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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.

5. REFERENCES

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