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ISSN 0894-9166

A PRECISE METHOD FOR SOLVING WAVE PROPAGATION IN HOLLOW SANDWICH CYLINDERS WITH PRISMATIC CORES⋆⋆ Junmiao Meng1

Zichen Deng1⋆

Xiaojian Xu1

Kai Zhang1

Xiuhui Hou2

1

( Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710072, China) (2 School of Civil Engineering, Chang’an University, Xi’an 710064, China) Received 12 March 2014, revision received 28 November 2014

ABSTRACT Wave propagation in infinitely long hollow sandwich cylinders with prismatic cores is analyzed by the extended Wittrick-Williams (W-W) algorithm and the precise integration method (PIM). The effective elastic constants of prismatic cellular materials are obtained by the homogenization method. By applying the variational principle and introducing the dual variables, the canonical equations of Hamiltonian system are constructed. Thereafter, the wave propagation problem is converted to an eigenvalue problem. In numerical examples, the effects of the prismatic cellular topology, the relative density, and the boundary conditions on dispersion relations, respectively, are investigated.

KEY WORDS extended Wittrick-Williams algorithm, cellular material, Hamiltonian system, dispersion relation

I. INTRODUCTION Sandwich structures have been extensively used in various engineering disciplines such as automotive engineering, civil engineering, and aerospace industry[1] . Because of their construction, sandwich structures exhibit many advantages including high strength/weight ratio, high stiffness/weight ratio, and energy absorption capabilities. Standard sandwich-type structures consist of two thin, stiff face sheets usually of the same thickness, separated by a lightweight, thick core[2] . Common cores can be categorized into two main types: foams and lattice truss materials. Whilst the former has random cellular morphologies, either open- or close-celled, the latter has periodically distributed cells including honeycombs, prismatic and truss cores. All metallic lightweight sandwich structures with periodic prismatic and truss cores have the potential for simultaneous load bearing and active cooling[3] with open channels in the cells. For example, an engine combustor, which is a typical hollow cylindrical structure, is subjected to internal pressure and high working temperature. In this situation, it is believed that replacing the solid tube with a prismatic core is supposed to be competent for the combustor[4, 5]. Corresponding author. E-mail: [email protected] Project supported by the National Basic Research Program of China (No. 2011CB610300), the 111 project (No. B07050), the National Natural Science Foundation of China (Nos. 11172239 and 11372252), the Doctoral Program Foundation of Education Ministry of China (No. 20126102110023), the Fundamental Research Funds for the Central Universities (310201401JCQ01001), China Postdoctoral Science Foundation (2013M540724) and Shaanxi postdoctoral research projects. The authors would also like to appreciate Associate Professor Qiang Gao, Dalian University of Technology, for innumerable helpful discussions. ⋆

⋆⋆

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Application of hollow sandwich structures has also motivated various studies of wave propagation and dynamic flexural deformation[6] . As a facet of the problem, the dispersion relations for the harmonic wave propagating in hollow sandwich cylinders are calculated. There are in general three approaches to solving the problem, namely, the elastodynamic theory[7, 8] , the numerical analytical method (NAM)[9] and sandwich shell theory[10] . The homogeneous cylindrical shell is a basic form of structure for many types of engineering structures, such as aeroplanes, marine crafts and construction buildings[11] . The earliest research can be traced back to Gazis[12] , who investigated the propagation of free harmonic waves for axisymmetric and asymmetric motions of infinitely long isotropic hollow cylinders and gave analytic results represented by Bessel functions on the basis of elastodynamic theory. Then, Markus and Mead[13] investigated the wave propagation in infinitely long orthotropic hollow cylinders to obtain analytic results. Meanwhile, Mirsky[14] proposed an approximate approach on the basis of the thick shell theory to solve the same problem. Elmaimouni et al.[7] presented a polynomial approach based on the elastodynamic theory for determining the guided waves in infinitely long hollow cylinders with anisotropic materials. Honarvar[15] took the advantage of the mathematical model developed in 1996[16] to derive the frequency equations of longitudinal and flexural wave propagation in free transversely isotropic cylinders. Akbarov[17] studied the torsional wave dispersion in a three-layered hollow cylinder with a pre-strained face (core) layer made from highly elastic materials. The wave motion in infinitely long functionally graded material (FGM) hollow cylinders was also investigated by Han et al.[9] and Elmaimouni et al.[8] . However, the orthogonal polynomial series method can only deal with layered hollow cylinders when the material properties of two adjacent layers do not change dramatically and is unable to lead to correct continuous normal stress and normal displacement profiles[18] . To overcome the limitations of the conventional orthogonal polynomial method, Gu et al.[18] proposed an improved orthogonal polynomial series method to investigate the harmonic wave propagation in multilayered spherical curved plates. Many works have been done for hollow cylinders, however, solutions for the guided waves in hollow sandwich cylinders with prismatic cores are rare. Zhou[10] investigated the propagation of axisymmetric free harmonic waves in infinitely long hollow sandwich cylinders, and obtained the critical velocity of moving loading by two methods: One is based on the sandwich shell theory and the other on the elastodynamics theory. Of the most common methods, the symplectic algorithm possesses its potential advantages of solving free wave propagation[19] and forced vibration induced by stationary harmonic loads of one-dimensional periodic structures[20] . A complete description of the dispersion relations with no restrictions on frequency and wavelength was provided by transforming the wave equations into the Hamiltonian system and using a transfer matrix approach for solving the problem in the Hamiltonian system[6] . The precise solutions for surface wave propagation in layered anisotropic material were analyzed by Zhong et al.[21] who also pointed out that the method can easily be extended to three-dimensional (3D) wave problems. After that, a systematic job was done by Gao et al.[22, 23] using a precise integration algorithm in conjunction with the extended W-W algorithm, where natural frequencies were calculated to an arbitrary precision. However, few of them took honeycomb cores into consideration. Later, Hou et al.[24] considered the axisymmetric wave propagation in hollow sandwich cylinders with hexagon prismatic cores. Motivated by the novelty of the topology of the cellular structures and numerical efficiency of the symplectic algorithm, the effects of the cellular topology, the relative density and the boundary conditions on the axisymmetric and asymmetric dispersion relations of hollow sandwich cylinders are treated by the precise numerical method[22, 23] to extend Hou’s[24] work. The present study is organized in five sections including the introduction above. §II describes the establishment of the canonical equations, which transform the wave propagation problem of infinitely long hollow sandwich cylinders from Lagrangian into Hamiltonian form. §III gives the procedure for calculating the angular frequencies by introducing the extended W-W algorithm, the piecewise-constant hypothesis and the PIM. Effects of the cellular topology, the relative density and the boundary conditions on the wave propagation properties of hollow sandwich cylinders with prismatic cores are studied in §IV. §V finally summarizes the main results of the work.

II. FORMULATION OF THE CANONICAL EQUATIONS Consider an orthotropic hollow sandwich cylinder shown in Fig.1(a), in which R1 is the inner radius, R2 is the outer radius, hf is the thickness of the face sheets and hc is the thickness of the sandwich

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core layer. The cylindrical coordinates Orθz, with the variables r along the radial direction, θ along the circumferential direction, and z along the axial cylinder are adopted to label an arbitrary material point of the cylinder. Figure 1(b) shows the cross-section in r-θ plane, which is divided by a large number of (say, 48) revolving periodic unit cells, of the hollow sandwich cylinder with rectangular cores. Figure 1(c) shows the cross-section in r-z plane.

Fig. 1. (a) Schematic diagram and cross-section in (b) r-θ and (c) r-z plane of a hollow sandwich cylinder under cylindrical coordinates.

It is assumed that the cylinder is infinitely long in the direction of the Oz axis and the lateral surfaces are stress free. The strain-displacement relations under infinitesimal deformations in cylindrical coordinates are given by ∂ur 1 ∂uθ ur ∂uz , εθ = + , εz = ∂r r ∂θ r ∂z 1 ∂ur ∂uθ uθ 1 ∂uz ∂uθ = + − , γθz = + , r ∂θ ∂r r r ∂θ ∂z

εr = γrθ

γrz =

∂uz ∂ur + ∂r ∂z

(1)

where ur , uθ , uz denote the displacement components in the radial, circumferential and axial directions, respectively. The generalized Hooke’s law for an orthotropic material is given by σr c11 c12 σ θ c13 σz = 0 τθz 0 τrz τrθ 0

c12 c22 c23 0 0 0

c13 c23 c33 0 0 0

0 0 0 c44 0 0

0 0 0 0 c55 0

0 εr εθ 0 0 εz 0 γθz 0 γrz c66 γrθ

(2)

where σr , σθ , σz , τθz , τrz , τrθ are the stress components; εr , εθ , εz , γθz , γrz , γrθ are the strain components; cij represents the elastic constants of the material.

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In the absence of the body force, the equilibrium equations can be expressed in cylindrical coordinates as 1 ∂τrθ ∂τrz σr − σθ ∂ 2 ur ∂σr + + + =ρ 2 ∂r r ∂θ ∂z r ∂t ∂τrθ 1 ∂σθ ∂τθz 2τrθ ∂ 2 uθ + + + =ρ 2 ∂r r ∂θ ∂z r ∂t ∂τrz 1 ∂τθz ∂σz τrz ∂ 2 uz + + + =ρ 2 ∂r r ∂θ ∂z r ∂t in which ρ is the density and may differ from face sheets to the core. Under the force boundary conditions pr = p¯r ,

pθ = p¯θ ,

pz = p¯z

(3)

(4)

and pr = nr σr + nθ τrθ + nz τrz pθ = nr τrθ + nθ σθ + nz τθz

(5)

pz = nr τrθ + nθ τθz + nz σz where p¯r , p¯θ , p¯z denote the given component forces per unit area on the boundary, nr , nθ , nz are the components of unit normal vector. Under the displacement boundary conditions ur = u ¯r ,

uθ = u ¯θ ,

uz = u¯z

where u ¯r , u ¯θ , u ¯z are prescribed. According to the virtual displacement principle within the volume of V , one can get Z ZZZ 1 ∂τrθ ∂τrz σr − σθ ∂ 2 ur ∂σr + + + − ρ 2 δur dV dt δΠ = − r ∂θ ∂z r ∂t Z t Z Z Z Ω ∂r 1 ∂σθ ∂τrθ ∂τθz 2τrθ ∂ 2 uθ + + + − ρ 2 δuθ dV dt − ∂r ∂z r ∂t Zt Z Z Z Ω r ∂θ 1 ∂τθz ∂σz τrz ∂ 2 uz ∂τrz + + + − ρ 2 δuz dV dt − r ∂θ ∂z r ∂t Zt Z Z Ω ∂r + [(pr − p¯r ) δur + (pθ − p¯θ ) δuθ + (pz − p¯z ) δuz ] dSdt t

(6)

(7)

Sσ

where dV = rdrdθdz and dS are the volume element and the area element, respectively. δur , δuθ and δuz are the virtual displacement components. The relations between the area element dS and the cylindrical coordinates can be expressed as dSr = nr dS = rdθdz dSθ = nθ dS = drdz

(8)

dSz = nz dS = rdrdθ Substituting Eqs.(1), (2) and (8) into Eq.(7), one can get Z ZZZ 1 2 2 2 + c55 γrz + c66 γrθ + c12 εr εθ δΠ = δ c11 ε2r + c22 ε2θ + c33 ε2z + c44 γθz 2 t Ω " 2 2 2 #) ρ ∂ur ∂uθ ∂uz +c13 εr εz + c23 εθ εz − + + rdrdθdzdt 2 ∂t ∂t ∂t ZZZ Z ZZ ∂ur ∂uθ ∂uz +ρ δur + δuθ + δuz rdrdθdz − (¯ pr δur + p¯θ δuθ + p¯z δuz ) dSdt ∂t ∂t ∂t Ω t Sσ (9)

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The total potential energy density, termed here Lagrange function, of the system can be obtained from Eq.(9): L (ur , uθ , uz , u˙ r , u˙ θ , u˙ z ) 2 2 2 2 1 1 ur 1 1 ∂uθ ∂ur 1 ∂uθ ∂uz 1 ∂uz = rc11 + rc22 + + rc33 + rc44 + 2 ∂r 2 r ∂θ r 2 ∂z 2 r ∂θ ∂z 2 2 1 ∂ur 1 ∂uθ uθ ur ∂uz 1 ∂ur ∂ur 1 ∂uθ + rc55 + + rc66 + − + rc12 + 2 ∂r ∂z 2 r ∂θ ∂r " r ∂r r ∂θ r 2 2 2 # ur ∂uz ρr 1 ∂uθ ∂ur ∂uθ ∂uz ∂ur ∂uz + rc23 + − + + +rc13 ∂r ∂z r ∂θ r ∂z 2 ∂t ∂t ∂t

(10)

˜ = {˜ where ( · ) = ∂/∂r. Let the displacement variables q q1 , q˜2 , q˜3 }T = {ur , uθ , uz }T and the dual T ˜ = ∂L/∂ ˜q˙ = {˜ variables p p1 , p˜2 , p˜3 } , one can calculate the dual variables from Eq.(10) as follows ∂L q˜1 ∂ q˜1 1 ∂ q˜2 ∂ q˜3 p˜1 = = rc11 + rc12 + + rc13 = rσr ˙ ∂r r ∂θ r ∂z ∂ q˜1 ∂ q˜2 q˜2 ∂L 1 ∂ q˜1 = rτrθ = rc66 + − p˜2 = ˙ ∂r r ∂ q˜2 r ∂θ ∂L ∂ q˜3 ∂ q˜1 p˜3 = = rc55 + = rτrz ˙ ∂r ∂z ∂ q˜3

(11)

From Eq.(11), one can get p˜1 c12 q˜1 c12 ∂ q˜2 c13 ∂ q˜3 q˜˙1 = − − − rc11 rc11 rc11 ∂θ c11 ∂z p ˜ ∂ q ˜ 1 q ˜ 2 1 2 q˜˙2 = − + rc66 r ∂θ r p ˜ ∂ q ˜ 3 3 q˜˙3 = − rc55 ∂z

(12)

By substituting Eq.(12) into Eqs.(1) and (2), other stress components can be obtained c12 c2 ∂q2 q1 c12 c13 ∂q3 p˜1 + c22 − 12 + + c23 − rc11 c11 r∂θ r c11 ∂z c13 c12 c13 ∂q2 q1 c2 ∂q3 σz = p˜1 + c23 − + + c33 − 13 rc11 c r∂θ r c ∂z 11 11 ∂q2 ∂q3 τθz = c44 + ∂z r∂θ σθ =

(13)

The expressions of p˜˙ 1 , p˜˙ 2 , p˜˙ 3 can be derived from Eqs.(2), (3), (12), (13) and the relations σr = p˜1 /r, τrθ = p˜2 /r, τrz = p˜3 /r. At this point, the canonical equations are obtained: ˙ q˜1 D11 D21 q˜˙2 ˙ q˜3 D = 31 D41 p˜˙ 1 D51 p˜˙ 2 ˙ D 61 p˜3

D12 D22 0 D42 D52 D62

D13 0 0 D43 D53 D63

D14 0 0 D44 D54 D64

0 D25 0 D45 D55 0

0 q˜1 q˜2 0 D36 q˜3 D46 p˜1 0 p˜2 0 p˜3

(14)

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where c1 ∂ c3 c1 ∂ , D12 = − , D13 = −c2 , D14 = r r ∂θ ∂z r 1 ∂ 1 c4 =− , D22 = , D25 = r ∂θ r r ∂ c5 ∂2 c6 c6 ∂ = − , D36 = , D41 = ρr 2 + , D42 = ∂z r ∂t r r ∂θ c1 1 ∂ ∂ c6 ∂ ∂ = c7 , D44 = , D45 = − , D46 = − , D51 = − ∂z r r ∂θ ∂z r ∂θ c6 ∂ 2 ∂2 ∂2 c1 ∂ 1 ∂2 − c8 r 2 , D53 = − (c7 + c8 ) , D54 = − , D55 = − = ρr 2 − ∂t r ∂θ2 ∂z ∂θ∂z r ∂θ r ∂2 ∂2 c8 ∂ 2 ∂2 ∂ ∂ , D63 = ρr 2 − − c9 r 2 , D64 = −c2 = −c7 , D62 = − (c7 + c8 ) ∂z ∂θ∂z ∂t r ∂θ2 ∂z ∂z

D11 = − D21 D31 D43 D52 D61 and

c12 , c11

1 1 1 , c4 = , c5 = c11 c66 c55 (15) c212 c12 c13 c2 c6 = c22 − , c7 = c23 − , c8 = c44 , c9 = c33 − 13 c11 c11 c11 Then the output Eq.(13) can be rewritten as ∂uz ur ∂uθ c1 + c7 + σθ = p˜1 + c6 r r ∂z r∂θ c2 ∂uθ ur ∂uz σz = p˜1 + c7 + + c9 (16) r r∂θ r ∂z ∂uθ ∂uz τθz = c8 + ∂z r∂θ For a free harmonic wave propagating in infinitely long orthotropic hollow sandwich cylinders, the displacement components and stress components are assumed to be in the form: c1 =

T

c2 =

c13 , c11

c3 =

T

{˜ q1 , q˜2 , q˜3 , p˜1 , p˜2 , p˜3 } = {q1 (r) , q2 (r) , q3 (r) , p1 (r) , p2 (r) , p3 (r)} exp [i (kz z + kθ θ − ωt)] (17) where q1 (r), q2 (r), q3 (r), p1 (r), p2 (r), p3 (r) are functions of r, kz , kθ are the wave numbers in the longitudinal and circumferential directions, respectively, and ω is the angular frequency. Substituting Eq.(17) into Eq.(14) gives the ordinary differential equations (ODEs) d V (r) = H(r)V (r) (18) dr where T V (r) = {q1 (r) , q2 (r) , q3 (r) , p1 (r) , p2 (r) , p3 (r)} (19) H11 H12 H13 H14 0 0 H21 H22 0 0 H25 0 H31 0 0 0 0 H 36 H(r) = (20) H H H H H H 42 43 44 45 46 41 H51 H52 H53 H54 H55 0 H61 H62 H63 H64 0 0 where 1 c3 c1 H11 = − , H12 = −ic1 kθ , H13 = −ic2 kz , H14 = r r r 1 1 c4 c5 c6 H21 = −ikθ , H22 = , H25 = , H31 = −ikz , H36 = , H41 = −ρrω 2 + r r r r r 1 c1 1 1 H42 = ic6 kθ , H43 = ic7 kz , H44 = , H45 = −ikθ , H46 = −ikz , H51 = −ic6 kθ r r r r 1 1 2 21 2 H52 = −ρrω + c6 kθ + c8 kz r, H53 = (c7 + c8 ) kθ kz , H54 = −ic1 kθ , H55 = − r r r 2 21 2 H61 = −ic7 kz , H62 = (c7 + c8 ) kθ kz , H63 = −ρrω + c8 kθ + c9 rkz , H64 = −ic2 kz r

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H

The matrix H (r) is a symplectic matrix in the sense that (JH (r)) = JH (r), where the superscript H means the complex conjugated transpose of a matrix, and 0 I3 J= (21) −I 3 0 in which I 3 is a 3 × 3 identity matrix. Based on the virtual displacement principle, the basic equations are transformed into the canonical Eq.(14) and the output Eq.(16). The novel arrangements greatly simplify the formulation and result in concise matrix equations in terms of the mixed variables instead of the displacement components. The Hamiltonian matrix of Eq.(20) can be block partitioned as A (r) D (r) H (r) = (22) B (r) −AH (r) in which A (r), B (r), D (r) are termed system matrices. So the Hamiltonian function can be expressed in terms of the dual variables[25]. q T Bq pT Dp V T (JH (r)) V =− + pT Aq + (23) H (q, p) = − 2 2 2 where T T q(r) = {q1 (r) , q2 (r) , q3 (r)} , p(r) = {p1 (r) , p2 (r) , p3 (r)} (24) It is clear that q (r) represents the displacement vector and p (r) represents the stress vector. The dual equations are then formed as follows ∂H (q, p) ∂H (q, p) q˙ = = Aq + Dp, p˙ = − = Bq − AH p (25) ∂p ∂q For orthotropic hollow sandwich cylinders as shown in Fig.1(a), the face sheets and the sandwich core layer can be divided into any appropriate number (say, nf = 10, nc = 80) of small-layers with equal thickness (small ∆ri = hi /ni , i = f, c), respectively. And then, each small-layer is divided into dozens of sub-layers with equal thickness (sub ∆ri = small ∆ri /2s ) (say, s = 6). When the thickness sub ∆ri is small enough, the piecewise constant hypothesis can be applied here. It is reasonable to choose the average radius rh = (ra + rb )/2 to replace r for the computation of H (r) within the sub-layer denoted by [ra , rb ], (rb > ra ). The first term ra in the square bracket corresponds to the left (or inner) side of the sub-layer, the second term rb corresponds to the right side (or outer) of the sub-layer. Therefore, [ra , rb ] can be used to represent the layer under study . The meaning of the square bracket in the following text represents the same meaning as [ra , rb ], unless otherwise stated. Thus, the system matrices A (r), B (r), D (r) in the interval [ra , rb ] can be regarded as constant. In §III, we will present the detailed numerical procedure for the calculation of the dispersion relations of orthotropic hollow sandwich cylinders.

III. FREQUENCY CALCULATION 3.1. The Interval Matrices and the Combination Let q a be the displacement vector at ra and pb the force vector at rb within an arbitrary layer [ra , rb ], (rb > ra ). If the vectors q a and pb along the interval [ra , rb ] are specified, the solutions of q b and pa in the interval [ra , rb ] can be expressed as linear functions of q a and pb , that is[26] q b = F q a + Gpb , pa = −Qq a + F H pb (26) in which F , G, Q are complex transfer matrices, also termed the interval matrices, to be determined hereafter. For calculation of the initialization of interval matrices by the Taylor series expansion method, the interval must be sufficient small. As mentioned in §II, the hollow sandwich cylinders have already been divided into plenty of sub-layers with the thickness of sub ∆ri = small ∆ri /2s (say, s = 6). Now, each of these sub-layers need to be further divided into 2N mini-layers with the extremely small thickness of τ = mini ∆ri = sub ∆ri /2N (say, N = 20). For such sufficient small interval τ , the Taylor series expansion method can be applied for the determination of the interval matrices F (τ ), G (τ ), Q (τ ). Then the interval combination method[26] is applied for the calculation of the overall interval matrices of hollow sandwich cylinders by integrating the interval matrices of the face sheets and the sandwich core layer. A detailed calculating process can be found in Refs.[22,23].

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3.2. The Extended W-W Algorithm It is inevitable to acquire the eigenvalues for solving wave propagation problems. The extended W-W algorithm[27] mainly used to achieve eigenvalue count Jm (ω), which can yield all the eigenvalues without missing any one. Jm (ω) represents the number of frequencies lower than a given frequency ω. No internal wave occurs within an extremely small layer with thickness τ , since the natural frequencies for such a small thickness must be almost infinitely high and so must exceed any sensibly chosen value of ω. In other words, the initial eigenvalue count Jm (ω) = 0 is set in this layer. The algorithm for combining the eigenvalue count of all the intervals into an overall interval Jmc is similar to the iterative procedure of interval matrices calculation. It is well known that there is a number of different boundary conditions associated with hollow sandwich cylinders. The eigenvalue counts of hollow sandwich cylinders with different boundary condiαβ tions are denoted as Jmc , where the superscript α = C or F represents the clamped and free boundary conditions of the inner surface, and β = C or F represents the clamped and free boundary conditions of the outer surface of sandwich hollow cylinders. Here we will discuss in detail about the eigenvalue CF (ω) , J FF (ω) , J FC (ω) , J CC (ω) for hollow sandwich cylinder subjected to CF, FF, FC counts Jmc mc mc mc and CC boundary conditions, respectively. CF (ω). (a) The eigenvalue count Jmc CF (ω) and The eigenvalue counts of two adjacent intervals [ra , rb ] and [rb , rc ] are denoted as Jm1 CF Jm2 (ω), respectively. Each of the intervals has a clamped condition in the inner surface and free condition in the outer surface. Then the two small intervals should be merged into a longer one [ra , rc ], which has a clamped condition in the inner surface and free condition in the outer surface. The eigenvalue count within the interval [ra , rc ] is given by[27] CF (ω) = J CF (ω) + J CF (ω) − s {G } + s G−1 + Q Jmc (27) 1 m1 m2 2 1

where G1 and Q2 are the interval matrices, s {· · · } is the sign count of the matrix within the brackets, namely, s {G1 } represents the number of negative sign in the diagonal matrix triangulated from G1 . FF (ω). (b) The eigenvalue count Jmc FF CF (ω), is equivalent to releasing the displacement constraints Jmc (ω), which can be derived from Jmc CF (ω). Whereas, Q represents the stiffness matrix of the inner surface when on the inner surface of Jmc FF (ω) can be obtained from Eq.(27) the outer surface is set free. The Jmc FF = J CF + s (Q) Jmc mc

(28)

FC (ω). (c) The eigenvalue count Jmc FC FF (ω), is equivalent to adding the displacement constraints Jmc (ω), which can be derived from Jmc FF (ω). We need to get the stiffness matrix of the outer surface when the inner on the outer surface of Jmc surface is free. Substituting pR1 = 0 into Eq.(26) yields q R2 = F q R1 + GpR2 ,

0 = −QqR1 + F H pR2

(29)

from which, one can obtain pR2 = F Q−1 F H + G q R2 = Rq R2

(30)

FC = J FF − s (R) = J CF + s (Q) − s (R) Jmc mc mc

(31)

where R represents the stiffness matrix of outer surface when the inner surface is free. As a result, the FC (ω) can be obtained as eigenvalue count Jmc

CC (ω). (d) The eigenvalue count Jmc CC (ω), which can be derived from J FC (ω), is equivalent to adding the displacement constraints Jmc mc on the inner surface of J FC (ω). We need to get the stiffness matrix of the inner surface when the outer mc

surface is clamped. Substituting q R2 = 0 into Eq.(26) yields 0 = F q R1 + GpR2 ,

pR1 = −QqR1 + F H pR2

(32)

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from which, one can obtain pR1 = − Q + F H G−1 F q R1 = −R′ q R1

(33)

CC = J FC − s (R′ ) = J CF + s (Q) − s (R) − s (R′ ) Jmc mc mc

(34)

where R′ represents the stiffness matrix of inner surface when the outer surface is clamped. As a result, CC (ω) can be obtained the eigenvalue count Jmc

For the above four different boundary conditions, the eigenvalues can be obtained by the eigenvalue counts and the bisection method. The eigenvalue counts for different boundary conditions can be converted to each other by releasing or adding constraints. In §IV, the effects of the above four boundary conditions on the dispersion relations of hollow sandwich cylinders are presented.

IV. NUMERICAL EXAMPLES AND DISCUSSIONS Based on the foregoing formulations, the dispersion behaviors of axisymmetric (kθ = 0) and asymmetric (kθ = 1) modes for hollow sandwich cylinders are numerically calculated. 4.1. Comparisons with the Polynomial Method To verify the validity of the formulations and programs, our approach is firstly implemented on the dispersion behaviors of infinitely long orthotropic hollow cylinders with both the internal surface and outer surface stress free. The material parameters of hollow cylinders are listed in Table 1 and the density is take as ρ = 1 kg/m3[7] . The geometric parameters are taken as: R1 = 70 mm, R2 = 90 mm. Then the mean thickness H and the average radius R of the hollow cylinder are found to be 20 mm and 80 mm, respectively. Table 1. Elastic constants of the infinitely long hollow cylinder (×1010 N/m2 )a

c11 28.1 a Ref.[7].

c12 12.6

c13 8.4

c22 34.9

c23 8.8

c33 29.4

c44 10.8

c55 13.2

c66 13.1

A comparison is made between the results obtained by Elmaimouni et al. using the polynomial approach[7] and the results obtained using the present method for a homogeneous hollow cylinder. In order to directly p compare the present results with those of Elmaimouni et al, the dimensionless frequency Ω = ωH/ π c55 /ρ , and the normalized wave number H/λz = H/ (2π/kz ) are defined. The dispersion relations of axisymmetric modes and asymmetric modes of a cylinder are presented in Fig.2, respectively. It is seen from Fig.2 that the excellent agreement between our numerical results and those of Elmaimouni’s polynomial method is observed, which indicates the efficiency and accuracy of the proposed numerical method. 4.2. Effect of the Prismatic Cellular Topology In this section, the proposed method is used to analyze the dispersion relations of hollow sandwich cylinders with different prismatic cores. Consider an orthotropic hollow sandwich cylinder subjected to free boundary conditions in the inner surface and clamped boundary conditions in the outer surface. The geometric parameters are taken as: R1 = 70 mm, R2 = 90 mm, hf = 2 mm hc = 16 mm. The face sheets 3 and the sandwich core layer are made of AISI340 stainless steel with mass density ρ = 8000 kg/m , Young’s modulus E = 193 GPa, the Poisson’s ratio ν = 0.23. The relative density of the cores is taken as γ = ρc /ρ = 0.1. The number of revolving periodic unit cells in a cross section is set to be 48 to avoid inducing unnecessary numerical errors[5] of using the homogenization method described in literature [28]. The effective elastic constants of a series of hollow sandwich cylinders with five different topologies (rectangular,

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Fig. 2. Dispersion curves.

Fig. 3. Schematic diagram of the unit cells of the prismatic cores. Table 2. Effective elastic constants of the prismatic cores and the face sheets (γ = 0.1)

cij /E

Face sheet

c11 /E c12 /E c13 /E c22 /E c23 /E c33 /E c44 /E c55 /E c66 /E

1.1592893 0.3462812 0.3462812 1.1592894 0.3462812 1.1592894 0.4065041 0.4065041 0.4065041

Rectangular 0.0356456 0 0.0081985 0.0699399 0.0160862 0.1055855 0.0067317 0.0034309 0.0001615

Sandwich cores Diamond Triangle-6 0.0144976 0.0090619 0.0244652 0.0153561 0.0089614 0.0056161 0.0421576 0.0658114 0.0153232 0.0186685 0.1055855 0.1055855 0.0064124 0.0078124 0.0037502 0.0023502 0.0245322 0.0153804

Triangle-8 0.0245218 0.0128790 0.0086022 0.0553058 0.0156825 0.1055855 0.0065628 0.0035998 0.0129061

Kagome 0.0091520 0.0152908 0.0056219 0.0658518 0.0186628 0.1055855 0.0078099 0.0023526 0.0153955

diamond, triangle-6, triangle-8 and Kagome) of unit cells, shown in Fig.3, are obtained based on the homogenization method, as tabulated in Table 2. Table 2 shows that the effective elastic constants of triangle-6 are close to those of Kagome cores. Consequently, conclusions can be drawn that the dispersion curves of hollow sandwich cylinders with triangle-6 cores are very similar to those of hollow sandwich cylinders with Kagome cores. The dispersion curves of hollow sandwich cylinders with rectangular, diamond, triangle-6 and triangle-8 topologies are therefore shown in Fig.4. Figure 4(a) shows the dispersion curves of sandwich cylinders with rectangular cores. Axisymmetric modes are a little lower than asymmetric modes, especially so for lower wave numbers. Each branch excluding the first three lowest branches starts horizontally for kz = 0. Generally frequency increases with the wave number slightly, while rising dramatically for certain wave numbers. In addition, there are also several curves interleaving amidst the dispersion diagram, rising so fast with the increase of wave number. Results of hollow sandwich cylinders with prismatic cores of diamond, triangle-6 and triangle-8 topologies are given in Figs.4(b), (c) and (d), respectively. It can be seen from these figures that the dispersion curves are similar to each other and starts horizontally. For each figure, the axisymmetric

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Fig. 4. Dispersion curves of hollow sandwich cylinders with four different topologies.

modes are very close to the asymmetric modes. The frequencies of hollow cylinders with triangle-6 cores are a bit lower those that of hollow cylinders with diamond and triangle-8 cores. In Fig.4, the slope of a line from the origin to any point on a dispersion curve is the dimensionless p phase velocity 2V ρ/c55 . For short wavelengths, the phase velocity of different modes approaches certain limits. From the aspect of the elasticity, the phase velocities of the first axisymmetric mode and the first asymmetric mode approach those of the Rayleigh surface waves. The phase velocities of the higher modes approach the phase velocities of shear waves[7]. It is seen from Fig.4 that three features of the dispersion relations are observed: (i) For each of the figures, each branch excluding the first three lowest branches starts horizontally for kz = 0. Similar tendencies have been observed in Refs.[6,29]. (ii) The dispersion curves of axisymmetric modes are very close to those of asymmetric modes, especially for larger wave numbers. This shows that the axisymmetric modes and the asymmetric modes approach the same asymptotic values. (iii) It is obvious that the tendencies of the dispersion curves shown in Fig.4(a) are different from those of the other three ones. It is possible that the diamond, triangle-6 and triangle-8 prismatic cellular topologies have isotropic in-plane effective properties, while the square is strongly anisotropic. So, we can conclude that the in-plane effective properties of prismatic cellular topologies exert sufficient influences on the dispersion relations.

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4.3. Effect of the Relative Density Being one of the most important structural characteristics of a cellular solid[30] , the relative density, γ = ρc /ρ (the density of the sandwich cores, ρc , divided by that of the solid of which it is made, ρ ) directly affects the effective elastic constants of the sandwich core layer, which will then have an effect on the dispersion relations of hollow sandwich cylinders. So, it is necessary to investigate the effect of the relative density γ on the dispersion relations of hollow sandwich cylinders. Generally speaking, cellular materials have a relative density less than 0.3[30] . The upper bound of the relative density is also taken as 0.3 in this study. The boundary conditions, geometric parameters and materials of hollow sandwich cylinders are the same as the example described in §4.2. And also, the normalized wave number in the longitudinal direction is taken as H/ (2π/kz ) = 0.1. First of all, a series of effective elastic constants of four different topologies (rectangular, diamond, triangle-6, triangle-8) of unit cells with the increase of the relative density are obtained based on the homogenization method described in literature [28]. Thereafter, the effect of the relative density on the first five lowest angular frequencies of hollow sandwich cylinders with four different prismatic cores is obtained using the proposed method shown in Fig.5. As shown in Fig.5(a), the axisymmetric modes are a little lower than the asymmetric modes, especially so at lower relative density. When the relative density is very small, the first five frequencies are very close to each other. It is also seen from Fig.5(a) that the angular frequencies increase dramatically

Fig. 5. Effect of the relative density on the first five lowest angular frequencies of hollow sandwich cylinders with four different prismatic cores.

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within a very small range of relative densities at first, and then the growth speed slows down. Figure 5(b) gives the results of hollow sandwich cylinders with diamond cores. It is shown that the second and third lowest branches of the dispersion curves corresponding to kθ = 0 cross each other when the relative density is near 0.15, whereas the feature does not appear in the asymmetric modes corresponding to kθ = 1. Figure 5(c) gives the results of hollow sandwich cylinders with triangle-6 cores. The slowly changing curves observed from Fig.5(c) indicates that the relative density has a small effect on the angular frequencies. Dispersion relations for hollow sandwich cylinders with triangle-8 cores are found similar to that of cylinders with diamond cores, except that the intersection occurs between the first and the second lowest angular frequencies, as shown in Fig.5(d). In general, the effect of the relative density on the dispersions of the hollow sandwich cylinders with rectangular cores is prominent compared with that of the other three topologies considered. It also illustrates that the in-plane effective properties (anisotropic or isotropic) of prismatic cores are very important for the dispersion relations. For the other three cores, which have isotropic in-plane effective properties, it can be seen that the angular frequencies increase slowly with the increase of the relative density and linear relations between the angular frequencies and relative densities are found for large relative densities.

Fig. 6. Effects of different boundary conditions (FF, CF, FC, CC) on the lowest frequency of hollow sandwich cylinders with four different prismatic cores.

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4.4. Effects of the Boundary Conditions The boundary conditions of the hollow sandwich cylinders include four different cases described in §3.2. To investigate the effects of the boundary conditions FF, CF, FC and CC on the angular frequencies of hollow sandwich cylinders with different prismatic cores, the dispersion curves of the lowest frequency are depicted in Fig.6. Figure 6(a) shows the first dispersion relation of hollow sandwich cylinders with rectangular cores subjected to four different boundary conditions. The inset is the zoom of Fig.6(a). For the FF boundary conditions, the differences of the axisymmetric and asymmetric dispersion curves are notable for small values of wave number, while becoming indistinguishable with the increase of the wave number. However, for the other three types of boundary conditions, CF, FC and CC, different behaviors are found. When the wave number is less than 0.1, the axisymmetric results of FC and CF boundary conditions are similar to the results of FF. Whereas the asymmetric results of FC and CF boundary conditions are similar to the results of CC. With the increase of the wave number, the axisymmetric dispersion curves are close to the asymmetric dispersion curves. Figures 6(b), (c) and (d) give the results of hollow cylinders with prismatic cores of diamond, triangle6 and triangle-8 cells. The figures indicate that the change tendencies of the dispersion curves are similar to each other. It is seen that the angular frequencies increase dramatically within a very small range of wave numbers at first, and then the growth speed slows down. In general, the results of sandwich cylinders with triangle-6 cores are a bit lower than those of sandwich cylinders with diamond cores, and the results of sandwich cylinders with diamond cores are a bit lower than those of sandwich cylinders with triangle-8 cores. It is also seen from Fig.6 that frequencies of different boundary conditions have the following relation: FF < CF < FC < CC. Thus, it can be concluded that the more constraints imposed on the boundary conditions, the higher angular frequency of hollow sandwich cylinders will be achieved.

V. CONCLUSIONS The dispersion relations of hollow sandwich cylinders with five different prismatic cores were studied. The mathematical model for the wave propagation was presented, and the axisymmetric and asymmetric modes of dispersion relations were investigated. The main results are summarized as follows: (a) The dispersion relations of hollow sandwich cylinders can be greatly affected by the in-plane effective properties of prismatic cellular topologies and the axisymmetric modes and the asymmetric modes approach to the same asymptotic values. (b) For the prismatic cellular cores, which have isotropic in-plane effective properties, linear relations between the angular frequencies and the relative densities are found for large relative densities. (c) The angular frequencies of hollow sandwich cylinders can be sorted in the ascending order for different boundary conditions: FF < CF < FC < CC. The great challenge to solving wave problems lies in efficient determination of the eigenvalue. A few simple problems can be solved analytically, but for more complex problems encountered in engineering, an efficient numerical method is badly needed. The proposed numerical method is likely to find application in areas of hollow sandwich cylinders and might serve as an efficient tool for future study of wave propagation in hollow sandwich cylinders with auxetic cellular cores and hierarchical cellular cores.

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