Strain Gradient Elasticity Theory Research Articles

Free Vibration of MEE Plate with Honeycomb Core Subjected to Thermomechanical Loads

PurposeThis paper examines the vibration buckling of a sandwich nanoplate. The top and bottom layers are piezoelectric Barium Titanate (BaTiO3) and Cobalt Ferrite (CoFe2O4), while the core is a metal (Ti6Al4V) honeycomb.MethodsNonlocal strain gradient elasticity and sinusoidal higher-order deformation theories were applied. The sandwich nanoplate’s motion equations were calculated using Hamilton’s principle and the piezoelectric surface plates’ magnetostrictive, electroelastic, and thermal properties. Next, Navier equations were solved. The study considered the geometric properties of the honeycomb-shaped core of the sandwich nanoplate, its nonlocal characteristics, temperature change, and the effects of electric and magnetic potentials. The study aimed to examine the sandwich nanoplate’s dimensionless fundamental natural frequencies.ResultsConsidering the given context, the natural frequencies decrease significantly when the temperature difference is applied to the sandwich nanoplate, which has a metal honeycomb structure at its core. Vibration buckling takes place at around 1980 K. In addition, when the thickness ratio of the honeycomb structure is increased, the natural frequencies decrease, whereas they increase with an increase in the edge ratio.ConclusionThis research presents innovative findings regarding the creation and utilization of nanosensors, transducers, and nanoelectromechanical systems (NEMS) engineered for high-temperature environments, enhancing the current state-of-the-art in nanoscale-free vibration analysis.

Investigation of the Critical Buckling Temperatures and Free Vibration Response of Porous Functionally Graded Magneto-Electro-Elastic Sandwich Higher-Order Nanoplates with Temperature-Dependent Material Properties

PurposeThis article aims to determine the critical buckling temperature and thermomechanical free vibration response of a porous functionally graded material (FGM) nanoplate sandwiched between porous magneto-electro-elastic (MEE) plate layers, taking into account temperature-dependent material properties.MethodsThe upper surface of the FGM host structure is considered zirconia (ZrO ), and the lower surface is stainless steel (SUS304). In addition, the MEE face layers are considered to be homogenous volumetric mixtures of cobalt-ferrite (CoFe O ) and barium-titanate (BaTiO ). It is assumed that the host structure’s effective mechanical and thermal material properties are graded in the thickness direction according to the power law distribution. The effective mechanical, electrical, thermal, and magnetic properties of the face plates are obtained by the rule of mixture. This study was conducted using sinusoidal higher-order shear deformation theory (SHSDT) and nonlocal strain gradient elasticity theory (NSGT). Hamilton's theory is utilized to derive the equations of motion of sandwich nanoplates. Closed-form solutions are obtained by the use of Navier's method.ResultsParametric simulations are carried out to examine the effects of the power law index, nonlocal parameters, electric, magnetic, and thermal loads, porosity volume fraction, porosity distribution and volume ratio of CoFe O and BaTiO on the free vibration and buckling behavior of the nanoplate. The results of the simulations are compared with published works to verify the accuracy of the proposed method.ConclusionAccording to the analysis results, it is determined that the power law index, porosity ratio, porosity distribution function, temperature dependent properties, magneto-electro-thermo-mechanical loads and nonlocal factors significantly affect the thermomechanical behavior of the nanoplate.

Periodic solutions for nonlinear deformation waves based on new Boussinesq-type equation within strain gradient elasticity and reductive perturbation technique

In this article, we present a detailed variational formulation to study and analyze the geometrically nonlinear behavior of elastic materials at small scales, where the classical theory of elasticity becomes inadequate. The presented technique is based on the variation of the internal energy of the elastic material and the virtual work of the external forces for the dynamical system in the framework of the first strain gradient theory of elasticity. This approach simultaneously determines the equilibrium equations in nonlinear form and the complete set of nonlinear boundary conditions. A new hyperbolic Boussinesq-type equation is derived based on the presented model in one-dimensional space and time to examine the impact of geometrical nonlinearity on wave propagation in an isotropic elastic material. The exponential reductive perturbation technique (ERPT) is employed to obtain an approximate and hierarchical solution for the new nonlinear partial differential equation (NPDE). The obtained results are plotted and analyzed, moreover, the results reveal that geometric nonlinearity significantly influences wave propagation in polycrystalline silicon.

The influence of bi-directional material gradation on a mode-III crack in functionally graded material via strain gradient elasticity theory

The influence of bi-directional material gradation on a mode-III crack in functionally graded material via strain gradient elasticity theory

Mixed FEM implementation of three-point bending of the beam with an edge crack within strain gradient elasticity theory

This paper considers the problem of symmetrical three-point bending of a prismatic beam with an edge crack. The solution is obtained by the mixed finite element method within the simplified Toupin–Mindlin strain gradient elasticity theory. A mixed variational formulation of the boundary value problem for displacements–strains–stresses and their gradients is applied, simplifying the choice of approximating functions. The concept of energy balance is adopted to calculate the energy release rate with a virtual increase in crack length. The increment of the potential energy of an elastic body is determined by accounting for the strain and stress gradient contribution. Numerical calculations were performed using a quasi-uniform triangular mesh of the cross-type. The mesh refinement was applied in the vicinity of the crack tip, at the concentrated support, and the point of application of the transverse force, and uniform mesh partitioning was utilized in the rest of the beam. The fine-mesh analysis was carried out on the successively condensed meshes in the stress concentration domain for different values of the length scale parameter. The crack opening displacements and the distribution of strains and Cauchy stresses for various values of the length scale parameter are presented. An increase in this parameter increases the stiffness of the crack, which leads to a decrease in the crack opening displacements and a smooth closure of its faces at the crack tip. In addition, accounting for the scale parameter reduces the calculated values of strains and stresses near the crack tip. Based on the energy balance criterion, local fracture parameters such as the release rate of elastic energy at the crack tip and the stress intensity factor are determined for different values of the mesh step. The numerical calculations indicate the convergence of the obtained approximations. The main feature of solutions, which includes the strain gradient contribution, is the decrease in the values of the calculated parameters associated with the fracture energy compared to the classical elasticity theory.

Eshelby's inhomogeneity model within Mindlin's first strain gradient elasticity theory and its applications in composite materials

Eshelby's inhomogeneity model within Mindlin's first strain gradient elasticity theory and its applications in composite materials

A size‐dependent model of strain gradient elastic laminated micro‐beams with a weak adhesive layer

AbstractA size‐dependent model for a laminated micro‐beam with a soft adhesive is developed in the framework of strain gradient elasticity theory. The layered beam is constituted by two Euler–Bernoulli strain gradient elastic isotropic beams, joined through a strain gradient spring‐type contact law at the adhesive level. The governing bending and extensional equations and boundary conditions are obtained by using the variational principle. The differential system shows a coupling between the flexural and axial behaviors of the upper and lower beams, due to the presence of interface terms related to shear and peeling stresses. Two benchmark problems have been presented through their closed‐form solutions, namely a simply‐supported laminated beam subjected to a uniform distributed load and a mode 2‐type loading configuration of a layered axially deformable beam. Size effects and non‐local phenomena, due to high strain concentrations, are highlighted. Though simple in their features, the examples prove the efficiency of the proposed approach in designing micro‐scale layered beams.

3D coupled vibrations of asymmetric nano-particle mass sensors: Two-phase integral nonlocal strain gradient model and MD simulation

Since it is possible for the nano-particle to have any asymmetric shape and may attach to every position of the resonator-based mass sensors, any type of mass eccentricity and subsequently coupled vibrations are possible. So, in the present work, the 3D coupled axial-torsional-flexural vibrations of the mass nano-sensors are investigated considering both in-plane and out-of-plane bending as well as axial and torsional vibrations. In addition, the integral form of the two-phase local/nonlocal strain gradient (LNSG), as the consistent type of the common nonlocal strain gradient (NSG) elasticity theory, is employed for the first time to consider the size effects in the mass nano-sensors. A quasi-3D coupled FE model is constructed with no shear-locking within the complicated environment of the integral LNSG. The impacts of the LNSG elasticity and diverse types of possible coupling in changing the vibrational behavior of the mass nano-sensor are scrutinized and their crucial role in modeling of the nano-sensors is indicated. Furthermore, molecular dynamic (MD) simulations are implemented to identify and analyze the coupled vibrations of a mass nano-sensor consisting of a fixed-free carbon nanotube. Comparison between the MD results with those of the present coupled FE model reveals that considering the coupling effects reduces the modeling error, especially in the cases in which the coupling influences intensify. The present work can help to provide more accurate models of mechanical resonance-based nanosensors to detect nanoparticles with different shapes, larger sizes, and high mass ratios, which can lead to achieving sensors with higher sensitivity.

Axisymmetric Hertzian contact problem accounting for surface tension and strain gradient elasticity

Axisymmetric Hertzian contact problem accounting for surface tension and strain gradient elasticity

The influence of material gradation parallel to two unequal mode-III cracks in functionally graded materials via strain gradient elasticity theory

The influence of material gradation parallel to two unequal mode-III cracks in functionally graded materials via strain gradient elasticity theory

Mixed finite element implementation of plane crack problems within strain gradient elasticity

AbstractAn approach is proposed and applied to solve boundary value problems within the strain gradient elasticity theory. A mixed variation formulation of the finite element method is used, where stresses, strains, and displacements are considered as independent variables. This significantly simplifies the pre‐requirement for approximating functions and allows one to use the simplest triangular finite elements with a linear approximation of displacements. Global unknowns in a discrete problem are nodal displacements, while the strains and stresses in nodes are treated as local unknowns. This discrete problem is solved by a modified iteration procedure, where the global stiffness matrix for classical elasticity problems is treated as a preconditioning matrix with fictitious elastic moduli. The modeling plane crack problem (tension of a square plate with a central crack) is solved. The obtained solution agrees with the results available in the literature. Good convergence is achieved by refining the mesh for all scale parameters.

Thermomechanical Vibration Response of Solid and Foam FGM Nano Actuator/Sensor Plates

PurposeIn this study, the effect of foam structure on the thermomechanical behaviour of high void ratio porous FGM piezoelectric smart nanoplates is investigated.MethodThe material of the smart nanoplate consists of PZT-4 on the bottom surface and BaTiO3 on the top surface and is formed by functional grading of these two materials along the thickness of the plate. Four different foam distribution models are modelled to examine the foam structure of the highly porous smart nanoplate, which has become widespread in biosensor applications. For this reason, uniform, symmetrical, top symmetrical and bottom symmetrical foam distribution models are created up to 75% void ratio. To determine the nano size, equations of motion are obtained by using nonlocal strain gradient elasticity and sinusoidal shear deformation theories together, and these equations are solved by the Navier method according to general boundary conditions.Result and ConclusionsAs a result of the analysis, it is observed that the applied external electric potential creates a softening effect on the plates with the piezoelectric elasticity effect and therefore reduces the thermal buckling temperatures. It is observed that the presence of the foam structure significantly improves the thermal resistance of the material and increases the buckling temperatures. It is also observed that the foam distribution model has significant effects on the thermomechanical behaviour.

3D wave dispersion analysis of graphene platelet-reinforced ultra-stiff double functionally graded nanocomposite sandwich plates with metamaterial honeycomb core layer

This research addresses the three-dimensional thermomechanical wave propagation behavior in sandwich composite nanoplates with a metamaterial honeycomb core layer and double functionally graded (FG) ultra-stiff surface layers. Due to its potential for high-temperature applications, pure nickel (Ni) is preferred for the honeycomb core layer, and an Al2O3/Ni ceramic-metal matrix is preferred for the surface layers. The functional distribution of graphene platelets (GPLs) in three different patterns, Type-U, Type-X, and Type-O, in the metal-ceramic matrix with a power law distribution provides double-FG properties to the surface layers. The mechanical and thermal material characteristics of the core and surface layers, as well as the reinforcing GPLs, are temperature-dependent. The pattern of temperature variation over the plate thickness is considered to be nonlinear. The sandwich nanoplate’s motion equations are obtained by combining the sinusoidal higher-order shear deformation theory (SHSDT) with nonlocal integral elasticity and strain gradient elasticity theories. The wave equations are established by using Hamilton’s principle. Parametric simulations and graphical representations are performed to analyze the effects of honeycomb size variables, wave number, the power law index, the GPL distribution pattern, the GPL weight ratio, and the temperature rise on three-dimensional wave propagation in an ultra-stiff sandwich plate. The results of the analysis reveal that the 3D wave propagation of the sandwich nanoplate can be significantly modified or tuned depending on the desired parameters and conditions. Thus, the proposed sandwich structure is expected to provide essential contributions to radar/sonar stealth applications in air, space, and submarine vehicles in high or low-temperature environments, protection of microelectromechanical devices from high noise and vibration, soft robotics applications, and wearable health and protective equipment applications.

Three-dimensional thermomechanical wave propagation analysis of sandwich nanoplate with graphene-reinforced foam core and magneto-electro-elastic face layers using nonlocal strain gradient elasticity theory

This article investigates the propagation of bending, longitudinal, and shear waves in a smart sandwich nanoplate with a graphene platelet (GPL)-reinforced foam core and magneto-electro-elastic (MEE) surface layers using sinusoidal higher-order shear deformation theory (SHSDT). The suggested nanoplate is comprised of a Ti–6Al–4V foam core placed between MEE surface layers. The MEE surface layers are composed of a volumetric combination of cobalt-ferrite (CoFe2O4) and barium-titanate (BaTiO3). The foam core and MEE face layers’ material characteristics are temperature dependent. In this study, three different core types are considered: metallic solid core (Type-I), GPL-reinforced solid core (Type-II) and GPL-reinforced foam core (Type-III), as well as three different foam distributions: symmetrical foam I (S-Foam I), symmetrical foam II (S-Foam II) and uniform foam (U-Foam). To derive the nanoplate's equations of motion and determine the system response, Hamilton's principle and Navier's method are employed. The effects of various parameters such as the wave number, nonlocal parameter, foam void coefficient and distribution pattern, GPL volume fraction, and thermal, electric, and magnetic charges, on the phase velocity and wave frequency are investigated via analytical calculations. The findings of the research indicate that the 3-D wave propagation characteristics of the sandwich nanoplate can be considerably modified or tuned with respect to external loads and material parameters. Thus, the proposed sandwich structure is expected to provide important contributions to radar stealth applications, protection of nanoelectromechanical devices from high frequency and temperature environments, advancement of smart nanoelectromechanical sensors characterized by lightweight and temperature sensitivity and wearable health equipment applications.

Postbuckling of functionally graded microbeams: a theoretical study based on a reformulated strain gradient elasticity theory

Postbuckling of functionally graded microbeams: a theoretical study based on a reformulated strain gradient elasticity theory

Modeling of size dependent buckling behavior of piezoelectric sandwich perforated nanobeams rested on elastic foundation with flexoelectricity

This article presents a size dependent mathematical model and an analytical solution methodology to accurately simulate and analyze the size dependent buckling behavior of piezoelectrically layered sandwich nanobeams with perforated core embedded in an elastic medium with flexoelectricity. The electric enthalpy energy function is expressed in terms of the electric and the flexoelectric effects. Regular squared cutouts perforation pattern is adopted for the perforated core. Closed forms for the equivalent geometrical parameters of perforated core are developed. The shear deformation effect is incroporated using the Timoshenko beam theory (TBT). To capture the nonlocality and the microstructure length scale effects, the nonlocal strain gradient elasticity theory is modified and adopted to include the electromechanical nonclassical effects. The Hamiltonian principle is utilized to derive the equilibrium equations. An analytical solution methodology is developed to derive closed forms for critical buckling loads. The developed solution procedure is implemented into a MATLAB software code. The accuracy of the developed procedure is verified by comparing obtained results with corresponding cases reported in the literature. Influences of different design variables on the electromechanical buckling behavior are explored through intensive parametric studies. Results obtained showed the significant effects of the flexoelectricity and the piezoelectric parameters on the size dependent buckling behavior of piezoelectric sandwich nanobeams with perforated core. Size dependent electromechanical as well as mechanical buckling behaviors could be controlled by adjusting these parameters. The developed procedure and the obtained numerical results are helpful in many technological and industrial applications as MEMS and NEMS.

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.

Inextensional vibrations of thin spherical shells using strain gradient elasticity theory

Inextensional vibrations of thin spherical shells using strain gradient elasticity theory

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.

Dislocations in nonlocal simplified strain gradient elasticity: Eringen meets Aifantis

Dislocations in nonlocal simplified strain gradient elasticity: Eringen meets Aifantis