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SIMULATION OF PLATE WITH CRACK USING EXTENDEDFINITE ELEMENT METHOD

Virender Kumar, I. V. Singh, B. K. Mishra and G. Bhardwaj

MIED, Indian Institute of Technology, Roorkee, India

E-mail: virenhp4!"#gmail$com ,ivsingh#gmail$com

ABSTRACT

In this paper, an extended finite element method (X!M" is emplo#ed to simulate a plate with

through thi$%ness $ra$%s. &he numeri$al formulation is done using Mindlin'eissner plate theor#.

) $ra$% in the plate is modelled *# using +eaiside and $ra$% tip enri$hment fun$tions. &he stress

intensit# fa$tor is $omputed using domain form of an intera$tion integral. -ifferent pro*lems of

through'the'thi$%ness $ra$%s are simulated and $ompared with aaila*le literature results. Seeral

pro*lems of plate with $ra$% under different *oundar# and loading $onditions are soled *# X!M.

Keywords: X!M, ra$%, /late *ending, Mindlin'eissner, Stress Intensit# a$tor

Inrod!"#on

ra$ture of plates is an area of great pra$ti$al as well as theoreti$al importan$e. Most of the stud#

on the fra$ture me$hani$s of shells and plates has *een fo$used on in'plane tensile loading as

des$ri*ed in literature *# Moes et al$ (0111". +oweer, there are man# pra$ti$al pro*lems where

loading is out'of'plane or a $om*ination of in'plane and out'of'plane. &he anal#ti$al approa$hes

used for soling these pro*lems are limited to simple geometries and *oundar# $onditions. /lates

when su*je$ted to high loading ma# deelop large stresses in the *od# ex$eeding material strength

and thus results in the progressie failure. &hese failures are generall# initiated *# surfa$e or near

surfa$e $ra$%s. &hese $ra$%s redu$e the strength of the material. &he presen$e of $ra$% generates

pro*lem, as $ra$% $an signifi$antl# affe$t the stru$tural integrit# of $omponent and shortens seri$e

life. &herefore, it is important to ealuate the stress distri*ution in the region near the $ra$% to predi$t the failure. &o predi$t the failure of $omponent, stress intensit# fa$tor (SI" needs to *e

$omputed a$$uratel#. ew anal#ti$al solutions are aaila*le in literature for simple geometries and

loads. 2umeri$al methods needs to *e applied for the $omplex geometries and loading $onditions.

In the present wor%, extended finite element method (X!M" is used for the simulation of plate in

the presen$e of $ra$%. &hin plate is modeled through Mindlin'eissner plate theor# to negle$t the

effe$t of shear lo$%ing. &he domain form of intera$tion integral is utili3ed to $al$ulate stress

intensit# fa$tor. 2umeri$al results o*tained using X!M are $ompared with aaila*le literature *#

Sosa and !i$hen (0145". &hese simulations show a good agreement in the results with literature

oer a wide range of $ra$% lengths and plate thi$%nesses. Seeral pro*lems of through thi$%ness

$ra$%s ($enter, left edge and dou*le edge $ra$%" in s6uare plate are soled su*je$ted *# X!M

su*je$ted to edge moments.

1/4

5th International Congress on Computational

Mechanics and Simulation, 10-13 December

2014, India

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"th International %ongress on %omputational Mechanics and &imulation, '-'( Decem)er *'4,

India

P$%e Mode$

&he Mindlin'eissner plate theor# is emplo#ed and is *ased on %inemati$ displa$ement model as in

-ol*ow et al$ (7888"( , , ) . ( , )

x u x y z z x y ψ = (0"

( , , ) . ( , ) y v x y z z x y ψ = (7"

0( , , ) ( , )w x y z w x y = (9"

where u, v and + are defined as displa$ement $omponent in , y and dire$tions respe$tiel#. &he

term . and . y represents aerage rotations to the referen$e plane along and y dire$tions

respe$tiel#. &he transerse displa$ement at the referen$e surfa$e is defined *# +.

XFEM &or P$%e

In X!M, the displa$ement approximation is lo$all# enri$hed to simulate dis$ontinuities. ewadditional degrees of freedom are added to the sele$ted nodes of elements whi$h are influen$ed *#

the $ra$% fa$e and $ra$% tip. &he e6uation for transerse displa$ement and rotational $omponent is

ta%en as des$ri*ed in -ol*ow et al$ (7888"

:

0 0 0 0

( " ( " ( " ( " ( " ( " α α

α

ξ ξ ξ ξ ξ β ξ = = = =

= + + ÷

∑ ∑ ∑ ∑

cf en ct nn n

h

i i / / k k

i / k

+ 0 u 0 1 a 0 ) (;"

:

0 0 0 0

( " ( " ( " ( " ( " ( " α α

α

ψ ξ ξ ξ ξ ξ ξ = = = =

= + + ÷

∑ ∑ ∑ ∑

cf en ct nn n

h

i i / / k k

i / k

0 u 0 1 a 0 2 ) (5"

<here ( "ξ 1 is the +eaiside fun$tion, andα α β 2 are $ra$% tip enri$hment fun$tions respe$tiel#

for transerse displa$ement and rotation.

1 if ( ) 0

( )1 otherwise

Hϕ ξ

ξ ≥

= −

(="

where ( )ϕ ξ represents the leel set fun$tion.

&he $ra$% tip enri$hment fun$tions are defined as in -ol*ow et al$ (7888">

9 9 9 99 9( " sin $os sin $os

7 7 7 7α

θ θ θ θ β ξ

== r r r r (4"

( " $os , sin , $os sin , sin sin7 7 7 7

α

θ θ θ θ ξ θ θ

=

2 r r r r (1"

where, and θ r are the lo$al $ra$% tip parameters.

Sress Inens#y F%"or

) most appropriate method for $omputing stress intensit# fa$tor is domain *ased intera$tion

integral approa$h. &he two states of field 6uantities are $onsidered i.e. present state ( M, 3, ., +"

and auxiliar# state ( M au , 3au , . au , +au" for deriing the intera$tion integral. &he intera$tion

integral is $omputed as des$ri*ed in -ol*ow et al . (7888"

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India

{ }

( ) ( ){ }

,1 ,1

,1,

int,1 ,1 1 ,

, 1 ,1 ,1

aux aux aux aux

A

aux aux aux aux aux s

A

I M M Q w Q w W q dA

M Q Q w qdA

α α αβ

α αβ β

αβ β β β β

α α α α α

ψ ψ δ

ψ ψ ε

= + + + − +

− + + −

∫

∫ (08"

9

7:π = +

au, au,

I I II II I

Eh (00"

<e set Iaux?0, II

aux?8 in e6. (00" to $ompute the alue of I.9

7:π =

I

Eh I (07"

&he alue of II is extra$ted in the same wa# as that of I.

Res!$s %nd D#s"!ss#on

In the present wor%, the anal#sis for plate in the presen$e of $ra$% has *een $arried out usingX!M under different *oundar# $onditions and loading $onditions. &he nine noded 6uadrilateral

elements are used for the purpose of anal#sis with regular mesh si3e of 30 30× . &he a$$ura$# of

solution de$reases due to the presen$e of $ra$% in an element. &herefore, a su*'triangulation

s$heme is emplo#ed for the integration in these elements. &he stress intensit# fa$tor is $omputed

using domain *ased intera$tion integral and are $ompared with aaila*le results in the literature. )

$enter $ra$%ed s6uare plate su*je$ted to edge moments is ta%en for anal#sis as shown in ig. 0.&he

following material properties are used for anal#sis, i.e. E ?0888, @?8.9, 5?0.

In $ase of $enter $ra$%ed pro*lem, the normali3ed moment intensit# fa$tor $omputed using X!M

for different 5Ah ratios i.e. 7 and 08 are $ompared with the aaila*le literature results *# Sosa and

!i$hen, (0145" as depi$ted in ig. 7. &he normali3ed moment intensit# fa$tor o*tained using

X!M is found to *e in good agreement with the aaila*le results. ) s6uare plate with dou*le edge

$ra$%s su*je$ted to edge moments is further ta%en for simulation as shown in ig. 9. In $ase of

dou*le edge $ra$% pro*lem, the normali3ed moment intensit# fa$tor $omputed are $ompared withthe aaila*le literature results in Sosa and !i$hen (0145" for 5Ah ? 7 and 08 as depi$ted in ig. :.

3/4

F#'( ). 2ormali3ed moment intensit# fa$tor for

s6uare plate $ontaining $entral $ra$%

8 8.0 8.7 8.9 8.: 8.; 8.5 8.= 8.4 8.1 0

8.5

8.4

0

0.7

0.:

0.5

0.4

7

7.7

a/L

K / M * s q r t ( a )

i

XFEM (L/h=2)

Literature (L/h=2)

XFEM (L/h=10)

Literature (L/h=10)

F#'( *. Geometr# and loading $onditions

for the $enter $ra$%ed plate

h

M

*a

M

*5

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M


India

Con"$!s#on

In this paper, through thi$%ness $ra$% plate is modeled and simulated *# X!M using Mindlin'

eissner plate theor#. &he alues of stress intensit# fa$tors are ealuated using domain *ased

intera$tion integral approa$h. Seeral pro*lems are soled for s6uare plate $ontaining through'the'

thi$%ness $ra$%s i.e. $enter, left edge and dou*le edge su*je$ted to edge moments. &he alues of

stress intensit# fa$tor are reported for a plate with $ra$%s under edge moment loading. rom the

simulations, it is o*sered that the results o*tained *# X!M for different 56h ratios for edge $ra$%,

$enter $ra$% and dou*le edge $ra$% are found in good agreement with the literature results.

Re&eren"es

-ol*ow, ., Moes, 2. and Bel#ts$h%o, &. (7888", CModeling fra$ture in Mindlin'eissner plates

with the extended finite element method,D International ournal of Solids and Stru$tures, 9=,

=050'=049.

+ui, .E., and 3ehnder, ).&., (0119", C) theor# for the fra$ture of thin plates su*je$ted to *ending

and twisting moments,D International ournal of ra$ture, 50,770'771.

Fasr#, ., enard, E., Salaun, M. (7808",D Stress Intensit# a$tors $omputation for *ending plates

with X!MD Int. . for 2umer. Meth. !ngng. 88, 0'70.

Moes, 2., -ol*ow, ., Bel#ts$h%o, &. (0111", C) finite element method for $ra$% growth without

remeshing,D Int. . 2umer. Meth. !ngng. :5,0900;8.

Sih, G.., and i$e, .. (0150",D &he Bending of /lates of -issimilar Materials <ith ra$%s,D

ournal of )pplied Me$hani$s$

Sosa, ).+., and !i$hen, <. . (0145", Computation of stress Intensit# fa$tors for plate *ending ia

a path'independent integral,D !ngineering ra$ture Me$hani$s ol. 7;, no. :, pp :;0':50.

4/4

F#'( +. Geometr# and loading $onditions for the

dou*le edge $ra$%ed plate

h

M

*5

*caa

F#'( ,. 2ormali3ed moment intensit# fa$tor for

s6uare plate $ontaining dou*le edge $ra$%

8 8.0 8.7 8.9 8.: 8.; 8.5 8.= 8.4 8.1 08.5

8.4

0

0.7

0.:

0.5

0.4

c/L

K / M * s q r t ( a )

XFEM (L/h=2)

Literature (L/h=2)

XFEM (L/h=10)

Literature (L/h=10)

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147-255-1-SM