A comment on “Nonlinear thermal buckling of axially functionally graded micro and nanobeams” [Composite Structures 168 (2017) 428–439]

Accepted Manuscript A comment on “Nonlinear thermal buckling of axially functionally graded micro and nanobeams” [Composite Structures 168 (2017) 428–439] Amir Mehdi Dehrouyeh-Semnani PII: DOI: Reference:

S0263-8223(17)31678-1 http://dx.doi.org/10.1016/j.compstruct.2017.07.002 COST 8657

To appear in:

Composite Structures

Received Date: Accepted Date:

27 May 2017 8 July 2017

Please cite this article as: Mehdi Dehrouyeh-Semnani, A., A comment on “Nonlinear thermal buckling of axially functionally graded micro and nanobeams” [Composite Structures 168 (2017) 428–439], Composite Structures (2017), doi: http://dx.doi.org/10.1016/j.compstruct.2017.07.002

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A comment on "Nonlinear thermal buckling of axially functionally graded micro and nanobeams" [Composite Structures 168 (2017) 428–439] Amir Mehdi Dehrouyeh-Semnani1 School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran

Abstract In a recent paper by Shafiei, Mirjavadi, Behzad, Afshari, Rabby, and Hamouda [1], for the first time the nonlinear mathematical formulations of axially functionally graded (AFG) micro/nano-beams under non-uniform temperature distribution in the thickness direction based on the classical theory as well as the non-classical theories (modified couple stress and nonlocal theories) were developed to study the thermal buckling of system. In this comment, it is indicated that the governing equations and associated boundary conditions of system were incompletely established not only for the non-classical models but also for the classical model. The main reason for the inaccuracy of mathematical formulations proposed by Shafiei et al. [1] is to neglect the influence of thermal moment at both the governing equations and boundary conditions. In addition, it is discussed that the size-dependent and –independent buckling temperature rise predicted by Shafiei et al. [1] is meaningless for both the simply supported and fully clamped case studies.

Keywords Axially functionally graded beam; Thermal loading; Mathematical formulation.

1. Governing equations and related boundary conditions In this section, the mathematical formulation of an axially functionally graded beam under immovable boundary conditions and in-plane non-uniform thermal loading in the thickness direction is derived based on the classical theory. It should be pointed out that the size-dependent governing equations and associated boundary conditions of system based on modified couple stress and non-local theories reduce to those obtained by using the classical theory when the material length scale parameter is set to zero. The nonlinear form of axial strain (
 ) of Euler-Bernoulli beam can be expressed as [2]: 2

ε xx =

∂u 1  ∂w  ∂ 2w +  − z  ∂x 2  ∂x  ∂x 2

(1)

1

Corresponding author: Email addresses: [email protected], [email protected] (A.M. Dehrouyeh-Semnani)

1

The axial stress ( ) in present of thermal loading can be written as [3, 4]:

σ xx = E (x ,T ) [ε xx − εT ]

(2)

where

εT = α (x ,T )∆T ( z )

(3)

in which ∆() can be obtained by (see Eq. 12 in Ref. [1]) αT

1 z  (4) ∆T ( z ) = T (z ) −T 0 = (T u −T L )  +  2 h where denotes the non-negative power index of temperature variation function. In addition,  and  stand for the temperature of the lower surface ( = 0) and the upper surface ( = ℎ), respectively. It should be pointed out that the formulation of (, ) and (, ) were given in Eqs. 2b-c of Ref. [1]. Moreover, the formulation of constituent materials ( ,  ,  , and  ) as a function of temperature were given in Eq. 3 of Ref. [1] with the corresponding coefficient in Table 1 of Ref. [1]. In respect of Eq. (4), (, ) and (, ) can be written as (, ()) and (, ()). The potential energy of system can obtained by [4]:

1 (σ xx ε xx − σ xx εT ) dV 2 ∫V Substitution of Eqs. (1) and (2) into Eq. (5) results in:

U =

(5)

2

1 L U = ∫  Ax 2 0 

2  ∂u 1  ∂w  2  ∂ 2w  ∂u 1  ∂w   +    +    -2B x   ∂x 2  ∂x 2  ∂x    ∂x 2  ∂x  

(6)

2   ∂u 1  ∂w  2  ∂ w  T T ∂ w 2 M N + C x  2  - 2N  +  + +  0  dx   ∂x 2  ∂x   ∂x 2  ∂x    where 2

{A ( x ) x

{N ( x ) T

2

B x ( x ) C x ( x )} = b ∫

h /2

− h /2

}

MT ( x )

N 0T ( x ) = b ∫

h /2

− h /2

{

E (x ,T (z )) 1 z

}

z 2 dz αT

1 z  = b (Tu −T L ) ∫ E (x ,T (z ))α (x ,T (z ))  +  − h /2 2 h  h /2

E (T (z ), x ) (α (T (z ), x )∆T

)

2

{1

z }dz

(7)

dz

The axial force  and the moment  as the stress resultants, respectively, can be obtained by:

2

 ∂u 1  dw 2  ∂ 2w N = ∫ σ xx dA =Ax  +  −N T   − Bx 2 A ∂x  ∂x 2  dx    ∂u 1  dw 2  ∂2w M = ∫ σ xx z dA =B x  +  −M T   −C x 2 A ∂x  ∂x 2  dx  

(8)

(9)

1.1. Mathematical formulation in terms of  and  The governing equations and associated boundary conditions of system can be obtained from Hamilton’s principle as follows: t

∫ δU 0

=0

(10)

Substituting Eq. (6) into Eq. (10) and integrating-by-parts with respect to x to relieve the virtual variations  and  of any differentiation, and then applying the fundamental lemma of calculus, one can obtain the governing equations and corresponding boundary conditions (at x=0 and L) as follows:   ∂u 1  ∂w 2  ∂ 2w  T − −  Ax  +  N B =0  x 2   ∂x 2  ∂x   ∂ x      2  ∂ 2   ∂u 1  ∂w   ∂ 2w T δw : - 2  B x  +  − C − M    x  ∂x   ∂x 2  ∂x   ∂x 2  2  ∂w  ∂   ∂u 1  ∂w   ∂ 2w T + − − =0  A B N     x x    ∂x  ∂x   ∂x 2  ∂x   ∂x 2   

δ u: -

∂ ∂x

 ∂u 1  dw 2  ∂ 2w Ax  +  − N T = 0 or u =u s  − Bx  2 ∂x  ∂x 2  dx   2  ∂   ∂u 1  dw   ∂ 2w T  Bx   +  − C − M  x  ∂x   ∂x 2  dx   ∂x 2  2 2   ∂u 1  dw    ∂w ∂w T +  Ax  +  − N  = 0 or w =w s  − Bx    ∂x 2  dx    ∂x ∂x 2    

(11)

(12)

(13)

(14)

 ∂u 1  dw  2  ∂ 2w ∂w ∂w s B xx  +  − C − M T = 0 or = (15)  xx  2 ∂x ∂x ∂x  ∂x 2  dx   It is worth noting that Eqs.(11)-(15) can be obtained by using the stress resultants  and  i.e., Eqs. (8) and (9) and the following relation [2]. t

t

L

0

0

0

∫ δU = ∫ ∫

  ∂δ u ∂w ∂δw  ∂δθ  ∂w  N  ∂x + ∂x ∂x  +M ∂x  dxdt = 0, θ = − ∂x    

3

(16)

Finally, it should be pointed out that the governing equations and boundary conditions can be easily rewritten in terms of  without any simplicity.

1.2. Mathematical formulation in terms of  by neglecting coupling terms Shafiei et al. [1] derived the governing equations and related boundary conditions of system in terms of  by neglecting the coupling terms between  and . For comparative study, the mathematical formulation of system in terms of  is derived by neglecting the coupling terms. The aforementioned mathematical formulation can be acquired by removing the axial displacement from Eqs. (12), (14), and (15).  1  ∂w  2  ∂w   A x      2  ∂x   ∂x  2 ∂ 2w  ∂w  ∂ 2M T ∂N T ∂w ∂ 2w ∂ 2  1  ∂w   ∂  − − B B = + +NT    x   x   ∂x 2  ∂x  ∂x 2 ∂x ∂x ∂x 2 ∂x 2  2  ∂x   ∂x  2  ∂  ∂ 2w  ∂  1  dw  − −M T  C B   x x   2   ∂x  ∂x  ∂x  2  dx   2 2  1  dw   ∂w ∂w = 0 or w =w s −  Ax  − Bx −N T   2  2  dx   ∂x ∂ x  

∂ 4w ∂C x ∂ 3w ∂ 2C x ∂ 2w ∂ Cx + + ∂x 4 ∂x ∂x 3 ∂x 2 ∂x 2 ∂x

2

Cx

∂ 2w 1  dw  − Bx  +MT =0  2 ∂x 2  dx 

or

∂w ∂w s = ∂x ∂x

(17)

(18)

(19)

2. On incompleteness of mathematical formulation proposed by Ref. [1] In order to be able to compare the current mathematical model with that proposed by Shafiei et al. [1], Eqs. (17)-(19) are used. It should be pointed out that the classical model based on Ref. [1] can be obtained by letting the material length scale parameter (ℓ) to zero in the modified couple stress theory-based mathematical formulation of system (see Eqs. 38 and 39 in Ref. [1]) or setting the nonlocal parameter ( ) to zero in the nonlocal theory-based mathematical formulation of system (see Eqs. 25 and 27 in Ref. [1]). The classical theory-based mathematical formulation based on Ref. [1] is as follows: ∂ 4w ∂C x ∂ 3w ∂ 2C x ∂ 2w ∂ Cx + + ∂x 4 ∂x ∂x 3 ∂x 2 ∂x 2 ∂x T

 1  ∂w 2  ∂w   A x      2  ∂x   ∂x 

(20)

2

∂N ∂w ∂w +NT =0 ∂x ∂x ∂x 2 ∂  ∂ 2w  C  xx  =0 or w =w s ∂x  ∂x 2 

+

(21)

4

∂ 2w

∂w ∂w s (22) = ∂x ∂x ∂x It should be pointed out that in Eqs. 25 and 38 in Ref. [1],  =  /(∆"# ) in which ∆"# =  −  (see Eqs. 11, 12, and 26 in Ref. [1] for definition of  and  ). C xx

2

= 0 or

By comparing the classical-based mathematical formulation proposed by Shafiei et al. [1] i.e., Eqs. (20)-(22) with that obtained in this study by neglecting the coupling terms i.e., Eqs. (17)-(19), it can be easily understood that Shafiei et al. [1] was incompletely derived the mathematical formulation of system. It can be seen that Shafiei et al. [1] neglected the thermal moment  and the axial-bending term % at both the governing equations and boundary conditions. In addition, some linear and nonlinear terms at the boundary conditions were ignored by Shafiei et al. [1]. Shafiei et al. [1] considered that the temperature-dependent (TD) system is under a nonuniform temperature rise in the thickness direction (see Eq. (4) or Eq. 12 in Ref. [1]). As a result, % and  are non-zero (see Eq. (7)). But, if the system to be under a uniform temperature rise or a non-uniform temperature rise in the length direction, % and  equals zero. In addition, if the system to be temperature-independent (TID) % is always equal to zero. The results of this discussion can be summarized as follows: T B ≠ 0, M ≠ 0 for T(z) or T(x,z) TD :  x T B x = 0, M = 0 for T = cte or T(x) (23) B x = 0, M T ≠ 0 for T(z) or T(x,z) TID:  T B x = 0, M = 0 for T = cte or T(x) Consequently, the mathematical formulation proposed by Shafiei et al. [1] is almost valid if the system under uniform temperature distribution or non-uniform temperature distribution in the length direction (). But, for a non-uniform temperature distribution in the thickness direction () the mathematical formulation is incomplete.

3. On inaccuracy of trend and results obtained by Ref. [1] Before starting discussion on the obtained results in Ref. [1], it is interesting to review the nonlinear stability response of a corresponding beam made of functionally graded in the thickness direction. In this case, the thermal moment doesn’t appear at the governing equations, but it exists at the boundary conditions related to the moment similar to the current model (see Eq. (19)). For fully clamped case study, the thermal moment doesn't appear at the boundary conditions of system; therefore, the system is stable and at a critical temperature rise buckles. But for a simply support case study, because of present of thermal moment at the boundary conditions, the system deflects at each arbitrary value of temperature rise. In other words, in this case the system doesn’t buckle. It is due to the fact that in this case the supports are incapable of handling the extra moment (thermal moment). The expressed facts can be found in Refs. [4-9]. It should be noticed that a simply supported homogeneous isotropic beam subjected to a nonuniform temperature distribution in thickness direction doesn’t buckle and there exists an

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equilibrium path for each prescribed temperature rise [4, 5]. Finally, it is interesting to note that Nickname et al. [10] investigated the nonlinear response of functionally graded (in thickness direction) tapered beam under thermal load and different boundary conditions. In this case the second derivation of thermal moment presents at the governing equations. It was indicated that the system independent of boundary conditions deflects for each arbitrary value of thermal load. As discussed in the previous section, the main difference of current model i.e., Eqs. (17)-(19) with that obtained by Shafiei et al. [1] i.e., Eqs. (20)-(22) is the present of thermal moment at the governing equations and boundary conditions. Eq. (17) shows that the second derivation of thermal moment ( &   ⁄&  ), which is a nonzero function of  (see Eq. (7) and Eqs. 2b-c in Ref. [1]), exists at the governing equations. It can be stated that this term acts as a distributed transversally loading over the beam. Hence, the system independent of boundary conditions deflects for each arbitrary value of temperature rise. So, for both the simply supported and fully clamped AFG beam under a non-uniform temperature distribution in the thickness direction, the system has a new equilibrium path for each temperature rise and doesn’t buckle. It can be expressed that for this system independent of boundary conditions the occurrence of buckling is meaningless like a simply supported beam functionally graded in the thickness direction and under a non-uniform temperature distribution in the thickness direction discussed in Refs. [4-9] or a FG tapered beam under thermal load argued in Ref. [10]. In addition, it can be easily understood that an identical scenario occurs for the microbeam based on modified couple stress theory and nanobeam based on nonlocal theory due to present of thermal moment at the sizedependent governing equations. Finally, it can be concluded that the size-dependent and – independent buckling temperature rise predicted by Shafiei et al. [1] is meaningless for both the simply supported and doubly clamped case studies under the non-uniform temperature distribution in the thickness direction.

References [1] [2] [3]

[4]

[5]

N. Shafiei, S. S. Mirjavadi, B. M. Afshari, S. Rabby, and A. M. S. Hamouda, "Nonlinear thermal buckling of axially functionally graded micro and nanobeams," Composite Structures, vol. 168, pp. 428-439, 2017. J. N. Reddy, "Microstructure-dependent couple stress theories of functionally graded beams," Journal of the Mechanics and Physics of Solids, vol. 59, pp. 2382-2399, 2011. M. R. Eslami, R. B. Hetnarski, J. Ignaczak, N. Noda, N. Sumi, and Y. Tanigawa. (2013). Theory of Elasticity and Thermal Stresses, Solid Mechanics and Its Applications, 197, DOI: 10.1007/978-94-007-6356-2_14, © Springer Science+Business Media Dordrecht 2013. D.-G. Zhang, "Thermal post-buckling and nonlinear vibration analysis of FGM beams based on physical neutral surface and high order shear deformation theory," Meccanica, vol. 49, pp. 283-293, 2014. F.-q. Zhao, Z.-m. Wang, and H.-z. Liu, "Thermal post-bunkling analyses of functionally graded material rod," Applied Mathematics and Mechanics, vol. 28, pp. 59-67, 2007.

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[6] [7]

[8]

[9] [10]

L. S. Ma and D. W. Lee, "A further discussion of nonlinear mechanical behavior for FGM beams under in-plane thermal loading," Composite Structures, vol. 93, pp. 831-842, 2011. M. Komijani, S. E. Esfahani, J. N. Reddy, Y. P. Liu, and M. R. Eslami, "Nonlinear thermal stability and vibration of pre/post-buckled temperature- and microstructuredependent functionally graded beams resting on elastic foundation," Composite Structures, vol. 112, pp. 292-307, 2014. A. Paul and D. Das, "Non-linear thermal post-buckling analysis of FGM Timoshenko beam under non-uniform temperature rise across thickness," Engineering Science and Technology, an International Journal, vol. 19, pp. 1608-1625, 2016. S. V. Levyakov, "Thermal elastica of shear-deformable beam fabricated of functionally graded material," Acta Mechanica, vol. 226, pp. 723-733, 2015. H. Niknam, A. Fallah, and M. M. Aghdam, "Nonlinear bending of functionally graded tapered beams subjected to thermal and mechanical loading," International Journal of Non-Linear Mechanics, vol. 65, pp. 141-147, 2014.

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