Elsevier

Composite Structures

Volume 280, 15 January 2022, 114897
Composite Structures

Investigation of delamination effect on nonlinear vibration behaviors of a composite plate resting on nonlinear elastic foundation

https://doi.org/10.1016/j.compstruct.2021.114897Get rights and content

Abstract

The aim of the present work is to investigate the delamination effect on nonlinear vibration behaviors of composite plate resting on nonlinear elastic foundation. The composite plate is assumed to be pre-delaminated with various delamination locations and modeled with an improved layerwise theory and finite element method. The delamination is simulated by the Heaviside unit step function allowing for the discontinuity of displacement fields. By means of the variational principle, the nonlinear governing equations can be obtained and further solved by the direct iterative method. The model takes into account three foundation parameters so as to investigate their effect on the nonlinear vibration characteristics. Three longitudinal locations and three interfacial locations are considered, and both clamped-free and clamped–clamped boundary conditions are considered to investigate the nonlinear vibration behaviors. The results show that different from the linear vibration, with the existence of delamination, the nonlinear fundamental frequencies may exceed the fundamental frequency of the healthy plate when the vibration amplitude ratio reaches a certain value. This finding may help engineers understand the nonlinear vibration behaviors and further detect the delamination in composite plate resting on nonlinear foundations in presence of delamination.

Introduction

Due to the discovery of new phenomena, the application of new materials and new structures, nonlinear theory of mechanics has been widely valued. Many scholars have modified classical theories from a geometric or physical point of view to varying degrees and put forward various nonlinear theories [1], [2], [3]. There are many factors that can cause nonlinear problems in structure engineering, including material nonlinearity, geometric nonlinearity, nonlinear loading, contact analysis, etc. Nonlinear analysis and accurate prediction of nonlinear response of structures are very important to their safety and service life from both static and dynamic aspects. However, nonlinear analysis is not only computationally much more expensive, but also much more difficult in mathematical modeling than linear analysis in most cases. Especially, when dealing with a system of anisotropic material properties, such as composite laminate which is a kind of layered material with different properties of each layer, it is much more complicate for nonlinear modeling and achieving a convergent result. Thus, nonlinear analysis of material and structural behaviors has been a big challenge to scholars and received more and more attentions since the last decades.

Nonlinear analysis of composite laminated plates and shells with general anisotropic material properties has been widely studied by various theories for large amplitude bending, free vibration, buckling, transient response, etc. A systematic study of nonlinear analysis of regular composite plates and shells for free vibration, bending and buckling can be found in Reddy’s book based on classical laminate theory, first-order shear deformation theory and higher-order shear deformation theory [4] and other refined four-variable theories [5], [6]. Nonlinear free vibration of composite plates has been widely studied during the last decades [7], [8], [9]. Ribeiro [10] investigated the nonlinear vibrations of laminated plate based on first-order shear deformation theory and finite element method. Nonlinear dynamic analysis of thick composite/sandwich laminates was investigated by using an higher-order theory [11]. Large amplitude free vibration of a shear deformable laminated composite parabolic plate with parabolically orthotropic plies [12] and polymer composite stiffened laminates [13] were also investigated. For nonlinear free vibration problems, nonlinear vibration of conical shell panel and cylindrical shells with cutouts were studied by Parhi [14] and Nanda [15]. Composite plates with random properties [16] and curvilinear fibers [9] were also investigated for nonlinear free vibration analysis. Nonlinear vibration of laminated composite skew plates was investigated by Singha [17] based on finite element method. Nonlinear vibration of functionally graded microbeams was also investigated considering the thermal environment and size effect [18], [19]. Nonlinear transient analysis of moderately thick laminated composite plates was studied by Nath [20] based on finite element method. Thermal post-buckling behavior of laminated composite cylindrical/hyperboloid shallow shell panel using nonlinear finite element method was studied by Panda [21]. The aforementioned literature works comprehensively introduced the modeling and calculation of nonlinear behaviors of composite laminated structures based on different approaches. These results provided important guidance for the design and optimization of laminated composite structures. Besides, there are many other relevant works introducing the nonlinear behaviors of composite structures, but they will not be introduced for sake of brevity.

These is another nonlinear factor that can affect the nonlinear behavior of composite structures, which is the nonlinear foundation. Composite plates and shells are usually surrounded or supported by different elastic or viscoelastic media in the serving environment. The surrounding materials can be soil, rubber, metal spring and other plastic materials for different usages and application fields. Particularly for composite structures with foundations, they can be used in fields of aerospace engineering and transportation engineering for lightweight structure design. The surrounded foundation can provide linear or nonlinear reaction forces to the structure so that it gives additional constraints to the structure and affect the structural behavior. Understanding the interaction between the structure and the foundation is very important in order to accurately predict the structural response. Since we only focus on the nonlinear foundation in this work, linear foundation problems have been widely studied and readers can refer to other existed references for composite structures resting on linear foundations [22], [23], [24], [25]. For nonlinear elastic foundations, many works have been fulfilled since the last decades. Tornabene [26] investigated the static analysis of laminated shells resting on nonlinear elastic foundation based on a posteriori stress and strain recovery procedure. Lopatin [27] studied the buckling of compressed rectangular orthotropic plate resting on elastic foundation with nonlinear change of transverse displacement over the thickness. Soares [28] studied the nonlinear vibrations of a rectangular hyperelastic membrane resting on a nonlinear elastic foundation with four different constitutive laws. Singh [29] studied the nonlinear bending response of laminated composite plates on nonlinear elastic foundation with uncertain system properties. Large amplitude vibration and post-buckling analysis of composite beams on nonlinear elastic foundations were investigated for variable cross-section [30] and unsymmetrical layer stacking sequence [31]. Dynamic analysis of beams on nonlinear elastic foundations under a moving oscillator was investigated by Rodrigues et al. [32] based on finite element method. Recently, Shariati et al. [33] studied the extremely large oscillation and nonlinear frequency of a multi-scale hybrid disk resting on nonlinear elastic foundation. Bidzard et al. [34] investigated the influences of pressure and thermal environment on nonlinear vibration characteristics of multilayer FG-GPLRC toroidal panels on nonlinear elastic foundation. Functionally graded circular plate resting on nonlinear elastic foundation was also studied by Javani [35]. Differential quadrature (DQ) large amplitude free vibration of composite beams on nonlinear elastic foundations with restrained edges was investigated by Malekzadeh [36]. Discrete singular convolution method and discrete singular convolution-differential quadrature coupled approach were also adopted to solve the nonlinear differential equations of laminated composite plates resting on nonlinear elastic foundations [37], [38]. Hygro-thermo-mechanical bending behavior of functionally graded ceramic plate resting on elastic and viscoelastic foundations has been recently investigated by Mudhaffar [39] and Merazka [40]. The porous functionally graded plates resting on elastic and viscoelastic foundations have also been studied by Tahir [41] and Guellil [42]. These literatures detailedly reported the recent progress of nonlinear behaviors of composite beams, plates, shells, membranes and functionally graded materials resting on nonlinear elastic foundations with various theories and approaches. Nevertheless, for the layered structures, the composite laminates are always assumed to be perfectly bonded in all these works without considering the bonding conditions between layers. Modeling of composite structures on nonlinear elastic foundation with imperfect bonding is also very important in order to understand its structural behaviors and further to detect the debonding at layer interfaces.

As we all know, delamination and crack are conventional failures occur in composite laminated structures which may be caused by over-loadings or initial imperfect bonding during the manufacturing process. Modeling and investigation of delamination effect in composite laminates have been widely studied for free vibration, transient analysis, frequency analysis, damage detection and other related topics in structural health monitoring [43], [44], [45], [46], [47], [48]. Aymerich et al. [49] predicted the impact-induced delamination in cross-ply composite laminates using cohesive interface elements. Kim et al. [50] developed a generalized layerwise approach for investigation of delamination effect on dynamic characteristics of composite laminates. Ghoshal et al. [51] developed a mathematical modeling of delamination in composite structures by incorporating the Fermi–Dirac distribution function and hybrid damage indicators. Raja et al. [52] analyzed the piezoelectric composite beams and plates with multiple delaminations using a four-node quadrilateral shear flexible plate element. Recently, Kapuria [53] developed a coupled efficient layerwise finite element model for free vibration analysis of smart piezo-bonded laminated shells featuring delaminations and transducer debonding. However, to the best of authors’ knowledge, modeling of delaminated composite plates and shells resting on nonlinear elastic foundation has not been studied yet. Relevant works regarding to the delaminated composite structures on linear elastic foundation are also rare [54]. Thus, it is worth studying and investigating the nonlinear vibration behaviors of composite plates resting on nonlinear elastic foundation in presence of interfacial delamination.

In this work, we will investigate the delamination effect on nonlinear vibration of composite laminate resting on three-parameter nonlinear elastic foundation, where the foundation considers the transverse normal stress, transverse shear stress and transverse nonlinear stress transferred from the foundation to the plate. The nonlinear vibration is analyzed by means of the improved layerwise theory with Heaviside unit step function and finite element implementation. With the Heaviside unit step function, it is possible to address the discontinuity of displacement fields. The nonlinear governing equations are solved by the direct iterative method. It should be noted that the proposed method is a quasi-3D approach which is computationally much more efficient than 3D approaches. Handling of delamination with additional degrees of freedom is much more convenient in the finite element modeling. The present work considers both longitudinal and interfacial delamination locations in clamped-free and clamed-clamped cross-ply laminates. The effect of nonlinear foundation parameters on the nonlinear free vibration of delaminated composite plates will be investigated. The results can help understand the nonlinear vibration behaviors of composite plates resting on nonlinear elastic foundations in presence of delamination.

Section snippets

Theoretical model

Fig. 1 presents a N-layer composite laminated plate with delamination at layer interface under investigation. The plate is in contact with the nonlinear elastic foundation and its two edges are clamped. In the figure, a, b and h represent the length, width and total thickness of the plate, respectively. The delaminated length is Ld and the longitudinal delamination location is Lx from the left end. In the following, firstly, the improved layerwise theory with Heaviside unit step function for

Results and discussion

In this section, the nonlinear vibration of a 16-layer cross-ply ([0/90]4s) laminate resting on nonlinear elastic foundation is investigated in presence of various delamination failures. The delaminated plate has the dimension of 30 cm × 5 cm × 0.2 cm in length, width and thickness directions, and the single layer thickness is 0.125 mm. The material properties of a single-layer laminae used for numerical analysis are from the reference [48] as below.E1 = 372GPa, E2 = E3 = 4.12GPa, G12 = G13

Conclusions

In this work, we present a mathematical model of delaminated composite plate resting on nonlinear elastic foundations. The model is based on the improved layerwise theory with finite element implementation. It is a quasi-3D theory with plate element which is computationally much more efficient than 3D approaches of delamination modeling. Simulation of delamination with additional degrees of freedom is convenient in mathematical calculation. Three-parameter foundation including the nonlinear

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11702150), Zhejiang Natural Science Foundation (Grant no. LY21A020003), Natural Science Foundation of Ningbo (Grant nos. 202003N4015 and 202003N4163), the project of Key Laboratory of Impact and Safety Engineering (Ningbo University), Ministry of Education (Grant no. CJ202009), the Technology Innovation 2025 Program of Municipality of Ningbo (Grant no. 2019B10122), and the research fund by the College of

References (56)

  • S.K. Panda et al.

    Thermal post-buckling behaviour of laminated composite cylindrical/hyperboloid shallow shell panel using nonlinear finite element method

    Compos Struct

    (2009)
  • S.S. Akavci

    Mechanical behavior of functionally graded sandwich plates on elastic foundation

    Compos Part B-Eng

    (2016)
  • J.L. Mantari et al.

    An original FSDT to study advanced composites on elastic foundation

    Thin Wall Struct

    (2016)
  • J.L. Mantari et al.

    Vibrational analysis of advanced composite plates resting on elastic foundation

    Compos Part B-Eng

    (2014)
  • F. Tornabene et al.

    A posteriori stress and strain recovery procedure for the static analysis of laminated shells resting on nonlinear elastic foundation

    Compos Part B-Eng

    (2017)
  • A.V. Lopatin et al.

    Buckling of compressed rectangular orthotropic plate resting on elastic foundation with nonlinear change of transverse displacement over the thickness

    Compos Struct

    (2021)
  • B.N. Singh et al.

    Nonlinear bending response of laminated composite plates on nonlinear elastic foundation with uncertain system properties

    Eng Struct

    (2008)
  • H. Asadi et al.

    Large amplitude vibration and post-buckling analysis of variable cross-section composite beams on nonlinear elastic foundation

    Int J Mech Sci

    (2014)
  • M. Baghani et al.

    Large amplitudes free vibrations and post-buckling analysis of unsymmetrically laminated composite beams on nonlinear elastic foundation

    Appl Math Model

    (2011)
  • C. Rodrigues et al.

    Finite element dynamic analysis of beams on nonlinear elastic foundations under a moving oscillator

    Eur J Mech A-Solid

    (2018)
  • A. Shariati et al.

    Extremely large oscillation and nonlinear frequency of a multi-scale hybrid disk resting on nonlinear elastic foundation

    Thin Wall Struct

    (2020)
  • A. Bidzard et al.

    Influences of pressure and thermal environment on nonlinear vibration characteristics of multilayer FG-GPLRC toroidal panels on nonlinear elastic foundation

    Compos Struct

    (2021)
  • M. Javani et al.

    Geometrically nonlinear free vibration of FG-GPLRC circular plate on the nonlinear elastic foundation

    Compos Struct

    (2021)
  • P. Malekzadeh et al.

    DQM large amplitude vibration of composite beams on nonlinear elastic foundations with restrained edges

    Commun Nonlinear Sci

    (2009)
  • A.K. Baltacıoğlu et al.

    Large deflection analysis of laminated composite plates resting on nonlinear elastic foundations by the method of discrete singular convolution

    Int J Pres Ves Pip

    (2011)
  • Ö. Civalek

    Nonlinear dynamic response of laminated plates resting on nonlinear elastic foundations by the discrete singular convolution-differential quadrature coupled approaches

    Compos Part B-Eng

    (2013)
  • I.M. Mudhaffar et al.

    Hygro-thermo-mechanical bending behavior of advanced functionally graded ceramic metal plate resting on a viscoelastic foundation

    Structures

    (2021)
  • B. Huang et al.

    PCA-based damage classification of delaminated smart composite structures using improved layerwise theory

    Comput Struct

    (2014)
  • Cited by (0)

    View full text