Nonlocal Stress Research Articles

Electromechanical Analysis of Flexoelectric Nanosensors Based on Nonlocal Elasticity Theory.

Flexoelectric materials have played an increasingly vital role in nanoscale sensors, actuators, and energy harvesters due to their scaling effects. In this paper, the nonlocal effects on flexoelectric nanosensors are considered in order to investigate the coupling responses of beam structures. This nonlocal elasticity theory involves the nonlocal stress, which captures the effects of nonlocal and long-range interactions, as well as the strain gradient stress. Based on the electric Gibbs free energy, the governing equations and related boundary conditions are deduced via the generalized variational principle for flexoelectric nanobeams subjected to several typical external loads. The closed-form expressions of the deflection and induced electric potential (voltage) values of flexoelectric sensors are obtained. The numerical results show that the nonlocal effects have a considerable influence on the induced electric potential of flexoelectric sensors subjected to general transverse forces. Moreover, the induced electric potential values of flexoelectric sensors calculated by the nonlocal model may be smaller or larger than those calculated by the classical model, depending on the category of applied loads. The present research indicates that nonlocal effects should be considered in order to understand or design basic nano-electromechanical components subjected to various external loads.

Cross-section effect on mechanics of nonlocal beams

In the current practice, the one-dimensional nonlocal constitutive relations are always employed to model beam-type structures, regardless of the inherent three-dimensional interactions between atoms, resulting in inaccurately predicted nonlocal structural behaviors. To improve modeling, the present work first reveals and establishes how the nonlocal interactions in beams’ width and height directions substantially affect the bending behaviors of nanobeams. Based on the new concept of a general three-dimensional nonlocal constitutive relation, a three-dimensional nonlocal Euler–Bernoulli beam model is developed for bending analysis. Closed-form solutions for the deflections of simply supported and cantilevered beams are derived. The cross-section effect on the nonlocal stress, the bending moment and the deflection is explored. Moreover, an effective calibration method is proposed for the determination of the intrinsic characteristic length based on molecular dynamics simulations. It has been found that the actually nonlocal physical dimension of beam-type structures is not in agreement with that from the one-dimensional geometric direction of classical beam mechanics, and the beams’ width and height must be taken into consideration, instead of just employing the beam’s length, a practice which has been somewhat blindly undertaken in the current practice without indeed much theoretical support and understanding. A beam problem at nanoscale has to be viewed and treated as a “three-dimensional” geometric and physical problem due to its geometric dimension in length and its physical dimensions in both the thickness and the width. When its intrinsic characteristic length is comparable to its width or thickness, a rigidity-softening effect of a beam can be observed. Further, it is established that the nonlocal effect is more sensitive in the thickness direction, as compared with the width direction.

Stress correlation function and linear response of Brownian particles

.We determine the non-local stress autocorrelation tensor in an homogeneous and isotropic system of interacting Brownian particles starting from the Smoluchowski equation of the configurational probability density. In order to relate stresses to particle displacements as appropriate in viscoelastic states, we go beyond the usual hydrodynamic description obtained in the Zwanzig-Mori projection-operator formalism by introducing the proper irreducible dynamics following Cichocki and Hess, and Kawasaki. Differently from these authors, we include transverse contributions as well. This recovers the expression for the stress autocorrelation including the elastic terms in solid states as found for Newtonian and Langevin systems, in case that those are evaluated in the overdamped limit. Finally, we argue that the found memory function reduces to the shear and bulk viscosity in the hydrodynamic limit of smooth and slow fluctuations and derive the corresponding hydrodynamic equations.Graphical abstract

Dynamic analysis of functionally graded beams with periodic nanostructures

It is known from experimental studies that softening or hardening material behaviours of nanostructures change with the microstructure of the considered material. However, most of the size dependent continuum models (nonlocal stress gradient, strain gradient and couple stress theories) predict only softening or hardening material behaviour. Except of these size dependent theories doublet mechanics predicts both softening and hardening responses of the material like experiments in micro/nano-structures. In the present study, free vibration analysis of functionally graded (FG) periodic structure nanobeams are investigated via doublet mechanics theory. Periodic FG nanobeams are modelled as a simple crystal lattice type. Micro strains and stresses are expanded in Taylor series and obtained micro relations transformed to macro stress-strain relations. Thus, by use of bottom-up approach yields the more physical and accurate analysis of nanostructures in the doublet mechanics model. After deriving the mathematical formulation of a periodic FG nanobeam, vibration problem is examined for general boundary conditions. Ritz method is used in the solution. Adjusting of softening and hardening responses of the material gives a beneficial optimization and design of nanostructures.

Fundamental solutions of two 3D rectangular semi-permeable cracks in transversely isotropic piezoelectric media based on the non-local theory

PurposeThe paper aims to present the non-local theory solution of two three-dimensional (3D) rectangular semi-permeable cracks in transversely isotropic piezoelectric media under a normal stress loading.Design/methodology/approachThe fracture problem is solved by using the non-local theory, the generalized Almansi's theorem and the Schmidt method. By Fourier transform, this problem is formulated as three pairs of dual integral equations, in which the elastic and electric displacements jump across the crack surfaces. Finally, the non-local stress and the non-local electric displacement fields near the crack edges in piezoelectric media are derived.FindingsDifferent from the classical solutions, the present solution exhibits no stress and electric displacement singularities at the crack edges in piezoelectric media.Originality/valueAccording to the literature survey, the electro-elastic behavior of two 3D rectangular cracks in piezoelectric media under the semi-permeable boundary conditions has not been reported by means of the non-local theory so far.

Universality in incompressible active fluid: Effect of nonlocal shear stress.

Phase transitions in active fluids attracted significant attention within the last decades. Recent results show [L. Chen etal., New J. Phys. 17, 042002 (2015)10.1088/1367-2630/17/4/042002] that an order-disorder phase transition in incompressible active fluids belongs to a new universality class. In this work, we further investigate this type of phase transition and focus on the effect of long-range interactions. This is achieved by introducing a nonlocal shear stress into the hydrodynamic description, which leads to superdiffusion of the velocity field, and can be viewed as a result of the active particles performing Lévy walks. The universal properties in the critical region are derived by performing a perturbative renormalization group analysis of the corresponding response functional within the one-loop approximation. We show that the effect of nonlocal shear stress decreases the upper critical dimension of the model, and can lead to the irrelevance of the active fluid self-advection with the resulting model belonging to an unusual long-range Model A universality class not reported before, to our knowledge. Moreover, when the degree of nonlocality is sufficiently high all nonlinearities become irrelevant and the mean-field description is valid in any spatial dimension.

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.

Effect of boundary conditions and constitutive relations on the free vibration of nonlocal beams

The free vibrations of nonlocal Euler and Timoshenko beams have been studied extensively, but there still remain some problems concerning boundary conditions and constitutive relations. For cantilever beams, a counterintuitive stiffening phenomenon is widely observed. And to the best of the authors’ knowledge, there is no work concerning about the Timoshenko beam with nonlocal shear force and local bending moment. This work reconsiders and explains these problems based on nonlocal differential theory. After obtaining the governing equation for nonlocal Euler beam, it is observed that the governing equation has the same form as that for the vibration of local beam with axial compressive force. Thus by comparing the nonlocal effects to the effects of end axial force, the stiffening phenomenon of cantilever beams is found to be caused by a meaningless term in boundary bending moment. This explanation is also suitable for nonlocal Timoshenko cantilever beams. Besides, for Timoshenko beams, the nonlocal effects on tensile and shear stress constitutive relations are considered separately. Three kinds of nonlocal Timoshenko beam models are solved. Numerical comparison shows that the effects of nonlocal tensile are bigger than that of nonlocal shear stress. Especially, for Timoshenko beams with nonlocal shear force only, the nonlocal effects lower the natural frequency for all boundary conditions. This finding can verify our explanation for the stiffening phenomenon of cantilever beams. In addition, it is also observed that, for both nonlocal beam theories, if all nonlocal internal forces are included in the governing equations, the eigenvalue equations and natural frequencies corresponding to local and nonlocal boundary conditions are the same.

Study of transversely isotropic nonlocal thermoelastic thin nano-beam resonators with multi-dual-phase-lag theory

This study deals with a novel model of forced vibrational analysis of nonlocal transversely isotropic thermoelastic nanobeam resonators due to ramp-type heating and due to time varying exponentially decaying load with multi-dual-phase-lag theory of thermoelasticity. The mathematical model is prepared for the nanobeam in a closed form with the application of Euler–Bernoulli (E–B) beam theory using nonlocal generalized thermoelasticity with multi-dual phase lags. The nonlocal nanobeam theory has a nonlocal parameter to depict small-scale effect. The Laplace Transform technique has been used to find the expressions for lateral deflection, conductive temperature, thermal moment, nonlocal axial stress and thermodynamic temperature for (i) clamped–clamped, (ii) simply supported–simply supported, (iii) clamped–simply supported, (iv) clamped–free, and (v) free–free nanobeam in the transformed domain. The general algorithm of the inverse Laplace Transform is developed to compute the results numerically in physical domain. The results exhibit that the amplitude of deflection and thermal moment is attenuated and depends upon the schematic design of the nanobeam being considered. Also, it can be found from both the numerical calculations and the analytic results that nonlocal multi-dual-phase-lag theory of thermoelasticity with two temperatures due to time varying exponentially decaying load has significant effect on deflection and thermal moment. The effect of different theories of nonlocal thermoelasticity, due to time varying exponentially decaying load, has been depicted on the various quantities. Some particular cases have also been deduced.

Nonlocal beam model and FEM of free vibration for pristine and defective CNTs

The natural frequencies of defective single-walled carbon nanotubes are studied and compared with the values for pristine structures. In order to capture the size-dependent effects in carbon nanotubes, nonlocal stress gradient theory within a Euler-Bernoulli beam model is involved. The analytical computations are compared with finite element analysis of a structural model of carbon nanotubes. The numerical analysis of free vibrations is applied to predict the value of the nonlocal parameter. The eigenfrequencies obtained from finite element analysis are much lower compared with analytical computations of local (classical) beam model. For defective structures having small vacancy, a slight decrease in the eigenfrequencies is observed. All studied boundary conditions indicated the higher the nonlocal parameter is, the lower the natural frequencies are.

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).

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.

Thermo-Electro-Mechanical Size-Dependent Buckling Response for Functionally Graded Graphene Platelet Reinforced Piezoelectric Cylindrical Nanoshells

An accurate buckling response analysis for functionally graded graphene platelet (GPL) reinforced piezoelectric cylindrical nanoshells subject to thermo-electro-mechanical loadings is presented by a rigorous symplectic expansion approach. Three types of GPL reinforced patterns are considered, and the modified Halpin–Tsai model is employed to determine their effective material properties. By using Eringen’s nonlocal stress theory and Reissner’s shell theory, new governing equations are established in the Hamiltonian form. Exact solutions are expanded into symplectic series and three possible forms are derived. A comparison with the existing study is presented to validate the solution and very good agreement is observed. The effects of material and geometrical properties of GPLs, electric voltage and temperature rise on critical buckling stresses are investigated and discussed in detail.

Weak solutions for the stationary anisotropic and nonlocal compressible Navier-Stokes system

In this paper, we prove the existence of weak solutions for the stationary compressible Navier-Stokes equations with an anisotropic and nonlocal viscous stress tensor in a periodic domain T3. This gives an answer to an open problem which is important for applications for instance in geophysics or in microfluidics. When dealing with weak solutions for such non-linear PDE system, the most delicate aspect is the stability analysis: Given a sequence of weak solutions for some well-chosen approximated systems, show that this sequence converges to a solution for the initial system. One of the key ingredients in the proof of stability is the adaptation of a new identity discovered by the authors [2] which was developed to study the quasi-stationary anisotropic compressible Brinkman system. This identity is used in order to recover strong convergence properties for the sequence of velocities and recover a posteriori strong convergence for the sequence of densities.

Axial dynamics of functionally graded Rayleigh-Bishop nanorods

Present study considers the longitudinal dynamic analysis of carbon nanotubes which has been modeled as an axially functionally graded Rayleigh-Bishop rod with the help of nonlocal stress gradient elasticity theory. Governing equation of motion and boundary conditions for the axial dynamics of nanorod have been obtained with variational formulation. Ritz method has been used in the solution of the problem. Effects of lateral inertia and transverse deformation of carbon nanotubes, material composition properties and grading power-law index on the axial dynamics response have been investigated. Mode shapes of functionally graded Rayleigh-Bishop nanorod have been depicted in various cases. Boundary conditions and power-law index have important effect on dynamics of axially graded Rayleigh-Bishop nanorod. Present results could be useful at designing of nano-electromechanical devices.

On the Computational Derivation of Bond-Based Peridynamic Stress Tensor

The concept of ‘contact stress’, as introduced by Cauchy, is a special case of a nonlocal stress tensor. In this work, the nonlocal stress tensor is derived through implementation of the bond-based formulation of peridynamics that uses an idealised model of interaction between points as bonds. The method is sufficiently general and can be implemented to study stress states in problems containing stress concentration, singularity, or discontinuities. Two case studies are presented, to study stress concentration around a circular hole in a square plate and conventionally singular stress fields in the vicinity of a sharp crack tip. The peridynamic stress tensor is compared with finite element approximations and available analytical solutions. It is shown that peridynamics is capable of capturing both shear and direct stresses and the results obtained correlate well with those obtained using analytical solutions and finite element approximations. A built-in MATLAB code is developed and used to construct a 2D peridynamic grid and subsequently approximate the solution of the peridynamic equation of motion. The stress tensor is then obtained using the tensorial product of bond force projections for bonds that geometrically pass through the point. To evaluate the accuracy of the predicted stresses near a crack tip, the J-integral value is computed using both a direct contour approximation and the equivalent domain integral method. In the formulation of the contour approximation, bond forces are used directly while the proposed peridynamic stress tensor is used for the domain method. The J-integral values computed are compared with those obtained by the commercial finite element package Abaqus 2018. The comparison provides an indication on the accurate prediction of the state of stress near the crack tip.

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.

Surface energy layers investigation of intelligent magnetoelectrothermoelastic nanoplates through a vibration analysis

Intelligent materials such as magnetoelectrothermoelastic (METE) nanoplates offer great potential to supply more efficient energy harvesting devices, transducers, sensors, and actuators. In this study, the influences of slanted angle of the METE nanoplate and orthotropic angle of Pasternak foundation on the magnitude of surface energy layers are investigated through a vibrational analysis. The nanoplate is exposed to outer thermal, electric, magnetic and in-plane loadings. For modeling the plate’s nonlocal behavior and surface stresses, the governing equations are constructed based on Hamilton’s principle. Galerkin method is implemented to solve the equilibrium motions, and also for the authenticity of the solution, the equations are resolved by the Navier’s method. Obtained results are deemed useful for the mechanical analysis and design of nano-/microelectromechanical system nanostructures constructed from the METE materials. The numerical examples indicated that after a particular value of slanted angle of the nanoplate, α ≥ 80, the magnitude of surface energy layers in the vibration behavior becomes opposite to the case when the slanted angles are 0° < α < 80°. In addition, the value of the slanted angle of the nanoplate can cause the harmonic responses of the vibration due to the orthotropic angle of Pasternak foundation to shift in harmony with the slanted angle degree.

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.

A new type of peridynamics: Element-based peridynamics

The general peridynamic theories are constructed using the interactions of particles in a horizon. The interactions of particles can be connected by using spring, rod and beam, etc. Element-based peridynamics (EBPD) is proposed in the present study, in which the interactions of particles are expressed by using elements in a horizon, and the basic element concepts in the continuum mechanics are reserved exactly. The force density of EBPD is described using 2-node rod elements for one-dimensional (1D) problems, 3-node triangle elements for two-dimensional (2D) problems and 4-node tetrahedron elements for three-dimensional (3D) problems. The EBPD equations are derived from the principle of the minimum potential energy. The treatments of surface effects, boundary conditions, failure criterion and solving process of the EBPD are provided in detail. The examples of the 1D singular bar, 2D stationary crack near-tip solution, 2D plate crack propagation and 3D cubic demonstrate the effectivity of the proposed EBPD. The proposed EBPD model is used for elasticity, and defined new nonlocal stress and strain expressions. In addition, no restriction of Poisson’s ratio and no zero energy modes are included in EBPD. Furthermore, EBPD has the potential to study the nonuniform discretization, varying horizons, plasticity, thermodynamic and anisotropic material. It is also convenient to couple with FEM to reduce the computational burden.