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Wave dispersion in functionally graded magneto-electro-elastic nonlocal rod
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Wave dispersion in functionally graded magneto-electro-elastic nonlocal rod ✩
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S. Narendar
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Defence Research and Development Laboratory, Dr. APJ Abdul Kalam Missile Complex, Kanchanbagh, Hyderabad-500 058, Telangana, India
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a r t i c l e
i n f o
a b s t r a c t
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Article history: Received 16 July 2015 Received in revised form 27 November 2015 Accepted 9 January 2016 Available online xxxx
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Keywords: Magneto–electro-elastic rod Functionally graded material Wave dispersion Nonlocal continuum mechanics Phase velocity Escape frequency
Dispersion of elastic waves in functionally graded magneto-electro-elastic (FG-MEE) rod is studied based on nonlocal continuum mechanics. Governing equations of motion for the rod element are formulated in the nonlocal field (i.e., stress, electric and magnetic fields). The exponentially varying material properties along the axial direction of the rod have also been considered. The formulated nonlocal equations of motion are solved using the harmonic waveforms of the displacement field. The resulting wavenumber relations are solved for studying the wave dispersion phenomena in nonlocal FG-MEE rods. The wave characteristics are studied with respect to the spectrum and dispersion curves. The effects of the gradation index, volume fraction and nonlocal scale parameter are captured and discussed in detail. The results presented in this work are useful for the design of the futuristic multi-functional nanoscale magneto-electro-elastic devices that make use of the axial wave propagation properties. © 2016 Elsevier Masson SAS. All rights reserved.
88 89 90 91 92 93 94 95 96 97 98
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1. Introduction
37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58
The structural members belonging to functionally graded materials (FGM) have received wide attention by the various fields research communities [1]. Because of the coupling of mechanical and electrical fields, extensive study has been made for magnetoelectro-elastic (MEE) materials. So, with increasing usage of MEE structures in various engineering fields, wave propagation in MEE material media has also attracted many researchers. Chen et al. [2] presented an analytical treatment for the propagation of harmonic waves in magneto-electro-elastic multi-layered plates. The results on dispersion curves, modal shapes, and natural frequencies are presented for layered plates made of orthotropic elastic, transversely isotropic, piezoelectric and magnetostrictive materials. They showed the influence of different stacking sequences on the field response of layered smart composites. By using the effective field method, Chen and Shen [3] simplified the multi-fiber scattering problem to the one-fiber scattering problem, and the analytical expressions of magneto-electro-elastic fields for the multi-fiber composites were obtained in the long-wave limit. Buchanan [4] studied the free vibration of fully coupled infinite magneto-electro-elastic cylinders. Annigeri et al. [5] used the constitutive equations of piezomagnetic medium to study the free vi-
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✩
62 63 64 65 66
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This work is dedicated to my son Mr. Saggam Kaushik Jayasurya. E-mail addresses: [email protected], [email protected]. Tel.: +91 8897 625977.
http://dx.doi.org/10.1016/j.ast.2016.01.012 1270-9638/© 2016 Elsevier Masson SAS. All rights reserved.
bration analysis of magneto-electro-elastic cylindrical shells. They developed a semi-analytic finite element model. They proved that magnetic effect will reduce the stiffness of system and also bring down the frequency of the system. Bin et al. [6] studied the propagation of harmonic waves in functionally graded magneto-electro-elastic plates composed of piezoelectric BaTiO3 and magnetostrictive CoFe2 O4 . They employed the Legendre orthogonal polynomial series expansion approach to determine the wave propagating characteristics in the plates. Du et al. [7] investigated the Love surface waves in layered piezomagnetic/piezoelectric structures with initial stress. They applied the magneto-electrically open and short conditions to solve the problem numerically. The effect of the initial stress on the wave velocity and the magneto-electromechanical coupling factor was studied in detail in their work. Li et al. [8] investigated the Buckling and free vibration analysis of magneto-electro-elastic nanoplate resting on Pasternak foundation via nonlocal Mindlin plate theory. They showed that for a magneto-electro-elastic nanoplate, the buckling load decreases with increasing the lateral load. They also showed that the buckling load and free vibration frequency decrease with increasing nonlocal scaling parameter. Kattimani and Ray [9] formulated a three dimensional finite element to investigate the geometrically nonlinear vibrations of the magneto-electro-elastic plates with the active constrained layer damping. Zhang et al. [10] studied the propagation behaviour of Love waves in layered magneto-electro-elastic structures with inhomogeneous initial stress. They found that the effects of the gradient coefficient of the inhomogeneous initial stress and the
102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
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magneto-electric coupling factor on the phase velocity of the Love waves. Xue et al. [11] derived the nonlinear solitary wave equation in a long MEE circular rod. They considered the geometric nonlinearity in the longitudinal direction. The wave equation was solved by the Jacobi elliptic function expansion method. Wu et al. [12] investigated the symmetric and antisymmetric Lamb waves propagation in an infinite magneto-electro-elastic plates based on the three-dimensional linear elastic equations and magnetoelectro-elastic constitutive relations. The electrically and magnetically open case and shorted case for both types of wave modes were discussed in their work. Xue and Pan [13] studied the axial wave propagation of functionally graded magneto-electro-elastic rod made of piezoelectric BaTiO3 and piezomagnetic CoFe2 O4 . They assumed the materials properties to vary exponentially along the rod longitudinal direction. They investigated the important influence of the gradient factor as well as material coupling on the wave features of the rod. Jiangong et al. [14] presented a dynamic solution for the propagation of harmonic waves in inhomogeneous magneto-electro-elastic hollow cylinders. The Legendre orthogonal polynomial series expansion approach was employed to determine the wave propagating characteristics in the hollow cylinders. In the recent year a lot of work has been carried out by many researchers in the area of the dynamics of the nonlocal rods [15–25]. To the best of authors’ knowledge, however, the wave propagation analysis in functionally graded magneto-electro-elastic nonlocal rods has not been considered in literature. Based on the nonlocal elasticity theory, the wave propagation analysis of a FG MME nanorods is investigated in the present work.
29 30
2. Mathematical formulations
31 32
2.1. Review of nonlocal continuum theory
35 36 37 38
The nonlocal continuum theory states that the stress at a reference point r is considered to be a function of the strain field at every point in the body. The basic equations for a linear, homogeneous, isotropic, and nonlocal elastic solids with zero body force are given as follows [26]:
39 40 41 42 43
σi j , j = 0 σi j (r) =
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62
α (|r − r |, τ )C i jkl εkl (r )dV (r ) ∀ r ∈
1
εi j = (u i , j + u j ,i )
(2)
(3)
2
where C i jkl is the elastic modulus tensor of classical isotropic elasticity, σi j and εi j are stress and strain tensors respectively, u i is the displacement vector and is the region occupied by the body. α = α (|r − r |, τ ) is the nonlocal modulus or attenuation function incorporating the nonlocal effects into the constitutive equations at the reference point r produced by local strain at the source r . |r − r | is the Euclidean distance, and τ = /, where = e 0 a, a is an internal characteristic length, e.g., length of C − C bond for carbon nanotubes, or lattice parameter, granular distance etc., and is an external characteristic length e.g., wavelength, crack length or size of the sample. In the published literature, for each material, an appropriate constant value has been assumed for the nonlocal scaling parameter, . As τ → 0, α must revert to the Dirac delta measure so that classical elasticity limit is included in the limit of vanishing internal characteristic length, that is,
63 64
44 45
(1)
lim
α (|r − r |, τ ) = δ(|r − r |)
65
τ →0
66
We therefore expect that
68 69 70 71 72 73 74 75 76 77 78 79 80 81
Fig. 1. Schematic of FG-MEE rod showing the geometry with radius R and also showing both poling and wave propagation directions. Here u r , u θ and u z are the mechanical displacements along r −, θ− and z− directions, respectively.
α is a delta sequence.
(4)
82 83 84 85
The integro-partial differential equations (1)–(3) representing the above linear nonlocal elasticity can be reduced to singular partial differential equations of a special class of physically admissible kernel. In addition, Hook’s law for the stress–strain relations is given by
86 87 88 89 90 91
σ (r) − 2 ∇ 2 σ (r) = E ε(r)
(5)
where E is the Young’s modulus of the material. Nonlocal elasticity in the limit → 0 reverts to classical elasticity, which can be seen by letting → 0 in Eq. (5), to obtain Hook’s law of classical elasticity.
92 93 94 95 96 97 98
2.2. Formulation of governing equations of motion
33 34
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99 100
The radius of FG-MME rod is R and is of circular cross sections as shown in Fig. 1. The longitudinal axis (z-axis) is perpendicular to the plane of the material isotropy. The material coefficients of this rod vary exponentially along the axial direction in a unified manner [27,28].
101 102 103 104 105 106
Elastic coefficients: c i j = c i0j e z ,
107
Piezoelectric coefficients: e i j = e 0i j e z ,
108
Piezomagnetic coefficients: qi j = q0i j e z ,
110
109 111
Dielectric coefficients: κi j = κi0j e z , Magnetoelectric
112 113
coefficients: di j = d0i j e z ,
114 115
0 z ije ,
Magnetic coefficients: μi j = μ
116
Density: ρi j = ρi0j e z
(6)
117 118
The superscript 0 in the above coefficients represents the constant factor of the material property and is the gradient index of the material. The nonlocal constitutive relations of the cylindrical rod are:
119 120 121 122 123
σ (r ) − ∇ 2 σ (r ) = c11 ε(r ) + c12 ε(θ ) + c13 ε(z) − e31 E (z) − q31 H (z) σ
(θ )
σ
( z)
2
2
(θ )
2
2
( z)
− ∇ σ − ∇ σ
(r )
+ c 11 ε
(r )
+ c 13 ε
= c 12 ε = c 13 ε
+ c 13 ε
(θ )
(θ )
+ c 33 ε
( z)
( z)
− e 31 E
( z)
− e 33 E
( z)
− q31 H
( z)
− q33 H
τ
2
2 (θ z)
− ∇ τ
= c 44 γ
(θ z)
τ (r θ ) − 2 ∇ 2 τ (r θ ) = c66 γ (r θ )
− e 15 E
(θ )
− q15 H
125 126
( z)
τ (rz) − 2 ∇ 2 τ (rz) = c44 γ (rz) − e15 E (r ) − q15 H (r ) (θ z)
124
127 128 129 130
(θ )
131
(7)
132
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1 2 3 4
D D
( z)
2
2
(θ )
2
2
( z)
− ∇ D − ∇ D
5
8 9 10 11
B
(r )
2
2
− ∇ B
(r )
14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
54 55 56 57 58 59 60 61 62 63 64 65 66
(r )
+ κ11 E
+ e 31 ε
(θ )
(θ )
+ d11 H
+ e 33 ε
( z)
(rz)
+ d11 E
+ μ11 H
B
( z)
2
2
− ∇ B
( z)
= q31 ε
(r )
+ q31 ε
(θ )
+ q33 ε
( z)
+ κ33 E
( z)
(r )
+ q33 E
( z)
+ μ33 H (z)
(9)
where σ (i ) and τ (i j ) are the normal and shear stresses in respective directions; ε (i ) and γ (i j ) are the normal and shear strains; E (i ) , H (i ) , D (i ) and B (i ) are respectively, the electric field, magnetic field, electric displacements and magnetic inductions along respective directions. Here is nonlocal parameter. In the absence of the body forces, electric and magnetic sources, the equations of motion of the FG-MEE rod are:
67
(16)
B (r ) = −ψ,r , B (θ ) = −r −1 ψ,θ , B (z) = −ψ,z
(17)
where φ and ψ are the electric and magnetic potentials, respectively. The simplifications made earlier are also applicable to the above mentioned relations. Using Eq. (13) in Eqs. (7)–(9) leads to
+
σ
−
2
σ,(zzz)
D
( z)
−
2
( z) D ,zz
ε
c 12
c 12
( z)
B (z) − 2 B ,zz = q31 −
+ d33 +
(r )
E
c 12
+ c 33 − ( z)
+
( z)
ε(r ) + q33 −
q31 c 13
E ( z) +
c 12
ε
H
ε(θ ) =
1 c 12
e 31 E (z) + q31 H (z) − c 11 ε (r ) − c 13 ε (z)
−
σ
( z)
−
2
σ,(zzz)
(19)
ε ( z)
−
q213
=
c 12
e 31 c 13 c 12
e 33 −
The finite elastic strain-displacement, electric field-potential and magnetic field-potential relations can be expressed as:
q33 −
e 231
− d33 +
c 12
φ,z − d33 +
c 12 q31 c 13 c 12
c 12
c 12 q31 c 11
(21)
( z) u ,z
φ,z − μ33 +
c 12
107
111 112 113 114 115 117 118 119 120 121 122 123
ψ, z
124
(22)
( z)
− νe q31 − u ,z c 12 q213
106
116
− q33 ψ,z
e 31 q13
105
110
c 12
104
109
e 31 c 11
98
108
( z)
u ,z
c 12
q31 c 13
97
102
c 12
91
103
q31 c 11
− νe e 31 −
q31 c 13
− e 33 φ,z −
c 12
− κ33 +
( z)
e 31 c 13
B (z) − 2 B ,zz =
90
101
H ( z)
c 12
89
100
c 12
(14)
c 12
e 31 c 11
( z) D (z) − 2 D ,zz
− νe c 12 −
2 c 11
85
99
− e 31 φ,z − − q31 ψ,z c 12
2 c 13 c 13 c 11 ( z) c 33 − u ,z − νe c 13 −
=
(13)
c 13 c 11
c 13 −
With the above simplifications, the equations of motion (Eqns. (10)–(12)) are reduced to: ( z) r −1 σ (θ ) = ρνe r u¨ (r ) = ρνe r u¨ ,z σ,(zz) = ρ u¨ (z) ( z) D ,z = 0 ( z) B ,z = 0
σ (θ ) − 2 σ,(θzz) =
84
96
Substituting Eqs. (15)–(17) into Eqs. (18)–(20) gives the following relations
83
95
(20)
79
94
• The cross section of the FG-MEE rod remains plane before and after the deformation. • The lateral surface of the rod has axial symmetry, which im∂ = 0. plies that u (θ) = 0 and ∂θ • To consider the Poisson’s effect, the gradient of the axial displacement u z and the radial displacement u r are related by (z) u (r ) = νe ru ,z , where νe is the effective Poisson’s ratio. • The problem is one-dimensional, the extended tractions on the lateral boundary of the rod should be zero. Therefore, σ (r ) = 0, τ (rz) = 0, τ (r θ) = 0, D (r ) = 0 and B (r ) = 0. From which the following relations are obtained: γ (rz) = 0, γ (θ z) = 0, E (r ) = 0, E (θ) = 0, H (r ) = 0, H (θ) = 0, D (θ) = 0, B (θ) = 0. From Eq. (7), we get
78
93
( z)
c 12
μ33 +
77
92
( z)
c 12
q31 c 13
( z)
(18)
e 31 q13
76
88
− q33 H
c 12
+ d33 +
75
87
e 31 c 13
+ e 33 −
74
86
( z)
ε
c 12
73
82
− q31 H (z)
q31 c 13
72
81
c 12
c 12
2 c 13
71
80
ε ( z)
q31 c 11 c 12
c 12
e 231
q31 c 11
(r )
− e 33 E
e 31 c 11
+ κ33 +
= e 31 − ε c 12
(12)
c 13 c 11
− e 31 E (z) +
c 13 c 11
e 31 c 13
(10)
c 12
e 31 c 11
= c 13 −
ε(r ) + c13 −
c 12
( z)
2 c 11
σ (θ ) − 2 σ,(θzz) = c12 −
+
Here the subscript , i represents the partial differentiation of the dependent variable with the independent variable i. We consider the wave propagation in a long FG-MEE circular rod. In the cylindrical coordinate system (r , θ, z), θ ∈ [0, 2π ], 0 ≤ r ≤ R. To facilitate the present wave propagation study in FG-MEE rod, the following assumptions are made:
70
E (r ) = −φ,r , E (θ ) = −r −1 φ,θ , E (z) = −φ,z
τ,(rr θ ) + r −1 σ,θ(θ ) + τ,(θz z) + r −1 τ (r θ ) = ρ u¨ (θ ) (11)
69
(15)
(θ z) ( z) + r −1 τ,θ + σ,z + r −1 τ (rz) = ρ u¨ (z) (r ) (θ ) ( z) D ,r + r −1 D ,θ + D ,z + r −1 D (r ) = 0 (r ) (θ ) ( z) B ,r + r −1 B ,θ + B ,z + r −1 B (r ) = 0
68
) ( z) (r ) (rz) γ (θ z) = r −1 u (,θz) + u (θ = r −1 u ,r + u ,z ,z , γ
σ,(rr ) + r −1 τ,θ(r θ ) + τ,(zrz) + r −1 σ (r ) − σ (θ ) = ρ u¨ (r ) τ,(rrz)
(r ) ) ε(z) = u (,zz) , γ (r θ ) = r −1 u ,θ − u (θ ) + u (θ , ,r
(8) (r )
(θ )
B (θ ) − 2 ∇ 2 B (θ ) = q15 γ (θ z) + d11 E (θ ) + μ11 H (θ )
52 53
= e 31 ε
= q15 γ
12 13
= e 15 γ
(θ z)
+ d33 H (z)
6 7
) (r ) ε(r ) = u ,(rr ) , ε(θ ) = r −1 u (θ , ,θ + u
D (r ) − 2 ∇ 2 D (r ) = e 15 γ (rz) + κ11 E (r ) + d11 H (r ) (θ )
3
125 126 127 128 129
ψ, z
130 131
(23)
132
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1 2 3
Introducing the equations of motion (Eqs. (14)) into the nonlocal constitutive relations (Eqs. (21)–(23)) leads to the following relations:
4 5
z) νe ρ r 2 u¨ (,zz) − 2 νe ρ r 2 u¨ (,zzz
6
=
7 8
c 13 −
9
−
10
c 12
13
z) ρ u¨ (z) − 2 ρ u¨ (,zz ={
15
−
16
c 33 − e 31 c 13 c 12
17 18 19 20
0={
22
− κ33 +
23
24 26
0={
q33 −
29 30 31 32 33 34 35 36 37 38 39 40 41 42 43
c 12
q31 c 13
− νe c 13 −
c 12
e 31 q13
( z) r 2 u¨ ,z
2
q213
( z)
( z)
( z) = ρ 0 u¨ (z) − 2 u¨ ,zz
60 61 62 63 64 65 66
ψ , z }, z
( z)
(26)
( z)
( z)
( z)
(30)
( z)
=0
( z)
(31)
( z)
0 0 0 0 0 0 T 10 u ,zz − T 11 φ,zz − T 12 ψ,zz + T 10 u ,z − T 11 φ,z − T 12 ψ, z
=0
(32)
The above equations govern the axial wave propagation in FG-MME nonlocal rod in terms of u z , φ and ψ . If = 0, the above equations will lead to the governing equations of classical wave propagation in FG-MME rods. For harmonic wave propagation in FG-MEE rod, the u z , φ, ψ field can be expressed in complex form as [29,30]
u
( z)
ˆ ( z, t ) = Ue
− jk z z j ωn t
e
ˆ e − jkz z e j ωn t ψ( z, t ) =
ˆe , φ( z, t ) =
− jk z z j ωn t
e
− T 10k2z
0
− 0.5νe ρ R
2
2 2 n kz
ω
=0
2
2 4 n kz
+ ω
−
j T 40 k z
−ρ
0
2 n
2
2 2 n kz
, (33)
ˆ = ˆ =
0 T 10 T 90 0 T 11 T 90
− −
0 T 12 T 70 0 T 12 T 80
0 0 T 11 T 70 − T 10 T 80
+ 0.5νe ρ R ω 0 T 20 T 70 T 12
73 74 76 77
82 83 84 85
(35)
86 87 88 89
(36)
90 91 92 93 95
ˆ U
(37)
96 97
98
ˆ U
(38)
0 T 12 T 80
0 T 20 T 90 T 10
99 100 101
−
0 T 11 T 90
0 T 30 T 80 T 10
102 103 104
0 0 0.5νe ρ 0 R 2 2 ωn2 T 12 T 80 − T 11 T 90 k2z
72
94
2 n
71
81
(34)
Substituting Eqs. (37) and (38) in Eqs. (34) leads to the following two relations:
2
70
80
0 0 T 11 T 90 − T 12 T 80
0
69
79
the following relations:
68
78
ˆ U −ω − ω
ˆ + T 60k2z + j T 60k z ˆ =0 + T 50k2z + j T 50k z
ˆ − T 70k2z − j T 70k z Uˆ + T 80k2z + j T 80k z
ˆ =0 + T 90k2z + j T 90k z
0 2 0 ˆ k z − j T 10 kz U − T 10
0 2 0 0 2 0 ˆ + T 12 ˆ =0 + T 11 k z + j T 11 kz k z + j T 12 kz − T 40k2z
67
75
ˆ + T 0k2z ˆ + T 30k2z ˆ U 2
ˆ and ˆ in terms of Uˆ from Eqs. (35) and (36) gives Solving for
T 70 u ,zz − T 80 φ,zz − T 90 ψ,zz + T 70 u ,z − T 80 φ,z − T 90 ψ,z
56
59
( z) u ,z
T 40 u ,zz − T 50 φ,zz − T 60 ψ,zz + T 40 u ,z − T 50 φ,z − T 60 ψ,z
53
58
T 10 u ,zz − T 20 φ,zz − T 30 ψ,zz = 0.5νe ρ 0 R 2 u¨ ,zz − 2 u¨ ,zzzz
52
57
(25)
( z) r 2 u¨ ,zzz
50
55
ψ, z }, z
where T i ( z) are given in Appendix A. Differentiating the first expression of Eq. (27) with respect to z and integrating the resulting equation over the cross section of the rod and the remaining equations can be simplified as
49
54
νe ρ − νe ρ = T 1 ( z)u (,zz) − T 2 ( z)φ,z − T 3 ( z)ψ,z z) ρ u¨ (z) − 2 ρ u¨ (,zz = { T 4 ( z)u (,zz) − T 5 ( z)φ,z − T 6 ( z)ψ,z },z (27) ( z) 0 = { T 7 ( z)u ,z − T 8 ( z)φ,z − T 9 ( z)ψ,z },z (28) ( z) 0 = { T 10 ( z)u ,z − T 11 ( z)φ,z − T 12 ( z)ψ,z },z (29)
48
51
(24)
Substituting graded material properties defined in Eq. (6) into the above equations gives the following relations:
46 47
− q33 ψ,z },z
( z)
q31 c 11
44 45
u ,z
c 12
c 12
( z)
u ,z
− νe q31 − c 12
c 12
c 12
c 12
e 31 c 11
φ,z − μ33 +
c 13 c 11
q31 c 13
− e 33 φ,z −
φ,z − d33 +
c 12
( z) u ,z
− q31 ψ,z
c 12
− νe e 31 −
c 12
− d33 +
28
e 231
q31 c 11
2 c 13
c 12
q31 c 13
27
c 12
21
25
e 31 c 13
e 33 −
− e 31 φ,z −
14
2 c 11
− νe c 12 −
c 12
e 31 c 11
11 12
c 13 c 11
ˆ , , ˆ ˆ are the frequency amplitudes, k z is the wavenumwhere U ber in z-direction, √ respectively, ωn is the frequency of the wave motion and j = −1. The nonlocal governing equations for the FG-MEE model are given in Eqs. (32). Our next step is to analyze the wave propagation in this system. Substitute the field defined in Eqs. (33) in the nonlocal governing equations of the FG-MEE rod (Eqs. (29)–(32)) yields
+
105 106
0 T 10 T 80 T 12
−
0 T 10 T 90 T 11
0 T 30 T 70 T 11
− + − + =0 (39) 0 0 0 0 0 0 0 2 2 0 0 0 0 − T 4 T 9 T 11 + T 4 T 8 T 12 + ρ ωn T 11 T 9 − T 12 T 8 0 0 0 0 − T 50 T 70 T 12 + T 50 T 90 T 10 − T 60 T 80 T 10 + T 60 T 70 T 11 k2z 0 0 0 0 0 + j − T 40 T 90 T 11 + T 40 T 80 T 12 − T 50 T 70 T 12 + T 50 T 90 T 10 − T 60 T 80 T 10 0 0 0 + T 60 T 70 T 11 k z + ρ 0 ωn2 T 11 T 90 − T 12 T 80 = 0 (40) Here, Eqn. (39) gives a simple relation between the wave frequency and wavenumber as a function of the material property of the (1) rod. The Eqn. (40) can be solved for the wavenumbers k z and (2) (1) k z = −k z . The phase velocity is defined as V p = ωn /k z . The group velocity of the wave mode can be obtained form the relation: V g = ∂∂ωkn .
107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126
3. Numerical results and discussion
127 128
We consider a functionally graded nanorod made up of BaTiO3 –CoFe2 O4 ume fraction of BaTiO3 . The effective BaTiO3 –CoFe2 O4 magneto-electro-elastic
magneto-electro-elastic with the variable volmaterial properties of rod are given in the
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Table 1 Effective material properties of BaTiO3 –CoFe2 O4 magneto-electro-elastic rod based on micro-mechanics approach [31,32]. v F : 0% [PM]
5
0 c 11
×109 N/m2
286
245
213
187
166
71
6
0 c 12
×109 N/m2
173
139
113
93
77
72
7
0 c 13
×109 N/m2
170
138
113
93.8
78
73
8
0 c 33
×109 N/m2
269.5
235
207
162
74
9
0 c 44
×109 N/m2
170
138
113
93.8
78
75
10
e 031
C/m2
0
−1.53
−2.71
−3.64
−4.4
76
11
e 033 e 015 0 11 0 33 0 11 0 33 q031 q033 q015 d011 d033 0
2
0
4.28
8.86
13.66
18.6
77
C/m2
0
0.05
0.15
0.46
11.6
78
×10−9 C2 /N m2
0.08
0.13
0.24
0.53
11.2
79
×10−9 C2 /N m2
0.093
3.24
6.37
9.49
12.6
×10−4 N s2 /C2
5 .9
3.57
2.01
0.89
0.05
×10−4 N s2 /C2
1.57
1.21
0.839
0.47
0.1
12 13 14 15 16 17 18 19 20 21 22 23
κ κ μ μ
ρ
C/m
50% [MEE]
75% [MEE]
100% [PE]
69
Units
4
25% [MEE]
68
Property
70
183
80 81 82 83
N/A m
580
378
222
100
0
N/A m
700
476
292
136
0
N/A m
550
331.2
185
79
0
86
−5.23
−6.72
0
87
1847.49
0
88
5.8
89
×10−12 N s/V C
0
−3.09
×10−12 N s/V C
0
2334.15
×103 kg/m3
5 .3
5.43
2750 5.55
5.66
84 85
24
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102
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104
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41
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42
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111
46
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47
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48
114
49
115
50
116
51
117
52
118
53
119
54
120
55
121
56
122
57
123
58
124
59
125
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126
Fig. 2. Spectrum curves for the rod made of 0% PM with different gradient index ( in 1/m) obtained from local [(a) and (b)] and nonlocal continuum theories [(c) and (d)]. Here figures (a) and (c) are plots of imaginary part of wavenumbers versus frequency and figures (b) and (d) are plots of real part of wavenumbers versus frequency.
63 64 65 66
127 128 129
Table 1. For the present wave propagation analysis, we have considered five different material combinations by taking the volume fractions of BaTiO3 as ν F = 0%, 25%, 50%, 75% and 100%, respec-
tively. When ν F = 0%, the composite is purely piezomagnetic (PM), whereas ν F = 100% corresponds to a purely piezoelectric (PE) material.
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Fig. 3. Spectrum curves for the rod made of 50% MEE with different gradient index ( in 1/m) obtained from local [(a) and (b)] and nonlocal continuum theories [(c) and (d)]. Here figures (a) and (c) are plots of imaginary part of wavenumbers versus frequency and figures (b) and (d) are plots of real part of wavenumbers versus frequency.
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Figs. 2–4 show the spectrum relation of the FG-MEE rod for volume fractions of 0%, 50% and 100%, respectively. In these figures, both the local and nonlocal continuum formulation results are plotted. The wavenumbers plotted in the figures (a) and (c) denote the imaginary part of the wavenumbers and that of (b) and (d) denote the real part of wavenumbers for different gradient index values. The frequency, where the real wavenumber appears is called the cut-off frequency. For zero gradient index, the rod is homogeneous and the wavenumbers are exactly the solutions of the spectrum equation known in literature (see [29]). This variation is known since a long time and more on this can be found for isotropic material. There is one property of the wavenumbers, where attention was not focused before, i.e., the wavenumbers do not possess nonzero real and imaginary parts, simultaneously. However, the situation changes dramatically, when nonzero values of gradient index ( ). As the figures suggest, with increase in the magnitude of , cut-off frequencies appear. The wavenumbers simultaneously possess both nonzero real and imaginary parts, which implies attenuation of the wave magnitude while propagation. At high frequencies, real part of the wavenumbers converges to their homogeneous counterpart, whereas, the imaginary parts take constant values. Also it is evident that the effect of volume fraction is more pronounced in the axial mode as shown by comparatively large shifting of the axial cut-off frequency. As the volume fraction of BaTiO3 increases the axial cut-off frequency decreases, whereas with the increase of gradient index the axial cut-off frequency increases. Compared to local continuum
results, the nonlocal continuum results are drastically different. A new frequency called escape frequency is observed in the axial wave mode (for homogeneous rod, such results are presented in Ref. [30]). This is the frequency at which the wavenumber tends to infinity. Similar effect is observed for the imaginary part of wavenumbers also for the first time. This result shows that the attenuation of the wave magnitude while propagation. The attenuation also produces similar escape frequency. The escape frequency also slightly altered by the volume fraction and the gradient index of the material. As the gradient index increases, the escape frequency decreases for a given volume fraction. Whereas, the difference between escape frequencies reduces as the volume fraction increases. This result shows that, as the volume fraction increases, the effect of the gradient index on the escape frequency is negligible (i.e., as the material changes from PM to PE via MEE). The dispersion relations in terms of phase and group velocities are plotted in Figs. 5 and 6, respectively. For a given frequency, presence of positive speed indicates propagation of the wave mode. As the figure suggests, for zero gradient index, the axial mode is propagating. However, nonzero introduces cut-off frequencies in axial mode, which means, if frequency content of the loading is less than the cut-off frequency there will be no response in the structure, i.e., gradation of materials will act as a filter. As the gradation increases, the velocities approach to the classical value in the local elasticity case. With the introduction of nonlocal scale parameter, the wave velocities tend to zero at some frequencies
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Fig. 4. Spectrum curves for the rod made of 100% PE with different gradient index ( in 1/m) obtained from local [(a) and (b)] and nonlocal continuum theories [(c) and (d)]. Here figures (a) and (c) are plots of imaginary part of wavenumbers versus frequency and figures (b) and (d) are plots of real part of wavenumbers versus frequency.
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(i.e. escape frequency). This means that there will be any response from the structures after the escape frequency. No propagation of waves occur beyond this frequency. There is a slight variation in cut-off frequency of the axial wave mode due to the nonlocal scale. As the gradient index and nonlocal scale parameter increase the cut-off frequency also increasing. The difference between the cutoff frequencies obtained from local and nonlocal theories decreases as the volume fraction increases. This effect can be clearly seen from the Figs. 5 and 6. The value of the gradation index cannot be increased indefinitely. If the is increased beyond certain value, there will be no response in the structure as all the wave modes will be effectively blocked or damped out, so gradation can be used effectively for selecting the wave modes of FG-MEE rods. All the results presented in this work are analyzed and considerable effects are observed as we started from PM material to reach PE material via MEE material. These interesting features are useful for the design of the futuristic multifunctional magnetoelectro-elastic devices that make use of the longitudinal wave propagation properties. 4. Conclusions
displacement. The wave behaviour in the rod is studied with respect to the spectrum and dispersion curves. Based on the present work, the following conclusions are drawn:
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1. With the presence of the gradation index ( ), cut-off frequencies appear in the axial wave modes. 2. The wavenumbers simultaneously possess both nonzero real and imaginary parts, which imply attenuation of the wave magnitude while propagation. This behaviour is observed for both classical and nonlocal continuum cases. 3. The effect of volume fraction is more pronounced in the axial mode by large shifting of the axial cut-off frequency. 4. The escape frequency of the FG-MEE rod is slightly altered by the volume fraction and the gradient index of the material. As the gradient index increases, the escape frequency decreases for a given volume fraction. No wave propagation occurs beyond this escape frequency. 5. With the introduction of nonlocal scale parameter, the wave velocities tend to zero at the escape frequencies.
62 63
104 105
60 61
103
111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
Axial wave propagation characteristics of a functionally graded magneto-electro-elastic (FG-MEE) rod are studied via nonlocal continuum mechanics. The nonlocal governing equations of the FGMEE rod are derived and solved using the spectral form for axial
The interesting features observed for FG-MEE nonlocal rod are useful for the design of the futuristic nanoscale multifunctional magneto-electro-elastic devices that make use of the wave propagation properties.
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Fig. 5. Dispersion curves in terms of phase velocity (V p ) for the rod with different gradient index ( in 1/m) obtained from local and nonlocal continuum theories for (a) 0% PM, (b) 50% MEE and (c) 100% PE materials.
39
40 41
Conflict of interest statement
T 6 ( z) =
42 43
None declared.
44 45
49 50 51 52 53 54 55
58
61
T 3 ( z) =
64 65 66
0 c 12 0 q031 c 11
0 c 12
0 c 33 −
T 4 ( z) =
62 63
0 − − νe c 12
0 e 031 c 11
T 2 ( z) =
59 60
0 0 c 11 c 11 0 c 12
e z
0 0 c 13 c 13
0 − − νe c 13
0 c 12
= T 401 − νe T 402 e z ≡ T 40 e z , T 5 ( z) =
0 c 12
− e 033
e
z
≡
T 50 e z ,
(41) 0 0 c 13 c 11 0 c 12
e z
e 031
− νe
0 c 12
d033
+
e 031 e 031 0 c 12
e 031 q013
− q031 e z ≡ T 30 e z ,
0 e 031 c 13
T 9 ( z) =
κ +
− e 031 e z ≡ T 20 e z ,
0 33
T 8 ( z) =
= T 101 − νe T 102 e z ≡ T 10 e z ,
56 57
0 c 13 −
T 1 ( z) =
0 0 c 13 c 11 0 c 12
−
106
− q033 e z ≡ T 60 e z , 0 e 31 c 13
=
01 T 10
−
(42)
T 11 ( z) = d033 +
T 12 ( z) =
0 33
μ +
z
0 e 031 c 11
e
e
110 111 112 113 114 115
T 80 e z ,
116 117
≡
e
− νe q031 −
z
≡
0 c 12
q013 q013
T 90 e z ,
0 q031 c 13
0 c 12
0 c 12
109
z
118
z
0 c 12 02 e T 10
≡
107 108
0 q031 c 13
−ν
e
0 c 12
q033 −
T 10 ( z) =
104 105
= T 701 − νe T 702 e z ≡ T 70 e z ,
The parameters used in Eqs. (27)–(29) are given below:
48
0 c 12
e 033
T 7 ( z) =
Appendix A
46 47
0 q031 c 13
103
(43) 0 q031 c 11 0 c 12
119 120 121 122
e z
123 124 125
0 z T 10 e ,
126 127
0 z e z ≡ T 11 e ,
128 129
130
e
z
≡
0 z T 12 e .
(44)
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Fig. 6. Dispersion curves in terms of group velocity (V g ) for the rod with different gradient index ( in 1/m) obtained from local and nonlocal continuum theories for (a) 0% PM, (b) 50% MEE and (c) 100% PE materials.
References [1] H.S. Shen, Functionally Graded Materials: Nonlinear Analysis of Plates and Shells, CRC press, Boca Raton, 2009. [2] J. Chen, E. Pan, H. Chen, Wave propagation in magneto-electro-elastic multilayered plates, Int. J. Solids Struct. 44 (2007) 1073–1085. [3] P. Chen, Y. Shen, Propagation of axial shear magneto-electro-elastic waves in piezoelectric–piezomagnetic composites with randomly distributed cylindrical inhomogeneities, Int. J. Solids Struct. 44 (2007) 1511–1532. [4] G.R. Buchanan, Free vibration of an infinite magneto-electro-elastic cylinder, J. Sound Vib. 268 (2003) 413–426. [5] A.R. Annigeri, N. Ganesan, S. Swarnamani, Free vibrations of clamped–clamped magneto-electro-elastic cylindrical shells, J. Sound Vib. 292 (2006) 300–314. [6] W. Bin, Y. Jiangong, H. Cunfu, Wave propagation in non-homogeneous magneto-electro-elastic plates, J. Sound Vib. 317 (2008) 250–264. [7] J. Du, X. Jin, J. Wang, Love wave propagation in layered magneto-electro-elastic structures with initial stress, Acta Mech. 192 (2007) 169–189. [8] Y.S. Li, Z.Y. Cai, S.Y. Shi, Buckling and free vibration of magnetoelectroelastic nanoplate based on nonlocal theory, Compos. Struct. 111 (2014) 522–529. [9] S.C. Kattimani, M.C. Ray, Smart damping of geometrically nonlinear vibrations of magneto-electro-elastic plates, Compos. Struct. 114 (2014) 51–63. [10] J. Zhang, Y.P. Shen, J.K. Du, The effect of inhomogeneous initial stress on Love wave propagation in layered magneto-electro-elastic structures, Smart Mater. Struct. 17 (2008) 025026. [11] C.X. Xue, E. Pan, S.Y. Zhang, Solitary waves in a magneto-electro-elastic circular rod, Smart Mater. Struct. 20 (2011) 105010. [12] X.H. Wu, Y.P. Shen, Q. Sun, Lamb wave propagation in magneto-electro-elastic plates, Appl. Acoust. 68 (2007) 1224–1240. [13] C.X. Xue, E. Pan, On the longitudinal wave along a functionally graded magneto-electro-elastic rod, Int. J. Eng. Sci. 62 (2013) 48–55. [14] Y. Jiangong, M. Qiujuan, S. Shan, Wave propagation in non-homogeneous magneto-electro-elastic hollow cylinders, Ultrasonics 48 (2008) 664–677. [15] M. Aydogdu, Axial vibration of the nanorods with the nonlocal continuum rod model, Physica E, Low-Dimens. Syst. Nanostruct. 41 (2009) 861–864.
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[16] S. Narendar, S. Gopalakrishnan, Nonlocal scale effects on ultrasonic wave characteristics of nanorods, Physica E, Low-Dimens. Syst. Nanostruct. 42 (2010) 1601–1604. [17] S. Narendar, S. Gopalakrishnan, Ultrasonic wave characteristics of nanorods via nonlocal strain gradient models, J. Appl. Phys. 107 (2010) 084312. [18] M. Aydogdu, Longitudinal wave propagation in nanorods using a general nonlocal unimodal rod theory and calibration of nonlocal parameter with lattice dynamics, Int. J. Eng. Sci. 56 (2012) 17–28. [19] S. Narendar, Terahertz wave propagation in uniform nanorods: a nonlocal continuum mechanics formulation including the effect of lateral inertia, Physica E, Low-Dimens. Syst. Nanostruct. 43 (2011) 1015–1020. [20] M. Simsek, Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nanorods, Comput. Mater. Sci. 61 (2012) 257–265. [21] M. Danesh, A. Farajpour, M. Mohammadi, Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method, Mech. Res. Commun. 39 (2012) 23–27. [22] M. Aydogdu, Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity, Mech. Res. Commun. 43 (2012) 34–40. [23] S. Narendar, Spectral finite element and nonlocal continuum mechanics based formulation for torsional wave propagation in nanorods, Finite Elem. Anal. Des. 62 (2012) 65–75. [24] S. Narendar, S. Gopalakrishnan, Spectral finite element formulation for nanorods via nonlocal continuum mechanics, J. Appl. Mech. 78 (2011) 061018. [25] S. Narendar, S. Gopalakrishnan, Axial wave propagation in coupled nanorod system with nonlocal small scale effects, Composites, Part B, Eng. 42 (2011) 2013–2023. [26] A.C. Eringen, Non-local Polar Field Models, Academic, New York, 1996. [27] R. Bhangale, N. Ganesan, Static analysis of simply supported functionally graded and layered magneto-electro-elastic plates, Int. J. Solids Struct. 43 (2006) 3230–3253. [28] L. Zhao, W.Q. Chen, Plane analysis for functionally graded magneto-electroelastic materials via the symplectic framework, Compos. Struct. 92 (2010) 1753–1761.
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[29] S. Gopalakrishnan, A. Chakraborty, D. Roy Mahapatra, Spectral Finite Element Method, Springer Verlag, London Ltd., 2008. [30] S. Gopalakrishnan, S. Narendar, Wave Propagation in Nanostructures: Nonlocal Continuum Mechanics Formulations, Springer Verlag, London Ltd., 2013.
[31] H.Y. Kuo, E. Pan, Effective magnetoelectric effect in multicoated circular fibrous multiferroic composites, J. Appl. Phys. 109 (2011) 104901. [32] H.Y. Kuo, Y.L. Wang, Optimization of magnetoelctricity in multiferroic fibrous composites, Mech. Mater. 50 (2012) 88–99.
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Wave dispersion in functionally graded magneto-electro-elastic nonlocal rod ✩
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S. Narendar
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Defence Research and Development Laboratory, Dr. APJ Abdul Kalam Missile Complex, Kanchanbagh, Hyderabad-500 058, Telangana, India
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a r t i c l e
i n f o
a b s t r a c t
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Article history: Received 16 July 2015 Received in revised form 27 November 2015 Accepted 9 January 2016 Available online xxxx
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Keywords: Magneto–electro-elastic rod Functionally graded material Wave dispersion Nonlocal continuum mechanics Phase velocity Escape frequency
Dispersion of elastic waves in functionally graded magneto-electro-elastic (FG-MEE) rod is studied based on nonlocal continuum mechanics. Governing equations of motion for the rod element are formulated in the nonlocal field (i.e., stress, electric and magnetic fields). The exponentially varying material properties along the axial direction of the rod have also been considered. The formulated nonlocal equations of motion are solved using the harmonic waveforms of the displacement field. The resulting wavenumber relations are solved for studying the wave dispersion phenomena in nonlocal FG-MEE rods. The wave characteristics are studied with respect to the spectrum and dispersion curves. The effects of the gradation index, volume fraction and nonlocal scale parameter are captured and discussed in detail. The results presented in this work are useful for the design of the futuristic multi-functional nanoscale magneto-electro-elastic devices that make use of the axial wave propagation properties. © 2016 Elsevier Masson SAS. All rights reserved.
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35 36
101
1. Introduction
37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58
The structural members belonging to functionally graded materials (FGM) have received wide attention by the various fields research communities [1]. Because of the coupling of mechanical and electrical fields, extensive study has been made for magnetoelectro-elastic (MEE) materials. So, with increasing usage of MEE structures in various engineering fields, wave propagation in MEE material media has also attracted many researchers. Chen et al. [2] presented an analytical treatment for the propagation of harmonic waves in magneto-electro-elastic multi-layered plates. The results on dispersion curves, modal shapes, and natural frequencies are presented for layered plates made of orthotropic elastic, transversely isotropic, piezoelectric and magnetostrictive materials. They showed the influence of different stacking sequences on the field response of layered smart composites. By using the effective field method, Chen and Shen [3] simplified the multi-fiber scattering problem to the one-fiber scattering problem, and the analytical expressions of magneto-electro-elastic fields for the multi-fiber composites were obtained in the long-wave limit. Buchanan [4] studied the free vibration of fully coupled infinite magneto-electro-elastic cylinders. Annigeri et al. [5] used the constitutive equations of piezomagnetic medium to study the free vi-
59 60 61
✩
62 63 64 65 66
1
This work is dedicated to my son Mr. Saggam Kaushik Jayasurya. E-mail addresses: [email protected], [email protected]. Tel.: +91 8897 625977.
http://dx.doi.org/10.1016/j.ast.2016.01.012 1270-9638/© 2016 Elsevier Masson SAS. All rights reserved.
bration analysis of magneto-electro-elastic cylindrical shells. They developed a semi-analytic finite element model. They proved that magnetic effect will reduce the stiffness of system and also bring down the frequency of the system. Bin et al. [6] studied the propagation of harmonic waves in functionally graded magneto-electro-elastic plates composed of piezoelectric BaTiO3 and magnetostrictive CoFe2 O4 . They employed the Legendre orthogonal polynomial series expansion approach to determine the wave propagating characteristics in the plates. Du et al. [7] investigated the Love surface waves in layered piezomagnetic/piezoelectric structures with initial stress. They applied the magneto-electrically open and short conditions to solve the problem numerically. The effect of the initial stress on the wave velocity and the magneto-electromechanical coupling factor was studied in detail in their work. Li et al. [8] investigated the Buckling and free vibration analysis of magneto-electro-elastic nanoplate resting on Pasternak foundation via nonlocal Mindlin plate theory. They showed that for a magneto-electro-elastic nanoplate, the buckling load decreases with increasing the lateral load. They also showed that the buckling load and free vibration frequency decrease with increasing nonlocal scaling parameter. Kattimani and Ray [9] formulated a three dimensional finite element to investigate the geometrically nonlinear vibrations of the magneto-electro-elastic plates with the active constrained layer damping. Zhang et al. [10] studied the propagation behaviour of Love waves in layered magneto-electro-elastic structures with inhomogeneous initial stress. They found that the effects of the gradient coefficient of the inhomogeneous initial stress and the
102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
magneto-electric coupling factor on the phase velocity of the Love waves. Xue et al. [11] derived the nonlinear solitary wave equation in a long MEE circular rod. They considered the geometric nonlinearity in the longitudinal direction. The wave equation was solved by the Jacobi elliptic function expansion method. Wu et al. [12] investigated the symmetric and antisymmetric Lamb waves propagation in an infinite magneto-electro-elastic plates based on the three-dimensional linear elastic equations and magnetoelectro-elastic constitutive relations. The electrically and magnetically open case and shorted case for both types of wave modes were discussed in their work. Xue and Pan [13] studied the axial wave propagation of functionally graded magneto-electro-elastic rod made of piezoelectric BaTiO3 and piezomagnetic CoFe2 O4 . They assumed the materials properties to vary exponentially along the rod longitudinal direction. They investigated the important influence of the gradient factor as well as material coupling on the wave features of the rod. Jiangong et al. [14] presented a dynamic solution for the propagation of harmonic waves in inhomogeneous magneto-electro-elastic hollow cylinders. The Legendre orthogonal polynomial series expansion approach was employed to determine the wave propagating characteristics in the hollow cylinders. In the recent year a lot of work has been carried out by many researchers in the area of the dynamics of the nonlocal rods [15–25]. To the best of authors’ knowledge, however, the wave propagation analysis in functionally graded magneto-electro-elastic nonlocal rods has not been considered in literature. Based on the nonlocal elasticity theory, the wave propagation analysis of a FG MME nanorods is investigated in the present work.
29 30
2. Mathematical formulations
31 32
2.1. Review of nonlocal continuum theory
35 36 37 38
The nonlocal continuum theory states that the stress at a reference point r is considered to be a function of the strain field at every point in the body. The basic equations for a linear, homogeneous, isotropic, and nonlocal elastic solids with zero body force are given as follows [26]:
39 40 41 42 43
σi j , j = 0 σi j (r) =
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62
α (|r − r |, τ )C i jkl εkl (r )dV (r ) ∀ r ∈
1
εi j = (u i , j + u j ,i )
(2)
(3)
2
where C i jkl is the elastic modulus tensor of classical isotropic elasticity, σi j and εi j are stress and strain tensors respectively, u i is the displacement vector and is the region occupied by the body. α = α (|r − r |, τ ) is the nonlocal modulus or attenuation function incorporating the nonlocal effects into the constitutive equations at the reference point r produced by local strain at the source r . |r − r | is the Euclidean distance, and τ = /, where = e 0 a, a is an internal characteristic length, e.g., length of C − C bond for carbon nanotubes, or lattice parameter, granular distance etc., and is an external characteristic length e.g., wavelength, crack length or size of the sample. In the published literature, for each material, an appropriate constant value has been assumed for the nonlocal scaling parameter, . As τ → 0, α must revert to the Dirac delta measure so that classical elasticity limit is included in the limit of vanishing internal characteristic length, that is,
63 64
44 45
(1)
lim
α (|r − r |, τ ) = δ(|r − r |)
65
τ →0
66
We therefore expect that
68 69 70 71 72 73 74 75 76 77 78 79 80 81
Fig. 1. Schematic of FG-MEE rod showing the geometry with radius R and also showing both poling and wave propagation directions. Here u r , u θ and u z are the mechanical displacements along r −, θ− and z− directions, respectively.
α is a delta sequence.
(4)
82 83 84 85
The integro-partial differential equations (1)–(3) representing the above linear nonlocal elasticity can be reduced to singular partial differential equations of a special class of physically admissible kernel. In addition, Hook’s law for the stress–strain relations is given by
86 87 88 89 90 91
σ (r) − 2 ∇ 2 σ (r) = E ε(r)
(5)
where E is the Young’s modulus of the material. Nonlocal elasticity in the limit → 0 reverts to classical elasticity, which can be seen by letting → 0 in Eq. (5), to obtain Hook’s law of classical elasticity.
92 93 94 95 96 97 98
2.2. Formulation of governing equations of motion
33 34
67
99 100
The radius of FG-MME rod is R and is of circular cross sections as shown in Fig. 1. The longitudinal axis (z-axis) is perpendicular to the plane of the material isotropy. The material coefficients of this rod vary exponentially along the axial direction in a unified manner [27,28].
101 102 103 104 105 106
Elastic coefficients: c i j = c i0j e z ,
107
Piezoelectric coefficients: e i j = e 0i j e z ,
108
Piezomagnetic coefficients: qi j = q0i j e z ,
110
109 111
Dielectric coefficients: κi j = κi0j e z , Magnetoelectric
112 113
coefficients: di j = d0i j e z ,
114 115
0 z ije ,
Magnetic coefficients: μi j = μ
116
Density: ρi j = ρi0j e z
(6)
117 118
The superscript 0 in the above coefficients represents the constant factor of the material property and is the gradient index of the material. The nonlocal constitutive relations of the cylindrical rod are:
119 120 121 122 123
σ (r ) − ∇ 2 σ (r ) = c11 ε(r ) + c12 ε(θ ) + c13 ε(z) − e31 E (z) − q31 H (z) σ
(θ )
σ
( z)
2
2
(θ )
2
2
( z)
− ∇ σ − ∇ σ
(r )
+ c 11 ε
(r )
+ c 13 ε
= c 12 ε = c 13 ε
+ c 13 ε
(θ )
(θ )
+ c 33 ε
( z)
( z)
− e 31 E
( z)
− e 33 E
( z)
− q31 H
( z)
− q33 H
τ
2
2 (θ z)
− ∇ τ
= c 44 γ
(θ z)
τ (r θ ) − 2 ∇ 2 τ (r θ ) = c66 γ (r θ )
− e 15 E
(θ )
− q15 H
125 126
( z)
τ (rz) − 2 ∇ 2 τ (rz) = c44 γ (rz) − e15 E (r ) − q15 H (r ) (θ z)
124
127 128 129 130
(θ )
131
(7)
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1 2 3 4
D D
( z)
2
2
(θ )
2
2
( z)
− ∇ D − ∇ D
5
8 9 10 11
B
(r )
2
2
− ∇ B
(r )
14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
54 55 56 57 58 59 60 61 62 63 64 65 66
(r )
+ κ11 E
+ e 31 ε
(θ )
(θ )
+ d11 H
+ e 33 ε
( z)
(rz)
+ d11 E
+ μ11 H
B
( z)
2
2
− ∇ B
( z)
= q31 ε
(r )
+ q31 ε
(θ )
+ q33 ε
( z)
+ κ33 E
( z)
(r )
+ q33 E
( z)
+ μ33 H (z)
(9)
where σ (i ) and τ (i j ) are the normal and shear stresses in respective directions; ε (i ) and γ (i j ) are the normal and shear strains; E (i ) , H (i ) , D (i ) and B (i ) are respectively, the electric field, magnetic field, electric displacements and magnetic inductions along respective directions. Here is nonlocal parameter. In the absence of the body forces, electric and magnetic sources, the equations of motion of the FG-MEE rod are:
67
(16)
B (r ) = −ψ,r , B (θ ) = −r −1 ψ,θ , B (z) = −ψ,z
(17)
where φ and ψ are the electric and magnetic potentials, respectively. The simplifications made earlier are also applicable to the above mentioned relations. Using Eq. (13) in Eqs. (7)–(9) leads to
+
σ
−
2
σ,(zzz)
D
( z)
−
2
( z) D ,zz
ε
c 12
c 12
( z)
B (z) − 2 B ,zz = q31 −
+ d33 +
(r )
E
c 12
+ c 33 − ( z)
+
( z)
ε(r ) + q33 −
q31 c 13
E ( z) +
c 12
ε
H
ε(θ ) =
1 c 12
e 31 E (z) + q31 H (z) − c 11 ε (r ) − c 13 ε (z)
−
σ
( z)
−
2
σ,(zzz)
(19)
ε ( z)
−
q213
=
c 12
e 31 c 13 c 12
e 33 −
The finite elastic strain-displacement, electric field-potential and magnetic field-potential relations can be expressed as:
q33 −
e 231
− d33 +
c 12
φ,z − d33 +
c 12 q31 c 13 c 12
c 12
c 12 q31 c 11
(21)
( z) u ,z
φ,z − μ33 +
c 12
107
111 112 113 114 115 117 118 119 120 121 122 123
ψ, z
124
(22)
( z)
− νe q31 − u ,z c 12 q213
106
116
− q33 ψ,z
e 31 q13
105
110
c 12
104
109
e 31 c 11
98
108
( z)
u ,z
c 12
q31 c 13
97
102
c 12
91
103
q31 c 11
− νe e 31 −
q31 c 13
− e 33 φ,z −
c 12
− κ33 +
( z)
e 31 c 13
B (z) − 2 B ,zz =
90
101
H ( z)
c 12
89
100
c 12
(14)
c 12
e 31 c 11
( z) D (z) − 2 D ,zz
− νe c 12 −
2 c 11
85
99
− e 31 φ,z − − q31 ψ,z c 12
2 c 13 c 13 c 11 ( z) c 33 − u ,z − νe c 13 −
=
(13)
c 13 c 11
c 13 −
With the above simplifications, the equations of motion (Eqns. (10)–(12)) are reduced to: ( z) r −1 σ (θ ) = ρνe r u¨ (r ) = ρνe r u¨ ,z σ,(zz) = ρ u¨ (z) ( z) D ,z = 0 ( z) B ,z = 0
σ (θ ) − 2 σ,(θzz) =
84
96
Substituting Eqs. (15)–(17) into Eqs. (18)–(20) gives the following relations
83
95
(20)
79
94
• The cross section of the FG-MEE rod remains plane before and after the deformation. • The lateral surface of the rod has axial symmetry, which im∂ = 0. plies that u (θ) = 0 and ∂θ • To consider the Poisson’s effect, the gradient of the axial displacement u z and the radial displacement u r are related by (z) u (r ) = νe ru ,z , where νe is the effective Poisson’s ratio. • The problem is one-dimensional, the extended tractions on the lateral boundary of the rod should be zero. Therefore, σ (r ) = 0, τ (rz) = 0, τ (r θ) = 0, D (r ) = 0 and B (r ) = 0. From which the following relations are obtained: γ (rz) = 0, γ (θ z) = 0, E (r ) = 0, E (θ) = 0, H (r ) = 0, H (θ) = 0, D (θ) = 0, B (θ) = 0. From Eq. (7), we get
78
93
( z)
c 12
μ33 +
77
92
( z)
c 12
q31 c 13
( z)
(18)
e 31 q13
76
88
− q33 H
c 12
+ d33 +
75
87
e 31 c 13
+ e 33 −
74
86
( z)
ε
c 12
73
82
− q31 H (z)
q31 c 13
72
81
c 12
c 12
2 c 13
71
80
ε ( z)
q31 c 11 c 12
c 12
e 231
q31 c 11
(r )
− e 33 E
e 31 c 11
+ κ33 +
= e 31 − ε c 12
(12)
c 13 c 11
− e 31 E (z) +
c 13 c 11
e 31 c 13
(10)
c 12
e 31 c 11
= c 13 −
ε(r ) + c13 −
c 12
( z)
2 c 11
σ (θ ) − 2 σ,(θzz) = c12 −
+
Here the subscript , i represents the partial differentiation of the dependent variable with the independent variable i. We consider the wave propagation in a long FG-MEE circular rod. In the cylindrical coordinate system (r , θ, z), θ ∈ [0, 2π ], 0 ≤ r ≤ R. To facilitate the present wave propagation study in FG-MEE rod, the following assumptions are made:
70
E (r ) = −φ,r , E (θ ) = −r −1 φ,θ , E (z) = −φ,z
τ,(rr θ ) + r −1 σ,θ(θ ) + τ,(θz z) + r −1 τ (r θ ) = ρ u¨ (θ ) (11)
69
(15)
(θ z) ( z) + r −1 τ,θ + σ,z + r −1 τ (rz) = ρ u¨ (z) (r ) (θ ) ( z) D ,r + r −1 D ,θ + D ,z + r −1 D (r ) = 0 (r ) (θ ) ( z) B ,r + r −1 B ,θ + B ,z + r −1 B (r ) = 0
68
) ( z) (r ) (rz) γ (θ z) = r −1 u (,θz) + u (θ = r −1 u ,r + u ,z ,z , γ
σ,(rr ) + r −1 τ,θ(r θ ) + τ,(zrz) + r −1 σ (r ) − σ (θ ) = ρ u¨ (r ) τ,(rrz)
(r ) ) ε(z) = u (,zz) , γ (r θ ) = r −1 u ,θ − u (θ ) + u (θ , ,r
(8) (r )
(θ )
B (θ ) − 2 ∇ 2 B (θ ) = q15 γ (θ z) + d11 E (θ ) + μ11 H (θ )
52 53
= e 31 ε
= q15 γ
12 13
= e 15 γ
(θ z)
+ d33 H (z)
6 7
) (r ) ε(r ) = u ,(rr ) , ε(θ ) = r −1 u (θ , ,θ + u
D (r ) − 2 ∇ 2 D (r ) = e 15 γ (rz) + κ11 E (r ) + d11 H (r ) (θ )
3
125 126 127 128 129
ψ, z
130 131
(23)
132
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1 2 3
Introducing the equations of motion (Eqs. (14)) into the nonlocal constitutive relations (Eqs. (21)–(23)) leads to the following relations:
4 5
z) νe ρ r 2 u¨ (,zz) − 2 νe ρ r 2 u¨ (,zzz
6
=
7 8
c 13 −
9
−
10
c 12
13
z) ρ u¨ (z) − 2 ρ u¨ (,zz ={
15
−
16
c 33 − e 31 c 13 c 12
17 18 19 20
0={
22
− κ33 +
23
24 26
0={
q33 −
29 30 31 32 33 34 35 36 37 38 39 40 41 42 43
c 12
q31 c 13
− νe c 13 −
c 12
e 31 q13
( z) r 2 u¨ ,z
2
q213
( z)
( z)
( z) = ρ 0 u¨ (z) − 2 u¨ ,zz
60 61 62 63 64 65 66
ψ , z }, z
( z)
(26)
( z)
( z)
( z)
(30)
( z)
=0
( z)
(31)
( z)
0 0 0 0 0 0 T 10 u ,zz − T 11 φ,zz − T 12 ψ,zz + T 10 u ,z − T 11 φ,z − T 12 ψ, z
=0
(32)
The above equations govern the axial wave propagation in FG-MME nonlocal rod in terms of u z , φ and ψ . If = 0, the above equations will lead to the governing equations of classical wave propagation in FG-MME rods. For harmonic wave propagation in FG-MEE rod, the u z , φ, ψ field can be expressed in complex form as [29,30]
u
( z)
ˆ ( z, t ) = Ue
− jk z z j ωn t
e
ˆ e − jkz z e j ωn t ψ( z, t ) =
ˆe , φ( z, t ) =
− jk z z j ωn t
e
− T 10k2z
0
− 0.5νe ρ R
2
2 2 n kz
ω
=0
2
2 4 n kz
+ ω
−
j T 40 k z
−ρ
0
2 n
2
2 2 n kz
, (33)
ˆ = ˆ =
0 T 10 T 90 0 T 11 T 90
− −
0 T 12 T 70 0 T 12 T 80
0 0 T 11 T 70 − T 10 T 80
+ 0.5νe ρ R ω 0 T 20 T 70 T 12
73 74 76 77
82 83 84 85
(35)
86 87 88 89
(36)
90 91 92 93 95
ˆ U
(37)
96 97
98
ˆ U
(38)
0 T 12 T 80
0 T 20 T 90 T 10
99 100 101
−
0 T 11 T 90
0 T 30 T 80 T 10
102 103 104
0 0 0.5νe ρ 0 R 2 2 ωn2 T 12 T 80 − T 11 T 90 k2z
72
94
2 n
71
81
(34)
Substituting Eqs. (37) and (38) in Eqs. (34) leads to the following two relations:
2
70
80
0 0 T 11 T 90 − T 12 T 80
0
69
79
the following relations:
68
78
ˆ U −ω − ω
ˆ + T 60k2z + j T 60k z ˆ =0 + T 50k2z + j T 50k z
ˆ − T 70k2z − j T 70k z Uˆ + T 80k2z + j T 80k z
ˆ =0 + T 90k2z + j T 90k z
0 2 0 ˆ k z − j T 10 kz U − T 10
0 2 0 0 2 0 ˆ + T 12 ˆ =0 + T 11 k z + j T 11 kz k z + j T 12 kz − T 40k2z
67
75
ˆ + T 0k2z ˆ + T 30k2z ˆ U 2
ˆ and ˆ in terms of Uˆ from Eqs. (35) and (36) gives Solving for
T 70 u ,zz − T 80 φ,zz − T 90 ψ,zz + T 70 u ,z − T 80 φ,z − T 90 ψ,z
56
59
( z) u ,z
T 40 u ,zz − T 50 φ,zz − T 60 ψ,zz + T 40 u ,z − T 50 φ,z − T 60 ψ,z
53
58
T 10 u ,zz − T 20 φ,zz − T 30 ψ,zz = 0.5νe ρ 0 R 2 u¨ ,zz − 2 u¨ ,zzzz
52
57
(25)
( z) r 2 u¨ ,zzz
50
55
ψ, z }, z
where T i ( z) are given in Appendix A. Differentiating the first expression of Eq. (27) with respect to z and integrating the resulting equation over the cross section of the rod and the remaining equations can be simplified as
49
54
νe ρ − νe ρ = T 1 ( z)u (,zz) − T 2 ( z)φ,z − T 3 ( z)ψ,z z) ρ u¨ (z) − 2 ρ u¨ (,zz = { T 4 ( z)u (,zz) − T 5 ( z)φ,z − T 6 ( z)ψ,z },z (27) ( z) 0 = { T 7 ( z)u ,z − T 8 ( z)φ,z − T 9 ( z)ψ,z },z (28) ( z) 0 = { T 10 ( z)u ,z − T 11 ( z)φ,z − T 12 ( z)ψ,z },z (29)
48
51
(24)
Substituting graded material properties defined in Eq. (6) into the above equations gives the following relations:
46 47
− q33 ψ,z },z
( z)
q31 c 11
44 45
u ,z
c 12
c 12
( z)
u ,z
− νe q31 − c 12
c 12
c 12
c 12
e 31 c 11
φ,z − μ33 +
c 13 c 11
q31 c 13
− e 33 φ,z −
φ,z − d33 +
c 12
( z) u ,z
− q31 ψ,z
c 12
− νe e 31 −
c 12
− d33 +
28
e 231
q31 c 11
2 c 13
c 12
q31 c 13
27
c 12
21
25
e 31 c 13
e 33 −
− e 31 φ,z −
14
2 c 11
− νe c 12 −
c 12
e 31 c 11
11 12
c 13 c 11
ˆ , , ˆ ˆ are the frequency amplitudes, k z is the wavenumwhere U ber in z-direction, √ respectively, ωn is the frequency of the wave motion and j = −1. The nonlocal governing equations for the FG-MEE model are given in Eqs. (32). Our next step is to analyze the wave propagation in this system. Substitute the field defined in Eqs. (33) in the nonlocal governing equations of the FG-MEE rod (Eqs. (29)–(32)) yields
+
105 106
0 T 10 T 80 T 12
−
0 T 10 T 90 T 11
0 T 30 T 70 T 11
− + − + =0 (39) 0 0 0 0 0 0 0 2 2 0 0 0 0 − T 4 T 9 T 11 + T 4 T 8 T 12 + ρ ωn T 11 T 9 − T 12 T 8 0 0 0 0 − T 50 T 70 T 12 + T 50 T 90 T 10 − T 60 T 80 T 10 + T 60 T 70 T 11 k2z 0 0 0 0 0 + j − T 40 T 90 T 11 + T 40 T 80 T 12 − T 50 T 70 T 12 + T 50 T 90 T 10 − T 60 T 80 T 10 0 0 0 + T 60 T 70 T 11 k z + ρ 0 ωn2 T 11 T 90 − T 12 T 80 = 0 (40) Here, Eqn. (39) gives a simple relation between the wave frequency and wavenumber as a function of the material property of the (1) rod. The Eqn. (40) can be solved for the wavenumbers k z and (2) (1) k z = −k z . The phase velocity is defined as V p = ωn /k z . The group velocity of the wave mode can be obtained form the relation: V g = ∂∂ωkn .
107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126
3. Numerical results and discussion
127 128
We consider a functionally graded nanorod made up of BaTiO3 –CoFe2 O4 ume fraction of BaTiO3 . The effective BaTiO3 –CoFe2 O4 magneto-electro-elastic
magneto-electro-elastic with the variable volmaterial properties of rod are given in the
129 130 131 132
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1 2 3
5
67
Table 1 Effective material properties of BaTiO3 –CoFe2 O4 magneto-electro-elastic rod based on micro-mechanics approach [31,32]. v F : 0% [PM]
5
0 c 11
×109 N/m2
286
245
213
187
166
71
6
0 c 12
×109 N/m2
173
139
113
93
77
72
7
0 c 13
×109 N/m2
170
138
113
93.8
78
73
8
0 c 33
×109 N/m2
269.5
235
207
162
74
9
0 c 44
×109 N/m2
170
138
113
93.8
78
75
10
e 031
C/m2
0
−1.53
−2.71
−3.64
−4.4
76
11
e 033 e 015 0 11 0 33 0 11 0 33 q031 q033 q015 d011 d033 0
2
0
4.28
8.86
13.66
18.6
77
C/m2
0
0.05
0.15
0.46
11.6
78
×10−9 C2 /N m2
0.08
0.13
0.24
0.53
11.2
79
×10−9 C2 /N m2
0.093
3.24
6.37
9.49
12.6
×10−4 N s2 /C2
5 .9
3.57
2.01
0.89
0.05
×10−4 N s2 /C2
1.57
1.21
0.839
0.47
0.1
12 13 14 15 16 17 18 19 20 21 22 23
κ κ μ μ
ρ
C/m
50% [MEE]
75% [MEE]
100% [PE]
69
Units
4
25% [MEE]
68
Property
70
183
80 81 82 83
N/A m
580
378
222
100
0
N/A m
700
476
292
136
0
N/A m
550
331.2
185
79
0
86
−5.23
−6.72
0
87
1847.49
0
88
5.8
89
×10−12 N s/V C
0
−3.09
×10−12 N s/V C
0
2334.15
×103 kg/m3
5 .3
5.43
2750 5.55
5.66
84 85
24
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Fig. 2. Spectrum curves for the rod made of 0% PM with different gradient index ( in 1/m) obtained from local [(a) and (b)] and nonlocal continuum theories [(c) and (d)]. Here figures (a) and (c) are plots of imaginary part of wavenumbers versus frequency and figures (b) and (d) are plots of real part of wavenumbers versus frequency.
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Table 1. For the present wave propagation analysis, we have considered five different material combinations by taking the volume fractions of BaTiO3 as ν F = 0%, 25%, 50%, 75% and 100%, respec-
tively. When ν F = 0%, the composite is purely piezomagnetic (PM), whereas ν F = 100% corresponds to a purely piezoelectric (PE) material.
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Fig. 3. Spectrum curves for the rod made of 50% MEE with different gradient index ( in 1/m) obtained from local [(a) and (b)] and nonlocal continuum theories [(c) and (d)]. Here figures (a) and (c) are plots of imaginary part of wavenumbers versus frequency and figures (b) and (d) are plots of real part of wavenumbers versus frequency.
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Figs. 2–4 show the spectrum relation of the FG-MEE rod for volume fractions of 0%, 50% and 100%, respectively. In these figures, both the local and nonlocal continuum formulation results are plotted. The wavenumbers plotted in the figures (a) and (c) denote the imaginary part of the wavenumbers and that of (b) and (d) denote the real part of wavenumbers for different gradient index values. The frequency, where the real wavenumber appears is called the cut-off frequency. For zero gradient index, the rod is homogeneous and the wavenumbers are exactly the solutions of the spectrum equation known in literature (see [29]). This variation is known since a long time and more on this can be found for isotropic material. There is one property of the wavenumbers, where attention was not focused before, i.e., the wavenumbers do not possess nonzero real and imaginary parts, simultaneously. However, the situation changes dramatically, when nonzero values of gradient index ( ). As the figures suggest, with increase in the magnitude of , cut-off frequencies appear. The wavenumbers simultaneously possess both nonzero real and imaginary parts, which implies attenuation of the wave magnitude while propagation. At high frequencies, real part of the wavenumbers converges to their homogeneous counterpart, whereas, the imaginary parts take constant values. Also it is evident that the effect of volume fraction is more pronounced in the axial mode as shown by comparatively large shifting of the axial cut-off frequency. As the volume fraction of BaTiO3 increases the axial cut-off frequency decreases, whereas with the increase of gradient index the axial cut-off frequency increases. Compared to local continuum
results, the nonlocal continuum results are drastically different. A new frequency called escape frequency is observed in the axial wave mode (for homogeneous rod, such results are presented in Ref. [30]). This is the frequency at which the wavenumber tends to infinity. Similar effect is observed for the imaginary part of wavenumbers also for the first time. This result shows that the attenuation of the wave magnitude while propagation. The attenuation also produces similar escape frequency. The escape frequency also slightly altered by the volume fraction and the gradient index of the material. As the gradient index increases, the escape frequency decreases for a given volume fraction. Whereas, the difference between escape frequencies reduces as the volume fraction increases. This result shows that, as the volume fraction increases, the effect of the gradient index on the escape frequency is negligible (i.e., as the material changes from PM to PE via MEE). The dispersion relations in terms of phase and group velocities are plotted in Figs. 5 and 6, respectively. For a given frequency, presence of positive speed indicates propagation of the wave mode. As the figure suggests, for zero gradient index, the axial mode is propagating. However, nonzero introduces cut-off frequencies in axial mode, which means, if frequency content of the loading is less than the cut-off frequency there will be no response in the structure, i.e., gradation of materials will act as a filter. As the gradation increases, the velocities approach to the classical value in the local elasticity case. With the introduction of nonlocal scale parameter, the wave velocities tend to zero at some frequencies
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Fig. 4. Spectrum curves for the rod made of 100% PE with different gradient index ( in 1/m) obtained from local [(a) and (b)] and nonlocal continuum theories [(c) and (d)]. Here figures (a) and (c) are plots of imaginary part of wavenumbers versus frequency and figures (b) and (d) are plots of real part of wavenumbers versus frequency.
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(i.e. escape frequency). This means that there will be any response from the structures after the escape frequency. No propagation of waves occur beyond this frequency. There is a slight variation in cut-off frequency of the axial wave mode due to the nonlocal scale. As the gradient index and nonlocal scale parameter increase the cut-off frequency also increasing. The difference between the cutoff frequencies obtained from local and nonlocal theories decreases as the volume fraction increases. This effect can be clearly seen from the Figs. 5 and 6. The value of the gradation index cannot be increased indefinitely. If the is increased beyond certain value, there will be no response in the structure as all the wave modes will be effectively blocked or damped out, so gradation can be used effectively for selecting the wave modes of FG-MEE rods. All the results presented in this work are analyzed and considerable effects are observed as we started from PM material to reach PE material via MEE material. These interesting features are useful for the design of the futuristic multifunctional magnetoelectro-elastic devices that make use of the longitudinal wave propagation properties. 4. Conclusions
displacement. The wave behaviour in the rod is studied with respect to the spectrum and dispersion curves. Based on the present work, the following conclusions are drawn:
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1. With the presence of the gradation index ( ), cut-off frequencies appear in the axial wave modes. 2. The wavenumbers simultaneously possess both nonzero real and imaginary parts, which imply attenuation of the wave magnitude while propagation. This behaviour is observed for both classical and nonlocal continuum cases. 3. The effect of volume fraction is more pronounced in the axial mode by large shifting of the axial cut-off frequency. 4. The escape frequency of the FG-MEE rod is slightly altered by the volume fraction and the gradient index of the material. As the gradient index increases, the escape frequency decreases for a given volume fraction. No wave propagation occurs beyond this escape frequency. 5. With the introduction of nonlocal scale parameter, the wave velocities tend to zero at the escape frequencies.
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Axial wave propagation characteristics of a functionally graded magneto-electro-elastic (FG-MEE) rod are studied via nonlocal continuum mechanics. The nonlocal governing equations of the FGMEE rod are derived and solved using the spectral form for axial
The interesting features observed for FG-MEE nonlocal rod are useful for the design of the futuristic nanoscale multifunctional magneto-electro-elastic devices that make use of the wave propagation properties.
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Fig. 5. Dispersion curves in terms of phase velocity (V p ) for the rod with different gradient index ( in 1/m) obtained from local and nonlocal continuum theories for (a) 0% PM, (b) 50% MEE and (c) 100% PE materials.
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Conflict of interest statement
T 6 ( z) =
42 43
None declared.
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58
61
T 3 ( z) =
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0 c 12 0 q031 c 11
0 c 12
0 c 33 −
T 4 ( z) =
62 63
0 − − νe c 12
0 e 031 c 11
T 2 ( z) =
59 60
0 0 c 11 c 11 0 c 12
e z
0 0 c 13 c 13
0 − − νe c 13
0 c 12
= T 401 − νe T 402 e z ≡ T 40 e z , T 5 ( z) =
0 c 12
− e 033
e
z
≡
T 50 e z ,
(41) 0 0 c 13 c 11 0 c 12
e z
e 031
− νe
0 c 12
d033
+
e 031 e 031 0 c 12
e 031 q013
− q031 e z ≡ T 30 e z ,
0 e 031 c 13
T 9 ( z) =
κ +
− e 031 e z ≡ T 20 e z ,
0 33
T 8 ( z) =
= T 101 − νe T 102 e z ≡ T 10 e z ,
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0 c 13 −
T 1 ( z) =
0 0 c 13 c 11 0 c 12
−
106
− q033 e z ≡ T 60 e z , 0 e 31 c 13
=
01 T 10
−
(42)
T 11 ( z) = d033 +
T 12 ( z) =
0 33
μ +
z
0 e 031 c 11
e
e
110 111 112 113 114 115
T 80 e z ,
116 117
≡
e
− νe q031 −
z
≡
0 c 12
q013 q013
T 90 e z ,
0 q031 c 13
0 c 12
0 c 12
109
z
118
z
0 c 12 02 e T 10
≡
107 108
0 q031 c 13
−ν
e
0 c 12
q033 −
T 10 ( z) =
104 105
= T 701 − νe T 702 e z ≡ T 70 e z ,
The parameters used in Eqs. (27)–(29) are given below:
48
0 c 12
e 033
T 7 ( z) =
Appendix A
46 47
0 q031 c 13
103
(43) 0 q031 c 11 0 c 12
119 120 121 122
e z
123 124 125
0 z T 10 e ,
126 127
0 z e z ≡ T 11 e ,
128 129
130
e
z
≡
0 z T 12 e .
(44)
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Fig. 6. Dispersion curves in terms of group velocity (V g ) for the rod with different gradient index ( in 1/m) obtained from local and nonlocal continuum theories for (a) 0% PM, (b) 50% MEE and (c) 100% PE materials.
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[16] S. Narendar, S. Gopalakrishnan, Nonlocal scale effects on ultrasonic wave characteristics of nanorods, Physica E, Low-Dimens. Syst. Nanostruct. 42 (2010) 1601–1604. [17] S. Narendar, S. Gopalakrishnan, Ultrasonic wave characteristics of nanorods via nonlocal strain gradient models, J. Appl. Phys. 107 (2010) 084312. [18] M. Aydogdu, Longitudinal wave propagation in nanorods using a general nonlocal unimodal rod theory and calibration of nonlocal parameter with lattice dynamics, Int. J. Eng. Sci. 56 (2012) 17–28. [19] S. Narendar, Terahertz wave propagation in uniform nanorods: a nonlocal continuum mechanics formulation including the effect of lateral inertia, Physica E, Low-Dimens. Syst. Nanostruct. 43 (2011) 1015–1020. [20] M. Simsek, Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nanorods, Comput. Mater. Sci. 61 (2012) 257–265. [21] M. Danesh, A. Farajpour, M. Mohammadi, Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method, Mech. Res. Commun. 39 (2012) 23–27. [22] M. Aydogdu, Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity, Mech. Res. Commun. 43 (2012) 34–40. [23] S. Narendar, Spectral finite element and nonlocal continuum mechanics based formulation for torsional wave propagation in nanorods, Finite Elem. Anal. Des. 62 (2012) 65–75. [24] S. Narendar, S. Gopalakrishnan, Spectral finite element formulation for nanorods via nonlocal continuum mechanics, J. Appl. Mech. 78 (2011) 061018. [25] S. Narendar, S. Gopalakrishnan, Axial wave propagation in coupled nanorod system with nonlocal small scale effects, Composites, Part B, Eng. 42 (2011) 2013–2023. [26] A.C. Eringen, Non-local Polar Field Models, Academic, New York, 1996. [27] R. Bhangale, N. Ganesan, Static analysis of simply supported functionally graded and layered magneto-electro-elastic plates, Int. J. Solids Struct. 43 (2006) 3230–3253. [28] L. Zhao, W.Q. Chen, Plane analysis for functionally graded magneto-electroelastic materials via the symplectic framework, Compos. Struct. 92 (2010) 1753–1761.
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[29] S. Gopalakrishnan, A. Chakraborty, D. Roy Mahapatra, Spectral Finite Element Method, Springer Verlag, London Ltd., 2008. [30] S. Gopalakrishnan, S. Narendar, Wave Propagation in Nanostructures: Nonlocal Continuum Mechanics Formulations, Springer Verlag, London Ltd., 2013.
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