Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory

In this study, non-linear free vibration of micro-plates based on strain gradient elasticity theory is investigated. A general form of Mindlin’s first-strain gradient elasticity theory is employed to obtain a general Kirchhoff micro-plate formulation. The von Karman strain tensor is used to capture the geometric non-linearity. The governing equations of motion and boundary conditions are obtained in a variational framework. The Homotopy analysis method is employed to obtain an accurate analytical expression for the non-linear natural frequency of vibration. For some specific values of the gradient-based material parameters, the general plate formulation can be reduced to those based on some special forms of strain gradient elasticity theory. Accordingly, three different micro-plate formulations are introduced, which are based on three special strain gradient elasticity theories. It is found that both geometric non-linearity and size effect increase the natural frequency of vibration. In a micro-plate having a thickness comparable with the material length scale parameter, the strain gradient effect on increasing the non-linear natural frequency is higher than that of the geometric non-linearity. By increasing the plate thickness, the strain gradient effect decreases or even diminishes. In this case, geometric non-linearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for micro-plates with some specific thickness to length scale parameter ratios, both geometric non-linearity and size effect have significant role on increasing the frequency of non-linear vibration.     

A nonlinear microbeam model based on strain gradient elasticity theory with surface energy

A microscale nonlinear Bernoulli–Euler beam model on the basis of strain gradient elasticity with surface energy is presented. The von Karman strain tensor is used to capture the effect of geometric nonlinearity. Governing equations of motion and boundary conditions are obtained using Hamilton’s principle. In particular, the developed beam model is applicable for the nonlinear vibration analysis of microbeams. By employing a global Galerkin procedure, the ordinary differential equation corresponding to the first mode of nonlinear vibration for a simply supported microbeam is obtained. Numerical investigations show that in a microbeam having a thickness comparable with its material length scale parameter, the strain gradient effect on increasing the beam natural frequency is higher than that of the geometric nonlinearity. By increasing the beam thickness, the strain gradient effect decreases or even diminishes. In this case, geometric nonlinearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for beams with some specific thickness-to-length parameter ratios, both geometric nonlinearity and size effect have significant role on increasing the frequency of nonlinear vibration.     

A NONLINEAR MICROBEAM MODEL BASED ON STRAIN GRADIENT ELASTICITY THEORY

A micro scale nonlinear beam model based on strain gradient elasticity is developed.Governing equations of motion and boundary conditions are obtained in a variational framework.As an example,the nonlinear vibration of microbeams is analyzed.In a beam having a thickness to length parameter ratio close to unity,the strain gradient effect on increasing the natural frequency is predominant.By increasing the beam thickness,this effect decreases and geometric nonlinearity plays the main role on increasing the natural frequency.For some specific ratios,both geometric nonlinearity and size effects have a significant role on increasing the natural frequency.     

A micro scale Timoshenko beam model based on strain gradient elasticity theory

A micro scale Timoshenko beam model is developed based on strain gradient elasticity theory. Governing equations, initial conditions and boundary conditions are derived simultaneously by using Hamilton's principle. The new model incorporated with Poisson effect contains three material length scale parameters and can consequently capture the size effect. This model can degenerate into the modified couple stress Timoshenko beam model or even the classical Timoshenko beam model if two or all material length scale parameters are taken to be zero respectively. In addition, the newly developed model recovers the micro scale Bernoulli–Euler beam model when shear deformation is ignored. To illustrate the new model, the static bending and free vibration problems of a simply supported micro scale Timoshenko beam are solved respectively. Numerical results reveal that the differences in the deflection, rotation and natural frequency predicted by the present model and the other two reduced Timoshenko models are large as the beam thickness is comparable to the material length scale parameter. These differences, however, are decreasing or even diminishing with the increase of the beam thickness. In addition, Poisson effect on the beam deflection, rotation and natural frequency possesses an interesting “extreme point” phenomenon, which is quite different from that predicted by the classical Timoshenko beam model.     

Non-local free and forced vibrations of graded nanobeams resting on a non-linear elastic foundation

We consider in this paper the free and forced vibration response of simply-supported functionally graded (FG) nanobeams resting on a non-linear elastic foundation. The two-constituent Functionally Graded Material (FGM) is assumed to follow a power-law distribution through the beam thickness. Eringen׳s non-local elasticity model with material length scales is used in conjunction with the Euler–Bernoulli beam theory with von Kármán geometric non-linearity that accounts for moderate rotations. Non-linear natural frequencies of non-local FG nanobeams are obtained using He׳s Variational Iteration Method (VIM) and the direct and discretized Method of Multiple Scales (MMS), while the primary resonance analysis of an externally forced non-local FG nanobeam is performed only using the MMS. The effects of the non-local parameter, power-law index, and the parameters of the non-linear elastic foundation on the non-linear frequency-response are investigated.     

A size-dependent Reddy–Levinson beam model based on a strain gradient elasticity theory

A size-dependent Reddy–Levinson beam model is developed based on a strain gradient elasticity theory. Governing equations and boundary conditions are derived by using Hamilton’s principle. The model contains three material length scale parameters, which may effectively capture the size effect in micron or sub-micron. This model can degenerate into the modified couple stress model or even the classical model if two or all material length scale parameters are taken to be zero respectively. In addition, the present model recovers the micro scale Timoshenko and Bernoulli–Euler beam models based on the same strain gradient elasticity theory. To illustrate the new model, the static bending and free vibration problems of a simply supported micro scale Reddy–Levinson beam are solved respectively; the results are compared with the reduced models. Numerical results reveal that the differences in the deflection, rotation and natural frequency predicted by the present model and the other two reduced Reddy–Levinson models are getting larger as the beam thickness is comparable to the material length scale parameters. These differences, however, are decreasing or even diminishing with the increase of the beam thickness. This study may be helpful to characterize the mechanical properties of small scale beam-like structures for a wide range of potential applications.     

Size-dependent vibration of functionally graded curved microbeams based on the modified strain gradient elasticity theory

On the basis of the modified strain gradient elasticity theory, the free vibration characteristics of curved microbeams made of functionally graded materials (FGMs) whose material properties vary in the thickness direction are investigated. A size-dependent first-order shear deformation beam model is developed containing three internal material length scale parameters to incorporate small-scale effect. Through Hamilton’s principle, the higher-order governing equations of motion and boundary conditions are derived. Natural frequencies of FGM curved microbeams corresponding to different mode numbers are evaluated for over a wide range of material property gradient index, dimensionless length scale parameter and aspect ratio. Moreover, the results obtained via the present non-classical first-order shear deformation beam model are compared with those of degenerated beam models based on the modified couple stress and the classical theories. It is found that the difference between the natural frequencies predicted by the various beam models is more significant for lower values of dimensionless length scale parameter and higher values of mode number.     

Initial thermo-mechanical post-buckling of beams with temperature-dependent physical properties

The objective of this work is to analyze the elastic buckling and initial post-buckling behavior of slender beams subjected to uniform heating. The beams are assumed to be double-hinged with fixed ends, preventing thermal expansion. Consequently, destabilizing compressive forces arise that may lead to beam buckling. When the temperature is further increased, the beam experiences finite displacements, with the result that the analysis is geometrically non-linear. The modulus of elasticity and the thermal induced strain, key material properties for this problem, are temperature-dependent. Thus, the coefficients of the governing equations are not constant. This suggests the physical non-linearity of the mathematical model. Hence, the analysis is geometrically and physically non-linear. The analysis is sensitive to the beam initial temperature, as the thermal strain is a function of the initial and final temperatures. The material is considered to be linear elastic, and consequently viscoelastic and plastic effects are not taken into account. Furthermore, the beam cross-section properties are assumed to be constant, which is consistent with the small strain formulation. A perturbation method is applied to the governing non-linear differential equations so that the initial post-buckling behavior may be analytically determined when temperatures above the critical temperature are applied to the beam. To illustrate the application of the formulation we present a case study for the aluminum 7075-T6 alloy, a material commonly used in aerospace and naval industries. Nonetheless, it is expected similar behavior for other metallic materials. The curves that define the variation of the modulus of elasticity, the thermal strain and the yield stress with temperature are considered in our analysis. The change in length, reaction forces at the supports and geometric configurations are obtained as a function of temperature and the beam slenderness ratio. The critical buckling loads and temperatures and the initial post-buckling analysis are also calculated in the context of the temperature-independent physical properties. Our results emphasize the importance of modeling the material's non-linearity if accuracy is required. However, from a practical application point of view results are acceptable if temperature-independent physical properties are employed, especially for large slenderness ratios.     

Vibration analysis of piezoelectric sandwich nanobeam with flexoelectricity based on nonlocal strain gradient theory

A nonlocal strain gradient theory(NSGT) accounts for not only the nongradient nonlocal elastic stress but also the nonlocality of higher-order strain gradients,which makes it benefit from both hardening and softening effects in small-scale structures.In this study, based on the NSGT, an analytical model for the vibration behavior of a piezoelectric sandwich nanobeam is developed with consideration of flexoelectricity. The sandwich nanobeam consists of two piezoelectric sheets and a non-piezoelectric core. The governing equation of vibration of the sandwich beam is obtained by the Hamiltonian principle. The natural vibration frequency of the nanobeam is calculated for the simply supported(SS) boundary, the clamped-clamped(CC) boundary, the clamped-free(CF)boundary, and the clamped-simply supported(CS) boundary. The effects of geometric dimensions, length scale parameters, nonlocal parameters, piezoelectric constants, as well as the flexoelectric constants are discussed. The results demonstrate that both the flexoelectric and piezoelectric constants enhance the vibration frequency of the nanobeam.The nonlocal stress decreases the natural vibration frequency, while the strain gradient increases the natural vibration frequency. The natural vibration frequency based on the NSGT can be increased or decreased, depending on the value of the nonlocal parameter to length scale parameter ratio.     

A size-dependent Kirchhoff micro-plate model based on strain gradient elasticity theory

A size-dependent Kirchhoff micro-plate model is developed based on the strain gradient elasticity theory. The model contains three material length scale parameters, which may effectively capture the size effect. The model can also degenerate into the modified couple stress plate model or the classical plate model, if two or all of the material length scale parameters are taken to be zero. The static bending, instability and free vibration problems of a rectangular micro-plate with all edges simple supported are carried out to illustrate the applicability of the present size-dependent model. The results are compared with the reduced models. The present model can predict prominent size-dependent normalized stiffness, buckling load, and natural frequency with the reduction of structural size, especially when the plate thickness is on the same order of the material length scale parameter.     

静电激励MEMS微结构吸合电压尺寸效应研究

基于应变梯度弹性理论,研究了静电激励MEMS微结构吸合电压的尺寸效应。利用最小势能原理分别推导出含尺寸效应的一维梁模型和二维板模型的高阶控制方程。采用广义微分求积法和拟弧长算法对控制方程进行了数值求解。结果表明,随着结构尺寸的降低,新模型所预测的归一化的吸合电压呈非线性增长,表现出尺寸效应(特别是当结构尺寸与内禀常数在同一数量级时尺寸效应更加强烈);而相应的经典理论模型并不能预测此尺寸效应。两种新模型可视为相应经典理论的推广。本文有助于研究MEMS微结构的特性并对微结构的设计有潜在的应用价值。     

Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem

A detailed variational formulation is provided for a simplified strain gradient elasticity theory by using the principle of minimum total potential energy. This leads to the simultaneous determination of the equilibrium equations and the complete boundary conditions of the theory for the first time. To supplement the stress-based formulation, the coordinate-invariant displacement form of the simplified strain gradient elasticity theory is also derived anew. In view of the lack of a consistent and complete formulation, derivation details are included for the tutorial purpose. It is shown that both the stress and displacement forms of the simplified strain gradient elasticity theory obtained reduce to their counterparts in classical elasticity when the strain gradient effect (a measure of the underlying material microstructure) is not considered. As a direct application of the newly obtained displacement form of the theory, the problem of a pressurized thick-walled cylinder is analytically solved. The solution contains a material length scale parameter and can account for microstructural effects, which is qualitatively different from Lamé’s solution in classical elasticity. In the absence of the strain gradient effect, this strain gradient elasticity solution reduces to Lamé’s solution. The numerical results reveal that microstructural effects can be large and Lamé’s solution may not be accurate for materials exhibiting significant microstructure dependence.     

Experiments and theory in strain gradient elasticity

Conventional strain-based mechanics theory does not account for contributions from strain gradients. Failure to include strain gradient contributions can lead to underestimates of stresses and size-dependent behaviors in small-scale structures. In this paper, a new set of higher-order metrics is developed to characterize strain gradient behaviors. This set enables the application of the higher-order equilibrium conditions to strain gradient elasticity theory and reduces the number of independent elastic length scale parameters from five to three. On the basis of this new strain gradient theory, a strain gradient elastic bending theory for plane-strain beams is developed. Solutions for cantilever bending with a moment and line force applied at the free end are constructed based on the new higher-order bending theory. In classical bending theory, the normalized bending rigidity is independent of the length and thickness of the beam. In the solutions developed from the higher-order bending theory, the normalized higher-order bending rigidity has a new dependence on the thickness of the beam and on a higher-order bending parameter, b_h. To determine the significance of the size dependence, we fabricated micron-sized beams and conducted bending tests using a nanoindenter. We found that the normalized beam rigidity exhibited an inverse squared dependence on the beam's thickness as predicted by the strain gradient elastic bending theory, and that the higher-order bending parameter, b_h, is on the micron-scale. Potential errors from the experiments, model and fabrication were estimated and determined to be small relative to the observed increase in beam's bending rigidity. The present results indicate that the elastic strain gradient effect is significant in elastic deformation of small-scale structures.     

Non-linear analysis of functionally graded plates in cylindrical bending under thermomechanical loadings based on a layerwise theory

A layerwise theory is used to analyze analytically displacements and stresses in functionally graded composite plates in cylindrical bending subjected to thermomechanical loadings. The plates are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity of the plate is assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. The non-linear strain–displacement relations in the von Kármán sense are used to study the effect of geometric non-linearity. The equilibrium equations are solved exactly and also by using a perturbation technique. Numerical results are presented to show the effect of the material distribution on the deflections and stresses.     

Non-linearities in rotation and thickness deformation in a new third-order thickness deformation theory for static and dynamic analysis of isotropic and laminated doubly curved shells

A geometrically non-linear theory is developed for shells of generic shape allowing for third-order thickness and shear deformation and rotary inertia by using eight parameters; geometric imperfections are also taken into account. The geometrically non-linear strain–displacement relationships are derived retaining full non-linear terms in all the 8 parameters, i.e. in-plane and transverse displacements, rotations of the normal and thickness deformation parameters; these relationships are presented in curvilinear coordinates, ready to be implemented in computer codes. Higher order terms in the transverse coordinate are retained in the derivation so that the theory is suitable also for thick laminated shells. Three-dimensional constitutive equations are used for linear elasticity. The theory is applied to circular cylindrical shells complete around the circumference and simply supported at both ends to study initially static finite deformation. Both radially distributed forces and displacement-dependent pressure are used as load and results for different shell theories are compared. Results show that a 6 parameter non-linear shell theory is quite accurate for isotropic shells. Finally, large-amplitude forced vibrations under harmonic excitation are investigated by using the new theory and results are compared to other available theories. The new theory with non-linearity in all the 8 parameters is the only one to predict correctly the thickness deformation; it works accurately for both static and dynamics loads.     

Nonlocal strain gradient beam model for nonlinear secondary resonance analysis of functionally graded porous micro/nano-beams under periodic hard excitations

Abstract

Functionally graded porous materials (FGPMs) have a wide range of applications as hollow members in biomedical and aeronautical engineering. In the FGPMs, the porosity is varied over the material volume because of the density change of pores. In the present work, an analytical treatment on the size-dependent nonlinear secondary resonance of FGPM micro/nano-beams subjected to periodic hard excitations is proposed in the simultaneous presence of the nonlocality and strain gradient size dependencies. Based upon the closed-cell Gaussian-random field scheme, the mechanical properties of the FGPM micro/nano-beams are extracted corresponding to the uniform and three different functionally graded patterns of the porosity dispersion. The nonlocal strain gradient theory of elasticity is applied to the classical beam theory to formulate a newly combined size-dependent beam model. Thereafter, an analytical solving methodology based on the multiple time-scales together with the Galerkin technique is adopted to achieve the nonlocal strain gradient frequency–response and amplitude–response curves associated with the subharmonic and superharmonic external excitations. For the subharmonic excitation, it is observed that the nonlocality causes to shift the junction point of the stable and unstable branches to the higher value of the detuning parameter. However, the strain gradient size dependency plays an opposite role. For the superharmonic one, it is illustrated that the nonlocal size effect makes an increment in the height of jump phenomenon and shifts the peak to higher value of the detuning parameter. However, the strain gradient small scale effect leads to decrease the height of the jump phenomenon and shifts the peak to lower value of the detuning parameter.     

Vibration and wave propagation analysis of twisted micro-beam using strain gradient theory

In this research, vibration and wave propagation analysis of a twisted microbeam on Pasternak foundation is investigated. The strain-displacement relations (kinematic equations) are calculated by the displacement fields of the twisted micro-beam. The strain gradient theory (SGT) is used to implement the size dependent effect at microscale. Finally, using an energy method and Hamilton’s principle, the governing equations of motion for the twisted micro-beam are derived. Natural frequencies and the wave propagation speed of the twisted micro-beam are calculated with an analytical method. Also, the natural frequency, the phase speed, the cut-off frequency, and the wave number of the twisted micro-beam are obtained by considering three material length scale parameters, the rate of twist angle, the thickness, the length of twisted micro-beam, and the elastic medium. The results of this work indicate that the phase speed in a twisted micro-beam increases with an increase in the rate of twist angle. Moreover, the wave number is inversely related with the thickness of micro-beam. Meanwhile, it is directly related to the wave propagation frequency. Increasing the rate of twist angle causes the increase in the natural frequency especially with higher thickness. The effect of the twist angle rate on the group velocity is observed at a lower wave propagation frequency.     

A nonlocal strain gradient shell model incorporating surface effects for vibration analysis of functionally graded cylindrical nanoshells

In this paper, a novel size-dependent functionally graded(FG) cylindrical shell model is developed based on the nonlocal strain gradient theory in conjunction with the Gurtin-Murdoch surface elasticity theory. The new model containing a nonlocal parameter, a material length scale parameter, and several surface elastic constants can capture three typical types of size effects simultaneously, which are the nonlocal stress effect, the strain gradient effect, and the surface energy effects. With the help of Hamilton's principle and first-order shear deformation theory, the non-classical governing equations and related boundary conditions are derived. By using the proposed model, the free vibration problem of FG cylindrical nanoshells with material properties varying continuously through the thickness according to a power-law distribution is analytically solved, and the closed-form solutions for natural frequencies under various boundary conditions are obtained. After verifying the reliability of the proposed model and analytical method by comparing the degenerated results with those available in the literature, the influences of nonlocal parameter, material length scale parameter, power-law index, radius-to-thickness ratio, length-to-radius ratio, and surface effects on the vibration characteristic of functionally graded cylindrical nanoshells are examined in detail.     

A robust hyper-elastic beam model under bi-axial normal-shear loadings

In this paper, a simple and robust constitutive model is proposed to simulate mechanical behaviors of hyper-elastic materials under bi-axial normal-shear loadings in the finite strain regime. The Mooney–Rivlin strain energy function is adopted to develop a two-dimensional (2D) normal-shear constitutive model within the framework of continuum mechanics. A motion field is first proposed for combined normal and shear deformations. The deformation gradient of the proposed field is calculated and then substituted into right Cauchy–Green deformation tensor. Constitutive equations are then derived for normal and shear deformations. They are two explicit coupled equations with high-level polynomial non-linearity. In order to examine capabilities of the developed hyper-elastic model, uniaxial tensile responses and non-linear stability behaviors of moderately thick straight and curved beams undergoing normal axial and transverse shear deformations are simulated and compared with experiments. Fused deposition modeling technique as a 3D printing technology is implemented to fabricate hyper-elastic beam structures from soft poly-lactic acid filaments. The printed specimens are tested under tensile/compressive in-plane and compressive out-of-plane forces. A finite element formulation along with the Newton–Raphson and Riks techniques is also developed to trace non-linear equilibrium path of beam structures in large defamation regimes. It is shown that the model is capable of predicting non-linear equilibrium characteristics of hyper-elastic straight and curved beams. It is found that the modeling of shear deformation and finite strain is essential toward an accurate prediction of the non-linear equilibrium responses of moderately thick hyper-elastic beams. Due to simplicity and accuracy, the model can serve in the future studies dealing with the analysis of hyper-elastic structures in which two normal and shear stress components are dominant.     

Special virtual fields for the direct determination of material parameters with the virtual fields method. 2––Application to in-plane properties

This paper deals with the direct identification of mechanical parameters that govern the in-plane constitutive law of orthotropic materials. Those parameters are extracted from heterogeneous strain fields that occur in a short beam specimen tested in a Iosipescu fixture. The procedure used is the virtual fields method with special virtual fields. The case of linear elasticity is first addressed. It is shown that the parameters are directly extracted with this method: no iterative calculations are required. The stability is also discussed in different cases. A non-linear shear response is then considered. The parameter that governs this non-linearity is also directly identified with the special virtual fields.