Konferans Bildirisi-- İstanbul Teknik Üniversitesi, Teorik ve Uygulamalı Mekanik Türk Milli Komitesi, 2017Conference Paper -- İstanbul Technical University, Theoretical and Applied Mechanical Turkish National Committee, 2017Lineer viskoelastik malzemelerden oluşan plakalarda Lamb dalgalarının dispersyonu incelenmiştir. Araştırmalar, lineer viskoelastik teorisinin kesin denklemlerini uygulayarak yapılmıştır. Dispersyon denklemi, malzemenin herediter operatörünün keyfi türü için elde ettikten sonra incelenen dalgaların dispersyonu üzerine sayısal sonuçlar elde etmek için bir çözüm algoritması geliştirilmiştir. Farklı atenüasyon durumları için dispersyon eğrileri elde edilmiş ve malzemenin viskozitesinin etkisi, elastik sabitlerle karakterize eden reolojik parametreler, yani, Q viskoelastik malzemenin karakteristik akma süresi, d viskoelastik malzemenin mekanik sabitlerinin uzun vadeli değeri ve α viskoelastik malzemenin deformasyonun başlangıç durumu etrafındaki mekanik davranışı aracılığıyla incelenemiştir. Malzemenin viskoelastisitesini Rabotnov'un kısmi eksponansiyel operatörü’nü uygulayarak sayısal sonuçlar sunularak tartışılmıştır. Özellikle, reolojik parametrelerin, incelenen plakada yayılan Lamb dalgalarının dispersyonuna nasıl etkilediği tespit edilmiştir.Dispersion of the Lamb waves propagating in a plate made of viscoelastic material is investigated. The investigations are made by utilizing the exact equations of the theory of linear viscoelasticity. The dispersion equation is obtained for an arbitrary type of hereditary operator of the materials of the constituents and a solution algorithm is developed for obtaining numerical results. Dispersion curves is presented for certain attenuation cases and the influence of the viscosity of the material is studied through the rheological parameters which characterize the characteristic creep time, long-term values and the mechanical behavior of the viscoelastic material around the initial state of the deformation. Numerical results are presented and discussed for the case where the viscoelasticity of the materials is described through fractional-exponential operators by Rabotnov. In particular, how the rheological parameters influence on the dispersion of the Lamb waves are established

Akbarov, Surkay D

English

Ulusal Üniversitelerarası Açık Erişim Sistemi - İstanbul Teknik Üniversitesi

 
 20. ULUSAL MEKANİK KONGRESİ 
 05 - 09 Eylül 2017, Uludağ Üniversitesi, Bursa 
 
97 
 
DISPERSION OF LAMB WAVES PROPAGATING IN A PLATE MADE OF 
VISCOELASTIC MATERIAL 
Surkay D Akbarov
1
 ve Masoud Negin
2
 
1
Yıldız Teknik Üniversitesi, İstanbul 
2
Bahçeşehir Üniversitesi, İstanbul 
 
ABSTRACT 
Dispersion of the Lamb waves propagating in a plate made of viscoelastic material is 
investigated. The investigations are made by utilizing the exact equations of the theory of 
linear viscoelasticity. The dispersion equation is obtained for an arbitrary type of hereditary 
operator of the materials of the constituents and a solution algorithm is developed for 
obtaining numerical results. Dispersion curves is presented for certain attenuation cases and 
the influence of the viscosity of the material is studied through the rheological parameters 
which characterize the characteristic creep time, long-term values and the mechanical 
behavior of the viscoelastic material around the initial state of the deformation. Numerical 
results are presented and discussed for the case where the viscoelasticity of the materials is 
described through fractional-exponential operators by Rabotnov. In particular, how the 
rheological parameters influence on the dispersion of the Lamb waves are established. 
 
ÖZET 
Lineer viskoelastik malzemelerden oluşan plakalarda Lamb dalgalarının dispersyonu 
incelenmiştir. Araştırmalar, lineer viskoelastik teorisinin kesin denklemlerini uygulayarak 
yapılmıştır. Dispersyon denklemi, malzemenin herediter operatörünün keyfi türü için elde 
ettikten sonra incelenen dalgaların dispersyonu üzerine sayısal sonuçlar elde etmek için bir 
çözüm algoritması geliştirilmiştir. Farklı atenüasyon durumları için dispersyon eğrileri elde 
edilmiş ve malzemenin viskozitesinin etkisi, elastik sabitlerle karakterize eden reolojik 
parametreler, yani, Q viskoelastik malzemenin karakteristik akma süresi, d viskoelastik 
malzemenin mekanik sabitlerinin uzun vadeli değeri ve α viskoelastik malzemenin 
deformasyonun başlangıç durumu etrafındaki mekanik davranışı aracılığıyla incelenemiştir. 
Malzemenin viskoelastisitesini Rabotnov'un kısmi eksponansiyel operatörü’nü uygulayarak 
sayısal sonuçlar sunularak tartışılmıştır. Özellikle, reolojik parametrelerin, incelenen plakada 
yayılan Lamb dalgalarının dispersyonuna nasıl etkilediği tespit edilmiştir. 
 
INTRODUCTION 
The research of Lamb wave propagation problems related to viscoelastic materials have been 
widely studied in literature [1-6]. Several mathematical models have been used by many 
authors to study the dispersion behavior of such waves in viscoelastic media. However, in 
most cases either they have described the viscoelasticity of the materials through some simple 
models such as the classical Kelvin-Voigt spring-dashpot models or they have used complex 
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98  XX. Ulusal Mekanik Kongresi 
elasticity modulus instead of the real one in the stress–strain relations of the viscoelastic 
materials. Consequently, in general, such a simple viscoelastic models and the numerical 
results obtained within these models cannot illustrate the real character of the influence of the 
rheological parameters of the viscoelastic materials on the corresponding wave dispersion and 
attenuation.   
These considerations led the authors to study the Lamb waves dispersion for a system 
consisting of a viscoelastic material utilizing more realistic mathematical viscoelastic model 
using Rabotnov [7] fractional exponential operator which are used for the first time in the 
papers [8-10], where dimensionless rheological parameters characterizing the characteristic 
creep time (denoted by Q), the long-term values of the elastic constants (denoted by d) and the 
form of the creep (or relaxation) function of the materials in the beginning region of 
deformations (denoted by α), are introduced, and through these parameters the viscoelasticity 
of the materials of the plate on the dispersion curves is studied. An increase in the values of 
these parameters means a decrease in the viscosity properties of the related material. The 
numerical results are presented and discussed mainly for the attenuation dispersion case. 
  
FORMULATION OF THE PROBLEM 
Figure 1 shows the geometry of the problem and the Lamb waves propagate in the positive 
direction of 1Ox  axis. 
 
Figure 1. Geometry of the considered problem. 
The mechanical relations of the model are as follows:  
*
11
*
112 ,        
*
22
*
222 ,        112
*
22 ,    
11 22
1
, , 1,2,
2
ji
ij
j i
uu
i
x x
   
 
   
   
                                        (1) 
and * , *  are the following viscoelastic operators: 
*
0 1
0
( ) ( ) ( ) ( ) ,
t
t t t d            
 
*
0 1
0
( ) ( ) ( ) ( ) .
t
t t t d                                                     (2) 
We use the fractional exponential operators by Rabotnov [7] for describing the viscoelasticity 
of the constituents: 
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 XX. Ulusal Mekanik Kongresi 99 
* 0 0
0
0 0
3 3
( ) ( ) ( ) ,
2(1 ) 2(1 )
t t R t

 
     
 


  
     
    
 
* 0 0
0
0 0
3
( ) ( ) ( ) ,
(1 ) 2(1 )
t t R t

 
     
 


  
     
    
                              (3) 
where, 
   
0
( ) , ( ) ,
t
R x t R x t d     
    
 
(1 )
0
( )
, ( ) ,
((1 )(1 ))
n n
n
x t
R x t t t
n


 





  
                                         (4) 
( )x  is the gamma function and  , 0  and   are the rheological parameters of the 
viscoelastic material [8].  
Representing the displacements and strains as 1( )
2( )
i kx t
i iu v x e

  and 
1( )
2( )
i kx t
ij ij x e
   , and doing mathematical manipulations as described in the papers [9, 10] 
and introducing notations 
   1 1 1 1
0 0
( ) cos , ( )sin .c ss s ds s s ds     
 
    
   1 1 1 1
0 0
( ) cos , ( )sin ,c ss s ds s s ds     
 
    
                        
(5) 
we obtain the following expressions for the stresses
 
         111 112 22 ,
i kx t
x M x e

  
       
         122 222 22 ,
i kx t
x M x e

  
       
     112 12 22 ,
i kx t
M x e

 

                                                              (6)
 
where 
     0 1 1 ,c si          
      0 1 1 .c sM i                                                    (7) 
Thus, we obtain the complex moduli    and  M  , instead of Lame constants in relation 
(1). 
Furthermore, according to [8, 9], we introduce the following dimensionless 
rheological parameters: 
 
2
1
0 1
0
, ,
c
d Q
R 


 



 

                                         (8) 
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100  XX. Ulusal Mekanik Kongresi 
The dimensionless rheological parameter d  characterizes the long-term value of the 
viscoelastic material and the rheological parameter Q  characterizes the creep time of the 
viscoelastic material, and finally the rheological parameter   characterizes the form of the 
creep (or relaxation) function for the viscoelastic material and the case where 0   
corresponds to the “standard solid body” model. Consequently, according to the foregoing 
expressions, the influence of the viscoelasticity of the material on the dispersion curves is 
estimated through these three dimensionless rheological parameters.  
Finally, considering the boundary conditions, we obtain the system of the linear 
algebraic equations where from the existence of the non-trivial solution we get the dispersion 
equation: 
                       det 0, , 1,2,3,4.ij i j                                                (9)                    
As the values of the determinant obtained in (9) are complex, the dispersion equation can be 
reduced to det 0,ij   where .  means the modulus of the complex number determinant. 
To find the roots of this dispersion equation here we use the algorithm which is based on 
direct calculation of the values of the modulus of the dispersion from the criterion 
9det 10ij
 , and the values of the wave dispersion velocity are determined under fixed 
values of the problem parameters.  
For solution to the dispersion equation (9) we employ the algorithm developed in [9] 
and under this solution procedure we use the relation 1 2 1(1 )k k ik k i      for the complex 
wavenumber. We analyze the case where the attenuation parameter  , according to [11], is 
presented as follows: 
1
0 1
( )1
.
2 ( ) ( )
s
c
 

   


                                                    (10) 
It is known that using the symmetry and asymmetry of the displacements of the plate, 
Lamb waves are divided into symmetric (extensional) and asymmetric (flexural) modes. In 
other words, the conditions 1 2 1 2( ) ( )u x u x  and 2 2 2 2( ) ( )u x u x   are used for determination 
of the symmetric Lamb waves, and the conditions 1 2 1 2( ) ( )u x u x   and 2 2 2 2( ) ( )u x u x  are 
used for determination of the asymmetric Lamb waves. 
 
NUMERICAL RESULTS AND DISCUSSIONS  
For numerical results we assume that 0.3  , 
37190 kg/m  , 3
0 74.2 10 MPa    and 
3
0 102.5 10 MPa   . Figure 2a (Figure 3a) shows the dispersion curves constructed for 
various values of the parameter Q under d = 10 for symmetric (asymmetric) Lamb waves and 
Figure 2b (Figure 3b) gives the results that illustrated the influence of the parameter d on the 
behavior of the dispersion curves in the case where Q = 10 for symmetric (asymmetric) Lamb 
waves. Note that these graphs are constructed in the case where 0.5  . 
 
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 XX. Ulusal Mekanik Kongresi 101 
 
 
(a) (b) 
Figure 2. Symmetric mode: (a) Dispersion curves for different values of parameter Q 
under d =10. (b) Dispersion curves for different values of parameter d under Q =10. 
 
  
(a) (b) 
Figure 3. Asymmetric mode: (a) Dispersion curves for different values of parameter Q 
under d =10. (b) Dispersion curves for different values of parameter d under Q =10. 
 
As discussed in the paper by Akbarov et al. [10], we can predict that for the all selected values 
of the parameter Q  and under a fixed value of the parameter d  the wave propagation velocity 
must have the same limit values as 1 0k h   and this limit velocity coincides with that 
obtained for the corresponding purely elastic case with long-term values of the elastic 
constants. As a result, we could say that these limit values of the wave propagation velocity 
depend on the rheological parameter d , not on the rheological parameters Q  and  . 
Therefore dispersion curves obtained under fixed values of the parameter d  are limited with 
the corresponding dispersion curves obtained for the purely elastic cases under instantaneous 
values of the elastic constants (upper limits), i.e. under 0t  , and under long-term values of 
the elastic constants (lower limits), i.e. under t  . It follows from Figure 2 and 3 that, first 
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102  XX. Ulusal Mekanik Kongresi 
of all, the viscoelasticity of the plate material causes a decrease in the wave propagation 
velocity. Moreover, these results show that the dispersion curves obtained for the viscoelastic 
case approach to the corresponding one obtained for the purely elastic case with instantaneous 
(long-term) values of the elastic constants at 0t   (at t  ) with increasing (decreasing) of 
the rheological parameters d  and Q . It should be noted that the mentioned increase 
(decrease) has monotonic character and considerable effect in this increasing (decreasing) are 
observed in the cases where 1 2.0k h  . 
Moreover, according to the character of the dispersion curves obtained for symmetric 
mode of Lamb wave given in Figure 2 we can conclude that for each value of the rheological 
parameter Q  and for each value of the rheological parameter d  there exist the case where  
2
1
( / )
0
( )
d c c
d k h
 .                                                       (11) 
The wave propagation velocity and dimensionless wavenumber related to this case we denote 
by crc  and 1( ) ,crk h respectively.  Note that for the dispersion curves related to the purely 
elastic waves there is not the case where the relation (11) takes place. Consequently, the 
appearing of the cases where the relation (11) takes place is caused namely with the 
viscoelasticity of the material of the plate under consideration. Thus, it follows from the 
foregoing discussions that, the viscoelasticity of the plate material influences on the 
dispersion curves of the Lamb waves not only in the quantitative sense but also in the 
qualitative sense.  
Now let us consider the results related to the effect of rheological parameters   on the 
wave dispersion curves. Thus, we consider graphs given in Figures 4 and 5 which illustrate 
the mentioned influence for symmetric and asymmetric modes of Lamb wave, respectively. 
Note that the graphs given in Figure 4 (Figure 5) show the effect of the rheological parameter 
  on the dispersion curves under for a fixed value of the rheological parameter Q (=10) and d 
(=10). Also note that the case where 0   coincides with the case where viscoelasticity of the 
plate material is described through the "standard solid body" model. 
 
Figure 4. Symmetric mode: Dispersion curves for different values of parameter α under Q =10 
and d =10.   
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 XX. Ulusal Mekanik Kongresi 103 
 
Figure 5. Asymmetric mode: Dispersion curves for different values of parameter α under Q 
=10 and d =10.   
 
It follows from the results given in Figure 4 that for symmetric mode of Lamb wave 
there exists such value of the dimensionless wavenumber 1k h  (denote it by 1( )*k h ) at which 
the change in the values of the rheological parameter   does not influence the values of the 
wave propagation velocity. However, in the cases where 1 1( )*k h k h  ( 1 1( )*k h k h ) an increase 
in the values of the parameter   causes a decrease (an increase) in the wave propagation 
velocities. According to the aforementioned numerical results, it can be concluded that the 
1( )*k h  depends on the values of the rheological parameters Q and d and an increase in the 
values of these parameters decrease the 1( )*k h . However, for asymmetric mode of Lamb 
wave such value of the dimensionless wavenumber 1k h  does not exist as shown in Figure 5 
where dispersion curves have monotonic character. 
We can also conclude that the change in the values of the rheological parameter   does 
not influence the limit values of the wave propagation velocity as 1 0k h  . However, in the 
near vicinity of this limit case, if to say more precisely in the region 1 10 ( )*k h k h  , the 
influence of the rheological parameter   on the dispersion curves is significant not only in 
the quantitative sense but also in the qualitative sense. So that under small values of  , for 
instance under 0.1  , the dispersion curves have well-defined minimum in the region 
1 10 ( )*k h k h   and at this minimum the relation (11) takes place. Moreover, in the near 
vicinity of this minimum the wave propagation velocity obtained for the viscoelastic cases 
become less than that obtained for the purely elastic case with long-term values of elastic 
constants at t  . Consequently, in the region 1 10 ( )*k h k h   a decrease in the values of   
causes to change the character of the dispersion curves. However, with increasing of   the 
aforementioned minimum disappears in the dispersion curves and wave propagation velocities 
are limited with the wave propagation velocities obtained for the purely elastic cases with 
instantaneous values of elastic constants at 0t   (upper limit) and with long-term values of 
elastic constants at t   (lower limit).  
 
CONCLUSIONS 
According to these numerical results, the following main conclusions can be drawn: 
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104  XX. Ulusal Mekanik Kongresi 
- Viscoelasticity of the material of the plate causes both symmetric and asymmetric 
modes of the Lamb wave propagation velocity to decrease and the magnitude of this 
decrease increases with a decrease in the aforementioned dimensionless rheological 
parameters d and Q; 
- For symmetric mode of propagation of the Lamb wave the character of the influence 
of the parameter α on the wave velocities and on the dispersion corves depends on the 
values of the dimensionless wavenumber 1k h  and on the values of the rheological 
parameters d and Q. 
- In general, the dispersion curves are limited by the purely elastic case with 
instantaneous values of the elastic constants and by those with long-term values of the 
elastic constants. 
- The high wavenumber limit values of the wave propagation velocity do not depend on 
the rheological parameters Q and α of the plate material. 
 
REFERENCES 
[1]  Simonetti, F., Lamb wave propagation in elastic plates coated with viscoelastic 
materials. The Journal of the Acoustical Society of America 115.5 (2004): 2041-2053. 
[2]  Castaings M, Hosten B., Guided waves propagating in sandwich structures made of 
anisotropic, viscoelastic, composite materials. The Journal of the Acoustical Society of 
America 113.5 (2003): 2622-34. 
[3] Sharma, J. N., Some Considerations on the Rayleigh–Lamb Wave Propagation in Visco-
thermoelastic Plates. Modal Analysis 11.10 (2005): 1311-1335. 
[4] Shorter, P.J., Wave propagation and damping in linear viscoelastic laminates. The 
Journal of the Acoustical Society of America 115. 5 (2004): 1917-1925. 
[5] Torres-Arredondo, Miguel A., and C-P. Fritzen, A viscoelastic plate theory for the fast 
modelling of Lamb wave solutions in NDT/SHM applications.  Ultrasound 66.2 (2011): 
7-13. 
[6] Taghipour Birgani, P., Naderan Tahan, K., Sodagar, S. and Shishesaz, M., Investigation 
of Lamb waves attenuation in elastic–viscoelastic three-layer adhesive joints in low and 
high frequencies: Theoretical modeling. Proceedings of the Institution of Mechanical 
Engineers, Part C: Journal of Mechanical Engineering Science 229.11(2015): 1939-
1952. 
[7] Rabotnov, Yu.N., Elements of Hereditary Solid Mechanics, Mir, Moscow, 1980. 
[8] Akbarov, S.D., Axisymmetric time-harmonic Lamb’s problem for a system comprising a 
viscoelastic layer covering a viscoelastic half-space. Mechanics of Time-Dependent 
Materials 18.1 (2014): 153-178. 
 [9] Akbarov, S.D., and T. Kepceler., On the torsional wave dispersion in a hollow sandwich 
circular cylinder made from viscoelastic materials. Applied Mathematical 
Modelling 39.13 (2015): 3569-3587. 
[10] Akbarov, S.D., Kocal, T. and Kepceler, T., On the dispersion of the axisymmetric 
longitudinal wave propagating in a bi-layered hollow cylinder made of viscoelastic 
materials. International Journal of Solids and Structures 100(2016): 195-210. 
[11] Ewing, W.M., Jazdetzky, W.S., Press, F., Elastic Waves in Layered Media, McGraw – 
Hill, New York, 1957. 
 


Dispersion Of Lamb Waves Propagating In A Plate Made Of Viscoelastic Material

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Dispersion Of Lamb Waves Propagating In A Plate Made Of Viscoelastic Material

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