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Micromechanics-based Determination of Effective Elastic Properties of Polymer Bonded Explosives Biswajit Banerjee and Daniel O. Adams Dept. of Mechanical Engineering, University of Utah, Salt Lake City, UT 84112 Polymer Bonded Explosives Polymer bonded explosives are particulate composites. The two primary components of these composites are explosive particles and a rubbery binder. The volume of particles in the composite is typically around 90% of the total volume. Table 1 Typical polymer bonded explosives. 0.87 0.92 0.90 KEL-F-3700 Estane/BDNPA-F KEL-F-800 Pressed PBX 9501 based microstructures 0.13 0.08 0.10 Modeling Particulate Composites PBX 9501 contains 92% by volume of HMX (high melting explosive) particles and 8% by volume of binder. The HMX particles are monoclinic and linear elastic. The binder is a 1:1 mixture of the rubber Estane 5703 and a plasticizer (BDNPA/F). The mechanical behavior of the binder is strain rate and temperature dependent. As a result, the response of PBX 9501 also depends on strain rate and temperature. V ol. Frac. = 0.1 17.7 0.32 4 10 Eeff /E binder 15.3 0.21 V ol. Frac. = 0.3 DEM 5 10 Poisson's ratio Expt. MD Simulation V ol. Frac. = 0.2 5 10 FE M DE M PB X 9501 (DE M) PB X 9501 (E xpt.) 10 Figure 1 Microstructure of PBX 9501. 3 2 1 25 C 22oC 16oC 0oC o -15 C -20oC -40oC 10 1 10 8 6 V ol. Frac. = 0.92 Figure 9 Models microstructures representing PBX 9501. 0 0.2 0.4 0.6 Particle V ol. Frac. 0.8 1 0 RCM-GMC 16x16 10 6000 5000 4 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Particle V ol. Frac. 1 8000 FE M GMC R CM-GMC 7000 4000 3 10 Pressed PBX 9501 FE M GMC R CM-GMC 5000 4000 3000 2 10 3000 2000 1 10 10 PBX 9501 dry blend 6000 2000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Particle V ol. Frac. 1 1000 0 Figure 6 Differential effective medium (DEM), Recursive Cell Method with FEM homogenizer (RCM-FEM), and Recursive Cell Method with GMC homogenizer (RCM-GMC) vs. finite element estimates. 55oC o 40 C 27oC 25oC o 17 C 0oC o -20 C -40oC -55oC 12 Y oung's Modulus (GPa) Y oung's Modulus (MPa) 2 14 o 10 PBX 9501 V ol. Frac. = 0.8 0 0 0 10 Contrast = 100000 FE M R CM Contrast = 10000 FE M R CM Contrast = 1000 FE M R CM Contrast = 100 FE M R CM Contrast = 10 FE M R CM 4 10 Binder V ol. Frac. = 0.7 5 10 1 10 RCM-FEM 16x16 10 10 3 V ol. Frac. = 0.6 10 3 2 10 V ol. Frac. = 0.5 10 10 4 V ol. Frac. = 0.4 Figure 5 Model particulate composite microstructures. Table 2 Elastic properties of HMX. Young's modulus (GPa) Expt. MD Simulation Figure 4 Schematic of recursive cell method (RCM). C 11 (MPa) RDX HMX TATB Vol. Frac C 11 (MPa) PBX 9010 PBX 9501 PBX 9502 Material Binder PBX 9501 dry blend based microstructures E ffective Y oung's Modulus (MPa) Particles Material Vol. Frac E ffective Y oung's Modulus (MPa) PBX Modeling PBX 9501 In the recursive cell method (RCM) the representative volume element (RVE) is divided into a regular grid of subcells. Instead of the whole RVE, small square blocks of subcells are homogenized at a time. The procedure is repeated recursively until a single homogeneous material remains. This material is the effective material. 1000 1 2 Model 3 4 0 1 2 Model 3 4 Figure 10 Effective properties of model RVEs from FEM, GMC, RCM-GMC. Three-D. vs. Two-D: 4 0 10 2 -1 10 -4 10 -2 10 0 10 Strain R ate (/s) 2 10 4 10 0 -4 10 -2 10 0 2 10 Strain R ate (/s) 10 4 10 Figure 7 Three-dimensional models of particulate composites. Figure 2 Young's modulus of binder and PBX 9501. Poisson's ratio is 0.49 for the binder and 0.35 for PBX 9501. Glass-Estane composites Micromechanics HMX-Binder composites 5 Exact relations and DEM are used to assess the accuracy of FEM and RCM Bounds can be observed to be widely separated. Direct GMC does not model stress bridging accurately. 4 10 Eeff /E binder Exact Relations Third-Order Bounds Differential Effective Medium (DEM) Finite Element (FEM) Approximations Generalized Method of Cells (GMC) Recursive Cell Method (RCM) 10 Figure 11 Finite element prediction vs. experimental data on the Young's modulus of PBX 9501. Upper B ound L ower B ound PB X 9501 Upper B ound PB X 9501 L ower B ound PB X 9501 E xpt. Conclusions 3 10 2 10 1 10 0 10 0 0.2 0.4 0.6 Particle V ol. Frac. 0.8 1 Figure 3 Third-order bounds and PBX 9501. Figure 8 Three-dimensional vs. two-dimensional finite element estimates. -- Two-dimensional finite element estimates are close to differential effective medium predictions. -- RCM-GMC estimates closer to finite element estimates than RCM-FEM predictions. -- Difference between two- and three-dimensional finite element estimates is small. - Predicted effective properties are strongly microstructure dependent. - Detailed numerical simulations are required for accurate estimates. - Recursive cell method with the generalized method of cells homogenizer can predict effective moduli that are close to finite element estimates with considerably greater computational efficiency. Acknowledgements Funded by the Department of Energy grant DE-FG03-02ER45914 and the Center for the Simulation of Accidental Fires and Explosions, University of Utah, Salt Lake City, UT 84112.