On the diffusion-strain coupling and dispersion of surface waves in transversely isotropic laser-excited solids

Author’s Accepted Manuscript On the diffusion-strain coupling and dispersion of surface WAVES in transversely isotropic laserexcited solids F. Mirzade www.elsevier.com/locate/physb

PII: DOI: Reference:

S0921-4526(15)30076-4 http://dx.doi.org/10.1016/j.physb.2015.05.028 PHYSB309007

To appear in: Physica B: Physics of Condensed Matter Received date: 27 February 2015 Revised date: 19 April 2015 Accepted date: 22 May 2015 Cite this article as: F. Mirzade, On the diffusion-strain coupling and dispersion of surface WAVES in transversely isotropic laser-excited solids, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2015.05.028 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

On the diffusion-strain coupling and dispersion of surface waves in transversely isotropic laser-excited solids F. Mirzade* Institute on Laser and Information Technologies, Russian Academy of Sciences, 140700, Moscow, Russia Abstract The present paper is aimed at studying the boundary value problem in elasticity theory concerning the propagation behavior of harmonic waves and vibrations on the surface of the transversely isotropic laser-excited crystalline solids with atomic defect generation. Coupled dynamical diffusion-deformation interaction model is employed to study this problem. The frequency equations of surface waves in closed form are derived and discussed. The three motions, namely, longitudinal, transverse, and diffusion of the medium are found to be dispersive and coupled with each other due to the defect concentration changes and anisotropic effects. The phase velocity and attenuation coefficient of the surface waves get modified due-to the defect-strain coupling and anisotropic effects, and are also influenced by the defect relaxation time. A softening of frequencies of surface acoustic waves (instability of frequencies) is obtained. Relevant results of previous investigations are deduced as special and limiting cases. Keywords: transversely isotropy, surface wave, laser-induced atomic defects, diffusion-strain coupling, frequency equation

1. Introduction Theory of elastic waves coupled to defect dynamics is an important generalization of the classical theory of elasticity. Problems concerning defect generation play a vital role in theoretical and practical problems of laser additive micro-and nanotechnologies, materials processing technology and etc. The theory of elasticity concerning the solid elastic material consisting of a distribution of defects is used widely for investigating such phenomena as laser annealing, fast recrystallization, selective laser sintering of powders and laser-assisted thin-film deposition process for which the use of the classical elasticity theory for mechanical behavior of materials is inadequate. All abovementioned processes can be accompanied by the generation of atomic point defects (interstitial atoms, vacancies, adatoms, electron-hole pairs). Also, efficient generation of nonequilibrium atomic defects may occur as a result of the action of intense impulse external energy fluxes (laser and corpuscular radiations) on condensed media or as a result of mechanical, thermo-chemical, and electric treatments of materials. The theory to include the effect of defect concentration change, known as diffusion-strain (DS) coupled theory, is well established [1,2]. This theory deals the deformational behavior of materials with a distribution of non-equilibrium atomic defects, where the concentration of defects is included among the kinematic (diffusional) variable. The theory reduces to the classical theory in the limiting case of concentration of defects tending to zero.

*

Corresponding author: Tel. +7(496)4522200 (118); fax: +7(496) 452 2532, E-mail address: [email protected] (F. Mirzade)

DS coupled theory has been receiving a lot of attention for the past two decades. Extensive theoretical efforts have been made so far to study the dynamical interaction between the defect concentration and mechanical fields in laser-excited solids in the context of DS coupled model. The problem of linear elasto-diffusive Rayleigh-type wave propagation on the surface of isotropic semiinfinitive solids and elastic thin plates with atomic defect generation has been considered by [1-5]. It has been shown that the surface waves in these types of media are dispersive in contrast to classical theory of elasticity in which Rayleigh-wave motions are not dispersive at any frequency. In general case, the surface waves are found to exhibit frequency dependent dispersion and are accompanied with attenuation or amplification. In Refs. [6,7], the small-scale effects on propagation of elasto-diffusive waves have been investigated in the context of coupled DS theory. For high intensities of incident laser radiation, the concentration of defects becomes so high that one should expect the appearance of cooperative effects in an ensemble of interacting (through the self-consistent elastic field of displacements in a medium) defects. The 1D and 2D self-organization of nonlinear coupled periodic and localized strain-defect structures (solitons or solitary waves) due to concentration-elastic instability were considered in [8,9]. A mechanism on the development of the instability is due to the coupling between defect dynamics and elastic field of the solids. Stabilization of this instability is due to the nonlinearity of the elastic continuum. The mathematical models of these studies were based on the wave type (hyperbolic) equations of motion for the displacement vector and diffusion type (parabolic) equation for atomic defect concentration accounting for elastic and concentration nonlinearities. The theory of mechanical waves coupled to atomic defect dynamics and including thermal change effects in solids under the action of laser pulses has been considered by Mirzade [10] and Bargmann and Favata [11]. Some features of the physical problems coupling diffusion, mechanics and thermal waves in geometrically linear and nonlinear solids has been studied in Refs. [12-14]. An overview of methods for analysis of elastic fields in solids from various structural defects (dislocation, inhomogeneous inclusions, grain boundaries and cracks) was presented by Mura [15] using the concept of eigenstrains in micromechanics. In the Mura's eigen-strain theory, the eigenstrains corresponding to each defect are conveniently expressed in terms of a defect density tensor. This theory has been applied successfully to modeling many important processes in crystalline solids such as diffusional phase transformations and microstructure coarsening, which involve diffusional redistribution of atoms under the influence of stresses arising from coherent compositional inhomogeneities as well as from structural defects [16,17]. The spatial distribution of defects in these models is described by the space-dependent eigenstrains. New perspectives on the phase field approach in modeling deformation and material defects as well as microstructural evolution (grain growth, precipitate evolution, solute segregation) are reviewed in [18]. Most of the studies on waves coupled to defect dynamics in elastic media discuss the propagation in isotropic media. Investigations of waves in anisotropic media are considerably more difficult than the classical and well-understood isotropic problem. The theory of elastic wave behavior propagation in anisotropic solids is well established [19,20]. Extensive theoretical efforts have been made so far to model the effect of heat conduction upon the propagation of plane harmonic waves in anisotropic elastic solids [21-26]. The study of wave propagation in a generalized thermoelastic anisotropic media with additional parameters like prestress, porosity, viscosity, thermal relaxation time, microstructure and other parameters allowed us to obtain vital information about existence of new or modified waves. The eigen-value problems of elastic waves in piezoelectric anisotropic solids were studied by Guo [27]. Valuable attempts have been carried out in [28] to investigate the propagation of waves in a homogeneous, transversely isotropic, piezo-thermoelastic plate. Acharya et al. investigated the general theory of transversely isotropic magneto-elastic interface waves in conducting media under initial hydrostatic tension or compression [29]. The growing applications of new anisotropic materials, especially in the various laser technologies, have encouraged the studies of impact and wave propagation in the anisotopic laserexcited materials and have become very important. In Ref. [30] the DS coupled model has been

extended to anisotropic laser-excited solids with atomic defect generation. Propagation of a body plane harmonic wave in an infinite elastic transverse isotropic solid was discussed in particular. It was found that four dispersive wave modes are possible namely, three quasi-elastic wave modes (E) and one quasi-defect concentration wave mode (N). All motions of the medium are found dispersive and coupled with each other due to the defect concentration changes and anisotropic effects. The present paper is a continuation of our previous work [30]. Its purpose is to study in the context of the DS coupled model the nature of Rayleigh-type surface wave propagation and vibrations in an anisotropic laser-excited solid with a distribution of atomic defects. The author believes that the problem in its present form has not been discussed so far. In the present investigation, frequency equations of coupled elasto-diffusive waves are obtained and some properties of their solutions are discussed. It is found that the surface waves are again found to be dispersive in character. The form of the frequency equations for small values of the diffusion-strain coupled parameter is also deduced and solved analytically. The phase velocity and attenuation coefficients of the waves are influenced by the anisotropic effects. Finally, the derived secular equations in various cases are solved numerically. The computer simulated results for a single crystal of Zn in respect of dispersion curves and attenuation coefficient are presented graphically. The obtained results are in agreement with the corresponding classical results of previous studies for limiting and special cases. 2. Basic governing equations Consider an elastic crystalline solid with hexagonal or transverse isotropic symmetry occupying the half-space z ≥ 0 in Cartesian coordinate system 0xyz . We assume that the planes of isotropy are perpendicular to z -axis. We choose x-axis in the direction of the propagation of waves so that all particles on a line parallel to y-axis are equally displaced. Thus the motion of the medium is
supposed to take place in the xz plane and for the assumed motion the displacement vector u has the component (u, 0, w) and all the other variables depend on x, z and t only. Let us assume that an external energy flux (e.g., laser radiation) generates high concentrations of non-equilibrium atomic defects (vacancies (V - defects) and interstitials (I - defects)) in the nearsurface layer (due to heating and renormalization of the defect formation energy). On one hand, the presence of defect density profile results in a force that may induce strain field in medium. On the other hand, when the strain waves propagate, the formation and migration energies of defects change in the compression and dilatation zones; this results in modulations of generation (g) and recombination (r) rates of defects. Therefore, the evolutions of strain and defect concentration fields are inherently coupled. The dynamical model that can describe the evolution of such a system should be based on: (i) the evolution of atomic defect concentration in a strained solid and (ii) the displacement field of a solid in the presence of a non-uniform defect concentration field. In formulating our theory, we limit our considerations to the case of only one type of atomic defects (for definiteness, I –type defects). Let the medium’s temperature be constant (with thermal strains neglected). Following [30], the constitutive strain-stress-defect relations and 2D field equations in terms of
the displacement vector u = (u, 0, w) and defect concentration fields n ( x, z , t ) for a linear transversely isotropic elastic medium, in the absence of the body forces, are σ xx = c11u, x + c13 w, z − ϑ1n , (1)

σ zz = c13u, x + c33 w, z − ϑ3 n , σ xz = 2c44 ( u, z + w, x ) ,

(2)

(3)

c11u, xx + c44u, zz + ( c13 + c44 ) wxz − ρ u = ϑ1n, x ,

(4)

( c13 + c44 ) u xz + c44 w, xx + c33 w, zz − ρ w = ϑ3n, z ,

(5)

τ n − D1n, xx − D3 n, zz + n = g 0 β ϑ1u, x + ϑ3 w, z  ,

(6)

−1

where cij are the components of the elasticity tensor, ρ is the density of the medium, D1 , D3 and

ϑ1 , ϑ3 , are respectively, the diffusivities and deformational potentials along and perpendicular to the plane of symmetry; τ −1 = r is the defect recombination rate ( τ , relaxation time); g 0 is the −1

defect generation rate constant; β = ( k BT ) . The comma notation is used for spatial derivatives and a superposed dot denotes time differentiation. The terms on the right-hand side of Eq. (6) account for the strain-induced generation of defects. The origin and the physical meaning of these terms are explained in Ref. [1]. We define the following dimensionless quantities v v2 ρ v3 ( x′, z′ ) = 1 ( x, z ) , t ′ = 1 t , ( u′, w′ ) = 1 ( u, w) , D1 D1 D1ϑ1n0 D c c c +1 c n r ′ = 21 r , n′ = , c1 = 33 , c2 = 44 , c3 = 13 2 44 , v1 n0 c11 2c11 c11

D=

D1 , D3

ϑ=

ϑ1 , ϑ3

ε=

g 0 βϑ12 . ρ v12 r

(7)

Here v1 = c11 ρ −1 is the velocity of compressional waves; ε is the strain-diffusion coupling constant. Introducing quantities (7) in Eqs. (1)-(6), we obtain (1a) σ xx = u, x + ( c3 − c2 ) w, z − n  ϑ1n0 ,

σ zz = ( c3 − c2 ) u, x + c1w, z − ϑ n  ϑ1n0 , σ xz = ( u, z + w, x ) ϑ1n0 c2 ,

(2a) (3a)

u, xx + c2u, zz + c3 wxz − u = n, x ,

(4a)

 = ϑ n, z , c3u xz + c2 w, xx + c1w, zz − w

(5a)

τ n − n, xx − Dn, zz + n = ε ( u, x + ϑ w, z ) . −1

(6a)

(the primes have been suppressed for convenience). The boundary conditions on the surface z=0 of the solid are given by (i) Mechanical conditions (stress-free surface) σ zz = 0 , σ xz = 0 .

(8a)

(ii) Diffusional conditions (diffusion flux of defects should vanish) n, z = 0 .

(8b)

2. Solution To solve the Eqns. (4a) to (6a), we consider

( u, w, n ) = U ( z ) ,W ( z ) , N ( z )  exp i ( kx − ωt )  , (9) where

ω = ΩD1 / v12 , k = qD1 / v1 , Ω is the frequency and q is the wave number. We define the phase velocity “ c ”and attenuation constant (damping coefficient) “ γ ”of the waves such that c = Ωr / q and γ = − Ω i , where Ωr and

Ωi mean respectively the real ( Re(Ω) ) and imaginary parts ( Im(Ω) ) of Ω . Substitution (9) into the Eqs. (1a), (2a) and (3a), yields

( c Dˆ 2

2

)

ˆ − ikN = 0, + ω 2 − k 2 U + ikc3 DW

(10a)

(

)

ˆ + c Dˆ 2 + ω 2 − c k 2 W − ϑ DN ˆ = 0, ikc3 DU 1 2

(

(10b) (10c)

)

ˆ + DDˆ 2 + iω − k 2 − τ −1 N = 0 , ikε U + εϑ DW

where Dˆ = d dz . Eqs. (10a-c), can be factorized as Dˆ 2 − α12 Dˆ 2 − α 22 Dˆ 2 − α 32 (U , W , N ) = 0 ,

(

)(

)(

)

where α i2 ( i = 1, 2, 3) are the roots of the characteristic equation

α 6 − Lα 4 + Jα 2 − P = 0 .

(11)

Here the coefficients L, J and P are given by L = ( Fk 2 − Eω 2 ) D + ( k 2 + τ −1 − iω ) c1c2 − εϑ 2 c2  ∆ ,

{

J = ( k 2 + τ −1 − iω )( Fk 2 − Eω 2 ) + ( k 2 − ω 2 )( c2 k 2 − ω 2 ) D −

} ∆,

− ε ( c1 − 2c3ϑ + ϑ 2 ) k 2 − ϑ 2ω 2 

{

P = ( c2 k 2 − ω 2 ) ( k 2 − ω 2 )( k 2 + τ −1 − iω ) − ε k 2

}

∆,

and

F = c1 + c22 − c32 , E = c1 + c2 , ∆ = c1c2 D . For sufficiently small values of the strain-diffusion coupling constant ε , the roots α 2j , j = 1, 2, 3 given by Eqs. (11) can be rewrite as

α 2j = α 2j + εα 2j + Ο ( ε 2 ) , j = 1, 2,3 ,

(12)

where 2 α1,2 =

M2 M1 ± 3 , 2c1c2 2c1c2

(13a)

α 32 = ( k 2 + τ −1 − iω ) D , 2 j

α =

(13b)

{

}

α 2j ( c1 − 2c3ϑ + ϑ 2 ) k 2 − ϑ 2ω 2 − c2ϑ 2α j2 − k 2 ( c2 k 2 − ω 2 ) ∆ (α − α 2 j

2 i

)(α

2 j

−α

2 k

)

,

(13c)

i ≠ j ≠ k = 1, 2, 3.

M 1 = Fk 2 − Eω 2 ,

M 2 = ( k 2 − ω 2 )( c2 k 2 − ω 2 ) , M 32 = M 12 − 4c1c2 M 2 .

From Eqs. (13a-c) we see that the root α 32 corresponds to the defect concentration waves, whereas the α12 and α 22 given by Eqns. (13a) and (13b) correspond to the coupled longitudinal (compressional) and transverse (distortional) elastic waves. Also from the expressions for the α12 and

α 22 it is clear that the elastic waves are not influenced by defect concentration variations but are affected due to anisotropy of the medium. If the strain and concentration fields are coupled with other then we see from Eqs. (13) that the roots α12 , α 22 and α 32 of Eq. (11) get modified due to strain-diffusion coupling effects as well as due to anisotropy. This is again in agreement with the corresponding result derived by Singh and Sharma [31] for surface waves in an anisotropic heat conducting anisotropic medium.

In general the roots α i2 ( i = 1, 2, 3) are complex and as we are considering surface waves only, we can assume that Re (α i ) ≥ 0 . Then the general solutions u , w and n are written as 3

u = ∑ Aj exp(−α j z ) exp i ( kx − ωt )  , j =1

(14a) 3

w = ∑ m j Aj exp(−α j z ) exp i ( kx − ωt )  , j =1

(14b) 3

n = ∑ l j Aj exp(−α j z ) exp i ( kx − ωt )  , j =1

(14c)

where

mj = i

c 2 α 2j ϑ + ( c3 − ϑ ) k 2 + ϑω 2 k α 2j ( c1 − c3ϑ ) + c2 k 2 + ω 2 

(c α = −i 2

lj

2 j

αj,

+ ω 2 − k 2 )( c 1α 2j + ω 2 − c2 k 2 ) + c32α 2j k 2 k α 2j ( c1 − c3ϑ ) + c2 k 2 + ω 2 

and Aj , j = 1, 2, 3 are the arbitrary constants.

3. Frequency equation Substituting values of u , w and n from Eqs. (14) into boundary conditions (8a,b) at the surface ( z = 0 ) gives 3

∑ α m c + ϑ l j

j 1

j

− ik ( c3 − c2 )  Aj = 0 ,

j =1 3

∑ α j − ikm j  Aj = 0 , j =1

3

∑α l A j j

j

=0.

(15)

j =1

The condition that non-trivial solutions for Aj , j = 1, 2, 3 of Eqns. (15) exist is d1α 2α 3 (α 22 − α 32 ) s23 + d 3α1α 2 (α12 − α 22 ) s12 = d 2α1α 3 (α12 − α 32 ) s13 ,

(16)

where

{

d j = c1c2ϑα12α 22 − (c3 − c2 )(c2 k 2 − ω 2 )k 2 − α 2j  c1 ( c3 − c2 − ϑ ) k 2 + ϑω 2 

}

+ c1c2ϑ (α12 + α 22 ) − ( c3 − c2 ) ( c1 − c3ϑ ) k 2  , j = 1, 2, 3,

{

sij = ( c1 + c2ϑ − c3ϑ ) (α i2α 2j − α12α 22 ) + ( c3 − c2 − ϑ ) k 2 + (1 + ϑ ) ω 2

}

× (α i2 + α 2j − α12 − α 22 ) , i ≠ j = 1, 2, 3 . This is the required equation characterizing the propagation of Rayleigh waves. From (16) we see that the wave velocity equation contains c and k as only unknown quantities and hence c can be expressed as a function of k in each case indicating the dispersive character of the waves.

If we take c11 = c33 = λ + 2µ , c13 = λ , c44 = µ ,

D1 = D3 = D , ϑ1 = ϑ3 = ϑ we have c1 = 1, c2 = δ 2 , c2 = 1 − δ 2 , ϑ = 1, D = 1, δ 2 = µ ( λ + 2 µ ) . Then the secular Eq. (16) reduces to

(k

2

2

− α12 ) (α 22 + α 32 + α 2α 3 − k 2 + ω 2 ) + 4 k 2α1α 2α 3 (α 2 + α 3 ) = 0 ,

(17)

where α1 , α 2 and α 3 are defined by the following equations:

δ 2 (α 2 − k 2 ) + ω 2 = 0 and

α 4 − ( 2k 2 + τ −1 − ω 2 − iω − ε ) α 2 + ( k 2 − ω 2 )( k 2 + τ −1 − iω ) − ε k 2 = 0 . Equation (17) can be identified as the phase velocity equation for Rayleigh waves in laser-excited isotropic half-space with Lame’s parameters λ and µ , diffusion coefficient D and the deformational potential ϑ , characterizing strain-defect interaction. This equation (non-dimensional) is exactly the same equation which has been obtained and discussed in [1]. 4. Dispersive elasto-diffusive surface waves The obtained dispersion equations are complex and discussion of their solutions in general presents considerable difficulties. This equation can be solved numerically or graphically, but in some cases of interest it can also be solved analytically. Equation (16) has solutions describing qualitatively different types of DS instability: 1) instability of frequencies of surface acoustic waves; 2) instability of amplitudes of acoustic waves; 3) generation of ordered surface (static) DS periodic structures. Below we limit our consideration to the case of only first type solution. The roots of the transcendental Eq. (16) are in general complex and the real part of an appropriate root measures the velocity of surface waves while the imaginary part gives the attenuation due to the defective nature of the medium. It is clear from this equation that the phase velocity depends on the diffusion-strain coupling factor ε. At sufficiently low values of ε , the expressions for l j and sij ( i, j = 1, 2,3 ) that appear in the dispersion Eq. (16), to a first approximation, then read d j = d j + ε d j + Ο ( ε 2 ) ,

(18a)

sij = sij + ε sij + Ο ( ε 2 ) ,

(18b) d j = −α 2j  c1c2α 2j − c1 ( k 2 − ω 2 ) + c3 ( c3 − c2 ) k 2  , j = 1, 2 ,

(18c) d 3 = c1c2α α − α c1c2 (α + α 2 1

2 2

2 3

2 1

2 2

) − c (k

{

1

2

−ω

2

) + c (c 3

d j = −α 2j c1c2ϑ (α12 + α 22 ) + c1 ( c3 − c2 − ϑ ) k 2 + ϑω 2  − ( c3 − c2 ) ( c1 − c3ϑ ) k 2  , j = 1, 2,3,

3

− c2 ) k  , 2

(18d)

} (18e)

{

sij = ( c1 + c2ϑ − c3ϑ ) (α i2α 2j + α 2j αi2 + ( c3 − c2 − ϑ ) k 2 (18f)

+ (1 + ϑ )ω 2 }(αi2 + α 2j ) , i ≠ j = 1.2.3, s31 = (α 32 − α 22 ) {( c2 − c3 ) α12 − k 2 + ω 2 } ,

(18g)

s23 = (α 32 − α12 ) {( c2 − c3 ) α 22 − k 2 + ω 2 } .

(18h)

Then the dispersion Eq. (16) can be rewritten as (on retaining the linear terms in ε ) R (ω , k ) = ε L (ω , k ) , (19)

R ( ω , k ) = α 2 (α − α 2 3

2 2

L (ω , k ) = α 2 (α 22 − α 33

)d s ) ( d s

1 23

− α 1 (α − α

1 23

2 3

2 1

)d s

2 31

,

{

)

+ d1s23 + d1 s23 α 2 (α 22 − α33 )

 α α 2 α 3   + (α 22 − α 33 )  2 23 + 2   + α1 (α 32 − α13 ) d2 s31 + d 2 s31 2α 2    2α 3

(

)

 α α 2 α 3    + d 2 s31 α1 (α 32 − α13 ) + (α 32 − α13 )  1 32 + 1   2α1    2α 3  + d 3 s12

α1α 2 2 α1 − α 22 ) . ( α3

If the defect concentration and strain fields are not coupled with each other, then the coupling constant ε is identically zero ( ε ≡ 0 ). Eq. (19) in this case splits into two equations iω = k 2 + τ −1 (20a) and (20b) α 2 (α 32 − α 22 ) d1s23 − α1 (α 32 − α12 ) d 2 s31 = 0 . Equation (20a) defines the attenuation constant for the defect-concentration waves. Clearly this is influenced by the defect relaxation time ( τ ). Further, if we define v 2 = ω 2 k 2 , then Eq. (20b) after some algebraic manipulations reduces to 2

c1 − ( c3 − c2 )2 − c1v 2  ( c2 − v 2 ) − c1c2 v 4 (1 − v 2 ) = 0 . (21)   This is the form of the well-known Rayleigh wave equation for a transversely isotropic half-space first derived by Stoneley. It reduces in the case of isotropy to the equation giving the velocity of Rayleigh waves. It was showed by Stoneley that the Eq. (21) has only real value for v 2 in the range 0 < v 2 < c2 . Eq. (21) also reveals that the elastic waves will be non-dispersive in character. Thus the left-hand side of Eq. (19) coincides with the Stoneley determinant which, being set equal to zero, determines the dispersion law for the Rayleigh acoustic wave. The right-hand side of Eq. (19) describes the force action ( ∝ ϑ ) of the defects, which deforms the surface and leads to the DS instability of various types. 2 2 In general case ( ε ≠ 0 ), introducing the variable ξ = ( ω k ) = ( Ω v1q ) , Eq. (19) becomes R (ξ ) = L (ξ , ε ) = ε L (ξ ) . (22)

Now we may consider that δ is the increment of the value ξ due to ε ≠ 0 . Then Eq. (22) can be written in the form of

R (ξ R + δ ) = L (ξ R + δ , ε )

(23)

( ξ R is a root of the equation R (ξ R ) = 0 ). Assuming that δ << 1 , we may expand both sides of Eq. (23) into a Taylor series in the vicinity of the point ξ = ξ R . Retaining only the first two terms, we obtain R ( ξ R ) + δ A (ξ R ) = L ( ξ R , ε ) + δ B ( ξ R , ε )

and

δ (ξ R , ε ) =

L (ξ R , ε ) L (ξ R ) ≈ ε = δ r + iδ i , A (ξ R ) − B (ξ R , ε ) A (ξ R ) (24)

where

A = ( ∂R ∂ξ ) ξ =ξ , B = ( ∂L ∂ξ ) ξ =ξ , δ r = Re (δ ) and δ i = Im (δ ) . R

R

The real part of δ characterizes the change of the phase velocity and its imaginary part ( δ i ) defines the attenuation of surface waves. Since 2 ξ = ( Ω v1q ) = ξ R + δ , we have  δ δ  Ω ≈ qvR  1 + r  + iqvR i = Ω r + iΩi , vR = v1 ξ R . 2ξ R  2ξ R  Hence the phase velocity and attenuation coefficient of surface waves are given as  Lr  c = cR  1 − ε  , 2ξ R A  

γ = − qvRε

Li 2 ξR A

,

(25a,b)

where Lr = Re ( L ) > 0 , Li = Im ( L ) > 0 . Since ε > 0 , in this case there is a softening of frequencies of surface acoustic waves (instability of frequencies Ωr → 0 and Ωi < 0 ) and this is related to taking into account the generation of atomic defects. It is necessary to notice, that reduction of frequency occurs not up to zero, and up to value Ωη = η v q 2 2 ρ << v1q ( ηv is the viscosity coefficient). Eq. (25b) describes attenuation of the amplitudes of surface waves. In the case of isotropic solids, expressions (25a,b) transform to results for the Rayleigh surface waves in elastic semi-space solids with atomic defect generation which were obtained and discussed in Refs. [1, 6].

5. Numerical results and discussion The obtained theoretical results indicate that the velocity and attenuation of elasto-diffusive surface waves depend on various material and defect subsystem parameters. Present analytical solutions can be used to find numerically the propagation characteristics of surface waves for a particular material modeled as anisotropic elastic material with atomic defect generation. For purpose of numerical calculations of elasto-diffusive Rayleigh wave velocity and attenuation we consider a single crystal of zinc (Zn) as transversely isotropic elastic solid for which the basic physical properties are c11 = 1.628 × 1011 N/m2, c13 = 0.508 ×1011 N/m2, c44 = 0.385 ×1011 N/m2, c33 = 0.627 × 1011 N/m2. Also we take ρ = 7140kgm−3 , ξ R = 0,1 , τ = 10−7 s and ϑ = 0.89 . The variations of non-dimensional phase velocity C = c cR and non-dimensional attenuation coefficient S = Ωi / Ω0 , Ω0 = 107 s −1 with respect to wave number Q = q / q0 , q0 = 106 m −1 (on a log-linear scale) have been computed and shown in Figs. 1 and 2 by using Eq. 19 for various values of diffusion-strain coupling parameter: ε = 0.2; 0.4; 0.6 . It is observed that, for smaller values of Q , the values of phase velocity increase monotonically, whereas for higher values of Q , the values of phase velocity increase smoothly and finally become dispersionless. From the obtained profiles we observe that the coupling between the elastic displacement and defect-concentration fields affects the phase velocity and attenuation of Rayleigh type surface waves in the transversely isotropic solids. Surface phase velocity decreases with the increase of values of ε (Fig.1). The influence of ε on S is significant in the wave number range 0.1 ≤ Q ≤ 10 and negligible small elsewhere in all cases. It is observed from Fig. 2 that the values of attenuation coefficients for smaller values of wave number (for Q = 0.1 to Q = 1 ) increase monotonically, decrease up to Q = 20 , and then remain constant. The maximum values of attenuation increase with increasing values of ε . For isotropic solids, these results are in agreement with the corresponding results obtained in [4].

Conclusions In the present paper, attempt has been made to investigate the problem of surface diffusive-strain wave propagation in a transversely isotropic solid half-space subjected to action of laser irradiation. The mathematical model of the problem is depicted by a set of partial differential equations of motion and defect-diffusion equation. We have derived secular equations of wave propagation in compact form after obtaining the analytical expressions for various field quantities that govern the wave motion. The complex secular equation has been solved using perturbation method. A combined effect of diffusion-strain coupling and anisotropy on propagation characteristics of waves has been considered. It has been observed that the surface waves are again found to be dispersive in character. The obtained phase velocity and attenuation coefficient of surface waves are found to be modified due to the anisotropy effects and strain-diffusion coupling phenomenon. A softening of frequencies of surface acoustic waves (instability of frequencies) has been obtained. When the defect concentration field and anisotropy effects are neglected, the frequency equations of surface waves have been deduced as particular cases. Finally, in order to illustrate the analytical developments, the secular equations have been solved numerically. The simulated results for a single crystal of Zn in respect of dispersion curves and attenuation coefficient profiles have been presented graphically. Other properties of solutions of the dispersion equation (in particular, instability of amplitudes of acoustic waves, generation of periodic (static) surface diffusion-strain structures) will be presented in future papers. The results for the material system of higher symmetry, such as cubic crystals can be obtained from our consideration as a special case.

The results obtained in this work are expected to be helpful in designing the mechanical behavior in small scale physical devices. The considered elastic waves propagating in laser-excited condensed media carry information about distortions of their form and energy and about the energy losses related to the defect structure; this information is needed for optical-acoustical diagnostics of various parameters and the structure of solids.

Acknowledgments The author wishes to thank reviewers for their valuable suggestions towards the improvement of the paper in its present form. The work was supported in part by Grant Program of the Presidium of RAS no 43-p

References [1] F. Mirzade, Physica Status Solidi (b), 246 (2009) 1597. [2] F. Mirzade, Physica B: Physics of Condensed Matter, 406 (2011) 119, doi:10.1016/j.physb.2010.10.037. [3] F. Mirzade, Physica B: Physics of Condensed Matter, 421 (2013) 28. [4] F. Mirzade, Physica B: Physics of Condensed Matter, 406 (2011) 4644. [5] D. Walgraef, N. M. Ghoniem, J. Lauzeral, Phys. Rev.B, 56 (1997), Article ID15361, 7 pages, [6] F. Mirzade, Advances in Condensed Matter Physics, 2013 (2013), Article ID 528208, 11 pages, doi:10.1155/2013/528208. [7] F. Mirzade, Journal of Nanoscience, 2014 (2014), pp.1-8, Article ID 513010, 8 pages, doi:10.1155/2014/513010. [8] F. Mirzade, Nonlinear strain waves interacting with laser induced carries of the local disorder, in Laser Technologies of Materials Treatment, V. Y. Panchenko, Ed., pp. 220–277, Moscow, Russia, (Fizmatlit, 2009), Russian. [9] F. Mirzade, Journal of Physics: Condensed Matter, 20 (2008), Article ID 275202. [10] F. Mirzade, Journal of Applied Physics, 103 (2008), Article ID 044904. [11] S. Bargmann, A. Favata, ZAMM · Z. Angew. Math. Mech., 94 (2014) 487, doi: 10.1002/zamm.201300116. [12] L. Anand, Int. J. of Solids and Structures, 48 (2011) 962. [13] P. Steinmann, A. McBride, S. Bargmann, and A. Javili, Int. J. Non-Linear Mech., 47 (2012) 215. [14] A. T. McBride, A. Javili, P. Steinmann, and S. Bargmann, J. Mech. Phys. Solids, 59 (2011) 2116. [15] T. Mura, Micromechanics of defects in solids, Dordrecht (Martinus Nijhoff Publishers, 1987). [16] S.Y. Hu, L.Q. Chen, Acta Materialia, 49 (2001) 463. [17] S.Y. Hu, L.Q. Chen, Computational Materials Science, 23 (2002) 270. [18] Y. Wang, J. Li, Acta Materialia, 58 (2010) 1212. [19] M.J.P. Musgrave, Crystal Acoustics, Holden-Day, San Francisco, 1970 [20] E. Dieulesaint and D. Royer, Elastic Waves in Solids, Wiley, New York, 1980. [21] P. Chadwick, L.T.C. Seet, Mathematica, 17 (1970) 255. [22] P. Chadwick, Journal of Thermal Stresses, 2 (1979) 193. [23] Rajneesh Kumar, Rajeev Kumar, Int. J. of Appl. Math. and Mech. 8 (2012) 76. [24] S. Sharma, K. Sharma, R.R. Bhargava, Materials Physics and Mechanics, 16 (2013)144. [25] R. Kumar, R. R. Gupta, Int. Commun. in Heat and Mass Transfer, 37 (2010) 1452. [26] B. Singh, Applied Mathematics and Computation, 217 (2010) 705. [27] S. H. Guo. The Eigen Theory of Waves in Piezoelectric Solids, Acoustic Waves, Don Dissanayake (Ed.), ISBN: 978-953-307-111-4, Sciyo, Croatia, 2010. [28] J.N. Sharma, M. Pal, Journal of Sound and Vibration, 270 (2004) 587.

[29] D.P. Acharya, I. Roy and S. Sengupta, Acta Mechanica, 202 (2009) 35. [30] F. Mirzade Physica B: Physics of Condensed Matter, 461 (2015) 17. [31] H. Singh and J.N. Sharma, J. Acoust. Soc. Am. 77 (1985) 1045.

Figure captions FIG.1. Dispersion curves for different values of diffusion-strain coupling parameter ( ε ) FIG.2. Attenuation coefficient (S) versus wave number (Q) with different diffusion-strain coupling parameters ( ε )

Ph ase vel ocit y, C

1

0.95

ɛ=0.2 ɛ=0.4 ɛ=0.6

0.9 0.1

100

10

1

Wave number, Q

FIG.1. Dispersion curves for different values of diffusion-strain coupling parameter (  )

3 At te nu ati on co eff ici en t, S

ɛ=0.2 ɛ=0.4 ɛ=0.6 2

1

0 0.1

1

10

100

Wave number, Q

FIG.2. Attenuation coefficient (S) versus wave number (Q) with different diffusion-strain coupling parameters (  )