2 Consttutve modelng Johnson-Cook model [14] s one of the most popular consttutve models for numercal smulatons of mpact and penetraton, and whch are usually assocated wth hgh stran rate. The model consders the effects of the stress hardenng, stran rate and the temperature evoluton. The yeld stress n Johnson-Cook model can be wrtten as n p * * m σ y = ( A+ Bε )(1 + Cln ε )(1 T ) (9) T T T = T T * room melt p where ε s the effectve plastc stran, * ε s a dmensonless stran rate, and T s the temperature. A, B, C, n and m are fve parameters n the Johnson-Cook model that need to be determned from the torson test under dfferent stran rate, the Hopknson Pressure Bar test wth dfferent temperatures, and the Standard statc tensle test. Detaled parameters n the Johnson-Cook model for dfferent metals can be found n [15]. 4 APPLICATIONS 4.1 Detonaton of a 1D ANFO bar The conventonal SPH method was successfully appled to model the detonaton of a 1D TNT bar, whch s of dea explosve wth a constant detonaton speed [1]. In ths work, the detonaton process of a 1D ANFO bar s modeled usng the method, whle ANFO s a nondeal explosve and the JWL++ equaton of state s used. The ANFO bar s 0.1m long, and room (10) 432

6 M. B. Lu, D. L. Feng and Z. M. Guo partcles are used n the smulaton. Fgure 1 shows the pressure profles along the slab at 1 µs nterval from 1 to 14 µs. The von Neumann spke can be well captured n ths smulaton by usng the JWL++ equaton of state. Instead, f usng the conventonal JWL equaton of state, t s only feasble to reach steady C-J pressure. The entre detonaton process costs 17 µs and the resultant detonaton speed s 5725 m/s, whch s very close to the expermental value of 5900 m/s. 25 Von Neumann Spke C-J State Pressure 20 P(GPa) x(m) Fgure 1: Pressure profles along the 1D ANFO slab durng the detonaton process. 4.2 Lnear shaped charge Fgure 2: Shaped charge et formaton and penetraton of a target plate. Shaped charge s a frequently used form of explosve charge for mltary and ndustral applcatons. It can produce powerful metal et and lead to stronger penetraton effects onto targets than normal charges. After the exploson of hgh explosve, the detonaton produced explosve gas can exert tremendous pressure on surroundng metal case and lner wth very large deformaton and even quck phase-transton. There are some prelmnary works of usng SPH to model shaped charges. For example, Lu et al. frst smulated the detonaton and exploson process of two-dmensonal shaped charges wth dfferent shapes of cavty [16] usng SPH method. It s found that SPH can effectvely model the explosve gas et formaton and dsperson. That work dd not consder surroundng metal case and lner whch present addtonal challenges n numercal smulaton due to the exstence of mult-materal 433

7 M. B. Lu, D. L. Feng and Z. M. Guo. (explosve-metal) and mult-phase (sold-gas-lqud). In ths work, a practcal lnear shaped charge s modeled wth metal case and lnear. Fgure 2 shows the entre process of HE detonaton and exploson, exploson-drven metal deformaton and et formaton as well the penetratng of a target alumnum plate at typcal nstants. The obtaned snapshots wth metal et and penetratng effects agree well wth expermental observatons. 4.2 Explosve weldng Explosve weldng can be used to bond two dssmlar metal plates to obtan a metal composte wth better performance. It s attractve when conventonal fuson weldng technques are not able to combne two metal plates together. Fgure 3: SPH modelng of the explosve weldng process. Fgure 3 shows the SPH smulaton of explosve (TNT) weldng of two steel plates. It s observed that wth the detonaton and exploson of TNT, the hgh pressure explosve gas drves the flyer plate, whch mpacts on and nteracts wth the substrate (base plate). As the detonaton wave moves rghtwards, the nteracton of the flyer plate and substrate also move rghtwards. A metal et produces between the two steel plates and a wave-lke weldng pattern can be obtaned durng the explosve weldng process, whch s also observable n laboratory experments. Fgure 4 shows a full scale vew of the wave-lke weldng pattern durng the explosve weldng process. Fgure 4: A full scale vew of the wave-lke weldng pattern. 12 CONCLUSIONS - A modfed SPH method wth kernel gradent correcton s developed for modelng exploson and mpact problems. - The modfed SPH method has been appled to a number of applcatons ncludng the detonaton of a 1D ANFO bar, lnear shaped charge et formaton and penetraton effects on a target plate, and the explosve weldng of two steel plates. The nherent physcs can be well captured and the obtaned numercal results are agreeable wth expermental observatons. 434

8 M. B. Lu, D. L. Feng and Z. M. Guo. Acknowledgement Ths work has been supported by the Natonal Natural Scence Foundaton of Chna ( ) and the 100 Talents Programme of the Chnese Academy of Scences. REFERENCES [1] M. B. Lu, G. R. Lu, Z. Zong and K. Y. Lam, Computer smulaton of hgh explosve exploson usng smoothed partcle hydrodynamcs methodology, Comput. Fluds, 32 (3) (2003): [2] J. A. Zukas. Hgh velocty mpact dynamcs. Wley-Interscence, (1990). [3] R. A. Gngold and J. J. Monaghan, Smoothed partcle hydrodynamcs-theory and applcaton to non-sphercal stars, Mon. Not. R. Astron. Soc., 181 (1977): [4] G. R. Lu and M. B. Lu. Smoothed partcle hydrodynamcs: A meshfree partcle method. World Scentfc, Sngapore, (2003). [5] M. B. Lu and G. R. Lu, Smoothed partcle hydrodynamcs (sph): An overvew and recent developments, Arch. Comput. Meth. Eng., 17 (1) (2010): [6] P. W. Randles and L. D. Lbersky, Smoothed partcle hydrodynamcs: Some recent mprovements and applcatons, Comput. Meth. Appl. Mech. Eng., 139 (1-4) (1996): [7] M. B. Lu, G. R. Lu and K. Y. Lam, Adaptve smoothed partcle hydrodynamcs for hgh stran hydrodynamcs wth materal strength, Shock Waves, 15 (1) (2006): [8] J. J. Monaghan, Smoothed partcle hydrodynamcs, Annu. Rev. Astron. Astr., 30 (1992): [9] J. J. Monaghan, Smoothed partcle hydrodynamcs, Rep. Prog. Phys., 68 (8) (2005): [10] M. B. Lu, G. R. Lu and K. Y. Lam, Constructng smoothng functons n smoothed partcle hydrodynamcs wth applcatons, J. Comput. Appl. Math., 155 (2) (2003): [11] J. R. Shao, H. Q. L, G. R. Lu and M. B. Lu, An mproved sph method for modelng lqud sloshng dynamcs, Comput. Struct., (2012): [12] E. L. Lee, H. C. Horng and J. W. Kury (1968) Adabatc expanson of hgh explosve detonaton products. Calforna Unv., Lvermore. Lawrence Radaton Lab. [13] M. Katayama, A. Takeba, S. Toda and S. Kbe, Analyss of et formaton and penetraton by concal shaped charge wth the nhbtor, Int. J. Impact Eng., 23 (1) (1999): [14] G. R. Johnson and W. H. Cook (1983) A consttutve model and data for metals subected to large strans, hgh stran rates and hgh temperatures, n Proceedngs of Seventh Internatonal Symposum on Ballstcs, Internatonal Ballstcs Commttee: The Hague, Netherlands. p [15] D. L. Feng, M. B. Lu, H. Q. L and G. R. Lu, Smoothed partcle hydrodynamcs modelng of lnear shaped charge wth et formaton and penetraton effects, Comput. Fluds, (In press) (2013). [16] M. B. Lu, G. R. Lu, K. Y. Lam and Z. Zong, Meshfree partcle smulaton of the detonaton process for hgh explosves n shaped charge unlned cavty confguratons, Shock Waves, 12 (6) (2003):