Nonlocal Stress Field Research Articles

A finite element based approach for nonlocal stress analysis for multi-phase materials and composites

AbstractThis study proposes a numerical method for calculating the stress fields in nano-scale multi-phase/composite materials, where the classical continuum theory is inadequate due to the small-scale effects, including intermolecular spaces. The method focuses on weakly nonlocal and inhomogeneous materials and involves post-processing the local stresses obtained using a conventional finite element approach, applying the classical continuum theory to calculate the nonlocal stresses. The capabilities of this method are demonstrated through some numerical examples, namely, a two-dimensional case with a circular inclusion and some commonly used scenarios to model nanocomposites. Representative volume elements of various nanocomposites, including epoxy-based materials reinforced with fumed silica, silica (Nanopox F700), and rubber (Albipox 1000) are subjected to uniaxial tensile deformation combined with periodic boundary conditions. The local and nonlocal stress fields are computed through numerical simulations and after post-processing are compared with each other. The results acquired through the nonlocal theory exhibit a softening effect, resulting in reduced stress concentration and less of a discontinuous behaviour. This research contributes to the literature by proposing an efficient and standardised numerical method for analysing the small-scale stress distribution in small-scale multi-phase materials, providing a method for more accurate design in the nano-scale regime. This proposed method is also easy to implement in standard finite element software that employs classical continuum theory.

Fracture behaviour of nanobeams through Two-Phase Local/Nonlocal Stress-Driven model

The aim of the present work is to employ the two-phase local/nonlocal Stress-Driven integral Model (SDM) to analyse the size-dependent Mode I fracture behaviour of Bernoulli-Euler cracked nanobeams, in terms of energy release rate, stress intensity factor and nonlocal stress field near the crack tip. Both edge- and centrally-cracked nanobeams, subjected to concentrated forces, are examined. The edge-cracked nanobeam is modelled as a pair of cantilever nanobeams, whereas the centrally-cracked nanobeam as a pair of double-clamped nanobeams with internal discontinuity due to concentrated loads. Moreover, a comparison with the results obtained from a gradient elasticity theory based model, available in the literature, is performed. From the present study, it is observed that the energy release rate decreases by increasing the nonlocality, showing the superior fracture performance of nanobeams with respect to large-scale beams.

Nonlocal hcp kernel functions based on ab initio calculations: Pertinent dislocation problems revisited

Eringen's nonlocal theory and an accurate determination of the nonlocal kernel functions for hexagonal close-packed (hcp) crystals are of interest. The kernel functions are closely related to the anisotropy as well as any crystalline symmetries. To this end, five new distinct nonlocal kernel functions which have the characteristics of discrete atomistic Green's functions in the stress space are obtained through consideration of the nonlocal dispersion relations associated with certain directions combined with ab initio Density Functional Perturbation Theory (DFPT) calculations of the pertinent phonon frequencies. This is the first work which provides the nonlocal hcp kernel functions pertinent to each component of the elastic moduli tensor of Eringen's nonlocal continuum theory based on quantum mechanical considerations. Moreover, the nonzero components of the classical elastic moduli as well as the equilibrium lattice parameters associated with the hcp crystals of Mg, Cd, Ti, and Zn have been computed using ab initio Density Functional Theory (DFT) and compared with the available experimental data. In order to show the importance of the anisotropy of the kernel functions, the nonlocal stress field of a straight screw dislocation and a straight edge dislocation inside an infinitely extended hcp medium have been examined for the case where the dislocation lies in the basal plane. Within this theory the unrealistic singular behavior of the classical stresses at the dislocation core is eliminated. In contrast to the classical anisotropy theory that predicts singular shear stress on the slip plane at the core of the dislocation, the corresponding nonlocal shear stress component predicted by the present theory is zero at the dislocation core. It is noteworthy to mention that the normalized nonlocal shear stress distribution and its maximum value are greatly influenced by the anisotropy of the crystalline solids.

Non‐local stress and electric displacement solution of a 3D semi‐permeable rectangular crack in infinite orthotropic piezoelectric materials

AbstractThe paper presents the non‐local stress and electric displacement solution of a 3D semi‐permeable rectangular crack in infinite orthotropic piezoelectric materials (OPMs). With the help of the generalized Almansi's theorem and 2D Fourier transform, the boundary problem is formulated by three pairs of dual integral equations, and the displacement jumps across the crack surfaces are defined. The Schmidt method is used to solve the dual integral equations. The non‐local stress field (NSF) and the non‐local electric displacement field (NEDF) along the crack edges are deduced. Numerical results are reported to explain the influence of the size of the rectangular crack, the lattice parameter and the electric permittivity of the air inside the crack on the NSF and the NEDF at the crack edges in OPMs in detail. The present non‐local solutions exhibit no stress and electric displacement singularities along the crack edges, and may be benefit future works.

A compressive study for porous FG curved nanobeam under various boundary conditions via a nonlocal strain gradient theory

This paper is concerned with the analysis of deflection, stresses, buckling and vibration analysis for a porous functionally graded (FG) curved nanobeam with different boundary conditions using a nonlocal strain gradient theory. The curved nanobeam is made of porous FG material. This property varies according to a power-law function over the thickness. The stresses can be calculated on the basis of the nonlocal strain gradient elasticity model which contains both the nonlocal stress and strain gradients stress field. The Hamilton’s principle is adopted in order to develop differential equation and boundary condition. Numerical results with various cases of boundary conditions are carried out with a view to discuss the influences of porosity factor, nonlocal, length-scale parameters and gradient index on the deflection, stresses, buckling and vibration of porous FG curved nanobeam.

Discrete Kernel Functions for fcc Crystals Within Eringen’s Nonlocal Theory of Elasticity

The dilemma with the deficiencies of the nonlocal kernel functions as the building blocks of the Eringen’s nonlocal theory has been of concern. The aim of the current work is to provide a remedy for the calculation of the components of the nonlocal moduli tensor pertinent to face center cubic (fcc) crystals accounting for their true symmetry group. To this end, three new distinct nonlocal kernel functions which are the discrete atomistic Green’s functions in the stress space are obtained through the nonlocal dispersion relations associated with the longitudinal and shear waves in fcc crystals combined with the corresponding ones calculated via ab initio based on density functional perturbation theory (DFPT). In other words, based on nonlocal continuum and quantum mechanical considerations different kernel functions pertinent to each component of the elastic moduli tensor are calculated for the first time. Moreover, the nonzero components of the classical elastic moduli as well as the equilibrium lattice parameter associated with Al, Cu, Ni, Pd, and Ag fcc crystals have been computed using ab initio density functional theory (DFT) and compared with the available experimental data. In order to show the importance of the anisotropy of the kernel functions, the nonlocal stress field of a screw dislocation inside an infinitely extended fcc medium has been calculated; the solutions for two cases where the dislocation line is along the $\left [ 0\ 0\ 1 \right ]$ and $\left [ 1\ 0\ 1 \right ]$ directions are compared. Within this theory the unrealistic singular behavior of the classical stresses at the dislocation core is eliminated.

Nonlocal strain gradient finite element analysis of nanobeams using two-variable trigonometric shear deformation theory

In the present paper, a new trigonometric two-variable shear deformation beam nonlocal strain gradient theory is developed and applied to investigate the combined effects of nonlocal stress and strain gradient on the bending, buckling and free vibration analysis of nanobeams. The model introduces a nonlocal stress field parameter and a length scale parameter to capture the size effect. The governing equations derived are solved employing finite element method using a 3-nodes beam element, developed for this purpose. The predictive capability of the proposed model is shown through illustrative examples for bending, buckling and free vibration of nanobeams. Comparisons with other higher-order shear deformation beam theory are also performed to validate its numerical implementation and assess its accuracy within the nonlocal context.

A new and efficient constitutive model based on fractional time derivatives for transient analyses of viscoelastic systems

In the open literature, many authors have used the fractional calculus in conjunction with the finite element method to model certain viscoelastic systems. The so-named fractional derivative model may be a better option for transient analyses of systems containing viscoelastic materials due to its causal behavior and its capability to fit accurately the viscoelastic damping properties and to represent properly their fading memory. However, depending on the situation, it leads to costly computations due to the integration of the non-local viscoelastic displacement and stress fields, especially for long time intervals. In this contribution, it is proposed a new and efficient general three-dimensional fractional constitutive formulation based on the use of a recurrence term to give a simplest and low-cost constitutive law to describe the frequency- and temperature-dependent behavior of viscoelastic materials, especially for complex systems. To demonstrate the efficiency and accuracy of the proposed formulation compared with those available in the literature, an academic example formed by a thin three-layer sandwich plate is performed and the main features and capabilities of the proposed methodology are highlighted.

Three-dimensional exact solution of layered two-dimensional quasicrystal simply supported nanoplates with size-dependent effects

A size-dependent plate model is developed to investigate the elastic responses of the multilayered two-dimensional quasicrystal nanoplates based on the nonlocal strain gradient theory for the first time. A nonlocal stress field parameter and a length scale parameter are taken into account in the new model to capture both stiffness-softening and stiffness-hardening size effects. The exact solution for a single-layer two-dimensional quasicrystal simply supported nanoplate is derived by utilizing the pseudo-Stroh formalism in conjunction with the nonlocal strain gradient theory. Afterward, a dual variable and position method is used to deal with the multilayered case. Numerical examples are presented to study the dependence of size-dependent effect on nanoplate length and the influences of scale parameters on the quasicrystal nanoplate subjected to a z-direction mechanical load on its top surface. The proposed model should be useful to verify various nanoplate theories and other numerical methods.

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 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.

Study on Dynamic Characteristics of Microchannel Fluid–Solid Coupling Systems in Nonlocal Stress Fields

Objective of the present work is predicting the dynamic behaviors of fluid-filled microchannels when considering the small-scale effects caused by microchannel and inner fluid. A dynamic Euler–Bernoulli beam model for fluid-filled microchannels is established when applying the nonlocal strain gradient constitutive equation and nonlocal fluid shear stress equation. The scale effects of microtubes are simulated by nonlocality of elastic stress and gradient effect of strain when the fluid nonlocality is applied for predicting the scale effects induced by fluid flow. The dynamic equilibrium equations and boundary conditions for the dynamic tube are derived. By solving the equilibrium equations, different types of scale effects on wave propagation behavior are analyzed. The numerical results indicate that the nonlocal effect induced by microchannels dampens propagation for waves of short wavelengths, whereas the strain gradient effect enhances wave propagation at all wavelengths. The scale effect contributed by fluid flow also leads to decaying of the wave, because the flow velocity is reduced by the nonlocality of the fluid.

Dynamic analytical solution of a limited-permeable mode-I crack in piezoelectric materials based on the non-local theory

This paper presents the dynamic non-local theory solution to a limited-permeable mode-I crack in piezoelectric materials (PMs) with the crack surfaces subjected to harmonic stress waves. The generalized Almansi’s theorem and the Schmidt method are used for this purpose. Based on the Fourier transform technique, this problem is formulated into coupled dual integral equations. The dynamic non-local stress and the dynamic non-local electric displacement fields at the crack tips are obtained by solving the derived dual integral equations. Numerical results are reported and discussed to illustrate the influences of the crack length, the lattice parameter, the characteristics of the harmonic wave and the electric permittivity of the air inside the crack on the dynamic non-local stress field and the dynamic non-local electric displacement field near the crack tips in PMs. As compared to the known electrically permeable crack mode, the proposed one exhibits more applicability.

Application of Chebyshev–Ritz method for static stability and vibration analysis of nonlocal microstructure-dependent nanostructures

This research develops a nonlocal couple stress theory to investigate static stability and free vibration characteristics of functionally graded (FG) nanobeams. The theory introduces two parameters based on nonlocal elasticity theory and modified couple stress theory to capture the size effects much accurately. Therefore, a nonlocal stress field parameter and a material length scale parameter are used to involve both stiffness-softening and stiffness-hardening effects on responses of FG nanobeams. The FG nanobeam is modeled via a higher order refined beam theory in which shear deformation effect is verified needless of shear correction factor. A power-law distribution is used to describe the graded material properties. The governing equations and the related boundary conditions are derived by Hamilton’s principle and they are solved applying Chebyshev–Ritz method which satisfies various boundary conditions. A comparison study is performed to verify the present formulation with the provided data in the literature and a good agreement is observed. The parametric study covered in this paper includes several parameters such as nonlocal and length scale parameters, power-law exponent, slenderness ratio, shear deformation and various boundary conditions on natural frequencies and buckling loads of FG nanobeams in detail.

Characteristics of elastic waves in radial direction of anisotropic solid sphere, a new closed-form solution

This study aims to study the radial wave dispersion of anisotropic solid sphere. In order to include the size-dependent effects, the nonlocal strain gradient theory in conjunction with three dimensional (3D) elasticity theory is used. This model presents both stiffness softening and hardening mechanisms of nanostructures using a nonlocal stress field parameter and also a strain gradient length scale parameter. The nanoparticles are made of different crystallinity (i.e. gold, beryllium oxide, rutile, and hematite), while the equations of wave motion are obtained based on mentioned theories in which an analytic technique is utilized to find the wave frequency, phase velocity and group velocity as a function of wave number. The reliability of present approach is verified for radial vibration by comparing obtained results with those provided in literature. It is demonstrated that, radius of nanoparticle, small-scale parameters, wave number, and crystallinity of nanoparticle have a remarkable effect on radial wave characteristics of anisotropic solid sphere. In addition, the possibility of replacing an anisotropic model with isotropic ones is studied.

An incompatibility tensor-based gradient plasticity formulation—Theory and numerics

In this work we discuss a gradient plasticity formulation which relies on the introduction of higher-order gradient contributions as additional arguments of the free energy function. These gradients can be interpreted in terms of the local geometrically necessary dislocation density. This gives rise to physically well-motivated kinematic-type hardening models in contrast to purely phenomenological approaches. At the same time, the framework results into a regularisation of the formulation such that localised plastic deformation processes in softening materials, e.g. the formation of shear bands, can be simulated. The employed theory is based on an extended nonlocal form of the Clausius–Duhem inequality, motivated by a possible energy exchange between particles at the microstructural level. This gives rise to the balance equation of a nonlocal stress tensor that is found to be work-conjugated to the plastic velocity gradient. The solution of the governing system of partial differential equations is approached by means of a multi-field finite element formulation with the solution of the Karush–Kuhn–Tucker conditions being addressed on a global level by means of Fischer–Burmeister complementary functions. We discuss a specific quadratic energy contribution in terms of the incompatibility, respectively dislocation density tensor that results into well-interpretable contributions to the nonlocal stress field and study the formation of shear bands induced by geometric imperfections as well as the constitutive response at a material interface where the yield limit exhibits a jump discontinuity.

A nonlocal strain gradient theory for scale-dependent wave dispersion analysis of rotating nanobeams considering physical field effects

This paper is concerned with the wave propagation behavior of rotating functionally graded temperature-dependent nanoscale beams subjected to thermal loading based on nonlocal strain gradient stress field. Uniform, linear and nonlinear temperature distributions across the thickness are investigated. Thermo-elastic properties of FG beam change gradually according to the Mori-Tanaka distribution model in the spatial coordinate. The nanobeam is modeled via a higher-order shear deformable refined beam theory which has a trigonometric shear stress function. The governing equations are derived by Hamilton\'s principle as a function of axial force due to centrifugal stiffening and displacement. By applying an analytical solution and solving an eigenvalue problem, the dispersion relations of rotating FG nanobeam are obtained. Numerical results illustrate that various parameters including temperature change, angular velocity, nonlocality parameter, wave number and gradient index have significant effect on the wave dispersion characteristics of the understudy nanobeam. The outcome of this study can provide beneficial information for the next generation researches and exact design of nano-machines including nanoscale molecular bearings and nanogears, etc.

Small scale effects on buckling and postbuckling behaviors of axially loaded FGM nanoshells based on nonlocal strain gradient elasticity theory

By means of a comprehensive theory of elasticity, namely, a nonlocal strain gradient continuum theory, size-dependent nonlinear axial instability characteristics of cylindrical nanoshells made of functionally graded material (FGM) are examined. To take small scale effects into consideration in a more accurate way, a nonlocal stress field parameter and an internal length scale parameter are incorporated simultaneously into an exponential shear deformation shell theory. The variation of material properties associated with FGM nanoshells is supposed along the shell thickness, and it is modeled based on the Mori-Tanaka homogenization scheme. With a boundary layer theory of shell buckling and a perturbation-based solving process, the nonlocal strain gradient load-deflection and load-shortening stability paths are derived explicitly. It is observed that the strain gradient size effect causes to the increases of both the critical axial buckling load and the width of snap-through phenomenon related to the postbuckling regime, while the nonlocal size dependency leads to the decreases of them. Moreover, the influence of the nonlocal type of small scale effect on the axial instability characteristics of FGM nanoshells is more than that of the strain gradient one.

Reissner stationary variational principle for nonlocal strain gradient theory of elasticity

Abstract 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.