Comparative study on the bending of sandwich FGM beams made up of different material variation laws using refined layerwise theory

doi:10.1016/j.mechmat.2020.103634

Mechanics of Materials

Volume 151, December 2020, 103634

https://doi.org/10.1016/j.mechmat.2020.103634 Get rights and content

Highlights

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Bending analysis of sandwich FGM beams using FE based HOZT.
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Formulation is free from any kind of penalty requirements.
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Modified form of exponential and sigmoidal laws are proposed.
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Comparative study between power, exponential and sigmoidal sandwich FGM beams.

Abstract

Comparative study is carried out in present work on sandwich functionally graded beams made up of different material property variation laws. For study, power, exponential and sigmoidal laws are used. A fourth-order zigzag theory is used for the analysis. Both in-plane and transverse displacement fields are considered to predict the behavior of thick beams more efficiently. A 3-noded 1-D finite element having 8 degrees of freedom per node is used during analysis. The present model satisfies inter-laminar transverse stress continuity conditions at interfaces along with zero value at the top and bottom surfaces of the beam for transverse shear stresses. The current model is free from the requirement of any kind of penalty or C-1 conditions and hence is computationally efficient. Present results are validated with those available in the literature. Results for exponential and sigmoidal law are new results in present work, which will serve as a benchmark for future studies. Results for stresses and deflection are presented in form of tables. For some cases, stress variation across the thickness of beam are also reported. A modified form of exponential and sigmoidal law named Type-E1 and Type-S2 are also presented in which central core is made up of FGM phase. Among all the material variation laws, sandwich FGM beam having ceramic face sheets and exponentially varying FGM core (C-Type-E1) is found to perform best among all the cases studied.

Introduction

Problems associated with laminated composite and sandwich structures such as delamination, matrix-cracking, matrix-fiber debonding, etc. give rise to functionally graded structures (Garg and Chalak, 2019). In functionally graded materials (FGM), material property varies across the thickness of the structure in a regular fashion. A detailed review of the analysis of sandwich FGM structures are given by (Liew et al., 2011; Jha et al., 2013; Tornabene and Reddy, 2013a; Swaminathan et al., 2015; Thai and Kim, 2015; Sayyad and Ghugal, 2019; Swaminathan and Sangeetha, 2017).

The behavior of sandwich FGM beams can be studied by using equivalent single layer theory (ESL), layerwise theory (LWT), or elasticity theory. Sankar (2001) carried out an analysis of FGM beams using elasticity theory. Later, Venkataraman and Sankar (2012) extended the same for carrying out analysis of sandwich FGM beams. In ESL models, the displacement functions are explained using unknow parameters of the reference plane, which is generally the middle plane. Simplest ESL model is known as classical laminated beam theory or Euler-Bernoulli beam theory (Akgöz and Civalek, 2013), which is an extension of the Euler-Bernoulli beam theory developed by Love (1888). However, this theory neglects transverse shear stresses and is not able to predict the behavior of sandwich FGM beams efficiently (Aydogdu and Taskin, 2007; Lee and Lee, 2017). First-order shear deformation beam theory (FOSDT) assumes constant transverse shear stresses across its thickness. Nguyen et al. (2013) carried out static and free vibration analysis of axially loaded sandwich FGM beams using FOSDT. Tornabene and his co-authors (Tornabene, 2009; Tornabene et al., 2011; Tornabene and Reddy, 2013b) carried out analysis of FGM plates and shells using generalised differential quadrature based FOSDT. Trabelsi et al. (2019) proposed FE based FOSDT for analysis of FGM shells. Civalek et al. (Trabelsi et al., 2019; Civalek et al., 2020) used FOSDT for analysis of reinforced beams using finite element (FE) method. The main problem with the shear correction factor is that its value depends upon a number of factors such that material property, the geometry of beam, end conditions, etc. (Pai, 1995; Birman and Bert, 2002; Nguyen and SabBonnet, 2007). The problem associated with shear correction factor is handled by higher-order shear deformation theory (HOSDT) in which displacement field is expressed as higher-order variation (Thai and Vo, 2012; Nguyen et al., 2013; Koutoati et al., 2019; Tornabene, 2009; Tornabene et al., 2011; Tornabene and Reddy, 2013b; Trabelsi et al., 2019; Civalek et al., 2020; Nguyen and SabBonnet, 2007; Kadoli et al., 2008; Kapuria et al., 2008; Ben-Oumrane et al., 2009; Li et al., 2010; Vo et al., 2014; Nguyen and Nguyen, 2015; Hadji and Safa, 2020; Ebrahimi et al., 2020; Akgöz and Civalek, 2014, 2016; Zghal et al., 2018). Thai and Vo, 2012 compared performance of different kinds of HOSDTs for predicting the bending and free vibration behavior of FG beams. Koutoati et al. (2019) employed FE based HOSDT for bending and free vibration analysis of sandwich FGM beams. Kadoli et al. (2008) used FE based HOSDT for static analysis of power-law FG beam having various end conditions. Kapuria et al. (2008) carried out bending and free vibration study on FG beams experimentally and proposed HOSDT based model for the same. Ben-Oumrane et al. (2009) carried out analysis of thick FG beams. Nguyen and Nguyen (2015) proposed Navier's solution based HOSDT for bending, free vibration and buckling analysis of sandwich FGM beams. Akgöz and Civalek (2014) carried out analysis of beams resting on elastic foundations using Navier's solution. Zghal et al. (2018) used FE based HOSDT for analysing FG plates. Most of the HOSDTs assume constant transverse deformation across the thickness of the sandwich FGM beam.

Regarding problems faced for the application of HOSDT, Chakrabarti et al. (2011) stated, “These theories predicted the continuous transverse shear strain across the thickness at interfaces with a discontinuity in the transverse shear stresses at interfaces. But in actuality, transverse shear stresses at interfaces must be continuous, although discontinuity in transverse shear strains may exist. Also, additional dependent unknowns are introduced in HOSDT with each new adding power of thickness coordinate. Even these theories do not satisfy the stress-free condition at the top and the bottom face of the structure”. Recently, Di Sciuva and Sorrenti (Di Sciuva and Sorrenti, 2019) and Dorduncu (2020) presented a comprehensive review of the application of zigzag theories and its higher-order forms for studying FGM structures. Dorduncu (2020) employed peridynamic differential operator for solving equilibrium equations. Neves et al. (2012) employed hyperbolic higher-order zigzag theory (HOZT) for the analysis of sandwich FGM plates. Tornabene et al., 2015, 2016 employed generalised differential quadrature method in framework of HOZT (in Carrera's unified formulation) for analysing sandwich FGM plates and shells under bending conditions. Iurlaro et al. (2014) carried out bending analysis of sandwich FGM plates using refined HOZT. Iurlaro et al. (2014) in their work has shown that the zigzag theories are able to predict the behavior of sandwich FGM plates more accurately as compared to HOSDTs and FOSDTs.

Recently, Garg and Chalak (2020a) proposed a HOZT for analysis of laminated composite plates in which zigzag effects are also introduced in transverse displacement field variation across the thickness of the plate which helps in predicting the behavior of thick sandwich plates efficiently. This theory has also been modified by Garg et al. (2020) for bending analysis of power and exponential law-based sandwich FGM plates. In present work, an attempt has been made to carry out the study on static analysis of sandwich FGM beams using recently proposed HOZT by Garg and Chalak (2020b). Formulation assumes fourth-order variation of in-plane and out of plane displacement fields. Zigzag effects are introduced with the help of linear unit Heaviside step function. Three noded C-0 FE having 8 degrees of freedom per node is used. Present formulation is free from requirement of any kind of penalty function or post-processing technique. A comparative study has been carried out between the sandwich FGM beams made up of different material property laws for both hardcore and softcore conditions. For exponential and sigmoidal material variation rules, new results are reported, which will serve as a benchmark for future studies.

The novelty of present work:

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Considering transverse displacement field variation across the thickness of the beam to predict the behavior of sandwich FGM beams more efficiently in present work.
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The present model satisfies interlaminar transverse stress continuity conditions along with zero value at the top and bottom of the beam for transverse shear stresses.
•
C-0 based finite element is used during analysis.
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The present model is free from the requirement of any kind of post-processing technique.
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A comparative study is carried out on the static analysis of sandwich FGM beam made up of different material property variation law across its thickness, namely power law, exponential, and sigmoidal law. Such a study is not available in the literature to the authors' best knowledge.

Section snippets

Assumptions involved during formulation

Following are the assumptions involved during theoretical formulation for analysis of sandwich FG beams:

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The beam is assumed to be made up of several layers.
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The displacement function consists of bending, extension, and shear components.
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The fourth-order displacement function is chosen along with the linear unit Heaviside step function.
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The present model satisfies zero transverse stress conditions at the top and bottom of the beam along with continuity condition at interfaces.
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1-D Hooke's law is

Results and discussion

Material properties used during the present study are: Metallic phase is made up of Aluminium ( $A l$ ) $E = 70 G P a, ν = 0.30$ . The ceramic phase is made up of Alumina $(A l_{2} O_{3})$ $E = 380 G P a, ν = 0.30$ . Following are the material property variation laws used in present work:

➢
Power law: Two different types of sandwich FGM beams are considered in which material property variation across the thickness of beam follows power-law variation, which are:

Type A: Top and bottom faces are made up of FGM and core is made up of $A l_{2} O$

Conclusion

In present work, a new higher-order zigzag theory is proposed for the analysis of sandwich FGM beams made up of different material laws. The fourth-order displacement field is assumed for both transverse and in-plane displacement fields. Zigzag effects are introduced by using linear unit Heaviside step function. The proposed theory satisfies interlaminar transverse stress continuity conditions along with zero values at the top and bottom of the beam for transverse shear stresses. The proposed

Credit author statement

Aman Garg: Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Roles/Writing: Writing - original draft, Review and editing, H.D. Chalak: Data curation, Formal analysis, Investigation, Methodology, Supervision, Validation, Roles/Writing: Review and editing, Anupam Chakrabarti: Conceptualization, Methodology, Supervision, Roles/Writing: Review and editing, Software.

Funding

The first author thanks MHRD, GoI, and Director, NIT Kurukshetra, for supporting the present work financially through a Ph.D. scholarship grant (2K17/NITK/PHD/6170004).

Declaration of competing interest

Authors declare there exists no conflict of interest in any form for the present manuscript.

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Comparative study on the bending of sandwich FGM beams made up of different material variation laws using refined layerwise theory

Highlights

Abstract

Introduction

Section snippets

Assumptions involved during formulation

Results and discussion

Conclusion

Credit author statement

Funding

Declaration of competing interest

Acta Astronaut.

Mater. Des.

Computat. Mater. Sci.

Compos. Struct.

Thin-Walled Struct.

Thin-Walled Struct.

Compos. Struct.

Appl. Math. Model.

Compos. Struct.

Int. J. Mech. Sci.

Compos. Struct.

Adv. Eng. Softw.

Compos. Part B Eng.

Int. J. Solid Struct.

Compos. Sci. Technol.

Compos. Struct.

Compos. Struct.

Compos. Struct.

Comput. Methods Appl. Mech. Engrg.

Int. J. Mech. Sci.

Compos. Struct.

Compos. Part B: Engng.

Eng. Struct.