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.