An atomistic and non-classical continuum field theoretic perspective of elastic interactions between defects (force dipoles) of various symmetries and application to graphene

Force multipoles are employed to represent various types of defects and physical phenomena in solids: point defects (interstitials, vacancies), surface steps and islands, proteins on biological membranes, inclusions, extended defects, and biological cell interactions among others. In the present work, we (i) as a prototype simple test case, conduct quantum mechanical calculations for mechanics of defects in graphene sheet and in parallel, (ii) formulate an enriched continuum elasticity theory of force dipoles of various anisotropies incorporating up to second gradients of strain fields (thus accounting for nonlocal dispersive effects) instead of the usual dispersion-less classical elasticity formulation that depends on just the strain (c.f. Peyla, P., Misbah, C., 2003. Elastic interaction between defects in thin and 2-D films. Eur. Phys. J. B. 33, 233-247). The fundamental Green's function is derived for the governing equations of second gradient elasticity and the elastic self and interaction energies between force dipoles are formulated for both the two-dimensional thin film and the three-dimensional case. While our continuum results asymptotically yield the same interaction energy law as Peyla and Misbah for large defect separations (∼1/rⁿ for defects with n-fold symmetry), the near-field interactions are qualitatively far more complex and free of singularities. Certain qualitative behavior of defect mechanics predicted by atomistic calculations are well captured by our enriched continuum models in contrast to classical elasticity calculations. For example, consistent with our atomistic calculations of defects in isotropic graphene, even two dilation centers show a finite interaction (as opposed to classical elasticity that predicts zero interaction). We explicitly find the physically consistent result that the self-energy of a defect is equivalent to half the interaction energy between two identical defects when they “merge” into each other. The atomistic, classical elastic and the enriched continuum predictions are thoroughly compared for two types of defects in graphene: Stone-Wales and divacancy.     

基于物理界面的双重介质有限裂隙渗流模型

基于连续介质或者离散裂隙假设,含裂隙的多孔介质渗流问题有多种数学力学模型。受物理界面的启发,提出一种新的有限裂隙连续介质力学模型,可以为宏观裂隙-多孔介质内的流体输运问题等提供近似计算方案。该模型属于一类双重介质模型,将曲面上低维度的流场转化为三维空间的流场,并且与连续的多孔介质的流场耦合,在数学上表示为统一的输运控制方程和初始边界条件。这个近似模型为不方便实施高维度-低维度耦合求解的数值计算方法提供新的模拟思路,如光滑粒子流体动力学等无网格粒子类方法。     

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.      

Force flux and the peridynamic stress tensor

The peridynamic model is a framework for continuum mechanics based on the idea that pairs of particles exert forces on each other across a finite distance. The equation of motion in the peridynamic model is an integro-differential equation. In this paper, a notion of a peridynamic stress tensor derived from nonlocal interactions is defined. At any point in the body, this stress tensor is obtained from the forces within peridynamic bonds that geometrically go through the point. The peridynamic equation of motion can be expressed in terms of this stress tensor, and the result is formally identical to the Cauchy equation of motion in the classical model, even though the classical model is a local theory. We also establish that this stress tensor field is unique in a certain function space compatible with finite element approximations.     

Elasto-plastic postbuckling of damaged orthotropic plates

Based on the elasto-plastic mechanics and continuum damage theory, a yield criterion related to spherical tensor of stress is proposed to describe the mixed hardening of damaged orthotropic materials. Its dimensionless form is isomorphic with the Mises criterion for isotropic materials. Furthermore, the incremental elasto-plastic damage constitutive equations and damage evolution equations are established. Based on the classical nonlinear plate theory, the incremental nonlinear equilibrium equations of orthotropic thin plates considering damage effect are obtained, and solved with the finite difference and iteration methods. In the numerical examples, the effects of damage evolution and initial deflection on the elasto-plastic postbuckling of orthotropic plates are discussed in detail.     

A generalized continuum theory with internal corner and surface contact interactions

We consider a classical derivation of a continuum theory, based on the fundamental balance laws of mass and momenta, for a body with internal corner and surface contact interactions. The balances of mass and linear and angular momentum are applied to the arbitrary parts of a continuum which supports non-classical internal corner and surface contact interactions. The form of the specific corner contact interaction force measured per unit length of the corner is derived. A generalized form of Cauchy’s stress theorem is obtained, which shows that the surface traction on an oriented surface depends in a specific way on both the oriented unit normal as well as the curvature of the surface. An explicit form of the surface-couple traction which acts on every oriented surface is obtained. Two fields in the continuum, which are denoted as stress and hyperstress fields, are shown to exist, and their role in representing the surface traction and the surface-couple traction is identified. Finally, the field equations for this theory are determined, and a fundamental power theorem is derived. In the absence of internal corner and surface-couple traction interactions, the equations of classical continuum mechanics are recovered. There is no appeal to any ‘principle of virtual power’ in this work.     

缺陷连续统理论及其在本构方程研究中的应用 Ⅰ.一般概述,运动学和变形几何学

缺陷连续统理论即缺陷场论是当代固体力学的一个重要分支,其主要任务是对物质的弹性和非弹性性质的宏、微观研究之间架起一座桥梁。它也被认为是由固体力学、近代物理和数学之间交互作用而发展起来的一门交缘学科。本文分三部分较系统地介绍了它的主要发展和最近结果。第Ⅰ部分讨论具有位错和旋错连续统的运动学和变形几何学,包括Nye,Kondo,Bilby和Krner等人的早期结果以及我们利用4维物质流形上Cartan结构方程推导出的非线性动力学方程的最近结果。第Ⅱ部分详细介绍了缺陷连续统的规范场理论,主要强调对该连续统动力学方程的发展。第Ⅲ部分研究缺陷场论对构造弹塑性物质本构关系的应用。      

Implication of wave velocities for porous medium mesostructure

For porous media whose constituents' properties vary appreciably, higher-order terms in the governing equations beyond those in classical continuum mechanics are required. Negative values of Poisson's ratio have thus been found and identified as a special feature of porous media since there lacks appropriate relations for the wave velocities in terms of the material properties.     

The State of Pure Shear

In classical continuum mechanics a state of pure shear is defined as one for which there is some orthonormal basis relative to which the normal components of the Cauchy stress tensor vanish. An equivalent characterization is that the trace of the Cauchy stress tensor must vanish. We give an elementary but complete discussion of this fundamental theorem here from both the geometric and algebraic points of view. This revised version was published online in August 2006 with corrections to the Cover Date.     

A tensor method for the derivation of the equations of rigid body dynamics

A tensor method for the derivation of the equations of rigid body dynamics,based onthe concepts of continuum mechanics,is presented.The formula of time derivative of theinertia tensor with zero corotational rate is used to prove the equivalences of five methods,namely,Lagrange’s equations,Nielsen’s equations,Gibbs-Appell’s equations,Kane’sequations and the generalized momentum type of Kane’s equations.Some differentialidentities on angular velocity and angular acceleration are given.     

A symmetric nonlocal damage theory

The paper presents a thermodynamically consistent formulation for nonlocal damage models. Nonlocal models have been recognized as a theoretically clean and computationally efficient approach to overcome the shortcomings arising in continuum media with softening. The main features of the presented formulation are: (i) relations derived by the free energy potential fully complying with nonlocal thermodynamic principles; (ii) nonlocal integral operator which is self-adjoint at every point of the solid, including zones near to the solid’s boundary; (iii) capacity of regularizing the softening ill-posed continuum problem, restoring a meaningful nonlocal boundary value problem. In the present approach the nonlocal integral operator is applied consistently to the damage variable and to its thermodynamic conjugate force, i.e. nonlocality is restricted to internal variables only. The present model, when associative nonlocal damage flow rules are assumed, allows the derivation of the continuum tangent moduli tensor and the consistent tangent stiffness matrix which are symmetric. The formulation has been compared with other available nonlocal damage theories.Finally, the theory has been implemented in a finite element program and the numerical results obtained for 1-D and 2-D problems show its capability to reproduce in every circumstance a physical meaningful solution and fully mesh independent results.     

Incorporation of polymer diffusivity and migration into constitutive equations

Given a general one-particle constitutive equation for the stress tensor, we discuss how to incorporate the additional effects of polymer diffusivity and migration into that constitutive equation within the framework of continuum mechanics. For the example of an upper-convected Maxwell model representing the polymer contribution to the stress tensor of a dilute polymer solution, we describe i) how to modify the constitutive equation for the stress tensor to include diffusion and migration effects, ii) how to formulate a balance equation for the polymer mass density in order to describe the nonhomogeneous composition of the polymer solution resulting from migration, and iii) how to close the extended set of coupled equations by means of further constitutive equations for the migration velocity and the diffusion tensor. In order to guarantee the material objectivity for all equations, we formulate them in the body tensor formulation of continuum mechanics (and then translate them into Cartesian space). The proposed equations are compared to results of a recent kinetic-theory approach.Dedicated to Professor Arthur S. Lodge on the occasion of his 70th birthday and his retirement from the University of Wisconsin.     

On a formulation for a multiscale atomistic-continuum homogenization method

The homogenization method is used as a framework for developing a multiscale system of equations involving atoms at zero temperature at the small scale and continuum mechanics at the very large scale. The Tersoff–Brenner Type II potential [Physical Review Letters 61(25) (1988) 2879; Physical Review B 42 (15) (1990) 9458] is employed to model the atomic interactions while hyperelasticity governs the continuum. A quasistatic assumption is used together with the Cauchy–Born approximation to enforce the gross deformation of the continuum on the positions of the atoms. The two-scale homogenization method establishes coupled self-consistent variational equations in which the information at the atomistic scale, formulated in terms of the Lagrangian stiffness tensor, concurrently feeds the material information to the continuum equations. Analytical results for a one dimensional molecular wire and numerical experiments for a two dimensional graphene sheet demonstrate the method and its applicability.     

On `best' shell models – From classical shells, degenerated and multi-layered concepts to 3D

Summary Problems of solid mechanics are most generally formulated within 3D continuum mechanics. However, engineering models favor reduced dimensions, in order to portray mechanical properties by surface or curvilinear approximations. Such attempts for dimensional reduction constitute interactions between theoretical formulations and numerical techniques. A classical reduced model for thin bodies is represented by shell theory, an approximation in terms of resultants and first-order moments. If the shell theory, with its inherent errors, is considered as qualitatively insufficient for a particular problem, a further improvement is given by solid shell models, which are gained by direct linear interpolation of the 3D kinematic relations. They improve considerably the analytic capabilities for shells, especially when their congenital locking effects are handled by variational `convergence tricks'. The next step towards 3D quality are layered shells or solid shell elements. The present paper compares these three approximation stages from the point of view of multi-director (integral) transformations of classical continuum mechanics. It offers physical convergence requirements for each of the treated models. Partial support to the present study by the German Science Foundation (DFG) within the Special Research Center (SFB) 398 is gratefully acknowledged.     

Theoretical derivation of the conservation equations for single phase flow in porous media: a continuum approach

This paper deals with the theoretical derivation of the conservation equations for single phase flow in a porous medium. The derivation is obtained within the framework of the continuum mechanics and classical thermodynamics. The adopted procedure provides the conservation equations of mass, momentum, mechanical energy, total energy, internal energy, entropy, temperature, enthalpy, Gibbs free energy and Helmholtz free energy. The obtained results highlight the connection between the basic equations of fluid mechanics and of fluid flow in porous media, as well as the restrictions and the limitations of Darcy’s law and Richards’ equation.