BACKGROUNDSBACKGROUNDS

Two Models of Deformable Bodycontinuum deformation expressed in terms of field variablesrigid-body spring assembly of rigid-bodies connected by spring

Distinct Element Method (DEM)simple treatment of failure: breakage of springnon-rigorous determination of spring constants

deformable body particle modeling DEM analysis

breakage of spring

spring constant need to be determined in terms of material parameters

MATHEMATICAL INTERPLETATION OF RIGID-BODY SPRING MODEL MATHEMATICAL INTERPLETATION OF RIGID-BODY SPRING MODEL

Non-Overlapping Functions for Discretization

smooth but overlapping functions are used for discretization

characteristic functions of domain are used for discretization

ordinary FEM DEMu1

u2

u1

u2

PARTICLE DISCRETIZATION FOR FUNCTION AND DERIVATIVE PARTICLE DISCRETIZATION FOR FUNCTION AND DERIVATIVE

Discretization of Function f(x) in terms of {ϕα (x)}

Discretization of Derivative f,i(x) in terms of {ψα (x)}

∑α

ααϕ= )(f)(f d xx

∫ −=α

V

2df ds))(f)(f(})f({E xx

∫ α

∫ ϕα ϕ= α

Vds1 dsff

V

∑α

ααψ= )(g)(g idi xx

( )∑ ∫β

ββα

∫ ψα ϕψ= α fdsg

V i,ds1

iV

minimize Ef

different from {ϕα}

form of discretization form of discretization

optimal coefficients for given Voronoi blocks {ϕα}

and Delaunay triangles {ψα}

derivative of {ϕα} is not bounded but integratable

1-D PARTICLE DISCRETIZATION1-D PARTICLE DISCRETIZATION

xn xn+1

y n-1

y n y n+1

∑α

ααϕ= )(f)(f d xx

∑α

ααψ== )(g)(g)(dxdf xxx

average value taken over interval for ϕα

is coefficient of characteristic function ϕα

average slope of interval for ψα

is coefficient of characteristic function ψα

mother points

middle point of neighboring mother points

2-D PARTICLE DISCRETIZATION2-D PARTICLE DISCRETIZATION

Voronoi blocks for function Delaunay triangles for derivative

dual domain decomposition

function and derivative are discretized in terms of sets of non-overlapping characteristic functions, such that function and derivative are uniform in Voronoi blocks and Delaunay triangles,

COMPARISON OF PARTICLE DISCRETIZATION WITH ORDINARY DISCRETIZATION COMPARISON OF PARTICLE DISCRETIZATION WITH ORDINARY DISCRETIZATION

x2 x3

x1

f 2

f 1

f 3

derivative of particle discretization coincides with slope of plane which is formed by linearly connecting Voronoi mother points

+ + =f 1 f 2 f 3

f 1 f 2 f 3

+ + =

ordinary discretization with linear function

particle discretization

particle discretization of derivative

PARTICLE DISCRETIZATION TO CONTINUUM MECHANICS PROBLEM PARTICLE DISCRETIZATION TO CONTINUUM MECHANICS PROBLEM

Conjugate Functional

FEM-β: FEM with Particle Discretizationstiffness matrix of FEM-β coincides with stiffness matrix of FEM with uniform triangular elementincluding rigid-body-rotation, FEM-β gives accurate and efficient computation for field with singularity

∑∑

∫

αα

αα

ψσ=σ

ϕ=

σσ−εσ=σ

ijij

ii

V klijklij21

ijij

uu

dsc),u(J

hypo-Voronoi for displacement

Delaunay for stress: coefficients are analytically obtained by stationarizing J

FAILURE ANALYSIS OF FEM-βFAILURE ANALYSIS OF FEM-β

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

]k[]k[]k[]k[]k[]k[]k[]k[]k[

333231

232221

131211

( )]k[]k[ ijT

ji =

]k[]k[]k[ indirect12

direct1212 +=

x3

x1

Ω3

Ω1

Ω2

x2

for direct interaction

for indirect interaction

stiffness matrix of FEM-β strain energy due to relative deformation of Ω1 and Ω2 through movement of Ω3

cut two springs of direct and indirect interaction together or separately, according to certain failure criterion of continuum

FAILURE MODELING BY BREAKING SPRINGSFAILURE MODELING BY BREAKING SPRINGS

BA

C

R

Q

P

O

broken edge

region with reduced strain energy

Φα1

Φα3

Φα2

0.000480

0.000490

0.000500

0.000510

0.000520

0 20000 40000 60000 80000 100000

Number of Elements

J-in

tegr

al

AnalyticalFEM-betaFEM

relative error of J-integral is around a few percents

EXAMPLE PROBLEMEXAMPLE PROBLEMSimulation of Crack Growth

Check of Convergencedisplacementstrain/strain energy

Pattern of Crack Growthcan small difference in initial configuration cause large difference in crack growth?

uniform tension

CONVERGENCE OF SOLUTIONCONVERGENCE OF SOLUTION

2.0 2.1 2.2 2.3 2.40.000

0.002

0.004

0.006

ξ

ηd

ηd

NDF

1021031041050.000

0.002

0.004

0.006

ξ...

...

=2.09

Best Performance

displacement norm w.r.t. NDFdisplacement norm w.r.t. mesh quality

∫∫ −

=η

B

2

B

2

ddv||u||

dv||uu||displacement norm:

CHECK OF CRACK GROWTHCHECK OF CRACK GROWTH

εyy

evolution of normal strain distribution

pattern of crack growth

EXAMPLE: PLATE WITH 3 HOLESEXAMPLE: PLATE WITH 3 HOLES

slight difference in location of 3rd hole

simulation of crack growth: crack stems from holes

case a case b

DIFFERENCE IN CRACK GROWTH: DISTRIBUTION OF NORMAL STRAIN DIFFERENCE IN CRACK GROWTH: DISTRIBUTION OF NORMAL STRAIN

case a

case b

SIMULATION OF BRAZILIAN TESTSIMULATION OF BRAZILIAN TEST

Brazilian test

SIMULATION OF FOUR POINT BENDING TESTSIMULATION OF FOUR POINT BENDING TEST

0.01

-0.01

0.00

σ11

four point bending test

FOUR POINT BENDING WITH IDEARLY HOMOGENEOUS MATERIALS FOUR POINT BENDING WITH IDEARLY HOMOGENEOUS MATERIALS

0.01

-0.01

0.00

σ11

FEM-β

puts two source of local heterogeneity, 1) mesh quality for particle discretization and 2) crack path along Voronoi boundary. An ideally homogeneous material which is modeled with best mesh quality sometimes fail to simulate crack propagation.

CONCLUDING REMARKSCONCLUDING REMARKS

Particle Discretizationdiscretization scheme using set of non-overlapping characteristic functions

Continuum Mechanics Problemessentially same accuracy as FEM with uniform strainapplicable to non-linear plasticity

Failure Analysissimple but robust treatment of failureMonte-Carlo simulation for studying local heterogeneity effects on failure- candidates of failure patterns are pre-determined by spatial discretization