NEW POINTS OF VIEW ON THE NONLOCAL FIELD THEORY AND THEIR APPLICATIONS TO THE FRACTURE MECHANICS(Ⅱ)——RE-DISCUSS NONLINEAR CONSTITUTIVE EQUATIONS OF NONLOCAL THERMOELASTIC BODIES

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A new nonlocal bending model for Euler–Bernoulli nanobeams

This paper is concerned with the bending problem of nanobeams starting from a nonlocal thermodynamic approach. A new coupled nonlocal model, depending on two nonlocal parameters, is obtained by using a suitable definition of the free energy. Unlike previous approaches which directly substitute the expression of the nonlocal stress into the classical equilibrium equations, the proposed approach provides a methodology to recover nonlocal models starting from the free energy function. The coupled model can then be specialized to obtain a nanobeam formulation based on the Eringen nonlocal elasticity theory and on the gradient elastic model. The variational formulations are consistently provided and the differential equations with the related boundary conditions are thus derived. Nanocantilevers are solved in a closed-form and numerical results are presented to investigate the influence of the nonlocal parameters.     

NEW POINTS OF VIEW ON THE NONLOCAL FIELD THEORY AND THEIR APPLICATIONS TO THE FRACTURE MECHANICS(Ⅱ)--RE-DISCUSS NONLINEAR CONSTITUTIVE EQUATIONS OF NON-LOCAL THERMOELASTIC BODIES

In this paper, nonlinear constitutive equations are deduced strictly according to the constitutive axioms of rational continuum mechanics. The existing judgments are modified and improved. The results show that the constitutive responses of nonlocal thermoelastic body are related to the curvature and higher order gradient of its material space, and there exists an antisymmetric stress whose average value in the domain occupied by thermoelastic body is equal to zero. The expressions of the antisymmetric stress and the nonlocal residuals are given. The conclusion that the directions of thermal conduction and temperature gradient are consistent is reached. In addition, the objectivity about the nonlocal residuals and the energy conservation law of nonlocal field is discussed briefly, and a formula for calculating the nonlocal residuals of energy changing with rigid motion of the spatial frame of reference is derived. Foundation item: the Natural Science Foundation of Province Jiangshu (BK97063)     

Symmetric Nonbarotropic Flows with Large Data and Forces

We prove the global existence of weak solutions of the Navier-Stokes equations of compressible, nonbarotropic flow in three space dimensions with initial data and external forces which are large, discontinuous, and spherically or cylindrically symmetric. The analysis allows for the possibility that a vacuum state emerges at the origin or axis of symmetry, and the equations hold in the sense of distributions in the set where the density is positive. In addition, the mass and momentum equations hold weakly in the entire space-time domain, but with a nonstandard interpretation of the viscosity terms as distributions. Solutions are obtained as limits of solutions in annular regions between two balls or cylinders, and the analysis allows for the possibility that energy is absorbed into the origin or axis, and is lost in the limit as the inner radius goes to zero.Research supported in part by the NSF under Grants DMS-9986658 and DMS-0305072 (Hoff) and Grant DMS-0206631 (Jenssen).     

NEW POINTSOF VIEW ON THE NONLOCAL FIELD THEORY AND THEIR APPLICATIONS TO THE FRACTURE MECHANICS( Ⅲ) ——RE_DISCUSS THE LINEAR THEORY OF NONLOCAL ELASTICITY

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Surface energy and adhesion energy of elastic bodies

We propose a method for computing the surface energy and the adhesion energy of elastic bodies in adhesion state. The method is based on a version of the elastic continuum model originating from the assumption about the multiparticle potential nonlocal interaction of infinitesimal particles of the medium.     

NEW POINTS OF VIEW ON THE NONLOCAL FIELD THEORY AND THEIR APPLICATIONS TO THE FRACTURE MECHANICS(Ⅲ)——RE-DISCUSS THE LINEAR THEORY OF NONLOCAL ELASTICITY

In this paper, it is proven that the balance equation of energy is the first integral of the balance equation of momentum in the linear theory of nonlocal elasticity. In other words, the balance equation of energy is not an independent one. It is also proven that the residual of nonlocal body force identically equals zero. This makes the transform formula of the nonlocal residual of energy much simpler. The linear nonlocal consitutive equations of elastic bodies are deduced in details, and a new formula to calculate the antisymmetric stress is given. Foundation item: the Natural Science Foundation of Jiangsu Province, China (BK97063)     

Modal interactions and energy transfers in isotropic turbulence as predicted by local energy transfer theory

Energy transfer processes in decaying, three-dimensional, isotropic turbulence are investigated using numerical results from local energy transfer (LET) theory. The study covers a wide range of evolved, microscale Reynolds numbers (5 < R_λ < 250). It is found that the energy transfer is mainly local (between scales of similar size), but there are also some signs of nonlocal transfer at higher Reynolds number. The nature of the underlying triad-wavenumber interactions, on the other hand, seems to depend on both the Reynolds number and the wavenumber range of interest. In the energy containing and dissipation ranges, both local (all three scales of the triad interaction are of comparable size) and nonlocal (one scale is much larger than the remaining two) interactions are important, with the latter becoming more dominant as the Reynolds number increases. But our nonlocal interactions tend to be less severe than those observed by Domaradzki and Rogallo. More significantly, in the inertial range of high Reynolds number flows, the LET theory predicts dominance of local and near-local interactions. While this is contrary to the result from eddy damped quasi-normal Markovian theory that the important triad interactions are mainly nonlocal, it is closer to the Kolmogorov picture of turbulence. Another interesting result is that, despite their inherent differences, the LET theory and the full simulation of Ohkitani and Kida predict inertial-range values for the energy transfer locality function in fairly good agreement, not only with each other, but also with the analytical closure theory result for infinite Reynolds number, stationary turbulence by Kraichnan. The calculated values reveal that the contributions to the net energy transfer are predominantly from near-local interactions (scale ratios ≈ 4), which is indicative of cancellation of large numbers of highly nonlocal interactions.     

Decaying Turbulence in the Generalised Burgers Equation

This work is concerned with the derivation of optimal scaling laws, in the sense of matching lower and upper bounds on the energy, for a solid undergoing ductile fracture. The specific problem considered concerns a material sample in the form of an infinite slab of finite thickness subjected to prescribed opening displacements on its two surfaces. The solid is assumed to obey deformation-theory of plasticity and, in order to further simplify the analysis, we assume isotropic rigid-plastic deformations with zero plastic spin. When hardening exponents are given values consistent with observation, the energy is found to exhibit sublinear growth. We regularize the energy through the addition of nonlocal energy terms of the strain-gradient plasticity type. This nonlocal regularization has the effect of introducing an intrinsic length scale into the energy. Under these assumptions, ductile fracture emerges as the net result of two competing effects: whereas the sublinear growth of the local energy promotes localization of deformation to failure planes, the nonlocal regularization stabilizes this process, thus resulting in an orderly progression towards failure and a well-defined specific fracture energy. The optimal scaling laws derived here show that ductile fracture results from localization of deformations to void sheets, and that it requires a well-defined energy per unit fracture area. In particular, fractal modes of fracture are ruled out under the assumptions of the analysis. The optimal scaling laws additionally show that ductile fracture is cohesive in nature, that is, it obeys a well-defined relation between tractions and opening displacements. Finally, the scaling laws supply a link between micromechanical properties and macroscopic fracture properties. In particular, they reveal the relative roles that surface energy and microplasticity play as contributors to the specific fracture energy of the material.     

A nonlocal species concentration theory for diffusion and phase changes in electrode particles of lithium ion batteries

A nonlocal species concentration theory for diffusion and phase changes is introduced from a nonlocal free energy density. It can be applied, say, to electrode materials of lithium ion batteries. This theory incorporates two second-order partial differential equations involving second-order spatial derivatives of species concentration and an additional variable called nonlocal species concentration. Nonlocal species concentration theory can be interpreted as an extension of the Cahn–Hilliard theory. In principle, nonlocal effects beyond an infinitesimal neighborhood are taken into account. In this theory, the nonlocal free energy density is split into the penalty energy density and the variance energy density. The thickness of the interface between two phases in phase segregated states of a material is controlled by a normalized penalty energy coefficient and a characteristic interface length scale. We implemented the theory in COMSOL Multiphysics\(^{\circledR }\) for a spherically symmetric boundary value problem of lithium insertion into a \(\hbox {Li}_x\hbox {Mn}_2\hbox {O}_4\) cathode material particle of a lithium ion battery. The two above-mentioned material parameters controlling the interface are determined for \(\hbox {Li}_x\hbox {Mn}_2\hbox {O}_4\), and the interface evolution is studied. Comparison to the Cahn–Hilliard theory shows that nonlocal species concentration theory is superior when simulating problems where the dimensions of the microstructure such as phase boundaries are of the same order of magnitude as the problem size. This is typically the case in nanosized particles of phase-separating electrode materials. For example, the nonlocality of nonlocal species concentration theory turns out to make the interface of the local concentration field thinner than in Cahn–Hilliard theory.     

An asymmetric theory of nonlocal elasticity—Part 1. Quasicontinuum theory

In this article, an asymmetric theory of nonlocal elasticity is developed on the basis of three dimensional atomic lattice model, the Galileo invariance for constitutive equations and by use of Fourier transformation of generalized function and energy method. It is shown that nonlocal characteristic functions (or constitutive parameters of internal elastic energy) can be explicitly expressed in terms of interacting forces connecting atoms, and the general model of nonlocal theory with rotation effects is asymmetric. Both symmetric stress and anti-symmetric stress is a nonlocal function of strain and local rotation for anisotropic materials. For isotropic materials, symmetric stress is only a nonlocal function of strain, while antisymmetric stress is only a nonlocal function of local rotation.     

On a theory of nonlocal elasticity of bi-Helmholtz type and some applications

A theory of nonlocal elasticity of bi-Helmholtz type is studied. We employ Eringen’s model of nonlocal elasticity, with bi-Helmholtz type kernels, to study dispersion relations, screw and edge dislocations. The nonlocal kernels are derived analytically as Green functions of partial differential equations of fourth order. This continuum model of nonlocal elasticity involves two material length scales which may be derived from atomistics. The new nonlocal kernels are nonsingular in one-, two- and three-dimensions. Furthermore, the nonlocal elasticity of bi-Helmholtz type improves the one of Helmholtz type by predicting a dispersion relationship with zero group velocity at the end of the first Brillouin zone. New solutions for the stresses and strain energy of screw and edge dislocations are found.     

Finite Scale Microstructures in Nonlocal Elasticity

In this paper we develop a simple one-dimensional model accounting for the formation and growth of globally stable finite scale microstructures. We extend Ericksen's model [9] of an elastic “bar” with nonconvex energy by including both oscillation-inhibiting and oscillation-forcing terms in the energy functional. The surface energy is modeled by a conventional strain gradient term. The main new ingredient in the model is a nonlocal term which is quadratic in strains and has a negative definite kernel. This term can be interpreted as an energy associated with the long-range elastic interaction of the system with the constraining loading device. We propose a scaling of the problem allowing one to represent the global minimizer as a collection of localized interfaces with explicitly known long-range interaction. In this limit the augmented Ericksen's problem can be analyzed completely and the equilibrium spacing of the periodic microstructure can be expressed as a function of the prescribed average displacement. We then study the inertial dynamics of the system and demonstrate how the nucleation and growth of the microstructures result in the predicted stable pattern. Our results are particularly relevant for the modeling of twined martensite inside the austenitic matrix.     

Variational formulations and a consistent finite-element procedure for a class of nonlocal elastic continua

The structural boundary-value problem in the context of nonlocal (integral) elasticity and quasi-static loads is considered in a geometrically linear range. The nonlocal elastic behaviour is described by the so-called Eringen model in which the nonlocality lies in the constitutive relation. The diffusion processes of the nonlocality are governed by an integral relation containing a recently proposed symmetric spatial weight function expressed in terms of an attenuation function. A firm variational basis to the nonlocal model is given by providing the complete set of variational formulations, composed by ten functionals with different combinations of the state variables. In particular the nonlocal counterpart of the classical principles of the total potential energy, complementary energy and mixed Hu–Washizu principle and Hellinger–Reissner functional are recovered. Some examples concerning a piecewise bar in tension are provided by using the Fredholm integral equation and the proposed nonlocal FEM.     

Data-driven solutions and parameter discovery of the nonlocal mKdV equation via deep learning method

In this paper, we systematically study the integrability and data-driven solutions of the nonlocal mKdV equation. The infinite conservation laws of the nonlocal mKdV equation and the corresponding infinite conservation quantities are given through Riccti equation. The data-driven solutions of the zero boundary for the nonlocal mKdV equation are studied by using the multilayer physical information neural network algorithm, which include kink soliton, complex soliton, bright-bright soliton and the interaction between soliton and kink-type. For the data-driven solutions with nonzero boundary, we study kink, dark, anti-dark and rational solution. By means of image simulation, the relevant dynamic behavior and error analysis of these solutions are given. In addition, we discuss the inverse problem of the integrable nonlocal mKdV equation by applying the physics-informed neural network algorithm to discover the parameters of the nonlinear terms of the equation.

     

A damage model with evolving nonlocal interactions

We present a damage model for softening materials with evolving nonlocal interactions. The thermodynamic implications and the material stability issue are addressed. The proposed nonlocal averaging scheme provides the obtained constitutive models with an evolving nonlocal interaction which is activated only when damage occurs. In the analysis of structures made of quasi-brittle materials, this feature helps not only to overcome some issues with the incorrect initiation of damage but also to better control the evolving size of the active fracture process zone. This is an essential feature that is usually not considered in depth in many existing nonlocal approaches to the continuum modelling of quasi-brittle fracture. Numerical examples are given to demonstrate features of the proposed modelling approach.     

NEW POINTS OF VIEW ON THE NONLOCAL FIELD THEORY AND THEIR APPLICATIONS TO THE FRACTURE MECHANICS (Ⅰ)──FUNDAMENTAL THEORY

In the linear nonlocal elasticity theory, the solution to the boundary-value problem of the crack with a constant stress boundary condition does not exist. This problem has been studied in this paper. The contents studied contain of examining objectivity of the energy balance, deducing the constitutive equations of nonlocal thermoelastic bodies, and determining nonlocal force and the linear nonlocal elasticity theory. Some new results are obtained. Among them, the stress boundary condition derived from the linear theory not only solves the problem mentioned at the beginning, but also contains the model of molecular cohesive stress on the sharp crack tip advanced by Barenblatt.     

Nonlocal integral elasticity: 2D finite element based solutions

A finite element based method, theorized in the context of nonlocal integral elasticity and founded on a nonlocal total potential energy principle, is numerically implemented for solving 2D nonlocal elastic problems. The key idea of the method, known as nonlocal finite element method (NL-FEM), relies on the assumption that the postulated nonlocal elastic behaviour of the material is captured by a finite element endowed with a set of (cross-stiffness) element’s matrices able to interpret the (nonlocality) effects induced in the element itself by the other elements in the mesh. An Eringen-type nonlocal elastic model is assumed with a constitutive stress–strain law of convolutive-type which governs the nonlocal material behaviour. Computational issues, as the construction of the nonlocal element and global stiffness matrices, are treated in detail. Few examples are presented and the relevant numerical findings discussed both to verify the reliability of the method and to prove its effectiveness.     

Modelling of localization and propagation of phase transformation in superelastic SMA by a gradient nonlocal approach

In this work, a nonlocal phenomenological behavior model is proposed in order to describe the localization and propagation of stress-induced martensite transformation in shape memory alloy (SMA) wires and thin films. It is a nonlocal extension of an existing local model that was derived from a micromechanical-inspired Gibbs free energy expression. The proposed model uses, besides the local field of the internal variable, namely the martensite volume fraction, a nonlocal counterpart. This latter acts as an additional degree of freedom, which is determined by solving an additional partial differential equation (PDE), derived so as to be equivalent to the integral definition of a nonlocal quantity. This PDE involves an internal length parameter, dictating the global scale at which the nonlocal interactions of the underlying micromechanisms are manifested during phase transformation. Moreover, to account for the unstable softening behavior, the transformation yield force parameter is considered as a gradually decreasing function of the martensite fraction. Possible material and geometric imperfections that are responsible for localization initiation are also considered in this analysis. The obtained constitutive equations are implemented in the Abaqus® finite element code in one and two dimensions. This requires the development of specific finite elements having the nonlocal volume fraction variable as an additional degree of freedom. This implementation is achieved through the UEL user’s subroutine. The effect of martensitic localization on the superelastic global behavior of SMA wire and holed thin plate, subjected to tension loading, is analyzed. Numerical results show that the developed tool correctly captures the commonly observed unstable superelastic behavior characterized by nucleation and propagation of martensitic phase. In particular, they show the influence of the internal length parameter, appearing in the nonlocal model, on the size of the localization area and the stress nucleation peak.     

A nonlocal continuum model for the buckling of carbon honeycombs

Using molecular dynamics (MD) simulations and Eringen’s nonlocal elasticity theory, in this paper we comprehensively study the small-scale effects on the buckling behaviours of carbon honeycombs (CHCs). The MD simulation results show that the small-scale effects stemming from the long-range van der Waals interaction between carbon atoms can significantly affect the buckling behaviours of CHCs. To incorporate the small-scale effects into the theoretical analysis of the buckling of CHCs, we develop a nonlocal continuum mechanics (CM) model by employing Eringen’s nonlocal elasticity theory. Our nonlocal CM model is found to fit MD simulations well by setting the nonlocal parameter in the nonlocal CM model as 0.67. It is shown in our MD-based nonlocal CM model that when the cell length of CHCs is smaller than 7.93 Å the influence of small-scale effects on the bucking of CHCs becomes unnegligible and the small-scale effects can greatly reduce the critical buckling stress of CHCs. This reduction in critical buckling stress induced by the small-scale effects becomes more significant as the length of the cell wall decreases. Moreover, CHCs are found to display two different buckling modes when they are under different states of loading. The critical condition for the transition between these two buckling modes of CHCs can be greatly affected by the small-scale effects when the vertical cell wall and the inclined cell wall of CHCs have different lengths.