Download PDF Investigation of the vibration problem of sandwich heterogeneous orthotropic plates

Investigation of the vibration problem of sandwich heterogeneous orthotropic plates

C. Bozkurt
,
H. Dilmac
&
A.H. Sofiyev

UNEC Journal of Engineering and Applied Sciences Volume 3, No 2 (2023), pages 14-20 Cite this article,  476 https://doi.org/10.61640/ujeas.2023.1202

Abstract

In this study, the solution of the free vibration problem of sandwich plates consisting of heterogeneous orthotropic layers is presented. First, the Kirchhoff-Love theory (KLT) for homogeneous plates is extended to sandwich plates composed of heterogeneous orthotropic layers. Within the framework of Donnell theory, after establishing the basic relations of multilayer plates, governing equations are derived based on Airy stress and deflection functions. The solution of the governing equations is carried out by the Galerkin method and the analytical expression is found for the linear frequency of sandwich plates consisting of heterogeneous orthotropic layers. Finally, the effects of various factors such as heterogeneity, number of layers and their arrangement on the free vibration frequency of sandwich plates are examined. 

Introduction

Sandwich structural elements, consisting of two surface layers and a core, are frequently used in different areas of contemporary engineering, from satellites to airplanes, from ships to automobiles, from rail vehicles to wind energy systems and bridge construction. The surface layers of sandwich structural elements may differ in thickness and material. The need for high-performance, low-weight structures ensures that sandwich structural elements consisting of heterogeneous materials continue to be in demand [1]. Since plates with various configurations are widely used in engineering structures, it is becoming increasingly important to study their dynamic behavior, especially their vibration behavior. The solution of dynamic problems of sandwich structural elements is associated with some difficulties. One of the most important of these difficulties is the heterogeneity of layers of sandwich structural elements with different properties. These factors makes it necessary in vibration problems of sandwich structural elements to take into account the heterogeneity of materials in the layers and study their influence on the vibration frequency. Although efforts to create analytical models that reflect the reality of the mechanical properties of heterogeneous orthotropic materials are always on the agenda, the number of modeling studies is limited [2,3].
One of the main goals when designing heterogeneous multilayer or sandwich structural elements is to make their vibration calculations. In most cases, in vibration calculations of structural elements consisting of homogeneous materials, it is seen that their frequencies are much different than in reality. One of the reasons for this phenomenon lies in the heterogeneous anisotropic nature of the layers forming the sandwich structural element. To date, some studies have been carried out on vibrations of multilayer or sandwich heterogeneous plates. Amon those, Fares and Zenkour [4] presented buckling and free vibration of non-homogeneous composite cross-ply laminated plates with various plate theories. Kuo and Shiau [5] studied buckling and vibration of composite laminated plates with variable fiber spacing. Orakdogen et al. [6] presented finite element analysis of functionally graded plates for coupling effect of extension and bending. Zerin [7] studied the vibration problem of laminated nonhomogeneous orthotropic shells. Dozio [8] investigated vibration behaviors of sandwich plates with FGM core via variable-kinematic 2-D Ritz models. Chen et al. [9] investigated vibration and stability of initially stressed sandwich plates with FGM face sheets in thermal environments. Haciyev et al. [10] solved the free bending vibration problem of thin non-homogenous orthotropic rectangular plates resting on viscoelastic foundations. Bacciocchi and Tarantino [11] presented natural frequency analysis of functionally graded orthotropic cross-ply plates based on the finite element method. Bouazza and Zenkour [12] studied vibration of inhomogeneous fibrous laminated plates using an efficient and simple polynomial refined theory. Alnujaie et al. [13] studied buckling and free vibration analysis of multi-directional functionally graded sandwich plates. Li et al. [14] examined free vibration of functionally graded sandwich plates in thermal environments. 
In this study, the vibration problem of sandwich plates consisting of heterogeneous layers will be discussed within the framework of classical plate theory. This theory is based on the Kirchhoff-Love theory, which assumes that the normal to the mid-plane remain straight and normal to the mid-plane remains normal before and after deformation and the size does not change.

Experiments

Consider a sandwich plate with total thickness h, lengths a and b on the x and y axes, respectively, which consists of heterogeneous orthotropic three layers. The geometry and coordinate system are shown in figure 1. The origin of the coordinate system Oxyz is in the upper left corner of the reference plane of the sandwich plate, while the x and y axes are on the z=0 reference plane, the z axis is perpendicular to the reference plane and directed inwards. 

Figure 1. The geometry, coordinate system and cross section of the sandwich plate

Figure 1. The geometry, coordinate system and cross section of the sandwich plate

It is assumed that the layers of sandwich plate are perfectly bonded to each other, they do not slip and remain elastic during deformation. The displacements in the x, y and z directions are indicated by U, V and W, respectively. Let Φ(x,y,t)  be the Airy stress function for the stress resultants, so that [15],
alt text(1)

Where  alt text are forces.
The mechanical properties such as Young moduli alt text , shear modulus  alt text and density alt text   of lamina alt text are linear and quadratic functions of the thickness coordinate and defined as follows [2-4,7,10]:
alt text (2)

where the symbols with “0” in the superscript indicate the mechanical properties of the homogeneous orthotropic material, μ indicates the heterogeneity parameter for the elasticity moduli and density in the layer of  sandwich plates, which characterizes its variation depending on the Z and alt text. In addition, μ=0 case indicates homogeneous orthotropic material.

Basic equations

Based on the KLT, the motion and strain compatibility equations for the layered plates composed of heterogeneous layers could be expressed as [15]:

alt text(3)

alt text(4)

where t is the time,   alt textare forces,   are strains in the reference plane, and alt text  is the density parameter of sandwich plates are defined, as follows [3,16]: 

alt text (5)

in which

alt text   (6)

The relationships between stresses alt text , and strains  alt text and alt text for the  alt text lamina contained in sandwich plates composed of heterogeneous layers can be written as follows [2,7],

alt text      (7)

where

alt text(8)

in which alt text are Young and shear moduli of heterogeneous orthotropic materials in the  lamina alt text  ,  alt text and  alt text are Poisson ratios and are considered constant since the effect of heterogeneity according to the thickness coordinate is very small and the following condition is satisfied: alt text.
    By using the relationships from (1) to (7), the motion and deformation compatibility equations for sandwich heterogeneous plates composed within KLT are obtained as follows:

alt text   (9)

alt text(10)

where alt text are coefficients depending on the material properties of sandwich plates composed of heterogeneous layers.

Solution procedure

Since all edges of the sandwich plate are assumed to be simply supported, the solution of (9) and (10) is sought as follows [2,15]:

alt text(11)

where alt text and  alt text are time dependent functions, alt text  and  alt text are the wave parameters in which (m, n) is the vibration mode.

          Substituting (11) into the set of eqs (9) and (10), and eliminating the unknown  alt text from the resulting equations, one gets,

alt text(12)

where the following definition applies:

alt text(13)

The dimensionless frequency parameter of sandwich plates composed of heterogeneous orthotropic layers within KLT is defined as:

alt text  (14)

Results and discussion

The numerical results are performed for free vibration frequency of sandwich rectangular plates made of heterogeneous orthotropic layers, using eq. (14). The heterogeneity parameters for elasticity moduli and density are used as   and is denoted as HT in figure. The μ=0  corresponds to homogeneous case and is denoted as H. The properties of homogeneous orthotropic material are taken from the study of Reddy [16]: alt text alt textand alt text. For subsequent examples use the following characteristics: a/b =0.5,1.0,1.5,2.0, a/h=15 and (m,n)=(1,1). 
The variations of the   for the H and HT-sandwich rectangular plates versus the a/b are presented in table 1 and figure 2. As a/b increases, the values of ω1 for single-layer and sandwich plates increase. 
When the a/b ratio increases from 0.5 to 2; the effect of the heterogeneous-linear profile on the frequency in (0°) plate is approximately 4.2% and is independent of the increase a/b. As the a/b ratio increases from 0.5 to 2; while the linear heterogeneity effect on frequency values decreases from 5.8% to 3.12 in the (0°/90°/0°)-array plate, this effect increases from 3.2% to 5.83% in the (90°/0°/90°) -array plate. When the  ratio increases from 0.5 to 2; the effect of the heterogeneous-quadratic profile on the frequency in (0°) plate is approximately 7.2% and is independent of the increase of a/b. As the a/b ratio increases from 0.5 to 2; while the effect of the heterogeneous-quadratic profile on the frequency values decreases from 7.47% to 6.44 in the (0°/90°/0°)-array plate, this effect increases from 6.4% to 7.42% in the (90°/0°/90°) -array plate.
The effects of layer arrangement on the frequency of sandwich plates vary. For example, while the effects of the homogeneous (0°/90°/0°)-arrangement on the frequency compared to the (0°)-single layer plate are 1.52%, 0%, 3.39% and 7.16%, respectively, those effects in the plate with the (0°/90°/0°)-arrangement are 52.14%, 0%, 67.82% and 120.45%, respectively, as  =0.5, 1.0, 1.5 and 2, respectively. In the heterogeneous-linear case, the effects of the (0°/90°/0°) arrangement on the frequency compared to the (0°)-single layer plate are 3.12%, 1.01%, 3.54% and 8.41%, respectively, while these effects in the (0°/90°/0°)-sequenced plate are 51.6%, 1.01%, 65.3% and 116.8%, as  =0.5, 1.0, 1.5 and 2, respectively. It can be seen that in the heterogeneous-quadratic case, the effects of the (0°/90°/0°)-arrangement on the frequency compared to the (0°)-single layer plate are 3.12%, 1.01%, 3.54% and 8.41%, respectively, while these effects are in the (0°/90°/0°)-aligned plate are 51.6%, 1.01%, 65.3% and 116.8%, when  =0.5, 1.0, 1.5 and 2, respectively.

Table 1. Variation of the ω1 for the homogenous and heterogeneous linear and quadratic profiled plates versus the a/b

Table 1. Variation of the ω1  for the homogenous and heterogeneous linear and quadratic profiled plates versus the a/b

Figure 2. Variation of the ω1 for the homogenous and heterogeneous linear and quadratic profiled plates versus the a/b

Figure 2. Variation of the ω1  for the homogenous and heterogeneous linear and quadratic profiled plates versus the a/b

Conclusion

In this study, the free vibration problem of sandwich plates consisting of heterogeneous orthotropic layers is solved within the framework of Kirchhoff-Love theory (KLT). Within the framework of Donnell type plate theory, the basic relationships of sandwich plates are established and the equation of motion is derived. By applying the Galerkin method, the analytical expression for the linear frequency of sandwich plates consisting of heterogeneous orthotropic layers is obtained. Finally, the importance of the effects of various factors such as inhomogeneity, number and arrangement of layers on the free vibration frequency of square and rectangular plates is examined.

References

1 J.R. Vinson, Applied Mechanics Review 54(3) (2001) 201. https://doi.org/10.1115/1.3097295

2 A.H. Sofiyev, E. Schnack, Acta Mechanica 162(1-4) (2003) 29. https://doi.org/10.1007/s00707-002-1001-2

3 Y. Ootao, Y. Tanigawa, Composite Structures 80(1) (2007) 10. https://doi.org/10.1016/j.compstruct.2006.02.028

4 M.E. Fares, A.M. Zenkour, Composite Structures 44(4) (1999) 279. https://doi.org/10.1016/S0263-8223(98)00135-4

5 S.Y. Kuo, L.C. Shiau, Composite Structures 90(2) (2009) 196. https://doi.org/10.1016/j.compstruct.2009.02.013

6 E. Orakdögen, S. Küçükarslan, A. Sofiyev, M.H. Omurtag, Meccanica 45(1) (2010) 63. https://doi.org/10.1007/s11012-009-9225-z

7 Z. Zerin, Meccanica 48(7) (2013) 1557. https://doi.org/10.1007/s11012-012-9684-5

8 L. Dozio, Composite Structures 96 (2013) 561. https://doi.org/10.1016/j.compstruct.2012.08.016

9 C.S. Chen, F.H. Liu, W.R. Chen, Steel and Composite Structures 23(3) (2017) 251. https://doi.org/10.12989/scs.2017.23.3.251

10 V.C. Haciyev, A.H. Sofiyev, N. Kuruoglu, Mechanics of Advanced Materials and Structures 26(10) (2019) 886. https://doi.org/10.1080/15376494.2018.1430271

11 M. Bacciocchi, A.M. Tarantino, Mathematical and Computational Applications 24(2) (2019) 52. https://doi.org/10.3390/mca24020052

12 M. Bouazza, A.M. Zenkour, Journal of Computational Applied Mechanics 52(2) (2021) 233. https://doi.org/10.22059/jcamech.2021.320751.605

13 A. Alnujaie, A.S. Sayyad, AS, L. Hadji, A. Tounsi, Structural Engineering and Mechanics 84(6) (2022) 813. https://doi.org/10.12989/sem.2022.84.6.813

14 D.D. Li, Z.B. Deng, G.P. Chen, International Journal of Mechanical System Dynamics 3(1) (2023) 39. https://doi.org/10.1002/msd2.12063

15 A.S. Volmir, The nonlinear dynamics of plates and shells. Science Edition, Moscow (1972) 543p.

16 J.N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. 2nd ed. CRC Press, New-York (2004) 858p. https://doi.org/10.1201/b12409

About this article

Cite this article

C. Bozkurt, H. Dilmac, A.H. Sofiyev, Investigation of the vibration problem of sandwich heterogeneous orthotropic plates , UNEC J. Eng. Appl. Sci. 3(2) (2023) 14-20 https://doi.org/10.61640/ujeas.2023.1202

  • Received04 Oct 2023
  • Accepted12 Dec 2023
  • Published26 Dec 2023

Share this article

Anyone you share the following link with will be able to read this content:

Keywords

  • heterogeneity
  • sandwich plates
  • vibration
  • frequency
Download PDF Investigation of the vibration problem of sandwich heterogeneous orthotropic plates
  1. J.R. Vinson, Applied Mechanics Review 54(3) (2001) 201. https://doi.org/10.1115/1.3097295

  2. A.H. Sofiyev, E. Schnack, Acta Mechanica 162(1-4) (2003) 29. https://doi.org/10.1007/s00707-002-1001-2

  3. Y. Ootao, Y. Tanigawa, Composite Structures 80(1) (2007) 10. https://doi.org/10.1016/j.compstruct.2006.02.028

  4. M.E. Fares, A.M. Zenkour, Composite Structures 44(4) (1999) 279. https://doi.org/10.1016/S0263-8223(98)00135-4

  5. S.Y. Kuo, L.C. Shiau, Composite Structures 90(2) (2009) 196. https://doi.org/10.1016/j.compstruct.2009.02.013

  6. E. Orakdögen, S. Küçükarslan, A. Sofiyev, M.H. Omurtag, Meccanica 45(1) (2010) 63. https://doi.org/10.1007/s11012-009-9225-z

  7. Z. Zerin, Meccanica 48(7) (2013) 1557. https://doi.org/10.1007/s11012-012-9684-5

  8. L. Dozio, Composite Structures 96 (2013) 561. https://doi.org/10.1016/j.compstruct.2012.08.016

  9. C.S. Chen, F.H. Liu, W.R. Chen, Steel and Composite Structures 23(3) (2017) 251. https://doi.org/10.12989/scs.2017.23.3.251

  10. V.C. Haciyev, A.H. Sofiyev, N. Kuruoglu, Mechanics of Advanced Materials and Structures 26(10) (2019) 886. https://doi.org/10.1080/15376494.2018.1430271

  11. M. Bacciocchi, A.M. Tarantino, Mathematical and Computational Applications 24(2) (2019) 52. https://doi.org/10.3390/mca24020052

  12. M. Bouazza, A.M. Zenkour, Journal of Computational Applied Mechanics 52(2) (2021) 233. https://doi.org/10.22059/jcamech.2021.320751.605

  13. A. Alnujaie, A.S. Sayyad, AS, L. Hadji, A. Tounsi, Structural Engineering and Mechanics 84(6) (2022) 813. https://doi.org/10.12989/sem.2022.84.6.813

  14. D.D. Li, Z.B. Deng, G.P. Chen, International Journal of Mechanical System Dynamics 3(1) (2023) 39. https://doi.org/10.1002/msd2.12063

  15. A.S. Volmir, The nonlinear dynamics of plates and shells. Science Edition, Moscow (1972) 543p.

  16. J.N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. 2nd ed. CRC Press, New-York (2004) 858p. https://doi.org/10.1201/b12409