Gradient Elasticity Theory Research Articles

Elastostatic analysis using the boundary element method and the Aifantis gradient elasticity theory

The increasing use of micro and nano-scale structures has sparked interest in theories incorporating the effect of scale since the classical continuum theory has limitations in capturing effects that depend on size. Therefore, three-dimensional elastostatic microstructure modeling is carried out in this work using the boundary element method (BEM).To account for microstructural effects, the simplified gradient theory proposed by Aifantis, a particularization of Mindlin's general theory, was employed. A variational argument was established to determine the governing equations and boundary conditions for the problem. This argument explains the fundamental solution of gradient elasticity, and the integral contour representation is constructed with the aid of the reciprocal identity. Curved triangular elements of Proriol spectral functions were used to approximate the geometry and the physical parameters for the discretization of the BEM. The presented formulation yielded results consistent with other analyses in the literature.

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

In contrast to classical mechanics, which primarily relies on continuum assumptions and neglects micro-structural effects, the strain gradient elasticity (SGE) theory represents a paradigm shift in understanding the mechanical behavior of materials at small length scales. In this article, the influence of the bi-directional material gradation on a mode-III crack in functionally graded material via SGE theory is studied. The SGE theory uses two material characteristic lengths, ℓ and ℓ′, to account for volumetric and surface strain-gradient factors, respectively. Our investigation is centered on a material gradation model assumed to vary exponentially, with the shear modulus represented as G(x,y)=G0eβx+γy, where β and γ are material gradation constants. To address the crack boundary value problem under consideration, we employ a methodology combining Fourier transforms and an innovative hyper-singular integro-differential equation approach. Using this approach, we systematically formulate a system of equations, which can be solved by selecting suitable collocation points. The closed-form analytical expressions are derived for the standard fracture parameters such as crack surface displacement (CSD), stress intensity factor (SIF), and energy release rate (ERR). Numerical studies are illustrated for the derived standard fractures, and the influence of these parameters β, γ, ℓ, ℓ′, and applied shear load is graphically presented. Through comprehensive analysis, our aim is to provide insights into the complex interplay between material parameters, loading conditions, and crack behavior in functionally graded materials.

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 problem is solved within the second form of Mindlin's first strain gradient elasticity theory for the prediction of the effective elastic properties of composites. Considering Green's function technique, an integral equation is established for an ellipsoidal inhomogeneity embedded in a homogeneous elastic medium and subjected to non-uniform boundary conditions. Within isotropic elasticity, the mean strain inside a spherical inhomogeneity is detailed to provide analytical results. In addition to the elastic properties of the inhomogeneity and the matrix, the strain localization depends on five gradient elastic constants, introduced by the first strain gradient elasticity theory. The effective bulk and shear moduli of a two-phase composite are predicted through Mori-Tanaka's scheme. The strain localization and the effective elastic moduli are then expressed within some simplified gradient elasticity theories. To test the relevance of the developed model, its predictions are compared with those of some investigations and the effective elastic properties are analyzed for a metal matrix composite. Finally, some comparisons with experimental data are performed to estimate the characteristic length scale parameters and gradient elastic constants of local phases.

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

In this paper, we investigate the axisymmetric Hertzian contact problem at micro-/nanoscale. The deformation of material bulk is described by a simplified theory of strain gradient elasticity, and the influence of surface tension is integrated based on the surface elasticity theory. Using the Mindlin's potential function method and double Fourier integral transform, the normal surface displacement induced by a concentrated force is derived in a closed form. Following this, the contact between a rigid sphere and an elastic half-space is formulated in terms of singular integral equation, which is numerically solved by applying the Gauss-Chebyshev method. The results indicate that the distribution of contact pressure is distinctly different from that in classical elasticity theory. The indented substrate tends to perform stiffer due to the effects of surface tension and strain gradient elasticity. When the contact radius is comparable with the material length parameter, the indentation force (depth) can be ten (three) times of that given by classical Hertz 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.

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

In this article, the strain gradient elasticity (SGE) theory is used to investigate the mechanical behaviour of a functionally graded material (FGM) weakened by two unequal collinear mode-III cracks. The SGE theory uses two material characteristic lengths, ℓ and ℓ′, to account for volumetric and surface strain-gradient factors, respectively. The directions of both cracks coincide with the material gradation, which is oriented parallel to the x-axis. The numerical outcomes of the problem are obtained by constructing the system of equations using the hyper-singular integral-differential equation technique. The influence of the material gradation parameter on various fracture parameters, including the stress intensity factor (SIF), strain and crack surface displacement (CSD), is numerically shown. In addition, different loads are considered while analysing CSD profiles in connection to the gradation parameter β. Furthermore, the effect of inter-crack distance on CSD profiles is comprehensively explored, showing information between crack geometry and material gradation.

Investigation of Thermal Impacts and Different Gradient Elasticity Theories on Wave Propagation through the Polymer Matrix Incorporated Carbon Nanotube Walls

Investigation of Thermal Impacts and Different Gradient Elasticity Theories on Wave Propagation through the Polymer Matrix Incorporated Carbon Nanotube Walls

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

We study inextensional vibrations of the spherical shell using strain gradient elasticity theory under Kirchhoff-Love hypotheses. For admissibility of the inextensional deformations the shell is punctured by making a tiny hole. The shell is assumed to be made of linearly elastic, homogeneous, and isotropic material. The closed form expression for the frequencies of inextensional modes of vibration of spherical caps of various polar angles and that of nearly complete spherical shell are derived using Rayleigh's method. Towards this, the displacement and velocity fields are obtained by setting the membrane strains to zero. Further, to facilitate the existence of inextensional vibrations, boundaries are assumed to be traction free. The frequencies are computed for the positive and negative signs in the strain gradient term of the constitutive law. The computed frequencies, employing negative sign are found to be higher than those computed using classical elasticity theory. The opposite effect is seen when the frequencies are computed with the positive sign. Further, the frequencies are found to saturate when the positive sign is used in the constitutive law. Parametric studies viz. variation in frequencies with the circumferential wavenumbers for different values of length scale parameter, variation in frequencies with circumferential wavenumbers for different values of polar angles and a given length scale parameter, variation of frequency with increasing polar angle for a given circumferential wavenumber, variation of saturation wavenumber with increasing polar angle, and increasing length scale parameter are carried out. The frequencies of nearly a spherical shell with increasing circumferential wavenumber are also computed.

Anisotropic functionally graded nano-beam models and closed-form solutions in plane gradient elasticity

This study delves into the investigation of exact analytical solutions for the plane stress and displacement fields within linear homogeneous anisotropic nano-beam models of gradient elasticity. It focuses on solving the Helmholtz equation, which encompasses a second-order non-homogeneous linear partial differential equation in plane gradient elasticity theory, utilizing polynomial series-type solutions. The analysis centers on the utilization of gradient Airy stress functions to derive stress fields in the gradient theory. The research yields closed-form analytical solutions for Airy stress functions, stress, and strain fields in both classical and gradient theories. The study considers five distinct types of two-dimensional functionally graded cantilever beams with various boundary conditions: a cantilever anisotropic nano-beam subjected to a concentrated force at the free end, a cantilever anisotropic nano-beam under a uniform load, a simple anisotropic nano-beam under a uniform load, a propped cantilever anisotropic nano-beam under a uniform load, and a fixed end cantilever anisotropic nano-beam under a uniform load. General analytical solutions for the gradient stress and displacement fields of two-dimensional and one-dimensional anisotropic nano-beams under different boundary conditions are provided. The study showcases significant strain gradient size effects at the nano-scale through the derived analytical solutions for anisotropic beams. Additionally, it demonstrates that the strain gradient theory results for the limit case of the gradient coefficient c precisely align with results for isotropic and anisotropic materials in elasticity theory and classical theory. Furthermore, real-world applications are discussed, considering stress and displacement fields in real anisotropic materials such as TiSi2 single crystals and orthotropic materials, which are special cases of anisotropic materials like wood and epoxy as documented in the literature.

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

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.