Torsion and bending periodic boundary conditions for modeling the intrinsic strength of nanowires

We present a unified approach for atomistic modeling of torsion and bending of nanowires that is free from artificial end effects. Torsional and bending periodic boundary conditions (t-PBC and b-PBC) are formulated by generalizing the conventional periodic boundary conditions (PBC) to cylindrical coordinates. The approach is simpler than the more general objective molecular dynamics formulation because we focus on the special cases of torsion and bending. A simple implementation of these boundary conditions is presented and correctly conserves linear and angular momenta. We also derive the virial expressions for the average torque and bending moment under these boundary conditions that are analogous to the virial expression for the average stress in PBC. The method is demonstrated by molecular dynamics simulation of Si nanowires under torsion and bending, which exhibit several modes of failure depending on their diameters.     

Mullins Effect for a Cylinder Subjected to Combined Extension and Torsion

We study the Mullins effect for a circular cylinder of incompressible, isotropic material under loading cycles of combined extension and torsion. The analysis is based on the constitutive model recently proposed in De Tommasi et al. (J. Rheol. 50: 495–512, 2006). This model assumes that the mechanical response at each material point results as a homogenized effect of a mixture of different materials with variable activation and breaking thresholds. We show the feasibility of this approach to treat complex, inhomogeneous deformations. In particular, we obtain for the generic loading path the analytical expressions of the stress field, of the axial force, and of the twisting moment. The proposed model exhibits the Mullins stress softening effect in the case of simple extension, simple torsion, and combined extension and torsion. We analyze in detail the path dependent behavior and the preconditioning effects.      

Numerical and experimental study of elastoplastic tension-torsion processes in axisymmetric bodies under large deformations

We consider a numerical solution technique for generalized axisymmetric problems with torsion for elastoplastic bodies of revolution of arbitrary shape under large strains, as well as simple or complex loading, and the conditions of inhomogeneous stress-strain state. The processes of elastoplastic deformation, strain localization, and fracture of solid axisymmetric steel samples of variable thickness are studied experimentally and numerically for the cases of proportional and nonproportional kinematic torsional and/or tensile loading until failure. The mutual influence of torsion and tension on the deformation and failure under large strains is estimated.     

Semi-inverse solution for Saint-Venant's problem in the theory of porous elastic materials

The purpose of this research is to study the Saint-Venant's problem for right cylinders with general cross-section made of inhomogeneous anisotropic elastic materials with voids. We reformulate the quasi-static equilibrium equations with the axial variable playing the role of a parameter. Two classes of semi-inverse solutions to Saint-Venant's problem are described in terms of five generalized plane strain problems. These classes are used in order to obtain a semi-inverse solution for the relaxed Saint-Venant's problem. An application of this results in the study of extension, bending, torsion and flexure of right circular cylinders in the case of isotropic materials is presented.     

Nonlinear effect in the torsion of a compressible cylindrical layer

We consider the finite deformation of a compressible composite elastic layer and investigate the nonlinear effect of the finite torsion of this layer, when it occupies an infinite region. Solving an equilibrium problem for this medium it is observed, using the complex variable method of elasticity, that the composite cylindrical layer experiences a reduction in radius when it is subjected to a torsion from the interior layer.     

A duality theorem for plastic torsion

Limit analysis of prismatic torsion bars was the earliest attempt to apply plasticity theory to a continuum. The simplicity of the problem made it feasible to use the two-dimensional Prandtl stress function, defined for the elastic torsion problems, for the plastic stress distributions as well. The gradient of the stress functions for plastic torsion has a constant magnitude, and hence a function of this type assumes the profile of a sand hill. This sand hill analogy of Nadai (1950, The Theory of Flow and Fracture of Solids. McGraw-Hill, U.K.) gave a visual sense of possible nonsmoothness of such stress functions and thus discontinuous stress fields. Many stress functions of plastic torsion for relatively simple cross-sections have been constructed graphically. However, collapse modes in terms of warping functions were much less reported. In this paper, we shall establish a duality theorem which relates the correct stress function to the correct warping function, thus providing the means to obtain complete static and kinematic solutions. This dual variational principle leads naturally to a general numerical algorithm which guarantees convergence and accuracy. In this paper, we shall only present three exact solutions to verify the theorem, to demonstrate the possible non-smooth feature of the solutions and to reiterate this effective dual variational approach to limit analysis in general.     

Elasticity solutions for a piezoelectric cone under concentrated loads at its apex

IntroductionTheproblemofaconesubjectedtoconcentratedloadsatitsapexisaclassicalprobleminthetheoryofelasticity.AnumberofscholarshavestUdiedtheproblem.LovereportedthesolutionstotheproblemofanisotropicconeunderconcentfatCdforcesatitsapex['].Lur'estudiedthisclassofproblemssystematicallybymeansofPapkovich-Neubergeneralsolution[2].LekniskiiandHu,byusingtheirrespectivegeneralsolutions,studiedcompressionandbendingproblemsofatransverselyisotropicconesubjectedtoaxialconcentfatedforcesandtfansverseconc…     

Saint-Venant problem for solids with helical anisotropy

We discuss the solution of Saint-Venant’s problem for solids with helical anisotropy. Here the governing relations of the theory of elasticity in terms of displacements were presented using the helical coordinate system. We proposed an approach to construct elementary Saint-Venant solutions using integration of ordinary differential equations with variable coefficients in the case of a circular cylinder with helical anisotropy. Elementary solutions correspond to problems of extension, of torsion, of pure bending and of bending of shear force. The solution of the problem is obtained using small parameter method for small values of twist angle and numerically for arbitrary values. Numeric results correspond to problems of extension–torsion. Dependencies of the stiffness matrix (in dimensionless form) on angle between the tangent to the helical coil and the axis of the cylinder corresponding to stiffness on stretching and torsion are illustrated graphically for different values of material and geometrical parameters.     

Generalized beam theory applied to shear stiffness

There is no consensus of opinion on the correct expressions for shear stiffness, even in the apparently simple case of rectangular sections made from homogeneous isotropic material. A general beam theory has been proposed which is applicable to all regular prismatic systems. This has been used to find the appropriate beam-like flexibilities for trusses. The same approach can be used for normal beams, giving values for the shear stiffnesses of various cross-sections as particular results of a general theory embracing torsion, binding, extension and shear of regular prismatic systems.     

Periodic Solutions of an Autonomous Reaction-Diffusion System—A Model System

We describe the dynamics of an autonomous system of two reaction-diffusion equations which can be looked at as a model system for more general reaction-diffusion systems. In our system all solutions tend to zero or to (finitely many) periodic orbits which can be fully described—including their stability properties. Furthermore, we construct invariant sets for the period map and show how a new invariant called torsion number is related to our model system.     

THE STABILITY OF AN AXIALLY ACCELERATING BEAM ON SIMPLE SUPPORTS WITH TORSION SPRINGS

The axially moving beams on simple supports with torsion springs are studied. The general modal functions of the axially moving beam with constant speed have been obtained from the supporting conditions. The contribution of the spring stiffness to the natural frequencies has been numerically investigated. Transverse stability is also studied for axially moving beams on simple supports with torsion springs. The method of multiple scales is applied to the partialdifferential equation governing the transverse parametric vibration. The stability boundary is derived from the solvability condition. Instability occurs if the axial speed fluctuation frequency is close to the sum of any two natural frequencies or is two fold natural frequency of the unperturbed system. It can be concluded that the spring stiffness makes both the natural frequencies and the instability regions smaller in the axial speed fluctuation frequency-amplitude plane for given mean axial speed and bending stiffness of the beam.     

A unified exact analysis for the Poynting effects of cylindrical tubes made of Hill’s class of Hookean compressible elastic materials at finite strain

A direct, natural extension of Hooke’s law to finite strain was achieved by R. Hill in 1978, employing the notion of work-conjugate measures of stress and strain. With Seth-Hill (Doyle-Ericksen) class of finite strain measures, this extension actually defines a broad class of compressible hyperelastic materials at finite strain, each of which retains the simple linear structure of Hooke’s law as stress–strain relationship. Several known simple elasticity models at finite strain are included as its particular examples. With a novel idea of utilizing a suitable parametric variable, here we present a unified study of the free-end torsion problem (Poynting effects) of thin-walled cylindrical tubes made of the foregoing Hill’s class of Hookean type hyperelastic materials. We show that it is possible to derive a unified exact solution to the nonlinear coupling equations relating the torque (the shear stress) and the controlling deformation quantities including, in particular, the axial length change. Discussions and comparisons concerning various Hookean type elasticity models are made based on the exact solution obtained.     

Second-order torsion problem of a cylinder consisting of different isotropic homogeneous compressible elastic materials

Some second-order solutions of the torsion problem for simply-connected regions are available based on the theory given by Green and others both for compressible and incompressible materials. Bhargava and Gupta [1] have recently extended the theory for torsion problem of multiply-connected regions. In the present paper these theories are extended further to account for the composite regions. The complex variable formulation is employed. As an illustration, results for the torsion problem of a composite cylinder of concentric circular cross-section are given.     

A SERIES METHOD FOR CALCULATION OF THE TORSION DEFORMATION OF THE SHAFT WITH TWO FIXED ENDS1)

轴类扭转是工程中常见的一种受载方式,由于其受载形式的多样性和边界约束情况的复杂性,其变形计算和内力分析按截面法和解除约束的方式常常是很繁琐的。本文针对两端固定扭转轴的变形计算问题利用广义函数和周期性延拓方法提出一种级数解法,该法无需截面法和解除约束,步骤简单,计算方便, 具有一定的应用推广意义。为了说明方法的有效性,本文用算例对材料力学一般处理方法和本文方法做了比较和分析,结果说明了本文方法是可行的。     

Effective properties of a nonuniform beam under torsion

Effective characteristics are considered in the pure torsion problem for a nonuniform beam. The Saint-Venant semi-inverse method is used. A torsion stress function is introduced; this function can be found by solving a cross-sectional boundary value problem for a partial differential equation with variable coefficients. Two special boundary value problems are formulated for such an equation; after solving these problems, some effective characteristics are calculated in the case of torsion. It is shown that these effective characteristics satisfy the conditions of symmetry and positive definiteness. The case of an infinite in-plane layer of nonuniform thickness is discussed.     

Original ArticleKinematic hardening rules in finite plasticity Part II: Some examples

We consider finite plasticity laws which are represented by means of dual variables and satisfy the intrinsic dissipation inequality. Within two families of dual variables, two different formulations for a kinematic hardening rule, referred to as Model 1 and Model 2, are proposed. In order to discuss basic properties of the two models, the predicted response for some simple deformations is calculated. It turns out that e.g. with respect to the simple shear and simple torsion the two models differ mainly in the effects of second order. Received June 14, 1995     

Effect of loading-path on the evolution of yield surface for anisotropic metals subjected to large pre-strain

A theory of plasticity previously formulated by the author to discuss the free-end torsion problem has been extended to the more general case of combined axial-torsion of a thin-walled tube. This paper is devoted to the discussion of evolution of the yield surface due to different (proportional and non-proportional) loading paths with pre-strains in the large strain range. Experimental yield surfaces with axial and torsion pre-strains of magnitudes up to 40 and 45%, respectively, are presented, and theoretical results are compared with the experimental data.     

Shear measurement using strain gages under large deformation and rotation

The measurement of shear strain under finite deformation and large rotation by using electrical-resistance metallic foil strain gages is studied both analytically and experimentally. Equations for calculating shear strain and axial and circumferential stretches are derived based on the kinematics of general tension-torsion deformation mode. These equations are applied to analyzing pure torsion experimental data. Comparison is made between results obtained with strain gages and a rotary transducer. It is shown that, in case of large rotation, one simple equation can be used to calculate the shear strain up to 30 percent with adequate accuracy.     

A continuum damage model for the high-cycle fatigue life prediction of styrene-butadiene rubber under multiaxial loading

In the present contribution, the relationship between the fatigue life of styrene-butadiene rubber (SBR) and the stretch amplitude was established. Focusing on the multiaxial loading effect on the life duration of SBR, experimental tests were conducted using cylindrical specimens subjected to tension and torsion loadings under constant and variable amplitudes. Based upon the continuum damage mechanics approach, a three-dimensional model was derived and coupled with the cracking energy density criterion to predict the fatigue life of SBR. The capabilities of the model, which requires only three damage parameters to be identified, were analysed and a good agreement between predicted values and experimental data were clearly highlighted for tension and torsion loadings both in constant and variable amplitudes.