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    15 16 15 16 Document Transcript

    • 15SHODH, SAMIKSHA AUR MULYANKAN International Indexed & Refereed Research Journal, ISSN 0974-2832, (Print), E- ISSN- 2320-5474, Aug-Oct, 2013 ( Combind ) VOL –V * ISSUE – 55-57 Introduction In this paper the problem of shear waves in linearnon-homogeneousviscoelasticinfiniteandfinite plates have been discussed. Laplace transfor and the WKBJ-method are applied to solve the problem. The cases of the non-homogeneity have been considered and their effects on the wave speed and propagation of the wave front are analysed. The numerical results hae been obtained for the instaneous shearing traction, applied uniformely at the boundary of the infinite plate and the outer boundary of the finite plate. An approximatesolutionoftheviscoelasticwaveproblem has been obtained by the application of the Rayleigh- Ritz and Laplace transform. A quantitative of the approximate solutions for the displacements and stresses between the non-homogeneous viscoelastic plates at various times have even made. 2. Formulation of The Problemn and Basic Equations. We use cylindrical polar coordinates (r, θ, z) and taketheorigin atthecentreoftheinfiniteplate.Let the plate under consideration occupy th eregion r0 < r < ∞, where r0 is the radius of the circular opening in the plate, and let t be the time. For radially are σrθ (r,t) ≡ σ(r,t),erθ (r,t) ≡ e(r,t)anduθ(r,t) ≡ u(r,t)respectively.. Sincethebehaviouroftheviscoelasticmaterial conforms to the simple Biot's model with a continuous spectrum of relaxation times,the constitutiveequation is: (2.1) whereµistheusualLeme'scoefficient,takenasafunction oftheradialdistancerfromthecentreofthecylindrical opening and the time t. The equation of motion and the strain - displacement relation for this case are ∂σ ∂ σ ρ ∂ ∂r r u t + = 2 2 2 , (2.2) e u r u r = −       1 2 ∂ ∂ , (2.3) whereρistheinitialdensitytakenasconstant. Multiplyingbothsidesof(2.1),(2.2)and(2.3)byexp(- pt) and integrating from θ to ∞ with respect to t, we obtain the transformed equations: ( )σ µ σ σ ρ = + = = −                2 2 1 2 2 p r p e d dr r p u e du dr u r , , , , (2.4) where use has been made of the quienscent initial conditions ( )u r o u t o t o , .=       = = ∂ ∂ (2.5) Research Paper -Mathematics Aug- Oct , 2013 Wave Propagation In Non Homnogeneous Viscoelastic Plates. * Dr. Hardeep Singh Teja * Associate Professor Govt. Mohindra College, Patala. ( ) ( )σ µ τ ∂ ∂τ τr t r t e d O t , , ,= −∫2 Eliminating e and u from the set of equations (2.4), we get ( ) ( ) d dr P r d dr Q r p 2 2 0 σ σ σ+ =, (2.6) where ( ) ( ) ( ) p r and Q r p r p r p = = − − 1 2 4 2 , , ρ µ (2.7) To remove the first order derivative in equation (2.6) we write ( ) ( )σ φr p r r p, ,/ = −1 2 (2.8) and obtain ( )φ φ, , ,rr g r p o+ = (2.9)
    • 16 International Indexed & Refereed Research Journal, ISSN 0974-2832, (Print), E- ISSN- 2320-5474, Aug-Oct, 2013 ( Combind ) VOL –V * ISSUE – 55-57 R E F E R E N C E 1. Bland, D.R. Theory of linear viscoelasticity Perganon Press, Ine. New York, 1960. 2. Christaneous, R.M.Theory of viscoelasticity, an introduction, academic press: New York and London 1971. 3. Kolexy, M. Stress waves in solids (Dower, Mxroyon 1963). 4. Moodie T.R. On the propagation, reflection and transmission transient cylindrical shear waves in non-homogeneous viscoelastic media. Bill Austral, Math Soc - 8 where g (r,p) = Q − − 1 2 1 4 2dP dr P (2.10) Nowthestressfieldhastobedeterminedfrom the equations (2.8) and (2.9) under the prescribed boundaryconditionandfortheparticularformof µ (r,t). Solution By Wkbj - Method We observe that equation (2.9) is in a form to which the WKBJ - method is applicable. Using this method we see that an approximate solution is given by φ = ±         − ∫c g Exp g dr r r o 1 1 4 1 2/ / (3.1) g (r,p) > o and | ( )d dr g− <<1 2 1/ | . (3.2) However, the solution is ( ) ( )φ = − ± −         − ∫c g Exp g dr r r O i 2 1 4 1 2/ / , (3.3) If g (r,p) < o. (3.4) In these equations c1 , c2 and r O i are arbitrary constants. Conclusion The mnethods of integral transforms and the WKBJ method were applied to find an approximate solution of the problem. A quantitative comparison of the approximate solutions for the displacements and stresses between the non-homogeneous and homogeneous viscoelastic plates at various times have been discussed.