A new3D elasticity numerical solution for nonlinear thermo-mechanical bending of orthotropic annular/circular micro/nanoplates using re-modified couple stress & SAPM
Subject Areas : Mechanical EngineeringAmirreza Golkarian 1 , Mehrdad Jabbarzadeh 2
1 - Ph. D. Student, Science & Research branch, Islamic Azad University, Tehran, Iran
2 - Department of Mechanical Engineering, Mashhad branch, Islamic Azad University, Mashhad, Iran
Keywords: Three dimensional elasticity theory, Re-modified couple stress theory, Semi-analytical polynomial method, Annular nanoplate, Nonlinear bending, Orthotropic material,
Abstract :
In this research, a new 3D elasticity numerical solution based on the Semi-Analytical Polynomial Method (SAPM) is presented for the nonlinear bending analysis of orthotropic annular/circular micro/nanoplates resting on a Winkler-Pasternak elastic foundation, utilizing re-modified couple stress theory. This is the first report of a 3D elasticity numerical solution specifically based on re-modified couple stress theory. Moreover, while previous 3D analytical solutions based on couple stress theory have primarily focused on the bending of rectangular plates with simply supported edges, this study investigates the nonlinear bending of circular plates under various boundary conditions. Additionally, the variation in thickness under different types of loading (mechanical, thermal, and thermo-mechanical) is reported for the first time. The effects of boundary conditions, couple stress scale parameters, aspect ratio, thickness, loading, and the elastic foundation are examined. Both the increasing and decreasing impacts of scale parameters on deflection are observed in this study.
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Journal of Solid Mechanics Vol. 16, No. 3 (2024) pp. 246-267 DOI: 10.60664/jsm.2024.1130439 |
Research Paper A New 3D Elasticity Numerical Solution For Nonlinear Thermo-Mechanical Bending of Orthotropic Annular/Circular Micro/Nano Plates Using Re-Modified Couple Stress & SAPM |
A.R. Golkarian 1, M. Jabbarzadeh 21 | |
1 Ph.D. Student, Science & Research Branch, Islamic Azad University, Tehran, Iran 2 Department of Mechanical Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran | |
Received 29 August 2024; Received in revised form 25 September 2024; Accepted 3 October 2024 | |
| ABSTRACT |
| In this research, a new 3D elasticity numerical solution based on the Semi-Analytical Polynomial Method (SAPM) is presented for the nonlinear bending analysis of orthotropic annular/circular micro/nano plates resting on a Winkler-Pasternak elastic foundation, utilizing re-modified couple stress theory. This is the first report of a 3D elasticity numerical solution specifically based on re-modified couple stress theory. Moreover, while previous 3D analytical solutions based on couple stress theory have primarily focused on the bending of rectangular plates with simply supported edges, this study investigates the nonlinear bending of circular plates under various boundary conditions. Additionally, the variation in thickness under different types of loading (mechanical, thermal, and thermo-mechanical) is reported for the first time. The effects of boundary conditions, couple stress scale parameters, aspect ratio, thickness, loading, and the elastic foundation are examined. Both the increasing and decreasing impacts of scale parameters on deflection are observed in this study.
|
| Keywords: Three dimensional elasticity theory; Re-modified couple stress theory; Semi-analytical polynomial method; Annular nanoplate; Nonlinear bending; Orthotropic material. |
1 INTRODUCTION
N
ANOTECHNOLOGY is one of the most intriguing research fields across various branches of science. The unique properties of nanomaterials and the challenges posed by experimental studies at the nanoscale motivate researchers to explore their mechanical properties through various theoretical or semi-experimental methods. As a result, theoretical approaches such as the Molecular Mechanics Method (MMM) [1-5] and the Molecular Dynamics Method (MDM) [6,7] have been introduced and employed to investigate these properties, both of which are grounded in molecular and mechanical sciences. Among the mechanical properties, only stretching properties and vibration frequencies can be experimentally evaluated due to the limitations of available instruments. Other mechanical properties, such as buckling behavior, must be assessed using MMM and MDM. However, certain analyses, such as bending, cannot be evaluated experimentally or even through molecular-based methods. Due to these limitations, alternative theoretical mechanical approaches, such as mechanical plate theories, are employed to examine the bending behavior of micro or nanoplates, some of which are mentioned here.
Experimental observations reveal that at the nanoscale, the mechanical behavior of structures is size-dependent. Since classical continuum theory neglects this size dependency, several non-classical continuum theories have been developed. These include classical couple stress theory [8–11], strain gradient theory [12], nonlocal elasticity [13], surface elasticity [14], and modified couple stress theory (MCST) [15]. Reddy and Berry [16] proposed nonlinear size-dependent models based on Classical Laminate Plate Theory (CLPT) and First-Order Shear Deformation Theory (FSDT) for the bending of circular plates. Reddy and Kim [17] utilized MCST and the nonlinear strains of von Karman to develop a size-dependent third-order plate model. Dastjerdi and Jabbarzadeh [18-23] employed Eringen's nonlocal theory and the Differential Quadrature Method (DQM), as well as introduced a new Semi-Analytical Polynomial Method (SAPM), to investigate the nonlinear thermo-mechanical bending of monolayer and bilayer graphene sheets based on FSDT and Third-Order Shear Deformation Theory (TSDT).Thai et al. [24–26] proposed new models based on Classical Plate Theory (CPT), FSDT, TSDT, and sinusoidal shear deformation plate theories for analyzing the bending and free vibration of rectangular microplates with simply-supported edges. Sahmani and Ansari [27] analyzed the free vibration of rectangular microplates using strain gradient theory and TSDT. Using nonlocal elasticity and a Higher-Order Shear Deformation Plate Theory (HSDT), Daneshmehr et al. [28] examined the size-dependent instability of nanoplates under biaxial in-plane loadings, solving the governing equations with the Generalized Differential Quadrature (GDQ) method. Also, there are some other valuable researches [29-35] which have examined nanomaterials from various aspects.
To the best of the authors' knowledge, no numerical study has been conducted on the bending analysis of circular plates (whether at the macro, micro, or nanoscale) based on 3D elasticity theory. In a few studies that utilize 3D analytical solutions, the bending of homogeneous or functionally graded (FG) rectangular plates subjected to sinusoidal transverse loading with simply supported edges has been examined [36-40]. Among these, the only research based on re-modified couple stress theory is the recent work by Salehipour et al. [40], which was the first to report a 3D analytical solution for the bending of FG micro/nanoplates. Additionally, the first report of finite element (FE) bending analysis of circular plates based on modified couple stress theory is by Reddy et al. [41], which is grounded in Classical Plate Theory (CPT) and First-Order Shear Deformation Theory (FSDT).
The absence of a numerical solution for 3D bending analysis in the literature is due to the challenges associated with developing such a solution for plates. Additionally, in existing analytical solutions, various restrictions on plate geometry (rectangular or circular), loading types, and boundary conditions are imposed to facilitate the solvability of the resulting governing equations. This study addresses this gap by introducing a new numerical approach to 3D elasticity solutions using the Semi-Analytical Polynomial Method (SAPM) recently presented by Dastjerdi and Jabbarzadeh [19]. For the first time, a novel numerical solution approach based on 3D elasticity theory is introduced for the nonlinear bending of annular/circular plates without imposing any restrictions on plate geometry or loading types, applicable to all boundary conditions.
This is also the first report of a 3D elasticity solution for the nonlinear bending analysis of orthotropic annular/circular micro/nanoplates based on re-modified couple stress theory. The analysis covers all boundary conditions, including clamped, simply supported, and free edges, under various types of loading such as mechanical, thermal, and thermo-mechanical. The effects of different elastic foundations, including Winkler, Pasternak, and Winkler-Pasternak, are also examined. Additionally, the influence of various parameters, particularly thermal loading, on the thickness variation of plates is investigated. The study explores the impact of all factors affecting plate deflection, including boundary conditions, couple stress scale parameters, aspect ratio, thickness, loading types, and elastic foundations.
2. GOVERNIN EQUATIONS
2.1 Re-modified couple stress theory
According to the Modified Couple Stress Theory (MCST) proposed by Yang et al. [42], the strain energy for an elastic and isotropic material within region V is [42]:
(1)
where
and
is the Cauchy stress tensor,
is the strain tensor, 
is the deviatory part of couple stress tensor and
is the symmetric curvature tensor which are defined respectively as [42]:
(2)
(3)
(4)
(5)
where
and
are Lame constants,
is the displacement vector,is the stress tensor,
is the material length scale parameters (MLSP),
is the rotation vector and
is Kronecker delta symbol which
andare defined as [42]:
(6)
(7)
where
is thermal diffusivity and
is the temperature difference.
Chen et al. [36] developed a re-modified couple stress theory for anisotropic elasticity, in which the curvature and couple stress tensors are redefined as [36]:
(8)
(9)
where
and
are MLSPs and shear modules in different directions respectively.
2.2 3D elasticity theory formulation
A schematic of an annular/circular micro/nanoplate is shown in Fig.1 with an inner radius
, an outer radius
, and a thickness 
subjected to uniform transverse loading
while resting on a Winkler-Pasternak elastic foundation characterized by two parameters.
and
are the Winkler and Pasternak stiffness coefficients of elastic foundation, respectively. According to the 3D elasticity theory of plates, the displacements filed can be expressed as follow:
(10)
(11)
(12)
where 
are the displacement components along the
directions respectively and
is equal to zero due to the symmetry. It is obvious that no assumption or simplification is employed in definition of displacement vectors. Substituting Eqs. (10)-(12) into Eq. (3),the non-zero components of the Von-Karman strain field can be written as [41]:
(13)
(14)
(15)
(16)
Due to the symmetry of the geometry and boundary conditions, other strain components are zero. Substituting Eqs. (10)–(12) into Eqs. (7) and (9) yields the non-zero components of curvature as [41]:
Fig. 1
The schematic view of an annular/circular graphitic plate under uniform loading and rested on elastic foundation.
(17)
(18)
(19)
Due to the orthotropic behavior of the employed micro/nanoplates, the stress-strain relations are derived as follows [43]:
(20)
(21)
(22)
(23)
where
is poison ratio,
is shear strain,
is the elastic module in
direction,
is the shear module of orthotropic micro/nanoplate and 
which 
, 
and
. The
and
are the elastic module in
and
directions.
2.3 Constitutive equations
In this study, the constitutive equations and boundary conditions are derived based on the principle of minimum total potential energy, as follows [43]:
(24)
where δ represents the variation symbol, and 
is the potential of applied forces, which includes the effects of transverse loadingand the Winkler-Pasternak elastic foundation on the surface of the micro/nanoplate. The components of the total potential energy are defined as follows:
(25)
(26)
where
is the Laplacian operator.
The constitutive relations for orthotropic elasticity can be expressed as:
(27)
where
,
and
is the stiffness matrix which based on re-modified coupled stress theory is defined as:
(28)
and
is the coefficients matrix of thermal strains as:
(29)
By substituting Eqs. (13)–(19) into Eq. (25) and neglecting body forces and external couples, the 3D equilibrium equations for an orthotropic annular/circular micro/nanoplate based on the re-modified couple stress theory are derived as follows:
(30)
(31)
For sake of generality and convenience, the following non-dimensional parameters are introduced:
(32)
As the final step, by substituting Eqs. (13)–(16) into Eqs. (20)–(23) to express the stress components, and Eqs. (17)–(19) into Eq. (8) to define the couple stress components in terms of the displacement fields, the stress terms in Eqs. (30) and (31) are replaced. Using non-dimensional parameters, two equilibrium equations are derived as follows:
(33)
(34)
3. SOLUTION PROCEDURE
In this study, based on SAPM [20], by considering each partial differential equation (two equilibrium equations) as follows:
(35)
where the function
is defined as:
(36)
where
is the number of grid points in
direction and
is the number of grid point in
direction, the partial differential equation is converted to the algebraic equation. The schematic view of grid points is illustrated in Fig. 2. The empty points must satisfy the equilibrium equations, while the filled points must satisfy the boundary conditions.
Fig. 2
The schematic view of grid points based on SAPM.
Based on the above explanations of SAPM, two displacement fields can be defined as follows:
(37)
(38)
The obtained algebraic equations are solved using numerical methods, such as the Newton-Raphson method.
4. BOUNDARY CONDITIONS
In the present study, all types of boundary conditions are categorized into three types: simply supported (S), clamped (C), and free edges (F). At the inner and outer radii (ri and ro), the boundary conditions can be defined as follows:
(39)
(40)
(41)
The displacement components should satisfy the following boundary conditions at the top and bottom surfaces of the plate:
(42)
(43)
(44)
(45)
5. NUMERICAL RESULTS AND DISCUSSIONS
To investigate the convergence of the SAPM results for different grid sizes, an orthotropic annular/circular plate with the following properties is considered for various boundary conditions. The results are reported in Table 1. The nonzero components of the stiffness matrix are as follows [44]:
(46)
Table 1.
Convergence checking of dimensionless deflections of an orthotropic annular/circular plate versus the number of grid points.
n (number of domain nodes) | w* | |||||||
Boundary Condition Type | ||||||||
C - C | S - S | C - S | S - C | C - F | F - C | S - F | F - S | |
3 | 0.000841 | 0.000841 | 0.000841 | 0.000841 | - | 0.01970 | - | 0.0839 |
5 | 0.003033 | 0.010096 | 0.006110 | 0.004381 | - | 0.04427 | - | 0.3823 |
7 | 0.003178 | 0.009951 | 0.005887 | 0.004519 | 0.02539 | 0.03438 | 0.3002 | 0.2077 |
9 | 0.003252 | 0.009828 | 0.005843 | 0.004549 | 0.04008 | 0.03242 | 0.1844 | 0.1847 |
11 | 0.003303 | 0.009801 | 0.005894 | 0.004574 | 0.04500 | 0.03198 | 0.1587 | 0.1795 |
13 | 0.003338 | 0.009820 | 0.005969 | 0.004597 | 0.04563 | 0.03180 | 0.1472 | 0.1782 |
Convergence is achieved with 11 grid points in the radial direction. For all cases, the number of grid points in the z direction is set to 5.
As the first step of validation, the deflection of an isotropic circular plate with the following properties and clamped edges is compared with results from Refs. [21, 45–47], as shown in Table 2:
(47)
Table 2.
Comparison of the non-dimensional deflection of an isotropic circular plate with clamped edges.
q* |
| ||||
[21] | [45] | [46] | [47] | Present study | |
0.0001 | 0.1685 | 0.1678 | 0.1687 | 0.1706 | 0.1774 |
0.0003 | 0.4642 | 0.4583 | 0.4655 | 0.5119 | 0.5322 |
0.001 | 1.0557 | 1.0509 | 1.0937 | 1.7069 | 1.7742 |
Good agreement is observed between the results, particularly with Ref. [47]. Additionally, the results for an isotropic annular plate with both clamped and simply supported edges are compared with those from Refs. [21, 48, 49] under the following conditions, as shown in Table 3:
(48)
Table 3.
Comparison of the non-dimensional deflection of an isotropic annular plate with clamped and simply supported edges.
Study | w* | |
Clamped | Simply supported | |
[21] | 2.810 | 10.633 |
[48] | 2.781 | 10.623 |
[49] | 2.774 | 10.572 |
Present study | 3.015 | 11.406 |
good agreement is observed.
To verify the results for orthotropic material behavior, and since deflection data for all boundary conditions considered in this study are not reported in the literature for an orthotropic annular/circular plate, the results are compared with those from a finite element model produced using ABAQUS software. The comparison, shown in Table 4, is based on the mechanical properties reported in Eq. (53), and the results demonstrate good agreement.
Table 4.
Comparison of the deflections for an orthotropic annular/circular plate with ABAQUS model for all boundary conditions.
Boundary condition type | w* | |
ABAQUS model | Present study | |
C - C | 0.00370 | 0.00330 |
S - S | 0.01048 | 0.00980 |
C - S | 0.00686 | 0.00589 |
S - C | 0.00470 | 0.00457 |
C - F | 0.04500 | 0.04500 |
F - C | 0.03392 | 0.03198 |
S - F | 0.14100 | 0.15870 |
F - S | 0.18302 | 0.17950 |
To validate the deflection of an orthotropic annular/circular plate with elastic foundations, the results for the following properties are compared with those from Ref. [21], as shown in Table 5:
(49)
Table 5.
Comparison of the dimensionless deflections with the literatures.
Boundary condition type | w* | ||
Ref. [21] | Present study (3D Elasticity) | ||
CPT | FSDT | ||
C – C | 0.0030 | 0.0034 | 0.0031 |
S – S | - | 0.0085 | 0.0085 |
C – S | - | 0.0065 | 0.0053 |
F – C | - | 0.026 | 0.0263 |
It is observed that the results show good agreement.
To verify the accuracy of the employed re-modified couple stress theory and the effect of scale parameters, the nonlinear bending analysis of both circular and annular micro/nanoplates under the following conditions is compared with the results reported by Reddy et al. [41], as shown in Table 6:
(50)
Table 6.
Comparison of the deflections for circular and annular micro/nanoplates with the literatures.
| w* | |||||||||
Study | Circular plate ( | Annular plate ( | ||||||||
F - C | F - S | F - C | C - F | F - S | ||||||
q = 100 Pa | q = 100 Pa | q = 100 Pa | q = 50 Pa | q = 50 Pa | ||||||
|
|
|
|
|
|
|
|
|
| |
[41] CPT | 0.01757 | 0.0071 | 0.050 | 0.048 | 0.01624 | 0.0055 | 0.02358 | 0.01663 | 0.04273 | 0.02963 |
[41] FSDT | 0.01829 | 0.0072 | 0.050 | 0.048 | 0.01685 | 0.0055 | 0.02358 | 0.01663 | - | - |
Present study | 0.01822 | 0.0076 | 0.072 | 0.052 | 0.01761 | 0.0053 | 0.01860 | 0.01381 | 0.04407 | 0.02557 |
In all three parts of the validation, the results fall within an accurate range. Additionally, it is observed that, in most cases, the 3D elasticity theory estimates higher deflections compared to other theories such as CPT, FSDT, and TSDT.
To investigate the influence of MLSP, its effect was examined using different combinations of MLSP values for a plate with conditions based on Eq. (46). The results are reported in Table 7.It is observed that for CC, SS, CS, and SC boundaries, the use of MLSP decreases the deflection, indicating that the plates become stiffer. This observation aligns with the results reported by Reddy et al. [41]. The minimum deflection is occurs in
and
.with one exception: when 
and
,the deflection of the plates increases, indicating that the plates become more flexible. This behavior was not reported by [41], which could be due to the use of the isotropic form of modified couple stress theory in Ref. [41].
Table 7.
The influence of MLSP on the non-dimensional deflection of an orthotropic annular/circular micro/nanoplate for different B.C.s.
| w* | ||||||||
B.C. |
|
|
| ||||||
|
|
|
|
|
|
|
|
| |
C – C | 0.00325 | 0.00312 | 0.00278 | 0.00325 | 0.00313 | 0.00278 | 0.00327 | 0.00314 | 0.00279 |
S – S | 0.00983 | 0.00959 | 0.00890 | 0.00988 | 0.00964 | 0.00895 | 0.01006 | 0.00982 | 0.00909 |
C – S | 0.00584 | 0.00553 | 0.00474 | 0.00585 | 0.00554 | 0.00475 | 0.00589 | 0.00558 | 0.00478 |
S – C | 0.00455 | 0.00441 | 0.00403 | 0.00457 | 0.00443 | 0.00404 | 0.00463 | 0.00449 | 0.00410 |
C – F | 0.04008 | 0.0412 | 0.0413 | 0.0404 | 0.0415 | 0.0417 | 0.0415 | 0.0427 | 0.0430 |
F – C | 0.0324 | 0.0326 | 0.0362 | 0.0326 | 0.0328 | 0.0369 | 0.0331 | 0.0336 | 0.0395 |
S – F | 0.1844 | 0.2027 | 0.2953 | 0.1866 | 0.2056 | 0.3038 | 0.1933 | 0.2174 | 0.3324 |
F – S | 0.1847 | 0.1877 | 0.2299 | 0.1870 | 0.1909 | 0.2439 | 0.1944 | 0.2016 | 0.3045 |
Additionally, for FS, SF, CF, and FC boundaries, an increasing trend in deflection is observed, with the maximum occurring in and. The results indicate that the effect of MLSP also depends on the boundary conditions. Some literature [18-23] reports a reduction in deflection for these cases using nonlocal theories. Moreover, Reddy et al. [41] observed a reduction in deflection for FC, FS, and CF boundaries using modified couple stress theory.
It should be noted that the results obtained in the current study, as shown in Table 6, are consistent with those of [41] and indicate a reduction in plate deflection. This suggests that other parameters might influence the effect of MLSP. To further illustrate this point, additional examinations were conducted. A comparison between the results of the current study and the literature suggests that parameters such as the mechanical properties of the plates, aspect ratio, or thickness may affect the outcomes.
To investigate the influence of material behavior, the nonlinear bending of plates with the same dimensions as specified in Eq. (50) was examined for both the circular case (with ri=0) and the annular case (with ri=0.25 nm), using the mechanical properties defined in Eq. (46) (see Table 8).For plates with simply supported edges, both circular and annular, a reduction in deflection is observed. However, for clamped boundaries (C-F or F-C), an increase in deflection occurs. Thus, it can be concluded that the effect of MLSP depends on the material behavior.
Table 8.
The influence of MLSP on the non-dimensional deflection of circular and annular plates for various B.C.s.
Annular/Circular plate | B.C. | w* | |
|
| ||
Circular | C | 1.8061´10-8 | 2.3300´10-8 |
Circular | S | 8.2966´10-8 | 6.7321´10-8 |
Annular | F-C | 1.3939´10-8 | 6.7850´10-8 |
Annular | F-S | 9.6360´10-8 | 5.6600´10-7 |
Annular | C-F | 1.5044´10-8 | 4.0009´10-8 |
Next, to investigate the influence of aspect ratio and thickness, the nonlinear bending of plates was examined under the conditions specified in Eq. (46) for two different aspect ratios (thick and thin plates) (see Table 9). The results indicate that the bending behavior of thick and thin plates under the same conditions differs. Specifically, the decreasing effect of MLSP observed for thick plates with S-S, C-S, and S-C boundaries changes to an increasing effect as the plate becomes thinner.
Table 9.
The influence of aspect ratio on the non-dimensional deflection of annular/circular micro/nanoplate.
| w* | |||
B.C. |
|
| ||
|
|
|
| |
C - C | 0.003178 | 0.002727 | 21.474 | 20.084 |
S - S | 0.009951 | 0.009026 | 65.394 | 66.893 |
C - S | 0.005887 | 0.004791 | 54.294 | 56.701 |
S - C | 0.004519 | 0.004041 | 33.604 | 34.040 |
Upon reviewing the literature, it is noted that some studies based on couple stress theory [41, 52, 53], nonlocal theory [19, 50, 51], or nonlocal strain gradient theory [54] report a decreasing effect on plate or beam deflection with an increase in the MLSP in couple stress theory or the nonlocal parameter in nonlocal theory. Conversely, other studies [53-57] observe an increasing effect. Additionally, Ref. [53] found that considering the nonlocal size effect results in greater deflection for the nonlinear bending of a micro/nanobeam, while accounting for strain gradient size dependency leads to the opposite observation.
Next, the effects of elastic foundation and temperature on deflection and thickness are investigated. From Table 7,it is deduced that C-C, S-S, C-S and S-C boundaries experiences their minimum deflection at
,and their maximum at,. Additionally, C-F, F-C, S-F and F-S boundaries show their minimum deflection at,and their maximum deflection at,. Therefore, the influence of elastic foundations and thermal effects is examined for these specific cases for a plate with conditions specified in Eq. (46) with
and
,
, under thermal loading of
.Table 10 presents the effect of elastic foundations under mechanical loading, Table 11 shows the effect of a combination of thermo-mechanical loading, and Table 12 illustrates the influence of thermo-mechanical loading in the presence of elastic foundations.
Table 10.
The effect of elastic foundation on the non-dimensional deflection of annular/circular micro/nanoplate under mechanical loading.
B.C. |
|
|
|
| ||||
w* | Δh* | w* | Δh* | w* | Δh* | w* | Δh* | |
C – C | 0.00311 | 0.0011 | 0.00268 | 0.0011 | 0.00313 | 0.0011 | 0.00269 | 0.001 |
S – S | 0.00865 | 0.0012 | 0.00795 | 0.0012 | 0.00883 | 0.0012 | 0.0081 | 0.0012 |
C – S | 0.00536 | 0.0012 | 0.00443 | 0.0012 | 0.00541 | 0.0012 | 0.00446 | 0.0012 |
S – C | 0.00428 | 0.0011 | 0.00382 | 0.0011 | 0.00435 | 0.0011 | 0.00388 | 0.0011 |
C – F | 0.03218 | 0.0011 | 0.0337 | 0.0016 | 0.03318 | 0.0011 | 0.03493 | 0.0016 |
F – C | 0.02696 | 0.0013 | 0.02964 | 0.0018 | 0.02752 | 0.0013 | 0.03199 | 0.0019 |
S – F | 0.1134 | 0.0015 | 0.1545 | 0.0018 | 0.1166 | 0.0015 | 0.1646 | 0.0018 |
F – S | 0.08852 | 0.0018 | 0.09803 | 0.0022 | 0.09105 | 0.0018 | 0.1109 | 0.0023 |
Table 11.
The effect of thermo-mechanical loading on the non-dimensional deflection of annular/circular micro/nanoplate.
B.C. |
|
|
|
| ||||
w* | Δh* | w* | Δh* | w* | Δh* | w* | Δh* | |
C – C | 0.00355 | 0.0038 | 0.00299 | 0.0038 | 0.00358 | 0.0038 | 0.0031 | 0.0038 |
S – S | 0.01361 | 0.0026 | 0.01187 | 0.0028 | 0.0140 | 0.0026 | 0.0122 | 0.0026 |
C – S | 0.00722 | 0.0026 | 0.00562 | 0.0026 | 0.00730 | 0.0026 | 0.0056 | 0.0026 |
S – C | 0.00509 | 0.0037 | 0.00446 | 0.0037 | 0.00520 | 0.0037 | 0.00454 | 0.0037 |
C – F | 0.0404 | 0.0038 | 0.0472 | 0.0038 | 0.0418 | 0.0025 | 0.0494 | 0.0038 |
F – C | 0.0398 | 0.0037 | 0.0478 | 0.0058 | 0.0409 | 0.0037 | 0.0544 | 0.0059 |
S – F | 0.1425 | 0.0025 | 0.2978 | 0.0031 | 0.1472 | 0.0025 | 0.3279 | 0.0031 |
Table 12.
The influence of thermo-mechanical loading on the non-dimensional deflection of annular/circular micro/nanoplate rested on elastic foundation.
B.C. |
|
|
|
| ||||
w* | Δh* | w* | Δh* | w* | Δh* | w* | Δh* | |
C – C | 0.0034 | 0.0039 | 0.00293 | 0.0039 | 0.00346 | 0.0039 | 0.0029 | 0.0039 |
S – S | 0.01149 | 0.0027 | 0.01030 | 0.0027 | 0.0118 | 0.0027 | 0.0105 | 0.0027 |
C – S | 0.00653 | 0.0027 | 0.00522 | 0.0027 | 0.0066 | 0.0027 | 0.00526 | 0.0027 |
S – C | 0.00482 | 0.0038 | 0.00428 | 0.0038 | 0.0049 | 0.0038 | 0.00435 | 0.0038 |
C – F | 0.0326 | 0.0026 | 0.0367 | 0.0038 | 0.0336 | 0.0026 | 0.03814 | 0.0038 |
F – C | 0.0322 | 0.0040 | 0.0373 | 0.0060 | 0.0330 | 0.0040 | 0.0414 | 0.0062 |
S – F | 0.1005 | 0.0027 | 0.01702 | 0.0036 | 0.1029 | 0.0027 | 0.1806 | 0.0036 |
F – S | 0.2184 | 0.0042 | 0.3079 | 0.0057 | 0.2313 | 0.0043 | 0.4722 | 0.0072 |
It is observed that employing an elastic foundation and increasing temperature do not cause significant changes in the effect of MLSP or the behavior of plate deflection. As expected, the use of an elastic foundation reduces the plate deflection, while an increase in temperature leads to greater deflection. The most significant increase in deflection due to thermal loading is observed under S-S boundary conditions, while the minimum increase occurs under C-F boundary conditions. Conversely, the most significant decrease in deflection due to the elastic foundation is observed under F-S and S-F boundary conditions, with the minimum decrease occurring under C-C boundary conditions.
The most significant thickness variation under mechanical loading and in the absence of an elastic foundation is observed for F-S and S-F conditions, where increasing MLSP has a notable effect on the plate thickness, especially for plates with one free edge. The inclusion of an elastic foundation does not show a considerable impact on plate thickness under mechanical loading. However, with thermo-mechanical loading, significant changes in plate thickness are observed, particularly in plates with no free edges. When an elastic foundation is added under thermo-mechanical loading, thickness variation becomes more pronounced for plates with one free edge, especially under the F-C boundary condition.
The effect of thickness variation on an annular plate with conditions of ,, and different boundary conditions, without an elastic foundation, is investigated and presented in Fig. 3. The plate properties are based on Eq. (46). It is observed that as the thickness increases, the deflection decreases, with the most noticeable reduction occurring in the S-S case. Additionally, it is noted that the effect of boundary conditions diminishes with increasing thickness.
The influence of temperature variation on deflection and thickness for the conditions specified in Eq. (46) and ,, with different boundary conditions and in the absence of an elastic foundation, is examined. The results are reported in Fig. 4 and Fig. 5, respectively. It is observed that increasing the temperature leads to increased deflection, with the most significant effect occurring in the S-S boundary condition which can be due to the most available degree of freedom in S condition. Additionally, temperature increases result in linear variation of thickness, which is more pronounced in C-C and S-C (plates with an outer clamped edge) compared to S-S and C-S boundaries which this linear variations shows direct effect between temperature and thickness variation.
As the final step, the influence of elastic foundation parameters on deflection is investigated. The variation of plate deflection under the conditions specified in Eq. (46) versus changes in elastic foundation parameters for different boundaries and ,is reported in Fig. 6 and Fig. 7. The results indicate that the effect of the Winkler parameter on decreasing deflection is much more pronounced compared to the Pasternak parameter. Nonlinear behavior is observed with the Winkler foundation, while linear behavior is noted with increasing stiffness coefficients of the Pasternak elastic foundation.
Fig. 3
The variation of non-dimensional deflection versus thickness of annular micro/nanoplate.
Fig. 4
The variation of non-dimensional deflection versus temperature variation of annular micro/nanoplate.
Fig. 5
The variation of non-dimensional thickness versus temperature variation of annular micro/nanoplate.
Fig. 6
The variation of non-dimensional deflection versus Winkler elastic foundation of annular micro/nanoplate.
Fig. 7
The variation of non-dimensional deflection versus Pasternak elastic foundation of annular micro/nanoplate.
6. CONCLUSION
In this study, a new numerical solution approach based on 3D elasticity theory and the SAPM is presented for the nonlinear thermo-mechanical bending analysis of orthotropic annular/circular micro/nanoplates resting on an elastic foundation, using re-modified couple stress theory. The effects of various parameters, including boundary conditions, couple stress scale parameters, aspect ratio, thickness, thermal stresses, and elastic foundation parameters, are investigated. Some of the key results are summarized below:
- The results obtained using 3D elasticity theory are higher compared to those obtained with CPT, FSDT, and TSDT under the same conditions.
- The influence of MLSP for isotropic materials with a single MLSP may differ from the effect of various combinations of MLSPs for anisotropic materials with three MLSPs.
- The influence of MLSPs may vary depending on the boundary conditions and different mechanical properties.
- Changing the plate aspect ratios, especially by making the plate thicker or thinner, can alter the effect of MLSPs.
- The influence of MLSP does not show significant changes when an elastic foundation is added or thermal loading is applied.
- The extent of the decreasing effect of the elastic foundation and the increasing effect of thermal loading on deflection depends on the boundary conditions.
- The most significant variation in plate thickness occurs in plates with one free edge, with thickness increasing under thermal loading.
- The effect of thermal loading on plate thickness is more pronounced in plates without free edges.
-Increasing the thickness decreases deflection, with the effect being more pronounced for the S-S boundary condition.
- The increasing effect of temperature on deflection is more significant for S-S boundaries.
- The influence of temperature on thickness variation increases linearly and is more pronounced in plates with an outer clamped edge.
- The decreasing effect of the Winkler parameter on deflection is more pronounced than that of the Pasternak parameter.
FUNDING STATEMENT
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
CONFLICT OF INTEREST
The author declares that there is no conflict of interest regarding the publication of this article.
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[1] Corresponding author. Tel. / Fax: +98 5136625046.
E-mail address: jabbarzadeh@mshdiau.ac.ir (M. Jabbarzadeh)