Explicit expressions for the plastic normality-flow rule associated to the Tresca yield criterion

The Tresca yield criterion is classical and important to the theory of plasticity. It is usually formulated in terms of the difference between the maximum and minimum principal stresses. Difficulties have been encountered in attempts to explicitly express the plastic normality-flow rule associated to it, because the difference between the maximum and minimum principal stresses is generally not differentiable but subdifferentiable with respect to the stress tensor. In this work, the corresponding subdifferential is determined and specified for all possible cases; the explicit mathematical expressions are obtained for the plastic normality-flow rule relative to the Tresca yield criterion.Received: November 30, 2004     

Eine Kuppel gleicher Festigkeit

Summary Thin shells of optimum strength are usually defined by the condition that the two principal stresses in any point are equal to the yield stress. This definition is too restrictive; it is sufficient that principal stresses satisfy the yield condition ofv. Mises or the one ofTresca.In this paper a proof is supplied for the existence of spherical cupolas of equal strength on the basis ofv. Mises' yield condition, and an exact solution is presented for the distribution of thickness.     

Calculation of the long-term strength of thin-walled polymeric tubes under biaxial loading

The problem on calculating the failure time of thin-walled polymer tubes in creep under internal pressure, internal pressure with tension, pure torsion, and torsion with tension is solved. The solution is constructed based on the concept of equivalent stresses. A mixed long-term failure criterion taking into account the sings of principal stresses is used as the equivalent stress. The calculation results are in a satisfactory agreement with experimental data.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 811–826, November–December, 2004.     

Determination of the temperature and mechanical fields in anisotropic elliptic plates

We propose an approach to solving the problem of determining the temperature fields and stresses in orthotropic elliptic plates whose principal axes of elasticity and thermal conduction do not coincide with the axes of the ellipse. Translated fromMatematichni Metodi i Fiziko-Mekhanichni Polya, Vol. 40, No. 1, pp. 19–23.     

Numerical Calculation of Fiber Orientation in Three-Dimensional Arterial Walls

In this contribution an approach for the fiber reorientation in three-dimensional arterial walls is presented. In detail the load-bearing capacity of the tissue is increased by re orienting the fibers with respect to the principal stresses, cf. [1]. The improved fiber reorientation algorithm is combined with the polyconvex nonlinear anisotropic material model presented in [3]. The results of a three-dimensional finite element simulation, where the reorientation approach is applied to a short segment of a patient-specific arterial geometry, are presented. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Creep equations for orthotropic material based on the theory of local deformations

By introducing a time factor into the local deformation function, the theory of local deformations can be used to unite the theories of plasticity and creep. The local deformation function is taken as a rheological dependence in the form of an integral equation, and it is assumed to vary in a way that depends on the direction relative to the principal isotropic axes. Thus, equations are obtained for an orthotropic material with nonlinear creep. The relations obtained also allow for the variation of Poisson's ratio with time. It is shown that special cases of these expressions were previously used to describe the creep of glass-reinforced plastics at low stresses.Mekhanika polimerov, Vol. 1, No. 1, pp. 44–49, 1965     

Wrinkle-free solutions in the theory of annular elastic membranes

Rotationally symmetric deformations of a flat annular elastic membrane under a gravitational force are studied, with prescribed radial stresses or horizontal displacements at the edges. The small-finitedeflection theory of Föppl-Hencky as well as a simplified version of Reissner's static first approximation theory of thin shells of revolution are applied which lead to consider a single, second-order, ordinary differential equation for the derivation of the principal stresses in the membrane. Using analytical methods, the range of those boundary data is determined for which the solutions of the differential equation are wrinkle free in the sense that both the radial and the circumferential stress components are nonnegative everywhere.     

Yield surfaces for non-homogeneous anisotropic shells of revolution

The state of stress in a rotationally symmetric shell is characterized by the direct stresses and moments in the circumferential and longitudinal directions. It is assumed that the material of the shell is rigid perfectly plastic and that the yield stress of the material varies over the thickness of the shell. The material has different yield stresses in tension and compression and the yield stresses in the principal directions have different values. The yield condition for the shell is obtained in terms of the stress resultants assuming that the material of the shell obeys the maximum shear stress criterion.     

Damage model for ferroelectric materials and its applications to multilayer actuators

In this paper a damage model for ferroelectric materials is presented. It is implemented in terms of a user element in the commercial FEM-code Abaqus. The model is based on micromechanical considerations of domain switching and its interaction with microcrack growth and coalescence. Finite element analysis of a multilayer actuator is performed, showing principal stresses leading to crack initiation and damage of the actuator. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

关于弹性力学平面问题中的主轴应力坐标

本文中讨论了弹性力学平面问题中由主轴应力曲线构成的正交曲线坐标系上的平衡方程,以及解的特性.同时,认为在弹性力学中还存在另一种构造解的方式,即可通过直接构造主轴应力正交网络获得主轴应力的解.     

Bending of a thin cantilevered orthotropic plate with a concentrated load at the free edge

The problem of a thin rectangular cantilevered plate of constant thickness with a concentrated load in the center of the free edge is considered. The plate is assumed to be orthotropic, the fixed edge coinciding with the principal direction of elasticity. An equation is obtained for the normal stresses at an arbitrary point on the plate. The theoretical results are compared with experiment.A. A. Zhdanov Gor'kii Polytechnic Institute. Translated from Mekhanika Polimerov, Vol. 4, No. 4, pp. 739–741, July–August, 1968.     

Effect of layer geometry and stiffness on the fields of normal stresses in multilayer beams under asymmetric bending

A mathematical model is suggested for calculating the bending stiffness and fields of normal stresses (strength) at any point in the cross section of a multilayer beam. It is found that the structure of the scalar field of normal stresses allows one to solve some optimization problems with multivariant parameters. The method is illustrated with an example of two-layer beams. The results of an investigation into the strength and stiffness of two-layer beams, with a geometric and (or) stiffness asymmetry, in asymmetric bending are presented. The kinetics of bending stiffness and strength in relation to variations in the geometric parameters of cross sections and in the ratio of elastic moduli of layers is examined. It is established that the normal stresses in multilayer beams under asymmetric bending considerably depend on the location of the flexural center, neutral plane, and bending stiffnesses relative to the principal axes of cross sections of the beams.     

Geometrically nonlinear dilation of an orthotropic plane with an elliptical hole

In a geometrically nonlinear formulation we obtain expressions for the stresses and displacements on the edge of an elliptical hole in an orthotropic plane dilated by a strain at infinity in the direction of the principal axes of anisotropy. We exhibit the dependence of the concentration of stresses on the magnitude of the load acting on the plane. We prove that the stresses are finite at the ends of the crack and we compute their values. We confirm the need to take account of the geometric nonlinearity of the problem for low-modulus orthotropic materials.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 90–96.     

界面弧形裂纹对混凝土开裂强度的影响研究

混凝土由于水分蒸发、干缩、泌水以及骨料与砂浆变形不一致等原因会导致骨料与砂浆的界面层中产生弧形裂纹,从而对混凝土开裂强度产生很大影响.从细观角度将混凝土视作由粗骨料和水泥砂浆组成的两相复合材料,并将界面层视为粗骨料与水泥砂浆的接触层进行分析.首先基于相互作用直推估计(interaction direct derivative, IDD)法,考虑混凝土中骨料颗粒的相互作用,将施加在混凝土表征体积元的远场外荷载等效为无限大基体中含单一骨料的等效外荷载.然后,将等效外荷载转化为最大和最小主应力,基于断裂力学理论得到界面层中弧形裂纹的应力强度因子,并根据复合型裂纹幂准则判断弧形裂纹是否发生开裂,进而来研究混凝土开裂强度的变化规律.通过与数值模拟结果的比较,验证了界面弧形裂纹应力强度因子解析解的有效性,参数分析结果表明,当裂纹与最大主应力垂直或与最小主应力呈45°夹角时,骨料周围弧形裂纹最易发生开裂破坏.随着裂纹长度增加,混凝土受拉和受压开裂强度先减小后增大,且均存在最不利的裂纹长度.混凝土开裂强度随着骨料体积分数的增加而增大,随着骨料粒径的增大而减小.在裂纹长度较小时,增大骨料的弹性模量有利于提高混凝土开裂强度.骨料周围承受同号应力可以提高混凝土的开裂强度,反之,异号应力会降低开裂强度.     

Construction of the strength surface of structurally inhomogeneous materials in complex states of stress

The article describes a method of constructing strength surfaces based on the approximation of experimental data. The surface is a nonrotational convex shape, symmetrical about the diagonal of the principal stresses. The suggested method is suitable for a broad class of materials with compressive and tensile strength differing from each other, and in some special cases it coincides with the classical conditions of strength. In the example of concrete the article shows that theoretical results are comparable with the experiments.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 20–24, 1988.     

Rheological properties of polymers in the fluid state

The possibilities and conditions of correlation are determined for the principal rheological characteristics of single-phase polymer systems measured for one-dimensional shear deformation in steady-state flow regimes, on transition from rest to steady-state flow, and in harmonic vibration regimes. Special significance attaches to the quantitative results of measuring the high-elastic properties of the polymer systems. It is shown that the Lodge theory, describing the flow behavior of high-elastic media, is well-founded in the linear region of deformation, i.e., for the limiting case of shear rates and shear stresses tending to zero, whereas the Mooney-Rivlin-Weissenberg theories are not in accord with the experimental data even in this limiting deformation regime.Topcheiv Institute of Petrochemical Synthesis, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, Vol. 5, No. 1, pp. 164–181, January–February, 1969.     

Investigation of internal stresses in glass-reinforced plastics

The strength properties of and internal stresses in epoxy and epoxyphenol resins and GRPs based on them are investigated using an optical method of determining internal stresses. The GRPs had tape and fabric reinforcement. Compared with the internal stresses in unplasticized specimens, the stresses in pure resin films and in GRPs based on plasticized resins are found to be smaller. It is shown that the distribution of internal stresses in GRPs is anisotropic. The highest internal stresses are observed in tape-reinforced GRPs in a direction normal to the fibers. Glass reinforcement in two directions at right angles reduces the internal stresses in GRPs as compared with pure resin films. In both reinforced and unreinforced films, the internal stresses depend on the curing conditions.Mekhanika polimerov, Vol. 1, No. 1, pp. 82–88, 1965     

Long-time strength of polyethylene in biaxial tension

The long-time strength of high-density polyethylene (HDPE) in biaxial tension has been experimentally investigated at ratios of the principal stress components =₁/₂=0, 0.5, 1, 4. The maximum duration of the experiments was 2500 h. The limit surface for HDPE has been constructed. It is shown that the limit surface for this material varies with time as the mode of fracture changes from quasibrittle at medium stresses to brittle at low stresses. In the case of quasibrittle fracture the condition of equivalence of simple and plane states of stress is satisfactorily described by the Malmeister and Gol'denblat-Kopnov criteria, and in the case of brittle fracture by the maximum normal stress criterion.All-Union Scientific-Research Institute of Hydromechanization, Sanitary Engineering, and Special Construction, Leningrad. Translated from Mekhanika Polimerov, No. 3, pp. 401–408, May–June, 1976.     

髋关节发育不良髋臼假体位置与稳定的三维有限元分析

目的利用三维有限元仿真的方法，研究全髋关节置换术治疗成人髋关节发育不良时3 种不同髋臼假体位置与髋臼 假体初始稳定的相关性，为临床手术操作提供参考依据。方法采用专门的生物力学有限元网格划分器，从髋关节发育不良的CT 扫描数据中，建立高度仿真的个性化髋骨三维有限元模型，并模拟在真臼位置安放髋臼假体的解剖位重建、将髋臼内壁打磨穿透内移的中心化重建、和在真性髋臼上方假臼高位重建3种常见的临床手术方案，模拟不同臼杯假体位置对全髋关节置换术后髋臼假体稳定的影响。结果髋臼假体在真臼位置安放的解剖重建，出现应力集中和大剪切应力的可能性最小，穿透内移安放的模拟应力分布略高于真臼重建，而在假体上移高位安放的模拟则出现预测应力大幅增加。结论成人髋关节发育不良全髋关节置换术中应尽可能在真臼位置重建和安放臼杯假体，以达到最优的应力分布和稳定性。