Nonlocal Strain Gradient Elasticity Research Articles

Magnetic field effects on the thermomechanical vibration behavior of functionally graded biocompatible material sandwich nanobeams

This study focuses on modeling and analyzing the thermomechanical vibration behavior of sandwich nanobeams made from biocompatible zirconia and Ti-6Al-4V materials. The sandwich nanobeam is composed of the materials mentioned earlier, featuring a porous structure in the core layer. The sandwich nanobeam experiences significant thermal load and is subjected to a horizontal external electromagnetic field. The equations of motion for the sandwich nanobeam were derived through the application of the sinusoidal high-order shear stress theorem and the constitutive equation based on nonlocal strain gradient elasticity. The equations were converted into equations of motion using Hamilton’s principle, incorporating the thermal load and the Lorentz force generated in the electromagnetic field, and solved using the Navier method. Analyses were conducted on the thermomechanical vibration behavior of the sandwich nanoplate, taking into consideration the properties of the sandwich layers, the impact of temperature, and the intensity of the applied horizontal magnetic field. The impact of the porous core layer in the sandwich nanoplate on the porosity volume ratio and porosity distribution function in thermomechanical behavior has been established. Research has demonstrated that the application of an external magnetic field can effectively mitigate the adverse impacts of thermal loads caused by high temperatures.

Dislocations in nonlocal simplified strain gradient elasticity: Eringen meets Aifantis

The nonlocal strain gradient elasticity theory is used to address mechanical problems at small scales where size effects and regularization cannot be neglected. In this work, dislocations are investigated in the framework of nonlocal simplified first strain gradient elasticity. It is shown that nonlocal simplified strain gradient elasticity is the unification of the theories of Eringen’s nonlocal elasticity of Helmholtz type and simplified first strain gradient elasticity. Nonlocal simplified strain gradient elasticity contains two characteristic lengths, namely the characteristic length of nonlocal elasticity of Helmholtz type and the characteristic length of simplified first strain gradient elasticity. The advantage of nonlocal simplified first strain gradient elasticity is that the displacement, elastic distortion, plastic distortion, total stress, Cauchy stress and double stress fields of screw and edge dislocations which are calculated here are nonsingular and finite everywhere. Moreover, the Peach-Koehler force of two screw dislocations and two edge dislocations is derived and it is shown that the Peach-Koehler force is also nonsingular. Numerical examples for all dislocation fields of screw and edge dislocations in aluminum are given.

The Casimir, Van der Waals, and electrostatic forces’ effects on the response of magneto-electro-elastic nanosensor/switch beams under thermal environment

This study investigates the impact of Casimir, Van der Waals, and electrostatic forces on nanomechanical switches’ thermomechanical and free vibration behavior. The analysis is conducted using a novel higher-order beam theory and the nonlocal strain gradient elasticity. The motion equations of the nanosensor/switch beam are derived using Hamilton’s principle and solved using Navier’s method for general boundary conditions. The nanoswitch is composed of electroelastic barium-titanate (BaTiO3) and magnetostrictive cobalt-ferrite (CoFe2O4) materials, which are modeled using a power-law approach to account for functionally graded material property variations across the beam’s thickness. The impact of different parameters, such as Casimir, Vander Waals, electrostatic forces, and variations in material composition, size parameters, and gap distance, on a nanoswitch system’s bucking and free vibration is comprehensively examined. With the intermolecular and electrostatic forces, the temperature dependency of barium-titanate and cobalt-ferrite nanoswitch materials, which have not been extensively studied in any previous research, is considered in the modeling of free vibration, and the buckling behavior of a nanoswitch for the first time. This research represents the first comprehensive analysis of these factors. Considering the investigated parameters, the study’s findings can provide helpful insights into developing micro/nano-electromechanical systems, including switches, sensors, and actuators.

The effect of the viscoelastic support and GRPL-reinforced foam material on the thermomechanical vibration response of piezomagnetic sandwich nanosensor plates

This paper investigates the vibration characteristics of a sandwich nanosensor plate composed of piezoelectric materials, specifically barium and cobalt, in the upper and lower layers, and a core material consisting of either ceramic (silicon nitride) or metal (stainless steel) foams reinforced with graphene (GPRL). The study utilized the novel sinosoidal higher-order deformation theory and nonlocal strain gradient elasticity theory. The equations of motion for nanosensor sandwich graphene were derived using Hamilton's principle, considering the thermal, electroelastic, and magnetostrictive characteristics of the piezomagnetic surface plates. These equations were then solved using the Navier method. The core element of the sandwich nanosensor plate can be represented using three distinct foam variants: a uniform foam model, as well as two symmetric foam models. The investigation focused on analyzing the dimensionless fundamental natural frequencies of the sandwich nanosensor plate. This analysis considered the influence of three distinct foam types, the volumetric graphene ratio, temperature variation, nonlocal parameters, porosity ratio, electric and magnetic potential, as well as spring and shear viscoelastic support. Furthermore, an analysis was conducted on the impact of the metal and ceramic composition of the central section of the sandwich nanosensor plate on its dimensionless fundamental natural frequencies. In this context, the use of ceramic as the central material results in a mean enhancement of 33% in the fundamental natural frequencies. In contrast, the incorporation of graphene into the core material results in an average enhancement of 27%. The thermomechanical vibration behavior of the nanosensor plate reveals that the presence of graphene-supported foam and a viscoelastic support structure in the core layer leads to an increase in thermal resistance. This increase is dependent on factors such as the ratio of graphene, porosity ratio of the foam, and parameters of the viscoelastic support. Metal foam or ceramic foam has been found to enhance thermal resistance when compared to solid metal or ceramic core materials. The analysis results showed that it is important to take into account the temperature-dependent thermal properties of barium and cobalt, which are piezo-electromagnetic materials, and the core layer materials ceramics and metal, as well as the graphene used to strengthen the core. The research is anticipated to generate valuable findings regarding the advancement and utilization of nanosensors, transducers, and nano-electromechanical systems engineered for operation in high-temperature environments.

Dynamic behavior of nanoplate on viscoelastic foundation based on spatial-temporal fractional order viscoelasticity and thermoelasticity

The nanodevices often work in a thermal environment and lead to the viscoelastic and thermoelastic coupling which becomes more complicated when further coupled with the size effects. A spatial-temporal fractional order model is proposed to study the dynamic behavior of thermoelastic nanoplate in the present work. The integer order differential is extended to the fractional order differential with the integer order differential as a special case. This extension makes the modeling of small-scale effects of mechanical behavior and generalized heat conduction more flexible. Non-local strain gradient elasticity and non-local heat conduction are used to describe the spatial non-local effects while the fractional order differential to describe the complicated history-dependent effects or the temporal nonlocal effects. The standard solid viscoelastic model and generalized heat conduction model with the fractional order differentials lead to the fractional-order differential governing equations. The fractional order differential equation is solved by the Laplace transform method, and the analytical solution of the response is expressed in terms of the Mittag-Leffler function. In order to verify the reliability of the analytical solution, the numerical solution is also provided for comparison. Based on the numerical results of dynamic response under the step load, the effects of thermoelastic fractional order parameters, viscoelastic fractional order parameters, and small-scale parameters are discussed. The new spatial-temporal fractional order model is a natural extension of the traditional integer order model which can be recovered from the present model.

Vibration Analysis of Single-Walled Carbon Nanotubes Embedded in a Polymer Matrix under Magnetic Field Considering the Surface Effect Based on Nonlocal Strain Gradient Elasticity Theory

Vibration Analysis of Single-Walled Carbon Nanotubes Embedded in a Polymer Matrix under Magnetic Field Considering the Surface Effect Based on Nonlocal Strain Gradient Elasticity Theory

Parametric-forced coupling resonance of core-shell nanowires with interfacial damage under weak viscoelastic boundary constraint

Revealing the synergistic effects of size-dependency and nonlinearity on dynamics of core-shell nanowires plays an important role in nonlinear design of nanoelectromechanical system (NEMS) maintaining stable working state. The nonlinear principal resonance of core-shell nanowires with weak interface under parametric-forced coupling excitations is studied. The nanowire is simply supported at both ends, with one end movable and under a weak viscoelastic boundary described by Kelvin-Voight model. The weak interface obeys the interfacial cohesive law coupled into the refined displacement field. Geometric nonlinearity due to large deformation is reflected in three aspects. Stiffness nonlinearity is modeled by nonlinear curvature. Longitudinal inertia nonlinearity is modeled based on an in-extensive beam assumption. Nonlinear constraint force provided by the viscoelastic boundary is reflected in nonlinear damping and stiffness. Size-dependency including nonlocal stress, strain gradient and surface elasticity are all incorporated. The governing equations are derived by Hamilton's variational principle and solved via a perturbation-incremental harmonic balance method (IHBM). Results provide the bifurcation diagram of coupled resonance and the stability boundary of parametric resonance for analyzing the change mechanism of bifurcation topology. Decisive ways are revealed that weak interface, size dependency and constraints from foundation and boundary affect the linear and nonlinear properties of the system.

Stability analysis of arbitrary restrained nanobeam embedded in an elastic medium via nonlocal strain gradient theory

A novel stability model is analytically reformulated for the nano-sized beam resting on a one-parameter elastic foundation. The stability solution is based on the nonlocal strain gradient elasticity theory. To corporate the small size effects, two small scale parameters are introduced. The six-order ordinary differential form of the buckling equation, together with two force boundary conditions, are utilized to examine the stability equation in terms of lateral deflection. The infinite terms of linear equations are discretized with the help of the Stokes’ transformation and Fourier sine series. The present work can investigate the effects of elastic spring parameters at the ends, nonlocal properties, elastic medium properties, strain gradient parameter, and buckling behavior of the nanobeam. The predictions of the proposed analytical model with deformable boundary conditions are in agreement with those available in the scientific literature for the nanobeam on elastic foundation based on a closed form of solution. The presence of the deformable conditions, elastic foundation, nonlocal, and strain gradient properties change the buckling loads and buckling mode shapes.

Thermomechanical vibration response of nanoplates with magneto-electro-elastic face layers and functionally graded porous core using nonlocal strain gradient elasticity

The thermal vibration and buckling behavior of a functionally graded nanoplate are examined in this study. The nanoplate is made up of a silicon nitride/stainless steel core plate and two cobalt-ferrite/barium-titanate face plates. Four alternative porosity models were used to simulate the porosity of the nanoplate, and numerous variables that can affect the nanoplate’s behavior were taken into account. The study found that the thermomechanical behavior of nanoplates with magneto-electro-elastic face layers and a functionally graded porous core plate is affected by material gradation indices, porosity ratios, nonlocal variables, and different core plate material porosity models.

Nonlinear stability analysis of embedded restrained nanobeams using the Stokes’ transformation

The geometrically nonlinear stability analysis of restrained nanobeams under an elastic medium is considered in the presented manuscript. The investigation is based on the nonlocal strain gradient elasticity theory, Euler-Bernoulli beam theory and geometrical nonlinearity which is a significant change in geometry. To the best of the authors’ knowledge, the nonlinear stability response of a nanobeam with elastic boundary conditions and on an elastic foundation has not been presented before via the theory of nonlocal strain gradient. The aim of this paper is to fill this gap in the literature by offering a method that can give a solution under general elastic boundary conditions. Using a Fourier sine series, a Fourier coefficient suitable for constructing an eigenvalue problem is computed. The constructed problem has been treated in the transformed region with the help of the Stokes’ transform, then the stability analysis is executed for arbitrary boundary conditions (rigid or restrained) based on the nonlocal strain gradient elasticity. The influences of various nonlinear parameters, deformable boundary conditions, small-scale parameters and elastic foundation parameters on the stability of the constrained nanobeam are studied. It is observed that the investigated variables produce significant changes in buckling loads.

On Bending of Piezoelectrically Layered Perforated Nanobeams Embedded in an Elastic Foundation with Flexoelectricity

Analysis of the electromechanical-size-dependent bending of piezoelectric composite structural components with flexoelectricity has been considered by many researchers because of the developments of nanotechnology and the applicability of piezoelectric composite nanobeam structures in Micro/Nano-Electro-Mechanical Systems (MEMS/NEMS). Therefore, the work investigates the size-dependent electromechanical bending of piezoelectrically layered perforated nanobeams resting on elastic foundations including the flexoelectric effect. Within the framework of the modified nonlocal strain gradient elasticity theory, both the microstructure and nonlocality effects are captured. The governing equilibrium equations including piezoelectric and flexoelectric effects are derived using Hamilton’s principle. Closed forms for the non-classical electromechanical bending profiles are derived. The accuracy of the proposed methodology is verified by comparing the obtained results with the available corresponding results in the literature within a 0.3% maximum deviation. Parametric studies are conducted to explore effects of perforation parameters, elastic foundation parameters, geometric dimensions, nonclassical parameters, flexoelectric parameters, as well as the piezoelectric parameters on the bending behavior of piezoelectrically layered perforated nanobeams. The obtained results demonstrate that incorporation of the nondimensional elastic foundation parameters, Kp = 2 and Kw = 20, results in a reduction in the relative percentage reduction in the maximum nondimensional mechanical transverse deflection due to increasing the perforation filling ratio from 0.2 to 1 from 199.86% to 91.83% for a point load and 89.39% for a uniformly distributed load. On the other hand, with Kp = 5 and Kw = 50, the relative percentage difference of the electromechanical bending deflection due to increasing the piezoelectric coefficient, e311, reaches about 8.7% for a point load and 8.5% for a uniformly distributed load at a beam aspect ratio of 50. Thus, the electromechanical as well as mechanical behaviors could be improved by controlling these parameters. The proposed methodology and the obtained results are supportive in many industrial and engineering applications, i.e., MEMS/NEMS.

A physically-based nonlocal strain gradient theory for crosslinked polymers

When the length scale of external stimulations is comparable to most chain lengths within a polymeric solid, the nonlocal and microstructure-dependent strain-gradient effects become significant. The present work proposes a physically-based nonlocal strain gradient theory of polymer networks, where the kernel functions and intrinsic length scales have unambiguous physical meanings. The main contribution lies in the establishment of the general framework that takes in an arbitrary complete set of microscopic descriptions (chain energetics, chain-length distribution, structure of interpenetrating network) and outputs a corresponding nonlocal strain gradient constitutive relation. Based on the hypotheses of interpolatory correlation, homogenization, and superposition, the physically-based construction of constitutive relation is specified to a degree to uniquely determine two linear functionals, which are used to evaluate the nonlocal stress and nonlocal hyper-stress. Application to eight-chain interpenetrating networks yields a specific theory of nonlocal strain gradient elasticity in which the explicit functional form of the kernel is analytically derived from the chain length distribution function. It is shown that the physically derived kernel for the strain gradient field results from the superposition of a spectrum of length scales. The proposed physically-based nonlocal strain gradient theory can be beneficial to clarify the relationship between the microstructure-dependent intrinsic lengths and properties (or responses) of crosslinked polymer network structures.

Vibration analysis of single-walled carbon nanotubes embedded in a polymer matrix under magnetic field considering the surface effect based on nonlocal strain gradient elasticity theory

Vibration analysis of single-walled carbon nanotubes embedded in a polymer matrix under magnetic field considering the surface effect based on nonlocal strain gradient elasticity theory

Thermomechanical vibration and buckling response of nonlocal strain gradient porous FG nanobeams subjected to magnetic and thermal fields

This study investigates the free vibration and thermal buckling behavior of functionally graded porous nanobeams in magnetic and thermal fields using high-order trigonometric shear stress and nonlocal strain gradient elasticity theories. The results demonstrated the effects of nonlocal differential and strain gradient elasticities on softening and stiffness enhancements, respectively. Additionally, the Lorentz force induced by the magnetic field makes nanobeam’s vibratory motion difficult, causing the natural frequencies to increase. This situation can contribute to the dynamic stability of nanobeams exposed to the nonlinear temperature distribution. This study’s results will assist in designing and implementing micro/nanoelectromechanical systems.

Traveling and Standing Flexural Waves in the Micro-Beam Based on the Fraction-Order Nonlocal Strain Gradient Theory

Abstract The traveling and standing flexural waves in the microbeam are studied based on the fraction-order nonlocal strain gradient elasticity in the present paper. First, the Hamilton’s variational principle is used to derive the governing equations and the boundary conditions with consideration of both the nonlocal effects and the strain gradient effects. The fraction-order derivative instead of the integer-order derivative is introduced to make the constitutive model more flexible while the integer-order constitutive model can be recovered as a special case. Then, the Euler–Bernoulli beam and the Timoshenko beam are both considered, and the corresponding formulations are derived. Two problems are investigated: (1) the dispersion of traveling flexural waves and the attenuation of the standing waves in the infinite beam and (2) the natural frequency of finite beam. The numerical examples are provided, and the effects of the nonlocal and the strain gradient effects are discussed. The influences of the fraction-order parameters on the wave motion and vibration behavior are mainly studied. It is found that the strain gradient effects and the nonlocal effect have opposite influences on the wave motion and vibration behavior. The fraction order also has evident influence on the wave motion and vibration behavior and thus can refine the prediction of the model.

Dynamic Analysis of a Piezoelectrically Layered Perforated Nonlocal Strain Gradient Nanobeam with Flexoelectricity

This study presents a mathematical size-dependent model capable of investigating the dynamic behavior of a sandwich perforated nanobeam incorporating the flexoelectricity effect. The nonlocal strain gradient elasticity theory is developed for both continuum mechanics and flexoelectricity. Closed forms of the equivalent perforated geometrical variables are developed. The Hamiltonian principle is exploited to derive the governing equation of motion of the sandwich beam including the flexoelectric effect. Closed forms for the eigen values are derived for different boundary conditions. The accuracy of the developed model is verified by comparing the obtained results with the available published results. Parametric studies are conducted to explore the effects of the perforation parameters, geometric dimensions, nonclassical parameters, flexoelectric parameters, as well as the piezoelectric parameters on the vibration behavior of a piezoelectric perforated sandwich nanobeam. The obtained results demonstrate that both the flexoelectric and piezoelectric parameters increased the vibration frequency of the nanobeam. The nonlocal parameter reduced the natural vibration frequency due to a decrease in the stiffness of the structures. However, the strain gradient parameter increased the stiffness of the structures and hence increased the natural vibration frequency. The natural vibration frequency based on the NSGT can be increased or decreased, depending on the ration of the value of the nonlocal parameter to the strain gradient parameter. This model can be employed in the analysis and design of NEMS, nanosensors, and nanoactuators.

Thermal vibration and buckling of magneto-electro-elastic functionally graded porous nanoplates using nonlocal strain gradient elasticity

This study models and examines the thermal vibration and buckling behaviours of a porous nanoplate made of functional grading of barium-titanate and cobalt-ferrite. The porosity in the nanoplate is modelled with uniform and symmetrical porosity distribution functions. In the constitutive equation of the nanoplate, the strains are assumed to originate from classical mechanics, thermal expansion, electroelastic and magnetostrictive properties, nonlocal elasticity, and strain gradient elasticity. The motion equations of magneto-electro-elastic (MEE) nanoplate is obtained by Hamilton's principle. The study investigated the effects of thermal stresses, magneto-electro-elastic coupling, externally applied electric and magnetic field potential, nonlocal properties (nonlocal and material size parameters), and porosity volume fraction and function of porosity variation across thickness in free vibration and buckling behaviour of the nanoplate. According to the analysis results, the dimensionless frequencies decrease as the barium-titanate ratio in the nanoplate increases, and the frequencies increase as the cobalt-ferrite ratio increases, depending on the material grading index. While temperature increased and porosity ratio decreased dimensionless frequencies, external magnetic potential increased dimensionless frequencies. On the other hand, the application of electric potential causes a slight increase in dimensionless frequencies compared to the effect of magnetic potential.

Nonlocal strain and stress gradient elasticity of Timoshenko nano-beams with loading discontinuities

A unified approach is applied for determining both strain- and stress-driven differential formulations of Timoshenko nano-beams in presence of loading discontinuities. The consequent models can simulate small scale effects with different types of constitutive laws (such as pure nonlocal, mixture of local and nonlocal phases, and nonlocal gradient). A specific novel feature of the proposed models is the ability to consider loading discontinuities, i.e. points of discontinuities for generalized internal forces occurring in presence of external supports, forces, or couples concentrated at internal points of the nano-beam. To this end, novel constitutive continuity conditions (CCCs) are imposed at the beam interior points of loading discontinuities. CCCs contain integral convolutions of generalized forces or displacements over suitable parts of the nano-beam; they represent a valid alternative to Dirac delta function and are different from the well-known constitutive boundary conditions (CBCs) imposed at the end-points of the nano-beam. Finally, the proposed models are applied for finding closed-form solutions to cases of practical interest.

Nonlocal strain gradient IGA numerical solution for static bending, free vibration and buckling of sigmoid FG sandwich nanoplate

For the first time, a numerical isogeometric numerical solution based on the nonlocal strain gradient elasticity theory for static bending, free vibration, and buckling of sigmoid functionally graded (S-FG) nanoplate is presented. Two configurations of S-FG sandwich material, including isotropic core and FG core, are considered. A parameter is proposed to define the location of the neutral axis through the cross-section. The simple Reissner-Mindlin plate theory, the nonlocal strain gradient theory, and Hamilton's principle are employed to establish the general equilibrium of S-FG nanoplate that contains two small size coefficients, including nonlocal and strain gradient parameters. The static bending, free vibration, and buckling responses of S-FG nanoplate are explored using isogeometric analysis with a NURBS basic function. The accuracy of the presented model is verified through the comparison with other solutions for nanoplate. As a result of numerical investigation studies, the static bending, free vibration, and buckling responses of S-FG are significantly affected by the material variation along the thickness direction, the neutral axis location, nonlocal parameter, strain gradient parameter, and material index.

Nonlinear forced vibrations of a slightly curved nanotube conveying fluid based on the nonlocal strain gradient elasticity theory

Nonlinear forced vibrations of a slightly curved nanotube conveying fluid based on the nonlocal strain gradient elasticity theory