Free vibration analysis of patch repaired plates with a through crack by p-convergent layerwise...

Research ArticleFree Vibration Analysis of Patch Repaired Plates witha Through Crack by 119901-Convergent Layerwise Element

Jae S Ahn Seung H Yang and Kwang S Woo

Department of Civil Engineering Yeungnam University 280 Daehak-Ro Gyeongsan Gyeongbuk 712-749 Republic of Korea

Correspondence should be addressed to Kwang S Woo kswooyuackr

Received 7 February 2014 Accepted 9 June 2014 Published 10 August 2014

Academic Editor Jose R drsquoAlmeida

Copyright copy 2014 Jae S Ahn et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The high-order layerwise element models have been used for damaged plates and shells in the presence of singularities such ascrack cutout and delamination In this study the extension of a proposed finite element model has been tested for free vibrationanalysis of composite laminated systems For the elements three-dimensional displacement fields can be captured by layer-by-layerrepresentation For the elements higher-order shape functions are derived by combination of one- and two-dimensional shapefunctions based on higher-order Lobatto shape functions not using pure higher-order three-dimensional shape functions Thepresent model can relieve difficulty of aspect ratios in modeling very thin thickness of bonding layer For verification of the modelnatural frequencies and corresponding mode shapes are calculated and then compared with reference values for uncracked andcracked plates Also the vibration characteristics of one-sided patch repaired plates with a through internal crack are investigatedwith respect to variation of crack length size and thickness of patch and shear modulus of adhesive respectively

1 Introduction

For enhancement of service life in structures with local dam-age or defect adhesively bonded technology has widespreadapplications in aircraft ship and other structures due to itslight weight and efficient load transfer characteristics Com-posite patches especially have shown to be a highly cost effec-tive method for extending the service life and maintaininghigh structural efficiency [1ndash3] To investigate the behaviorof damaged structures repaired by composite patches stressintensity factors obtained from stress analysis have oftenbeen considered which are reduced by the presence of thepatches For the stress analysis some authors have addressedvarious analytical numerical and experimental aspects Asthe analytical solutions [4ndash6] could not effectively handle thecomplexities of real-life patch-repair problems the emphasishas been on experimental and numericalmethods In the caseof numerical methods most authors [7ndash9] have relied uponconventional finite element analysis based on ℎ-refined meshdesign utilizing hexahedral elements for three-dimensionalmodeling and plate elements for two-dimensional modelingIn three-dimensional models discretization of the extremelythin adhesive layer with hexahedral elements with acceptable

aspect ratios led to models with unacceptable large numberof elements Ahn and Basu [10] proposed a mixed-modelapproach to analyze cracked metal plates with patch repairwhich shows a strong robustness of the element with respectto very large aspect ratios of 1200 for extremely thin adhesivemodeling Some other authors [11 12] have focused theirattention on the optimal design of the bonded patches byfinite element models

Although static analyses of patch repaired plates with acrack have intensively been performed to determine stressintensity factors investigations on their free vibrations ofpatch repaired plates with a crack are rather little Researcheson vibration analysis of some plates including through inter-nal cracks have often been implemented Stahl and Keer [13]reported vibration phenomenonof cracked rectangular plateswith simple supports whose analysis is based on a dual seriesequation Solecki [14] studied vibration of rectangular plateswith a crack parallel to one edge using a finite Fourier seriestransformation in conjunction with the generalized Green-Gauss theorem Liew et al [15] used domain decompositionmethod in determining frequencies of cracked plates Wuand Shih [16] studied dynamic instability of rectangularplates with an edge crack Bachene et al [17] used extended

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 427879 8 pageshttpdxdoiorg1011552014427879

2 The Scientific World Journal

finite element method based on Mindlin plate theory forvibration analysis of cracked plates Recently Ritz methodconsidering shear deformation was applied for determiningfrequencies and nodal patterns of thick cracked rectangularplates [18] To the authorrsquos knowledge it is nearly impossibleto find the published literatures about the free vibrationanalysis of patch repaired plateswith a through internal crackNatural frequency is one of the significant characteristicsin engineering applications and dynamic responses of thecracked structure that may change after patching Henceit is necessary to understand the variation of the naturalfrequencies of the patched and unpatched cracked structurefor effective patching design

Meanwhile the quest for the robust finite elements for awide class of practical problems involving stress singularitieshas triggered researchers to develop higher-order finite ele-ments Mathematical justification showing the advantages ofhigher-order approximations of the field variables has beenreported [19] regarding high accuracy high convergence ratecoarse mesh and improved performance in handling stresssingularity problems As previously worked fracture analysisusing the 119901-convergent layerwise elements [20] based onhierarchical shape functions was implemented in whichthree-dimensional displacement fields can be captured bylayer-by-layer representation For the elements higher-ordershape functions are derived by combination of one- and two-dimensional shape functions based on Lobatto shape func-tions not using pure higher-order three-dimensional shapefunctionsThen stress intensity factors of cracked plates witha patch repair [10 21] were obtained by the 119901-convergentlayerwise elements In this paper the proposed elements areapplied to the free vibration analysis of cracked plates witha patch repair For verification of the proposed elementsat first natural frequencies and the corresponding modeshapes are compared with reference values for uncracked andcracked plates Then vibration characteristics of one-sidedpatch repaired plates are investigated on natural frequenciesin terms of crack length size and thickness of patch and shearmodulus of adhesive

2 119901-Convergent Layerwise Element Model

21 Displacement Fields In this approach [20] to three-dimensional modeling of patch repaired systems each layupis treated discretely with deformation of a point in thelayup in terms of three-displacement components defined foreach layer separately Displacement field at bottom and topsurfaces within a layer is approximated by two-dimensionalshape functions Then the two surfaces are connected byinterpolating technique using one-dimensional shape func-tions which are first-order or more variations across thick-ness The one-dimensional hierarchical shape functions canbe classified into two groups as nodalmodes (119865) and nodelessmodes (119861) For two-dimensional hierarchical shape func-tions three-mode groups belong to the quadrilateral elementsuch as nodal modes (119873) and nodeless modes (119872) includingside and internal modes Generally the nodal modes havephysical meaning while the nodeless modes with respect tothe increase of order of the Lobatto shape function do not

Layer 1 Layer 2

Nodal modesSide nodeless modes

Internal nodeless modesThickness nodeless modes

(a) Laminated plates with two layers (b) Finite elements modeling

Figure 1 Modeling scheme of laminate plates with two layers using119901-convergent layerwise element

have physical meaning but improve accuracy of analysis Thedisplacement field Φ consisting of three components (119880119881and119882) at a point (119909 119910 119911) can be written as

119880 = 119873119894119865119895119906119895

119894+ 119873119894119861119904119886119904

119894+119872119896119865119895119887119895

119896+119872119896119861119904119888119904

119896

119894 = 1 2 3 4 119895 = 1 2

119881 = 119873119894119865119895V119895119894+ 119873119894119861119904119889119904

119894+119872119896119865119895119890119895

119896+119872119896119861119904119891119904

119896

119896 = 1 2 119901 (119901 + 3)

2minus 1

119882 = 119873119894119865119895119908119895

119894+ 119873119894119861119904119892119904

119894+119872119896119865119895ℎ119895

119896+119872119896119861119904119902119904

119896

119904 = 1 2 1199011015840minus 1

(1)

where for the sake of brevity the Einstein summation con-vention has been introduced for a repeated index 119906119895

119894 V119895119894

and 119908119895

119894are the nodal variables and 119886

119904

119894 119887119895119896 119888119904119896 119889119904119894 119890119895119896 119891119904119896

119892119904

119894 ℎ119895119896 and 119902

119904

119896are nodeless variables The number of the

nodeless variables depends on order of the approximationfunctions 119901 and 119901

1015840 (ge2) which are independent of eachother Figure 1 depicts the modeling scheme with presentelements for a laminated system with two layers If thereare no gaps and empty spaces between interfaces of layerscompatibility conditions can be applied at the layer interfacesEach layer has eight nodal modes Also it takes side internaland thickness nodeless modes of which numbers depend onthe order of the approximation functions adopted 119901 and 1199011015840

22 Shape Functions For the functions stated above at firstone-dimensional shape functions with higher-order degreesare adopted from the Lobatto shape functions [22] definedwithin the space (minus1 le 119909 le 1) that are given by

1198651 (119909) =

1 minus 119909

2 119865

2 (119909) =1 + 119909

2

119861119904 (119909) =

11003817100381710038171003817119871 119904

1003817100381710038171003817

int

119909

minus1

119871119904 (120585) 119889120585 119904 ge 119901

1015840minus 1

(2)

where

1003817100381710038171003817119871 1199041003817100381710038171003817 =

radic2

2119904 + 1 (3)

The Scientific World Journal 3

The higher-order Legendre polynomials 119871119904 can be defined

by differential relations as follows

119871119904 (119909) =

1

2119904119904

119889119904

119889119909119904(1199092minus 1)119904

for 119904 = 0 1 2 (4)

Their orthogonal relationship is exactly specified by

int

119909

minus1

119871119894 (119909) 119871119895 (119909) 119889119909 =

2

2119894 + 1for 119894 = 119895

0 otherwise(5)

The one-dimensional Lobatto shape functions derived fromthe higher-order integrals of Legendre polynomials play anessential role in the design of two-dimensional hierarchicalshape functions for this discrete layer model The two-dimensional shape functions associated with the values ofnodes are given by

119883119894119895= 119865119894 (119909) 119865119895 (119910) 119894 119895 = 1 2 (6)

where

1198731= 11988311 119873

2= 11988321 119873

3= 11988322 119873

4= 11988312

(7)

In any 119901-levels (119901 ge 2) two-dimensional shape functionsassociated with nodeless variables are as follows

11987205119894(119894+3)+120572

= 119861119894 (119909) 1198651 (119910)

11987205(119894+1)(119894+2)+120572

= 1198652 (119909) 119861119894 (119910)

11987205(1198942+3119894+4)+120572

= 119861119894 (119909) 1198652 (119910)

11987205(1198942+3119894+6)+120572

= 1198651 (119909) 119861119894 (119910)

119894 = 1 2 119901 minus 1 120572 = minus1 in 119894 = 1

0 otherwise

(8)

For 119901 ge 4 the additional shape functions of nodeless vari-ables are obtained by

11987205(1198952+119895+1)+119894

= 119861119894 (119909) 119861119895minus119894minus2 (119910) 119894 = 1 2 119895 minus 3

for 119895 = 4 5 119901(9)

23 Strain Fields For a typical layer 119897 stress-strain relation-ships which are based on three-dimensional elasticity theoryare linear as follows

120590119909119910119911

119897

6times1= [119863]

119897

6times6120576119909119910119911

119897

6times1 (10)

Here [119863] is a general elasticity matrix of orthotropic materi-als and strain matrix is given by

120576119909119910119911

= lfloor120597119880

120597119909

120597119881

120597119910

120597119882

120597119911

120597119880

120597119910+120597119881

120597119909

120597119880

120597119911+120597119882

120597119909

120597119881

120597119911+120597119882

120597119910rfloor

119879

(11)

24 Equation of Motion Lagrange equations for most struc-turalmechanics problemsmay be derived from considerationof Hamiltonrsquos principle that is well known in the text For freevibration problemswithout damping the governing equationof motion requires the functional to satisfy the condition asfollows

int119905

120575 (119879 minus 119880) 119889119905 = 0 (12)

where119879 is total kinetic energy119880 is potential energy includingboth strain energy and potential energy of any conservativeexternal forces and 120575 is a variation taken during the indicatedtime interval 119905 The displacement fields Φ defined in (1) canbe written by the following general form

Φ = [119867] 119889 (13)

where all nodal and nodeless variables are included in thematrix 119889 and the matrix [119867] indicates hierarchical shapefunctions defined in (7)ndash(9) First-order derivative of thedisplacement fields with respect to time is given by

Φ = [119867] 119889 (14)

Then the kinetic energy 119879 can be written by

119879 =1

2int119881

120588Φ119879

Φ 119889119881 (15)

Also from the strain vector 120576 and the stress vector 120590defined in (10) and (11) the potential energy119880 can be writtenas

119880 =1

2int119881

120576119879120590 119889119881 (16)

Thus the energy functional expressed in matrix form can beobtained as follows

int119905

120575 (1

2int119881

120588 119889119879

[119867]119879[119867] 119889 119889119881

minus1

2int119881

119889119879[119861]119879[119863] [119861] 119889 119889119881)119889119905 = 0

(17)

where [119861] is the strain-displacement matrix with respectto layer reference axes [119863] is an elasticity matrix with anorthotropicmaterialThe total kinetic energy the first term of(17) is a functional with respect to displacements and veloc-ities while the potential energy is a functional with respectto only displacements The velocity-related term in (17) isintegrated by parts and then the minimization of energyfunctional is appliedThen by differentiating (17) with respectto time the final equation of motion for free vibrationproblems for undamped system can be expressed in matrixform as

[119872] 119889 + [119870] 119889 = 0 (18)

where

[119872] = int119881

120588[119867]119879[119867] 119889119881

[119870] = int119881

[119861]119879[119863] [119861] 119889119881

(19)


Table 1 Comparison of nondimensional frequencies with respect to variation of crack lengths

119888119886 Model Mode1 2 3 4 5

0 Huang et al (2011) [18] 19312 46050 46050 70641 86052Present 19346 46242 46242 71070 86683






Natural vibration is nothing but the periodic motion withany natural circular frequencies 119908 By assuming the properperiodic motion (20) can be obtained

[[119870] minus 1199082[119872]] 119889 = 0 (20)

When (20) has a nontrivial solution characteristic matrixof 119889 should be singular matrix to satisfy the condition asfollows

10038161003816100381610038161003816[119870] minus 119908

2[119872]

10038161003816100381610038161003816= 0 (21)

Using a commercial package like MATLAB characteristicequation (21) to find natural circular frequencies and thecorresponding mode shapes can be solved

3 Numerical Examples

31 Cracked Square Plates The free vibration of simply sup-ported square plates with a center crack is considered when119886119905 ratio is fixed as 10 where 119886 and 119905 represent the sideand thickness of square plate respectively The plates arediscretized into 3 times 2 elements like in Figure 2 Based onconvergence tests the orders of polynomial approximationare kept to 6 and 3 in plane and along thickness respectivelyTo facilitate comparison of natural circular frequencies (120596)the nondimensional frequency parameter 120582 is considered as

120582 =1205961198862

1205872119905

radic12120588 (1 minus ]2)

119864

(22)

where 120588 is material density of the plates 119864 is Youngrsquos mod-ulus and ] is Poissonrsquos ratio The first five nondimensionalfrequency parameters are presented in Table 1 for differentcrack lengths (119888119886 = 01 02 03 04 and 05) where 119888 is thecrack length of plates and then are compared with referencevalues [18] It should be pointed out that the present resultsare in good agreement with the reference values within therelative error of plusmn2 for all cases It is true that frequenciesare reduced with the increase of crack The fundamental

x

y

1 2 3

4 5 6Crack

a

ac

Figure 2 Modeling of cracked plates by the 119901-convergent layerwiseelements

frequency of the cracked plate with 119888119886 = 05 is reduced upto 12 as compared with that of the uncracked plate with119888119886 = 0 Also it is seen that the reductions of frequencies aremuch larger for the second and the fifth modes than for theother modes The frequencies may respectively be reducedby about 19 for the secondmode and 17 for the fifthmodewhile reductions of the third and fourthmodes are within 5It means that first second and fifth mode shapes of first fivemodes aremore dependent on crack size than the othermodeshapes Figure 3 shows the first five vibration mode shapesof uncracked and cracked plates to present the influence ofa crack It is observed how the cracks split the plates into twoparts according to mode shapes

32 Cracked Square Plates with a Patch Repair A repairmethod using perfectly bonded composite patch coveringa structural defect can be used to enhance the service Inthis study the center-cracked steel plates with a single-sidedpatch repair are considered as shown in Figure 4 To obtainfundamental frequencies of the patched problems and toinvestigate effect of some parameters the present model is


Table 2 Material properties

Material 1198641(GPa) 119864

2 1198643(GPa) 119866

12 11986613(GPa) 119866

23(GPa) ^12 ^13 ^23 120588 (kgmm3)

Steel 200 200 769 769 03 27 times 10minus6

Film adhesive 3068 3068 1138 1138 035 033 times 10minus6

Boronepoxy 2234 2413 8481 5275 023 21 times 10minus6

1 2 3 4 5Modes

Uncracked(ca = 0)

(ca = 05)

Cracked

Figure 3 Mode shapes of uncracked and cracked plates

y

x

a

a b

b

c

CrackPlate

Patch

Adhesive

z

ta

ts

tp

Figure 4Geometry of center-cracked plateswith externally bondedrepairs

applied The plates have the following dimensions length119886 = 300mm thickness 119905

119904= 30mm patching length 119887 =

180mm 119905119901

= 10mm and 119905119886

= 15mm respectivelyFor patching material composite material with combinationof boron and epoxy is adopted The elastic properties of thesteel film adhesive and boronepoxy are given in Table 2 Ifthere are no additional conditions of geometry and materialsin specific cases those values aforementioned are chosen Forfinite element meshing work the steel plates are discretizedinto 5 times 4 elements and 3 times 2 mesh design is used for patch

Patch layer

Crack

Adhesive layer

Center-cracked plate

Figure 5 Modeling of cracked plates with a single-sided patchrepair using present elements

and adhesive layers as shown in Figure 5 Like the perviousproblem the orders of polynomial approximation are kept to6 and 3 in plane and along thickness respectively through theconvergence tests

At first Figure 6 shows the variation of natural frequen-cies with crack size for cracked plates with and without patchrespectively in which the values of the cracked plates arecompared with the value of an uncracked plate It is seenfrom the results that the natural frequencies of all crackedplates with and without patch are smaller than that of theuncracked plateThe natural frequencies of the patched platesare somewhat larger than those of the cracked plates withoutpatch below 119888119886 = 03 When crack size is small total masshas more influence on natural frequencies than stiffnessThe figure also shows that decreasing ratio of the valuesin the patched plates is smaller than that of the unpatched


26

27

28

29

30

31

01 02 03 04 05ca

Uncracked plateCracked plateCracked plate with patch

times10minus3

(1205962120587)

Figure 6 Variation of natural frequencies with crack size

26

27

28

29

30

31

01 02 03 04 05ca

times10minus3

tp (5mm)tp (10mm)

tp (20mm)tp (30mm)

(1205962120587)

Figure 7 Variation of natural frequencies with different patchthickness

plates with crack It is noted that the relatively large stiffnesseffect by patching can lower the decreasing ratio of naturalfrequencies

Figure 7 illustrates the variation of natural frequenciesaccording to different patch thickness varying from 119905

119901=

5mm to 119905119901

= 30mm as a central crack propagates Inthis case the adhesive thickness 119905

119886is fixed as 075mm and

other dimensions and material properties are exactly thesame as the values mentioned in Table 2 As we are aware ofit the patch repair generally reduces stress intensity factorssignificantly up to a certain level of patch thickness sincethe stiffness of cracked plates enforced by patching effect isincreased However it is noted from Figure 7 that growth of

bc


27

28

29

30

31

10 15 20 25 30

times10minus3

(1205962120587)

Figure 8 Variation of natural frequencies with patch length

patch thickness decreases the natural frequencies of patchrepaired plates when the plates have the same crack lengthThis is why the mass increment is more dominant to naturalfrequency than to the increase of stiffness Figure 8 showsvariation of natural frequencies with patch length 119887 from100mm to 280mm where crack length 119888 is fixed to 90mmWhen patch length is smaller than double length of cracksize it is seen that patched plates have smaller frequenciesthan cracked plates without patch The phenomenon occursbecause stiffness intension by patching is smaller than massincrease by adding patching materials boronepoxy andadhesiveWhenpatching effect is enough natural frequenciesare close to those of the uncracked plates It can be told thatvariation of patch length hasmore positive influence than thatof patch thickness as illustrated by the results of Figures 7 and8

Next influence of crack length and thickness of adhesivelayer is given in Figure 9 It can be told that the increase of theadhesive thickness decreases the fundamental frequencies Itis why the increase of the adhesive thickness causes a massincrement The natural frequencies may be slightly reducedto approximately 27 for 119888119886 = 01 and 42 for 119888119886 = 05

although octuple increase of the adhesive thickness is givenfrom 119905

119886= 0375mm to 119905

119886= 3mm Therefore in practical

cases of patching problems effect of adhesive thickness maybe negligible for frequency values since variation of adhesivethickness is very small Figure 10 presents influence of shearmodule of adhesives depending on variation of crack lengthThe increase in the natural frequencies due to single patchingeffect can be approximately between 75 for 119888119886 = 01 and117 for 119888119886 = 05 when the adhesive shear modulus isvaried from 100MPa to 2000MPa It is also noted that thenatural frequencies decrease as the crack length is increasedFrom Figures 9 and 10 it is observed that the adhesiveshear modulus has more significant effect on variation of thenatural frequencies as compared with the adhesive thickness


26

27

28

29

30

01 02 03 04 05ca

times10minus3

ta (0375mm)ta (075mm)

ta (15mm)ta (3mm)

(1205962120587)

Figure 9 Variation of natural frequencies with adhesive thickness

24

25

26

27

28

29

30

01 02 03 04 05ca

times10minus3

Ga (100MPa)Ga (200MPa)Ga (400MPa)Ga (800MPa)

Ga (1200MPa)Ga (1600MPa)Ga (2000MPa)

(1205962120587)

Figure 10 Variation of natural frequencies with different adhesiveshear modulus

4 Conclusions

The aim of this study is to show the efficiency of the proposed119901-convergent layerwise model for the free vibration analysisof cracked square plates without and with patch Also thisstudy is extended to single patching effect of cracked platesThe obtained results deduce the following conclusions

(1) Since the proposed 119901-convergent layerwise modeltolerates the large aspect ratio the number of meshescan be drastically reduced as compared with theconventional solid element especially in the case ofconsidering very thin adhesive and patch

(2) Frequency of each mode is reduced with increase ofthe crack length because stiffness of cracked platesdecreases Particularly it can be told that frequenciesof first second and fifth modes are largely decreasedmore than those of third and fourth modes

(3) The patching effect can help decreasing ratios ofnatural frequencies reduce as crack size increases

(4) Increase of patch length has more positive effect thanincrease of patch thickness in order to be close to thenatural frequencies of original plates prior to damage

(5) It is observed that the shear modulus of adhesive hasmore influence on the natural frequency as comparedwith the adhesive thickness

(6) From these results in future it is necessary to investi-gate the suitable size and thickness of patch before thedesign of optimal patching systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the National Research Founda-tion of Korea (NRF)Grant funded by the Korean government(MEST) (no 2011-0017108)

References

[1] A A Baker R J Callinan M J Davis R Jones and J GWilliams ldquoRepair of mirage III aircraft using the BFRP crack-patching techniquerdquo Theoretical and Applied Fracture Mechan-ics vol 2 no 1 pp 1ndash15 1984

[2] L Molent R J Callinan and R Jones ldquoDesign of an allboronepoxy doubler reinforcement for the F-111C wing pivotfitting structural aspectsrdquo Composite Structures vol 11 no 1pp 57ndash83 1989

[3] A A Baker ldquoRepair efficiency in fatigued-cracked aluminumcomponents reinforced with boronepoxy patchesrdquo Fatigue andFracture of Engineering Materials amp Structures vol 16 no 7 pp753ndash765 1993

[4] D W Oplinger ldquoEffects of adherend deflections in single lapjointsrdquo International Journal of Solids and Structures vol 31 no18 pp 2565ndash2587 1994

[5] M Y Tsai and J Morton ldquoAn evaluation of analytical andnumerical solutions to the single-lap jointrdquo International Jour-nal of Solids and Structures vol 31 no 18 pp 2537ndash2563 1994

[6] A Barut J Hanauska E Madenci and D R Ambur ldquoAnalysismethod for bonded patch repair of a skin with a cutoutrdquoComposite Structures vol 55 no 3 pp 277ndash294 2002

[7] E Oterkus A Barut E Madenci and D R Ambur ldquoNonlinearanalysis of a composite panel with a cutout repaired by a bondedtapered composite patchrdquo International Journal of Solids andStructures vol 42 no 18-19 pp 5274ndash5306 2005

[8] M R Ayatollahi and R Hashemi ldquoMixed mode fracture inan inclined center crack repaired by composite patchingrdquoComposite Structures vol 81 no 2 pp 264ndash273 2007


[9] F Ellyin F Ozah and Z Xia ldquo3-D modelling of cyclicallyloaded composite patch repair of a cracked platerdquo CompositeStructures vol 78 no 4 pp 486ndash494 2007

[10] J S Ahn and P K Basu ldquoLocally refined p-FEM modeling ofpatch repaired platesrdquo Composite Structures vol 93 no 7 pp1704ndash1716 2011

[11] A M Kumar and S A Hakeem ldquoOptimum design of sym-metric composite patch repair to centre cracked metallic sheetrdquoComposite Structures vol 49 no 3 pp 285ndash292 2000

[12] J Wang A N Rider M Heller and R Kaye ldquoTheoretical andexperimental research into optimal edge taper of bonded repairpatches subject to fatigue loadingsrdquo International Journal ofAdhesion and Adhesives vol 25 no 5 pp 410ndash426 2005

[13] B Stahl and L M Keer ldquoVibration and stability of cracked rect-angular platesrdquo International Journal of Solids and Structuresvol 8 no 1 pp 69ndash91 1972

[14] R Solecki ldquoBending vibration of a simply supported rectangu-lar plate with a crack parallel to one edgerdquo Engineering FractureMechanics vol 18 no 6 pp 1111ndash1118 1983

[15] K M Liew K C Hung andM K Lim ldquoA solution method foranalysis of cracked plates under vibrationrdquo Engineering FractureMechanics vol 48 no 3 pp 393ndash404 1994

[16] G Wu and Y Shih ldquoDynamic instability of rectangular platewith an edge crackrdquo Computers amp Structures vol 84 no 1-2 pp1ndash10 2005

[17] M Bachene R Tiberkak and S Rechak ldquoVibration analysisof cracked plates using the extended finite element methodrdquoArchive of Applied Mechanics vol 79 no 3 pp 249ndash262 2009

[18] C S Huang A W Leissa and R S Li ldquoAccurate vibrationanalysis of thick cracked rectangular platesrdquo Journal of Soundand Vibration vol 330 no 9 pp 2079ndash2093 2011

[19] I Babuska and B Szabo ldquoOn the rates of convergence of thefinite element methodrdquo International Journal for NumericalMethods in Engineering vol 18 no 3 pp 323ndash341 1982

[20] J S Ahn P K Basu andK SWoo ldquoHierarchic layermodels foranisotropic laminated platesrdquoKSCE Journal of Civil Engineeringvol 15 no 6 pp 1067ndash1080 2011

[21] J S Ahn P K Basu and K S Woo ldquoAnalysis of cracked alu-minum plates with one-sided patch repair using p-convergentlayered modelrdquo Finite Elements in Analysis and Design vol 46no 5 pp 438ndash448 2010

[22] P Solin K Segeth and I Dolezel Higher-Order Finite ElementMethods Chapman and Hall 2004

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finite element method based on Mindlin plate theory forvibration analysis of cracked plates Recently Ritz methodconsidering shear deformation was applied for determiningfrequencies and nodal patterns of thick cracked rectangularplates [18] To the authorrsquos knowledge it is nearly impossibleto find the published literatures about the free vibrationanalysis of patch repaired plateswith a through internal crackNatural frequency is one of the significant characteristicsin engineering applications and dynamic responses of thecracked structure that may change after patching Henceit is necessary to understand the variation of the naturalfrequencies of the patched and unpatched cracked structurefor effective patching design

Meanwhile the quest for the robust finite elements for awide class of practical problems involving stress singularitieshas triggered researchers to develop higher-order finite ele-ments Mathematical justification showing the advantages ofhigher-order approximations of the field variables has beenreported [19] regarding high accuracy high convergence ratecoarse mesh and improved performance in handling stresssingularity problems As previously worked fracture analysisusing the 119901-convergent layerwise elements [20] based onhierarchical shape functions was implemented in whichthree-dimensional displacement fields can be captured bylayer-by-layer representation For the elements higher-ordershape functions are derived by combination of one- and two-dimensional shape functions based on Lobatto shape func-tions not using pure higher-order three-dimensional shapefunctionsThen stress intensity factors of cracked plates witha patch repair [10 21] were obtained by the 119901-convergentlayerwise elements In this paper the proposed elements areapplied to the free vibration analysis of cracked plates witha patch repair For verification of the proposed elementsat first natural frequencies and the corresponding modeshapes are compared with reference values for uncracked andcracked plates Then vibration characteristics of one-sidedpatch repaired plates are investigated on natural frequenciesin terms of crack length size and thickness of patch and shearmodulus of adhesive

2 119901-Convergent Layerwise Element Model

21 Displacement Fields In this approach [20] to three-dimensional modeling of patch repaired systems each layupis treated discretely with deformation of a point in thelayup in terms of three-displacement components defined foreach layer separately Displacement field at bottom and topsurfaces within a layer is approximated by two-dimensionalshape functions Then the two surfaces are connected byinterpolating technique using one-dimensional shape func-tions which are first-order or more variations across thick-ness The one-dimensional hierarchical shape functions canbe classified into two groups as nodalmodes (119865) and nodelessmodes (119861) For two-dimensional hierarchical shape func-tions three-mode groups belong to the quadrilateral elementsuch as nodal modes (119873) and nodeless modes (119872) includingside and internal modes Generally the nodal modes havephysical meaning while the nodeless modes with respect tothe increase of order of the Lobatto shape function do not

Layer 1 Layer 2

Nodal modesSide nodeless modes

Internal nodeless modesThickness nodeless modes

(a) Laminated plates with two layers (b) Finite elements modeling

Figure 1 Modeling scheme of laminate plates with two layers using119901-convergent layerwise element

have physical meaning but improve accuracy of analysis Thedisplacement field Φ consisting of three components (119880119881and119882) at a point (119909 119910 119911) can be written as

119880 = 119873119894119865119895119906119895

119894+ 119873119894119861119904119886119904

119894+119872119896119865119895119887119895

119896+119872119896119861119904119888119904

119896

119894 = 1 2 3 4 119895 = 1 2

119881 = 119873119894119865119895V119895119894+ 119873119894119861119904119889119904

119894+119872119896119865119895119890119895

119896+119872119896119861119904119891119904

119896

119896 = 1 2 119901 (119901 + 3)

2minus 1

119882 = 119873119894119865119895119908119895

119894+ 119873119894119861119904119892119904

119894+119872119896119865119895ℎ119895

119896+119872119896119861119904119902119904

119896

119904 = 1 2 1199011015840minus 1

(1)

where for the sake of brevity the Einstein summation con-vention has been introduced for a repeated index 119906119895

119894 V119895119894

and 119908119895

119894are the nodal variables and 119886

119904

119894 119887119895119896 119888119904119896 119889119904119894 119890119895119896 119891119904119896

119892119904

119894 ℎ119895119896 and 119902

119904

119896are nodeless variables The number of the

nodeless variables depends on order of the approximationfunctions 119901 and 119901

1015840 (ge2) which are independent of eachother Figure 1 depicts the modeling scheme with presentelements for a laminated system with two layers If thereare no gaps and empty spaces between interfaces of layerscompatibility conditions can be applied at the layer interfacesEach layer has eight nodal modes Also it takes side internaland thickness nodeless modes of which numbers depend onthe order of the approximation functions adopted 119901 and 1199011015840

22 Shape Functions For the functions stated above at firstone-dimensional shape functions with higher-order degreesare adopted from the Lobatto shape functions [22] definedwithin the space (minus1 le 119909 le 1) that are given by

1198651 (119909) =

1 minus 119909

2 119865

2 (119909) =1 + 119909

2

119861119904 (119909) =

11003817100381710038171003817119871 119904

1003817100381710038171003817

int

119909

minus1

119871119904 (120585) 119889120585 119904 ge 119901

1015840minus 1

(2)

where

1003817100381710038171003817119871 1199041003817100381710038171003817 =

radic2

2119904 + 1 (3)




119871119904 (119909) =

1

2119904119904

119889119904

119889119909119904(1199092minus 1)119904

for 119904 = 0 1 2 (4)


int

119909

minus1

119871119894 (119909) 119871119895 (119909) 119889119909 =

2

2119894 + 1for 119894 = 119895

0 otherwise(5)


119883119894119895= 119865119894 (119909) 119865119895 (119910) 119894 119895 = 1 2 (6)

where

1198731= 11988311 119873

2= 11988321 119873

3= 11988322 119873

4= 11988312

(7)


11987205119894(119894+3)+120572

= 119861119894 (119909) 1198651 (119910)

11987205(119894+1)(119894+2)+120572

= 1198652 (119909) 119861119894 (119910)

11987205(1198942+3119894+4)+120572

= 119861119894 (119909) 1198652 (119910)

11987205(1198942+3119894+6)+120572

= 1198651 (119909) 119861119894 (119910)

119894 = 1 2 119901 minus 1 120572 = minus1 in 119894 = 1

0 otherwise

(8)


11987205(1198952+119895+1)+119894


for 119895 = 4 5 119901(9)


120590119909119910119911

119897

6times1= [119863]

119897

6times6120576119909119910119911

119897

6times1 (10)


120576119909119910119911

= lfloor120597119880

120597119909

120597119881

120597119910

120597119882

120597119911

120597119880

120597119910+120597119881

120597119909

120597119880

120597119911+120597119882

120597119909

120597119881

120597119911+120597119882

120597119910rfloor

119879

(11)


int119905

120575 (119879 minus 119880) 119889119905 = 0 (12)


Φ = [119867] 119889 (13)


Φ = [119867] 119889 (14)


119879 =1

2int119881

120588Φ119879

Φ 119889119881 (15)


119880 =1

2int119881

120576119879120590 119889119881 (16)


int119905

120575 (1

2int119881

120588 119889119879

[119867]119879[119867] 119889 119889119881

minus1

2int119881

119889119879[119861]119879[119863] [119861] 119889 119889119881)119889119905 = 0

(17)


[119872] 119889 + [119870] 119889 = 0 (18)

where

[119872] = int119881

120588[119867]119879[119867] 119889119881

[119870] = int119881

[119861]119879[119863] [119861] 119889119881

(19)



119888119886 Model Mode1 2 3 4 5








[[119870] minus 1199082[119872]] 119889 = 0 (20)


10038161003816100381610038161003816[119870] minus 119908

2[119872]

10038161003816100381610038161003816= 0 (21)




120582 =1205961198862

1205872119905

radic12120588 (1 minus ]2)

119864

(22)


x

y

1 2 3

4 5 6Crack

a

ac






Material 1198641(GPa) 119864

2 1198643(GPa) 119866

12 11986613(GPa) 119866

23(GPa) ^12 ^13 ^23 120588 (kgmm3)




1 2 3 4 5Modes

Uncracked(ca = 0)

(ca = 05)

Cracked


y

x

a

a b

b

c

CrackPlate

Patch

Adhesive

z

ta

ts

tp




180mm 119905119901

= 10mm and 119905119886


Patch layer

Crack

Adhesive layer






26

27

28

29

30

31

01 02 03 04 05ca


times10minus3

(1205962120587)


26

27

28

29

30

31

01 02 03 04 05ca

times10minus3

tp (5mm)tp (10mm)

tp (20mm)tp (30mm)

(1205962120587)




119901=

5mm to 119905119901




bc


27

28

29

30

31

10 15 20 25 30

times10minus3

(1205962120587)





119886= 0375mm to 119905




26

27

28

29

30

01 02 03 04 05ca

times10minus3

ta (0375mm)ta (075mm)

ta (15mm)ta (3mm)

(1205962120587)


24

25

26

27

28

29

30

01 02 03 04 05ca

times10minus3



(1205962120587)


4 Conclusions










Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119871119904 (119909) =

1

2119904119904

119889119904

119889119909119904(1199092minus 1)119904

for 119904 = 0 1 2 (4)


int

119909

minus1

119871119894 (119909) 119871119895 (119909) 119889119909 =

2

2119894 + 1for 119894 = 119895

0 otherwise(5)


119883119894119895= 119865119894 (119909) 119865119895 (119910) 119894 119895 = 1 2 (6)

where

1198731= 11988311 119873

2= 11988321 119873

3= 11988322 119873

4= 11988312

(7)


11987205119894(119894+3)+120572

= 119861119894 (119909) 1198651 (119910)

11987205(119894+1)(119894+2)+120572

= 1198652 (119909) 119861119894 (119910)

11987205(1198942+3119894+4)+120572

= 119861119894 (119909) 1198652 (119910)

11987205(1198942+3119894+6)+120572

= 1198651 (119909) 119861119894 (119910)

119894 = 1 2 119901 minus 1 120572 = minus1 in 119894 = 1

0 otherwise

(8)


11987205(1198952+119895+1)+119894


for 119895 = 4 5 119901(9)


120590119909119910119911

119897

6times1= [119863]

119897

6times6120576119909119910119911

119897

6times1 (10)


120576119909119910119911

= lfloor120597119880

120597119909

120597119881

120597119910

120597119882

120597119911

120597119880

120597119910+120597119881

120597119909

120597119880

120597119911+120597119882

120597119909

120597119881

120597119911+120597119882

120597119910rfloor

119879

(11)


int119905

120575 (119879 minus 119880) 119889119905 = 0 (12)


Φ = [119867] 119889 (13)


Φ = [119867] 119889 (14)


119879 =1

2int119881

120588Φ119879

Φ 119889119881 (15)


119880 =1

2int119881

120576119879120590 119889119881 (16)


int119905

120575 (1

2int119881

120588 119889119879

[119867]119879[119867] 119889 119889119881

minus1

2int119881

119889119879[119861]119879[119863] [119861] 119889 119889119881)119889119905 = 0

(17)


[119872] 119889 + [119870] 119889 = 0 (18)

where

[119872] = int119881

120588[119867]119879[119867] 119889119881

[119870] = int119881

[119861]119879[119863] [119861] 119889119881

(19)



119888119886 Model Mode1 2 3 4 5








[[119870] minus 1199082[119872]] 119889 = 0 (20)


10038161003816100381610038161003816[119870] minus 119908

2[119872]

10038161003816100381610038161003816= 0 (21)




120582 =1205961198862

1205872119905

radic12120588 (1 minus ]2)

119864

(22)


x

y

1 2 3

4 5 6Crack

a

ac






Material 1198641(GPa) 119864

2 1198643(GPa) 119866

12 11986613(GPa) 119866

23(GPa) ^12 ^13 ^23 120588 (kgmm3)




1 2 3 4 5Modes

Uncracked(ca = 0)

(ca = 05)

Cracked


y

x

a

a b

b

c

CrackPlate

Patch

Adhesive

z

ta

ts

tp




180mm 119905119901

= 10mm and 119905119886


Patch layer

Crack

Adhesive layer






26

27

28

29

30

31

01 02 03 04 05ca


times10minus3

(1205962120587)


26

27

28

29

30

31

01 02 03 04 05ca

times10minus3

tp (5mm)tp (10mm)

tp (20mm)tp (30mm)

(1205962120587)




119901=

5mm to 119905119901




bc


27

28

29

30

31

10 15 20 25 30

times10minus3

(1205962120587)





119886= 0375mm to 119905




26

27

28

29

30

01 02 03 04 05ca

times10minus3

ta (0375mm)ta (075mm)

ta (15mm)ta (3mm)

(1205962120587)


24

25

26

27

28

29

30

01 02 03 04 05ca

times10minus3



(1205962120587)


4 Conclusions










Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119888119886 Model Mode1 2 3 4 5








[[119870] minus 1199082[119872]] 119889 = 0 (20)


10038161003816100381610038161003816[119870] minus 119908

2[119872]

10038161003816100381610038161003816= 0 (21)




120582 =1205961198862

1205872119905

radic12120588 (1 minus ]2)

119864

(22)


x

y

1 2 3

4 5 6Crack

a

ac






Material 1198641(GPa) 119864

2 1198643(GPa) 119866

12 11986613(GPa) 119866

23(GPa) ^12 ^13 ^23 120588 (kgmm3)




1 2 3 4 5Modes

Uncracked(ca = 0)

(ca = 05)

Cracked


y

x

a

a b

b

c

CrackPlate

Patch

Adhesive

z

ta

ts

tp




180mm 119905119901

= 10mm and 119905119886


Patch layer

Crack

Adhesive layer






26

27

28

29

30

31

01 02 03 04 05ca


times10minus3

(1205962120587)


26

27

28

29

30

31

01 02 03 04 05ca

times10minus3

tp (5mm)tp (10mm)

tp (20mm)tp (30mm)

(1205962120587)




119901=

5mm to 119905119901




bc


27

28

29

30

31

10 15 20 25 30

times10minus3

(1205962120587)





119886= 0375mm to 119905




26

27

28

29

30

01 02 03 04 05ca

times10minus3

ta (0375mm)ta (075mm)

ta (15mm)ta (3mm)

(1205962120587)


24

25

26

27

28

29

30

01 02 03 04 05ca

times10minus3



(1205962120587)


4 Conclusions










Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Material 1198641(GPa) 119864

2 1198643(GPa) 119866

12 11986613(GPa) 119866

23(GPa) ^12 ^13 ^23 120588 (kgmm3)




1 2 3 4 5Modes

Uncracked(ca = 0)

(ca = 05)

Cracked


y

x

a

a b

b

c

CrackPlate

Patch

Adhesive

z

ta

ts

tp




180mm 119905119901

= 10mm and 119905119886


Patch layer

Crack

Adhesive layer






26

27

28

29

30

31

01 02 03 04 05ca


times10minus3

(1205962120587)


26

27

28

29

30

31

01 02 03 04 05ca

times10minus3

tp (5mm)tp (10mm)

tp (20mm)tp (30mm)

(1205962120587)




119901=

5mm to 119905119901




bc


27

28

29

30

31

10 15 20 25 30

times10minus3

(1205962120587)





119886= 0375mm to 119905




26

27

28

29

30

01 02 03 04 05ca

times10minus3

ta (0375mm)ta (075mm)

ta (15mm)ta (3mm)

(1205962120587)


24

25

26

27

28

29

30

01 02 03 04 05ca

times10minus3



(1205962120587)


4 Conclusions










Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










26

27

28

29

30

31

01 02 03 04 05ca


times10minus3

(1205962120587)


26

27

28

29

30

31

01 02 03 04 05ca

times10minus3

tp (5mm)tp (10mm)

tp (20mm)tp (30mm)

(1205962120587)




119901=

5mm to 119905119901




bc


27

28

29

30

31

10 15 20 25 30

times10minus3

(1205962120587)





119886= 0375mm to 119905




26

27

28

29

30

01 02 03 04 05ca

times10minus3

ta (0375mm)ta (075mm)

ta (15mm)ta (3mm)

(1205962120587)


24

25

26

27

28

29

30

01 02 03 04 05ca

times10minus3



(1205962120587)


4 Conclusions










Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










26

27

28

29

30

01 02 03 04 05ca

times10minus3

ta (0375mm)ta (075mm)

ta (15mm)ta (3mm)

(1205962120587)


24

25

26

27

28

29

30

01 02 03 04 05ca

times10minus3



(1205962120587)


4 Conclusions










Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Free vibration analysis of patch repaired plates with a through crack by p-convergent layerwise...

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Transcript of Free vibration analysis of patch repaired plates with a through crack by p-convergent layerwise...