Material Length Scale Parameter Research Articles

Size-dependent vibration analysis of porous 3D-FG microshells in complex thermal environments using a neural network enhanced finite element model

This work aims to investigate the vibration behavior of porous three-directional functionally graded (3D-FG) microshells in thermal environments, where both mechanical and thermal material properties vary along three spatial directions following a power-law distribution, based on the modified couple stress theory (MCST). An isoparametric thermal element is employed for uncoupled heat conduction analysis, followed by dynamic analysis using a penalty solid element with three translation degrees of freedom (DOFs) and three rotation DOFs per node. However, due to the strong nonlinearity of the temperature field in 3D-FG microshells, obtaining exact temperature values at integration points in dynamic analysis requires a pretty dense mesh of the solid element. To overcome this issue, a back propagation (BP) neural network is utilized to predict the temperature field for various gradient indexes, enhancing the efficiency and precision of temperature distribution calculation. Numerical results demonstrate the effectiveness of the neural network enhanced finite element model. Furthermore, the effects of the material length scale parameter (MLSP), temperature difference, and gradient indexes on the natural frequencies are investigated. It is shown that higher structural stiffness weakens the impact of temperature difference on the frequency.

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.

Dynamic Response of Advanced Lightweight Porous Plates Integrated with Nanocomposite Face Sheets Resting on Elastic Substrate

This study centers on the analysis of a scale-dependent microplate configuration characterized by a porous core enveloped by nanocomposite patches embedded with graphene nanoplatelets. The microplate’s behavior is explored within a humid environment to comprehend the effects of moisture variations on its dynamic performance. Additionally, the microplate rests upon a Kerr foundation, a three-parameter elastic substrate. The material properties of all three layers are contingent upon thickness. To enhance precision, an innovative quasi-three-dimensional shear and normal trigonometric theory is employed, elucidating the kinematic interrelations of the microstructure. Notably, this novel theory accommodates the presence of transverse normal strain. For a comprehensive analysis of size influences, the modified couple stress theory is harnessed. This theory integrates a material length-scale parameter to anticipate outcomes at the micro-scale. By invoking Hamilton’s principle, differential motion equations are deduced and subsequently solved analytically. The investigation centers on probing the consequences of varied parameters on the natural frequencies. The findings underscore that the incorporation of GPLs amplifies the microplate’s stiffness, thus elevating its natural frequencies. In contrast, an escalation in the porosity index leads to a reduction in natural frequencies.

Vibration of embedded restrained composite tube shafts with nonlocal and strain gradient effects

Torsional vibration response of a circular nanoshaft, which is restrained by the means of elastic springs at both ends, is a matter of great concern in the field of nano-/micromechanics. Hence, the complexities arising from the deformable boundary conditions present a formidable obstacle to the attainment of closed-form solutions. In this study, a general method is presented to calculate the torsional vibration frequencies of functionally graded porous tube nanoshafts under both deformable and rigid boundary conditions. Classical continuum theory, upgraded with nonlocal strain gradient elasticity theory, is employed to reformulate the partial differential equation of the nanoshaft. First, torsional vibration equation based on the nonlocal strain gradient theory is derived for functionally graded porous nanoshaft embedded in an elastic media via Hamilton’s principle. The ordinary differential equation is found by discretizing the partial differential equation with the separation of variables method. Then, Fourier sine series is used as the rotation function. The necessary Stokes' transformation is applied to establish the general eigenvalue problem including the different parameters. For the first time in the literature, a solution that can analyze the torsional vibration frequencies of functionally graded porous tube shafts embedded in an elastic media under general (elastic and rigid) boundary conditions on the basis of nonlocal strain gradient theory is presented in this study. The results obtained show that while the increase in the material length scale parameter, elastic media and spring stiffnesses increase the frequencies of nanoshafts, the increase in the nonlocal parameter and functionally grading index values decreases the frequencies of nanoshafts. The detailed effects of these parameters are discussed in the article.

On nonlinear buckling of microshells

Investigation of the geometrical nonlinear action of doubly curved shell panels (DCSPs) in micro scale is the main target of this paper. The proposed microshell panels (MSPs) are assumed to be made of an auxetic honeycomb core (AHOC), leading to negative magnitudes of Poisson's ratio, covered by two nanocomposite enriched coating layers (NCECLs). To conduct the size-dependent nonlinear analysis and achieve the corresponding nonlinear equilibrium path (EQP) of the proposed MSPs, the nonlocal strain gradient theory (NLSGT) is utilized. The governing equations containing the equilibrium and compatibility nonlinear partial differential equations in terms of the deformation components are analytically solved based on the Galerkin technique for different types of simply-supported panels. The achieved results of the present solution exhibit the fact that nonlocal and material length scale parameters significantly affect the EQP of the proposed MSPs especially at their post-buckling stage during their snap-through instability. By solving several numerical examples, the effects of various parameters on the size-dependent EQP of the proposed MSPs are investigated. The results indicate that the influences of size-dependency are significantly affected by the curvature and also boundary conditions of the microshells.

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.

An investigation on the torsional vibration of a FG strain gradient nanotube

AbstractIn this work, an attention is paid to the prediction of torsional vibration frequencies of functionally graded porous nanotubes based on the Lam strain gradient elasticity theory. The nanotubes are formed of functionally graded porous nanomaterials that vary in the radial direction. This study also aims to obtain the analytical solution of the strain gradient model presented by Lam for torsional vibration response, in a simple manner, for different rigid or restrained boundary conditions. The torsion angle of a functionally graded nanotube is defined by an infinite Fourier series. Then, the Stokes’ transformation is applied to force the boundary conditions to the desired state. An eigenvalue problem is established with the help of the two systems of equations obtained. This eigenvalue problem, which includes deformable springs at both ends of the nanotube, appears as a general analytical solution that can find torsional vibration frequencies. It is shown that the vibrational responses can be significantly influenced by the through‐radius gradings of material, material length scale parameters and deformable springs of the functionally graded nanotubes and consequently can be predicted by giving proper values to torsional spring parameters.

Nonlinear oscillation of microscale fiber-reinforced composite laminated beams under a thermal loading

The work analyzes the nonlinear dynamic response of fiber-reinforced composite laminated microbeams in thermal environment, accounting for the dependence of the material properties on temperature. The governing equations of the nonlinear partial differential equations are based on the Euler-Bernoulli beam theory, and the modified couple stress theory involving the von Kármán geometrical nonlinearity. The governing nonlinear equations are reduced to a single equation by neglecting the axial inertia, which is discretized in the form of nonlinear ordinary differential equation, according to the Galerkin method, and it is solved analytically using the multiple time scales (MTS) method. A systematic investigation checks for the effect of the material length scale parameter, temperature increase, and damping parameter of the microscale beam on its time history and phase plane trajectory. Different stacking sequences are also considered within the parametric investigation. Based on results, it is noticed that the system is in a damped period-doubling bifurcation for four cycle oscillation states. It is also noticed that the stacking sequence affects significantly the nonlinear dynamic behavior of the system. It is found that the microscale composite laminated beam with [0/0/0] lay-ups has the highest dynamical deflection and the highest nonlinear frequency, followed by [0/90/0], [30/-30/30] (angle ply), [90/0/90], [45/-45/45] (angle ply) and [60/-60/60] (angle ply). However, the microscale composite laminated beam with [90/90/90] lay-ups has the lowest dynamical deflection and the lowest nonlinear frequency.

Dynamic stability of the euler nanobeam subjected to inertial moving nanoparticles based on the nonlocal strain gradient theory

This research studied the dynamic stability of the Euler-Bernoulli nanobeam considering the nonlocal strain gradient theory (NSGT) and surface effects. The nanobeam rests on the Pasternak foundation and a sequence of inertial nanoparticles passes above the nanobeam continuously at a fixed velocity. Surface effects have been utilized using the Gurtin-Murdoch theory. Final governing equations have been gathered implementing the energy method and Hamilton's principle alongside NSGT. Dynamic instability regions (DIRs) are drawn in the plane of mass-velocity coordinates of nanoparticles based on the incremental harmonic balance method (IHBM). A parametric study shows the effects of NSGT parameters and Pasternak foundation constants on the nanobeam's DIRs. In addition, the results exhibit the importance of 2T-period DIRs in comparison to T-period ones. According to the results, the Winkler spring constant is more effective than the Pasternak shear constant on the DIR movement of nanobeam. So, a 4 times increase of Winkler and Pasternak constants results in 102 % and 10 % of DIR movement towards higher velocity regions, respectively. Furthermore, the effect of increasing nonlocal and material length scale parameters on the DIR movement are in the same order regarding the magnitude but opposite considering the motion direction. Unlike nonlocal parameter, an increase in material length scale parameter shifts the DIR to the more stable region.

Rayleigh wave propagation in centrosymmetric materials with micro-stiffness, flexoelectric and micro-inertia effects

A theoretical investigation of Rayleigh waves propagation in polarized media has been carried out using a reformulated flexoelectric theory for isotropic dielectrics with micro-inertia effect. Within this non-classical theory, the internal energy density is the functional of the strain tensor, dilatation gradient, deviatoric part of stretch gradient and rotation gradient tensors, polarization vector, and polarization gradient. The obtained system of governing equations additionally contains three material length-scale parameters to account the micro-stiffness effect, one material constant to capture the micro-inertia effect, two flexoelectric constants to describe the flexoelectric effect and three length scale parameters related to the polarization gradient. To solve the coupled governing equations, the method of Lamé-type potentials for mechanical displacement and electric polarization vectors is used. The influences of various factors such as micro-stiffness, flexoelectricity, electric quadrupoles and micro-inertia effects on the phase velocity of the Rayleigh waves in a homogeneous isotropic half-space are studied. It is found that above effects become significant with the increase of the wavenumber. This study can be important for the investigation of high frequency surface acoustic waves in dielectric materials.

Symmetric/Asymmetric Vibrational Characteristics of Bi-Directional Functionally Graded Annular Microplates

In this study, a free vibration model of bi-directional functionally graded (BD-FG) annular microplates is established based on the modified couple stress theory (MCST) and the first-order shear deformation theory (FSDT), and the corresponding governing equations are derived from Hamilton’s principle. The annular microplates are assumed to be composed of metal and ceramic materials, and the equivalent material properties are a combination of power-law function and exponential ones. The generalized differential quadrature method (GDQM) is used to obtain the vibration frequencies and mode displacements for the C–C, S–C, C–S, and S–S (S: simply supported; C: clamped) annular microplates. The convergence and validity of the proposed model are verified through selective numerical examples. Further, the effects of gradient index, material length scale parameter, boundary conditions, and geometrical dimensions on the symmetric/asymmetric vibration frequencies of the annular microplate are explored. The modal assurance criterion is applied to estimate the influence of the radial gradient index and material length scale parameter on different-order mode displacements of the annular microplate. The results show that: (1) the vibration frequencies of the annular microplate are sensitive to changes in the geometry and boundary conditions; (2) compared with the axial gradient index, the radial gradient index not only affects the vibration frequencies of the annular microplate, but also impacts the distribution of the highest and lowest points of the vibration mode displacements as well as the vibration mode nodal lines; (3) as the material length scale parameter increases, the vibration frequencies are significantly increased, and the higher-order vibration modes of the annular microplate also change.

Application of Modified Couple-Stress Theory to Nonlinear Vibration Analysis of Nanobeam with Different Boundary Conditions

PurposeIn the present study, the nonlinear vibration analysis of a nanoscale beam with different boundary conditions named as simply supported, clamped-clamped, clamped-simple and clamped-free are investigated numerically.MethodsNanoscale beam is considered as Euler-Bernoulli beam model having size-dependent. This non-classical nanobeam model has a size dependent incorporated with the material length scale parameter. The equation of motion of the system and the related boundary conditions are derived using the modified couple stress theory and employing Hamilton’s principle. Multiple scale method is used to obtain the approximate analytical solution.ResultNumerical results by considering the effect of the ratio of beam height to the internal material length scale parameter, h/l and with and without the Poisson effect, υ are graphically presented and tabulated.ConclusionWe remark that small size effect and poisson effect have a considerable effect on the linear fundamental frequency and the vibration amplitude. In order to show the accuracy of the results obtained, comparison study is also performed with existing studies in the literature.

Free Vıbratıon of a Carbon Nanotube-Reınforced Nanowıre/Nanobeam wıth Movable Ends

ObjectiveThe purpose of this work is to investigate the size-dependent free vibrations of carbon nanotube-reinforced nanowires/nanobeams with movable ends. For this purpose, two movable end conditions are considered for carbon nanotube-reinforced nanowires/nanobeams with different carbon nanotube distributions. The size effect is addressed with the modified couple stress theory, which includes a material length scale parameter.MethodsIn this study, a solution approach based on the Fourier sine series and Stokes' transformation is used. With the help of this approach, both rigid and movable end conditions can be investigated. Firstly, equation sets consisting of infinite series and movable end parameters are derived. Then, eigenvalue problems are obtained for the free vibration of carbon nanotube-reinforced nanowires/nanobeams. The eigenvalues of these problems give the frequencies of the carbon nanotube-reinforced nanowires/nanobeams with movable ends.ConclusionsThe eigenvalue problems obtained in this study can be used to examine both rigid and movable end conditions. The accuracy of the problems obtained for solution is proven by various comparison studies. Then, a series of analyses are carried out for nanowire/nanobeam reinforced with carbon nanotubes, including both the size effect and the rotary inertia effect. When the frequencies of nanowire/nanobeams reinforced with carbon nanotubes are examined, it is understood that the material length scale parameter, carbon nanotube distribution, carbon nanotube volume fraction, rotational and lateral spring parameters can cause significant changes in free vibration.

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.

On the variable length scale parameter in functionally graded non-porous and porous microplate/nanoplate

In this article, the solution for free vibration and static bending of nonporous and porous square microplate/nanoplate is studied by practicing the modified couple stress theory (MCST). The impact of incorporating the length scale parameter of MCST is confirmed by comparing the results to that of molecular dynamics simulations. This study is aimed at investigating the influence of implementing a variable length scale parameter on the static and dynamic behavior of the small-scaled plates. Results are obtained adopting the analytical technique and compared to cases of presupposing constant numerical values of material length scale throughout each plate. The novelty of this theoretical study is the approach employed to achieve the formulation of the material length scale parameter, which varies along the thickness of both porous and nonporous plates. The results indicate that the inclusion of the variable length scale parameter increases the fundamental frequency and stiffness of plates. When comparing two cases of variable length scale parameter, one approximating the scale parameter of each constituent based on the bending parameter and Poisson’s ratio of the material, and the other adopting the numerical values estimated by prior experimental studies, reveals that the latter gives higher results and is chosen correspondingly.

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.

A novel isogeometric model for dynamic buckling analysis of doubly curved two-directional functionally graded porous shallow microshells in thermal environments via variable length-scale parameters

In this study, the thermo-mechanical dynamic buckling analysis of two-directional functionally graded porous (2D-FGP) shallow microshells resting on elastic foundations with arbitrary boundary conditions is performed using the isogeometric approach. The isogeometric approach takes advantage of non-uniform rational B-spline basic functions to exactly represent the structure geometry models and the attainment of higher order approximation conditions. The characteristics of the material vary in both the thickness and axial x orientation. Especially, a new point in this material length-scale parameter of microshell is analyzed as a function of spatial coordinates and as a function of the material gradient parameters. Using Hamilton’s principle, the modified couple stress theory, and Kirchhoff-Love’s shell theory, the governing equations of a 2D-FGP microshell lying on a Pasternak medium are obtained. A computer program that employs an isogeometric analysis method to analyze the static and dynamic buckling of a 2D-FGP doubly curved shallow microshell with various boundary conditions. The numerical results for thin cylindrical, spherical, and hyperbolic paraboloidal shallow microshells with various planforms, such as rectangular and circular, are presented. Several special cases are used to demonstrate the accuracy and efficacy of the developed model by comparing the numerical results obtained by the proposed formulations with other published data. In addition, the effects of certain parameters, such as the length scale parameters, the power-law indexes, the thickness-to-sides ratio, and the radius ratio, on the dynamic thermo-mechanical buckling response of the 2D-FGP shallow microshells with double curvature are explored in depth.

Vibration analysis of functionally graded carbon nanotubes reinforced composite nanoplates

This work presents the analytical analysis for free linear vibration behavior of functionally graded-carbon nanotubes reinforced composite (FG-CNTRC) nanoplates in the framework of nonlocal strain gradient theory (NSGT) and the first-order shear deformation plate theory (FSDPT). The nanoplate is considered made of a mixture of an isotropic polymer matrix and reinforced carbon nanotubes (CNTs). Four different distributions of CNTs are examined including uniformly distributed and FG reinforcements (FG-O, FG-X, and FG-V). The governing equations of motion are established based on the Hamilton’s principle. The closed-form analytical solution for the natural frequency of FG-CNTRC nanoplates with simply supported all edges is carried out by using the Navier-type solution. The impact of some key parameters on the natural frequencies of FG-CNTRC nanoplates is also studied and discussed. The result shows that FG-CNTRC nanoplates reveal the softening- or hardening-stiffness effects depending on the relationship between the nonlocal parameter and the material length scale parameter. The aspect ratios of FG-CNTRC nanoplates, the volume fraction, and the distribution pattern of CNTs have also an important impact on the vibration behavior of FG-CNTRC nanoplates.

3D Size-Dependent Dynamic Instability Analysis of FG Cylindrical Microshells Subjected to Combinations of Periodic Axial Compression and External Pressure Using a Hermitian C2 Finite Layer Method Based on the Consistent Couple Stress Theory.

This work develops a three-dimensional (3D) weak formulation, based on the consistent couple stress theory (CCST), for analyzing the size-dependent dynamic instability behavior of simply-supported, functionally graded (FG) cylindrical microshells that are subjected to combinations of periodic axial compression and external pressure. In our formulation, the microshells are artificially divided into nl layers. The displacement components of each individual layer are selected as the primary variables, which are expanded as a double Fourier series in the in-plane domain and are interpolated with Hermitian C2 polynomials in the thickness direction. Incorporating the layer-wise displacement models into our weak formulation, we develop a Hermitian C2 finite layer method (FLM) for addressing the current issue. The accuracy and the convergence rate of our Hermitian C2 FLM are validated by comparing the solutions it produces with the accurate two-dimensional solutions of critical loads and critical pressures of FG cylindrical macroshells and single-walled carbon nanotubes, which were reported in the literature. The numerical results show the effects of the material length-scale parameter, the inhomogeneity index, the radius-to-thickness and length-to-radius ratios, the load magnitude ratio, and the static and dynamic load factors on the first principal and first secondary instability regions of parametric resonance of simply-supported FG cylindrical microshells are significant.

A three-dimensional micropolar beam model with application to the finite deformation analysis of hard-magnetic soft beams

The main purpose of this contribution is to develop a three-dimensional (3D) nonlinear beam model based on the micropolar continuum theory. To do so, a kinematic model based on the deformation of three directors and accounting for the micro-rotation tensor of the micropolar theory is introduced. One of the main characteristics of the present beam model is that 3D constitutive equations without any modification can be directly used in the formulation. Furthermore, it is known that a body couple field is induced in hard-magnetic soft materials (HMSMs) when subjected to external magnetic fluxes. Therefore, the stress tensor in HMSMs is asymmetric, in general. Since the asymmetry of stress is one of the main features of the micropolar theory, the present formulation can be used for analyzing the deformation of beams made of HMSMs. Accordingly, the virtual external work of the present model is formulated so that it accounts for the contribution from uniform or constant-gradient external magnetic fluxes on the beam. Moreover, a Total Lagrangian (TL) nonlinear finite element (FE) formulation to provide numerical solutions of the related problems is developed. Several numerical examples are solved to investigate the capability of the developed formulation. It is shown that the present formulation can model the size-dependent behavior of beam-like structures if the material length-scale parameter of the micropolar constitutive model is comparable to the thickness of the beam. Moreover, the proposed model can successfully predict the finite deformation of 3D beams made of HMSMs subjected to magnetic loading.