Three-dimensional vibrations of cracked rectangular parallelepipeds of functionally graded material

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Three-dimensional vibrations of cracked rectangular parallelepipeds

of functionally graded material

C.S. Huang

a

, O.G. McGee III

b,n

, K.P. Wang

a a

Department of Civil Engineering, National Chiao Tung University, 1001 Ta-Hsueh Rd., Hsinchu, Taiwan 30050, Republic of China (R.O.C.) b_{Department of Mechanical Engineering Howard University Washington DC, U.S.A.}

a r t i c l e

i n f o

Article history:

Received 12 July 2011 Received in revised form 11 May 2012

Accepted 15 May 2012 Available online 26 June 2012 Keywords:

Vibrations

Cracked rectangular parallelepiped Ritz method

3-D elasticity

Functionally graded material (FGM)

a b s t r a c t

A novel examination of the three-dimensional (3-D) vibrations of rectangular parallelepipeds of functionally graded material (FGM) having side cracks is summarized. Employing 3-D theory of elasticity and a variational Ritz methodology, new hybrid series of mathematically complete orthogonal polynomials and crack functions as the assumed displacement fields are proposed to enhance the convergence modeling of the stress singular behavior of the crack terminus edge front in a rectangular FGM parallelepiped. The proposed admissible hybrid series properly describe theWð1=pffiffiffirÞ3-D stress singularities at the terminus edge front of the crack, allowing for displacement discontinuities across the crack sufficient to explain the most general 3-D ‘‘mixed modes’’ of local crack-edge deformation and stress fields typically seen in fracture mechanics. The correctness and validity of the vibration analysis are confirmed through comprehensive convergence studies and comparisons with published results for cracked rectangular FGM parallelepipeds modeled as homogeneous rectangular plates with side cracks and FGM rectangular plates with no cracks based on various plate theories. Two types of FGM parallelepipeds, Al/Al2O3and Al/ZrO2, are included in the study. The locally effective material properties are estimated by a simple power law and the effects of the volume fraction on the frequencies are investigated. For the first time in the published literature, this work reports frequency data and nodal patterns for FGM rectangular parallelepipeds modeled as moderately thick plates with several combinations of hinged, clamped, and completely free kinematic and stress conditions along the four side faces, and having side cracks with varying crack size effects implying flaw-size influence in FGM parallelepiped vibration and fracture, including crack length ratios (d/a and d/b), crack positions (cx/a and cy/b), and crack inclination angles (a).

1. Introduction

Laminated composite materials are prevalent in engineering systems, particularly in aeronautical vehicles and aerospace structures. However, the abrupt change in material properties across the interface between material layers can cause large inter-laminar stresses even some de-laminations. Functionally graded materials (FGMs)[1]are found to overcome these adverse interlamination stress and delamination effects associated with conventional laminated composite builds. Material properties of FGMs vary continuously by gradually changing the volume fraction of constituent material properties. FGMs have been extensively explored in the last two decades along a variety of interdisciplinary

fronts, including electronics, chemistry, optics, biomedicine, aeronautical and mechanical engineering.

Rectangular parallelepipeds modeled as plates are employed in a wide range of mechanical and structural system components in civil, mechanical and aeronautical engineering. Such rectangular parallelepipeds are oftentimes subjected to irregular loads generated by waves or subjected to cyclic loads induced by machinery. Consequently, fatigue cracks may be initiated in the parallelepiped components. Vibrations of fractured FGM parallelepipeds require dynamic stress analysis of their sensitivity to ﬂaws or crack-liked defects with high local stresses progressing through crack propagation mechanisms lengthening even inclining fatigue cracking.

Redistribution of stresses in cracked FGM parallelepipeds causes dynamic characteristics markedly different from those for an intact parallelepiped, requiring linear-elastic stress analysis, including elevated stresses local to the crack terminus edge. Here, the free surfaces of the crack dynamically moving relative to each other signiﬁcantly inﬂuence the distribution of Contents lists available atSciVerse ScienceDirect

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

International Journal of Mechanical Sciences

n

Corresponding author. Tel.: þ1 614 404 5427.

E-mail addresses: [email protected] (C.S. Huang), [email protected] (O.G. McGee III).

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stresses local to the crack terminus edge. Various combinations of clamped, free, and hinged boundary conditions of a cracked parallelepiped affect the intensity of the local dynamic stress field at the crack terminus edge. The dynamic stress field near the crack terminus edge within a FGM parallelepiped may be classi-fied as three basic dynamic responses each associated with a local mode of crack deformation. First, a crack opening mode in FGM parallelepiped vibration is associated with local displacement in which the crack surfaces move directly apart; second, a crack shearing or edge-sliding mode in FGM parallelepiped vibration is characterized by deformations in which the crack surfaces slide over one another perpendicular to the leading terminus edge of the crack; third, a crack tearing mode in FGM parallelepiped vibration finds the crack surfaces sliding with respect to one another parallel to the leading terminus edge of the crack. Crack opening and crack shearing or edge-sliding in FGM parallelepiped vibration can be modeled as two-dimensional plane-extension theory of elasticity, classified as symmetric (crack opening) and skew-symmetry (crack shearing/sliding) with respect to the leading edge of the crack. Crack tearing in FGM parallelepiped vibration may be modeled as two-dimensional pure shear (or torsion). Well-known superposition of crack opening, shearing or sliding, and tearing modes or ‘‘mixed mode’’ cracking in FGM parallelepiped vibration is sufficient to describe the most general three-dimensional dynamic aspects of local crack-edge deforma-tion and stress fields in cracked parallelepipeds. Such local crack-edge deformation and stress fields involve trigonometric distribu-tions in a circumferential coordinate (

y

) (x¼ cos

y

, y¼sin

y

, see

Fig. 1) local to the crack terminus edge. The crack-edge stress fields are dominated by the order of an inverse square root of a local polar coordinate ðr ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2_þy2_{, see Fig: 1Þ,}

_W

_ð1=pffiffiffi_r_Þ, emanat-ing from the crack terminus edge, whereas the crack-edge deformations are dominated by the order of a square root of the local polar coordinate, r,

W

ð1=pffiffiffirÞ.

Hence, a need arises to establish these linear-elastic crack-edge stress (and displacement) ﬁelds in the dynamic character-istics of cracked parallelepipeds, speciﬁcally modeled herein as cracked thick plates to provide reference three-dimensional (3-D)

solutions that delineate validity and accuracy of a growing database of cracked plates in the literature recently offered by Huang and co-workers using classical Kirchhoff thin-plate theory

[2,3], proposed by Li [4] using ﬁrst-order shear deformable Mindlin plate theory, and reported by Huang and Chang using

[5]higher third-order shear deformable plate theory.

Lynn and Kumbasar [6], Stahl and Keer [7], Aggarwala and Ariel[8], Neku [9], Solecki[10], Hirano and Okazaki [11], Qian et al.[12], Krawczuk[13], Yuan and Dickinson [14], Liew et al.

[15], and Huang and Leissa[2,3] proposed various approximate solution techniques using classical Kirchhoff thin-plate theory to study the vibrations of cracked rectangular parallelepipeds, mod-eled as homogeneous thin plates. Lee and Lim[16]employed a simpliﬁed Reissner theory and the Ritz method along with a subdomain technique to examine the vibrations of cracked rectangular parallelepipeds modeled as thick plates including transverse shear deformation and rotary inertia. Maruyama and Ichinomiya [17], Ma and Hsieh [18], and Ma and Huang [19]

established an experimental bench test database of cracked thin homogeneous plates.

Documented in the published literature are substantial ﬁnd-ings on vibrations of parallelepipeds modeled as thick homoge-neous plates and FGM plates with no cracks. Yang and Shen[20]

and He et al.[21]used the classical plate theory, while Zhao et al.

[22]and Reddy[23]adopted the ﬁrst-order and the third-order shear deformation plate theories, to investigate vibrations of FGM plates with no crack. Matsunaga [24] and Qian et al. [25]

proposed solutions for vibrations and stability of FGM rectangular shear deformable plates incorporating higher-order transverse shear effects. Vel and Batra[26]proposed a three-dimensional exact vibration solution of simply-supported thick FGM plates and Reddy and Cheng[27]presented a three-dimensional asymp-totic solution for vibrations of simply-supported FGM plates.

However, there appears to be no published ﬁndings of accurate vibration frequencies and nodal patterns for arbitrarily-oriented and positioned cracked FGM parallelepipeds having various combinations hinged, clamped or simply-supported face condi-tions. In the above-mentioned literature, the solutions, except for

a a d b r c_y cx r b d 2 h 2 h

Fig. 1. A rectangular Functionally Graded Material (FGM) parallelepiped modeled as a thick plate having a side crack showing position coordinates (cxand cy), crack length (d), and crack orientation (a); (a) top view of a plate with a crack intersecting x¼ a; (b) top view of aplate with a crack intersecting y ¼b and (c) side view of plate.

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the finite element solutions, by no means considered the char-acteristic of the stress singularities at the crack terminus edge. The present work proposes a novel 3-D elasticity-based Ritz variational procedure to investigate the vibrations of side-cracked rectangular parallelepipeds modeled as thin and thick plates. The novelty of the analysis incorporates into the displacement fields of the dynamical energies, a hybrid series of mathematically complete admissible orthogonal polynomials [28] and newly-developed crack functions accounting for stress singularities at the front of the crack and allowing for displacement disconti-nuities across the crack (sufficient to describe the most general 3-D mixed modes of local crack-edge deformation and stress fields), altogether accurately predicting the vibrations of FGM parallelepipeds with side cracks.

Based on 3-D elasticity theory, Hartranft and Sih[29], Chaud-huri and Xie [30], Williams [31], and Erdogan and Sih [32]

established eigenfunction expansions showing stress singularities in statically-loaded cracked homogenous parallelepipeds. The most direct approach in determining the stress and displacement ﬁelds associated with each crack-edge mode draws upon funda-mentals of fracture put forth early-on by Irwin [33] using methodologies originally proposed by Westergaard[34]. Recently, the free vibrations of cracked FGM parallelepipeds modeled as Kirchhoff thin plates[35]and higher-order Reddy shear deform-able thick plates[36]have been addressed.

The present 3-D Ritz methodology is validated by comprehen-sive convergence studies and by comparisons of published solu-tions obtained using alternative theories and methods for cracked homogeneous and FGM parallelepipeds modeled as thin[35]and shear deformable thick plates[36]. An extensive amount of non-dimensional frequencies and nodal patterns are reported herein for the ﬁrst time for FGM rectangular parallelepipeds modeled as moderately thick plates having various combinations hinged, clamped or simply-supported face conditions, and having side cracks with different crack length ratios (d/a and d/b), crack positions (cx/a and cy/b), and crack inclination angles (

a

). The

effects of the volume fraction in a power law for describing material properties of FGM on the frequencies are also examined. These results can serve as the benchmark values for future numerical techniques in plate vibrations and for establishing simpliﬁed FGM plate theories and asymptotic perturbation solutions.

2. Theoretical formulation

Consider inFig. 1the rectangular coordinates (x,y,z) assumed originating at the mid-plane of a rectangular FGM parallelepiped modeled as a thick plate with a side crack. Seen also therein are the polar coordinates (r,

y

) assumed emanating at the tip of crack on the mid-plane, which are used to describe in the limit as r approaches zero the inﬁnite stress at the crack front.

The material properties (i.e., elastic modulus, E¼E(z), Poisson’s ratio, u ¼ uðzÞ and mass density,

r

¼

r

(z)) vary as a simple power law in the parallelepiped thickness (i.e., the z direction inFig. 1), as follows:

PðzÞ ¼ PbþVðzÞ

D

P ð1Þ

where VðzÞ ¼ z hþ12 m^

, Pb denotes the properties at the bottom parallelepiped face z¼ h/2

D

P, is the difference between Pband the corresponding property at the top parallelepiped face (z¼ h/ 2); h is the parallelepiped thickness, and ^m is the parameter of volume fraction that governs the material variation proﬁle in the thickness direction. In the present study, the FGM parallelepipeds under consideration are made of aluminum (Al) and ceramic

(zirconia (ZrO2) or alumina (Al2O3)), whose material properties

are given inTable 1.

In using the Ritz method, the dynamic characteristics of the parallelepiped are predicted by minimizing the energy functional

P

¼VmaxTmax, ð2Þ

where Vmax is the maximum strain energy and Tmax is the

maximum kinetic energy in simple harmonic motion, epffiffiffiffiffi1ot_-1: Based on 3-D elasticity theory, a parallelepiped vibrating harmo-nically with circular frequency

o

and amplitudes Ui (x, y, z)

(i ¼1,2,3) along the x, y and z coordinate directions, respectively, is described by the following maximum strain and kinetic energy expressions at the peak displacements and velocities of the vibratory cycle: Vmax¼ 1 2 Z V n

l

ðzÞðU1,xþU2,yþU3,zÞ2þGðzÞ 2ðU1,xÞ2þ2ðU2,yÞ2 h

þ ðU2,zþU3,yÞ2þ ðU3,xþU1,zÞ2 io dV, ð3aÞ Tmax¼

o

2 2 Z V

r

ðzÞðU21þU 2 2þU 2 3ÞdV, ð3bÞ where

l

ðzÞ ¼ uðzÞEðzÞ ð1 þ uðzÞÞð12uðzÞÞand GðzÞ ¼ EðzÞ

½2ð1 þ uðzÞ, with the subscript comma in Eq. (3) denoting partial differentiation with respect to the coordinate deﬁned by the variable after the comma.

Displacement amplitude functions, Ui(x,y,z), are expressed in

terms of admissible functions, ~Uijðx,y,zÞ, as

Uiðx,y,zÞ ¼ XNi j ¼ 1

aijU~ijðx,y,zÞ ði ¼ 1,2,3Þ, ð4Þ

wherein the ~Uijðx,y,zÞ, which is formulated shortly, are con-structed in mathematical completeness to satisfy minimally the geometric (kinematical) conditions corresponding to the various combinations of hinged, clamped, and completely free face con-ditions of the cracked parallelepiped and additionally, the stress singularities at the front of the crack and displacement disconti-nuities across the crack. Substituting Eq. (4) into Eqs. (2) and (3) and minimizing the energy functional (Eq. (2)) with respect to the generalized coefﬁcients, aij, (Eq. (4)), yields the following matrix set of linear homogeneous algebraic equations:

K11 _K12 _K13 K22 _K23 sym K33 2 6 4 3 7 5 a1j a2j a3j 8 > < > : 9 > = > ;¼

o

2 M11 _M12 _M13 M22 _M23 sym M33 2 6 4 3 7 5 a1j a2j a3j 8 > < > : 9 > = > ;, ð5Þ which is expressed in the form of a standard eigenvalue problem—the eigenvalues being the circular frequencies of free vibration

o

of a cracked FGM parallelepiped. The associated eigenvectors of generalized coefﬁcients, aij, may be substituted back into Eq. (4) to obtain the physical vibration mode shapes of the cracked FGM parallelepiped corresponding to each circular frequencies of vibration

o

.

An appropriately enhanced Ritz procedure proposed herein yields accurate solutions of cracked FGM parallelepiped vibrations

Table 1

Material properties of the FGM components. Material Properties

E (GPa) Poisson’s ratio (n) r(kg/m3 )

Aluminum (Al) 70 0.3 2702

Alumina (Al2O3) 380 0.3 3800

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with the accuracy and efﬁciency of the approximate solutions largely depending on the appropriate choice of ~Uijðx,y,zÞ in Eq. (4). A hybrid series of mathematically complete admissible orthogonal polynomials and newly-developed crack functions accounting for stress singularities at the front of the crack, while permitting displacement discontinuities across the crack, are used to approx-imate each of displacement amplitudes, Ui(x, y, z). In Eq. (4), the

displacement amplitude functions are expressed as

Ui¼ ^Uipþ ^Uic ð6Þ

where Ûipis an assumed finite series of mathematically complete polynomials; and Ûicis an assumed finite series of crack functions, supplementing the assumed polynomial series, Ûip, to appropri-ately describe the essential singular stresses and displacement discontinuities across the crack.

Orthogonal polynomials are adopted to expand ^Uipin Eq. (6) as ^ Uipðx,y,zÞ ¼ fiðzÞ XNiz l ¼ 1 XNix j ¼ 1 XNiy k ¼ 1

AðiÞ_jklPðiÞ_j ðxÞQðiÞ_kðyÞzl1 _{ði ¼ 1,2,3Þ,} _ð7Þ

where PðiÞ_j ðxÞand QðiÞ_kðyÞ are orthogonal polynomials in the x and y directions, respectively, and are generated by using a standard Gram–Schmidt orthogonalization process[28]. Imbedded inside the adopted Gram–Schmidt orthogonalization procedure are associated boundary functions in which PðiÞ_j ðxÞ satisfy the geo-metric boundary conditions for Ui on the parallelepiped faces,

x¼ 0 and x ¼a, while QðiÞ_kðyÞ satisfy the geometric boundary conditions for Uion the parallelepiped faces, y¼0 and y¼b. In

Eq. (7), the assumed series of algebraic polynomials, zi 1_,

resem-bles the kinematical assumption of displacements in the thick-ness (z) coordinate incorporated in various well-established Mindlin and higher-order shear deformable plate theories. Boundary functions, fiðzÞ, are used to satisfy the geometric boundary conditions for Uion the top and bottom parallelepiped

faces. In this work, completely stress free top and bottom parallelepiped faces are assumed so that fiðzÞ ¼ 1.

To enhance the convergence accuracy of the proposed Ritz procedure due to the presence of a crack, stress singularities at the front of the crack and displacement discontinuities across the crack are considered in constructing admissible crack functions,

^

Uic, augmenting the assumed series of orthogonal polynomials, ^

Uip. Based on 3-D elasticity theory, Hartranft and Sih [29]and Chaudhuri and Xie [30] established eigenfunction expansions showing a well-known stress singularity order of 1/2 for a cracked homogenous parallelepiped having completely free side cracks. Huang and Chang [35] and Huang et al. [36] further established that the vibrations of cracked FGM parallelepipeds modeled as Kirchhoff thin and Reddy shear deformable thick plates have analogous stress singularity order as a homogenous plate. Hence, the following set of crack functions is proposed for a cracked FGM parallelepiped: rð2n1Þ=2_cos2m þ 1 2

y

and r ð2n1Þ=2_sin2m þ 1 2

y

9m ¼ 0,1,2,. . .,n and n ¼ 1,2,3,. . . ð8Þ and ^Uicin Eq. (6) is expressed as

^ Uicðr,

y

,zÞ ¼ giðx,y,zÞ XNiz l ¼ 1 XNi n ¼ 1 Xn m ¼ 0 BðiÞ_nmlrð2n1Þ=2_cos2m þ1 2

y

þCðiÞ_nmlrð2n1Þ=2_sin2m þ 1 2

y

zl1 _ð9Þ

where giðx,y,zÞ (i ¼ 1, 2, 3) are boundary functions to satisfy the geometric boundary conditions for Uion the parallelepiped faces.

The admissible functions in Eq. (9) yield the well-known fracture mechanics fact that (for n ¼1) the order of the 3-D stress tensor

components are

s

ij

W

ð1= ffiffiffir p

Þ as r approaches zero. Yet, the gradients of Eq. (9) are not continuous across a FGM crack ð

y

¼7

p

Þ. Such displacement discontinuities across the crack are fully accounted for in establishing the admissibility of the crack functions through the boundary functions giðx,y,zÞ (i¼1, 2, 3). Note that the associated eigenvectors of generalized coefﬁcients, aij, (Eqs. (4)–(5)) comprise the A

ðiÞ

jkl(from Eq. (7)), and B ðiÞ nmland C

ðiÞ nml (from Eq. (9)).

Near the crack tip, where the terms corresponding to n ¼1 dominate, the assumed crack functions (Eq. (9)) possess a neces-sary positive square root in the polar coordinate r, (dominated by the order of square root of r,

W

ðpffiffiffirÞ), and the associated 3-D stress fields possess the well-established square root singularity as r goes to zero (dominated by the order of inverse square root of r,

W

ð1=pffiffiffirÞ, emanating from the crack-edge), thus, yielding the appropriate admissibility required regardless of the FGM paralle-lepiped geometry and normal mode response and stress distribu-tion. By incorporating appropriate gradients of the displacement fields (Eqs. (7) and (9)), the associated 3-D stress fields of the cracked FGM parallelepiped are

s

x¼

l

ðzÞðU1,xÞ ¼

l

ðzÞ½ ^U1p,xþ ^U1c,x

s

y¼

l

ðzÞðU2,yÞ ¼

l

ðzÞ½ ^U2p,yþ ^U2c,y

s

z¼

l

ðzÞðU3,zÞ ¼

l

ðzÞ½ ^U3p,zþ ^U3c,z

t

xy¼ ðGðzÞ=2ÞðU1,yþU2,xÞ ¼ ðGðzÞ=2Þ½ð Û1p,yþ Û1c,yÞ þ ð Û2p,xþ Û2c,xÞ

t

yz¼ ðGðzÞ=2ÞðU2,zþU3,yÞ ¼ ðGðzÞ=2Þ½ð Û2p,zþ Û2c,zÞ þ ð Û3p,yþ Û3c,yÞ

t

xz¼ ðGðzÞ=2ÞðU3,xþU1,zÞ ¼ ðGðzÞ=2Þ½ð Û3p,xþ Û3c,xÞ þ ð Û1p,zþ Û1c,zÞ ð10Þ wherein the above usual transformations between the global Cartesian (x,y,z) coordinates of the FGM parallelepiped and the local polar (r,

y

,z) coordinates at the crack terminus edge are assumed in the present 3-D calculations.

Values of the generalized constants, BðiÞ_nmland CðiÞ_nml, in Eqs. (9)–(10), depend on the cracked FGM parallelepiped geometry, boundary conditions, and the associated normal mode response; however, the polar coordinate dependence of the corner functions sequence (Eq. (8)) do not change with the cracked FGM parallelepiped geometry, boundary conditions, and normal mode response. Nonetheless, for n¼1,2,3,y, the generalized constants, BðiÞ_nmland CðiÞ_nml, in Eqs. (9)–(10), may be generalized as directly proportional to crack-edge stress (field) intensity factors[33,34] (n¼1 being associated with the most dominant 3-D stresses in Eq. (10), seeAppendix) through a propor-tionality constantpffiffiffiffiffiffi2

p

for crack opening responses ðpffiffiffiffiffiffi2

p

BðiÞ_nmlÞ and crack shearing or sliding responses ðpffiffiffiffiffiffi2

p

CðiÞ_nmlÞ, including more gen-eralized 3-D ‘‘mixed-mode’’ crack opening-shearing/sliding-tearing responses (seeAppendix), depending on the complexity of cracked FGM parallelepiped vibratory behavior. Givenpffiffiffiffiffiffi2

p

BðiÞ_nmlandpffiffiffiffiffiffi2

p

CðiÞ_nml are independent of r and

y

, they embody the strength of the stress fields surrounding the crack edge. Alternatively, pffiffiffiffiffiffi2

p

BðiÞ_nml and

ffiffiffiffiffiffi 2

p

CðiÞ_nml may be mathematically viewed as the strengths of the

W

ð1=pffiffiffirÞstress singularities at the crack edge most dominate for n¼1. The BðiÞ_nmland CðiÞ_nmlare evaluated in the present Ritz procedure by the clamped, hinged, and stress free boundaries of the cracked FGM parallelepiped, consequently, formulae for their evaluation may be understood from a complete stress analysis (seeAppendix) of a given cracked FGM parallelepiped configuration. In the present analysis, such stress fields near the crack tip are assumed to be linearly dependent on such stress intensities,pffiffiffiffiffiffi2

p

CðiÞ_nml. That is, scaling pffiffiffiffiffiffi2

p

BðiÞ_nml and pffiffiffiffiffiffi2

p

CðiÞ_nml also directly scales the stress and displacement ﬁelds near the crack edge of the FGM parallelepiped,

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whereas the stresses at the crack edge of the FGM parallelepiped remain unbounded.

Physically,pffiffiffiffiffiffi2

p

CðiÞ_nml may be regarded as inten-sities of vibratory stresses through the crack edge region caused by the introduction of a crack into the FGM parallelepiped. Corre-spondingly, formulae forp2ffiffiffiffiffiffi

p

CðiÞ_nml(seeAppendix) may be regarded as reflecting strain redistributions for transmission of linear-elastic stress past the crack edge, even though small degrees of nonlinearity at the crack edge is embedded within the stress field and do not significantly impact the overall correctness of the present analysis. Hence, ‘‘mixed mode’’ cracking in FGM parallelepiped vibration is sufficient to describe the most general three-dimensional dynamic ‘‘small-scale’’ yielding of local crack-tip deformation and stress fields in cracked parallelepipeds. Because fracture processes of a functionally graded material may be regarded as ‘‘caused’’ by this surrounding crack-edge stress field nature, the intensitiespffiffiffiffiffiffi2

p

BðiÞ_nml andpffiffiffiffiffiffi2

p

CðiÞ_nml may be interpreted in a generalized sense as fracture correlation coefﬁcients in current practice. It should be noted that the intensities have units of (force) (length)3/2[33,34], and since the intensities are linear factors in elastic stress, they must be propor-tional to the vibratory force, and other characteristic lengths, such as crack size, i.e., determined in the present analysis by different crack length ratios (d/a and d/b), crack positions (cx/a and cy/b), and crack

inclination angles (

a

) (seeFig. 1). These crack size effects addressed in the present analysis imply flaw-size influence in FGM parallelepiped vibration and fracture, suggesting that these crack size effects can be fully analyzed only if crack-edge stress singularity effects and allow-ing for displacement discontinuities across the crack (sufficient to describe the most general 3-D mixed modes of local crack-edge deformation and stress fields) are included.

Since the 3-D stress values near a crack edge are unbounded, a strength-of-materials approach of FGM failure prediction that the FGM fails, when the 3-D stresses exceed some critical Von-Mises measure or ultimate/yield value is inadequate. As a cracked FGM parallelepiped responds in a normal mode of vibration, in spite of the fact that the stress field near the crack tip is unbounded, the FGM parallelepiped may not necessarily fail. Beyond such critical stress limit, however, such fatigue and failure mechanisms are possible. Instead of a conventional approach of referencing a maximum FGM stress value with a critical FGM stress values, in a fracture mechanics approach FGM parallelepiped failure may be predicted by theoretically comparing the stress intensity con-stants, pffiffiffiffiffiffi2

p

CðiÞ_nml, with some critical value, thus, establishing a generalized spectral accuracy of the present Ritz methodology, especially within the crack region of a normal mode response of a cracked FGM parallelepiped. Such a critical value of generalized stress intensity may be considered a critical stress intensity or fracture toughness of the FGM. Theoretically speak-ing, a cracked FGM parallelepiped may fail under high-cycle fatigue and fracture as pffiffiffiffiffiffi2

p

CðiÞ_nml values increase proportionately (depending on the cracked FGM parallelepiped geometry and normal mode response) beyond some critical stress intensity or fracture toughness of the FGM. Such fracture tough-ness is a material property, analogous to elastic homogeneous materials ultimate stress or yield stress.

In the present Ritz analysis, the material properties (i.e., elastic modulus, E, Poisson’s ratio, u, and mass density,

r

) vary as a simple power law in the parallelepiped thickness (see Fig. 1), as described in Eq. (1), which exponentially depends a parameter

^

m being the volume fraction that governs the material varia-tion proﬁle in the FGM parallelepiped thickness including the thickness at the crack tip. The FGM parallelepipeds under consideration are made of aluminum (Al) and ceramic (zirconia (ZrO2) or almina (Al2O3)), whose material properties are given in

Table 1.

The values of Nix, Niy, Nizin Eq. (7) and Niare assumed to vary for various i. For simplicity, Nix¼ ^Nx, Niy¼ ^Ny, Niz¼ ^Nz, and Ni¼ ^Nc for i¼1, 2, and 3 in the present study. Substituting Eqs. (7) and (9) into Eqs. (2) and (3) and minimizing the energy functional

P

yield 3 ð ^Nx ^Ny ^Nz ^Nc ð ^Ncþ3Þ ^NzÞ simultaneous algebraic solution matrix equations in Eq. (5).

3. Convergence and comparison studies

Leissa[37]described about 500 publications which appeared before 1966, and more than 1500 papers have been published since then. Relatively few published results are available for cracked rectangular plates, and most of them considered plates with simply-supported (SSSS) boundary conditions at all sides or at two opposite sides. Because exact analytical solutions exist for such plates with no crack, semi-analytical solutions[38]can be constructed for such plates with cracks along a straight line perpendicular to the simply-supported edges.

In examining the vibrations of homogeneous SSSS cracked rectangular parallelepipeds modeled as cracked plates, Lynn and Kumbasar[6]used Green’s functions to represent the transverse displacements of plates, resulting in homogeneous Fredholm integral equations of the ﬁrst kind, while Stahl and Keer [7]

formulated such problems as dual series equations which reduced to homogeneous Fredholm integral equations of the second kind. Aggarwala and Ariel [8] applied Stahl and Keer’s approach to analyze the vibration of parallelepipeds having various crack length ratios (d/a or d/b) and positions (cy/b) along its symmetry

axes. Neku [9] modiﬁed Lynn and Kumbasar’s approach [6] by establishing the needed Green’s functions via Levy’s form of solution. Solecki[10] constructed a solution for vibrations of a cracked plate by using a Navier’s solution, along with ﬁnite Fourier transformation of discontinuous functions for the displa-cement and slope across the crack. Recently, Khadem and Rezaee

[39] used so-called modiﬁed comparison functions constructed from Levy’s solution as the admissible functions of the Ritz method to analyze a SSSS rectangular cracked parallelepiped with arbitrary crack length ratio (d/a or d/b) and position (cy/b) parallel

to one side of the parallelepiped. Due to the specialized construc-tion of their assumed Levy’s soluconstruc-tions as admissible transverse displacement ﬁeld, the Khadem and Rezaee [39]procedure can only be applied to rectangular parallelepipeds with two opposite edges simply-supported. Hirano and Okazaki[11]also developed solutions for vibrations of cracked rectangular parallelepipeds with two opposite edges simply-supported by utilizing a Levy’s solution transverse displacement ﬁeld and further matching the boundary conditions by means of a weighted residual method.

The convergence and comparison studies for natural frequen-cies of SSSS homogeneous and FGM rectangular parallelepipeds, having a side crack of various length ratios (d/a or d/b), crack positions (c/b), and crack orientation (

a

), are summarized now, not only to addresses the importance of using crack functions to accelerate the convergence of crack FGM parallelepiped vibra-tions, but also to appropriately describes the behaviors of stress singularities at the crack tip and show the discontinuities of displacement and slope crossing the crack, which are character-istics of the true solutions and the 3-D mode shapes described in

Section 5. The present numerical results are compared with the published results and show better accuracy than those obtained by the Ritz method combining with different domain decomposi-tion techniques (i.e., Yuan and Dickinson[14]and Liew et al.[15]). The present Ritz procedure yields upper bounds to the exact values as the number of terms retained increases in the assumed hybrid series, Eq. (4), yielding in Eq. (5) a solution matrix size, 3 ð ^Nx ^Ny ^Nzþ ^Nc ð ^Ncþ3Þ ^NzÞ. A convergence study

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Table 2

Convergence ofoðb2=hÞpffiffiffiffiffiffiffiffiffir=Efor a homogeneous, cracked SSSS rectangular parallelepiped modeled as a thin plate with a horizontal side crack@

(a/b¼ 2.0, h/b¼ 0.01, cy/b¼ 0.5, d/a¼ 0.5,a¼01).

Mode no. Crack functions ( ^Nc)

Polynomial solution size (I J)@

[ ] ^Nz¼2; ( ) ^Nz¼3; { } ^Nz¼4 Stahl & Keer

n [7] Huang et al.þ [2,3] 4 4 5 5 6 6 7 7 8 8 9 9 1 0 [4.132] [4.132] [4.131] [4.131] [4.131] [4.131] 3.05n 3.053þ (3.733) (3.733) (3.733) (3.733) (3.733) (3.733) {3.733} {3.733} {3.733} {3.733} {3.733} {3.733} 2 [3.877] [3.648] [3.618] [3.464] [3.452] [3.368] (3.551) (3.384) (3.363) (3.247) (3.238) (3.173) {3.551} {3.383} {3.363} {3.247} {3.238} {3.173} 4 [3.234] [3.234] [3.220] [3.217] [3.217] [3.216] (3.065) (3.057) (3.054) (3.052) (3.051) (3.051) {3.064} {3.057} {3.053} {3.051} {3.051} {3.050} 5 [3.225] [3.217] [3.217] [3.216] [3.216] [3.215] (3.058) (3.052) (3.051) (3.051) (3.050) (3.050) {3.057} {3.052} {3.051} {3.050} {3.050} {3.049} 6 [3.220] [3.216] [3.216] [3.215] [3.215] [3.214] (3.054) (3.051) (3.050) (3.050) (3.050) (3.049) {3.054} {3.050} {3.050} {3.049} {3.049} {3.048} 2 0 [7.198] [6.618] [6.618] [6.609] [6.609] [6.609] 5.507n 5.506þ (6.539) (5.981) (5.980) (5.972) (5.972) (5.971) {6.539} {5.980} {5.980} {5.971} {5.971} {5.971} 2 [6.919] [6.209] [6.189] [6.102] [6.096] [6.060] (6.347) (5.699) (5.685) (5.614) (5.609) (5.578) {6.346} {5.699} {5.685} {5.613} {5.609} {5.578} 4 [5.840] [5.799] [5.780] [5.773] [5.769] [5.767] (5.554) (5.517) (5.512) (5.507) (5.506) (5.505) {5.554} {5.516} {5.512} {5.506} {5.506} {5.505} 5 [5.779] [5.765] [5.761] [5.759] [5.758] [5.757] (5.539) (5.508) (5.505) (5.504) (5.504) (5.504) {5.539} {5.507} {5.504} {5.504} {5.504} {5.503} 6 [5.765] [5.761] [5.758] [5.758] [5.756] [5.756] (5.532) (5.506) (5.504) (5.504) (5.504) (5.503) {5.532} {5.505} {5.504} {5.503} {5.503} {5.503} 3 0 [13.75] [13.74] [10.84] [10.84] [10.74] [10.74] 5.570n 5.570þ (12.78) (12.61) (9.801) (9.799) (9.703) (9.703) {12.77} {12.61} {9.801} {9.798} {9.703} {9.703} 2 [13.51] [12.93] [10.75] [10.74] [10.64] [10.63] (12.51) (11.90) (9.747) (9.733) (9.642) (9.637) {12.50} {11.90} {9.747} {9.732} {9.642} {9.636} 4 [6.027] [5.973] [5.967] [5.960] [5.960] [5.958] (5.631) (5.598) (5.584) (5.578) (5.574) (5.573) {5.630} {5.597} {5.583} {5.577} {5.573} {5.572} 5 [6.006] [5.961] [5.957] [5.956] [5.956] [5.956] (5.583) (5.571) (5.568) (5.566) (5.564) (5.564) {5.582} {5.570} {5.566} {5.565} {5.563} {5.562} 6 [5.996] [5.958] [5.956] [5.956] [5.956] [5.955] (5.570) (5.567) (5.564) (5.564) (5.562) (5.562) {5.569} {5.566} {5.563} {5.562} {5.561} {5.560} 4 0 [17.49] [14.09] [14.09] [14.04] [14.04] [14.04] 9.336n 9.336þ (16.04) (12.74) (12.74) (12.69) (12.69) (12.68) {16.04} {12.74} {12.73} {12.68} {12.68} {12.68} 2 [14.32] [13.34] [12.51] [12.45] [12.10] [12.07] (13.40) (12.26) (11.59) (11.54) (11.27) (11.24) {13.40} {12.26} {11.59} {11.54} {11.27} {11.24} 4 [11.44] [10.32] [10.16] [10.15] [10.12] [10.12] (10.52) (9.545) (9.382) (9.370) (9.343) (9.341) {10.52} {9.544} {9.381} {9.369} {9.342} {9.340} 5 [10.77] [10.13] [10.12] [10.11] [10.10] [10.10] (9.884) (9.359) (9.349) (9.341) (9.333) (9.333) {9.883} {9.359} {9.348} {9.340} {9.332} {9.332} 6 [10.62] [10.11] [10.11] [10.10] [10.10] [10.10] (9.766) (9.341) (9.339) (9.332) (9.332) (9.331) {9.765} {9.340} {9.338} {9.331} {9.331} {9.330} 5 0 [19.96] [16.57] [16.57] [16.51] [16.51] [16.51] 12.76n 12.87þ (18.25) (14.97) (14.97) (14.92) (14.92) (14.92) {18.25} {14.97} {14.97} {14.92} {14.92} {14.92} 2 [18.51] [15.46] [15.30] [15.24] [15.15] [15.15] (17.01) (14.06) (13.90) (13.84) (13.76) (13.75) {17.01} {14.06} {13.90} {13.84} {13.75} {13.75} 4 [14.73] [14.65] [14.57] [14.50] [14.48] [14.46] (13.33) (13.27) (13.21) (13.15) (13.14) (13.13) {13.33} {13.27} {13.21} {13.15} {13.14} {13.13}

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was conducted for rectangular parallelepipeds modeled as thick plates with different h/b to verify the correctness of the proposed solutions. The rectangular parallelepipeds are assumed simply-supported on the faces x¼ 0, x¼ a, y¼0, and y¼b. The correspond-ing geometric boundary conditions are U2¼ U3¼0 at x ¼0 and x ¼a, and U1¼U3¼0 at y¼0 and y ¼b. Thus, in Eq. (9), g1ðx, y, zÞ ¼ yðbyÞ, g2ðx, y, zÞ ¼ xðaxÞ, and g3ðx, y, zÞ ¼ xðaxÞyðbyÞ. Poisson’s ratio (

n

) is set equal to 0.3 for the homogeneous parallelepipeds analyzed.Table 2summarizes the convergence studies of the ﬁrst ﬁve non-dimensional frequency parameters for a rectangular (a/b ¼2), homogeneous SSSS paral-lelepiped modeled as a thin plate (h/b¼0.01) with a side crack, having position ratio, cy/b ¼0.5, and having crack length ratio, d/

a ¼0.5, and crack orientation horizontally at

a

¼01. Posted values of the ﬁrst ﬁve

o

ðb2=hÞpffiffiffiffiffiffiffiffiffi

r

=E were obtained using increasing orthogonal polynomial solution size, ^Nx ^Ny¼ (4 4), (5 5),y, (9 9), crack functions, ^Nc¼0, 2, 4, 5, 6, and algebraic polynomials solution size, ^Nz¼2, 3, 4, assuming correspondingly first-order, second-order, or cubic-order transverse shear deformation through the parallelepiped thickness. For an intact SSSS parallelepiped, using only polynomials as assumed displacement fields in the present Ritz procedure gives insufficiently converged solutions, as the polynomial solution sizes, ^Nx ^Nyand ^Nz increases. Augmenting the assumed displacement of admissible polynomials with crack functions, i.e., for a solution matrix size, 3 ð ^Nx ^Ny ^Nzþ ^Nc ð ^Ncþ3Þ ^NzÞ, yields upper bounds to converge to the values in excellent agreement with the published results of Stahl and Keer[7]and Huang and Leissa[2,3] using the classical thin plate theory (seeTable 2). Stahl and Keer[7]

used an accurate Fredholm integration approach, while Huang and Leissa[2,3] used the Ritz method using classical thin-plate theory assuming hybrid series of plate’s transverse displacement ﬁeld of algebraic polynomials and special corner functions that appropriately described the stress singularities at the crack tip and discontinuities of transverse displacement and slope crossing the crack.

It can be seen inTable 2that using an orthogonal polynomial solution size, ^Nx ^Ny¼9 9, corner functions, ^Nc¼6, and alge-braic polynomial solution size, ^Nz¼4, taking on cubic-order transverse shear flexibility through the parallelepiped thickness (for a 1620-term solution matrix size) yields converged frequen-cies at least to three significant figures, slightly lower than those solutions obtained using the classical thin plate theory (Stahl and Keer[7]and Huang and Leissa[2,3]), mainly because of the effects of shear deformation and rotary inertia inherent to the present 3-D solutions. As contrasted inTable 2, the 3-D solutions obtained assuming constant transverse shear flexibility through the paral-lelepiped thickness ( ^Nz¼2) are markedly insufficient in regards to overall solution accuracy compared to 3-D solutions obtained assuming linear or parabolic transverse shear flexibility through the parallelepiped thickness ( ^Nz¼3 or 4).

Table 3describes the convergence of the ﬁrst ﬁve non-dimen-sional frequencies for SSSS square FGM parallelepiped modeled as a moderately thick (h/b¼0.1), square FGM plate having a horizon-tal side crack (

a

¼01) positioned at cy/b¼ 0.5 with crack length

ratio, d/a¼0.5. The parallelepiped is made of aluminum (Al) and alumina (Al2O3), and the material properties linearly vary ( ^m ¼ 1 in

Eq. (1)) along the thickness direction. The non-dimensional fre-quency parameter,

o

ðb2=hÞpffiffiffiffiffiffiffiffiffiffiffiffi

r

c=Ec, in which subscript ‘‘c’’ refers to a reference ceramic material, is employed. Reported values of ð

o

Þðb2=hÞ ffiffiffiffiffiffiffiffiffiffiffiffi

r

c=Ec

p

were obtained using increasing orthogonal poly-nomial solution size, ^Nx ^Ny¼ ð4 4Þ, ð5 5Þ, . . ., ð9 9Þ, crack functions, ^Nc¼0, 2, 4, 5, 6, and algebraic polynomial solution size,

^

Nz¼4, 5, 6, taking on correspondingly 3rd, 4th, or 5th higher-order transverse shear deformation through the parallelepiped thickness. In contrast to conventional laminated composite mate-rials, which have abrupt change in material properties, causing large interlaminar stresses and delaminations across the interface between differing material layers with different materials, func-tionally graded materials do not have these adverse interlamina-tion stress and delaminainterlamina-tion effects, as material properties of functionally graded materials vary continuously by gradually changing the volume fraction of constituent properties.

Mac and Huang [19]employed element-free kp-Ritz method based on Mindlin plate theory, reporting the ﬁrst four non-dimensional frequencies

o

r

=E of an intact SSSS FGM plate, as follows: Mode 1: 4.3474, Mode 2: 10.416, Mode 3: 10.416, Mode 4: 15.936. These Mac and Huang[19]intact Mindlin SSSS FGM plate solutions appear to be in proximity agreement with the predicted 3-D solutions of a cracked SSSS FGM paralle-lepiped modeled as a moderately thick, cracked FGM plate (h/b ¼0.1) using no crack functions ( ^Nc¼0) reported inTable 3 (that is, Mode 1: 4.426, Mode 2: 10.63, Mode 3: 10.63, Mode 4: 16.20, Mode 5: 16.20), albeit the present 3-D solutions using no crack functions ( ^Nc¼0) appear to be converging to slightly higher upper-bounds on the exact solutions above the upper-bound Mindlin FGM SSSS plate solutions of Mac and Huang [19]. However, by incorporating crack functions into the present 3-D calculations, one substantially enhances the convergence of solu-tions. Indeed, using an orthogonal polynomial solution size,

^

Nx ^Ny¼9 9, corner functions, ^Nc¼6, and algebraic polyno-mial solution size, ^Nz¼6, assuming 5th-order transverse shear flexibility through the parallelepiped thickness (for a 2430-term solution matrix size) yields 3-D solutions posted inTable 3that can be described as exact to at least three significant figures. The Mac and Huang[19]Mindlin SSSS FGM plate solutions are indeed not in close agreement with the predicted 3-D solutions of a cracked SSSS FGM parallelepiped modeled as a moderately thick FGM plate using six crack functions ( ^Nc¼6) reported inTable 3, which are as follows: Mode 1: 4.270, Mode 2: 9.963, Mode 3: 10.34, Mode 4: 13.95, Mode 5: 14.45, predicted using four or six terms in the parallelepiped thickness (z) direction. Moreover, due to the presence of the horizontal crack, the converged cracked

Table 2 (continued ) Mode no. Crack

functions ( ^Nc)

Polynomial solution size (I J)@_{[ ] ^}_N

z¼2; ( ) ^Nz¼3; { } ^Nz¼4 Stahl & Keer

n [7] Huang et al.þ [2,3] 4 4 5 5 6 6 7 7 8 8 9 9 5 [14.11] [13.98] [13.95] [13.94] [13.93] [13.93] (12.88) (12.81) (12.79) (12.78) (12.78) (12.77) {12.88} {12.81} {12.79} {12.78} {12.77} {12.77} 6 [13.97] [13.93] [13.92] [13.92] [13.91] [13.91] (12.79) (12.77) (12.77) (12.76) (12.76) (12.76) {12.79} {12.77} {12.76} {12.76} {12.76} {12.76}

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Table 3

Convergence ofoðb2=hÞprffiffiffiffiffiffiffiffiffiffiffiffi_c=Ecfor a Al/Al2O3FGM cracked SSSS square parallelepiped modeled as a Al/Al2O3FGM SSSS square, moderately thick plate with a horizontal side crack@

(h/b ¼0.1, cy/b¼0.5, d/a¼ 0.5,a¼01, ^m¼ 1). Mode no. Crack

functions ( ^Nc)

Polynomial solution size (I J)@

[ ] ^Nz¼4; ( ) ^Nz¼5; { } ^Nz¼6 4 4 5 5 6 6 7 7 8 8 9 9 1 0 [4.427] [4.427] [4.426] [4.426] [4.426] [4.426] (4.427) (4.427) (4.426) (4.426) (4.426) (4.426) {4.427} {4.427} {4.426} {4.426} {4.426} {4.426} 2 [4.304] [4.292] [4.289] [4.284] [4.283] [4.281] (4.303) (4.292) (4.289) (4.284) (4.823) (4.280) {4.303} {4.292} {4.289} {4.284} {4.283} {4.280} 4 [4.275] [4.273] [4.273] [4.272] [4.272] [4.271] (4.275) (4.273) (4.273) (4.272) (4.271) (4.271) {4.275} {4.273} {4.273} {4.272} {4.271} {4.271} 5 [4.274] [4.271] [4.271] [4.271] [4.271] [4.270] (4.273) (4.271) (4.271) (4.271) (4.270) (4.270) {4.273} {4.271} {4.271} {4.271} {4.270} {4.270} 6 [4.272] [4.271] [4.271] [4.270] [4.270] [4.270] (4.272) (4.271) (4.270) (4.270) (4.270) (4.270) {4.272} {4.271} {4.270} {4.270} {4.270} {4.270} 2 0 [12.49] [10.66] [10.66] [10.63] [10.63] [10.63] (12.49) (10.66) (10.66) (10.63) (10.63) (10.63) {12.49} {10.66} {10.66} {10.63} {10.63} {10.63} 2 [10.36] [10.16] [10.12] [10.10] [10.08] [10.08] (10.36) (10.15) (10.12) (10.10) (10.08) (10.08) {10.36} {10.15} {10.12} {10.10} {10.08} {10.07} 4 [9.994] [9.987] [9.978] [9.974] [9.970] [9.969] (9.994) (9.986) (9.977) (9.973) (9.969) (9.968) {9.993} {9.986} {9.976} {9.973} {9.969} {9.968} 5 [9.981] [9.977] [9.971] [9.969] [9.967] [9.966] (9.980) (9.976) (9.970) (9.968) (9.966) (9.965) {9.980} {9.975} {9.970} {9.968} {9.965} {9.965} 6 [9.972] [9.970] [9.968] [9.966] [9.965] [9.964] (9.972) (9.970) (9.967) (9.965) (9.964) (9.963) {9.971} {9.969} {9.966} {9.965} {9.963} {9.963} 3 0 [12.49] [10.66] [10.66] [10.63] [10.63] [10.63] (12.49) (10.66) (10.66) (10.63) (10.63) (10.63) {12.49} {10.66} {10.66} {10.63} {10.63} {10.63} 2 [10.77] [10.39] [10.38] [10.36] [10.36] [10.35] (10.77) (10.39) (10.38) (10.36) (10.36) (10.35) {10.77} {10.39} {10.38} {10.36} {10.36} {10.35} 4 [10.36] [10.35] [10.34] [10.34] [10.34] [10.34] (10.35) (10.35) (10.34) (10.34) (10.34) (10.34) {10.35} {10.35} {10.34} {10.34} {10.34} {10.34} 5 [10.35] [10.34] [10.34] [10.34] [10.34] [10.34] (10.35) (10.34) (10.34) (10.34) (10.34) (10.34) {10.35} {10.34} {10.34} {10.34} {10.34} {10.34} 6 [10.34] [10.34] [10.34] [10.34] [10.34] [10.34] (10.34) (10.34) (10.34) (10.34) (10.34) (10.34) {10.34} {10.34} {10.34} {10.34} {10.34} {10.34} 4 0 [16.20] [16.20] [16.20] [16.20] [16.20] [16.20] (16.20) (16.20) (16.20) (16.20) (16.20) (16.20) {16.20} {16.20} {16.20} {16.20} {16.20} {16.20} 2 [14.02] [14.01] [14.00] [14.00] [13.99] [13.99] (14.02) (14.01) (14.00) (14.00) (13.99) (13.99) {14.02} {14.01} {14.00} {14.00} {13.99} {13.99} 4 [13.96] [13.95] [13.95] [13.95] [13.95] [13.95] (13.96) (13.95) (13.95) (13.95) (13.95) (13.95) {13.96} {13.95} {13.95} {13.95} {13.95} {13.95} 5 [13.95] [13.95] [13.95] [13.95] [13.95] [13.95] (13.95) (13.95) (13.95) (13.95) (13.95) (13.95) {13.95} {13.95} {13.95} {13.95} {13.95} {13.95} 6 [13.95] [13.95] [13.95] [13.95] [13.95] [13.95] (13.95) (13.95) (13.95) (13.95) (13.95) (13.95) {13.95} {13.95} {13.95} {13.95} {13.95} {13.95} 5 0 [16.20] [16.20] [16.20] [16.20] [16.20] [16.20] (16.20) (16.20) (16.20) (16.20) (16.20) (16.20) {16.20} {16.20} {16.20} {16.20} {16.20} {16.20} 2 [16.20] [14.95] [14.88] [14.83] [14.79] {14.78} (16.20) (14.95) (14.88) (14.83) (14.79) (14.78) {16.20} {14.95} {14.88} {14.83} {14.79} {14.78} 4 [14.51] [14.49] [14.48] [14.47] [14.46] [14.46] (14.51) (14.49) (14.48) (14.46) (14.46) (14.46) {14.50} {14.49} {14.47} {14.46} {14.46} {14.46}

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SSSS FGM parallelepiped solutions possess no repeated frequency Modes 2–3 and Modes 4–5.

Symmetric and anti-symmetric,

o

r

=E, associated to independent normal mode responses are obtained, when the cracked FGM parallelepiped vibrates symmetrically and anti-symmetrically (as seen inFigs. 2–6to be discussed in more detail subsequently in Section 5). When the vibratory stress is in a direction perpendicular to the crack surface, then the normal mode response sees an opening mode of the crack. When the vibratory stress is parallel to the crack surface, then the normal mode response sees a shearing or sliding mode of the crack. In some coupled normal mode responses, the vibratory stress creates both opening and shearing/sliding mode components of the crack inducing a mixed modal vibration along the crack. Each of these cracked normal response negate any kinds of repeated mode

o

r

=Eassociated with an intact FGM parallelepiped (Mac and Huang[19]).

To the authors’ knowledge, there are no published results on the vibrations of cracked parallelepipeds modeled as plates. To further verify the accuracy of the present 3-D vibration solutions,

Table 4contrasts

o

r

=E predictions obtained by various theories for SSSS homogeneous cracked square parallelepipeds with horizontal side cracks (

a

¼01) having various length ratios (d/a) and positioned at cy/b¼0.5. The present 3-D Ritz

o

r

=E solutions are posted against

o

r

=E calcu-lated for homogeneous cracked parallelepipeds modeled as classi-cal thin plates and Mindlin plates. Square parallelepipeds modeled as plates having three different thickness-length ratios (h/b¼0.002 (very thin), 0.05 (thin), 0.1 (moderately thick)) and two different horizontal crack length ratios (d/a¼0.2 and 0.4) are considered. Only

o

r

=Efor the out-of-plane modes are listed inTable 4. Huang and Leissa[2,3] applied the Ritz method with the admis-sible functions, including a set of crack functions possessing admissibility analogous to that of the present 3-D analysis, yet suitable for incorporation with classical thin plate theory, while Li

[4]employed a similar methodology based on Mindlin plate theory with the shear correction factor equal to

p

2_{=12. The present 3-D} results for very thin plates (h/b¼0.002) were obtained using orthogonal polynomial solution size, ^Nx ^Ny¼9 9 (Eq. (7)), crack functions, ^Nc¼6 (Eq. (9)), and algebraic polynomial solution size (Eq. (7)), ^Nz¼4 and 6, taking on correspondingly a cubic-order and 5th-order transverse shear deformation through the paralle-lepiped thickness. This yields a 1620-term solution matrix size for

^

Nx ^Ny¼9 9, ^Nc¼6, ^Nz¼4, and a 2430-term solution matrix size for ^Nx ^Ny¼9 9, ^Nc¼6, ^Nz¼4.

Considering stress singularities in cracked parallelepiped vibration in ﬂexure, a ﬁrst-order (Mindlin) shear deformable analysis[4]should yield lower

o

r

=Evalues than classical thin-plate solutions [2,3]. Shear deformation reduces the

o

r

=Esolutions. Rotary inertia reduces the higher modes of

o

r

=E. As

o

r

=Eis non-dimensionalized with respect to

r

, E, and explicitly, h, comparisons inTable 4between ﬁrst-order (Mindlin) and higher-order shear deformable[4]and

classical thin-plate

o

r

=Esolutions[2,3] are largely inde-pendent of parallelepiped shape being square (a/b¼1) or rectan-gular (a/ba1), showing primarily inﬂuences of crack length ratio (d/a) at position (cy/b ¼0.5) and horizontal orientation (

a

¼01). As

the plate becomes thicker or inTable 4 as the frequency mode number increases, distinctions inTable 4between shear deform-able[4]and classical plate[2,3]

o

r

=Esolutions are more apparent.

As expected, the differences between the frequencies of very thin plates (h/b¼0.002) obtained based on the classical thin plate theory[2,3] and Mindlin plate theory[4]are negligible, and both are consistent with the present 3-D elasticity-based predicted solutions up to at least three signiﬁcant ﬁgures. For the thin (h/b¼ 0.05) and moderately thick (h/b ¼0.1) plates, the predicted

o

r

=Esolutions based on the classical thin plate theory are considerably stiffer than the predicted shear deformable

o

r

=Esolutions based on Mindlin plate theory [4] and present 3-D elasticity theory, especially for homogeneous SSSS parallelepipeds modeled as moderately thick plates (h/b ¼0.1) and for the higher

o

r

=E modes. The homogeneous SSSS parallelepiped

o

r

=E solutions modeled from shear deformable Mindlin plate theory[4]are slightly over-correcting in reducing the classical thin plate theory

o

r

=Esolutions

[2,3] for transverse shear effects compared to the corrections in reducing the classical thin plate theory

o

r

=Esolutions for transverse shear effects inherent to the present 3-D

o

r

=E solutions. Percentage differences between the shear deformable Mindlin

o

r

=E solutions and the present 3-D

o

r

=Esolutions are less than 1%. Generally speaking, this 1% difference does not signiﬁcantly increase with increasing crack length ratio (d/a).

4. Numerical results and discussion

The vibrations of cracked rectangular parallelepipeds with arbitrary boundary conditions are typically modeled as plates using both the finite element method and the Ritz method. Qian et al. [12]developed a finite element solution by deriving the stiffness matrix for an element including the crack tip from the integration of the stress intensity factor. Krawczuk[13]proposed a solution similar to that of Qian et al. [12], except that the stiffness matrix for an element including the crack tip was expressed in a closed form. Yuan and Dickinson[14]decomposed a rectangular plate into several domains and introduced artificial springs at the interconnecting boundaries between the domains so that the Ritz method with regular admissible functions can be applied to find the solutions. Similar to the approach used by Yuan and Dickinson [14], Liew et al. [15] required the continuities of displacement and slope in a sense of integration along the interconnecting boundaries. In the approach of Liew et al., the continuities of displacement and slope are not satisfied at every point along the interconnecting boundaries. Notably, the

Table 3 (continued ) Mode no. Crack

functions ( ^Nc)

Polynomial solution size (I J)@_{[ ] ^}_N

z¼4; ( ) ^Nz¼5; { } ^Nz¼6 4 4 5 5 6 6 7 7 8 8 9 9 5 [14.48] [14.47] [14.46] [14.46] [14.45] [14.45] (14.48) (14.47) (14.46) (14.46) (14.45) (14.45) {14.48} {14.47} {14.46} {14.45} {14.45} {14.45} 6 [14.46] [14.46] [14.46] [14.45] [14.45] [14.45] (14.46) (14.46) (14.45) (14.45) (14.45) (14.45) {14.46} {14.46} {14.45} {14.45} {14.45} {14.45}

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solutions of Yuan and Dickinson[14]and Liew et al.[15]destroy the good characteristics of providing upper-bound solutions for vibration frequencies, normally associated with the Ritz method. These published solutions, except for the ﬁnite element solutions, by no means address the stress singularities at the crack terminus

edge of FGM parallelepipeds. Addressed in this section is 3-D elasticity-based Ritz predictions including such stress singularities to investigate the vibrations of side-cracked rectangular FGM paralle-lepipeds, particularly bringing forth the possible discontinuities of displacement and slope across the crack. Since a crack is a special case

Fig. 2. (a) Mode shapes, nodal patterns andoðb2=hÞpffiffiffiffiffiffiffiffiffir=Efor a SSSS homogeneous square parallelepiped modeled as a moderately thick plate (cx=a ¼0, d/b¼ 0, h/b ¼ 0.1); (b) mode shapes, nodal patterns andoðb2_=hÞ ffiffiffiffiffiffiffiffiffiffiffiffi_r

c=Ec p

for SSSS homogeneous cracked square parallelepipeds modeled as moderately thick cracked plates (cx=a¼ 0, d/b ¼0.2,0.5, h/b ¼ 0.1, cy/b ¼0.5,a¼01) and (c) nodal patterns andoðb2=hÞ

ffiffiffiffiffiffiffiffiffi

r=E p

for SSSS homogeneous cracked square parallelepipeds modeled as classically-thin cracked plates (Huang and Leissa[2]) ( ^m ¼0, d/b ¼ 0.2,0.5, h/b ¼0.01, cy/b ¼0.5,a¼01).

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of a V-notch, the asymptotic solutions (or crack functions) derived by Williams [31] were shown to be particularly effective for this, according to Kim and Jung [40]who applied this methodology to investigate the vibrations of rhombic plates with V-notches, and according to the authors’ experiences in studying vibrations of a circular plate with a V-notch (Leissa et al.[41]). However, Huang and

Leissa [2,3] have demonstrated that using Williams’ asymptotic solutions and regular polynomials as admissible functions does not yield as rapidly convergent solutions for parallelepipeds having a deep (d/a or d/b¼ 0.5) compared to a shallow (d/a or d/b¼0.2) side crack.

The present 3-D Ritz technique is used to predict

o

ðb2=hÞ ffiffiffiffiffiffiffiffiffiffiffiffi

r

c=Ec

p

for FGM rectangular parallelepipeds with various

mˆ Mode 1 2 3 4 5 5 0 (3.768) (8.909) (8.909) (12.64) (12.64) 0.2 (3.756) (8.851) (8.867) (12.04) (12.64) 0.5 (3.511) (7.379) (8.621) (10.49) (11.17) mˆ d / b d / b d / b 1 2 3 4 5 5 0 (3.772) (8.927) (8.927) * (12.64) * (12.64) mˆ 1 2 3 4 5 5 0.2 * * 0.5 * Mode Number Mode Number (12.64) (12.05) (8.885) (8.867) (3.759) (3.513) (7.334) (8.635) (10.49) (11.18)

Fig. 3. (a) Mode shapes, nodal patterns andoðb2=hÞ ffiffiffiffiffiffiffiffiffiffiffiffir_c=Ec p

for a SSSS square Al/Al2O3FGM parallelepiped modeled as a moderately thick Al/Al2O3FGM plate ( ^m¼ 5, d/b ¼0, h/b ¼0.1); (b) mode shapes, nodal patterns andoðb2=hÞpffiffiffiffiffiffiffiffiffiffiffiffir_c=Ecfor SSSS cracked Al/Al2O3FGM square parallelepipeds modeled as moderately thick cracked Al/Al2O3 FGM plates ( ^m¼ 5, d/b ¼ 0.2,0.5, h/b ¼ 0.1, cy/b¼ 0.5,a¼01) and (c) nodal patterns and ^mfor SSSS cracked Al/Al2O3FGM square parallelepipeds modeled as a moderately thick cracked Al/Al2O3FGM plate using Reddy thick plate theory (Huang et al.[32]) ( ^m¼ 5, d/b ¼0,0.2,0.5, h/b ¼ 0.1, cy/b¼ 0.5,a¼01).

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face conditions modeled as plates with various thickness ratios (h/a) and having side cracks at various locations (cx/a and cy/b),

inclination angles (

a

), and length ratios (d/a and d/b). Three types of conditions on side faces 1,2,3,4 (see Fig. 1) are considered, namely SSSS, FFFF and CFCF, where S, F and C denote simply-supported, free, and clamped face conditions, respectively. The

SSSS rectangular parallelepipeds are assumed simply-supported on the faces x ¼0, x¼a, y¼0, and y¼b. The corresponding geometric face conditions are Uc¼U3¼0 at x ¼0 and x ¼a, and U1¼U3¼0 at y ¼0 and y¼b. Thus, in Eq. (9), g1(x, y, z)¼y(b y),

g2(x, y, z)¼x(a x), and g3(x, y, z) ¼x(a x)y(b y). The FFFF

rectangular parallelepipeds are assumed stress free on the faces

c_x / a d / b Mode Number 1 2 3 4 5 0.5 0º 0 (2.512) (3.746) (4.608) (6.270) (6.270) cx / a d / b Mode Number 5 4 3 2 1 0.5 0º 0º 0.2 (2.369) (3.536) (4.431) (5.760) (5.918) 0.5 (1.525) (2.666) (4.085) (4.149) (4.212) c_x / a d / b Mode Number 1 2 3 4 5 0.75 0.2 0.5 * (2.420) (3.639) (4.488) (5.642) (6.149) (4.622) (3.987) (3.655) (2.769) (1.593)

for a FFFF square Al/Al2O3FGM parallelepiped modeled as a moderately thick Al/Al2O3FGM plate ( ^m¼ 5, d/b ¼0, h/b ¼ 0.1, cx/a ¼0.5,a¼01); (b) mode shapes, nodal patterns andoðb2=hÞ

r=E p

for FFFF cracked square Al/Al2O3FGM parallelepipeds modeled as moderately thick cracked Al/Al2O3FGM plates ( ^m¼5, d/b ¼ 0.2,0.5, h/b¼ 0.1, cx/a ¼0.5,a¼01); (c) mode shapes, nodal patterns andoðb2=hÞ

r=E p

for FFFF cracked square Al/Al2O3FGM parallelepipeds modeled as moderately thick cracked Al/Al2O3FGM plates ( ^m ¼5, d/b¼ 0.2,0.5, h/b¼ 0.1, cx/a¼ 0.75,a¼01); (d) mode shapes, nodal patterns and

oðb2_=hÞpffiffiffiffiffiffiffiffiffi_r_=E_{for FFFF cracked square Al/Al}

2O3FGM parallelepipeds modeled as moderately thick cracked Al/Al2O3FGM plates ( ^m¼ 5, d/b ¼0.2,0.5, h/b ¼0.1, cx/a¼ 0.75,

a¼301) and (e) nodal patterns andoðb2_=hÞpffiffiffiffiffiffiffiffiffi_r_=E_{for FFFF homogeneous cracked square parallelepipeds modeled as classically-thin cracked plates (Huang and Leissa}_[2]₎ ( ^m¼ 0, d/b¼ 0.2,0.5, h/b ¼ 0.01, cy/b ¼ 0.5,0.75,a¼01, 301).

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x ¼0, x ¼a, y¼0, and y¼b. Thus, in Eq. (9), g1(x, y, z)¼g2(x, y,

z) ¼g3(x, y, z) ¼1. The CFCF rectangular parallelepipeds are

assumed clamped on the faces x¼0 and x¼ a, and stress free on the faces y¼0 and y ¼b. The corresponding clamped face condi-tions are U1¼U2¼U3¼0 at x ¼0, and U1¼U2¼U3¼0 at x ¼a. Thus,

in Eq. (9), g1(x, y, z)¼1, g2(x, y, z)¼g3(x, y, z)¼x(a x). As stated

earlier, imbedded inside the adopted Gram–Schmidt orthogona-lization procedure in Eq. (7) are associated face functions in which PðiÞ_j ðxÞ satisfy the above SSSS, FFFF, and CFCF face conditions for Ui

on the parallelepiped faces, x ¼0 and x ¼a, while QðiÞ_kðyÞ satisfy the

c_x / a α d / b Mode Number 5 4 3 2 1 0.75 30° 0° 30° 0.2 (2.428) (3.603) (4.506) (5.758) (6.112) 0.5 (1.722) (2.471) (4.169) (4.497) * (4.986) Mode Number α c / b d / a 1 2 3 4 5 0 (4.076) (5.934) (7.348) (10.54) (10.54) 0.5 0.2 (3.849) (5.698) (7.127) (9.705) (10.03) 0.5 (2.496) (4.470) (6.649) (6.990) (9.654) 0.75 0.2 (3.925) (5.825) (7.223) (9.563) (10.38) 0.5 (2.615) (4.630) (6.385) (7.584) (10.06) 0.75 0.2 (3.946) (5.768) (7.223) (9.772) (10.30) 0.5 (2.840) (4.113) (6.712) (7.520) (9.793) Fig. 4. (continued)

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SSSS, FFFF, and CFCF face conditions for Uion the parallelepiped

faces, y¼0 and y¼b. Poisson’s ratio (

n

) is set equal to 0.3 for the homogeneous parallelepipeds analyzed.

The results for cracked plates were obtained using orthogonal polynomial solution size, N^x ^Ny¼9 9 (Eq. (7)), crack

functions, ^Nc¼6 (Eq. (9)), and algebraic polynomial solution size (Eq. (7)), ^Nz¼4 (for h/b less than 0.05) and 6, taking on correspondingly cubic-order and 5th-order transverse shear deformation through the parallelepiped thickness (yielding a 1620-term solution matrix size).

cx / a d / b Mode Number 90° cx / a d / b Mode Number 1 2 3 4 5 90° 0.25 0.2 (1.628) (2.520) (4.303) (5.705) (7.689) 0.5 (1.479) (2.228) (3.876) (5.511) (6.024) 5 4 3 2 1 0.5 0.2 (1.598) (2.560) (4.355) (5.592) * (7.654) 0.5 (1.453) (2.521) (3.197) (4.915) (7.206)

Fig. 5. (a) Mode shapes, nodal patterns andoðb2_=hÞpffiffiffiffiffiffiffiffiffi_r_=E_{for CFCF homogeneous cracked rectangular parallelepipeds modeled as moderately thick cracked plates ( ^}_{m ¼0,} a/b¼ 2, d/b¼ 0.2,0.5, h/b ¼ 0.1, cx/a¼ 0.25,a¼901); (b) mode shapes, nodal patterns andoðb2=hÞ

r=E p

for CFCF homogeneous cracked rectangular parallelepipeds modeled as moderately thick cracked plates ( ^m¼0, a/b ¼ 2, d/b¼ 0.2,0.5, h/b ¼ 0.1, cx/a ¼0.5,a¼901); (c) mode shapes, nodal patterns andoðb2=hÞ

r=E p

for CFCF homogeneous cracked rectangular parallelepipeds modeled as moderately thick cracked plates ( ^m¼ 0, a/b ¼2, d/b¼ 0.2,0.5, h/b ¼0.1, cx/a ¼0.25,a¼1351) and (d) mode shapes, nodal patterns andoðb2=hÞpffiffiffiffiffiffiffiffiffir=Efor CFCF homogeneous cracked rectangular parallelepipeds modeled as moderately thick cracked plates ( ^m¼0, a/b¼ 2, d/b ¼0.2,0.5, h/b ¼ 0.1, cx/a¼ 0.5,a¼1351).

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Tables 5 and 6show the ﬁrst ﬁve non-dimensional frequency parameters for SSSS square Al/Al2O3 FGM parallelepipeds

mod-eled as moderately thick (h/b ¼0.1) and thick (h/b¼ 0.2) plates having horizontal side cracks (

a

¼01) with various length ratios (d/b) positioned at cy/b¼ 0.5. Comparisons of

o

ðb

2

=hÞpffiffiffiffiffiffiffiffiffiffiffiffi

r

c=Ec3-D solutions exact to at least three signiﬁcant ﬁgures using an orthogonal polynomial solution size ^Nx ^Ny¼11 11 (Eq. (7)), and algebraic polynomial solutions size (Eq. (7)), ^Nz¼6, for intact SSSS parallelepipeds having no cracks (i.e., d/a ¼0) with pre-viously published solutions by Mac and Huan[19]and He et al.

[21]are also shown inTable 5. Mac and Huang[19]employed a ﬁrst-order shear deformation plate theory and He et al. [21]

utilized a higher-order plate theory. The present 3-D elasticity-based

o

r

c=Ec

p

upper bounds on the exact solutions agree favorably with the

o

r

c=Ec higher-order shear deformable plate solutions of He et al. [21]up to three or four signiﬁcant ﬁgures. Yet, the present 3-D

o

r

c=Ec

p

upper bounds on the exact solutions are slightly larger than the

o

r

c=Ec solutions of Mac and Huang[19], mainly due to the full transverse shear ﬂexibilities inherent to the present 3-D

cx / a d / b cx / a d / b Mode Number 5 4 3 2 1 135° 0.25 0.2 (1.629) (2.553) (4.334) (5.737) (7.703) 0.5 (1.483) (2.333) (4.013) (5.177) (5.565) Mode Number 135° 5 4 3 2 1 0.5 0.2 (1.616) (2.568) (4.377) (5.703) * (7.770) 0.5 (1.551) (2.475) (3.378) (4.897) (6.960) Fig. 5. (continued)

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elasticity-based Ritz formulation (showing analogous trends observed and previously discussed inTable 4).

Generally speaking,

o

r

c=Ec decrease with increasing ^

m—the volume fraction parameter that governs the material

variation proﬁle in the thickness direction (see Eq. (1)). Decreasing

o

r

c=Ec with increasing ^m is not only because increasing

^

m reduces the stiffness more than it does the mass of plate, but also because

o

r

c=Ec

p

decreases with increasing

cx / a d / b cx / a d / b 5 4 3 2 1 90° 90° 0.25 0.2 (1.525) (2.353) (4.016) (5.314) * (7.139) 0.5 (1.383) (2.082) (3.618) (5.136) (5.601) Mode Number Mode Number 5 4 3 2 1 0.5 0.2 (1.497) (2.391) (4.064) (5.207) * (6.914) 0.5 (1.361) (2.355) (2.977) (4.585) * (6.592)

for CFCF cracked rectangular Al/ ZrO2FGM parallelepipeds modeled as moderately thick cracked Al/ ZrO2FGM plates ( ^m ¼5, a/b ¼ 2, d/b¼ 0.2,0.5, h/b ¼0.1, cx/a¼ 0.5,a¼901); (b) mode shapes, nodal patterns andoðb2=hÞ

ffiffiffiffiffiffiffiffiffiffiffiffi

r_c=Ec p

for CFCF cracked rectangular Al/ ZrO2 FGM parallelepipeds modeled as moderately thick cracked Al/ ZrO2FGM plates ( ^m¼5, a/b ¼ 2, d/b ¼0.2,0.5, h/b ¼ 0.1, cx/a ¼0.5,a¼901); (c) mode shapes, nodal patterns and

oðb2_=hÞ ffiffiffiffiffiffiffiffiffiffiffiffi_r c=Ec p

for CFCF cracked rectangular Al/ ZrO2FGM parallelepipeds modeled as moderately thick cracked Al/ ZrO2FGM plates ( ^m ¼5, a/b ¼ 2, d/b ¼ 0.2,0.5, h/b ¼ 0.1, cx/a¼ 0.25,a¼1351) and (d) mode shapes, nodal patterns andoðb2=hÞ

ffiffiffiffiffiffiffiffiffiffiffiffi

rc=Ec p

for CFCF cracked rectangular Al/ ZrO2FGM parallelepipeds modeled as moderately thick cracked Al/ ZrO2FGM plates ( ^m ¼5, a/b¼ 2, d/b ¼0.2,0.5, h/b¼ 0.1, cx/a¼ 0.5,a¼1351).

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parallelepiped thickness (h). As the crack length ratio (d/a) increases,

o

r

c=Ec decrease, due to the reduction in the parallelepiped stiffness. There is no clear trend between the reduction of

o

r

c=Ec

p

for SSSS parallelepipeds due to a crack and the value of m. However, two extremes in^

o

r

c=Ec trends can be seen across the wide spectrum of SSSS cracked FGM parallelepiped data reported inTables 5 and 6. A shallow crack of length ratio, d/a¼0.1, only reduce the ﬁrst ﬁve

o

r

c=Ec less than 2% relative to those for an intact SSSS parallelepiped (d/a¼0). In contrast, a deep crack of length ratio, d/a¼0.5, decreases up to 18% the second mode

o

r

c=Ecof the SSSS FGM ( ^m¼10) parallelepiped modeled as moderately thick plate (h/b¼ 0.1).

Tables 7 and 8 list the ﬁrst ﬁve nonzero

o

r

c=Ec for completely free (FFFF) square FGM parallelepipeds modeled as thin (h/b¼0.02) and moderately thick (h/b¼0.1) plates. The FGM parallelepipeds are assumed aluminum (Al) and alumina (Al2O3),

and the material properties vary as Eq. (1) with ^m ¼ 5 along the thickness direction. The effects of crack length ratios (d/a), crack positions (cx/a and cy/b), and crack inclination angles (

a

) on

o

r

c=Ecare studied. Not listed are the rigid body vibration modes (zero

o

r

c=Ec) for the FFFF parallelepipeds exam-ined. Similar to the ﬁndings ofTables 5 and 6, the

o

r

c=Ec

p inTables 7 and 8decrease with increasing FFFF FGM parallelepiped thickness and crack length (d/a). When the location of crack changes from cy/b¼0.5 to 0.75,

o

r

c=Ec p generally increases cx / a d / b c_x / a d / b 5 4 3 2 1 135° 0.25 0.2 (1.526) (2.384) (4.046) (5.344) * (7.105) 0.5 (1.388) (2.179) (3.748) (4.817) (5.181) Mode Number Mode Number 5 4 3 2 1 135° 0.5 0.2 (1.514) (2.399) (4.086) (5.310) * (7.019) 0.5 (1.453) (2.312) (3.148) (4.563) (6.479) Fig. 6. (continued)