Length Scale Parameter Research Articles

Size-dependent behaviours of functionally graded sandwich thin-walled beams based on the modified couple stress theory

This paper examines the static and vibration analysis of functionally graded sandwich thin-walled microbeams using the modified couple stress theory. The material properties in both web and flanges vary continuously through their thickness. The equations of motion are derived and solved by the Ritz method. Verification studies are performed on the bending and vibration responses of FG sandwich thin-walled beams. The effects of material distribution, span-to-height's ratio, and material length scale parameters on the responses of microbeams for various boundary conditions are investigated. The results of the analysis indicate that the deflections and natural frequencies of functionally graded sandwich thin-walled microbeams are greatly influenced by the variations of parameters mentioned earlier. The microbeams with the size effect are predicted to be much stiffer than their macro counterparts. Some benchmark results can be used as a reference for future studies.

Thermal stability and effects of thermal fluctuations on the static and spherically symmetric hairy black hole by gravitational decoupling

In this paper, we explore the thermodynamic aspects of the hairy black hole solution developed by gravitational decoupling in the spherical symmetry, considering its energy emission, Gibbs free energy, and thermal fluctuations. Our analysis involves the calculation of various quantities, including the Hawking temperature, geometric mass, and heat capacity to discuss the thermodynamic stability at local and global scales. To determine the temperature of the black hole, we utilize the first law of thermodynamics. Additionally, we evaluate the energy emission rate and establish its proportionality to the decoupling parameter α. By calculating the Gibbs free energy, we investigate the phase transition behavior through the swallowing tails of the hairy black holes. Furthermore, we derive the corrected entropy to examine the impact of thermal fluctuations on small and large black holes. We also calculated horizon radius r0, inner stable circular orbit rISCO , photon sphere radius rph, and shadows radius Rsh. Remarkably, the length scale parameters α (decoupling parameter) and β exhibit substantial influences on the thermodynamic properties of under consideration hairy black holes.

Data-driven solution for nonlinear wave dispersion of sandwich nanoshell with piezoelectric patch layers

It’s all about speed and scale in the quickly evolving corporate world of today, and business executives are looking to cutting-edge technology to help them find answers to their most difficult problems. A data-driven strategy provides the means to more thoroughly comprehend, confirm, and quantify decision-making outcomes. It is possible to feel greater confidence in the decisions made by the designers in a data-driven business. So, using this essential technique, and via mathematical modeling, the nonlinear wave dispersion of a sandwich nanoshell made of a Bi-directional functionally graded (BD-FG) core, and piezoelectric patch is investigated. In the mathematical modeling, first-order shear deformation theory with moderately thick hypothesis is used to model the displacement fields of the current work for both layers. After that, compatibility conditions are presented to model interface equations between the layers. For correct and exact modeling of the current nanosystem, nonlocal strain gradient theory with length scale and nonlocal parameters is presented. For solving the nonlinear equations, the harmonic solution technic for the displacement domain, and the multiple scale method for the time domain are presented in the solution procedure section. After that, to decrease the computational cost of the current work, the data-driven solution is presented for solving the nonlinear vibration problem. Comparing the results of current work with the outcomes of data-driven solutions and another published article can verify the results. Finally, in the results section, the influences of various factors such as length scale, nonlocal, applied voltage, area of piezoelectric patch, FG power index, and geometry condition of the structure on the nonlinear phase velocity of the BD-FG nanoshell coupled with piezoelectric patch is presented.

Free vibration of porous FG magneto-electro-elastic microbeams in the hygrothermal environment based on differential transformation method

Free vibration of the porous functionally graded magneto-electro-elastic (FGMEE) microbeams in hygrothermal environment is investigated. Four kinds of distributions of porosities including uniform porosity, “O,”“X,” and “V” type porosities are assumed in this paper. For the functionally graded microbeam, the physical neutral surface is considered. The modified couple stress theory is adopted to capture the size effect. The equations of motion for the porous FGMEE microbeam are derived and solved by differential transformation method. In the numerical examples, the effect of electric and magnetic loadings, material length scale parameter, the temperature rise, the moisture concentration, material gradient index, and porosity volume fraction on the natural frequency are analyzed and discussed in detail. The results indicate that different porous distribution modes have different effects on natural frequency. Moisture concentration/temperature rise can reduce the natural frequency regardless of what the temperature and moisture modes is. These results will be useful for the design of the porous FGMEE structures.

Non-Linear Non-Classical Modeling of Microparticles Manipulation Based on Finite Element Method

A comprehensive model was presented using the finite element method to analyze the dynamics of cylindrical microparticles, considering non-linear and size effects. The governing equations of motion have been derived based on the weak form method. The stiffness and mass matrices, along with the force vectors, have been presented for non-classical non-linear models of both the Euler–Bernoulli and Timoshenko types. The manipulation dynamics of cylindrical gold microparticles was simulated using FEM. In the simulations, four models of Euler–Bernoulli and Timoshenko beams were utilized. A mesh independence study was investigated. In non-linear models, the stiffness matrix takes into account the effects of higher-order strain terms, leading to a decrease in the maximum bending of the microparticle. The examination of the effects of aspect ratio showed that decreasing it reduces the deflection. After investigating the effects of length scale parameter and aspect ratio, a gold microparticle was manipulated by 200 nm using various models, and the final position of the particle was recorded. To validate the results, the deformations of polystyrene microrods were compared with the available data. The results can be used to make accurate estimates for the positioning of cylindrical microparticles, such as nanowires, nanorods and nanotubes.

Optimization of time-dependent displacements of the axially graphene nanoplatelets reinforced micro-curved panels via swarm and genetic algorithm approaches

Micro-curved panels with moving micro/nanoparticles included are essential microstructures in the engineering field. So, improving its stability for use in various situations is an essential key for biomechanical engineers. There is potential for using graphene nanoplatelets (GPLs)-reinforced epoxy as the matrix phase in a fiber composite as it has been shown that GPLs successfully improve the properties of epoxy materials when compared to clean epoxy. So, in the current work, dynamic deflection of the axially reinforced thick micro-curved panel under a moving load is presented. Optimizing the geometry of composite materials is essential in structural design if you want to achieve the highest possible levels of strength or stability. In the present study, we use the genetic algorithm and particle swarm optimization to find the optimal radius ratio for lowering the displacement fields. For modeling the micro-size structure, modified couple stress theory with one length scale parameter is presented. The displacement fields are modeled using Taylor’s series higher-order terms considering strain in the thickness direction. Using forced vibration analysis, the space and time-dependent equations are solved via double Fourier series and state-space method, respectively. It is shown that reinforcing a microstructure with GPLs in the axial direction has lower deflection than in other directions. The results show that velocity and external loading of moving micro/nanoparticle, geometry properties, and length scale parameters have an important role in the time-dependent deflections of the axially reinforced thick micro-curved panel under moving micro/nanoparticles. The current outputs can be used in the biomechanical engineering field for modeling a micro-curved panel under moving micro/nanoparticles.

Vibration mode shifting due to scale-dependency for cylindrical, conical, and spherical microshells

Microshells structures are broadly utilized in micro-electro-mechanical systems (MEMS) such as microresonator, microgyroscope, microsensor, microactuator, microantenna, and microgripper. Taking into consideration the scale-dependency in such a scale and the importance of vibration characteristics in design of such microstructures, the current study aims to analyze the role of scale-decency in mode shifting of common types of thin microshells in MEMS. A scale-dependent quadrilateral finite shell element based on the modified couple stress theory is developed to compute the frequencies and mode shapes of the system. Verification studies are presented to display the accuracy of the established numerical model for the prediction of the vibrational response of microshells with and without scale-dependency. A comprehensive examination is then carried out to disclose the impact of material length scale parameters together with geometrical parameters and boundary conditions on the mode shifting of microscale cylindrical, conical, and spherical shells. It is believed that the results of the present study are handy in the design of the microshells when the scale-decency is of great importance.

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.

Dynamic finite element modeling of axially functionally graded Timoshenko microbeams under a moving mass

This paper presents a finite element model for dynamic analysis of axial functionally graded (AFG) microbeams subjected to amovingmass. Thematerial properties of themicrobeams are considered to vary in the axial direction by a power-law function, and they are evaluated byMori-Tanaka scheme. The first-order shear deformation theory is employed in conjunction with the modified couple stress theory to establish the differential equations of motion for the AFG microbeams. A two-node beam element with six degrees of freedom was derived and used in combination with Newmark method to solve the equations of motion and to compute the dynamic response of the microbeams. Numerical investigations are carried out on AFG microbeam with simply supported ends. The proposed method is validated by comparing with results in previous work. The effects of dimension-less material length scale parameters, the material distribution and the moving mass parameters on the dynamic characteristics of the AFG microbeams are studied and discussed in detail.

A size-dependent quasi-3D model for bending and buckling of porous functionally graded curved nanobeam

In this work, we provide an analysis of the bending and buckling properties of functionally graded (FG) nanobeams with trigonometric function-dependent porosity distribution. Herein, a high-order curved nanobeam model considering both shear deformations and thickness stretching effect is established. Two-phase local/nonlocal stress-driven gradient theory is utilized to capture small-scale effects. Hamilton's principle is adopted to obtain the governing equation and boundary conditions, and the analytical solutions, such as exact formulas for lateral displacement and critical buckling load are obtained by employing Navier's method for simply supported boundary conditions. Furthermore, the effects of local volume fraction, nonlocal parameter, material length scale parameter, shear deformation, material distribution, and porosity distribution pattern in stress-driven models are analyzed in detail. The parametric studies indicate that the thickness stretching effect has a different influence on bending and buckling. In addition, the thickness stretching effect and the opening angle show different coupling mechanisms during bending and buckling. On the other hand, the higher-order buckling mode leads to a more significant scale effect and thickness tensile effect, but reduces the influence of the opening angle. Finally, we find that due to the coupling effect between the pore distribution pattern and the orientation bias of the FG nanobeam, the variation of the material gradient index leads to a different order of the dimensionless critical buckling loads predicted by the three pore distributions.

A Hermitian Cn finite cylindrical layer method for 3D size‐dependent buckling and free vibration analyses of simply supported FG piezoelectric cylindrical sandwich microshells subjected to axial compression and electric voltages

AbstractWithin the framework of the consistent couple stress theory (CCST), we develop a Hermitian Cn (n = 1, 2) finite cylindrical layer method (FCLM) for carrying out the three‐dimensional (3D) analysis of the size‐dependent buckling and free vibration behaviors of simply supported, functionally graded (FG) piezoelectric cylindrical sandwich microshells. The microshells of interest are placed under closed‐circuit surface conditions and subjected to axial compression and electric voltages. We derive a 3D weak formulation based on Hamilton's principle for this study. In the resulting formulation, the microshell is artificially divided into nl microlayers, with the elastic displacement components and the electric potential selected as the primary variables. By incorporating a layer‐wise kinematic model into our weak formulation, we develop a Hermitian Cn FCLM, which can be used for analyzing FG piezoelectric cylindrical sandwich microshells. Each primary variable is expanded as a double Fourier series in the in‐surface domain and is interpolated in the thickness direction using Hermitian Cn polynomials. The accuracy and the convergence rate of our Hermitian Cn FCLMs are validated by comparing the solutions they produce for FG piezoelectric cylindrical macroshells and FG elastic cylindrical microshells with the relevant exact and approximate 3D solutions which have been reported in the literature. The material length scale parameter of our FCLMs is set at zero in the comparison made with the FG piezoelectric macroshells. In contrast, the piezoelectric and flexoelectric effects are ignored in the comparison made with the FG elastic microshells. The impact of some essential factors on the critical load, critical voltage, and natural frequency of simply supported FG piezoelectric cylindrical sandwich microshells is assessed. The important factors are identified as piezoelectricity, flexoelectricity, the material length scale parameter, the inhomogeneity index, the radius‐to‐thickness ratio, the length‐to‐radius ratio, and the magnitude of the applied voltage and the applied load.

Thermoelastic dissipation of circular-cross-sectional ring including nonlocal and dual-phase-lagging effects

In the high-frequency region, thermoelastic damping (TED) is essential to ensure the accuracy of micro- or nano-precision sensors or actuators. The size-effect of molecules should be considered in the design and analysis since the atomic volume is not negligible in the nanoscale model. Moreover, a toroidal ring is easy to manufacture, so there is a high possibility that it will be developed in the near future. And thus, the 3-dimensional (3D) TED model of a circular cross-sectional ring is investigated by adopting dual-phase-lagging (DPL) and modified couple stress (MCS) theories based on nonlocal elasticity. Additionally, the size-dependent influences are specified by length-scale parameters in the elastic and thermal fields. Then the TED results are predicted as a quality factor (Q) with the peaks of the spectra and are compared separately by the causes of physical quantity. Moreover, the phenomenon is analyzed in terms of how the properties and length-scale parameters correlate with each other. Finally, it is confirmed that nonlocal elasticity should be considered when the size of the resonator decreases to the scale of the energy-carrying mean-free path. In this regard, this work provides an idea for the analysis of DPL TED of the toroidal ring, including nanoscale effects based on nonlocal elasticity and MCS theories.

Thermal buckling of piezoelectric graphene platelets reinforced cylindrical microshells using Navier’s and meshless methods

Thermal buckling analysis of a simply supported sandwich cylindrical microshell with piezoelectric layers reinforced with graphene platelets is presented in the current study. The motion equations of the piezoelectric microshell are derived in reference to the first-order shear deformation hypothesis alongside the modified couple stress theory. The critical buckling temperature difference of piezoelectric graphene platelets reinforced cylindrical microshells is derived from adopting Navier’s approach and the meshless method. The material properties of every layer for microcomposite shell layers reinforced by graphene platelets are estimated using Halpin–Tsai model. The comparison study is conducted to verify upcoming results. The goal is to investigate the effects of the applied voltage, the thickness of the piezoelectric layers, the distribution pattern, the weight fraction, the material length scale parameter, the length to radius ratio, and the thickness to radius ratio on the buckling of piezoelectric graphene platelets cylindrical microshell.

Nonlocal elasticity of Klein‐Gordon type with internal length and time scales: Constitutive modelling and dispersion relations

AbstractIn this work, a nonlocal elasticity theory with nonlocality in space and time is presented by considering nonlocal constitutive equations with a dynamical scalar nonlocal kernel. Based on the proposed theory, we consider the isotropic nonlocal elasticity of Klein‐Gordon type including a characteristic internal time scale parameter in addition to the characteristic internal length scale parameter. The dispersion relations for homogeneous isotropic media in the framework of nonlocal elasticity of Klein‐Gordon type are analytically determined. The obtained results reveal the ability of the presented nonlocal elasticity model to predict, for the first time in the framework of nonlocal elasticity, in addition to the acoustic modes (low‐frequency modes), optic modes (high‐frequency modes) as well as frequency band‐gaps. The phase and group velocities for all four modes (acoustic and optic branches of longitudinal and transverse waves) are determined showing that all four modes exhibit normal dispersion with positive group velocity. The presented model allows for physically realistic dispersive wave propagation.

Closed-Form Wave Dispersion Relationships for Ordinary State-Based Peridynamics

Peridynamics (PD) is a new continuum mechanics formulation introduced to overcome limitations of classical continuum mechanics (CCM). This is mainly achieved by using integro-differential equations rather than partial differential equations. Another important difference of PD is its nonlocal nature with respect to local characteristic of CCM. Moreover, it has a length scale parameter, horizon, defining the range of nonlocal interactions between material points. This nonclassical behaviour also shows itself for dispersion relationships. As opposed to linear dispersion relationships for CCM, PD dispersion relationships are non-linear similar to the observed in experiments. In this study, closed-form dispersion relationships are provided for ordinary state-based peridynamics which is one of the most common PD formulations. Finally, derived closed-form solutions are used to demonstrate the dispersion relationships for various material systems including copper, gold, silver and platinum.

A phase field approach to model crack-interface interaction in ceramic matrix composites

Ceramic matrix composites are widely applied in coatings, engines, and space vehicles because of their superior thermo-mechanical properties. Various failure modes can be possible in ceramic matrix composites, notable ones are fiber breakage, matrix cracking and fiber matrix debonding. In order to predict the failure mechanisms at microstructural level (matrix-fiber system) of such composites, understanding crack nucleation and propagation across different phases, namely, fiber, and matrix is significant. To assess structural integrity, the material parameters that govern the crack growth should be determined. Nonlocal approaches consider the interactions between infinitesimal material volumes by considering a material length scale parameter. Phase field method is one of the popular nonlocal approach to model failure in cracked solids. In the present work, phase field approach is adopted to model the crack propagation through different phases of a ceramic matrix composite at microstructural level. The performance of the proposed formulation is illustrated through representative numerical examples. The model proposed is implemented in the framework of the finite element method. Crack propagations are studied by considering two ceramic matrix composites, namely, carbon fibers in carbon matrix (C/C) composite, and carbon fibers in silicon carbide (C/SiC) composite. For the case of C/SiC composite, parametric study is conducted by varying the fiber volume fraction.

Various homogenisation schemes for vibration characteristics of axially FG core multilayered microbeams with metal foam face layers based on third order shear deformation theory

This study assesses the significance of homogenisation models in theoretically analysing the vibrations of three-layered microbeams with a core composed of axially functionally graded (AFG) material and with upper and lower face layers composed of metal foam material, especially when the third order shear deformation beam theory and the modified couple stress theory are used. The material of the core is assumed to vary following a power gradation profile. The range of homogenisation schemes, including the Voigt model, Reuss model, Voigt-Reuss-Hill model, Tamura model, Mori-Tanaka model, lower and upper bounds of Hashin-Shtrikman, and the local representative volume element (LRVE) model, are utilised to approximate the material properties of the AFG core. The upper and lower layers of metal foams are modelled using the closed-cell and open-cell solid porosity models. The coupled motion equations are solved for the natural frequencies of the system for the transverse, rotational and axial directions. A software-based finite element analysis (FEA) study of two simplified macrobeam systems was undertaken and the agreement between the FEA and numerical results, using the presented theoretical method, is found to be excellent. It has been confirmed that the homogenisation schemes have a notable effect on the mechanical properties of the AFG core, and hence on the coupled translational-rotational motion's natural frequencies. The largest difference in natural frequencies between different homogenisation schemes is found with a power gradation constant of approximately 0.53. A larger disparity of natural frequencies between different homogenisation methods was seen with an increase in the thickness of the AFG core and the microbeam's length-scale parameter of multilayered porous microbeam. The results of this study highlight the significance of homogenisation scheme employed when investigating the vibrations of a multilayered microbeam with an AFG core and metal foam porous face layers.

Analysis of Flexoelectric Solids With a Cylindrical Cavity

Abstract Flexoelectricity, a remarkable size-dependent effect, means that strain gradients can give rise to electric polarization. This effect is particularly pronounced near defects within flexoelectric solids, where large strain gradients exist. A thorough understanding of the internal defects of flexoelectric devices and their surrounding multiphysics fields is crucial to comprehend their damage and failure mechanisms. Motivated by this, strain gradient elasticity theory is utilized to investigate the mechanical and electrical behaviors of flexoelectric solids with cylindrical cavities under biaxial tension. Closed-form solutions are obtained under the assumptions of plane strain and electrically impermeable defects. In particular, this study extends the Kirsch problem of classical elasticity theory to the theoretical framework of higher-order electroelasticity for the first time. Our research reveals that different length scale parameters of the strain gradient and bidirectional loading ratios significantly affect the hoop stress field, radial electric polarization field, and electric potential field near the inner cylindrical cavity of the flexoelectric solid. Furthermore, we validate our analytical solution by numerical verification using mixed finite elements. The congruence between the two methods confirms our analytical solution’s accuracy. The findings presented in this paper provide deeper insights into the internal defects of flexoelectric materials and can serve as a foundation for studying more complex defects in flexoelectric solids.

Phase-field modeling of stochastic fracture in heterogeneous quasi-brittle solids

Owing to the random nature of heterogeneity, damage and fracture behavior of quasi-brittle materials exhibits a considerable degree of uncertainty. Computational modeling of stochastic fracture in quasi-brittle materials has become an indispensable tool for analysis and design of engineering structures. To this end, we present in this paper a computational framework to capture probabilistic fracture in heterogeneous quasi-brittle solids by combining the random field theory and the phase-field cohesive zone model (PF-CZM). The spatial variation of the material strength and fracture energy is represented by a cross-correlated bivariate random field generated by the Karhunen–Loève expansion. The recently proposed PF-CZM is employed to simulate the stochastic crack nucleation and propagation in quasi-brittle solids. The objectivity of the Monte-Carlo simulation is achieved by imposing a specific condition on the phase-field length scale parameter and the correlation length of the random field. In particular, upon this condition the width of the fracture process zone (FPZ) is considerably smaller than the correlation length of the random field such that the material inside the FPZ does not exhibit significant spatial variations of mechanical properties. As the fracture energy is intrinsically incorporated in the PF-CZM, it is unnecessary in this case to explicitly consider the FPZ width. The resulting probabilistic PF-CZM together is applied to the Monte-Carlo simulations of fracture in concrete structures of different geometries. It is shown that the stochastic simulation results are insensitive to both the phase-field length scale parameter and the finite element mesh discretization as in the previous deterministic analyses. Enhanced with the specific condition on the involved characteristic lengths, the PF-CZM provides a viable tool for stochastic simulations of damage and fracture in quasi-brittle structures.

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.