Thermo-mechanical post-critical analysis of nonlocal orthotropic plates

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Thermo-mechanical post-critical analysis of nonlocal orthotropic plates Manjur Alam, Sudib K. Mishra∗ Department of Civil Engineering, Indian Institute of Technology Kanpur, 208016, UP, India

a r t i c l e

i n f o

Article history: Received 3 July 2019 Revised 21 September 2019 Accepted 8 October 2019 Available online xxx Keywords: Nonlocal Perturbation Buckling Post-critical Thermal loading

a b s t r a c t A closed-form analytical solution for critical temperature and nonlinear post-critical temperature-deflection behaviour for nonlocal orthotropic plates subjected to thermal loading is presented. The long-range molecular interactions are represented by a nonlocal continuum framework, including orthotropy. The Von-Karman nonlinear strains are employed in deriving the governing equations. An approximate solution to the system of nonlinear partial differential equations is obtained using a perturbation type method. Series expansions up to second order of the associated field variables and the load parameter, dictating nonlinearity are employed. The behaviour in the post-critical regime is illustrated numerically by adopting an example of orthotropic Single Layer Graphene Sheet (SLGS), a widely acclaimed nano-structure, often modelled as plate. Post-critical temperature-deflection paths are presented with special emphasis on their post-critical reserve in strength and stiffness. Influence of aspect ratio and behaviour in higher modes are demonstrated. Implications of nonlocal interactions on the redistribution of in-plane forces are presented to show striking disparity with the classical plates. The obtained solution may serve as benchmark for verification of numerical solutions and may be useful in formulating simple design guidelines for plate type nanostructures liable to the thermal environment. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Significant progress in miniaturisation has taken place over the last few decades in mechanical, optical and electronic systems/devices which has contributed to the development of nanotechnology. Efficient analysis and design of such systems are important to ensure reliable performance in critical applications. Design of such systems requires accurate analysis of stress, strains and their functions. Due to small characteristics dimension(s), mechanics of these systems is dominated by a variety of forces of molecular origin, such as inter-atomic and intra-atomic interactions. Molecular Dynamics (MD) simulation is often employed for the simulation of such systems. However, MD simulation is computationally intensive for routine analysis/design purpose. As an alternative,the principles of continuum mechanics were extended to describe the mechanics at molecular level. Molecular interaction forces on an atom do not necessarily restrict to its neighbouring atoms only but also on atoms away from it. Such interactions are referred as long-range interactions [1,2]. In presence of long-range interactions, the stresses at a point depend not only on the strains at that point (as in classical solid) but on strains away from that point. In order to accommodate such long-range molecular interactions in the continuum framework, the notion ∗

Corresponding author. E-mail address: [email protected] (S.K. Mishra).

https://doi.org/10.1016/j.apm.2019.10.018 0307-904X/© 2019 Elsevier Inc. All rights reserved.

Please cite this article as: M. Alam and S.K. Mishra, Thermo-mechanical post-critical analysis of nonlocal orthotropic plates, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.018

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of nonlocal field theory was introduced by Eringen et al. [3–5]. The stress-strain constitutive equations are expressed either in an integral (over a characteristics length scale) form [1–3] or in terms of higher-order gradients of stresses (differential form) [6–8]. The link between the integral and differential form was established [7] to show that the differential formulation corresponds to specific form of the kernel in the integral formulation. The nonlocal parameters in the constitutive model were estimated by matching the wave dispersion relationship obtained from atomic level to the wave dispersion relationship from the equivalent continuum level [9]. Using the nonlocal field theory, Eringen et al. [10] studied Griffith crack to show that the stress singularity at the crack tip can be eliminated. A nonlocal solution for the screw dislocation problem was also obtained [6]. These studies helped in bridging the gap between the continuum and molecular level behaviour using the notion of nonlocal continua. A number of nanomaterials (e.g. Graphene,Carbon Nanotubes, Boron nitrides, Zinc Sulphide) were developed in last few decades to offer remarkable strength [11,12], partly attributed to their structures and sizes [13,14]. Although higher strength allows slender nano-structures, it also enhances their vulnerability to geometric instability, such as buckling. Therefore, geometric instability of nano-structures forms an important area of research. Wang et al. [15] studied the elastic buckling of micro/nano-rods to demonstrate that nonlocal interaction reduces the critical load, analogous to the effect of shear deformation. Reddy [16] formulated the theories of bending, buckling and vibration in nonlocal beams. Pradhan and Murmu [17] studied the nonlocal effect on the buckling of a bi-axially compressed single layer Graphene sheet. The differential quadrature method was employed to numerically solve the governing equations. Pradhan [18] applied higher-order shear deformation theory (HSDT) to demonstrate the effect of shear deformations on the buckling characteristics of nano-plates using Navier’s solution approach. Hashemi and Samaei [19] showed the dependence of critical load on nonlocal parameter for bi-axially loaded Mindlin plates. Aksencer and Aydogdu [20] presented a levy type solution for vibration and buckling of nano-plates using nonlocal elasticity. An explicit expression for critical stress in single layer Graphene was provided by Ansari and Rouhi [21]. Farajpour et al. [22] studied buckling of orthotropic nonlocal plate subjected to varying in-plane loading. Nonlocal elasticity was also employed by Hosseini and Tourki [23] to investigate buckling of plates at small length scales. Abadi and Daneshmehr [24] employed modified coupled stress theory to investigate size-dependent buckling of micro-beams. Two parameter refined nonlocal plate theory was employed by Narendar [25] to study buckling of nonlocal plate. Although major studies are focused on single layer Graphene, buckling of multi-layer Graphene was also presented by Kim and Im [26]. An expression for critical load was exploited by Zhang et al. [27] to estimate the nonlocal parameters as an alternative to conventional dispersion matching procedure. Axisymmetric buckling of carbon nano-tube was studied by Farajpour et al. [28] using nonlocal theory. The effect of anisotropy on the buckling behaviour of the nonlocal plate was investigated by Narendar and Gopalakrishnan [29]. Samaei et al. [30] studied buckling of Graphene embedded in an elastic substrate using nonlocal theory in view of its analogy to nano-composite laminate. A Galerkin based solution for nonlocal buckling was presented by Babaei and Shahidi [31]. Fluid–structure interaction analysis in fluid conveying nanotube was investigated by Ghazavi and Molki [32] using second-order strain gradient theory. Exact mode shapes for magnetic field induced buckling of nano-beams was presented by Dai et al. [33]. Yang et al. [34] researched the elastodynamics of fluidfilled nanotube using nonlocal strain gradient theory [34]. An analytical solution for thermal vibration of nano-beams was proposed by Jiang and Wang [35]. Iso-geometric analysis of vibration for functionally graded nano-plates was explored by Norouzzadeh and Ansari [36]. Further, the wave propagation analysis in a rotating thermo-elastically actuated nano-beams was conducted by Ebrahimi and Haghi [37]. The in-plane load carrying capacity for classical continuum plates do not necessarily get depleted on reaching their critical value but they can sustain further in-plane forces of significantly higher magnitude beyond critical loads [38–41]. Such additional post-critical reserve is exploited in design [42]. In fact, the post-critical behaviour of continuum plate is well established in the literature [38–42]. However, the post-critical behaviour of nonlocal plates beyond critical load was not investigated extensively, although the post-critical reserve may as well be exploited for designing plate type nano structures. Among the limited number of studies on the post-critical behaviour of nonlocal plates, Mahdavi et al. [43] studied postbuckling of single layer Graphene sheet embedded in polymer matrix, which is modelled as thin nonlocal plate. Naderi and Saidi [44] investigated the nonlinear behaviour of Graphene in polymer matrix subjected to in-plane loading using nonlocal theory. Vaz et al. [45] studied thermal buckling of a beam, mimicking carbon nano-tubes. Studies on post-critical behaviour of nonlocal plates, otherwise, are scanty. It appears that more research works are necessary to optimally exploit the potential post-buckling reserve in nonlocal plates for efficient design of nano structures. With this being the eventual goal, in this study, the post-critical behaviour of nonlocal thin plates under thermal loading is analyzed and several aspects of thermal buckling are touched on in the presence of nonlocal interactions. Thermallyinduced in-plane biaxial loading in the restrained nonlocal plate is considered. The governing equations are formulated considering von Karman nonlinear strains that allow for in-plane redistribution of stresses and consequent post-critical reserve. The long-range molecular interactions are incorporated through nonlocal stress-strain relationship. In contrast to simple isotropic behaviour, adopted by Mahdavi et al. [43]; the orthotropic behaviour of Graphene [44] is considered to take into account the directional dependency. The resulting nonlinear boundary value problem and linearized eigenvalue problem are formulated and solved using a perturbation type method involving convergent expansion of the relevant field variables and the load parameter, dictating the degree of nonlinearity. An approximate closed-form solution for the relevant field variables is obtained. The behaviour at the onset of buckling and in the post-critical regime are elucidated numerically on Single-Layer Graphene Sheets (SLGS) using the obtained closed-form solution. Alternative configurations of SLGSs (Zigzag and Armchair) are considered for illustration. In particular, the critical temperature and the post-critical temperature-deflection paths are Please cite this article as: M. Alam and S.K. Mishra, Thermo-mechanical post-critical analysis of nonlocal orthotropic plates, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.018

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emphasized. Implications of important parametric variations are highlighted. A comparative assessment with the classical local continuum plates is also presented. 2. Formulation of the governing equations Eringen’s nonlocal elastic field theory [3–6] considers the stress at any point x to be a function of the strain field at other points x in the body. This accords the atomic theory of lattice dynamics and experimental observations on phonon dispersion [6–9]. Systems with tiny characteristic dimensions (comparable to atomic dimension) may be modelled with equivalent nonlocal continuum. Nano-materials employed as reinforcement in nano-composites may also be modelled so. Respective nonlocal parameters are estimated by matching the dispersion characteristics (wave-number (κ ) vs. frequency (ω) relationship) obtained from the atomistic simulation (molecular/lattice/quantum-mechanical dynamics) with the one obtained from the equivalent nonlocal continuum. This methodology was proposed by Eringen himself [4] and also employed with several modifications by other investigators [9]. The dispersion (κ vs. ω ) relationship is obtained by carrying out elastodynamic wave propagation simulation on a domain (generally uni-axial) with specific input waveform to obtain the respective output. The output waveform is Fourier transformed (in both space (for the domain) and time) in order to identify the dominant frequency-wave-number (κ − ω) pairs for identifying waves permitted through the domain. The frequency-wavenumber is then fitted with a relation or may be retained as it is. This dispersion relation is then matched with the theoretical dispersion for nonlocal continuum, obtained in close form. The nonlocal parameters are employed as the design variables for fitting by minimising the error between the two dispersion relations. Although the stresses depend on strains at all points in nonlocal continua, in limiting case, as the effect of strains at points other than x is neglected, the classical “local” theory is reclaimed. The relationship between the classical “local” and nonlocal stress tensor is expressed as [4,10]

σi j ( x ) =



v

       α x − x , τ σiclj x dv x

(1)

In which σ ij is the nonlocal stress tensor. The symbol σiclj is the classical stress tensor at a point x . The kernel function

α (|x − x |, τ ) (also known as “nonlocal modulus”) depends on the distance |x − x | . The symbol τ =

e0 a0 l

is a constant with (a0 /l) as the characteristic length ratio. In this expression, a0 is an internal characteristic length (e.g. lattice parameter, intergranular distance) and l is an external characteristic length, such as wave length, crack length or the width of localisation band. The symbol e0 refers to a material constant. The underlying constitutive relationship can be expressed in differential form as [6]



1 − e20 a20 ∇ 2



σi j = Ci jkl εkl

(2)

where ∇ is the Laplacian, σ ij is the nonlocal stress tensor, ε ij is the strain tensor and Cijkl are the elastic constants. Eq. (2) differentiates the nonlocal theory from the theory of classical continuum. A rectangular thin nonlocal plate is considered for analysis with its length (a), width (b) and thickness (h) aligned along x, y and z axes, respectively. The origin is assigned to be at a corner of the mid-plane of the plate. The plate is supported by out-of-plane and in-plane boundary restraints. The in-plane stresses due to temperature rise (T) are induced due to in-plane restraints. The strain-displacement relationship with the Von Karman nonlinear terms are written as

1 2

(3a)

εyy = v,y + (w,y )2

1 2

(3b)

γxy = u,y + v,x + w,x w,y

(3c)

εxx = u,x + (w,x )2

where εxx , εyy are the normal strain along x and y, respectively and γ xy is the in-plane shear strain. The displacement components along x, y and z directions are u, v and w, respectively. Considering a change in temperature T, the relationship between the strain and the in-plane forces per unit width (Nx , Ny , Nxy ) are expressed as

εxx =

A12 A22 (1 − e20 a20 ∇ 2 )Nxx − (1 − e20 a20 ∇ 2 )Nyy + αx T A11 A22 − A212 A11 A22 − A212

(4a)

εyy =

A12 A11 (1 − e20 a20 ∇ 2 )Nyy − (1 − e20 a20 ∇ 2 )Nxx + αy T A11 A22 − A212 A11 A22 − A212

(4b)

γxy =

1 (1 − e20 a20 ∇ 2 )Nxy A66

(4c)

In these expressions, the coefficients are given as

A11 =

E1 h , 1 − ν12 ν21

A22 =

E2 h , 1 − ν12 ν21

A12 =

ν12 E2 h , 1 − ν12 ν21

A66 = G12

(5)

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where E1 , E2 , G12 are the elastic constants and ν 12 , ν 21 are the Poisson ratios for the orthotropic plate material. The stress resultants (Nxx , Nyy , Nxy ) per unit width of the plate are obtained by integrating the respective stresses over the plate thickness (h) as

(1 − e20 a20 ∇ 2 )Nxx = A11 εxx + A12 εyy − A11 αx T − A12 αy T

(6a)

(1 − e20 a20 ∇ 2 )Nyy = A12 εxx + A22 εyy − A22 αy T − A12 αx T

(6b)

(1 − e20 a20 ∇ 2 )Nxy = A66 γxy

(6c)

The equilibrium equations along the in-plane and out-of-plane directions can be expressed as

 



1 − e20 a20 ∇ 2 (Nxx,x + Nxy,y ) = 0

(7a)



1 − e20 a20 ∇ 2 (Nxy,x + Nyy,y ) = 0

(7b)





D11 w,xxxx + 2(D12 + 2D66 )w,xyxy + D22 w,yyyy − 1 − e20 a20 ∇ 2 (Nxx w,xx + 2Nxy w,xy + Nyy w,yy ) = 0

(7c)

Eqs. (7a) and (7b) express the equilibrium along in-plane and Eq. (7c) expresses the equilibrium along the out-of-plane direction. Subscripts (,) denote differentiation with respect to the respective coordinates (x, y). 3. Perturbation approximation through convergent series expansions A perturbation type method is employed for successive simplification of the governing differential Eqs. (7a)–(7c). The displacement components (u, v, w) are expanded using a series expansion of an arbitrary parameter (ξ ). These are in order to convert the nonlinear equations into a series of linear partial differential equations. However, it may be noted that the methodology is not restricted by the criteria of “small” perturbation parameter, as the higher-order displacement components associated with the higher powers of (ξ ) becoming increasingly smaller. The displacement components (u, v, w) are expanded around their critical values, i.e. corresponding to the arbitrary parameter ξ = 0 at buckling. The arbitrary parameter is chosen to be in terms of the change in temperature as

ξ 2 = (T − Tcr )/Tcr

(8)

where Tcr is the critical temperature for buckling and T refers to the temperature of the plate. In the present discussion T ≥ Tcr . The displacement fields are expanded as

u = u0 + ξ 2 u2 + ξ 4 u4 ...

(9a)

v = v0 + ξ 2 v2 + ξ 4 v4 ...

(9b)

w = ξ w1 + ξ 3 w3 + ξ 5 w5 ...

(9c)

The asymptotic expansions for u and v start with zeroth power of the perturbation parameter (ξ ) and continue with even powers only. Contrastingly, the expansion for w starts with first power and continue with odd powers only. At the onset of buckling, the deflection profile of the plate may take either convex or concave (out-of-plane) configuration and the postbuckling deflection shape (w) also follows the same. Therefore, only the odd powers of (ξ ) are included in the expansion of w to retain the sign of w. However, the in-plane displacements (u and v) do not change sign with changes (convex/concave) in the buckling induced out-of-plane deflection. Hence, only even powers of (ξ ) are included in the expansions of (u, v). If all the powers of (ξ ) were included in the expansions, they would have eventually get cancelled by forming a homogeneous system of equations. The details on these expansions may be obtained from the literature [41]. The solution methodology also requires the expansion of the loading parameter (T) as

T = T0 + ξ 2 T2 + ξ 4 T4 ...

(10)

Since the direction of buckled configurations (convex/concave) are independent of the change in temperature, only the even terms are retained in the expansion (T). Substituting the asymptotic series of the displacements and temperature field ((9a–c) and (10)) in the in-plane forces vs. strain expressions (6a–c); the expansions for the in-plane forces can be obtained as

(1 − e20 a20 ∇ 2 )Nxx =

∞  n=0,2

(1 − e20 a20 ∇ 2 )Nxn ξ n +

∞ ∞  

(1 − e20 a20 ∇ 2 )Nxmn ξ m+n

(11a)

n=1,3 m=1,3

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(1 − e20 a20 ∇ 2 )Nyy =

∞  n=0,2

(1 − e20 a20 ∇ 2 )Nxy =

∞ 

∞ ∞  

(1 − e20 a20 ∇ 2 )Nyn ξ n +

5

(1 − e20 a20 ∇ 2 )Nymn ξ m+n

(11b)

n=1,3 m=1,3 ∞ ∞  

(1 − e20 a20 ∇ 2 )Nxyn ξ n +

n=0,2

(1 − e20 a20 ∇ 2 )Nxymn ξ m+n

(11c)

n=1,3 m=1,3

where

(1 − e20 a20 ∇ 2 )Nxn = A11 un,x + A12 vn,y − A11 αx Tn − A12 αy Tn

(11d)

(1 − e20 a20 ∇ 2 )Nyn = A12 un,x + A22 vn,y − A12 αx Tn − A22 αy Tn

(11e)

(1 − e20 a20 ∇ 2 )Nxyn = A66 (un,y + vn,x )

(11f)

(1 − e20 a20 ∇ 2 )Nxmn = (1/2)(A11 wm,x wn,x + A12 wm,y wn,y )

(11g)

(1 − e20 a20 ∇ 2 )Nymn = (1/2 )(A12 wm,x wn,x + A22 wm,y wn,y )

(11h)

(1 − e20 a20 ∇ 2 )Nxymn = A66 wm,x wn,y

(11i)

The subscripts (n = 1, ..., 4) refer to the varying power of the approximate expansions. The equilibrium equations (7a–c) are substituted with the expansions of the in-plane forces from Eqs. (11a–c). The linearised equations for successive approximation are obtained by equating the coefficients of different power of ξ to zero. This provides a set of linear partial differential equations for obtaining the displacements of varying order (u0 , u2 , u4 , v0 , v2 , v4 , w1 , w3 ) as

 

1 − e20 a20 ∇ 2 1 − e20 a20 ∇ 2

 



Nx0,x + Nxy0,y = 0



Nxy0,x + Ny0,y = 0

(12a)

D11 w1,xxxx + 2(D12 + 2D66 )w1,xyxy + D22 w1,yyyy



− 1 − e20 a20 ∇ 2

 

1 − e20 a20 ∇ 2

1 − e20 a20 ∇ 2

 





Nx0 w1,xx + 2Nxy0 w1,xy + Ny0 w1,yy = 0













Nx2,x + Nxy2,y = − 1 − e20 a20 ∇ 2

Nxy2,x + Ny2,y = − 1 − e20 a20 ∇ 2

Nx11,x + Nxy11,y Nxy11,x + Ny11,y



D11 w3,xxxx + 2(D12 + 2D66 )w3,xyxy + D22 w3,yyyy − 1 − e20 a20 ∇ 2



= 1 − e20 a20 ∇ 2

 

1 − e20 a20 ∇ 2 1−

e20 a20

∇

2

 











Nx4,x + Nxy4,y = − 1 − e20 a20 ∇ 2 Nxy4,x + Ny4,y = − 1 −

e20 a20

∇

2





(12c)

Nx0 w3,xx + 2Nxy0 w3,xy + Ny0 w3,yy

2Nx13,x + Nxy13,y + Nxy31,y

 



2Ny13,y + Nxy13,x + Nxy31,x



(12d)



(12e)



Nx0 w5,xx + 2Nxy0 w5,xy + Ny0 w5,yy

    (Nx2 + Nx11 )w3,xx + Ny2 + Ny11 w3,yy + 2 Nxy2 + Nxy11 w3,xy      + (Nx4 + Nx13 )w1,xx + Ny4 + Ny13 w1,yy + 2 Nxy4 + Nxy13 + Nxy31 w1,xy

= 1 − e20 a20 ∇ 2





     (Nx2 + Nx11 )w1,xx + Ny2 + Ny11 w1,yy + 2 Nxy2 + Nxy11 w1,xy

D11 w5,xxxx + 2(D12 + 2D66 )w5,xyxy + D22 w5,yyyy − 1 − e20 a20 ∇ 2





(12b)

(12e)



(12f)

These approximations retain the terms of until third-order (power) for the out of plane deflection (w), which leads to second-order accuracy in the post-critical deflection. Eqs. (12a) and (12b) govern the small deflection of plates and can be readily solved. The succeeding equations are solved by progressively substituting the solutions obtained from the previous equations. Please cite this article as: M. Alam and S.K. Mishra, Thermo-mechanical post-critical analysis of nonlocal orthotropic plates, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.018

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A rectangular nonlocal plate, simply supported (in out-of-plane directions) along four edges is considered to be subjected to uniform temperature rise. Buckling is triggered by compressive in-plane forces induced by the in-plane restraints along the edges. The boundary conditions are written as

zero out-of-plane deflection : w(0, y ) = w(a, y ) = w(x, 0 ) = w(x, b) = 0

(13a)

zero bending moment : w,xx (0, y ) = w,xx (a, y ) = w,yy (x, 0 ) = w,yy (x, b) = 0

(13b)

zero in-plane displacements : u(0, y ) = u(a, y ) = v(x, 0 ) = v(x, b) = 0

(13c)

zero in-plane shear stress : v,x (0, y ) = v,x (a, y ) = u,y (x, 0 ) = u,y (x, b) = 0

(13d)

These boundary conditions also hold for individual components un ,vn and wn (i = 0, 1, ...4) in the convergent expansions. The behaviour of the nonlocal plate in the buckling and post-buckling regime are obtained by solving the set of linear partial differential Eqs. (12a–f) subjected to the boundary conditions (13a–d). The first set of Eqs. (12a) furnish the small deflection solution in the pre-buckling regime. The critical temperature for buckling is obtained by solving Eq. (12b). The solutions of the rests (12c–12f) provide the post-critical deflection behaviour. An alternative formulation for finding out the critical load can be obtained from consideration of vibration of the plates using the nonlocal theory. This involves consideration of the equations of motion of the plate in out of plane direction in presence of the in-plane forces induced by temperature rise. Consideration of inertial forces is crucial in this. On solving the free vibration equations of motion, the natural frequencies may be obtained as a function of the temperature. It is generally noted that, with rising temperature intensity (and development of the in-plane compression), the frequencies decrease due to the softening effect induced by the in-plane forces on transverse stiffness. The value of the temperature, at which the frequency becomes zero provides the critical load for buckling. The critical temperatures for higher modes are also obtained by equating the higher frequencies to zero along the spectra [46]. The buckling mode shapes may also be obtained from the corresponding eigenvectors. The present study, however, does not adopt this route and rather follows the static approach to thermal buckling and post-buckling analysis. Nevertheless, the post-critical behaviour gets complicated as the vibration dynamics of the plate starts getting affected significantly by the incipient degree of nonlinearity. This might even lead to chaotic behaviour reported in the literature for classical plates [47]. This, however, is beyond the scope of the present investigation and left for future study. 4. Solutions of the linearised equations The first set of Eqs. (12a) and Eq. (12b) are solved to obtain the pre-buckled configuration of the plate. Eqs. (12a) for the in-plane forces are subjected to the boundary conditions (u0 = v0 = 0) can be simplified as

  



1 − e20 a20 ∇ 2 Nx0 = −(A11 αx + A12 αy )T0

(14a)



1 − e20 a20 ∇ 2 Ny0 = −(A12 αx + A22 αy )T0

(14b)



1 − e20 a20 ∇ 2 Nxy0 = 0

(14c)

A solution satisfying Eq. (12b) and boundary conditions for the out-of-plane deflection may be adopted as

w1 = W1 sin (mx ) sin (ny )

(15a)

m = (mπ /a ), n = (nπ /b)

(15b)

where

Substituting (14a–c) and (15a) and (15b) in Eq. (12b), the first approximation (T0 ) of temperature (T) is obtained as

T0 =



1 1 + η2



4

2 2

D11 m + 2(D12 + 2D66 )m n + D22 n

4

(A11 αx + A12 αy )m2 + (A12 αx + A22 αy )n2

(15c)

in which

  η2 = e20 a20 m2 + n2

(15d)

This (T0 ) is the critical temperature (Tcr ) for buckling of the plate. The parameter (η) bears the influence of nonlocal parameter. Neglecting the nonlocal interactions (e0 a0 = 0), expression (15c) reclaims the critical temperature for the classical continuum plate. It may be noted that depending upon the values of m and n (and their combinations that imply different mode shapes for buckling), T0 takes different values for plates with varying aspect ratios. The buckling mode shapes and aspect ratio have important implications on the critical temperature as demonstrated through numerical illustration. Please cite this article as: M. Alam and S.K. Mishra, Thermo-mechanical post-critical analysis of nonlocal orthotropic plates, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.018

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In successive steps, solutions (15c) and (15d) are employed to obtain the second-order approximation for the in-plane displacements (u2 ,v2 ) in terms of W1 by solving the set of Eqs. (12c) as



u2 = −

 W1 2 1  2 2 A11 m − A12 n sin (2mx ) − m sin (2mx ) cos (2nx ) 16 mA11

v2 = −

 W12 1  2 2 A22 n − A12 m sin (2ny ) − n cos (2my ) sin (2ny ) 16 nA22



(16a)

(16b)

These expressions for the in-plane displacements are substituted in (12c) to obtain the next higher-order approximations for the in-plane forces as





1 − e20 a20 ∇ 2 (Nx2 + Nx11 ) =

 W1 2  2  2 m A12 − A11 A22 cos (2ny ) 8A22 

2

+ A11 m + A12 n



1 − e20 a20 ∇ 2





Ny2 + Ny11 =

2

1 − e20 a20 ∇ 2



− T2 (A11 αx + A12 αy )

(17a)

 W1 2  2  2 n A12 − A11 A22 cos (2mx ) 8A11 

2

2

+ A22 n + A12 m







− T2 (A12 αx + A22 αy )

(17b)



Nxy2 + Nxy11 = 0

(17c)

Substituting Eqs. (17a, 17b) and (14a–c) in Eq. (12d), the governing equation for first-order out-of-plane post-critical deflection may be obtained as





D11 w3,xxxx + 2(D12 + 2D66 )w3,xyxy + D22 w3,yyyy + (A11 αx + A12 αy ) 1 − e0 2 a0 2 ∇ 2 w3,xx





+ (A12 αx + A22 αy ) 1 − e0 2 a0 2 ∇ 2 w3,yy = −



1 + η2





m (A11 αx + A12 αy ) + n (A12 αx + A22 αy ) W1 T2 2

2

 4     W13  2 2 4 1 + η2 m 3A11 − A212 A22 −1 + 4A12 m n + n 3A22 − A212 A11 −1 16

   W13 η4 m2 n2  2  + sin mx sin ny   m A11 − A212 A22 −1 + n2 A22 − A212 A11 −1 4 m2 + n2 4

+

W13 m 16

+

W13 n 16

4



A11 − A212 A22 −1



 

A22 − A212 A11 −1





1 + η 2 − 4η 4 n m + n

 

2



2





2 −1



2 −1

1 + η 2 − 4η 4 m m + n 2

sin mx sin 3ny

2

sin 3mx sin ny

(18)

Eq. (18) will have a meaningful solution, only if, the coefficient of sin mx sin ny on the right-hand side of the equation is made equal to zero to avoid secular terms in the solution. This provides



W12

=



1 + η2



4

m



16T2 1 + η2 3A11 −

A212 A22







m (A11 αx + A12 αy ) + n (A12 αx + A22 αy ) 2

2 2

+ 4A12 m n + n

2

4



3A22 −

A212 A11







4 2 2 2 − 4η 2m n2 m A11 −

(

m +n

)

A212 A22



2

+n



A22 −

A212 A11

 (19)

Eq. (19) furnishes the first-order approximation for the post-critical deflection (W). On enforcing the expression for W12 (Eq. (19), the solution for Eq. (18) is obtained as

w3 = W3 sin (mx ) sin (ny ) + W13 sin (mx ) sin (3ny ) + W31 sin (3mx ) sin (ny ) The terms W13 and W31 in the expression (20a) are evaluated as



W13 =



W12 m A11 − A212 A22 −1





1 + η 2 − 4η 4 n

2



2

(20a)

2

m +n



16 D11 (m ) + 2(D12 + 2D66 )(m ) (3n ) + D22 (3n ) − (A11 αx + A12 αy )(m ) − (A12 αx + A22 αy )(3n ) 4

2

2

4

2

2



(20b)

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W31 =

W12 n A22 − A212 A11 −1







1 + η 2 − 4η 4 m

2



2

2

m +n



16 D11 (3m ) + 2(D12 + 2D66 )(3m ) (n ) + D22 (n ) − (A11 αx + A12 αy )(3m ) − (A12 αx + A22 αy )(n ) 4

2

2

4

2

2



(20c)

The next higher order of approximation for the deflection (W3 ) is determined similar to W1 . Further higher order approximation for the in-plane forces (N s) in Eq. (12e) are determined subsequently by substituting for w3 . The following approximations for the in-plane displacements (u4 ,v4 ) are adopted as

u4 = f1 sin (2mx ) + f2 sin (4mx ) + f3 sin (2mx ) cos (2ny ) + x f4 sin (2mx ) cos (4ny ) + f5 sin (4mx ) cos (2ny )

(21a)

v4 = g1 sin (2ny ) + g2 sin (4ny ) + g3 cos (2mx ) sin (2ny ) + g4 cos (2mx ) sin (4ny ) + g5 cos (4mx ) sin (2ny )

(21b)

In these expressions, the following quantities are defined to be as 2

f1 =

W m W1 A12 n (W3 − W31 ) − 1 (W3 − 3W31 ) 8mA11 8

W1W31 f2 = 16m f3 = −



2

A12 n 2 − 3m A11



(21d)

  2 2 W1 m  4 A11 A66 (W13 − 3W31 −W3 )m + A11 A22 − A212 (W13 − 3W31 − W3 ) − 2A12 A66 (3W13 − W31 − W3 ) m n 8 4

+ A22 A66 (5W13 + W31 − W3 )n f4 =

(21c)



4





2 2

A11 A66 m + A11 A22 − A212 − 2A12 A66 m n + A22 A66 n4

−1



 4  4   −1 W1W13 m 2 2 2 2 4 1 + 4n A22 A66 − m n A12 A66 m A11 A66 + 4m n A11 A22 − A212 − 2A12 A66 + 16n A22 A66 8 



4



2 2

A11 A66 m + A11 A22 − A212 − A12 A66 m n W1W31 m f5 = 1−   2 2 4 4 16A11 A66 m4 + 4A11 A22 − 4A212 − 8A12 A66 m n + A22 A66 n

(21e)

(21f)

 (21g)

2

g1 =

W n W1 A12 m (W3 − W13 ) − 1 (W3 − 3W13 ) 8A22 n 8

g2 =

 W1W13  2 2 A12 A22 −1 m − 3n 16n

g3 =

  2 2 W1 n  4 −A11 A66 (W13 + 5W31 − W3 )m + A11 A22 − A212 (3W13 − W31 + W3 ) − 2A12 A66 (W13 − 3W31 + W3 ) m n 8

(21i)

+ A22 A66 (3W13 − W31 + W3 )n



(21h)

4



4





2 2

A11 A66 m + A11 A22 − A212 − 2A12 A66 m n + A22 A66 n

4

2 2





4n A22 A66 − m n A11 A22 + A212 + A12 A66 W1W13 n g4 = 1− 4   2 2 4 4 m A11 A66 + 4m n A11 A22 − A212 − 2A12 A66 + 16n A22 A66



4

2 2

W1W31 n 4m A11 A66 − m n A12 A66 g5 = 1+   2 2 4 4 8 16m A11 A66 + 4 A11 A22 − A212 − 2A12 A66 m n + n A22 A66



4 −1

(21j)

 (21k)

 (21l)

The obtained expressions for the previous order of in-plane and out-of-plane displacements are subsequently substituted in Eqs. (11a–11c) to obtain the following equations for the in-plane forces of next higher order as





1 − e0 2 a20 ∇ 2 (Nx4 + 2Nx13 ) =

 W1 m   W1W3  2 2 m A11 + n A12 − (W3 − W13 ) A11 − A212 A22 −1 cos (2ny ) 4 4 2

2

−

W1W13 m 4





A11 − A212 A22 −1 cos (4ny ) 2 4

−





W1 (W13 + W31 )m n A11 A22 − A212 cos (2mx ) cos (2ny )





m A11 + m n A11 A22 − A212 − 2A12 A66 A66 −1 + n A22 4

2 2

4

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+





W1W13 m n A11 A22 − A212 cos (2mx ) cos (4ny )

 2



m A11 + 4m n A11 A22 − A212 − 2A12 A66 A66 −1 + 16n A22 4

2

2 4

+

9



4



W1W31 m n A11 A22 − A212 cos (4mx ) cos (2ny )

 2



− (A11 αx + A12 αy )T4

64m A11 + 16m n A11 A22 − A212 − 2A12 A66 A66 −1 + 4n A22 4

2

4

(22a)



1 − e0 2 a20 ∇ 2





 W1 n   W1W3  2 2 m A12 + n A22 − (W3 − W31 ) A22 − A212 A11 −1 cos (2mx ) 4 4 2

Ny4 + 2Ny13 =

2

−

W1W31 n 4





A22 − A212 A11 −1 cos (4mx )



4 2

−





m A11 + m n A11 A22 − A212 − 2A12 A66 A66 −1 + n A22 4

2 2

4 2

+



4



W1W13 m n A11 A22 − A212 cos (2mx ) cos (4ny )





4m A11 + 16m n A11 A22 − A212 − 2A12 A66 A66 −1 + 64n A22 4

2 2

 2

4

+



W1 m n (W13 + W31 ) A11 A22 − A212 cos (2mx ) cos (2ny )

4



W1W31 m n A11 A22 − A212 cos (4mx ) cos (2ny )





16m A11 + 4m n A11 A22 − A212 − 2A12 A66 A66 −1 + n A22 4

2 2

4

− (A12 αx + A22 αy )T4 (22b)



1−

e0 2 a20

∇

2







3 3

Nxy4 + Nxy13 + Nxy31 = −





m A11 + m n A11 A22 − A212 − 2A12 A66 A66 −1 + n A22 4

2 2

3 3

+



4



W1W13 m n A11 A22 − A212 sin (2mx ) sin (4ny )





2m A11 + 8m n A11 A22 − A212 − 2A12 A66 A66 −1 + 32n A22 4

2 2

3 3

+



W1 m n (W13 + W31 ) A11 A22 − A212 sin (2mx ) sin (2ny )



4



W1W31 m n A11 A22 − A212 sin (4mx ) sin (2ny )





(22c)

32m A11 + 8m n A11 A22 − A212 − 2A12 A66 A66 −1 + 2n A22 4

2 2

4

Finally, the expressions in (14a–c), (17a–c) and (22a–c) are substituted back in the expansion Eqs. (11a–c), which provide the estimate of the in-plane forces (Nxx , Nyy , Nxy ) in the post-critical regime. Finally, the second order of approximation (W3 ) for the post-critical out-of-plane deflection is obtained following a similar procedure as in W1 . This involves solving Eq. (12f). On solving, the following expression is obtained as





W3 = 2W12W13 m A211 A212 A−1 11 − A22 4



−16W1 T4 A11 A22 1 + η2









2

2

 

2 −1

1 + η 2 − 4η 4 n m + n

4



m (A11 αx + A12 αy ) + n (A12 αx + A22 αy ) 2

2





+ 2W12W31 n A222 A212 A−1 22 − A11



3W12 1 + η2





A212 − 3A11 A22

 −1  2   2 2 η 4 m 2 + n2 A12 − A11 A22 A11 m + A12 n   2 −1 2 +16T2 A11 A22 1 + η2 m (A11 αx + A12 αy ) + n (A12 αx + A22 αy )



1 + η 2 − 4η 4 m m + n 2



4

4

2

m A11 + n A22

 

2 −1



−12W12 m n A11 A22 A12 1 + η2 − 12W12 m n 2 2

2 2

(23)

Total deflection is obtained by summing up each order of approximation as in (9c). This, however, is not explicitly mentioned here for brevity. Following Naderi and Saidi [44] (for classical plate), the magnitude of the out of plane deflection (w) is restricted within the thickness of the plate (h) for accuracy of the solution. Thus, the respective normalised deflection (w/h ) is kept within unity for numerical illustration. It is important to note that an eigenvalue problem for buckling analysis of plates is solved herein to obtain the critical loads and associated eigen buckling modes. Individual critical loads and the modes are specific to the choiceof m and n and their combinations. Subsequent post-critical responses are obtained through nonlinear analysis, which is also specific to individual modes, i.e. for specific m and n only. Therefore, the convergence analysis for varying (m, n) is not relevant to the present analysis. Nevertheless, the convergence of the series expansion is ensured. 5. Verification of the obtained solution The solution obtained above is first verified with respect to already established solutions in literature for its legitimacy. This includes a comparison with several existing solutions, both numerical as well as closed-form analytical solutions. The Please cite this article as: M. Alam and S.K. Mishra, Thermo-mechanical post-critical analysis of nonlocal orthotropic plates, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.018

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Table 1 Minimum normalised critical temperature (α Tcr ) for simply supported isotropic plate with (a/b) = 1,

ν = 0.3,

Thickness ratio (a/h )

Present solution

Zhao et al. [48] (kp-Ritz method)

Noor and Burton [49] (Navier’s solution)

10 20 100

0.012650 0.0031633 0.0001265

0.011830 0.003089 0.0001271

0.011830 0.003109 0.0001264

E = 1.0 × 106 N/m2 . Matsunaga [50] (Navier’s solution) 0.011830 0.003109 0.0001264

Table 2 Comparison among thecritical temperature Tc r (K) of a simply supported nonlocal beam. Nonlocal parameter (μ = e0 a0 )

Present solution

0 1 2 3

70.5941 64.2526 58.9565 54.4670

Ebrahimi and Salari [51] (Navier’s series solution) 68.6671 62.4988 57.3473 52.9803

Fig. 1. Single Layer Graphene Sheet (SLGS) with (a) Armchair and (b) Zigzag configurations in reference coordinate system.

first verification is made against an existing numerical solution by Zhao et al. [48]. In this study, the buckling of a functionally graded plate under thermo-mechanical loading is numerically investigated with an element free kp-Ritz method. A stabilised, conforming nodal integration scheme was employed to evaluate the plate bending stiffness, whereas the shear and membrane terms were estimated by direct nodal integration. The minimum critical temperature is employed for comparison in Table 1. A reasonable degree of match is observed. An additional comparison is also presented with two other analytically obtained solutions [49,50], in which, the Navier’s series solution approach was employed to solve for thermal buckling in multilayered anisotropic plate [49] and composite/sandwiched plate [50]. Once again, the obtained critical loads therein are observed to be very similar to that predicted by the present solution for varying thickness ratio. The obtained solution is also compared with an existing solution [51] for nonlocal beam buckling, for which, the obtained solution is specialised to a limiting case of much longer length comparing its width to essentially mimic a beam. The comparison is also shown in Table 2. Reasonable match of the present solution with the solution given by Ebrahimi and Salari [51] is observed for varying nonlocal length scales. Taken together, these comparisons point out to the legitimacy of the proposed solution, which may be adopted for further illustration. It is important to note that the present solution explores (beyond the previous solutions) the behaviour along much higher temperature range with a significantly higher degree of geometric nonlinearity that results in redistribution of in-plane stresses and reserved post-critical behaviour, illustrated subsequently. 6. Numerical illustration The solution for the critical temperature and post-critical temperature-deflection behaviour for the nonlocal plate obtained previously is numerically demonstrated by adopting an example of SLGS, a well-acclaimed nanostructure often modelled as nonlocal plate [52–54]. Representative geometric dimensions of SLGS, pertinent boundary restraints and alternative configurations of Graphene (armchair and zigzag) are considered for illustration. Two alternate configurations of SLGS (Armchair and Zigzag) are shown in Fig. 1a and b, respectively. The plate idealisation with the x, y and z axes aligned along their length (a), width (b) and thickness (h) are shown, respectively. The SLGS is subjected to a uniform rise in temperature to induce compressive force to cause buckling at the critical temperature (Tcr ). Beyond critical temperature, the redistribution of stresses occurs to result in post-critical reserve stiffness. Please cite this article as: M. Alam and S.K. Mishra, Thermo-mechanical post-critical analysis of nonlocal orthotropic plates, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.018

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Table 3 Critical temperature for alternative SLGSs with varying combination of scale parameters (e0 a0 ) and ambient temperature (T). Temperature T (K)

Nonlocal parameters e0 a0 (n m)

Critical temperature Tcr (K)

Armchair Sheet I: a = 9.519 nm, b = 4.844 nm, h = 0.129 nm 300 0 298.84 500 0 331.375 700 0 332.286 Armchair Sheet II: a = 6.995 nm, b = 4.847 nm, h = 0.143 nm 0 408.759 300 500 0 435.671 700 0 522.819 Armchair Sheet III: a = 4.888 nm, b = 4.855 nm, h = 0.156 nm 300 0 712.785 500 0 668.696 700 0 641.029 Zigzag Sheet IV: a = 9.496 nm, b = 4.877 nm, h = 0.145 nm 300 0 490.820 500 0 432.503 700 0 384.198 Zigzag Sheet V: a = 7.065 nm, b = 4.887 nm, h = 0.149 nm 300 0 573.057 500 0 504.722 700 0 536.119 Zigzag Sheet VI: a = 4.855 nm, b = 4.888 nm, h = 0.154 nm 300 0 693.749 500 0 650.428 700 0 623.798

Temperature T (K)

Nonlocal parameters e0 a0 (n m)

Percentage disparity

0.67 0.67 0.67

241.423 267.732 268.468

19.20 19.20 19.20

0.47 0.47 0.47

359.393 383.056 459.679

12.08 12.08 12.08

0.27 0.27 0.27

672.034 630.466 604.38

5.72 5.72 5.72

0.47 0.47 0.47

439.866 387.603 344.313

10.38 10.38 10.38

0.32 0.32 0.32

539.315 475.003 504.552

5.88 5.88 5.88

0.22 0.22 0.22

666.901 625.256 599.657

3.87 3.87 3.87

It may be mentioned that the present article is not specifically limited to the buckling/post-buckling of Graphene, rather the adoption of SLGSs are purely for demonstration purpose. The closed-form solution presented herein might have also been illustrated by adopting other plate type nanostructures. 6.1. Critical temperature, the effect of nonlocality, aspect ratio and behaviour in higher modes The effect of nonlocality on the buckling and post-critical temperature-deflection behaviour is presented first. The nonlocal parameters and other properties are adopted from Shen et al. [52] for demonstration. Six different SLGS configurations are considered, out of which, three are armchair and three are zigzag type. The nonlocal constitutive properties for these are listed in Table 3. The equations and the obtained solution are compared with the ones reported [41] for classical (e0 a0 = 0) (“local”) isotropic plate. Specifically, Eq. (15c) is compared with the one given by Stein [41]. The expressions for critical temperature exactly matches with the classical solution as the nonlocal parameter (e0 a0 ) is set to be zero, assuming isotropy. The in-plane forces and displacements are also matched with the classical solution. The values of the critical temperatures are also presented in Table 3. The critical temperatures are observed to be sensitive to the nonlocal scale parameter. Nonlocal interactions are seen to reduce the critical temperature, compared with those obtained using classical continuum theory. This trend is in line with the observations made by the previous investigators [53–55]. The reductions of critical temperatures due to nonlocal interactions appear to be higher for armchair configurations than the zigzag. For instance, the reductions in the critical temperature range from 6% to 19% for armchair, which changes to 4−10% for zigzag type Graphene sheet. Reductions for specific cases may also be noted in Table 3. Eqs. (15c) and (15d) show the dependency of critical temperature on aspect ratio (a/b), which is also shown in Fig. 2. The variations are also contrasted with the respective estimate based on “local” theory. However, the behaviour of only the armchair SLGS case III is presented in this Figure. The critical temperature (Tcr ) is normalised with respect to their corresponding value for the fundamental mode (Tcrf ) in this plot. The behaviour of critical temperatures in higher modes (m = 2, 3) is also presented. The critical temperature decreases in a significant proportion with increasing aspect ratio. However, Tcr values in different modes (fundamental and higher) asymptotically approach certain value for increasingly higher aspect ratio. This is in contrast with the buckling behaviour under pure compression, in which, increasing aspect ratio leads to mode shifting (from lower to higher modes) [40]. Contrastingly, no changes/shift among the buckling modes is observed with increasing aspect ratio for thermal buckling, which is typical to thermal buckling [40]. The effect of nonlocal interactions may also be observed in Fig. 2. The nonlocal scale effect is more prominent for buckling in higher modes. However, for a specific mode, the reductions in critical temperature by the nonlocal interactions become less with higher aspect ratios. These observations may be explained from the fact that, higher buckling mode shapes or lower aspect ratios, both are conducive to the enhanced gradient of displacement fields that contribute to enhanced nonlocal interactions. Please cite this article as: M. Alam and S.K. Mishra, Thermo-mechanical post-critical analysis of nonlocal orthotropic plates, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.018

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Fig. 2. Critical temperature vs. aspect ratio under a varying combination of nonlocal parameters and buckling modesfor case III (armchair).

Fig. 3. Post-critical temperature-deflection path for armchair SLGS case-I.

6.2. Post-critical temperature-deflection behaviour The temperature-deflection behaviour of a nonlocal plate in the post-critical regime is illustrated in this section. Six different cases of SLGSs enlisted in Table 3 are considered for illustration. It may be noted that the properties of Graphene are temperature-dependent [52], although, the specific pattern of such dependence is yet to be established. However, within the considered range of temperature increase (beyond the critical temperature), the change in the nonlocal parameters are relatively insignificant. Therefore, slight changes in the nonlocal properties due to the temperature increase are ignored while studying the post-critical behaviour under thermal loading. Please cite this article as: M. Alam and S.K. Mishra, Thermo-mechanical post-critical analysis of nonlocal orthotropic plates, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.018

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Fig. 4. Post-criticaltemperature-deflectionpathfor (a) armchair SLGS case II and (b) armchair SLGS case III.

Fig. 5. Post-critical temperature-deflection path for zigzag SLGS case (a) IV (b) V and (c) VI.

6.2.1. Effect of nonlocal interactions on the temperature-deflection path The post-critical temperature-deflection paths are obtained for all six scenarios. The effect of nonlocal scale parameter is studied first. The solutions obtained for the post-critical temperature-deflection path are shown in Fig. 3 for armchair sheet case-I. The equilibrium paths predicted by the nonlocal theory are contrasted with the corresponding paths predicted by the classical continuum theory in order to pinpoint the effect of nonlocal interactions. Other than reducing the critical temperature, the nonlocal effect is observed to increase the deflection (at identical temperature) in initial stages of postPlease cite this article as: M. Alam and S.K. Mishra, Thermo-mechanical post-critical analysis of nonlocal orthotropic plates, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.018

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Fig. 6. Nonlocal effects onthe post-critical paths in higher modes for (a) armchair (case II) and (b) zigzag (case V) type SLGS.

critical regime shown in Fig. 3. However, after a certain threshold, the nonlocal theory predicts lower deflection than the classical (“local”) theory. In effect, the nonlocal interactions lead to reduced stiffness in the vicinity of buckling but offer enhanced effective stiffness after substantial post-critical deformation. The local and nonlocal paths intersect at around (T /Tcr = 2.5). The solution methodology employs convergent expansion of varying degree for the relevant field variables (u, v, w). Accuracy generally increases with the inclusion of higher-order terms in the expansion. Approximation until first order (ε for w and ε 2 for u and v) generally provides sufficient accuracy. However, in several instances, an approximation of secondorder (ε 3 in w and ε 4 for u and v) is sought to achieve acceptable level of accuracy. The solutions obtained using first and second order of approximation are contrasted in Fig. 3. Significant improvements are observed on the incorporation of higher order approximation. Approximation of first-order overestimates the post-buckling deflection (thus underestimating the post-critical stiffness). In fact, stiffening induced by nonlocal effect at later stages can only be explored through secondorder approximation. Moreover, overestimation errors are also noted to be amplified by nonlocal interactions. Thus, nonlocal interactions reduce the rate of convergence of the solution. The preceding discussion explains the usefulness of the higher order approximation in assessing the correct post-critical paths for the armchair SLGS case-I. However, it is also shown in subsequent discussions that the higher-order terms are not equally useful for other SLGS configurations. The post-critical behaviour for the alternative armchair configurations (cases – II and III) are shown in Fig. 4a and b, respectively. The equilibrium paths are found to be mostly similar to those observed earlier. Similar to armchair-(case I), armchair-(case II) also results in higher deflection (i.e. softening) at initial stages and lower deflection (i.e. stiffening) at latter stages while compared with the respective classical (“local”) paths. Of course, the extent of initial softening and subsequent (case specific) stiffening are significantly reduced from that observed in armchair type-I (Fig. 3). The disparity between the solutions from the first and higher order of approximation is also diminished. Fig. 4b shows the post-critical paths for armchair SLGS case-III. The post-critical path follows a similar trend as in Fig. 4a. However, unlike the previous two cases (armchair case I and case II), further stiffening due to nonlocal interactions at larger deflection is not observed. The nonlocal theory rather predicts more deflection, leading to softening at later stages of postcritical behaviour. While the second-order approximate nonlocal post-critical path intersects the second-order approximate local path due to nonlocality induced stiffening in both armchair case I and case II (Figs. 3 and 4a), this is not so for armchair case-III presented in Fig. 4b. Thus, although the softening at initial post-critical regime due to nonlocal interactions are consistent, further stiffening at latter stages of post-critical deflection are not available for other configurations of Graphene. The relative insignificance (except case I) of higher order approximation in case II and III are also noted. A first-order approximation appears to be good enough to adequately trace the post-critical path for these cases. In next, the post-critical behaviour is presented for zigzag configurations of Graphene. The post-critical paths are shown in Fig. 5a, b and c for zigzag SLGS case IV, V and VI, respectively. The paths are contrasted with the ones obtained for classical (“local”) plate. Unlike armchair SLGS (particularly case I and II), the zigzag SLGSs do not show stiffening at large post-critical deflection. Nevertheless, the softening at the initial post-critical regime remains intact. It is also important to note that except zigzag case IV, in other two cases (V and VI), the higher order convergent approximation may not be necessary; as the post-critical paths obtained by the first and second-order approximations do not differ significantly. With an identical increase in post-critical temperature, the influence of nonlocal interactions is seemingly more pronounced for the armchair than the zigzag configurations. Please cite this article as: M. Alam and S.K. Mishra, Thermo-mechanical post-critical analysis of nonlocal orthotropic plates, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.018

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Fig. 7. Effect of aspect ratio on the post-critical paths corresponding to (a) armchair (case I, II and III) and (b) zigzag (case IV,V,VI) SLGS.

Fig. 8. Redistribution of longitudinal stresses in classical and nonlocal (zigzag Graphene) (a/b = 1.947) plates at (a) 800 K (b) 1000 K (c) 1200 K and (d) 1400 K.

6.2.2. Effect of aspect ratio and behaviour in higher modes In this section, the effect of nonlocal interactions is demonstrated on the post-critical behaviour of SLGS in higher modes. Fig. 6a and b presents the post-critical paths for the armchair and zigzag type SLGS, respectively. Case II for armchair and case V for zigzag are only considered for this illustration. The temperature-deflection paths for the higher modes are contrasted with the fundamental mode, keeping the aspect ratios identical. The disparity in critical temperatures for higher modes due to nonlocal interactions are remarkably higher than the corresponding increase in the fundamental mode. For higher modes, the post-critical paths obtained from nonlocal theory are mostly parallel to the respectivepaths predicted by classical theory and the paths never intersect (at least within the considered deflection range i.e. w ≈ h). This is in conPlease cite this article as: M. Alam and S.K. Mishra, Thermo-mechanical post-critical analysis of nonlocal orthotropic plates, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.018

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Fig. 9. Redistribution of transverse stresses in classical and nonlocal (zigzag Graphene) (a/b = 1.947) plates at (a) 80 0 K (b) 10 0 0 K (c) 120 0 K and (d) 140 0 K.

trast to the post-critical behaviour at fundamental mode, in which, the nonlocal path is either too close or even intersects with the path obtained using “local” theory at larger post-critical deflection. Generally, deflection at higher modes is significantly amplified due to nonlocal interactions. In effect, this leads to significant post-critical softening at higher modes. The temperature-deflection behaviour for both types of Graphene configurations is mostly identical in higher modes, except that the zigzag SLGS experiences slightly higher deflections comparing the armchair. The importance of aspect ratio in influencing the critical temperature is demonstrated in the earlier section. The influence of the same on the post-critical behaviour is investigated herein. Fig. 7a and b shows the temperature-deflection paths for nonlocal plates (armchair and zigzag type SLGSs) considering three different aspect ratios. The equilibrium paths are contrasted with the paths obtained from classical “local” theory. Higher aspect ratio is observed to reduce the critical temperature in the previous discussion. Higher aspect ratios are also noted to increase the post-critical deflection. For an identical increase in post-critical temperature, the level of deflections in zigzag sheets appear to be considerably higher than the armchair. This trend is in reverberation to earlier trends shown in Figs. 3, 4a–c (for armchair) and in Fig. 5a–c (for zigzag). Thus, it may be commented that higher aspect ratio leads to post-critical softening. Furthermore, such trends are more prominent for the armchair than zigzag type SLGS. Taken together, it may be concluded that the load-carrying capacity of nonlocal plates does not necessarily deplete on buckling due to the substantial reserve stiffness in the post-critical regime. Such reserve may be exploited in designing as done for classical plates. This investigation reveals the post-critical reserve for the nonlocal plates. The nonlocal interactions of molecular origin are shown to have important implications on such reserve. Several other parameters, such as aspect ratio, nonlocal constitutive properties and alternative configurations of nanostructures are also observed to influence the post-critical behaviour. Furthermore, if the nonlocal plates areadequately constricted to buckle in higher modes, the nonlocal influences are shown to be amplified. 6.3. Redistribution of stresses: key to post-critical reserve Having the post-critical reserve been demonstrated, the possible mechanism behind the reserve is discussed in this section. The redistribution of in-plane stresses was identified to be the key to such reserve [38–42], which is relooked herein in presence of nonlocal interactions in plates under thermally induced compression. Please cite this article as: M. Alam and S.K. Mishra, Thermo-mechanical post-critical analysis of nonlocal orthotropic plates, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.018

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Fig. 10. Redistribution of longitudinal stresses in classical and nonlocal (armchair Graphene) (a/b = 1.947) plates at (a) 800 K, (b) 10 0 0 K, (c) 120 0 K, (d) 1400 K.

Distribution of longitudinal in-plane forces in the nonlocal (Zigzag Graphene) plate are shown in Fig. 8a–d for four representative temperatures, representing different post-critical regimes depending on the temperature difference from the respective critical value. The edgewise distribution of the in-plane forces remains uniform in the pre-buckled regime, which gradually starts redistributing with temperature increase beyond critical. The redistributions are characterised by sharing of enhanced forces by particular strips of the plate, whereas reducing the share in other strips. Higher is the temperature beyond its critical value, larger is the extent of redistribution. The redistribution maybe characterised by the ratio of the peaks at the two different strips. At sufficiently large temperature, the longitudinal in-plane compression may essentially be shared by the end strips with negligible contribution of its share by the middle strips. It may be recalled that the concept of equivalent width was formulated to simplifying the design guideline for classical plates to effectively exploit the post-critical reserve by taking advantage of the redistribution for uniaxially compressed plates [42]. The present scenario, however, is not uniaxial compression but rather biaxial, as the compression develops along both the directions due to restricted (thermally induced) expansions along both the directions. Although due to difference among length and width, the force along one direction would be predominant to show some degree of similarity with that of uniaxially loaded plate. The nonlocal interaction seems to hinder the extent of redistribution by reducing the ratios of the magnitude of peak forces between the middle and end strip. The trend becomes even more predominant at higher temperature for zigzag type SLGS, as noted from Fig. 8a–d. The redistribution is also illustrated for the transverse in-plane forces for zigzag type SLGS under identical temperature increase, as shown in Fig. 9a–d. The longitudinal and transverse in-plane forces are of identical magnitude in the pre-buckled regime. However, redistribution results larger transverse in-plane forces at middle strip and much lower values along the edge strips. It is important to note that unlike purely compressive longitudinal forces, the redistributed transverse forces may even change to tensile along the middle strip, which substantially contributes to tension stiffening. The trend shows that the transition to tensile transverse forces may occur at larger temperature. Further, the nonlocal interactions are also shown to prevent such redistribution to certain extent. The disparity of behaviour from uni-axially loaded plate may be noted, in which, the transverse forces immediately reflect the tensile redistribution in the post-critical regime. In present case, however, redistributive tensile forces show up at relatively latter post-critical regime (at much higher temperature), which may be attributed to the bi-axial nature of the incipient thermal compression, which might be masking the redistributive tensile forces. The nature of redistribution of in-plane stresses presented in Figs. 8a–d and 9a–d for zigzag Graphene shows that although nonlocal interactions somewhat prevent the development of redistributive stresses, the qualitative nature of the Please cite this article as: M. Alam and S.K. Mishra, Thermo-mechanical post-critical analysis of nonlocal orthotropic plates, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.018

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Fig. 11. Redistribution of transverse stresses in classical and nonlocal (armchair Graphene) (a/b = 1.947) plates at temperature of (a) 80 0 K (b) 10 0 0 K (c) 1200 K and (d) 1400 K.

redistribution remains mostly unaffected. This, however, does not hold for armchair SLGS. The redistribution of longitudinal and transverse in-plane forces are presented subsequently in Figs. 10a–d and 11a–d for various values of temperature beyond critical value. Redistribution of longitudinal stresses for armchair Graphene, shown in Fig. 10a–d indicates that the nonlocal interactions not only prevent redistribution but completely change the nature of redistribution with enhanced sharing by the end strips and reduced sharing by the middle strips. Further, the extent of redistribution amplifies with increasing temperature beyond critical value. However, particularly interesting is the oscillatory distribution of the transverse forces of tensile nature, shown in Fig. 11a–d, which are in striking disparity with the previous case of Zigzag type SLGS. 7. Conclusion The critical temperature for buckling and post-critical behaviour of nonlocal plates (SLGSs) are studied analytically. Owing to molecular characteristic dimensions, the long-range atomic interactions are crucial and are taken into account following a nonlocal stress-strain relationship in an equivalent continuum framework. The constitutive relation includes orthotropic nature of nonlocal plate. Whereas the buckling is adequately described in a linearised framework, the post-critical behaviour requires consideration of von-Karmann geometric nonlinear strains.The system of nonlinear partial differential equations is formulated and solved analytically using a perturbation type method. The workings of the proposed solution are numerically demonstrated by adopting a well-acclaimed plate type nanostructure, SLGS. The constitutive properties of SLGS are adopted from literature. Two alternate configurations of Graphene, namely, the armchair and zigzag are presented. It is observed that the nonlocal interactions significantly reduce the critical temperature, and such reductions become more prominent in higher modes. However, the disparity reduces for increasing aspect ratio. The nonlocal interactions are shown to significantly influence the post-critical behaviour as well. The nonlocal interactions generally amplify the post-critical deflection (leads to softening) around the vicinity of the critical temperature. However, in certain instances (such as armchair case-I), nonlocal interactions are shown to reduce the post-critical deflection to result in effective stiffening at relatively larger level of post-critical deflection. A first-order approximation appears to be good enough for the solutions, except in cases of further stiffening at later stages, as in armchair case-I. The effect of nonlocal interactions is more pronounced in armchair than the zigzag configurations. Please cite this article as: M. Alam and S.K. Mishra, Thermo-mechanical post-critical analysis of nonlocal orthotropic plates, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.018

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The amplification in post-critical deflection due to nonlocal interactions become more prominent in higher modes, indicating significant softening. Higher aspect ratios are observed to play synergistic role for such behaviour. The redistribution of the in-plane stresses is presented as key to the post-critical reserve. Nonlocal interactions are generally noted to prevent redistribution, to an extent that it may even alter the nature of the redistribution, as in armchairtype SLGS. The analytical solution may serve as a benchmark for verification of numerical solutions (such as those based on Finite Element Method) involving more complex geometries, boundary and loading conditions. The solutions may also be useful in developing design guidelines for plate-like nano-structures, for which, the post-critical reserve is intended to be exploited efficiently. References [1] P. Zhang, H. Jiang, Y. Huang, P.H. Geubelle, K.C. 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