Strain Gradient Elasticity Theory Research Articles

Application of the Green's function method for static analysis of nonlocal stress-driven and strain gradient elastic nanobeams

This paper focuses on the static analysis of a generally restrained Euler-Bernoulli nanobeam (GRNB), which is subjected to arbitrary distributed or concentrated loads. The small length scale effect is introduced through a strongly nonlocal integral elastic law called the stress-driven nonlocal Euler-Bernoulli beam model, and a weakly nonlocal elastic law called the strain-gradient Euler-Bernoulli beam model. The stress-driven nonlocal theory and the strain gradient elasticity theory are both employed, to formulate the differential equation governing the Euler-Bernoulli nanobeams. Both theories are ruled by a similar 6th order differential equation, with distinct higher-order boundary conditions. The paper demonstrates that the strain gradient theory can be also reformulated as a stress-driven nonlocal theory with unsymmetrical kernel, in contrast to the initial stress-driven nonlocal theory which has symmetrical exponential kernel. Additionally, some theoretical backgrounds about both nonlocal elasticity theorems are provided. Exact solutions for both nanobeam models are presented, using Green's function method. The parameters of the Green's functions are derived for each beam theory, and then utilized to establish the static displacement function of both nanobeam theories. Exact solutions are provided for the bending of both nanobeams with various end boundary conditions, that account for the small length scale effects. The paper also compares the responses of the stress-driven Euler-Bernoulli beam and the strain gradient Euler-Bernoulli beam, showing close deflections and similar effects of increasing the nonlocal parameter on beam stiffness. A major difference highlighted between the two theories is that the strain gradient theory predicts a homogeneous strain state for homogeneous stress state, as opposed to the stress-driven theory.

Non-singular straight dislocations in anisotropic crystals

A non-singular dislocation theory of straight dislocations in anisotropic crystals is derived using simplified anisotropic incompatible first strain gradient elasticity theory. Based on the non-singular theory of dislocations for anisotropic crystals, all dislocation key-formulas of straight dislocations are derived in generalized plane strain, for the first time. In this model, the singularity of the dislocation fields at the dislocation core is regularized owing to the nonlocal nature of strain gradient elasticity. The non-singular dislocation fields of straight dislocations are obtained in terms of two-dimensional anisotropic Green functions of simplified anisotropic strain gradient elasticity. All necessary Green functions, including the two-dimensional Green tensor of the twofold anisotropic Helmholtz-Navier operator and the two-dimensional F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{F}$$\\end{document}-tensor of generalized plane strain, are derived as sum of the classical part and a gradient part in terms of Meijer G-functions. Among others, we calculate the regularization of the Barnett solution for the elastic distortion of straight dislocations in an anisotropic crystal. In the framework of simplified anisotropic first strain gradient elasticity, the necessary material parameters are computed for cubic materials including aluminum (Al), copper (Cu), iron (Fe) and tungsten (W) using a second nearest-neighbour modified embedded-atom-method interatomic potential. The elastic distortion and stress fields of screw and edge dislocations of 12⟨111⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{1}{2} \\langle 111\\rangle$$\\end{document} Burgers vector in bcc iron and bcc tungsten and screw and edge dislocations of 12⟨110⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{1}{2} \\langle 110\\rangle$$\\end{document} Burgers vector in fcc copper and fcc aluminum have been computed and presented in contour plots.

Peridynamic correspondence model with strain gradient elasticity for microstructure dependent size effects

This study presents a peridynamic (PD) correspondence model with strain gradient elasticity (SGE) to capture size effect on strength of nano- and micro-scale structures. The classical elasticity theory lacking a length scale parameter does not account for the microstructure-dependent size effects. Strain Gradient (SG) elasticity is an extension of the classical elasticity with a length scale parameter(s) which can be linked to the microstructure of the material. Peridynamic theory introduces damage into the constitutive relations in a natural way by allowing for interactions of a material point within its horizon which is referred to as the internal length parameter. The correspondence model of the PD theory permits the incorporation of the constitutive relationship of SG theory. Therefore, the present approach combines the effects of PD and SGE length scale parameters on the stiffness and strength of the material. The resulting equations present two length parameters: the horizon of a material point in PD theory and the characteristic length in SGE theory. The PD equation of motion with SGE (PDSG) is free of spatial derivatives and allows for the imposition of classical and nonclassical boundary conditions. The PDSG is first applied to investigate the deformation response of a one-dimensional nanoscale film subjected to a quasi-static tensile load and a dynamic load through an initial constant strain. The static response is compared with the analytical solution for varying ratio of SG length parameter to PD horizon. The dynamic response is compared with a computational solution while satisfying the nonclassical boundary conditions. Subsequently, the PDSG is applied to study the deformation response of a two-dimensional nanoscale film subjected to a quasi-static axial load and a specified tangential end displacement. These two cases are considered under special displacement constraints to compare with the available analytical solutions. Subsequently, a two-dimensional nanoscale film with or without an internal crack is subjected to uniform tensile loading. Finally, crack propagation is simulated under incremental displacement loading. The results capture the analytical predictions and the expected increase in stiffness with increasing ratio of SG length parameter to PD horizon.

On mechanics of piezocomposite shell structures

This study presents an original and novel investigation into the mechanics of piezo-flexo-magneto-elastic nanocomposite doubly-curved shells (PFMDCSs) and the ability to detect the lower and higher levels of electro-magnetic fields. In this context, by utilizing the first-order shear deformation shell model, stresses and strains are acquired. By imposing Hamilton's principle and the von Kármán approach, the governing equations have been obtained. The intelligent shell model consists of size-dependent influences, viz., strain gradients. This will take place via Mindlin's strain gradient elasticity theory and the subsequent re-establishing of the mathematical framework by incorporating this concept. The strain gradient results in a flexoelectric/flexomagnetic effect. The converse effect of the magnetic field on the basis of a close circuit has been assumed. The developed bending equations have been transferred into the algebraic ones by substituting an analytical technique based on homogeneous immovable simple support for the four edges. The problem has been solved according to the Newton-Raphson iteration scheme, and transverse deflections have been computed. For researching the rightness and precision of the shell models together with the solution process, a comparison is prepared by the finite element method (FEM) results for simplified shells, and a good correlation has been observed. At last, by examining several factors governing the problem, the conditions under which the magnetic effects can be noticeable and dominant in doubly-curved shells have been sought. This study could serve as a benchmark reference for piezoceramic-DCSs, as the presented governing equations are original. The most interesting outcome of this research is that the electro-magnetic response of intelligent structures can be entirely geometry-dependent.

The strain gradient viscoelasticity full field solutions for Mode-I and Mode-II crack problems

The impacts of size and viscosity become distinctly evident when considering micro- and nano-scale phenomena for advanced materials with micro- and nanostructures. In this study, the mechanical behavior of the advanced material is characterized by using the strain gradient viscoelasticity theory, and a novel solution is presented for the mode-I and mode-II cracks, which is formulated based on the strain gradient viscoelasticity theory, employing the Wiener-Hopf method. Besides, the gradient-dependent viscoelastic crack solutions are directly derived by applying the correspondence principle that aligns strain gradient viscoelasticity with strain gradient elasticity within the Maxwell's standard linear solid model. Relative to the influence of elastic strain gradient effects, the involvement of viscous gradient effects instigates a reinforcement to the stress field around the crack tip, thereby offering a more reasonable representation of advanced materials crack behavior. When the viscosity effect is omitted or as time tends to infinity, the solutions based on strain gradient viscoelasticity theory converges to those on the classical strain gradient elasticity theory.

In-plane surface waves propagating in a coated half-space based on the strain-gradient elasticity theory

By employing the strain gradient elasticity theory, we investigate the propagation of in-plane surface waves in a coated half-space with microstructures. We first investigate the general case of the present problem, that is, both the surface layer and the half-space are described by the strain-gradient elasticity theory. We formulate the boundary and continuity conditions of the general case and derive the dispersion relations of the surface waves. Then we investigate two special cases: (1) the surface layer is described by the strain-gradient elasticity theory, while the half-space by the classical elasticity theory; (2) the surface layer is described by the classical elasticity theory while the half-space by the strain-gradient elasticity theory. We examine the effects of strain-gradient characteristic lengths on the dispersion curves of surface waves in all cases. This study helps to further understand the propagation characteristics of elastic waves in materials with microstructures.

Plane crack problems within strain gradient elasticity and mixed finite element implementation

AbstractAn alternative 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 (FEM) based on the concept of the Galerkin method is used. To construct finite-dimensional subspaces separate approximations of displacements, deformations, stresses, and their gradients are implemented by choosing the different sets of piecewise polynomial basis functions, interrelated by the stability condition of the mixed FEM approximation. This significantly simplifies the pre-requirement for approximating functions to belong to class C1 and allows one to use the simplest triangular finite elements with a linear approximation of displacements under uniform or near-uniform triangulation conditions. Global unknowns in a discrete problem are nodal displacements, while the strains and stresses and their gradients are treated as local unknowns. The conditions of existence, uniqueness, and continuous dependence of the solution on the problem’s initial data are formulated for discrete equations of mixed FEM. These are 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. This avoids the need to form a global stiffness matrix for the problem of strain gradient elasticity since it is enough to calculate only the residual vector in the current approximation. A set of modeling plane crack problems is solved. The obtained solutions agree with the results available in the relevant literature. Good convergence is achieved by refining the mesh for all scale parameters. All three problems under study exhibit specific qualitative features characterizing strain gradient solutions namely crack stiffness increase with length scale parameter and cusp-like closure effect.

Effect of the magnetic field on the thermomechanical flexural wave propagation of embedded sandwich nanobeams

This work examines thermo-mechanical bending wave propagation in a sandwich nanobeam using advanced sandwich nanobeam and nanolocal strain gradient elasticity theories. The sandwich nanobeam is a unique structure with biocompatible ceramic ZrO2 and metal Ti6Al4V on the top and bottom sides. Sandwich nanobeam cores have functionally graded materials. This combination gives the nanobeam distinctive qualities and opens up many uses in diverse industries. The wave propagation equation is computed by applying the Navier method to the medium’s thermal, Lorentz, and viscoelastic equations of motion. The sandwich nanobeam is analyzed using four distinct models, taking into account its composition of ceramic and metal materials. The various factors that affect sandwich nanobeam bending wave propagation have been extensively studied. In the scenario where the magnetic field intensity is Hm  = 0, an increase in temperature difference causes the wave frequency of all models (except Model 2) to decrease to zero, resulting in buckling. In Model 2, the sandwich nanobeam exhibits a phase velocity of 0.43 Km/s at ΔT = 0, which subsequently decreases by ∼9% to 0.39 km/s at ΔT = 500. These factors include the strength of the magnetic field, the impact of thermal loads, the nonlocal effect, the dimensions of the sandwich nanobeam, and the foundation’s influence. The findings of this research will help build nanosensor systems that can work in aerospace applications under extreme temperatures. These findings will contribute to the optimization of the design process, ensuring the reliability and functionality of the nanosensors under severe thermal conditions.

Nonlinear dynamic behavior of carbon nanotubes incorporating size effects

This research endeavors to investigate the geometrically nonlinear dynamic characteristics of Carbon Nanotubes (CNTs) while incorporating size effects based on the Second Strain Gradient (SSG) elasticity theory. To this end, the nonlinear governing equations and its corresponding boundary conditions are deduced in alignment with the Green–Lagrange strain tensor and Hamilton’s principle. Concurrently, the weak form is expounded through the utilization of the C3 continuum Hermite interpolation functions, which ensure the continuity and smoothness of higher-order strain and displacement fields. Subsequently, a perturbation methodology is introduced, incorporating nonlinear phenomena into the context of linear wave propagation within the framework of periodic structures theory. The wave propagation characteristics manifest a pronounced disparity between the models based on linear SSG theory and those through nonlinear SSG theory with stiffness hardening phenomenon. In contrast to the zigzag CNTs, armchair CNTs evince an elevated aptitude for wave control. The application of nonlinear SSG theory in combination with the wave finite element method is significant for comprehending the wave propagation characteristics of complex CNTs.

Plane strain problem of flexoelectric cylindrical inhomogeneities

In nanotechnology, flexoelectric solids exhibit notable electrical polarization induced by internal strain gradients, rendering them promising for various applications. However, inherent material imperfections are inevitable. Particularly within flexoelectric solids, substantial strain gradients exist in proximity to internal defects, resulting in localized concentration of electrical polarization and potential structural failure. Among the various defect types, circular-shaped inhomogeneity is prevalent. This paper comprehensively investigates the plane strain problem on cylindrical inhomogeneities within flexoelectric solids. The full-field analytical solution is derived for this problem for the first time. Given that flexoelectric theory encompasses pure strain gradient elasticity theory, the strain gradient elasticity solution for plane strain cylindrical inhomogeneities is also established for the first time. This study reveals that the stiffness and size of the inhomogeneity, along with the loading ratio in two directions, exert a noteworthy influence on the local electromechanical coupling behavior near the inhomogeneity. Finally, the mixed finite element method is utilized to approximate the solution numerically, and the close agreement between the finite element results and the analytical solution demonstrates this study’s reliability and rigor. Therefore, this investigation imparts valuable insights into examining defects in flexoelectric solids and serves as a foundation for studying more intricate defect typologies.

Size effects on a one-dimensional defective phononic crystal sensor

The influence of size effects on one-dimensional defective phononic crystal (PnC) sensors based on simplified strain gradient elasticity theory (SSGET) is studied in this paper. PnCs have been widely used in high-sensitivity gas and liquid sensors by introducing defects to disrupt the perfect PnC modes. In comparison with classical elasticity theory, the SSGET includes two microstructure-related material parameters that can accurately reflect the size effects of the structure. In this paper, the stiffness matrix method was used to calculate the transmission coefficients of the proposed model, avoiding the numerical instability of the transfer matrix method. The results show that the size effects at the microscale affect the perfect PnC bandgap’s frequency range, and the microstructure constants impress the resonant frequency while detecting liquids. Consequently, the accuracy of the sensor is reduced. These findings provide a theoretical basis for designing microscale PnC sensors.

Bending and wave propagation analysis of axially functionally graded beams based on a reformulated strain gradient elasticity theory

Bending and wave propagation analysis of axially functionally graded beams based on a reformulated strain gradient elasticity theory

Airy stress based nonlinear forced vibrations and internal resonances of nonlocal strain gradient nanoplates

This paper presents a novel investigation of nonlinear forced vibrations and internal resonance in nonlocal strain gradient nanoplates, taking into account in-plane motions. The study comprehensively models the nanoplate structure, employing Kirchhoff's plate theory, nonlocal elasticity theory, and strain gradient elasticity theory. The incorporation of the Airy stress function allows for the consideration of in-plane displacements. The large amplitude deformations in the nanoplate are described using the von Kármán geometric nonlinearity model, and the equations of motion are derived using the force-moment dynamic balance method. The coupled equations of motion are solved using the Galerkin scheme and a dynamic equilibrium technique. Detailed discussions are provided on the influence of nonlocal and strain gradient parameters on the nonlinear vibration response of the Airy stress nanostructure. Moreover, it is demonstrated that specific combinations of nonlocal and strain gradient parameters lead to various types of internal resonance (including two-to-one and three-to-one internal resonances), significantly affecting the nonlinear frequency responses of the nanoplate, resulting in rich nonlinear behaviours. This study contributes to the understanding of the complex dynamics of nanoplates subjected to external time-dependant loads and offers valuable insights for the design of diverse nanoplate systems, including graphene sheets, which are relevant to nanoelectromechanical systems (NEMS) applications.

Analysis of Flexoelectric Solids With a Cylindrical Cavity

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

Gradient elasticity solutions of 2D nano-beams

In this study, the exact analytical solutions of a two-dimensional linear homogeneous isotropic nano-beam in gradient elasticity are studied. Four different types of two-dimensional cantilever beams and related boundary conditions are considered. The cases are a cantilever beam under a concentrated force at the end, a cantilever beam under a uniform load, a propped cantilever beam under a uniform load, and a fixed-end beam under a uniform load. The two-dimensional stress gradient fields are investigated and obtained from the analytical solutions of a linear second-order partial differential equation written in terms of the classical and the gradient Airy stress functions. Additionally, the micro-size effects in the displacement components for different loads and support conditions for the two-dimensional cantilever beams by using strain gradient elasticity theory are investigated. Furthermore, for one-dimensional Euler–Bernoulli beam model, the associated stress and strain elasticity solutions are obtained from two-dimensional analytical solutions. The graphical presentations of the exact closed-form solutions are provided and discussed.

Second-order homogenization of 3-D lattice materials towards strain gradient media: numerical modelling and experimental verification

The literature in the field of higher-order homogenization is mainly focused on 2-D models aimed at composite materials, while it lacks a comprehensive model targeting 3-D lattice materials (with void being the inclusion) with complex cell topologies. For that, a computational homogenization scheme based on Mindlin (type II) strain gradient elasticity theory is developed here. The model is based on variational formulation with periodic boundary conditions, implemented in the open-source software FreeFEM to fully characterize the effective classical elastic, coupling, and gradient elastic matrices in lattice materials. Rigorous mathematical derivations based on equilibrium equations and Hill–Mandel lemma are provided, resulting in the introduction of macroscopic body forces and modifications in gradient elasticity tensors which eliminate the spurious gradient effects in the homogeneous material. The obtained homogenized classical and strain gradient elasticity matrices are positive definite, leading to a positive macroscopic strain energy density value—an important criterion that sometimes is overlooked. The model is employed to study the size effects in 2-D square and 3-D cubic lattice materials. For the case of 3-D cubic material, the model is verified using full-field simulations, isogeometric analysis, and experimental three-point bending tests. The results of computational homogenization scheme implemented through isogeometric simulations show a good agreement with full-field simulations and mechanical tests. The developed model is generic and can be used to derive the effective second-grade continuum for any 3-D architectured material with arbitrary geometry. However, the identification of the proper type of generalized continua for the mechanical analysis of different cell architectures is yet an open question.

An analytical model for nanoscale flexoelectric doubly curved shells

The phenomenon of flexoelectricity, wherein the generation of electrical polarization results from strain gradients, has garnered significant attention in electromechanics. This phenomenon holds immense potential for diverse applications in nanoelectromechanical systems. Doubly curved shells present a particularly compelling structure for implementing flexoelectric devices, representing the most general scenario. To comprehensively understand the mechanical behavior of nanoscale flexoelectric doubly curved shells, we present a novel analytical model in this study. Our model incorporates a reformulated strain gradient elasticity theory, a modified expression of electric enthalpy density considering Maxwell’s self-field gradient, and a moderately thick shell configuration. To further elucidate this model, we establish complete formulations and analytical solutions of static bending and free vibration problems. This study reveals the crucial roles that various parameters such as principal curvature radii, length scale parameters, transverse structural sizes, and the Winkler–Pasternak foundation play in controlling the mechanical behavior of nanoscale flexoelectric doubly curved shells. These theoretical findings offer valuable insights into the design of nanodevices based on the direct flexoelectric effect.

On Flamant–Boussinesq problem with dynamical flexoelectric effect and micro-inertia effect in dielectrics subjected to dynamical wave loading

This work investigates the Flamant–Boussinesq problem for a half-space made of a homogeneous and isotropic dielectric material. The dynamical flexoelectric effect and the dynamical flexocoupling between displacement and polarization, due to mechanical and electrical states, are taken in consideration. The mechanical loading is taken as a wave of a decaying behavior in time at the surface of a half-space, while the electric potential is considered in an open circuit with no charge on the terminals. The first strain gradient theory of elasticity is used as a mathematical frame in the problem formulation. The equation of motion for the representative volume element additionally accounts for the micro-inertia effect because of the intrinsic structure of the dielectrics at the nanoscale. The governing equations and the boundary conditions for homogeneous, isotropic dielectric material are presented with reference to previous work, using a variational technique for internal energies and external forces. An analytical harmonic wave solution is obtained for the problem under consideration, involving different coupling parameters arising from the mechanical and electrical loadings. The results are analyzed and discussed. The solutions for the quantities of practical interest are represented graphically with different choices of material parameters and flexocoupling parameters. The solution is finite everywhere. The existing damping phenomenon arises, not only from the various physical parameters in the governing field equations as shown in the figures, but also through the boundary conditions.

Peridynamics with strain gradient for modeling carbon nanotube under static and dynamic loading

This study couples peridynamic (PD) theory with strain gradient elasticity (SGE) theory to investigate the combined effect of PD and SGE length scale parameters on size effect. The SGE theory introduces a length scale parameter in the stress–strain relations to account for size effect. The solution to the classical form of SGE equation of motion requires two additional non-classical boundary conditions arising from the presence of fourth-order spatial derivatives. The wave dispersions level off as the wave number increases as observed in real materials. The PD theory allows for nonlocal interactions of a material point within its horizon which serves as the PD length scale parameter. The equation of motion emerges in the form of an integral equation and the internal forces are expressed through nonlocal interactions (bonds) between the material points within a continuous body. It is a reformulation of classical continuum mechanics which is free of spatial derivatives. The numerical results concern a previously considered carbon nanotube (CNT). Under quasi-static axial loading, the results indicate an increased strengthening effect along the length of the tube. Under dynamic loading, its longitudinal vibration response is captured through explicit time integration. The numerical predictions capture the reference solutions corresponding to the classical SGE equation.

Isogeometric analysis of small-scale effects on the vibration of functionally graded porous curved microbeams based on the modified strain gradient elasticity theory

Isogeometric analysis of small-scale effects on the vibration of functionally graded porous curved microbeams based on the modified strain gradient elasticity theory