Refined hierarchical kinematics quasi-3D Ritz models for free vibration analysis of doubly curved FGM shells and sandwich shells with FGM core

Journal of Sound and Vibration 333 (2014) 1485–1508

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Refined hierarchical kinematics quasi-3D Ritz models for free vibration analysis of doubly curved FGM shells and sandwich shells with FGM core Fiorenzo A. Fazzolari a,n, Erasmo Carrera b a b

City University London, Northampton Square, London EC1V 0HB, United Kingdom Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

a r t i c l e i n f o

abstract

Article history: Received 2 March 2013 Received in revised form 23 October 2013 Accepted 25 October 2013 Handling Editor: S. Ilanko Available online 27 November 2013

In this paper, the Ritz minimum energy method, based on the use of the Principle of Virtual Displacements (PVD), is combined with refined Equivalent Single Layer (ESL) and Zig Zag (ZZ) shell models hierarchically generated by exploiting the use of Carrera's Unified Formulation (CUF), in order to engender the Hierarchical Trigonometric Ritz Formulation (HTRF). The HTRF is then employed to carry out the free vibration analysis of doubly curved shallow and deep functionally graded material (FGM) shells. The PVD is further used in conjunction with the Gauss theorem to derive the governing differential equations and related natural boundary conditions. Donnell–Mushtari's shallow shelltype equations are given as a particular case. Doubly curved FGM shells and doubly curved sandwich shells made up of isotropic face sheets and FGM core are investigated. The proposed shell models are widely assessed by comparison with the literature results. Two benchmarks are provided and the effects of significant parameters such as stacking sequence, boundary conditions, length-to-thickness ratio, radius-to-length ratio and volume fraction index on the circular frequency parameters and modal displacements are discussed. & 2013 Elsevier Ltd. All rights reserved.

1. Introduction Functionally graded materials (FGMs) represent a class of heterogeneous composite materials made up of a mixture of ceramics and metals that are characterized by the smooth and continuous variation in properties from the bottom to the top of the considered structural element. The material properties of FGMs are controlled by the variation of the volume fraction of the constituent materials. FGMs are ultrahigh temperature-resistant materials suitable for aerospace applications such as aircraft, space vehicles, barrier coating and propulsion systems. Moreover, common laminated composite materials are characterized by a strong discontinuity of mechanical properties across the interfaces of two layers, and because of the low ratio of transverse shear modulus to in-plane modulus the failure due to delamination occurs. This drawback can be easily overcome by employing FGMs. Since the main applications of FGMs have been in high-temperature environments, in the past most of the research was oriented and restricted to thermal stress analysis. On the contrary, during the last recent years a huge amount of works have been devoted to free vibration, buckling,

n

Corresponding author. Tel.: þ44 2070408483; fax: þ44 2070408566. E-mail address: [email protected] (F.A. Fazzolari).

0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2013.10.030

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bending, fracture mechanics and many other analyses of structural components made up of FGMs. Amongst these, Loy et al. [1] and Pradhan et al. [2] studied the free vibration behavior of functionally graded cylindrical shells using Love's shell theory [3,4] and the Ritz method. The natural frequencies of simply supported functionally graded shallow shells were investigated by Matsunaga [5] using 2D higher order theory. Neves et al. [6] dealt with free vibration of functionally graded shells by a Higher order Shear Deformation Theory (HSDT) accounting for through-the-thickness deformations and radial basis functions collocation. A C0 finite element formulation based on a HSDT was presented by Pradyumna and Bandyopadhyay [7] to cope with free vibration analysis of functionally graded curved panels. Zhao et al. [8] analyzed the free vibration of functionally graded shells using the element-free Kp-Ritz method. Tornabene et al. [9] used the First-order Shear Deformation Theory (FSDT) and Generalized Differential Quadrature (GDQ) method for the analysis of functionally graded conical and cylindrical shell structures. Yang and Shen [10] investigated the free vibration and paramentric resonance of shear deformable functionally graded cylindrical panels. Free vibration characteristics of functionally graded elliptical cylindrical shells were analyzed by Patel et al. [11] using Finite Element Method (FEM) based on the theory with higher-order through-the-thickness approximations of both in-plane and transverse displacements. An exact elasticity solution for free vibration analysis of functionally graded anisotropic cylindrical shells was provided by Vel [12]. Closed-form solutions of free vibration problems of simply supported multilayered shells made of functionally graded materials were presented by Cinefra et al. [13] where variablekinematics shell models based on Carrera's Unified Formulation (CUF) were employed in conjunction with Reissner's Mixed Variation Theorem (RMVT). Free vibration analysis of functionally graded material cylindrical shells with holes was studied by Zhi-yuan and Hua-ning [14]. A three-dimensional vibration analysis of fluid-filled orthotropic FGM cylindrical shells was performed by Chen et al. [15]. Sofiyev [16] studied the vibration and stability behavior of freely supported FGM conical shells subjected to external pressure. A review on the recent research on meshless methods for functionally graded shells has been presented by Liew et al. [17]. In order to perform accurate analysis of FGM shells advanced kinematics shell models are required. During the years many theories have been developed to analyze the dynamic behavior of doubly curved shell structures such as those provided by Donnell [18,19], Mushtari [20,21], Love [3,4], Timoshenko [22], Gol'denveizer [23], Novozhilov [24], Flügge [25,26], Lur'ye [27], Byrne [28], Reissner [29], Naghdy-Berry [30], Sanders [31] and Vlasov [32,33] others were obtained expressly for circular cylindrical shells such as those given by Arnold and Warburton [34,35] and Houghton and Jhons [36] (see Leissa [37] for a comprehensive coverage). Although these theories have been successfully used for years, they are not able to describe properly the dynamic behavior of advanced FGM structures when 3D effects appear. This problem can be somehow overcome by referring to CUF-based refined ESL and ZZ shell models (see [38,39] for more details). However, on the other hand, a refinement in results cannot be only achieved by virtue of an enhancement in the kinematics description of the displacement model, but combining it with an adequate description of the curvature terms h=Ri with i ¼ α; β. In this paper the HTRF, extensively employed in the analysis of laminated composite plates and shells [40–42,38,39,43] has been extended, for the first time, to FGM shells. In order to prove the accuracy of the proposed shell models, two different challenging benchmarks, including the investigation of FGM shells and sandwich shells with FGM core, have been provided. Convergence analysis of the HTRF is carefully analyzed. Furthermore, the effects of several parameters such as stacking sequence, length-to-thickness ratio, boundary conditions, radius-to-length ratio and volume fraction index on the circular frequency parameters and modal displacements are commented.

2. FGM shell geometries The salient features of FGM shell geometries are shown in Fig. 1. Two different configurations of FGM shells are considered. More specifically, cylindrical and spherical FGM shells and cylindrical and spherical sandwich shells made up of isotropic face sheet and FGM core. The integer k, used as superscript or subscript, denotes the layer number which starts from the bottom of the shell. The layer geometry is denoted by the same symbols as those used for the whole multilayered shell and vice versa. With αk and βk the curvilinear orthogonal coordinates (coinciding with the lines of principal curvature) on the layer reference surface Ωk (middle surface of the k layer). The zk denotes the rectilinear coordinate in the normal direction with respect to the layer middle surface Ωk. The angle ϕ is commonly referred to as shallowness angle. The Γk is the g m Ωk boundary: Γk and Γk are those parts of Γk on which the geometrical and mechanical boundary conditions are imposed, respectively. These boundaries are herein considered parallel to αk or βk. For convenience the further dimensionless thickness coordinate is introduced ζ k ¼ 2zk =hk where hk denotes the thickness in Ak domain ðAk A ½zk ; zk þ 1 Þ. The following relationships hold in the given orthogonal system of curvilinear coordinates [44,45]: 1. Square of line elements ds2k ¼ ðH kα Þ2 dα2k þ ðH kβ Þ2 dβ2k þ ðH kz Þ2 dz2k

(1)

2. Area of an infinitesimal rectangle on Ωk dΩk ¼ H kα H kβ dαk dβk

(2)

F.A. Fazzolari, E. Carrera / Journal of Sound and Vibration 333 (2014) 1485–1508

Ceramic

1487

Metal

Metal Ceramic FGM core

Fig. 1. Doubly curved FGM shells and doubly curved sandwich shells with FGM core.

3. Infinitesimal volume dV k ¼ H kα H kβ H kz dαk dβk dzk

where H kα ¼ Ak1 1 þ

zk Rkα

! ;

H kβ ¼ Ak2 1 þ

zk Rkβ

(3)

! ;

H kz ¼ 1 k

(4) k

Rkα and Rkβ are the radii of curvature in the αk and βk directions, respectively; A1 and A2 are the coefficients of the first fundamental form of Ωk and are also frequently referred to as Lamé parameters on dΩk and H kα and H kβ are the Lamé parameters at zk. Attention is herein focused on shells with a constant curvature, i.e., doubly curved shells (cylindrical, spherical, hyperbolic paraboloidal geometries) for which Ak1 ¼ Ak2 ¼ 1. 3. Geometrical relationships and constitutive equations for FGMs The notation for the displacement vector is u ¼ ½uα uβ uz T

(5)

Superscript T represents the transposition operator. The stresses, r, and the strains, ε, are expressed as follows: rkpH ¼ ½skαα skββ τkαβ T ;

εkpG ¼ ½ɛ kαα ɛ kββ γ kαβ T

rknH ¼ ½τkαz τkβz skzz T ;

εknG ¼ ½γ kαz γ kβz ɛkzz T

(6)

The subscripts n and p denote transverse (out-of-plane) and in-plane components, respectively, whilst the subscripts H and G state that Hooke's law and geometric relations are used. The strain–displacement relations are εkpG ¼ Dp uk þ Ap uk εknG ¼ Dn uk þδD An uk ¼ Dnp uk þ δD An uk þ Dnz uk

(7)

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where δD is a tracer used to introduce Donnell–Mushtari's shallow shell-type approximation, Dp , Dn , Dnp and Dnz are differential matrix operators: 2∂ 3 2∂ 3 ∂ 0 0 0 ∂α ∂α ∂z 6 0 ∂ 07 6 ∂ ∂ 7 7; Dn ¼ 6 0 ∂z Dp ¼ 6 ∂β ∂β 7 4 5 4 5; ∂ ∂ ∂ 0 0 0 ∂β ∂α ∂z 2

0

6 Dnp ¼ 4 0 0

0 0 0

3

∂ ∂α ∂ 7 ∂β 5;

0

2

∂ ∂z

0

6 Dnz ¼ 4 0 0

0

07 5

∂ ∂z

0 

(8)

∂ ∂z

0

Ap and An are matrices containing geometrical parameter operators: 2 3 2 1 0 0  k1 k Hkα Rkα 7 6 6 Hα Rα 6 7 6 1 Ap ¼ 6 0 0 7; A n ¼ 6 0 H kβ Rkβ 5 4 4 0 0 0 0

3

0 1

H kβ Rkβ

0

3

7 7 07 5 0

(9)

The 3D constitutive equations according to Hooke's law are given as rk ¼ Ck εk

(10)

By using Eq. (6), the previous equation becomes k k rkpH ¼ C~ pp ðzÞεkpG þ C~ pn ðzÞεknG k k rknH ¼ C~ np ðzÞεkpG þ C~ nn ðzÞεknG

where matrices

k C~ pp ðzÞ,

k C~ nn ðzÞ,

k C~ pn ðzÞ

and

2~

C 11 ðzÞ k 6 C~ pp ðzÞ ¼ 4 C~ 12 ðzÞ

k C~ np ðzÞ

are

C~ 12 ðzÞ C~ 22 ðzÞ

0

0 2

(11)

0 k 6 C~ pn ðzÞ ¼ 4 0

0 0

0

0

0 0

3k 7 5 ;

C~ 66 ðzÞ 3k C~ 13 ðzÞ 7 C~ 23 ðzÞ 5 ;

2~ C 55 ðzÞ

k 6 C~ nn ðzÞ ¼ 4

2 k 6 C~ np ðzÞ ¼ 4

0

0

0

0

C~ 44 ðzÞ

0

0

0

0

0

0

0 C~ 23 ðzÞ

C~ 13 ðzÞ

C~ 33 ðzÞ 3 0 k 07 5 0

3k 7 5 ;

(12)

The elastic coefficients in Eq. (12) C~ ij are computed by the following three steps (see [6]): 1. Computation of volume fraction of ceramic and metal phases. 2. Computation of elastic properties, Young's modulus Ek and Poisson's coefficient νk. 3. Computation of elastic coefficients C~ ij . The volume fraction of the ceramic and metal phases is defined according to the following power-law:     1 z p h h þ ; zA  ; V kc ðzÞ ¼ 2 h 2 2

(13)

h is the thickness of the shell and the exponent p is the volume fraction index indicating the material variation through the thickness direction. The volume fraction of the metal phase is given as V km ðzÞ ¼ 1  V kc ðzÞ. In the case of sandwich shells with FGM core the volume fraction of the FGMs is assumed to obey a power-law function along the thickness direction: 8 k V ðzÞ ¼ 0; z A ½h1 ; h2  > > > m  p > < z  h2 k V ðzÞ ¼ ; z A ½h2 ; h3  (14) h3  h2 > m > > > : V k ðzÞ ¼ 1; z A ½h ; h  3

m

4

hc ¼ h3 h2 is the thickness of the FGM core. The volume fraction of the ceramic phase is given as V kc ðzÞ ¼ 1  V km ðzÞ. Young's modulus Ek , Poisson's coefficient νk and density ρk are computed by the law-of-mixtures: 8 k > E ðzÞ ¼ ðEm Ec ÞV km ðzÞ þ Ec > < νk ðzÞ ¼ ðνm  νc ÞV km ðzÞ þ νc (15) > > : ρk ðzÞ ¼ ðρ  ρ ÞV k ðzÞ þρ m

c

m

c

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Finally the elastic coefficients C~ ij are given as 8 Ek ðzÞ½1  ðνk ðzÞÞ2  > > ~k ~k ~k > > C 11 ðzÞ ¼ C 22 ðzÞ ¼ C 33 ðzÞ ¼ > 1  3ðνk ðzÞÞ2  2ðνk ðzÞÞ3 > > > > < k k k Ek ðzÞ½νk ðzÞ  ðνk ðzÞÞ2  C~ 12 ðzÞ ¼ C~ 12 ðzÞ ¼ C~ 23 ðzÞ ¼ > 1  3ðνk ðzÞÞ2  2ðνk ðzÞÞ3 > > > > > k > k k Ek ðzÞ > > : C~ 44 ðzÞ ¼ C~ 55 ðzÞ ¼ C~ 66 ðzÞ ¼ 2ð1 þ νk ðzÞÞ

1489

(16)

4. Refined hierarchical shell models In the analysis of metallic and composite plates and shells the 3D elastic problem is generally reduced to a 2D one by exploiting the use of axiomatic assumptions coming from some pioneering insights due to eminent scientists and researchers. The simplest plate/shell theory is based on Kirchhoff/Love's hypothesis and it is usually referred to as Classical Lamination Theory (CLT) [4,46]. Both transverse shear strains and transverse normal strain are discarded, being in many practical applications negligible with respect to the in-plane ones, 8 ∂uz0 ðα; β; tÞ > > > uα ðα; β; z; t Þ ¼ uα0 ðα; β; t Þ  z > > ∂α < ∂uz0 ðα; β; tÞ (17) u ð α; β; z; t Þ ¼ u ð α; β; t Þ z β β0 > > ∂β > > > : uz ðα; β; z; tÞ ¼ uz0 ðα; β; tÞ The inclusion of transverse shear strains, in the aforementioned theory, leads to Reissner–Mindlin theory even known as FSDT [29,47], 8 > < uα ðα; β; z; tÞ ¼ uα0 ðα; β; tÞ þ zuα1 ðα; β; tÞ uβ ðα; β; z; tÞ ¼ uβ0 ðα; β; tÞ þ zuβ1 ðα; β; tÞ (18) > : u ðα; β; z; tÞ ¼ u ðα; β; tÞ z

z0

Although these theories have been successfully employed for years, due to their inconsistency in discarding the transverse normal stress in the material constitutive equations, are no longer valid when 3D local effects appear. To remove the inconsistency completely, higher-order expansion of the unknown with respect to the z coordinate is needed. According to the above considerations CUF, well known in the static and dynamics analysis of layered beams, plates and shells, overcomes the drawback generating a large variety of 2D and quasi-3D hierarchical plate/shell models using a unified approach. Its accuracy has been proved in many applications ranging from multifield to aeroelastic problems and it turned out to be a powerful tool to deal with metallic and composite laminated beams, plates and shells. In this paper the capability to expand each displacement variable in the displacement field at any desired order independently from the others and regarding the accuracy and the computational cost has been introduced. Such artifice permits to treat each variable independently from the others and this becomes extremely useful when multifield problems such as thermoelastic and piezoelectric applications [38,48] are investigated. Thereby, following this approach the displacement field can be written as 8 u ðα; β; z; tÞ ¼ F τuα ðzÞuατuα ðα; β; tÞ; τuα ¼ 0; 1; …; N uα > > < α uβ ðα; β; z; tÞ ¼ F τuβ ðzÞuβτuβ ðα; β; tÞ; τuβ ¼ 0; 1; …; Nuβ (19) > > : uz ðα; β; z; tÞ ¼ F τu ðzÞuzτu ðα; β; tÞ; τuz ¼ 0; 1; …; Nuz z z and in the compact form u ¼ Fτ uτ ;

τ ¼ τuα ; τuβ ; τuz

(20)

where 2

F τ uα 6 0 Fτ ¼ 6 4 0

0 F τ uβ 0

0

3

7 0 7; 5 F τ uz

8 u > < ατuα uτ ¼ uβτuβ > :u zτuz

9 > = > ;

(21)

F τuα , F τuβ , F τuz are the thickness functions. The functions uατuα , uβτuβ , uzτuz are the displacement vector components and N uα , Nuβ and N uz are the orders of expansion. According to Einstein's notation, the repeated subscripts τuα , τuβ , τuz indicate summation. An example of a possible ESL displacement field according to the unified formulation in Eq. (19) is given below, the

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expansion indexes are Nuα ¼ 6, Nuβ ¼ 2, N uz ¼ 4: 8 2 3 4 5 6 > < uα ¼ uα0 þzuα1 þz uα2 þ z uα3 þ z uα4 þz uα5 þ z uα6 uβ ¼ uβ0 þ zuβ1 þ z2 uβ2 > : u ¼ u þ zu þ z2 u þz3 u þ z4 u z

z0

z1

z2

z3

(22)

z4

Classical ESL shell models do not describe the ZZ form of the displacement field in the shell thickness direction. Such a limitation could somehow be overcome by referring to Murakami's idea. Murakami [49] proposed adding the ZZ function MðzÞ ¼ ð 1Þk ζk to Eq. (19). This function is generally referred to as Murakami's Zig Zag function (MZZF). Such a function permits one to reproduce the discontinuity of the first derivative of the displacement variables in the z-direction which physically comes from the intrinsic transverse anisotropy of multilayer structures. An exhaustive and comprehensive coverage along with some clarifying illustrations about the ESL, ZZ as well as LW (Layer Wise) assembly procedure from layer to multilayer is available in [41,42,50–52]. 4.1. The hierarchical trigonometric Ritz formulation The PVD is the variational statement employed both to develop the HTRF and to derive the governing differential equations with natural boundary conditions of the doubly curved shell structures. The explicit form of the PVD at multilayer level can be written as Nl Z Z Nl T T ∑ ðδεkpG rkpC þδεknG rknC Þ dΩk dz ¼  ∑ δLkF in (23) k¼1

Ωk Ak

k¼1

The HTRF, herein proposed for doubly curved shells, is based on the so-called Ritz fundamental primary and secondary nuclei, which can be developed in a systematic manner in the following four steps [43]: 1. 2. 3. 4.

The The The The

choice of the variational statement. introduction of the stress–strain constitutive relationships. choice of the Ritz functions. use of the geometric relations.

The first step is fulfilled by using the PVD as variational statement (see Eq. (23)), the second step by selecting the stress– strain constitutive relationships given in Eq. (11). The third step is the definition of the displacement field in terms of Ritz functions. In particular, in the Ritz method the displacement amplitude vector components uατuα , uβτu and uzτuz individuate β the maximum amplitude in the oscillation that maximizes the related virtual internal and inertial works. They are expressed in series expansion as N

uατuα ¼ ∑ U ατuα i ψ αi eiωij t ; i

N

uβτu ¼ ∑ U βτuβ i ψ βi eiωij t ; β

i

N

uzτuz ¼ ∑ U zτuz i ψ zi eiωij t

(24)

i

pffiffiffiffiffiffiffiffi where N indicates the order of expansion in the approximation, i ¼  1, t is the time and ωij the circular/angular frequency. U ατuα i , U βτuβ i , U zτuz i are the unknown coefficients, ψ αi , ψ βi , ψ zi are the Ritz functions appropriately selected making reference to the features of the analyzed problem. Convergence to the exact solution is guaranteed if the basis functions are admissible functions in the used variational principle [40,41,53,54]. The harmonic displacement field is given as N

N

uα ¼ ∑ F τuα U ατuα i ψ αi eiωij t ;

uβ ¼ ∑ F τuβ U βτuβ i ψ βi eiωij t ;

i

i

N

uz ¼ ∑ F τuz U zτuz i ψ zi eiωij t

(25)

i

or in the compact form u ¼ Fτ Uτi Ψi

(26)

where 2

U ατuα i eiωij t

3

6 7 iω t 7 6 Uτi ¼ 6 U βτuβ i e ij 7; 4 5 U zτuz i eiωij t

2

ψ αi

6 Ψi ¼ 4 0 0

0 ψ βi 0

0

3

0 7 5; ψ zi

2

F τ uα 6 0 Fτ ¼ 6 4 0

0 F τ uβ 0

0

3

7 0 7 5 F τ uz

(27)

In the fourth step, by writing stresses and strains in terms of displacement components given in Eq. (7) and substituting them in Eq. (23) the explicit expressions of the internal virtual work and the virtual work done by the inertial forces in terms

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1491

of Ritz functions and unknown coefficients are obtained: Z Z k δðLkint Þmax ¼ δUTτi  ð½ðDp ðFτ Ψi ÞÞT þðAp ðFτ Ψi ÞÞT ½C~ pp Dp ðFs Ψj Þ Ωk Ak

k k k k þ C~ pp Ap ðFs Ψj Þ þ C~ pn Dnp ðFs Ψj Þ þ C~ pn δD An ðFs Ψj Þ þ C~ pn Dnz ðFs Ψj Þ k þ ½½Dnp ðFτ Ψi ÞT þ½δD An ðFτ Ψi ÞT þ ½Dnz ðFτ Ψi ÞT ½C~ np Dp ðFs Ψj Þ k k k k þ C~ np Ap ðFs Ψj Þ þ C~ nn Dnp ðFs Ψj Þ þ C~ nn δD An ðFs Ψj Þ þ C~ nn Dnz ðFs Ψj ÞÞ

Usj dΩk dz

δðLkFin Þmax ¼

Z Z Ωk Ak

€ sj dΩk dz δUTτi ½ρk ðFτ Ψi ÞT ðFs Ψj ÞU

(28)

The above quadratic forms lead to the Ritz fundamental primary stiffness and mass nuclei: Z Z k ð½ðDp ðFτ Ψi ÞÞT þ ðAp ðFτ Ψi ÞÞT ½C~ pp Dp ðFs Ψj Þ Kkτsij ¼ Ωk Ak

k k þ C~ pp Ap ðFs Ψj Þ þ C~ pn Dnp ðFs

k k Ψj Þ þ C~ pn δD An ðFs Ψj Þ þ C~ pn Dnz ðFs Ψj Þ

k þ ½½Dnp ðFτ Ψi ÞT þ ½δD An ðFτ Ψi ÞT þ½Dnz ðFτ Ψi ÞT ½C~ np Dp ðFs Ψj Þ k k k k þ C~ np Ap ðFs Ψj Þ þ C~ nn Dnp ðFs Ψj Þ þ C~ nn δD An ðFs Ψj Þ þ C~ nn Dnz ðFs Ψj ÞÞ dΩk dz

Mkτsij ¼

Z Z Ωk Ak

ðρk ½ðFτ Ψi ÞT ðFs Ψj ÞÞ dΩk dz

(29)

At this stage it is useful to introduce the following thickness integrals in order to write in a concise manner the explicit form of the Ritz fundamental secondary nuclei. 8 ! k k > R > kτs kτs kτs kτs kτs kτs k k Hα Hβ k k > > ; J ; J ; J ; J ; J ¼ F F 1; H ; H ; ; ; H H J k τ s > α β α=β β=α αβ α β α β dz A > > H kβ H kα > > > ! > > k k > R ∂F τ > kτz s kτz s kτz s kτz s kτz s kτz s k k Hα Hβ k k > > F ; J ; J ; J ; J ; J ¼ 1; H ; H ; ; ; H H J k > s α α β α β dz β αβ α=β β=α A ∂z > < H kβ H kα ! (30) k > R ∂F s > Hkβ kτsz kτsz kτsz kτsz k k Hα k k > z > 1; H J kτsz ; J kτs ; J ; J ; J ; J ¼ F ; H ; ; ; H H dz k τ k > α α β α β β αβ α=β β=α A > ∂z > H kβ Hα > > > ! > k > k > R ∂F τ ∂F s > kτz sz kτz sz kτz sz kτz sz kτz sz kτz sz k k Hα Hβ k k > > 1; H ; J ; J ; J ; J ; J ¼ ; H ; ; ; H H J k > α α β α β dz β αβ α=β β=α A ∂z ∂z > : Hkβ H kα where τ ¼ τuα ; τuβ ; τuz and s ¼ suα ; suβ ; suz . To the same aim, it is convenient to rewrite the Ritz functions involved in the inplane integrals, namely ψ αi , ψ βi , ψ zi , as 8 M N > > u u > > ψ αi ðα; βÞ ¼ ∑∑ϕmα ðαÞϕnα ðβÞ > > m n > > > < M N u u ψ βi ðα; βÞ ¼ ∑∑ϕmβ ðαÞϕnβ ðβÞ (31) > m n > > > > M N > > u u > > ψ zi ðα; βÞ ¼ ∑∑ϕmz ðαÞϕnz ðβÞ : m n by using Eq. (31) the general expressions of the in-plane integrals can be written as Z a ξ ζ d ϕm ðαÞ d ϕp ðαÞ i ξζ dα m ¼ 1; …; M; p ¼ 1; …; P; j I mp ¼ ξ dα dαζ 0 Z b ξ ζ d ϕn ðβÞ d ϕq ðβÞ i ξζ dβ n ¼ 1; …; N; q ¼ 1; …; Q ; j I nq ¼ dβζ dβξ 0

(32)

where i; j ¼ uα ; uβ ; uz and ξ, ζ are differentiation orders. The Ritz functions in Eq. (31) are chosen to satisfy the simply supported and fully clamped boundary conditions, following indicated, respectively: SSSS

α ¼ 0; a; ) uβ ¼ uz ¼

∂uz ¼ 0; ∂α

β ¼ 0; b; ) uα ¼ uz ¼

∂uz ¼ 0; ∂β

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mπα ; a m   M mπα u ; ϕmβ ðαÞ ¼ ∑ sin a m mπα M ; ϕumz ðαÞ ¼ ∑ sin a m M

ϕumα ðαÞ ¼ ∑ cos

  nπβ ; b n   N nπβ u ϕnβ ðβÞ ¼ ∑ cos ; b n   N nπβ ϕunz ðβÞ ¼ ∑ sin ; b n N

ϕunα ðβÞ ¼ ∑ sin

(33)

∂uz ∂uz ¼ 0; β ¼ 0; b; ) uα ¼ uβ ¼ uz ¼ ¼ 0; ∂α ∂β   mπα M N nπβ ; ϕunα ðβÞ ¼ ∑ sin ; ϕumα ðαÞ ¼ ∑ sin a b m n   mπα M N nπβ u u ; ϕnβ ðβÞ ¼ ∑ sin ; ϕmβ ðαÞ ¼ ∑ sin a b m n   mπα M N nπβ ; ϕunz ðβÞ ¼ ∑ sin ; ϕumz ðαÞ ¼ ∑ sin a b m n

α ¼ 0; a; ) uα ¼ uβ ¼ uz ¼

CCCC

(34)

Therefore, considering Eqs. (30) and (32), the explicit forms of the Ritz fundamental secondary stiffness nuclei are reported in the following: k kτ s uα 00uα 11uα ~ k kτuα suα uα I 10uα u uα I 01uα u þ C~ k J kτuα suα K τuuααusαuα ¼ C~ 11 J β=αuα uα uα 16 uα I mp uα uα I nq uα þ C 16 J α uα nq α uα mp

þ C~ 66 J α=βuα k

kτ

11 α;z ~ I 00 mp I nq þ C 55 J αβ k

s uα

kτu

suα;z uα 00uα uα 00uα uα I mp uα uα I nq uα þδD

 C~ 55

1

k

uα 10uα uα 01uα uα I mp uα uα I nq uα

kτuα;z suα uα 00uα uα 00uα uα I mp uα uα I nq uα

Jβ

Rkα

k 1 kτu su 1 kτ s uα 00uα uα 00uα 00uα 00uα ~k J β=αuα uα uα  C~ 55 k J β α α;z uα uα I mp uα uα I nq uα þ C 55 uα I mp uα uα I nq uα Rα ðRkα Þ2

τ uα s u

!

kτuα suβ uα 11uβ uα 00uβ uα 01uβ uα 10uβ β β ~ k kτuα suβ uα I 10u ~ k kτuα suβ uα I 01u uβ I mp uα uβ I nq uα þ C 12 J uβ mp uα uβ I nq uα þ C 66 J uβ mp uα uβ I nq uα

K uα uβ β ¼ C~ 16 J β=α k

kτuα suβ

þ C~ 26 J α=β k

kτuα suβ uα 00uβ uα 00uβ uβ I mp uα uβ I nq uα þδD

11 ~ I 00 mp I nq þ C 45 J αβ k

 C~ 45 k

1

kτuα suβ;z uα 10uβ uα 00uβ ~k uβ I mp uα uβ I nq uα þ C 45

Jβ

Rkα

1 Rkα Rkβ

 C~ 45

1

k

Rkβ

J kτuα;z suβ;z

kτuα;z suβ uα 00uβ uα 00uβ uβ I mp uα uβ I nq uα

Jα

!

uα 00uβ uα 00uβ uβ I mp uα uβ I nq uα

k τ s kτ s uα;z suz uα 00uz uα 00uz uα 01uz 10uz ~ k kτuα;z suz uα I 01uz u uα I 00uz u þ C~ k J kτ K uuααuzuz ¼ C~ 13 J β uα uz uα uz I mp uα uz I nq uα þ C 55 J β uz mp 45 α α uz nq α uz I mp uα uz I nq uα k kτu su uα 01uz 00uz ~ k 1 J kτuα suz uα I 10uz u uα I 00uz u þ C~ k 1 J kτuα suz uα I 10uz u uα I 00uz u þ C~ 36 J α α;z z uα uz mp 12 α uz nq α α uz nq α uz mp uz I mp uα uz I nq uα þ C 11 β=α Rkα Rkβ

þ C~ 16 k

1

uα 00uz 01uz ~ J kτuα suz uα uz I mp uα uz I nq uα þ C 26

1

k

Rkα

Rkβ

kτ

J α=βuα

suz uα 01uz uα 00uz uz I mp uα uz I nq uα

k 1 kτ s uα 10uz 00uz ~ k 1 J kτuα suz uα I 00uz u uα I 01uz u δD C~ 55 k J β=αuα uz uα uz I mp uα uz I nq uα þ C 45 α uz nq α uz mp Rα Rkα

τ u s uα kτu suα k uβ 00uα 11uα ~ k kτuβ suα K uββuα ¼ C~ 16 J β=αβ uβ uα I mp uβ uα I nq uβ þ C 12 J

uβ 10uα uβ 01uα ~ k kτuβ suα uβ I 10uα u uβ I 01uα u β uα nq β uα I mp uβ uα I nq uβ þ C 66 J uα mp

kτu suα uβ 00u uβ 11uα α ~ k kτuβ suα uβ I 00uα u uβ I 00uα u þ δD uα I mp uβ uα I nq uβ þ C 45 J αβ β uα nq β uα mp

þ C~ 26 J α=ββ k

k 1 kτu  C~ 45 k J α β Rβ

τ u su

!

suα;z uβ 00u uβ 00uα α ~k uα I mp uβ uα I nq uβ þ C 45

 C~ 45 k

1 Rkα

kτuβ;z suα uβ 00u uβ 00uα α uα I mp uβ uα I nq uβ

Jβ

!

1

uβ 00uα 00uα J kτuβ;z suα;z uβ uα I mp uβ uα I nq uβ Rkα Rkβ

kτu suβ uβ 11uβ uβ 00uβ uβ 10uβ uβ 01uβ β β ~ k kτuβ suβ uβ I 01u ~ k kτuβ suβ uβ I 10u uβ I mp uβ uβ I nq uβ þ C 26 J uβ mp uβ uβ I nq uβ þ C 26 J uβ mp uβ uβ I nq uβ

K uββuβ β ¼ C~ 66 J β=αβ k

kτu suβ uβ 00uβ uβ 11uβ uβ 00uβ β ~ k kτuβ;z suβ;z uβ I 00u uβ I mp uβ uβ I nq uβ þ C 44 J αβ uβ mp uβ uβ I nq uβ

þ C~ 22 J α=ββ k

þ δD

k 1 kτu  C~ 44 k J α β;z Rβ

suβ uβ 00uβ uβ 00uβ ~k uβ I mp uβ uβ I nq uβ  C 44

1

kτu su 00uβ uβ 00uβ ~k J α β β;z uβ uβ I mp uβ uβ I nq uβ þ C 44 Rkβ

1

kτu su 00uβ uβ 00uβ J β=ββ β uβ uβ I mp uβ uβ I nq uβ ððRkβ Þ2

!

F.A. Fazzolari, E. Carrera / Journal of Sound and Vibration 333 (2014) 1485–1508 τ u s uz

1493

kτuβ;z suz uβ 01u uβ suz;z uβ 00u uβ 00uz uβ 10uz z z ~ k kτuβ;z suz uβ I 00uz u uβ I 01uz u þ C~ k J kτ uz I mp uβ uz I nq uβ þ C 44 J α 23 α β uz nq β uz mp uz I mp uβ uz I nq uβ

K uββuz ¼ C~ 45 J β k

1

þ C~ 12 k

Rkα

uβ 10uz uβ 00uz ~k uz I mp uβ uz I nq uβ þ C 22

J kτuβ suz

þ C~ 16

1

k

k kτu su τ s K uuzzuαuα ¼ C~ 13 J β z;z α

1

k

1 Rkβ

uβ 00uz uβ 01uz ~k uz I mp uβ uz I nq uβ þ C 44

J kτuβ suz

uβ 10uz uβ 00uz uz I mp uβ uz I nq uβ

!

1

kτu suz;z uβ 00u uβ 01uz z J α=ββ uz I mp uβ uz I nq uβ Rkβ

uz suα;z uz 00uα uz 00uα uz 10uα uz 10uα ~ k kτuz;z suα uz I 00uα u uz I 01uα u þ C~ k J kτ 45 α z uα nq z uα mp uα I mp uz uα I nq uz uα I mp uz uα I nq uz þ C 36 J α

1

þ C~ 11 k

Rkα

1 Rkβ

kτ suα uz 00uα uz 10uα ~ k kτuz suα;z uz I 10uα u uz I 00uα u uα I mp uz uα I nq uz þ C 55 J β z uα nq z uα mp

J α=βuz

1

kτ suα uz 00uα uz 10uα ~k uα I mp uz uα I nq uz þ C 12

J β=αuz

k 1 kτ s  δD C~ 55 k J β=αuz uα Rα

τ uz s u

kτu suz uβ 00u uβ 10uz z ~ k kτuβ suz;z uβ I 10uz u uβ I 00uz u β uz nq β uz mp uz I mp uβ uz I nq uβ þ C 36 J β

J α=ββ

kτu suz uβ 00u uβ 10uz z ~k uz I mp uβ uz I nq uβ þ C 26

uz 10uα uz 00uα ~k uα I mp uz uα I nq uz þ C 26

J kτuz suα

Rkα

Rkβ

J β=αβ

k 1 C~ 45 k J kτuβ suz Rβ

 δD

þ C~ 16

Rkα

1

Rkβ

uz 01uα uz 00uα ~k uα I mp uz uα I nq uz þ C 45

J kτuz suα

uz 10uα uz 00uα uα I mp uz uα I nq uz

!

1

uz 01uα uz 00uα uα I mp uz uα I nq uz

J kτuz suα;z

Rkα

kτuz;z suβ uz 01uβ uz 00uβ uz 01uβ uz 10uβ β β ~ k kτuz;z suβ uz I 00u ~ k kτuz suβ;z uz I 00u uβ mp uz uβ I nq uz þ C 44 J α uβ mp uz uβ I nq uz uβ I mp uz uβ I nq uz þ C 23 J α

K uz uβ β ¼ C~ 36 J β k

þ C~ 12 k

1 Rkα

J kτuz suβ

uz 10uβ uz 00uβ ~k uβ I mp uz uβ I nq uz C 22

þ C~ 16

1

k

Rkα

1 Rkβ

kτuz suβ uz 00uβ uz 10uβ uz 00uβ β ~ k kτuz suβ;z uz I 10u uβ I mp uz uβ I nq uz þ C 45 J β uβ mp uz uβ I nq uz

J α=β

1

kτuz suβ uz 00uβ uz 10uβ ~k uβ I mp uz uβ I nq uz þ C 26

J β=α

k 1 kτuz su δD C~ 45 k J β=α β Rα

k τ s kτ s 11uz K uuzzuzuz ¼ C~ 55 J β=αuz uz uz uz I mp uz

Rkβ

uz 00uβ uz 01uβ ~k uβ I mp uz uβ I nq uz þ C 44

J kτuz suβ

1 Rkα

uz 10uβ uz 00uβ uβ I mp uz uβ I nq uz

! J kτuz suβ;z

uz 01uβ uz 00uβ uβ I mp uz uβ I nq uz

uz 00uz ~ k kτuz suz uz I 10uz u uz I 01uz u þ C~ k J kτuz suz uz I 01uz u uz I 10uz u 45 uz I nq uz þ C 45 J z uz nq z z uz nq z uz mp uz mp

þ C~ 44 J α=βuz k

þ C~ 23 k

1 Rkβ

kτ suz uz 00uz uz 11uz ~ k kτuz;z suz;z uz I 00uz u uz I 00uz u þ C~ k uz I mp uz uz I nq uz þ C 33 J αβ 13 z uz nq z uz mp kτuz;z suz uz 00uz uz 00uz ~k uz I mp uz uz I nq uz þ C 13

Jα

þ C~ 11 k

þ C~ 12 k

1 ðRkα Þ2 1 ðRkα Rkβ Þ

1 Rkα

kτuz suz;z uz 00uz uz 00uz ~k uz I mp uz uz I nq uz þ C 23

Jβ

kτ suz uz 00uz uz 00uz ~k uz I mp uz uz I nq uz þ C 12

J β=αuz

J kτuz suz

1 Rkα

uz 00uz uz 00uz ~k uz I mp uz uz I nq uz þ C 22

1 ðRkα Rkβ Þ 1 ðRkβ Þ2

J kτuz suz kτ

J α=βuz

kτuz;z suz uz 00uz uz 00uz uz I mp uz uz I nq uz

Jβ

1 Rkβ

kτuz suz;z uz 00uz uz 00uz uz I mp uz uz I nq uz

Jα

uz 00uz uz 00uz uz I mp uz uz I nq uz

suz uz 00uz uz 00uz uz I mp uz uz I nq uz

(35)

and the Ritz fundamental secondary mass nuclei are given as

¼ 0;

τuα su uα 00uα uα 00uα M uα uβ β ¼ 0; uα I mp uα uα I nq uα τ u su 00uβ uβ 00uβ Muββuβ β ¼ ρk J kτuβ suβ uβ uβ I mp uβ uβ I nq uβ

¼ 0;

τ uz s u M uz uβ β

τ

M τuuααusαuα ¼ ρk J kτuα suα

M uuααuzuz ¼ 0

τ u s uα M uββuα

τ u s uz M uββuz

τ s M uuzzuαuα

¼ 0;

τ s M uuzzuzuz

¼

s

¼0

uz 00uz 00uz ρk J kτuz suz uz uz I mp uz uz I nq uz

(36)

The Ritz method leads to the discrete form of the governing differential equations in terms of the Ritz primary fundamental nuclei: T

δUkτi : ½Kkτsij  ω2ij Mkτsij Uksj ¼ 0

(37)

The free-vibration response of the multilayered shells leads to the following eigenvalues problem: J Kkτsij  ω2ij Mkτsij J ¼ 0 The double bars denote the determinant.

(38)

1494

F.A. Fazzolari, E. Carrera / Journal of Sound and Vibration 333 (2014) 1485–1508

5. Governing differential equations of doubly curved shells In order to derive the governing differential equations and natural boundary conditions the Gauss theorem is applied: Z Z Z ððDp Þδak ÞT ak dΩk ¼  δak ððDp ÞT δak Þ dΩk þ δak ððIp ÞT δak Þ dΓ k Ωk Ωk Γk Z Z Z ððDnp Þδak ÞT ak dΩk ¼  δak ððDnp ÞT δak Þ dΩk þ δak ððInp ÞT δak Þ dΓ k (39) Ωk

Ωk

Γk

where a can be displacement or stress variables and the introduced Ip and Inp arrays are 2 nα 3 2 3 0 0 0 0 nαk Hk Hα 6 α n 7 β 60 7 6 0 7; I ¼ 6 0 0 n β 7 7 Ip ¼ 6 Hkβ np 6 7 4 Hkβ 5 4 nβ nα 5 0 0 0 0 H kβ Hkα The normal to the boundary of domain Ωk is

"

nα

n^ ¼

nβ

#

" ¼

cos ðφα Þ

(40)

# (41)

cos ðφβ Þ

where φα and φβ are the direction cosines, namely the angles between the normal n^ and the directions α and β, respectively. m The governing differential equations and natural boundary conditions (Neumann-type) on Γk , for a doubly curved anisotropic composite shell at multilayer level can be written as Nl Z Z k k k ∑ δuτ ð  ½ðFτ ÞT ðDp ÞT þ ðFτ ÞT ðAp ÞT ½C~ pp Dp ðFτ Þ þ C~ pp Ap ðFτ Þ þ C~ pn Dnp ðFτ Þ Ωk Ak

k¼1

k k þ C~ pn δD An Dnp ðFτ Þ þ C~ pn Dnz ðFτ Þ ½ðFτ ÞT ðDnp ÞT þ ðFτ ÞT ðδD An ÞT þðFτ ÞT ðDnz ÞT 

þ ½C~ np Dp ðFτ Þ þ C~ np Ap ðFτ Þ þ C~ nn Dnp ðFτ Þ þ C~ nn δD An Dnp ðFτ Þ þ C~ nn Dnz ðFτ ÞÞus dΩk dz Nl Z Z k k k k þ ∑ δuτ ððFτ ÞT ðI p ÞT ½C~ pp Dp ðFτ Þ þ C~ pp Ap ðFτ Þ þ C~ pn Dnp ðFτ Þ þ C~ pn δD An ðFτ Þ k

k

k

k

k

Γ k Ak

k¼1

k k k k k þ C~ pn Dnz ðFτ Þ þ ðFτ ÞT ðI np ÞT ½C~ np Dp ðFτ Þ þ C~ np Ap ðFτ Þ þ C~ nn Dnp ðFτ Þ þ C~ nn δD An ðFτ Þ Nl

þ C~ nn Dnz ðFτ ÞÞus dΩk dz ¼ ∑ k

k¼1

Z Z

Ωk Ak

δuτ ðρk ðFτ ÞT ðFs ÞÞu€ s dΩk dz

(42)

and in compact form are kτs € Kkτs u u us þMu u u s ¼ 0

δuτ : Γm k : where

Z Kkτs uu ¼

Ak

kτs Πkτs u u us ¼ Π u u u s ;

Γ gk : us ¼ u s

(43)

k k k ð  ½ðFτ ÞT ðDp ÞT þ ðFτ ÞT ðAp ÞT ½C~ pp Dp ðFτ Þ þ C~ pp Ap ðFτ Þ þ C~ pn Dnp ðFτ Þ

k k þ C~ pn δD An Dnp ðFτ Þ þ C~ pn Dnz ðFτ Þ ½ðFτ ÞT ðDnp ÞT þ ðFτ ÞT ðδD An ÞT þðFτ ÞT ðDnz ÞT  k k k k k þ ½C~ np Dp ðFτ Þ þ C~ np Ap ðFτ Þ þ C~ nn Dnp ðFτ Þ þ C~ nn δD An Dnp ðFτ Þ þ C~ nn Dnz ðFτ ÞÞH kα H kβ dz

Z Πkτs uu¼

ððFτ ÞT ðI p ÞT ½C~ pp Dp ðFτ Þ þ C~ pp Ap ðFτ Þ þ C~ pn Dnp ðFτ Þ þ C~ pn δD An ðFτ Þ k

Ak

k

k

k

þ C~ pn Dnz ðFτ Þþ ðFτ ÞT ðI np ÞT ½C~ np Dp ðFτ Þ þ C~ np Ap ðFτ Þ þ C~ nn Dnp ðFτ Þ þ C~ nn δD An ðFτ Þ k

k

k

k

k

þ C~ nn Dnz ðFτ ÞÞH kα H kβ dz k

Z Mukτsu ¼

Ak

ρk ðFτ ÞT ðFs ÞH kα H kβ dz

The nine components of the fundamental primary differential nucleus Ku u are reported in the following:         k k ∂ ∂ ∂ ∂ kτ s uα suα ¼  C~ 11 J β=αuα uα  C~ 16 J kτuα suα K kτ uu ∂α suα ∂α τuα ∂α τuα ∂β suα

(44)

F.A. Fazzolari, E. Carrera / Journal of Sound and Vibration 333 (2014) 1485–1508

1495

Table 1 Doubly curved FGM shells. 

Ec ½GPa

ρc

Alumina 380 Silicon nitride (Si3N4) 322.2715 Em ½GPa

Aluminium 70 Stainless steel (SUS304) 207.7877

Kg m3



νc

3000

0.3

2370   kg ρm m3

0.24

2707

0.3

8166

0.31776

νm

Table 2 Doubly curved sandwich shells with FGM core. 

Ec ½ GPa

ρc

Alumina 380 Em ½ GPa

Aluminium 70

kg m3



3800   kg ρm m3

0.3

2707

0.3

p=10

0.4

0.4

p=10

0.3

0.3 p=5

0.2

0.2 0.1

0

p=0

p=1

p=2

-0.1

z/h

0.1

z/h

νm

0.5

0.5

p=2 p=1

p=0.5

-0.2

p=0.5

-0.3

p=5

0 -0.1

-0.2

-0.3

p=0

-0.4

-0.4 -0.5

νc

0

0.2

0.4

0.6

0.8

Vc

1

-0.5

0

0.2

0.4

0.6

0.8

Vc

Fig. 2. Fractions volume Vc, (a) FGM shells and (b) sandwich shells with FGM core.

Table 3 List of acronyms used in tables to denote three-dimensional and two-dimensional shell theories. Acronym

Description

HSDT RBFC-1

HSDT and radial basis functions collocation including transverse normal deformation by Neves et al. [6] HSDT and radial basis functions collocation discarding transverse normal deformation by Neves et al. [6] HSDT finite element formulation by Pradyumna and Bandyopadhyay [7] HSDT semi-analytical approach based on Galerkin by Yanga and Shen [10] 3D-Ritz formulation by Li et al. [56]

HSDT RBFC-2 HSDT FEM HSDT SAG 3D-Ritz

1

1496

F.A. Fazzolari, E. Carrera / Journal of Sound and Vibration 333 (2014) 1485–1508

 C~ 16 J kτuα suα k

þ C~ 55 J α βαz k

kτu



suαz

       k ∂ ∂ ∂ ∂ kτ s  C~ 66 J α=βuα uα ∂α suα ∂β τuα ∂β suα ∂β τuα 

δD kτuαz suα δD kτuα suαz δD kτ s J  J þ J β=αuα uα Rαk β Rαk β ðRαk Þ2

!

Table 4

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Convergence analysis of the first four dimensionless frequency parameters ω^ ¼ ωa2 ρm =Dm of CCCC square FGM cylindrical shells made up of Si3 and SUS304, a/h ¼ 10, R/a¼ 100, ED555 shell model and varying the fraction volume index p. p

Mode

M, N 2

4

6

8

10

12

0

1 2 3 4

126.5096 285.2364 287.1863 288.9511

82.1215 152.9115 153.0656 212.6437

77.6619 147.0326 147.1917 205.9157

76.2159 144.9725 145.1336 203.4797

75.5810 144.0234 144.1854 202.3251

75.2498 143.5110 143.6735 201.6888

0.2

1 2 3 4

103.1057 231.6428 233.4424 236.9678

66.9435 124.5854 124.7316 173.1906

63.3170 119.8093 119.9560 167.7427

63.3170 119.8093 119.9560 167.7427

61.6149 117.3508 117.4977 164.8199

61.3404 116.9275 117.0744 164.2966

2

1 2 3 4

67.6907 151.4478 152.3110 157.0841

44.8083 83.1066 83.1998 115.2999

42.4629 80.0449 80.1353 111.8074

42.4629 80.0449 80.1353 111.8074

41.3398 78.4267 78.5154 109.8863

41.1511 78.1359 78.2242 109.5277

10

1 2 3 4

57.5851 128.7062 129.1122 134.6243

38.7365 71.6810 71.7559 99.3026

36.7684 69.1332 69.2080 96.3975

36.7684 69.1332 69.2080 96.3975

35.8173 67.7701 67.8447 94.7848

35.6545 67.5201 67.5946 94.4779

1

1 2 3 4

54.1602 121.2777 121.5049 127.5036

36.1620 67.0491 67.1167 92.9227

34.3039 64.6318 64.7013 90.1920

34.3039 64.6318 64.7013 90.1920

33.4001 63.3306 63.4013 88.6673

33.2433 63.0894 63.1603 88.3744

Table 5 First four dimensionless frequency parameters ω^ ¼ ωa2 varying the fraction volume index p. Mode

Theory

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρm =Dm of CCCC square FGM cylindrical shells made up of Si3 and SUS304, a/h¼ 10, R/a¼ 100 and

p 0

0.2

2

10

1

1

HSDT HSDT HSDT HSDT ED555

FEM SAG RBFC-1 RBFC-2

72.9613 74.518 74.2634 74.5821 75.2498

60.0269 57.479 60.0061 60.3431 61.3404

39.1457 40.750 40.5259 40.8262 41.1511

33.3666 35.852 35.1663 35.4229 35.6545

32.0274 32.761 32.6108 32.8593 33.2433

2

HSDT HSDT HSDT HSDT ED555

FEM SAG RBFC-1 RBFC-2

138.5552 144.663 141.6779 142.4281 143.5110

113.8806 111.717 114.3788 115.2134 116.9275

74.2915 78.817 76.9725 77.6639 78.1359

63.2869 69.075 66.6482 67.1883 67.5201

60.5546 63.314 61.9329 62.4886 63.0894

3

HSDT HSDT HSDT HSDT ED555

FEM SAG RBFC-1 RBFC-2

138.5552 145.740 141.8485 142.6024 143.6735

114.0266 112.531 114.5495 115.3665 117.0744

74.3868 79.407 77.0818 77.7541 78.2242

63.3668 69.609 66.7332 67.2689 67.5946

60.6302 63.806 62.0082 62.5668 63.1603

4

HSDT HSDT HSDT HSDT ED555

FEM SAG RBFC-1 RBFC-2

195.5366 206.992 199.1566 200.3158 201.6888

160.6235 159.855 160.7355 162.0337 164.2966

104.7687 112.457 107.9484 108.9677 109.5277

89.1970 98.386 93.3350 94.0923 94.4779

85.1788 90.370 86.8160 87.6341 88.3744

F.A. Fazzolari, E. Carrera / Journal of Sound and Vibration 333 (2014) 1485–1508

kτuα suβ

K uu

¼  C~ 12 J kτuα suβ k

kτuα su k  C~ 26 J α=β β



kτuαz suβ

þ C~ 45 J αβ k

kτ

K uuuα

suz



z



∂ ∂β



       kτuα su k ∂ ∂ ∂ ∂  C~ 16 J β=α β ∂α τuα ∂β su ∂α suα ∂α τu





suβ

∂ ∂β

β



τ uα

k  C~ 66 J kτuα suβ



   ∂ ∂ ∂α su ∂β τuα β

δD kτuαz suβ δD Jα  J Rβk Rαk β

kτuα su

βz

þ

δD J kτuα suβ Rαk Rβk

1497

β



      k k kτu su 1 kτuα suz ∂ 1 kτuα suz ∂ ∂ J β=α  C~ 12 J  C~ 13 J β α zz Rαk ∂α τuα Rβk ∂α τuα ∂α τuα       k k k 1 kτuα suz ∂ 1 kτuα suz ∂ ∂ uα suz  C~ 16 J  C~ 26 J α=β  C~ 36 J kτ α Rαk ∂β τuα Rβk ∂β τuα ∂β τuα ¼  C~ 11 k

Table 6

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dimensionless fundamental circular frequency parameter ω^ ¼ ωa2 ρm =Dm , of square FGM cylindrical shells, length-to thickness ratio a/h ¼ 10, ED shell models and varying the radius-to-length ratio and the graduation index. Rβ =a

0.5

1

5

a

Theory

p 0

0.2

0.5

1

2

10

1

HSDT FEM HSDT RBFC-1 HSDT RBFC-2

68.8645 70.1594 69.9872

64.4001 65.3889 65.2100

59.4396 60.4255 60.2422

53.9296 54.8909 54.7074

47.8259 48.7807 48.6005

37.2593 38.2792 38.1172

31.9866 31.7000 31.6222

ED111 ED222 ED333 ED444 ED555 ED666 ED777 ED888 ED999

Present ED models C-Ea 71.1957 66.4638 70.0091 65.4235 69.9703 65.3946 69.9700 65.3944 69.9700 65.3944 69.9700 65.3944 69.9700 65.3944 69.9700 65.3944 69.9700 65.3944

61.2320 60.3451 60.3245 60.3243 60.3243 60.3243 60.3243 60.3243 60.3243

55.4724 54.7299 54.7149 54.7146 54.7146 54.7146 54.7146 54.7146 54.7146

49.1866 48.5420 48.5264 48.5250 48.5250 48.5250 48.5250 48.5250 48.5250

38.5606 37.8928 37.8410 37.8408 37.8405 37.8401 37.8401 37.8401 37.8401

32.1683 31.6321 31.6146 31.6145 31.6145 31.6145 31.6145 31.6145 31.6145

HSDT FEM HSDT RBFC-1 HSDT RBFC-2

51.5216 52.1938 52.1101

47.5968 47.9338 47.8590

43.3019 43.6883 43.6239

38.7715 39.1753 39.1246

34.3338 34.7654 34.7289

28.2757 28.8072 28.7611

24.1988 23.5827 23.5448

ED111 ED222 ED333 ED444 ED555 ED666 ED777 ED888 ED999

Present ED models C-Ea 55.1181 50.7952 52.1912 48.1762 52.1006 48.1083 52.1003 48.1080 52.1003 48.1080 52.1003 48.1080 52.1003 48.1080 52.1003 48.1080 52.1003 48.1080

46.0892 43.7965 43.7477 43.7474 43.7474 43.7474 43.7474 43.7474 43.7474

41.1787 39.1991 39.1636 39.1631 39.1631 39.1630 39.1630 39.1630 39.1630

36.4806 34.7272 34.6901 34.6870 34.6869 34.6869 34.6869 34.6869 34.6869

30.3554 28.6947 28.5698 28.5698 28.5684 28.5673 28.5673 28.5673 28.5673

24.9039 23.5815 23.5406 23.5404 23.5404 23.5404 23.5404 23.5404 23.5404

HSDT FEM HSDT RBFC-1 HSDT RBFC-2

42.2543 42.6701 42.7172

40.1621 38.7168 38.7646

37.2870 34.8768 34.9273

33.2268 30.9306 30.9865

27.4449 27.5362 27.5977

19.3892 24.2472 24.2839

19.0917 19.2796 19.3008

ED111 ED222 ED333 ED444 ED555 ED666 ED777 ED888 ED999

Present ED models C-Ea 47.0658 43.0159 42.8458 39.1636 42.7164 39.0645 42.7160 39.0642 42.7160 39.0642 42.7160 39.0642 42.7160 39.0642 42.7160 39.0642 42.7160 39.0642

38.6110 38.6110 35.0813 35.0811 35.0811 35.0811 35.0811 35.0811 35.0811

34.1605 31.0966 31.0420 31.0414 31.0414 31.0414 31.0414 31.0414 31.0414

30.3746 27.6265 27.5678 27.5636 27.5634 27.5634 27.5634 27.5634 27.5634

26.7237 24.3109 24.1271 24.1258 24.1245 24.1228 24.1228 24.1228 24.1228

21.2657 19.3590 19.3005 19.3003 19.3003 19.3003 19.3003 19.3003 19.3003

C-E (Complete equations).

1498

F.A. Fazzolari, E. Carrera / Journal of Sound and Vibration 333 (2014) 1485–1508

þ C~ 45 J α k

kτuαz suz

k kτu su þ C~ 55 J β αz z

kτu suα

K uu β

¼  C~ 12 J kτuβ suα k

kτu suα

 C~ 26 J α=ββ k



kτu suαz

þ C~ 45 J αβ βz k





∂ ∂β



∂ ∂α

  suz

  ! 1 kτuα suz ∂ J Rαk ∂β suz

  ! 1 kτuα suz ∂  J Rαk β=α ∂α suz suz





∂ ∂β

       k kτu suα ∂ ∂ ∂ ∂  C~ 16 J β=αβ ∂α suα ∂β τu ∂α suα ∂α τu







suα

∂ ∂β



τ uβ

β

 C~ 66 J kτuβ suα k





∂ ∂α



τ uβ

∂ ∂β



β

suα

δD kτuβ suαz δD kτuβz suα δD kτuβ suα Jα  J þ J Rβk Rαk β Rαk Rβk



Table 7

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dimensionless fundamental circular frequency parameter ω^ ¼ ωa2 ρm =Dm , of square FGM cylindrical shells, length-to thickness ratio a/h ¼10, ED shell models and varying the radius-to-length ratio and the graduation index. Rβ =a

10

50

Plate

a

Theory

p 0

0.2

0.5

1

2

10

1

HSDT FEM HSDT RBFC-1 HSDT RBFC-2

41.9080 42.3153 42.3684

39.8472 38.3840 38.4368

36.9995 34.5672 34.6219

32.9585 30.6485 30.7077

27.1789 27.2979 27.3616

19.1562 24.1063 24.1444

18.9352 19.1193 19.1433

ED111 ED222 ED333 ED444 ED555 ED666 ED777 ED888 ED999

Present ED models C-Ea 46.7741 42.7473 42.4992 38.8387 42.3680 38.7380 42.3677 38.7377 42.3677 38.7377 42.3677 38.7377 42.3677 38.7377 42.3677 38.7377 42.3677 38.7377

38.3646 34.8507 34.7764 34.7761 34.7761 34.7761 34.7761 34.7761 34.7761

33.9395 30.8185 30.7627 30.7622 30.7621 30.7621 30.7621 30.7621 30.7621

30.1936 27.3904 27.3302 27.3260 27.3258 27.3258 27.3258 27.3258 27.3258

26.6256 24.1746 23.9876 23.9862 23.9848 23.9832 23.9832 23.9832 23.9832

21.1339 19.2023 19.1431 19.1429 19.1429 19.1429 19.1429 19.1429 19.1429

HSDT FEM HSDT RBFC-1 HSDT RBFC-2

41.7963 42.2008 42.2560

39.7465 38.2842 38.3384

36.9088 34.4809 34.5365

32.8750 30.5759 30.6355

27.0961 27.2423 27.3055

19.0809 24.0762 24.1125

18.8848 19.0675 19.0924

ED111 ED222 ED333 ED444 ED555 ED666 ED777 ED888 ED999

Present ED models C-Ea 46.6802 42.6696 42.3874 38.7407 42.2557 38.6394 42.2553 38.6391 42.2553 38.6391 42.2553 38.6391 42.2553 38.6391 42.2553 38.6391 42.2553 38.6391

38.3019 34.7656 34.6907 34.6904 34.6904 34.6904 34.6904 34.6904 34.6904

33.8910 30.7459 30.6896 30.6891 30.6890 30.6890 30.6890 30.6890 30.6890

30.1616 27.3335 27.2726 27.2684 27.2683 27.2682 27.2682 27.2682 27.2682

26.6126 24.1445 23.9560 23.9545 23.9532 23.9515 23.9515 23.9515 23.9515

21.0914 19.1518 19.0923 19.0922 19.0922 19.0922 19.0922 19.0922 19.0922

HSDT FEM HSDT RBFC-1 HSDT RBFC-2

41.7917 42.1961 42.2513

39.7426 38.2827 38.3368

36.9057 34.4820 34.5376

32.8726 30.5792 30.6386

27.0937 27.2472 27.3102

19.0778 24.0802 24.1171

18.8827 19.0654 19.0903

ED111 ED222 ED333 ED444 ED555 ED666 ED777 ED888 ED999

Present ED models C-Ea 46.6763 42.6695 42.3827 38.7389 42.2510 38.6376 42.2507 38.6373 42.2507 38.6373 42.2507 38.6373 42.2507 38.6373 42.2507 38.6373 42.2507 38.6373

38.3050 38.3050 34.6914 34.6914 34.6912 34.6912 34.6912 34.6912 34.6912

33.8968 30.7489 30.6924 30.6919 30.6919 30.6918 30.6918 30.6918 30.6918

30.1693 27.3378 27.2767 27.2726 27.2725 27.2724 27.2724 27.2724 27.2724

26.6185 24.1480 23.9594 23.9578 23.9565 23.9548 23.9548 23.9548 23.9548

21.0897 19.1497 19.0902 19.0901 19.0901 19.0901 19.0901 19.0901 19.0901

C-E (Complete equations).

F.A. Fazzolari, E. Carrera / Journal of Sound and Vibration 333 (2014) 1485–1508

kτu suβ

K uu β

kτu suβ

¼  C~ 22 J α=ββ k

k  C~ 26 J kτuβ suβ





       k ∂ ∂ ∂ ∂  C~ 26 J kτuβ suβ ∂α su ∂α τu ∂α su ∂β τu β

β

β

kτu suβ

þ C~ 44 J αβ βz k

kτu suz

K uu β

¼  C~ 12 k

β

       k kτu su ∂ ∂ ∂ ∂  C~ 66 J β=αβ β ∂α τu ∂β su ∂α suα ∂β τu

z



β

δD kτuβz suβ δD kτuβ suβz δD kτuβ suβ Jα  Jα þ J Rβk Rβk ðRβk Þ2 α=β

β

β

!

      k k kτu suz 1 kτuβ suz ∂ 1 kτuβ suz ∂ ∂ J  C~ 22 J α=β  C~ 23 J α β z Rαk ∂β τu Rβk ∂β τu ∂β τu β

k  C~ 16

β

      k k kτu suz 1 kτuβ suz ∂ 1 kτuβ suz ∂ ∂ J β=α  C~ 26 J  C~ 36 J β β Rαk ∂α τu Rβk ∂α τu ∂α τuα β

1499

β

β

Table 8

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dimensionless fundamental circular frequency parameter ω^ ¼ ωa2 ρm =Dm , of square FGM spherical shells ðRα ¼ Rβ ¼ RÞ, length-to thickness ratio a/h¼ 10, ED shell models and varying the radius-to-length ratio and the graduation index. R=a

0.5

1

Theory

0

0.2

0.5

1

2

10

1

HSDT FEM HSDT RBFC-1 HSDT RBFC-2

124.1581 126.2994 126.0882

115.7499 117.3053 117.0197

106.5014 108.0044 107.6572

96.2587 97.6938 97.2968

84.8206 86.2288 85.8028

65.2296 66.7088 66.3594

57.2005 57.0657 56.9702

ED111 ED222 ED333 ED444 ED555 ED666 ED777 ED888 ED999

Present ED models C-Ea 126.3528 117.5725 126.0874 117.3494 126.0687 117.3362 126.0684 117.3360 126.0684 117.3360 126.0684 117.3360 126.0684 117.3360 126.0684 117.3360 126.0684 117.3360

107.9837 107.8050 107.7954 107.7952 107.7952 107.7952 107.7952 107.7952 107.7952

97.4508 97.3135 97.3060 97.3056 97.3056 97.3056 97.3056 97.3056 97.3056

85.7904 85.6818 85.6732 85.6723 85.6723 85.6723 85.6723 85.6723 85.6723

66.0831 65.9482 65.9311 65.9306 65.9306 65.9305 65.9305 65.9305 65.9305

57.0898 56.9699 56.9615 56.9613 56.9613 56.9613 56.9613 56.9613 56.9613

72.6343 73.2663 73.0034

66.5025 67.1623 66.9033

59.8521 60.5121 60.2636

52.7875 53.4659 53.2311

41.6702 42.4365 42.2155

36.2904 35.8131 35.6948

Present ED models C-Ea 80.6930 74.7524 79.0398 73.3077 78.9801 73.2635 78.9797 73.2631 78.9797 73.2631 78.9797 73.2631 78.9797 73.2631 78.9797 73.2631 78.9797 73.2631

68.2825 67.0560 67.0245 67.0242 67.0242 67.0242 67.0242 67.0242 67.0242

61.3339 60.3120 60.2892 60.2888 60.2888 60.2888 60.2888 60.2888 60.2888

54.0671 53.1839 53.1610 53.1592 53.1592 53.1591 53.1591 53.1591 53.1591

42.9288 42.0121 41.9378 41.9378 41.9373 41.9368 41.9368 41.9368 41.9368

36.4594 35.7124 35.6855 35.6853 35.6853 35.6853 35.6853 35.6853 35.6853

41.7782 40.3936 40.4211

38.7731 36.4453 36.4782

34.6004 32.3691 32.4101

28.7459 28.7833 28.8329

20.4691 25.0772 25.1038

19.8838 20.0818 20.0927

Present ED models C-Ea 48.6357 44.4880 44.5919 40.8117 44.4675 40.7169 44.4671 40.7166 44.4671 40.7166 44.4671 40.7166 44.4671 40.7166 44.4671 40.7166 44.4671 40.7166

39.9815 36.6994 36.6300 36.6297 36.6297 36.6297 36.6297 36.6297 36.6297

35.4103 32.5166 32.4651 32.4645 32.4645 32.4645 32.4645 32.4645 32.4645

31.4469 28.8590 28.8039 28.7998 28.7996 28.7996 28.7996 28.7996 28.7996

27.4219 25.1179 24.9426 24.9415 24.9403 24.9389 24.9388 24.9388 24.9387

21.9750 20.1479 20.0917 20.0915 20.0915 20.0915 20.0915 20.0915 20.0915

HSDT FEM HSDT RBFC-1 HSDT RBFC-2 ED111 ED222 ED333 ED444 ED555 ED666 ED777 ED888 ED999

5

HSDT FEM HSDT RBFC-1 HSDT RBFC-2 ED111 ED222 ED333 ED444 ED555 ED666 ED777 ED888 ED999 a

p

C-E (Complete equations).

78.2306 79.2626 79.0008

44.0073 44.4455 44.4697

1500

F.A. Fazzolari, E. Carrera / Journal of Sound and Vibration 333 (2014) 1485–1508

   ! ∂ 1 kτuβ suz ∂  J ∂α suz Rβk ∂α suz     ! kτu suz ∂ 1 kτuβ suz ∂ J α βz  J ∂β suz Rβk α=β ∂β suz kτuβ suz

þ C~ 45 J β k

þ C~ 44 k

kτ suα

K uuuz



z

      k k kτu su 1 kτuz suα ∂ 1 kτuz suα ∂ ∂ J β=α þ C~ 12 J þ C~ 13 J β zz α Rαk ∂α suα Rβk ∂α suα ∂α suα       k k 1 kτuz sαz ∂ 1 kτuz sα ∂ ∂ kτu sα J þ C~ 26 J α=β þ C~ 36 J α zz z Rαk ∂β suα Rβ k ∂β suα ∂β suα     ! k ∂ 1 kτuz suα ∂ uz sαz  C~ 45 J kτ  J α ∂β τuz Rαk ∂β τuz     ! k ∂ 1 kτuz sαz ∂ kτu su  J  C~ 55 J β z αz ∂α τuz Rαk β=α ∂α τuz

¼ C~ 11 k

þ C~ 16 k

(45)

Table 9

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dimensionless fundamental circular frequency parameter ω^ ¼ ωa2 ρm =Dm , of square FGM spherical shells ðRα ¼ Rβ ¼ RÞ, length-to thickness ratio a/h¼ 10, ED shell models and varying the radius-to-length ratio and the graduation index. R=a

Theory

10

50

Plate

a

p 0

0.2

0.5

1

2

10

1

HSDT FEM HSDT RBFC-1 HSDT RBFC-2

42.3579 42.7709 42.8180

40.2608 38.8074 38.8551

37.3785 34.9574 35.0080

33.3080 31.0012 31.0572

27.5110 27.5984 27.6602

19.4357 24.3034 24.3401

19.1385 19.3251 19.3464

ED111 ED222 ED333 ED444 ED555 ED666 ED777 ED888 ED999

Present ED models C-Ea 47.1749 43.1147 42.9470 39.2552 42.8172 39.1559 42.8169 39.1556 42.8169 39.1556 42.8169 39.1556 42.8169 39.1556 42.8169 39.1556 42.8169 39.1556

38.6988 35.2357 35.1625 35.1622 35.1622 35.1622 35.1622 35.1622 35.1622

34.2371 31.1676 31.1129 31.1123 31.1122 31.1122 31.1122 31.1122 31.1122

30.4421 30.4421 27.6301 27.6260 27.6258 27.6258 27.6258 27.6258 27.6258

26.7847 24.3672 24.1830 24.1817 24.1803 24.1790 24.1788 24.1788 24.1787

21.3150 19.4047 19.3461 19.3459 19.3459 19.3459 19.3459 19.3459 19.3459

HSDT FEM HSDT RBFC-1 HSDT RBFC-2

41.8145 42.2192 42.2741

39.7629 38.2988 38.3528

36.9234 34.4922 34.5478

32.8881 30.5840 30.6437

27.1085 27.2474 27.3109

19.0922 24.0791 24.1168

18.8930 19.0759 19.1006

ED111 ED222 ED333 ED444 ED555 ED666 ED777 ED888 ED999

Present ED models C-Ea 46.6963 38.7552 42.4054 38.6541 42.2738 38.6541 42.2735 38.6538 42.2735 38.6538 42.2735 38.6538 42.2735 38.6538 42.2735 38.6538 42.2735 38.6538

38.3098 34.7769 34.7021 34.7018 34.7018 34.7018 34.7018 34.7018 34.7018

33.8955 30.7544 30.6981 30.6976 30.6975 30.6975 30.6975 30.6975 30.6975

30.1630 27.3392 27.2784 27.2743 27.2741 27.2741 27.2741 27.2741 27.2741

26.6129 24.1477 23.9595 23.9580 23.9567 23.9553 23.9551 23.9551 23.9550

21.0987 19.1600 19.1005 19.1004 19.1004 19.1004 19.1004 19.1004 19.1004

HSDT FEM HSDT RBFC-1 HSDT RBFC-2

41.7917 42.1961 42.2513

39.7426 38.2827 38.3368

36.9057 34.4820 34.5376

32.8726 30.5792 30.6386

27.0937 27.2472 27.3102

19.0778 24.0802 24.1171

18.8827 19.0654 19.0903

ED111 ED222 ED333 ED444 ED555 ED666 ED777 ED888 ED999

Present ED models C-Ea 46.6763 42.6695 42.3827 38.7389 42.2510 38.6376 42.2507 38.6373 42.2507 38.6373 42.2507 38.6373 42.2507 38.6373 42.2507 38.6373 42.2507 38.6373

38.3050 38.3050 34.6914 34.6914 34.6912 34.6912 34.6912 34.6912 34.6912

33.8968 30.7489 30.6924 30.6919 30.6919 30.6918 30.6918 30.6918 30.6918

30.1694 27.3379 27.2768 27.2726 27.2725 27.2724 27.2724 27.2724 27.2724

26.6185 24.1480 23.9594 23.9578 23.9565 23.9551 23.9549 23.9549 23.9548

21.0897 19.1497 19.0902 19.0901 19.0901 19.0901 19.0901 19.0901 19.0901

C-E (Complete equations).

F.A. Fazzolari, E. Carrera / Journal of Sound and Vibration 333 (2014) 1485–1508

kτuz suβ

K uu

k

k

kτ suz

      k k kτuz su 1 kτuz suβ ∂ 1 kτuz suβ ∂ ∂ J þ C~ 22 J α=β þ C~ 23 J α z β Rαk ∂β su Rβk ∂β su ∂β su β β β       kτuzz suβ k k 1 kτuz suβ ∂ 1 kτuz suβ ∂ ∂ ~ ~ J þ C 26 J þ C 36 J β Rαk β=α ∂α su Rβk ∂α su ∂α su β β β !     kτuz suβ k ∂ 1 ∂ kτ s z  C~ 45 J β  J uz uβ ∂α τuz Rβk ∂α τuz     ! kτuz suβ k ∂ 1 kτuz suβ ∂ z  J  C~ 44 J α ∂β τuz Rβk α=β ∂β τuz

¼ þ C~ 12

þ C~ 16

K uuuz

1501

k 1 kτuz suz ~ k kτuzz suzz 1 J α=β þ C 33 J α β þ2C~ 12 J kτuz suz Rαk Rβk ðRβk Þ2   k k 1  kτuzz suz 1  kτuzz suz kτu su kτu su þ C~ 13 Jβ þ J β z zz þ C~ 23 Jα þJ α z zz Rαk Rβ k         k kτuz suz k kτuz suz ∂ ∂ ∂ ∂  C~ 55 J β=α  C~ 44 J α=β ∂β suz ∂β τuz ∂α suz ∂α τuz         k k ∂ ∂ ∂ ∂  C~ 45 J kτuz suz  C~ 45 J kτuz suz ∂α suz ∂β τuz ∂α τuz ∂β suz

¼ þ C~ 11 k

1

kτ suz

ðRαk Þ2

J β=αuz

þ C~ 22 k

Table 10

pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Dimensionless fundamental circular frequency parameter ω ¼ ðωb =hÞ ρ0 =E0 , of square cylindrical sandwich shells with FGM core, length-to thickness ratio a/h ¼10, M ¼ N ¼ 1 unless differently indicated and varying the radius-to-length ratio and the graduation index. p

Theory

Rβ =a 0.5

1

5

10

50

Plate

0.5

ED111 ED222 ED333 ED999 EDZ111 EDZ333

2.32117 2.28377 2.28278 2.28277 2.32117 2.28278

1.76665 1.67395 1.67177 1.67174 1.76665 1.67177

1.45493 1.32165 1.31868 1.31861 1.45493 1.31868

1.44139 1.30678 1.30379 1.30373 1.44139 1.30379

1.43582 1.30109 1.29811 1.29804 1.43582 1.29811

1.43517 (1.29751)a 1.30054 1.29757 1.29751 1.43517 1.29757

1

ED111 ED222 ED333 ED999 EDZ111 EDZ333

2.49415 2.45737 2.45675 2.45673 2.49415 2.45675

1.86654 1.77327 1.77193 1.77191 1.86654 1.77193

1.51129 1.37393 1.37211 1.37208 1.51129 1.37211

1.49611 1.35715 1.35533 1.35529 1.49611 1.35531

1.49611 1.35715 1.35533 1.35529 1.49611 1.35533

1.48943 (1.34847)a 1.35031 1.34850 1.34846 1.48943 1.34850

2

ED111 ED222 ED333 ED999 EDZ111 EDZ333

2.68342 10.496 10.496 10.496 10.496 10.496

1.78148 1.68774 1.68681 1.68678 1.78148 1.68681

1.57392 1.43243 1.43105 1.43101 1.57392 1.43105

1.56016 1.41645 1.41645 1.41501 1.56016 1.41505

1.55466 1.41029 1.40889 1.40885 1.55466 1.40887

1.55408 (1.40828)a 1.40970 1.40831 1.40827 1.55408 1.40831

5

ED111 ED222 ED333 ED999 EDZ111 EDZ333

2.79349 2.75509 2.75452 2.75446 2.79349 2.75447

2.07096 1.97159 1.97035 1.97022 2.07096 1.97035

1.67026 1.52054 1.51879 1.51861 1.67026 1.51879

1.65396 1.50210 1.50033 1.50016 1.65395 1.50032

1.64792 1.49551 1.49373 1.49356 1.64792 1.49373

1.64741 (1.49309)a 1.49501 1.49324 1.49307 1.64741 1.49324

10

ED111 ED222 ED333 ED999 EDZ111 EDZ333

2.86078 2.82071 2.81997 2.81989 2.86078 2.81997

2.13111 2.02780 2.02610 2.02594 2.13111 2.02610

1.73266 1.57749 1.57505 1.57482 1.73266 1.57505

1.71675 1.55934 1.55687 1.55664 1.71675 1.55687

1.71099 1.55293 1.55046 1.55023 1.71099 1.55046

1.71053 (1.54980)a 1.55248 1.55001 1.54978 1.71053 1.55001

a

3D-Ritz.

1502

F.A. Fazzolari, E. Carrera / Journal of Sound and Vibration 333 (2014) 1485–1508

The nine components of the fundamental primary boundary nucleus Πu u can be written as follows: kτ

Π uuuα

s uα

kτuα suβ

Π uu

¼ nα C~ 11 J β=αuα k

kτ

suα



kτuα su k ¼ nα C~ 16 J β=α β

       k kτ s k k ∂ ∂ ∂ ∂ þnβ C~ 66 J α=βuα uα þnβ C~ 16 J kτuα suα þnα C~ 16 J kτuα suα ∂α suα ∂β suα ∂α suα ∂β suα



       k kτuα su k k ∂ ∂ ∂ ∂ þnβ C~ 26 J α=β β þnβ C~ 66 J kτuα suβ þnα C~ 12 J kτuα suβ ∂α su ∂β su ∂α su ∂β su β

kτ

Π uuuα

suz

¼ nα

β

kτu suα

kτu suα k ¼ nα C~ 16 J β=αβ

kτu suβ

Π uu β

β

k 1 ~ k kτuα suz 1 ~ k kτuα suz 1 ~ k kτuα suz kτu su þ nα þ nα C~ 13 J β α zz þ nβ C J C J C J Rαk 11 β=α Rβk 12 Rαk 16 α=β

þ nβ

Π uu β

β

k kτu su 1 ~ k kτuα suz þ nβ C~ 36 J α α zz C J Rβk 26 α=β

(46)



k kτu su ¼ nα C~ 66 J β=αβ β

       kτu suα k k k ∂ ∂ ∂ ∂ þnβ C~ 26 J α=ββ þnβ C~ 12 J kτuβ suα þ nα C~ 66 J kτuβ suα ∂α suα ∂β suα ∂α suα ∂β suα



       k kτu su k k ∂ ∂ ∂ ∂ þ nβ C~ 22 J α=ββ β þ nβ C~ 26 J kτuβ suβ þnα C~ 26 J kτuβ suβ ∂α su ∂β su ∂α su ∂β su β

β

β

β

Table 11

2 pffiffiffiffiffiffiffiffiffiffiffiffiffi Dimensionless fundamental circular frequency parameter ω ¼ ðω=b hÞ ρ0 =E0 , of square cylindrical sandwich shells with FGM core, length-to thickness ratio a/h ¼100, M ¼ N ¼ 1 unless differently indicated and varying the radius-to-length ratio and the graduation index.

p

Theory

Rβ =a 0.5

1 a

5

10

50

Plate

b

5.98076 5.77987 5.77975 5.77975 5.98076 5.77975

2.84171 2.77028 2.77026 2.77026 2.84171 2.77028

1.91580 1.80757 1.80755 1.80755 1.91580 1.80755

1.50213 1.36121 1.36118 1.36118 1.50213 1.36118

1.48233 (1.33931)c 1.33934 1.33931 1.33931 1.48233 1.33931

0.5

ED111 ED222 ED333 ED999 EDZ111 EDZ333

8.40568 7.85653 7.85585 7.85583 8.40567 7.85585

1

ED111 ED222 ED333 ED999 EDZ111 EDZ333

8.82764a 8.26867 8.26825 8.26825 8.82763 8.26825

6.37163b 6.16988 6.16981 6.16981 6.37163 6.16981

3.03892 2.96739 2.96739 2.96739 3.03892 2.96739

2.01975 1.90984 1.90982 1.90982 2.01975 1.90982

1.55711 1.41141 1.41139 1.41139 1.55711 1.41139

1.53476 (1.38669)c 1.38672 1.38670 1.38669 1.53476 1.38669

2

ED111 ED222 ED333 ED999 EDZ111 EDZ333

9.26293a 8.68653 8.68620 8.68619 9.26292 8.68620

6.73374b 6.52687 6.52682 6.52681 6.73374 6.52682

3.21745 3.14415 3.14415 3.14415 3.21745 3.14415

2.12380 2.01039 2.01038 2.01038 2.12380 2.01038

1.62347 1.47177 1.47176 1.47176 1.62347 1.47176

1.59918 (1.44491)c 1.44493 1.44492 1.44492 1.59918 1.44492

5

ED111 ED222 ED333 ED999 EDZ111 EDZ333

9.78926a 9.17889 9.17849 9.17846 9.78925 9.17849

7.10299b 6.88305 6.88298 6.88298 7.10299 6.88298

3.39206 3.31397 3.31396 3.31396 3.39206 3.31396

2.24410 2.12354 2.12353 2.12353 2.24410 2.12353

1.72031 1.55950 1.55948 1.55948 1.72030 1.55948

1.69492 (1.53143)c 1.53145 1.53143 1.53143 1.69492 1.53144

10

ED111 ED222 ED333 ED999 EDZ111 EDZ333

10.10569a 9.46860 9.46805 9.46800 10.10569 9.46805

7.29229b 7.06118 7.06109 7.06108 7.29229 7.06108

3.47738 3.39515 3.39513 3.39513 3.47738 3.39513

2.31382 2.18756 2.18754 2.18754 2.31381 2.18754

1.78637 1.61921 1.61918 1.61918 1.78637 1.61918

1.76093 (1.59105)c 1.59109 1.59106 1.59106 1.76094 1.59106

a b c

Indicates that the used half-wave numbers are M ¼1, N¼ 3. Indicates that the used half-wave numbers are M ¼ 1, N ¼2. 3D-Ritz.

F.A. Fazzolari, E. Carrera / Journal of Sound and Vibration 333 (2014) 1485–1508

kτu suz

Π uu β

¼ nα

k kτu suz 1 ~ k kτuβ suz 1 ~ k kτuβ suz 1 ~ k kτuβ suz þnα þnα C~ 36 J β β z þ nβ C J C J C J Rαk 16 β=α Rβk 26 Rαk 12 α=β

þ nβ kτ suα

Π uuuz

¼  nα

k kτuz suβz k kτuz suβ 1 ~ k kτuz suβ 1 ~ k kτuz sβ þnα C~ 45 J β  nβ þnβ C~ 44 J α C J C J Rβk 45 Rβk 44 α=β

¼  nα

kτ suz

k kτ s ¼ nα C~ 55 J β=αuz uz

Π uuuz

k kτu suz 1 ~ k kτuβ suz þ nβ C~ 23 J α β z C J Rβk 22 α=β

k kτu su k kτu su 1 ~ k kτuz suα 1 ~ k kτuz suα þnα C~ 55 J β z αz  nβ þnβ C~ 45 J α z αz C J C J Rαk 55 β=α Rαk 45

kτuz suβ

Π uu

1503



     k kτ s k ∂ ∂ ∂ þnβ C~ 44 J α=βuz uz þ nβ C~ 45 J kτuz suz ∂α suz ∂β suz ∂α suz   k ∂ þ nα C~ 45 J kτuz suz ∂β suz

Finally the fundamental primary mass nucleus Mu u components can be written as uα suα ¼ ρk J kτuα suα Mkτ uu

s uz

¼0

kτu suz

¼0

Muuuα

kτu suβ

¼ ρk J kτuβ suβ

Muu β

kτuz suβ

¼0

¼0

Muu β

kτ suα

¼0

Muu

Muuuz

kτ

¼0

kτu suα

Muu β

kτuα suβ

Muu

kτ suz

Muuuz

¼ ρk J kτuz suz

(47)

Table 12

pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Dimensionless fundamental circular frequency parameter ω ¼ ðωb =hÞ ρ0 =E0 , of square spherical sandwich shells ðRα ¼ Rβ ¼ RÞ with FGM core, length-to thickness ratio a/h¼ 10, M¼ N ¼1 unless differently indicated and varying the radius-to-length ratio and the graduation index. p

Theory

R/a 0.5

1

5

10

50

Plate

0.5

ED111 ED222 ED333 ED999 EDZ111 EDZ333

4.30293 4.29423 4.29327 4.29324 4.30293 4.29327

2.69021 2.63773 2.63615 2.63613 2.69020 2.63615

1.51646 1.38839 1.38549 1.38544 1.51646 1.38549

1.45835 1.32483 1.32185 1.32179 1.45835 1.32185

1.43690 1.30211 1.29912 1.29906 1.43690 1.29913

1.43517 (1.29751)a 1.30054 1.29757 1.29751 1.43517 1.29757

1

ED111 ED222 ED333 ED999 EDZ111 EDZ333

4.63882 4.63051 4.62996 4.62992 4.63882 4.62996

2.87826 2.82667 2.82567 2.82565 2.87826 2.82567

1.57983 1.44847 1.44670 1.44667 1.57982 1.44670

1.51483 1.37722 1.37541 1.37537 1.51483 1.37541

1.49121 1.35196 1.35015 1.35011 1.49121 1.35015

1.48943 (1.34847)a 1.35031 1.34850 1.34846 1.48943 1.34850

2

ED111 ED222 ED333 ED999 EDZ111 EDZ333

4.91618 4.90810 4.90774 4.90770 4.91618 4.90774

3.04166 2.98994 2.98915 2.98912 3.04166 2.98915

1.65055 1.51509 1.51373 1.51369 1.65055 1.51373

1.58077 1.43834 1.43694 1.43690 1.58077 1.43694

1.55582 1.41137 1.40997 1.40993 1.55582 1.40997

1.55408 (1.40828)a 1.40970 1.40831 1.40827 1.55408 1.40831

5

ED111 ED222 ED333 ED999 EDZ111 EDZ333

5.14630 5.13808 5.13773 5.13768 5.14630 5.13773

3.19205 3.13821 3.13730 3.13721 3.19205 3.13730

1.74591 1.60351 1.60182 1.60165 1.74591 1.60182

1.67415 1.52416 1.52240 1.52223 1.67415 1.52240

1.64899 1.49661 1.49483 1.49466 1.64899 1.49483

1.64741 (1.49309)a 1.49501 1.49324 1.49307 1.64741 1.49324

10

ED111 ED222 ED333 ED999 EDZ111 EDZ333

5.23759 5.22903 5.22863 5.22857 5.23759 5.22863

3.26121 3.20507 3.20507 3.20378 3.26121 3.20389

1.80793 1.66029 1.65795 1.65773 1.80793 1.65795

1.73669 1.58123 1.57879 1.57856 1.73669 1.57879

1.71200 1.55400 1.55152 1.55129 1.71200 1.55152

1.71053 (1.54980)a 1.55248 1.55001 1.54978 1.71053 1.55001

a

3D-Ritz.

1504

F.A. Fazzolari, E. Carrera / Journal of Sound and Vibration 333 (2014) 1485–1508

6. Numerical results and discussion Numerical results are proposed in terms of dimensionless circular/angular frequency parameters and modal displacements. The following two benchmarks have been carried out: 1. Benchmark 1: Doubly curved FGM shells made up of materials given in Table 1 and CCCC and SSSS boundary conditions. 2. Benchmark 2: Doubly curved sandwich shells with FGM core made up of material given in Table 2 and SSSS boundary condition. The trend of the fraction volume Vc through the thickness direction for both shell structure typologies in Benchmarks 1 and 2 is given in Fig. 2. When the Ritz functions involved in the case of SSSS boundary condition (see Eq. (33)) are employed as solution functions in Eq. (45) the Navier-type closed-form solution is obtained. Then when simply supported boundary condition is investigated, the present HTRF leads to the Navier-type closed-form solution. This phenomenon is usually referred to as one-term Ritz solution. The results are given using the usual acronyms system adopted in CUF [39,55]. Therefore, the ESL theories are indicated as EDNuα Nuβ Nuz where E means the ESL approach, D means that the PVD has been employed and Nuα , N uβ , Nuz are the three expansion orders used in the displacement field. Similarly the acronym used to describe the ZZ theories is EDZ Nuα Nuβ Nuz , where Z states that MZZF has been introduced. 6.1. Circular frequency parameters of fully clamped cylindrical FGM shells An accurate convergence analysis of the first four modes of CCCC square FGM cylindrical shells made up of Si3 (ceramic) and SUS304 (metal) (see Table 1), a/h¼ 10, R/a¼100 and varying the volume fraction index p is carried out in Table 4. The convergence is monitored varying the half-wave numbers M and N. As it is possible to note in the same table, a good convergence is reached for M ¼N ¼12. The rate of convergence increases when increasing the volume fraction index. Table 13

pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Dimensionless fundamental circular frequency parameter ω ¼ ðω b =hÞ ρ0 =E0 , of square spherical sandwich shells ðRα ¼ Rβ ¼ RÞ with FGM core, length-to thickness ratio a/h¼ 100, M ¼ N ¼1 unless differently indicated and varying the radius-to-length ratio and the graduation index. p

Theory

R/a 0.5

1

5

10

50

Plate

0.5

ED111 ED222 ED333 ED999 EDZ111 EDZ333

41.28730 41.28646 41.28644 41.28644 41.28730 41.28644

23.37136 23.36544 23.36544 23.36544 23.37136 23.36544

5.07458 5.03515 5.03514 5.03514 5.07457 5.03514

2.84591 2.77429 2.77427 2.77427 2.84591 2.77427

1.56005 1.42486 1.42483 1.42483 1.56004 1.42483

1.48233 (1.33931)a 1.33934 1.33931 1.33931 1.48233 1.33931

1

ED111 ED222 ED333 ED999 EDZ111 EDZ333

44.65246 44.65164 44.65163 44.65163 44.65246 44.65163

25.27412 25.26829 25.26829 25.26829 25.27412 25.26829

5.46970 5.43053 5.43052 5.43052 5.46970 5.43052

3.04332 2.97160 2.97159 2.97159 3.04332 2.97159

1.62238 1.48310 1.48308 1.48308 1.62238 1.48308

1.53476 (1.38669)a 1.38672 1.38670 1.38669 1.53476 1.38669

2

ED111 ED222 ED333 ED999 EDZ111 EDZ333

47.51043 47.50959 47.50959 47.50959 47.51043 47.50959

26.89257 26.88662 26.88662 26.88662 26.89257 26.88662

5.81213 5.77213 5.77212 5.77212 5.81213 5.77213

3.22203 3.14853 3.14852 3.14852 3.22203 3.14852

1.69433 1.54957 1.54956 1.54956 1.69433 1.54956

1.59918 (1.44491)a 1.44493 1.44492 1.44492 1.59918 1.44492

5

ED111 ED222 ED333 ED999 EDZ111 EDZ333

49.97623 49.97532 49.97532 49.97532 49.97623 49.97532

28.29280 28.28641 28.28641 28.28641 28.29280 28.28641

6.11980 6.07714 6.07714 6.07713 6.11980 6.07714

3.39681 3.31851 3.31851 3.31850 3.39681 3.31850

1.79438 1.64085 1.64083 1.64083 1.79438 1.64083

1.69492 (1.53143)a 1.53145 1.53143 1.53143 1.69492 1.53144

10

ED111 ED222 ED333 ED999 EDZ111 EDZ333

50.98897 50.98799 50.98799 50.98799 50.98897 50.98799

28.86999 28.86321 28.86321 28.86321 28.86999 28.86321

6.25407 6.20901 6.20900 6.20900 6.25407 6.20900

3.48224 3.39978 3.39977 3.39976 3.48224 3.39976

1.86072 1.70089 1.70086 1.70086 1.86072 1.70086

1.76093 (1.59105)a 1.59109 1.59106 1.59106 1.76094 1.59106

a

3D-Ritz.

F.A. Fazzolari, E. Carrera / Journal of Sound and Vibration 333 (2014) 1485–1508

1505

In Table 5 for the same shell configuration, the results in terms of the first four dimensionless circular frequency parameters are compared to those provided in the literature (see Table 3), varying the volume fraction index.

6.2. Circular frequency parameters of simply supported cylindrical and spherical FGM shells In Tables 6 and 7 a wide assessment the proposed ESL models has been carried out for the dimensionless fundamental pof ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3 circular frequency parameter ω^ ¼ ωa2 ρm =Dm with Dm ¼ Em h =12ð1 ν2m Þ of SSSS square FGM cylindrical shells made up of Alumina (ceramic) and Aluminium (metal) (see Table 1), length-to thickness ratio a/h ¼10 and varying the radius-to-length ratio ðR=aÞ and the volume fraction index. Results are in excellent agreement with those proposed in the literature, in

0.5

0.5

p=0.5 p=1 p=5 p=10

0.4 0.3 0.2

0.3 0.2 0.1

z/h

z/h

0.1 0

0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4

−0.4

−0.5 −0.2 −0.15 −0.1 −0.05

0

0.05

0.1

0.15

p=0.5 p=1 p=5 p=10

0.4

−0.5 −0.04

0.2

−0.03

−0.02

0.5

0.5

p=0.5 p=1 p=5 p=10

0.4 0.3 0.2

0.3 0.2 0.1

z/h

z/h

0

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4

−0.4 0.05

0.1

0.15

−0.5 0.03

0.2

0.04

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

−0.2 −0.3 −0.4 −0.5 0.9

−0.2 −0.3 −0.4 0.91

uy [m]

0.06

0.07

0.08

0 −0.1

p=0.5 p=1 p=5 p=10 0.905

0.05

uy [m]

z/h

z/h

uy [m]

−0.1

0.01

0

−0.1

0

0

p=0.5 p=1 p=2 p=10

0.4

0.1

−0.5 −0.2 −0.15 −0.1 −0.05

−0.01

uy [m]

uy [m]

0.915

0.92

−0.5 0.9972

p=0.5 p=1 p=5 p=10 0.9974

0.9976

0.9978

uy [m]

Fig. 3. Modal displacements ux ((a) a/h ¼10, (b) a/h ¼100), uy ((c) a/h ¼ 10, (d) a/h¼ 100) and uz ((e) a/h ¼10, (f) a/h¼ 100) for cylindrical sandwich shells with FGM core, R/a¼ 5, ED555 shell model and varying the length-to-thickness ratio.

1506

F.A. Fazzolari, E. Carrera / Journal of Sound and Vibration 333 (2014) 1485–1508

particular, a HSDT FEM formulation given by Pradyumna and Bandyopadhyay [7] and two HSDTs proposed by Neves et al. [6] and here defined as HSDT RBFC-1 and HSDT RBFC-2. The transverse normal deformation ɛzz is included in the former and discarded in the latter. As can be observed in both tables the dimensionless circular frequency parameter decreases when increasing both the radius-to-length ratio and the volume fraction index. By increasing the expansion orders in the proposed shell models convergence is quickly reached. Based on the above consideration the ED444 is the most accurate shell model with less demanding computational cost. In Tables 8 and 9 a similar assessment has been undertaken for the dimensionless fundamental circular frequency parameter of SSSS square FGM spherical shells using the same material. Understandably, as expected, the dimensionless fundamental circular frequency parameters are globally higher than those computed in the previous analysis. From an overall point of view, the same main considerations drawn from the analysis of FGM cylindrical shells are found in the case of spherical shells.

0.5

0.3 0.2 0.1

p=0.5 p=1 p=5 p=10

0.4 0.3 0.2 0.1

0

z/h

z/h

0.5

p=0.5 p=1 p=5 p=10

0.4

0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4

−0.4

−0.5 −0.2 −0.15 −0.1 −0.05

0

0.05

0.1

0.15

−0.5 0.02

0.2

0.03

ux [m]

0.5

0.3 0.2

0.3 0.2

z/h

z/h

0.1

0

0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4

−0.4

−0.5 −0.2 −0.15 −0.1 −0.05

0

0.05

0.1

0.15

−0.5 0.02

0.2

0.03

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

−0.4

p=0.5 p=1 p=5 p=10

−0.5 0.9

0.905

−0.2 −0.3 −0.4 0.915

uz [m]

0.05

0.06

0.9977

0.998

0 −0.1

0.91

0.04

uy [m]

z/h

z/h

uy [m]

−0.3

0.06

p=0.5 p=1 p=5 p=10

0.4

0.1

−0.2

0.05

0.5

p=0.5 p=1 p=5 p=10

0.4

−0.1

0.04

ux [m]

0.92

0.925

−0.5 0.997

p=0.5 p=1 p=5 p=10 0.9972

0.9975

uz [m]

Fig. 4. Modal displacements ux ((a) a/h¼ 10, (b) a/h ¼100), uy ((c) a/h ¼ 10, (d) a/h ¼100) and uz ((e) a/h ¼10, (f) a/h ¼100) for spherical sandwich shells with FGM core, R/a¼ 5, ED555 shell model and varying the length-to-thickness ratio.

F.A. Fazzolari, E. Carrera / Journal of Sound and Vibration 333 (2014) 1485–1508

1507

6.3. Circular frequency parameters of simply supported cylindrical and spherical sandwich shells with FGM core pffiffiffiffiffiffiffiffiffiffiffiffi 2 In Tables 10 and 11 the dimensionless fundamental circular frequency parameter ω ¼ ðωb =hÞ ρ0 =E0 with ρ0 ¼ 1 kg=m3 and E0 ¼ 1 GPa of square cylindrical sandwich shells with FGM core (see Fig. 1), material in Table 2, length-to thickness ratio a/h¼10 and a/h¼100 respectively and varying the radius-to-length ratio and the graduation index are investigated. The analysis is carried out considering the face sheets thickness t c ¼ t m ¼ 0:1 h and the core thickness hc ¼ 0:8 h being h the total shell thickness. Several refined ESL theories and ZZ theories have been assessed comparing the results with 3D Ritz solution given by Li et al. [56]. The ED999 exactly match the 3D Ritz solution whilst the introduction of MZZF in the ESL fields does not affect significantly the results accuracy, both for moderately thick and thin cylindrical shells. The dimensionless fundamental circular frequency parameters increase when increasing the volume fraction index and decrease when increasing the radius-to-length ratio. In Tables 12 and 13 the same analysis has been performed for square spherical sandwich shells with FGM core (see Fig. 1). Once again the dimensionless fundamental circular frequency parameters are higher, but on the whole similar considerations to the cylindrical shells analysis can be inferred. 6.4. Modal displacements of simply supported cylindrical and spherical sandwich shells with FGM core In Figs. 3 and 4 the modal displacements ux, uy and uz of simply supported sandwich cylindrical and spherical shells with FGM core have been depicted varying the volume fraction index and for both moderately thick and thin cylindrical and spherical sandwich shells. The ED555 has been employed along with a radius-to-length ratio R/a¼5. 7. Conclusions The HTRF based on the use of advanced ESL and ZZ shell models with hierarchical capabilities has been employed to cope with the free vibration analysis of FGM shells. Two benchmarks considering FGM cylindrical and spherical shells and cylindrical and spherical sandwich shells with FGM core have been studied. The accuracy of the presented formulation has been widely demonstrated. The effects of several parameters such as stacking sequence, length-to-thickness ratio, boundary conditions, radius-to-length ratio and volume fraction index on the circular frequency parameters and modal displacements have been commented. From the analysis carried out the following main conclusions can be drawn: 1. The developed refined quasi-3D shell models lead to the 3D exact solution. 2. From the proposed advanced quasi-3D shell models the ED444 turned out to be the most accurate with less demanding computational cost. 3. Boundary conditions remarkably affect the free vibration characteristics of doubly curved FGM shells. 4. The dimensionless circular frequency parameters decrease when increasing the volume fraction index in the Benchmark 1 and increase when increasing the volume fraction index in the Benchmark 2 this happens due to the fact that two different power-laws are used to describe the volume fraction of the ceramic and metal phases. 5. The dimensionless circular frequency parameters decrease when increasing the radius-to-length ratio. 6. In the present analysis the inclusion of MZZF does not affect significantly the results accuracy.

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