Plane Waves on Rotating Microstretch Elastic Solid with Temperature Dependent Elastic Properties

Transcription

1 Appl. Math. Inf. Sci. 9, No. 6, (1) 963 Applied Mathematics & Information Sciences An International Journal Plane Waves on Rotating Microstretch Elastic Solid with Temperature Dependent Elastic Properties Mohammad I. A. Othman 1,, and Adnan Jahangir 3 1 Faculty of Science, Department of Mathematics, Zagaig University, P.O. Box 19, Zagaig, Egypt. Department of Mathematics, Faculty of Science, Taif University 888, Saudi Arabia 3 Department of Mathematics, COMSATS Institute of Information Technology, Wah Campus, Pakistan Received: 1 Feb. 1, Revised: 1 Apr. 1, Accepted: Apr. 1 Published online: 1 Nov. 1 Abstract: Plane waves propagating in generalied thermoelastic half space rotating with specific angular frequency has been investigated. The appropriate model of the problem in context of Green and Naghdi theory is generated, while modulus of elasticity is taken as a linear function of reference temperature. An exact approach of normal mode analysis method is implemented to obtain the expression for the displacement components, stresses and temperature distribution functions. The variations of the considered variables against the vertical distance are illustrated graphically and a comparison is made for results predicted by both the theories in presence and absence of rotation effect we have also encountered the effect because of temperature involved in elastic parameters. Keywords: Green-Naghdi theory, thermoelasticity, heat dependent elastic constants, microstretch, rotation, normal mode method 1 Introduction The linear theory of elasticity is of paramount importance in the stress analysis of steel, which is the commonest engineering structural material. To a lesser extent, linear elasticity describes the mechanical behavior of the other common solid materials, e.g. concrete, wood and coal. However, the theory does not apply to the behavior of many of the new synthetic materials of the clastomer and polymer type. The linear theory of micropolar elasticity is adequate to represent the behavior of such materials. For ultrasonic waves the influence of the body microstructure becomes significant, this influence of microstructure results in the development of new types of waves is not in the classical theory of elasticity. Metals, polymers, composites, solids, rocks, and concrete are typical media with microstructures. More generally, most of the natural and man-made materials including engineering, geological and biological media possess a microstructure. Eringen and Suhubi[1] and Eringen[]- []developed the linear theory of micropolar elasticity and microstretch elastic solids. This theory of microstretch thermoelasticity is the generaliation of the theory of micropolar elasticity and a special case of the micromorphic theory. The material points of microstretch elastic solids can stretch and contract independently of their translations and rotations. The basic results in the theory of micro stretch elastic solids were obtained in the literature[]-[8]. The theory of thermo-micro-stretch elastic solids was introduced by Eringen[9]. In the framework of this theory Eringen established a uniqueness theorem for the mixed initial-boundary value problem. The theory was illustrated through the solution of one-dimensional waves and compared with lattice dynamical results. The asymptotic behavior of solutions and an existence result were presented by Bofill and Quintanilla[1]. A reciprocal theorem and a representation of Galerkin type were presented by De Cicco and Nappa[11]. De Cicco and Nappa[1] extended a linear theory of thermo-microstretch elastic solids that permits the transmission of heat as thermal waves at finite speed. The theory is based on the entropy production inequality proposed by Green and Laws[13]. The basic results and an extensive review of the theory of thermo-microstretch elastic solids can be found in the book of Eringen[]. Most of the investigations were done under the assumption of temperature-independent material properties, which limit the applicability of the solutions obtained to certain ranges of temperature. Modern Corresponding author m i a othman@yahoo.com

2 96 M. Othman, A. Jahangir: Plane Waves on Rotating Microstretch Elastic Solid... structural elements are often subjected to temperature change of such magnitude that their material properties may be longer be regarded as having constant values even in an approximate sense. At high temperature the material characteristics such as modulus of elasticity, thermal conductivity and the coefficient of linear thermal expansion are no longer constants[1]. The thermal and mechanical properties of the materials vary with temperature, so the temperature-dependent on the material properties must be taken into consideration in the thermal stress analysis of these elements. Othman and Kumar[1] investigated the dependence of modulus of elasticity on reference temperature in generalied magneto-thermo-elasticity and obtained interesting results. Youssef[16] used the equation of generalied thermoelasticity with one relaxation time with variable modulus of elasticity and the thermal conductivity to solve a problem of an infinite material with spherical cavity. Green and Naghdi[17]-[19] proposed another three models, which are subsequently referred to as GN-I, II and III models. The linearied version of model-i corresponds to the classical thermoelastic model-ii the internal rate of production of entropy is taken to be identically ero implying no dissipation of thermal energy. This model admits un-damped thermoelastic waves in a thermoelastic material and is best known as the theory of thermoelasticity without energy dissipation. Model-III includes the previous two models as special cases, and admits dissipation of energy in general. Othman [] constructed a model of the two-dimensional equations of generalied thermoelasticity with two relaxation times in an isotropic elastic medium with the modulus of elasticity being dependent on the reference temperature. Lotfy and Othman [1] studied the effect of rotation on generalied thermo-microstretch elastic medium under different theories. Recently some authors discussed different types of problems in generalied thermo-microstretch elastic medium (Marin[]-[3] Baljeet[], Kumar[]-[6] Othman et al. [7]-[8], Ellahi et al. [9]-[33]). This paper presents an attempt to examine the temperature dependency of elastic modulus and rotation on the behavior of two-dimensional solution in a generalied thermo-microstretch elastic medium. We have also encountered the fascinating theories of generalied thermoelasticity presented by Green and Naghdi. We have adopted normal mode analysis method and find the expressions for field variables. Formulation of the Problem We obtain the constitutive and the field equations for a linear isotropic generalied thermo-microstretch elastic solid in the absence of body forces. We use a rectangular coordinate system(x,y,) having originated on the surface y = and -axis pointing vertically into the medium. The thermoelastic plate is rotating uniformly with an angular frequency Ω = Ω n, where n is a unit vector representing the direction of the axis of rotation. The basic governing equations of linear generalied thermoelasticity with rotation in the absence of body forces and heat sources are σ ji, j = ρ[ü+ Ω (Ω u)+ω u] i, (1) ε i jr σ jr + m ji, j = jρ φ i t, () α φ 1 3 λ 1φ 1 3 λ (.u)+ 1 3 ˆγ 1T = 3 ρ j φ K T + K Ṫ = ρ C E T + ˆγ T ü i,i + ˆγ 1 T φ t, (3) t, () σ il =(λ φ + λ u r,r )δ il +( µ+ k)u l,i + µ u i,l k ε ilr φ r ˆγ T δ il, () m il = α φ r,r δ il + β φ i,l + γ φ 1,i, (6) λ i = α φ,i, (7) e= u x + w Where T is the temperature above the reference temperaturet chosen so that (T T ) / T << 1,λ, µ are the counterparts of Lame parameters, the components of displacement vectoru areu i,tis the time, σ i j are the components of stress tensor, eis the dilatation, e i j are the components of strain tensor, j the micro inertia moment,φ is the scalar microstretch,φ is the rotation vector m i j is the couple stress tensor, δ i j is the Kronecker delta, ε i jr is the alternate tensor, the mass density isρ,the specific heat at constant strain isc E,, the thermal conductivity is K and Kmaterial characteristic of the theory. The state of plane strain parallel to the xy-plane is defined by (8) u 1 = u(x,,t),u =,u 3 = w(x,,t),ϕ 1 = ϕ 3 =, (9) φ = φ (x,,t),φ = φ (x,,t), Ω =(,Ω,). We assume that λ = λ f (T), µ = µ f (T),γ = γ f (T),λ = λ f (T), k=k f (T),β = β f (T),α = α 1 f (T), ˆγ = ˆγ f(t), α = α f (T),λ 1 = λ 3 f (T), ˆγ 1 = ˆγ 1 f (T). Where f(t) is a non-dimensional function depending on temperature, during the case of temperature independent

3 Appl. Math. Inf. Sci. 9, No. 6, (1) / 96 modulus of elasticity, f(t) = 1 Equations (1)-(9) will become, ρ[ü+ Ω (Ω u)+ω u] i = f(t)[λ ϕ,i +(λ + µ )u j, ji +(µ + k )u i, j j + k ε i jl ϕ l, j γ T,i ] + f, j [λ ϕ δ i j + λ u r,r δ i j + µ u j,i +(µ + k )u i, j + k ε i jl ϕ l γ T δ i j ] (1) f(t)[γ φ, j j k φ + k (u 1,3 u 3,1 )]+γ f, j φ, j = jρ φ, (11) f (T)(α φ, j j 1 3 λ 3φ 1 3 λ u j, j γ 1 T) = 3 ρ j φ, (1) K T, j j + K Ṫ, j j = ρ C E T + f(t)( ˆγ T ë + ˆγ 1T φ ). (13) Where ˆγ = (3λ + µ + k)α t1, ˆγ 1 = (3λ + µ + k)α t and, = x + (1) The constants ˆγ and ˆγ 1 depend on the mechanical as well as the thermal properties of the body and the dot denote the partial derivative with respect to time, α t1, α t are the coefficients of linear thermal expansions. For convenience, the following non-dimensional variables are used: x i = ω c x i, ū i = ρc ω γ T u i, t = ω t, T = T T, σ i j = σ i j γ T, m i j = ω c γ T m i j, φ = ρc γ T φ, λ 3 = ω c γ T λ 3, φ = ρ c γ T φ, ω = ρ C Ec K, Ω = Ω ω, c = µ ρ. (1) Using (1), Eqs. (1)-(13) become (dropping the bar for convenience) [ü+ Ω (Ω u)+ω u] i = f(t) c [λ ϕ,i +λ + µ u j, ji +(µ + k )u i, j j ]+k ε i jl ϕ l, j ρc T,i ] + f, j ρc [λ ϕ δ i j + λ u r,r δ i j + µ u j,i +(µ + k )u i, j + k ε i jl ϕ l ρc Tδ i j ] (16) f(t) jρc [γ φ, j j k c ω φ + k c ω (u 1,3 u 3,1 )] f(t)[ α c + γ jρc f, j φ, j = φ, (17) φ, j j 1 λ 3 3 ω φ 1 λ 3 ω u j, j+ 1 γ1 ρc 3 γ ω T]= 3 ρ j φ, (18) ε ( T x + T ) + ε 3 ( Ṫ x + Ṫ ) Where φ = T + f(t)(ε 1 ë+ ε ). (19) t ε 1 = γ T (1 β T ) ρ C E c, ε = K ρc E c, ε 3 = Kω ρc E c and ε = ˆγ γ1 T (1 β T ) ρ C E ω c. We have taken the special case of (T T ) / T << 1i.e., infinitesimal temperature deviation from reference temperature are considered. Therefore f(t)can be taken in the form f (T) = 1 β T, whereβ is an empirical material constant. Also introducing the dimension less scalar potential functionsφ(x,, t)and ψ(x,, t)defined by the relations, u= R x + ψ, w= R ψ x Equations (16) in component form will become, ( + a Ω a ( + a Ω a t )R+Ωa t )ψ Ωa ( a a (a 6 a 7 () ψ t + a 1ϕ a T = (1) R t a 3ϕ = () t )φ + a ψ =, (3) t )φ a 8 R+a 9 T =, () [(ε + ε 3 t ) t ] T = ε 1 ë+ε φ t. ()

4 966 M. Othman, A. Jahangir: Plane Waves on Rotating Microstretch Elastic Solid... Where, c 1 = (λ + µ + k ) ρ c = λ 3(1 β T ) 9ρ j a = a = c c 1 (1 β T ),a = c c 1, c 3 = α (1 β T ), 3ρ j, ;c = λ (1 β T ), 9ρ j,a 1 = ρ c (µ + k )(1 β T ), a 3 = λ λ + µ + k, k (µ + k ) a = k c γ ω, ρ jc a = γ (1 β T ), a 6 = c 3 c, a 7 = c ω, a 8 = c ω and a 9 = ˆγ 1c (1 β T ) 9 ˆγ jω. 3 The Solution of the Problem The solution of the considering physical variables can be decomposed in terms of normal mode as given in the following form: [R, ψ, φ,φ,σ il,m il,t,λ ](x,, t) = [ R, ψ, φ, φ, σ il, m il, T, λ ] () exp(ω t+ i bx) (6) Where[ R, ψ, φ, ϕ, σ il, m il, T, λ ]()the amplitude of the functions, ωis a complex and b is the wave number in thexdirection. Using (6), then (1)-() become (D A 1 ) R a T + a 1 φ + A ψ =, (7) (D A 3 ) ψ a 3 φ A R=, (8) (D A ) φ + a (D b ) ψ =, (9) (a 6 D A 6 ) φ a 8 (D b ) R+a 9 T =, (3) [ε(d b ) ω ] T ε 1 ω (D b ) R ε ω φ =. (31) Where D= d d, A 1 = b + a (ω Ω ), A = Ω a ω, A 3 = b + a (ω Ω ), A = Ω a ω, A = b + a + a ω, A 6 = b a 6 + a 7 + ω,ε =(ε +ε 3 ω). Eliminating, φ, ψ, R, T and φ in (7)-(31), we get the following tenth order ordinary differential equation [D 1 AD 8 +BD 6 CD +ED F]{ φ, ψ, R, T, φ }()= (3) Equation (3) can be factored as ( D k 1) (D k )( D k 3)( D k )( D k ) { φ, ψ, R, T, φ }()=. (33) Where, A= g 18 /g 17, B=g 19 /g 17, C=g /g 17, E = g 1 /g 17, F = g /g 17,g 1 = ε ω a 1 ε 1 ω, g = ε ωa 1 + a 1 ε 1 ω b, g 3 = a 1 (εb + ω )+a ε ω, g = A ε ω, g = A 3 + A a 3 a, g 6 = A 3 A a 3 a b, g 7 = g g 1 g,g 8 = g g + g 1 g 6 + g A,g 9 = g g 6 g A A, g 1 = g 3 a 1 εg, g 11 = g 3 g + a 1 εg 6, g 1 = g 3 g 6, g 13 = a 6 (εb + ω )+A 6 ε, g 1 = ε a 9 ω+ A 6 (εb + ω ), g 1 = ε 1 a 6 ω b + A 6 ε 1 ω a 8 ε ω, g 16 = ε a 8 ω b ε 1 ω A 6 b, g 17 = ε(a 6 g 1 + a 1 ε 1 ω a 6 ), g 18 = a 6 (εg 7 + ε 1 ω g 1 )+g 13 g 1 + a 1 ε g 1, g 19 = a 6 (εg 8 + ε 1 ω g 11 ) g 13 g 7 + g 1 g 1 a 1 εg 16 g 1 g 1, g = a 6 εg 9 + g 13 g 8 g 1 g 7 + g 1 g 16 + g 11 g 1 + g 1 a 6 ε 1 ω, g 1 = g 13 g 9 + g 1 g 8 g 11 g 16 + g 1 g 1. g = g 1 g 9 g 1 g 16. The solution of (33), has the form T = ψ = R= φ = φ = M n e k n, (3) H 1n M n e k n, (3) H n M n e k n, (36) H 3n M n e k n, (37) H n M n e k n. (38) WhereM n are some parameters andk n, (n = 1,,3,,) are the roots of the characteristic equation of (33). Here, H 1n =[k n a 6ε 1 ω k n g 1 g 16 ] / [k n a 6ε k n g 13+ g 1 ], H n =[A (k n A )] / [k n k n g + g 6 ], H 3n =[ a A (k n b )] / [k n k n g + g 6 ], H n =[a 8 (k n b ) a 9 H 1n ] / [a 6 k n A 6 ]. The Boundary Conditions The plane boundary subjects to an instantaneous normal point force and the boundary surface is isothermal, the boundary conditions at = are

5 Appl. Math. Inf. Sci. 9, No. 6, (1) / The mechanical boundary condition is that the surface of the half-space obeys, σ xx = p(x,t),σ = σ x = λ =. (39).The thermal boundary condition is that the surface of the half-space is subjects to a thermal shock, T = f(x,t). () We obtain the non-dimensional expressions for the displacement components, force stress, coupled stress and temperature distribution of the microstretch generalied thermoelastic medium as follows, where a 1 = λ ρ c ū= w= λ = σ xx = σ = σ x = σ x = (ib k n H n )M n e k n ( k n ibh n )M n e k n (1) () H n M n e k n, (3) H 6n M n e k n, () H 7n M n e k n, () H 8n M n e k n, (6) a 1 k n H n M n e k n. (7), a 11 = c 1 c, a 1 = λ ρ c a 1 = k ρ c, a 1 = α ω ρ c, H n = (1 β T )[a 1 H n + iba 11 (ib k n H n ) +k n a 1 (k n + ibh n ) H 1n ], H 6n = (1 β T )[a 1 H n + k n a 11 (k n + ibh n ) +iba 1 (ib k n H n ) H 1n ],, a 13 = (µ + k ) ρ c, H 7n = (1 β T )[ k n (ib k n H n ) a 13 ib(k n + ibh n ) +a 1 H 3n ], H 8n = (1 β T )[ ib(k n + ibh n ) k n a 13 (ib k n H n ) +a 1 H 3n ]. Applying the boundary conditions (39) and () at the surface = of the plate, we obtain a system of five equations. After applying the inverse of matrix method, M 1 M M 3 M M = H H H 3 H H H 61 H 6 H 63 H 6 H 6 H 71 H 7 H 73 H 7 H 7 k 1 H 1 k H k 3 H 3 k H k H H 11 H 1 H 13 H 1 H 1 1 p f (8) We obtain the values of the five constantsm n,n = 1,,3,,. Hence, we obtain the expressions for the displacements, the force stress, the coupled stress and the temperature distribution of the microstretch generalied thermoelastic medium. Particular Cases Case 1: The corresponding equations for the generalied micropolar thermoelasticity elastic medium without stretch can be obtained from the above mentioned cases by taking: α = λ = λ 1 = ϕ = (9) After substituting (9) in (1)-(7) and using (), (8) and (3) we get (D A 1 ) R+ A ψ a T =, () (D A 3 ) ψ A R a 3 φ =, (1) [D A ] φ + a (D b ) ψ =, () [ε(d b ) ω ] T = ε 1 ω (D b ) R. (3) Eliminating φ, ψ, Rand T in ()-(3), we get the following eight order ordinary differential equations for φ, ψ, R and T. [D 8 AD 6 + BD C D + E]{ φ, ψ, R, T}() = () Equation () can be factored as ( D k 1 )(D k )( D k 3 )( D k ){ φ, ψ, R, T}()=. () Where A=g 6/ ε, B=g 7 / ε, C= g 8 / ε, E = g 9 / ε, g 1 = a ε 1 ω + εb + ω + εa 1,g = a ε 1 ω b +εb A 1 + ω A 1,g 3 = εb + ω,g = A 3+ A a 3 a, g = A 3A a 3 a b, g 6 = εg + g 1, g 7 = εg + g 1 g + g + A A ε, g 8 = g 1g + g g + A A (g 3+ εa ), g 9 = g g + A A A g 3

6 968 M. Othman, A. Jahangir: Plane Waves on Rotating Microstretch Elastic Solid... The solution of (6) has the form T = ψ = R= φ = Z n e k n, (6) H 1nZ n e k n, (7) H nz n e kn, (8) H 3nZ n e k n. (9) Where n are some parameters and k n, (n = 1,,3,) are the roots of the characteristic equation of () and H 1n =[ε 1 ω (k n b )] / [ε(k n b ) ω ], H n =[A (k n A )] / [k n k n g + g ], H 3n =[ a A (k n b )] / [k n k n g + g ]. This gives the required expressions for the displacement components, force stress, coupled stress and temperature distribution of the medium as follows Where ū= w= σ xx = σ = σ x = σ x = (ib k n H n )Z ne k n, (6) ( k n ibh n )Z ne k n, (61) H n Z ne k n, (6) H n Z ne kn, (63) H 6n Z ne k n, (6) H 7nZ n e k n. (6) H n = iba 11 ( ib kn H n) + kn a 1 ( kn + ibh n) H 1n, H n = k na 11 ( kn + ibh n) + iba1 ( ib kn H n) H 1n, H 6n = k n(ib k n H n) a 13 ib ( k n + ibh n) + a1 H 3n, H 7n = ib(k n+ ibh n ) k na 13 ( ib kn H n) + a1 H 3n. Applying the boundary conditions σ xx = p(x,t), σ = σ x =, T = f(x,t) at the surface = of the plate, we obtain a system of four equations. After applying the inverse of matrix method, Z 1 Z H 1 H H 3 H H 1 H H 3 H Z = 3 H 61 H 6 H 63 H 6 Z H 11 H 1 H 13 H 1 p (66) f 1 We obtain the values of the four constants Z n,,,3,. Case : The corresponding equations for the displacement, stresses and temperature distribution functions are obtained for generalied thermoelastic medium with stretch can be derived by using k=α = β = γ = in (3)-(38). Case 3: The equations for displacement, stresses and temperature distribution function for generalied thermo microstretch elastic medium without rotating medium will be obtained by assumeω. T GN II (Ω=) GN II (Ω=1) GN III (Ω=) GN III (Ω=1) Fig. 1: Temperature distribution against for rotational frequency 6 Numerical Results and Discussions In order to illustrate our theoretical results obtained in the preceding section and to compare theories of thermoelasticity, we now present some numerical results. In the calculation, we take a magnesium crystal, the micropolar parameters are taken as Othman and Lotfy[8], thermal characteristic as Prafitt and Eringen [3]and stretch parameters as Lotfy and Othman. As the material subjected to mechanical and thermal disturbances,

7 Appl. Math. Inf. Sci. 9, No. 6, (1) / GN II (Ω=) GN II (Ω=1) GN III (Ω=) GN III (Ω=1).8.6. GN II (Ω=) GN II (Ω=1) GN III (Ω=) GN III (Ω=1) σ 1.. u GN II (Ω=) GN II (Ω=1) GN III (Ω=) GN III (Ω=1) Fig. 3: Displacement distribution against for rotational frequency σ xx λ. GN II (Ω=) GN II (Ω=1) GN III (Ω=) GN III (Ω=1) Fig. : Normal Stress distribution against for rotational frequency sinceωis a complex constant,we take ω = ω + iζwith ω = 1and ζ = 1.The physical constants used are ρ = kmm 3, j = m, λ = N m, T = 98K, t =.1s, µ =. 1 1 Nm,k = Nm, γ = N,α t1 =. 1 3 K 1, f =., α t =. 1 3 K 1, K = K = Jm 1 s 1 K 1, λ = Nm, C E = J kg 1 K 1 λ 3 =.7 11 Nm, α = N, ε 1 = 1.8, ε = 1.7, ε 3 = 1, p=.. The change in amplitudes of field variables against a vertical component of distance in the context of Green and Naghdi theory of both types II and III for generalied thermo-micro-stretch medium is represented graphically. Figs. 1- show the variation in field variables by the angular frequency in these figures we have adjusted reference temperature by fixingβ at.16, while Figs. -8 represent the variation in field variables for different Fig. : Microstress distribution against for rotational frequency value of two temperature parameter. In these figures solid lines and dashed lines for GN-II, dashed with dot and dotted lines are for GN-III at Ω = and Ω = 1respectively, it is more precisely explained in each figure. Fig. 1 explains the amplitude of temperature distribution against a vertical component of distance. In this figure we have considered the effect of rotational frequency on T.It is observed that rotational frequency increases the amplitude oft for GN-II, but for GN-III the effect of rotation is negligible in the present selected parameters. Both the curves converge to ero for sufficient large values of vertical distance. Fig. depicts the stress distribution functions against vertical distance components. In both the components of stress σ xx and σ rotation is having an increasing effect on both the theories of GN. Maximum amplitude forσ xx is obtained for with energy dissipation and in the presence of rotational frequency but for σ maximum amplitude of the stress

8 97 M. Othman, A. Jahangir: Plane Waves on Rotating Microstretch Elastic Solid T σ Fig. : Temperature distribution against for different values of β =.16,., distribution function is obtained from without energy dissipation. Fig. 3 shows the distribution of the horiontal component of displacement becomes oscillating in the presence of rotational frequency. It can also be seen that in the presence of rotational frequency curves of displacement distribution function for GN-III are higher than the curves for GN-II. All curves converge to ero but the curves without rotation converge to ero faster. In the graphs of microstretch distribution function rotational frequency reduces the micro-stress component for GN-II but in the presence of energy dissipation rotation have increasing effect. Fig. depicts the distribution of micro-stress, it can be seen that in the presence of rotational frequency, curves of micro-stress distribution function for GN-III are higher than the curves for the GN-II. All curves converge to ero but the curves without rotation converge to ero faster. Figs. -8 are representing the behavior of field variables for different value of empirical constant. In these sets of figures we have fixed the medium on rotation with angular frequency Ω = 1. The figures contain the curves with following presentations - is for GN-IIβ =.16;...is for GN-IIβ =.; is for GN-IIβ =.1; is for GN-IIIβ =.16; * * *is for GN-III β =. o o o is for GN-III β =.1 Fig. depicts the variations in temperature distribution function for different values of empiric material constant. For GN-II and GN-III amplitude of T is directly proportional to that of β i.e., by increasing the intensity of β amplitude of the temperature distribution function also increases. It can also be seen that starting point of temperature T for each value of β is same for both theories of the GN with curves of GN-II while, having great amplitude as compared to those of GN-III. It indicates that energy dissipation is having decreasing effect on temperature distribution. All curves converge to 3 σ xx Fig. 6: Stresses distribution functions against for different values ofβ =.16,.,.1 ero as the vertical distance from the surface increases satisfying the condition of surface waves. The variation in field variables σ xx,σ xx,u,w and λ is represented in Fig. 6,7 and 8.It can be seen clearly from these figures that curve for each variable increases while increasing the value of intrinsic material constantβ For both components of the normal stresses and displacement distribution function maximum amplitude of stress distribution functions for are obtained under GN-III for β =.but for micro-stress distribution function maximum amplitude is obtained under GN-II for β =.1 7 Conclusion In this article the effect of rotational frequency on plane waves in a generalied thermo-microstretch elastic media is studied in addition we have also encountered the influence of reference temperature on field variables. By analysing the graphical behaviour we have concluded the following important pionts

9 Appl. Math. Inf. Sci. 9, No. 6, (1) / u 1 8 λ Fig. 8: Microstress distribution against for different values ofβ =.16,.,.1 3 w Fig. 7: Displacement distribution function against for different values ofβ =.16,.,.1 1- In curves of temperature distribution function it is being observed that the amplitude for obtained by GN-II is less than the amplitude obtained from the GN- III in the absence of rotational frequency, while in the presence of rotational effect behavior of temperature distribution changes. Rotation of medium plays a significant role in the propagation of the wave through the medium. - Rotation is having an increasing effect on the distribution function of each field Variable accept micro-stress distribution function in which it has small decreasing effect. 3- By increasing the intensity ofβ effect of energy dissipation also increases on field variables. Hence we can say that reference temperature has very significant impact on wave propagation through the medium.. Greater the value ofβ greater the amplitude of normal stress distribution functions and micro-stress distribution function against. It also increases the exponential behavior of stress propagating through the medium. - All the curves converges to ero as distance from surface of medium increases, this satisfies the condition for surface wave propagation. References [1] Eringen, A.C. and Suhubi, E.S., Non linear theory of simple micropolar solids. Int. J. Eng. Sci.,, 1-18 (196). [] Eringen, A.C., Linear theory of micropolar elasticity. J. Math Mech., 1, (1966). [3] Eringen, A.C., Theory of micropolar elasticity. In: Liebowit H, editor. Fracture, vol. II, Academic Press. p (1968). [] Eringen, A.C., Micropolar elastic solids with stretch. Ari Kitabevi Matbassi, Istanbul,, 1-18 (1971). [] Eringen, A.C., Microcontinuum field theories I: foundation and solids. New York, Berlin, Heidelberg: Springer-Verlag; (1999). [6] Iesau, D. and Nappa L., On the plane strain of microstretch elastic solids. Int. J. Eng. Sci., 39, (1). [7] Iesau, D. and Pompei, A., On the equilibrium theory of microstretch elastic solids. Int. J. Eng. Sci., 33, 3991 (199). [8] De Cicco. S., Stress concentration effects in microstretch elastic bodies. Int. J. Eng. Sci, 1, (3). [9] Eringen, A.C., Theory of thermo-microstretch elastic solids. Int. J. Eng. Sci., 8, (199). [1] Bofill, F. and Quintanilla, R., Some qualitative results for the linear theory of thermo-microstretch-elastic solids. Int. J. Eng. Sci., 33, 11-1 (199). [11] De Cicco, S. and Nappa, L., On the theory of thermomicrostretch elastic solids. J.Thermal Stresses,, 6-8 (1999). [1] De Cicco, S. and Nappa, L., Some results in the linear theory of thermo-micro-stretch elastic solids. J. Math. Mech.,, 67-8 (). [13] Green, A.E. and Laws, N., On the entropy production inequality. Arch Ration Mech. Anal., 7-9 (197). [1] Lomakin, A., The Theory of Elasticity of Non- Homogeneous Bodies. Ncew, (1976).

10 97 M. Othman, A. Jahangir: Plane Waves on Rotating Microstretch Elastic Solid... [1] Othman, M.I.A. and Kumar, R., Reflection of magnetothermoelasticity waves with temperature dependent properties in generalied thermoelasticity, Int. Comm. In Heat and Mass Transfer., 36, (), 13- (9). [16] Youssef, H.M., Dependence of modulus of elasticity and thermal conductivity on reference temperature in generalied thermoelasticity for an infinite material with a spherical cavity, Applied Mathematics and Mechanics, 6, (), 7-7 (). [17] Green, A.E. and Naghdi, P.M., A re-examination of the basic postulate of thermo-mechanics. Proc. Roy. Soc. London.,3, (1991). [18] Green, A.E. and Naghdi, P.M., On undamped heat waves in an elastic solid. J. Thermal stresses, 1, 3-6 (199). [19] Green, A.E. and Naghdi, P.M., Thermoelasticity without energy dissipation, J. Elasticity, 31, (1993). [] Othman, M.I.A., State-space approach to generalied thermoelasticity plane waves with two relaxation times under dependence of the modulus of elasticity on the reference temperature. Can. J. Phys., 81, (3). [1] Lotfy, Kh. and Othman, M.I.A., The effect of rotation on plane waves in generalied thermo-micro-stretch elastic solid with one relaxation time for a mode-i crack problem. Chin. Phys. B.,, 1-13 (11). [] Marin, M., Modeling a microstretch thermoelastic body with two temperatures. Abstract and Applied Analysis, 7, (13). [3] Marin, M., A domain of influence theorem for microstretch elastic materials. Nonlinear Analysis: RWA, 11: () 36-3 (1). [] Singh, B., Reflection of plane waves from free surface of a microstretch elastic solid. Proc. Indian Acad. Sci., 111, 9-37 (). [] Kumar, R., Garg S.K., and Ahuja S., Propagation of plane waves at the interface of an elastic solid half space and a microstretch thermoelastic diffusion solid half space. Latin American J. Solids and Structures, 1, (13). [6] D. Ieana, R. Quintanillab, On a Theory of Thermoelastic Materials with a Double Porosity Structure, Journal of Thermal Stresses, 37, (1) [7] Othman, M.I.A., Sarhan, Y.A., Jahangir, A. and Khan, A., Generalied magneto- thermo-microstretch elastic solid under gravitational effect with energy dissipation, Multidisciplinary Modeling in Materials and Structures., 9, (13). [8] Othman, M.I.A., Lotfy, Kh. and Farouk, R.M., Generalied thermo-microstretch elastic medium with temperature dependent properties for different theories, Eng. Ana. with Boundary Elements, 3, 9-37 (1). [9] Hayat, T., Mumta, S., and Ellahi, R., MHD unsteady flows due to non-coaxial rotations of a disk and a fluid at infinity. Acta Mech. Sinica, 19, 3- (3). [3] Hayat, T., R. Ellahi, Asghar, S., and Siddiqui, A.M., Flow induced by non-coaxial rotation of a porous disk executing non-torsional oscillating and second grade fluid rotating at infinity, Applied Mathematical Modelling, 8, 91-6 (). [31] Hayat, T., Ellahi, R., and Asghar, S., Unsteady magnetohydrodynamic non-newtonian flow due to noncoaxial rotations of a disk and a fluid at infinity. Chemical Engineering Communications, 19, 37-9 (7). [3] Ellahi, R. and Asghar, S., Couette flow of a Burgers fluid with rotation, Int. J.Fluid Mech. Research, 3, 8-61 (7). [33] Hayat, T., Ellahi, R., and Asghar, S., Hall effects on unsteady flow due to noncoaxially rotating disk and a fluid at infinity. Chemical Eng. Comm., 19, (8). [3] Prafitt, V.R. and Eringen, A.C., Reflection of plane waves from the flat boundary of a micropolar elastic half-space. J. Acoust. Soc. Am.,, (1969). Adnan Jahangir Did PhD from CIIT, Islamabad Pakistan and working as Assistant Professor in Wah Campus of CIIT. Research Intereset: Thermoelasticity; Microstretch Solids; Solid Mechanics. Author of about 1 published articles in well reputed journals. M. I. A. Othman Research interests: Finite element method; Fluid mechanics; Thermoelasticity; Magneto-thermoelasticity; Thermoelastic diffusion; Fiber-reinforced; Thermo-viscoelasticity; Heat and mass transfer; Thermoelasticity with voids; Micropolar thermoelasticity; 1- Member in the American Mathematical Society; - A member of the Standing Committee for the Promotion of Professors and Assistant Professors 13-1; 3- Member of the Egyptian Mathematical Society; - Considered for inclusion in World Marquis Who s Who in the World, 1 (9th Edition); - Considered for inclusion in Who s Who in the Thermal-Fluid, 11; 6- Considered for inclusion in the Encyclopedia of Thermal Stresses 11; 7- An Associated Editor of ISRN Applied Mathematics; 8 Member of Editorial Board of Journal of Thermoelasticity; 9- Reviewer for the 6 International Journals; 1- Have about 18 published papers in defined fields.