Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 54, 2, pp. 579-592, Warsaw 2016 DOI: 10.15632/jtam-pl.54.2.579 P-WAVE INTERACTION WITH A PAIR OF RIGID STRIPS EMBEDDED IN AN ORTHOTROPIC STRIP Sanjoy Basu Haldia Institute of Technology, Department of Applied Science, Haldia, India; e-mail: basu1982@gmail.com S.C. Mandal Jadavpur University, Department of Mathematics, Kolkata, India; e-mail: scmandal@math.jdvu.ac.in The present paper is concerned with the problem of scattering of the P-wave by two co- -planer finite rigid strips placed symmetrically in an infinitely long orthotropic strip. Using the Hilbert transform technique, the mixed boundary value problem has been reduced to the solution of dual integral equations which has finally been reduced to the solution of a Fredholm integral equation of the second kind. Solving this integral equation numerically, stress intensity factors have been calculated at the inner and outer edges of the rigid strips, and the vertical displacement outside the strips has been calculated and plotted graphically to show the effect of material orthotropy. Keywords:P-wave, Fourier transform,Hilbert transform, Fredholm integral equation, stress intensity factor 1. Introduction The dynamic interaction of rigid strips with an elastic isotropic or orthotropic medium is a subject of considerable interest in mechanics. Dynamical analysis of this kind is of importance to earth-quake engineering, machine, vibrations and seismology. The performance of engineered systems is affected by inhomogeneities such as cracks and inclusions present in the material. Cracks and rigid inclusions in an elastic material have become the subject of investigations. Presently, the use of anisotropicmaterials is increasing due to their strength. The increasing use of anisotropic media demands that the study should be extensive. A detailed reference of work done on the determination of the dynamic stress field around a crack or inclusion in an elastic solid was given by Sih (1977), Sih andChen (1981), Chen (1978), Cinar (1983). However, in the presence of finite boundaries, the problem becomes complicated since they involve additional geometric parameters, describing the dimension of the solids. Forced vertical vibration of a single stripwas treated byWickham (1977). Singh et al. (1983) solved the problemof diffraction of a torsional wave by a circular rigid disc at the interface of two bonded dissimilar elastic solids. In that paper, they discussed an iterativemethod to solve the Fredholm integral equation of the second kind and described the stress intensity factor with the wave number. Mandal et al. (1997, 1998) solved the problem of forced vibration of two and four rigid strips on a semi-infinite elastic medium. Mandal et al. (1998) also treated the diffraction problem by four rigid strips in an orthotropic medium. Interaction of elastic waves with a periodic array of the coplanar Griffith crack in an orthotropic mediumwas discussed byMandal et al. (1994). Das et al. (1998) solved the problem of determining the stress intensity factor for an interfacial crack between two orthotropic half planes bonded to a dissimilar orthotropic layer with a punch. They reduced the problem to a system of simultaneous integral equations which were solved by Chebyshev polynomials. The problem of two perfectly bonded dissimilar orthotropic strips with an interfacial crack was studied by Li (2005). He derived the analytical expression for the 580 S. Basu, S.C.Mandal stress intensity factor. Sarkar et al. (1995) solved the problem of diffraction of elastic waves by three coplanar Griffith cracks in an orthotropic medium. Das (2002) solved the problem of interaction between line cracks in an orthotropic layer. An elastostatic problem of an infinite row of parallel cracks in an orthotropicmediumwas analyzed by Sinharoy (2013).Monfared and Ayatollahi (2013) investigated theproblemof determining the dynamic stress intensity factors of multiple cracks in an orthotropic stripwith a functionally gradedmaterials coating. They solved the problem by reducing it to a singular integral equation of the Cauchy type. The problem of interaction of three interfacial Griffith cracks between bonded dissimilar orthotropic half planes was studied byMukherjee and Das (2007). Das et al. (2008) solved the problem of determining the stress intensity factors due to symmetric edge cracks in an orthotropic strip under normal loading. They derived an analytical expression for the stress intensity factor at the crack tip. The problem of finding the stress intensity factors for two parallel interface cracks between a nonhomogeneous bonding layer and two dissimilar orthotropic half-planes under tension was studied by Itou (2012). Shear wave interaction with a pair of rigid strips in elastic strip was analyzed byPramanick et al. (1999).WUDa-zhi et al. (2006) considered the torsional vibration problemof a rigid circular plate on a transversely isotropic saturated soil. Very recentlyMorteza et al. (2010a,b) considered the vibration problemof a rigid circular disc on transversely isotropic media. Diffraction of elastic waves by two parallel rigid strips in an infinite orthotropic medium was analyzed by Sarkar et al. (1995). In this paper, the diffraction of the elastic P-wave by two rigid strips embedded in an infinite orthotropic strip is analyzed. Using the Hilbert transform technique, themixed boundary value problem has been reduced to the Fredholm integral equation of the second kind which has been solved numerically by the Fox and Goodwin method (1953). Stress intensity factors at both the edges of the strips have been calculated and shown graphically for different parameters and materials. Finally, vertical displacement has been calculated outside the strips and shown by 3D-graphs. 2. Formulation of the problem Let us consider an infinitely long orthotropic elastic strip of width 2h containing two coplanar rigid strips embedded in it. The location of the strips are b ¬ |X| ¬ a, Y = 0, |Z|< ∞, with reference to the cartesian co-ordinate axes (X,Y,Z). Normalizing all lengths with respect to a and putting X/a = x, Y/a = y, Z/a = z, b/a = c, the locations of the rigid strips are defined by c¬ |x| ¬ 1, y=0, |z|<∞ (Fig. 1). Fig. 1. Geometry of the strips Let a time harmonic wave given by u = 0 and v = v0e i(ky−ωt), where k = aω/(cs √ c22), cs = √ µ12/ρwithρbeing thedensity of thematerial,ω the circular frequency and v0 a constant, travelling in the direction of the positive y-axis and be incident normally on the strips. P-wave interaction with a pair of rigid strips... 581 The non-zero stress components τyy, τxy and τxx are given by τyy µ12 = c12 ∂u ∂x +c22 ∂v ∂y τxy µ12 = ∂u ∂y + ∂v ∂x τxx µ12 = c11 ∂u ∂x + c12 ∂v ∂y (2.1) whereu and v are displacement components and cij (i,j =1,2) are non-dimensional parameters related to the engineering elastic constants Ei, µij and νij (i,j =1,2,3) by the relations c11 = E1 µ12 ( 1− ν 2 12E2 E1 ) c22 = E2 µ12 ( 1− ν 2 12E2 E1 ) = c11 E2 E1 c12 = ν12E2 µ12 ( 1− ν212E2 E1 ) = ν12c22 = ν21c11 (2.2) for the generalized plane stress and c11 = E1 ∆µ12 (1−ν23ν32) c22 = E2 ∆µ12 (1−ν13ν31) c12 = E1 ∆µ12 ( ν21+ ν13ν32E2 E1 ) = E2 ∆µ12 ( ν12+ ν23ν31E1 E2 ) (2.3) where ∆=1−ν12ν21−ν23ν32−ν31ν13−ν12ν23ν31−ν13ν21ν32 for the plane strain. The constants Ei and νij satisfy Maxwell’s relation νij Ei = νji Ej (2.4) Therefore, substitutingu(x,y,t)=u(x,y)e−iωt and v(x,y,t) = v(x,y)e−iωt, our problem reduces to the solution of the equations c11 ∂2u ∂x2 + ∂2u ∂y2 +(1+ c12) ∂2v ∂x∂y +k2su=0 c22 ∂2v ∂y2 + ∂2v ∂x2 +(1+ c12) ∂2u ∂x∂y +k2sv=0 (2.5) where k2s = a 2ω2/c2s. Thus the problem is to find the stress distribution near the edges of the strips subject to the following boundary conditions v(x,0+)= v(x,0−) =−v0 c¬ |x| ¬ 1 (2.6) τyy(x,0)= 0 |x| 0 τxx µ12 = 2 π { ∞ ∫ 0 [( c11ξ− c12α1ν1 ξ ) e−ν1|y|− ( c11ξ− c12α2ν2 ξ ) e−ν2|y| ] A1(ξ)cos(ξx) dξ + ∞ ∫ 0 [ (c11ν3− c12α3)A3(ζ)cosh(ν3x) +(c11ν4− c12α4)A4(ζ)cosh(ν4x) ] sin(ζy) dζ } y> 0 (2.16) Boundary conditions (2.6) and (2.7) yield the following pair of dual integral equations ∞ ∫ 0 1 ξ [1+H(ξ)]A(ξ)cos(ξx) dξ= p(x) c¬ |x| ¬ 1 ∞ ∫ 0 A(ξ)cos(ξx) dξ=0 |x| 1 This problem has been analyzed in detail by Sarkar et al. (1995). 4. Quantities of physical interest The stress τyy(x,y) for y → 0 in the neighbourhood of the strip can be found from equation (2.16)1, and is given by τyy(x,0±)=∓ 2µ12c22 π ∞ ∫ 0 A(ξ)cos(ξx) dξ c¬ |x| ¬ 1 (4.1) Now ∆τyy(x,0)= τyy(x,0+)− τyy(x,0−) (4.2) then ∆τyy(x,0)=− 4 π µ12c22 ∞ ∫ 0 A(ξ)cos(ξx) dξ (4.3) Substituting the value of A(ξ) from equation (3.1) into equation (4.3), we get ∆τyy(x,0)= 2µ12c22 h(x2) x (4.4) Since h(x2)= 1 √ (x2− c2)(1−x2) H(x2) x2 = c2cos2φ+sin2φ equation (4.4) becomes ∆τyy(x,0)= 2µ12c22G(φ) x √ (x2− c2)(1−x2) (4.5) 588 S. Basu, S.C.Mandal So the stress intensity factors Nc andN1 at the two tips of the strip can be expressed as Nc = lim x→c+ [∆τyy(x,0) πc22µ12 √ x− c ] = 2 π G(0) c √ 2c(1− c2) (4.6) and N1 = lim x→1− [∆τyy(x,0) πc22µ12 √ 1−x ] = 2 π G ( π 2 ) √ 2(1− c2) (4.7) Making c tend to zero, the two strips merge into one, and in that case N1 = √ 2 π G (π 2 ) Now from equation (2.15)2 after substituting the value of A1(ξ) and using equation (3.1), we get the vertical displacement outside the strip as v(x,y) = 2 π 1 ∫ c h(t2) t dt { ∞ ∫ 0 (α1e −ν1y−α2e−ν2y) [1− cos(ξt)]cos(ξx) α1ν1−α2ν2 dξ + ∞ ∫ 0 1 ζ [α3A5(ζ)cosh(ν3x)+α4A6(ζ)cosh(ν4x)]cos(ζy) dζ } (4.8) 5. Numerical calculations and discussions It is important to choose a numerical method of solving the Fredholm integral equation. The Fox andGoodwinmethods require that the definite integrals should be calculable by numerical quadrature, using known formulae in the theory of finite differences, andFredholm equations are conveniently treated by solving simultaneous equations. Themethods enable accurate solutions to be obtained without a prohibitive expenditure of time and energy. The choice of an interval is of course rather arbitrary. We want to keep to a minimum number of linear equations, but the interval must not be large that the finite-difference equations are meaningless. Since the differences are examined, the method guards against the possibility of obtaining wrong results from this case. It also ensures that neither too few nor too many differences are retained in the quadrature formulae. The method of Fox and Goodwin (1953) has been used to solve integral equation (3.12) numerically for different values of the dimensionless frequency ks, material strip width 2h and separating distance of the strips c. The integral in (3.12) has been represented by a quadrature formula involving values of the desired functionG at pivotal points in the range of integration, which leads to a set of algebraic linear simultaneous equations. The solution of the set of linear algebraic equations gives the first approximation of the required pivotal values of G which has been improved by the use of the difference correction technique. After solving integral equation (3.12) for different values of engineering elastic constants of several orthotropic materials listed in Table 1, the stress intensity factors (SIF), kc and k1 at both ends of the strip given by equations (4.6) and (4.7) has been plotted against ks for different values of h and c and for different materials. Instead of the real part of SIF, its mod value is taken because both shows the same type of results. In Fig. 2a and 4a, Nc (SIF, at the inner edge of the strip) and N1 (SIF, at the outer edge of the strip) have been plotted against ks for h=2.0 and h=2.5 and for different strip lengths (c= 0.2, 0.4, 0.6) for material type I. In Fig. 3a and 5a, Nc and N1 have been plotted against P-wave interaction with a pair of rigid strips... 589 Table 1.Engineering elastic constants E1 [Pa] E2 [Pa] µ12 [Pa] ν12 Type I E-type glass-epoxi composite 9.79 ·109 42.3 ·109 3.66 ·109 0.063 Type II Stainless steel-aluminium composite 79.76 ·109 85.91 ·109 30.02 ·109 0.31 ks for c=0.4 and c=0.6 and for different material strip widths (h=2.0, 2.5, 3.0) for material type I. The same set of parameters stated above for the graphs ofNc andN1 have been plotted in Figs. 2b, 4b, 3b, 5b for material type II. For a particular value of material strip width h (=2.0, 2.5), the value of Nc decreases initially and, after increasing again, it decreases with an increase in ks for material type I (Fig.2a), whereas for material type II, it is slowly decreasing with an increase in ks (Fig. 2b) for different values of strip length c (=0.2, 0.4, 0.6). It is also observed that with an increase in c, the value of Nc increases.When strip length c is fixed, the value ofNc is higher for higher values ofh (=2.0, 2.5, 3.0) (Fig. 3a andFig. 3b) for both types of materials. Figure 4 and 5 show thatN1 has initial decreasing tendency and then increases with an increase in ks for both the materials. For fixed c, Nc is higher when material strip width h is higher. In all the cases, it is seen that as the length of the strip increases the value of N1 decreases. Fig. 2. Stress intensity factorN c verses frequency k s Fig. 3. Stress intensity factorN c verses frequency k s Finally, in Fig. 6 and 7 the vertical displacement v(x,y) has been plotted outside the strips (0 < x < c, 1 < x < h) for fixed values of h = 2.5, ks = 0.4 and c = 0.6 for both the materials. In Fig. 6, v(x,y) has been plotted for the inner side of the strip (0 < x < c) and in Fig. 7 for the outer side of the strip (1 < x < h). In Fig. 6a and 7a, it is observed that the vertical displacement v(x,y) increases initially with an increment of the values of x and y, then it decreases for material I. But in the case of Fig. 6b and 7b, it is seen that the vertical displacement v(x,y) increases slowly with an increase in the values of x and y, then it decreases 590 S. Basu, S.C.Mandal Fig. 4. Stress intensity factorN1 versus frequency ks Fig. 5. Stress intensity factorN1 versus frequency ks Fig. 6. Displacement |v(x,y)| versus distances (x,y) Fig. 7. Displacement |v(x,y)| versus distances (x,y) P-wave interaction with a pair of rigid strips... 591 formaterial II. In all cases, thewave like nature has been observed, and finally the displacement tends to zero as (x,y)→∞, which satisfies the radiation condition. 6. Conclusions Thediffraction of the elasticP-wave by two rigid strips embedded in an infinite orthotropic strip is investigated on two types ofmaterials by using the integral equation technique. The governing differential equation with constant coefficients with the boundary conditions becomes a mixed boundary value problem.Then, themixed boundary value problem is transformed into a pair of dual integral equations with an unknown constant A(ξ). To reduce the dual integral equations (2.17)1 and (2.17)2 to a singleFredholm integral equation,weassume theunknownconstantA(ξ) in the form of equation (3.1), so that equation (2.17)2 can be automatically satisfied. Also, it has been found that the normal stress component τyy(x,0) at the two tips of the strip has a square root singularity at x= c and x=1. The form of (3.1) has a square root type singularity in it, which can be utilized to find stress singularities at the tips of the strips. From all the graphs of SIF, it can be concluded that the SIF decreases gradually with an increment of the frequency (ks), after reaching the minimum value, it increases slowly. In all suggested cases, it is noted that the maximum value of the SIF at both tips of the strip for material II is little higher than that formaterial I.TheSIFcanbearrestedwithina certain range, which is very important with respect to growth of the crack. Finally, the vertical displacement v(x,y) has been calculated outside the strips for both the materials. It has been observed the wave like nature from all the 3D figures, which finally decreases as the distance increases. 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WuD.,CaiY.,XuC., ZhanH., 2006,Torsional vibrations of rigid circular plate on transversely isotropic saturated soil,Applied Mathematics and Mechanics, 27, 1541-1548 Manuscript received October 28, 2013; accepted for print September 30, 2015