S.P.H. : A SOLUTION TO AVOID USING EROSION CRITERION? Célne GALLET ENSICA 1 place Emle Bloun 31056 TOULOUSE CEDEX e-mal :cgallet@ensca.fr Jean Luc LACOME DYNALIS Immeuble AEROPOLE - Bat 1 5, Avenue Albert Durand 31700 BLAGNAC Tel : +33-5-61 16 42 18 e-mal :dynals@wanadoo.fr
Abbrevatons: S.P.H. Smoothed Partcle Hydrodynamcs Keywords: Partcle methods S.P.H. Smoothed Partcle Hydrodynamcs Eroson crteron ABSTRACT A new partcle element has been added to LS-DYNA. It s based on Smoothed Partcle Hydrodynamcs theory. SPH s a meshless lagrangan numercal technque used to modelze the flud equatons of moton. SPH has proved to be useful n certan class of problems where large mesh dstortons occur such as hgh velocty mpact, crash smulatons or compressble flud dynamcs. Frst, the bass prncples of the SPH method wll be ntroduced. Then, the model of perforaton of a bullet through a thn plate wll be presented. Two models are realsed: one s made of lagrangan brck elements only, and the second one uses SPH elements for the plate. Fnally, a dscusson s proposed on the dfferent methods used to deal wth the penetraton problem. INTRODUCTION Meshfree methods have known szeable developments these last years for solvng conservaton laws. S.P.H. (Smoothed Partcle Hydrodynamcs) s a meshfree lagrangan method developed ntally to smulate astrophyscal problems. But, the easy way wth whch t s possble to ntroduce sophstcated phenomena, has made of SPH a very nterestng tool to resolve other physc problems: resoluton of contnuum mechancs, crash smulatons, ductle and brttle fractures n solds. The hgh handness of SPH allows the resoluton of many problems that are hardly reproducble wth classcal methods. It s very smple to obtan a frst approach of the knematcs of the problem to study. For nstance, due to the absence of mesh, one can calculate problems wth large rregular geometry. From a computatonal pont of vew, we represent a flud wth a set of movng partcles evolvng at the flow velocty. Each SPH partcle represents an nterpolaton pont on whch all the propertes of the flud are known. The soluton of the entre problem s then computed on all the partcles wth a regular nterpolaton functon, the so-called smoothng length. The equatons of conservaton are then equvalent to terms expressng flux or nter-partcular forces. In ths paper, the bass prncples of the method are frst gven. Then, the penetraton of a bullet through a thn plate s consdered. In order to smulate ths problem, two knds of methods are tested. Frstly, the target s meshed wth fnte elements and an eroson crteron s appled. Comparson s done wth the second model where the target s meshed wth SPH elements.
BASIS PRINCIPLES OF THE SPH METHOD Partcle methods are based on quadrature formulas on movng partcles ( x ( t ), w ( t )). P s the set of the partcles, x(t) s the locaton of partcle and w(t) s the weght of the partcle. We classcally move the partcles along the characterstc curves of the feld v and also modfy the wegths wth the dvergence of the flow to conserve the volume : P d d ( ) x = v( x, t) ( ) w = dv( v( x, t)) w dt dt (1.1) We can then wrte the followng quadrature formula : Ω f ( x) dx w ( t ) f ( x ( t)) P (1.2) Partcle approxmaton of functon The prevous quadrature formula together wth the noton of smoothng kernel leads to the defnton of the partcle approxmaton of a functon. To defne the smoothng kernel, we need frst to ntroduce an auxlary functon q. The most useful functon used by the SPH communty s the cubc B-splne whch has some good propertes of regularty. It s defned by: 3 2 3 1 y + y 2 4 1 3 θ ( y) = C ( 2 y) 4 0 3 for for for y 1 1 < y 2 y > 2 (2.1) where C s the constant of normalzaton that depends on the space dmenson. We have then enough elements to defne the smoothng kernel W: W ( x 1 x x, h ) = θ h h x (2.2) W ( x x, h) δ when h, where δ s the Drac functon. h s a functon of x and x and s the so-called 0 smoothng length of the kernel. We can now defne the partcle approxmaton Π h u of the functon u, by approxmatng the ntegral (1.2): h Π u ( x ) = w ( t ) u( x ) W ( x x, h ) Ω (2.3)
The approxmaton of gradents s obtaned by applyng the operator of dervaton on the smoothng length. We then obtan : h Π u ( x ) = w ( t) u( x ) W ( x x, h) Ω (2.4) PERFORATION OF A BULLET THROUGH A THIN PLATE The case of a bullet mpactng a thn target s consdered here. The proectle s a 5.56 mm calber bullet and the target s a 1,5mm thck ttanum plate. The proectle has several angles of mpact wth a velocty of 1010 m/s. However, n ths paper only the mpact wth a 90 angle of ncdence s dsplayed. The dffculty of ths test s the good modelsaton of the plate's falure. Two soluton seems to be convenent: on the frst hand, the lagrangan method wth an eroson crteron, on the other hand the use of the SPH elements. For ths model, two comparson parameters have been chosen : - the resdual velocty of the proectle - the shape of the deformed plate Moreover, an expermental database for the resdual velocty s avalable. As shown durng expermental trals, very few deformaton of the proectle have been notced, the proectle s then assumed to be rgd n the whole calculaton. Numercal model usng brck elements wth eroson Two dfferent models are compared. The frst one s only realsed wth brck elements. The proectle s modeled wth 33250 brck elements. The target s modeled wth 72900 elements. We apply some non-reflectng boundary elements around the plate to smulate an nfnte plate. The materal data are summed up n the followng lst: Proectle Densty Young's modulus Posson rato 11340 kg/m 3 112.82 GPa 0.30 Target Densty Shear modulus Yeld stress 4510 kg/m 3 43.4 GPa 0.85 GPa The proectle s modeled as a rgd body. The plate s modeled usng the materal elastc plastc hydrodynamc. In order to represent the perforaton of the proectle through the plate, an eroson crteron on the plate s appled. Two tests are performed. One s realsed by settng the falure stran for eroson at 1.5, and the other one by settng ths value to 0.5. Numercal model usng SPH elements In ths model, the center of the plate s meshed wth SPH elements. Around these elements a set of brck elements s defned. A ted node to surface contact s used between the SPH elements and the brck elements, and an automatc node to surface contact s defned between the partcles and the proectle. As n the lagrangan model, some non-reflectng boundares are defned. The default values are used for the SPH control and secton cards.
The model s realzed wth 44100 partcles n the center of the plate wth 9 partcles n the thckness of the plate. Some prevous tests have shown that 9 partcles are enough to represent the deformatons of the plate durng the perforaton. The man dfference wth the lagrangan model s that n ths case no eroson crteron s necessary. Indeed, the proectle s expected to push away the partcles and then to create the crack n the target. Results The dfferent models have been run untl the proectle has reached a constant velocty. Ths termnaton tme s 30 mcroseconds for the two lagrangan models and 25 mcroseconds for the SPH model. In both models, the resdual veloctes obtaned are n good agreement wth the expermental data. Nevertheless, n the lagrangan models, the fnal shape of the target s hghly related to the value of the eroson crteron as shown on fgures 1 and 2. As no eroson crteron s necessary wth the SPH technque to smulate falure phenomena, the problem of modelsaton that occurs wth the lagrangan formalsm s not rased here (fgure 3). Conclusons A new partcle element has been added to LS-DYNA. It s based on Smoothed Partcle Hydrodynamcs theory. In ths paper, the use of SPH for modellng perforaton problems where falure occurs s descrbed. If the only evaluaton crteron was based on the resdual velocty of the proectle, both lagrangan and SPH methods used here would be convenent. However, f the shape of the plate was another evaluaton pont, wthout expermental trals data, the SPH seems to be more sutable. Acknowledgments The authors would lke to thank Mrs. Nathale Soule from CEA-CESTA for provdng the tests data.
Fgure 1: Lagrange model, Falure stran = 0.5 Fgure 2: Lagrange model, Falure stran = 1.5 Fgure 3: SPH model