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The mixture stress gradient theory of elasticity is conceived via consistent unification of the classical elasticity theory and the stress gradient theory within a stationary variational framework. The boundary-value problem associated with a functionally graded nano-bar is rigorously formulated. The constitutive law of the axial force field is determined and equipped with proper non-standard boundary conditions. Evidences of well-posedness of the mixture stress gradient problems, defined on finite structural domains, are demonstrated by analytical analysis of the axial displacement field of structural schemes of practical interest in nano-mechanics. An effective meshless numerical approach is, moreover, introduced based on the proposed stationary variational principle while employing autonomous series solution of the kinematic and kinetic field variables. Suitable mathematical forms of the coordinate functions are set forth in terms of the modified Chebyshev polynomials, satisfying the required classical and non-standard boundary conditions. An excellent agreement between the numerical results of the axial displacement field of the functionally graded nano-bar and the analytical solution counterpart is confirmed on the entire span of the nano-sized bar, in terms of the mixture parameter and the stress gradient characteristic parameter. The effectiveness of the established meshless numerical approach, demonstrating a fast convergence rate and an admissible convergence region, is hence ensured. The established mixture stress gradient theory can effectively characterize the peculiar size-dependent response of functionally graded structural elements of advanced ultra-small systems.

The unified gradient elasticity theory with applications to nano-mechanics of torsion is examined. The Reissner stationary variational principle is invoked to detect the differential and boundary conditions of equilibrium along with the consistent form of the constitutive laws. An efficient meshless numerical approach is established by making recourse to the Reissner variational functional wherein independent series solution of the kinematic and kinetic field variables are proposed. Suitable forms of the coordinate functions, in terms of the Chebyshev polynomials, are introduced to fulfill a set of kinematic and higher-order boundary conditions in the elastic torsion of nano-bars with practical kinematic constraints. Torsional behavior of the unified gradient elastic bar is studied for structural schemes of applicative interest. An excellent agreement between the torsional responses of the nano-bar detected based on the established meshless method and obtained exact analytical solution is realized. The proposed meshless numerical approach is confirmed to have a fast convergence rate and an admissible convergence region in determination of the torsional rotation field with high accuracy. The introduced meshless method is demonstrated to be highly efficacious in characterizing both the softening and stiffening structural behaviors at nano-scale. The presented numerical approach therefore paves the way ahead in mechanics of nano-structures.

A size-dependent elasticity theory, founded on variationally consistent formulations, is developed to analyze the wave propagation in nanosized beams. The mixture unified gradient theory of elasticity, integrating the stress gradient theory, the strain gradient model, and the traditional elasticity theory, is invoked to realize the size effects at the ultra-small scale. Compatible with the kinematics of the Timoshenko–Ehrenfest beam, a stationary variational framework is established. The boundary-value problem of dynamic equilibrium along with the constitutive model is appropriately integrated into a single function. Various generalized elasticity theories of gradient type are restored as particular cases of the developed mixture unified gradient theory. The flexural wave propagation is formulated within the context of the introduced size-dependent elasticity theory and the propagation characteristics of flexural waves are analytically addressed. The phase velocity of propagating waves in carbon nanotubes (CNTs) is inversely reconstructed and compared with the numerical simulation results. A viable approach to inversely determine the characteristic length-scale parameters associated with the generalized continuum theory is proposed. A comprehensive numerical study is performed to demonstrate the wave dispersion features in a Timoshenko–Ehrenfest nanobeam. Based on the presented wave propagation response and ensuing numerical illustrations, the original benchmark for numerical analysis is detected.

Two frameworks of the nonlocal integral elasticity and the modified strain gradient theory are consistently merged to conceive the nonlocal modified gradient theory. The established augmented continuum theory is applied to a Timoshenko–Ehrenfest beam model. Nanoscopic effects of the dilatation, the deviatoric stretch, and the symmetric rotation gradients together with the nonlocality are suitably accommodated. The integral convolutions of the constitutive law are restored with the equivalent differential model subject to the nonclassical boundary conditions. Both the elastostatic and elastodynamic flexural responses of the nano-sized beam are rigorously investigated and the well posedness of the nonlocal modified gradient problems on bounded structural domains is confirmed. The analytical solution of the phase velocity of flexural waves and the deflection and the rotation fields of the nano-beam is detected and numerically illustrated. The transverse wave propagation in carbon nanotubes is furthermore reconstructed and validated by the molecular dynamics simulation data. Being accomplished in revealing both the stiffening and softening structural responses at nano-scale, the proposed nonlocal modified gradient theory can be beneficially implemented for nanoscopic examination of the static and dynamic behaviors of stubby nano-sized elastic beams.

The unified higher-order theory of two-phase nonlocal gradient elasticity is conceived via consistently introducing the higher-order two-phase nonlocality to the higher-order gradient theory of elasticity. The unified integro-differential constitutive law is established in an abstract variational framework equipped with ad hoc functional space of test fields. Equivalence between the higher-order integral convolutions of the constitutive law and the nonlocal gradient differential formulation is confirmed by prescribing the non-classical boundary conditions. The strain-driven and stress-driven nonlocal approaches are exploited to simulate the long-range interactions at nano-scale. A range of generalized continuum models are restored under special ad hoc assumptions. The established unified higher-order elasticity theory is invoked to analytically examine the wave dispersion phenomenon. In contrast to the first-order size-dependent elasticity model, the higher-order two-phase nonlocal gradient theory can efficiently capture the wave dispersion characteristics observed in experimental measurements. The precise description of nano-scale wave phenomena noticeably underlines the importance of applying the proposed higher-order size-dependent elasticity theory. A viable approach to tackle peculiar dynamic phenomena at nano-scale is introduced.

Practical vibration-based nano-devices are normally subject to disturbances where intense vibrations reveal significant nonlinear characteristics. Efficient implementation of nonlinear nano-systems...

The thick bar model, accounting for the lateral deformation, shear stiffness, and lateral inertia effect, is the most comprehensive structural theory to study the axial deformation of carbon nanotubes. Physically motivated definition of the axial force field and associated higher order boundary conditions are determined applying a consistent variational framework. The effects of long-range interactions are suitably realized in the framework of the nonlocal integral elasticity. The integral convolutions of the nonlocal constitutive law are determined and suitably resorted with the equivalent nonlocal differential model equipped with non-standard boundary conditions. Preceding contributions on the elastodynamic analysis of the elastic thick bar are, therefore, amended by properly taking into account the higher order and non-standard boundary conditions. The established size-dependent thick bar model is demonstrated to be exempt from the inherent drawbacks of the nonlocal differential formulation and leads to well-posed elastodynamic problems. The wave desperation response and free vibrational behavior of elastic thick bars with kinematic constraints of nano-mechanics interest are rigorously investigated by making recourse to a viable solution approach. New numerical benchmarks are detected for the elastodynamic response of nonlocal thick nano-bars. A consistent approach for nanoscopic study of the field quantities in the nonlocal mechanics is proposed that is capable of properly confirming the smaller-is-softer phenomenon.

In the present study, residual stress is measured in fiber-reinforced SWCNT/epoxy at weight fractions of 0.1% and 0.5% with a cross-ply layup on a micro-scale. The mechanical properties of the SWCNT/epoxy composites were determined by tensile testing and the Young's modulus of the epoxy increased moderately with the addition of CNTs. The micro-residual stress of the cross-ply CF/epoxy and CNF-reinforced CF/epoxy laminates were measured using a new experimental approach. The micro-hole was milled by laser beam and the surface displacement was recorded by SEM after milling. In order to determine the residual stress from the recorded strain, the calibration matrix was calculated using the finite element method. The residual stress was obtained at a certain hole depth of specimens. The reliability of this approach was assessed by comparing the residual stress measurements from this method and from the standard hole-drilling method. The experimental results of the present approach confirmed that laser hole drilling SEM-DIC has excellent potential as a reliable method for measuring residual stress in polymer nanocomposites. Generally, CNT agglomerates, especially in high weight fractions, increased the micro-residual stress. An analytical method based on classical theory was used to calculate the residual stress and was compared with the experimental results. Good agreement was found between the results of the analytical methods and the experimental measurement.

In this paper, the size-dependent static behavior of Bishop rods is investigated by Lam and Aifantis strain gradient formulations of elasticity. Appropriate constitutive boundary conditions are established for both the theories by making recourse to a variational approach. Unlike contributions of literature, no higher-order kinematic and static boundary conditions, which have not a clear physical meaning, are required to close the relevant gradient problems. The proposed methodology leads to mathematically well-posed elastostatic problems and is illustrated by examining size effects in selected thick rods of nanotechnological interest. Exact solutions of Bishop nano-rods are detected for a variety of loading systems and kinematic boundary conditions. Peculiar properties, merits, and implications of both the strain gradient formulations, equipped with the proper boundary conditions, are illustrated and commented. The outcomes can be useful for the design and optimization of rod-like thick components of nanoelectromechanical systems.

The stress-driven nonlocal integral model of elasticity for 1D nano-structures (Romano & Barretta, 2017a) is extended in this paper to Kirchhoff axisymmetric nano-plates. The nonlocal formulation, relating elastic principal flexural curvatures and moments, provides an effective methodology to assess size effects in 2D nano-structures. The associated elastostatic problem of nano-plates is conveniently expressed by differential relations equipped with constitutive boundary conditions involving nonlocal curvature fields. The proposed approach is illustrated by examining case-studies of engineering interest. In particular, nonlocal displacement solutions of axisymmetric nano-plates are detected for a variety of loading systems and kinematic boundary conditions. Merits and implications of the stress-driven strategy are elucidated by comparing the achieved results with those of the strain gradient model of elasticity generated by Reissner's variational principle. The outcomes can be useful for design and optimization of plate-like components of ground-breaking Nano-Electro-Mechanical-Systems (NEMS).

The nonlocal integral elasticity and the modified strain gradient theory are consistently integrated in the framework of the nonlocal modified gradient theory of elasticity. The equivalent differential formulation of the constitutive law, equipped with appropriate nonstandard boundary conditions, is introduced. The size-dependent effects of the dilatation gradient, deviatoric stretch gradient, and symmetric rotation gradient in addition to the nonlocality are beneficially captured in the flexure problem of nano-beams. The well posedness of the proposed nonlocal modified gradient problem is demonstrated via analytical examination of the elastostatic flexure and the wave dispersion phenomenon in nano-beams. The dispersive behavior of flexural waves is verified in comparison with the molecular dynamics simulation. The dominant stiffening effect of the gradient characteristic parameters associated with the nonlocal modified gradient elasticity is confirmed. Both the stiffening and softening responses of nano-structured materials are effectively realized in the framework of the introduced augmented elasticity theory. The conceived nonlocal modified gradient elasticity theory can accordingly provide a practical approach for nanoscopic study of the field quantities.

Forecasting spare parts requirements is a challenging problem, because the normally intermittent demand has a complex nature in patterns and associated uncertainties, and classical forecasting approaches are incapable of modeling these complexities. The present study introduces a hybrid model that can impressively overcome the limitations of classical models while simultaneously using their unique advantages in dealing with the complexities in intermittent demand. The strategy of the proposed hybrid model is to use the three individual autoregressive moving average (ARMA), single exponential smoothing (SES), and multilayer perceptron (MLP) models simultaneously. Each of them has the potential of modeling a different structure and patterns of behavior among the data. The accuracy in forecasting ability is also increased by the suitable examination of these in the intermittent data. Croston’s method is the backbone of the suggested model. The proposed hybrid model is based on CV2 and ADI criteria, which improve its efficacy in examining inappropriate structures by reducing the cost of inappropriate modeling while increasing the prediction model accuracy. Using these results prevents the hybrid model from being confused or weakened in the modeling of all groups and reduces the risk of choosing the disproportionate model. The accuracy of prediction models was evaluated and compared using mean absolute percentage error (MAPE) by implementing an example, and promising results were achieved.

Nonlocal gradient mechanics of elastic beams subject to torsion is established by means of a variationally consistent methodology, equipped with suitable functional spaces of test fields. The proposed elasticity theory is the generalization of size-dependent models recently contributed in literature to assess size-effects in nano-structures, such as modified nonlocal strain gradient and strain- and stress-driven local/nonlocal elasticity formulations. General new ideas are elucidated by examining the torsional behavior of elastic nano-beams. Equivalence between nonlocal integral convolutions and differential problems subject to variationally consistent boundary conditions is demonstrated for special averaging kernels. The variational procedure leads to well-posed engineering problems in nano-mechanics. Elasto-static responses and free vibrations of nano-beams under torsion are analyzed applying an effective analytical solution technique. Nonlocal strain- and stress-driven gradient models of elasticity can efficiently predict both stiffening and softening structural responses, and thus, notably characterize small-scale phenomena in structures exploited in modern Nano-Electro-Mechanical-Systems (NEMS).

Nonlinear flexural behavior of elastic nano-beams under static loads is investigated in the framework of the Eringen’s nonlocal integral elasticity. The integro-differential and boundary conditions of equilibrium for inflected Timoshenko–Ehrenfest elastic beams with von Karman nonlinear strains are established by using the minimum total potential energy principle. Eringen’s nonlocal differential law, consequent (but not equivalent) to Eringen’s nonlocal integral convolution, is well established to be inconsistent when applied to bounded domains of nano-mechanics interest. Accordingly, nonlocal integral elasticity formulation is utilized to capture scale effects in the nonlinear flexure of nano-beams. An analytical series solution in terms of Chebyshev polynomials is proposed to suitably approximate the displacement field of inflected nano-beams. Selected case studies of applicative interest are investigated, and the approximated nonlinear solutions of nonlocal nano-beams are detected. The demonstrated benchmark examples can be effectively employed in the assessment of beam-type elements of nano-electromechanical systems.

Numerous contributions can be found in the recent literature exploiting the nonlocal strain gradient model, introduced in consequence of unification of the differential relation (consequent but not equivalent to Eringen nonlocal integral law) and strain gradient elasticity. In the present paper, Eringen nonlocal integral and Lam modified strain gradient theories are coupled to formulate a nonlocal Lam strain gradient model of elasticity. Three scale parameters, describ-ing nonlocality, dilatation, and stretch gradient, are utilized to significantly estimate size-dependent responses of 1D nanocontinua. The governing constitutive law is established via a variationally consistent approach, based on suitably selected test fields, projected for formulating well-posed static and dynamic problems of engineering interest. The non-local Lam strain gradient model, developed for nanorods, provides axial force fields in terms of integral convolutions involving elastic axial strain fields. The integral law, equivalent to an expedient set of constitutive differential and boundary conditions, is exploited for studying static and free vibration behaviors of simple nanostructural schemes. Exact analytical solutions are gotten in terms of nonlocal and gradient characteristic parameters. Validation of the proposed strategy is carried out by comparing the contributed results with those obtained by the modified nonlocal strain gradient theory.

Thick rods are employed in nanotechnology to build modern electromechanical systems. Design and optimization of such structures can be carried out by nonlocal continuum mechanics which is computationally convenient when compared to atomistic strategies. Bishop’s kinematics is able to describe small-scale thick rods if a proper mathematical model of nonlocal elasticity is formulated to capture size effects. In all papers on the matter, nonlocal contributions are evaluated by replacing Eringen’s integral convolution with the consequent (but not equivalent) differential equation governed by Helmholtz’s differential operator. As notorious in integral equation theory, this replacement is possible for convolutions, defined in unbounded domains, governed by averaging kernels which are Green’s functions of differential operators. Indeed, Eringen himself, in order to study nonlocal problems defined in unbounded domains, such as screw dislocations and wave propagation, suggested to replace integro-differential equations with differential conditions. A different scenario appears in Bishop rod mechanics where nonlocal integral convolutions are defined in bounded structural domains, so that Eringen’s nonlocal differential equation has to be supplemented with additional boundary conditions. The objective is achieved by formulating the governing nonlocal equations by a proper variational statement. The new methodology provides an amendment of previous contributions in the literature and is illustrated by investigating the elastostatic behavior of simple structural schemes. Exact solutions of Bishop rods are evaluated in terms of nonlocal parameter and cross section gyration radius. Both hardening and softening structural responses are predictable with a suitable tuning of the parameters.

The nonlocal strain gradient theory of elasticity is the focus of numerous studies in literature. Eringen’s nonlocal integral convolution and Lam’s strain gradient model are unified by a variational methodology which leads to well-posed structural problems of technical interest. The proposed nonlocal Lam strain gradient approach is presented for functionally graded (FG) beams under torsion. Static and dynamic responses are shown to be significantly affected by size effects that are assessed in terms of nonlocal and gradient length parameters. Analytical elastic rotations and natural frequencies are established by making recourse to a simple solution procedure which is based on equivalence between integral convolutions and differential equations supplemented with variationally consistent (but non-standard) nonlocal boundary conditions. Effects of Eringen’s nonlocal parameter and stretch and rotation gradient parameters on the torsional behavior of FG nano-beams are examined and compared with outcomes in literature. The illustrated methodology is able to efficiently model both stiffening and softening torsional responses of modern composite nano-structures by suitably tuning the small-scale parameters.

Carbon nanotubes (CNTs) are principal constituents of nanocomposites and nano-systems. CNT size-dependent response assessment is therefore a topic of current interest in Mechanics of Advanced Materials and Structures. CNTs are modeled here by a variationally consistent nonlocal strain gradient approach for Timoshenko nano-beams, extending the treatment in [Int. J. Eng. Science 143 (2019) 73–91] confined to slender structures. Scale effects are described by integral convolutions, conveniently replaced with differential and boundary nonlocal laws. The theoretical predictions, exploited to analytically estimate the reduced Young elastic modulus of CNTs, are validated by molecular dynamics simulations.

The higher-order unified gradient elasticity theory is conceived in a mixed variational framework based on suitable functional space of kinetic test fields. The intrinsic form of the differential and boundary conditions of equilibrium along with the constitutive laws is consistently established. Various forms of the gradient elasticity theory, in the sense of stress or strain gradient models, can be retrieved as particular cases of the introduced generalized elasticity theory. The proposed stationary variational principle can effectively realize the nanoscopic structural effects while being exempt of restrictions typical of the nonlocal gradient elasticity model. The well-posed generalized gradient elasticity theory is invoked to study the mechanics of torsion and the torsional behavior of elastic nano-bars is analytically examined. The closed-form analytical formulae of the size-dependent shear modulus of nano-sized bar is determined and efficiently applied to reconstruct the shear modulus of SWCNTs with dissimilar chirality in comparison with the numerical simulation data. A practical approach to calibrate the characteristic lengths associated with the higher-order unified gradient elasticity theory is introduced. Numerical results associated with the torsion of higher-order unified gradient elastic bars are demonstrated and compared with the counterpart size-dependent elasticity theories. The conceived generalized gradient elasticity theory can beneficially characterize the nanoscopic response of advanced nano-materials.

Dispersion characteristics of flexural waves in a functionally graded (FG) porous nanobeam are analytically examined. Kinematics of the nano-sized beam is assumed to be consistent with the Timoshenko-Ehrenfest beam model. Material behavior of the FG porous beam is considered to vary symmetrically along the beam thickness while appropriate symmetric porosity models are applied to account for two diverse distributions of porosity with variable volume of voids. For the first time, size-dependent response of the symmetric FG porous nanobeam is realized within the framework of the higher-order nonlocal gradient elasticity theory. The integro-differential constitutive laws of the stress resultant fields are established and reinstated with the equivalent differential relations equipped with non-standard boundary conditions. The closed-form solution of the phase velocity of flexural waves is analytically determined. The ensuing numerical results of the flexural wave dispersion detect new benchmarks for numerical analysis and can be effectively exploited in design and optimization of composite nano-structural elements of advanced nano-electro-mechanical systems.

This research deals with the nonlinear forced vibration behavior of embedded functionally graded double layered nanoplates with all edges simply supported via nonlocal strain gradient elasticity theory based on the Mindlin plate theory along with von Karman geometric nonlinearity. The interaction of van der Waals forces between adjacent layers is included. For modeling surrounding elastic medium, the nonlinear Winkler–Pasternak foundation model is employed. The exact solution of the nonlinear forced vibration for primary and secondary resonance through the Harmonic Balance method is then established. For the double layered nanoplate, uniform distribution and sinusoidal distribution of loading are considered. It is assumed that the material properties of functionally graded double layered nanoplates are graded in the thickness direction and estimated through the rule of mixture. The Galerkin-based numerical technique is employed to reduce the set of nonlinear governing equations into a time-varying set of ordinary differential equations. The effects of different parameters such as length scale parameters, elastic foundation parameters, dimensionless transverse force and gradient index on the frequency responses of functionally graded double layered nanoplates are investigated. The results show that the length scale parameters give nonlinearity of the hardening type in nonlinear forced vibration and the increase in strain gradient parameter causes to increase in maximum response amplitude, whereas the increase in nonlocal parameter causes to decrease in maximum response amplitude.

The second-order integro-differential nonlocal theory of elasticity is established as an extension of the Eringen nonlocal integral model. The present research introduces an appropriate thermodynamically consistent model allowing for the higher-order strain gradient effects within the nonlocal theory of elasticity. The thermodynamic framework for third-grade nonlocal elastic materials is developed and employed to establish the Helmholtz free energy and the associated constitutive equations. Establishing the minimum total potential energy principle, the integro-differential conditions of dynamic equilibrium along with the associated classical and higher-order boundary conditions are derived and comprehensively discussed. A rigorous formulation of the third-grade nonlocal elastic Bernoulli–Euler nano-beam is also presented. A novel series solution based on the modified Chebyshev polynomials is introduced to examine the flexural response of the size-dependent beam. The proposed size-dependent beam model is demonstrated to reveal the stiffening or softening flexural behaviors, depending on the competitions of the characteristic length-scale parameters. The higher-order gradients of strain fields are illustrated to have more dominant effects on the beam stiffening.

The general form of Reissner stationary variational principle is established in the framework of the nonlocal strain gradient theory of elasticity. Including two size-dependent characteristic parameters, the nonlocal strain gradient elasticity theory can demonstrate the significance of the strain gradient as well as the nonlocal elastic stress field. Based on the Reissner functional, the governing differential and boundary conditions of dynamic equilibrium and differential constitutive equations of the classical and first-order nonlocal stress tensor are derived in the most general form. Additionally, the boundary congruence conditions are formulated and discussed for the nonlocal strain gradient theory. To exhibit the application value of Reissner variational principle, it is employed to examine the nonlinear vibrations of size-dependent Bernoulli-Euler and Timoshenko beams. In the case of immovable boundary conditions, employing the weighted residual Galerkin method, the homotopy analysis method is also utilized to determine the closed form analytical solutions of the geometrically nonlinear vibration equations. Consequently, the analytical expressions for the nonlinear natural frequencies of Bernoulli-Euler and Timoshenko nonlocal strain gradient beams are derived.

This paper comprehensively studies the nonlinear vibration of functionally graded nano-beams resting on elastic foundation and subjected to uniform temperature rise. The small-size effect, playing an essential role in the dynamical behavior of nano-beams, is considered here applying the innovative stress driven nonlocal integral model due to Romano and Barretta. The governing partial differential equations are derived from the Bernoulli–Euler beam theory utilizing the von Karman strain–displacement relations. Using the Galerkin method, the governing equations are reduced to a nonlinear ordinary differential equation. The closed form analytical solution of the nonlinear natural frequency for four different boundary conditions is then established employing the Homotopy Analysis Method. The nonlinear natural frequencies, evaluated according to the stress-driven nonlocal integral model, are compared with those obtained by Eringen differential model. Finally, the effects of different parameters such as length, elastic foundation parameter, thermal loading and nonlocal characteristic parameter are investigated. The emergent results establish that when the nonlocal characteristic parameter increases, the nonlinear natural frequencies obtained by the stress-driven nonlocal integral model reveal a stiffness-hardening effect. On the other hand, Eringen's differential law reveals a stiffness-softening effect excepting the case of cantilever nano-beam. Also, increase in temperature and the elastic foundation parameter leads to increase in the nonlinear frequency ratios in Eringen differential model but decrease in the frequency ratios in the stress-driven nonlocal integral model.

Reissner mixed variational principle is employed for establishment of the nonlinear differential and boundary conditions of dynamic equilibrium governing the flexure of beams when the effects of true shear stresses are included. Based on the Reissner mixed variational principle, the nonlinear size-dependent model of the Reissner nano-beam is derived in the framework of the nonlocal strain gradient elasticity theory. Furthermore, the closed form analytical solutions for the geometrically nonlinear flexural equations are derived and compared to the nonlinear flexural results of the Timoshenko size-dependent beam theory. The profound differences in the assumptions and formulations between the Timoshenko and the Reissner beam theory are also comprehensively discussed. The Reissner beam model is shown not to be a first-order shear deformation theory while comprising the influences of the true transverse shearing stress and the applied normal stress. Moreover, it is exhibited that the linear and nonlinear deflections obtained based on the Reissner beam theory are consistently lower than their Timoshenko counterparts for various gradient theories of elasticity.

Delamination is a well-known defect mode that can arise in the manufacturing process of laminated composite plates. Due to the importance of analyzing the destructive effects of delamination on the mechanical behavior of composite plates, the flexural analysis of circular laminated composite plates with circular delamination is presented here. The governing equilibrium equations of laminated plates are first determined using third-order shear deformation theory. Both the linear and geometrically nonlinear strain states are considered, and the variational approach with a moving boundary is employed to derive the equilibrium equations. The governing equations in terms of the displacement field are then solved using the Galerkin method and the spectral homotopy analysis method to obtain the linear and nonlinear strain states, respectively. The effect of the variations of the elastic material properties on the strain energy release rate is also comprehensively studied. The results of the present study are consistent with the results of other methods found in the literature.

In this paper, the fully-coupled rolling contact problem between a rigid cylinder and an orthotropic medium is investigated. The governing equations are developed analytically; then, a numerical method, the Gauss-Chebyshev method, is used to solve the coupled integral equations. The rolling contact problem is solved by assuming a central stick zone accompanied with two slip regions. The main purpose of this study is to obtain the surface stresses in the contact area and investigate the effect of material orthotropic properties such as shear parameter and stiffness ratio and also the coefficient of friction on these contact stresses. In addition, the subsurface stresses in the orthotropic medium are determined and a parametric study, was subsequently executed. The current paper shows that the orthotropic parameters and the coefficient of friction significantly influence the contact surface stresses and the distribution of the interior field stresses. By appropriately choosing the values of these parameters, the stresses will decrease; thus, the failure behavior of the orthotropic medium improves.

Nonlocal strain gradient continuum mechanics is a methodology widely employed in literature to assess size effects in nanostructures. Notwithstanding this, improper higher-order boundary conditions (HOBC) are prescribed to close the corresponding elastostatic problems. In the present study, it is proven that HOBC have to be replaced with univocally determined boundary conditions of constitutive type, established by a consistent variational formulation. The treatment, developed in the framework of torsion of elastic beams, provides an effective approach to evaluate scale phenomena in smaller and smaller devices of engineering interest. Both elastostatic torsional responses and torsional free vibrations of nano-beams are investigated by applying a simple analytical method. It is also underlined that the nonlocal strain gradient model, if equipped with the inappropriate HOBC, can lead to torsional structural responses which unacceptably do not exhibit nonlocality. The presented variational strategy is instead able to characterize significantly peculiar softening and stiffening behaviours of structures involved in modern Nano-Electro-Mechanical-Systems (NEMS).

In elastic continuous structures defined on bounded domains, Eringen strain-driven integral model leads to ill-posed elastostatic problems since the constitutive stress field, got by convoluting th...

In order to study the intrinsic size-effects, the stress gradient theory is implemented to a nano-scale beam model in nonlinear flexure. The nonlocal integral elasticity model is considered as a suitable counterpart to examine the softening behavior of nano-beams. Reissner variational principle is extended consistent with the stress gradient theory and applied to establish the differential, constitutive and boundary conditions of a nano-sized beam in nonlinear flexure. The nonlinear integro-differential and boundary conditions of inflected beams in the framework of the nonlocal integral elasticity are determined utilizing the total elastic strain energy formulation. A practical series solution approach in terms of Chebyshev polynomials is introduced to appropriately estimate the kinematic and kinetic field variables. A softening structural behavior is observed in the flexure of the stress gradient and the nonlocal beam in terms of the characteristic parameter and the smaller-is-softer phenomenon is, therefore, confirmed. The flexural response associated with the stress gradient theory is demonstrated to be in excellent agreement with the counterpart results of the nonlocal elasticity model equipped with the Helmholtz kernel function. The nonlocal elasticity theory endowed with the Error kernel function is illustrated to underestimate the flexural results of the stress gradient beam model. Detected numerical benchmark can be efficiently exploited for structural design and optimization of pioneering nano-engineering devices broadly utilized in advanced nano-electro-mechanical systems.

In this research, Araldite 2011 has been reinforced using different weight fractions of Reduced Graphene Oxide (RGO). Fourier Transform Infrared Spectroscopy (FTIR), Scanning Electron Microscopy (SEM) and Transmission Electron Microscopy (TEM) analyses were conducted and it has been shown that introduction of the RGO greatly changes the film morphology of the neat adhesive. Uni-axial tests were carried out to obtain the mechanical characteristics of the adhesive-RGO composites. It has been observed that introducing 0.5 wt% RGO enhances the ultimate tensile strength of the composites by 30%. In addition, single lap joints using neat adhesive and adhesive-RGO composites were fabricated to investigate the effect of the added RGO on the lap shear strength of the joints. Results show that the joints with added 0.5 wt RGO exhibited 27% higher lap shear strength compared to the joints bonded with neat adhesive. Finally, Finite Element (FE) numerical solutions using Cohesive Zone Modeling (CZM) have been carried out to simulate the failure behavior of the joints, and it has been shown that the FE models can predict the joint’s failure load.

The flexure problem of Saint-Venant's elastic beam under lateral traction is revisited. First an engineering stress field is proposed with the explicit closed form solution that results in shear stress distribution determined by mechanics of materials theory. Afterward a modified stress field is introduced for Saint-Venant's flexure problem with uniaxial symmetric cross-sections that recovers the solutions available from the theory of elasticity. The modified stress field which is a unified formulation of Saint-Venant's solution confirms the main features of shear stress distribution found in the earlier investigations and has an excellent agreement with the results of the theory of elasticity. Also the shear flexibility factor is comprehensively discussed and an explicit solution form is presented based on the modified stress field.

A consistent variational theory of the higher–order nonlocal gradient elasticity is conceived to appropriately introduce the nonlocality to the higher-order strain gradient theory. The abstract variational approach, based on appropriate functional spaces of test fields, is applied to establish the higher–order nonlocal gradient mechanics of elastic beams in flexure. Two nonlocal and two gradient characteristic lengths are exploited to describe the size–dependent response of continua with nano–structures. Integral convolutions of the higher–order constitutive law are restored to the equivalent differential problem endowed with non–standard boundary conditions of constitutive–type. The higher–order strain gradient theory, higher–order nonlocal elasticity and modified nonlocal strain gradient theory, extensively adopted in the community of Engineering Science, are demonstrated to be particular cases of the introduced higher-order nonlocal gradient theory. The well-posedness of the developed higher-order nonlocal gradient problem is revealed by studying the flexural response of structures with wide-ranging applications in nano-engineering. Exact analytical solution for elastostatic deflections of nano-beams is derived and new benchmark examples of nano-mechanics interest are detected. The higher-order nonlocal gradient elasticity theory can effectively characterize nanoscopic phenomena in advanced nano–composites and nano–structures.

Size-dependent vibrational behavior of functionally graded (FG) Timoshenko nano-beams is investigated by strain gradient and stress-driven nonlocal integral theories of elasticity. Hellinger-Reissner's variational principle is preliminarily exploited to establish the equations governing the elastodynamic problem of FG strain gradient Timoshenko nano-beams. Differential and boundary conditions of dynamical equilibrium of FG Timoshenko nano-beams, with nonlocal behavior described by the stress-driven integral theory, are formulated. Free vibrational responses of simple structures of technical interest, associated with nonlocal stress-driven and strain gradient strategies, are analytically evaluated and compared in detail. The stress-driven nonlocal model for FG Timoshenko nano-beams provides an effective tool for dynamical analyses of stubby composite parts of Nano-Electro-Mechanical Systems.

Size-dependent structural behavior of nano-beams under torsion is investigated by two-phase integral elasticity. An effective torsional model is proposed by convexly combining the purely nonlocal integral stress-driven relation with a local phase. Unlike Eringen's strain-driven mixture, the projected model does not exhibit singular behaviors and leads to well-posed elastostatic problems in all cases of technical interest. The new theory is illustrated by studying torsional responses of cantilever and doubly-clamped nano-beams under simple loading conditions. Specifically, the integral convolution of the two-phase mixture is done by considering the special bi-exponential kernel. With this choice, the stress-driven two-phase model is shown to be equivalent to a differential problem equipped with higher-order constitutive boundary conditions. Exact solutions are established and comparisons with pertinent results obtained by the Eringen strain-driven two-phase mixture and by the strain gradient theory of elasticity are carried out. The outcomes could be useful for the design and optimization of nano-devices and provide new benchmarks for numerical analyses.

A modified stress function approach is developed for reconstruction of residual stress and eigenstrain fields from limited experimental measurements. The modified approach is successfully applied to three experimental case studies where residual stresses are introduced by surface peening. The smooth reconstructed residual fields not only minimize the deviation of measurements from its approximation but also satisfy all continuum mechanics requirements. Furthermore, a comprehensive discussion is performed on regularity of the solution in Tikhonov scheme and the regularization parameter is then determined utilizing Morozov discrepancy principle. Newton iterative method is also shown to have an excellent fast convergence to the regularization parameter while effectively reduces the computational cost.

A smoothed inverse eigenstrain method is developed for reconstruction of residual field from limited strain measurements. A framework for appropriate choice of shape functions based on the prior knowledge of expected residual distribution is presented which results in stabilized numerical behavior. The analytical method is successfully applied to three case studies where residual stresses are introduced by inelastic beam bending, laser-forming and shot peening. The well-rehearsed advantage of the proposed eigenstrain-based formulation is that it not only minimizes the deviation of measurements from its approximations but also will result in an inverse solution satisfying a full range of continuum mechanics requirements. The smoothed inverse eigenstrain approach allows suppressing fluctuations that are contrary to the physics of the problem. Furthermore, a comprehensive discussion is performed on regularity of the asymptotic solution in the Tikhonov scheme and the regularization parameter is then exactly determined utilizing Morozov discrepancy principle. Gradient iterative regularization method is also examined and shown to have an excellent convergence to the Tikhonov–Morozov regularization results.