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Exact free vibration study of rectangular Mindlin plates with all-over

part-through open cracks

Sh. Hosseini-Hashemi a,*, Heydar Roohi Gh. a, Hossein Rokni D.T. b

a Impact Research Laboratory, School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iranb School of Engineering, University of British Columbia Okanagan, Kelowna, BC, Canada V1V 1V7

a r t i c l e i n f o

Article history:

Received 21 January 2010

Accepted 11 June 2010

Available online 13 July 2010

Keywords:

Free vibration

Exact solution

Rectangular plate

Part-through crack

Mindlin theory

a b s t r a c t

Based on Mindlin plate theory (MPT), a set of exact closed-form characteristic equations incorporating

shear deformation and rotary inertia are proposed for the first time to analyze free vibration problem

of moderately thick rectangular plates with an arbitrary number of all-over part-through cracks. The pro-

posed rectangular plates have two opposite edges simply supported while six possible combinations of

free, simply supported and clamped boundary conditions are taken into account for two other edges.

The crack is assumed to be open, non-propagating and perpendicular to two opposite simply supported

edges. A continuously distributed line-spring model is used to describe the elastic behavior of an all-over

part-through crack. The accuracy of the current approach is investigated through comparing the present

exact natural frequencies with those of 3D finite element method obtained by ABAQUS software package.

A parametric study is undertaken to show the effect of crack depth, crack location, number of cracks and

thickness-to-length ratio on natural frequencies of rectangular moderately thick plates with different

boundary conditions in tabular and graphical forms. Finally, the effect of shearing and tearing modes

on the modeling of cracks located at the nodal line is shown.

2010 Elsevier Ltd. All rights reserved.

1. Introduction

Structures consisting of plates with different shapes, sizes,

thickness variations and boundary conditions are widely observed

in aerospace, civil, optical, electronic, automotive, mechanical, and

shipbuilding industries. Owing to their wide practical applications,

it is very likely that cracks, which are one of the most common

damages, appear in the vibrating plates. It is well known that the

crack can lead to changes in dynamic and stability characteristics

of plates. Therefore, examining the effect of crack damage on trans-

verse free vibration of thin and moderately thick plates is crucial

for engineers and designers.

Depending on the value of plate thickness, two main theoriesmay be considered for modeling a rectangular plate containing

an open crack. The classical plate theory, referred to as Kirchhoffs

theory [1], must be employed for thin plates due to ignoring the ef-

fect of shear deformation through the plate thickness. A consider-

able research work pertaining to the free vibration of thin cracked

plates has been performed. An extensive literature survey of the

vibration of cracked structures, reported until 1996, can be found

in the work of Dimarogonas[2].

Early research effort for free vibration analysis of thin rectangu-

lar plates with cracks dates back to the work of Lynn and Kumbasar

[3], who used Greens functions to represent the transverse dis-

placements of simply supported rectangular plates, resulting in

Fredholmintegral equations of the first kind. Stahl and Keer [4] for-

mulated freely vibrating thin simply supported rectangular plates

as dual series equations, reducing homogeneous Fredholm integral

equations from the first kind to the second one. Hirano and

Okazaki [5] utilized Levys form of solution and further matched

the boundary conditions using a weighted residual method to

investigate free vibrations of cracked rectangular plates with two

opposite edges simply supported and the remaining sides free or

clamped. Aggarwala and Ariel[6]determined the natural frequen-cies of simply supported cracked plates by using the homogeneous

Fredholm integral of the second kind by taking the stress singular-

ity at the crack tips into account. Neku [7]analyzed free vibration

of a simply supported rectangular plate with straight through cen-

tral or side cracks by means of finite Fourier transformation. The

flexural vibration of a simply supported rectangular plate with

arbitrarily located rectilinear cracks were studied by Solecki [8]

using the finite Fourier transformation of discontinuous functions.

At the beginning of 21st century, Khadem and Rezaee [9] devel-

oped an analytical approach for crack detection in rectangular thin

plates with an all-over part-through crack and subjected to uni-

form external loads using vibration analysis.

0045-7949/$ - see front matter 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.compstruc.2010.06.004

* Corresponding author. Tel.: +98 2177 240 190; fax: +98 2177 24 0488.

E-mail addresses: [email protected] (Sh. Hosseini-Hashemi), [email protected]

(Hossein Rokni D.T.).

Computers and Structures 88 (2010) 10151032

Contents lists available at ScienceDirect

Computers and Structures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p s t r u c
http://dx.doi.org/10.1016/j.compstruc.2010.06.004mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.compstruc.2010.06.004http://www.sciencedirect.com/science/journal/00457949http://www.elsevier.com/locate/compstruchttp://www.elsevier.com/locate/compstruchttp://www.sciencedirect.com/science/journal/00457949http://dx.doi.org/10.1016/j.compstruc.2010.06.004mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.compstruc.2010.06.004

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The dynamic characteristics of centrally located cracked plates

in tension were analyzed by Petyt[10]by the use of the finite ele-

ment (FE) displacement method. A finite element method (FEM) of

cracked plates was employed by Qian et al. [11]to investigate the

vibration analysis of simply supported and cantilevered square

thin plates having a through-thickness crack, parallel to one side

of the plate, using the integral of stress intensity factor. Prabhakara

and Datta [12] examined the static stability and vibrations of dam-

aged rectangular thin plates. Krawczuk[13]presented a FE model

based on the stiffness matrix to investigate the influence of the

crack location and its length on the changes of the natural frequen-

cies of the simply supported and cantilever rectangular thin plates.

Later, Krawczuk and Ostachowicz[14]used the same approach as

Krawczuk [13]did in order to investigate the influence of the crack

length and its location upon the amplitudes of the transverse

forced vibration for aluminum cantilevered cracked plates. Han

and Ren[15]analyzed transverse vibration of cracked rectangular

plates by means of a model of zero dimension elements with crack.

Su et al.[16]extended the two-level FEM to analyze the free vibra-

tion of a thin cracked plate with arbitrary boundary conditions.

Krawczuk et al.[17]studied elasto-plastic FE model of simply sup-

ported and cantilever square plates with a through crack. Fujimoto

et al.[18]reported a vibration analysis of rectangular plates sub-

jected to a tensile load and containing a centrally located crack

along the line of symmetry perpendicular to the direction of the

tensile load using the FEM. Very recently, Saito et al. [19]investi-

gated the linear and nonlinear vibration response of a cantilevered

rectangular plate with a transverse crack using a FE model.

By decomposing a rectangular plate into various domains and

introducing artificial springs at the interconnecting boundaries be-

tween the domains, Yuan and Dickinson [20] used the Rayleigh

Ritz method with regular admissible functions to find the solutions

for the flexural vibration of rectangular plates. Lee[21]employed

the Rayleigh method to obtain the fundamental frequency of annu-

lar plates with internal concentric cracks. Lee and Lim [22] used

the RayleighRitz approach for the study of the free vibration of

rectangular plates with a centrally located crack by consideringtransverse shear deformation and rotary inertia. The decomposi-

tion method was used by Liewet al. [23] to determine the vibration

frequencies of cracked plates with any combination of boundary

conditions. They assumed the cracked plate domain to be an

assemblage of small subdomains with the appropriate functions

formed and led to a governing eigenvalue equation. Ramamurti

and Neogy [24] applied the generalized RayleighRitz method to

determine the natural frequency of cracked cantilevered square

plates. Khadem and Rezaee[25]employed the modified compari-

son functions to obtain the natural frequencies of a simply sup-

ported rectangular cracked plate using the RayleighRitz method.

Very recently, free vibrations of simply supported and completely

free rectangular thin plates with a side crack were studied by

Huang and Leissa[26]using the famous Ritz method with special

displacement functions.

By applying the Galerkins method to von Karmans plate the-

ory, Wu and Shih [27] determined natural frequencies of simply

supported square and rectangular cracked plates subjected to the

pulsating in-plane forces by using incremental harmonic balance

method. Hu and Fu[28]introduced linear free vibration of visco-

elastic simply supported rectangular plates with a crack using

the Galerkins method. Transverse free vibration of viscoelastic

rectangular thin plate with linearly varying thickness and multiple

all-over part-through cracks was studied by Wang and Wang[29]

using the differential quadrature method. Very recently, a new

analytical model was presented by Israr et al.[30]for the vibration

analysis of cracked rectangular plates subjected to transverse load-

ing at some specified position with different sets of boundary con-

ditions using Galerkins method.

Adams and Cawley[31]developed an experimental technique

to estimate the location and depth of a crack from changes in nat-

ural frequencies of a rectangular plate. Maruyama and Ichinomiya

[32]obtained experimentally the natural frequencies for rectangu-

lar plates with straight narrow slits, which the effects of lengths,

positions and inclination angles of slits on natural frequency and

mode shape are discussed. Variations of natural frequencies and

associated mode shapes with respect to changes in crack length

for a square plate with an edge crack were reported by Ma and

Huang[33]using both experiments and FEM. A baseline finite ele-

ment model along with modal test data were utilized by Wu and

Law [34]to obtain natural frequencies of a completely free rectan-

gular Aluminum thin plate with an inclined through crack.In order to eliminate the deficiency of the classical plate theory

for moderately thick plates, the first-order shear deformation the-

ory was proposed by Reissner [35], and developed further for the

deformable plates in statics and dynamics by Mindlin et al. [36].

Although considerable papers exist for vibration analysisof Mindlin

Fig. 1. Strip with edge crack under bending moment.

Table 1

Values for coefficient Cnbb

.

n Cnbb

0 1.9710

1 4.42772 34.4952

3 165.73214 626.3926

5 2144.46516 7043.4169

7 19003.21998 37853.3028

9 52595.468110 48079.2948

11

25980.1559

12 6334.2425Fig. 3. Geometry and dimension of a rectangular cracked plate.

Fig. 2. A Mindlin plate with coordinate convention.

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rectangular plates [3740], according to the authors knowledge,

only one publication has been devoted to the free vibration analy-

sis of their cracked counterparts. Bachene et al. [41]adopted the

extended finite element method (X-FEM) to analyze free vibrations

of rectangular plates containing through edge and central cracks

based on the MPT. The formulation was first implemented numer-

ically using two types of quadrilateral elements. Then, they devel-

oped a FORTRAN computer code on the basis of the X-FEM

formulation and consequently calculated natural frequencies of

the cracked plates.

The line-spring model, adopted in this study to solve part-

through crack plates, has been widely used in the fracture mechan-

ics analysis of plate components containing surface or internal

cracks. The line-spring model was first proposed by Rice and Levy

[42] to give an approximate treatment for three-dimensional prob-

lem of a surface crack penetrating partly through the thickness of

an elastic plate. The problem was simplified to a two-dimensional

through crack problem in plate theory, in which the constraint ef-

fects of net ligaments from the three-dimensional problem were

incorporated in the form of a membrane load and bending moment

imposed on the through crack. Delale and Edrogan[43]reformu-

lated the line-spring model in the context of the Reissner plate the-

ory to include transverse shear effects.

The aim of this paper is to extract exact closed-form character-

istic equations for rectangular plats with all-over part-through

cracks based on the Mindlin plate theory. The plate has a pair of

opposite simply supported edges, called the levy plate [44], while

the other two edges may be taken any combinations of free, simply

supported andclamped boundary conditions. It is assumed that the

crack is open, non-propagating and perpendicular to two opposite

simply supported edges. The local flexibility, induced by an all-over

part-through crack in the plate, is modeledby the mode (I) fracture,

which is generated by the bending moment. Using the well-known

commercial software ABAQUS [45], 3D finite element method is

Fig. 4. A meshed rectangular plate with an all-over part-through crack whena= 0.5,f = 0.5,g = 2/3 andd = 0.1. Total number of elements: 32306; total number of singularelement: 1472; total number of nodes 141347.

Table 2

First seven frequency parameter (b) for SFSF cracked rectangular thin plate when g = 2/3.

f a Methods Frequency parametersmn

d= 0.05

0 0.5 Present 9.646311 12.841612 22.535513 38.395521 39.356814 41.761522 52.693823

3D FEM 9.646411 12.840612 22.533313 38.400121 39.356614 41.763722 52.694523

% Diff 0.00 0.01 0.01 0.01 0.00 0.01 0.000.5 0.25 Present 9.626511 12.797012 22.430013 37.381314 38.288521 41.630622 52.665723

3D FEM 9.626811 12.760412 22.304413 37.389114 38.302921 41.605722 52.460723

% Diff 0.00 0.29 0.56 0.02 0.04 0.06 0.390.5 0.5 Present 9.619911 12.841612 21.665513 38.286921 39.356814 41.761522 51.829023

3D FEM 9.621511 12.791712 21.708413 38.386921 39.127814 41.701222 51.928923

% Diff 0.02 0.39 0.20 0.26 0.58 0.14 0.19

d= 0.1

0 0.5 Present 9.515711 12.571112 21.749013 36.594121 37.301914 39.588622 49.291223

3D FEM 9.516511 12.569012 21.746313 36.609521 37.307614 39.597022 49.303623

% Diff 0.01 0.02 0.01 0.04 0.02 0.02 0.030.5 0.25 Present 9.4849 12.506412 21.556913 34.332814 36.446921 39.420722 49.242423

3D FEM 9.4860 12.438912 21.331613 34.239914 36.481921 39.384622 48.899423

% Diff 0.01 0.54 1.05 0.27 0.10 0.09 0.700.5 0.5 Present 9.4740 12.571112 20.442313 36.443221 37.301914 39.588622 48.147023

3D FEM 9.4786 12.477612 20.530413 36.483621 37.017914 39.495822 48.330623

% Diff 0.05 0.74 0.43 0.11 0.76 0.23 0.38

d= 0.2

0 0.5 Present 9.066611 11.750512 19.556113 31.578221 31.942714 33.788222 40.887023

3D FEM 9.069011 11.747412 19.552013 31.601521 31.957414 33.810222 40.917923

% Diff 0.03 0.03 0.02 0.07 0.05 0.07 0.080.5 0.25 Present 9.028711 11.681912 19.213213 28.391214 31.433021 33.641022 40.802423

3D FEM 9.032011 11.561812 18.832413 28.187614 31.496521 33.596122 40.314523

% Diff 0.04 1.03 1.98 0.72 0.20 0.13 1.200.5 0.5 Present 9.012111 11.750512 17.957613 31.419021 31.942714 33.788222 39.758623

3D FEM 9.020511 11.589712 18.030613 31.494021 31.591214 33.652622 39.962923

% Diff 0.09 1.37 0.41 0.24 1.10 0.40 0.51

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adopted to show the merit of the present analytical approach. The

influence of crack depth, crack location, and thickness-to-length ra-

tio on frequency parameters of cracked rectangular plates is inves-

tigated for different boundary conditions. In addition, the effect of

second and third modes of the crack on natural frequencies of the

plate is discussed. Finally, fundamental frequency parameters of

rectangular plates with an arbitrary number of cracks are given

for six possible combinations of boundary conditions.

2. Governing equations of cracked Mindlin plate

2.1. Modeling of crack in vibrating plates

Consider an edge-cracked elastic strip subjected to a bending

momentMb, as shown inFig. 1. The stress intensity factor of sucha loaded cracked strip is given by[42]:

KI H12rbgb; 1

where His the strip thickness andrbis the nominal stress at a pointfar from the crack tip which is imposed by the bending moment

(rb= 6Mb/H2). Functiongb can be expressed in dimensionless form

as[46]:

where f is the ratio of crack depth to strip thickness (f= h/H). Eq. (2)

is valid when the parameter f changes within the range of

0 < f< 0.8. Also, the relationship between stress intensity factor

(KI) and the strain energy release rate (G), in a state of plane strain,

is expressed as

G 1 m2

E K2I; 3

where m and E are Poissons ratio and Youngs modulus, respec-

tively. By substituting Eq.(1)into Eq.(3), the strain energy releaserate is derived as

Table 3

First seven frequency parameter (b) for SSSF cracked rectangular thin plate when g = 2/3.


d= 0.05

0 0.5 Present 10.602811 18.079412 33.064613 39.376221 47.294222 56.005914 62.727823

3D FEM 10.602811 18.079612 33.065613 39.379721 47.299122 56.013714 62.737823

% Diff 0.00 0.00 0.00 0.01 0.01 0.01 0.020.5 0.25 Present 10.589411 17.616112 31.577513 39.342721 46.770022 55.131014 61.481523

3D FEM 10.575011 17.620312 31.715513 39.335221 46.810122 55.057114 61.588623

% Diff 0.14 0.02 0.44 0.02 0.09 0.13 0.170.5 0.5 Present 10.559811 17.672712 32.644313 39.256421 46.896722 53.371714 62.247523

3D FEM 10.551911

17.668012

32.566713

39.264421

46.897422

53.456814

62.118823

% Diff 0.07 0.03 0.24 0.02 0.00 0.16 0.210.5 0.75 Present 10.549511 18.078912 31.888713 39.171121 47.277122 52.575714 62.105723

3D FEM 10.543211 17.969312 31.835713 39.188721 47.141022 52.633614 61.978823

% Diff 0.06 0.61 0.17 0.04 0.29 0.11 0.20

d= 0.1

0 0.5 Present 10.440411 17.603312 31.634513 37.480621 44.603322 52.323114 58.165223

3D FEM 10.440911 17.603812 31.643013 37.496221 44.625322 52.357814 58.202723

% Diff 0.00 0.00 0.03 0.04 0.05 0.07 0.060.5 0.25 Present 10.418911 16.878312 29.532013 37.433721 43.883322 51.173914 56.574123

3D FEM 10.392411 16.898312 29.761213 37.431121 43.974122 50.952614 56.757123

% Diff 0.25 0.12 0.78 0.01 0.21 0.43 0.320.5 0.5 Present 10.372511 16.968712 31.029413 37.314321 44.052922 48.806514 57.558723

3D FEM 10.360311 16.969912 30.881613 37.339721 44.075422 48.777214 57.355623

% Diff 0.12 0.01 0.48 0.07 0.05 0.06 0.350.5 0.75 Present 10.359311 17.600412 29.745213 37.204121 44.587022 47.799414 57.255723

3D FEM 10.360311 17.418112 29.616313 37.243421 44.364322 47.783914 57.012523

% Diff 0.01 1.04 0.43 0.11 0.50 0.03 0.42

d= 0.2

0 0.5 Present 9.897211 16.154612 27.647613 32.248521 37.575822 43.181314 47.283923

3D FEM 9.899311 16.161112 27.681313 32.294221 37.641022 43.270914 47.388823

% Diff 0.02 0.04 0.12 0.14 0.17 0.21 0.220.5 0.25 Present 9.867811 15.223912 25.371713 32.198621 36.833422 42.089214 45.806423

3D FEM 9.822711 15.243612 25.639013 32.219321 36.955322 41.741514 45.960523

% Diff 0.46 0.13 1.05 0.06 0.33 0.83 0.340.5 0.5 Present 9.808311 15.332012 26.980313 31.936721 36.998122 39.829514 46.734023

3D FEM 9.788911 15.318912 26.719713 31.971221 37.043022 39.955514 46.429823

% Diff 0.20 0.09 0.97 0.11 0.12 0.32 0.650.5 0.75 Present 9.800311 16.135812 25.117013 31.983521 37.571022 38.739914 46.202023

3D FEM 9.781111 15.870812 24.869813 32.042721 37.273522 38.701214 45.725923

% Diff 0.20 1.64 0.98 0.19 0.79 0.10 1.03

gbf ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffipf1:1202 1:8872f 18:0143f2 87:3851f3 241:9124f4 319:9402f5 168:0105f6

q ; 2


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G H1 m2

E r2bg

2b : 4

One may consider two identical elastic strips, one of which has

an edge crack, as shown in Fig. 1. If both strips are subjected to

equal moment, the presence of the crack will cause the rotation

of one end relative to the other to increase more than the

rotation of an uncracked strip. Accordingly, one can write G as

the sum of potential energy release rates due to bending momentas[42]:

G 12 rb

o

oh

H2h

6

!" #: 5

By comparing Eqs.(4) and (5), one can obtain the added slope

(h) as[42]:

h 121 m2

E rbabb; 6

whereabb is the non-dimensional bending compliance factor givenby[46]:

abb

f2 X12

n0C

nbbf

n;

7

and the values ofC

nbb are listed inTable 1.

As an approximation, one may assume that the stress intensity

factor at a point along the crack front of a plate with an all-over

part-through constant depth crack, is identical to the stress inten-

sity factor for an edge-cracked strip in-plane strain state with the

same loading conditions and crack depth[42].

2.2. Governing equations of Mindlin plate

Consider a moderately thick rectangular plate of width a, length

b, and uniform thickness H, oriented so that its undeformed middle

surface contains thex1 andx2 axes of a Cartesian coordinate system

(x1,x2, x3), as shown inFig. 2.

The displacements along the x1 and x2 axes are denoted by U1andU2, respectively, while the displacement in the direction per-

pendicular to the undeformed middle surface is denoted by U3.

In the Mindlin plate theory, the displacement components are as-

sumed to be given by

U1 x3w1 x1;x2; t ;U2 x3w2 x1;x2; t ;U3 w3 x1;x2; t ;

8ac

wheretis the time variable, w1 andw2 are the rotational displace-ments about the x2 and x1 axes at the middle surface of the plate,

Table 4

First seven frequency parameter (b) for SFSC cracked rectangular thin plate when g = 2/3.


d= 0.05

0 0.5 Present 10.895111 20.025712 37.060313 39.501821 48.484122 61.992414 65.519523

3D FEM 10.896011 20.034012 37.086213 39.506521 48.497722 62.048014 65.551623

% Diff 0.01 0.04 0.07 0.01 0.03 0.09 0.050.5 0.25 Present 10.834511 20.002212 35.295913 39.287321 48.478822 58.500814 64.591223

3D FEM 10.824911 19.877512 35.303113 39.304021 48.319922 58.578914 64.535723

% Diff 0.09 0.62 0.02 0.04 0.33 0.13 0.090.5 0.5 Present 10.860811 19.365912 36.985813 39.387821 47.929722 58.330814 65.286323

3D FEM 10.847411

19.403412

36.870513

39.392921

47.969222

58.546014

65.067523

% Diff 0.12 0.19 0.31 0.01 0.08 0.37 0.340.5 0.75 Present 10.895011 19.830012 35.399113 39.483421 48.070522 59.537114 63.977423

3D FEM 10.847411 19.815012 35.568213 39.471921 48.082822 59.688614 64.190023

% Diff 0.44 0.08 0.48 0.03 0.03 0.25 0.33

d= 0.1

0 0.5 Present 10.709911 19.349812 35.019213 37.579321 46.530222 56.975414 60.229323

3D FEM 10.711811 19.363812 35.063513 37.595821 46.551622 57.074914 60.297823

% Diff 0.02 0.07 0.13 0.04 0.05 0.17 0.110.5 0.25 Present 10.618711 19.303612 32.373813 37.292421 45.525922 52.549714 58.974923

3D FEM 10.602311 19.062612 32.318913 37.330721 45.268222 52.200614 58.882723

% Diff 0.15 1.25 0.17 0.10 0.57 0.66 0.160.5 0.5 Present 10.655911 18.368612 34.889413 37.420321 44.800622 52.326414 59.900923

3D FEM 10.633611 18.446012 34.675813 37.440221 44.879622 52.454114 59.567623

% Diff 0.21 0.42 0.61 0.05 0.18 0.24 0.560.5 0.75 Present 10.709811 19.040212 32.701413 37.552621 44.952122 54.169614 58.327123

3D FEM 10.683011 19.031312 33.017813 37.545621 45.010522 54.030614 58.681023

% Diff 0.25 0.05 0.97 0.02 0.13 0.26 0.61

d= 0.2

0 0.5 Present 10.106011 17.388812 29.670713 32.300321 38.044222 45.411014 48.207023

3D FEM 10.110011 17.412012 29.745913 32.346521 38.121022 45.581914 48.343823

% Diff 0.04 0.13 0.25 0.14 0.20 0.38 0.280.5 0.25 Present 10.000511 17.300212 26.560613 32.028921 38.043722 41.229114 46.910023

3D FEM 9.972111 16.969112 26.382213 32.086321 37.710622 40.737614 46.709723

% Diff 0.28 1.91 0.67 0.18 0.88 1.19 0.430.5 0.5 Present 10.036011 16.248812 29.428213 32.131821 37.349722 41.368314 47.829523

3D FEM 10.003411 16.322012 29.023513 32.181821 37.439822 41.493714 47.430023

% Diff 0.32 0.45 1.38 0.16 0.24 0.30 0.840.5 0.75 Present 10.106011 16.984712 27.377613 32.270121 37.444822 43.518414 46.618723

3D FEM 10.063111 16.980612 27.785013 32.288421 37.560422 43.252114 47.014323

% Diff 0.42 0.02 1.49 0.06 0.31 0.61 0.85


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respectively, andw3is the transverse displacement. For a harmonicsolution, the rotational and transverse displacements are assumed

to be:

w1 x1;x2; t ~w1 X1;X2 eixt;w2 x1;x2; t ~w2 X1;X2 eixt;

w3 x1;

x2;

t 1

a~

w3 X1;

X2 eixt

;

9ac

wherex denotes the natural frequency of vibration in radians andi

ffiffiffiffiffiffiffi1

p . It should be noted that each parameter having the over-

tilde is non-dimensional. Introducing the dimensionless

parameters:

X1x1a

; X2x2b

; d Ha

; g ab

; b xa2ffiffiffiffiffiffiffiqHD

r ; 10ad

governing equations of motion in dimensionless form can be ex-

pressed as[37]:

~w1;11 g2~w1;22 m2m1~w1;11 g~w2;12

12K2

d2

~w1 ~w3;1

b2d2

12m1~w1;

~w2;11 g2~w2;22 m2m1 g ~w1;12 g~w2;22

12K2

d2~w2 g~w3;2

b

2d2

12m1~w2;

~w3;11 g2~w3;22 ~w1;1 g~w2;2

b2d2

12K2m1~w3:

11ac

where comma-subscript convention represents the partial differen-

tiation with respect to the normalized coordinates, b is the fre-

quency parameter, m1= (1 m)/2, m2= ( 1 + m)/2, K2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:86667

p is

the shear correction factor to account for the fact that the

transverse shear strains are not truly independent of the thickness

coordinate. Finally, The general solutions to Eq. (11), in terms of

the three dimensionless potentials W1, W2 and W3, may be ex-

pressed as

~w1 C1W1;1 C2W2;1 gW3;2;~w2 C1gW1;2 C2gW2;2 W3;1;~w3 W1 W2;

12ac

where

C1 1 a22

m1a23; C2 1 a

21

m1a23;

a21;a22

b2

2

d2

12

1

K2m1 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2

12

!21

K2m1 1

2 4b2

vuut264

375;

a23 12K2

b2d

2a21a

22 12K

2

d2

b2

d4

144K2m1 1 !:

13

Based on these dimensionless potentials, the governing equa-

tions of motion may now be given by

W1;11 g2W1;22 a21W1;W2;11 g2W2;22 a22W2;W3;11 g2W3;22 a23W3:

14ac

Table 5

First seven frequency parameter (b) for SSSS cracked rectangular thin plate when g = 2/3.


d= 0.05

0 0.5 Present 14.166211 27.086612 43.033821 48.300613 55.634122 76.336023 77.361614

3D FEM 14.167011 27.089012 43.041421 48.310013 55.648222 76.360523 77.387914

% Diff 0.01 0.01 0.02 0.02 0.03 0.03 0.030.5 0.25 Present 13.991611 25.821212 42.802121 46.969813 54.567522 75.261123 77.361614

3D FEM 13.969111 25.883812 42.797721 46.955013 54.694922 75.203123 77.185414

% Diff 0.16 0.24 0.01 0.03 0.23 0.08 0.230.5 0.5 Present 13.837311 27.086612 42.593921 45.484513 55.634122 74.082823 77.361614

3D FEM 13.860111 26.957312 42.652821 45.649413 55.406222 74.351823 77.187014

% Diff 0.16 0.48 0.14 0.36 0.41 0.36 0.23

d= 0.1

0 0.5 Present 13.908511 26.180312 40.846721 45.584613 52.100122 70.021923 70.892914

3D FEM 13.911611 26.191712 40.874721 45.619713 52.146222 70.104823 70.978314

% Diff 0.02 0.04 0.07 0.08 0.09 0.12 0.120.5 0.25 Present 13.624611 24.315712 40.516621 43.822413 50.688822 68.704423 70.892914

3D FEM 13.598111 24.340712 40.530621 43.749413 50.844622 68.609923 70.550214

% Diff 0.19 0.10 0.03 0.17 0.31 0.14 0.480.5 0.5 Present 13.393811 26.180312 40.236821 41.736513 52.100122 67.195423 70.892914

3D FEM 13.437511 25.938912 40.362621 41.954713 51.733822 67.560323 70.560914

% Diff 0.33 0.92 0.31 0.52 0.70 0.54 0.47

d= 0.2

0 0.5 Present 13.025011 23.400512 34.855821 38.384713 43.123622 55.586023 56.173114

3D FEM 13.036611 23.437212 34.936821 38.482813 43.248022 55.785723 56.378414

% Diff 0.09 0.16 0.23 0.26 0.29 0.36 0.370.5 0.25 Present 12.634811 21.246912 34.496621 36.678513 41.746722 54.436023 56.173114

3D FEM 12.574711 21.349512 34.545321 36.514113 41.939122 54.254523 55.654014

% Diff 0.48 0.48 0.14 0.45 0.46 0.33 0.920.5 0.5 Present 12.351911 23.400512 34.212921 34.523613 43.123622 53.057223 56.173114

3D FEM 12.408911 23.100512 34.379421 34.728513 42.671822 53.464223 55.456414

% Diff 0.46 1.28 0.49 0.59 1.05 0.77 1.28


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One set of solutions toEq. (14)are:

W1 A1 sink1X2 A2 cosk1X2 sinl1X1 B1 sink1X2 B2 cosk1X2 cosl1X1;

W2 A3 sinhk2X2 A4 coshk2X2 sinl2X1 B3 sinhk2X2 B4 coshk2X2 cosl2X1;

W3

A5

sinhk

3X

2 A

6cosh

k

3X

2cos

l3X

1 B

5sinh

k

3X

2 B6 coshk3X2 sinl3X1;15

provided thata21> 0;a22< 0;a

23< 0. These conditions must hold in

order to avoid imaginary roots in the characteristic equations of

the plate.Ai andBi are the arbitrary constants.ki andli are also re-lated to theai by

a21 l21 g2k21;a22 l22 g2k22;a23 l23 g2k23:

16ac

On the assumption of a simply supported edge at both X1= 0

and,X1= 1 Eq.(15)may be written as[37]:

W1 A1 sin k1X2 A2 cos k1X2 sin mpX1 ;W2 A3 sinh k2X2 A4 cosh k2X2 sin mpX1 ;W3 A5 sinh k3X2 A6 cosh k3X2 cos mpX1 :

17ac

In this theory, the bending moments, twisting moments, and

the transverse shear forces in terms ofw1,w2 andw3 are obtainedby integrating the stresses and moment of the stresses through the

thickness of the plate. These are given by

M11Z H

2

H2

r11x3dx3 D w1;1 mw2;2

;

M22Z H

2

H2

r22x3dx3 D w2;2 mw1;1

;

M12Z H

2

H2

r12x3dx3 D2

1 m w1;2 w2;1

;

Q1Z H

2

H2

r13dx3 K2GH w1 w3;1

;

Q2 Z H

2

H

2

r23dx3 K2GH w2 w3;2 ;

18ae

Table 6

First seven frequency parameter (b) for SSSC cracked rectangular thin plate when g = 2/3.


d= 0.05

0 0.5 Present 15.449511 30.561812 43.674921 53.856813 57.825522 80.383423 84.733614

3D FEM 15.456011 30.584412 43.686821 53.907913 57.856122 80.447023 84.826514

% Diff 0.04 0.07 0.03 0.09 0.05 0.08 0.110.5 0.25 Present 15.162711 29.049512 43.383521 52.819213 56.613122 79.462623 84.562714

3D FEM 15.150411 29.135512 43.387121 52.801513 56.764522 79.360323 84.431414

% Diff 0.08 0.30 0.01 0.03 0.27 0.13 0.160.5 0.5 Present 15.040411 30.419912 43.183721 50.935113 57.774922 77.978823 84.102614

3D FEM 15.074411

30.316012

43.252121

51.148513

57.546622

78.272023

84.021414

% Diff 0.23 0.34 0.16 0.42 0.40 0.38 0.100.5 0.75 Present 15.414011 29.471312 43.520321 51.251213 56.725922 78.415323 83.617914

3D FEM 15.377111 29.583312 43.495821 51.466313 56.879822 78.601323 83.598314

% Diff 0.24 0.38 0.06 0.42 0.27 0.24 0.02

d= 0.1

0 0.5 Present 15.088411 29.199612 41.359221 50.023813 53.771422 72.894423 76.203414

3D FEM 15.100411 29.240912 41.394121 50.116213 53.842122 73.027323 76.377814

% Diff 0.08 0.14 0.08 0.18 0.13 0.18 0.230.5 0.25 Present 14.637711 27.049712 40.956421 48.636213 52.210222 71.745223 76.046314

3D FEM 14.621311 27.205012 40.984221 48.559513 52.478022 71.594123 75.771914

% Diff 0.11 0.57 0.07 0.16 0.51 0.21 0.360.5 0.5 Present 14.460111 29.017812 40.688321 46.090213 53.720922 69.920923 75.624914

3D FEM 14.538911 28.816912 40.826121 46.387813 53.354722 70.396323 75.346314

% Diff 0.55 0.69 0.34 0.65 0.68 0.68 0.370.5 0.75 Present 15.028511 27.586212 41.132321 46.850613 52.322722 70.718123 75.241014

3D FEM 14.967811 27.835612 41.119221 47.106513 52.623722 70.943723 75.003014

% Diff 0.40 0.90 0.03 0.55 0.58 0.32 0.32

d= 0.2

0 0.5 Present 13.911311 25.323512 35.127321 40.658313 43.918322 56.749523 58.307414

3D FEM 13.934711 25.395612 35.216221 40.818313 44.067122 57.000023 58.601114

% Diff 0.17 0.28 0.25 0.39 0.34 0.44 0.500.5 0.25 Present 13.343511 22.992712 34.716221 39.257713 42.463722 55.702323 58.259414

3D FEM 13.307411 23.101712 34.776221 39.010513 42.758222 55.522323 57.979614

% Diff 0.27 0.47 0.17 0.63 0.69 0.32 0.480.5 0.5 Present 13.133311 25.200912 34.443421 36.772813 43.897222 54.150623 58.116914

3D FEM 13.262511 24.797912 34.642521 37.122813 43.437422 54.505623 57.263414

% Diff 0.98 1.60 0.58 0.95 1.05 0.66 1.470.5 0.75 Present 13.823411 23.546412 34.871721 38.221013 42.578822 55.250923 58.069314

3D FEM 13.735911 23.841812 34.908421 38.585713 42.943422 55.300023 57.605014

% Diff 0.63 1.25 0.11 0.95 0.86 0.09 0.80


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whereG = E/2(1 + m) is the shear modulus, D = EH3/12(1 m2) is theflexural rigidity andrij(i,j = 1, 2 and 3) represents normal stresseswheni = j and the shear stresses when i j.

2.3. Geometrical configuration of cracked plate

The problem considered in this study is an elastic flat rectangu-

lar moderately thick plate with an arbitrary number of all-over

part-through cracks. They are perpendicular to one pair of opposite

simply supported edges, and their location is defined by the

parameter a , as shown in Fig. 3. Since an exact solution may beachieved for rectangular plates having at least two opposite simply

supported edges, central and side cracks, either oblique or parallel

to one edge of the plate, cannot taken into account in this study. In

addition, an oblique all-over part-through crack divides the rectan-

gular plate into two trapezoidal plates which, to the best knowl-

edge of the authors, no exact solution has been proposed for

these types of plates. For the sake of convenience, following formu-

lations are given for the plate with only one crack. As shown in

Fig. 3, the plate has thickness H, widtha and length b, while the

crack has a uniform depthh and constant lengtha. The plate is di-

vided by the crack line into two regions (I) and (II). The cracked

section is represented as continuously distributed line-spring

model so that the plate vibrates as an entire system. In order to ap-

ply dynamic and geometric boundary conditions, two Cartesian

coordinate systems (x1, x2, x3) and x01;x

02;x

03

are needed to define

for regions (I) and (II), respectively, which are related to each other

by the following form:

x01 x1; x022x2 1

2 ; x03x3; 19

x2= 0 andx02 0 are located in the middle of the plate. It should be

pointed out that considering the Cartesian coordinate systems like

those shown inFig. 3nullifies ill-conditioning phenomenon in the

mass and the stiffness matrices of the cracked plate.

2.4. Compatibility conditions for connection of two regions

At the crack location (X2a2or,X02

a

1

2 ), continuity conditionscan be written along the crack line as

w1I w1II;w3I w3II;M22I M22II;M12I M12II;Q2I Q2II;

20ae

where Eqs. 20(a)(e) express the equality of the slope of the plate in

X1 direction (w1), transverse displacement (w3), bending moment(M22), twisting moment (M12) and shearing force (Q2) at the two

sides of the crack location, respectively. The slope compatibility

condition of the crack in the X2 direction at the crack location

X2a

2

is satisfied by taking the following form:w2I w2II h; 21

wheretheaddedslope angle (h) induced bythe crackis givenby Eq.(6).

2.5. Classical boundary conditions

In this study, plates may take any classical boundary conditions

at non-simply supported edges, including free, simply supported

Table 7

First seven frequency parameter (b) for SCSC cracked rectangular thin plate when g = 2/3.


d= 0.05

0 0.5 Present 17.178011 34.574812 44.459021 59.884113 60.353522 84.836423 90.433031

3D FEM 17.193811 34.624812 44.478721 59.986613 60.408122 84.946723 90.477131

% Diff 0.09 0.14 0.04 0.17 0.09 0.13 0.050.5 0.25 Present 17.091411 33.044812 44.255221 57.298913 59.023222 82.881723 90.105731

3D FEM 17.062111 33.230212 44.236721 57.530113 59.230922 83.055023 90.142931

% Diff 0.17 0.56 0.04 0.40 0.35 0.21 0.040.5 0.5 Present 16.611011 34.574812 43.891021 56.179613 60.353522 81.956423 89.696631

3D FEM 16.671911 34.460112 43.974921 56.527413 60.095322 82.360223 89.843331

% Diff 0.37 0.33 0.19 0.62 0.43 0.49 0.16

d= 0.1

0 0.5 Present 16.645511 32.587612 41.970021 54.672813 55.645022 75.955523 81.572614

3D FEM 16.672711 32.670412 42.014021 54.833213 55.746222 76.147323 81.844814

% Diff 0.16 0.25 0.10 0.29 0.18 0.25 0.330.5 0.25 Present 16.509211 30.451112 41.682421 51.672413 53.969522 73.859123 81.119614

3D FEM 16.465311 30.812412 41.680621 51.933813 54.337722 74.049723 80.626514

% Diff 0.27 1.19 0.00 0.51 0.68 0.26 0.610.5 0.5 Present 15.805211 32.587612 41.213021 49.963213 55.645022 72.573923 80.843431

3D FEM 15.935111 32.365812 41.375221 50.485813 55.213622 73.169723 81.119531

% Diff 0.82 0.68 0.39 1.05 0.78 0.82 0.34

d= 0.2

0 0.5 Present 15.014711 27.340012 35.434621 42.893613 44.762422 57.930323 60.357614

3D FEM 15.056311 27.454512 35.531913 43.120413 44.939722 58.234823 60.745614

% Diff 0.28 0.42 0.27 0.53 0.40 0.53 0.640.5 0.25 Present 14.845211 25.235112 35.134821 40.719313 43.319322 56.520423 60.277014

3D FEM 14.780611 25.620812 35.182921 40.986613 43.721422 56.810623 59.675414

% Diff 0.44 1.53 0.14 0.66 0.93 0.51 1.000.5 0.5 Present 14.061211 27.340012 34.697321 38.660213 44.762422 55.184223 60.357614

3D FEM 14.254511 26.937912 34.914121 38.990713 44.277022 55.604623 59.584914

% Diff 1.37 1.47 0.62 0.85 1.08 0.76 1.28


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and clamped. The boundary conditions along the edges X2 a2andX02 1a2 are satisfied by the following relations:

(a) for a free edge

M11 M12 Q1 0: 22(b) for a simply supported edge

M11 ~w2 ~w3 0: 23(c) for a clamped edge

~w1 ~w2 ~w3 0: 24

By application of the aforementioned boundary conditions and

the crack continuity conditions, a set of homogenous equation are

obtained. The coefficient matrix of rectangular moderately thick

plates with an all-over part-through crack under six possible com-

binations of boundary conditions has been given in Appendix A.

Vanishing the determinant of the coefficient matrix results in the

frequency equation of the cracked plate.

3. Results and discussion

By solving the closed-form characteristic equations for six pos-

sible combinations of boundary conditions, frequency parameters

of rectangular moderately thick plates with an all-over part-

through crack are obtained as a function of crack location a,relative crack depth f, aspect ratio g and wave number m. For

the brevity, SCSF denotes that the edges X1= 0,X2= 0,X1= 1, andX2 = 1 are simply supported, clamped, simply supported and free,

respectively. All frequencies are expressed in terms of the dimen-

sionless parameterb xa2 ffiffiffiffiffiffiffiffiffiffiffiffiffiqH=D

p .

For the sake of comparison, ABAQUS software package was used

to demonstrate the merit and high accuracy of the present exact

solution procedure. All the specimens were modeled and analyzed

with quadratic hexahedral 3D stress elements of type C3D20in or-

der to consider the transverse shear deformation effect. Quarter-

point singular elements of type C3D20 were used at the first ring

of the elements around the crack tip to bring models closer to real-

ity. A mesh sensitivity analysis was carried out to ensure indepen-

dency of FEM results from the number of elements. For example, a

total number of 32306 3D elements and 1472 singular elements

produce sufficiently converged results for a rectangular Mindlin

Fig. 5. The variation ofbC/bUCwith respect to the crack locationa for an SFSF rectangular plate (g= 0.5, d = 0.1).

http://-/?-http://-/?-

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plate (d= 0.1,g = 2/3) with an all-over part-through crack (a= 0.5,f= 0.5), as shown inFig. 4.

3.1. Effect of crack depth

The first seven natural frequency of cracked rectangular plates

(g= 2/3), obtained by the present analytical solution and 3D FEmodel, are listed in Tables 27 for SFSF, SSSF, SFSC, SSSS, SSSC

and SCSC boundary conditions, respectively. These results are gi-

ven for two relative crack depths (f= 0, 0.5), three relative thick-nesses (d = 0.05, 0.1 and 0.2) and three crack locations (a= 0.25,0.5 and 0.75). Needless to say that f= 0 indicates an uncracked

plate. It should be noted that when the plate adopts symmetrical

boundary conditions abouta= b/2 (i.e., SFSF, SSSS and SCSC cases),natural frequencies fora= 0.25anda= 0.75arethe same.In Tables27, the corresponding mode shapes m and n denote the number of

half-sine waves in the X1 andX2 direction, respectively. The per-

centage difference given inTables 27is defined as follows:

%DiffAnalytical method 3D FEM Analytical method 100:

It is evident fromTables 27that the discrepancy between the

results of the present methodand 3D FEM is small and does not ex-

ceed 2% for the worst case. It is worth noting that the reduction ofnatural frequencies in the vibrating modes (n,m) = (1, 4) and (3, 1)

for SFSF and SCSC cracked plates, respectively, is so dominant that

the phenomenon of mode switching occurs. The vibrating mode (n,

m) = (1, 4) switches from the fifth frequency to fourth one and the

vibrating mode (n, m) = (3, 1) from the seventh frequency to sixth

one. InTables 2 and 7, these types of modes are labeled in bold.

3.2. Effect of crack location

The variation ofbC/bUC(subscriptsCand UC refer to cracked and

uncracked, respectively) against the crack location a for the firstfour mode shapes of rectangular moderately thick plates (g= 0.5)is shown inFigs. 510for six possible combinations of boundary

conditions when f= 0.2, 0.4 and 0.6. To gain further insight into

the effect of the crack location on frequency parameters of cracked

plates, 3D mode shapes of their uncracked counterparts are also

plotted.

It can be observed from Figs. 510that when the crack location

is close to the nodal line (i.e., the line of zero displacement), the ef-

fect of crack on the natural frequencies decreases and frequency

parameters of cracked plates approach to those of the plate with-

out crack (i.e., bC/bUC? 1). In contrast, if the crack is located at

the middle of two successive nodal lines, this effect can be maxi-

mized. It is also evident fromFigs. 7, 9 and 10that when the crack

gets much closer to the clamped edge of the plate, a considerablereduction in the natural frequencies of the cracked plate occurs.

Fig. 6. The variation ofbC/bUCwith respect to the crack location a for an SSSF rectangular plate (g= 0.5, d = 0.1).


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This may be attributed to the fact that the additional rotation in-

duced by the crack violates the clamped boundary condition,

where the slope is assumed to be zero. It is worthy to note from

Figs. 510that the natural frequencies decrease with the increase

of the relative crack depth f. This is due to the fact that the pres-

ence of the deeper crack causes a more reduction in the overall

stiffness of the plate.

3.3. Effect of stress intensity factor

In Fig. 11, the discrepancy between the present analytical meth-

od and 3D FEM for the first seven natural frequency of an SSSS

cracked rectangular plate (g= 2/3) is plotted in terms of the modenumber when a= 0.5,f= 0.5 andd= 0.05, 0.1 and 0.2. It can be ob-served fromFig. 11 that the most discrepancy takes place in the

modes (1, 2), (2, 2) and (1, 4), where the crack is located at the no-

dal line. This is due to the fact that only the first stress intensity

factor (KI) is considered in the present analytical solution for mod-

eling the crack. However, it seems that the second and third stress

intensity factors (KIIandKIII) should play a key role when the crack

lies on the nodal line.

A crack in a body may be subjected to three types of loading,which involve displacement of the crack surfaces (see Fig. 12).

The mechanical behavior of a solid containing a crack of a specific

geometry and size can be predicted by evaluating the stress inten-

sity factors (SIFs) KI, KIIand KIIIwhich are related to deformation

modesI,IIandIII. Hence, calculating SIFs is the first step for eval-

uating various crack deformations.

A lot of methods have been developed for determining the SIFs.

However, a very simple method for extracting these parameters is

called displacement extrapolation. In this approach, SIFs are com-

puted from the nodal displacements near the crack tip. In fact,

the nodal displacements in the vicinity of the crack are fitted to

the displacement equations and consequently the SIFs can be com-

puted from simple equations. By the use of collapsed-node isopara-

metric quadrilateral elements in the first ring around the crack tip,

the values of the SIFs are directly computed from nodal-point dis-

placements on opposite sides of the crack plane through the fol-

lowing relations[47]:

KIffiffiffiffiffiffiffi

2pp

2G1k jDVj / jDVj

KIIffiffiffiffiffiffiffi

2pp

2G1k jDUj / jDUj

KIIIffiffiffiffiffiffiffi

2pp

2GjDWj / jDWj;

k 3

4v For plane strain or axisymmetric;

3v1v For plane stress;

( 25ac

Fig. 7. The variation ofbC/bUCwith respect to the crack location a for an SFSC rectangular plate (g= 0.5,d = 0.1).


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whereG = E/2(1 m) is the shear modulus and DV, DUand DWarethe difference of the nodal displacements in opposite sides of the

crack inX1,X2 andX3 directions, respectively. For instance, the no-

dal displacement of opposite sides of the crack inX2 direction (i.e.,

DV) is shown inFig. 13. According toEq. (25), the magnitude ofKI,

KIIandKIIIis directly proportional to the values ofDV, DUand DW,

respectively. Therefore, the displacement differences DV, DU and

DW are properly utilized to compare the values of KI, KII and KIII

with each other. For example, for deformation mode I, the valueofDV is significantly greater than the other displacement differ-

ences DUand DW.

The aforementioned method is used to compare the deforma-

tion modesI, IIandIIIat the crack tip. In order to show the effect

of KII and KIII on natural frequencies of plates, first two shape

modes of an SSSS cracked rectangular plate (g= 2/3) are illustratedinFig. 14fora = 0.5 andf = 0.5. The values ofDV, DUand DWforthree locationsX1= 0,X1= 0.25andX1= 0.5 are listed in Table 8 for

the first two modes of the SSSS cracked plate. Note that these val-

ues have been calculated for the maximum deflection of the vibrat-

ing plate at the crack location.

It can be seen from Table 8 that the stress intensity factor (KI) of

mode Iis zero when the plate with a crack located at the nodal line

is vibrating in the second mode (1, 2). Under this circumstance, thepresent analytical method gives incorrect results and the natural

frequencies of cracked and uncracked plates become identical. It

can be attributed to the fact that the deformation modes II and

IIIare dominant for the cracks located at the nodal line. It is note-

worthy that in fracture mechanics, modeling of deformation modes

II and III are much more complicated than that of deformation

modeI. That is why the function gb, given by Eq.(2)for the defor-

mation mode I [46], has not been reported in literature for the

deformation modes IIand III. If this function was available, it could

be used in the present solution to model other two fracture modes.

3.4. Effect of multiple cracks

The present analytical approach can be extended to the plates

with several all-over part-through cracks. There are six unknown

coefficients in Eq. (17) which can be determined by applying

boundary conditions at two non-simply supported edges. As earlier

mentioned, one crack divides the plate into two segments resulting

in 12 unknown coefficients. They can be determined by applying

six boundary conditions at two non-simply supported edges and

other six boundary conditions defined by the continuity equations

(i.e.,Eqs. (20) and (21)) at the crack line. Similarly, a 12 12 coef-ficient matrix for one crack, for example, turns into an 18 18coefficient matrix for two cracks. Table 9shows the fundamentalfrequency parameters of a cracked rectangular plate (g= 2/3,

Fig. 8. The variation ofbC/bUCwith respect to the crack locationa for an SSSS rectangular plate (g= 0.5, d = 0.1).


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d= 0.1,f = 0.5) with six possible combinations of boundary condi-

tions, while there exist either one, two or three cracks in the plate.

It is seen fromTable 9that the more the cracks are introduced in

the plate, the more the natural frequencies decrease, as expected.

4. Conclusion

Based on the Mindlin plate theory, free vibration analysis of

rectangular moderately thick plates with an all-over part-throughcrack was investigated for different classical boundary conditions

of free, simply supported and clamped. The present solution was

given for the case where the cracks were perpendicular to two

opposite edges simply supported and remained open during the

entire cycle of motion. They were also treated as the mode (I) frac-

ture. ABAQUS software package was used to demonstrate the high

accuracy of the present analytical method. The influence of the

crack parameters, number of cracks, thickness-to-length ratios

and boundary conditions on natural frequencies of cracked rectan-

gular plates was investigated in the tabular and graphical form. It

was shown that the effect of cracks on the reduction of natural fre-

quencies is so tangible for some vibrating modes and boundary

conditions that the mode switching occurs. It was also found that

the natural frequencies decrease with the increase of the relativecrack depth and the number of cracks. The crack which is very

close to the clamped edge, results in a considerable reduction in

the natural frequencies of the cracked plate. Finally, it was ob-

served that the closer the crack is to the nodal line, the more dom-

inant the effect of the second and third stress intensity factors (KIIandKIII) on the frequency parameter will be. The significant merit

of the present paper lies in the fact that the exact closed-form char-

acteristic equations for six cases have been given in an explicit

form which is easy to compute natural frequencies of cracked rect-

angular plates.

Appendix A

There exist closed-form exact solutions to the characteristic

equations of cracked rectangular Mindlin plates under six combi-

nations of free, simply supported and clamped boundary condi-

tions. For the sake of clarity, the characteristic equations are

represented in a matrix form and the determinant will not be ex-

panded. For each case, an exact solution can be obtained by setting

the determinant of the matrices equal to zero. Roots of the deter-

minant are the natural frequencies of cracked rectangular Mindlin

plates for given wave number and dimensions of the plate and

crack. The characteristic determinants for the six boundary condi-

tions (i.e., SFSF, SSSF, SFSC, SSSS, SSSC and SCSC) are expressed asfollows:

Fig. 9. The variation ofbC/bUCwith respect to the crack locationa for an SSSC rectangular plate (g= 0.5,d = 0.1).


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Case 1. SFSF

Case 2. SSSF

Case 3. SFSC

C11Z4 S11Z4 C12Z5 S12Z5 S13Z6Z0 C13Z6

Z00 0 0 0 0 0

S11Z1 C11Z1 S12Z2 C12Z2 C13Z3Z0S13Z3

Z00 0 0 0 0 0

C11Z7

S11Z7 C12Z8 S12Z8

S13lZ0

C13lZ0

0 0 0 0 0 0

S21c1l C21c1l S22c2l C22c2l C23gk3Z0 S23gk3

Z0S31c1l C31c1l S32c2l C32c2l C33gk3Z0 S33g

k3Z0

S21 C21 S22 C22 0 0 S31 C31 S32 C32 0 0S21Z1 C21Z1 S22Z2 C22Z2 C23Z3Z0

S23Z3Z0

S31Z1 C31Z1 S32Z2 C32Z2 C33Z3Z0 S33Z3

Z0

C21Z7 S21Z7 A10Z8 A9Z8 A11lZ0 A12l

Z0C31Z7 S31Z7 C32Z8 S32Z8 S33lZ0

C33lZ0

C21Z4 S21Z4 A10Z5 A9Z5 A11Z6Z0 A12Z6

Z0C31Z7 S31Z7 C32Z8 S32Z8 S33Z6Z0

C33Z6Z0

Z9 Z10 Z11 Z12 Z13 Z14 C31c1gk1 S31c1gk1 C32c2gk2 S32c2gk2 S33lZ0C33l

Z0

0 0 0 0 0 0 C41Z4 S41Z4 C42Z5 S42Z5 S43Z6Z0 C43Z6

Z0

0 0 0 0 0 0 S41Z1 C41Z1 S42Z2 C42Z2 C43Z3Z0S43Z3

Z0

0 0 0 0 0 0 C41Z7 S41Z7 C42Z8 S42Z8 S43lZ0 C43l

Z0

0: A:1


Z00 0 0 0 0 0

S11 C11 S12 C12 0 0 0 0 0 0 0 0

S11Z1 C11Z1 S12Z2 C12Z2 C13Z3Z0S13Z3

Z00 0 0 0 0 0


Z0S31c1l C31c1l S32c2l C32c2l C33gk3Z0

S33gk3Z0


S23Z3Z0

S31Z1 C31Z1 S32Z2 C32Z2 C33Z3Z0 S33Z3

Z0

C21Z7 S21Z7 C22Z8 S22Z8 S23lZ0 C23lZ0

C31Z7 S31Z7 C32Z8 S32Z8 S33lZ0C33l

Z0

C21Z4

S21Z4 C22Z5 S22Z5

S23Z6

Z0

C23Z6

Z0

C31Z7 S31Z7

C32Z8

S32Z8

S33Z6Z0

C33Z6Z0


Z0

0 0 0 0 0 0 C41Z4 S41Z4 C42Z5 S42Z5 S43Z6Z0 C43Z6

Z0

0 0 0 0 0 0 S41Z1 C41Z1 S42Z2 C42Z2 C43Z3Z0S43Z3

Z0

0 0 0 0 0 0 C41Z7 S41Z7 C42Z8 S42Z8 S43lZ0 C43l

Z0

0: A:2

C11Z4 S11Z4 C12Z5 S12Z5 S13Z6Z0 C13Z6

Z00 0 0 0 0 0

S11

Z1

C11

Z1

S12

Z2

C12

Z2

C13Z3

Z0

S13Z3

Z00 0 0 0 0 0

C11Z7 S11Z7 C12Z8 S12Z8 S13lZ0 C13lZ0

0 0 0 0 0 0



S33gk3Z0


S23Z3Z0

S31Z1 C31Z1 S32Z2 C32Z2 C33Z3Z0 S33Z3

Z0

C21Z7 S21Z7 C22Z8 S22Z8 S23lZ0 C23lZ0

C31Z7 S31Z7 C32Z8 S32Z8 S33lZ0C33lZ0

C21Z4 S21Z4 C22Z5 S22Z5 S23Z6Z0 C23Z6

Z0C31Z7 S31Z7 C32Z8 S32Z8 S33Z6Z0

C33Z6Z0

Z9 Z10 Z11 Z12 Z13 Z14 C31c1gk1 S31c1gk1 C32c2gk2 S32c2gk2 S33lZ0C33lZ0

0 0 0 0 0 0 S41c1l C41c1l S42c2l C42c2l C43gk3Z0 S43gk3

Z0

0 0 0 0 0 0 C41c1gk1 S41c1gk1 C42c2gk2 S42c2gk2 S43lZ0C43lZ0

0 0 0 0 0 0 S41 C41 S42 C42 0 0

0: A:3


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Case 4. SSSS

Case 5. SSSC

Case 6. SCSC


Z00 0 0 0 0 0

S11 C11 S12 C12 0 0 0 0 0 0 0 0

S11Z1 C11Z1 S12Z2 C12Z2 C13Z3Z0S13Z3

Z00 0 0 0 0 0

S21c1l C21c1l S22c2l C22c2l

C23gk3

Z0 S23gk3

Z0 S31c1l

C31c1l

S32c2l

C32c2l

C33gk3

Z0

S33gk3

Z0


S23Z3Z0

S31Z1 C31Z1 S32Z2 C32Z2 C33Z3Z0 S33Z3

Z0

C21Z7 S21Z7 C22Z8 S22Z8 S23lZ0 C23lZ0

C31Z7 S31Z7 C32Z8 S32Z8 S33lZ0C33lZ0

C21Z4 S21Z4 C22Z5 S22Z5 S23Z6Z0 C23Z6

Z0C31Z7 S31Z7 C32Z8 S32Z8 S33Z6Z0

C33Z6Z0


0 0 0 0 0 0 S41c1l C41c1l S42c2l C42c2l C43gk3A0 S43gk3

A0

0 0 0 0 0 0 S41 C41 S42 C42 0 0

0 0 0 0 0 0 S41Z1 C41Z1 S42Z2 C42Z2 C43Z3A0S43Z3

A0

0: A:4


Z00 0 0 0 0 0

S11 C11 S12 C12 0 0 0 0 0 0 0 0

S11Z1 C11Z1 S12Z2 C12Z2 C13Z3Z0S13Z3

Z00 0 0 0 0 0



S33gk3Z0


S23Z3Z0

S31Z1 C31Z1 S32Z2 C32Z2 C33Z3Z0 S33Z3

Z0

C21Z7 S21Z7 C22Z8 S22Z8 S23lZ0 C23lZ0

C31Z7 S31Z7 C32Z8 S32Z8 S33lZ0C33lZ0

C21

Z4

S21

Z4

C22

Z5

S22

Z5

S23Z6

Z0 C23Z6

Z0 C

31Z

7 S

31Z

7 C

32Z

8 S

32Z

8

S33Z6

Z0

C33Z6

Z0



Z0

0 0 0 0 0 0 S41 C41 S42 C42 0 0

0 0 0 0 0 0 C41c1gk1 S41c1gk1 C42c2gk2 S42c2gk2 S43lZ0 C43lZ0

0: A:5


Z00 0 0 0 0 0

S11 C11 S12 C12 0 0 0 0 0 0 0 0

C11c1gk1 S11c1gk1 C12c2gk2 S12c2gk2 S13lZ0 C13lZ0 0 0 0 0 0 0S21c1l C21c1l S22c2l C22c2l C23gk3Z0

S23gk3Z0

S31c1l C31c1l S32c2l C32c2l C33gk3Z0S33gk3

Z0


S23Z3Z0

S31Z1 C31Z1 S32Z2 C32Z2 C33Z3Z0 S33Z3

Z0

C21Z7 S21Z7 C22Z8 S22Z8 S23lZ0 C23lZ0

C31Z7 S31Z7 C32Z8 S32Z8 S33lZ0C33l

Z0

C21Z4 S21Z4 C22Z5 S22Z5 S23Z6Z0 C23Z6

Z0C31Z7 S31Z7 C32Z8 S32Z8 S33Z6Z0

C33Z6Z0


Z0


Z0

0 0 0 0 0 0 S41 C41 S42 C42 0 0

0 0 0 0 0 0 C41c1gk1 S41c1gk1 C42c2gk2 S42c2gk2 S43lZ0 C43lZ0

0:

A:6


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where

c1 a2

; c2a2

; c3a 1

2 ; c4

1 a2

;

Si1 sincik1Ci1 coscik1

;

Sij sinhcikjCij coshcikj

; for i 1; 2; 3 and 4;

j 2; 3:

Z0 sinhc1k3; Z1 c1 g2k21 ml2

; Z2 c2 g2k22 ml2

;

Z3 g1 mlk3; Z4 2c1gk1l; Z5 2c2gk2l;

Z6 g2k23 l2; Z7 c1 1gk1; Z8 c2 1gk2;

Z9 c1gk1C21 6Habba

Z1S21; Z10 c1gk1S21 6Habba

Z1C21;

Z11 c2gk2C22 6Habba

Z2S22;

Z12 c2gk2S22 6Habba

Z2C22; Z13 lZ23 6Habba

Z3C23

Z0;

Z14 lC23 6Habba

Z3S23

Z0:

It should be pointed out that all elements, including cosh

(ci k3) and sinh (ci k3) (i= 1, 2, 3, 4), take very large values

Fig. 10. The variation ofbC/bUCwith respect to the crack location a for an SCSC rectangular plate (g= 0.5, d = 0.1).

Fig. 11. The percent of discrepancy for the first seven mode shapes of an SSSS

cracked rectangular plate (g= 2/3) whena = 0.5 andf = 0.5.


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in comparison with other elements excluding the aforementioned

parameters, whend< 0.1. In order to eliminate the ill-conditioning

problem in the matrices, given by Eqs. (A.1)(A.6), the Cartesian

coordinate systems should be considered as those shown in

Fig. 3 so that the elements with large values appear in the

fifth, sixth, eleventh and twelfth columns. Then, dividing all ele-

ments in those four columns by Z0 results in well-conditioning

matrices.

Fig. 12. Stress loading modes: (a) mode I (opening mode); (b) modeII (shearing mode); (c) modeIII(tearing mode).

Fig. 13. The nodal-point displacement for a crack.

Fig. 14. The effect ofKIIand KIIIfor first two shape modes of an SSSS cracked rectangular plate ( g= 2/3) witha = 0.5 andf = 0.5.

Table 8

The values ofDV, DUand DWfor the first two modes of an SSSS cracked rectangular

plate (g= 2/3, d = 0.1) when a = 0.5 and f = 0.5.

X1 First mode (1, 1) Second mode (1, 2)

DV 102 DU 102 DW 102 DV 102 DU 102 DW 102

0 0 0 0 0 0 1.109

0.25 0.884 0 0 0 0.443 0.912

0.5 1.310 0 0 0 0.658 0.148


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Table 9

Fundamental frequency parameters (b) of a rectangular plate (g = 2/3, d = 0.1) with multiple cracks when f = 0.5.

Crack location (a) Boundary conditions

SFSF SSSF SFSC SSSS SSSC SCSC

Uncracked plate 9.5157 10.4404 10.7099 13.9085 15.0884 16.6455

0.5 9.4740 10.3725 10.6559 13.3938 14.4601 15.8052

1/3 and 2/3 9.4472 10.3247 10.6131 13.2023 14.3147 15.8258

0.25, 0.5 and 0.75 9.4261 10.2907 10.5762 13.0518 14.1518 15.6941


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