Lattice with long-range interaction of power-law type for fractional non-local elasticity

Lattice models with long-range interactions of power-law type are suggested as a new type of microscopic model for fractional non-local elasticity. Using the transform operation, we map the lattice equations into continuum equation with Riesz derivatives of non-integer orders. The continuum equations that are obtained from the lattice model describe fractional generalization of non-local elasticity models. Particular solutions and correspondent asymptotic of the fractional differential equations for displacement fields are suggested for the static case.     

A non-local implicit gradient-enhanced model for unstable behaviors of pseudoelastic shape memory alloys in tensile loading

In this paper, a gradient-enhanced 3-D phenomenological model for shape memory alloys using the non-local theory is developed based on a 1-D constitutive model. The method utilizes a non-local field variable in its constitutive framework with an implicit gradient formulation in order to achieve results independent of the finite element discretization. An efficient numerical approach to implement the non-local gradient-enhanced model in finite element codes is proposed. The model is used to simulate stress drop at the onset of transformation, and its performance is evaluated using different experimental data. The potential of the presented numerical approach for behavior of shape memory alloys in eliminating mesh-dependent simulations is validated by conducting various localization problems. The numerical results show that the developed model can simulate the observed unstable behaviors such as stress drop and deviation of local strain from global strain during nucleation and propagation of martensitic phase.     

Non-local continuum mechanics and fractional calculus

The present work introduces fractional calculus into the continuum mechanics area describing non-local constitutive relations. Considering a one-dimensional body and assuming total stored energy depending not only upon the local strain but also upon a fractional derivative of the stain, an elastic model with non-local stress–strain behavior is introduced. Fractional calculus provides a natural framework for describing non-local constitutive relations and requires no assumptions for the interval of non-local influence. Furthermore, the proposed method works in finite intervals contrary to the existing theories requiring infinite domains.     

A novel Volume-Compensated Particle method for 2D elasticity and plasticity analysis

A novel Volume-Compensated Particle model (VCPM) is proposed for the modeling of deformation and fracture in solids. In this proposed method, two potentials are introduced to model the interactions between material particles, i.e., a local pair-wise potential and a non-local multi-body potential. The local pair-wise potential is utilized to account for the constitutive relationship within the connecting bonds between particles while the non-local multi-body potential is employed for considering the volumetric effects under general mechanical loadings. The potential coefficients are determined by matching the potential energy stored in a discrete unit cell to the strain energy at the classical continuum level. A volume conservation scheme is proposed to model the plastic deformation. The validity of the proposed model is tested against the classical elasticity and elasto-plasticity benchmarks before its application to fracture problems. Several conclusions are drawn based on the proposed study.     

Non-local effects on the non-linear modes of vibration of carbon nanotubes under electrostatic actuation

Carbon nanotubes (CNTs) based NEMS with electrostatic sensing/actuation may be employed as sensors, in situations where it is fundamental to understand their dynamic behaviour. Due to displacements that are large in comparison with the thickness and to the non-linearity of the electrostatic force, these CNT based NEMS operate in the non-linear regime. The knowledge of the modes of vibration of a CNT provides a picture of what one may expect from its dynamic behaviour not only in free, but also in forced vibrations. In this paper, the non-linear modes of vibration of CNTs actuated by electrostatic forces are investigated. For that purpose, a p-version finite element type formulation is implemented, leading to ordinary differential equations of motion in the time domain. The formulation takes into account non-local effects, which influence the inertia and the stiffness of CNTs, as well as the electrostatic actuation. The ordinary differential equations of motion are transformed into algebraic equations of motion via the harmonic balance method (HBM) and then solved by an arc-length continuation method. Several harmonics are considered in the HBM. The importance of non-local effects, combined with the geometrical non-linearity and with the action of the electrostatic force, is analysed. It is found that different combinations of these effects can result in alterations of the natural frequencies, variations in the degrees of softening or hardening, changes in the frequency content of the free vibrations, and alterations in the mode shapes of vibration. It is furthermore found that the small scale, here represented by the non-local theory, has an effect on interactions between the first and higher order modes which are induced by the geometrical and material non-linearities of the system.     

Surface and non-local effects for non-linear analysis of Timoshenko beams

In this paper, we present a non-local non-linear finite element formulation for the Timoshenko beam theory. The proposed formulation also takes into consideration the surface stress effects. Eringen׳s non-local differential model has been used to rewrite the non-local stress resultants in terms of non-local displacements. Geometric non-linearities are taken into account by using the Green–Lagrange strain tensor. A C⁰ beam element with three degrees of freedom has been developed. Numerical solutions are obtained by performing a non-linear analysis for bending and free vibration cases. Simply supported and clamped boundary conditions have been considered in the numerical examples. A parametric study has been performed to understand the effect of non-local parameter and surface stresses on deflection and vibration characteristics of the beam. The solutions are compared with the analytical solutions available in the literature. It has been shown that non-local effect does not exist in the nano-cantilever beam (Euler–Bernoulli beam) subjected to concentrated load at the end. However, there is a significant effect of non-local parameter on deflections for other load cases such as uniformly distributed load and sinusoidally distributed load (Cheng et al. (2015) [10]). In this work it has been shown that for a cantilever beam with concentrated load at free end, there is definitely a dependency on non-local parameter when Timoshenko beam theory is used. Also the effect of local and non-local boundary conditions has been demonstrated in this example. The example has also been worked out for other loading cases such as uniformly distributed force and sinusoidally varying force. The effect of the local or non-local boundary conditions on the end deflection in all these cases has also been brought out.     

Non-local elastic effects in electroactive polymers

In this work we study the onset of inhomogeneous deformations in thin electroactive polymers (EAPs) under voltage control. In order to account for the regularizing effects due to both the constitutive nature of the film and to its mechanical interaction with the compliant electrodes, we introduce a non-local energy term depending on the second gradient of deformation. We prove that very small non-local effects are sufficient to find realistic inhomogeneous deformations at the onset of the bifurcation, which are characterized by periodic thickness undulations with finite wavelength. Finally we prove that strong regularizing effects can suppress the onset of inhomogeneous deformations.     

Non-local continuum modeling of carbon nanotubes: Physical interpretation of non-local kernels using atomistic simulations

The longitudinal, transverse and torsional wave dispersion curves in single walled carbon nanotubes (SWCNT) are used to estimate the non-local kernel for use in continuum elasticity models of nanotubes. The dispersion data for an armchair (10,10) SWCNT was obtained using lattice dynamics of SWNTs while accounting for the helical symmetry of the tubes. In our approach, the Fourier transformed kernel of non-local linear elastic theory is directly estimated by matching the atomistic data to the dispersion curves predicted from non-local 1D rod theory and axisymmetric shell theory. We found that gradient models incur significant errors in both the phase and group velocity when compared to the atomistic model. Complementing these studies, we have also performed detailed tests on the effect of length of the nanotube on the axial and shear moduli to gain a better physical insight on the nature of the true non-local kernel. We note that unlike the kernel from gradient theory, the numerically fitted kernel becomes negative at larger distances from the reference point. We postulate and confirm that the fitted kernel changes sign close to the inflection point of the interatomic potential. The numerically computed kernels obtained from this study will aid in the development of improved and efficient continuum models for predicting the mechanical response of CNTs.     

On the physical interpretation of fractional diffusion

Even if the diffusion equation has been widely used in physics and engineering, and its physical content is well understood, some variants of it escape fully physical understanding. In particular, anormal diffusion appears in the so-called fractional diffusion equation, whose main particularity is its non-local behavior, whose physical interpretation represents the main part of the present work.     

Numerical implementation of non-local polycrystal plasticity using fast Fourier transforms

We present the numerical implementation of a non-local polycrystal plasticity theory using the FFT-based formulation of Suquet and co-workers. Gurtin (2002) non-local formulation, with geometry changes neglected, has been incorporated in the EVP-FFT algorithm of Lebensohn et al. (2012). Numerical procedures for the accurate estimation of higher order derivatives of micromechanical fields, required for feedback into single crystal constitutive relations, are identified and applied. A simple case of a periodic laminate made of two fcc crystals with different plastic properties is first used to assess the soundness and numerical stability of the proposed algorithm and to study the influence of different model parameters on the predictions of the non-local model. Different behaviors at grain boundaries are explored, and the one consistent with the micro-clamped condition gives the most pronounced size effect. The formulation is applied next to 3-D fcc polycrystals, illustrating the possibilities offered by the proposed numerical scheme to analyze the mechanical response of polycrystalline aggregates in three dimensions accounting for size dependence arising from plastic strain gradients with reasonable computing times.     

On fractional non-local bodies with variable length scale

This paper discusses the application of the variable length scale concept in the framework of non-local fractional model. The considerations are motivated by the fact that real material characteristic dimension is never uniform and simultaneously the problem of existence of the virtual boundary layer in the boundary value problems, discussed in previous papers, is removed. The considerations are illustrated with a series of analyses of 1D elasticity problems. Nonetheless, the conclusions are applicable for an arbitrary configurations.     

Investigation of the Dynamic Behavior of a Griffith Crack in a Piezoelectric Material Strip Subjected to the Harmonic Elastic Anti-plane Shear Waves by Use of the Non-local Theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material strip subjected to the harmonic anti-plane shear waves is investigated by use of the non-local theory for impermeable crack surface conditions. To overcome the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and the electric displacement near at the crack tip. By means of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the thickness of the strip, the circular frequency of incident wave and the lattice parameter.     

Investigation of the crack problem in non-local piezoelectric materials under combined electromechanical loadings

The present investigation of the crack problem in piezoelectric materials is performed based on the non-local theory. After some manipulations, the impermeable crack, the permeable crack （the crack gap is full of NaCI solution）, and the semi-permeable crack （the crack gap is full of air or silicon oil） are reduced to a uniform formulation by assuming the normal electric displacement on the crack surfaces to be an unknown variable. Thus, a triple integral equation with the unknown normal electric displacement is established. By using the Newton iterative method and solving the triple integral equation, it is found that the normal electric displacement on the crack surfaces is no longer a constant as determined by previous studies, rather, it depends upon the remote combined electromechanical loadings. Numerical results of the stresses and electric displacement fields show that there are no singularities at the crack tips so that the stresses remain finite. It is of great significance that the concrete electric boundary condition on the crack surfaces exerts significant influence on the near-tip fields and in this way plays an important role in evaluating the crack stability in the non-local piezoelectric materials. More specifically, the impermeable crack model always overestimates the finite stresses at the crack tips, whereas the permeable crack model always underestimates them.     

Exact Averaging of Stochastic Equations for Flow in Porous Media

It is well-known that at present, exact averaging of the equations for flow and transport in random porous media have been proposed for limited special fields. Moreover, approximate averaging methods—for example, the convergence behavior and the accuracy of truncated perturbation series—are not well-studied, and in addition, calculation of high-order perturbations is very complicated. These problems have for a long time stimulated attempts to find the answer to the question: Are there in existence some, exact, and sufficiently general forms of averaged equations? Here, we present an approach for finding the general exactly averaged system of basic equations for steady flow with sources in unbounded stochastically homogeneous fields. We do this by using (1) the existence and some general properties of Green’s functions for the appropriate stochastic problem, and (2) some information about the random field of conductivity. This approach enables us to find the form of the averaged equations without directly solving the stochastic equations or using the usual assumption regarding any small parameters. In the common case of a stochastically homogeneous conductivity field we present the exactly averaged new basic non-local equation with a unique kernel-vector. We show that in the case of some type of global symmetry (isotropy, transversal isotropy, or orthotropy), we can for three-dimensional and two-dimensional flow in the same way derive the exact averaged non-local equations with a unique kernel-tensor. When global symmetry does not exist, the non-local equation with a kernel-tensor involves complications and leads to an ill-posed problem.     

Non-local and numerical formulations for dry sliding friction and wear at high velocities

Severe contact stress problems generate high temperature and create thermomechanical gouging and wear due to high velocity sliding between two materials staying in contact. In order to improve the facilitation of the design of particular components and improve performance of these engineering applications, it is necessary to better understand the physical behavior of high speed environment. As presented here this environment is made up of two components in contact. Therefore, basing on the experimental approach ( [Lodygowski, 2010] and [Lodygowski et al., submitted for publication]) the major consideration of this paper is aimed to develop an experimental/theoretical model for the material constitutive behavior in order to better characterize and predict the internal failure surrounding the gouging and wear events.This research is to be carried out in two stages. First, by investigating the phenomenon of wear and later it will be extended to incorporate gauging problems. The principle of virtual power is used by introducing the contributions from damage and its corresponding gradients as a measure of micro motion of damage within the bulk. In addition two internal state variables are introduced on the frictional contact interface, one measuring the tangential slip and another measuring the wear. By using these internal state variables together with displacement and temperature, the constitutive model is formulated with state laws based on the free energies and the complimentary laws based on the dissipation potentials. The proposed theoretical model is implemented as user defined subroutine VUMAT in the explicit finite element code ABAQUS to analyze the structural response of the ultra high speed sliding experiment between Steel and VascoMax steel at Ecole de’Nationale Institut der Mechanic, at Metz France.This model provides a potential feature for enabling one to relate the non-local continuum plasticity and damage of the bulk material to friction and wear at the contact interfaces. The findings of this research effort is invaluable in providing a multiscale material model and numerical procedure that will be used within a hydrocode to better facilitate the design components of the severe contact stress applications.     

Comparative FE-studies of shear localizations in granular bodies within a polar and non-local hypoplasticity

Paper presents a FE-analysis of shear localizations in granular bodies with a finite element method based on a hypoplastic constitutive law. The law can reproduce essential features of granular bodies in dependence on the void ratio, pressure level and deformation direction. To simulate the formation of a spontaneous shear zone inside of cohesionless sand during plane strain compression, a hypoplastic law was extended by polar and non-local terms. The effect of both models on the thickness of a shear zone was compared.     

Transport of black carbon aerosols from non-local sources: A case study in Shanghai

Black carbon (BC) aerosol mass carried by winds of varying directions from non-local sources was estimated based on hourly measured data of BC mass concentration (C_BC) and meteorological parameters from January 2008 to December 2012 in Shanghai, and the relationship between annual average C_BC and wind speed was analyzed. The results show that the annual average C_BC decreased with wind speed for speeds exceeding 0.3 m/s. The relationship between the two was determined by a linear fit with correlation coefficient 0.88. Assuming BC aerosol mass of non-local sources transported by a southeast wind was zero, annual average BC concentrations (μg/m³) carried by winds of variable direction were 1.99 (southwest), 1.95 (west), 1.15 (northwest), 0.54 (south), 0.39 (north), 0.01 (northeast), and 0.01 (east). BC aerosol mass of non-local sources transported by wind to Shanghai was about 6404.05 t per year, among which the total contribution of southwest, west, and northwest winds was nearly 84%. The aerosol mass transported to Shanghai in winter accounted for 35% that of the entire year, and was greater than that of the other seasons.     

Length scale effects on the shear localization process in metallic glasses: A theoretical and computational study

Some recent experiments on sub-micron and nano-sized metallic glass (amorphous alloy) specimens have shown that the shear localization process becomes more stable and less catastrophic when compared to the response exhibited by large sample sizes. This leads to the discovery that the shear localization process and fracture can be delayed by decreasing sample volume. In this work we develop a non-local and finite-deformation-based constitutive model using thermodynamic principles and the theory of micro-force balance to study the causes for the aforementioned observations. The constitutive model has also been implemented into a commercially available finite-element program by writing a user-material subroutine. With the aid of finite-element simulations, our constitutive model predicts that metallic glass samples have the intrinsic ability to exhibit: (a) the delaying of (catastrophic) shear localization with decreasing sample size, and (b) homogeneous deformation behavior for sample volumes smaller than the shear band nucleus.The cause for the observations listed above is the increasing influence of a non-local interaction stress with decreasing sample volume. This interaction stress has energetic origins and it affects plastic deformation due to the strong coupling between plastic shearing and free-volume generation. Akin to strain-gradient plasticity theory, the role of the interaction stress is to strengthen the material at locations where the defect density/free volume is higher compared to the rest of metallic glass sample.     

Crack bifurcation predicted for dynamic anti-plane collinear cracks in piezoelectric materials using a non-local theory

A non-local theory of elasticity is applied to obtain the dynamic interaction between two collinear cracks in the piezoelectric materials plane under anti-plane shear waves for the permeable crack surface boundary conditions. Unlike the classical elasticity solution, a lattice parameter enters into the problem that make the stresses and the electric displacements finite at the crack tip. A one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and electric displacement near the crack tips. By means of the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations in which the unknown variable is the jump of the displacement across the crack surface. The solutions are obtained by means of the Schmidt method. Crack bifurcation is predicted using the strain energy density criterion. Minimum values of the strain energy density functions are assumed to coincide with the possible locations of fracture initiation. Bifurcation angles of ±5° and ±175° are found. The result of possible crack bifurcation was not expected before hand.     

Versatile code DLAYZ for investigating population kinetics and radiative properties of plasmas in non-local thermodynamic equilibrium

A versatile code DLAYZ based on collisional-radiative model is developed for investigating the population kinetics and radiative properties of plasmas in non-local thermodynamic equilibrium. DLAYZ is implemented on the detailed level accounting (DLA) approach and can be extended to detailed configuration accounting (DCA) and hybrid DLA/DCA approaches. The code can treat both steady state and time-dependent problems. The implementation of the main modules of DLAYZ is discussed in detail including atomic data, rates, population distributions and radiative properties modules. The complete set of basic atomic data is obtained using relativistic quantum mechanics. For dense plasmas, the basic atomic data with plasma screening effects can be obtained. The populations are obtained by solving the coupled rate equations, which are used to calculate the radiative properties. A parallelized version is implemented in the code to treat the large-scale rate equations. Two illustrative examples of a steady state case for carbon plasmas and a time-dependent case for the relaxation of a K-shell excited argon are employed to show the main features of the present code.