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: Jie.voo@gmail.com

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.