Modified Strain Gradient Elasticity Research Articles

On the roles of strain gradient tensors in quasi‐3D nonlinear dynamics of microplate‐type piezoelectric systems of energy harvesting under triangular mechanical actuation

AbstractThe small‐scale‐dependent nonlinear dynamical attributes of microplate‐type laminated piezoelectric systems of harvesting energy subjected to a triangular mechanical actuation are explored in the current exploration. The core of harvesting systems is supposed to be made of nanocomposites reinforced by randomly oriented carbon nanotubes (CNTs) having various degrees of agglomeration coated by piezoelectric layers. In this regard, the microplate‐type systems of harvesting energy are modeled via a quasi‐3D plate theory combined with the modified strain gradient elasticity in the presence of various microsize‐dependent strain gradient tensors. Thereafter, the solution of the developed coupled electromechanical size‐dependent nonlinear problem is obtained numerically by utilizing the meshless collocation approach employing incorporation of the polynomial and radial basis functions to remove any feasible singularity for the associated moment matrices. It comes to the conclusion that by elevating the quantity of CNTs disclosed within the clusters, the significance of the microsize dependency in the achieved voltage enhances from 22.06% to 22.73% toward the simply supported bridge‐type context, and from 33.00% to 33.92% toward the clamped bridge‐type context.Highlights Development of a microstructural‐dependent quasi‐3D shear flexible energy harvester model. Incorporating the roles of various strain gradient tensors in the nonlinear dynamics of microharvesters. Size‐dependent time‐histories of achieved voltage from triangular mechanical‐loaded microharvesters. Influence of the CNT reinforcement clustering on the nonlinear dynamic performance.

Isogeometric analysis of small-scale effects on the vibration of functionally graded porous curved microbeams based on the modified strain gradient elasticity theory

Isogeometric analysis of small-scale effects on the vibration of functionally graded porous curved microbeams based on the modified strain gradient elasticity theory

A simple size-dependent isogeometric approach for bending analysis of functionally graded microplates using the modified strain gradient elasticity theory

In this study, a simple size-dependent isogeometric approach for bending analysis of functionally graded (FG) microplates using the modified strain gradient theory (MSGT), simple first-order shear deformation theory (sFSDT) and isogeometric analysis is presented for the first time. The present approach reduces one variable when comparing with the original first-order shear deformation theory (FSDT) within five variables and only considers three material length scale parameters (MLSPs) to capture size effects. Effective material properties as Young’s modulus, Poisson’s ratio and density mass are computed by a rule of mixture. Thanks to the principle of virtual work, the essential equations which are solved by the isogeometric analysis method, are derived. Rectangular and circular FG microplates with different boundary conditions, volume fraction and material length scale parameter are exampled to evaluate the deflections of FG microplates.

Hearing distributed mass in nanobeam resonators

One-dimensional vibrating nanostructures show remarkable performance in detecting small adherent masses added to a referential configuration. The mass sensing principle is based on measuring the resonant frequency shifts caused by the unknown attached masses. In spite of its important application in several fields, few studies have been devoted to this inverse eigenvalue problem. In this paper we have developed a distributed mass reconstruction method for initially uniform nanobeams based on measurements of the first lower resonant frequencies of the free bending vibration. Two main inverse problems are addressed. In the first problem, the mass variation is determined by using the first lower eigenfrequencies of a supported nanobeam, under the a priori assumption that the mass variation has support contained in half of the axis interval. In the second problem, we show that the a priori assumption can be removed, provided that the spectral input data include an additional set of first lower eigenfrequencies belonging to a second spectrum associated to different end conditions. The nanobeam is modelled using the modified strain gradient elasticity accounting for size effects. The reconstruction is based on an iterative procedure which takes advantage of a closed-form solution when the mass change is small, and shows to be convergent under this assumption and for smooth mass variation. The accuracy of the reconstruction deteriorates in presence of discontinuous mass variation. For these cases, a constrained least-squares optimization filtering shows to be very effective to reduce the spurious oscillations around the target coefficient. Numerical simulations show that the identification method performs well even for not necessarily small mass changes and it is stable in presence of errors on the data. An experimental validation of the method has provided encouraging results, despite the fact that only the first four eigenfrequencies under cantilever end conditions were used.

Vibration of functionally graded Mindlin plate based on a modified strain gradient elasticity theory

Vibrational behavior of functionally graded (FG) microplates is investigated by a new modified strain gradient Mindlin plate (MSGMP) model. With the help of Hamilton’s principle, the dynamic equation is easily obtained. Furthermore, the general forms of boundary conditions are gotten by using coordinate transformation. The MSGMP model can be degenerated to a couple stress elastic Mindlin plate model or the classical Mindlin plate (CMP) model. Analytical solutions of vibrational problem of a rectangular microplate with four simply supported edges are gotten. Numerical results reveal significant effects of the dimensionless nonlocal parameters, the power law index and vibration mode on the free vibration behavior of FG plate.

Monitoring mass changes using nanoresonator sensors

Nanoresonators consisting in a one-dimensional vibrating structure have remarkable performance in detecting small adherent masses. The mass sensing principle is based on monitoring the resonant frequency shifts caused by unknown attached masses. In spite of its important application, few studies are available on this inverse problem. In this work, we have developed a distributed mass reconstruction method in an initially uniform nanobeam under bending vibration, by using finite eigenfrequency data belonging to one spectrum corresponding to supported end conditions. To avoid trivial non-uniqueness due to the symmetry of the initial configuration of the nanobeam, it is assumed that the mass variation has support contained in half of the axis interval. The nanobeam is modelled using the modified strain gradient elasticity accounting for size effects. The reconstruction is based on an iterative procedure, which takes advantage of a closed-form solution when the mass change is small, and shows to be convergent under this assumption. The identification method performs well even for not necessarily small mass changes, and in presence of errors on the data.

Size-dependent analysis of FG-CNTRC microplates based on modified strain gradient elasticity theory

We present in this study a size-dependent computational approach based on the modified strain gradient theory (MSGT) and higher-order shear deformation theory for static bending and free vibration analyses of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) microplates. Three material length scale parameters (MLSPs) are taken into account in MSGT to capture size effects of microplate behavior. The effective material properties of FG-CNTRC microplates are obtained by an extended rule of mixture. Four types of carbon nanotube distributions, which are either uniform or functionally graded (FG) through the plate thickness, are considered. The governing equations are derived from the principle of virtual work and are then solved by isogeometric analysis (IGA). The IGA is suitable for a numerical implementation of the size-dependent models since it requires higher-order gradients in the weak form. The inclusion of geometrical parameters, boundary conditions, distributed types of carbon nanotube and material length scale parameters are studied to evaluate the displacement and natural frequency of FG-CNTRC microplates. In addition, the present size-dependent model can be retrieved into the modified couple stress model or classical model when a few MLSPs are ignored.

Dynamic analysis of microbeams based on modified strain gradient theory using differential quadrature method

ABSTRACTDynamic analysis of microbeams based on the modified strain gradient elasticity theory (MSGT) is carried out in this study. MSGT theory comprises additional material length scale parameters to effectively capture the size effect. Beams with fixed-fixed, simply supported and fixed-free boundary conditions are analysed. Additionally, frequency analysis for beams based on modified coupled stress and continuum theory is also presented by neglecting one or more length scale parameters. Results obtained for various theories in the present analysis are compared with those available in literature. Differential quadrature method (DQM) is employed to perform the analysis. Two different techniques are presented for implementing different boundary conditions of beam. It is shown that frequencies obtained from the strain gradient theory are higher when compared to the frequencies predicted by modified coupled stress theory and classical theory, when beam thickness becomes comparable to the length scale paramet...

Design of pressure-sensing diaphragm for MEMS capacitance diaphragm gauge considering size effect

MEMS capacitance diaphragm gauge with a full range of (1∼1000) Pa is considered for its wide application prospect. The design of pressure-sensing diaphragm is the key to achieve balanced performance for this kind of gauges. The optimization process of the pressure-sensing diaphragm with island design of a capacitance diaphragm gauge based on MEMS technique has been reported in this work. For micro-components in micro scale range, mechanical properties are very different from that in the macro scale range, so the size effect should not be ignored. The modified strain gradient elasticity theory considering size effect has been applied to determine the bending rigidity of the pressure-sensing diaphragm, which is then used in the numerical model to calculate the deflection-pressure relation of the diaphragm. According to the deflection curves, capacitance variation can be determined by integrating over the radius of the diaphragm. At last, the design of the diaphragm has been optimized based on three parameters: sensitivity, linearity and ground capacitance. With this design, a full range of (1∼1000) Pa can be achieved, meanwhile, balanced sensitivity, resolution and linearity can be kept.

Isogeometric analysis of size-dependent isotropic and sandwich functionally graded microplates based on modified strain gradient elasticity theory

This paper presents a size-dependent four-unknown shear deformable model for static bending, free vibration and buckling analyses of isotropic and sandwich functionally graded (FG) microplates based on the modified strain gradient theory (MSGT). The MSGT requires three material length scale parameters instead of five parameters as well known in the original strain gradient theory. The material parameters of isotropic and sandwich FG microplates are directly derived from a rule of mixture. The governing equations are derived from the principle of virtual work. Since the present method contains the higher-order gradients in the weak form, NURBS-based isogeometric analysis is suitable for the solution procedure. The effects of geometrical parameters, boundary conditions, volume fraction and material length scale parameters are investigated through isotropic and sandwich FG microplate examples. Obtained results indicate that the consideration of strain gradients leads to a rise of the plate stiffness, and a reduction of displacement and an increase in natural frequency as well as critical buckling load of FG microplates are therefore remarked. Moreover, the present model can be degenerated into the modified couple stress model or classical model when a few material length-sale parameters are neglected.

Thermoelastic damping in microbeam resonators based on modified strain gradient elasticity and generalized thermoelasticity theories

The present study aimed to investigate the size-dependent quality factor of thermoelastic damping in a microbeam resonator based on modified strain gradient elasticity theory. The governing differential equation of motion is derived by using the principle of virtual displacements. The generalized thermoelasticity theory of the well-known Lord–Shulman model is also utilized to derive the equation of coupled thermoelasticity. The quality factor of thermoelastic damping in the microbeam is obtained based on both the complex frequency and the entropy generation methods. Further, by applying the generalized thermoelasticity theory and entropy generation method, explicit formulae for the quality factor of thermoelastic damping in the microbeam resonators are developed in the frameworks of both classical and modified couple stress elasticity theories. The effects of several parameters including size effect are investigated on thermoelastic damping in cantilever and clamped–clamped silicon microbeams as case studies. The results are validated by observing an excellent agreement regarding the comparison of the results of the present study and with previously published results.

Size-dependant behaviour of functionally graded microplates based on the modified strain gradient elasticity theory and isogeometric analysis

This paper presents a robust numerical model, which takes into account both size-dependent and shear deformation effects, for the bending, buckling and free vibration analyses of functionally graded microplates. The size-dependent effect is captured by using the modified strain gradient elasticity theory with three length scale parameters, whilst the shear deformation effect is accounted by using the third-order shear deformation theory. The rule of mixture is employed to describe the distributions of material phrases through the plate thickness. By using Hamilton’s principle, the governing equations are derived and then discretized by employing an Isogeometric Analysis (IGA) approach, where the Non-Uniform Rational B-Splines (NURBS) basis functions are adopted to meet the C2-continuity requirement. Physical mesh convergence and verification studies are performed to prove the accuracy and reliability of the present model. In addition, parametric studies are also carried out to investigate the size effect in conjunction with the influences of gradient index, shear deformation effect and boundary conditions on the responses of microplates.

Variational formulations and general boundary conditions for sixth-order boundary value problems of gradient-elastic Kirchhoff plates

Abstract Sixth-order boundary value problems for gradient-elastic Kirchhoff plate bending models are formulated in a variational form within an H 3 Sobolev space setting. Existence and uniqueness of the weak solutions are then established by proving the continuity and coercivity of the associated symmetric bilinear forms. Complete sets of boundary conditions, including both the essential and the natural conditions, are derived accordingly. In particular, the gradient-elastic Kirchhoff plate models feature two different types of clamped and simply supported boundary conditions in contrast to the classical Kirchhoff plate model. These new types of boundary conditions are given additional attributes singly and doubly ; referring to free and prescribed normal curvature, respectively. The formulations and results of the analyzed strain gradient plate models are compared to two other generalized Kirchhoff plate models which follow a modified strain gradient elasticity theory and a modified couple stress theory. It is clarified that the results are extendable to these model variants as well. Finally, the relationship of the natural boundary conditions to external loadings is analyzed.

Bending analysis of FG microbeams resting on Winkler elastic foundation via strain gradient elasticity

In this study, bending response of non-homogenous microbeams embedded in an elastic medium is investigated based on modified strain gradient elasticity theory in conjunctions with various beam theories. The governing differential equations and related boundary conditions are derived with the aid of minimum total potential energy principle. Elastic medium is modeled by Winkler foundation model. Bending problem of simply supported microbeams made of functionally graded materials is solved by Navier’s solution procedure. A detailed parametric study is performed to investigate the effects of the material length scale parameters-to-thickness ratio l/h, material property gradient index k, slenderness ratio L/h, Winkler modulus kw and shear correction factor on the bending behavior of embedded functionally graded microbeams.

Size Dependent Vibration Behavior of an AFM with Sidewall and Top-Surface Probes Based on the Strain Gradient Elasticity Theory

In this paper, the size-dependent vibration behavior of an atomic force microscope with assembled cantilever probe (ACP) is analyzed utilizing the modified strain gradient elasticity theory. The proposed ACP comprises a horizontal cantilever, a vertical extension and two tips located at the free ends of the cantilever and extension. Because the vertical extension is located between the clamped and free ends of the microcantilever, the cantilever is modeled as two beams. The results of the current model are compared to those evaluated by both modified couple stress and classical beam theories. The results indicate that the resonant frequency and sensitivity of the proposed ACP is strongly size-dependent especially when the contact stiffness is very low or it is very high. The results also declare that utilizing the strain gradient theory is essential in the analysis of the vibration behavior of the proposed ACP.

Size-Dependent Resonant Frequency and Flexural Sensitivity of Atomic Force Microscope Microcantilevers Based on the Modified Strain Gradient Theory

In the present study, the resonant frequency and flexural sensitivity of atomic force microscope (AFM) microcantilevers are predicted incorporating size effects. To this end, the modified strain gradient elasticity theory is applied to the classical Euler-Bernoulli beam theory to develop a non-classical beam model which has the capability to capture size-dependent behavior of microcantilevers. On the basis of Hamilton's principle, the size-dependent analytical expressions corresponding to the frequency response and sensitivity of AFM cantilevers are derived. It is observed that by increasing the contact stiffness, the resonant frequencies of AFM cantilevers firstly increase and then tend to remain constant at an especial value. Moreover, the resonant frequencies of AFM cantilevers obtained via the developed non-classical model is higher than those of the classical beam theory, especially for the values of beam thickness close to the internal material length scale parameter.

A discussion on different non-classical constitutive models of microbeam

In this study, static bending of thin plane-strain microbeam is investigated based on modified couple stress and modified strain gradient elasticity constitutive beam models. It is indicated that the results are obtained based on modified couple stress constitutive beam model are in excellent agreement with those observed experimentally, like modified strain gradient elasticity one. Comparison between the results of the constitutive beam models for static bending behavior with common boundary conditions (i.e. clamped-free, clamped–clamped, clamped–pined and pined–pined) reveals that these beam models are in very good agreement. In addition, the constitutive beam models are compared with modified couple stress and modified strain gradient elasticity Euler–Bernoulli beam models.

Bending, buckling and free vibration analysis of size-dependent functionally graded circular/annular microplates based on the modified strain gradient elasticity theory

Abstract A Mindlin microplate model based on the modified strain gradient elasticity theory is developed to predict axisymmetric bending, buckling, and free vibration characteristics of circular/annular microplates made of functionally graded materials (FGMs). The material properties of functionally graded (FG) microplates are assumed to vary in the thickness direction. In the present non-classical plate model, the size effects are captured through using three higher-order material constants. By using Hamilton's principle, the higher-order equations of motion and related boundary conditions are derived. Afterward, the generalized differential quadrature (GDQ) method is employed to discretize the governing differential equations along with various types of edge supports. Selected numerical results are given to indicate the influences of dimensionless length scale parameter, material index and radius-to-thickness ratio on the deflection, critical buckling load and natural frequency of FG circular/annular microplates.

Size-dependent axial buckling analysis of functionally graded circular cylindrical microshells based on the modified strain gradient elasticity theory

In this paper, a size-dependent first-order shear deformable shell model is developed based upon the modified strain gradient theory (MSGT) for the axial buckling analysis of functionally graded (FG) circular cylindrical microshells. It is assumed that the material properties of FG materials, which obey a simple power-law distribution, vary through the thickness direction. The principle of virtual work is utilized to formulate the governing equations and corresponding boundary conditions. Numerical results are presented for the axial buckling of FG circular cylindrical microshells subject to simply-supported end conditions and the effects of material length scale parameter, material property gradient index, length-to-radius ratio and circumferential mode number on the size-dependent critical buckling load are extensively studied. For comparison purpose, the critical buckling loads predicted by modified couple stress theory (MCST) and classical theory (CT) are also presented. Results show that the size effect plays an important role for lower values of dimensionless length scale parameter. Moreover, it is observed that the critical buckling loads obtained based on MSGT are greater than those obtained based on MCST and CT.

Size-Dependent Thermal Buckling and Postbuckling of Functionally Graded Annular Microplates Based on the Modified Strain Gradient Theory

This article reports on the thermal instability of functionally graded (FG) annular microplates with different boundary conditions. The modified strain gradient elasticity theory is employed to capture size effects. The non-linear governing equations and boundary conditions are derived based on the first-order shear deformation theory (FSDT) and virtual displacements principle. The generalized differential quadrature technique is implemented so as to discretize. To obtain the critical buckling temperature, the set of linear discretized governing equations is solved as an eigenvalue problem. Also, the non-linear problem of thermal postbuckling is solved by the pseudo arc-length continuation method. The effects of boundary conditions, length scale parameter, and the variation of material through the thickness and geometrical properties on both critical buckling temperature and thermal postbuckling behavior are studied.