Nonlocal Constitutive Equations Research Articles

Buckling and vibration analysis of axially functionally graded nanobeam based on local stress- and strain-driven two-phase local/nonlocal integral models

This paper presented an efficient finite element formulation for analysis of the buckling and free vibration of the axial functionally graded (AFG) Euler-Bernoulli nanobeam based on minimum total potential energy principle with linear constraints. The stress- and strain-driven two-phase local/nonlocal integral models are utilized to address small scale effects. The material properties of the AFG nanobeam vary exponentially along the axial direction. The nonlocal integral constitutive equation can be equivalently transformed into a differential equation with two constitutive boundary conditions. The governing equation and standard boundary conditions are derived via Hamilton's principle. The weak form associated with total potential energy is derived and the constitutive boundary conditions can be converted to external forces at the boundaries. By employing a numerical solution methodology on the basis of the finite element method (FEM) together with Lagrangian multiplier method (LMM), a unified finite element formulation based on stress- and strain-driven two-phase local/nonlocal elasticity is constructed to obtain the critical buckling load and vibration frequency. Meanwhile, the numerical results obtained through generalized differential quadrature method (GDQM) validate the efficiency and accuracy of the present results. The influence of gradient index, nonlocal parameter and local phase parameter on buckling and free vibration of AFG Euler-Bernoulli nanobeam is investigated in detail under different nonlocal models and boundary conditions. It is demonstrated that the gradient index has more effect on mode shapes than the buckling load and vibration frequency between the local stress- and strain-driven two-phase local/nonlocal models.

Direct modeling of non-uniform strain field of heterogeneous materials

Engineering materials have a heterogeneous microstructure that induces non-uniform deformations (NUDs) even under uniform stress. Therefore, the development of NUDs in heterogeneous materials under a macroscopically non-uniform stress is characterized by the interactions between micro- and macroscopic NUDs. In this study, we attempt to directly describe microscopic NUDs by introducing a constitutive equation for the strain field. Using polycrystalline metal, which is a well-known heterogeneous material, we performed the uniaxial tensile tests on specimens with curved gauge sections. During modeling, the constitutive equation without nonlocal effect was first fitted based on the distribution of the local equivalent stress and strain. The strain calculated using the fitted equation was regarded as the strain for the homogeneous material, whereas the strain deviating from the fitted equation was assumed to be induced by the microscopic heterogeneity of the material. The respective strains were represented using different constitutive models, and the parameters employed in the models were fitted separately using the experimentally measured strain field. The plastic compliance gradient was employed as a material parameter quantifying the microscopic NUDs. Subsequently, a multiaxial elastoplastic nonlocal constitutive equation was formulated based on the nonlocally extended J2 flow theory. Finite element method simulations introducing the proposed nonlocal effect indicated that a scale-dependent NUD characterized by the interactions between micro- and macro-NUDs could be represented using the proposed model. Furthermore, the proposed model could predict the experimentally observed strain field characterized by a polycrystalline microstructure, such as the dispersion of the strain concentration at the curved gauge section and the appearance of NUDs even under uniform stress.

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.

Microscopic origin of granular fluidity: An experimental investigation.

Granular fluidity has been central to the development of nonlocal constitutive equations, which are necessary for characterizing nonlocal effects observed in experimental granular flow data. However, validation of these equationshas been largely computational due to challenges in laboratory experiments. Specifically, the origin of the fluidity on a microscopic, single-particle level is still unproven. In this work, we present an experimental validation of a microscopic definition of granular fluidity, and show the importance of basal boundary conditions to the validity of the theory.

Nonlocal elasticity of Klein‐Gordon type with internal length and time scales: Constitutive modelling and dispersion relations

AbstractIn this work, a nonlocal elasticity theory with nonlocality in space and time is presented by considering nonlocal constitutive equations with a dynamical scalar nonlocal kernel. Based on the proposed theory, we consider the isotropic nonlocal elasticity of Klein‐Gordon type including a characteristic internal time scale parameter in addition to the characteristic internal length scale parameter. The dispersion relations for homogeneous isotropic media in the framework of nonlocal elasticity of Klein‐Gordon type are analytically determined. The obtained results reveal the ability of the presented nonlocal elasticity model to predict, for the first time in the framework of nonlocal elasticity, in addition to the acoustic modes (low‐frequency modes), optic modes (high‐frequency modes) as well as frequency band‐gaps. The phase and group velocities for all four modes (acoustic and optic branches of longitudinal and transverse waves) are determined showing that all four modes exhibit normal dispersion with positive group velocity. The presented model allows for physically realistic dispersive wave propagation.

Effective nonlocal behavior of peridynamic random structure composites subjected to body forces with compact support and related prospective problems

We consider a static problem for statistically homogeneous matrix linear bond-based peridynamic composite materials (CMs) subjected to body force with compact support. Estimation of the effective displacements is performed by the exploitation of the most popular tools and concepts used in conventional local elasticity of composite materials (CMs) with their adaptation to peridynamics. The method is based on estimation of a perturbator introduced by one inclusion inside the infinite peristatic matrix subjected to the body force. The statistical averages of both the displacements and stresses are estimated by summation of these perturbators for all possible locations of inclusions. It leads to obtaining of a new general integral equation (GIE) where a renormalizing procedure is not required for the involved absolutely convergent integrals. It allows us to estimate not only the effective nonlocal constitutive equation but also to evaluate the statistical field averages (inhomogeneous and nonlocal) inside the phases at the fine scale that is critically important for advanced modeling (e.g. for any nonlinear phenomena potentially considered in the future). In particular, in the generalized effective field method (EFM) proposed, the effective field is evaluated from self-consistent estimations using closing of a corresponding integral equation in the framework of the quasi-crystalline approximation. In so doing, the classical effective field hypothesis is relaxed, and the hypothesis of the ellipsoidal symmetry of the random structure of CMs is not used. The method proposed is a promising ingredient at the construction of provably robust data-driven surrogate equations describing both the effective behavior and effective displacement concentration operator inside inclusions. Some opportunities for generalizations of the method proposed are considered. Numerical results for the estimation of effective deformations are obtained for one-dimensional (1D) statistically homogeneous peridynamic composite bar with the prescribed self-equilibrated body forces.

Nonlocal elasticity of Klein–Gordon type: Fundamentals and wave propagation

A nonlocal elasticity theory with nonlocality in space and time is developed by considering nonlocal constitutive equations with a dynamical scalar nonlocal kernel function. The proposed theory is specified to isotropic nonlocal elasticity of Klein–Gordon type, which is an extension of nonlocal elasticity of Helmholtz type, and it possesses one characteristic internal length scale parameter as well as one characteristic internal time scale parameter in addition to the elastic material parameters. Dynamical nonlocal kernels are analytically derived as retarded Green functions of the Klein–Gordon operator. The dispersion relations of the considered isotropic nonlocal elasticity model of Klein–Gordon type are analytically determined. The obtained results reveal the advantage of the proposed nonlocal elasticity model, that is, its ability to predict for the first time in the framework of nonlocal elasticity in addition to the acoustic modes (low-frequency modes), optic modes (high-frequency modes) as well as frequency band-gaps between the acoustic and optic modes. A qualitative examination of the phase and group velocities for all four modes (acoustic and optic branches of longitudinal and transverse waves) shows that the proposed model allows for physically realistic dispersive wave propagation.

Modeling of buckling of nanobeams embedded in elastic medium by local-nonlocal stress-driven gradient elasticity theory

A novel buckling model is formulated for the Bernoulli-Euler nanobeam resting on the Pasternak elastic foundation. The formulation is based on the local-nonlocal stress-driven gradient elasticity theory. In order to incorporate the size-dependency, the strain at each point is defined as the integral convolutions in terms of the stresses and their first-order gradients in all the points, accounting also for the local contribution. The differential form of the nonlocal constitutive equation, together with a set of constitutive boundary conditions, are used to define the buckling equation in terms of transverse displacement, which is solved in closed form. Both variationally consistent and the constitutive boundary conditions are imposed to calculate the buckling loads and the corresponding mode shapes. The predictions of the present model are in agreement with the results available in the literature for the carbon nanotubes based on the molecular dynamics simulations. Insightful results are presented for the first three buckling modes of local-nonlocal nanobeams considering the gradient effects. The distinctive feature of the present model is its capability to capture both stiffening and softening behaviors at the small-scales, which result in, respectively, higher and lower buckling loads of the nanobeams with respect to those of the large-scale beams.

On Nonlocal Vertical and Horizontal Bending of a Micro-Beam

The vertical and horizontal bending of micro-beams subjected to axial compressive and transverse concentrated loadings is a common lateral deformation in micro-/nano-engineering that plays a significant role in the design and optimization of micro-/nano-devices. The present study aims to investigate it using the nonlocal theory. For this purpose, the simplified mathematical model is developed, and the nonlocal differential constitutive equation is applied. Since the mechanical properties of micro-beams are different from those of macro-beams, the non-classical nonlocal bending moment is introduced to improve the classical bending formulation in order to adapt to the vertical and horizontal bending of micro-beams. The effects of the external load, external size, structural stiffness, and internal characteristic scale on the vertical and horizontal bending deformation including the midpoint deflection and critical compression are presented. The present analytical model and results are validated by the finite element method. It is shown that the critical compression decreases with increasing the internal characteristic scale. Moreover, the midpoint deflection varies remarkably with respect to the axial and transverse loadings, structural stiffness, internal characteristic scale, and external size. An obvious nonlocal scale effect is found, in which the internal characteristic scale cannot be neglected compared with the external size. Besides, a threshold value of the structural stiffness is determined, and the connotation of the nonlocal interaction requires that the structural stiffness shall not be lower than that threshold. A mutual restriction between structural stiffness and external loadings is observed in the vertical and horizontal bending. In particular, it is further proved that the classical continuum mechanics can not be used in micro-/nano-scaled mechanics through a strange phenomenon that is contrary to mechanical common sense in the calculation example. The study is expected to be beneficial to the design and application of micro-beams subjected to the vertical and horizontal bending.

Free transverse vibrations of nanobeams with multiple cracks

A nonlocal model is formulated to study the size-dependent free transverse vibrations of nanobeams with arbitrary numbers of cracks. The effect of the crack is modeled by introducing discontinuities in the slope and transverse displacement at the cracked cross-section, proportional to the bending moment and the shear force transmitted through it. The local compliance of each crack is related to its stress intensity factors assuming that the crack tip stress field is undisturbed (non-interacting cracks).The kinematic field is defined based on the Bernoulli-Euler beam theory, and the small-scale size effect is taken into account by employing the constitutive equation of the stress-driven nonlocal theory of elasticity. In this manner, the curvature at each cross-section is defined as an integral convolution in terms of the bending moments at all the cross-sections and a kernel function which depends on a material characteristic length parameter. The integral form of the nonlocal constitutive equation is elaborated and converted into a differential equation subjected to a set of mathematically consistent boundary and continuity conditions at the nanobeam's ends and the cracked cross-sections. The equation of motion in each segment of the nanobeam between cracks is solved separately and the variationally consistent and constitutive boundary and continuity conditions are imposed to determine the natural frequencies. The model is applied to nanobeams with different boundary conditions and the natural frequencies and the mode shapes are presented at the presence of one to four cracks. The results of the model converge to the experimental results available in the literature for the local cracked beams and to the solutions of the intact nanobeams when the crack length goes to zero. The effects of the crack location, crack length, and nonlocality on the natural frequencies are investigated, also for the higher modes of vibrations. Novel findings including the amplification and shielding effects of the cracks on the natural frequencies are presented and discussed.

微視的な不均質構造によって生じる巨視的な不均一変形のモデル化に関する研究

Engineering materials typically have a heterogeneous microstructure, and it induces microscopic non-uniform deformation even under the macroscopically uniform stress state. With the decrease in the dimension of engineering structural components, such a microscopic non-uniform deformation affects the macroscopic deformation state. In this study, the development of the microscopic heterogeneity-induced non-uniform deformation is modeled using the rate-form nonlocal constitutive equation. First, the inelastic strain is decomposed into local and nonlocal inelastic strains. Then, the nonlocal strain is assumed to be developed by the distribution of plastic compliance, which is characterized by an intrinsic size of the material microstructure. Finally, the multiaxial rate-form constitutive equation, in which the stress on the material points is calculated by both the local and nonlocal strains, is formulated. The finite element simulations of the tensile tests of specimens with curved and parallel gauge sections are performed using the proposed model. The obtained results clarified that the mechanical response and the non-uniform deformation are characterized by the microscopic heterogeneity of the material.

The nonlocal, local and mixed forms of the SPH method

From its early days the SPH method has been criticised for its shortcomings namely tensile instability and consistency. Without thorough understanding of the method attempts were made to make the classical SPH method consistent and stable which resulted in the local and Total Lagrangian forms of SPH similar to the finite element method. In this paper we derived and analysed a consistent nonlocal SPH which has similarity with Bazant’s imbricate continuum. In addition, the paper provides comparison and discussion of different SPH forms including: Classical SPH, Nonlocal, Local and Mixed SPH. The partition of unity approach was used to define the following two mixed forms: Local–Nonlocal and Local–Classical SPH. These mixed forms were intended for modelling of physical processes characterised with local and nonlocal effects (local and nonlocal constitutive equations), e.g. progressive damage and failure. The stabilising effect of the Local form on the Classical SPH, which is inherently unstable (tensile instability), are also illustrated. The stability analysis, presented in appendices A and B, demonstrate stability of the continuous and discrete form of the nonlocal SPH based on Eulerian kernels for elastic continuum.

A bi-Helmholtz type of two-phase nonlocal integral model for buckling of Bernoulli-Euler beams under non-uniform temperature

It is well-acknowledged by the scientific community that Eringen’s nonlocal integral theory is not applicable to nanostructures of engineering interest due to conflict between equilibrium and constitutive requirements. In this paper, a well-posed two-phase nonlocal integral elasticity with the bi-Helmholtz kernel is developed to study the size-dependent buckling response of Bernoulli-Euler beams under non-uniform temperatures. The governing equation is derived by invoking the variational principle of virtual work, and the temperature effect is equivalent to the thermal load along the axial direction, which is determined by nonlocal heat conduction. The two-phase nonlocal integral constitutive equation is transformed into a differential one equipped with four constitutive boundary conditions, and then exact solutions for the buckling loads of the beam with various boundary edges are obtained. Numerical results are validated by comparing them with those from local elasticity. Moreover, the effects of parameters related to the two-phase nonlocal elastic model and the nonlocal heat conductive model are investigated.

Nonlocal integral static problems of nanobeams resting on an elastic foundation

Eringen’s nonlocal theory has been widely used for the study of micro- and nano-structures exhibiting size effect phenomena. Eringen’s nonlocal differential constitutive equation has been employed to structural engineering applications, and a number of inconsistencies and paradoxes has been raised. This work centers on exploring static engineering benchmark problems of a nanobeam resting on a Pasternak-type elastic foundation by means of the nonlocal integral elasticity for the first time. The modified kernel’s model and the two phase stress model leading to well-posed problems are used. The static responses of the integral models show to have a flexible behavior in comparison with those of the classic and the nonlocal differential models for all the investigated problems. Neither paradoxes nor inconsistencies are raised for the integral models. The modified kernel highlights the model’s robustness from a physical point of view. The conclusions are promising to spur the applications of nanomaterials, nanocomposites and biomaterials.

A generalized integro-differential theory of nonlocal elasticity of n-Helmholtz type—part II: boundary-value problems in the one-dimensional case

This paper is the second in a series of two that deal with a generalized theory of nonlocal elasticity of n-Helmholtz type. This terminology is motivated by the fact that the attenuation function (kernel) of the integral type nonlocal constitutive equation is the Green function associated with a generalized Helmholtz differential operator of order n. In the first paper, the governing equations have been derived and supported by suitable thermodynamic arguments. In this second paper, the proposed nonlocal model is specialized for the one-dimensional case to solve boundary-value problems. First, the relevant higher-order nonstandard boundary conditions in the differential (or, more precisely, integro-differential) version of the theory are derived. These boundary conditions are consistent with the particular family of attenuation functions adopted in the integral formulation. Then, some simple applications to statics and dynamics problems are presented. In particular, the theory is used to capture the static response and to perform free vibration analysis of a discrete lattice model with periodic microstructure (mass-and-spring chain) featured by nearest neighbor and next nearest neighbor particle interactions. In the latter case, boundary effects arise at the two lattice ends that are well captured by the proposed nonlocal continuum formulation. The nonlocal material parameters are identified a priori by matching the dispersion curve of the discrete lattice model, and a comparison in terms of attenuation function is also presented.

A nonlocal Layerwise theory for free vibration analysis of nanobeams with various boundary conditions on Winkler-Pasternak foundation

In this study, a nonlocal Layerwise theory is presented for free vibration analysis of nanobeams resting on an elastic foundation. Eringen's nonlocal elasticity theory is used to consider the small-scale effect on behavior of nanobeam. The governing equations are obtained by employing Hamilton's principle and Layerwise theory of beams and Eringen's nonlocal constitutive equation. The presented theory takes into account the in-plane and transverse normal and shear strain in the modeling of the nanobeam and can predict more accurate results. The governing equations of the beam are solved by Navier's method for Simple-Simple boundary conditions and semi-analytical methods to obtain the natural frequency for various boundary conditions including Clamped-Simple (C-S), Clamped-Clamped (C-C) and Free-Free (F-F) boundary conditions. Predictions of the present theory are compared with benchmark results in the literature. Effects of nonlocal parameter, Pasternak shear coefficient, Winkler spring coefficient, boundary conditions, and the aspect ratio on the free vibration of nanobeams are studied. The flexural mode and thickness mode natural frequencies of the nanobeam are predicted. It is shown that the predictions of present method are more accurate than the equivalent single layer theories. The theoretical developments and formulation presented herein should also be served to analyze the mechanical behavior of various nanostructures with various loading and boundary conditions.

A non-local plasticity model for porous metals with deformation-induced anisotropy: Mathematical and computational issues

A non-local (gradient) plasticity model for porous metals that accounts for deformation-induced anisotropy is presented. The model is based on the work of Ponte Castañeda and co-workers on porous materials containing randomly distributed ellipsoidal voids. It takes into account the evolution of porosity and the evolution/development of anisotropy due to changes in the shape and the orientation of the voids during plastic deformation. A “material length” ℓ is introduced and a “non-local” porosity is defined from the solution of a modified Helmholtz equation with appropriate boundary conditions, as proposed by Geers et al. (2001); Peerlings et al. (2001). At a material point located at x, the non-local porosity f(x) can be identified with the average value of the “local” porosity floc(x) over a sphere of radius R≃3ℓ centered at x. The same approach is used to formulate a non-local version of the Gurson isotropic model. The mathematical character of the resulting incremental elastoplastic partial differential equations of the non-local model is analyzed. It is shown that the hardening modulus of the non-local model is always larger than the corresponding hardening modulus of the local model; as a consequence, the non-local incremental problem retains its elliptic character and the possibility of discontinuous solutions is eliminated. A rate-dependent version of the non-local model is also developed. An algorithm for the numerical integration of the non-local constitutive equations is developed, and the numerical implementation of the boundary value problem in a finite element environment is discussed. An analytical method for the required calculation of the eigenvectors of symmetric second-order tensors is presented. The non-local model is implemented in ABAQUS via a material “user subroutine” (UMAT or VUMAT) and the coupled thermo-mechanical solution procedure, in which temperature is identified with the non-local porosity. Several example problems are solved numerically and the effects of the non-local formulation on the solution are discussed. In particular, the problems of plastic flow localization in plane strain tension, the plane strain mode-I blunt crack tip under small-scale-yielding conditions, the cup-and-cone fracture of a round bar, and the Charpy V-notch test specimen are analyzed.

Generalized elastodynamic model for nanophotonics

A self-consistent theory for the classical description of the interaction of light and matter at the nano-scale is presented, which takes into account spatial dispersion. Up to now, the Maxwell equations in nanostructured materials with spatial dispersion have been solved by the introduction of the so-called Additional Boundary Conditions. In this paper, we derive an approach where non-local effects are studied in a precise and uniquely defined way, thus allowing the treatment of all solid-solid interfaces (metals, semiconductors or insulators), as well as solid-vacuum interfaces in the same framework. The theory is based on the derivation of a potential energy for an ensemble of electrons in a given potential, where the deformation of the ensemble is treated as in a solid, including both shear and compressional deformations, instead of a fluid described only by a bulk compressibility like in the hydrodynamical approach. The derived classical equation of motion for the ensemble describes the deformation vector and the corresponding polarization vector as an elastodynamic field, including viscous forces, from which a generalized non-local constitutive equation for the dielectric constant is derived. Boundary conditions are identical to that of elastodynamics and they emerge in a natural way, without any physical hypothesis outside the current description, as it is commonly required in other non-local approaches. This description does not require the discontinuity of any component of the electric, magnetic or polarization fields and, consequently, no bounded currents or charges are present at the interface, which is a more suitable description from the microscopic point of view. It is shown that the method converges to the local boundary conditions in the low spatial dispersion limit for insulators and conductors, quantified by means of a parameter defined as the "characteristic length".

Nonlocal thermo-elastic constitutive relation of fibre-reinforced composites

The effective properties of composite materials have been predicted by various micromechanical schemes. For composite materials of constituents which are described by the classical governing equations of the local form, the conventional micromechanical schemes usually give effective properties of the local form. However, it is recognized that under general loading conditions, spatiotemporal nonlocal constitutive equations may better depict the macroscopic behavior of these materials. In this paper, we derive the thermo-elastic dynamic effective governing equations of a fibre-reinforced composite in a coupled spatiotemporal integral form. These coupled equations reduce to the spatial nonlocal peridynamic formulation when the microstructural inertial effects are neglected. For static deformation and steady-state heat conduction, we show that the integral formulation is superior at capturing the variations of the average displacement and temperature in regions of high gradients than the conventional micromechanical schemes. The approach can be applied to analogous multi-field coupled problems of composites.

An accurate nonlocal bonded discrete element method for nonlinear analysis of solids: application to concrete fracture tests

We present a numerical procedure for elastic and nonlinear analysis (including fracture situations) of solid materials and structures using the discrete element method. It can be applied to strongly cohesive frictional materials such as concrete and rocks. The method consists on defining nonlocal constitutive equations at the contact interfaces between discrete particles using the information provided by the stress tensor over the neighbor particles. The method can be used with different yield surfaces, and in the paper, it is applied to the analysis of fracture of concrete samples. Good comparison with experimental results is obtained.