The formation of a one-dimensional residual stress field in the neighbourhood of a cylindrical defect in the continuity of an elastoplastic medium

The one-dimensional finite deformations of the elastoplastic material of a thin-walled tube accompanying the loading of its external cylindrical surface are investigated. It is observed that, under conditions of total unloading when the external pressure is removed, the onset of a repeated plastic flow is possible, which changes the form of the distribution of the residual stresses considerably. The effect of the adaptability of an elastoplastic body with a cylindrical surface to repeated loading is pointed out.     

Deformation of tape-wound cylindrical shells under internal pressure

The state of stress of a cylindrical shell would from glass tape at arbitrary angles to the generator is examined. The pressure dependence of the strains and stresses is obtained for uniform internal pressure, and the question of rationalization of the structure of the shell material is investigated. To describe the deformation characteristics associated with failure of the resin under load, a perfectly plastic model is introduced. The results of an experimental determination of the relative elongations are presented for the case of active loading and unloading.Mekhanika Polimerov, Vol. 3, No. 5, pp. 915–920, 1967     

Anisotropy in the flow processes of a thin plastic layer

The theory of the flow of a thin layer of plastic material over surfaces developed by Il’yushin is extended to the case of an anisotropic ideally plastic material and anisotropic flow on the surface. Particular attention is given to determining the contact pressure. Two methods are proposed for solving this problem: a variational method and reduction to a Cauchy problem. The effect of anisotropy is revealed using specific examples.     

Pressure distribution in molding GRP parts of complex shape

This paper presents the results of an experimental study of the pressure distribution in the mold in molding parts consisting of cylindrical and conical components. The effect of the weight of molding compound introduced into the mold and the loading conditions on the pressure distribution is investigated. It is shown that the hypothesis of hydrostatic pressure transmission to the walls of the mold is incorrect in relation to the manufacture of glass-reinforced plastic (GRP) parts of complex shape.Mekhanika Polimerov, Vol. 1, No. 4, pp. 130–135, 1965     

On the critical amplitudes of acceleration wave to shock wave transition in white noise random media

The problem of transition of acceleration waves into shock waves is addressed in the context of weakly random media. A class of random media is modeled by a vector white noise random process representing two material coefficients appearing in the Bernoulli equation governing the evolution of acceleration waves. The problem of shock formation, which involves a stochastic competition of dissipation and elastic nonlinearity, is treated using a diffusion formulation for the Markov process of the inverse amplitude. The first four moments of the critical inverse amplitude are derived explicitly as functions of the means and crosscorrelations of the underlying vector random process. It is found that the Stratonovich as well as the Itô interpretation of the stochastic Bernoulli equation lead to an increase of the average critical amplitude of the random medium problem over the critical amplitude of the deterministic homogeneous medium problem. Probability distribution of the critical inverse amplitude is found to be, in general, of Pearson's Type IV.     

A Method to Estimate the Inelastic Response in Notched Structures of Anisotropic Materials with Elastic ‐ Perfectly Plastic Behavior

The fatigue strength of engineering structures is often limited by notches which arise from constructive features or manufacturing defects. The constitutive behavior in notch regions is then characterized by small plastic zones which are contained in an elastic region. To avoid costly plastic calculations, approximate methods have been developed to estimate the inelastic stress‐strain response at the notch tip. One of the first and best known approximate models is that of Neuber which, over the last 40 years, has received considerable attention particularly in connection with fatigue life prediction. Numerous studies have been conducted to the verification and the generalization of the Neuber approach which respect to multiaxiality, cyclic loading and creep conditions. Recently an extension of the Neuber method to anisotropic materials has been proposed in [1] and applied to directionally solidified and single crystal Nickel based superalloys as they are used in high temperature material applications. In this short notice we modify the approach in [1] for the special case of an elastic ‐ perfectly plastic anisotropic material.     

Nonconservative scheme with the isentropic condition in rarefaction waves as applied to the compressible Euler equations

A previously developed second-order accurate quasi-monotone scheme is tested using the Riemann problem with high initial pressure and density ratios. For shock waves, the scheme is conservative, while, in rarefaction waves, the isentropic condition along the trajectory of a Lagrangian particle is used instead of conservativeness in energy. It is shown that the shock front position produced by the scheme has no considerable errors typical of a representative set of conservative quasi-monotone schemes of various orders of accuracy. The numerical accuracy is significantly improved in the case of moving grids with a contact discontinuity explicitly introduced in the form of a grid node. It is shown how the method can be extended to cover the multidimensional case and the presence of additional terms in the original equations.     

Deformation and stress distribution in a layered plastic

The stress distribution over the unidirectionally reinforced layers is investigated in relation to the layer thickness ratio and the direction of loading of a two-way reinforced plastic. An expression is obtained for the modulus of elasticity of the layered plastic in an arbitrary direction relative to the directions of reinforcement. The effect of the geometry of the structure of the layered material on its deformation properties is experimentally illustrated.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 563–570, May–June, 1972.     

Asymptotic Results for the Sum of Dependent Non-identically Distributed Random Variables

In this paper we extend some results about the probability that the sum of n dependent subexponential random variables exceeds a given threshold u. In particular, the case of non-identically distributed and not necessarily positive random variables is investigated. Furthermore we establish criteria how far the tail of the marginal distribution of an individual summand may deviate from the others so that it still influences the asymptotic behavior of the sum. Finally we explicitly construct a dependence structure for which, even for regularly varying marginal distributions, no asymptotic limit of the tail of the sum exists. Some explicit calculations for diagonal copulas and t-copulas are given. Dominik Kortschak was supported by the Austrian Science Fund Project P18392.     

Joined dissimilar thin elastic tubes containing steady perfect flow

Two tubes of circular cross-section and of the same radius initially, but composed of different elastic materials, are joined together to form a single straight tube with discontinuous properties. Stretching in the axial direction causes radial displacement that varies along the length of the tube. In a perfect inviscid incompressible flow through the tube of variable cross-section the internal pressure varies as described by Bernoulli's equation, and the variable pressure also causes variation of the radial displacement. The equations of coupled finite deformation and fluid pressure problem can be integrated explicitly (using membrane theory for the tube) for arbitrary material properties, but determination of the integration constants is not trivial. The results are interpreted numerically for Mooney materials. Also considered in this context is the similar problem where two semi-infinite cylindrical membranes of the same material are separated by a cuff of different material. Numerical illustrations are obtained for various upstream velocities. The results obtained here thus solve the problem of steady internal pressure loading of this type in extended dissimilar thin isotropic tubes. The tube will become unstable if the fluid velocity is too large. Applications to engineering structures are possible.     

Plane-stress creep of a glass-reinforced plastic

The creep behavior of an orthotropic glass-reinforced plastic based on Polimal'-109 resin and 19-S4 glass fabric is described. A special device for testing tubular specimens in plane-stress creep has been designed. The loading conditions include combined torsion and tension, internal pressure, and loading along the principal directions of anisotropy. Constitutive equations describing creep and recovery are proposed for the two-dimensional case.Warsaw. Translated from Mekhanika Polimerov, No. 3, pp. 398–410, May–June, 1971.     

Some problems of the theory of plasticity for metals with strength differential

Within the framework of the problem of describing the plastic deformation of polycrystalline metals with strength differential, we consider problems of selection of universal constants and functions of the material that control the influence of the first and third invariants of the stress tensor. The possibility of using different types of experiments for the determination of these constants and verification of the theory is analyzed. It is shown that the traditionally used combined tension–torsion experiments on a thin-walled tube do not enable one to distinguish the influence of hydrostatic pressure and the Lode angle. The expedience of using biaxial tension and other experiments on simple and complex loading of a thin-walled tube by an axial force and internal pressure is justified.     

The effect of microdamage on fatigue-crack growth

The theory of fatigue-crack growth, based on a synthesis of fracture mechanics and continuum mechanics of microdamage accumulation, is applied to the problem of crack growth under cyclic loading, taking into account the plastic deformations in the tip zones. The model of a thin plastic zone, which is a region of considerable plastic deformations at the crack tip, is supplemented by taking into account the effect of microdamage on the value of the specific fracture work and the limit stresses in the tip zone. Governing equations which describe fatigue-crack growth taking these factors into account are derived. The effect of the material characteristics and the load parameters on the growth rate and the distribution of microdamage in the tip zone and on its extensions is investigated by a computational experiment. Particular attention is given to the initial stage when crack growth may occur abruptly and the growth rate depends substantially on the initial conditions     

Optimal design of structures with unspecified loading distribution

The problem of the optimal distribution of loading on a structure that corresponds to the minimum of the elastic compliance or the maximum of the safety factor for plastic collapse is considered. Optimality criteria are derived, and their applicability is illustrated in the case of beams. Besides the optimally varying cross section, also the support positions and the load distribution are determined from the optimal solution.     

The limit equilibrium of an anisotropic medium under the general plasticity condition

A theory of the limit equilibrium of an anisotropic medium under the general plasticity condition in the plane strain state is developed. The proposed yield criterion (the limit equilibrium condition) is obtained by combining the von Mises–Hill yield criterion of an ideally plastic anisotropic material and Prandtl's limit equilibrium condition for a medium under the general plasticity law. It is shown that the problem is statically determinate, i.e., if the boundary conditions are specified in stresses, the stress state in plastic region can only be obtained using equilibrium equations. It is established that the equations describing the stress state are hyperbolic and have two families of characteristic curves that intersect at variable angles. In deriving the equations describing the velocity field, the material is assumed to be rigid plastic, and the associated law of flow is applied. It is shown that the equations for the velocities are also hyperbolic, and their characteristic curves are identical with those of the equations for stresses. However, the directions of the principal values of the stress and strain rate tensors are different due to the anisotropy of the material. The characteristic directions differ from the isotropic case in that the normal and tangential components of the stress tensor do not satisfy the limit conditions. It is established that the equations obtained allow of partial solutions, and in this case, at least one family of characteristic curves consists of straight lines. The conditions along the lines of discontinuity of the velocity are investigated, and it is shown that, as in the isotropic case, these are characteristic curves of the system of governing equations. In the anisotropic formulation, the well-known Rankine problem of the limit state of a ponderable layer is solved. From an analysis of the velocity field it is shown that plastic flow of the entire layer is possible only for a slope angle equal to the angle of internal friction. For slope angles less than the angle of internal friction, the solutions obtained are solutions of problems of the pressure of the medium on the retaining walls. The change in this pressure as a function of the parameters of anisotropy is investigated, and turns out to be significant.     

The problem of the stability of quasilinear flows with respect to perturbations of the hardening function

A formulation of the linearized boundary-value problem of the stability of a deformation process with respect to small perturbations of the hardening function (of the scalar constitutive relation of the material) is presented. The characteristic vector relations of the medium are assumed to be linear. The occurrence of rigid zones in the domain of the solid and the change in their boundaries in the perturbed motion are taken into account. A perfect rigid plastic deformation and the flow of a Newtonian fluid are considered explicitly as the basic flow. In the latter case, the equation of the asymptotic boundary of the rigid zone, which appears when there is a small variation in the yield stress and a transition to a viscoplastic material, is derived.     

率相关晶体塑性模型的塑性各向异性分析

在Sarma和Zacharia的工作基础上,改进了单晶晶体弹粘塑性本构模型的积分算法,并采用改进的欧拉法结合迭代方法求解,特点是稳定性好计算效率较高．然后用上述模型及算法研究了:1）在单向拉伸和平面应变压缩变形下单晶塑性各向异性的特点;2）晶体模型中的主要材料参数（应变率敏感指数m和潜硬化比率q）和加载应变率对单晶塑性各向异性的影响;3）沿不同的晶体方向加载对滑移系启动的影响．     

Analytical solutions in elasto-plastic bending of beams with rectangular cross section

Deflection analysis of beams with rectangular cross section is considered under specific loading conditions, resulting in at most quadratic bending moment distribution, and assuming elasto-plastic behaviour with no hardening. Within the framework of small strain and small displacement approach analytical solutions are derived, which enable elasto-plastic analyses of beams to be performed in a closed analytical form. In consequence, clear tracing of the elasto-plastic response evolution with a propagation of the plastic zone through the volume, i.e. its spreading along the beam’s longitudinal axis as well as its penetration through the cross section, is enabled as loads increase, from the appearance of a first plastic yielding in a structure till its collapse. With the derivation of the general solution, listed explicitly by Eqs. (21), (22) and (23), which was never presented in any article or book before, the presented article fills the gap in the analytical non-linear mechanics of beams.     

The propagation of shock waves in anisotropic materials

The propagation of shock waves in anisotropic materials (aluminium alloys), the state of which obeys a non-linear relation, which generalizes Grüneisen's equation for isotropic materials, is investigated. The concept of total generalized pressure and the pressure corresponding to the thermodynamic response is proposed. A modification of the anisotropic Hill criterion in the case of non-associated plastic flow, for which the yield surface is independent of the generalized hydrostatic stress, is considered.     

Modeling the nonlinear deformation of composite laminates based on plasticity theory

The plasticity theory has been successfully used for describing the nonlinear deformation of laminated composite materials under a monotonically increasing loading. Generally, several tests are needed to determine the parameters of the plastic potential for a laminate. We explore an alternative approach and obtain the plastic potential by using theoretical considerations based on a laminate analysis. The model is shown to provide an accurate prediction for the response of a cross-ply glass/epoxy laminate under uniaxial tensile loading at different angles to the material orthotropy axes. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 3, pp. 309–318, May–June, 2007.