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European Journal of Mechanics A/Solids 50 (2015) 100e111

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European Journal of Mechanics A/Solids

journal homepage: www.elsevier .com/locate/ejmsol

Analytical modelling for vibration analysis of partially crackedorthotropic rectangular plates

P.V. Joshi*, N.K. Jain, G.D. RamtekkarDepartment of Mechanical Engineering, National Institute of Technology, Raipur, C.G., India

a r t i c l e i n f o

Article history:Received 4 June 2014Accepted 9 November 2014Available online 20 November 2014

Keywords:PlateCrackOrthotropic

* Corresponding author.E-mail addresses: psad@rediffmail.com (P.V. Josh

(N.K. Jain), gdramtekkar.ce@nitrr.ac.in (G.D. Ramtekka

http://dx.doi.org/10.1016/j.euromechsol.2014.11.0070997-7538/© 2014 Elsevier Masson SAS. All rights re

a b s t r a c t

An analytical model is presented for vibration analysis of a thin orthotropic plate containing a partialcrack located at the centre of the plate. The continuous line crack is parallel to one of the edges of theplate. The equation of motion of an orthotropic plate is derived using the equilibrium principle based onClassical Plate Theory. The crack terms are formulated using the Line Spring Model. Using Berger'sformulation for the in-plane forces and Galerkin's method for solution, the derived equation involving acubic nonlinear term is converted into the Duffing equation. The effect of nonlinearity is established byderiving the frequency response equation for the cracked plate using the method of multiple scales. Theinfluence of crack length and boundary conditions on the fundamental frequency of square and rect-angular plate is demonstrated for three boundary conditions. It is found that the vibration characteristicsare affected by the presence of crack. Further, it is deduced that the presence of a crack across the fibresaffects the fundamental frequency more as compared to a crack along the fibres. The effect of varyingelasticity ratio on fundamental frequency of cracked plate is also established.

© 2014 Elsevier Masson SAS. All rights reserved.

1. Introduction

Orthotropic plates find many applications in the areas ofstructural mechanics, shipbuilding, civil and mechanical in-stallations. As a simplification, composite plates are often modelledas orthotropic. The orthotropic behaviour can be by the use ofmaterials having orthotropic stress strain relationships. In case ofisotropic plate, unequal stiffening in the two mutually perpendic-ular directions makes it orthotropic (Biancolini et al., 2005; Xingand Liu, 2009). In order to cater the wide applications of suchplates, dynamic analysis, especially, knowledge of vibration char-acteristics becomes important. An exhaustive literature review onvibration of plates has been given by Leissa (1969) in his mono-graph. RayleigheRitz method, one of the well known methods iswidely used to provide approximate solutions to the natural fre-quencies of orthotropic plates and many researchers have effec-tively applied this method. The conventional Navier and Levy typesolution methods (Xing and Liu, 2009; Szilard, 2004) can also beextended to orthotropic plates having either all edges simply sup-ported or two opposite edges simply supported. Kshirsagar and

i), nkjmanit@rediffmail.comr).

served.

Bhaskar (2008) applied superposition approach developed byBhaskar and Sivaram (2008), for free vibration of orthotropic platesinvolving mixed boundary conditions. Xing and Liu (2009) ob-tained exact natural frequencies for thin orthotropic rectangularplates by the method of separation of variables. Biancolini et al.(2005) deduced an approximate way of finding fundamentalmodes of free vibration of thin orthotropic plates starting with ageneral approximate formula. Although the Classical Plate Theoryover predicts the fundamental frequencies, literature shows thatmany researchers have employed it for orthotropic thin plates(Biancolini et al., 2005; Xing and Liu, 2009). The presence of adefect in the form of crack or hole affects the vibration character-istics. Rice and Levy (1972) represented a continuous line crack bydistributed line spring with stretching and bending compliancesthus formulating the Line Spring Model (LSM) using Classical PlateTheory. Gutierrez et al. (2000) considered circular and rectangularholes in their work on transverse vibration of isotropic andorthotropic plates. They concluded that, for a clamped plate thefundamental frequency increases with increase in hole diameterdue to dynamic stiffening. Huang et al. (2011) presented results forfundamental frequencies, applying the Ritz method for an isotropicrectangular plate with a through crack. They proposed new ad-missible functions to consider the singularities at the crack tips andconcluded that increase of crack length decreases the fundamental

P.V. Joshi et al. / European Journal of Mechanics A/Solids 50 (2015) 100e111 101

frequency. Extending their work, Huang et al. (2012) consideredthrough internal cracks for three dimensional vibration analysis offunctionally graded rectangular plates. Bachene et al. (2009)applied extended finite element method for the solution ofcracked plates. Viola et al. (2013) applied differential quadraturefinite element method to investigate dynamics of thick compositeplates containing a through crack. Natarajan et al. (2011a) appliedthe extended finite element method proposed by Bachene et al.(2009) and considered 20 degrees of freedom for a 4 node quad-rilateral plate element in their application for cracked functionallygraded plates. The authors provided extensive results on cracklength, orientation, crack location and for multiple cracks. Baiz et al.(2011) studied linear buckling of cracked isotropic plates bysmoothed curvatures and extended finite element method.Recently Ismail and Cartmell (2012) presented an analyticalapproach for partially cracked isotropic plate, wherein the crack isinclined to the edges of the plate. Their approach is based on LSMand crack compliances are used to represent the surface crack.More recently, Bose and Mohanty (2013) considered arbitrary po-sition and orientation of a part through crack in a thin isotropicplate for vibration analysis and deduced that the orientation ofcrack affects the vibration characteristics of the plate. Khadem andRezaee (2000) presented vibration analysis of cracked rectangularplate considering bending compliance only and proposed newcomparison functions. They deduced that presence of crack at aspecific location affects the natural frequencies differently.Krawczuk et al. (2001) considered elasticeplastic through crack intheir formulation by finite element scheme. Israr et al. (2009)developed an approximate analytical model for vibrations of acracked isotropic plate using LSM to represent bending and tensileforces by net ligament at crack location. In their work, a surfacecrack located at the centre is parallel to one of the edges of theplate. The authors introduced bending and stretching force effectsdue to the crack. Their study is based on Classical Plate Theory andconcluded that the fundamental frequency decreases as cracklength increases. Recently, Huang and Chan (2014), in their appli-cation of Ritz method used moving least squares interpolationfunctions for vibration of cracked plate. Jha et al. (2013) appliedhigher order shear and normal deformation theory for free vibra-tion analysis of functionally graded plates.

Vel and Batra (2002) presented an exact solution for three-dimensional thermo-elastic deformations of a simply supportedfunctionally graded thick rectangular plate using the power se-ries method. They reduced the partial differential equationsgoverning the thermoelastic deformations to a set of ordinarydifferential equations by suitable displacement and temperaturefunctions. The authors compared their results with classical, firstorder and third order shear deformation theories for functionallygraded plates. Extending their work (Vel and Batra, 2004) to freeand forced vibrations of functionally graded plates (FGP), theauthors presented exact natural frequencies for FGP. They alsoconsidered the effect of varying microstructure in the thicknessdirection. Batra and Jinb (2005) combined the finite elementmethod with the first order shear deformation theory to analysefree vibrations of functionally graded anisotropic plates. Ferreiraet al. (2006) presented results for natural frequencies of FGP by ameshless method based on first and third order theories. Theyused the global collocation method with radial basis functions intheir approach. Ferreira et al. (2009) extended the radial basisfunction approach to analyse natural frequencies of orthotropic,monoclinic and hexagonal material thick plates. The authorsdemonstrated the efficacy of the pseudospectral method foreigenvalue problems. Ahmad Akbari et al. (2010) applied theMeshless Local PetroveGalerkin method to simulate the propa-gation of a thermoelastic wave through FGP and presented

results for dynamic displacement and stress fields under tran-sient thermal field. The work of Rahimabadi et al. (2013) showsthe effect of cutouts and cracks on vibrations of FGP. The authorsused an enriched shear flexible 4-noded quadrilateral elementwith a Heaviside function to affect the displacement across thediscontinuity and asymptotic branch functions to accommodatethe singularity around the crack tip. They presented results fornatural frequencies of FPG affected by cutout size and cracklengths in the presence of thermal environment. The work ofNatarajan et al. (2011b, 2013, 2014, 2012) shows the applicationof finite element method to study static bending, free vibration,mechanical and thermal buckling of intact and cracked FGP. Theauthors used 8-noded shear flexible element (Natarajan et al.,2011b), enriched 4-noded quadrilateral element (Natarajanet al., 2013), 3-noded triangular element (Natarajan et al.,2014) and NURBS basis functions based iso-geometric finiteelement method (Natarajan et al., 2012), based on the work ofValizadeh et al. (2013), Shojaee et al. (2012). Shear correctionfactors are evaluated using the energy equivalence principle. Theplate kinematics are based on first order shear deformationtheory and they concluded that the natural frequencies of FGPdecrease as a result of increase in gradient index, through cracklength and temperature gradients. The supersonic flutterbehaviour of simply supported thin, cracked FGP submerged insupersonic flow is also studied by Natarajan et al. (2013)wherein, the crack is modelled independent of the underlyingmesh. They concluded that the critical frequency and criticalpressure are at a minimum when the crack is aligned to flowangle. A shear locking free, simple and efficient plate elementformulation for FGP is presented by Yin et al. (2014) to studybending, buckling and free vibration. Erdogan and Wu (1996,1997) presented mode I stress intensity factors for cracked FGP.They also considered the effect of thermal environment (Erdoganand Wu, 1996). Reddy (2000) presented deflection and stressanalysis of FGP based on third order shear deformation theorywith linear and nonlinear finite element models. Ganapathi et al.(2006) studied the critical buckling of simply supported func-tionally graded skew plates subjected to mechanical loads. Theirfinite element formulation is based on first order shear defor-mation theory and they presented results for critical bucklingload as affected by gradient index, aspect ratio and skew angle.The literature (Huang et al., 2011; Ismail and Cartmell, 2012;Khadem and Rezaee, 2000; Krawczuk et al., 2001; Israr et al.,2009; Natarajan et al., 2011b, 2013, 2014, 2012; Wu and Law,2004; Wu and Shih, 2005) shows that the natural frequenciesof isotropic plate depend on several factors such as crack length,plate geometry, crack orientation and crack location. Thus, it isimportant to analyse the effect of partial crack on the funda-mental frequencies of thin orthotropic plates. The present workaddresses this by proposing an analytical model.

The present work references the methods and analytical modeldeduced by Israr et al. (2009) for an isotropic plate containing asurface crack, extends and applies it to a thin orthotropic plate. Theline spring model is modified to accommodate the orthotropicnature of the plate. Results for fundamental frequencies are pre-sented considering the influence of boundary conditions and cracklength. The presence of a hole in a plate reduces both mass andstiffness, in the present work the crack is in the form of continuousline and is parallel to one of the edges of the plate. Such a line crackaffects only the stiffness of the plate and not its mass. The config-uration of the cracked plate for analysis is shown in Fig. 1. Fig. 1(a)shows a partial crack parallel to the x axis and Fig. 1(b) shows acrack parallel to the y axis. L1 and L2 are the plate dimensions alongx and y axes respectively. The plate of thickness h has a partial crackof length 2a along the fibres and is parallel to x axis, whereas its

Fig. 1. Orthotropic plate with a part through crack. (a) Crack parallel to x axis, (b) crack parallel to y axis.

P.V. Joshi et al. / European Journal of Mechanics A/Solids 50 (2015) 100e111102

length is 2bwhen it is across the fibres and is parallel to the y axis.Let “s” be the ratio of depth of the crack to the plate thickness. Acubic nonlinear system is obtained by expressing the in-planeforces by Berger's formulation. Crack terms are devised using thecrack compliance coefficients in the LSM. Three boundary condi-tions, (i) All edges simply supported (SSSS) (ii) Two adjacent edgesclamped and two free (CCFF) and (iii) Two adjacent edges clampedand two simply supported (CCSS) are considered to study the effectof the length of the crack on the vibration characteristics. Thenonlinear effect of bending hardening or softening in the frequencyresponse is established using the method of multiple scales. Theinfluence of elasticity ratio Ex/Ey on vibration characteristics is alsoillustrated.

2. Governing equation of cracked plate

The governing differential equation of motion of the orthotropicrectangular plate is derived considering two mutually perpendic-ular principle directions of orthotropy to coincide with x and y axes.The standard form of such an equation for an orthotropic intactplate using the equilibrium principle is treated by Szilard (2004).The equation for cracked plate is derived by the equilibrium prin-ciple, wherein a line surface crack of length 2a is parallel to x axis asshown in Fig. 1.

The assumptions involved are 1) The plate is thin and ortho-tropic requiring four elastic constants nx, ny, Ex, Ey, for stressestrainrelations. 2) The thickness h of the plate is small as compared to theother two dimensions of the plate. 3) The effect of rotary inertia,shear deformation and normal stress component in transverse di-rection are neglected. 4) Crack terms are formulated based on LSM.Applying these simplifying assumptions, the equation of motion ofan orthotropic intact plate considering simultaneous stretching andbending is

Dxv4wvx4

þ2Bv4w

vx2vy2þDy

v4wvy4

¼�rhv2wvt2

þNxv2wvx2

þ Nyv2wvy2

þ2Nxyv2wvxvy

þ Pz (1)

Where, Dx ¼ Exh3/12(1 � nxny) and Dy ¼ Eyh

3/12(1 � nxny) are theflexural rigidities, Pz is the transverse load, nx and ny are Poisson'sratios, the effective torsional rigidity is B ¼ ½(nyDx þ nxDy þ 4 Dt).For uniform thickness of the orthotropic plate, torsional rigidity isgiven by Dt ¼ 1=2ð1� ffiffiffiffiffiffiffiffiffi

nxnyp Þ ffiffiffiffiffiffiffiffiffiffiffi

DxDyp

. r is density and w is trans-verse deflection of the plate. Nx, Ny and Nxy are membrane or in-plane forces per unit length.

Employing a treatment of equilibrium similar to that employedin deriving Eq. (1), for the cracked orthotropic plate element oneobtains the equation of motion of cracked plate as

Dxv4wvx4

þ 2Bv4w

vx2vy2þ Dy

v4wvy4

¼ �rhv2wvt2

þ v2My

vy2þ Nx

v2wvx2

þ Nyv2wvy2

þ Pz (2)

Due to discontinuity across y axis, the stretching or in-planeforces in y direction Ny and Nxy are neglected (Israr et al., 2009) inEq. (2). The analytical model proposed by Ismail and Cartmell(2012) for an inclined surface crack shows that the in-plane shearforce Nxy ¼ Nyx appears only when the crack is inclined to the edgeof the plate and hence this is neglected in Eq. (2). Two new termsMy and Ny are introduced in the equilibrium principle in derivingEq. (2), which compensates for the moment and membrane forcedue to the crack of length 2a. Also the bendingmomentsMx,My andMxy ¼ Myx are expressed in terms of lateral deflection as

P.V. Joshi et al. / European Journal of Mechanics A/Solids 50 (2015) 100e111 103

Mx ¼ �Dx

v2wvx2

þ nyv2wvy2

!

My ¼ �Dy

v2wvy2

þ nxv2wvx2

!

Mxy ¼ �2Dt

v2wvxvy

!(3)

3. Crack terms My and Ny

The LSM approximates the crack problem by reducing the netligament stresses to the neutral plane of the plate as an unknownmembrane and bending load and assumes that the constraining ef-fect caused by the net ligament stresses may be accounted byapplying bending and membrane stress resultants on the crack sur-faces. The stress intensity factor along the crack front may beapproximated by the corresponding stress intensity factor for anedge crack strip under plane strain is another assumption of the LSM.Thus the crack terms can be represented by crack compliance co-efficients. Israr et al. (2009) converted the relation between stressesat the far side of the plate and stresses at the crack surface into forceeffects for an isotropic plate. To derive such relations for an ortho-tropicplate shown inFig.2, it is necessary toobtainnewrelationshipsbetweenNy&Ny andMy&My compensating the orthotropic nature ofthe plate. For the cracked plate model shown in Fig. 2, the boundaryconditions (assuming the origin at the centre of the plate) are

at y / ∞ sx¼ 0, and sy ¼ spqat y / ∞ Mx¼ 0, and My ¼mpq

Thus the plate element shown in Fig. 2 is subjected to normaltensile stress and bending moment. The bending and tensilestresses at the far edges of the plate are (Rice and Levy, 1972; Ismailand Cartmell, 2012; Israr et al., 2009)

spq ¼ Npq

h¼ 1

h

Zþh=2

�h=2

tpqðx; y; zÞdz (4)

mpq ¼ 6h2

Mpq ¼ 6h2

Zþh=2

�h=2

ztpqðx; y; zÞdz (5)

Fig. 2. An orthotropic plate with a crack of length 2awith tensile and bending stresses.

where tpqðx; y; zÞ is stress state, Npq and Mpq are force andbending moment per unit length in the direction perpendicularto crack length at the far edges of the plate respectively, p and qare intermediate variables. Applying the LSM proposed by Riceand Levy (1972), Israr et al. (2009) presented expressionsinvolving tensile and bending stresses at the crack location andat the far sides of the plate for an isotropic plate. Zeng and Dai(1994) proposed such expressions to investigate the stress in-tensity factors for an isotropic plate element with variably ori-ented crack under biaxial loading. The LSM represents the cracksection as a continuous line spring with its compliance matchedto that of the edge cracked strip under plane strain using twodimensional plane stress and plate bending, thus giving therelationship between spq; spq and mpq; mpq. Where spqand spq

are tensile stresses at far sides of the plate and at crack locationrespectively. Similarly mpq and mpq are the bending stresses.These relations for an orthotropic plate element can be proposedas

spq ¼ 2að6ctb þ cttÞ

�1� n2x

�hþ 2a

spq (6)

mpq ¼ 2a3ðctb6 þ cbbÞð3þ nxÞð 1� nxÞhþ 2a

mpq (7)

The literature (Rice and Levy, 1972; Ismail and Cartmell, 2012;Israr et al., 2009) shows that these tensile and bending stressescan be converted into tensile and bending force effects. Thus, for anorthotropic plate the new relations between tensile and bendingforce at crack location and at the far sides of the plate can bewrittenas

Ny ¼ 2að6ctb þ cttÞ

�1� n2x

�hþ 2a

Ny (8)

My ¼ 2a3ðctb6 þ cbbÞð3þ nxÞð 1� nxÞhþ 2a

My (9)

cbb, ctb ¼ cbt and ctt are bending, stretchingebending and tensilecompliance coefficients at the centre of the crack. Eqs. (8) and (9)are for a crack parallel to x axis, for a crack parallel to y axis theserelations become

Nx ¼ 2b

ð6ctb þ cttÞ�1� n2y

�hþ 2b

Nx (10)

Mx ¼ 2b

3�ctb6 þ cbb

��3þ ny

��1� ny

�hþ 2b

Mx (11)

The crack compliance coefficients in the crack terms are func-tions of crack depth, thickness of the plate and can be found bysuitable curve fitting the shape functions defining the stress in-tensity factor as deduced in the LSM. Such treatment for compli-ance coefficients is presented in the work of Aksel and Erdogan(1985). They found the stress intensity factors in case of internaland multiple cracks and established polynomials Bi and Ci for crackcompliance coefficients. The crack compliance coefficients can befound as

cij ¼1h

Zx0

gigjdx

P.V. Joshi et al. / European Journal of Mechanics A/Solids 50 (2015) 100e111104

i,j ¼ b,t can be used to find tensile, bending and tensile-bendingcoefficients for depth of crack x. gt(s) and gb(s) are shape func-tions for tension and bending respectively

gtðsÞ ¼ffiffiffiffiffiffips

p Xni¼1

Bis2ði�1Þ

gbðsÞ ¼ffiffiffiffiffiffips

p Xni¼1

Cisði�1Þ (12)

where, s is the ratio of depth of crack to plate thickness.In this work these coefficients, originally deduced for finding

stress intensity factors are used to evaluate the constraining effectof the net ligament, which reduce the stiffness of the plate. Thedamage in the form of crack causes a reduction in the stiffness ofthe plate, therefore Eqs. (8) and (9) are used in Eq. (2) with anegative sign, a phenomenon found in literature (Ismail andCartmell, 2012; Khadem and Rezaee, 2000; Israr et al., 2009). Us-ing Eqs. (2), (8) and (9), the equation of motion of cracked platebecomes

Dxv4wvx4

þ 2Bv4w

vx2vy2þ Dy

v4wvy4

¼

� rhv2wvt2

þ2a Dy

v4wvy4 þ nx

v4wvx2vy2

!

3�cbt6 þ cbb

�ð 3þ nxÞð1� nxÞhþ 2a

þ Nxv2wvx2

� 2að6cbt þ cttÞ

�1� n2x

�hþ 2a

Nyv2wvy2

þ Pz

(13)

4. Solution of equation of motion

The general solution for transverse deflection by Galerkin'smethod is

wðx; y; tÞ ¼X∞n¼1

X∞m¼1

AmnXmYn∅mnðtÞ (14)

Where Xm and Yn are the characteristic modal functions treatedby different researchers satisfying the boundary conditions. Amn

is arbitrary amplitude and ∅mn(t) is time dependent modal co-ordinate. By neglecting the strain energy due to second invariantof mid-surface strains, Berger (1955) showed that the form of in-plane forces obtained works well for combinations of simplysupported and clamped boundary conditions. Many researchers(Ismail and Cartmell, 2012; Israr et al., 2009) have appliedBerger's formulation for nonlinear vibrations of rectangularplate. This can be extended to orthotropic plate. Thus expressingthe membrane forces in terms of mid-surface strains, oneobtains

Ny ¼ Eyh�1� nxny

��εy þ nxεx�

Nx ¼ Exh�1� nxny

��εx þ nyεy�

Applying the definition of w(x, y, t), integrating over plate areaand multiplying both sides by XmYn

phX∞ X∞

Amn

ZL1 ZL2X2mY

2ndxdy

v2∅mnðtÞ2 þ

X∞

n¼1 m¼1 0 0

vt n¼1

�X∞m¼1

Amn∅mnðtÞZL10

ZL20

8>><>>:Dx

�XivmYn

�þ 2B

�XiimY

iin

�

þ Dy

�Yivn Xm

��

2aDy

�vxXii

mYiin þ Yiv

n Xm

�3�

cbt6 þ cbb

�ð3þ xð1� vxÞhþ 2aÞ

9>>=>>;XmYndxdy

þX∞n¼1

X∞m¼1

Amn3∅mnðtÞ3

ZL10

ZL20

(� DxB1mmXmY2

nXiim

þ 2aDyB2mnYiinX

2m

ð6cbt þ cttÞ�1� v2x

�hþ 2a

)dxdy

¼ Pz(15)

where,

B1mn ¼ 6h2L1L2

X∞n¼1

X∞m¼1

Z L1

0

Z L2

0

(�vXm

vx

�2

Yn2

þ ny

�vYnvy

�2

Xm2

)dxdy

B2mn ¼ 6h2L1L2

X∞n¼1

X∞m¼1

Z L1

0

Z L2

0

(�vYnvy

�2

Xm2

þ nx

�vXm

vx

�2

Yn2)dxdy

In Eq. (15), the middle surface strains εx and εy are expressed interms of lateral deflection and in-plane forces are integrated overplate area. The modal peak amplitude Amn is normalized to unity.The lateral load Pz is neglected here for free vibrations. Eq. (15) maybe expressed in the form of Duffing equation as

Mmn

v2∅mn

�t�

vt2þ Kmn∅mnðtÞ þ Gmn∅mnðtÞ3 ¼ 0 (16)

where,

Mmn ¼ rhX∞n¼1

X∞m¼1

Amn

ZL10

ZL20

Xm2Yn2dxdy (17)

Kmn¼X∞ X∞

Amn∅mnðtÞZL1ZL2n

Dx

�XivYn

�þ2B

�Xii Yii

�

n¼1m¼1 0 0

m m n

þDy

�Yivn Xm

��

2aDy

�nxXii

mYiin þYiv

n Xm

�3�cbt6 þcbb

�ð3þnxÞð1�nxÞhþ2a

oXm Yndxdy

(18)

Gmn ¼X∞ X∞

Amn3∅mnðtÞ3

ZL1 ZL2n� DxB1mnXm Yn2Xii

m

n¼1 m¼1 0 0

þ 2aDyB2mnYiinYnX

2m

ð6cbt þ cttÞ�1� n2x

�hþ 2a

odxdy (19)

From Eq. (16), the natural frequency u can be evaluated asu2mn ¼ Kmn=Mmn.

Table 1Fundamental frequency parameter for an intact and cracked orthotropic plate.

Boundary conditions Frequency parameter, F

Intact plate Cracked plate (a/L1 ¼ 0.01)

CCSS Sakata et al. (1996) 17.18 e e

Present 17.18 e e

SSSS Szilard (2004) 10.99 ANSYS 10.91Present 10.99 Present 10.92

P.V. Joshi et al. / European Journal of Mechanics A/Solids 50 (2015) 100e111 105

5. Frequency response

The method of multiple scales (Murdock, 1999) applied to studythe frequency response of an isotropic plate with a surface crack inIsrar et al. (2009) is extended here in the case of the crackedorthotropic plate. Eq. (16) is rearranged by considering the appro-priate delta function for the external load as given in the work ofFan (2001).

€∅mnðtÞ þ u2mn∅mnðtÞ þ dmn∅mnðtÞ3 ¼ lmnp0

�t�

(20)

Where dmn¼Gmn/Mmn is the nonlinear term and the loadPz ¼ lmnp0(t). The term lmn¼ Xm(x0)Yn(y0 )/Mmn can be obtained forthe location of Pz at (x0, y0).

With the idealization of the nonlinear damping toweak classicallinear viscous damping and the external load to be harmonic,p0(t) ¼ p0cosqmn(t) the equation for cracked plate becomes

€∅mn tð Þ þ 2m∅:

mn tð Þ þ u2mn:mn tð Þ þ dmn:mn tð Þ3

¼ lmnp0 cos qmn tð Þ (21)

Literature (Ismail and Cartmell, 2012; Israr et al., 2009;Murdock, 1999) shows the use of detuning parameter, emn whichdescribes the closeness of excitation frequency qmn, to umn and canbe written as qmn ¼ umn þ gemn, where g is perturbation parameter.Restricting the solution to first order perturbation as higher orderperturbations does not contribute much to the solution. Perturbingthe damping, cubic nonlinearity, external excitation and expanding∅mn(t) and its derivatives as

∅mnðt;gÞ ¼ ∅0mnðT0; T1Þ þ g∅1mnðT0; T1Þ (22)

ddt

¼ v

vT0þ g

v

vT1

d2

dt2¼ v

vT0þ 2g

v

vT0

v

vT1(23)

where T0¼ t is fast time and T1 ¼ gt is slow time. Eqs. (22) & (23)can be applied to Eq. (21) and separating the like terms of g0 andg1, one obtains,

v2∅0mn

vT02 þ u2

mn∅0mn ¼ 0 (24)

v2∅1mn

vT02 þ u2

mn∅1mn ¼ �2v

vT0

�v∅0mn

vT1

�� 2m

v∅0mn

vT0� dmn∅3

0mn

þ lmnp0 cos

umnT0 þ emnT1

!

(25)

Eqs. (24) and (25) can be solved by assuming a general solutioninvolving unknown complex amplitude and its complex conjugate.Expressing the amplitude in polar form, frequency responseequation in case of steady state motion is obtained which can bestated as

emn ¼ 3 dmn J28umn

±

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� l2mn

4 u2mnJ2

p02 � m2�s

(26)

Eq. (26) is a frequency response relation for geometricallynonlinear cracked orthotropic plate and flexural rigidity is not

present in the second term on the right hand side of this equationwhich makes this relation different from its isotropic counterpartobtained by Israr et al. (2009). J is modal amplitude of response as afunction of detuning parameter emn and initial excitation amplitudep0. It can also be shown that the peak amplitude (Jp) is independentof the cubic nonlinearity and is given as

Jp ¼ lmn

2umnmp0 (27)

Where, p0 is the amplitude of excitation and m is the dampingfactor. In Eq. (27) flexural rigidity is not present, which makes itdifferent from its isotropic counterpart derived in Israr et al. (2009).

6. Results and discussion

This section presents new results for fundamental frequencies ofa cracked orthotropic thin plate. The effect of crack length on thesignificant first mode natural frequency is examined. The Ist modenatural frequency is expressed as non dimensional frequencyparameter, F ¼ uL21

ffiffiffiffiffiffiffiffiffiffiffiffiffirh=Dx

p. Three boundary conditions and two

plate aspect ratios (L1/L2 ¼ 1, 2) are considered for analysis.Although the crack compliance coefficients are available in theliterature on stress intensity factors (Joseph and Erdogan, 1989) forcrack depth of 0he0.95h, Literature (Rice and Levy,1972; Ismail andCartmell, 2012; Israr et al., 2009; Joshi et al., 2014) shows that theapplication of the crack terms based on LSM when the plate is thinand hence in this work, the plate thickness and crack depth aretaken as 10 mm and 6 mm respectively. Thus, the ratio of crackdepth to plate thickness (“s”) is 0.6. The crack lengths parallel to xand y axes are 2a and 2b respectively. Results for various half cracklengths (“a” and “b”) are presented in Tables 3e8. A point load of10 N is considered to be acting at a point (0.375, 0.75) for frequencyresponse. The material for the orthotropic plate is taken as BoronEpoxy with properties, Ex ¼ 208 � 109 Pa, Ey ¼ 18.9 � 109 Pa,nx ¼ 0.23, ny ¼ 0.0208, Gxy ¼ 5.7 � 109 Pa, r ¼ 2000 kg/m3, m ¼ 0.08.Table 1 shows the non-dimensional frequency parameter for anintact and cracked orthotropic plate of aspect ratio L1/L2 ¼ 1 andvalidates the present results for intact plate with known results.The results obtained from the present model for cracked plate arecompared with the numerical results obtained using ANSYS for aplate of aspect ratio 1 and Boron Epoxy material.

A plate of thickness 10 mm is modelled as a 3D quarter plate,with given geometric and mechanical properties. Partial line crackof depth 6 mm is introduced in the plate model using suitablesymmetric boundary conditions. Partial crack is difficult to modelusing 2D elements and hence the plate is modelled as a 3D prob-lem. 8 node Brick element and Block Lanczos solver are employedfor modal analysis. Number of checks and convergence tests arecarried out to select the proper element type and the element edgelength after the convergence test is taken as 0.002 m. The FEMresults show good agreement.

To the best of the author's knowledge, the literature lacks inresults for partially cracked orthotropic/functionally graded thin

Table 2First mode frequency parameter l for cracked FGP, crack depth ¼ 0.95h.

Frequency parameter�l ¼ u L21

h

ffiffiffiffircEc

q �

Absence of thermal environment TC ¼ TM ¼ 300 K

Half cracklength (a)

Natarajan et al.(2011a) (throughcrack)

Present (partialcrack withs ¼ 0.95)

Natarajan et al.(2011b) (throughcrack)

Present(partial crackwith s ¼ 0.95)

0 3.3376 (3.3875a) 3.4597 3.3279 (3.3279b) 3.39680.01 e 3.3255 3.2296 3.31350.02 3.0452 3.1325 3.0442 3.12850.03 2.8657 2.9512 2.8644 2.9383

a Natarajan et al. (2012).b Natarajan et al. (2014).

P.V. Joshi et al. / European Journal of Mechanics A/Solids 50 (2015) 100e111106

plates. Results for functionally graded plates with a through crackare available in Natarajan et al. (2011a, 2011b, 2014, 2012). Thepresent model for partial crack is based on LSM involving the crackcompliance coefficients which represent the net ligament (intactportion) stresses. The compliance coefficients can be obtained fromliterature on stress intensity factors (Rice and Levy,1972; Ismail andCartmell, 2012; Israr et al., 2009; Aksel and Erdogan, April 1985;Joseph and Erdogan, 1989) for the ratio of crack depth to platethickness varying from 0 to 0.95 (crack depth of 0he0.95h). Hence,for the purpose of validation, the present model is applied togeneral functionally graded plate (FGP) with crack depth to platethickness ratio of 0.95 (crack depth of 0.95h) and the results arecompared with the results for through crack (Natarajan et al.,2011a, 2011b, 2014, 2012). The FGP material considered is siliconnitride and stainless steel (Si3N4/SUS304), the properties of whichare given in Natarajan et al. (2011a, 2011b, 2014, 2012).

The present model can also be extended to incorporate envi-ronment effects such as rise in temperature of FGP. Neglecting thein plane forces due to lateral deflection considered in recent liter-ature (Israr et al., 2009; Joshi et al., 2014) and considering the inplane forces due to uniform rise in temperature of the plate, onecan write the equation of motion as

D�V4w

�¼ �rh

v2wvt2

þ Pz þ v2My

vy2� v2MT

vx2� v2MT

vy2� NTx

v2wvx2

� NTyv2wvy2

� NTyv2wvy2

(28)

Where D is the effective flexural rigidity, MT is the resultantmoment due to temperature gradient, NTx and NTy are in planeforces due to uniform rise in temperature when the in-plane de-flections of the plate are restrained, which can be expressed as(Jones, 2006; Birman, 2011) NTx ¼ NTy¼ EahTc/1� n. Where E, a andn are the effective properties of FGP. The crack terms based on LSM(Eqs. (8) and (9)) can be applied to in plane force NTy. Hence, thestiffness of a cracked FGP subjected to uniform thermal environ-ment becomes (thermalmomentMT is neglected for uniform rise intemperature of the plate)

Kmn ¼ DAmn

ZL10

ZL20

�XivmYn þ 2Xii

mYiin þ Yiv

n Xm

�XmYndxdyþ EahTc

1� nAmn

ZL10

ZL20

(XiimY2

nXm þ YiinX

2mYn

þ 2aYiinX

2mYn

ð6ctb þ cttÞ�1� n2

�hþ 2a

)dxdy� 2a

3�ctb6 þ cbb

�ð3þ nÞð1� nÞhþ 2a

DAmn

ZL10

ZL20

�Yivn Xm þ nXii

mYiin

�XmYndxdy (29)

Table 2 shows the first mode non dimensional frequencyparameter (l) for SSSS FGP, for an aspect ratio of L1/L2 ¼ 1, L1/h ¼ 10and gradient index n ¼ 1. Results are given for two cases, (1)Presence of thermal environment (Ceramic andmetal temperature,TC ¼ TM ¼ 300 K) (2) Absence of the thermal environment. Thepresent results are slightly higher than published results because oftwo reasons, (1) the present results are for a partial crackwith crackdepth equal to 0.95h whereas results from Natarajan et al. (2011a,2011b, 2014, 2012) are for a through crack. (2) The present modelis based on classical plate theory, whereas Natarajan et al. (2011a,2011b, 2014, 2012) are based on finite element formulations.

Considering these two facts, the present results are in goodagreement with the available results.

The present analytical approach has obvious advantages of be-ing simple, very fast, ease of parametric study, improving physicalunderstanding of the problem and efficient computation timewhen compared to finite element models. The variation of firstmode natural frequency parameter for the three boundary condi-tions, two plate aspect ratios (L1/L2 ¼ 1, 2) and various half cracklengths ‘a’ is shown in Table 3. Whereas, Table 4 shows the same forvarious half crack lengths ‘b’. It is seen that the presence of a crackdecreases the natural frequencies. For all plate aspect ratios thenatural frequencies go on decreasing as the crack length increases.Comparing the decrease in frequencies in Tables 3 and 4, it can bededuced that the presence of crack perpendicular to the fibres in anorthotropic plate affects the vibration characteristics more than thecrack along the fibres which satisfies one's intuition. As the naturalfrequencies are influenced by the geometry of the plate; this isequally true for the cracked plate also. Thus, it is concluded that thenatural frequencies are affected by crack length. Although thereduction in natural frequencies is different for CCFF and CCSS, it isseen that the findings for SSSS plate are principally true for CCFFand CCSS boundary condition.

The effect of varying elasticity ratio Ex/Ey on the natural fre-quencies is studied for various crack lengths ‘a’ and ‘b’. The values

Ex ¼ 208 � 109 Pa, nx ¼ 0.23 are kept constant and Ey and ny arevaried so that the elasticity ratio varies from 1 to 100. The naturalfrequencies in Table 5e7 are expressed in terms of non-dimensional frequency parameter ‘F’. Table 5 shows the effect ofelasticity ratio on the natural frequencies for an intact plate ofaspect ratio L1/L2 ¼ 1. The results for the three boundary conditionsare in agreement with known results. Table 6 shows the variation offundamental frequency parameter with elasticity ratio for varioushalf crack lengths ‘a’ for a plate of aspect ratio L1/L2 ¼ 1. Whereas,this variation for half crack length ‘b’ is shown in Table 7. For all halfcrack lengths, there is a decrease in fundamental frequency with

Table 3Frequency parameter ‘F’ for the three boundary conditions with various half cracklength ‘a’ (s ¼ 0.6).

a/L1 Boundary condition

SSSS CCFF CCSS

L1/L2 L1/L2 L1/L2

1 2 1 2 1 2

0 10.995 17.253 3.687 5.547 17.179 26.5330.01 10.916 16.541 3.662 5.279 17.058 25.3710.05 10.751 14.982 3.610 4.685 16.807 22.8120.1 10.665 14.118 3.584 4.350 16.678 21.3870.2 10.596 13.378 3.562 4.059 16.572 20.1570.3 10.566 13.045 3.553 3.926 16.526 19.6020.4 10.549 12.855 3.547 3.850 16.500 19.2850.5 10.538 12.733 3.544 3.801 16.484 19.080

Table 4Frequency parameter ‘F’ for the three boundary conditions with various half cracklength ‘b’ (s ¼ 0.6).

b/L2 Boundary condition

SSSS CCFF CCSS

L1/L2 L1/L2 L1/L2

1 2 1 2 1 2

0 10.995 17.253 3.687 5.547 17.179 26.5330.01 10.350 17.023 3.446 5.462 16.127 26.1620.05 8.751 16.352 2.842 5.216 13.508 25.0750.1 7.720 15.831 2.441 5.023 11.802 24.2310.2 6.713 15.251 2.037 4.807 10.116 23.2880.3 6.209 14.936 1.828 4.688 9.263 22.7730.4 5.905 14.737 1.698 4.614 8.742 22.4490.5 5.700 14.600 1.608 4.563 8.389 22.227

Table 6Variation of frequency parameter ‘F’ with Ex/Ey for various ‘a’ (s ¼ 0.6).

a/L1 Ex/Ey F

SSSS CCFF CCSS

0.01 1 19.24 4.91 27.535 14.14 3.88 20.65

10 12.91 3.72 19.1350 11.24 3.57 17.22

100 10.84 3.55 16.790.05 1 18.19 4.47 25.77

5 13.86 3.77 20.1910 12.76 3.66 18.8950 11.21 3.56 17.17

100 10.82 3.55 16.760.1 1 17.63 4.24 24.83

5 13.72 3.72 19.9610 12.68 3.64 18.7750 11.19 3.55 17.14

100 10.82 3.54 16.75

Table 7Variation of frequency parameter with Ex/Ey for various ‘b’ (s ¼ 0.6).

b/L2 Ex/Ey l

SSSS CCFF CCSS

0.01 1 19.24 4.91 27.535 13.77 3.70 19.98

10 12.44 3.51 18.3250 10.65 3.33 16.22

100 10.22 3.30 15.750.05 1 18.19 4.47 25.77

5 12.57 3.14 17.8910 11.14 2.92 16.0550 9.13 2.70 13.66

100 8.63 2.67 13.100.1 1 17.63 4.24 24.83

5 11.86 2.79 16.6210 10.36 2.53 14.6550 8.17 2.28 11.99

100 7.61 2.24 11.36

P.V. Joshi et al. / European Journal of Mechanics A/Solids 50 (2015) 100e111 107

increase in the elasticity ratio for the three boundary conditions.This decrease is naturally more for crack lengths ‘b’ as compared to‘a’.

This decrease in fundamental frequencies is due to decrease inEy and constant Ex in varying elasticity ratio. Thus, the effect ofelasticity ratio is different for crack along the two mutuallyperpendicular directions. It can also be deduced that the effect ofelasticity ratio increase is more on the CCSS plate as compared tothe other two boundary conditions. The CCFF plate being lessaffected. Thus, it is concluded that the effect of increase in elasticityratio is more for cracks along the direction of low modulus ofelasticity. Employing the method of multiple scales, the peakamplitude of the cracked vibrating plate is examined. Table 8 showsthe peak amplitude for a plate of aspect ratio L1/L2 ¼ 1 and forvarious crack lengths ‘a’ or ‘b’. For all the three boundary conditions,insertion of crack increases the peak amplitude and this increase issubstantial as the crack length increases.

Table 5Variation of frequency parameter, ‘F’ for intact plate.

Edge conditions Ex/Ey

1 5 10 50 100

SSSS 19.7392(19.7391){19.9527}

14.2834(14.2830){14.4623}

12.9907(12.9906){12.9349}

11.2648(11.2642){11.2108}

10.8566(10.8564){10.8004}

CCFF 5.1184{5.2888}

3.9362{3.9634}

3.7503{3.7780}

3.5802[3.6003]

3.5543{3.5865}

CCSS 28.3483{28.5645}

20.8726{20.9877}

19.2570{19.3589}

17.2520{17.3764}

16.8092{16.9813}

( ) Szilard (2004), { } ANSYS.

The crack terms in the present model are based on LSM whichassumes the crack to be a continuous line crack. Thus, it is difficultto apply this model to a plate with cutouts/holes/kinks. It can beextended to a realistic case of multiple cracks (two or more con-current line cracks at the centre) by following the superposition ofcrack terms as deduced in the recent work of Joshi et al. (2014).

Using Eq. (26), geometrically linear (dmn¼ 0) and geometricallynonlinear (dmn< 0, soft spring and dmn> 0, hard spring) responsecurves are plotted for a square plate of side 1 m and half cracklength of a ¼ 0.01 m for a given damping m and excitation. Theselinear and nonlinear response curves for the three edge conditionsare shown in Fig. 3. The response curves for the same square platewith half crack length b ¼ 0.01 m are shown in Fig. 4. The linearcurves are symmetric around the resonant frequency. The effect ofthe cubic nonlinear term dmn, is to bend the curve towards the rightor left depending on whether it is hard or soft spring This bendinghardening nonlinearity for a ¼ 0.01 m, bends the frequencyresponse curve towards the right for CCSS and SSSS boundary

Table 8Peak amplitude (mm) for various half crack lengths (s ¼ 0.6).

Crack length a/L1 ¼ 0 a/L1 ¼ 0.01 a/L1 ¼ 0.05 b/L2 ¼ 0 b/L2 ¼ 0.01 b/L2 ¼ 0.05

SSSS 25.16 25.34 25.73 25.16 26.73 31.61CCFF 15.47 15.58 15.80 15.47 16.55 20.08CCSS 33.06 33.30 33.79 33.06 35.22 42.05

P.V. Joshi et al. / European Journal of Mechanics A/Solids 50 (2015) 100e111108

conditions and towards left for CCFF condition. Similar to isotropicplate (Ismail and Cartmell, 2012; Israr et al., 2009), Fig. 3(b) showsthat for orthotropic plate also the hardening is more for SSSS platethan CCSS. For a crack across the fibres (b ¼ 0.01 m), it is interestingto note that the SSSS and CCSS plate behaves like a soft spring,which satisfies one's physical understanding, whereas the CCFFplate behaves like a hard spring. The transformation of SSSS platefrom bending hardening (for a ¼ 0.01 m, Fig. 3(b)) to bendingsoftening (for b¼ 0.01 m, Fig. 4(b)) shows that there might be someinclination of the partial crack for which the softening and

Fig. 3. Linear and non linear response curves for

hardening characteristics balance each other. Comparing the linearresponse curves from Figs. 3 and 4 which are approximately same,the significance of the nonlinear term dmn is established.

7. Extension of the model to incorporate couple stress theory

The behaviour of micro-plates has been proven to be sizedependent and hence implementation of strain gradient theories(containing internal material length scale parameters) is essentialdue to inadequacy of the classical continuum theory. The higher

the three boundary conditions, a ¼ 0.01 m.

Fig. 4. Linear and non linear response curves for the three boundary conditions, b ¼ 0.01 m.

P.V. Joshi et al. / European Journal of Mechanics A/Solids 50 (2015) 100e111 109

order theories in their simple form contain two additional con-stants (Lame constants, Lam et al., 2003). Many researchers(Lazopoulos, 2004; Papargyri-Beskou and Beskos, 2008) have useda simplified strain gradient theory containing only one strain co-efficient of length squared dimension. Yang et al. (2002) proposed amodified couple stress theory in which only one material lengthparameter captures the size effect. Papargyri-Beskou and Beskos(2008) presented sixth order partial differential equation of mo-tion for Kirchhoff plates, including the strain gradient coefficientand constant in plane forces.

Tsiatas (2009) developed a new Kirchhoff plate model based ona simplified couple stress theory proposed by Yang et al. (2002).They presented a static analysis of micro plates of complex geom-etries and boundary conditions using only one internal materiallength scale parameter to capture the size effect. Yin et al. (2010)implemented the non-classical Kirchhoff plate model developedby Tsiatas (2009), for dynamic analysis of micro-plates usingmodified couple stress theory. They presented results for first twonatural frequencies of intact isotropic plates and concluded that theinternal material length scale parameter increases bending rigidity

P.V. Joshi et al. / European Journal of Mechanics A/Solids 50 (2015) 100e111110

thereby giving higher values of frequency as compared to classicalresults. They also showed that when the material length scaleparameter is twice the plate thickness, then the frequency increasesby a factor of 4 when compared to classical results thus capturingthe strong dependence of size effect and Poisson's ratio on very thinplates. The present model of cracked plate can also be extended toincorporate the single internal material length scale parameterproposed by Yin et al. (2010), thus making it the present modelbased onmodified couple stress theory. The equation of motion of acracked FGP capturing the effect of a material length parameter canbe written as Dþ Dl

!(v4wvx4

þ v4wvx2vy2

þ v4wvy4

)

¼ �rhv2wvt2

þ v2My

vy2þ Nx

v2wvx2

þ Nyv2wvy2

þ Pz (30)

Where Dl¼ El2h/2(1þ n), l is the material length scale parameter. Dl

is the contribution of the rotation gradients to bending rigidity.Eq. (30) can be further solved by substituting the crack terns

using LSM and it can be shown that the frequency values, thusobtained are slightly higher than the classical values. It would thusbe easy to carry out a parametric study to capture the effect ofmaterial length size parameter on the fundamental frequencies ofcracked FGP.

8. Conclusion

An analytical model is proposed for a partially cracked thinorthotropic plate. The effect of length of crack on the first modefrequencies for three boundary conditions is deduced. The effect ofgeometric nonlinearity in the frequency response is also estab-lished using the method of multiple scales. The bending hardeningphenomenon is more pronounced for a simply supported plate ascompared to other two boundary conditions for a given plate aspectratio. It is observed that the presence of crack decreases the fre-quencies and increases peak amplitude. Also the presence of crackacross the fibres decreases the frequency more as compared to acrack along the fibres. It is verified that the geometry of the plateaffects the natural frequencies of cracked orthotropic plate similarto isotropic plate. The effect of an increase in the elasticity ratio onthe fundamental frequencies is also established. It would beinteresting to formulate the response of this model using someshear deformation theory and including the internal materiallength size parameter.

Acknowledgements

The authors are thankful for the constructive comments fromthe reviewers. This research work is not funded by anyorganization.

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