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Virtual Crack Closure Technique Based on Meshless Shepard Interpolation Method (MSIM ). Ph.D. Feng Su 1,2 , Jie Wu 1,2,* , Prof. Yongchang Cai 1 ,2. 1 State Key Laboratory for disaster reduction in Civil Engineering - PowerPoint PPT Presentation
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Virtual Crack Closure Technique Based on Meshless Shepard Interpolation Method (MSIM)
1State Key Laboratory for disaster reduction in Civil Engineering
2Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education
Tongji UniversityJune 17, 2013Presented to:
13th International Conference on Fracture
Ph.D. Feng Su1,2, Jie Wu1,2,*, Prof. Yongchang Cai1,2
*Corresponding author: [email protected]
MotivationsFractures widely exist in materials such as
Rock mass and concrete.The accurate calculation of the stress
intensity factor is critical important and is hence of great interest to researchers.
An simple and efficient numerical calculation method for crack is desirable to the practical modeling of the complex geotechnical engineering.
OutlineGeneral review of representative methods
modeling fracturesMeshless Shepard interpolation method
(MSIM)Virtual crack closure technique & J integralNumerical investigation conclusions
How to numerically simulate the discontinuities?
General review of representative methods modeling fractures
Finite element method (FEM): ‘joint element’ or ‘interface element’ mesh coincides with the fractures, meshing
complicated; remeshing, simulation tedious, time-consuming
Modifications to the FEM XFEM: incorporate enrichment functions to represent
discontinuities; GFEM: incorporate high-order terms or handbook
functions to tackle multiple corners, voids, cracks, etc.
Boundary element method (BEM): not efficient in dealing with material heterogeneity
and non-linearity ;
Numerical manifold methodThe numerical manifold method was first proposed by Shi (1991)
A regular mesh is adopted throughout the calculation The discontinuity can be treated in a straightforward
manner The generation of the finite cover system is complex
which is hard to be applied to 3D analysis
Meshless methodAdvantage
h-adaptivity is simpler to incorporate in MMs than in mesh-based methods,
Problems with moving boundaries can be treated with ease
Large deformation can be handled more robustly
Higher-order continuous shape functionsDisadvantage
Higher computational cost compared with FEM Difficulties with the treatment of essential
boundary
Meshless Shepard interpolation method
Interpolations/weight functions& discrete equations
Subdomain:
}:{ mii rC ixxx
0 1
100
0 r
i(r)
uΓ
Node i
mir
x
Node j
Neighboring ofpoint x
C iC j
PU-based interpolation
n
i
lii
h uu1
0 xxx
xuxxuxn
j
lbjj
n
iii
21
1
0
1
ln0
Cover Ci not on the boundary
Shepard function Cover Cj on the boundary
n
ii
ii
w
w
1
0
x
xx
mii
miimi
i
i
mi
i
rr
rrrr
rr
w0
,2cos+
22
2 x
Singular at Xi
uΓ
mir
x
Neighboring ofpoint x
C iC j
Cover interpolation for Node ixyayaxaau iiiii 4321
ln )( x04321
ln )( iiiiiiiiiii uyxayaxaaxu
Similar procedure is implemented in y-direction )()()()( 4320
lniiiiiiiii yxxybyybxxbvv x
)()()()( 4320ln
iiiiiiiii yxxyayyaxxauu x
ii
i
ii
v
uΤΨ
x
xxu
ln
lnln
iiii
iiiii
yxxyyyxxyxxyyyxx
0001000001
Ψ
Tiiiiiiiii bababavu 44332200Τ
where:
0
0ln ,i
iiii v
uyxu
It means that the cover function of the node is interpolated in terms of nodal displacements .
uΓ
mir
x
Neighboring ofpoint x
C iC j
Cover interpolation for Node j
J=∑𝑗=1
𝑀
[𝑢 𝑗 0−∑𝑘=1
𝑚
𝑝𝑘 (𝐗 )𝒂𝒌 ]
Minimization
𝐚=𝐀−𝟏𝐁𝐔𝟎𝐣
where: ,
𝐚=𝐀−𝟏𝐁𝐔𝟎𝐣
m
kkk
li apu
1
T )()()()( xxaxPx This approximation does not fit the nodal displacement values
A modification is made like this:
𝛗 𝑗 (𝐗 )=[𝜑1𝑗 (𝐱 )𝜑 2
𝑗 (𝐱 )…𝜑𝑀𝑗 (𝐱 )]
𝑢 𝑗𝑙𝑏 (𝐗 )=𝛗 𝑗 (𝐗 )𝐔0
𝑗
𝑢 𝑗𝑙𝑏 (𝐗 )=𝛗 𝑗 (𝐗 )𝐔0
𝑗
¿ [𝜑1𝑗 (𝐱 )−𝜑1
𝑗 (𝐱 𝑗 )…𝜑 𝑗𝑗 (𝐱 )−𝜑 𝑗
𝑗 (𝐱 𝑗 )+1…𝜑𝑀𝑗 (𝐱 )−𝜑𝑀
𝑗 (𝐱 𝑗 )]
𝜑 𝑗𝑗 (𝐱 𝑗 )=𝟏
𝜑𝑖𝑗 (𝐱 𝑗 )=𝟎
∑𝑘=1
𝑀
𝜑 𝑘𝑗 (𝐗 )=𝟏
Treatment of discontinuity
Crack 1
Crack 2
Crack 1
Crack 2
Gauss point
Crack 1
Concepts of the mathematical and physical cover in NMM is employed to express the discontinuity
Virtual crack closure technique & J integral
Differences are investigated by a numerical example
J integralDeveloped by Cherepanov, 1967 & by Jim Rice, 1968,independentlyAccurate and widely used in the
calculation of stress intensity factorPath-independent
y
xcrack
Contour path However, the broken crack cannot be properly simulated by this method
Decreasing the radius of the contour path and enrich function will often be employed.
Virtual crack closure technique (VCCT)Assumption:• The energy released when crack
extended from i to j is identical to the energy required to close the crack between i and j
• The two displacements are approximately the same.
y
x
a ∆a ∆a
i j m
aBvFG jjym
I
2
',
aBuFG jjxm
II
2
',
• Only the node displacement and force are required• Fracture mode separation is determined explicitly• The calculation results are free of the affect of crack
length• Always incorporated in FEM, and the remeshing cannot
be avoided
Implementation VCCT in MSIMCalculate the constructed MSIM model;Set the assistant mesh near the crack tipAcquire the nodal displacement U in the
assistant mesh based on the MSIM calculation result.
Construct the global stiffness matrix K of the assistant mesh
Get the nodal force near the crack tip by F=K*UCalculate the SERRTransform SERR into SIF by
Numerical investigationsCracked plate under remote tension
& A star-shaped crack in a square plate under bi-axial tension
Cracked plate under remote tension
Geometric and MSIM model
=1kPa=1*107 Pa, u=0.3h=5m, W=5m
Vary a from 0.1W to 0.8W
Collocation methodJ integral (linear basis)VCCT (linear basis)VCCT (enrich basis)
J integral (linear basis)VCCT (linear basis)VCCT (enrich basis)
J integral V.S. VCCT
SIF F1
Relative error of F1
Accurate results can be get from both of the methods
The relative error get from J integral seen a big change with the increase of the crack length, while that get from VCCT is more stable
Remark:
A star-shaped crack in a square plate under bi-axial tension
Geometric and MSIM model
=1kPa=1*107 Pa, u=0.3W=2m, =60
aKF
aKF
aKF
BII
BII
BI
BI
AI
AI
Normalized SIF:
VCCT (linear basis)
VCCT (Enriched basis)
Relative error of FI
A
Star-shaped crack (cont.)
Relative error of FII
B
VCCT (linear basis)
VCCT (Enriched basis)
VCCT (Enriched basis)
VCCT (linear basis)
Relative error of FI
B
Conclusions The essential boundary could be easily imposed in MSIM due
to delta property The discontinuity problem could be well addressed in the
MSIM Only the node displacement and force are required in VCCT,
and fracture mode separation is determined explicitly The relative error get from J integral seen a big change with
the increase of the crack length, while that get from VCCT is more stable
Further investigation results show that the VCCT in the framework of MSIM is prominent in modeling complex crack problem.
Thanks you very muchThe authors gratefully acknowledge the support of:• Program for New Century Excellent Talents (NCET-12-
0415)• National Science and Technology Support Program
(2011BAB08B01)• & Fundamental Research Funds for the Central
Universities.