Stress-driven Nonlocal Integral Model Research Articles

Stress-driven nonlocal integral model with discontinuity for size-dependent buckling and bending of cracked nanobeams using Laplace transform

Due to the frequent occurrence of cracks as structures age, examining the size-dependent responses of cracked micro/nanobeams is critical for safely designing and optimizing high-tech devices such as those used in MEMS/NEMS. With this understanding, we present the size-dependent bending and buckling studies of cracked nanobeams with different boundary edges adopting the well-posed stress-driven nonlocal integral model with discontinuity. Since the presence of a crack, the beam is divided into two segments connected by a rotary spring and a linear spring, whose elastic coefficients can be determined through compliance calibration methods. We propose an efficient method to solve the original integral model directly by converting the integral form of the nonlocal constitutive equation into a general equation through the Laplace transform technique, equipped with three extra constraint conditions. Subsequently, the governing equations of two beam segments are solved by considering the standard boundary and compatibility conditions as well as the extra constraint conditions, while the bending deformation and buckling loads of various bounded nanobeams are presented analytically in the presence of a crack. After verifying the derived formulation, the influence of crack length and location as well as nonlocality on the beam deflections and buckling loads of the first several modes are investigated in detail. Communicated by Rossana Dimitri.

A generalized functions based approach for stress-driven nanobeam bending problems subjected to point loads

The research community has widely accepted the stress-driven nonlocal integral model to study static and dynamic characteristics of nanostructures. Although closed-form solutions for stress-driven point-loaded nanobeam bending problems that find potential for applications in the design of real-world MEMS/NEMS-based sensors have been explored recently, the static bending behavior of stress-driven Bernoulli-Euler and Timoshenko nanobeams subjected to point-loading conditions in various end configurations are studied in the present work wherein, analytical solutions for the same are arrived at by expressing the input bending and shear fields in terms of generalized functions for the first time. Equipped with the Helmholtz bi-exponential kernel function to model the nonlocality, the obtained solutions have demonstrated their ability to capture size-effects consistently. An overall stiffening effect on the nanobeam is observed as a consequence of these size-effects.

Bi-Helmholtz kernel based stress-driven nonlocal integral model with discontinuity for size-dependent fracture analysis of edge-cracked nanobeam

Mathematical formulation is proposed to deal with average bi-Helmholtz kernel (BHK) based stress-driven nonlocal integral model (SDNIM) with discontinuity, which is converted into equivalent differential form with constitutive constraints. Based on average BHK-SDNIM with discontinuity, mathematical model for size-dependent fracture behavior of mode I and II cracks is established for edge-cracked nanobeam. For different load conditions, the analytical bending deflections are deduced firstly, and the corresponding external works and energy release rates can be calculated explicitly. Numerical study shows that energy release rates decrease with the increase of nonlocal parameter κ , which explains the superior fracture performance of nano-materials.

Linear and nonlinear free vibration analysis of functionally graded porous nanobeam using stress-driven nonlocal integral model

Linear and nonlinear free vibration analysis of functionally graded porous (FGP) nanobeam with four different porous distribution patterns is performed on the basis of stress-driven two-phase local/nonlocal integral model. The strain is defined according to von Karman nonlinearity, and the differential governing equations as well as standard boundary conditions are derived through the Hamilton’s principle. The integral relation between nonlinear strain and nonlocal stress is transformed equivalently into differential form with constitutive boundary conditions. Neglecting the nonlinear terms, the bending deflection and moment are derived and expressed explicitly through the Laplace transformation with six unknown constants for linear free vibration, and the nonlinear characteristic equation for vibration frequency and linear vibration mode shape can be calculated through standard and constitutive boundary conditions. Two analytic methods including Ritz–Galerkin and perturbation methods based on both local and nonlocal linear vibration mode shape are applied to study the linear and nonlinear free vibration of nonlocal FGP nanobeams under different boundary conditions Meanwhile, direct numerical method based on general differential quadrature method and Newton’s iterative process are utilized to obtain the numerical solutions of nonlinear vibration frequencies. The influence of nonlocal parameters and vibration amplitude on the free vibration frequencies is investigated numerically. The numerical results show that influence of nonlocal parameters on linear vibration mode shape is not significant. Local linear vibration mode shape based Ritz–Galerkin​ and perturbation methods would overestimate and underestimate the linear free vibration of nonlocal simply-supported and clamped–clamped nanobeams. Nonlocal linear vibration mode shape based Ritz–Galerkin and perturbation methods could provide more accurate prediction on nonlinear vibration frequencies than local mode shape based methods when W(0.5) is small. However, the relative error increases quickly with the increase of W(0.5). Based on the results obtained in this study, local and nonlocal linear vibration mode shape based Ritz–Galerkin and perturbation methods cannot provide accurate prediction on nonlinear vibration frequencies when the vibration magnitude is big, since linear vibration mode shape assumption is not suitable for strong nonlinear vibration.

Flexibility-based stress-driven nonlocal frame element: formulation and applications

In the present work, a novel flexibility-based nonlocal frame element for size-dependent analyses of nano-sized frame-like structures is proposed. The material small-scale effect is consistently represented by the stress-driven nonlocal integral model and the element equation is constructed within the framework of flexibility-based finite element formulation. The merits of both stress-driven nonlocal integral model and flexibility-based finite element formulation render the proposed nonlocal frame element “consistent” and “exact”. Therefore, the “one-element-per-member” modeling approach is applicable. The modified Tonti’s diagram is utilized to show the overview of the element formulation and to summarize the formulation step. The element state determination process as well as the displacement recovery procedure are also discussed. Three numerical examples are employed to show accuracy, characteristics, and applications of the proposed nonlocal frame element. The first example shows the model capability to eliminate the “constant-force” paradoxical responses inherent to the Eringen’s nonlocal frame model and the simplified strain-gradient frame model; the second presents and characterizes the essence of the material small-scale effect on global and local responses of a propped-cantilever nanobeam; the third investigates the material small-scale effect on the tensile response of an auxetic metamaterial. All analysis results demonstrate that the material nonlocality associated with the stress-driven nonlocal integral model consistently yields a stiffer system response.

Differences between stress-driven nonlocal integral model and Eringen differential model in the vibrations analysis of carbon nanotubes conveying magnetic nanoflow

Differences between stress-driven nonlocal integral model and Eringen differential model in the vibrations analysis of carbon nanotubes conveying magnetic nanoflow

Nonlinear free and forced vibration of carbon nanotubes conveying magnetic nanoflow and subjected to a longitudinal magnetic field using stress-driven nonlocal integral model

In this paper primary and secondary resonance of carbon nanotube conveying magnetic nanofluid and subjected to a longitudinal magnetic field resting on viscoelastic foundation with different boundary conditions is investigated. To investigate the small scale effects, stress driven nonlocal integral model has been used and to show the more correctness of stress driven nonlocal integral model response, in studying the behavior of carbon nanotube with different boundary conditions, its results are compared with strain gradient model. The governing partial differential equations are derived from the Bernoulli–Euler beam theory utilizing the von Kármán strain–displacement relations. Using the Galerkin method, the governing equations are reduced to a nonlinear ordinary differential equation. The nonlinear natural frequencies are obtained from the perturbation method and the divergence and flutter instability due to the increase in nanofluid velocity is investigated. Then the frequency response for primary, subharmonic and superharmonic resonance is obtained using the method of multiple scales.Finally, the effects of length small scale parameters, longitudinal magnetic field, magnetic nanofluid and boundary conditions on nonlinear free and forced vibration of carbon nanotube are investigated. As the most important results, as the intensity of the magnetic field increases, the critical flow velocity increases and divergence and flutter occur later. But the critical flow velocity decreases with increasing the intensity of the magnetic field for a carbon nanotube conveying magnetic nanofluid. In forced vibration, increasing the intensity of the magnetic field increases the amplitude of the response for all boundary conditions in primary and secondary resonance.

Bending and Buckling Analysis of Functionally Graded Euler–Bernoulli Beam Using Stress-Driven Nonlocal Integral Model with Bi-Helmholtz Kernel

Static bending and elastic buckling of Euler–Bernoulli beam made of functionally graded (FG) materials along thickness direction is studied theoretically using stress-driven integral model with bi-Helmholtz kernel, where the relation between nonlocal stress and strain is expressed as Fredholm type integral equation of the first kind. The differential governing equation and corresponding boundary conditions are derived with the principle of minimum potential energy. Several nominal variables are introduced to simplify differential governing equation, integral constitutive equation and boundary conditions. Laplace transform technique is applied directly to solve integro-differential equations, and the nominal bending deflection and moment are expressed with six unknown constants. The explicit expression for nominal deflection for static bending and nonlinear characteristic equation for the bucking load can be determined with two constitutive constraints and four boundary conditions. The results from this study are validated with those from the existing literature when two nonlocal parameters have same value. The influence of nonlocal parameters on the bending deflection and buckling loads for Euler–Bernoulli beam is investigated numerically. A consistent toughening effect is obtained for stress-driven nonlocal integral model with bi-Helmholtz kernel.

Torsional static and vibration analysis of functionally graded nanotube with bi-Helmholtz kernel based stress-driven nonlocal integral model

A torsional static and free vibration analysis of the functionally graded nanotube (FGNT) composed of two materials varying continuously according to the power-law along the radial direction is performed using the bi-Helmholtz kernel based stress-driven nonlocal integral model. The differential governing equation and boundary conditions are deduced on the basis of Hamilton’s principle, and the constitutive relationship is expressed as an integral equation with the bi-Helmholtz kernel. Several nominal variables are introduced to simplify the differential governing equation, integral constitutive equation, and boundary conditions. Rather than transforming the constitutive equation from integral to differential forms, the Laplace transformation is used directly to solve the integro-differential equations. The explicit expression for nominal torsional rotation and torque contains four unknown constants, which can be determined with the help of two boundary conditions and two extra constraints from the integral constitutive relation. A few benchmarked examples are solved to illustrate the nonlocal influence on the static torsion of a clamped-clamped (CC) FGNT under torsional constraints and a clamped-free (CF) FGNT under concentrated and uniformly distributed torques as well as the torsional free vibration of an FGNT under different boundary conditions.

Vibration analysis of stress-driven nonlocal integral model of viscoelastic axially FG nanobeams

Finite element method (FEM) and generalized differential quadrature method (GDQM) are developed for damping vibration analysis of viscoelastic axially functionally graded (VAFG) nanobeams within the framework of stress-driven nonlocal integral model (SDM). The equivalent differential law of SDM and two constitutive non-classic boundary conditions (CBCs) are utilized to construct FE and GDQ models. In FEM, three different types of beam elements are derived: two different types for the both ends of the nanobeam and another type for elements located in the middle of nanobeam. Convergence and accuracy of the present FEM and GDQM are evaluated, and it is shown that the present results and formulations are efficient and reliable. Also, in the damping vibration analysis of VAFG nanobeams by SDM, variations of the mechanical properties are considered as a power-law function. After validation of present results and mathematical modeling, various benchmark results are presented to determine the influences of several parameters, such as nonlocal SDM parameter, FG index and damping factor on both parts of size-dependent complex frequencies of VAFG nanobeams with different boundary conditions.

Buckling analysis of curved sandwich microbeams made of functionally graded materials via the stress-driven nonlocal integral model

Size-dependent buckling analysis for slightly curved sandwich microbeams made of functionally graded (FG) materials is performed via a stress-driven nonlocal model. The Fredholm integral constitutive equations are transformed into the Volterra type of the first kind and then solved analytically using the Laplace transformation and its inversion under different boundary conditions. The exact solutions are validated against those existing results. The effect of the nonlocal parameter, thickness ratio of core-to-skin layers, FG power-law index, and length-height ratio on the buckling loads, as well as on the ratio of the result predicted by two common beam-theories is investigated.

Nonlinear free vibrations analysis of geometrically imperfect FG nano-beams based on stress-driven nonlocal elasticity with initial pretension force

Size-dependent flexural nonlinear free vibrations of geometrically imperfect straight Bernoulli-Euler functionally graded nano-beams are investigated by the stress-driven nonlocal integral model (SDM). By using the Galerkin method, the governing equations is reduced to a nonlinear ordinary differential equation. The closed form analytical solution of the nonlinear natural flexural frequency for Simply-Supported, Clamped-Simply Supported and Clamped-Clamped nano-beams is then established using the Hamiltonian approach to nonlinear oscillators. Effects of nonlocal scale parameter and an initial axial tension force on fundamental frequencies are examined and compared with those obtained by Eringen’s nonlocal model. It is shown that the nonlinear approach based on nonlocal stress model, with the appropriate constitutive boundary conditions, is capable of capturing the dynamical responses of the nano-beams and provides an advantageous method for the design and the optimization of a wide range of nano-scaled beam-like components of Nano-Electro-Mechanical-Systems (NEMS).

On size-dependent mechanics of nanoplates

In this paper, a two-dimensional stress-driven nonlocal integral model is introduced for the bending and transverse vibration of rectangular nanoplates for the first time. An appropriate kernel function, which satisfies all essential properties, is proposed for two-dimensional problems in the Cartesian coordinate system. Using Leibniz integral rule and Hamilton's principle, the curvature-moment relations, classical and constitutive boundary conditions, as well as the equations of motion of rectangular small-scale plates are derived. Two differential quadrature techniques are utilised to implement both classical and non-classical boundary conditions and obtain an accurate numerical solution. The solution is used to simulate the bending and vibration of nanoplates. The Laplacian-based nonlocal strain gradient model of plates is also developed for the sake of comparison. It is found that the stress-driven integral model can better estimate the size-dependent mechanical characteristics of small-scale rectangular plates with various boundary conditions.

One-dimensional stress-driven nonlocal integral model with bi-Helmholtz kernel: Close form solution and consistent size effect

In this paper, stress-driven nonlocal integral model with bi-Helmholtz kernel is applied to investigate the elastostatic tensile and free vibration analysis of microbar. The relation between nonlocal stress and strain is expressed as first type of Fredholm integral equation which is transformed to first type of Volterra integral equation. The general solution to the axial displacement of nonlocal microbar is obtained through the Laplace transformation with four unknown constants. Taking advantage of boundary and constitutive constraint equations, one can obtain the exact tensile displacements of microbar under different boundary and loading conditions, and the nonlinear characteristic equations about vibration frequency of clamped-free and clamped-clamped nonlocal microbars. Numerical results show that the nonlocal microbar model can be degraded to local bar model when the nonlocal parameters approach to 0, and a consistent toughening response for elastostatic tension and free vibration can be obtained for different boundary and loading conditions.

Exact solutions for bending of Timoshenko curved nanobeams made of functionally graded materials based on stress-driven nonlocal integral model

Size-dependent static bending analysis of functionally graded (FG) curved nanobeams based on the Timoshenko beam theory is performed with the application of a stress-driven nonlocal integral model. The governing equations and corresponding boundary conditions are derived via Hamilton’s principle. Through detailed derivation, the integral constitutive equation is equivalent to a differential form equipped with two extra boundary conditions. By using the Laplace transform technique and its inversion, the coupling equations are solved analytically for different boundary and loading conditions. According to the numerical results, it is revealed that for FG Timoshenko curved beam theory, increasing the nonlocal parameter has a consistently stiffening effect on the bending of the curved nanobeams subjected to different boundary and loading conditions. The non-dimensional deflections decrease with the increase of the FG gradient index for all boundary cases, significantly when FG power-law index p < 4. Moreover, through comparing with the results of the Euler-Bernoulli theory, an interesting finding is that for Simply-Clamped, Clamped-Clamped and Clamped-Free nanobeams, the shear-deformable effect becomes more significant with the increase of nonlocal parameter, which cannot be ignored even for larger length-to-height ratio.

Theoretical Analysis of Free Vibration of Microbeams under Different Boundary Conditions Using Stress-Driven Nonlocal Integral Model

Previous studies have shown that Eringen’s differential nonlocal model leads to some inconsistencies for both the Euler–Bernoulli and Timoshenko beams subjected to different boundary conditions. In this paper, the free vibration analysis of the Euler–Bernoulli and Timoshenko beams is performed theoretically using the stress-driven nonlocal integral model. The Fredholm-type integral constitutive equations of the first kind are transformed to Volterra integral equations of the first kind by simply adjusting the limit of integrals. Also, the general solutions to the deflection, bending moment and so on are derived by solving the integro-differential governing equations by the Laplace transformation, of which the unknown constants are determined by the boundary conditions and extra constraint equations related to the constitutive relationship. Then the characteristic equations for free vibration of the Euler–Bernoulli and Timoshenko beams are derived, from which the vibration frequency for different boundary conditions can be determined. The effects of nonlocal parameter and vibration order on the natural frequency of the Euler–Bernoulli and Timoshenko beams are investigated numerically. The results from the present model are validated against those existing in the literature, and demonstrated to be theoretically consistent.

Theoretical analysis on elastic buckling of nanobeams based on stress-driven nonlocal integral model

Several studies indicate that Eringen’s nonlocal model may lead to some inconsistencies for both Euler-Bernoulli and Timoshenko beams, such as cantilever beams subjected to an end point force and fixed-fixed beams subjected a uniform distributed load. In this paper, the elastic buckling behavior of nanobeams, including both Euler-Bernoulli and Timoshenko beams, is investigated on the basis of a stress-driven nonlocal integral model. The constitutive equations are the Fredholm-type integral equations of the first kind, which can be transformed to the Volterra integral equations of the first kind. With the application of the Laplace transformation, the general solutions of the deflections and bending moments for the Euler-Bernoulli and Timoshenko beams as well as the rotation and shear force for the Timoshenko beams are obtained explicitly with several unknown constants. Considering the boundary conditions and extra constitutive constraints, the characteristic equations are obtained explicitly for the Euler-Bernoulli and Timoshenko beams under different boundary conditions, from which one can determine the critical buckling loads of nanobeams. The effects of the nonlocal parameters and buckling order on the buckling loads of nanobeams are studied numerically, and a consistent toughening effect is obtained.

Stress-driven nonlocal integral elasticity for axisymmetric nano-plates

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).

Nonlocal inflected nano-beams: A stress-driven approach of bi-Helmholtz type

Size-dependent bending behavior of Bernoulli-Euler nano-beams is investigated by elasticity integral theory. Eringen’s strain-driven nonlocal integral formulation, providing the bending moment interaction field as output of the convolution between elastic curvature and bi-Helmholtz averaging kernel, is examined. Corresponding higher-order constitutive boundary conditions are established for the first time, showing inapplicability of strain-driven nonlocal integral law of bi-Helmholtz type to continuous nano-structures defined on bounded domains. As a remedy, an innovative stress-driven nonlocal integral elastic model of bi-Helmholtz type is presented for inflected Bernoulli-Euler nano-beams, by swapping input and output of Eringen’s nonlocal integral elastic law. The proposed stress-driven nonlocal integral constitutive law is proven to be equivalent to a fourth-order linear differential equation in the elastic curvature, fulfilling suitable homogeneous constitutive boundary conditions. The corresponding elastic equilibrium problem of an inflected nano-beam is well-posed and numerically solved for statics schemes of nano-technological interest, under standard loading and kinematic boundary conditions. New benchmarks for nonlocal computational mechanics are also detected.

Stress-driven nonlocal integral model for Timoshenko elastic nano-beams

Size-dependent structural behavior of inflected Timoshenko elastic nano-beams is investigated by nonlocal continuum mechanics. An innovative stress-driven integral model of elasticity is conceived by swapping source and output fields of Eringen's strain-driven theory. Unlike Eringen's model, the stress-driven nonlocal integral formulation leads to well-posed structural problems. Solution uniqueness and continuous dependence on data are thus ensured. Selected case-studies of technical interest are examined and exact nonlocal solutions of Timoshenko nano-beams are provided, detecting thus also new benchmarks for numerical analyses. The contributed results are compared with those obtained by the gradient theory of elasticity and by the differential constitutive model consequent (not equivalent) to Eringen's strain-driven theory equipped with Helmholtz's kernel.