Simulation of Impact and Fragmentation with the Material Point Method

Transcription

1 Simulation of Impact and Fragmentation with the Material Point Method Biswajit Banerjee J. Guilkey, T. Harman, J. Schmidt, P. McMurtry Center for the Simulation of Accidental Fires and xplosions University of Utah March 21, 2005 C SAF

2 Outline The Problem. The Tool: Material Point Method. Failure Simulation: The Approach. Simulations and Results. Conclusions and Future Work. C SAF 2

3 Problem: The xperiment C SAF 3

4 Problem: The Outcome C SAF 4

5 Problem: The Goal Predict fragment velocities and fragment size distributions in various fire scenarios C SAF 5

6 Tools: Material Point Method Sulsky et al.,1995, Computer Physics Communications, 87, C SAF 6

7 Tools: Material Models - I Additive decomposition of the rate of deformation. d = d e + d th + d p (1) Mie-Grüneisen equation of state. p = ρ 0C0ζ [ ( 1 Γ ) ] 0 2 ζ [1 (S α 1)ζ] 2 + Γ 0 (2) where p = pressure, C 0 = bulk speed of sound, ζ = (ρ/ρ 0 1), = internal energy, Γ 0 = Gruneisen s gamma at reference state, S α = linear Hugoniot slope coefficient. (Zocher et al., 2000, COMAS, Barcelona.) Hypoelastic law for deviatoric stress. dev( σ) = 2 µ(p, T ) dev( ε e ) (3) 7

8 Tools: Material Models - II Hüber-von Mises yield condition. 2 f(σ, ε p, ε p, T ) = dev(σ) 3 σ y(ε p, ε p, T ) 0 Associative rate-independent plasticity. ε p = λ f(σ, εp, ε p, T ) σ Johnson-Cook plasticity model. (4) (5) σ y (ε p, ε p, T ) = [A + Bε n p][1 + C ln( ε p)][1 T m H ] (6) where ε p = 2 3 dp, ε p = ε p / ε p0, ε p = t 0 ε p(τ) dτ, T H = T T r T m T r, and A, B, C, n, m are material constants. (Johnson and Cook, 1983, Proc. 7th Intl. Symp. Ballistics, The Hague.) 8

9 Tools: Material Models - III Temperature- and pressure-dependent shear modulus. µ(p, T ) = 1 J where ˆT = T T m, η = ( ρ ρ 0 ) 1/3, J = 1+exp[ [ (1 ˆT )[µ 0 + µ ] p (p ρkt )] + η Cm ˆT 1 ζ(1 ˆT 1+ζ ) (Nadal and Le Poac, 2003, J. Appl. Phys., 93(5), ). Pressure-dependent melt temperature. [ ( T m (p) = T m0 exp 2a 1 1 )] η (7) ], and ζ, C, m are material constants. η 2(Γ 0 a 1/3) (8) where T m0 is the melt temperature at ρ = ρ 0 and a is a correction to Grüneisen s gamma Γ 0. (Steinberg et al., 1980, J. Appl. Phys., 51(3), ). 9

10 Tools: Heating Isotropic thermal expansion rate. d th = α T 1 (9) where α is the thermal expansion coefficient. Plastic work converted into a plastic heating rate. T p = χ σ : d p (10) ρc p where χ is the Taylor-Quinney coefficient, and C p (T ) is the specific heat at constant pressure. Heat conduction (summed over grid points). T g = T g s + T g p κ g p T (11) ρc v κ = 0 for adiabatic heating. 10

11 Failure Simulation: Approach Determine failed material point. volve porosity and a scalar damage variable and check TPLA-F criterion. Check loss of hyperbolicity of the incremental governing equations. Check for melting. If failed. Incrementally lower the material point stress to zero. Assign a separate velocity field to failed material points. Allow failed and unfailed material points to interact via contact. 11

12 Failure Simulation: volution Rules Porosity evolution. f = f nucl + f grow (12) [ ] where f grow = (1 f)tr(d p ), f f n nucl = exp 1 (ε p ε n ) 2 (s n 2π) 2 ε s 2 p, f n is the n volume fraction of void nucleating particles, ε n is the mean of the distribution of nucleation strains, and s n is the standard deviation of the distribution. (Chu and Needleman, 1980, ASM J. ngg. Mater. Tech., 102, ). Scalar damage evolution (Johnson-Cook model). Ḋ = ε p /ε f p (13) where ε f p = [ D 1 + D 2 exp ( D3 3 σ )] [1 + D 4 ln( ε p )] [1 + D 5 T H ], σ = Tr(σ), D is the scalar damage variable, ɛ f p is the fracture strain, and D 1, D 2, D 3, D 4, D 5 are constants. (Johnson and Cook, 1985, Int. J. ng. Fract. Mech., 21, 31-48). σ eq 12

13 Failure Simulation: Failed? TPLA-F failure criterion satisfied. (f/f c ) 2 + ( ε p /ε f p) 2 = 1 (14) (Johnson and Addessio, 1988, J. Appl. Phys., 64(12), ). Drucker stability postulate violated. σ : d p 0 (15) (Drucker, 1959, J. Appl. Mech., 26, ) Loss of hyperbolicity of the incremental equations. det(a) 0 (16) where A = n M n + n σ (n1), M is the incremental tangent modulus tensor, and n is the normal to the localization band. (Perzyna, 1998, Localization and Fracture Phenomena in Inelastic Solids, ). 13

14 Failure Simulation: Approach C SAF 14

15 Simulations: Impact Chhabildas et al, 1998, Int. J. Impact ngrg., 23, C SAF 15

16 Simulations: Impact L1 Aluminum Sphere Velocity = 1480 m/s S1 S2 S3 Aluminum Sphere 9.52 VISAR Reading of Axial Velocity Axial Strain Gages X (0.12,2.5) Y S4 S5 S6 Aluminum Plate Hollow Aluminum Cylinder 78 Aluminum Plate Z (All dimensions are in mm. Not to scale) Z C SAF 16

17 Simulations: Impact L1 Failed Material Points. Stress Distribution. C SAF 17

18 Results: Impact L1 nergy (J) Kinetic nergy Strain nergy Total nergy Axial Velocity (m/s) xpt. Inner Circle 2nd Circle 3rd Circle 4th Circle Time (µ sec) 2 nergy Time (µ sec) 8 x Velocity. xpt. Simulation Momentum (kg m/s) Momentum (mag) X Momentum Y Momentum Z Momentum Axial Strain Time (µ sec) Momentum Time (µ sec) Strain at S4. 18

19 Simulations: Impact L S1 S2 S3 Aluminum Sphere Aluminum Sphere Velocity = 1470 m/s 9.52 VISAR Reading of Axial Velocity X (11.4, 3.8) Y Axial Strain Gages S4 S5 S6 Aluminum Plate Hollow Aluminum Cylinder 78 Aluminum Plate Z (All dimensions are in mm. Not to scale) Z C SAF 19

20 Results: Impact L3 nergy (J) Kinetic nergy Strain nergy Total nergy Axial Velocity (m/s) xpt. Inner Circle 2nd Circle 3rd Circle 4th Circle Time (µ sec) 2 nergy Time (µ sec) Velocity. xpt. Simulation Momentum (kg m/s) Momentum (mag) X Momentum Y Momentum Z Momentum Axial Strain Time (µ sec) Momentum Time (µ sec) Strain at S4. 20

21 Simulations: Hot Jet: 2D C SAF 21

22 Simulations: Pool Fire: 3D C SAF 22

23 Simulations: Heat Tape: 3D C SAF 23

24 Results: Fragmentation: 2D Coarse Grid - no tensile stresses in failed particles. Fine Grid - no tensile stresses in failed particles. C SAF 24

25 Results: Fragmentation: 2D Coarse Grid - mass of failed particles removed. Fine Grid - mass of failed particles removed. C SAF 25

26 Results: Fragmentation: 2D Coarse Grid - zero deviatoric stress in failed particles. Fine Grid - zero deviatoric stress in failed particles. C SAF 26

27 Results: Fragmentation: 2D Coarse Grid - quation of State for PBX9501. Fine Grid - quation of State for PBX9501. C SAF 27

28 Conclusions/Future Work Wave arrival times and velocity peaks captured. Reasonable fragment distribution in 2D. Mesh dependence of results. Viscoplastic regularization. Nonlocal plasticity/continuum damage. Strong discontinuity approach to fracture. C SAF 28

29 Questions? C SAF O F 29