INVESTIGATION OF PLANE LONGITUDINAL WAVES IN A MICRO-ISOTROPIC, MICRO-ELASTIC SOLID

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In the frame work of plane harmonic wave solution, we investigate the phase velocities of longitudinal micro-rotational (MR), micro-isotropical (MI), and micro strain (MS) waves. Two sets of MR waves, one set of MI and MS waves are obtained. These waves are dispersive in nature. Micro non-rotational (MNR) waves are obtained as a particular case of MR waves. Micro-isotropical shear (MIS) and micro strain shear (MSS) waves are obtained as a particular case of MI and MS waves. MNR waves are dispersive in nature, while MIS and MSS waves are non-dispersive and they depend only on elastic constants. With the help of MATLAB programme, numerical example is considered, and speeds are plotted against frequency ratios and wave number.


INTRODUCTION
Eringen and his co-researchers [1,2] are developed the non linear theory of micro elasticity.
Eringen [3] modified the linear theory of micropolar elasticity. The basic difference between the theory of micropolar elasticity and classical theory of elasticity is only the introduction of an independent micro rotation vector. With this reason, the motion in the micropolar solids is characterized by six degrees of freedom, namely three of translation and three of rotation. The introduction between two parts of micropolar body is transmitted not only by a force vector but also by a couple resulting in a symmetric force stress tensor, and couple stress tensor. Propagation of waves in micro polar elastic solids was studied by many authors like Parfitt and Eringen [4], Tomar and Gogna [5], Tomer et.al. [6] and Tomer and Gogna [7,8]. Singh and Tomer [9] are studied the plane wave propagation in thermo elastic solids with voids. Pochhmer [10] and Chree [11] discussed the longitudinal waves in a cylinder within the frame work of classical theory of elasticity. The velocity of longitudinal waves in a cylindrical bars was derived by Bancroft [12]. Oliver [13] investigated the wave propagation in a cylindrical rods under the pulse techniques. In recent, Somaiah [14] studied the plane wave propagation in micro stretch elastic solids.

BASIC EQUATIONS
The basic governing equations of displacements, micro-rotations and micro-strains for a micro-isotropic, micro-elastic solid in the absence of body forces and body couples are given by Nowacki [15] as follows: where λ , µ are Lames constants, K,τ 1 , τ 2 are elastic constants,ρ is the density, J is the moment of micro inertia.α, β , γ are micro-rotational constants,σ 1 , σ 2 are respectively isotropical and strain parameters, δ i j is the Kronecker delta and double dot on right hand side is the second order partial derivative with respect to time t .

FORMULATION AND SOLUTION OF THE PROBLEM
Equations (1) and (2) can be written as While equation (3) written by Somaiah.K and Sambaiah.K [16] as follows: The equations (4) and (5) are coupled in u and φ , while equations (6) and (7) are uncoupled in For plane harmonic solution in the positive direction of unit vectorn, we may seek the solution of equations (4) to(7) as, where A, B are vector amplitudes,C pp , D (i j) are scalar amplitudes, q is the wave number, r is the position vector, v is the phase velocity. Thus, where l is the wave length, x k are the components of position vector and ω is the angular frequency of the solid.

LONGITUDINAL MICRO-ROTATIONAL WAVES
On using eq. (9) into eq. (4) and eq. (5) we obtain, Taking scalar product of equation (11) with vector A we obtain, Solving eq. (12) for B , we get On using eq.(15) into eq. (13) we obtain, For plane longitudinal wave, take q = ω v in equation (17) we obtain the following quadratic equation in v 2 ; The roots of equation (18) are given by Equation (20) represents the speed of two sets longitudinal micro-rotational (MR) waves and they are not encountered in any classical theory of elasticity and they are influenced by microrotational parameters α, β , γ . Also they are dispersive in nature. The classical results can be obtained as a particular case if and only if K → 0(i.e., ω 2 0 = c 2 6 = c 2 7 = 0) in equation (17) we obtained the speeds of micro-rotational (MR) waves as, and these are non dispersive in nature.

LONGITUDINAL MICRO-ISOTROPICAL AND MICRO-STRAIN WAVES
On using eqution (9) in the equations (6) and (7), we obtained the speed of micro-isotropical (MI) wave v p and the speed of micro strain (MS) wave v (i j) are given by From equations (23) and (24), we observed that the speed of two distinct micro-isotropical (MI) and two distinct micro-strain (MS) waves are also depends on the wave number, so they are dispersive in nature, and they are influenced by isotropical and strain parameters σ 1 , σ 2 .

Particular case.
In the absence of isotropical and strain parameters(i.e, σ 1 = 0; σ 2 = 0), the eq.(6) and eq.(7) are reduces to Equations (25) and (26) represents micro-isotropical shear (MIS) waves and micro-strain shear (MSS) waves respectively, and their speeds v * p and v * (i j) are given by It is observed that isotropical and strain shear waves are non-dispersive and they are depends on elastic constants.
From these figures we observed that the MNR waves are fall down to low speed at frequency ratio less than 1 × 10 4 , and suddenly jumped to high speed at frequency ratio 1 × 10 4  micro-isotropic, micro-elastic solid, we concluded that: • Two sets of MR waves and one set of MI and MS waves are propagate with distinct speeds.
• MR, MNR, MI and MS waves are dispersive in nature, while MIS and MSS waves are non-dispersive.
• MIS and MSS waves are constant and they are only elastic constant dependent waves.
• MS waves slower than MI waves.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.