On the strain gradient effects on buckling of the partially covered laminated microbeam

doi:10.1016/j.apm.2021.10.002

Applied Mathematical Modelling

Volume 102, February 2022, Pages 472-491

https://doi.org/10.1016/j.apm.2021.10.002 Get rights and content

Highlights

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A size-dependent buckling response model of the laminated microbeam with a partially covered elastic layer is established.
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The relations among the classical/anti-symmetric/symmetric couple stress theory and the general theory are studied.
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Compared with the general theory, the reduced theories underestimate the strain gradient effects.
•
The size dependency of the buckling response with different boundary conditions is analyzed.
•
The effects of the position of laminated area and material length-scale parameters on the buckling load are discussed.

Abstract

The buckling responses of the micro-components show size dependency. The general strain gradient theory is applied to explain the size effects. In this paper, we derive the theoretical relations among the classical couple stress theory, the modified couple stress theory and the general theory. The general theory includes all strain gradients and can respectively reduce to the classical couple stress theory and the modified couple stress theory when certain strain gradients are ignored. Subsequently, we further compare the difference of the ability of the general theory and the reduced theories to capture the size-dependent buckling response of the beam. The buckling load of the beam predicted by the general theory is larger than that predicted by the reduced theories. The general theory can predict the size dependency more accurately. In addition, the variation law of buckling load with the location of the upper elastic layer is also clarified. The location variation of the upper elastic layer leads to the change of buckling load, and thus affects the ability of the beam to resist the deformation.

Introduction

The mechanical responses of the micro components show size dependency [1], [2], [3], [4], [5]. Classical theory does not include the strain gradients and fails to descripe the size effects. The strain gradient elasticity theory with the strain gradient effects is applied to capture the size effects.

Different versions of strain gradient elasticity theories have been established to solve the size-dependent problems [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. The development of the strain gradient theories was reviewed by Polizzotto [15]. Subsequently, these theories are used to study the size-dependent behaviors of monolayer or bilayer micro-beams/plates. For monolayer beam, Shaat [16] applied the couple stress theory [7] and the modified versions to solve the bending problem of the beam, and found the modified couple stress theories [8], [9] underestimate the size-dependent bending response. Zhang et al. [17] compared the effects of loading and boundary conditions on the bending of Euler–Bernoulli beams and Timoshenko beams. The size-dependent vibration response of microbeam was studied by Kong et al. [18]. Kong et al. [19] afterwards proposed the non-conventional model to investigate the dynamic and static behavior of microbeam. The nonlinear dynamic and static responses were subsequently explained [20]. Akgöz and Civalek [21] performed the buckling analysis. Afterwards, this solution was extended to buckling analysis of the carbon nanotubes under various boundary conditions [22], [23]. Zaera et al. [24] performed the bending and buckling analysis of the beam. Wang et al. [25] considered the microporosity defect effects when they analysed the vibration and bending behavior of microbeam. The influences of thermal effects on the vibration behavior of the microbeam were further analysed by Babaei et al. [26]. In addition, Fu et al. [27] further investigated the strain gradients and surface parameters on the dynamic and static responses of the microbeam.

For bilayer beam, Chen et al. [28] established model to predict the bending response of the bilayer beam. Li et al. [29] proposed the displacement variation modeling method for the laminated micro-components, and established the bilayer beam model. Subsequently, Sidhardh and Ray [30] applied the general theory [14] to analyse the effects of various material paramters on the bilayer beam. The vibration analysis of the microbeam was performed by Dehrouyeh-Semnani [31]. Ghadiri et al. [32] further considered thermal effects when they analyse the dynamic response of the bilayer Euler–Bernoulli beams and Timoshenko beams. Abadi and Daneshmehr [33] further performed the buckling analysis. Mohammadabadi et al. [34] considered the thermal effects when they analysed the buckling response. Fu et al. [35] further clarified the influences from the position of laminated region.

For monolayer plate, the bending problem of the microplate was solved by Tsiatas [36]. Afterwards, Ji et al. [37] applied the general strain gradient theory to derive the strain gradient solution for the bending of microplate. The size-dependent vibration response of microplate under various boundary conditions was studied by Ke et al. [38]. Thai and Vo [39] extended the solution to the vibration and bending behavior of the functional graded microplate. Ramezani [40] performed the nonlinear bending analysis of microplates with different boundary conditions. The nonlinear dynamic and static problems were subsequently solved [41], [42]. Malikan [43] performed the buckling analysis of the microplates. Subsequently, Mohammadi and Mahani [44] derived the strain gradient solution of the buckling load of microplates. The size-dependent bending deflection, dynamic vibration and buckling load of the microplates were further studied by Papargyri-Beskou and Beskos [45].

For bilayer plate, Li et al. [46] proposed the size-dependent bilayer microplate bending model with strain gradient effects, solved the high-order partial differential equation, derived the solution of the bending deflection, and discussed the effects of material length-scale parameter on the deflection. Vidal et al. [47] extended the solution to the microplate made of porous materials. The size dependency of free vibration problem of microplates with various boundary conditions was solved by Nematollahi and Mohammadi [48]. Thai et al. [49] derived the strain gradient solution to the deflection and frequency of the functionally graded microplate with different loading and boundary conditions. Ghayesh [50] subsequently solved the nonlinear dynamic problem of the bilayer microplate. The influences from the temperature effect and surface parameter on the vibration of microplates were further considered by Allahyari and Asgari [51]. Arefi et al. [52] compared the effects of different boundary conditions on the buckling response of bilayer plates. The influence from the temperature on the buckling bahaviour of functionally graded microplate was studied by Fallah and Khorshidi [53]. Shiva et al. [54] further considered the surface effects when they performed the buckling analysis of microplate. In addition, Wang et al. [55] reviewed the development of the theoretical modeling of beams and plates.

The research above is concentrated on the buckling response of monolayer/bilayer beams and plates, while little attention is paid to the buckling behavior of the partially covered laminated beam. Beam of this type is widely used as the structural components in the buckling-based micro-devices. The work status of the micro-devices is decided by the property of the beam. Wang [56], [57] applied the classical theory to describe the buckling response of the beam. Maleki and Mohammadi [58] further performed the buckling analysis of the cracked functionally graded beam. However, without the strain gradients, the classical theory underestimates the size dependency of buckling response obviously.

To describe the buckling behavior of the beam accurately, there is an urgency for application of the general theory to establish the size-dependent partially covered laminated beam model. In this paper, the general strain gradient theory [14] is applied to establish the size-dependent laminated beam buckling model. The buckling analysis of the laminated beam under various boundary conditions is performed. The influence of location and geometric parameters of the laminated region on the buckling load is discussed.

The paper is organized as follows. The equations of the general strain gradient theory were reviewed in Section 2. The boundary conditions and governing equation of the laminated beam were derived in Section 3. Subsequently, we perform the buckling analysis of the laminated beam. Finally, the conclusion is given in Section 4.

Section snippets

Strain gradient elasticity theory

Zhou et al. [14] established the general strain gradient theory with three independent material length-scale parameters, and give the strain energy density $w_{g e n}$ as follows $\begin{matrix} w_{g e n} = & \frac{1}{2} λ ε_{i i} ε_{j j} + μ ε_{i j} ε_{i j} + (\frac{9}{5} μ l_{0}^{2} - \frac{4}{15} μ l_{1}^{2} - μ l_{2}^{2}) η_{i i k} η_{j j k} \\ - (\frac{6}{5} μ l_{0}^{2} + \frac{4}{15} μ l_{1}^{2} - 2 μ l_{2}^{2}) η_{i i k} η_{k j j} + (\frac{6}{5} μ l_{0}^{2} - \frac{1}{15} μ l_{1}^{2} - μ l_{2}^{2}) η_{k i i} η_{k j j} \\ + (\frac{1}{3} μ l_{1}^{2} + 2 μ l_{2}^{2}) η_{i j k} η_{i j k} + (\frac{2}{3} μ l_{1}^{2} - 2 μ l_{2}^{2}) η_{k i j} η_{i j k} \end{matrix}$ $λ$ and $μ$ are the Lamé constants. The material length-scale parameters are denoted as $l_{i} (i = 0, 1, 2)$ . The strain tensor $ε_{i j}$ is defined as $ε_{i j} = \frac{1}{2} (u_{i, j} + u_{j, i})$ where $u_{i}$ is the

Solutions of the Buckling problem

Based on the Eq. (23), the non-laminated region ( $0 < x < L_{1}$ ) deflection is derived as $w_{g e n - l e f t} (x) = c_{1} + c_{2} x + c_{3} \sin (B x) + c_{4} \cos (B x) + c_{5} \sinh (D x) + c_{6} \cosh (D x)$ with $B = \sqrt{\frac{- s + \sqrt{s^{2} + 4 k P}}{2 k}} D = \sqrt{\frac{s + \sqrt{s^{2} + 4 k P}}{2 k}}$ Similarly, the non-laminated region ( $L_{2} < x < L$ ) deflection is derived as $w_{g e n - r i g h t} (x) = c_{7} + c_{8} x + c_{9} \sin (B x) + c_{10} \cos (B x) + c_{11} \sinh (D x) + c_{12} \cosh (D x)$ Based on Eqs. (28) and (29), the deflection and axial displacement of the laminated region ( $L_{1} < x < L_{2}$ ) are written as $\begin{matrix} w_{g e n - l a m i n a t e d} (x) = c_{13} + c_{14} x + c_{15} e^{r_{1} x} + c_{16} e^{r_{2} x} + c_{17} e^{r_{3} x} + c_{18} e^{r_{4} x} \\ + c_{19} e^{r_{5} x} + c_{20} e^{r_{6} x} \end{matrix}$ $\begin{matrix} u_{g e n} (x \end{matrix}$

Numerical results

The comparison of the buckling response of the beam is studied. The substrate elastic layer is silicon, $E_{s u b}$ = 130 Gpa. $ν_{s u b}$ = 0.35. The geometric parameters are $L = 25 h_{s u b}$ , $b_{s u b} = 2 h_{s u b}$ , $h_{s u b} = 1$ µm. The material parameters of upper elastic layer satisfy: $E_{u p p}$ = 0.5 $E_{s u b}$ . $ν_{u p p}$ =0.5 $ν_{s u b}$ . $b_{u p p} = b_{s u b}$ . The total thickness of the beam is $h = h_{u p p} + h_{s u b}$ . We define $T_{R}$ as the thickness ratio, $T_{R} = h_{u p p} / h_{s u b}$ . $L_{R}$ is defined as the length ratio, $L_{R} = (L_{2} - L_{1}) / L$ . In addition, the length parameters satisfy: $l_{i (s u b)} = l$ , $l_{i (u p p)}$

Conclusions

In this paper, the relations and differences among the anti-symmetric couple stress theory, the symmetric couple stress theory, the classical couple stress theory and the general theory are discussed. The anti-symmetric/symmetric couple stress theory and the classical couple stress theory are the reduced theory of the general theory. Then, the size dependency of buckling response of partially covered laminated microbeam is analysed. The expression of the buckling load of the beam is derived.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the Taishan Scholars Program of Shandong Province (tsqn201909108), the Open Fund of State Key Laboratory of Applied Optics (SKLAO2020001A16) and Shandong Provincial Key RESEARCH and Development Plan (public welfare) (2019GGX104033) and the Natural Science Fund of Shandong Province of China (ZR202102230795).

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On the strain gradient effects on buckling of the partially covered laminated microbeam

Highlights

Abstract

Introduction

Section snippets

Strain gradient elasticity theory

Solutions of the Buckling problem

Numerical results

Conclusions

Declaration of Competing Interest

Acknowledgments

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