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Abstract: The goal of this study is to understand the physical meaning and evaluate the intrinsic length scale parameters, featured in the theories of gradient elasticity, by deploying the analytical treatment and experimental measurements of the dispersion of elastic waves. The developments are focused on examining the propagation of longitudinal waves in an aluminum rod with periodically varying cross-section. First, the analytical solution for the dispersion relationship, based on the periodic cell analysis of a bi-layered laminate and Bloch theorem, is compared to two competing models of gradient elasticity. It is shown that the customary gradient elastic model with two length-scale parameters is able to capture the dispersion accurately up to the beginning of the first band gap. On the other hand, the gradient elastic model with an additional length scale (affiliated with the fourth-order time derivative in the field equation) is shown to capture not only the first dispersion branch before the band gap, but also the band gap itself and the preponderance of the second branch. Closed form relations between the microstructure parameters and the intrinsic length scales are obtained for both gradient elasticity models. By way of the asymptotic treatment in the limit of a weak contrast between the laminae, a clear physical meaning and scaling of the length-scale parameters was established in terms of: (i) the microstructure (given by the size of the unit cell and the contrast between the laminae), and (ii) thus induced dispersion relationship (characterized by the location and the width of the band gap). The analysis is verified through an experimental observation of wave dispersion, and wave attenuation within the band gap. A comparison between the analytical treatment, the gradient elastic model with three intrinsic length scales, and experimental measurements demonstrates a good agreement over the range of frequencies considered. [Copyright &y& Elsevier]

Abstract: A gradient-enriched shell formulation is introduced in the present study based on the first order shear deformation shell model and the stress gradient and strain-inertia gradient elasticity theories are used for dynamic analysis of single walled carbon nanotubes. It provides extensions of the first order shear deformation shell formulation with additional higher-order spatial derivatives of strains and stresses. The higher-order terms are introduced in the formulation by using the Laplacian of the corresponding lower-order terms. The proposed shell formulation includes two length scale size parameters related to the strain gradients and inertia gradients. The effects of the transverse shear, aspect ratio, circumferential and half-axial wave numbers and length scale parameters on different vibration modes of the single-walled carbon nanotubes are elucidated. The results are also compared with those obtained from a classical shell theory with Sanders–Koiter strain-displacement relationships. [Copyright &y& Elsevier]

The buckling of a bar is studied analytically on the basis of a simple linear theory of gradient elasticity in the frame of the method of initial values. The method of initial values provides the values of the displacements and stress resultants throughout the bar once the initial displacements and initial stress resultants are known. We use probably for the first time the method of initial values to get critical loads of a strain gradient beam under completely different boundary conditions at the two end faces of the beam. Exact carryover matrix is presented for the classical beam and gradient beam analytically. The first mode shapes of classical beam and gradient beam are plotted. The method of initial values is also applied to the beams with variable cross-section. The priorities of the method of initial values are depicted. The variational approach gives a sixth-order ordinary differential equation for a beam in buckling. The additional boundary conditions are used to obtain critical loads. It is observed that critical loads increase dramatically for increasing values of the gradient coefficient. [ABSTRACT FROM AUTHOR]

According to Gradient Mechanics (GM), stress fields have to be determined by directly incorporating into the stress analysis a length scale which that takes into account the material microstructural features. This peculiar modus operandi results in stress fields in the vicinity of sharp cracks which are no longer singular, even though the assessed material is assumed to obey a linear-elastic constitutive law. Given both the geometry of the cracked component being assessed and the value of the material length scale, the magnitude of the corresponding gradient enriched linear-elastic crack tip stress is then finite and it can be calculated by taking full advantage of those computational methods specifically devised to numerically implement gradient elasticity. In the present investigation, it is first shown that GM's length scale can directly be estimated from the material ultimate tensile strength and the plane strain fracture toughness through the critical distance value calculated according to the Theory of Critical Distances. Next, by post-processing a large number of experimental results taken from the literature and generated by testing cracked ceramics, it is shown that gradient enriched linearelastic crack tip stresses can successfully be used to model the transition from the short- to the long-crack regime under Mode I static loading. [ABSTRACT FROM AUTHOR]

Highlights: [•] Double cantilever nanobeam model via the gradient elasticity theory is presented. [•] Stress intensity factors are obtained and exhibit size effects. [•] Stress intensity factors are reduced if the size effect is considered. [•] Fracture toughness is enhanced for nanobeams compared to large scale beams. [Copyright &y& Elsevier]

The buckling of higher-order shear plates is studied in this paper with a unified formalism. It is shown that usual higher-order shear plate models cm be classified as gradient elasticity Mindlin plate models, by augmenting the constitutive law with the shear strain gradient. These equivalences are useful for a hierarchical classification of usual plate theories comprising Kirchhoff plate theory, Mindlin plate theory and third-order shear plate theories. The same conclusions were derived by Challamel [Mech. Res. Commun. 38 (2011) 388] for higher-order shear beam models. A consistent variational presentation is derived for all generic plate theories, leading to meaningful buckling solutions. In particular, the variationally-based boundary conditions are obtained for general loading configurations. The buckling of the isotropic or orthotropic composite plates is then investigated analytically for simply supported plates under uniaxial or hydrostatic in-plane loading. An analytical buckling formula is derived that is common to all higher-order shear plate models. It is shown that cubic-based interpolation models for the displacement field are kinematically equivalent, and lead to the same buckling load results. This conclusion concerns for instance the plate models of Reddy [J. Appl. Mech. 51 (1984) 745] or the one of Shi [Int. J. Solids Struct. 44 (2007) 4299] even though these models are statically distinct (leading to different stress calculations along the cross-section). Finally, a numerical sensitivity study is made. [ABSTRACT FROM AUTHOR]

Abstract: Finite element analysis of the gradient enhanced elastic continuum requires continuity across the element boundaries. However the use of finite elements is usually avoided due to the complexity in their implementation and the excessive number of primary unknowns at each node. Hence except for a very few solid mechanics problems such as plate bending, the potential capabilities of finite elements are not widely explored. This paper presents a series of analyses where two benchmark problems of gradient elasticity are tackled by using finite elements in which displacements and their first and second derivatives are the primary nodal unknowns. It is shown that very accurate solutions are achieved with relatively coarse finite element meshes. Moreover, it is shown that finite elements are very useful for elastoplastic analyses where a large number of elements are usually needed to achieve accurate results. An elastoplastic analysis of bending in a deep beam shows that even a very coarse mesh can provide highly accurate results. Successful performance of finite elements in the above mentioned problems suggests that elements are useful alternatives for numerical investigation of the issues of size effect and strain localization in the response of elastic and elastoplastic systems. [Copyright &y& Elsevier]

Abstract: We derive generic formulae for all possible C 1 continuous polynomial interpolations for triangular elements, by considering individual shape functions, without the need to prescribe the type of the degrees of freedom in advance. We then consider the possible ways in which these shape functions can be combined to form finite elements with given properties. The simplest case of fifth-order polynomial functions is presented in detail, showing how two existing elements can be obtained, as well as two new elements, one of which shows good numerical behaviour in numerical tests. [Copyright &y& Elsevier]

Abstract: The equivalence between nonlocal and gradient elasticity models is investigated by making reference to one-dimensional boundary value problems equipped with two integral stress–strain laws proposed by Eringen (Nonlocal Continuum Field Theories (2002)). Corresponding closed-form solutions are derived through a procedure for the reduction of integral to differential equations. The reproduction of size effects in micro/nano rods is discussed. The differential formulation associated with the local/nonlocal model is shown to correspond to the strain-gradient formulation proposed by Aifantis (Mech. Mater. 35 (2003) 259–280). [Copyright &y& Elsevier]

Abstract: This paper contains a study of the problem of torsion of chiral bars with arbitrary cross-sections in the context of the linear theory of gradient elasticity. The solution is expressed in terms of solutions of four auxiliary plane problems characterized by loads which depend only on the constitutive coefficients. It is shown that, in general, the torsion produces extension (or contraction) and bending effects. The results are used to investigate the torsion of a homogeneous circular bar. In contrast with the case of achiral circular cylinders, the torsion and extension cannot be treated independently of each other. [Copyright &y& Elsevier]

Abstract: The behavior of transverse waves propagating in carbon nanotubes (CNTs) in a free space and in an elastic matrix is investigated. The CNTs are modeled as Timoshenko beams of hybrid gradient elasticity theory and the elastic matrix is modeled as a bi-parameter Pasternak foundation. A governing equation with two scale factors is derived for Timoshenko beams where shear deformation and rotary inertia are taken into account. The dispersion relation of flexural waves in CNTs is given and confirmed by molecular dynamics simulations. A comparison of the phase velocity of single-walled CNTs is made when neglecting shear deformation and/or rotary inertia. The wave speed of acoustic branch is especially focused for multi-walled CNTs and the wave speed is dependent on van der Waals interaction. The effects of the surrounding medium and scale parameters on the velocity of bending waves are discussed, in particular for acoustic mode. [Copyright &y& Elsevier]

Abstract: A dynamic analysis of an elastic gradient-dependent polymeric fiber subjected to a periodic excitation is considered. A nonlinear gradient elasticity constitutive equation with strain–dependent gradient coefficients is first derived and the dispersive wave propagation properties for its linearized counterpart are briefly discussed. For the linearized problem a variational formulation is also developed to obtain related boundary conditions of both classical (standard) and non-classical (gradient) type. Analytical solutions in the form of Fourier series for the fiber''s displacement and strain fields are provided. The solutions depend on a dimensionless scale parameter (the diameter to length radio d = D/L) and, therefore, size effects are captured. [Copyright &y& Elsevier]

Abstract: The fundamental problem of non-singular dislocations in the framework of the theory of gradient elasticity is presented in this work. Gradient elasticity of Helmholtz type and bi-Helmholtz type are used. A general theory of non-singular dislocations is developed for linearly elastic, infinitely extended, homogeneous, and isotropic media. Dislocation loops and straight dislocations are investigated. Using the theory of gradient elasticity, the non-singular fields which are produced by arbitrary dislocation loops are given. ‘Modified’ Mura, Peach–Koehler, and Burgers formulae are presented in the framework of gradient elasticity theory. These formulae are given in terms of an elementary function, which regularizes the classical expressions, obtained from the Green tensor of the Helmholtz–Navier equation and bi-Helmholtz–Navier equation. Using the mathematical method of Green’s functions and the Fourier transform, exact, analytical, and non-singular solutions were found. The obtained dislocation fields are non-singular due to the regularization of the classical singular fields. [Copyright &y& Elsevier]

ABSTRACT The Theory of Critical Distances (TCD) is a bi-parametrical approach suitable for predicting, under both static and high-cycle fatigue loading, the non-propagation of cracks by directly post-processing the linear-elastic stress fields, calculated according to continuum mechanics, acting on the material in the vicinity of the geometrical features being assessed. In other words, the TCD estimates static and high-cycle fatigue strength of cracked bodies by making use of a critical distance and a reference strength which are assumed to be material constants whose values change as the material microstructural features vary. Similarly, Gradient Mechanics postulates that the relevant stress fields in the vicinity of crack tips have to be determined by directly incorporating into the material constitutive law an intrinsic scale length. The main advantage of such a method is that stress fields become non-singular also in the presence of cracks and sharp notches. The above idea can be formalized in different ways allowing, under both static and high-cycle fatigue loading, the static and high-cycle fatigue assessment of cracked/notched components to be performed without the need for defining the position of the failure locations a priori. The present paper investigates the existing analogies and differences between the TCD and Gradient Mechanics, the latter formalized according to the so-called Implicit Gradient Method, when such theories are used to process linear-elastic crack tip stress fields. [ABSTRACT FROM AUTHOR]

Abstract: The present work aims at investigating the vibrational characteristics of single-walled carbon nanotubes (SWCNTs) based on the gradient elasticity theories. The small-size effect, which plays an essential role in the dynamical behavior of nanotubes, is captured by applying different gradient elasticity theories including stress, strain and combined strain/inertia ones. The theoretical formulations are established based upon both the Euler–Bernoulli and the Timoshenko beam theories. To validate the accuracy of the present analysis, molecular dynamics (MDs) simulations are also conducted for an armchair SWCNTs with different aspect ratios. Comparisons are made between the aforementioned different gradient theories as well as different beam assumptions in predicting the free vibration response. It is shown that implementation of the strain gradient elasticity by incorporating inertia gradients yields more reliable results especially for shorter length SWCNTs on account of two small scale factors corresponding to the inertia and strain gradients. Also, the difference between two beam models is more prominent for low aspect ratios and the Timoshenko beam model demonstrates a closer agreement with MD results. [Copyright &y& Elsevier]

Abstract: Generalized theories of continuum mechanics, such as gradient and nonlocal elasticity, have been widely used to account for the small scale effects on materials’ behavior when dealing with structures at the micro- or nano- scale. It has been demonstrated that these enhanced theories provide better approximations that are closer to experimental observations than classical ones for problems in the field of fracture mechanics, dislocations, and wave propagation. The present work investigates the capability of one-dimensional elastic models -gradient, nonlocal and mixed- to predict the dispersive behavior of traveling waves, in comparison with the Born–Karman model of lattice dynamics. The linear theories adopted herein are limited to Mindlin’s first (grade 2) and second (grade 3) strain gradient theories in elasticity with two and three intrinsic parameters and Eringen’s nonlocal elasticity theory with one and two intrinsic parameters. Mixed models of nonlocal and gradient theories with up to three intrinsic parameters are also considered. More specifically, seven 1D models are considered: one grade 2 elastic bar with micro-inertia, one grade 3 elastic 1D model, three nonlocal elastic bars -two with Helmholtz operator, and one with bi-Helmholtz operator after , one mixed nonlocal/grade 2 elastic bar with Helmholtz operator, and the mixed nonlocal model after . Only three models, under specific assumptions for their intrinsic parameters, result in matching satisfactorily the dispersion curve of Born–Karman’s atomic model. The rest suffer violation of their fundamental thermodynamic restrictions. This violation is naturally explained by further analyzing the mathematical structure of the obtained dispersion relations, via Padé approximants, whose coefficients are directly related to each model’s intrinsic parameters. [Copyright &y& Elsevier]

ABSTRACT As the dimensions of structures are scaled down to the micro- and nano-domains, the mechanical behaviour becomes size dependent and thus, we cannot expect the classical elasticity solutions to hold. In particular, recent experimental investigations of fatigue strength of metals show pronounced strengthening due to the influences of small geometrical dimensions. Based on second gradient elasticity framework as particularized on beams, closed form solutions to idealized problems of elastic cantilever bending, elastic three-point bending and elasto-plastic torsion have been found, showing considerable stiffening, toughening and hardening, respectively, compared to the classical theory predictions. In these models, the intrinsic material length scale was taken to be constant. Furthermore, we describe a gradient solid with a characteristic length which is not a fixed material parameter but depends on the amount of plastic effective strain amplitude, as obtained from cyclic strain hardening. A respective evolution law is suggested and discussed. [ABSTRACT FROM AUTHOR]

The use of gradient elasticity has been advocated in the past decades because of its ability to predict singularity-free stresses and strains around the tips of sharp cracks. However, identification and validation of the intrinsic length scales that appear in gradient elasticity is in a less advanced state, which hampers a more widespread use of gradient elasticity. In this short note, we provide validation of the gradient elasticity length scales for two polymers, namely PMMA and PVC, using static fracture experiments from the literature. The obtained length scales are in the order of magnitude of several tens of microns. [ABSTRACT FROM AUTHOR]

Abstract: Second-grade elastic materials featured by a free energy depending on the strain and the strain gradient, and a kinetic energy depending on the velocity and the velocity gradient, are addressed. An inertial energy balance principle and a virtual work principle for inertial actions are envisioned to enrich the set of traditional theoretical tools of thermodynamics and continuum mechanics. The state variables include the body momentum and the surface momentum, related to the velocity in a nonstandard way, as well as the concomitant mass-accelerations and inertial forces, which do intervene into the motion equations and into the force boundary conditions. The boundary traction is the sum of two parts, i.e. the Cauchy traction and the Gurtin–Murdoch traction, whereas the traction boundary condition exhibits the typical format of the equilibrium equation of a material surface (as known from the principles of surface mechanics) whereby the Gurtin–Murdoch traction (incorporating the inertial surface force) plays the role of applied surfacial force density. The body’s boundary surface constitutes a thin boundary layer which is in global equilibrium under all the external forces applied on it, a feature that makes it possible to exploit the traction Cauchy theorem within second-grade materials. This means that a second-grade material is formed up by two sub-systems, that is, the bulk material operating as a classical Cauchy continuum, and the thin boundary layer operating as a Gurtin–Murdoch material surface. The classical linear and angular momentum theorems are suitably extended for higher order inertia, from which the local motion equations and the moment equilibrium equations (stress symmetry) can be derived. For an isotropic material featured by four constants, i.e. the Lamé constants and two length scale parameters (Aifantis model), the dynamic evolution problem is characterized by a Hamilton-type variational principle and a solution uniqueness theorem. Closed-form solutions of the wave dispersion analysis problem for beam models are presented and compared with known results from the literature. The paper indicates a correct thermodynamically consistent way to take into account higher order inertia effects within continuum mechanics. [Copyright &y& Elsevier]

Abstract: The standard shear lag theory for elastic aligned short-fibre composites is extended to allow for a gradient of overall strain. The result is a one-dimensional strain gradient theory of the Toupin–Mindlin type. All parameters are the same as in the standard theory, and in the limits of weak strain gradients, large fibre aspect ratios or low elastic modulus ratios, the standard theory is recovered. The gradient effect is illustrated by a simple one-dimensional boundary value problem: a vertical composite rod fixed at both ends and loaded by gravity. The fibre length significantly affects the solution when the fibres are rigid and their length is near the rod length; but otherwise the effect is weak. [Copyright &y& Elsevier]

Abstract: The increasing structural use of cables and cable-systems has given rise to a substantial technical literature concerning their static and dynamic response. However, the majority of these works is based on classical elasticity theory, according to which the rigidity of the cable is neglected. The present work re-examines in detail the behavior of steel cables under in-plane loading, taking into account the effect of the material microstructure on the overall stiffness, on the basis of the well-established simplified gradient elasticity theory. Accounting for two-dimensional deformations the generalized equilibrium equations under dipolar action are assessed and the corresponding boundary value problem is solved. Numerical results obtained for a characteristic cable segment indicate a much stiffer behavior than the one predicted by classical elasticity theory. This phenomenon is observed in the rod or cable-actuated braking mechanism of bicycles and can be used in novel structural applications. [Copyright &y& Elsevier]

Abstract: A micromechanics-based approach for the derivation of the effective properties of periodic linear elastic composites which exhibit strain gradient effects at the macroscopic level is presented. At the local scale, all phases of the composite obey the classic equations of three-dimensional elasticity, but, since the assumption of strict separation of scales is not verified, the macroscopic behavior is described by the equations of strain gradient elasticity. The methodology uses the series expansions at the local scale, for which higher-order terms, (which are generally neglected in standard homogenization framework) are kept, in order to take into account the microstructural effects. An energy based micro–macro transition is then proposed for upscaling and constitutes, in fact, a generalization of the Hill–Mandel lemma to the case of higher-order homogenization problems. The constitutive relations and the definitions for higher-order elasticity tensors are retrieved by means of the “state law” associated to the derived macroscopic potential. As an illustration purpose, we derive the closed-form expressions for the components of the gradient elasticity tensors in the particular case of a stratified periodic composite. For handling the problems with an arbitrary microstructure, a FFT-based computational iterative scheme is proposed in the last part of the paper. Its efficiency is shown in the particular case of composites reinforced by long fibers. [Copyright &y& Elsevier]

Abstract: This paper presents a novel three-dimensional finite element for gradient elasticity. The new finite element BR153L9 is a straightforward extension of the two-dimensional element QU34L4 developed by Shu et al. (1999) , which is based on a mixed-type formulation. Within this paper the derivation of the mixed-type finite element scheme is reviewed and details of the implementation are given. Finally, numerical results of an extended patch test and a benchmark test with the three-dimensional finite element are presented in order to validate the formulation and prove the convergence. [Copyright &y& Elsevier]

Abstract: A gradient enhanced model is formulated which simulates micro-structurally induced wave dispersion and is capable of achieving mesh objective results when modelling strain-softening materials. Length scale parameters are introduced to incorporate both micro-inertial effects and strain smoothing. The model is formulated such that all the gradient enhancement terms are contained within the kinetic energy functional, whilst all non-linear terms are linked to the strain energy functional. In addition, discretisation of the governing equations is performed in a manner such that only -continuity is required and is symmetric in the elastic range. Dispersion analysis is performed to show the correct format that a constitutive model must conform to in order that regularisation of strain-softening material models will occur. This dispersion analysis is then used to explore how the two length scales may influence the width of the zone in which strain localisation takes place. The efficacy of the model in simulating strain-softening behaviour in a mesh objective manner and the effects of wave dispersion on damage initiation and accumulation are demonstrated in numerical examples. [Copyright &y& Elsevier]

Singularities occurring at a crack tip are a well known phenomenon; gradient elasticity has been proven to have the ability to remove singularities through additional length scale parameters which at the same time provides extra microstructural information of the material. This study uses gradient elasticity to simulate singularity-free stresses at bi-material bone-implant interfaces and outlines the relationship between the fracture orientation and maximum stresses at crack tip depicted by this model. [ABSTRACT FROM AUTHOR]

The plane strain problem is analyzed in detail for a class of isotropic, compressible, linearly elastic materials with a strain energy density function that depends on both the strain tensor ε and its spatial gradient ∇ ε. The appropriate Airy stress-functions and double-stress-functions are identified and the corresponding boundary value problem is formulated. The problem of an annulus loaded by an internal and an external pressure is solved. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we discuss various formats of gradient elasticity and their performance in static and dynamic applications. Gradient elasticity theories provide extensions of the classical equations of elasticity with additional higher-order spatial derivatives of strains, stresses and/or accelerations. We focus on the versatile class of gradient elasticity theories whereby the higher-order terms are the Laplacian of the corresponding lower-order terms. One of the challenges of formulating gradient elasticity theories is to keep the number of additional constitutive parameters to a minimum. We start with discussing the general Mindlin theory, that in its most general form has 903 constitutive elastic parameters but which were reduced by Mindlin to three independent material length scales. Further simplifications are often possible. In particular, the Aifantis theory has only one additional parameter in statics and opens up a whole new field of analytical and numerical solution procedures. We also address how this can be extended to dynamics. An overview of length scale identification and quantification procedures is given. Finite element implementations of the most commonly used versions of gradient elasticity are discussed together with the variationally consistent boundary conditions. Details are provided for particular formats of gradient elasticity that can be implemented with simple, linear finite element shape functions. New numerical results show the removal of singularities in statics and dynamics, as well as the size-dependent mechanical response predicted by gradient elasticity. [Copyright &y& Elsevier]

Abstract: A general non-local approach to regularise strain-softening continua is presented. The key idea is to introduce the gradient-type enrichment at the level of displacements (rather than some internal variable), so the model is formulated with two distinct displacement fields. In fact, gradient models based on two displacement fields are usual in non-local elasticity, where the goal is to avoid the shortcomings of classical (local) elasticity (i.e. strain singularities in statics, non-dispersive behaviour in dynamics). We show that such a gradient elasticity backbone model can be combined with any standard nonlinear constitutive driver to render a regularised model for softening inelasticity. To illustrate the generality of the approach, two prototype models (isotropic damage and von Mises plasticity) are discussed. The numerical examples show that the regularised models exhibit all of the desired features: mesh insensitivity, imperfection size insensitivity and description of size effects. [Copyright &y& Elsevier]

In the present contribution the concept of isogeometric analysis is extended towards the numerical solution of the problem of gradient elasticity in two dimensions. In gradient elasticity the strain energy becomes a function of the strain and its derivative. This assumption results in a governing differential equation which contains fourth order derivatives of the displacements. The numerical solution of this equation with a displacement-based finite element method requires the use of C-continuous elements, which are mostly limited to two dimensions and simple geometries. This motivates the implementation of the concept of isogeometric analysis for gradient elasticity. This NURBS based interpolation scheme naturally includes C and higher order continuity of the approximation of the displacements and the geometry. The numerical approach is implemented for two-dimensional problems of linear gradient elasticity and its convergence behavior is studied. [ABSTRACT FROM AUTHOR]

Abstract: Chirality is, generally speaking, the property of an object that can be classified as left- or right-handed. Though it plays an important role in many branches of science, chirality is encountered less often in continuum mechanics, so most classical material models do not account for it. In the context of elasticity, for example, classical elasticity is not chiral, leading different authors to use Cosserat elasticity to allow modelling of chiral behaviour. Gradient elasticity can also model chiral behaviour, however this has received much less attention than its Cosserat counterpart. This paper shows how in the case of isotropic linear gradient elasticity a single additional parameter can be introduced that describes chiral behaviour. This additional parameter, directly linked to three-dimensional deformation, can be either negative or positive, with its sign indicating a discrimination between the two opposite directions of torsion. Two simple examples are presented to show the practical effects of the chiral behaviour. [ABSTRACT FROM AUTHOR]

Abstract: Gradient elastic flexural Kirchhoff plates under static loading are considered. Their governing equation of equilibrium in terms of their lateral deflection is a sixth order partial differential equation instead of the fourth order one for the classical case. A variational formulation of the problem is established with the aid of the principle of virtual work and used to determine all possible boundary conditions, classical and non-classical ones. Two circular gradient elastic plates, clamped or simply supported at their boundaries, are analyzed analytically and the gradient effect on their static response is assessed in detail. A rectangular gradient elastic plate, simply supported at its boundaries, is also analyzed analytically and its rationally obtained boundary conditions are compared with the heuristically obtained ones in a previous publication of the authors. Finally, a plate with two opposite sides clamped experiencing cylindrical bending is also analyzed and its response compared against that for the cases of micropolar and couple-stress elasticity theories. [ABSTRACT FROM AUTHOR]

Abstract: This paper initiates a theoretical analysis of wave propagation of fluid-conveying single-walled carbon nanotubes based on strain gradient elasticity theory with consideration of both inertia and strain gradients, in which two small-scale parameters are accounted for. For comparison purpose, the stress gradient theory for fluid-conveying carbon nanotubes is also discussed. Both theories are formulated using either the Euler–Bernoulli or the Timoshenko beam assumptions. It is found that the results predicted by these beam models are quite different. From a continuum-based point-of-view, the combined strain/inertia gradient Timoshenko beam model and its conclusion regarding wave propagation may be more reliable. Results show that the effect of internal fluid flow on the phase velocity of upstream-travelling wave is significant when the wave number is relatively low. However, this effect may be ignored when the wave number is sufficiently high. Moreover, the two small-scale parameters related to the inertia and strain gradients are shown to significantly affect the phase velocity at higher wave numbers. The present theoretical study highlights the significance of the effects of fluid flow and small scale related to inertia gradients on wave propagation in carbon nanotubes conveying fluid. [ABSTRACT FROM AUTHOR]

The present works aims at solving the equations of dipolar gradient elasticity via the finite volume method. Initially, a general, mixed, finite volume formulation is stated. As a first approach, the equations of 1D and 2D gradient elasticity are solved via the proposed method. Numerical implementation shows that the suggested model is in excellent agreement with analytic solutions. The approach seems to be promising for extension to 3D problems also. [ABSTRACT FROM AUTHOR]

We study the dynamic behavior of a first strain gradient elastic bar with micro-inertia by means of the finite element method. The partial differential equation describing the motion of the bar expressed in terms of displacement is of fourth order with respect to the spatial variable, therefore, for the standard Galerkin formulation, Hermite elements are required. Consistent mass matrices are employed. The results are validated by comparison with some special cases where the exact solutions are derivable. Dispersion relations for the longitudinal waves are also derived. The effect of micro-inertia in the dynamic response of the bar is analyzed and comparisons are made with the classical elastic case. It is found that the micro-inertia parameter considerably affects the dynamic response of the bar and the dispersion characteristics of longitudinal waves. [ABSTRACT FROM AUTHOR]

Abstract: An asymptotic crack-tip solution under conditions of plane strain is developed for a material that obeys a special form of linear isotropic strain gradient elasticity. In particular, an elastic constitutive equation of the form is considered, where are the stress and strain tensors, are the Lamé constants, and ℓ is a material length. Both symmetric (mode-I) and antisymmetric (mode-II) solutions are developed. The asymptotic solution predicts finite strains at the crack-tip. The mode-I crack-tip displacement field u is of the formwhere are crack-tip Cartesian and polar coordinates, respectively, ν is Poisson’s ratio, and are dimensionless constants determined by the complete solution of a boundary value problem. The A- and B-terms above correspond to uniform normal strains parallel and normal to the crack line, which do not contribute to the crack-tip “energy release rate” (J-integral). Detailed finite element calculations are carried out for an edge-cracked-panel (ECP) loaded by point forces and the asymptotic solution is verified. The region of dominance of the asymptotic solution for the ECP geometry analyzed is found to be of order ℓ/10. The “energy release rate” is found to decrease with increasing ℓ. [Copyright &y& Elsevier]

Abstract: We present a micromechanically motivated form of the curvature energy in infinitesimal isotropic gradient elasticity. The basis is a homogenization/averaging scheme using a micro-randomness assumption imposed on a directional higher gradient interaction term. These directional interaction terms are matrix-valued allowing to apply the standard orthogonal Cartan Lie-algebra decomposition. Averaging over all (subgrid) directions leads to three quadratic curvature terms, which are conformally invariant when neglecting volumetric effects. Restricted to rotational inhomogeneities we motivate thereby a symmetric couple stress tensor in the infinitesimal indeterminate couple stress model of Koiter–Mindlin–Toupin-type. Relations are established to a novel conformally invariant linear Cosserat model. [Copyright &y& Elsevier]

Abstract: Analytical wave propagation studies in gradient elastic solids and structures are presented. These solids and structures involve an infinite space, a simple axial bar, a Bernoulli–Euler flexural beam and a Kirchhoff flexural plate. In all cases wave dispersion is observed as a result of introducing microstructural effects into the classical elastic material behavior through a simple gradient elasticity theory involving both micro-elastic and micro-inertia characteristics. It is observed that the micro-elastic characteristics are not enough for resulting in realistic dispersion curves and that the micro-inertia characteristics are needed in addition for that purpose for all the cases of solids and structures considered here. It is further observed that there exist similarities between the shear and rotary inertia corrections in the governing equations of motion for bars, beams and plates and the additions of micro-elastic (gradient elastic) and micro-inertia terms in the classical elastic material behavior in order to have wave dispersion in the above structures. [Copyright &y& Elsevier]

Abstract: In this paper a new Kirchhoff plate model is developed for the static analysis of isotropic micro-plates with arbitrary shape based on a modified couple stress theory containing only one material length scale parameter which can capture the size effect. The proposed model is capable of handling plates with complex geometries and boundary conditions. From a detailed variational procedure the governing equilibrium equation of the micro-plate and the most general boundary conditions are derived, in terms of the deflection, using the principle of minimum potential energy. The resulting boundary value problem is of the fourth order (instead of existing gradient theories which is of the sixth order) and it is solved using the Method of Fundamental Solutions (MFS) which is a boundary-type meshless method. Several plates of various shapes, aspect and Poisson’s ratios are analyzed to illustrate the applicability of the developed micro-plate model and to reveal the differences between the current model and the classical plate model. Moreover, useful conclusions are drawn from the micron-scale response of this new Kirchhoff plate model. [Copyright &y& Elsevier]

The discretisation aspects are investigated of a recently developed gradient elasticity model that is capable of describing wave dispersion and microstructural effects. The model includes higher-order stiffness terms alongside higher-order inertia terms. To avoid the need for $${\fancyscript{C}^1}$$ -continuous interpolations in a numerical implementation, the field equations are rewritten using a recently developed operator split. The equations are discretised in space using finite elements and discretised in time using the Newmark time integrator. Firstly, the critical time step is derived for use with conditionally stable time integrators. Next, the dispersion behaviour of the continuum is compared with the dispersion behaviour of the discretised medium, which allows the formulation of rules on how to select the element size and the time step size. These rules are then verified in a one-dimensional bar example and in a two-dimensional example of wave propagation. It follows that the element size must be chosen roughly equal to the smallest of the intrinsic length scales, while the optimal time step follows from the ratio of element size and wave velocity and from the ratio of the length scales. [ABSTRACT FROM AUTHOR]

Abstract: The phenomenon of wave dispersion in dry sand is studied both by purely analytical studies and by analytical–numerical experiments on the basis of gradient elastic and viscoelastic material models. These material models are employed in order to simulate the microstructural characteristics of dry sand. The analytical studies treat the material body as a one-dimensional (for the viscoelastic case) and three-dimensional (for the gradient elastic case) and for both material models provide explicit expressions for the velocity of propagation of harmonic compressional (P) and shear (S) waves. These velocities are found to be functions of frequency, i.e., dispersive. The analytical–numerical studies treat the material as a one-dimensional one and try to simulate P and S wave propagation along the axial direction of cylindrical dry sand specimens. Thus, a sinusoidal pulse with a specific frequency is applied at one end of the specimen and the response is determined at some other point by solving a transient dynamic boundary value problem with the aid of a numerical Laplace transform. This analytical–numerical experiment is repeated for various frequencies. Thus, one determines the velocities of P and S waves as functions of frequency, thereby proving again that wave propagation in dry sand is dispersive. [Copyright &y& Elsevier]

Bi-material interfaces are studied with cracks that end perpendicular to the interface. As is well-known, singularities in the stresses appear when classical elasticity is used. Moreover, the nature of the singularity depends on the difference in elastic constants of the two materials. In this paper, the gradient elasticity theory of Aifantis is used to remove these singularities. This is demonstrated for a range of ratios between the two Young’s moduli. [ABSTRACT FROM AUTHOR]

Smart composite materials have been an active field of research in the last few decades. Magnetorheological elastomers (MREs) are examples of such smart composites and they show a coupling between magnetism and elasticity. MREs are heterogeneous materials and they consist of magnetic particles and a silicone-based elastomer. To describe and predict the behaviour of the macroscopic continuum accurately, microstructural information needs to be taken into account in the analysis of heterogeneous materials. Therefore, a combined approach called multi-scale analysis is used to consider various scales of observation simultaneously. The concept of Representative Volume Element (RVE) is typically employed by multi-scale approaches to describe the micro scale, and thus the size of RVE becomes a model parameter in such techniques. This has motivated the determination of the RVE size and the derivation of magnetoelastic constitutive relations in terms of the RVE sizes in our paper. The finite element method and a statistical analysis based on the coefficient of variation have been used to determine the RVE size of MREs. The results show that it is possible to determine a lower bound of the RVE size for an MRE. Furthermore, a parametric study has been conducted to examine the sensitivity of the RVE size on the different material properties of the constituents. It was found that the RVE size is primarily set by the contrast of the different material properties, i.e. the stiffness, permeability and magnetoelastic coupling coefficients. • A statistical analysis followed to determine lower bound of the RVE sizes for an MRE. • Contrast in material properties of the constituents can affect the minimum RVE sizes. • For practical purposes one may assume same RVE size for all RVEs. • A piezomagnetic continuum model developed with gradients in terms of RVE sizes. [ABSTRACT FROM AUTHOR]

In the first part of our review paper, we consider the problem of approximating the Green's function of the Lagrange chain by continuous analogs. It is shown that the use of continuous equations based on the two-point Padé approximants gives good results. In the second part of the paper, the problem of singularities arising in the classical theory of elasticity with affecting concentrated loadings is considered. To overcome this problem, instead of a transition to the gradient theory of elasticity, it is proposed to change the concept of concentrated effort. Namely, the Dirac delta function is replaced by the Whittaker–Shannon–Kotel'nikov interpolating function. The only additional parameter that characterizes the microheterogeneity of the medium is used. An analog of the Flamant problem is considered as an example. The found solution does not contain singularities and tends to the classical one when the microheterogeneity parameter approaches zero. The derived formulas have a simpler form compared to those obtained by the gradient theory of elasticity. [ABSTRACT FROM AUTHOR]

The current study focuses on the wave dispersion analysis of granular systems by investigating a paradigmatic one-dimensional discrete chain. Most of the analysis of the discrete problem are based on the discrete granular chain without additional confinement. This work is motivated by the detection of standing waves and negative velocity of acoustic and optical waves observed in discrete granular models. This study allows to understand better the incorporation between the theoretical models, granular materials and the included effective parameters. Concerning wave propagation, granular elements create a structured wave-guide network through which mechanical energy is transferred. In the presence of internal (microstructural) length scales, the elastic wave propagation problem involves an interplay between wave dispersion and structural features. The unidimensional granular chain is composed of uniform spherical grains elastically connected with both shear and rotational springs. The presented structural system can be considered as an elastic lattice model or a Cosserat chain model with shear interaction. We show how the elastic foundation affects the wave dispersion response of the system. The length scale (grain diameter) at which the system is probed, is an important issue bridging together multi-scale behavior and heterogeneity. In a first step, the wave dispersion of the discrete system is derived from the uncoupled mixed differential-difference equation of deflection. We further show that for infinite number of grains the dispersion equation of the discrete model converges toward the continuum model of Bresse-Timoshenko. The wave dispersive properties of this discrete model are investigated also in the Brillouin zone. Next, the discrete to continuous model is approximated by the methods of Taylor polynomial development and Padé rational expansion. Finally, a comprehensive dispersive analysis is done through the dispersion equations of the discrete model, the discrete to continuous approximation beam and the continuous one. Some experimental results for Carbon Nanotubes (CNTs) are replicated by the model. We show that the model is able to predict accurately the positive and negative group of wave velocity as well as the standing wave. Scale effects and wave dispersive characteristics of the granular chain are captured by the continuous gradient elasticity model. [ABSTRACT FROM AUTHOR]

Gradient elasticity theories can be used to simulate dispersive wave propagation as it occurs in heterogeneous materials. Compared to the second-order partial differential equations of classical elasticity, in its most general format gradient elasticity also contains fourth-order spatial, temporal as well as mixed spatial-temporal derivatives. The inclusion of the various higher-order terms has been motivated through arguments of causality and asymptotic accuracy, but for numerical implementations it is also important that standard discretization tools can be used for the interpolation in space and the integration in time. In this paper, we will formulate four different simplifications of the general gradient elasticity theory. We will study the dispersive properties of the models, their causality according to Einstein and their behavior in simple initial/boundary value problems. [ABSTRACT FROM AUTHOR]

Abstract: The present study investigates the microstructural size effect on the strength of a bar under axial loading, and on the toughness and crack growth of a beam under three-point bending within the framework of strain gradient elasticity. The gradient responses have been found considerably tougher as compared to the classical theory predictions and the observed deviation increases with increasing values of the non-dimensional parameter g/L (microstructural length over structural length). Based on the analytical solution of the strain energy release rate for the three-point bending case, a new, simple and universal, strain gradient elasticity, brittle fracture criterion and a new, size adjusted fatigue crack growth law have been established. Finally, the analytical predictions of the current modeling compare well with previous experimental data, based on three-point bending tests on single-edge notched concrete beams. [Copyright &y& Elsevier]

Displacement and strain fields of a screw dislocation in a nanowire are considered within the theory of gradient elasticity. The gradient solution of the corresponding boundary value problem is derived and discussed in detail. It is shown that the dislocation fields do not contain classical jumps and singularities at the dislocation line. The maximum values of the dislocation displacement and elastic strain strongly depend on both the dislocation position and nanowire radius, thus demonstrating a nonclassical size effect. [Copyright &y& Elsevier]