Buckling Analysis of Functionally Graded Sandwich Beam Based on Third-Order Zigzag Theory

1. Gupta ^*, HD. Chalak

Department of Civil Engineering, National Institute of Technology, Kurukshetra, 136119, India

1.     Introduction

The Buckling phenomenon is a very different structural response than in-plane compression and can lead to catastrophic failure at critical load. It is also a principal mode of failure for slender components like laminated sandwich beams. Recently sandwich (SW) structures having a three-layered architecture are used in abundance because of their attractive properties of high strength-to-weight ratios, high energy absorption, etc [1, 2]. SW structures use the advantages of two or more distinct materials at a time and in one place. For example, the most common SW structure comprising metal and ceramic exhibits both strength and temperature resistance properties. The main drawback of using SW laminated structures is related to the interface of the layers like de-bonding and stress concentration [3, 4]. A proper solution to these problems is using functionally graded material (FGM) having smoothly varied properties in between two widely varied properties of constituted layers. FGM layer(s) can be used as middle or edge layers of SW beams to produce functionally graded sandwich (FGSW) beams.

FGMs are a new class of composites and have found numerous applications in many engineering and biomedical fields such as nuclear projects, the aviation industry, the aerospace sector, defense, the automobile industry, electronics, manufacturing, the energy sector, dentistry, orthopedics [5], etc. As FGM is a combination of distinct materials so, through suitable tailoring, a designer can draw the most complicated requisite properties from them. It provides a better option than conventional composites. There are numerous examples of graded structures present in nature such as bones, teeth, skin, leaves, etc [6].  It is a well-known fact that a structural element built by nature carry out all its functions effectively and is irreplaceable in any manner. So, it can be said that FGM is an ideal material for assigning a structure. Although FGM is ideally suited material for all requirements, can’t be used everywhere because of the difficulty and cost of production. Earlier it was impossible to attain the variable microstructure, but now with the advent of high-edge technologies, it is possible to generate an FGM. Mostly volume gradient structure is made for an FGM.

The highly heterogeneous structure of FGM causes inflation of shear deformation effects in the thickness direction of SW beams. So its behaviour becomes complex which creates reliability issues in practical applications. So, response data of FGSW structures should be generated and compiled to help in increasing the practical application in various fields.

Many theories are developed for getting the true behaviour of SW beams. An elaborate review of different theories used for analyzing FGSW structures is given by [7]. Elasticity theories [8-12] provide benchmark data by achieving the highest degree of accuracy but they are cumbersome, so many simplified theories based on some assumptions are developed by researchers. When going through the literature two broad categories can be identified for the theories developed. They are equivalent single layer (ESL) and layer-wise theories (LWT). ESL theory assumes the primary variable with reference to the central/neutral layer while LWT assumes the same layer-wise. The order of displacement in independent variable assumption creates different theories like classical plate theories CPT [13-16], First order shear deformation theory FSDT [17-19], third-order shear deformation theory TOT [20], higher-order shear deformation theory HOT [21-23], refined HOT [24-30] and quasi-3D theory [31]. CPT ignores shear deformation effects and overestimates the buckling loads. FSDT requires a correction factor for satisfying parabolic variation of shear deformation. TOT and HOT do not give traction-free shear stresses. These displacement-based theories are applied using ESL or LWT approach. The ESL approach is simple so, is mostly used by researchers.

LWT is more accurate than ESL theory because it has a more realistic approach for multilayered beams having an uneven distribution of dependent variables resulting from a layer-wise construction. LWT is further classified as discrete LWT and zigzag LWT. Discrete LWT [32] assumes the variables with the individual layers, so the solution becomes difficult to achieve for a larger number of layers. Also, most of the discrete LWT does not satisfy the interlaminar stress continuity, since two different values of stresses are built at the interface of these layers from the different elastic modulus of two layers. Zigzag LWT [33-43] describes the nonuniform variation in the dependent variable by including an additional term in the displacement field while adopting the ESL assumption. The superiority of zigzag LWT in getting static and dynamic results over FSDT and TOT is listed by Kapuria et al. [37]. By comparing zigzag LWT with benchmark elasticity results for SW beams, he found it gives the least percentage of errors. The manner by which the zig-zag LWT is able to satisfy the interlaminar displacement continuity of laminated structures is well cited by [41].

Pandey and Pradyumna [32] analyzed the FGSW plate by employing eight noded C⁰ isoparametric finite elements having 13DOF using a layer-wise expansion for displacement fields. Chakrabarty et al [38] defined the issue of C1 continuity associated with implementing the finite element method using the zig-zag theory and solved the buckling problem of the SW beam. Averil [33] developed first-order zig-zag formulations for laminated beams and employed a penalty function to alleviate the C1 requirement for two noded finite elements. Later Cho and Averill [34] used sublaminate approximation to avoid the C1 requirement for four noded elements. Vo et al [24] used a linear interpolation function in generalized displacements to solve the C1 continuity issue. Neves [36] used the meshless method in place of FEM to solve the buckling problem of FGM sandwich beams. Kapuria and Ahmed [42] used interdependent interpolation. In the present work problem of C0 continuity arising from the use of zigzag theory in combination with FEM for getting solutions is avoided by expressing the derivation terms of shear stress in terms of other variables and solving the equations. It does not require any post-processing method.

Solving a problem involves two main steps: consideration of a theory and the type of solution method. Until now most researchers have used ESL theories for simplicity. This paper uses an advanced theory that overcomes all the deficiencies of earlier proposed theories as overestimating the buckling loads (CPT); requiring a shear correction factor for correctly assessing the variation of shear stresses (FSDT); providing the actual parabolic variation of transverse shear stresses, but not fulfilling stress-free boundary conditions (HOT);  providing a solution which is dependent on the number of layers and giving two values of shear stress at the interface because of the two different displacement equations in adjacent layers (LWT). The zigzag theory used in this work is supreme to all the above-stated theories as it does not need any shear correction factors, provides the actual parabolic variation of transverse shear stresses, fulfills stress-free boundary conditions, and is not dependent on the number of layers for the solution. The only issue in using Zigzag theory is the problem of C0 continuity due to the inclusion of additional terms in displacement approximation, which is avoided here as discussed earlier.

Based on the literature review, authors found that plenty of study is available for the structural response of FG sandwich beams based on analytical approach such as Navier’s solution, because of its accuracy and ease in finding solutions, but has restraint in terms of boundary conditions, material law, loading conditions, etc. These restraints are overcome by using numerical methods like FEM, meshfree method, etc. In this study a numerical method: FEM is employed to study the response of symmetric and asymmetric FGSW beams subjected to various end conditions and material laws. Sayyad and Ghugal [43] reported that an ample amount of research is available for the analysis of plates and bending and vibration response SW structures while buckling analysis of FGSW beams is very less studied. Although a sufficient amount of literature is present on the zig-zag analysis of SW structures; but till now very less authors have taken up the zig-zag method for analyzing FGSW beams because of the complexity in arriving at the solutions due to the inclusion of the zig-zag factor along with FGM layer(s). Among the zig-zag analysis-based studies, to the author’s best knowledge, buckling analysis is not available and the bending and vibration studies do not adopt the numerical solution method, nor take into account the thickness stretching effects. The novelty of the present paper is presenting the buckling analysis results for the FGSW beam using a zig-zag theory with thickness stretching through a numerical method (FEM). Present paper deals with finding buckling responses of FGSW beams for various end conditions, aspect ratios, and homogenization laws based on the recently proposed zig-zag theory [38]. The present theory satisfies interlaminar stress continuity. A C0 continuous FEM formulation of 3 noded elements with 7DOF/node is used. The present paper also gives new results which will serve as a benchmark for future works with a similar vision.

2.     Modeling of Material Properties

A three-layered SW beams (Fig 1) of two types: Type P and Type E are synthesized. These are further classified according to the core material: Type P-H, E-H (hardcore) (Fig 2a), and Type P-S, E-S (softcore) (Fig 2b) for analysis. These beams have FGM as face sheets and cores built as ceramic (Type H) and metallic (Type S). As it is established that the material property consideration has a great effect on the analysis results, so this study uses two types of material modeling (power law and exponential law) and the results were compared. Material property variation is modeled in two ways:

2.1. Power law:

Garg et al. [44] presented a review of the analysis of FGM sandwich structures and prepared a list of literature based on the type of material law used. They found that more than 90% of the FGM-related literature used power law for estimating the material properties as it is simplest as per the ease of use. A similar observation was made by Swaminathan et al. [45]. In this paper, material properties are graded in the thickness direction according to the power-law distribution in terms of the volume fractions of the constituents of the material, and the effective material properties are estimated on the basis of the Voigt model as the homogenization method. Although the Voigt rule does not consider the interaction among adjacent inclusions, Mori–Tanaka method considered these interactions of neighboring phases at the microscopic level as done in [46,47]. Using Voigt model material property in FGM, P(z) is expressed as:

where  and  are material properties of metal and ceramic and their variation is shown in Fig. 3 for a lamination scheme of 2-2-1.  is the volume fraction of the ceramic part which is written as: (For Type P beams)

2.2. Exponential law:

The FGM part is made to obey exponential law and the variation of material property by this law is shown in Fig. 4 for a lamination scheme 2-2-1and is written as:

Fig. 1. Geometry of sandwich plate in the Cartesian coordinate system

Fig. 2. Layer configurations of FGSW beam of various types (a) Type P-H and Type E-H (b) Type P-S and Type E-S

where  is given as: (For Type E beams)

3.     Theoretical Formulations

Consider a beam in the assumed coordinate system shown in Fig 1. The displacement in x ( ) direction is assumed as:

where is mid-plane displacement along the x-axis for any point in the SW beam,  is the rotation of normal to mid-plane and  depict the number of upper and lower layers, respectively. ,  are higher-order unknowns, and ,  are the slope of a-th and b-th layers corresponding to the upper and lower layers respectively. denotes unit step function. The displacement in z-direction ( ) is assumed as taken in [38] as:

where ,  and  are the values of the transverse displacement at the top, middle and bottom lamina of the core, respectively, and , and  are Lagrangian interpolation functions in the z-direction. By taking the reference of [38], the constitutive relation for stress in local coordinates ( ) of k-th lamina is given as:

where  is transformed rigidity matrix of k-th lamina and  appeared in the above equation can be converted into a global coordinate system by using the transformed compliance matrix  as:

Now using the conditions of zero transverse shear stress at z=h/2 and z=-h/2 and transverse shear stress continuity at interfaces of layers with at z=h/2,  at z=-h/2, .

The terms ,  and  are expressed in terms of displacement  and  as:

Fig. 3. Variation of material property across the thickness of 2-2-1 FGSW beam (a) Type-P-H (b) Type P-S

Fig. 4. Variation of material property across the thickness of 2-2-1 FGSW beam (a) Type E-H (b) Type E-S

where ,
 and material properties determine elements of . Through Equation (7) we are able to write the differentiation of transverse displacements in terms of other unknowns, thus avoiding the C1 continuity issue. Now Equation (3) is rewritten as:

where the coefficients of s are determined by values of z, H, and material properties. Now, when all the coefficients of higher order terms of Equation (3) are eliminated so, we can write the generalized displacement with the help of Equations (4) and (8) as:

Writing strain field in terms of unknowns as a combination of linear and nonlinear parts by using strain-displacement connection and Equations (3-6) as:

where the linear part of the strain is

and nonlinear part of strain is:

where,  or
 and

And the elements of matrices , and  are dependent on z and unit step functions. The data  and the elements of  can be accessed by the corresponding author through the mail.

Now applying the virtual work principle on the same lines as done in [35], the total potential energy of the system is given as :

where  is the strain energy and is the energy due to externally applied load. Utilizing equations (5) and (9), the strain energy is

where

 is computed as:

where  and  is the stress matrix of the i-th layer generated from the external in-plane loads which is given as
 

4.     Finite element Formulations

A numerical method i.e., FEM is used for the solution of buckling problems. A quadratic element with three nodes and seven degrees of freedom is considered.

The generalized displacement vector δ at any point can be expressed in terms of displacement  and shape functions  related to i-th node.

Here, n is no. of nodes in one element. From Equation (17), the strain vector  used in Equation(11) is given as:

where [B] is the strain displacement matrix.

The potential energy given in Equation (13) can be rewritten by using Equations (14-16) as:

where,

Finally minimizing with respect to {δ}

where and  are stiffness matrix and geometrical stiffness matrix and λ is the buckling load factor. A flow chart is made by incorporating all the steps needed to be followed for determining λ, which is given in the appendix. A code is written in FORTRAN for calculating the λ. A simultaneous iteration method is utilized for solving the buckling Equation (22).

5.     Results and discussions

Buckling analysis for four types of FGSW beams is presented here for the materials having properties: Ceramic (Al2O3) Ec=380GPA, µ=0.3 and Metal (Al) Ec=70 GPA, µ=0.3. The Non-dimensional factor used in this study is given as:

where L and h are illustrated in Fig. 1 and  is the transverse modulus of elasticity of the face sheet. Six-layer configurations of FGSB are used in this study (Table 1), wherein h_1, h_2, etc are measured from the central layer of SW.

Table 1. Thickness coordinates of different lamination schemes

A convergence study was performed for present models (1-2-1), Type P-H (Fig. 5), and Type P-S (Fig. 6) at L/h=5 and n=2 for using different mesh divisions of 4, 8, 16, and 32. As the results converged at a mesh size of 16 so, it is adopted throughout this study. Table 2 presents the buckling response of the FGSW beam for the six lamination schemes and different power law factors. The present results are compared with those reported earlier: Kahya et al [17] used FSDT, Nguyen et al [19] used HOT, Vo et al [21] used refined HOT, Vo et al [28] used quasi-3D theory for getting the responses and the present results are in good agreement with these.

Fig. 5. Variation of buckling load with mesh divisions for a SS 1-2-1 Type P-H FGSW beam

Fig. 6. Variation of buckling load with mesh divisions for
a SS 1-2-1 Type P-S FGSW beam

So, the present theory can be applied to buckling solution of FGSW beams. As expected, FSDT results [17] are yielding lower values of non-dimensional buckling load in comparison to all other theories. Beams with lower power-law factors were found to withstand higher buckling loads irrespective of the lamination schemes for both Type-H beams owing to the variation of the material property of FGSW (Fig. 3a, 4a) in the thickness direction, while beams with higher power-law factor were found to withstand higher buckling loads irrespective of lamination schemes for both homogenization rule of Type S owing to the material property variation FGSW (Fig. 3b, 4b).

A high drift in buckling response is seen for a change of 1 unit (0 to 1) in the power law factor, which can be attributed to a change in material properties with a change of the power-law factor (Figs. 3a, 4a). However, for a change of 5 units (5-10) of the power-law, the change in the buckling resistance is found to be very less in comparison to that found an increase of 1 unit of power law factor from 0 to 1. The above-stated variation of buckling resistance with a change in power law factor is valid for both Type S and Type H beams and all lamination schemes considered in the present study. The 1-2-1 lamination scheme was found to have the highest buckling resistance for Type H beams for both homogenization schemes and for all power law factors, which can be attributed to the highest core thickness of the 1-2-1 scheme which is made up of ceramic material. End conditions were also found to have a significant effect on buckling strength.

Table 3 presents the buckling response of Type S FGSW beams for various lamination schemes and power-law factors. With an increase in the power-law factor, an enhanced buckling response is observed (Figs. 3b, 4b) owing to the dependency of the material property on the power-law factor. Lamination scheme 1-0-1 was found to have the highest buckling resistance for Type S beams for both homogenization rules and for all power law factors, which can be attributed to the lowest core thickness made of metal material (having lower strength in comparison to ceramic material). Again, lamination scheme 1-2-1 was found to have the highest buckling resistance among all lamination schemes for type H beams, for both homogenization rules and for all power law factors, which can be attributed to the highest core thickness made of ceramic material (having high strength in comparison to metallic material).

Tables 4 and 5 provide the variation of buckling response with an augment in the aspect ratio of the beam. Tables 4 and 5 are represented again in terms of graphs for greater clarity of the buckling strength variation. The effect of an increase in the length-to-height ratio on buckling response was found to be significant up to a value of 20, after which there was a little change in buckling response for both Type H and S FGSW beams (Figs. 7-12).

Fig. 7. Variation of buckling load with aspect ratio for Type P-S FGSW beam with the CC end condition

Table 2. Variation of non-dimensional buckling load for Type-H FGSW beam for SS end condition (L/h=5)

Table 3. Variation of non-dimensional buckling load for Type-S FGSW beam for SS end condition (L/h=5)

Table 4. Variation of non-dimensional buckling load for Type-H FGSW beam at n=2

Table 5. Variation of non-dimensional buckling load for Type-S FGSW beam at n=2

Fig. 8. Variation of buckling load with aspect ratio for Type P-H FGSW beam with the CC end condition

Fig. 9. Variation of buckling load with aspect ratio for Type P-S FGSW beam with CF end condition

Fig. 10. Variation of buckling load with aspect ratio for Type P-H FGSW beam with CF end condition

Fig. 11. Variation of buckling load with aspect ratio for Type P-S FGSW beam with the SS end condition

Fig. 12. Variation of buckling load with aspect ratio for Type P-H FGSW beam with SS end condition

6.     Conclusions

This paper presents buckling responses of the FGSW beams made of power law and exponential law using zigzag theory. Higher-order terms are assumed for displacement approximations. Numerical results are arrived at by using the FEM of three noded elements having 7DOF/node. The present model is C0 continuous and does not require any post-processing method. The locking phenomenon which is associated with FEM is avoided here. Results of the present model are compared with the existing ones and are found to be consistent, which describes the suitability of the present model in deriving results for FGSW beams. It is found that buckling response is dependent on the power-law factor, aspect ratio, lamination schemes, and end conditions. Many new results are given which will pose as a benchmark for parallel studies. The main inferences drawn from the study are:

1) The buckling strength was improved by increasing the power-law factor for Type S beams while the opposite behavior was seen in type H beams for all types of lamination schemes and end conditions.

2) The end conditions played a major role in deciding the buckling response of FGSW beams. 3) Two types of laws were used in this paper to synthesize the FGM part of FGSW beams. The difference in buckling load resistance on using these two laws is small, but its trend is different for the two types: Type S and Type H.

4) It is found that exponential law-governed FGSW beams show a little higher buckling resistance behavior in comparison to power law-governed FGSW beams for Type S while the opposite behavior is seen for Type H beams for all types of end conditions and lamination schemes.

Acknowledgments

The first author of this paper was financially supported jointly by MHRD, GoI, and Director, NIT Kurukshetra, through a Ph.D. scholarship grant (2K18/NITK/PHD/6180093).

Conflicts of Interest

The author declares that there is no conflict of interest regarding the publication of this manuscript. In addition, the authors have entirely observed the ethical issues, including plagiarism, informed consent, misconduct, data fabrication and/or falsification, double publication and/or submission, and redundancy.

Appendix

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