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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.

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.

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).

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.

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.

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.

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 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.

Purpose – The linear regression technique is widely used to determine empirical parameters of fatigue life profile while the results may not continuously depend on experimental data. Thus Tikhonov-Morozov method is utilized here to regularize the linear regression results and consequently reduces the influence of measurement noise without notably distorting the fatigue life distribution. The paper aims to discuss these issues. Design/methodology/approach – Tikhonov-Morozov regularization method would be shown to effectively reduce the influences of measurement noise without distorting the fatigue life distribution. Moreover since iterative regularization methods are known to be an attractive alternative to Tikhonov regularization, four gradient iterative methods called as simple iteration, minimum error, steepest descent and conjugate gradient methods are examined with an appropriate initial guess of regularized coefficients. Findings – It has been shown that in case of sparse fatigue life measurements, linear regression results may not have continuous dependence on experimental data and measurement error could lead to misinterpretations of the solution. Therefore from engineering safety point of view, utilizing regularization method could successfully reduce the influence of measurement noise without significantly distorting the fatigue life distribution. Originality/value – An excellent initial guess for mixed iterative-direct algorithm is introduced and it has been shown that the combination of Newton iterative approach and Morozov discrepancy principle is one of the interesting strategies for determination of regularization parameter having an excellent rate of convergence. Moreover since iterative methods are known to be an attractive alternative to Tikhonov regularization, four gradient descend methods are examined here for regularization of the linear regression problem. It has been found that all of gradient decent methods with an appropriate initial guess of regularized coefficients have an excellent convergence to Tikhonov-Morozov regularization results.

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.

Two elastostatic approaches are presented in order provide a simple, but technically effective, assessment of shear coefficients in Timoshenko beam theory. First the elasticity solution of ...

The stress function approach is revisited for the inverse determination of residual stresses and eigenstrains from limited pointwise data in spherically symmetric stress state. The robust least squares technique is utilized to minimize the deviation of the measurement data from the model predictions while a full range of continuum mechanics requirements are satisfied. The application of the newly proposed spherical stress function is effectively demonstrated for two cases of analytical and numerical solutions considering different material behavior models. Also, the eigenstrains are inversely determined satisfying the equation of strain compatibility and a closed form analytical solution is presented.

The smoothed inverse eigenstrain method is revisited for the reconstruction of residual fields and eigenstrains from limited strain measurements within axially symmetric tubes. The application of the present approach is successfully demonstrated for two cases of analytical solution and experimental measurements. The well-known advantage of the smoothed inverse eigenstrain approach is that it not only minimizes the deviation of measurements from the model predictions but also will result in an inverse solution satisfying all of the continuum mechanics requirements. As a result, less number of experimental measurements is required to reconstruct the complete residual fields. Consequently, the distribution of residual stresses is obtained without requiring the details of the hardening behavior of the material. Furthermore, the eigenstrain field is inversely determined satisfying the total strain compatibility equations, and a closed form analytical solution is presented for the distribution of eigenstrains.