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
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 : 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
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