Rayleigh lamb waves in micropolar isotropic elastic plate - Springer Link

Applied Mathematics and Mechanics (English Edition), 2006, 27(8):1049–1059 c Editorial Committee of Appl. Math. Mech., ISSN 0253-4827

RAYLEIGH LAMB WAVES IN MICROPOLAR ISOTROPIC ELASTIC PLATE ∗ Rajneesh Kumar1 , Geeta Partap2 (1. Department of Mathematics, Kurukshetra University, Kurukshetra 136119, Haryana, India; 2. Department of Applied Mathematics, Dr. B. R. Ambedkar National Institute of Technology (Deemed University), Jalandhar 144011, Punjab, India) (Communicated by DAI Tian-min, Original Member of Editorial Committee, AMM)

Abstract: The propagation of waves in a homogeneous isotropic micropolar elastic cylindrical plate subjected to stress free conditions is investigated. The secular equations for symmetric and skew symmetric wave mode propagation are derived. At short wave limit, the secular equations for symmetric and skew symmetric waves in a stress free circular plate reduces to Rayleigh surface wave frequency equation. Thin plate results are also obtained. The amplitudes of displacements and microrotation components are obtained and depicted graphically. Some special cases are also deduced from the present investigations. The secular equations for symmetric and skew symmetric modes are also presented graphically. Key words: micropolar elastic plate; circular crested waves; secular equations; phase velocity; amplitudes Chinese Library Classification: O326 2000 Mathematics Subject Classification: 74A35; 74B20; 74F20; 74J15; 74K20 Digital Object Identifier(DOI): 10.1007/s 10483-006-0805-z

Introduction Modern engineering structures are often made up of materials possessing an internal structure. Polycrystalline materials, materials with fibrous or coarse grain structure come in this category. Classical elasticity is inadequate to represent the behavior of such materials. The analysis of such materials requires incorporating the theory of oriented media. For this region, micropolar theories were developed by Eringen[1−3] for elastic solids, fluid and further for nonlocal polar fields. A micropolar continuum is a collection of inter connected particles in the form of small rigid bodies undergoing both translational and rotational motion. A comprehensive work in the micropolar theory of elastic solid and nonlocal continuum field theory is given by Eringen[4−6] in his recent books. Due to theoretical and practical importance, many problems of waves and vibration of micropolar elasticity have been investigated by different researchers. Propagation of monochromatic waves in an infinite micropolar elastic plate has been studied by Nowacki and Nowacki[7]. Benerji and Sengupta[8] discussed propagation of waves in a micropolar elastic layer immersed in an infinite liquid. Rao and Rao[9] and Rao[10] investigated the problems of longitudinal wave propagation in micropolar wave guide. Rao and Reddy[11] studied the Rayleigh type wave propagation in a micropolar cylindrical surface. Kumar and Singh[12] investigated propagation of waves in a micropolar generalized thermo elastic layers with stretch. Kumar and Deswal[13] discussed wave propagation through cylindrical bore contained in a microstretch elastic medium. Kumar and Tomer[14] studied the elastic wave propagation in cylindrical bore situated in a ∗ Received Aug.21, 2005; Revised Feb.13, 2006 Corresponding author Rajneesh Kumar, E-mail: rajneesh kuk@rediffmail.com

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Rajneesh Kumar and Geeta Partap

micropolar elastic medium with stretch. Kumar and Choudhary[15] studied the response of orthotropic micropolar elastic medium due to time harmonic source. Kumar and Ailawalia[16] investigated the effects of viscosity with moving load at micropolar boundary surface. The cylindrical plates and panels are frequently used as structural components and their vibration characteristics are important for practical design. A large number of previous studies on the vibration of isotropic curved panels have performed. Thick composite shells and plates can be used in applications involving aerospace, off-shore, submarine structures, pressure vessels, civil engineering structures, chemical pipe and automotive suspension components. The present investigation is aimed to study the propagation of plane crested waves in an infinite homogeneous, isotropic micropolar elastic cylindrical plate of thickness 2d. It is noticed that the motion for circular crested waves is also governed by Rayleigh-Lamb type secular equations in case of rectangular plate. At short wave limit, the secular equations for symmetric and skew symmetric waves in a stress free circular plate reduce to Rayleigh surface frequency equations.

1

Basic Equations

The equations of motion and the constitutive relations in a micropolar elastic medium without body forces and body couples given by Eringen[17] are ∂2u , ∂t2 ∂ 2φ (α + β + γ)∇(∇ · φ) − γ∇ × (∇ × φ) + K∇ × u − 2Kφ = ρj 2 , ∂t tkl = λur,r δkl + μ(uk,l + ul,k ) + K(ul,k − klr φr ),

(3)

mkl = αφr,r δkl + βφk,l + γφl,k ,

(4)

(λ + 2μ + K)∇(∇ · u) − (μ + K)∇ × ∇ × u + K∇ × φ = ρ

(1) (2)

where u = (ur , uθ , uz ) is the displacement vector, φ = (φr , φθ , φz ) is the microrotation vector, λ, μ, α, β, γ, K are material constants, ρ is the density, j is the microinertia, tkl and mkl are components of force stress and couple stress, respectively.

2

Formulation of Problem and Solution

We consider an infinite, homogeneous isotropic micropolar cylindrical plate of thickness 2d. The plate is axisymmetric with the z-axis as the axis of the symmetry. The origin of the coordinate system (r, θ, z) is taken as the middle surface of the plate and z-axis normal to it along the thickness. The surface z = ±d is subjected to different boundary conditions. We take r-z plane as the plane of incidence. For two dimensional problem, we take u = (ur , 0, uz ) and φ = (0, φθ , 0). We define the non-dimensional quantities: ⎧ 2 2 ω∗ ω∗ ω∗ ω∗ K ⎪ ∗ ⎪ r , = r, u = u , u = u , t = ω t, φ = jφθ , ω ∗ = ⎪ r z r z θ 2 ⎪ ⎪ c c c c ρj 1 1 1 ⎪ 1 ⎪ ⎨ 1 ω∗j ω2 λ + 2μ + K μ+K tij = tij , mij = , c22 = , mij , ω 2 = ∗2 , c21 = ⎪ μ γc ρ ρ ω ⎪ 1 ⎪ ⎪ ⎪ ⎪ 2 K γ c2 c2 K ⎪ ⎩ c24 = , p = 2 , δ 2 = 22 , δ22 = 12 , δ ∗ = 2 , ρj ρc1 c1 c4 ρc4 where ω ∗ is the characteristic frequency of the medium.

(5)

Rayleigh Lamb Waves in Micropolar Isotropic Elastic Plate

Upon using Eq.(5) in Eqs.(1) and (2), we obtain ∂ 2 ur 1 ∂φθ 2 ∂e 2 2 + δ ∇ − 2 ur − = , (1 − δ ) ∂r r ∂z ∂t2 ∂e 1 ∂ ∂ 2 uz + δ 2 ∇2 uz + (rφθ ) = (1 − δ 2 ) , ∂z r ∂r ∂t2 2 ∂ur ∂uz ∂ 2 φθ 1 − ∇2 − 2 φθ − 2δ22 φθ + δ ∗ = δ22 2 , r ∂z ∂r ∂t

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(6) (7) (8)

where

∂ur ur ∂uz ∂2 ∂2 1 ∂ + + , ∇2 = 2 + + 2. ∂r r ∂z ∂r r ∂r ∂z Here primes have been suppressed for convenience. Introducing the potential functions φ and ψ in Eqs.(6), (7) and (8) through the relations, e=

ur =

∂φ ∂ψ + , ∂r ∂z

uz =

∂φ ∂ψ ψ − − , ∂z ∂r r

(9)

we obtain ∂2φ = 0, ∂t2 ψ φθ 1 ∂2ψ ∇2 ψ − 2 − 2 − 2 2 = 0, δ δ ∂t r 2 2 1 1 ∗ 2 2 ∂ φθ 2 2 δ + 2δ2 φθ − ∇ − 2 φθ . ∇ − 2 ψ = δ2 r ∂t2 r ∇2 φ −

(10) (11) (12)

We assume the solutions of Eqs.(10)–(12) have the form: (φ, ψ, φθ ) = [f (z)J0 (ξr), g(z)J1 (ξr), h(z)J1 (ξr)]e−iωt ,

(13)

where ω is the frequency, ξ is the wave number and J0 (ξr) and J1 (ξr) are, respectively, the Bessel functions of order zero and one. Using Eq.(13) in Eqs.(10), (11) and (12) and solving the resulting differential equations, the expressions for φ, ψ and φθ are obtained as φ = (A cos m1 z + B sin m1 z)J0 (ξr)e−iωt ,

(14)

ψ = (C cos m2 z + D sin m2 z + E cos m3 z + F sin m3 z)J1 (ξr)e 2

2

φθ = δ [(β −

m22 )(C

2

cos m2 z + D sin m2 z) + (β −

m23 )(E

−iωt

,

cos m3 z + F sin m3 z)]J1 (ξr)e

(15) −iωt

, (16)

where c2 m21 = ξ 2 (c2 − 1) = α2 , m22 = ξ 2 (c2 a22 − 1), m23 = ξ 2 (c2 a23 − 1), β 2 = ξ 2 ( 2 − 1), δ

2 ∗2 2 2 ∗2 2 2 1 1 1 δ 2δ2 δ δ 2δ2 δ 2 2 2 2 (a2 , a3 ) = δ2 + 2 + 2 2 (1 − ∗2 ) ± − δ2 + 2 2 (1 − ∗2 ) 2 δ ω δ δ2 ω δ δ δ 12 2 4δ 2 + 2 22 {δ ∗ − 2(δ22 δ 2 − 1)} , ω δ and c =

ω ξ

is the phase velocity of the waves.

(17)

(18)

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The displacement components ur and uz are obtained from Eqs.(9), (14),(15) and (16) as ur =[−ξ(A cos m1 z + B sin m1 z) − m2 (C sin m2 z − D cos m2 z) − m3 (E sin m3 z − F cos m3 z)]J1 (ξr)e−iωt , uz =[−m1 (A sin m1 z − B cos m1 z) − ξ(C cos m2 z + D sin m2 z + E cos m3 z + F sin m3 z)]J0 (ξr)e−iωt .

(19) (20)

The nondimensional mechanical boundary conditions at z = ±d are given by tzz = 0,

tzr = 0,

mzθ = 0,

(21)

where tzz , tzr and mzθ are given by ∂uz + λe tzz = (2μ + K) ∂z ∂ur ∂uz ∂ur tzr = μ + − φθ , +K ∂z ∂r ∂z ∂φθ . mzθ = γ ∂z

3

Derivation of Secular Equations

Invoking boundary conditions (21) on the surfaces z = ±d of the plate and using Eqs.(14)– (20), we obtain a system of six simultaneous equations: P (Ac1 + Bs1 ) + Q{m2 (Cs2 − Dc2 ) + m3 (Es3 − F c3 )} = 0,

(22)

P (Ac1 − Bs1 ) + Q{m2 (−Cs2 − Dc2 ) + m3 (−Es3 − F c3 )} = 0, Q(−m1 s1 A + m1 c1 B) + P (Cc2 + Ds2 + Ec3 + F s3 ) = 0,

(23) (24)

Q(m1 s1 A + m1 c1 B) + P (Cc2 − Ds2 + Ec3 − F s3 ) = 0, f2 (Cm2 s2 − Dm2 c2 ) + f3 (Em3 s3 − F m3 c3 ) = 0,

(25) (26)

f2 (Cm2 s2 + Dm2 c2 ) + f3 (Em3 s3 + F m3 c3 ) = 0,

(27)

where ⎧ ⎨ ⎩

P = β2 − ξ2 + si = sin mi z,

pξ 2 p , Q = 2ξ(1 − 2 ), fi = β 2 − m2i , 2 δ 2δ ci = cos mi z, i = 1, 2, 3.

i = 2, 3,

(28)

The system of Eqs.(22)–(27) has a non-trivial solution if the determinant of the coefficients of amplitudes [A, B, C, D, E, F ]T vanishes. After lengthy algebraic reductions and manipulations leads to the secular equations for the plate:

±

± −4ξ 2 (1 − 2δp2 )2 m1 m2 (m23 − m22 ) tan m1 d m2 (β 2 − m22 ) tan m1 d . (29) − = 2 2 2 2 tan m2 d m3 (β 2 − m23 ) tan m3 d (β 2 − ξ 2 + pξ δ 2 ) (m3 − β ) Here the superscript +1 refers to skew symmetric and −1 refers to symmetric modes. Equation (29) is the secular equation for the propagation of modified micropolar elastic waves in the plate. These equations can be recognized as Rayleigh-Lamb equations for symmetric and anti-symmetric waves in an infinite rectangular plate in micropolar elastic solid. We refer to such waves rather than Lamb waves whose properties were derived by Lamb[18] for isotropic elastic solids in elastokinetics. Thus Rayleigh-Lamb type equation also governs circular crested


1053

micropolar elastic waves in a plate. Although the frequency wave number relationship holds whether the waves are straight or circularly crested. The displacement and stress vary according to Bessel functions rather than trigonometric functions as far as the radial coordinate is concerned. For large value of r, we have J0 (ξr) →

sin ξr + cos ξr √ , πξr

J1 (ξr) →

sin ξr − cos ξr √ . πξr

(30)

Thus, far from the origin the motion becomes periodic in r. Actually, “far” occurs rather rapidly, within four to five zeros of Bessel function. As r becomes very large, the straight crested behavior is the limit of the circular crested waves. Particular Case: Elastic Plate In the absence of micropolarity effect (K = p = j = 0), we have m21 = α2 ,

a22 =

1 , δ2

a23 = 0,

m22 = β 2 ,

m23 = −ξ 2 ,

and consequently, the secular equation (29) reduces to

tan m1 d tan m2 d

± =−

4ξ 2 αβ . (β 2 − ξ 2 )

(31)

The equation (31) agrees with that of Graff[19] discussing for homogeneous, isotropic, stress free elastic plate.

4

Discussion of Secular Equation

Regions of Secular Equation Here depending on whether c < δ, 1, |a1i | , i = 2, 3, we may have β, mi (i = 1, 2, 3) being purely imaginary, zero or real. Then for example the frequency equation (29) is correspondingly altered as follows: Region I This region is characterized by c < δ, 1, |a1i | , i = 2, 3. In this case, we replace β with iβ and mi with iαi , i = 1, 2, 3, respectively. The secular equation (29) becomes

th α1 d th α2 d

±

± 4ξ 2 (1 − 2δp2 )2 α1 α2 (α23 − α22 ) α2 (β 2 − α22 ) th α1 d . − = 2 2 2 2 2 2 2 α3 (β 2 − α3 ) th α3 d ( pξ δ 2 − β − ξ ) (α3 − β )

(32)

Region II This region is characterized by δ < c < 1. In this case, we have β = iβ , m1 = iα1 , m2 = m2 , m3 = m3 and the secular equation becomes

th α1 d tan m2 d

±

± 4ξ 2 (1 − 2δp2 )2 α1 m2 (m23 − m22 ) m2 (β 2 − m22 ) th α1 d − = − . 2 m3 (β 2 − m23 ) tan m3 d ( pξ2 − β 2 − ξ 2 )2 (m23 − β 2 )

(33)

δ

Region III This region is characterized by ξ < ω, c > 1 and the secular equation is given by Eq.(29).

5

Thin Plate Results

Let us consider the case when transverse wavelength with respect to thickness of the plate 2π 2π 2π 2π is quite large, so that m , , , β ≥ d. Regions I and II yield the results of interest in this 1 m2 m3 case. In Region I, the symmetric case has no roots. The skew symmetric case, upon retaining

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the first two terms in the expansion of hyperbolic tangents, the secular equation (32) reduces to 2 2 pξ p 2 2 β 2 d2 α22 d2 2 2 2 −β −ξ 1− = 4ξ 1 − 2 α2 1 − . (34) δ2 3 2δ 3 If we ignore the micropolarity effect, i.e., (K = p = j = 0), then Eq.(34) reduces to 2 β 2 d2 α22 d2 2 2 2 2 1− = 4ξ α2 1 − . β +ξ 3 3

(35)

In Region II, skew symmetric case has no roots. The secular equation for symmetric case reduces to 4ξ 2 (1 − 2δp2 )2 α23 α22 . (36) (α23 + α22 − β 2 ) = 2 2 (β 2 + ξ 2 − pξ δ2 ) If we ignore the micropolarity effect, i.e., (K = p = 0), then Eq.(36) reduces to (β 2 + ξ 2 )2 = 4ξ 2 β 2 .

6

(37)

Waves of Short Wavelength

Some information on the asymptotic behavior is obtained by letting ξ → ∞. If we take ξ > ωδ , it follows that c < δ, 1 .The roots of the secular equation lie in Region I. In this case, we replace β with iβ and mi with iαi , i = 1, 2, 3, respectively, and secular equation (29) reduces to

±

± 4ξ 2 (1 − 2δp2 )2 α1 α2 (α23 − α22 ) th α1 d α2 (β 2 − α22 ) th α1 d . (38) − = 2 th α2 d α3 (β 2 − α23 ) th α3 d ( pξ2 − β 2 − ξ 2 )2 (α23 − β 2 ) δ

For ξ → ∞,

th α1 d th αi d

→ 1, i = 2, 3, so the frequency equation (38) reduces to

2 2 pξ p 2 2 2 4ξ 2 1 − 2 α1 α2 α3 (α2 + α3 ) = − β − ξ (α22 + α23 + α2 α3 − β 2 ) 2δ δ2

(39)

for symmetric and skew symmetric cases. These are merely Rayleigh surface wave equations. The Rayleigh results enter here since, for such small wavelengths, the finite thickness plate appears as a half-space. Hence vibrational energy is transmitted mainly along the surface of the plate. If the micropolarity effect is neglected, i.e., (K = p = 0), in Eq.(39), we get 4ξ 2 α1 β = (β 2 + ξ 2 )2 .

7

(40)

Lame Modes

A special class of exact solutions, called the Lame modes, but evidently first identified by Lamb[18] can be obtained by considering the special case β 2 = ξ 2 (1 − δp2 ), the roots for this case are in Region II and the frequency equation (29) reduces to: Symmetric modes: tan m2 d = ∞ ⇒ m2 = nπ 2d , n = 1, 3, 5, · · · ; Anti-symmetric modes: tan m2 d = 0 ⇒ m2 = nπ 2d , n = 0, 2, 4, · · · . Here, the frequency is given by 4β 2 d2 + n2 π 2 (1 − δp2 ) ω= . 2da2 1 − δp2 However in the absence of micropolarity effect (p = 0), β = m2 and consequently the frequency nπδ is given by ω = √ which agrees with Graff[19] . It is obvious that these modes depend upon 2d the micropolar parameter (K or p) and thickness of the plate.


8

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Amplitudes of Displacements and Microrotation

In this section the amplitudes of displacement components and microrotation for symmetric and skew symmetric modes of plate waves, have been computed. Upon using Eqs.(19) and (20) and (22)–(27), we obtain (ur )sym = −{ξ cos m1 z + Lm2 cos m2 z + M m3 cos m3 z}AJ1 (ξr)e−iωt , (ur )asym = −{ξ sin m1 z + L m2 sin m2 z + M m3 sin m3 z}BJ1 (ξr)e−iωt , (uz )sym = −{m1 sin m1 z − ξ(L sin m2 z + M sin m3 z)}AJ0 (ξr)e−iωt , (uz )asym = {m1 cos m1 z − ξ(L cos m2 z + M cos m3 z)}BJ0 (ξr)e−iωt , f 2 m 2 s2 (φθ )sym = δ 2 (β 2 − m22 ) cos m2 z − (β 2 − m23 ) cos m3 z CJ1 (ξr)e−iωt , f 3 m 3 s3 f 2 m 2 c2 sin m3 z DJ1 (ξr)e−iωt , (φθ )asym = δ 2 (β 2 − m22 ) sin m2 z − (β 2 − m23 ) f 3 m 3 c3 where P 2 T1−1 − Q2 m1 m3 T3−1 s1 Q2 m1 m2 T2−1 − P 2 T1−1 s1 , M = − , P Q(m2 T2−1 − m3 T3−1 ) s2 P Q(m2 T2−1 − m3 T3−1 ) s3 Q 2 m 1 m 3 T 3 − P 2 T 1 c1 P 2 T 1 − Q 2 m 1 m 2 T 2 c1 L = , M = , P Q(m2 T2 − m3 T3 ) c2 P Q(m2 T2 − m3 T3 ) c3 Ti = tan mi d, i = 1, 2, 3.

L=−

9

Example Results

With the view of illustrating theoretical results obtained in the preceding sections, we now present some numerical results. The material chosen for this purpose is Magnesium crystal (micropolar elastic solid), the physical data for which is given below: ρ = 1.74 × 103 kg/m3 ,

λ = 9.4 × 1010 N/m2 ,

K = 1.0 × 1010 N/m2 ,

γ = 0.779 × 10−9 N,

μ = 4.0 × 1010 N/m2 , j = 0.2 × 10−19 m2 ,

d = 0.01 m.

The non-dimensional phase velocity of symmetric and skew symmetric modes of wave propagation has been computed for various values of non-dimensional wave number from secular equation (29). The corresponding numerically computed values of phase velocity are shown graphically in Figs.1 and 2 for different modes (n = 0 to n = 6). The solid curves correspond to micropolar elastic plate (MEP) and dashed curves refer to elastic plate (EP). The amplitudes of displacements and microrotation components for symmetric and skew symmetric modes are presented graphically in Figs.3–8. The phase velocity of lowest symmetric mode ( n = 0 ) remains constant with the variation in wave number, whereas the phase velocity of lowest skew symmetric mode varies at lower wave number and becomes constant at higher wave number. The phase velocity of higher modes of wave propagation, symmetric and skew symmetric attains quite large values at vanishing wave number, which sharply slashes down to become steady with increasing wave number. It is observed that the phase velocities of different modes of wave propagation start from large values at vanishing wave number and then exhibit strong dispersion until the velocity flattens out to the value of the micropolar Rayleigh wave velocity of the material at higher wave numbers. The reason for this asymptotic approach is that for short wavelengths (or high frequencies) the material plate behaves increasingly like a thick slab and hence the coupling between upper

1056


and lower boundary surfaces is reduced and as a result the properties of symmetric and skew symmetric waves become more and more similar. It is observed that for symmetric mode n = 1, the values of phase velocity are smaller in micropolar than in elastic plate (EP) for wave number ξd ≤ 3 and for wave number ξd ≥ 3 phase velocities in MEP and EP are nearly same. In case of symmetric mode n = 2, for wave number lying between 1.5 and 3.0, phase velocity for EP is more than in case of MEP and phase velocity profiles coincide in respect of EP and MEP for wave number ξd ≤ 1.5 and ξd ≥ 3.0. At higher symmetric mode n = 4, the phase velocity profiles in respect of MEP and EP coincide for wave number ξd ≥ 2.0 and for ξd ≤ 2.0 the values of phase velocity are smaller in EP. For symmetric modes n = 3, 5, 6, phase velocity profiles for EP and MEP coincide. For skew symmetric modes of wave propagation, we observe the following: (a) for mode n = 1 and wave number ξd ≤ 1.0, phase velocity in elastic is more than in micropolar elastic and for wave number lying between 1.0 and 2.0, phase velocities in MEP and EP are nearly same, for wave number lying between 2.0 and 4.0, phase velocity for MEP is more than in EP, and for wave number ξd ≥ 4.0 phase velocities in MEP and EP are nearly same; (b) for modes n = 0, 2, 3, 4, 5, 6 phase velocity profiles for micropolar elastic and elastic coincide.

Fig.1

Phase velocity profile for symmetric modes of wave propagation

Fig.3

Amplitude of symmetric displacement (ur )

Fig.2

Fig.4

Phase velocity profile for skew symmetric modes of wave propagation

Amplitude of skew symmetric displacement (ur )


Fig.5

Fig.7

Amplitude of symmetric displacement (uz )

Amplitude of symmetric microrotation (φθ )

Fig.6

Fig.8

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Amplitude of skew symmetric displacement (uz )

Amplitude of skew symmetric microrotation (φθ )

Figures 3–6 depict the variations of symmetric and skew symmetric amplitudes of displacements (ur ) and (uz ) in case of micropolar elastic plate (MEP) and elastic plate (EP). The displacement (ur ) of the plate is minimum at the centre and maximum at the surfaces for symmetric mode and zero at the centre and maximum at the surfaces for skew symmetric mode as can be seen from Figs.3 and 4, respectively. From Figs.5 and 6, it is noticed that the values of the displacement (uz ) of the plate is zero at the centre and maximum at the surfaces for symmetric mode and maximum at the centre and minimum at the surfaces for skew symmetric mode. Figures 7 and 8 represent the variations of symmetric and skew symmetric amplitudes of microrotation (φθ ) in case of micropolar elastic plate (MEP). It is evident from Figs.7 and 8 that the values of microrotation (φθ ) of the plate is minimum at the centre and maximum at the surfaces for symmetric mode and zero at the centre and maximum at the surfaces for

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skew symmetric mode. (ur )sym , (ur )asym , (uz )sym , (uz )asym , (φθ )sym and (φθ )asym correspond to the values of (ur ), (uz ) and (φθ ) for symmetric and skew symmetric modes respectively. It is observed that behavior and trend of variations (ur )asym resembles with (uz )sym and that of (ur )sym is opposite to (uz )asym . The values of the displacements (uz ) of the plate are more in elastic plate (EP) in comparison to micropolar elastic plate (MEP) for symmetric and skew symmetric modes. The values of the displacements (ur ) of the plate are more in elastic plate (EP) in comparison to micropolar elastic plate (MEP) for symmetric mode, whereas the values of the displacements (ur ) of the plate are smaller in elastic plate (EP) in comparison to micropolar elastic plate (MEP) for skew symmetric mode.

10

Conclusions

The propagation of circular crested waves in infinite homogeneous, isotropic micropolar elastic cylindrical plates was investigated after deriving the secular equations. It is noticed that the motion of circular crested waves is governed by the Rayleigh-Lamb type secular equations. Various regions of the secular equations are also explored. At short wave limit, the secular equations in case of symmetric and skew symmetric modes of wave propagation in a stress free circular plate reduce to the Rayleigh surface frequency equations because in such situation the finite thickness plate appears as a half space and the vibration energy is transmitted along the surface of the plate. When the transverse wave length with respect to the thickness of the plate is quite large, regions I and II yield the results of the interest. In region I, the secular equations for the skew symmetric case pertain to flexural vibration and represent only a single vibration mode in a limited frequency range in the overall frequency spectrum. For the secular equations in the symmetric case in region II, the phase velocity expression reduces to the thin plate or plate stress analogue of the bar velocity of longitudinal rod theory which also depends upon the micropolarity effect and as a special case reduces to the result obtained by Graff[19] .

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[14] Kumar R, Tomer S K. Elastic wave propagation in a cylindrical bore situated in a micropolar elastic medium with stretch[J]. Proc Indian Acad Sci, Math Sci, 1999, 109:425–433. [15] Kumar R, Choudhary S. Response of orthotropic micropolar elastic medium due to time harmonic source[J]. Sadhana, 2004, 29(1):83–92. [16] Kumar R, Ailawalia P. Effects of viscosity with moving load at micropolar boundary surface[J]. Internat J of Applied Mechanics and Engineering, 2005, 10(1):95–108. [17] Eringen A C. Theory of micropolar elasticity[M]. In:Leibowitz H (ed). Fracture. Vol II, Academic Press, New York, 1968. [18] Lamb H. On waves in an elastic plate[J]. Phil Trans Roy Soc, London, Ser A, 1917, 93:114–128. [19] Graff K F. Wave Motion in Elastic Solids[M]. Dover Publications, New York, Inc, 1991.