Simulation of Hypervelocity Spacecrafts And Orbital Debris Collisions using Smoothed Particle Hydrodynamics in LS-DYNA

Transcription

1 Smulaton of Hypervelocty Spacecrafts And Orbtal Debrs Collsons usng Smoothed Partcle Hydrodynamcs n LS-DYNA J.L.Lacome a, Ch. Espnosa b, C. Gallet b, a Consultant LSTC Rue des Pyrénées, F Grenade s/ Garonne b ENSICA - 1 Place Emle Bloun, F Toulouse Cedex 5 Abstract Ths paper s devoted to the results of smulatons of hypervelocty mpacts on thn alumnum plates usng the Smoothed Partcle Hydrodynamcs (SPH) opton of the LS-DYNA hydrocode. SPH s a meshless Lagrangan numercal technque used to model 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. The smulaton of an alumnum sphere mpactng two alumnum plates at a velocty of 5.5 km/s s presented. Introducton The rsk of collson between spacecraft and orbtal debrs s becomng more and more mportant. The sze of the most numerous fragments ranges from 1 to 10 mm. The velocty of orbtal debrs can reach 15 km/s. Today, no expermental features are avalable for such hgh velocty and such bg debrs. Only smulatons can be used to understand the events and desgn shelds. 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. In ths paper, SPH algorthm s brefly descrbed. Then, we present the results of the mpact of an alumnum sphere of 7.9 mm dameter mpactng a set of alumnum plates.

2 Basc prncples of the SPH method A flud s represented 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 nterpartcular forces. SPH method s 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 : d d ( ) x = v( x, t) ( ) w = dv( v( x, t)) w (1) dt dt We can then wrte the followng quadrature formula : x) dx Partcle approxmaton of functon Ê f ( w ( t) f ( x ( t)) (2) 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. P P It s defned by: Ñ Ô1 - y + y for y ˆ Ô 1 3 q ( y) = C Ò ( 2 - y) for 1 < y ˆ 2 (3) Ô 4 Ô 0 for y > 2 ÔÓ where C s the constant of normalzaton that depends on the space dmenson. We have then enough elements to defne the smoothng kernel W:

3 1 Ë x - x Û W ( x - x h = Ì, ) q Ü (4) h Í h Ý W ( x - x, h) d when h 0, where d s the Drac functon. h s a functon of x and x and s the so-called smoothng length of the kernel. We can now defne the partcle approxmaton P h u of the functon u, by approxmatng the ntegral (2): h P u ( x ) = w ( t) u( x ) W ( x - x, h) (5) Ê The approxmaton of gradents s obtaned by applyng the dervaton operator on the smoothng length. We then obtan : Impact confguraton h P u ( x ) = w ( t) u( x ) W ( x - x, h) (6) Ê The test conssts of a 7.9 mm dameter 2017-T4 alumnum sphere mpactng frst an alumnum 6061-T6 plate, 2mm thck, at 5.5 km/s. After the frst mpact, the debrs cloud created mpacts a second plate, 6.4 mm thck, of the same materal than the frst plate. The dstance between the two plates s 10 cm. A wtness plate s placed 15 cm away from the second plate to check f the debrs cloud went through the second plate. 260 mm 0.79 cm Al v = 5.5 km/s 0 deg 10 cm 15 cm 2.0 mm Al bumper 6.4mm Al rearwall 1.6 mm Al WP Fgure 1 - Impact confguraton

4 Smulaton wth LS-DYNA and SPH Modellng procedure The total smulaton tme for ths mpact s 95ms. A complete 3D model has been bult. Fgure 2 - Target and proectle meshes In order to mesh the frst plate wth an effcent way, t s necessary to dvde t n two parts; the central part and the rest of the plate. The central part wll nteract a lot wth the sphere, and wll be therefore completely dstorted after mpact. SPH elements are used to model ths central part. The rest of the plate can be modelled wth standard Lagrangan brck elements snce deformatons are very low n that regon. Ths way of modellng allows us to save some cpu tme for the smulaton snce the Lagrangan elements have a cheaper cost than the SPH elements. SPH partcles of the central are lnked to the Lagrangan outer part by knematcal constrants. A total of partcles are used to model the sphere and the two plates. A classcal sldng nterface s used to treat the contact between SPH partcles of the cloud and Lagrangan fnte elements of the target plates durng mpact. Materal models To represent the hydrodynamc behavour of alumnum durng the mpact, the Johnson-Cook materal model s used for both the proectle and the target. The Me-Grünesen equaton of state s used to determne the pressure. Data for the materals are brefly descrbed n the followng Table 1. JOHNSON-COOK Al 2024-T3 Al 6061-T6 Densty (g/cm 3 ) 2,785 2,703 Shear modulus (Mbar) 0,286 0,276

5 A (Mbar) 0, ,00275 B (Mbar) 0, ,00255 c 0,015 0,0 n 0,34 0,3 m 1 1 Specfc heat 0, , Melt température (K) Room température 273 K 273 K Table 1 - Materal data In order to smulate the falure of the materal, the hydrodynamc pressure s lmted to 0.02 Mbar. Below ths pressure lmt, spallng effects occur n the materal. Results The analyss of the results of the mpact s dvded n two parts. Frst, we compare the mpact of the sphere on the frst plate wth theoretcal calculaton. Then, the result of the complete smulaton s presented. Analytcal results The results of the smulaton are compared wth analytcal calculaton, by the mean of several representatve parameters: mpact pressure, axal velocty, and densty. In order to valdate the SPH formulaton of LS-DYNA, a one-dmenson calculaton usng the Hugonot relatons s done to compute the analytcal mpact pressure. In the target, the pressure s gven by: P Hc = r U (7) 0c Scv 1 r 0c s the densty of the target, v 1 s the velocty of the partcles at the mpact pont, U Sc s the relatve velocty of the shock wave n the target. In the proectle, the pressure s gven by: P ( v0 1) Hp 0pU Sp -v where v 0 s the mpact velocty. =r (8) These two pressures are equal. We have a supplementary relaton:

6 U S C + S v = 0 (9) C 0 s the sound speed n the materal. S s a coeffcent dependng on the materal. We have enough equatons to fnd the mpact pressure. Ths pressure s compared wth the one gven by the smulaton (Fgure 3 and Table 2). Results of the smulaton exhbt slghtly lower a pressure than the analytcal result. Numercal results Analytcal results P Hc (Mbar) 0,646 0,683 P Hp (Mbar) 0,61 0,683 V 1c (cm/ms) 0,229 0,277 V 1p (cm/ms) 0,326 0,277 r 1c (g/cm 3 ) 2,75 3,883 r 1p (g/cm 3 ) 2,78 4,001 Table 2 - Numercal and analytcal values of the representatve parameters Debrs cloud d2 V1 V2 h V3 b d1 a Fgure 3 - Correlated parameters Specfc geometrc parameters summarsed n Table 3 below and representatve knematcal parameters of the cloud are observed (Fgure 3). Snce expermental results are not yet avalable, we show the characterstc parameters that we are nterested n.

7 Axal velocty V1 Axal velocty V2 Axal velocty V3 Smulaton 6.5 km/s 4.6 km/s 3.3 km/s a b d1 d2 h Smulaton cm 6.5 cm 7.95 cm Table 3 Numercal characterstcs of the partcle cloud Fgure 4 presents the result of the mpact after the frst plate, and Fgure 5 shows the result of the mpact on the set of plates. Fgure 4-3D vew of debrs cloud after smulated mpact on the frst plate Even f hgh deformatons can be seen on the rear of the second plate, we don t notce the complete fracture of t. However, some lttle fragments of the second plate reach and mpact the wtness plate. After mpact on the second plate, the average velocty at the front of the cloud s around 200 m/s, whch s stll mportant.

8 Fgure 5-3D vew of debrs cloud Concluson The smulaton of a hgh mpact velocty on one plate gves results that are n good correlaton wth the expected results or analytcal results.

9 The mpact on a set of dfferent stacked plates seems, however, to be less precse. Indeed, for the complete model on the set of plates, we reached the lmtaton of our workstaton n term of maxmum number of partcles for a 3D complete model. More partcles are necessary for ths model to obtan reasonable good and relable results for ths knd of mpact. The use of the coupled opton of SPH n LS-DYNA, allows us to use standard Lagrangan elements, less tme consumng, that could help to have a fner model, but t looks lke that opton s not suffcent for our requrements. The model presented here can only gve us a qualtatve result of ths mpact but not a quanttatve answer. References [1] W. Schonberg, E. Mohamed, Analytcal hole dameter and crack length models for mult-wall systems under hypervelocty proectle mpact, Internatonal Journal of Impact Engneerng [2] C.Loupas : Rapport technque CEG.T96-01, [3] J.L. Lacome, Analyse de la méthode partculare SPH. Applcatons à la détonque, Thèse INSA, Janver [4] J-D Frey, F.Jancot, X. Garaud, P. Groenenboom, The valdaton of hydrocodes for orbtal debrs mpact smulaton, Internatonal Journal of Impact Engneerng, [5] R. F. Stellngwerf, C.A. Wngate, Impact modellng wth smoothed partcle hydrodynamcs, Internatonal Journal of Impact Engneerng, [6] W. Reschauer and K. Thoma, Vsualsaton of a hypervelocty mpact, Technsche Unverstat Munch,Germany [7] M. Faraud, R Destefans, D. Palmer, M. Marchett, SPH smulaton of debrs mpact usng two dfferent computer codes, Internatonal Journal of Impact Engneerng, [8] K. Holan, M. Burkett, Senstvty of hypervelocty mpact smulaton to equaton of state, Unversty of Calforna,1987. [9] Petkutowsk, Characterstcs of debrs clouds produced by hypervelocty mpact of alumnum, Unversty of Dayton Research Insttute, [10] W. Herrmann, J.S.Wlbeck, Revew of hypervelocty penetraton theores, Sanda Natonal Laboratores, 1987.