NONLINEAR BULK ELASTIC WAVES IN LAYERED SOLID WAVE GUIDES
I.V.Semenova, G.V.Dreiden, A.M.Samsonov and K.R.Khusnutdinova*
A.F.Ioffe Physical Technical Institute,
St.Petersburg, 194021, Russia
*Loughborough University, Loughborough, LE11 3TU, UK
ABSTRACT
During the last decade we have performed theoretical investigations and successful experimental research demonstrating the
existence and main properties of bulk strain solitary waves in nonlinearly elastic solid wave guides. Following the topic we
present preliminary results on strain soliton propagation in layered wave guides made of PMMA. The obvious physical
relevance of data on strain soliton formation and evolution is primarily due to tentative generation of such waves in elastic
structures, that was not considered before in estimations of their strength, plasticity and damage threshold. The parameters of
these waves in layered structures may be of crucial importance for their operational integrity and robustness.
Introduction
A considerable attention has been paid recently to the development of various approaches for dynamic nondestructive testing
to be used for small to large laminated structural components in physics and engineering. Some methods have been studied
extensively, and their limitations initiated both analytical and experimental research into novel approaches.
The major problem in laminated structure behaviour consists in a sudden and irreversible delamination under dynamic loading
of any nature, and in a non-destructive detection of a delaminated area. The delamination may cause total failure of a structure
with no visible signal up to catastrophe. The laminated (or layered) material may be manufactured with layers oriented along or
across the elastic pulse propagation direction, bonded with a glossy or rubber-like glue located between (non-)linearly elastic
layers etc.
A simplest model of a bi-layer as a combination of two different materials is widely used to model a layered structure, which
integrity is defined by the quality of an interface. Delamination (cracking) may be caused by an abrupt change in material
properties and/or deformation under loading and, thus, in considerable stress concentration near interface.
A laminated wave guide may be made of elastic layers glued together with either hard substance, e.g. glossy adhesive, or with
a soft glue. In both cases it may cause very unusual behaviour of a structure due to a considerable difference in elastic moduli
of materials. Most of industrial laminated wave guides may be considered being between these two limiting cases.
The problem of the interface fracture is often considered on the base of representation of a solid as a lattice of elements, being
much larger than molecules, therefore the fracture might be described as vanishing links between elements, see [1-3]. In [1] a
laminated composite was modeled as a lattice with periodic structure, namely, a modified Toda lattice with an external linear
elastic term. This latter term was shown to lead to a new effect: an external spring does not allow the propagation of a Toda
soliton, but a wave packet only. These wave packets are stable with respect to mutual collisions as an “envelope soliton”, and,
consequently, the energy concentration on bulk soliton can be avoided by introduction of an external spring, that may be of
interest for engineering applications.
The results obtained already indicate the applicability of non-linear waves to detection of delamination areas in layered
structures and lead to the necessity of detailed analytical and experimental studies. An analysis becomes somewhat easier,
while the wave impact – more impressive, when a wave guide is considered instead of a (semi-)infinite structure. One of the
reasons consists in the fact that strain solitary bulk longitudinal waves in solids, being the least decaying elastic waves, see [4],
may provide delamination at any arbitrary and/or prescribed area of a wave guide due to considerable and well localized
concentration of stress energy.
Only a few papers have been published up to date devoted to theoretical analysis of solitary waves behaviour in layered media
and wave guides, see, e.g., [2] and [5], in which layers were assumed to be oriented across the wave propagation. In the
model it constitutes a Toda-lattice–like structure having well known properties.
Commonly used methods for nondestructive testing of adhesively bonded structures are based on ultrasonic technology and
include normal and oblique ultrasonic scans, resonant ultrasonic spectroscopy and Lamb-wave NDT [6]. These methods allow
to detect voids, delaminations, porosity, cracks, and poor adhesion in bonding layers. We are aimed to study the problem of
delamination in physical experiments with bulk elastic solitary longitudinal waves, which may cause delamination when
propagating in layered wave guides along an interface (bond layer).
What is a bulk elastic strain soliton?
Nowadays a term “soliton” becomes associated not only with the well known localized shallow water or light waves, but also
with their mechanical analog – a strain solitary wave in solids. Bulk strain soliton is a nonlinear long quasi stationary localized
strain wave that can be formed in a solid wave guide (having certain elastic and geometrical characteristics) from an initial
pulse (also having certain parameters). Being formed the soliton propagates along the homogeneous wave guide with almost
no changes of amplitude, shape and velocity.
Bulk soliton generation in a nonlinearly elastic wave guide can be initiated by a short and strong wave of deformation (a weak
shock wave, for instance) propagating along it. However, the curvature of a wave front may increase rapidly right up to
irreversible deformations, if the nonlinear elasticity is not balanced with wave dispersion inside the wave guide, having small,
finite but not an infinitesimal cross section. Moreover, in absence of balance caused by various factors, e.g., by the nonlinearity
of an opposite sign with respect to compression wave propagation, the initial pulse decays very rapidly. The balance of an
elastic nonlinearity of a wave guide material and a dispersion in a wave guide results in the bulk soliton generation, producing
a wave which is resistive to main mechanisms of elastic energy decay, quite stable and keeping a permanent shape when
moving for a considerable distance. Therefore bulk solitary waves may transfer elastic energy with almost no losses for long
distances.
Till now we have performed a synthetic theoretical and experimental research of the generation and succeeding propagation
of bulk strain solitary waves in various polymeric (made of polystyrene (PS) and plexiglas (PMMA)) wave guides (rods and
plates), see [7-13]. We have demonstrated the successful generation and propagation of strain solitons in rods, bars and
plates [7,8], observed the soliton focusing in a tapered wave guide [9], and the process of soliton reflection from free and
clamped ends of the rod [10]. The process of soliton dissipation has been studied in long and thin bars (over half a meter long)
[11], and the potential of application of solitons for measuring the 3d order elastic moduli of materials has been shown [12].
However, the research activities so far were focused mostly on fundamental aspects of the phenomenon related to soliton
formation and succeeding propagation in homogeneous wave guides. The influence of some particular inhomogeneities on the
wave parameters were analyzed mainly via numerical simulations (see [13] and references therein). Nevertheless, at present
the soliton behaviour in real structures gains major importance and attention.
Experimental details
The experimental technique used for generation and observation of strain solitary waves in different wave guides was
described in details in our previous papers (see, e.g. [7,8]) and is based on laser generation and optical (namely, by
holographic interferometry) recording of the waves under study.
The set-up allows to record a wave pattern inside and outside the transparent wave guide due to the wave induced density
variations, which lead to shifts of carrier fringes on the resulting holographic interferogram. Using this fringe shift value the
soliton amplitude A is then calculated by means of the following formula (for a bar as a waveguide):
A =
∆Kλ0
h ( n 1 − 1)( 1 − ν )
(1)
where ∆K is the fringe shift measured on the interferogram, λ0 is the recording light wavelength, n1 is the refractive index of a
solid, h is the bar thickness along the recording light path, ν is the Poisson ratio.
In experiments with homogeneous wave guide we used a 60 cm long PMMA bar with 1x1 cm cross section. For the
experiments with composite wave guides specific rod/bar combinations were applied, as it will be discussed below. The energy
of the laser pulse generating the shock wave was kept constant and controlled for each pulse to make sure that laser energy
variations would not be a source of soliton parameter variations.
For correct interpretation of data obtained for complex, inhomogeneous wave guides, it is necessary to understand in every
detail the behaviour and properties of the wave under study in homogeneous wave guides as well as on boundaries and
interfaces. That is why we will first outline briefly the main results obtained for the uniform homogeneous bar.
The results on soliton evolution in lengthy wave guides made of PMMA were first reported in [11]. Interferograms recorded in
different areas of the wave guide allowed to observe the evolution of the wave pattern in the bar and to visualize the process
of soliton formation and its consequent propagation along the wave guide. Figure 1 presents the typical interferogram of the
soliton in the PMMA bar at the long distance (345-400 mm) from its input. The soliton looks like a rather extended symmetrical
trough-shaped longitudinal wave, which is not followed by any tensile wave. The soliton propagates for a long distance
undergoing a very low dissipation (see Table 1).
∆K
Figure 1. Holographic interferogram of a strain soliton in the PMMA bar at the distance 345-400 mm from its input. Wave
moves from left to right. Fringe shift, representing the wave, is shown below the interferogram.
Table 1 summarizes the data obtained for solitons in lengthy PMMA bars: the fringe shift and soliton width measured on the
interferograms, the corresponding soliton amplitude calculated using eq. (1) and calculated magnitudes of soliton dissipation
decrement α.
Table 1. Soliton parameters variations due to dissipation in a bar made of PMMA.
Distance
(mm)
1: 70-125
2: 345-400
3: 545-600
mean value
Fringe shift
(∆
∆K)
1.00
0.88
0.65
Max strain amplitude,
-4
10
2.15
1.89
1.39
Width
(mm)
38.4
43.2
54.4
α 10 , cm
(distance for e-times decay)
1-2: 4.9 (204.1)
1-3: 8.6 (116.3)
2-3: 12.6 (79.4)
8.7 (114.94)
.
-3
-1
Thus, the solitary wave propagating in a uniform homogeneous polymeric bar is amazingly stable in comparison with any
conventional elastic bulk wave in polymers. Long nonlinear solitary strain waves exhibit extremely low decay, and may become
a convenient tool for nondestructive testing of inhomogeneous (layered, in particular) wave guides. The advantage of these
waves is due to their stability: since in the absence of inhomogeneity the wave parameters undergo no changes while it
propagates for long distances, then any variations observed can be directly attributed to the influence of inhomogeneity.
Layered bars can be considered as one type of inhomogeneous wave guides. And on the other hand they can be used to
study how the solitary wave propagating in these composite constructions affects their mechanical stability and bonding
characteristics.
In our experiments we used a complex wave guide, shown schematically in Figure 2. The soliton was first formed in a uniform
part (rod), 50 mm long, and then entered a layered part (bar) being glued to the rod by a thin layer of cyanide glue.
Transverse bonding layer
Longitudinal
bonding layer
Rod, 10 mm in diameter
Bar, with 10x10 mm cross section made of two
identical layers glued (or clenched) together.
Figure 2. Schematic of a layered wave guide
This first layer of glue exhibits a transversal bonding layer. Earlier ([14]) we have reported our studies on soliton propagation
through such layers. It was shown that the bonding layer seems to be “invisible” for the bulk strain soliton, which propagates
into the second PMMA rod as if the structure would be a homogeneous wave guide. This is predictable since the layer
thickness is much smaller than the soliton wave length. Perhaps the acoustical wave resistance of glue comparable with that
of PMMA contributes to this effect, too. Numerous (of the order of hundreds) shots performed on soliton propagation through
this layer showed that the transverse layer of cyanide glue is resistant to loads induced by a head-on impact of the solitary
wave with given above parameters (see Table 1). No signs of layer fracture or delamination have been observed.
Results and discussion
The following Figures 3-6 present strain soliton at the same distance from the input (70-120 mm), but in different conditions. In
Figure 3 – soliton in the homogeneous bar, in Figure 4 – soliton in 2-layer glued bar, in Figure 5 – soliton in a 2-layer bar, in
which layers are not glued, but clenched together, the initial uniform part (rod) was glued to both the layers symmetrically, and
in Figure 6 – soliton in a 2-layer bar similar to that in Figure 5, but the separation of layers is provided by thin (1 mm in
diameter) wires placed between layers in transversal direction.
Air
Bar
Air
Figure 3. Soliton in the homogeneous bar
Air
1st layer
Glue
2nd layer
Air
Figure 4. Soliton in the 2-layer glued bar
Air
1st layer
Contact
2nd layer
Air
Figure 5. Soliton in the 2-layer bar with layers clenched together
Air
1st layer
Air
2nd layer
Air
Figure 6. Soliton in the 2-layer bar with completely separated layers
The following tendencies in soliton characteristics variation during its propagation in the above mentioned complex layered
wave guides were driven from the analysis of the series of interferograms recorded. Table 2 presents the averaged values of
soliton parameters for the four specified configurations of wave guides:
Table 2. Averaged values of soliton parameters in layered bars.
Type of wave guide
Homogeneous bar
2-layer glued bar
2-layer bar with clenched layers
2-layer bar with separated layers
-4
Amplitude, 10
2.15
2.15
2.37
2.58
Width (FWHM), mm
20.0
20.2
19.2
18.5
The solitons in homogeneous and glued bars are identical (within the measurement accuracy), that means that a thin bonding
layer of cyanide glue between the layers does not introduce any inhomogeneity into the wave guide structure that would be
noticeable to the bulk soliton propagation. On the contrary, when the layers are just clenched, the soliton is slightly
amplificated , its width decreases. When the layers are well separated, the soliton amplitude rises more noticeably, and its
width decreases further. This effect is clearly due to the distribution of soliton energy into two wave guides with smaller cross
sections. These experimental results are in agreement as a whole with the soliton behaviour in wave guides with abrupt
change of cross section obtained earlier in numerical simulations.
Three consequent graphs of soliton propagation along a rod with abruptly changed cross section are shown in Fig.7 as a result
−2
of numerical simulation. An initial bell-shaped pulse of strain U = A cosh k ( x − Vt ) with the unit amplitude, A = 1 (part
I), approaches a sudden two-fold decrease of the cross section area. It leads to the soliton amplification in 19% and to the
formation of a second soliton with much lower amplitude (part II). The soliton velocity is proportional to its amplitude, and that
is why at longer distances the second soliton lags behind the first one and they become well separated (part III).
On the contrary, a smooth variation of cross section in a tapered rod provides the 47% increase of the soliton amplitude with
no second pulse generation, see [9,13].
There is a remarkable difference between the data provided by an asymptotic analysis of soliton propagation in a rod with
slowly varying cross section and in a rod with abrupt decrease of cross section area, as shown in Fig. 7. An asymptotics based
on smooth variation analysis and being formally applied to the soliton variation in layered bar, would result in amplitude
doubling and in considerable decrease of width, and seems hardly be applicable to our cases of delaminated, glued or splitted
wave guides. The results of numerical simulation shown in Fig.7 lead to the more realistic estimation of the amplitude value,
however a further theoretical analysis is required to achieve better agreement with experiments.
Clearly the bonded wave guide strength is defined at most by properties of a glue/solid contact. Two different contact models
were considered in a problem of wave propagation in a thin layer placed over an elastic half space, see [15]. It was found that
for a sliding contact an equation governed nonlinear longitudinal strain waves may be reduced to the well known BenjaminOno equation, while in a full contact problem an equation is integro-differential one and can be analysed numerically.
Conditions were found to generate nonlinear stresses in a contact zone sufficient to demolish a thin layer glued to the half
space, similar to a phenomenon observed in [16, 17 and 18]. A bond covered by elastic wave guides on both sides may
provide a similar behaviour, and the more rigid (glossy) a glue is, the less resistive it is with respect to any shear deformation.
Rubber–like bonds are expected to be more elastic and delamination resistive. Bonding layer thickness, its own elasticity,
contact properties, etc. are of considerable importance and their influence on delamination prevention should be studied in
detail.
Conclusions
The first results of experimental observation of bulk solitary wave behaviour in layered and delaminated wave guides allows to
conclude that the difference in amplitude values is detectable (ca. 20%) and may be used for delamination area recognition.
The soliton usage seems to be more attractive because it is the only bulk elastic wave which does not decay even at long
distances. However, the parameters of solitons in layered structures are to be refined in further theoretical and experimental
studies, because the accuracy of measurements and estimations may be of crucial importance for operational integrity and
robustness of layered structures.
Figure 7. Numerical simulation of the soliton evolution due to its transition
from thick to thin part of rod (from [13, p.193]).
Acknowledgments
The support of joint research in the framework of the Research agreement between the A.F.Ioffe Physical Technical Institute
and the University of Loughborough (project code MAJCE) is gratefully acknowledged.
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