Nonlocal Stress Research Articles

Size-dependent buckling analysis of Euler–Bernoulli nanobeam under non-uniform concentration

Nano-sized batteries composed of nanostructured electrode materials stand as one of the most promising candidates for next-generation rechargeable charging devices which have been widely used in the energy storage systems for advanced power applications. Buckling analysis of nanobeam under non-uniform concentration is significantly important for the design of nano-sized batteries under the rapid charging mode. In this work, such issue is investigated in the framework of the size-dependent mechanical–diffusion model, and size effect of mass transfer on buckling property is considered for the first time. By using the eigenvalue method, the critical buckling loads of Euler–Bernoulli nanobeam under the conditions of clamped–clamped, clamped–free, simply supported–simply supported and clamped–simply supported are analytically obtained. The derived results are compared with those of non-gradient nonlocal elastic stress theory, classical elasticity theory and classical theory of mass transfer. It is also found that the value of critical buckling load will be reduced if diffusive nonlocal parameter becomes larger.

Exact analytical solutions to the problem of relative post-buckling stiffness of thin nonlocal graphene sheets

Abstract The main purpose of this research is to obtain analytical formulations for exact calculation of relative post-buckling stiffness of nonlocal graphene sheets. In addition to calculating the post-buckling stiffness reduction, the buckling and initial post-buckling responses of these structures, when they are subjected to end-shortening strain, have also been studied. To investigate these phenomena, a new technique called semi-Galerkin technique is used in which the out-of-plane deflection function is firstly postulated as the only displacement field and then, exact nonlocal stress function is calculated through a complete solution of the von-Karman compatibility equation. Finally, Galerkin's method is used to solve the unknown parameter considered in the proposed technique. The nano-sheets are modeled as an orthotropic layer with Kirchhoff assumptions and nonlocal differential elasticity theory is employed to achieve the buckling loads and exact relative stiffness values. For in-plane movements of the longitudinal edges of the nano-sheets, two essential and natural boundary conditions are adopted to be “straightly movable” or “freely movable”. The effects of aspect ratio and nonlocal parameter have been studied for each type of boundary conditions and for graphene sheets with different materials. The stress distribution along the length and the width of the nano-sheets is also investigated and discussed by the two local and nonlocal theories and for various values of nonlocal parameter.

Snap-buckling of functionally graded multilayer graphene platelet-reinforced composite curved nanobeams with geometrical imperfections

We in this paper dedicate to study the snap-buckling of functionally graded multilayer graphene platelet-reinforced composite curved beams with geometrical imperfections on the nanometer length. It is supposed that graphene platelets (GPLs) are uniformly distributed and randomly oriented in each layer, while its weight fraction varies from one layer to another based on various functionally graded patterns. The effective material properties of functionally graded multilayer GPLRC beams are evaluated via the model of the Halpin-Tsa. In the theoretical framework of the nonlocal strain gradient theory (NSGT) and the surface elasticity theory, a generalized nonlinear size-dependent curved beam model is established. The novel model includes not only a nonlocal parameter and a material length scale parameter of NSGT but also three surface constants of the surface elasticity theory, which can explore the coupling effect of nonlocal stress, strain gradient and surface energy on the snap-buckling of nanoarches. Furthermore, the present model can be transformed into the Euler-Bernoulli shallow arch model, the Timoshenko shallow arch model as well as the Reddy's higher-order shear deformation shallow arch model by choosing the appropriated shape function. Next, On the basis of the principle of Hamilton, the nonlinear governing equations of curved nanobeams are derived. Then to obtain the analytical solution of snap-buckling of curved beams, the nonlinear equations are solved by a two-step perturbation method. Finally, a detailed parametric study is carried out based on the analytical solution, aiming at analyzing the influences of respective physical parameters on the snap-buckling of such nanostructure.

Variationally consistent dynamics of nonlocal gradient elastic beams

The variational static formulation contributed in [International Journal of Engineering Science 143, 73–91 (2019)] is generalized in the present paper to model axial and flexural dynamic behaviors of elastic nano-beams by nonlocal strain and stress gradient approaches. Appropriate forms of non-standard boundary conditions are detected and properly prescribed. Equivalence between differential laws and integral convolutions is elucidated and invoked to analytically evaluate size-dependent axial and flexural fundamental frequencies of cantilever and fully-clamped beams which significantly characterize new-generation nano-actuators. The proposed methodology and ensuing results are tested by pertinent outcomes in literature. Advantageously, in comparison with available nonlocal gradient models, the developed formulation of elasticity leads to well-posed dynamic structural problems of nano-mechanics. Outcomes obtained by the well-established strain-driven and stress-driven nonlocal and gradient theories of Engineering Science are recovered under special ad hoc assumptions. The present study offers a simple and effective strategy to predict peculiar stiffening and softening dynamic responses of nano-scaled components of advanced technological devices, such as Nano-Electro-Mechanical-Systems (NEMS), and modern composite nano-structures.

Eigenfrequencies of microtubules embedded in the cytoplasm by means of the nonlocal integral elasticity

A biologically microscopic system presenting a highly scientific interest is the microtubule (MT). Our research endeavor revolves around the eigenfrequencies’ analysis of an MT embedded in the cytoplasm of the cell by means of the nonlocal integral elasticity for the first time. The MT is simulated as a beam and the cytoplasm as a Pasternak-type elastic foundation, respectively. The responses of the nonlocal integral stress models show to have a softening behavior in comparison with that of the classic model. Unlike the nonlocal differential model, no paradoxes and inconsistencies are raised for the nonlocal integral models. Our research conclusions are a hopeful sign for the applications of biomaterials and bioengineering structures.

Theoretical analysis for static bending of Euler–Bernoulli using different nonlocal gradient models

In this work, the bending response of Euler–Bernoulli beam is analyzed with three nonlocal strain gradient model with high-order boundary conditions and modified nonlocal strain and stress gradient models, in which the integral constitutive equation is transformed to an equivalent differential equation with two extra constraint equations. The corresponding differential equations are solved uniquely by the Laplace transformation and its inverse version. Under different boundary and loading conditions, the inconsistency of a nonlocal strain gradient model is examined extendedly, and consistent softening and toughening responses can be observed while using modified nonlocal strain and stress gradient model.

Semi-Implicit Numerical Strategy for Implementing a Nonlocal Finite-Deformation Gurson Model for Glassy Polymer Using Finite Element Analysis

Post-peak strain softening for some thermoplastic resins often leads to localized deformation with plastic flow instability, spurious energy dissipation, and numerical solutions with mesh dependency using finite element analysis (FEA) due to the lack of a length scale. This paper introduces the nonlocal integral theory with a length scale into the Gurson model to capture objective void evolution and shear bands for glassy polymer under quasi-static loading. Nonlocal numerical technique using the finite-deformation Gurson model is developed, where the nonlocal implicit stress return algorithm and the nonlocal consistent tangent stiffness are presented in the corotational configuration by performing nonlocal averaging on the void volume fraction in a semi-implicit way. The numerical strategy is implemented by combining the finite element software ABAQUS-UMAT, USDFLD, and UEXTERNALDB subroutines. For plane tension problem, the developed nonlocal numerical technique circumvents mesh dependency well by studying the void growth, the evolution of shear bands, and the load responses.

A meshless collocation method for band structure simulation of nanoscale phononic crystals based on nonlocal elasticity theory

In this paper, the band structures of nanoscale phononic crystals based on the nonlocal elasticity theory are calculated by using a meshfree local radial basis function collocation method (LRBFCM). The direct method is applied to enhance the stability of the derivative calculations in the LRBFCM. A simple summation in the LRBFCM is proposed to deal with the integration related to the nonlocal stresses or tractions. The LRBFCM for the band structure calculations is validated by the results obtained with the first-principle and the transfer matrix (TM) method for one-dimensional (1D) phononic crystals, as well as the comparison of the frequency responses of the two-dimensional (2D) periodic structures.

On transient hygrothermal vibration of embedded viscoelastic flexoelectric/piezoelectric nanobeams under magnetic loading

This paper investigates the vibration characteristics of flexoelectric nanobeams resting on viscoelastic foundation and subjected to magneto-electro-viscoelastic-hygro-thermal (MEVHT) loading. In this regard, the Nonlocal strain gradient elasticity theory (NSGET) is employed. The proposed formulation accommodates the nonlocal stress and strain gradient parameter along with the flexoelectric coefficient to accurately predict the frequencies. Further, with the aid of Hamilton\'s principle the governing differential equations are derived which are then solved through Galerkin-based approach. The variation of the natural frequency of MEVHT nanobeams under the influence of various parameters such as the nonlocal strain gradient parameter, different field loads, power-law exponent and slenderness ratio are also investigated.

Bending Analysis of Nonlocal Functionally Graded Beams

In this paper, we study the nonlocal linear bending behavior of functionally graded beams subjected to distributed loads. A finite element formulation for an improved first-order shear deformation theory for beams with five independent variables is proposed. The formulation takes into consideration 3D constitutive equations. Eringen’s nonlocal differential model is used to rewrite the nonlocal stress resultants in terms of displacements. The finite element formulation is derived by means of the principle of virtual work. High-order nodal-spectral interpolation functions were utilized to approximate the field variables, which minimizes the locking problem. Numerical results and comparisons of the present formulation with those found in the literature for typical benchmark problems involving nonlocal beams are found to be satisfactory and show the validity of the developed finite element model.

On scale–dependent stability analysis of functionally graded magneto-electro-thermo-elastic cylindrical nanoshells

The present paper employs nonlocal strain gradient theory (NSGT) to study buckling behavior of functionally graded magneto-electro-thermo-elastic (FG-METE) nanoshells under various physical fields. NSGT modeling of the nanoshell contains two size parameters, one related to nonlocal stress field and another related to strain gradients. It is considered that mechanical, thermal, electrical and magnetic loads are exerted to the nanoshell. Temperature field has uniform and linear variation in nanoshell thickness. According to a power-law function, piezo-magnetic, thermal and mechanical properties of the nanoshell are considered to be graded in thickness direction. Five coupled governing equations have been obtained by using Hamilton's principle and then solved implementing Galerkin's method. Influences of temperature field, electric voltage, magnetic potential, nonlocality, strain gradient parameter and FG material exponent on buckling loads of the FG-METE nanoshell have been studied in detail.

Experimental Investigation and Numerical Simulation on the Crack Initiation and Propagation of Rock With Pre-Existing Cracks

Rock mass behavior is determined not only by the properties of the rock matrix but also mostly by the pre-existing cracks in the rock mass. Before the overall failure of rock, the crack initiation and propagation around the tip of pre-existing cracks (i.e., pre-crack) will occur and contribute to rock failure. In this paper, the deep granite from a gold mine is taken and made to specimens with the pre-crack of 0.3 mm thickness. Uniaxial compression tests are carried out on the pre-cracked specimens. The acoustic emission (AE) sensors and digital image correlation (DIC) system are employed to record the failure characteristics of the specimens. The extended finite element method (XFEM) with the non-local stress field calculation is used to simulate the crack initiation and propagation of pre-cracks. The crack patterns, opening and shearing displacements of the cracked surface, and the crack length development are obtained from numerical simulations. Finally, the effects of friction of crack surface on the crack pattern and crack propagation are investigated and discussed. It has been found that, for pre-cracked specimens, crack initiation and propagation will occur when the stress is much smaller than the rock compressive strength. And in the range of pre-crack angle 30-60°, the larger the pre-crack angle is, the larger the compressive strength is. The crack patterns from numerical simulations have a good agreement with those from experimented DIC results. Moreover, the order of crack propagation speed is consistent with the order of the compressive strength. The crack pattern and crack propagation are affected by the friction coefficient of the cracked surface.

A single-variable first-order shear deformation nonlocal theory for the flexure of isotropic nanobeams

In this paper, a displacement-based single-variable first-order shear deformation nonlocal theory for the flexure of isotropic nanobeams is presented. In the present theory, the beam axial displacement consists of a bending component, whereas the beam transverse displacement consists of a bending component and a shearing component. Bending components do not take part in the cross-sectional shearing force, and the shearing component does not take part in the cross-sectional bending moment. The present theory utilizes linear strain–displacement relations. The displacement functions of the present theory give rise to the constant transverse shear strain through the beam thickness. Hence as is the case with the nonlocal Timoshenko beam theory, the present theory also requires a shear correction factor. In the present theory, nonlocal differential stress–strain constitutive relations of Eringen are utilized in order to take into account the size-dependent behaviour of nanobeams. In the present theory, beam gross equilibrium equations are utilized to derive the governing differential equation. As against other first-order shear deformation nonlocal beam theories, the present theory involves only one governing differential equation of fourth order and has only one unknown function. There exists a strong resemblance between expressions of the cross-sectional shearing force, cross-sectional bending moment and governing differential equation of the present theory and corresponding expressions of the nonlocal Bernoulli–Euler beam theory based on the nonlocal elasticity theory of Eringen. For the present theory, boundary conditions are described. The present theory also describes two distinct and physically meaningful clamped boundary conditions. Illustrative examples pertaining to the flexure of shear deformable isotropic nanobeams demonstrate the efficacy of the present theory.

A nonlocal strain gradient shell model incorporating surface effects for vibration analysis of functionally graded cylindrical nanoshells

In this paper, a novel size-dependent functionally graded (FG) cylindrical shell model is developed based on the nonlocal strain gradient theory in conjunction with the Gurtin-Murdoch surface elasticity theory. The new model containing a nonlocal parameter, a material length scale parameter, and several surface elastic constants can capture three typical types of size effects simultaneously, which are the nonlocal stress effect, the strain gradient effect, and the surface energy effects. With the help of Hamilton’s principle and first-order shear deformation theory, the non-classical governing equations and related boundary conditions are derived. By using the proposed model, the free vibration problem of FG cylindrical nanoshells with material properties varying continuously through the thickness according to a power-law distribution is analytically solved, and the closed-form solutions for natural frequencies under various boundary conditions are obtained. After verifying the reliability of the proposed model and analytical method by comparing the degenerated results with those available in the literature, the influences of nonlocal parameter, material length scale parameter, power-law index, radius-to-thickness ratio, length-to-radius ratio, and surface effects on the vibration characteristic of functionally graded cylindrical nanoshells are examined in detail.

Overview of the geometrical influence on the fatigue strength of steel butt welds by a nonlocal approach

AbstractThe paper deals with the high‐cycle fatigue behaviour of steel butt‐welded joints in the as‐welded condition, under constant amplitude loading. It tackles the problem of strength assessment by changing the geometrical shape and features of the weldment. This investigation considers the recommendations available in design standards, particularly referring to the Eurocodes, and it re‐analyses experimental data where a clear profile of the actual weld shape is available. For the available weld profiles, numerical investigations are shown by applying the Implicit Gradient approach for the evaluation of the nonlocal effective stress in the fatigue strength assessment. The results are very promising since the nonlocal approach significantly agrees with design standard recommendations in conventional cases, but it is also able to provide specific indications on the influence of the geometrical features, as well as the tip radius and opening angles, specifying their influence when no explicitly detailed indications are available in the design standards.

Dynamic analysis of two collinear permeable Mode-I cracks in piezoelectric materials based on non-local piezoelectricity theory

Purpose Based on the non-local piezoelectricity theory, this paper is concerned with two collinear permeable Mode-I cracks in piezoelectric materials subjected to the harmonic stress wave. The paper aims to discuss this issue. Design/methodology/approach According to the Fourier transformation, the problem is formulated into two pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. Findings Finally, the dynamic non-local stress and the dynamic non-local electric displacement fields near the crack tips are obtained. Numerical results are provided to illustrate the effects of the distance between the two collinear cracks, the lattice parameter and the circular frequency of the incident waves on the entire dynamic fields near the crack tips, which play an important role in designing new structures in engineering. Originality/value Different from the classical solutions, the present solution exhibits no stress and electric displacement singularities at the crack tips in piezoelectric materials. It is found that the maximum stress and maximum electric displacement can be used as a fracture criterion.

A framework and finite element formulation for incompatibility tensor‐based gradient plasticity

AbstractThe present contribution deals with a gradient plasticity formulation that accounts for energetic contributions of plastic incompatibilities that are interpreted in terms of dislocation densities. At the outset of the theory, an extended form of the dissipation inequality in terms of a nonlocality residual is assumed in accordance with [1]. Taking into account dislocation density tensors as additional arguments of the free energy density function, this form naturally gives rise to the balance equation of a generalised (quasi) nonlocal stress field that is energetically conjugated to the plastic deformation. Against this background one furthermore observes that the energetic dual to the dislocation density tensor is closely related to a back stress‐type part of the generalised stress field. For the solution of the resulting system of coupled partial differential equations, a multi‐field finite element approach is taken, with the plastic consistency conditions being accounted for on global level by means of Fischer‐Burmeister complementarity functions.

Dynamic non-local stress analysis of two collinear semi-permeable mode-I cracks in a piezoelectric medium

This article investigates the dynamic non-local stress analysis of two collinear semi-permeable mode-I cracks in a piezoelectric medium under the harmonic waves by using the generalized Almansi theorem and the Schmidt method. Based on the Fourier transform technique, this problem is formulated into coupled dual integral equations. The dynamic stress and the dynamic electric displacement fields at the crack tips are obtained by solving the derived dual integral equations. Numerical examples are provided to show the effects of the crack length, the distance between the two collinear cracks, the lattice parameter, the electric permittivity of the air inside the crack, and the characteristics of the harmonic wave on the dynamic stress field and the dynamic electric displacement field near the crack tips. The dynamic stress and electric displacement decrease with increasing the distance between two collinear cracks and lattice parameter in a piezoelectric medium. Meanwhile, the dynamic field will impede or enhance crack propagation in piezoelectric medium depending on the circular frequency of the incident wave. Different from the classical solutions, the present solutions exhibit no stress and electric displacement singularities at the crack tips. This work is expected to be helpful for theoretical modeling of piezoelectric medium at nanoscale.

A non-local fractional stress–strain gradient theory

A generalized non-local stress–strain gradient theory is presented using fractional calculus. The proposed theory includes as a special case: the classical theory; the non-local strain gradient theory; the Eringen non-local theory; the strain gradient theory; the general Eringen non-local theory; and the general strain gradient theory. This new formulation is therefore more comprehensive and more complete to model physical phenomena. Its application has been shown in free vibration, buckling and bending of simply supported (S–S) nano-beams. The non-linear governing equations have been solved by the Galerkin method. Furthermore the effects of different (additional) model parameters like: the length scale parameter; the non-local parameter; and different orders (integer and non-integer) of strain and stress gradients have been shown.

Buckling and frequency analysis of the nonlocal strain–stress gradient shell reinforced with graphene nanoplatelets

In this study, buckling and vibrational characteristics of a nanoshell reinforced with graphene nanoplatelets under uniform axial load are investigated. The material properties of the piece-wise graphene-reinforced composites (GPLRCs) are assumed to be graded in the thickness direction of a nanoshell and are estimated using a nanomechanical model. The effects of the small scale are analyzed based on nonlocal stress–strain gradient theory (NSGT). The governing equations and boundary conditions (BCs) are developed using Hamilton’s principle and are solved with assistance of the generalized differential quadrature method. The novelty of the current study is the consideration of GPLRC and size effect as well as satisfying various boundary conditions implemented on the proposed model using NSGT. The results show that, nonlocal parameter, graphene platelet (GPL) distribution pattern, length scale parameter, number of layers, and GPL weight function have significant influence on the buckling and natural frequency of the GPLRC nanoshell. Another significant result is that nonlocal parameter does not have any effect on the buckling load for each BC. The results of the current study are useful for design of the nanoactuators and nanosensors.