Analytical solutions for the hygro-thermo-mechanical bending of FG beams using a new fifth order shear and normal deformation theory

A new analytical solution is presented for functionally graded (FG) beams to investigate the bending behaviour under the hygro-thermo-mechanical loading using a new fifth order shear and normal deformation theory (FOSNDT). The material properties of the FG beam are varied along the thickness direction according to the power law index. In the present theory, a polynomial shape function is expanded up to fifth-order in terms of thickness coordinate to consider the effects of transverse shear and normal deformations. The present theory is free from the shear correction factor. Using the Navier’s solution technique the closed-form solution is obtained for simply supported FG beams. All the results are presented in non-dimensional form and validated it by developing the classical beam theory (CBT), first order shear deformation theory (FSDT by Mindlin) and third order shear deformation theory (TSDT by Reddy) considering the hygro-thermo-mechanical loading effects which is mostly missing in the literature. It is noticed that the presented FOSNDT is very simple and accurate to predict the bending behaviour of FG beams under linear and non-linear hygro-thermo-mechanical loadings. c © 2020 University of West Bohemia. All rights reserved.


Introduction
The composites are usually a mixture of two or more than two materials in combination. The materials having the proper and significant chemical properties are normally chosen to get the typical mixture and required strength. Functionally graded materials (FGMs) are similar kinds of composite material having unique characteristics. Nowadays, FG materials are used in many engineering and structural applications because of its novel properties. The greatest advantage of FG material is flexibility in design, which makes it more suitable to grade the material properties in any specific direction as per requirement. Currently, the main focus of scientists and researchers is to suggest and introduce a new type of material which is light in weight, suits for any environmental conditions, should be reliable in service performance and able to minimize the size of structural elements. The available materials are not much suitable for all the engineering applications, hence the FG beams are used in main structural components of aerospace vehicles, defense sector units, shipbuilding industries, seashore structures, many automobile and chemical industries. In fact, the beams made from FG materials are very strong to resist the thermal stresses produced in spacecrafts and aircrafts operations. FG beams are even most suitable to resist hygro stresses generally developed in seashore and marine structures located in humid environment.
Hence, before using FG beam for important engineering applications, it is absolutely necessary to study the bending behaviour of FG beam under hygro-thermo-mechanical loading. Laminated composites may fail due to delamination failure or stress concentration at the layer interfaces during the life span. To overcome the delamination failure one can use FG material which is free from layers and the properties of such a material can be tailored in specific direction using a power law index. The ceramic and metal are the two major constituents used to form the FG materials. The ceramic constituents are having low thermal conductivity and good for wearing resistance, whereas the metals having good ductile performance, which sustains the deformations and can prevent the fractures caused due to stress. Looking into current scenario, the present research is an attempt towards fulfillment of the industry demands by developing the new solution technique to analyze important weight sensitive structures where safety and accuracy is on top propriety.
The several analytical techniques, numerical methods and few elasticity solutions are available in the literature which is addressed by various researchers across the globe, to study the static and dynamic response of the FG beams under mechanical and thermo-mechanical loading. But there is acute shortage of literature on the FG beams subjected to hygro-thermo-mechanical loadings. Hence, the main aim of this study is to present the bending analysis of FG beams considering the linear and non-linear variations of hygro-thermo-mechanical loadings. In this study, non-linearity is related to loadings and not a geometric.
The CBT (classical beam theory) which is developed by Euler-Bernoulli ignores the effect of shear deformation. The CBT is a simple beam theory used for the analysis of thin beams hence it not recommended for the analysis of thick beams due to neglect of the shear deformation effect. In the thick beam analysis, CBT underestimates the stresses and deformations. In 1921, Timoshenko developed the FSDT which assumes the first order variation in the axial displacement. But this theory needs the problem dependent shear correction factor to address the correct behaviour shear deformation. FSDT also fails to satisfy the zero transverse shear stress conditions at the top and bottom surfaces of the beam. To address the deficiencies of CBT and FDST, many researchers are trying to develop the new kinds of higher order theories.
During the space plane project in 1984, Japanese scientists introduced the FG materials to resist the ultra-high temperature. The FG sheets tested under high temperature fluctuations across the thin cross-sectional thickness. The use and applications of the FG materials can be found in the literature by Koizumi [59,60], Muller et al. [69], Birman and Byrd [25], etc. The detailed information on the FG beams and plates is available in the review articles published by Jha et al. [53], Swaminathan et al. [90], Swaminathan and Sangeeta [91], Sayyad and Ghugal [85,87,88]. This reviews given the good insights over the available literature which influences the many researchers to extend their efforts to analyze the FG beams under linear and non-linear hygro-thermo-mechanical loading which is very rarely addressed in the literature. Sankar [80], Ding et al. [38], Zhong and Yu [99], Daouadji et al. [36], Chu et al. [31], Ying et al. [95] and Xu et al. [94] presented an elasticity solution for the FG beams. The 3-D elasticity solutions are analytically very complicated, cumbersome and difficult to solve. Therefore, researchers are taking interest to propose a new analytical techniques and numerical methods for the FG beams in which the numerical results are closer to the exact solutions obtained by elasticity solutions.
The beam made from FG materials are most suitable in thermal environments due to low thermal conductivity of ceramic materials. The response of the FG beam using different models under thermal and thermo-mechanical load is reported by a few researchers like Aboudi et al. [3], Chin and Chen [30], Giunta et al. [48], Megharbel [66], Sankar and Tzeng [81], etc. The effect of non-linear thermo-mechanical loads on bending behaviour of FG beams is reported by Shen [89], Ma and Lee [63,64], Ma and Wang [65], Esfahani et al. [39], Arbind et al. [9], Nirmala and Upadhyay [100], etc. Toudehdehghan et al. [82] studied the effect of thermal environment on FG coated beams using clamped-clamped end condition. Zhau et al. [82] and Sator et al. [75] used the meshless method to address the thermal effects.
The hygro-thermo-mechanical analysis for the FG plate is studied by Zidi et al. [101], Zenkour et al. [97], Daouadji et al. [32], Zenkour and Radwan [86] and Sayyad and Ghugal [98] using different higher order theories and mathematical approaches. But the given literature is limited to only for FG plates; there is acute shortage of literature on FG beams subjected to non-linear hygro-thermo-mechanical loads. Recently, a fifth order theory has been used by Ghumare and Sayyad [43][44][45][46] for the analysis of FG beams and plates and also by Naik and Sayyad [70][71][72][73] for the analysis of laminated composite beams and plates subjected to mechanical and thermo-mechanical loadings.
The main focus of the present study is to investigate the bending behaviour of FG beam using fifth order shear and normal deformation theory under the non-linear hygro-thermo-mechanical loadings. In the present theory, a polynomial shape functions are expanded to fifth-order in terms of the thickness coordinates. This theory includes the effect of thickness stretching. The theory involves only six independent field variables. This theory does not require shear correction factor. To obtain the closed-form solution, the Navier's solution technique is used for simply supported FG beam. Using power-law the material properties are graded. Using the principle of virtual work and the fundamental lemma of calculus, the variationally consistent governing equations and associated boundary conditions are evolved. To validate the present theory authors have developed CBT, FSDT and TSDT for the non-linear hygro-thermo-mechanical load and the dimensionless results are compared with these theories. It is found that the present FOSNDT is very simple and accurate to predict the bending response under the non-linear hygro-thermomechanical loadings.

Geometry of the FG Beam
The dimensions and geometry of the FG beam used for the analytical solution are shown in Fig. 1. The y-directional width of the FG beam is assumed as unity. The top surface of the FG beam is made up of metal, whereas the bottom surface is made up of ceramic material.  considering the thickness coordinate up to third-order polynomial is not sufficient. Therefore, in the present theory thickness coordinate is expanded up to fifth-order polynomial to get the accurate displacements and stresses. 4. Transverse normal stress/strain plays an important role in the modeling of thick beams which is neglected by many theories available in the literature. The present theory considers the effects of both transverse shear and normal deformations. 5. The displacement field of the theory enforces the realistic variation of the transverse shear stresses (parabolic) across the thickness of the FG beam. 6. To grade the material a simple mixture rule, i.e. power-law is used.

The displacement field
The displacement field of the present theory is as follows: where, u and w are the axial and transverse displacement of any point on the FG beam. u 0 and w 0 are the x and z-directional displacements of any point on the neutral surface of the FG beam. The terms φ x and ψ x are the shear slopes associated with the transverse shear deformation, whereas φ z and ψ z are the shear slopes associated with the transverse normal deformations.
Using the theory of elasticity, the strain components are obtained:

The power-law for material gradation
The power law is used to change the volume fraction of the constituent materials continuously along the thickness direction. The power-law is stated as where E represents the elasticity modulus. Subscripts m and c represent the metal and ceramic constituent's, respectively, p represents the power law index. The value of p is equal to zero represents the fully ceramic phase, whereas p equal to infinity represents a fully metallic phase. The effect of variation of the Poisson's ratio ν on the bending response of FG beam is very small, hence the Poisson's ratio is assumed as constant. The variation of thermal and moisture loads are also assumed to vary according to power-law index through the thickness of a FG beam. Fig. 2 shows the variation of volume fractions across the thickness of the FG beam.

The constitutive law for the FG beam
The FG beam follows the linear constitutive relations at a point can be written as follows, where α x , α z and β x , β z represent the coefficients of thermal and moisture concentration coefficients, respectively. The temperature and moisture variation profiles are assumed as follows, where T 0 represents constant temperature, T 1 represents a linear temperature variation, T 2 and T 3 represents non-linear temperature variation (see Fig. 3). The similar meaning and variations are assumed for moisture concentration terms C 0 , C 1 , C 2 , C 3 .

The governing equations
The six variationally consistent governing equations are obtained using the principle of virtual displacement which equates external and internal work: Substituing strains from (2) and stresses from (5) into the equation (8), one will get where N x is the resultant axial force, M b X represents the resultant moment due to bending, M S 1 X , M S 2 X are the resultant moments due to shear deformation and The governing equations mentioned in (11)-(16) are obtained by integrating (11) by parts and setting the coefficients of δu 0 , δw 0 , δφ x , δψ x , δφ z , δψ z and equating it to zero: The governing equations involve mechanical, thermal and moisture constants which are mentioned in (17)- (19) as follows: Mechanical constants Thermal constants

The boundary conditions
The boundary conditions obtained at x = 0 and x = L are in the following form:

The analytical solution
Using the Navier's solution technique, the analytical solution is obtained for the simply supported FG beam. The displacement variables are expressed in the single trigonometric series where u m , w m , φ xm , ψ xm , φ zm , ψ zm are the unknown coefficients and α = mπ/L. The top surface of the beam is transversely loaded with q(x) and load is expressed in a single trigonometric form as where q m = q 0 (sinusoidal load).

Illustrative examples, numerical results and discussion
To verify the accuracy of the present theory, bending response under thermo-mechanical and hygro-thermo-mechanical loadings are presented and discussed in this section.

Illustrative example
To validate the present theory, the following examples are solved.
Example 4: Transverse displacements and stresses in FG beam under non-linear hygro-thermomechanical load (T 1 = T 2 = 10, C 1 = C 2 = 100, q 0 = 100). Table 1 shows the comparison of maximum dimensionless axial displacementū, transverse deflectionw, normal stressσ x and transverse shear stressτ xz for FG beams subjected to linear thermo-mechanical load with the power index p. The transverse and axial displacements obtained by the present theory are marginally less for ceramic and metal phases when p = 0, ∞, whereas for p = 1, 2, 5, 10 the transverse and axial displacements are in close agreement with the other theories. It is observed that the displacement increases with an increase in power law index. This is due to the decrease in the stiffness of the FG beam. It is noted that neglecting the transverse shear deformation effect, the CBT underestimates the displacements. Also, it is important to observe that the numerical results for normal and transverse shear stresses are having a close agreement with other refined theories. Table 2 shows maximum values of dimensionless transverse deflectionw under linear thermo-mechanical loads by varying the aspect ratio and power law index. As the aspect ratio increases, the dimensionless deflection found decreased, whereas deflection increases with power law. Table 3 shows the comparison of non-dimensional  Table 5. Non-dimensional deflections in FG beam for different models subjected to linear and non-linear thermo-mechanical load (L/h = 10, T 1 = 10, p = 10, q 0 = 100) axial displacementū, transverse deflectionw, normal stressσ x and transverse shear stressτ xz for FG beams subjected to non-linear thermo-mechanical loads. The observations are similar to the thermo-mechanical case except some marginal improvement in numerical values due to the inclusion of non-linear effect. Table 4 shows transverse deflectionw for FG beam subjected to non-linear thermo-mechanical loads by varying aspect ratios. Tabulated results clearly show that transverse displacements are found larger in the case of non-linear thermo-mechanical loads compared to the linear load case. Table 5 shows the transverse displacements using different models when power law index p = 10. It is observed that the present FOSNDT gives accurate results compared to other theories. Fig. 4 shows the through thickness variation of axial displacementū for FG beam under linear and non-linear thermo-mechanical loadings, respectively. It is noticed that for the fully ceramic case (p = 0) the axial stresses are compressive in nature, whereas for fully metallic case (p = ∞) it is tensile in nature. Fig. 5 shows the variation of transverse displacementw for FG beam under linear and non-linear thermo-mechanical load using different aspect ratios. In non-linear cases the displacements are slightly higher than in linear cases. Fig. 6 shows the variation of bending stress for the FG beam under linear and non-linear thermo-mechanical loads. The bending stress is zero at a neutral plane for ceramic and metal phases due to isotropic properties of the material and when p = 1, 2, 5, 10 there is a shift of the neutral plane due to continuous change in volume fraction across the thickness. It is important to observe that in the case of non-linear thermo-mechanical loads (T 1 = T 2 = 10) for p = ∞, the bending stress varies non-linearly. Fig. 7 compares the bending stress by CBT, TSDT and present FOSNDT to understand the accuracy of the present theory when compared with other theories taking into   account the effect of linear and non-linear thermo-mechanical loads. It is observed that bending stresses are higher in the case of non-linear loading case compared to the linear loading case. Fig. 8 shows the distribution of transverse shear stress through the thickness of FG beam. When p = 1, 2, 5 and 10 due to the variation of volume fractions, the maximum values of transverse shear stress are found shifted above the neutral surface. For ceramic and metal phases, the transverse shear stress is a maximum at a neutral surface because of isotropic properties of materials. Table 6 shows the comparison of displacements and stresses for FG beams subjected to linear hygro thermo-mechanical load. The comparison of thermo-mechanical and hygro-thermomechanical loading cases shows that the variations in stresses and displacements are similar in both cases, but due to the inclusion of moisture load the numerical values are greater for the hygo-thermo-mechanical case. It shows that the moisture load cannot be ignored to know the accurate behaviour of the FG beams. Table 7 shows the maximum non-dimensional transverse deflection under linear hygro-thermo-mechanical loads considering different aspect ratios. As the aspect ratio increases, the deflection decreases. Table 8 shows the comparison of maximum non-dimensional axial displacementū, transverse deflectionw, normal stressσ x and transverse shear stressτ xz for FG beams subjected to non-linear hygro-thermo-mechanical loads. In case of axial displacementū, it is noticed that the axial displacement values obtained by present  Table 9 shows maximum deflection for FG beam subjected to non-linear hygro-thermo-mechanical loading for various aspect ratio and power law index. As the aspect ratio increases, the non-dimensional deflection decreases and as the power law index increases, the deflection increases due to increase in flexibility of the FG beams. The displacements are noticeably greater in non-linear hygro-thermo-mechanical load than in the linear load case. Table 10 shows the results of transverse displacements for different models for power law index p = 10, it is observed that all the displacement values are having a good agreement with other theories. Fig. 9 shows the axial displacementsū across the FG beams subjected to linear and non-linear hygro-thermo-mechanical loadings, respectively. In the non-linear case, the axial displacements are larger. Fig. 10 shows the variation of transverse displacementsw for FG beams subjected to linear and non-linear hygro-thermo-mechanical loads. Due to low stiffness when p = ∞, the displacements are noticeably larger. Fig. 11 shows the bending stress variation for FG beams subjected to linear and non-linear hygro-thermo-mechanical loads. The variations are exactly similar to the hygro-thermo-mechanical loading case. It is important to notice that in non-linear hygro-thermo-mechanical loads when p = ∞, the bending stress varies non-linearly. Fig. 12 shows the distribution of transverse shear stress through the thickness of the FG beam for thermo-mechanical and hygro-thermo-mechanical loading cases. For ceramic and metal phases,

Conclusions
The present study includes the effect of non-linear hygro-thermo-mechanical loading and thickness stretching to predict the accurate bending response of the FG beams using a fifthorder shear and normal deformation theory (FOSNDT). FOSNDT includes the fourth order variation of transverse displacement and fifth-order variation of axial displacements. This theory does not need the shear correction factor to satisfy the traction free boundary conditions of the FG beam. Using the principle of virtual displacement, the variationally consistent governing equations and associated boundary conditions are obtained. Using the Navier's solution technique, the analytical solution is obtained for simply supported FG beam. Several important observations from the present study can be formulated: 1. Displacements are increasing with an increase in the power law index due to the increase in the flexibility with an increase in the power law index.
2. The analysis of obtained results shows that the displacements and stresses are marginally higher in non-linear hygro-thermo mechanical loading cases than in cases with linear loading.
3. It is important to notice that in the case of non-linear hygro-thermo-mechanical loads when p = ∞, the bending stress varies non-linearly. 4. The present theory is very simple to predict the accurate bending response of FG beams in the presence of both thermal and hygro loads.
5. The numerical results by the present FOSNDT are in excellent agreement with other theories.

Important recommendations
From the numerical results and observations, authors recommendations are as follows: 1. For hygro-thermo-mechanical analysis Carrera recommended that thickness cocordinate must be expanded at least up to the fifth order to get the accurate results for FG beams and plates.
2. Since the temperature and moisture effects are changing through the thickness, it is recommended to consider the thickness stretching effect to get the accurate results when FG beams are subjected to hygro-thermo-mechanical load.
3. As per the authors' knowledge, the results obtained by the present investigation with the hygro-thermo-mechanical loading will be the benchmark for future researchers and scientists in the area which is addressed for FG beam first time.

Scope for the future work
It will be very interesting to address the effect of hygro-thermo-mechanical load on FG beam considering the following points: 1. Dynamic and buckling response of FG beams using FOSNDT under non-linear hygrothermo mechanical loadings considering the thickness stretching.
2. To analyze the FG beams for other boundary conditions using FOSNDT considering the thickness stretching and non-linear hygro-thermo mechanical loading.
3. To study the bending the behaviour of the curved beams and shells using FOSNDT with the thermal and moisture effect.