Displacement Solution for a Static Vertical Rigid Movement of an Interior Circular Disc in a Transversely Isotropic Tri-Material Full-Space

Transcription

1 Vo:8, No:, 4 Dispacement Soution for a Static Vertica Rigid Movement of an Interior Circuar Disc in a Transversey Isotropic Tri-Materia Fu-Space D. Mehdizadeh, M. Rahimian, M. Eskandari-Ghadi Internationa Science Index, Aerospace and Mechanica Engineering Vo:8, No:, 4 waset.org/pubication/ Abstract This artice is concerned with the determination of the static interaction of a verticay oaded rigid circuar disc embedded at the interface of a horizonta ayer sandwiched in between two different transversey isotropic haf-spaces caed as tri-materia fuspace. The axes of symmetry of different regions are assumed to be norma to the horizonta interfaces and parae to the movement direction. With the use of a potentia function method, and by impementing Hanke integra transforms in the radia direction, the government partia differentia equation for the soey scaar potentia function is transformed to an ordinary 4th order differentia equation, and the mixed boundary conditions are transformed into a pair of integra equations caed dua integra equations, which can be reduced to a Fredhom integra equation of the second kind, which is soved anayticay. Then, the dispacements and stresses are given in the form of improper ine integras, which is due to inverse Hanke integra transforms. It is shown that the present soutions are in exact agreement with the existing soutions for a homogeneous fu-space with transversey isotropic materia. To confirm the accuracy of the numerica evauation of the integras invoved, the numerica resuts are compared with the soutions exists for the homogeneous fuspace. Then, some different cases with different degrees of materia anisotropy are compared to portray the effect of degree of anisotropy. Keywords Transversey isotropic, rigid disc, easticity, dua integra equations, tri-materia fu-space. B I. INTRODUCTION EACAUSE of mathematica difficuties and engineering appication, the static interaction of an embedded rigid disc with an eastic medium has been a subect of active research for many years. The first research in the topic is probaby reated to the work done by Bousinesq in 888 on indentation probem. The probem was reinvestigated by Harding and Sneddon [], who used Love's stress function. Some other static or dynamic soutions were obtained on the response of a rigid disc resting on the surface of a haf-space, as in Sneddon [], Keer [3], Arnod et a. [4], Bycroft [5], Awaobi and Grootenhuis [6], Robertson [7], Gadwe [8], and Pak and Gobert [9]. In a cases, an integra transform has been used, which resuted in some dua integra equations, D. Mehdizadeh is with the Schoo of Civi Engineering, Coege of Engineering, University of Tehran, P.O. Box , Tehran, Iran (emai: d_mehdizadeh67@yahoo.com). M. Rahimian is with the Schoo of Civi Engineering, Coege of Engineering, University of Tehran, P.O. Box , Tehran, Iran (phone: ; e-mai: rahimian@ut.ac.ir). M. Eskandari-Ghadi is with the Schoo of Civi Engineering, Coege of Engineering, University of Tehran, P.O. Box , Tehran, Iran (phone: ; e-mai: whose soution has been given in the numerica form in genera case. The soution has been given anayticay for some simper cases. This paper investigates the axisymmetric interaction of verticay oaded rigid disc embedded at the interface of a transversey isotropic tri-materia fu-space. The axes of symmetry of different regions are norma to the horizonta interfaces and parae to the movement direction. Because of its competeness and simpicity, the potentia function introduced in [] is used to uncoupe the equations of motions. Using the Hanke integra transforms [], the reaxed mixed boundary-vaue probem considered here is transformed into a pair of integra equations named as dua integra equations in the iteratures. With the aid of Nobe s mutipying factor procedure [], the dua integra equations obtained in this paper are changed to the form of Fredhom integra equation of the second kind, which is numericay soved for the genera ayered fu-space. It is shown that the numerica soution for homogeneous transversey isotropic fu-space which is degenerated from the genera soution presented here is identica to the numerica soution given in [3].. BOUNDARY VALUE PROBLEM AND THE SOLUTION A massess rigid disc of radius a, embedded at an interface of a ayered, transversey isotropic, ineary eastic fu-space is considered (see Fig. ). The disc is assumed to be undergoing a forced rigid body transation Δ in the vertica direction. In view of the axia symmetry of the probem, a cyindrica coordinate system (, r θ, z ) is instaed at the center of the disc. Considering this coordinate system, the arrangement of different ayers of fu-space may be defined as an upper haf-space with z< h caed as Region I, an intermediate ayer with h< z< denoted as Region and a ower haf-space with z > showing by Region. In the absence of body force, the axisymmetric equations of motion in terms of dispacements in each region can be expressed as []. u u u u w q q q q q A ( + ) + A + ( A + A ) = (a) q q44 q3 q44 r r r r z r z w w w u q q q q uq A ( + ) + A + ( A + A )( + ) = (b) q44 q33 q3 q44 r r r z r z r z Internationa Schoary and Scientific Research & Innovation 8() 4 8 schoar.waset.org/37-689/

2 Vo:8, No:, 4 Internationa Science Index, Aerospace and Mechanica Engineering Vo:8, No:, 4 waset.org/pubication/ q = I, and, u and w are the dispacement components in r and z directions, respectivey and A are the easticity constants. For a transversey isotropic materia, five independent eastic constants are needed to describe its behavior []. These easticity constants which are different in different region are connected to the engineering constants E, E, υ, υ, G and G through some reations (see [] and [4] for detais). Considering the vertica movement of the disc, a reaxed treatment of the mixed boundary-vaue probem can be stated in terms of the components of the Cauchy stress tensor σ (, i = r, z) and the dispacement components u and w as foows wr (, z= ) = Δ, r a () w (, r z = ) w (, r z = ) =, r (3) u (, r z = ) u (, r z = ) =, r (4) zz σ (, r z = ) σ (, r z = ) = R() r, r a (5) zz zz σ (, r z = ) σ (, r z = ) =, r > a (6) rz zz σ (, r z = ) σ (, r z = ) =, r (7) rz wi(, r z = h ) w(, r z = h ) =, r (8) ui(, r z = h ) u(, r z = h ) =, r (9) σzzi( r, z = h ) σzz( r, z = h + ) =, r () σrzi(, r z = h ) σrz(, r z = h ) =. r () Here, the function R() r is the resutant vertica stress appied on the rigid disc. Utiizing the scaar potentia function, F, introduced by Lekhnitskii [] as the static case of the generaized potentia function given by Eskandari-Ghadi [5], the difficuty in deaing with the couped partia differentia equations in () is avoided. The dispacement- and the stress-potentia function reationships are given in Hanke integra transformed space shorty. Thus, the soutions for the potentia functions in different regions in Hanke space are [6]: () siξz siξz ξ ξ ξ I I I F (, z) = A( ) e + C ( ) e, z < h () () s ξ z s ξz s ξz ( ξ, ) = ( ξ) + ( ξ) + ( ξ) s ξ z + D ξ e h < z < F z A e B e C e ( ), (3) () s ξz s ξz ξ ξ ξ F (, z) = B ( ) e + D ( ) e. z > (4) the reguarity condition has been appied in both vertica and horizonta directions. It is emphasized that in the above soutions, ξ is the Hanke s parameter, A ( ξ ) to D ( ξ ) I are unknown functions to be determined using both the continuity and boundary conditions, and s q and sq are the roots of (5). Fig. Axisymmetric transversey isotropic tri-materia fu-space containing a rigid disc A A s + ( A + A A A A ) s + A A = (5) 4 q33 q44 q q3 q3 q44 q q33 q q q44 The dispacement components and stresses in each region can be expressed in terms of the potentia function F, as foows [6]. () df () q () () uq = αq3ξ, wq = [ αq ξ ( + α q)] Fq, (6) dz z () d () σqzr = Aq 44ξ ( αq3 αq ) + ξ ( + α q) Fq, dz () d σqzz = q3aq 3 Aq 33 ( q) dz α ξ ξ + α d () + Aq33α q F, q dz q q66 q44 q3 q66 q q q3 Aq66 Aq66 Aq66 (7) α A = + A, α A =, α A = + A, (8) and q = I, and. Using ()-(4) in (6) and (7), it is possibe to write the continuity and boundary conditions in the matrix form as [7]. Internationa Schoary and Scientific Research & Innovation 8() 4 83 schoar.waset.org/37-689/

3 Vo:8, No:, 4 T c I AI CI A B C D B D s ( ξ ) [ ( ξ) ( ξ) ( ξ) ( ξ) ( ξ) ( ξ) ( ξ) ( ξ)] = ( ξ) (9) s ξ = R ξ () c () T ( ) [ ( ) ] η η η η η η ν ν ν ν ν ν siξh siξh sξh sξh sξh sξh αi3siξe αi3sie α3 se α3se α3λe α3se siξh siξh sξh sξh sξh sξh ϕie ϕie ϕe ϕe ϕe ϕe I( ξ ) = siξh siξh sξh sξh sξh sξh ηie ηie ηe ηe ηe η e siξh siξh sξh sξh sξh sξh νie νie νe νe νe νe α3sξ α3sξ α3sξ α3 sξ α3sξ α3sξ ϕ ϕ ϕ ϕ ϕ ϕ () Internationa Science Index, Aerospace and Mechanica Engineering Vo:8, No:, 4 waset.org/pubication/ ( ) ηqi = A q44 αq3 αq sqi + αq + ϕqi = αqsqi αq ν = ( + α α s ) A α A s qi q q qi q33 q3 q3 qi Soving (9) it is possibe to find () A( ξ) C ( ξ) A ( ξ) I I = I ( ξ), = I ( ξ), = I ( ξ), () () () 3 R ( ξ) R ( ξ) R ( ξ) B ( ξ) C ( ξ) D ( ξ) = I ( ξ), = I ( ξ), = I ( ξ), (3) () 4 () 5 () 6 R ( ξ) R ( ξ) R ( ξ) B ( ξ) D ( ξ) = I ( ξ), = I ( ξ), () 7 () 8 R ( ξ) R ( ξ) I ( ) i ξ = the eement at the i th row and the second coumn of the inverse matrix of I( ξ ) and i=,,,8. Equations () and (6) resut in a pair of integra equations caed dua integra equations. For the treatment of these equations, the procedure introduced by Nobe [] is used. To do so, it is convenient to write the equations in the foowing form Δ ξ [ + H( ξ)] B( ξ) J ( rξ) d ξ =, r a (4a) B( ξ) J ( rξ) d ξ =, r > a (4b) B ( ξ) D ( ξ) ξ H( ξ) = + ϕ + ϕ, () () R ( ξ) R ( ξ) ϑ ( ξ) (5) () B( ξ ) = ξϑ( ξ ) R ( ξ ), (6) A ( ξ) B ( ξ) C ( ξ) ϑξ ( ) = ν + ν () () () R ( ξ) R ( ξ) R ( ξ) D ( ξ) B ( ξ) D ( ξ) + ν ν () () (), R ( ξ) R ( ξ) R ( ξ) (7) and is a modifier needed for the function H ( ξ ) to make the condition at infinity to be satisfied [8]. This modifier is defined as = + H( ). By soving the dua integra equations (4) for R () ( ξ ), this function is known, and the remaining functions A I ( ξ ) to D ( ) ξ are then known from (3).. FREDHOLM INTEGRAL EQUATION As seen in [], with the aid of Hanke inversion theorem, the governing dua integra equations (4) can be reduced to a Fredhom integra equation as a Δ θ( r) + M( r, ρ) θ( ρ) dρ, r a π = < (8) M ( r, ρ) = H( ξ)cos( rξ)cos( ρξ) dξ. r, ρ < a(9) and the function R () ( ξ ) is expressed as [9] ( ξ ) = θ( ρ) ( ρξ) ρ. πϑ( ξ ) (3) () R a Cos d Substituting (3) in (3), the functions A I ( ξ ) to D ( ξ ) can be found. Furthermore, the dispacements and stresses can be obtained in terms of θ() r by substituting these eight functions into the expression of the potentia function F, and then using the dispacement- and stress- potentia reations. Equations (8) with (9) can be numericay soved for θ() r. Internationa Schoary and Scientific Research & Innovation 8() 4 84 schoar.waset.org/37-689/

4 Vo:8, No:, 4 Internationa Science Index, Aerospace and Mechanica Engineering Vo:8, No:, 4 waset.org/pubication/ IV. NUMERICAL EVALUATION OF THE FREDHOLM INTEGRAL EQUATION In the previous section, the Fredhom integra equation was expressed in terms of θ() r. Generay, the Fredhom integra (9) cannot be anayticay soved. As a resut, the equation may be converted to a set of inear agebraic equations of the form Δ Kθ =, i, =,,..., n (3) K = δ + WM( ri, ρ ), i, =,,..., n (3) π moreover, W is the weight function for transforming an integra to a summation, and n is the number of points seected on the disc for numerica evauation []. To evauate M = M( ri, ρ ) from (9), an adaptive numerica quadrature method is adopted and coded in MATHEMATICA software. After determining M, the function θ is found from (3) at the seected points. Then, repacing θ = θ( ρ) in (9) and integrating, the function θ is found as a function of horizonta distance, r, after which R () ( ξ ) is obtained from (3). In order to vaidate the present study, numerica soutions presented in [3] for a homogeneous fu-space are evauated here and used for comparison. To understand the importance of reative materia modui in the probem, 5 sets of materia constants as isted in Tabe I are considered. TABLE I SYNTHETIC MATERIAL ENGINEERING CONSTANTS Materia E E G G (N/mm ) (N/mm ) (N/mm ) (N/mm ) ν ν (Transversey Isotropic) 45,,,,.5.5 (Transversey Isotropic) 5, 5,,, (Transversey Isotropic) 5, 55,,, (Transversey Isotropic) 5,,,, (Transversey Isotropic) 5, 5,,,.5.5 With these materias, three different cases are defined as beow to demonstrate the soution: Case : AI = Mat5 A = Mat5 A = Mat5 Case : AI = Mat A = Mat A = Mat5 : AI = Mat A = Mat3 A = Mat5 : AI = Mat A = Mat4 A = Mat5 The ast three cases are defined with different ratio of E / E, to portrait the effect of degree of anisotropy. As it is E / E for Mat, Mat3 and Mat4 is.5, ~ and, seen respectivey. On the other hand, the first case is a homogenous fu-space, which is used for verification. For generaity, a numerica resuts presented here are dimensioness. Therefore, the stress σ zz and the vertica dispacement are presented in the form of dimensioness parameters as σ zz /( F / πa ) and w / Δ, respectivey. Here F is the vertica resutant force, appied on the disc, which itsef a is evauated by the integra F = π Rrrdr ( ). F is aso evauated directy in terms of the Fredhom equation as a F = 4 a θ( ρ) dρ []. Fig. iustrates the stress difference R() r for the different cases isted above. As indicated in the figure, there exists a singuar behavior at the vicinity of the edge of the disc. A very precise attention has been paid in the numerica evauation, however, there exists a sma error in the evauation of the stress difference outside the disc, the stress difference shoud be zero. This error is due to the number of points seected for evauations of the stress in (3), and the truncation point seected in (9). Fig. 3 indicates a depth wise variation of the stress σ zz for different cases. As seen in each curve, there exists a ump at z =, the disc is ocated, which is equa to Rr ( =, z= ). Since E in is the argest among the tri-materia cases, the stress σ zz in this case for constant Δ is the argest one. The soution presented by Ardeshir-Behrestaghi and Eskandari-Ghadi [3] is used as a benchmark, to provide a comparison with the resuts in this paper and proof vaidity of the numerica evauations. To this end, Case with the same materia properties for a the ayers is defined. Fig. 3 aso shows the resuts for a homogenous fuspace for both studies, an exceent agreement is discovered between the two soutions R(r)/(F/πa ) r/a Case Fig. Normaized resutant vertica stress appied on the rigid disc in terms of horizonta distance Internationa Schoary and Scientific Research & Innovation 8() 4 85 schoar.waset.org/37-689/

5 Vo:8, No:, z/a z/a Region I Case Region Internationa Science Index, Aerospace and Mechanica Engineering Vo:8, No:, 4 waset.org/pubication/ Case 3 Case (Ref [3]) Case (Present study) 4 σ zz (r=,z)/(f/πa ) Fig. 3 The stress σ zz at r = in terms of depth z Fig. 4 iustrates the normaized vertica dispacement at z = in terms of radia distance. The vertica dispacement from zero to r = a shoud be equa to Δ as inferred from (). As seen in Fig. 5, athough the dispacement is continuous at z =, its derivative with respect to depth is not continuous as indicated in (5) and the strain-dispacement and stress-strain reationships. Since E in is the argest, the vertica dispacement in this case for constant Δ is the argest one E' w(r,z=)/δ Case r/a Fig. 4 Variation of vertica dispacement at z= in terms of horizonta distance 3 Region 4 w(r=,z)/δ Fig. 5 Depth-wise variation of vertica dispacement at r= V. CONCLUSION With the use of a scaar potentia function and Hanke integra transforms, the reaxed mixed boundary vaue probem for a vertica movement of a rigid disc embedded in a tri-materia fu-space considered in this paper have been transformed to a Fredhom integra equation of the second kind, which have been soved numericay. The soution is coapsed on the homogenous fu-space reported in the iterature, which proves the vaidity of the soution presented in this paper. Some different arrangements have been compared to see the effect of different degree of anisotropy. REFERENCES [] J. W. Harding, and I. N. Sneddon, The Eastic Stresses Produced by the Indentation of the Pane Surface of a Semi-Infinite Eastic Soid by a Rigid Punch, in Proc. Camb. Phi. Soc., 4, pp. 6-6, 954. [] I. N. Sneddon, The Reissner-Sagoci Probem, in Proc. Gasg. Math. Assoc., 7, pp , 966. [3] L. M. Keer, Mixed Boundary Vaue Probems for an Eastic Haf- Space, in Proc. Camb. Phi. Soc., 63, pp , 967. [4] R. N. Arnod, G. N. Bycroft, and G. B. Warburton, Forced Vibrations of a Body on an Infinite Eastic Soid, J. App. Mech., ASME, vo., pp. 39 4, 955. [5] G. N. Bycroft, Forced Vibrations of a Rigid Circuar Pate on a Semi- Infinite Eastic Space and on an Eastic Stratum, Phi. Trans. Roy. Soc. Lond, vo. 48, A(948), pp , 956. [6] A. O. Awaobi, and P. Grootenhuis, Vibrations of Rigid Bodies on Semi-Infinite Eastic Media, in Proc. Roy. Soc., 87, A, pp 7-63, 965. [7] I. A. Robertson, Forced Vertica Vibration of a Rigid Circuar Disc on a Semi-Infinite Eastic Soid, in Proc. Camb. Phi. Soc., 6 A, , 966. [8] G. M. L. Gadwe, Forced Tangentia and Rotatory Vibration of a Rigid Circuar Disc on a Semi-Infinite Soid, Int. J. Engng. Sci., vo. 6, pp , 968. [9] R. Y. S. Pak, and A. T. Gobert, Forced Vertica Vibration of Rigid Discs with an Arbitrary Embedment, J. Eng. Mech., vo. 7(), pp , 99. [] S. G. Lekhnitskii, Theory of Anisotropic Eastic Bodies. Hoden-Day Pubishing Co., San Francisco, Caif. 98. [] I. N. Sneddon, Fourier Transforms. McGraw-Hi, New York, N. Y. 95. [] B. Nobe, The Soution of Besse Function Dua Integra Equations by a Mutipying-Factor Method, in Proc. Camb. Phi. Soc., 59, pp , 963. Internationa Schoary and Scientific Research & Innovation 8() 4 86 schoar.waset.org/37-689/

6 Vo:8, No:, 4 Internationa Science Index, Aerospace and Mechanica Engineering Vo:8, No:, 4 waset.org/pubication/ [3] A. Ardeshir-Behrestaghi, M. Eskandari-Ghadi, Dynamic Interaction of a Rigid Pate with Transversey Isotropic Fu-Space, C. E. I. J. (J. of Facuty of Engineering), vo. 45(7), pp ,. (In Persian) [4] M. Eskandari-Ghadi, R. Y. S. Pak, and A. Ardeshir-Behrestaghi, Transversey Isotropic Eastodynamic Soution of a Finite Layer on an Infinite Subgrade under Surface Loads, Soi. Dyn. Earthq. Eng., vo. (), 8, pp , 8. [5] M. Eskandari-Ghadi, A Compete Soution of the Wave Equations for Transversey Isotropic Media, J. of Easticity, vo. 8, pp. -9, 5. [6] M. Eskandari-Ghadi, S. Sture, R. Y. S. Pak, and A. Ardeshir- Behrestaghi, A Tri-Materia Eastodynamic Soution for a Transversey Isotropic Fu-Space, Int. J. Soids.Struct., vo. 46(5), pp. -33, 9. [7] A.Khoasteh, M. Rahimian, and M. Eskandari, Three-Dimensiona Dynamic Green s Functions in Transversey Isotropic Tri-Materias, App. Math. Mode., vo. 37(5), pp , 3. [8] M. Eskandari-Ghadi, and A. Ardeshir-Behrestaghi, Forced Vertica Vibration of Rigid Circuar Disc Buried in an Arbitrary Depth of a Transversey Isotropic Haf Space, Soi. Dyn. Earthq. Eng., vo. 3(7), pp ,. [9] A. Erdeyi, and I. N. Sneddon, Fractiona Integra Equation and Dua Integra Equations, Can. J. Math, vo. 4, pp , 96. [] M. Eskandari-Ghadi, M. Faahi, and A. Ardeshir-Behrestaghi, Forced Vertica Vibration of Rigid Circuar Disc on a Transversey Isotropic haf-space, J. Eng. Mech., vo. 36(7), pp. 93-9, 9. [] A. A. Katebi, A. Khoasteh, M. Rahimian, and R. Y. S. Pak, Axisymmetric Interaction of a Rigid Disc with a Transversey Isotropic Haf Space, Int. J. Numer. Ana. Met., vo. 34(), pp. -36,. Internationa Schoary and Scientific Research & Innovation 8() 4 87 schoar.waset.org/37-689/