Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 49, 4, pp. 1183-1201, Warsaw 2011 NON-PLANE WAVE SCATTERING FROM A SINGLE ECCENTRIC CIRCULAR INCLUSION – PART I: SH WAVES Wolfgang Weber Bernd W. Zastrau Technische Universität Dresden, Dresden, Germany e-mail: wolfgang.weber@tu-dresden.de In this contribution, an analytical formalism for investigating the scat- tering behaviour of a single multi-layered inclusion in a homogeneous isotropicmatrix under the influence of non-plane elastic SHwaves is de- rived. Herein, the inclusion is assumed to have a circular shape with its layers being eccentrically arranged. Numerical examples are performed with Textile Reinforced Concrete, a new building material in civil engi- neering.Within these investigations both shielding and amplification of SHwaves is shown. Key words: non-plane waves, scattering, eccentric scatterer, SH wave, elastic wave, TRC 1. Introduction In many applications in civil, mechanical, biological engineering and related disciplines, investigations concerning the dynamical behaviour of structures or parts of them are regularly necessary as they are exposed to dynamic lo- ads. Thesemay be time-harmonic loads as they occur with rotatingmachines mounted at floor slabs or transient loads due to e.g. impacts. The correspon- ding engineer is then interested in the structures response. For these investi- gations at the so-called macro- or structural-scale, profound insight into the materials behaviour at the micro- andmeso-scale is necessary in order to de- velop adequate mechanical models which allow efficient calculations with a specified accuracy. In real-life applications these forecasts or simulations are made complicated by the fact that both thematerial and geometric properties of the structure and its constituents as well as the dynamic loads themselves 1184 W. Weber, B.W. Zastrau may vary. Hence, capable solutions including their efficient numerical imple- mentation are required to fit the needs arising from numerical studies on the micro- or meso-scale. In what follows, the wave scattering behaviour of a single inhomogeneous inclusion embedded in a homogeneousmatrix of infinite extent is investigated. Herein, thematrixmaterial is supposed to behave linear elastic and isotropic. The calculations are performed with elastic waves propagating through the textile reinforced concrete (TRC), a new construction material in civil engi- neering consisting of a fine grained concretematrixwith amaximumgrain size of approx. 0.6mm and fibres made of glass or carbon which are warp-knitted to uni- or multiaxial reinforcement clutches and then are worked in the fine grained concrete matrix, cf. Ortlepp and Curbach (2009). These fibres, which are often called rovings as well, consist of 800 up to 2000 filaments, each with a diameter of approx. 13.5µm. A view in the direction of the fibres as shown in Fig.1 gives rise to the assumption that the single filaments do participate differently in the load bearing behaviour. In detail, the outer filaments have Fig. 1. Setup of a roving a better interconnection to the surrounding matrix than the inner ones. For some models concerning the decreasing degree of bonding see e.g. Lepenies (2007). Additionally, the matrix behaviour in the vicinity of the rovings will differ from the undisturbedmatrix. Consequently, the inclusion – which acts as an obstacle for the incoming (non-plane) wave and thus will scatter it – is modeled as a double-layered scatterer. Herein the inner layer is the roving corresponding to the inner filaments. Thus, it can be called the (inner) core as well. On the other hand, the outer layer is motivated by the interphase between the outer filaments and thematrix in the vicinity of the fibre.Among other things, due to the production and curing process of TRC the inner and outer layer will be eccentric in most cases, see Weber and Zastrau (2010). In the following, an analytical solution for the scattering of non-plane shear waves by eccentric layered inclusions is derived and some numerical results for arbitrarily chosen deterministic frequencies are discussed. Other parameters Non-plane wave scattering from a single eccentric... 1185 influencing the wave scattering behaviour of the example dealt with here are uncertainties in thematerial properties (e.g.Young’smodulus,Poisson’s ratio) and the dynamic loads (i.e. amplitude, frequency). 2. Some general remarks on elastic waves Based on Navier’s equation without body forces µ∆u+(λ+µ)∇∇·u= ̺ü (2.1) with ∆ = ∇2, u = (u,v,w) displacement vector, λ, µ Lamé constants and ̺ the density of the material at hand, two general wave types can be distin- guished according to Fig.2: the so called S wave is connected to the direction of particle motion d perpendicular to the direction of wave propagation (the wave normal n) whereas in the P wave case the direction of particle motion is parallel to the wave normal. Hence, the P wave is referred to as pressure wave and the S wave as shear wave. Both waves have different propagation speeds. The (longitudinal) P wave propagates with cL = √ (λ+2µ)/̺, the (transverse) S wave with cT = √ µ/̺, respectively. As cL >cT the P wave is called primary wave as well – and hence S wave can be understood as secon- dary wave, see e.g. Achenbach (1973). Obviously these propagation speeds c are characteristic for a certain material and so is the wave number k k= ω c (2.2) where ω is the circular frequency of the wave, cf. Cai (2004). In general, the impinging of one wave type – either P or S wave – upon the surface of an elastic body or an obstacle induces both wave types in the reflection problem (or additional twowaves in the case of refractedwaves), see Fig.2. The Swave Fig. 2. Mode conversion, using the example of reflection of waves 1186 W. Weber, B.W. Zastrau can be decomposed into a SV and a SH wave, where SV means a vertically polarised shear wave and SH a horizontally polarised shear wave, respectively. If the direction of particle motion is parallel to the axis of the obstacle, a SH wave is at hand, see Fig.2. It is characteristic that a SHwave is reflected (and refracted) as a SHwave only. For the sake of brevity, this special case will be evaluated throughout this paper. An eccentric layered elastic circular cylinder which is embedded in a linear elastic and isotropic medium of infinite extent as depicted in Fig.3b with different wave numbers km, kiph, and kfi is considered. Herein andwith other parameters the subscripts m, iph, and fi holdwith thematrix, the interphase and the fibre, respectively. The resulting waves are sought due to the acting of a time-harmonic incident SHwave. Fig. 3. Problem configuration, eccentric circular cylinder case; (a) 3D-view Asmentioned above, the only nontrivial displacement component for a SH wave scattering problem is the out of plane displacement w, if the SH wave propagates in the xy-plane and the axis of the obstacle is parallel to the z-axis as shown in Fig.3b. This displacement takes the form w(x,y,t) =w0cos(ωt+α)= Re{φ(x,y)e −iωt} (2.3) where the complex amplitude φ with |φ| = w0, Re{φ} = w0cosα satisfies the Helmholtz equation ∆φ+ ω2 c2T φ=0=∆φ+k2φ (2.4) which follows from introducing the ansatz Eq. (2.3) into Navier’s equation (2.1). Herein the imaginary unit i and the circular frequency ω were introdu- ced. By means of the shear modulus µ the stress components can be derived from the complex amplitude φ. In the following, the temporal factor e−iωt will be omitted. Non-plane wave scattering from a single eccentric... 1187 3. An analytical approach to non-plane SH wave scattering 3.1. Concentric circular scatterer The single scatterer problemat handaswell as the characterising variables in cylindrical coordinates are shown in Fig.3b. The treatment of the single rovings as layered inclusions (and hence as scatterers for the impingingwaves) wasmotivated in Section 1. In order to calculate the waves in thematrix, the interphase and the fibre, the solution of the Helmholtz equation in a circular coordinate system is sought. With the Laplacian in this coordinate system, equation (2.4) yields ∂2φ ∂r2 + 1 r ∂φ ∂r + 1 r2 ∂2φ ∂θ2 +k2φ=0 (3.1) cf. Morse and Feshbach (1953). Using the ansatz φ(r,θ) = R(r)Θ(θ) and separation of constants leads to ∂2Θ ∂θ2 +n2Θ=0 r2 ∂2R ∂r2 +r ∂R ∂r +(k2r2−n2)R=0 (3.2) where the latter equation is known as the Bessel differential equation. The solution of theHelmholtz equation in polar coordinates is a linear combination of cylindrical wave functions consisting of simple harmonics as the angular factor Θn(θ)= c1ncos(nθ)+s1n sin(nθ)=C1ne inθ+C2ne −inθ (3.3) and the Bessel functions of various kinds as the radial factor Rn(r)=A1nJn(kr)+ iA2nYn(kr)=C3nH (1) n (kr)+C4nH (2) n (kr) (3.4) cf.Pao andMow (1971). Herein H (1),(2) n (kr)= Jn(kr)±iYn(kr) are theHankel functionsof thefirstand secondkind, respectively.Due to the timedependency e−iωt used in this contribution theHankel functions of the first kind H (1) n (kr) represent outgoingwaves –whereas in e.g. Skelton and James (1997), the time dependency is e+iωt and thus waves moving outward are calculated bymeans of the Hankel functions of the second kind H (2) n (kr). For a more detailed description of the Bessel and Hankel functions, see e.g. Watson (1922). By means of the shear modulus µ, the stress components can be derived from the complex amplitude of the displacement. 1188 W. Weber, B.W. Zastrau Often, planewaves are dealt with. As this special incomingwave is regular in the entire xy-plane and the only function in the general ansatz of Eq. (3.4) being regular throughout this plane is the Bessel function of the first kind, it is φin = ∞ ∑ n=−∞ AnJn(kmr)e inθ (3.5) where the index m of the wave number km induces that the function is eva- luated in the matrix region. The same holds for kiph (inside the interphase) and kfi (inside the fibre). According to Eq. (2.3), the Hankel functions of the first kind represent outgoing waves whereas the Hankel functions of the se- cond kind count for waves moving inward. Hence, the scattered wave – which is directed outward with respect to the scatterer – can be described by the Hankel functions of the first kind φsc = ∞ ∑ n=−∞ BnH (1) n (kmr)e inθ (3.6) In the interphase,waves propagating inbothdirections occur and the complete ansatz including the Hankel functions of the first and second kind is needed φiph = ∞ ∑ n=−∞ [ DnH (1) n (kiphr)+EnH (2) n (kiphr) ] einθ (3.7) The Hankel functions of the first and second kind are singular at the ori- gin1. Obviously, the refracted wave in the inner layer is regular, hence it is represented by the Bessel functions of the first kind φfi = ∞ ∑ n=−∞ CnJn(kfir)e inθ (3.8) As can be seen from Eqs. (3.5)-(3.8) all wave fields are composed of so the calledwave expansion coefficients (An, . . . ,En) andwave basis functions, that is Bessel functions of various kinds. The entire field outside the scatterer, which will be evaluated in the given examples, is expressible as φent =φin+φsc (3.9) For the problem at hand, a multi-layered elastic inclusion, the boundary conditions to be fulfilled are that the stress component σrz and the displace- ment φare continuous across the interfaces.WithinSection4, adouble-layered 1Due to the singularity of theBessel functions of the secondkind Yn(kr) at kr=0 Non-plane wave scattering from a single eccentric... 1189 elastic inclusion with boundaries at r = r0 and r = r1 will be investigated. Hence, the boundary conditions yield µm (∂φin ∂ξ + ∂φsc ∂ξ ) ∣ ∣ ∣ ∣ r=r1 =µiph ∂φiph ∂r ∣ ∣ ∣ ∣ r=r1 µiph ∂φiph ∂r ∣ ∣ ∣ ∣ r=r0 =µfi ∂φfi ∂r ∣ ∣ ∣ ∣ r=r0 (φin+φsc) ∣ ∣ ∣ r=r1 =φiph ∣ ∣ ∣ r=r1 φiph ∣ ∣ ∣ r=r0 =φfi ∣ ∣ ∣ r=r0 (3.10) The spatial distribution of the entire wave according to equation (3.9) for a certain time t, that is, Re{φente −iωt}, is exemplarily shown in Fig.4a. Herein andwith the other plots use of thematerial properties given inTable 1 is made. In general, the main interest is finding maximum exposures of the material at hand – independently of time t. This information is provided by mapping the amplitude |φent| as done in Fig.4b for the present case of a concentric scatterer. In these figures, a plane wave of unit amplitude acts upon the concentric obstacle, the inclusions geometric configuration hereby is plotted within the diagram. Fig. 4. Spatial distribution of displacement Re{φente −iωt} (a) and |φent| (b), concentric scatterer (r0 =0.6r1, d=0, and θd =0) 1190 W. Weber, B.W. Zastrau 3.2. Specialisation to an eccentric circular scatterer In real-life applications, the obstacles which scatter the incoming waves will not be concentric in general. Thus, the general ansatz developed in the precedent Section has to be extended with respect to eccentric scatterers. To fit the needs of eccentricities, for an n-layered scatterer n coordinate systems will benecessary. In this contribution n=2hadbeen chosen.Herein the inner layer – the so called core – is shifted by dwith respect to the outer layer. The line connecting the centres of both layers is then rotated by θd as shown in Fig.3b. Hence, two (polar) coordinate systems are needed. The coordinate system connected with the inner layer (the fibre) is denoted with (rfi,θfi) whereas the one of the interphase, i.e. the outer layer, is (riph,θiph). The latter one will be used as the global coordinate system, e.g. for evaluating the waves in thematrix aswell. Equations (3.5)-(3.8) for calculating the incoming, scattered and refracted waves thus have to be rewritten in terms of different coordinate systems. The ansatz functions for the fields inside the matrix, the interphase and the fibre then follow to φin = ∞ ∑ n=−∞ AnJn(kmriph)e inθiph φsc = ∞ ∑ n=−∞ BnH (1) n (kmriph)e inθiph φiph = ∞ ∑ n=−∞ [ DnH (1) n (kiphrfi)e inθfi +EnH (2) n (kiphriph)e inθiph ] φfi = ∞ ∑ n=−∞ CnJn(kfirfi)e inθfi (3.11) It is emphasized that for the wave within the interphase both coordinate sys- tems are used. To fulfil the boundary and transition conditions as given with Eqs. (3.10), a formulation of all fields bymeans of the same coordinate system is adjuvant. With the addition theorem proved inMeixner and Schaefke (1954) it follows B (j) n (krfi)e in(θfi−θiph) = ∞ ∑ m=−∞ B (j) n+m(kriph)Jm(kd)e in(θiph−θ) (3.12) where B (j) n (kr) with j =1, . . . ,4 is the Bessel, Neumann or Hankel function of the first or second kind and order n. It is also known as Graf’s addition theorem. By means of this addition theorem the two coordinate systems can be converted to each other. Non-plane wave scattering from a single eccentric... 1191 The spatial distribution of the amplitude |φent| due to a time-harmonic planewave interactingwith the present case of an eccentric scatterer is plotted in Fig.5. Again, the scatterers geometric configuration is plotted within the diagram and the material properties according to Table 1 are used. For the eccentric case, especially in the first quadrant, enormous changes are observa- ble in comparison with the concentric case shown in Fig.4b. In these regions, the amplitude of the total wave by far exceeds the one of the incoming wave; the affected areas from which cracks may originate are plotted dark. Hence, the investigation of influences of the scatterers eccentricities to the behaviour of brittle matrices such as fine grained concrete is necessary. Fig. 5. Spatial distribution of the displacement amplitude |φent|, eccentric scatterer (r0 =0.6r1, d=0.2r1, and θd =4π/5) The extension to an inclusion of amore general shape, that is, an elliptical shape, is possible and was shown in e.g. Weber and Zastrau (2009). Substitution of the wave expansions for different coordinate systems into boundary and transition conditions (3.10) yields a linear system of equations for the coefficients Bn, . . . ,En in terms of the expansion coefficients An of the incoming non-plane wave which will be determined in the next Section. 3.3. Modeling the incoming non-plane wave This contribution deals with the scattering of non-plane SH waves. As an example, a cylindrical wave whose source is parallel to the axis of the scatterer is being looked at, cf. Fig.3a. For determining the wave expansions 1192 W. Weber, B.W. Zastrau of the incoming wave φin its geometry is provided with Fig.6, where the source of the incoming wave has the coordinates (rs,θs) with respect to the global cylinder coordinate system. There, the obstacle – or rather, the center of the interphase – is situated. In general, a cylindrical wave propagating in the chosen matrix material follows the relation φin =H (1) 0 (km‖r−rs‖) =H (1) 0 ( km √ r2+r2s −2rrscos(θ−θs) ) =H (1) 0 (kmR) (3.13) with R being the distance between the particular point of observation and the source of the wave, see also Fig.6. Again, the time dependency has been suppressed as throughout the whole paper. Fig. 6. Geometry of the source of a non-plane SHwave This term has to be evaluated with respect to the global coordinate sys- tem as shown in Figs.3b and 6. Two cases have to be taken into account: (i) r < rs and (ii) r ­ rs. For the first case, the admissible wave functions are Jn(kmr)e inθ, as φin has to be finite at r = 0 and periodic in 2π on θ. For the latter case, the admissible functions yield H (1) n (kmr)e inθ, as φin has to represent waves propagating outward. Thus φin =            ∞ ∑ n=−∞ anH (1) n (kmrs)Jn(kmr)e in(θ−θs) for r < rs ∞ ∑ n=−∞ anJn(kmrs)H (1) n (kmr)e in(θ−θs) for r­ rs (3.14) where the expansion coefficients an have to be determined now. For this, the source of the cylindrical wave is set into infinity, e.g. (rs = +∞, θs = 0). Non-plane wave scattering from a single eccentric... 1193 Asymptotic formulas for the Hankel functions as given in Harrington (2001) may be used to obtain φin =H (1) n (km‖r−rs‖) −→ rs=+∞,θs=0 √ 2i πrs e−irseircosθ (3.15) and, consequently φin −→ rs=+∞,θs=0 √ 2i πrs e−irs ∞ ∑ n=−∞ ani nJn(kmr)e inθ (3.16) Herein, use of the expansion of a plane wave propagating in the positive x-direction eircosθ = ∞ ∑ n=−∞ AnJn(kmr)e inθ = ∞ ∑ n=−∞ inJn(kmr)e inθ (3.17) was made, cf. Weyrich (1937). Obviously an =1, thus leading φin =            ∞ ∑ n=−∞ H(1)n (kmrs)Jn(kmr)e in(θ−θs) for r < rs ∞ ∑ n=−∞ Jn(kmrs)H (1) n (kmr)e in(θ−θs) for r­ rs (3.18) Referring back to the ansatz given with Eq. (3.5), the wave expansion coeffi- cients An for a cylindrical wave situated at (rs,θs) are An =    H (1) n (kmrs)e in(−θs) for r < rs Jn(kmrs)e in(−θs) for r­ rs (3.19) which implies changes within the wave basis functions of the incoming wave as well. As the final step, these An may be scaled in such a way that φin has unit amplitude at a certain point2 in free space – e.g. at (r= r1, θ=π), that is, at the nearest point of the obstacle facing the source of the incoming wave φin, at the origin (r = 0, θ = 0) or at (r = r1, θ = 0) – the point of the scatterer having the longest distance to the source of the cylindrical wave. Figures 7a,b show the spatial distribution of both the current displace- ment field Re{φente −iωt} at an arbitrary time t and the displacement ampli- tude |φent| for an eccentric double-layered scatterer due to a cylindrical wave. To ensure a better comparability with Figs.4 and 5, the incoming cylindri- cal wave φin has unit amplitude at the centre of the obstacle. The material properties refer to Table 1. 2In fact all points within the xy-plane lying in the material and having the same distance from the source will have equal amplitudes 1194 W. Weber, B.W. Zastrau Fig. 7. Spatial distribution of the displacement Re{φente −iωt} (a) and |φent| (b), eccentric scatterer (r0 =0.6r1, d=0, and θd =4π/5), due to a non-plane SHwave 4. Numerical example In this Section, some numerical results for TRC based on the analytical so- lutions of the precedent Sections are presented. The investigations deal with the displacement distribution in the vicinity of a double-layered obstacle. The radius of the scatterers outer layer is r1, whereas the radius of the inner layer is r0 = 0.6r1, see also Figs.4, 5 and 7. Between both, the interphase is ar- ranged, the fibre is represented by the inner layer. Carbon fibres were used as reinforcement of the fine-grained concrete. The constituents properties are listed in Table 1 und were taken from SFB528. The scatterer is subjected to a non-planar incident SHwave as given with Eq. (3.18) whose source is set at (rs =5a, θs =π). The intensity of the source is scaled in such a way that the incoming non-plane wave would reach unit amplitude at (r=0, θ=0), thus at the centre of the obstacle. Table 1.Material properties of a certain type of TRC Constituent Matrix Carbon fibre Interphase ̺ [kg/m3] 2160 1760 1500 µ [GPa] 12 97 1.17 ν 0.2 0.0 0.49 cT [m/s] 2340 7440 880 Non-plane wave scattering from a single eccentric... 1195 Inmechanical sense, the scattering of non-plane elastic SHwaves is influen- ced by three geometric parameters, that is, the core size r0 with respect to r1, the offset d and the orientation angle θd as introduced inFig.3b.As the inves- tigations in this contributionweremotivated by the requirements of TRC, the range of the core size is limited3 and so is the offset4. Hence, the focus of the investigations lies on varying angles θd, the orientation angle. Additionally, the behaviour of the chosen scatterer for different frequency regimes is being looked at. For both fixed eccentricity and frequency, the total displacement field Re{φente −iωt} for a certain time t as well as the spatial distribution of its amplitude |φent| due to a cylindrical wave is shown in Figs.7a,b. In addition to these aforementioned evaluations, the scattering behaviour can be exploited well by investigating the amplitude of the entire wave at a certain distance from the scatterer for different frequency regimes. In this contribution, the chosen distance is r=4r1 referring to the centre of the outer layer. In Figs.8 and 9, these amplitudes are sketchedwithin a polar plot. Each polar plot contains the (normalised) amplitudes of the entirewave φent for the dimensionless frequencies kmr1 = {1,5,10}. Please note that these diagrams do not provide classical directivity patterns, as not the scattered wave φsc is plotted but the entire wave φent. Fig. 8. Polar plot of |φent| at a distance r=4r1 with d=0 (concentric scatterer) Ascanbe readily reconstructed fromFig.7a, theamplitudeof the incoming wave (of course) decreases with increasing distance from its source – thus showing a main difference between planar waves as given in Fig.4a. Hence, 3The core size is set constant to r0 = 0.6r1, a useful range should be 0.6r1 ¬ r0 ¬ r1 4The offset is d = 0.2r1 for all calculations performed, a useful range should be 0¬ d¬ (r1−r0) 1196 W. Weber, B.W. Zastrau Fig. 9. Polar plot of |φent| at a distance r=4r1 with d=0.2r1 Non-plane wave scattering from a single eccentric... 1197 a normalisation of the entire field at each point in free space lying on the circumferential line with distance r = 4r1 from the scatterers origin by the respective maximum value – that is |φent(r=4r1,θ,km)| |φent(r=4r1,θ,km)|max as done inWeber and Zastrau (2010) is not helpful if the relation of the entire wave to the incoming one is of interest. Due to this fact the authors decided to normalise each field point circumscribing the scatterer at a distance from r=4r1 measured from the origin by the amplitude of the incoming non-plane wave at this particular point |φent(r=4r1,θ,km)| |φin(r=4r1,θ,km)| Thus, if no obstacle was present5 this quotient would equal 1 for all field points, hence yielding to a circle with radius 1 in the polar plots in Figs.8 and 9.As a reference, Fig.8 shows the normalised amplitude of the entirewave on a circumferential line around the concentric scatterer (d=0). To show the remarkable influence of eccentricities to the wave scattering behaviour of TRC, the inner layer is then shifted by d = 0.2r1 and rotated by θd = {0, 1 5 π, 2 5 π, 3 5 π, 4 5 π,π} with respect to the positive x-axis, as defined in Fig.3b. As can be seen from Figs.9a-9f for a fixed core size and offset the scattering behaviour strongly depends on the frequency of the incoming non-plane SH wave. For low (dimensionless) frequencies, e.g. kmr1 = 1, the normalised amplitude of the entire wave φent oscillates around 1 with θ, where the oscillating “frequency”6 is quite low. As expected. the normalised amplitude does not vary that much for kmr1 =1. Additionally it clearly can be seen that the changes in the graphs curvature even decreasewith increasing orientation angle θd.This is dueto the fact that for increasing θd the scattering gets more and more forward, see Weber and Zastrau (2010), thus reducing the peaks in the backward direction at θ ≈ ±3 4 π. Investigating the medium frequency regime kmr1 = 5, yields that the “frequency” of the oscillatory structure of the plotted graph is higher than in the regime kmr1 =1. It is also apparent that the ratios |φent(r=4r1,θ,km)| |φin(r=4r1,θ,km)| > 1.5 5Or the inclusion would have the same material properties as the surrounding matrix 6Of the varying normalised amplitude 1198 W. Weber, B.W. Zastrau mainly occur for the orientation angles −1 5 π ¬ θd ¬ 1 5 π and 4 5 π ¬ θd ¬ 6 5 π. Thus, small deviances of the position of the inner core with respect to the x-axis should be omitted. For kmr1 = 10, it can be stated that the oscilla- tions within the scattering pattern have an even higher “frequency” as in the two cases investigated before. The second observation is that there seems no clear connection between small orientation angles θd and enlarged amplitudes of φent whichwas revealed for kmr1 =5. It is conjectured that the higher the dimensionless frequency kmr1 the more chaotic the scattering patterns are. Hence, further investigations are necessary. Concerning the shielding of the incoming wave, it can be stated that the entire waves amplitude in the forward direction (θ=0) as conjectured decre- ases with increasing frequency of the incoming wave. The previous numerical results further showed that within the design process the engineer should as- sume displacement amplitudes in the vicinity (r=4r1) of a single obstacle of radius r1 due to scattering approximately reaching twice the amplitude of the incoming wave, at least in the region θ=±(1 6 π,. . . , 5 6 π). Fig. 10. Polar plot of |φsc| at a distance r=4r1 with d=0.2r1, θd =π; (a) non-plane wave, (b) plane wave Obviously, the directivity pattern – that is, the polar plot of only the scat- teredfield φsc (normalisedby its respectivemaximumvalue)which is provided in Fig.10a – for the example of a non-plane wave highly differs from the one obtained for the plane wave case as given in Fig.10b. For this, two main re- asons can be identified: firstly, thewave chosen in this contribution propagates cylindrically outward with respect to its centre, thus more and more decre- asing its amplitude due to energy conservation. Hence, the incoming wave at the surface of the obstacle in the forward direction of the scatterer will ha- ve a lower amplitude than at the surface in the backward direction – and so Non-plane wave scattering from a single eccentric... 1199 will the scattered wave. These difficulty can be overcome by means of proper normalisation as shown in Figs. 9a-9f. The second reason for the remarkable changes in the directivity patterns and related graphs is the drastically diffe- rent distribution of the incoming wave field at the surface of the obstacle due to the additional curvature of the wavefront of the incoming cylindrical wave. On the other hand, this is exactly the reason for the investigations presented here. It is evident that with increasing distance R of the scatterer from the source of the incoming wave, with decreasing size of the obstacle or with in- creasing km the differences related to the curvature of the wavefront of the incoming wave will reduce. First signs of which can be seen from Fig.10a for the high-frequency regimewhen comparing themwith the results obtained for the plane wave case in Fig.10b. 5. Conclusions Efficient investigations concerning the scatteringbehaviourofnon-planeelastic SHwaves due to a single butmulti-layered eccentric obstacle can beperformed by the derived analytical approach. It opens up the possibility for fast calcu- lations of different geometric configurations as they occur with stochastic or fuzzy-stochastic analyses. Additionally, the obtained results may be coupled to standard procedures as BEM, thus avoiding – for special cases – the use of FEM or XFEM. Numerical calculations were performed for TRC and showed regions with a tremendous increase of the materials exposures due to an incoming wave even in the single scatterer case. These exposures exceed the ones known from concentric scatterers with comparable material properties by far. The conjec- ture that the scattering of non-planewaves leads to different results compared with incomingplanewaves could beconfirmed.Thesedifferencesdecreasewith the increasing ratio R/r1, where R is the distance of the scatterer from the source of the incomingwave and r1 the outer radius of the obstacle itself, and with increasing km. Hereby, the geometric property R/r1 in the limiting case R/r1 → ∞ directly leads to plane waves. It is thus apparent that especially in the vicinity of non-plane wave sources the procedure proposed in this con- tribution is essential for studying the wave scattering behaviour, whereas in the far field region a “reduced” approach with incoming plane waves – which means reduced numerical effort – may be accurate enough. Aprofoundknowledge concerning thewave scattering behaviour of compo- site materials at themicro- andmeso-scale, whichmay be obtained bymeans 1200 W. Weber, B.W. Zastrau of the presented procedure, allows the engineer to derive propermaterial mo- dels at the macro- or structural scale. On the other hand, materials having certain wave-scattering-properties may be constructed within an inverse pro- cess. However, this implies manufacturing of thesematerials under laboratory conditions. Hence, this work is a contribution to prevent structures from ha- zardous responses to dynamic loads. References 1. Achenbach J.D., 1973, Wave Propagation in Elastic Solids, North-Holland, NewYork 2. Cai L.-W., 2004,Multiple scattering in single scatterers, J. Acoust. Soc. Am., 115, 3, 986-995 3. Harrington R.F., 2001, Time-Harmonic Electromagnetic Fields, Wiley & Sons, NewYork 4. Lepenies I., 2007, Zur hierarchischen und simultanen Multi-Skalen-Analyse von Textilbeton, Ph.D. Thesis, Technische Universität Dresden 5. Meixner J., Schaefke F.W., 1954, Mathieusche Funktionen und Sphäroidfunktionen, Springer, Heidelberg 6. Morse P.M., Feshbach H., 1953,Methods of Theoretical Physics, McGraw- Hill, NewYork 7. Ortlepp R., Curbach M., 2009, Verstärken von Stahlbetonstützen mit te- xtilbewehrtemBeton,Beton- und Stahlbetonbau, 104, 10, 681-689 8. Pao Y.-H., Mow C.-C., 1971, Diffraction of Elastic Waves and Dynamic Stress Concentrations, Crane Russak& Co, NewYork 9. SkeltonE.A., James J.H., 1997,Theoretical Acoustics of Underwater Struc- tures, Imperial College Press, London 10. WatsonG.N., 1922,ATreatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge 11. Weber W., Zastrau B.W., 2009, On SH wave scattering in TRC – Part I: Concentric elliptical inclusion,Machine Dynamics Problems, 33, 2, 105-118 12. Weber W., Zastrau B.W., 2010, On the influence of asymmetric roving configurations on the wave scattering behaviour of TRC, 2nd ICTRC Textile Reinforced Concrete, 271-281 13. Weyrich R., 1937,Die Zylinderfunktionen und ihre Anwendungen, Teubner, Leipzig Non-plane wave scattering from a single eccentric... 1201 Niepłaskie rozpraszanie fal od kołowego mimośrodowo osadzonego wtrącenia w materiale – część I: fale SH Streszczenie W pracy przedstawiono analityczny formalizm zagadnienia rozpraszania fal na pojedynczym,wielowarstwowymwtrąceniumateriałowymw izotropowej, jednorodnej strukturze osnowypoddanej działaniuniepłaskich fal spolaryzowanychwpłaszczyźnie poziomej (fale typu SH). Założono, że wtrącenie ma kołowy kształt, a jego warstwy są rozmieszczone mimośrodowo. Zamieszczono rezultaty symulacji numerycznych na przykładzienowegomateriałubudowlanego, jakimjestbetonwzmacnianytekstyliami. Pokazano efekt ekranowania i wzmacniania fal SH w takiej strukturze. Manuscript received November 28, 2010; accepted for print February 18, 2011