Discrete and non-local elastica

In this paper, the buckling and post-buckling behavior of an elastic lattice system referred to as the discrete elastica problem is investigated using an equivalent non-local continuum approach. The geometrically exact post-buckling analysis of the elastic chain, also called Hencky system, is first numerically solved using the shooting method. This discrete physical model is also mathematically equivalent to a finite difference formulation of the continuum elastica. Starting from the exact difference equations of the discrete problem, a continualization method is applied for approximating the difference operators by differential ones, in order to better characterize the discrete system by an enriched continuous one. It is shown that the new continuum associated with the discrete system exactly fits the discrete elastica post-buckling problem, where the non-locality is of Eringen׳s type (also called stress gradient non-local model). An asymptotic expansion is performed for both the discrete and the non-local continuum models, in order to approximate the post-buckling branches of the discrete system. Some numerical investigations show the efficiency of the non-local approach, especially for capturing the scale effects inherent to the cell size of the lattice model.     

Network Approximation in the Limit of¶Small Interparticle Distance of the¶Effective Properties of a High-Contrast¶Random Dispersed Composite

We consider a high-contrast two-phase composite such as a ceramic/polymer composite or a fiberglass composite. Our objective is to determine the dependence of the effective conductivity (or the effective dielectric constant or the effective shear modulus) of the composite on the random locations of the inclusions (ceramic particles or fibers) when the concentration of the inclusions is high. We consider a two-dimensional model and show that the continuum problem can be approximated by a discrete random network (graph). We use variational techniques to provide rigorous mathematical justification for this approximation. In particular, we have shown asymptotic equivalence of the effective constant for the discrete and continuum models in the limit when the relative interparticle distance goes to zero. We introduce the geometrical interparticle distance parameter using Voronoi tessellation, and emphasize the relevance of this parameter due to the fact that for irregular (non-periodic) geometries it is not uniquely determined by the volume fraction of the inclusions. We use the discrete network to compute numerically. For this purpose we employ a computer program which generates a random distribution of disks on the plane. Using this distribution we obtain the corresponding discrete network. Furthermore, the computer program provides the distribution of fluxes in the network which is based on Keller's formula for two closely spaced disks. We compute the dependence of on the volume fraction of the inclusions V for monodispersed composites and obtaine results which are consistent with the percolation theory predictions. For polydispersed composites (random inclusions of two different sizes) the dependence is not simple and is determined by the relative volume fraction V _r of large and small particles. We found some special values of V _r for which the effective coefficient is significantly decreased. The computer program which is based on our network model is very efficient and it allows us to collect the statistical data for a large number of random configurations.     

A nonlocal constitutive model generated by matrix functions for polyatomic periodic linear chains

We establish a discrete lattice dynamics model and its continuum limits for nonlocal constitutive behavior of polyatomic cyclically closed linear chains being formed by periodically repeated unit cells (molecules), each consisting of \({n \geq 1}\) atoms which all are of different species, e.g., distinguished by their masses. Nonlocality is introduced by elastic potentials which are quadratic forms of finite differences of orders \({m \in \mathbf{N}}\) of the displacement field leading by application of Hamilton’s variational principle to nondiagonal and hence nonlocal Laplacian matrices. These Laplacian matrices are obtained as matrix power functions of even orders 2m of the local discrete Laplacian of the next neighbor Born-von-Karman linear chain. The present paper is a generalization of a recent model that we proposed for the monoatomic chain. We analyze the vibrational dispersion relation and continuum limits of our nonlocal approach. “Anomalous” dispersion relation characteristics due to strong nonlocality which cannot be captured by classical lattice models is found and discussed. The requirement of finiteness of the elastic energies and total masses in the continuum limits requires a certain scaling behavior of the material constants. In this way, we deduce rigorously the continuum limit kernels of the Laplacian matrices of our nonlocal lattice model. The approach guarantees that these kernels correspond to physically admissible, elastically stable chains. The present approach has the potential to be extended to 2D and 3D lattices.     

A nonhomogeneous nonlocal elasticity model

Nonlocal elasticity with nonhomogeneous elastic moduli and internal length is addressed within a thermodynamic framework suitable to cope with continuum nonlocality. The Clausius–Duhem inequality, enriched by the energy residual, is used to derive the state equations and all other thermodynamic restrictions upon the constitutive equations. A phenomenological nonhomogeneous nonlocal (strain difference-dependent) elasticity model is proposed, in which the stress is the sum of two contributions, local and nonlocal, respectively governed by the standard elastic moduli tensor and the (symmetric positive-definite) nonlocal stiffness tensor. The inhomogeneities of the elastic moduli and of the internal length are conjectured to be each the cause of additional attenuation effects upon the long distance particle interactions. The increased attenuation effects are accounted for by means of the standard attenuation function, but with the standard spatial distance replaced by a suitably larger equivalent distance, and with the spatially variable internal length replaced by the largest value within the domain. Formulae for the computation of the equivalent distance are heuristically suggested and illustrated with numerical examples. The solution uniqueness of the continuum boundary-value problem is proven and the related total potential energy principle given and employed for possible nonlocal-FEM discretizations. A bar in tension is considered for a few numerical applications, showing perfect numerical stability, provided the free energy potential is positive definite.     

The Nonlinear Heat Equation on W-Random Graphs

For systems of coupled differential equations on a sequence of W-random graphs, we derive the continuum limit in the form of an evolution integral equation. We prove that solutions of the initial value problems (IVPs) for the discrete model converge to the solution of the IVP for its continuum limit. These results combined with the analysis of nonlocally coupled deterministic networks in Medvedev (The nonlinear heat equation on dense graphs and graph limits. ArXiv e-prints, 2013) justify the continuum (thermodynamic) limit for a large class of coupled dynamical systems on convergent families of graphs.     

Modified virtual internal bond model based on deformable Voronoi particles

In last time, the series of virtual internal bond model was proposed for solving rock mechanics problems. In these models, the rock continuum is considered as a structure of discrete particles connected by normal and shear springs(bonds). It is well announced that the normal springs structure corresponds to a linear elastic solid with a fixed Poisson ratio, namely, 0.25 for threedimensional cases. So the shear springs used to represent the diversity of the Poisson ratio.However, the shearing force calculation is not rotationally invariant and it produce difficulties in application of these models for rock mechanics problems with sufficient displacements. In this letter, we proposed the approach to support the diversity of the Poisson ratio that based on usage of deformable Voronoi cells as set of particles. The edges of dual Delaunay tetrahedralization are considered as structure of normal springs(bonds). The movements of particle's centers lead to deformation of tetrahedrals and as result to deformation of Voronoi cells. For each bond, there are the corresponded dual face of some Voronoi cell. We can consider the normal bond as some beam and in this case, the appropriate face of Voronoi cell will be a cross section of this beam. If during deformation the Voronoi face was expand, then, according Poisson effect, the length of bond should be decrees. The above mechanism was numerically investigated and we shown that it is acceptable for simulation of elastic behavior in 0.1–0.3 interval of Poisson ratio. Unexpected surprise is that proposed approach give possibility to simulate auxetic materials with negative Poisson's ratio in interval from –0.5 to –0.1.     

关于采用粗粒化提高颗粒材料多尺度模拟守恒特性的研究

连续体-颗粒耦合方法常用来描述连续-非连续颗粒行为或解决颗粒材料与其他可变形构件间相互作用问题。粗粒化coarse-graining (CG)是基于统计力学的均匀化方法,由离散的颗粒运动定义连续的宏观物理场。本文利用粗粒化(CG)推导有限元-离散元(FEM-DEM)表面和体积耦合的一般性表达式。对于表面耦合,CG可以将耦合力分布到颗粒-单元接触点以外的位置,如相邻的积分点;对于体积耦合,CG可以将颗粒尺度的运动均匀化到耦合单元上。由粗粒化推导出的耦合项仅包含一个参数,即粗粒化宽度,为均匀化后的宏观场定义了一个可调整的空间尺度。当粗粒化宽度为零时,表面和体积耦合表达式简化为常规局部耦合。本文通过弹性立方体冲击颗粒床和离散-连续介质间波传播两个数值算例,展示使用粗粒化方法提高耦合系统能量守恒的优势,并结合其他耦合参数(如体积耦合深度)讨论了粗粒化参数对数值稳定性和计算效率的影响。     

Derivation of Mindlin’s first and second strain gradient elastic theory via simple lattice and continuum models

Mindlin, in his celebrated papers of Arch. Rat. Mech. Anal. 16, 51–78, 1964 and Int. J. Solids Struct. 1, 417–438, 1965, proposed two enhanced strain gradient elastic theories to describe linear elastic behavior of isotropic materials with micro-structural effects. Since then, many works dealing with strain gradient elastic theories, derived either from lattice models or homogenization approaches, have appeared in the literature. Although elegant, none of them reproduces entirely the equation of motion as well as the classical and non-classical boundary conditions appearing in Mindlin theory, in terms of the considered lattice or continuum unit cell. Furthermore, no lattice or continuum models that confirm the second gradient elastic theory of Mindlin have been reported in the literature. The present work demonstrates two simple one dimensional models that conclude to first and second strain gradient elastic theories being identical to the corresponding ones proposed by Mindlin. The first is based on the standard continualization of the equation of motion taken for a sequence of mass-spring lattices, while the second one exploits average processes valid in continuum mechanics. Furthermore, Mindlin developed his theory by adding new terms in the expressions of potential and kinetic energy and introducing intrinsic micro-structural parameter without however providing explicit expressions that correlate micro-structure with macro-structure. This is accomplished in the present work where in both models the derived internal length scale parameters are correlated to the size of the considered unit cell.     

Generalized continuum modeling of 2-D periodic cellular solids

A generalized continuum representation of two-dimensional periodic cellular solids is obtained by treating these materials as micropolar continua. Linear elastic micropolar constants are obtained using an energy approach for square, equilateral triangular, mixed triangle and diamond cell topologies. The constants are obtained by equating two different continuous approximations of the strain energy function. Furthermore, the effects of shear deformation of the cell walls on the micropolar elastic constants are also discussed. A continuum micropolar finite element approach is developed for numerical simulations of the cell structures. The solutions from the continuum representation are compared with the “exact” discrete simulations of these cell structures for a model problem of elastic indentation of a rectangular domain by a point force. The utility of the micropolar continuum representation is illustrated by comparing various cell structures with respect to the stress concentration factor at the root of a circular notch.     

Application of quasi-continuum models for perturbation analysis of discrete kinks

Here, we study a relation between discrete and continuum models on an example of the sine-Gordon and ?? ⁴ equations. The analysis of various receptions of continualization in a linear case is carried out. The best approach allowing describing all spectrum of the discrete one-dimensional medium is chosen. Also, the nonlinear discrete sine-Gordon and ?? ⁴ models are analyzed. The possibility of improvement of the known continuum approximations of these equations is shown.     

Magnetostrictive composites in the dilute limit

We calculate the effective properties of a magnetostrictive composite in the dilute limit. The composite consists of well separated identical ellipsoidal particles of magnetostrictive material, surrounded by an elastic matrix. The free energy of the magnetostrictive particles is computed using the constrained theory of DeSimone and James [2002. A constrained theory of magnetoelasticity with applications to magnetic shape memory materials. J. Mech. Phys. Solids 50, 283-320], where application of an external field causes rearrangement of variants rather than rotation of the magnetization or elastic strain in a variant. The free energy of the composite has an elastic energy term associated with the deformation of the surrounding matrix and demagnetization terms. By using results from the constrained theory and from the Eshelby inclusion problem in linear elasticity, we show that the energy minimization problem for the composite can be cast as a quadratic programming problem. The solution of the quadratic programming problem yields the effective properties of Ni₂MnGa and Terfenol-D composite systems. Numerical results show that the average strain of the composite depends strongly on the particle shape, the applied stress, and the elastic modulus of the matrix.     

离散颗粒集合体-Cosserat连续体模型的连接尺度方法

基于针对分子动力学-Cauchy连续体模型提出的连接尺度方法(BSM)[1,2],发展了耦合细尺度上基于离散颗粒集合体模型的离散单元法(DEM)和粗尺度上基于Cosserat连续体模型的有限元法(FEM)的BSM。仅在有限局部区域内采用DEM以从细观层次模拟非连续破坏现象,而在全域则采用花费计算时间和存储空间较少的FEM。通过连接尺度位移(包括平移和转动)分解,和基于作用于Cosserat连续体有限元节点和颗粒集合体颗粒形心的离散系统虚功原理,得到了具有解耦特征的粗细尺度耦合系统运动方程。讨论和提出了在准静态载荷条件下粗细尺度域的界面条件,以及动态载荷条件下可以有效消除粗细尺度域界面上虚假反射波的非反射界面条件(NRBC)。本文二维数值算例结果说明了所提出的颗粒材料BSM的可应用性和优越性,及所实施界面条件对模拟颗粒材料动力学响应的有效性。     

Modeling of strongly nonlinear effects in diatomic lattices

A previously developed strongly nonlinear continuum model for diatomic crystals is examined using continuum limit of a discrete diatomic model. It suggested suitable expression for the forces in the discrete model; however, its continuum limit not only explains the nonlinear terms continuum model but also gives rise to some additional terms. It turns out that one of them supports bell-shaped localized variations in the diatomic material but suppresses kink-shaped variations.     

Effects of structural and material length scales on stress-induced martensite macro-domain patterns in tube configurations

This paper studies the effects of structural and material length scales on the equilibrium domain patterns in thin-walled long tube configurations during stress-induced elastic phase transition under displacement-controlled quasi-static isothermal stretching. A nonconvex and nonlocal continuum model is developed and implemented into a finite element code to simulate the domain formation and evolution during the phase transition. The morphology and evolution of the macro-domains in different tube geometries are investigated by both analytical (energy analysis) and numerical (nonlocal finite element) methods. Energy minimization is used as the principle to explain the experimentally observed macroscopic domain patterns in a NiTi polycrystal tube. It is found that the domain pattern, as the minimizer of the system energy, is governed by the relative values of the material length scale g, tube-wall thickness h and tube radius R through two nondimensional factors: h/R and g/R. Physically, h/R and g/R serve as the weighting factors of bending energy and domain-wall energy over the membrane energy in the minimization of the total energy of the tube system. Theoretical predictions of the effects of these length scales on the domain pattern are quantified and confirmed by the computational parametric study. They all agree qualitatively well with the available experimental observations.     

Bifurcations in granular media: macro- and micro-mechanics approaches

Various failure modes related to different kinds of bifurcations occur in nonassociated elastoplastic materials such as geomaterials. After presenting experimental evidence, we study this question by means of phenomenological constitutive relations and direct numerical simulations based on the discrete element method. The second-order work criterion related to diffuse failure modes is particularly considered within the framework of continuum and discrete mechanics. The equations of the bifurcation domain boundary and unstable stress direction cones are established. Diffuse failure is simulated numerically by perturbing bifurcation states. To cite this article: F. Darve et al., C. R. Mecanique 335 (2007).     

Elastostatic resonances—a new approach to the calculation of the effective elastic constants of composites

A new method is presented for a systematic evaluation of the effective elastic tensor C^(e) in a two-component composite. Both C^(e) and local strain field are expanded in terms of a complete set of elastostatic resonances. The resonances are found by calculating eigenstates of a certain integral operator, and this can be carried out in stages. First one finds the eigenstates of individual, isolated grains or fibers, and only then does one attempt to calculate eigenstates of the entire composite. We apply this procedure to 2D periodic arrays of cylinders—both hexagonal and square. Using simple matrix perturbation techniques we obtain exact expansions for the elastic constants in powers of p, the volume fraction of the cylinders, that go up to the order p¹¹ in the case of bulk modulus of the hexagonal array.     

Hydrodynamic Limit for a Hamiltonian System with Boundary Conditions and Conservative Noise

We study the hyperbolic scaling limit for a chain of N coupled anharmonic oscillators. The chain is attached to a point on the left and there is a force (tension) τ acting on the right. In order to provide good ergodic properties to the system, we perturb the Hamiltonian dynamics with random local exchanges of velocities between the particles, so that momentum and energy are locally conserved. We prove that in the macroscopic limit the distributions of the elongation, momentum and energy converge to the solution of the Euler system of equations in the smooth regime.     

Multiscale mass-spring models of carbon nanotube foams

This article is concerned with the mechanical properties of dense, vertically aligned CNT foams subject to one-dimensional compressive loading. We develop a discrete model directly inspired by the micromechanical response reported experimentally for CNT foams, where infinitesimal portions of the tubes are represented by collections of uniform bi-stable springs. Under cyclic loading, the given model predicts an initial elastic deformation, a non-homogeneous buckling regime, and a densification response, accompanied by a hysteretic unloading path. We compute the dynamic dissipation of such a model through an analytic approach. The continuum limit of the microscopic spring chain defines a mesoscopic dissipative element (micro-meso transition) which represents a finite portion of the foam thickness. An upper-scale model formed by a chain of non-uniform mesoscopic springs is employed to describe the entire CNT foam. A numerical approximation illustrates the main features of the proposed multiscale approach. Available experimental results on the compressive response of CNT foams are fitted with excellent agreement.     

Convergence of Peridynamics to Classical Elasticity Theory

The peridynamic model of solid mechanics is a nonlocal theory containing a length scale. It is based on direct interactions between points in a continuum separated from each other by a finite distance. The maximum interaction distance provides a length scale for the material model. This paper addresses the question of whether the peridynamic model for an elastic material reproduces the classical local model as this length scale goes to zero. We show that if the motion, constitutive model, and any nonhomogeneities are sufficiently smooth, then the peridynamic stress tensor converges in this limit to a Piola-Kirchhoff stress tensor that is a function only of the local deformation gradient tensor, as in the classical theory. This limiting Piola-Kirchhoff stress tensor field is differentiable, and its divergence represents the force density due to internal forces. The limiting, or collapsed, stress-strain model satisfies the conditions in the classical theory for angular momentum balance, isotropy, objectivity, and hyperelasticity, provided the original peridynamic constitutive model satisfies the appropriate conditions.