INFLUENCE OF POINT DEFECTS ON ULTRASONIC WAVES PROPAGATING IN THE THIN PLATE

Transcription

1 Materials Physics and Mechanics (5) -4 Received: March 7 5 INFLUENCE OF POINT DEFECTS ON ULTRASONIC WAVES PROPAGATING IN THE THIN PLATE MV Belubeyan VI Erofeev * AV Sheoyan Institute of Mechanics National Academy of Sciences of the Republic of Armenia Marshal Baghramyan Ave 4-B Yerevan Armenia Mechanical Engineering Research Institite of RAS Belinsogo str 85 Nizhny Novgorod 64 Russia Lobachevsy State University of Nizhni Novgorod Prospet Gagarina (Gagarin Avenue) BLDG Nizhny Novgorod 695 Russia * erf4@sinnru Abstract In the linear formulation a two-dimensional self-consistent problem of the propagation of elastic (ultrasonic) waves in the plate taing into account its interaction with point defects present in its material is provided We study the effect of point defects on the dispersion laws of planar and bending elastic waves Introduction Since the 8s of the last century the effect of radiation (including laser) for materials is intensively studied Theoretically and experimentally it has been shown that under the influence of the laser beam in the materials produced numerous point defects (vacancies interstitials) created in the surface layer of the stress-strain state [] The surface layer is modeled by a thin elastic plate undergoing a planar or bending vibrations interacting with the point defects [-6] Two-dimensional equations of vibrations of plates obtained from the three-dimensional equations of elasticity theory by applying Kirchhoff hypotheses [7] The question about how to obtain two-dimensional inetic equations for point defects is usually passed over in silence For obtaining of two-dimensional inetic equations describing the behavior of point defects can use two ways In the first case it can be assumed that the point defects changing along the thicness of the plate changes in a linear manner However experimentally confirm this hypothesis is impossible Therefore more correctly in our opinion proceed from the value of the number of point defects on the plate boundaries which can be measured experimentally The aim of the paper is to pose and study the self-consistent problem of propagation of elastic waves in a plate with regard to their interaction with point defects present in its material Statement of the problem and the general equations In three dimensions the linear system of equations describing the above process is as follows: u t x i () 5 Institute of Problems of Mechanical Engineering

2 Influence of point defects on ultrasonic waves propagating in the thin plate n t n t q q D n n n q q D n n n () () The definitions are conventional [ 8] volumetric deformation i - deformations q and q the rate of defects generation before the disturbances that without loss of generality we can tae q q Boundary conditions are the following: z h n n x y h t n n x y h t (4) z h n n x y h t n n x y h t (5) z h (6) For the sae of simplicity coordinates x x x are changed to x y z respectively By analogy with the thermoelasticity problems (for example [9]) for point defects we consider the following approximations: n n n z n n n z (7) n n n n n n n n where n n n n In Kirchhoff theory it is assumed that the movements of the plate have the following form: w w u u z u z u w (8) x y where functions u w do not depend on z coordinate Similarly as it is done in the theory of plates and shells we can obtain equations for planar vibrations of plates and shells Equations for u и i [] have the following form: x x y x t u v u u C xy x y y t u v C For bending vibrations we have: w D w D h t where v E v C x y v v E Eh d n d n D v (9) () () 4 4 d n d n 4 4 x x y y v is Poisson's ratio E is Young's modulus

3 MV Belubeyan VI Erofeev AV Sheoyan Averaging procedure can also be applied to the inetic equations () and () Expressions for n and n from (7) as well as are substituted to equations () and () In accordance with the Kirchhoff s theory has the following form: vu z w E z v x y After substitution we nullify the coeficients for equations from () and () for z : z and z Thus we get the following n v u q D n q n q n t v x y n v u q D n q n q n t v x y n v for z : q w D n q n q n t v t n v q w D n q n q n v () () (4) where q q E d q q E d q q E d q q E d Equations (9) () () () define planar vibrations of the plate taing into account its interaction with point defects while equations () (4) (5) - bending vibrations of a plate For one-dimension vibrations ( ) we get the following equations of planar vibrations: u u E c e c t c e x E x t x t n u n q D q n qn t x x n u n q D q n qn dn dn t x x while for bending vibrations we get: y (5) (6) 4 w w D D 4 dn dn h x x t n w n q D q n qn t x x n w n q D q n qn t x x (7) Distribution of one-dimensional planar elastic-vacancy waves v does not interact with vacancies and From system (6) it follows that shear waves interstitial atoms For the longitudinal elastic wave solution is represented as exp n B it x n C it x u A i t x exp exp (8)

4 Influence of point defects on ultrasonic waves propagating in the thin plate Substitution of (8) to the first third and fourth equations of system (6) lead us to the following system of algebraic equations with respect to arbitrary constants A B C Cl A icl db dc q q q A id i q B i C q q q A i B id i In (9) we have introduced the following definitions v v v () E v The condition that the determinant of the system (9) equals to zero leads us to the equations which allows us to define the phase velocity of the wave: F F q q 4 i F 4 F C 4 l ic l F F q d d Cl q q F F q q d q d d F d F where F D q F D q () From the () follows that elastic vibration and vacancy (internode) vibrations do not interact in three cases: a) v 5 or b) q or c) d d () Taing into account () the equation () will loo as follows C l if F F F qq In case of the first two equalities in () when q or v 5 but d d means that defects do not feel the wave and in a layer as it follows from (4) propagate two waves The first has velocityc l and the frequency of the other can be defined from the second multiplicand in (4) The second wave decays if F F q q (5) In the inverse inequality (5) the oscillation amplitude will increase unlimitedly In this case the linear theory is not suitable it is necessary to tae into account the nonlinear terms in the equations (6) When d d а q и v 5 then defect feel the wave but the wave does not feel defects In this case the second multiplier in (4) describes the process Increasing or decreasing the number of all the aforementioned defects and remains valid q q (6) Consider the general case when all the coefficients are different from zero but the (9) () (4)

5 4 MV Belubeyan VI Erofeev AV Sheoyan influence of defects on the elastic wave is wea For planar vibration plate we see a solution in the form (8) we obtain the system of equations (9) The dispersion equation can be conveniently represented in the following form: i d d Cl i where D D q q d q d q Fd d F D D D q q D q q (7) The frequency is a complex value ie i is the small disperse is the small absorption coefficient and is the main frequency Сl d d d d In the case when bending waves described by equations (7) propagate in the plate the solution is sought in the form (8) Then the dispersion equation taes the form: D T i d d h T i F F 4 Dh D If q q D D q q d d then h v Here T d F d F q d d q T F F q q D D T F F TT d d F F 4 D T d d T F F 4h T F F Acnowledgements One of the authors (Erofeev VI) received support from the Russian Science Foundation for the wor (grant ) References [] FH Mirzoev VJ Pancheno LA Shelepin // Advances in Physical Sciences 66 (996) [] VI Emel yanov IF Uvarova // Journal of Experimental and Theoretical Physics 94(/8) (988) 55 [] VI Emelyanov AA Sumbatov // Physics Chemistry and Mechanics of Surface 7 (988) [4] VI Emelyanov IF Uvarova // Metallofizia (989) (in Russian) [5] FH Mirzoyev VI Emelyanov LA Shelepin // Quantum Electronics (994) 769 [6] FH Mirzoyev LA Shelepin // Technical Physics 46(8) () 95 [7] The vibrations in the art Directory Vol (Mechanical Engineering Moscow 978) [8] LD Landau EM Lifshitz Theory of Elasticity (Naua Moscow 987) (in Russian) [9] V Novatsy Questions of thermoelasticity (IzdAN the USSR Moscow 96) (in Russian) [] MV Belubeyan In: Proceedings Problems in mechanics of thin deformable bodies (Yerevan ) (8)