Advances in Copyright 4 Tech Science Press 5 Point Load Generated Surface Waves in a Transversely Isotropic Half-Space using Elastodynamic eciprocity A. C. Wijeyewickrema 1, M. Hattori 1, S. Leungvichcharoen 1 Summary Surface waves generated by sub-surface time-harmonic point-loads in a transversely isotropic elastic half-space are considered. The numerical results computed are compared with the Green s function solutions obtained previously. Introduction Point-load problems are usually solved by an application of integral transform techniques, and an evaluation of the related inverse transforms by contour integration and residue calculus. However, the resulting integrals in general are not easy to evaluate analytically. ecently elastodynamic reciprocity has been used to study time-harmonic point-loads in an isotropic elastic half-space [1, ]. This paper extends the use of elastodynamic reciprocity in [1, ] to a transversely isotropic half-space, using the same techniques. However, as mentioned in [1, ] the calculation does not include a consideration of the body waves generated by the point-loads. The numerical results has been compared with the Green s function solutions [3], and when the material is reduced to the isotropic elastic case, the solutions of [1, ] are recovered. asic Equations The homogeneous transversely isotropic elastic half-space and the Cartesian coordinate system Ox1xx 3 are shown in Fig. 1. The x 1 x - plane coincides with the surface of the half-space and the x 3 -axis is perpendicular to the plane of isotropy. In Fig. 1, the time harmonic point-load F has a vertical component P and a horiontal component Q. Considering the formulation of [4], solutions for displacement components are taken as, 1 ϕ ( x, x ) i t x, u3( x, t) = W( x3) ϕ ( x1, x) e ω, (1) 1 iωt uα (, t) = V( x3 ) e k xα i t where α = 1, and k= ω / c. In the following analysis, the factor e ω is omitted, and Greek indices refer to x1, x. Solutions of the form given by Eq. (1) satisfy. 1 Department of Civil Engineering, Tokyo Institute of Technology, O-okayama, Meguro-ku, Tokyo, Japan Proceedings of the 4 International Conference on 6-9 July, 4, Madeira, Portugal
Advances in Copyright 4 Tech Science Press 51 x = 3 θ Q r x 1 x P F the elastodynamic equations of motion, if the dimensionless function ϕ ( x1, x) is taken as the solution of ϕ( x, x ) + k ϕ( x, x ) =, 1, αα 1 where repeated indices indicate summation, and V( x3 ) and W( x3 ) are solutions of the following system of ordinary differential equations, C44V ( x3) ( k C11 ρω ) V( x3 ) + k( C13 + C44 ) W ( x3) =, (3) C W ( x ) ( k C ρω ) W( x ) k( C + C ) V ( x ) =. Here, 33 3 44 3 13 44 3 Cij are the elastic constants of transversely isotropic medium, ρ is the mass density and prime indicates d dx 3. For the elastic half-space x 3, solutions of Eq. (3) that decay exponentially with the depth can be written as V( x ) = A e + e, W( x ) = m A e + m e, (4) ks1x3 ksx3 ks1x3 ksx3 3 3 1 where A, are constants of length dimension and k = ω/ c, m = [ ρc C + s C ] [ s ( C + C )], in which x 3 Fig. 1. Transversely isotropic elastic half-space subjected to a subsurface time-harmonic point-load. α 11 α 44 α 13 44 1 1, = [ 3 ± ( 3 4 33 44 1 ) ] [ 33 44 ], s s C C C C = ρc C, = ρc C, = ( C + C ) + C + C. (6) 1 11 44 3 13 44 1 33 44 From Eq. (4) the solutions which satisfy the traction free conditions at x 3 = plane, can be written as, (5) Proceedings of the 4 International Conference on 6-9 July, 4, Madeira, Portugal
Advances in Copyright 4 Tech Science Press 5 ( ) [ s m ], ( ) [ s V x = A e e W x = A e m e ], (7) k 1 3 1 1 3 1 3 1 1 3 s x ks x k 3 3 s x s m m m ks x 1 s m where c is the surface wave speed calculated from ρc ss[ C + C ] =. (8) 1 1 33 1 13 Surface Wave Motion Generated by P Equation (1) can be rewritten in the cylindrical coordinate system (, r θ,) as 1 ϕ ( r, θ ) ur (,) r = V(), k r and Eq. becomes ϕ() r 1 ϕ() r + + k (). ϕ r = r r r u (,) r = W() ϕ (, r θ ), (9) For this axially symmetric case, the relevant solution of Eq. (1) for an outgoing wave is then a Hankel function, (1) ϕ () r = H ( k r). (11) Then Eq. (9) is simplified to u (,) r = AV() H ( k r), r 1 u (,) r = AW() H ( k r), (1) where A V( ) = V( ) and A W( ) = W( ). Expressions for the corresponding stresses can be obtained from Hooke s law as σ r ( r, ) = AC 44[ V ( ) + kw ( )] H1 ( kr ), σ ( r, ) = A[ C13kV ( ) CW 33 ( )] H ( kr ), (13) σ ( r, ) = A [ C kv( ) CW ( )] H ( kr) CV( H ) ( kr) r. { } rr 11 13 66 1 Use of Elastodynamic eciprocity Considering two distinct time-harmonic states of the same frequency denoted by the superscripts A and, the reciprocity relation is given by ( f A A ) ( A A i ui fi ui dv = ui σij ui σij ) nj ds, (14) V S A A A where fi, f i are body forces, σ ij, σ ij are stresses and ui, u i are displacements, and n j are the components of the outward normal. Here repeated indices imply summation. For V, the regions are defined by r a, <, Proceedings of the 4 International Conference on 6-9 July, 4, Madeira, Portugal
Advances in Copyright 4 Tech Science Press 53 θ π. For state A, actual displacements and stresses are given by Eqs. (1) and (13), while for state, dummy solutions consisting of the sum of an outgoing wave and a converging wave are selected as follows [], u (,) r = V()[ H ( k r) + H ( k r)], u (,) r W()[ H ( k r) H ( k r)]. (1) r 1 1 (1) = + (15) Since the displacements defined by Eq. (15) is bounded at r =, it can be verified that the left-hand side of Eq. (14) becomes P u (, ) = PW( ), (16) where the force P is applied at =. Hence, Eq. (14) is expressed as follows π A A A A A A ( r rr r rr ) ( r r ) ( θ rθ θ rθ ) PW( ) = a u σ u σ + u σ u σ + u σ u σ dθ d. (17) Substitution of the corresponding expressions for displacements and stresses for state A and into Eq. (17) yields (1) (1) PW( ) = πaai H ( kah ) 1 ( ka ) H1 ( kah ) ( ka ), (18) where I is defined as { 11 13 44 } I = [ C k V( ) + C W ( )] V( ) C [ V ( ) + k W( )] W( ) d, (19) and Eq. (18) can be rewritten as PW AiI k ( ) = 4 /. y using Eqs. (1) and, the displacement components at position ( r, ) are obtained as ur (,) r = ikpw( ) V() H1 ( kr)(4), I (1) u (,) r = ik PW( ) W() H ( k r)(4). I In a similar manner, surface wave displacements due to the horiontal component Q are obtained as u (, r θ,) = ik QV( ) V()[ H ( k r) ( k r) H ( k r)]cos θ /(4), I 1 r 1 u (, r θ,) = ik QV( ) W() H ( k r)cos θ /(4), I 1 u (, r θ,) = iqv( ) V() H ( k r)sin θ /(4 ri). θ 1 () Proceedings of the 4 International Conference on 6-9 July, 4, Madeira, Portugal
Advances in Copyright 4 Tech Science Press 54 Numerical esults The elastic constants of transversely isotropic and isotropic materials used in [3] are given in Table 1. Table 1 Elastic constants of materials. Materials 4 C C C C 11 1 13 33 C ( 1 N / mm) 44 Isotropic ( ν =.5 ) 3. 1. 1. 3. 1. eryl rock 4.13 1.47 1.1 3.6 1. Graphite/epoxy composite.4.683.73 1.17.41 note: Cij = Cij / C44 The results of the present study for a vertical load P are compared with the Green s function solutions of [3]. For a prescribed frequency ω = ω ρ/ C44 = 1., the non-dimensional vertical displacement along the free surface, u(,) r = u(,) r C44 / P and the non-dimensional normal stress at =, σ (, r) = σ (, r) / P are plotted in Figs. and 3, respectively. It can be seen in Figs. and 3 that when the real parts of the displacements and stresses of the present study, are infinite while the imaginary parts are finite. Since the point load P is applied below the surface at =, the displacement should be bounded at the free surface when. ut when the real part of the displacement e[ u ( r,)] are unbounded in the present study (Fig. ). This may be due to the fact that only surface waves are considered in this solution. When? the present solution agrees well with ef. [3]..15.1.5.1.15.5.. -.5 -.5 -.1 -.1 -.15 4 6 8 1 -.15 4 6 8 1 Fig.. Displacement in -direction; lines - present study, symbols - ef. [3]. Proceedings of the 4 International Conference on 6-9 July, 4, Madeira, Portugal
Advances in Copyright 4 Tech Science Press 55.5.5.5.5.. -.5 -.5 -.5 4 6 8 1 -.5 4 6 8 1 Fig. 3. Normal stress σ (, r); lines - present study, symbols - ef. [3]. Conclusions Generally, it is well known that the surface wave dominates wave fields far from the applied load. Figures and 3 show that as the radial distance increases, the present solution agrees well with the Green s function solution [3]. Therefore the accuracy of expressions for surface wave displacements is confirmed. eferences 1 Achenbach, J. D., : Calculation of wave fields using elastodynamic reciprocity, International Journal of Solids and Structures, Vol. 37, pp. 743-753. Achenbach, J. D., : Calculation of surface wave motions due to a subsurface point force: An application of elastodynamic reciprocity, Journal of the Acoustical Society of America, Vol. 17, pp. 189-1897. 3 ajapakse,. K. N. D. and Wang, Y., (1993): Green s functions for transversely isotropic elastic half space, Journal of Engineering Mechanics, ASCE, Vol. 119, pp. 175-1744. 4 Achenbach, J. D., (1998): Explicit solutions for carrier waves supporting surface waves and plate waves, Wave Motion, Vol. 8, pp. 89-97. Proceedings of the 4 International Conference on 6-9 July, 4, Madeira, Portugal