Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

51, 1, pp. 183-194, Warsaw 2013

ANALYTICAL DESCRIPTION OF FRC SUBJECTED TO

TRANSIENT LOADS

Wolfgang Weber, Bernd W. Zastrau

Technische Universität Dresden, Dresden, Germany

e-mail: wolfgang.weber@tu-dresden.de

In many real-world applications, such as in civil engineering, non-periodic composite ma-
terials are used, whose dynamical behaviour is still not deeply understood, especially con-
cerning its wave scattering properties. In this paper, the scattering of a transient non-plane
elastic SH wave by an arbitrary arrangement of identical multi-layered and thus inhomo-
geneous obstacles is investigated. The obstacles are embedded in a homogeneous, isotropic,
and linear elasticmatrix of infinite extent.The solutionprocedure is analytical andwill then
be evaluated numerically for investigating a smallmaterial clipping of a real-world problem.

Key words: randommedia, multiple scattering, transient loads

1. Introduction

In many engineering applications, the dynamical behaviour of systems or whole structures is of
interest. In recent times, materials having certain designated properties have been developed.
These include compositematerials as they are used in e.g. optics and aerospace engineering.Due
to the production process,which takes place under laboratory conditions, these compositemate-
rials often are regular even at themicroscale, cf.Wächter andMichaelis (2010). This regularity
allows both analytical and numerical investigations of a single inclusion within the particular
matrix material. Concerning the interaction of the scatterer and the surrounding medium, ap-
propriate mechanical models have to be developed. The single obstacle and the surrounding
matrix in its vicinity are then put into a unit- or periodic-cell, where numerous unit-cells re-
present the whole material aperture, Parnell and Abrahams (2006). For doing this, appropriate
boundary conditions between adjacent cells have to be fulfilled, cf. Liu and Su (2010). For com-
bining these unit-cells, several theorems exist, of which the Bloch theorem may be one of the
mostwidespread ones, cf.Datta andShah (2008). Also, theories dealingwith effective (dynamic)
properties of periodic media exist. Apparently the first with respect to effective elastic moduli
goes back to Voigt (1910), others followed, e.g. Reuß (1929). For problems arising from these
theories cf. Achenbach (1973).
On the other hand, particularly in civil engineering some compoundmaterials should have a

periodic configuration–butdonothave in general.Mainly, these irregularities are inducedat the
building site, as these newmaterials are not produced under laboratory conditions, thus giving
the main difference to the periodic media mentioned before. One example of a new compound
material gaining more and more attention in civil engineering praxis but being produced on-
site surely is textile reinforced concrete (TRC), which will shortly be introduced in Section 2.
Although some models for evaluating the influence of the aforementioned irregularities exist,
these are limited to only a small number of so-called defects, cf. Maurel and Pagneux (2008).
Additionally, these are currently restricted to one-dimensional problems. Thus, they are not
applicable to the present problem.
In real-life applications, forecasts or simulations concerning the dynamic behaviour of com-

pound materials are made complicated by the fact that both the material and geometrical



184 W.Weber, B.W. Zastrau

properties of the structure and its constituents as well as the dynamic loads themselves may
vary, see the experimental investigations in Hummeltenberg et al. (2011), for example. The-
oretical investigations of such large scale problems will be carried out by means of numerical
procedures in most cases. These strategies often involve homogenization techniques including
their efficient numerical implementation. For recent developments see e.g. Parnell and Weber
(2010). However, in order to adequately describe the dynamic behaviour of a homogenized ma-
terial, profound insight into its behaviour at the micro- and meso-scale is necessary. As could
be shown in e.g. Weber and Zastrau (2009), the scattering behaviour of elastic SH waves de-
pends on the cross section of the respective single scatterer. Additionally, a layered build-up
of the reinforcement element does influence the scattering behaviour, too, see also Weber and
Zastrau (2009). Experimental investigations showed that the layered built-up of reinforcement
elements often includes eccentricities. This was the motivation for the investigations in Weber
and Zastrau (2011). The aforementioned contributions thus allow a precise description of the
(SH) wave scattering behaviour of a reinforcement element of several cross sections and setups.

What is still missing is the embedding of this analytical method in a framework allowing
the investigation of the wave scattering behaviour of an arrangement of several reinforcement
elements. Hence, in this paper the wave scattering behaviour of an arbitrary number of inho-
mogeneous inclusions in a homogeneous linear elastic and isotropic matrix is analyzed. Hereby
the scatterers aremodelled as double-layered obstacles, whereas both layers are concentric with
respect to each other. For this, additional motivations are given in Section 2. Afterwards, the
scattering of shear waves by means of a single scatterer is investigated in Section 3 by an ana-
lytical approach using time-harmonic waves. The extension to multiple scattering, that is, for
dealing with an arbitrary number of scatterers, which allows further analyses especially con-
cerning the interaction of these scatterers with each other, is given in Section 4. So far, the
investigations are based on time-harmonic waves, the numerical example given in Section 6 de-
als with transient loads. Themissing link is provided by Section 5, where the Fourier integral is
introduced and adapted to the needs within this contribution. In Section 6, all results obtained
so far are put together to numerically evaluate a small real-world example. Special focus is set
on the interactions of the several inclusions with each other and the embeddingmatrix. Finally,
a conclusion and future tasks are given in Section 7.

2. Textile reinforced concrete

As motivated in Section 1, the scattering of waves is of interest in many areas of engineering.
We will focus on civil engineering and herein especially on textile reinforced concrete (TRC), a
material having a non-periodic structure due to its production process on-site.

TRC is anewconstructionmaterial developedparticularly tofit theneedsof civil engineering
praxis. It consists of afine-grained concretematrixwith amaximumgrain size of approx. 0.6mm
andfibresmade of glass or carbonwhich are looped to uni- ormulti-axial reinforcement clutches
and then are worked into the fine-grained concrete matrix, cf. Schladitz et al. (2011). Hence,
the rovings of TRC allow a well-directed reinforcement matching the respective load cases as
examined in Ortlepp et al. (2011), thus giving the main difference to fibre reinforced concrete,
that is, a concrete matrix with arbitrarily orientated short fibres of steel, glass, or synthetics.
These rovings, which are often called fibres as well, consist of 800 up to 2000 filaments, each
with a diameter of approx. 13.5µm.Aview in the direction of the fibres as shown inFig. 1 gives
rise to the assumption that the single filaments do participate differently in the load bearing
behaviour. In detail, outer filaments have a better interconnection to the surrounding matrix
than the inner ones, cf. Scheffler et al. (2009). For somemodels concerning the decreasing degree
of bonding see Lepenies (2007). The glass or carbon fibres and their specific bond behaviour



Analytical description of FRC subjected to transient loads 185

Fig. 1. Setup of a roving with elliptical cross section as discussed inWeber and Zastrau (2009)

ensure an improved ductility at least for static load cases. Also, TRC has a high strength.
Hence, slender structures as plates show (i) an increased load bearing capacity and (ii) higher
deflections than conventionally reinforced structures, see also Fig. 2. The former mentioned
properties clearly are an improvement concerning civil engineering applications. The latter has
the advantage of announcing critical exposures which may lead to collapse if further increased.
With the conventional concrete, the failure is more or less abrupt due to the quasi brittle
behaviourof this constructionmaterial.With the sponsorshipof theGermanScienceFoundation,
the static behaviour of TRCwas thoroughly investigated within the twoCollaborative Research
Centres SFB 528 and SFB 532, see e.g. Schladitz et al. (2011). However, the dynamic behaviour
of TRC is still not understood completely, and so is its behaviour concerning the scattering of
waves.

Fig. 2. Quasi-ductile behaviour of TRC

3. Wave scattering by a single inhomogeneous obstacle

The specific build-up of TRC accounts for the investigations within this contribution. Hence, an
appropriatemechanical model has to be developed which fits the needs occuring with the given
task. As introduced in Section 2, TRC is a composite material made of fine-grained concrete
and reinforced with glass or carbon fibres. Due to the radially stratified structure of these
rovings, which is due to a decreasing degree of bonding between the filaments constituting the
fibres, they aremodelled asmulti-layered and thus inhomogeneous concentric circular inclusions.
Additionally, thematrix behaviour in the vicinity of the rovings will differ from the undisturbed
matrix whichmay be amotivation for introducing an additional layer, commonly known as the
interphase, cf. Scheffler et al. (2009). For showing the general approach, it is sufficient tomodel
the single roving – which acts as an obstacle for the incoming (non-plane) wave and thus will
scatter it – as a double-layered scatterer. The extension to a multi-layered obstacle is straight
forward. Herein the inner layer is the roving corresponding to the inner filaments, the outer
layer is motivated by the interphase between the outer filaments and thematrix in the vicinity
of the fibre. When dealing with other materials, e.g. mortar and steel fibres, this zone is also
called interfacial transition zone. As mentioned before, the focus in this contribution is set on



186 W.Weber, B.W. Zastrau

concentric scatterers. For the behaviour of an eccentric scatterer see e.g. Weber and Zastrau
(2011). As the concentric scatterer is a special case of an eccentric scatterer, we will restrict
ourselves to the basic equations in what follows.
The so-called Navier’s equation under absence of (time-dependent) body forces reads as

µ∆u+(λ+µ)∇∇·u= ̺ü (3.1)

with ∆ = ∇2, u = [u,v,w]T displacement vector, λ, µ Lamé constants and ̺ density of the
material at hand. It is valid both for pressure (P) and shear (S)waves. For a detailed description
of these wave types see e.g. Graff (1991).
In the next Sections, a SH wave is being looked at. Hereby, a polarization is assumed such

that the particle motion is parallel to the axis of the matrix inclusion, whose properties do not
change along its axis as shown in Fig. 3a. A concentric layered elastic circular cylinder which is
embedded in a linear elastic and isotropic medium of infinite extent as depicted in Figs. 3a,b, is
thus considered.

Fig. 3. Problem configuration, concentric circular cylinder case; (a) 3D-view, (b) 2D-view

The resulting waves are sought due to the acting of a time-harmonic incident SHwave. The
only nontrivial displacement component for a SH wave scattering problem is the out of plane
displacement w, if theSHwavepropagates in the xy-plane and the axis of the obstacle is parallel
to the z-axis as shown in Fig. 3a. Hence, a time-harmonic ansatz also for the displacement is
advantageous. Additionally, a so-called complex amplitude φ is introduced. Its absolute value
gives the amplitude of the displacement w, whereas its real part gives the displacement at a
certain time instance, Graff (1991). Introducing this ansatz into Navier’s equation (3.1) yields
the Helmholtz equation

∆φ+
ω2

c2T
φ=0=∆φ+k2φ (3.2)

Herein the wave number k was introduced.
In order to calculate the waves in the matrix, the interphase and the fibre, the solution

to the Helmholtz equation in a circular coordinate system is sought. Using the Laplacian in
this coordinate system and the ansatz φ(r,θ) =R(r)Θ(θ), which separates the radial and the
angular part of the solution, two differential equations are obtained, cf. Morse and Feshbach
(1953). The solution to the Helmholtz equation in polar coordinates is a linear combination
of cylindrical wave functions consisting of simple harmonics as the angular factor and Bessel
functions of various kinds as the radial factor

Rn(r)= b1nJn(kr)+ ib2nYn(kr)= c3nH
(1)
n (kr)+ c4nH

(2)
n (kr) (3.3)

cf. Weyrich (1937). Herein H
(1),(2)
n (kr)= Jn(kr)± iYn(kr) are the Hankel functions of the first

and second kind, respectively. With the time-dependency exp(−iωt) used in this contribution



Analytical description of FRC subjected to transient loads 187

and the well-known asymptotic expressions of the Hankel functions according to e.g. Weyrich
(1937), Harrington (2001), it is

exp(−iωt)H(1)n (kmr)∼

√
2

πkmr
exp
[
i
(
kmr−

π

4
−n
π

2
−ωt
)]
[1−O(r−1)]

exp(−iωt)H(2)n (kmr)∼

√
2

πkmr
exp
[
i
(
−kmr+

π

4
+n
π

2
−ωt
)]
[1+O(r−1)]

(3.4)

Thus, waves moving outward are represented by Hankel functions of the first kind H
(1)
n (kr).

For amore detailed description of Bessel- and Hankel functions seeWeyrich (1937). Herein and
with other parameters the subscripts m, iph, and fi hold with the matrix, the interphase and
the fibre, respectively.
The treatment of the single roving as a layered inclusion (and hence as a scatterer for the im-

pingingwaves) wasmotivated in Sections 1 and 2.Based on this, radially stratified configuration
ansatz functions have to be derived now. They are composed of wave basis functions describing
the general behaviour of the respective waves and wave expansion coefficients which scale the
contribution of each basis function. For a detailed discussion of the respective ansatz function(s)
within eachmaterial involved in the wave scattering process one is referred to e.g. Graff (1991),
Cai (2004). In Section 6 we will focus on a transient wave propagating through an arbitrary ar-
rangement of identical but inhomogeneous scatterers. As only the waves in the matrix material
will be looked at, only the equations describing thewave fields in thematrix are of interest here.
In detail, these are φin for the incoming wave and φsc for the scattered one

φin(km,r,θ)=
∞∑

n=−∞

AnJn(kmr)e
inθ

φsc(km,r,θ)=
∞∑

n=−∞

BnH
(1)
n (kmr)e

inθ

(3.5)

However, for determining thewave expansion coefficients An and Bn, boundary conditions have
to be fulfilled across all interfaces r= {R1,R2}, which also involve the wave fields in the other
materials, that is, within all layers of the single scatterer, cf. Weber and Zastrau (2011). This
yields a system of equations for each n and each frequency ω.
Theentirefieldoutside the scatterer,whichwill beevaluated in thenext section, is expressible

as

φent =φin+φsc =
∞∑

n=−∞

[AnJn(kmr)+BnH
(1)
n (kmr)]e

inθ (3.6)

If the single obstacle is not concentric, extensions as shown inWeber and Zastrau (2011) are
necessary. Bymeans of a similar procedure, the extension to inclusions of amore general shape,
that is, ellipses, is possible and was shown in e.g.Weber and Zastrau (2009).

4. Wave scattering by an arbitrary arrangement of obstacles

In the precedent Section, the scattering of elastic SHwaves by a single inhomogeneous obstacle
was derived. Inmost cases, real-world applications involve numerous scatterers, of course. Thus,
a proper procedure for using the insights gained from the single-scatterer-case for the solution
of more general tasks is needed. As motivated in Section 2, the focus of this contribution is set
on aperiodic media which do not allow the use of the concept of periodically arranged unit-



188 W.Weber, B.W. Zastrau

-cells. One of the first – if not the first – approaches to deal with several obstacles goes back to
von Ignatowsky (1914), where an infinite grating of rigid obstacles was looked at. Also, concepts
involving average values of wave functions due to a couple of obstacles have been suggested,
see Foldy (1945). The restriction to isotropic scattering was overcomewith e.g. Twersky (1962).
The concept of infinite gratings by von Ignatowsky (1914) was put on a more general basis
by Waterman (1976). The concept presented there implies that for an arbitrary time t all
scatterers i do have a non-vanishing incoming field φ̂in,i. Thus, this concept is formulated for
the steady-state which actually is not a problem as shall be seen in Section 5.
In general, the total wave field in thematrix material consists of the incomingwave and the

waves scattered by all N scatterers

φent =φin+
N∑

i=1

φsc,i(φ̂in,i) (4.1)

cf. Waterman (1976). For the special case N =1, equation (3.6) is obtained. Herein, the wave
scattered from scatterer i is dependent on all the waves impinging upon this scatterer i. The-
se are the global incoming wave φin,i at the scatterer i and the scattered waves of all other
(N −1) scatterers. Thus,

φ̂in,i =φin,i+
N∑

j=1,j 6=i

φsc,j(φ̂in,j) (4.2)

Hence, the associated single scatterer solution of each obstacle has to be known. As all
scatterers are assumed to be identical, these solutions are identical, too. In fact, the respective
solutionwas presented in Section 3.Within the expressions for thewave fields exemplarily given
with equations (3.5), the global coordinate system has to be replaced by the local coordinate
system of the scatterer i. Additionally, the wave expansion coefficients An of the incoming
wave have to be expressed in this local coordinate system as well to take into account different
phases of the incoming wave. With reference to equation (4.2), it is necessary to (re-)express
the scattered wave of the scatterer j in the local coordinate system of the scatterer i. Thus,
coordinate transformations are inevitable, see also Fig. 4. For this purpose, use of the addition
theorem of Bessel functions

Z
(p)
n (kri)e

in(θi−θij) =
∞∑

m=−∞

Z
(p)
n+m(krj)Jm(kd)e

in(θji−θj) (4.3)

proved inMeixner and Schaefke (1954) ismade.Herein, Z
(p)
n (kr) with p=1, . . . ,4 is theBessel,

Neumann or Hankel function of the first or second kind and order n. It is also known as Graf’s
addition theorem.Bymeans of this addition theorem, the particular two coordinate systems can
be converted to each other. Within equation (4.3) the distance between two scatterers i and j
is measured by d. The line connecting both scatterers has an angle θij in the scatterers i local
coordinate system and θji in the coordinate system of the scatterer j, respectively. As can be
seen from Fig. 4, the same holds with the coordinates of an arbitrary field point P which are
(ri,θi) and (rj,θj), respectively. With these transformations and the boundary conditions, the
wave expansion coefficients of the scattered and refracted waves can be determined. The entire
field in the matrix region due to the incoming wave and the scattered waves of all N obstacles
then follows from equation (4.1).
The precedent equations for describing the scattering of waves by an arbitrary distribution

of N identical inclusions are based on the assumption that the positions of single obstacles are
known deterministically. If this is not the case or if the inner structure of each obstacle may
vary, more detailed investigations have to be made. One mathematical framework to perform
these investigations is the fuzzy analysis. It was applied to the scattering behaviour of a single
eccentric obstacle inWeber et al. (2013).



Analytical description of FRC subjected to transient loads 189

Fig. 4. Definition of the variables for Graf’s addition theorem

5. Modelling the transient wave

For given circular frequencies ω, the scattering of time-harmonic elastic SHwaves byanarbitrary
arrangement of N identical inhomogeneous concentric circular obstacles was derived in the
precedent Sections. These steady-state solutions are connected with both the free vibrations
and transient responses of the systemvia theFourier transformmethod, cf.Morse andFeshbach
(1953). Hence, once the time-harmonic problems are solved by themethods of Sections 3, 4, the
propagation of a pulse may be obtained by a Fourier integral.

In general, the load case “pulse” consists of a single principal impulse of an arbitrary shape,
which additionally is of a relatively short duration. Although the response of the system to an
impulsive load is sought in the time-domain generally, it is often advantageously to perform the
respective analyses in the frequency-domain.The investigations of thewave scattering behaviour
presented within Sections 3, 4 took place in the frequency-domain on a natural way. Thus, the
results obtained so far can be used one by one for the following investigations. The general ap-
proach is to (i) express the applied loading in terms of harmonic components, then (ii) evaluate
the response of the structure to each of those components (what was done in the precedent Sec-
tions), and finally (iii) superpose the harmonic responses to obtain the total structural response,
see e.g. Clough andPenzien (1995). Obviously at least thematerial behaviour at the frequencies
dominating the impulse-load has to be known. For real-life applications, this knowledgemay be
gained from both time-harmonic experiments and frequency analyses of the materials rebound
to well-defined short-time exposures under laboratory conditions, see de Andrade Silva et al.
(2011).

In order to apply the periodic-loading technique to arbitrary loadings the Fourier series
concept obviously has to be extended, thus allowing the treatment of non-periodic functions,
cf.CloughandPenzien (1995). Theclue is to let theperiodof the load, overwhich the integration
has to take place, go against infinity so that no spurious repetitive loading occurs. In the limit
case

p(t)=
1

2π

∞∫

−∞

P(ω)e−iωt dω P(ω)=

∞∫

−∞

p(t)eiωt dt (5.1)

is obtained, where the first equation is known as the inverse (exponential) Fourier transform
and the latter one as the direct (exponential) Fourier transform, respectively, see also Bronstein
et al. (2000). Herein, the arbitrary loading p(t) can be expressed as an infinite series of simple
harmonics having known complex amplitudes. The complex amplitude again contains informa-
tion concerning the amplitude and phase of the associated oscillation, see also e.g. Morse and
Feshbach (1953).



190 W.Weber, B.W. Zastrau

The necessary conditions for the existence of the direct Fourier transform are given with the
Dirichlet-Jordan criteria, one of which is that the integral

∞∫

−∞

|p(t)| dt<∞ (5.2)

is finite – which clearly is the case if the loading p(t) has a finite duration. Concerning the
modulated signal, several waveforms are applied in engineering praxis. For some pulse shapes
that are used in Structural Health Monitoring see Schulte (2011), Hennings and Lammering
(2010). For these practical applications, the infinite (exponential) Fourier transforms have to be
expressed in their approximate finite forms.

6. Numerical example

In the following, the scattering of a SH pulse by an arbitrary arrangement of identical elastic
circular concentric obstacles is evaluated bymeans of the insights of the precedent sections. As
the focus is set on the interaction of the scatterers, a plane SHwave is applied. For a planewave,
the wave expansion coefficients An as introduced with equation (3.5)1 follow from the Jacobi-
Anger expansion to An = i

n. The material parameters forming the basis for the calculations
presented here are given with Table 1 andwere taken from SFB 528. Concerning the number of
obstacles, N =3was chosen to maintain a small example allowing basic insights.

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

The transient load was chosen as

p(t)=

2ω0∫

0

sin2
(πω
2ω0

)
w(ω)e−iωt dω (6.1)

which gives rather distinct pulses and uses as much as possible of the computed data from the
frequency-domain according to Boström et al. (1994). The corresponding signal in the time-
domain is plotted in Fig. 5a and is quite similar to a sinc-pulse, the frequency spectrum of the
transient load is plotted in Fig. 5b.

The plane SH pulse impinging upon the surface of the arrangement of scatterers has unit
a amplitude and propagates in thepositive x-direction as can be seen from Fig. 6. The N =3
identical elastic circular concentric scatterers of radius R2 =1 are located at

x̂scatterers = {(−5,−5),(0,5),(5,0)} (6.2)

wherein normalizationwith thewavelengthΛ0 =2πcT/ω0 connectedwith themid-frequency ω0
of equation (6.1) was applied. Figure 6a shows the geometry of the problem at hand at the time
instant τ =2. Herewith, τ is a dimensionless time given with

τ =
cT
R2
t (6.3)



Analytical description of FRC subjected to transient loads 191

Fig. 5. (a) Applied pulse (Boström-pulse) in the time-domain; (b) frequency spectrum of the applied
pulse

Fig. 6. φent due to a pulse for different dimensionless times τ =(cT/R2)t

With respect to the forthcoming analyses it is advantageous to enumerate the obstacles. In this
respect, the sequence follows the arrival time of the pulse, see equation (6.2).

Within Fig. 6, the contour plots of the displacement field for various values of τ are shown.
The pulse is applied at τ =0 at the left hand side of the plot. At τ =2, the so-far undisturbed



192 W.Weber, B.W. Zastrau

pulse impinges upon the surface of the 1st obstacle thus inducing both a reflected and refracted
wave. As can be seen fromFig. 6b, at τ =7 the reflected wave radially moves outward. On the
other hand, the former undisturbed pulse has a decreased amplitude within the shadow region
of scatterer 1. The current time instant τ = 7 is also characterized by the impinging of the
impulsive load upon the surface of scatterer 2. The reflected wave of scatterer 2 can clearly be
identified fromFig. 6c at τ =10. It is also apparent that the reflectedwave of scatterer 1 radially
moves outward further, thus decreasing its amplitude due to energy conservation. In the next
Fig. 6d, the weakened pulse has already met scatterer 3 at τ =14. Please note the remarkable
interference of the reflected waves of scatterers 1, 2, and 3 directly in front of the 3rd scatterer
leading to exposures exceeding the one of the incoming pulse enormously. If the incident wave
leads to exposures comparable to the strength of the material, these interference effects could
induce cracks and hence could cause a local failure.

7. Conclusion

The derived analytical approach is capable of investigating wave scattering problems as they
arise in e.g. civil engineering, mechanical engineering and related disciplines. It allows arbitrary
positioning of the single obstacles. Hence, the method is applicable in particular to materials
producedon-site.Asnomeshingandother time-consumingproceduresare required fornumerical
investigations, this approach is very efficient and may be – for special cases – an addendum or
even an alternative to numerical methods such as FEM or XFEM. Thus, the approach given
here allows in-depth analyses of compositematerials such as TRC,which lead to deeper insights
into the material behaviour at the micro- and meso-scale. By means of these insights, proper
mechanicalmodels allowing efficient simulationsat themacro- or structural-scalemaybederived.
With engineering structures, impulsive loads are events occurring seldomly, thus giving re-

spective security margins with respect to the strength of the chosen material. On the other
hand, the numerical example within this contribution showed that within the scattering process
of waves arising from impulsive loads the already huge exposures to the material at hand may
even be further increased due to the interference of the incoming and scattered waves. Hence,
a detailed investigation of the material is vital. The proposedmethod also allows substantiated
forecasts concerning testing of both damaged and undamaged materials. These forecasts are
necessary to evaluate the measuring results obtained from non-destructive testing (NDT) and
themonitoring data from non-destructive evaluation (NDE) processes, e.g. in Structural Health
Monitoring. Concerning the investigation of damaged structures, the voids/cracks are model-
led as circular cavities, see Deng and Yang (2011). These cavities are degenerated cases of the
inhomogeneous elastic obstacles dealt with here, cf. Weber and Zastrau (2009).
Future investigations are necessary to allow predictions concerning the wave scattering be-

haviour of an arbitrary number of different inclusions. These different inclusions may be caused
by chemical processes within concrete hardening which may additionally lead to eccentricities
within the fibre and the surrounding interphase. Also, focus was set on SH waves so far. Hen-
ce, the scattering of both SV and P waves by arbitrary reinforcement arrangements has to be
studied in the future.

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Analityczny opis właściwości kompozytów wzmacnianych włóknami poddanych

obciążeniom zmiennym

Streszczenie

W rzeczywistych obiektach, np. w inżynierii lądowej, zastosowanie znajdują materiały kompozytowe
o strukturze nieperiodycznej. Dynamika takich elementów nadal pozostaje niedostatecznie rozpoznana,
zwłaszcza w kontekście właściwości rozpraszania fal. W prezentowanej pracy omówiono problem roz-
praszania przejściowych, niepłaskich fal sprężystości spolaryzowanych poziomo (typu SH) w dowolnym
układzie identycznych, wielowarstwowych, i przez to niejednorodnych, ekranów tłumiących. Założono,
że ekrany te zostały wbudowane w jednorodną, izotropową, liniowo-sprężystą osnowę o nieskończenie
wielkich rozmiarach. Zaproponowano analityczne rozwiązanie problemu, które posłuży symulacjom nu-
merycznym efektu słabego obcinania fali wmateriale wybranego obiektu rzeczywistego.

Manuscript received January 11, 2012; accepted for print April 23, 2012