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Stewart SillingPaul Demmie
Multiscale Dynamic Material Modeling DepartmentSandia National Laboratories
Thomas L. WarrenConsultant
2007 ASME Applied Mechanics and Materials Conference, Austin, TXJune 6, 2007
Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,
for the United States Department of Energy under contract DE-AC04-94AL85000.
Peridynamic Simulation of High-Rate Material Failure
SAND2007-3464C
SAND2007-3464C • frame 2
Background:A better way to model cracks
• Problem
– Develop a general tool for modeling material and structural failure due to cracks.
• Motivation
– Standard mathematical theory for modeling deformation cannot handle cracks.
• PDE’s break down if a crack is present.
• Finite elements and similar methods inherit this problem.
• Approach
– Develop a mathematical theory in which:
• The same equations apply on or off of a crack.
• Cracks are treated like any other type of deformation.
• Cracks are self-guided: no need for supplemental equations.
– Implement the theory in a meshless Lagrangian code called EMU.
SAND2007-3464C • frame 3
Peridynamic theory – the basic idea
•PDEs are replaced by the following integral equation:
•Compare classical PDE:
where
u = displacement; f = force density that x’ exerts on x;
b = prescribed external force density; H = neighborhood of xwith fixed radius δ.
∫ −−−=H
tdVttt ),(')'),,(),((),( xbxxxux'ufxu&& ρρρρ
x
x’f
δ
H
),(),(),( ttt xbxxu −•∇= σσσσρρρρ &&
SAND2007-3464C • frame 4
Material models
• A peridynamic material model gives bond force density as a function of bond stretch.
• Can include dependence on rate and history of stretch.
• Notation:
ξξξξ
ξξξξηηηηξξξξ −+Bond stretch
Loading
Unloading
Bond force ),( ξξξξηηηηf
xxuu −′=−′= ξξξξηηηη
SAND2007-3464C • frame 5
Microelastic materials
• A body is microelastic if f is derivable from a scalar micropotential w, i.e.,
• Interactions (“bonds”) can be thought of as elastic (possibly nonlinear) springs.
• Strain energy density at x is found by summing the energies of all springs connected to x’:
),(),( ξηη
ξη∂
∂=
wf uu −= 'η xx −= 'ξ
∫ −−=R
dVxxuuwxW ')','(2
1)(
f
x
x’
SAND2007-3464C • frame 6
What if you really want a stress tensor?
• Stress tensors (and strain tensors) play no role in the theory so far.
• However, define the peridynamic stress tensor field by
where S is the unit sphere and Ω is solid angle.
• This field satisfies the classical equation of motion:
( ) ( )
( ) ( )qpqupuqp
mxmxx
−−=
Ω−++= ∫ ∫ ∫∞ ∞
),()(,ˆ
,ˆ2
1)(
0 0
2
ii
mji
S
ij
ff
dzdydmzyfzyσσσσ
ijiji bu += ,σσσσρρρρ &&
fx
x-zm
x+ym
Bond through xin the direction m
m
SAND2007-3464C • frame 7
• Damage is introduced at the bond level.
• Bond breakage occurs irreversibly according to some criterion such as exceeding a prescribed critical stretch.
• In practice, bond breakages tend to occur along 2D surfaces (cracks).
Material modeling:
Damage
Bond stretch
Bond force
s*
SAND2007-3464C • frame 8
Energy required to advance a crackdetermines the bond breakage stretch
• Adding up the work needed to break all bonds across a crack yields the energy release rate:
∫ ∫+
=δδδδ
0
02R
dVdzwhG
z
ξ
R+Crack
There is also a version of the J-integral that applies in this theory.
*sξξξξ
f
w0
w0 = work to break one bond
Bond elongation = sξξξξ
SAND2007-3464C • frame 9
EMU numerical method
• Integral is replaced by a finite sum.
• Resulting method is meshless and Lagrangian.
),() ,( tV iiik
Hk
n
i
n
k
n
i xbxxuufu +∆−−= ∑∈
&&ρ
δ
ik f
H
SAND2007-3464C • frame 10
EMU numerical method:Relation to SPH
• Both are meshless Lagrangian methods.
• Both involve integrals.
• But the basic equations are fundamentally different:
– SPH relies on curve fitting to approximate derivatives that appear in the classical PDEs.
– Peridynamics does not use these PDEs, relies on pair interactions.( )
bx
u
dVxKxx
x
v
x
v
dVxKxvx
v
T
+∂
∂=
=∂
∂
=
∂
∂+
∂
∂=
=∂
∂
∫
∫
σσσσρρρρ
σσσσσσσσ
εεεεσσσσσσσσ
εεεε
&&
&
')'()'(
2
1
')'()'(
SPH
x
σσσσx∂∂ /σσσσ
( ) ( )( ) ( )xbdVxxxuxufxu +−−= ∫ ''),('&&ρρρρ
Emu
x'x
f
SAND2007-3464C • frame 11
• Assume a homogeneous deformation.
Bulk response with damage
Undeformed circle
Deformed
Broken bonds
ξFξ
λ1
σ11
Bulk response
1 λ
f
µ=0
µ=1
Bond response
Expect instability
SAND2007-3464C • frame 12
A validation problem:
Center crack in a brittle panel (3D)
Bond stretch
Bond force
s0
w0
Bulk strain (µ)
Fo
rce
(kN
)
Based on s0=0.002, find G=384 J/m2. Full 3D calculation shows crack growth
when σ=46.4 MPa. Use this in
22
J/m 371==E
aG
πσ
SAND2007-3464C • frame 13
Example: dynamic fracture in steel
SAND2007-3464C • frame 14
Isotropic materials:Other examples
Impact and fragmentation
Crack turning in a 3D feature
Defect
Spiral crack due to torsion
Defect
Transition to unstable crack growth
BANG!Defect
SAND2007-3464C • frame 15
Polycrystals: Mesoscale model (courtesy F. Bobaru, University of Nebraska)
• Vary the failure stretch of interface bonds relative to that of bonds within a grain.
• Define the interface strength coefficient by
Large β favors intra(trans)-granular fracture.β =sinterface
*
sgrain
*
β = 1 β = 4β = 0.25
•What is the effect of grain boundaries on the fracture of a polycrystal?
SAND2007-3464C • frame 16
Example: dynamic fracture in PMMA
SAND2007-3464C • frame 17
Applications: Fragmentation of a concrete sphere
• 15cm diameter concrete sphere against a rigid plate, 32.4 m/s.
– Mean fragment size agrees well with experimental data of Tomas.
SAND2007-3464C • frame 18
Example: Concrete sphere drop, ctd.
• Cumulative distribution function of fragment size (for 2 grid spacings):
– Also shows measured mean fragment size*
*J. Tomas et. al., Powder Technology 105 (1999) 39-51.
SAND2007-3464C • frame 19
Complex materials:Prediction of composite material fracture
• Bonds in different directions can have different properties• Can use this principle to model anisotropic materials and their failure.
Crack growth in a notched panel Delamination caused by impact
SAND2007-3464C • frame 20
Statistical distribution of critical stretches
• Vary the failure stretch of interface bonds relative to that of bonds within a grain.
• Define the interface strength coefficient by• We can introduce fluctuations in s* as a function of position and bond orientation according to Weibull or other distribution.• This is one way of incorporating the statistical nature of damage evolution.
s* s
f
s*
P(s*)
SAND2007-3464C • frame 21
Peridynamic states:A more general theory
• Limitations of theory described so far:
– Poisson ratio = 1/4.
– Can’t enforce plastic incompressibility (can’t decouple deviatoric and isotropic response).
– Can’t reuse material models from the classical theory.
• More general approach: peridynamic states.
– Force in each bond connected to a point is determined collectively by the deformation
of all the bonds connected to that point.
f
Bond
f
State
SAND2007-3464C • frame 22
Peridynamic deformation statesand force states
• A deformation state maps any bond ξ into its deformed image Y(ξ).
• A force state maps any bond ξ into its force density T (ξ).
• Constitutive model: relation between T and Y .
ξξξξ
x
)(ξξξξY
x
Deformation state
)(ξξξξT
x
Force state
SAND2007-3464C • frame 23
Peridynamic states:Volume term in strain energy
• One thing we can now do is explicitly include a volume-dependent term in the strain energy density… can get any Poisson ratio.
ξξξξ)(ξξξξY
x x
Undeformed state Deformed state
∫
∫
∫
=
+=
−=
dVwxW
dVwxW
dVY
),,(~)(
)(),()(
)(
ϑϑϑϑξξξξηηηη
ϑϑϑϑψψψψξξξξηηηη
ξξξξ
ξξξξξξξξϑϑϑϑDilatation:
Strain energy density:
or:
SAND2007-3464C • frame 24
Peridynamic states:Using material models from classical theory
• Map a deformed state to a deformation gradient tensor.
• Use a conventional stress-strain material model.
• Map the stress tensor onto the bond forces within the state.
F
σ σ σ σ ((((F))))
f=T ((((ξξξξ))))
Y(ξξξξ)=ξ ξ ξ ξ + ηηηη
SAND2007-3464C • frame 25
State-based equation of motion
• A force state at x gives the force in each bond connected to x:
• Bond-based:
• State-based:
),(')','(),( tdVtR
xbxxuufxu +−−= ∫&&ρρρρ
),(')'](,'[)'](,[2
1),( tdVttt
R
xbxxxTxxxTxu +−−−= ∫&&ρρρρ
)(ξξξξTf =
Force states at x and x’ combine
x
x’
],[ txT ],[ tx'T
SAND2007-3464C • frame 26
Elastic state-based materials
• Suppose W is a scalar valued function of a state such that
• Bodies composed of this elastic state-based material conserved energy in the usual sense of elasticity.
∫ ∫ ⋅=∂
∂dVubWdV
t&
YYWYT Y allfor )()( =)
Frechet derivative (like a gradient)
SAND2007-3464C • frame 27
Peridynamic state model:
Effect of Poisson ratio on crack angle
α
Thick, notched brittle linear elastic plate is subjected to combined tension and shear
V
V
-V
-V
Defect2a
SAND2007-3464C • frame 28
Peridynamic state model:
Effect of Poisson ratio on crack angle, ctd.
•Hold E constant and vary ν.•Larger ν means smaller µ.
•Does this change the crack angle?•Approximate analysis near initial crack tip based on Mohr’s stress circle:
•Find orientation θ of plane of max principal stress.
),( 1222 σσσσσσσσ
Normal stress
Shear stress
2θ o
o
E
aVtaEVt
8.164.0
1.211.0
))1/(1(tan)/2(tan2
)1(2/
//0
1
2212
1
122211
=⇒=
=⇒=
+==
+=
===
−−
θθθθνννν
θθθθνννν
ννννσσσσσσσσθθθθ
ννννµµµµ
µµµµσσσσσσσσσσσσ
SAND2007-3464C • frame 29
Peridynamic state model:
Effect of Poisson ratio on crack angle, ctd.
•Predicted crack angles near initial crack tip are close to orientation of max principal stress
o1.21=θθθθ
ν= 0.1
Model result
o8.16=θθθθ
ν= 0.4
Model result
SAND2007-3464C • frame 30
Peridynamic states vs. FE:
Elastic-plastic solid
•Direct comparison between a finite-element code and Emuwith a conventional material model.
SAND2007-3464C • frame 31
Summary:Peridynamic vs. conventional model
• Peridynamic
– Uses integral equations.
– Same equations hold on or off discontinuities.
– Nonlocal.
– Force state (or deformation state) has “infinite degrees of freedom.”
• Classical
– Uses differential equations.
– Discontinuities require special treatment (methods of LEFM, for example).
– Local.
– Stress tensor (or strain tensor) has 6 degrees of freedom.