M. L. L. Wijerathne (Lalith)Hide Sakaguchi

Kenji OguniMuneo Hori

3D crack propagation analysis withPDS-FEM

Japan Agency for Marine-Earth Science and Technology (JAMSTEC)The University of Tokyo

Motivation

uniform tension

0.0 0.3-0.3 0.0 0.3-0.3…

Minor heterogeneity play a significant role

Necessary to consider minor local heterogeneitySize and distribution local heterogeneity cannot be measuredMonte-Carlo simulations with randomly distributed heterogeneity

Meshless or adaptive methods are to complicated for stochastic studies Difficult to introduce heterogeneity to the numerical modelSophisticated and computationally intensive

PDS-FEM provides simple means of modeling size and distribution of local heterogeneity with numerically efficient failure treatments

No two experiments generate identical crack profiles

Organization

Discretization scheme and formulation of PDS-FEM

Failure treatment with torsional failure as an example

Dynamic model and kidney stone breaking as an example

Smooth, overlapping interpolation functions

u1

u2

u3

Background: two models of deformable body

fracture

continuum

BVP

FEM / BEM / FDM DEM

equivalence to continuum is not verified and springs are mysterious

cont

inuu

m

Rig

id-b

ody

sprin

g

difficult to deal with failure

Numerically intensive failure treatment Efficient failure treatment

spring properties ?

Discretization schemes of FEM and DEM

Ordinary FEM

Smooth and overlapping shape functions

DEM

Particles can be interpreted as non-overlapping shape functions

u1

u2

u1

u2

PDS-FEM: numerical method to solve BVP of a continuum with particle discretization

u3

discontinuities

1-D particle discretization

αu∑=α

ααϕ )()( xx uud

fα is the average value taken over the domain ϕα

1−αϕ αϕ

mother points

)(xu

Domains between neighboring mother points

βψ 1+βψ

∑==α

ββψ )()()( xxx ggdxdf

An average value for derivative is obtained on a conjugate geometry ψα

βg

Function and derivative are discretized using conjugate geometries.

βψ

2D-Particle discretization

∑=α

ααϕ )()( xuxud∑==β

ββψ )()()( xgxgdx

xduj

dj

j

d

Voronoi tessellation for function u(x) Delaunay tessellation for derivative u,i(x)

αϕ

Function and derivative are discretized using conjugate geometries, Voronoi and Delaunay tessellations

1u2u

Particle discretization for continuum: PDS-FEMBoundary Value Problem for Linear Elasticity

FunctionalOrdinary FEM functional

Functional used in PDS-FEM

∫ −−−=B jjijijijklijklijijijj dvubucuJ )(),,( ,2

1 εσεεεσ

⎩⎨⎧

∂==+

BuxuBxbxuc

ii

jilkijkl

on )(in 0)()),( ( ,

uB

B∂

∫ −+−+−−= dvucbuJ ijijijklijklijijjiijj )()()( ,, εδσεσδεσδδ

first variation

∫ −=B iilkijklji dvubucuI

21)( ,,u

u2

u3

u1

Particle discretization for continuum: PDS-FEM

∑∫

∫∫∑′

′

′

′

′′ ==βαα

ααβ

βαββα

αα

αααα

ψ

ψϕψϕ

,,21

,21

,,ki

B

B lijklB jkiik uu

dv

dvcdvuuKJ

1. Functional

2. Conjugate discretization

3. determination of uα

4. With Voronoi and Delaunay tessellations, coincides with stiffness matrix of FEM with linear characteristic functions

∫ −−−=B jjijijijklijklijijijj dvubucuJ )(),,( ,2

1 εσεεεσ

αϕ ∑= )()( xuxu iiααϕ

∑= )()( xx ijijββψσσ

∑= )()( xbxb iiααϕ ∑= )()( xx ijij

ββψεε

∑= )()( xcxc ijklijklββ ψ

βψ

Voronoi Delaunay

αα ′ikK

Organization

Discretization scheme and formulation of PDS-FEM

Failure treatment with torsional failure as an example

Dynamic model and kidney stone breaking as an example

Failure treatment

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

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

333231

232221

131211

]k[]k[]k[ indirect12

direct1212 +=

ϕ3

ϕ1

ϕ2

for direct interaction

for indirect interaction

stiffness matrix of FEM-βSpring properties are rigorously determined with material properties; E and ν

Failure is modeled by appropriately modifying the components of element stiffness matrix

ψβ

ϕ1 ϕ2

ϕ3

ψβ

Failure treatment: modeling brittle failure

( )∑ Ψ= ′′

β

βαβββαααlijkljik bcbK

∫Ψ=

B jj dvb βαβ

βα ψϕ,1

CBBK T=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=33

3

3

22

2

2

11

1

1

12

2

1

12

2

1

12

2

1

βαβα

βα

βα

βαβα

βα

βα

βαβα

βα

βα

bbb

b

bbb

b

bbb

bB

Candidate crack paths

No new DOFs or elements are introduced to accommodate the new crack surfaceComputational overhead is almost equal to re-computation of element stiffness matrix

an infinitesimally thin crack

ψβγ2

ψβγ1

ψβγ3

∫Ψ=

B jj dvd βγαβ

βγα ψϕ,1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−000

0

2121

21

21

1111

11

11

12

2

1

12

2

1

αβγαβγ

αβγ

αβγ

αβγαβγ

αβγ

αβγ

ddd

d

ddd

d

ϕ3

ϕ3

Failure treatment of PDS-FEM is approximate

No new DOFs or elements are introduced to accommodate the new crack surfaceCannot guarantee the satisfaction of BCs on new crack surface

PDS-FEM

new crack surfaces

No new nodes are introduced

ϕ2

FEM

One new node ad four elements are introduced

new crack surfaces

Accuracy of approximate failure treatment: problem setting

Far field stress σyy

Accuracy of crack tip stress field with J-integral

∫ ∂∂

−=s

ji

ijijij dsnxuJ σεσ

21

yyσ

yyσ

Accuracy of crack tip stress field

PDS-FEM crack tip stress filed is as accurate as the FEM solution, regardless of approximate failure treatment. The accuracy of crack tip stress filed can be improved by including rotational DOF

Accuracy of rotational component

PDS-FEM estimates the rotational component fairly accurately, leading to better estimation of crack tip stress filed

Example problem: torsion testing

Problem setting: torsion testing

E =70GPaν = 0.3

80 mm

φ15 mm

Real experiment Helical spiral fracture surface

Example problem: torsion testing

Helical spiral fracture surfaceGhost layer

(computed using MPI)

Organization

Discretization scheme and formulation of PDS-FEM

Failure treatment with torsional failure as an example

Dynamic model and kidney stone breaking as an example

Time integration approaches for PDS-FEM

FEM like continuum representation N-body problem like particle representation

( ) ( ) 0 ., , 1

0

1

0

=+ ∫∫ dtqqqFdtqqLt

t

t

tδδ &&

∑∑ ′′+=+=α

αααα

α

αααijijii uuKuumVTH

21

21

&&

Candy’s method: 4th order symplectic

41)()(

11

11

, ..,ktatb

kkkk

kkkk

=Δ+=

Δ+=−−

−−

pPqqqFpp

qqqF

∂∂

−=)()( V

pppP

∂∂

=)()( T

ααii uq = ααα

ii ump &=

∑∑ ′′−=−=α

αααα

α

αααijijii uuKuumVTL

21

21

&&

Hamiltonian principle

Lagrangian based Hamiltonian based

Second order explicit algorithm(a range of Variational integrators)

Simulating failure of concrete wall under tsunami load

Snaps of the experiment Damaged concrete wall

Conc. wall

Conducted at the LHC facility at the Port and Airport Research Institute , Japan

Model details

Material parameters

2.45

m

2.7m

100mm

300x300mm100mmFixed boundaries

Concrete SteelE /(GPa) 30 210ν 0.2 0.3σt /(MPa) 5 400

Reinforcement mesh (φ6@200mm x 200mm)

A A

Section A-A

Input data: pressure time histories at 5 heights

Locations of pressure gauges

P1

P2

P3

P4

P5

Pressure histories are interpolated for the intermediate points

Crack patterns : front side of 100mm thick wall

Crack patterns : back side of 100mm thick wall

Semicircular crack patterns of the front side are reaching the back

Shockwave Lithotripsy: kidney stone breaking

Complete phenomena is not well explained Simulation of this mechanism would help further development of this technology

Ultrasonic pulse generator

X-ray source

Typical pressure history in water induced by a single ultrasonic pulse

Pressure pulse is focused into kidney

Current studies of Shockwave Lithotripsy (SWL)

Xufeng Xi and Pei Zhong, J of Acoust. Soci. Am. 2001

P-wave in Epoxy

High speed photoelasticity and ray tracing are used to find the possible high stress regions and the locations of crack initiationPredicting the crack path of this dynamic phenomena has not yet been done

Interesting crack patterns in plaster of Paris samplesCylindrical Rectangular prism

Plane wave Plane wave

Crack initiation and propagation is due to a dynamic state of stressThis could be one of the toughest crack propagation problem to be simulated

Simulation of SWL: problem setting

60

0 42 6 8 Time μs

0

20

40

MPa

4 μs

Input pressure pulse

48mm

50mm 48mm

12.7mm

14mm

Pressure pulse applied on this plane

~ 6 million elements~ 3.5 million DOFs

E = 8.875 GPaν = 0.228Vp = 2478 m/sVs = 1471 m/s

Plaster of ParisΚ= Vp = 1483 m/s

Water

Yufeng Zhou and Pei Zhong, J. Acoust. Soc. Am.; 119(6) 2006

Stress waves of σyy

vertical plane

horizontal plane

x

y

z

cylindrical sample

Crack patterns at different sections of cylinder

f

t

t Kdt ≥−∫0

21 )( σσ

Tuler and Butcher failure criterion :

tσ tensile strength

-5mm -4mm -3mm -2mm -1mm

5mm 4mm 3mm 2mm 1mm

Summary

Particle discretization for continuum mechanics problemsuses a set of non-overlapping characteristic functions on conjugate geometriesnumerically efficient approximate failure treatmentaccuracy of crack tip stress field can be improved with rotational DOFparticle physics type dynamic simulations (i.e. a simplified N body problem)

We simulated several 3D crack profiles with complicated geometries