Micromechanics-based determination of effective. elastic properties of polymer bonded explosives

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1 Miromehanis-based determination of effetive elasti roerties of olymer bonded exlosives Biswajit Banerjee and Daniel O. Adams Det. of Meh. Eng., Univ. of Utah, 50 S Central Camus Dr., Salt Lake City, UT Abstrat Polymer bonded exlosives are artiulate omosites ontaining a high volume fration of stiff elasti exlosive artiles in a omliant visoelasti binder. Sine the volume fration of artiles an be greater than 0.9 and the modulus ontrast greater than 0 000, rigorous bounds on the elasti moduli of the omosite are an order of magnitude different from exerimentally determined values. Analytial solutions are also observed to rovide inaurate estimates of effetive elasti roerties. Diret finite element aroximations of effetive roerties require large omutational resoures beause of the omlexity of the mirostruture of these omosites. An alternative aroah, the reursive ells method (RCM) is also exlored in this work. Results show that the degree of disretization used in finite element models of PBXs an signifiantly affet the estimated Youngs moduli. Key words: Effetive roerties. High volume fration. High modulus ontrast. Corresonding author: Fax: , banerjee@eng.utah.edu Prerint submitted to Elsevier Siene 14 July 00

2 1 ntrodution Mehanial roerties of olymer bonded exlosives (PBXs) have traditionally been determined exerimentally. However, the hazardous nature of these materials makes mehanial testing exensive. With imrovement in omutational ower, numerial determination of mehanial roerties of PBXs has beome feasible. Elasti roerties of a omosite an be obtained using miromehanis based methods if the elasti roerties of the omonents are known from moleular dynamis simulations. n this work, rigorous bounds, effetive medium aroximations and finite element aroximations of elasti roerties are exlored. A less omutationally intensive aroah, alled the reursive ells method (RCM) is also investigated. The roerties redited by these aroahes are omared with exerimental data for PBX PBX materials and PBX 9501 PBXs are artiulate omosites omosed of exlosive artiles and a rubbery binder. PBX 9501 ontains 9% by volume of HMX (high melting exlosive) artiles and 8% by volume of binder. The HMX artiles are monolini and linear elasti. The exerimentally determined value of Youngs modulus of HMX is around 15. GPa [1] while that from moleular dynamis (MD) simulations is around 17.7 GPa []. The Poissons ratio from exeriments is 0. and that from MD simulations is 0.1. The binder is a 1:1 mixture of the rubber Estane 570 and a lastiizer (BDNPA/F). The mehanial behavior of the binder is strain rate and temerature deendent. As a result, the resonse of PBX 9501 also deends on strain rate and temerature. At or near room temerature and at low strain rate, the

3 # 0 # 0 * * * Youngs modulus of the binder is around 0.7 MPa and the Poissons ratio is The Youngs modulus of PBX 9501 under these onditions is around 1 GPa and the Poissons ratio is 0.5. The modulus ontrast between HMX and the binder is to under these onditions. Miromehanis aroahes.1 Third-order bounds Third-order bounds [5] on the effetive roerties of two-omonent olydiserse artiulate omosites an be written as U b b (1) U b b "! () #%$ #%$ #%$ L #%$ & b b #%$ #%$ () #%$ L #%$ #%$ #'$ & b b #'$ (4) )( where, is a volume fration, is a bulk modulus and is a shear modulus. The subsrits, b, and denote the artile, binder, and omosite resetively. The suersrits U and L denote the uer and lower bounds resetively., For any quantity *, * * b b, + * b * b, * - * b-. b, and * * / * b/ b. The quantities - and / for olydiserse omosites are given by - - b 014 and / / b Also, ( # # #%$ 768 #%$ 9 #'$. $ : 9"; and!,< 76=,< >?. $ 976.

4 X F m l B B [ZZZZZZZZZZZZY G A J G A O l [ZZZZZZZZZZZZY l K. Differential effetive medium aroximation Effetive elasti moduli an be alulated using the differential effetive medium aroximation (DEM) [6] from E E FHG F D F D G O E O E J G KLHM=N=O KLHM=N)O P O K7Q G K7Q P where the same symbols are used as in Eq.(1-4) and Q O E. O G KWV)O G (5) (6) K. Finite element aroximation Finite element (FEM) aroximations of two-dimensional effetive elasti moduli of a omosite an be obtained by determining the average stresses and strains in a reresentative volume element (RVE) under normal and shear dislaement boundary onditions using Eq. (7) \]_^`^ba \]d`ea \ fg^ ha ikj jjjjjjjjjjj M=m Bqn ^ M=m ^`^ ` Bon d^ M=m M=m ` ^`^ A M8O ^ ikj jjjjjjjjjjj ikj \rs^`^da \rg`_a jjjjjjjjjjj \],w4w asx%\ ryw4w a \ fg^ hasx%\ t%^ _a where are the volume averaged normal strains and stresses; wzw x ^ x{n w} are the volume averaged shear strain and stress; m \utv^ _a (7) are the two-dimensional effetive Youngs moduli, shear modulus, and Poissons ratios. These two-dimensional moduli are onverted to three-dimensional moduli using the relations n D n D eff E and m D eff D eff ~ ACB@n D eff E d. 4 eff n D eff

5 Y First teration Partiles X Seond teration Third teration Binder.4 Reursive ells method Fig. 1. Shemati of the reursive ells method. n the reursive ells method (RCM) the RVE is divided into a regular grid of subells. nstead of determining the effetive roerties of the whole RVE at a time, smaller bloks of subells are homogenized and the roedure is reeated reursively until the effetive roerty of the RVE is obtained as shown in Fig. 1. This aroah is similar to real-sae renormalization tehniques used to redit effetive ondutivities of random omosites [7]. The urrent imlementation of RCM uses finite element analyses to determine the effetive roerties of a blok of subells. 4 Results and disussion Third-order bounds on the Youngs modulus of PBX 9501 omuted using Eq. (1-4) are an order of magnitude different from the exerimentally determined value. 5

6 f = 0.1 f = 0. f = 0. f = 0.4 f = 0.5 f = 0.6 f = 0.7 f = 0.8 f = 0.9 Fig.. RVEs ontaining 10% to 9% irular artiles. A better estimate is obtained from the DEM aroximation - around 1/5 th the exerimental Youngs modulus of PBX Fig. shows nine two-dimensional RVEs with artile volume frations from 0.1 to 0.9, irular artiles, and no artile-artile ontat that have been used for FEM and RCM alulations. Fig. shows the effetive Youngs modulus of these RVEs alulated by FEM, using about 70,000 six-noded triangular elements, omared to DEM reditions. The DEM and FEM reditions math losely. Sine the FEM estimate for a volume fration of 0.9 is also 1/5 the exerimental Youngs modulus of PBX 9501, this imlies that some stress-bridging has to be inororated in a RVE that models PBX One aroah to inororating stress-bridging is to use aroximate irles that have ragged edges by dividing eah RVE into subells/elements. Fig. 4 shows omarisons of FEM and RCM alulations on the nine models using four-noded square elements. RCM tends to overestimate the effetive Youngs modulus for all volume frations and modulus ontrasts. However, the FEM estimate of Youngs modulus is quite lose to the exerimental value. Models based on the atual size distribution of PBX 9501 have also been simulated using FEM and RCM. Eight models, eah ontaining about 86% by volume 6

7 FEM DEM PBX 9501 (DEM) PBX 9501 (Ext.) E eff /E b Partile Vol. Fra. Fig.. Comarison of DEM and FEM estimates of Youngs modulus RCM 16x16 FEM PBX 9501 (Ext.) E eff /E b Partile Vol. Fra. Fig. 4. Comarisons of FEM and RCM estimates of Youngs modulus. of artiles, were generated and eah model was divided into 56 ± 56 square subells from RCM ( ± subells/blok) and FEM alulations. Subells were assigned HMX roerties if they ontained more than 50% artiles by area. The binder was dirty, i.e., effetive roerties from DEM were assigned to the binder to bring the volume fration of artiles u to 9%. FEM and RCM reditions for these models are shown in Fig. 5. For these mirostrutures, the FEM estimates 7

8 Youngs modulus (GPa) FEM Dry Blend PBX 9501 RCM Dry Blend PBX 9501 FEM Pressed PBX 9501 RCM Pressed PBX 9501 Ext. PBX No. of Partiles in RVE Fig. 5. FEM and RCM reditions PBX of Youngs modulus vary from to 6 times the exerimental Youngs modulus of PBX Though the RCM estimates are onsiderably higher than the FEM reditions, these onverge to the FEM results with inreasing subells er blok. Aetable auray in the RCM aroximation is obtained when bloks of 16 ² 16 subells are used for the RCM alulations. 5 Conlusions Third-order bounds on effetive elasti moduli are too far from the atual moduli of PBX 9501 to be of use. DEM estimates are lose to FEM estimates on mirostrutures without artile-artile ontat but underestimate the Youngs modulus of PBX RCM aroximations overestimate the effetive roerties but onverge towards FEM estimates with inrease in the number of subells er blok. FEM estimates an vary from as low as 1/5 th of the Youngs modulus of PBX 9501 to as high as 6 times the modulus deending on the mirostruture and the degree of disretization used in model RVEs for the same volume fration of artiles. 8

9 Aknowledgements This researh was suorted by the University of Utah Center for the Simulation of Aidental Fires and Exlosions (C-SAFE), funded by the Deartment of Energy, Lawrene Livermore National Laboratory, under subontrat B4149. Referenes [1] J.M. Zaug, in: Pro. 11th ntl. Detonation Sym., Snowmass, Colorado, 1998, [] T.D. Sewell, R. Menikoff, Personal Communiation, [] C.M. Cady, W.R. Blumenthal, G.T. Gray, D.J. dar, in: Pro. ntl. Conf. on Fund. ssues and Al. of Shok-Wave and High-Strain-Rate Phenomena (EXPLOMET 000), 000, Albuquerque, New Mexio. [4] G.T. Gray, D.J. dar, W.R. Blumenthal, C.M. Cady, P.D. Peterson, in: Pro. 11th ntl. Detonation Sym., Snowmass, Colorado, 1998,. 76. [5] G.W. Milton, Phys. Rev. Lett. 46 (1981) 54. [6] K.Z. Markov, in: K.Z. Markov, L. Preziosi (Eds.) Heterogeneous Media: Miromehanis Modeling Methods and Simulations, Birkhauser, Boston, 000,.1. [7] N. Shah, J.M. Ottino, Chemial Eng. Si. 41 (1986) 8. 9

arxiv:cond-mat/ v1 [cond-mat.mtrl-sci] 13 Oct 2005

arxiv:cond-mat/ v1 [cond-mat.mtrl-sci] 13 Oct 2005 arxiv:cond-mat/0510367v1 [cond-mat.mtrl-sci] 13 Oct 2005 Effective elastic moduli of polymer bonded explosives from finite element simulations Abstract Biswajit Banerjee Dept. of Mechanical Engineering,