Simulation of Impact andFragmentation with theMaterial Point Method

Biswajit BanerjeeJ. Guilkey, T. Harman, J. Schmidt, P. McMurtry

Center for the Simulation of Accidental Fires and ExplosionsUniversity of Utah

March 21, 2005

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Outline

�The Problem.

�The Tool: Material Point Method.

�Failure Simulation: The Approach.

�Simulations and Results.

�Conclusions and Future Work.

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Problem: The Experiment

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Problem: The Outcome

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Problem: The Goal

Predict fragment velocities and

fragment size distributions

in various fire scenarios

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Tools: Material Point Method

Sulsky et al.,1995, Computer Physics Communications, 87, 236-252

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Tools: Material Models - I� Additive decomposition of the rate of deformation.

d = de + dth + dp (1)

� Mie-Gruneisen equation of state.

p =ρ0C

20ζ[1 +

(1− Γ0

2

)ζ]

[1− (Sα − 1)ζ ]2 + Γ0E(2)

where p = pressure, C0= bulk speed of sound, ζ = (ρ/ρ0 − 1), E = internal energy,

Γ0 = Gruneisen’s gamma at reference state, Sα = linear Hugoniot slope coefficient.

(Zocher et al., 2000, ECOMAS, Barcelona.)

� Hypoelastic law for deviatoric stress.

dev(σ) = 2 µ(p, T ) dev(εe) (3)

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Tools: Material Models - II� Huber-von Mises yield condition.

f (σ, εp, εp, T ) = ‖dev(σ)‖ −√

2

3σy(ε

p, εp, T ) ≤ 0

(4)

� Associative rate-independent plasticity.

εp = λ∂f (σ, εp, εp, T )

∂σ(5)

� Johnson-Cook plasticity model.

σy(εp, εp, T ) = [A + Bεn

p ][1 + C ln(ε∗p)][1− TmH ] (6)

where εp =√

23‖d

p‖, ε∗p = εp/εp0, εp =∫ t

0 εp(τ ) dτ , TH = T−TrTm−Tr

, and A, B, C, n,m

are material constants. (Johnson and Cook, 1983, Proc. 7th Intl. Symp. Ballistics, The Hague.)

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Tools: Material Models - III� Temperature- and pressure-dependent shear modulus.

µ(p, T ) =1

J

[(1− T )[µ0 +

∂µ

∂p(p

η)] +

ρkT

Cm

](7)

where T = TTm

, η = ( ρρ0

)1/3, J = 1+exp[ T−1

ζ(1− T1+ζ )

], and ζ, C,m are material constants.

(Nadal and Le Poac, 2003, J. Appl. Phys., 93(5), 2472-2480).

� Pressure-dependent melt temperature.

Tm(p) = Tm0 exp

[2a

(1− 1

η

)]η2(Γ0−a−1/3) (8)

where Tm0 is the melt temperature at ρ = ρ0 and a is a correction to Gruneisen’s

gamma Γ0. (Steinberg et al., 1980, J. Appl. Phys., 51(3), 1498-1504).

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Tools: Heating� Isotropic thermal expansion rate.

dth = αT1 (9)

where α is the thermal expansion coefficient.

� Plastic work converted into a plastic heating rate.

T p =χ

ρCpσ : dp (10)

where χ is the Taylor-Quinney coefficient, and Cp(T ) is the specific heat at constant

pressure.

� Heat conduction (summed over grid points).

Tg = T sg + T p

g −κ

ρCv∇g ·∇pT (11)

κ = 0 for adiabatic heating.

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Failure Simulation: Approach�Determine failed material point.

•Evolve porosity and a scalar damage variable andcheck TEPLA-F criterion.

•Check loss of hyperbolicity of the incrementalgoverning equations.

•Check for melting.

� If failed.• Incrementally lower the material point stress tozero.

•Assign a separate velocity field to failed materialpoints.

•Allow “failed” and unfailed material points to in-teract via contact.

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Failure Simulation: Evolution Rules� Porosity evolution.

f = fnucl + fgrow (12)

where fgrow = (1 − f )Tr(dp), fnucl =fn

(sn

√2π)

exp

[−1

2

(εp − εn)2

s2n

]εp, fn is the

volume fraction of void nucleating particles, εn is the mean of the distribution of

nucleation strains, and sn is the standard deviation of the distribution.

(Chu and Needleman, 1980, ASME J. Engg. Mater. Tech., 102, 249-256).

� Scalar damage evolution (Johnson-Cook model).

D = εp/εfp (13)

where εfp =

[D1 + D2 exp

(D3

3σ∗

)][1 + D4 ln(εp

∗)] [1 + D5TH ], σ∗ =Tr(σ)

σeq, D

is the scalar damage variable, εfp is the fracture strain, and D1, D2, D3, D4, D5 are

constants. (Johnson and Cook, 1985, Int. J. Eng. Fract. Mech., 21, 31-48).

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Failure Simulation: Failed ?� TEPLA-F failure criterion satisfied.

(f/fc)2 +(εp/ε

fp

)2= 1 (14)

(Johnson and Addessio, 1988, J. Appl. Phys., 64(12), 6699-6712).

� Drucker stability postulate violated.

σ : dp ≤ 0 (15)

(Drucker, 1959, J. Appl. Mech., 26, 101-106.)

� 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, 99-241).

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Failure Simulation: Approach

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Simulations: Impact

Chhabildas et al, 1998, Int. J. Impact Engrg., 23, 101-112.

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Simulations: Impact L1

Z Z

Aluminum SphereVelocity = 1480 m/s

9.52

90

VISAR Reading ofAxial Velocity

31.8

Aluminum Plate

S5 S6

S1 S2 S3

4819

Axial Strain Gages

78 Aluminum Plate

Hollow Aluminum Cylinder

13.614

28.6

2

X

S4

Y

(All dimensions are in mm. Not to scale)

(0.12,2.5)

Aluminum Sphere

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Simulations: Impact L1

Failed Material Points. Stress Distribution.

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Results: Impact L1

0 20 40 60 80 1000

500

1000

1500

Time (µ sec)

Ene

rgy

(J)

Kinetic EnergyStrain EnergyTotal Energy

Energy.

0 20 40 60 80 1000

0.5

1

1.5

2

Time (µ sec)

Mom

entu

m (k

g m

/s)

Momentum (mag)X−MomentumY−MomentumZ−Momentum

Momentum.

0 10 20 30 40 50 60 70 80 90 100

−50

0

50

100

150

200

250

300

Time (µ sec)

Axi

al V

eloc

ity (m

/s)

Expt.Inner Circle2nd Circle3rd Circle4th Circle

Velocity.

0 10 20 30 40 50 60 70 80 90 100−8

−6

−4

−2

0

2

4

6

8x 10−3

Time (µ sec)

Axi

al S

trai

n

Expt.Simulation

Strain at S4.18

Simulations: Impact L3

Z Z

9.52

90

VISAR Reading ofAxial Velocity

31.8

Aluminum Plate

S5 S6

S1 S2 S3

4819

Axial Strain Gages

78 Aluminum Plate

Hollow Aluminum Cylinder

X

S4

Y

(All dimensions are in mm. Not to scale)

Aluminum Sphere

13.919

28.6

Aluminum SphereVelocity = 1470 m/s

(11.4,−3.8)

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Results: Impact L3

0 20 40 60 80 1000

500

1000

1500

Time (µ sec)

Ene

rgy

(J) Kinetic Energy

Strain EnergyTotal Energy

Energy.

0 20 40 60 80 1000

0.5

1

1.5

2

Time (µ sec)

Mom

entu

m (k

g m

/s)

Momentum (mag)X−MomentumY−MomentumZ−Momentum

Momentum.

0 10 20 30 40 50 60 70 80 90 100−10

0

10

20

30

40

50

Time (µ sec)

Axi

al V

eloc

ity (m

/s)

Expt.Inner Circle2nd Circle3rd Circle4th Circle

Velocity.

0 10 20 30 40 50 60 70 80 90 100−0.012

−0.01

−0.008

−0.006

−0.004

−0.002

0

Time (µ sec)

Axi

al S

trai

n

Expt.Simulation

Strain at S4.

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Simulations: Hot Jet: 2D

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Simulations: Pool Fire: 3D

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Simulations: Heat Tape: 3D

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Results: Fragmentation: 2D

Coarse Grid - no tensile stresses in failed particles.

Fine Grid - no tensile stresses in failed particles.

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Results: Fragmentation: 2D

Coarse Grid - mass of failed particles removed.

Fine Grid - mass of failed particles removed.

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Results: Fragmentation: 2D

Coarse Grid - zero deviatoric stress in failed particles.

Fine Grid - zero deviatoric stress in failed particles.

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Results: Fragmentation: 2D

Coarse Grid - Equation of State for PBX9501.

Fine Grid - Equation of State for PBX9501.

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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.

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Questions?

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