Bending-induced extension in two-dimensional crystals

We find by ab initio simulations that significant overall tensile strain can be induced by pure bending in a wide range of two-dimensional crystals perpendicular to the bending moment, just like an accordion being bent to open. This bending-induced tensile strain increases in a power law with bent curvature and can be over 20% in monolayered black phosphorus and transition metal dichalcogenides at a moderate curvature of 2 nm?1 but more than an order weaker in graphene and hexagon boron nitride. This accordion effect is found to be a quantum mechanical effect raised by the asymmetric response of chemical bonds and electron density to the bending curvature.     

Twist buckling and the foldable cylinder: an exercise in origami

Basic mechanisms for the buckling of a thin cylindrical shell under torsional loading are reviewed from a post-buckling perspective. Deflections are considered so far into the large-deflection range that the shell is allowed to fold to a flat two-dimensional form, in a mechanism reminiscent of a deployable structure. Critical and initial post-buckling effects are explored through concepts of energy minimization and hidden symmetries. For comparisons with the final large-deflection folded shape, a truss element program is employed. It is shown that, as buckling develops, the mode shape must change to accommodate both the symmetry-breaking aspects of the predominately inwards deflection, and the rotation of peak and valley lines of the buckle pattern necessary to accommodate the geometry of the final folded shape.     

Characterization of Elasticity-Tensor Symmetries Using SU(2)

A symmetry class of an elasticity tensor, c, is determined by the variance of this tensor with respect to a subgroup of the special orthogonal group, SO(3). Using the double covering of SO(3) by the special unitary group, SU(2), we determine the subgroups of SU(2) that correspond to each of the eight symmetry classes. A family of maps between C² and R³ that preserve the action of the two groups is constructed. Using one of these maps and three associated polynomials, we derive new methods for characterizing the symmetry classes of elasticity tensors. Mathematics Subject Classifications (2000) 74B05, 74E10.     

CORC电缆中超导带材在缠绕、弯曲和扭曲变形下的应变分析

CORC(conductor on round core)超导电缆是一种基于REBCO(rare-earth barium copper oxide)导体的高温超导电缆,已经成为迄今为止最灵活的高温超导(HTS)导体之一.然而,由于超导REBCO陶瓷材料的脆性,临界电流表现出敏感的应变依赖性,即临界电流与应变密切相关.该材料的应变一旦超过临界值,就会引发明显的衰减.除此之外,由于带材缠绕和电缆绞合后的复杂变形,切实存在一个实际性的挑战,即如何准确预测电缆超导带材在各种变形下的轴向应变.在本文中,我们建立了分析模型,可以用来准确计算CORC电缆的超导带材在电缆内芯上缠绕、电缆弯曲和扭曲变形等实际条件下的轴向应变.并且通过比较分析结果与有限元数值结果和实验数据,验证了分析模型,即超导带的临界电流随CORC电缆的核心直径而变化.在轴向应变结果的基础上,进一步确定了临界参数,即超导层的轴向应变恰好达到临界应变时的参数,如临界核心直径、临界弯曲直径和临界绞距等.对CORC电缆中REBCO超导带材的应变分析研究将有助于CORC电缆的加工和进一步设计高水平的超导线圈和磁体.     

Dynamical analysis of multi-stage cyclic structures

This paper deals with the dynamical analysis of a multi-stage assembly of cyclic structures such as, for example, turbomachinery compressor or turbines. If such assemblies are traditionally modelled stage by stage, the inter-stage coupling effect can sometime be important. As an answer to this issue, we propose a new method which combines a cyclic modelling of each stage with a realistic inter-stage coupling. Study cases are presented to evaluate the efficiency of the method.     

Fluid pressure response in poroelastic materials subjected to cyclic loading

When cyclic loading is applied to poroelastic materials, a transient stage of interstitial fluid pressure occurs, preceding a steady state. In each stage, the fluid pressure exhibits a characteristic mechanical behavior. In this study, an analytical solution for fluid pressure in two-dimensional poroelastic materials, which is assumed to be isotropic, under cyclic axial and bending loading is presented, based on poroelasticity. The obtained analytical solution contains transient and steady-state responses. Both of these depend on three dimensionless parameters: the dimensionless stress coefficient; the dimensionless frequency; and, the axial-bending loading ratio. We focus particularly on the transient behavior of interstitial fluid pressure with changes in the dimensionless frequency and the axial-bending loading ratio. The transient properties, such as half-value period and contribution factor, depend largely on the dimensionless frequency and have peak values when its value is about 10. This suggests that, under these conditions, the transient response can significantly affect the mechanical behavior of poroelastic materials.     

求解含缺陷一维周期结构动力响应的高效算法

基于周期结构的动力特性和群理论,建立了一种高效求解含缺陷一维周期结构动力响应的数值方法。在求解结构动力响应时,高效求解结构对应的线性代数方程组最为关键。采用凝聚技术,可减小结构对应线性代数方程组的规模。基于周期结构动力系统中线性代数方程组的特性,通过一个小规模含缺陷结构和一维周期结构的响应分析,可得到含缺陷一维周期结构的动力响应。同理,一维周期结构的动力响应可通过一系列小规模结构的响应分析得到,且小规模结构的动力响应可基于群理论高效求解。数值算例表明,本文算法有较高的求解效率。     

Solving viscoelastic problems with cyclic symmetry via a precise algorithm and EFGM

The paper combines a self-adaptive precise algorithm in the time domain with Meshless Element Free Galerkin Method (EFGM) for solving viscoelastic problems with rotationally periodic symmetry. By expanding variables at a discretized time interval, the variations of variables can be described more precisely, and iteration is not required for non-linear cases. A space-time domain coupled problem with initial and boundary values can be converted into a series of linear recursive boundary value problems, which are solved by a group theory based on EFGM. It has been proved that the coefficient matrix of the global EFG equation for a rotationally periodic system is block-circulant so long as a kind of symmetry-adapted reference coordinate system is adopted, and then a partitioning algorithm for facilitating parallel processing was proposed via a completely orthogonal group transformation. Therefore instead of solving the original system, only a series of independent small sub-problems need to be solved, leading to computational convenience and a higher computing efficiency. Numerical examples are given to illustrate the full advantages of the proposed algorithm. The project supported by the National Natural Science Foundation of China ((10421002, 10472019 and 10172024); NKBRSF (2005CB321704) and the Fund of Disciplines Leaders of Young and Middle Age Faculty in Colleges of Liaoning Province. The English text was polished by Yunming Chen.     

Generalized lie symmetries and the integrability of generalized Holmes-Rand non-linear oscillator

Generalized Lie symmetries and the integrability of generalized Holmes-Rand non-linear oscillator (GHRNO) are considered. The constraint which the variable-coefficient functions must satisfy for the GHRNO to have infinite-dimensional symmetry algebras is derived. The integral of motion for the GHRNO under this condition can be read off from the symmetry vector fields. The structure of the symmetry algebras is also presented.     

Global and bifurcation analysis of a structure with cyclic symmetry

This article combines the application of a global analysis approach and the more classical continuation, bifurcation and stability analysis approach of a cyclic symmetric system. A solid disc with four blades, linearly coupled, but with an intrinsic non-linear cubic stiffness is at stake. Dynamic equations are turned into a set of non-linear algebraic equations using the harmonic balance method. Then periodic solutions are sought using a recursive application of a global analysis method for various pulsation values. This exhibits disconnected branches in both the free undamped case (non-linear normal modes, NNMs) and in a forced case which shows the link between NNMs and forced response. For each case, a full bifurcation diagram is provided and commented using tools devoted to continuation, bifurcation and stability analysis.     

Efficient meshless SPH method for the numerical modeling of thick shell structures undergoing large deformations

The objective of this paper is to present an extension of the Lagrangian Smoothed Particle Hydrodynamics (SPH) method to solve three-dimensional shell-like structures undergoing large deformations. The present method is an enhancement of the classical stabilized SPH commonly used for 3D continua, by introducing a Reissner–Mindlin shell formulation, allowing the modeling of moderately thin structure using only one layer of particles in the shell mid-surface. The proposed Shell-based SPH method is efficient and very fast compared to the classical continuum SPH method. The Total Lagrangian Formulation valid for large deformations is adopted using a strong formulation of the differential equilibrium equations based on the principle of collocation. The resulting non-linear dynamic problem is solved incrementally using the explicit time integration scheme, suited to highly dynamic applications. To validate the reliability and accuracy of the proposed Shell-based SPH method in solving shell-like structure problems, several numerical applications including geometrically non-linear behavior are performed and the results are compared with analytical solutions when available and also with numerical reference solutions available in the literature or obtained using the Finite Element method by means of ABAQUS^© commercial software.     

Nonequilibrium molecular dynamics for bulk materials and nanostructures

We describe a method of constructing exact solutions of the equations of molecular dynamics in non-equilibrium settings. These solutions correspond to some viscometric flows, and to certain analogs of viscometric flows for fibers and membranes that have one or more dimensions of atomic scale. This work generalizes the method of objective molecular dynamics (OMD) (Dumitric? and James, 2007). It allows us to calculate viscometric properties from a molecular-level simulation in the absence of a constitutive equation, and to relate viscometric properties directly to molecular properties. The form of the solutions is partly independent of the form of the force laws between atoms, and therefore these solutions have implications for coarse-grained theories. We show that there is an exact reduction of the Boltzmann equation corresponding to one family of OMD solutions. This reduction includes most known exact solutions of the equations of the moments for special kinds of molecules and gives the form of the molecular density function corresponding to such flows. This and other consequences leads us to propose an addition to the principle of material frame indifference, a cornerstone of nonlinear continuum mechanics. The method is applied to the failure of carbon nanotubes at an imposed strain rate, using the Tersoff potential for carbon. A large set of simulations with various strain rates, initial conditions and two choices of fundamental domain (unit cell) give the following unexpected results: Stone-Wales defects play no role in the failure (though Stone-Wales partials are sometimes seen just prior to failure), a variety of failure mechanisms is observed, and most simulations give a strain at failure of 15-20%, except those done with initial temperature above about 1200 K and at the lower strain rates. The latter have a strain at failure of 1-2%.     

STABILITY CHARACTERISTICS OF SINGLE-LAYERED ZINC OXIDE NANOSHEETS UNDER UNIAXIAL LOADING

This paper is aimed to propose a three-dimensional model which would be used for investigation on the mechanical behavior of single-layered zinc oxide nanosheets. To develop this model, molecular mechanics is coupled with the density functional theory. Simulating the hexagonal lattices of nanosheets as a hexagonal mechanical structure composed of structural beam elements, the buckling behavior of zinc oxide nanosheets is studied. Effects of different parameters on the stability of armchair and zigzag nanosheets are examined. It is shown that the buckling forces of zigzag nanosheets are slightly greater than those of armchair ones. However, with increasing size of nanosheets the effect of atomic structure on the stability of nanosheets diminishes.By studying the effect of end conditions on the buckling behavior of nanosheets, it is shown the stability of nanosheets is affected significantly by boundary conditions.     

A note on the classification of anisotropy of bodies defined by implicit constitutive relations

In this note we consider the definition of anisotropy with regard to the response of bodies described by implicit constitutive relations. The class of response relations under considerations in this work is implicit relations between the history of the stress, the history of the density, and the history of the deformation gradient. It is shown that the work of Noll [4] defining the anisotropy of bodies in terms of symmetry groups for Simple Materials can be very easily extended to define the anisotropy in terms of symmetry groups for materials whose response is described by relations between the histories of the stresses and the deformation gradient. While symmetry groups are defined, the more arduous task of developing representation theorems for bodies defined through implicit response relations is an important open task.     

Renormalization group symmetry method and gas dynamics

A new scenario in the renormalization group symmetry method is introduced to solve an initial value problem for a system of partial differential equations. As a specific example, we give an exact solution for the adiabatic perfect gas dynamics, which describes a contracting and expanding localized mass of gas.     

基于拉格朗日乘子法和群论的具有任意位移边界条件的旋转周期对称结构有限元分析

对任意位移边界条件下的旋转周期对称结构,由拉格朗日乘子法建立有限元方程。在对称适应的坐标系下,由结构刚度矩阵的块循环性质,利用群变换给出一种新的求解方法。数值验证给出令人满意的结果。     

On Compressible Materials Capable of Sustaining Axisymmetric Shear Deformations. Part 4: Helical Shear of Anisotropic Hyperelastic Materials

Conditions on the form of the strain energy function in order that homogeneous, compressible and isotropic hyperelastic materials may sustain controllable static, axisymmetric anti-plane shear, azimuthal shear, and helical shear deformations of a hollow, circular cylinder have been explored in several recent papers. Here we study conditions on the strain energy function for homogeneous and compressible, anisotropic hyperelastic materials necessary and sufficient to sustain controllable, axisymmetric helical shear deformations of the tube. Similar results for separate axisymmetric anti-plane shear deformations and rotational shear deformations are then obtained from the principal theorem for helical shear deformations. The three theorems are illustrated for general compressible transversely isotropic materials for which the isotropy axis coincides with the cylinder axis. Previously known necessary and sufficient conditions on the strain energy for compressible and isotropic hyperelastic materials in order that the three classes of axisymmetric shear deformations may be possible follow by specialization of the anisotropic case. It is shown that the required monotonicity condition for the isotropic case is much simpler and less restrictive. Restrictions necessary and sufficient for anti-plane and rotational shear deformations to be possible in compressible hyperelastic materials having a helical axis of transverse isotropy that winds at a constant angle around the tube axis are derived. Results for the previous case and for a circular axis of transverse isotropy are included as degenerate helices. All of the conditions derived here have essentially algebraic structure and are easy to apply. The general rules are applied in several examples for specific strain energy functions of compressible and homogeneous transversely isotropic materials having straight, circular, and helical axes of material symmetry.     

烷基化芳胺润滑添加剂抗氧抗腐蚀机理的密度泛函理论研究

采用量子化学的密度泛函理论计算了8种烷基化芳胺抗氧抗腐蚀添加剂与烷氧自由基(C6H13O·)的结合能以及与铁原子簇的化学吸附作用能,探讨了化合物的结构特征、作用机理、授受电子的性质和取代基效应.结果表明:这些添加剂的HOMO均为带有杂原子的孤对电子的π-分子轨道,HOMO可以与金属原子的LUMO发生相互作用,HOMO的电子转移到金属原子的LUMO上形成配位键和稳定的吸附态;添加剂的LUMO均为苯环的π-共轭体系组成,可与RO·的SOMO相互作用,LUMO接受RO·的电子生成稳定的加成产物,添加剂具有授受电子性质;烷基化芳胺添加剂抗氧抗腐蚀性能与取代基的供电子效应或共轭效应有关,当供电子效应强时可以增加添加剂与RO·的结合能以及与铁原子簇的化学吸附作用能.依据计算结果可以推测8种化合物的抗氧抗腐性能由高到低顺序为:化合物Ⅰ＞Ⅵ＞Ⅷ＞Ⅶ＞Ⅲ＞Ⅴ＞Ⅳ＞Ⅱ,计算结果与实验结果一致.     

A constitutive model in finite viscoelasticity

A new constitutive model is suggested for the viscoelastic behavior of rubber-like materials at finite strains. The model treats a viscoelastic medium as a system with a variable number of purely elastic links, which can arise and collapse due to micro-Brownian motion of molecules.Assuming that the processes of birth and death for elastic links are independent of stresses, we obtain operator linear constitutive equations in finite viscoelasticity. According to this model, elastic and viscous effects may be distinguished and described independently of each other by a relaxation measure and a strain energy density.The potential energy of deformations is assumed to depend on the principal invariants of the relative Finger tensor of strains. Unlike the standard approach, we do not suggest any expression for the strain energy densitya priori, but suppose that this function is presented as a sum of two functions of one variable which are found by fitting experimental data.The proposed approach allows results of several experiments (uniaxial tension, biaxial tension, and torsion) for styrene butadiene rubber and butyl rubber to be predicted correctly.