Modified Strain Gradient Theory Research Articles

The free vibration of the MEE sandwich microplate with the FG-CNTRC core using the MSGT

This study deals with the free vibration of the sandwich microplate with the core made of functionally graded carbon nanotube-reinforced composites (FG-CNTRC) and magneto-electro-elastic (MEE) face sheets. The governing equation of the microplates is derived by using the refined plate theory (RPT) with two variables and the modified strain gradient theory (MSGT). The Non-Uniform Rational B-Splines (NURBS) basis function of the isogeometric approach (IGA) is used for the approximation of the displacement and electric and magnetic fields of the microplates. The paper studies the effect of the length scale parameters (LSPs), CNTs distributions, CNTs volume fraction, magnetic and electric loads and geometry on the vibrational frequency of the MEE sandwich microplate.

A meshfree formulation for size-dependent thermal buckling and post-buckling behaviour of porous microplates on elastic foundation subjected to localized heating

The semi-analytical framework is developed for in-plane thermal stresses within the microplates under localized heating. Localized heating is modelled using Fourier Series expansions over the complete domain of the plate. These stresses within the plate are estimated after solving the in-plane thermoelasticity problem using Airy's stress approach. Utilizing these stresses, present work also studies the buckling and post-buckling behaviour of a porous metal foam microplate resting on Winkler and Pasternak elastic foundations. The plate is modelled using Reddy's third-order shear deformation theory (TSDT) and von Kármán geometric nonlinearity, and the size-dependent effects are included using the modified strain gradient theory (MSGT). Galerkin's weighted residual method converts the partial differential equations (PDEs) into algebraic equations. The post-buckling equilibrium path is obtained using the modified Newton-Raphson method. The mode shapes of the microplate for the various configurations of the plate with different boundary conditions due to localized thermal loading are plotted. Also, these mode shapes are compared with the mode shapes of the microplate due to in-plane mechanical loading. The parametric effect of porosity, elastic foundation parameters, thickness of plate, size of plate, boundary conditions and loading concentrations on the buckling and post-buckling behaviour of the plate is studied.

Computer Simulation for Determination of Critical Buckling Temperatures of Sandwich Microplates with Cellular Core and CNT-RC Piezoelectric Patches via MSGT

This study introduces a computational simulation which leads to obtaining the critical buckling temperatures of a sandwich microplate which is modeled based on the modified strain gradient theory. This theory includes three material length scale parameters. The core of the microstructure is made from cellular materials which is treated with piezoelectric carbon nanotubes-reinforced composite patches. An advanced trigonometric hyperbolic shear deformation theory, eliminating the need for shear correction factors, is employed to analyze the microstructure. Thickness-dependent variations in structural characteristics are observed based on predefined functions. The equilibrium equations are derived using the virtual displacement principle. Fourier series functions provide an analytical way of solving these equations. The findings are verified by comparison with earlier research that was published and used fewer complex combinations. The study looks at how several important factors affect the structure’s reaction to thermal buckling.

Three-phase-lag thermoelastic damping analysis of graphene-reinforced laminated composite microplate resonators based on modified strain gradient theory

Three-phase-lag thermoelastic damping analysis of graphene-reinforced laminated composite microplate resonators based on modified strain gradient theory

An intelligent computational iBCMO-DNN algorithm for stochastic thermal buckling analysis of functionally graded porous microplates using modified strain gradient theory

The authors propose an intelligent computational method using the deep feedforward neural network integrated with an improved balance composite motion optimization algorithm to formulate a so-called iBCMO-DNN for solving the stochastic thermal buckling problems of functionally graded porous microplates. In the present approach, the deterministic behaviors of the functionally graded porous microplates were firstly analyzed by the combination of a unified higher-order shear deformation theory, modified strain gradient theory, and Ritz-type series solutions, and then stochastic responses under uncertainty of material properties are obtained by the iBCMO-DNN algorithm. The deep neural network with the long short-term memory model is used as a surrogate method to replace the time-consuming computational model, while the improved balance composite motion optimization is used to search for the optimal solutions. The obtained numerical results for various boundary conditions, uncertainty parameters, and three types of temperature distribution indicated that the proposed method achieves its accuracy and effectiveness in predicting stochastic thermal buckling of the functionally graded porous microplates. The improved balance composite motion optimization algorithm demonstrates a computational time ∼1.7 times faster than its predecessor. Furthermore, integrating a deep neural network into the improved algorithm reduces computational time to about 2/5 compared to the method without it.

An enhanced strain gradient model for temperature-dependent sound transmission loss in double-walled functionally graded microplates

With the growing use of functionally graded (FG) microplates in structural acoustic metamaterials for aerospace and automotive applications, accurate modeling of sound transmission loss is critical for effective vibration and noise control engineering. In this study, a theoretical model is formulated based on the first-order shear deformation theory (FSDT) to estimate sound transmission loss (STL) through air-filled rectangular double-walled FG microplates with simply supported boundary conditions. The microplates are subjected to a nonlinear thermal environment, and the modified strain gradient theory (MSGT) is employed, incorporating three material length scale parameters to account for the size effect. The material properties are temperature-dependent and vary across the thickness following a power-law distribution. Size-dependent coupled vibroacoustic equations are derived utilizing the sound velocity potential, normal velocity continuity conditions, and Hamilton's principle, and are subsequently solved using the Galerkin method. Comparative analysis is performed by contrasting the results obtained from the MSGT model with those from the modified couple stress theory (MCST) and classical continuum theory (CCT) models, allowing for the assessment of the accuracy and precision of the proposed solution. Additionally, the influence of various parameters, such as gradient index, length scale parameters, temperature variation, incident angles, and acoustic cavity depth, on the STL is investigated. Key findings demonstrate that in the stiffness-controlled region, length scale parameters significantly enhance STL, while power-law index and temperature variation reduce it.

Strain gradient theory-based vibration analyses for functionally graded microbeams reinforced by GPL

On the basis of modified strain gradient theory (MSGT), the free vibration of graphene nanoplatelets (GPL) reinforced functionally graded (FG) microbeam on elastic foundation is studied. Firstly, material properties of the composites are calculated by Halpin-Tsai model and linear mixed model. Then, the MSGT and Reddy beam theory are adopted to establish the fundamental equations of the microbeam on the elastic foundation. And the governing differential equations related to the size effect are derived by the Hamiltonian principle and solved by the Navier method. Finally, the influences of the graded distribution, mass fraction, geometric shape of GPL and size effect on the vibration behaviors are discussed in detail. Results show that these parameters have significant influence on the natural frequencies. The methods to improve the natural frequencies of the microbeams are proposed considering the influence of internal microstructure on the overall characteristics. The research results are of great significance to design this kind of micro-nano structures.

Vibration frequency and mode localization characteristics of strain gradient variable-thickness microplates

Based on the modified strain gradient theory and Kirchhoff–Love assumptions, a free vibration model is established for variable-thickness microplates with three material length scale parameters, and the corresponding Euler–Lagrange equation is derived. Keeping the microplate volume constant, three kinds of thickness variations along the X-direction are considered: linear, parabolic, and cubic variations. Further, a C2-type four-node 36-degree-of-freedom variable-thickness differential quadrature finite element is developed to solve the resulting variable-coefficient boundary value problem by combining the Gauss–Lobatto and differential quadrature rules. Galerkin approximate solution for fully simply supported microplates is provided as a benchmark for numerical method verification. The convergence and accuracy of the model are verified through several examples. Finally, the vibration frequencies of the microplates under different thickness variation types, taper ratios, thickness ratios, material length scale parameters, and boundary conditions are examined through parametric analysis. Also, the influence of each factor on the mode localization of the microplates is quantitatively characterized by the modal assurance criterion (MAC) for the first time. The numerical results show that the thickness variation has a minor effect on the vibration frequency but a significant influence on the mode shape when the volume of the microplate is constant. The strain gradient, taper ratio, and discontinuous boundary have a remarkable effect on the distribution of higher-order mode shape contour lines and the regions where vibration localization occurs.

Static and free vibration responses of nanobeams considering flexoelectricity and surface effect

In this paper, the electro-mechanical behavior of a flexoelectric nanobeam, considering the surface effect, is studied with an induced electric potential. Based on the modified strain gradient theory and Hamilton’s variational principle, the governing differential equations of the nanobeam and the corresponding boundary conditions are obtained. The exact deflection solution of the flexoelectric nanobeam is derived from static bending. The relationship between the induced electric potential, including the surface effect, and the angle of the end of the cantilever beam is presented. Moreover, the characteristic equation of natural frequency is obtained by using a separate variable method under an open circuit with surface electrodes and an induced electric potential condition (OCI). The simulated results indicate the electro-mechanical response of the cantilever beam can be controlled by adjusting the flexoelectric coefficient, residual surface stress, and material length scale parameters of strain gradient theory. The free end of the beam with a platform phenomenon is also found by setting the appropriate parameters. The results also show that the residual surface stress and the ratio of beam thickness to material length scale parameters have a more significant effect on the effective frequency shift of the flexoelectric nanobeam. Therefore, considering the induced electric potential and surface effect is of great significance for the study of flexoelectric nanobeam sensors.

Nonlinear thermal buckling analysis of temperature-dependent porous annular and circular microplates reinforced by graphene platelets by using isogeometric analysis method

In this research, by using isogeometric analysis (IGA) method, nonlinear thermal buckling analysis of porous circular and annular microplates reinforced by graphene platelets (GPLs) under uniform temperature rise is performed by taking both the temperature dependence of the material properties and the size effects into account. The closed-cell metal foam scheme combined with the Halpin-Tsai micromechanics model and the rule of mixture is utilized to identify the effective material properties of the porous nanocomposites for three GPL dispersion patterns and three porosity distributions. With the aid of the virtual work principle, the classical and the non-classical motion equilibrium formulation based on the higher-order shear deformation theory in conjunction with the modified strain gradient (MSG) theory for the nonlinear temperature-dependent buckling behaviour are established. The higher-order NURBS basis functions capable of accomplishing easily the smoothness of arbitrary continuity order are successfully utilized to solve the nonlinear governing equations by satisfying the C2-continuity of the displacement field required by the MSG theory. The thermal buckling behaviour is assessed in detail by using various parameters such as the thickness-to-radius ratio, the porosity coefficient, the GPL weight fraction and the material length scale as well as the porosity distribution scheme and the GPL dispersion pattern. The results show the significance of incorporating the temperature dependency of the material properties when analyzing the thermal buckling behaviour and present the combination scheme of porosity and GPL distributions for enhancing the thermal buckling resistance.

Vibration characteristics of multilayer functionally graded microplates with variable thickness reinforced by graphene platelets resting on the viscoelastic medium under thermal effects

Due to their improved mechanical properties and adaptability, microplates with tailored variable thickness profiles are becoming essential parts of advanced micro- and nanoelectromechanical systems (MEMS and NEMS). This study conducts a thorough analytical analysis of the vibration properties of thermally loaded, multilayer functionally graded graphene platelet-reinforced composite (FG-GPLRC) microplates of linearly or parabolically varying thickness resting on viscoelastic medium under different boundary conditions. The Halpin–Tsai micromechanical model and the law of mixtures are employed to calculate the effective material characteristics for various reinforcement distributions in the microplate. These distributions encompass uniformly symmetric and asymmetric arrangements. The study utilized the first-order shear deformation theory (FSDT) in conjunction with the modified strain gradient theory (MSGT) and Hamilton's principle to generate the dynamic governing equations for the structure, accounting for size-dependent effects. The resulting equations are afterwards solved using the utilization of the Galerkin technique. This enables the evaluation of the proposed solution's correctness and precision. The impact of various factors on vibration behavior is investigated through numerical analysis. These factors encompass length scale parameters, temperature fluctuations, temperature distribution profiles, boundary conditions, the distribution pattern of the GPL, taper constants in both unidirectional and bidirectional scenarios, the weight fraction of the GPL.

Transient nonlinear responses of a sandwich plate with micro-cellular auxetic core based on modified strain gradient theory under impact loads

This study investigates the nonlinear dynamic response of an auxetic sandwich structure under impact load. The core is made of a micro-cellular auxetic structure. The main point of interest is studying the micro-cellular core properties’ effects on the structure’s nonlinear dynamic response. First, the mechanical properties of the micro-cellular auxetic core are introduced based on modified strain gradient theory (MSGT), followed by experimental and numerical validation. Then, Hamilton’s principle is applied to derive the sandwich structure’s nonlinear governing partial differential equations. Lastly, Galerkin’s method paved the way to achieving the final solution. So as to perform frequency-response analysis, the four-dimensional averaged equations are obtained using the perturbation method. In order to carry out experimental validation, a nanosecond laser system is used for micro-cellular auxetic sample fabrication, and the tensile tests are carried out per ISO 6892 standard. The comparison between the experimental and the theoretical results indicates an excellent agreement among the results. It can be concluded that MSGT plays a significant role in the mechanical properties and the nonlinear dynamic response of the micro-cellular auxetic structures. The present study is applicable in the field of protective panels, sensors, and energy harvesters.

Elastic size effect of single crystal copper beams under combined loading of torsion and bending

Among various strain gradient theories, the modified couple stress theory, which introduces a single length scale parameter as an additional material property, has garnered significant interest owing to its simplified portrayal of the material behavior. In this study, we investigated whether a single length scale parameter is sufficient to predict the mechanical behavior under two different type of strain gradients: torsion and bending. L-shaped beams made of single crystal copper with thicknesses ranging from 2.4μm to 9.1μm were fabricated, and loads were applied using an indenter. The contributions of bending and torsion were controlled by adjusting the loading position. Through these experiments, we demonstrated the existence of elastic size effect of single crystalline materials under strain gradients. Specifically, size effect was observed in both bending and torsion, with a larger effect observed in cases closer to pure bending. Moreover, we report that the modified couple stress theory and the modified strain gradient theory are not applicable for simulating size effect under combined loading. This discovery highlights the necessity for the development of a new theory capable of adequately simulating size effect under the intricate loading scenarios encountered in practical applications.

Scale-Dependent MSG Formulations for Vibrational Study of Honeycomb Microplates Covered by Piezoelectric Nanocomposite Patches

This study investigates free vibration analysis of a microplate featuring a honeycomb (HC) core and incorporates two nanocomposite piezoelectric (NP) layers. Carbon nanotubes (CNTs) are hired to enhance the electro-mechanical performance of the piezoelectric patches, which are subjected to an externally applied electric voltage. All layers are tightly bonded together and are supported by an elastic substrate according to the Pasternak model, capable of withstanding both normal and shear loads. To establish the kinematic relations, a trigonometric plate theory is employed. The governing equations of motion are deduced through the application of Hamilton’s principle and variational technique. The modified strain gradient theory, which includes three material length-scale parameters (MLSPs), is used to account for the scale effect. An analytical approach based on Fourier series functions is employed to solve the differential equations of motion. Subsequently, the impact of various factors, such as the geometrical characteristics of the HC core, distribution patterns of CNTs, and other significant parameters, on the normalized frequencies of the model is assessed after validating the accuracy of the results. The outcomes of this research have the potential to facilitate the advancement and production of lightweight and intelligent structures and devices, ultimately enhancing their efficiency.

Nonlinear dynamic stability analysis of CNTs reinforced piezoelectric viscoelastic composite nano/micro plate under multiple physical fields resting on smart foundation

This study analyzes the nonlinear dynamic stability of carbon nanotube reinforced composite (CNTRC) piezoelectric viscoelastic nano/micro plate under time dependent harmonic compressive biaxial mechanical loading. The nano/micro plate is single-layer with a uniform and rectangular cross section. Polyvinylidene fluoride (PVDF) is the material used for the nano/micro plate. Also, it is located on a nonlinear viscoelastic piezoelectric foundation made of Zinc Oxide (ZnO). Single-walled carbon nanotubes (SWCNTs) have been used to strengthen nano/micro plate. Using Kelvin-Voigt model, the material properties of the system are assumed to be viscoelastic. Nano/micro plate is subjected to 2D magnetic field and electric potential. Magnetic field effects are incorporated using Maxwell’s relations. An alternating and direct voltage is applied to the nano/micro plate and also smart foundation in thickness direction. Eshelby-Mori-Tanaka approach is used to estimate the effective elastic properties. Additionally, follower force and uniform thermal gradient is applied to nano/micro plate in this study. The nonlinear strain–displacement relations are based on the Von-Kármán theory. According to classical, first order, third order, sinusoidal, parabolic, hyperbolic and exponential shear deformation plate theories, a new formulation is proposed incorporating surface stress effects through the Gurtin-Murdoch elasticity theory. In order to consider small scale parameters, this research is based on modified strain gradient theory (MSGT). Using the energy method and Hamilton’s principle, the non classical governing equations are derived. For nano/micro plate, simply supported boundary conditions are considered. For the purpose of solving differential equations, an analysis based on Galerkin procedure and finally the Bolotin method will be used to obtain the dynamic instability region (DIR). It is the objective of this study to investigate the impact of such parameters as small-scale parameters, alternating and direct applied voltage, intensity of magnetic field, surface effects, thermal environment and static load factor on the DIR. This study revealed that taking into account the smart foundation reduces the area of dynamic instability region by more than 60% for a constant intensity of magnetic field. The stability responses are also more convergent with the smart foundation. Additionally, the range is moved toward higher excitation frequencies and greater system stability. Validation of answers based on reliable scientific articles is one of the final stages of this research. These results can be used to design nano/micro electromechanical systems.

Prediction of nonlocal elasticity parameters using high-throughput molecular dynamics simulations and machine learning

Higher-order elasticity theories are able to model the mechanics at nanoscale. However, these theories have length-scale parameters that need to be evaluated either through experiments or molecular dynamics (MD) simulations. In the current literature, these length-scale parameters are assumed to be constant for a chosen material. However, this is not true, as shown in this paper through a set of high-throughput MD simulations on a variety of boundary value problems (BVP). For carbon and boron nitride nanotubes, the length-scale parameter in the modified strain gradient theory is found to vary with the absolute dimensions, type of the BVP, and the deformation level. This complex dependence makes prediction of the length-scale parameter l0 challenging. To address this issue, a supervised machine learning (ML) based framework is developed here. In this new framework, MD, continuum formulation, and ML are used in tandem to predict the length-scale parameter for a given material, dimension, and boundary condition. It is a three-step process. First, a number of MD simulations are performed for varying chirality, length, and boundary conditions. Then, the values of l0 are found using these MD output in the continuum formulation. Finally, an ML-based regression model is used to capture the variability of l0 and subsequently predict its value for any chirality, length, and boundary condition. Here three regression models – Gaussian process regression, support vector machine, and neural network – are used to make the predictions.This predictive tool eliminates need for expensive MD simulations for further continuum scale analysis. In a broader context, this novel three-step framework opens an accurate yet inexpensive door for applying non-classical continuum theories to nanoscale mechanics problems. The accuracy is imposed by the MD simulations, while the computational cost reduction is achieved by machine learning.

Natural frequencies of submerged microplate structures, coupled to stationary fluid, using modified strain gradient theory

This paper investigates the free vibration analysis of a microplate, interacting with a stationary fluid. For the fluid part, the potential flow assumptions are considered, and it is supposed to be incompressible, irrotational and inviscid. Different positions for the fluid relative to the structure are studied: a structure in contact with a fluid with a free surface, a submerged microplate and a floating structure. For the structure, the mechanical properties of the microplate such as density, Young modulus and shear modulus are assumed to be a function of structure thickness. Besides, the modified strain gradient theory on the basis of higher-order plate theory is adopted to capture the size effects of the microplate. Using Bernoulli’s relation and fluid velocity potential function, the fluid pressure affecting the microplate is calculated, and then Hamilton’s principle is employed to derive coupled fluid–solid equations of motion. After validation of the presented formulations, a comprehensive parametric study is conducted to exhibit the response of the system under variations of some parameters such as structure mechanical properties, fluid parameters and size-effect parameters.

Analytical solution for size-dependent nonlinear behavior of AFM microcantilever with assembled probe in liquid environments

Using modified strain gradient theory, this study aims to examine the size-dependent nonlinear behavior of atomic force microscopy (AFM) with an assembled cantilever probe (ACP) in various liquid environments. The largest contribution of this theory lies in its consideration of the material length scale parameter (MLSP) to analyze the effect of size on physical characteristics. When studying ACP in liquids, the hydrodynamic force creates a different behavior compared to an ACP beam enclosed in air. In this study, Hamilton’s principle is used to derive the nonlinear partial differential equation (PDE) of motion and associated boundary conditions (BCs) of the ACP in liquid environments. To achieve this, a combination of Euler-Bernoulli’s theory and strain gradient concepts are employed. As the next step, a reduced order model of the system is developed using the Galerkin method. To assess the nonlinear frequency response of the ACP in various liquid environments, perturbation technique, including the method of multiple scales (MMS), is applied to the developed reduced order model. The study also calculates resonance frequencies, phase planes, stability, and the potential function of the system analytically. The results of this study indicate that the system becomes nonlinear as the ratio of beam thickness to MLSP decreases. Several factors, including the ratio of beam thickness to MLSP as well as the viscous damping coefficient, can influence the jump phenomenon. Furthermore, the results of the phase and amplitude of the ACP in liquid environments demonstrate that an increase in the (h/l) ratio results in a decrease in response amplitude and a leftward shift in the phase of the response diagram. This phenomenon is indicative of a softening behavior exhibited by the system. Lastly, the paper examines the validity and accuracy of the MMS solution by comparing it to a numerical solution.

Analysis of thermoelastic damping in a microbeam following a modified strain gradient theory and the Moore-Gibson-Thompson heat equation

Analysis of thermoelastic damping in a microbeam following a modified strain gradient theory and the Moore-Gibson-Thompson heat equation

Isogeometric bending and free vibration analyses of carbon nanotube-reinforced magneto-electric-elastic microplates using a four variable refined plate theory

The classical continuum mechanics theory is inadequate for modeling the mechanical responses of the microstructures due to its failure to account for size effects. Instead, modified strain gradient theory (MSGT) is considered as one of the most accurate theories due to its ability to adjust three length scale parameters (LSPs). Isogeometric approach (IGA) is a computational technique that is capable of accurately solving complex problems. For those reasons, the size-dependent analysis of carbon nanotube-reinforced magneto-electric-elastic microplates based on the MSGT are firstly proposed. The present approach uses the refined plate theory (RPT) with four variables, the MSGT and IGA. The magnetic and electric potentials are assumed to be a combination of a half-cosine and linear variation to satisfy Maxwell’s equations. Different types of functionally graded carbon nanotubes (CNTs) including UD, FG-X, FG-O and FG-V are reinforced in the magneto-electric-elastic matrix of the microplates. Governing equations are derived using the extended virtual work principle and then solved by IGA to determine the deflection and natural frequency of the microplates. The influence of the LSPs, CNT distributions, CNT’s volume fraction, matrix’s volume fraction, magnetic potential, electric voltage and geometry on the deflection and natural frequency of the microplates are studied and discussed.