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COMPARISON OF EQUIVALENT PLATE MODELS USING
WAVENUMBER APPROACH
U. Arasan1,3,4Fabien MARCHETTI1Fabien CHEVILLOTTE1
Franc¸ois-Xavier BECOT1Kerem EGE2Dimitrios CHRONOPOULOS3
Emmanuel GOURDON4
1Matelys - Research Lab, F-69120 Vaulx-en-Velin, France
2Universit´
e de Lyon, INSA-Lyon, 69621 Villeurbanne, France
3Institute for Aerospace Technology, University of Nottingham, NG7 2RD, UK
4Universit´
e de Lyon, ENTPE, 69518 Vaulx-en-Velin, France
arasan.uthayasuriyan@matelys.com, fabien.chevillotte@matelys.com
ABSTRACT
Mutli-layered structures (such as sandwich panels) are
commonly used in engineering applications for the im-
proved sound comfort and noise reduction. Finite Element
modelling of these layered structures would often results in
increased total number of degrees of freedom which would
lead to high computation time. An alternative to the full
modelling of multi-layered system is the use of equivalent
plate models to simulate the multi-layered system as a sin-
gle layer. These models would aims at reducing the num-
ber of degrees of freedom in finite element models. Ap-
plicability of these models would be limited up to certain
frequency range due to underlying assumptions of wave
propagation in each layer. In this work, these equivalent
plate models are compared using wavenumber analysis.
Sandwich panels of different types (soft and stiffer cores)
are considered for this comparison study and behaviour of
each model are analysed.
1. INTRODUCTION
Multi-layer or sandwich composite panels which exhibit
high stiffness with light weight are commonly used in the
civil and transportation industries. Since the multi-layered
system comprises of diversified materials, FE modelling
requires suitable mesh types based on the material used
in each layer, which would increase total mesh count and
thereby increasing the computer power required to do the
calculation. Equivalent plate models are used as an alterna-
tive which condense the behaviour of the multi-layer sys-
tem in to a single layer material. This would reduce the
computational power significantly when it is used in FE
modelling.
In this paper, two frequently used equivalent plate mod-
els : Guyader model [1, 2] and RKU model [3–6], are
discussed briefly with their underlying assumptions. Us-
ing wavenumber analysis, these models are compared to
discuss their validity over different sandwich plate types.
Finally, the influences of symmetric and antisymmetric
motions of the sandwich plates are studied as the present
equivalent plate models exhibit only antisymmetric motion
of the system.
2. WAVENUMBER ANALYSIS OF PLATE
MODELS
2.1 Single isotropic layer
Wave propagation with respect to thin and thick plate the-
ories for a single isotropic layer is briefly explained in this
section. Following the usual notations, the propagating
wave number solution for the thin and thick plate theories
are given as:
kthin =kb=sωrms
D,(1)
kthick =±s1
2k2
s+k2
r+q4k4
b+ (k2
s−k2
r)2,(2)
where, ks=ωpms
G∗hand kr=ωqIz
Drepresent the
corrected shear and membrane wavenumbers respectively.
Here, ω= 2πf is the circular frequency, Dis the bend-
ing stiffness, msis the mass per unit area, G∗is the cor-
rected shear modulus and Izis mass moment of inertia of
the layer.
From Eqn. (2), it can be seen that kthick approaches
kthin and kswhen ω→0and ω→ ∞ respectively. The
same can be observed, for example, in Fig. 1 for a plas-
terboard plate of 25 mm. From these two asymptotes of
kthick, it is inferred that the thin plate assumption is valid
up to certain frequency after which the thick plate theory
must be considered to include the shear motion of the plate
along with bending motion. If the plate is considerably
thick and/or soft in terms of shearing motion, the thick
plate theory need to be applied. In this context, an ana-
lytical frequency limit for the thin plate theory is given in
the Ref. [7].
2.2 Equivalent plate theories
By condensing the free vibration behaviour of the multi-
layer system, equivalent plate models provide the sin-
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Figure 1. Propagating wavenumbers of a Reissner-
Mindlin plate and its asymptotic behaviours. The plate
is made of plasterboard with h= 25 mm, E= 3 GPa,
ρ= 700 kg/m3and ν= 0.22.
gle layer material properties to simulate the same vibro-
acoustic behaviour of the multi-layers. Guyader [1, 2] and
RKU [3–6] are the two equivalent plate models which are
widely used in engineering applications. Though both the-
ories provide equivalent isotropic thin plate properties of
a single layer, Guyader model assumes both bending and
shear wave propagation in each layer whereas RKU model
assumes propagation of bending and shear waves only in
the outer and core layers respectively. On the applicabil-
ity front, Guyader model can be used for any number of
multi-layer system while RKU model is used mainly on
three-layer sandwich plates.
Equivalent dynamic bending rigidity (Deq(ω)) is com-
puted from these models and substituted in Eqn. (1) to find
the propagating wavenumber as Deq corresponds to equiv-
alent single layer isotropic thin plate. In the following sub-
sections, these equivalent plate models are compared in the
wavenumber domain for different configurations of three-
layer sandwich plates. Sandwich plates from Tab. 1 are
considered for comparing these two models.
Skin Soft core Stiff core
h(mm) 5 10 50
ρ(kg/mm3)2780 200 2300
E(GPa) 71 0.1 30
ν0.3 0.3 0.3
Table 1. Material properties of isotropic layers used in this
article.
In Tab. 1, h, ρ, E and νare thickness, density, Young’s
modulus and Poission’s ratio of the isotropic layer respec-
tively.
2.2.1 Sandwich plate with stiff core
In this section, a three-layer sandwich plate with stiff core
(concrete) bonded by two aluminium skins are consid-
ered for the comparison study and the equivalent bend-
ing wavenumber computed by different models are pre-
sented in Fig. 2. Along with Guyader and RKU models,
added stiffness model is also presented here for compari-
son. In this model, Deq is computed by adding the indi-
vidual bending stiffness of each layer that is obtained with
respect to the neutral axis of the multi-layer plate. This
is similar to the approach followed for beams with multi-
layers, as described in the Ref. [8].
Figure 2. Wavenumbers of equivalent plate theories: sand-
wich plate with stiff core.
From Fig. 2, it is observed that the low frequency re-
sponse of Guyader model matches well with the added
stiffness model whereas RKU model fails to predict the
correct low frequency response. This is mainly due to the
assumption made in RKU model that the core is assumed
to have shearing motion and not bending motion. Since the
core is made of stiffer material in this example, the bend-
ing contribution from the core layer is ignored by the RKU
model which results in deviated response in low frequen-
cies with other equivalent plate models.
2.2.2 Sandwich plate with soft core without
thickness-stretching effect
For the sandwich panel with soft core, the wavenumbers
obtained from the equivalent plate theories are presented
in Fig. 3. Although the core material is soft, the thickness-
stretching effect (or compressional effect) is ignored for
this example. In other words, only anti-symmetric motions
(bending, shear and membrane) are considered for this ex-
ample and contribution from symmetric motion is ignored.
It is observed from Fig. 3 that both Guyader and RKU
models are in good agreement with each other through-
out the entire frequency range. Another observation (from
Figs. 1 and 3) is that the asymptotic behaviour of the prop-
agating wavenumber of a three-layer sandwich plate is dif-
ferent from that of the isotropic single layer plate. Fur-
ther, three types of region, associated with asymptotic be-
haviours, are also observed as described by Boutin and
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Figure 3. Wavenumbers of equivalent plate theories: sand-
wich plate with soft core.
Viverge [9]. At low frequencies, the sandwich wave prop-
agation is mainly controlled by the global bending [9]
whereas higher frequency region is controlled by the in-
ner bending behaviour of the skins. The transition phase
from lower asymptote to upper asymptote is influenced by
the shearing behaviour of the core. The bending rigidity
expressions to calculate these wavenumber asymptotes are
given in Eqn. (3) for symmetric sandwich plate with soft
core [9].
Dlow =D18 + 12h2
h1
+6h2
2
h2
1;Dhigh = 2D1,(3)
In the above equation, ‘1’ and ‘2’ represents skin and
core respectively. The equivalent shear wavenumber is ex-
pressed as,
keqshear =ωrM
G2ht
,(4)
where Mand htare the total mass per unit area and total
thickness of the sandwich plate respectively.
Based on these asymptotic behaviours at three regions
(low, high and transition), a simple equivalent plate model
can be formulated to compute the dynamic bending stiff-
ness (which would be valid for the entire frequency range)
of a three-layer sandwich plate. This model will be pro-
posed in a journal article in the near future [10] and it will
be easier for implementation compared to other equivalent
plate models.
2.2.3 Sandwich plate with soft core with
thickness-stretching effect
In this section, wavenumbers corresponding to anti-
symmetric and symmetric motions of three-layer sand-
wich plates are compared to analyse the influencing effect
of symmetric motions in the vibro-acoustic computations.
For this purpose, first anti-symmetric (A0) and symmetric
(S0) wavenumber solutions for the three-layer sandwich
panel are computed from the Lamb wave theory [11].
For the three-layer sandwich plate with stiff core, it is
observed from Fig. 4 that symmetric wavenumber (corre-
spond to compressional or dilatational motion) starts to
propagate only after 20 kHz whereas the anti-symmetric
wavenumber (correspond to bending and shear motions) is
propagative throughout the frequency range. The reader
may note that first anti-symmetric mode solution from
Lamb wave theory is well captured by the Guyader model.
Figure 4. Anti-symmetric (A0) and symmetric (S0)
wavenumbers of the first mode of a sandwich plate with
soft and stiff cores, obtained from Lamb wave theory.
Therefore, for the audible frequency range, the existing
equivalent plate models are sufficient to compute the vibro-
acoustic indicators if the sandwich core is made of stiffer
materials.
In case of a three-layer sandwich plate with soft core, it
is observed from Fig. 4 that first symmetric wavenumber
starts to propagate from around 4 kHz and progressively
reaches the first anti-symmetric wavenumber (or equiv-
alent bending wavenumber from Guyader model) in the
higher frequencies.
As the existing equivalent plate models assume constant
transverse velocity throughout the plate thickness, these
models would not be valid after the frequency at which the
symmetric wavenumber starts to propagate. If the core is
very soft, the symmetric wavenumber would start to prop-
agate from the low frequencies and therefore, for this kind
of system, the existing equivalent plate models would not
be suitable candidates to condense or compute the vibro-
acoustic behaviours of the multi-layer plates.
In this context, a new condensed model could be devel-
oped by including both the symmetric and anti-symmetric
admittances of the multi-layer plates by following the work
by Dym and Lang [12, 13] to compute the admittances.
This model will also be proposed in a journal article [14]
in the near future.
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3. CONCLUDING REMARKS
Three-layer sandwich panels of different types (stiff and
soft cores) are taken for the comparison study of existing
equivalent plate models in the wavenumber domain. It is
observed that RKU model fails in low frequency region
in comparison with Guyader model. Negligence of bend-
ing wave propagation inside the core in the RKU model
leads to this mismatch at lower frequencies. In case of
soft core, both Guyader and RKU model provide well-
matched results. Further, from the observations made on
this comparison study, two new models would be proposed
in the journal articles on the following topics: first, a sim-
ple equivalent plate model for three-layer sandwich system
that would be formulated based on the physical behaviours
observed in the low, high and transition frequency regimes;
second, a new condensed model which would take contri-
butions from both symmetric and anti-symmetric motions
of the symmetric multi-layer plate to compute the equiva-
lent dynamic response.
4. ACKNOWLEDGEMENT
The authors would like to gratefully acknowledge Marie-
Skłodowska Curie Actions (MSCA) Project 765472 ‘N2N:
No2Noise’ for the financial support.
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