This document outlines the use of peridynamics to model impact studies. It begins with an introduction to peridynamics as a way to model discontinuities compared to continuum mechanics. The document then summarizes the basic formulation of peridynamics using integral equations. It describes the prototype microelastic brittle model and its implementation in LAMMPS. Applications to plate impact and cluster-cluster impact are shown and parameters for glass and silica impacts are provided. The document concludes with examples of damage from impacts.

1. Outline Motivation Basic Formulation The PMB Model LAMMPS Applications Conclusions
Impact Studies with Peridynamics
C.J.Ruestes 12 D.R.Tramontina 12 E.M.Bringa 13
1 Instituto de Ciencias B´sicas, UNCuyo, Mendoza, Argentina
a
2 ANPCyT
3 CONICET
MECOM 2012 - Salta, Argentina - 15 de Noviembre de 2012
dtramontina@uncu.edu.ar
https://sites.google.com/site/simafweb
Funding: PICT-PRH-2008-0092, PICT2009-1325, 06/M035 SeCTyP-UNCuyo
Impact Studies with Peridynamics Instituto de Ciencias B´sicas, UNCuyo, Mendoza, Argentina, ANPCyT, CONICET
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2. Outline Motivation Basic Formulation The PMB Model LAMMPS Applications Conclusions
Outline
Motivation
Basic Formulation
The PMB Model
LAMMPS Implementation
Applications
Plate Impact
Cluster-cluster Impact
Conclusion
Impact Studies with Peridynamics Instituto de Ciencias B´sicas, UNCuyo, Mendoza, Argentina, ANPCyT, CONICET
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3. Outline Motivation Basic Formulation The PMB Model LAMMPS Applications Conclusions
Motivation
Continuum Mechanics Peridynamics
DE’s Integral Equations
Spontaneous formation of Non-local Models
discontinuities
Phase changes Objective: To compare obtained
Cracks results with experimental/MD
data.
Impact Studies with Peridynamics Instituto de Ciencias B´sicas, UNCuyo, Mendoza, Argentina, ANPCyT, CONICET
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4. Outline Motivation Basic Formulation The PMB Model LAMMPS Applications Conclusions
Basic Formulation
S.A.Silling, J.Mech Phys S., 48,175-209(1999)
ρ¨(x, t) =
u f (u(x , t) − u(x, t), x − x)dVx + b(x, t) , t 0
H
η =u −u , ξ =x −x
Conditions:
linear admissibility:
f (−η, −ξ) = −f (η, ξ) ∀η, ξ
angular admissibility:
(ξ + η) × f (η, ξ) = 0 ∀η, ξ
Impact Studies with Peridynamics Instituto de Ciencias B´sicas, UNCuyo, Mendoza, Argentina, ANPCyT, CONICET
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5. Outline Motivation Basic Formulation The PMB Model LAMMPS Applications Conclusions
Prototype Microelastic Brittle (PMB)
δw
Microelastic Material f (η, ε) = δη (η, ε)
Isotropic material
Bond strength depends only on the bond stretch (s).
Critical stretch s0
Non-recoverable bond rupture if s > s0
Impact Studies with Peridynamics Instituto de Ciencias B´sicas, UNCuyo, Mendoza, Argentina, ANPCyT, CONICET
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6. Outline Motivation Basic Formulation The PMB Model LAMMPS Applications Conclusions
Prototype Microelastic Brittle (PMB)
|ε + η| − |ε| y − |ε|
s= =
|ε| |ε|
f (y (t), ε) = g (s(t, ε))µ(t, ε)
g (s) = cs
(
1 si s(t , ε) < s0 ; ∀0 t t
µ(t, ε) =
0 otherwise
δw 9 s
∃f (η, ε) = (η, ε) ∀ε, η,>
= 18k 5G0
δη c = , s0 =
2
>
; πδ 4 9kδ
W = 9ks /2,
Impact Studies with Peridynamics Instituto de Ciencias B´sicas, UNCuyo, Mendoza, Argentina, ANPCyT, CONICET
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7. Outline Motivation Basic Formulation The PMB Model LAMMPS Applications Conclusions
LAMMPS (http://lammps.sandia.gov/)
Some good reasons to use LAMMPS:
Free, open source (GNU License)
Easy to learn.
Responsible developers and user community
Extensive docs
(http://lammps.sandia.gov/doc/Section commands.html/)
It runs efficiently in my laptop (2 cores) and in BlueGeneL (100 K cores),
including parallel I/O, with the same input script. Also efficient for GPUs.
Very efficient parallel energy minimization.
Includes many-body, bond order, & reactive potentials. Can simulate
inorganic & bio systems, and granular systems.
Extensive set of analysis routines. Coordination, Centro, CNA, etc.
Impact Studies with Peridynamics Instituto de Ciencias B´sicas, UNCuyo, Mendoza, Argentina, ANPCyT, CONICET
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8. Outline Motivation Basic Formulation The PMB Model LAMMPS Applications Conclusions
Prototype Microelastic Brittle (PMB)-LAMMPS
[Silling et.al, Comp.& Str., 83(17-18), 1526-1535(2005)]
For fragile materials, is better to avoid the s0 oversimplification
assuming the following definition:
so (t) = s00 − αsmin (t)
y (t) − |ε|
smin (t) = min{ }
ε ε
1
α=
4
Damage:
H µ(t, ε, η)dVx
φ(x, t) = 1 −
H dVx
Impact Studies with Peridynamics Instituto de Ciencias B´sicas, UNCuyo, Mendoza, Argentina, ANPCyT, CONICET
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9. Outline Motivation Basic Formulation The PMB Model LAMMPS Applications Conclusions
Plate Impact. Parameters
Case 1*: Case 2:
Projectile: Rigid Projectile: Glass
Target: Glass
r = 10mm , v = 100m/s
Target: Glass
D = 74mm , esp = 2,5mm
Material:
ρ = 2x103 kg /m3 ; G0 = 10J/m2 ; k = 14,9GPa
α = 0,25; s00 = 5x10−4 ; c = 1,68x1022
δ = 1,5mm; tsim = 0,2ms
(*)Parks M.L et.al., Comp Phys Comm. 179,777-783(2008)
Impact Studies with Peridynamics Instituto de Ciencias B´sicas, UNCuyo, Mendoza, Argentina, ANPCyT, CONICET
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10. Plate Impact
Case 1 Case 2
(c) y (d) S.Bouzid et.al, Int J Impact Eng. 25(9), 831-845(2001)

11. Plate Impact. Damage

12. Cluster-cluster Impact
N.Ohnishi, E.Bringa, J. Phys. Conf. Series 112, 042017(2008). Run in 256 cores (LLNL)
C.Ringl,E.Bringa et.al.,Astrophys. J. 752(2),
151-164(2012). Run in 48 cores(Germany).
Parameters
Velocity
Impact parameter
Radius
Structure orientation
Kalweit et.at.,Phys.Rev.B 74, 235415(2006)

13. Outline Motivation Basic Formulation The PMB Model LAMMPS Applications Conclusions
Cluster-cluster Impact. Silica. Parameters
d = 10mm
ρ = 5x103 kg /m3
v = 30m/s
k = 100GPa
α = 0,25; ∆x = 0,5mm
δ = 1,5mm
s00 = 5x10−4 ; G0 = 67,5J/m2
Impact Studies with Peridynamics Instituto de Ciencias B´sicas, UNCuyo, Mendoza, Argentina, ANPCyT, CONICET
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14. Cluster-cluster Impact. Silica
C.Ringl,E.Bringa et.al., Astrophys. J. 752(2), 151-164(2012)

15. Cluster-cluster Impact. Silica. Damage

16. Outline Motivation Basic Formulation The PMB Model LAMMPS Applications Conclusions
Conclusions
Peridynamics with PMB can be used for damage studies.
Good agreement with experimental and MD data.
Despite the simplicity of the Peridynamics with PBM, our
simulations reproduce fracture and damage patterns seen in
experiments and granular simulations.
Possible application in other fields, as impacts of several types
involving fracture for astrophysics, safety design, etc.
Further constitutive model developing is needed
Impact Studies with Peridynamics Instituto de Ciencias B´sicas, UNCuyo, Mendoza, Argentina, ANPCyT, CONICET
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17. Outline Motivation Basic Formulation The PMB Model LAMMPS Applications Conclusions
Thanks!
dtramontina@uncu.edu.ar
https://sites.google.com/site/simafweb
Impact Studies with Peridynamics Instituto de Ciencias B´sicas, UNCuyo, Mendoza, Argentina, ANPCyT, CONICET
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