Plastic deformation of titanium alloys under simple loading and its mathematical modeling

Results of an experimental investigation of the regularities of plastic deformation of titanium alloys in a plane stress state are analyzed. Tests were performed by loading thin-walled tubular specimens by an axial force and internal pressure under conditions of a proportional increase in the loads. The alloys are found to be transversely isotropic materials whose isotropic surface coincides with the cylindrical surface of the specimen. The process of plastic deformation of the alloys under simple loading is shown to be described well by equations of a previously proposed deformation theory of the plasticity of transversely isotropic media. Translated from Problemy Prochnosti, No. 5, pp. 27–35, September–October, 1999.     

Problems of the mathematical theory of plasticity

This paper gives the results of investigations aimed at the development and elaboration of a theory for the relationships governing plastic-deformation processes in the contemporary mathematical theory of plasticity. We discuss its two main classical directions: the theory of flow and the theory of processes. The first direction is based on the concept of existence of a limiting surface and the possibility of decomposition of strains into elastic and plastic components. On the contrary, the second direction, which is progressing rapidly, does not use the concept of existence of limiting surfaces and does not allow decomposition of strains into elastic and plastic components except for the cases of simple loading and simple unloading. It is thought that in complex loading and complex unloading the deformation is elastoplastic (partially plastic or partially elastic). We believe that these two directions in the theory of plasticity in the case of complex loading will eventually approach each other because they are intended for the investigation of the mechanisms of the same physicomechanical processes of plastic deformation of various media. This work shows one possibility of such approach. Tver' State Technical University, Tver', Russia. Translated from Problemy Prochnosti, No. 1, pp. 22–41, January–February, 2000.     

Prediction of the fracture of metals in processes of cold plastic deformation. Part 2. Effect of anisotropic hardening and experimental verification of the model of plastic deformation and fracture

On the basis of the experimental investigation of the Bauschinger effect, in steels with developed prestrains, we generalize the model of plastic deformation to the case of anisotropic hardening. Within the framework of the model, we show that for high strains, the Bauschinger effect is caused by dislocations. We present the results of the experimental verification of the developed semiphenomenological model of the joint process of plastic deformation and fracture of metals under the conditions of cold deformation. It is shown that this model gives adequate predictions (in good agreement with the experimental data) of the probability of fracture of the metal caused by exhaustion of the plasticity margin in the processes of plastic deformation realized under the conditions of both simple and complex loading. Ufa State Aviation Technical University, Ufa, Russia. Translated from Problemy, Prochnosti, No. 2, pp. 74–84, March–April, 1999.     

Investigation of the effect of surface treatment on the cyclic strength characteristics of nickel alloy by the acoustic emission method

Using the acoustic emission method, we study deformation processes in a plastic structural material with different technological inheritance properties under static and cyclic loading condition. The dependence of the total acoustic emission on applied stresses is shown to contain a special point characterizing the moment of change in the stress-strain state of the surface layer of the material. We derive relationships for the evaluation of fatigue limits of structural materials by acoustic emission inspection results. Zaporozh'e State Technical University, Zaporozh'e, Ukraine. Translated from Problemy Prochnosti, No. 3, pp. 139–145, May–June, 1999.     

Prediction of asymmetric cyclic loading of materials in the framework of the endochronic theory of plasticity

We analyze the efficiency of constitutive equations of the endochronic theory of plasticity for the prediction of strains in plastically incompressible materials under asymmetric cyclic loading. As an example, we analyze the mechanical behavior of 1020 steel subjected to cyclic deformation either with constant amplitude of total strain or with constant amplitude of applied stresses and conclude that the results of numerical calculations are in good agreement with experimental data. We also outline the prospects of subsequent development of constitutive equations for the case of asymmetric cyclic loading in a broad range of amplitudes and mean stresses. Translated from Problemy Prochnosti, No. 5, pp. 46–50, September–October, 1997.     

Analytical solution of boundary value problems of heat conduction in composite regions with arbitrary convection boundary conditions

Summary A mathematical model is developed describing the mechanical behavior of an elastic-plastic composite on the basis of phase properties and the micro-structural geometry. The overall yield condition and hardening rule are obtained in general form for two-phase composites with different phase properties. The constituents may be elastic-perfectly plastic or strain hardening ones. The case of a plastic state of both components is investigated in detail. The loading surface in the space of macro stresses has singular points. Plastic micro strains cannot be eliminated from the system of equations, and must be considered as additional hardening parameters. The hardening of composites with ideally plastic constituents is limited. The limiting surface exists in the space of macro stresses which defines the condition of perfect plasticity. However, it is not associated with the plastic flow law.     

Analytic correction of an inadmissible stressed state in the numerical analysis of elastoplastic deformation of rigid bodies

We propose closed analytic expressions reducing the stresses typical of a physically inadmissible state obtained as a result of the numerical solution of a quasistatic elastoplastic problem to the state corresponding to the yield surface. The Lagrange functional is modified to remove the excess stresses at the points where the condition of yield is violated, which enables one to maintain the equilibrium state of the elastoplastic body in each stage of loading. Technical University of Moldova, Kishinev, Moldova. State University of Moldova, Kishinev, Moldova. Translated from Problemy Prochnosti, No. 6, pp. 100–115, November – December, 1999.     

On the theory of plasticity with smooth surfaces of loading

The well-known theories of plastic yield with smooth surfaces of loading are developed and generalized on the basis of a model of a nonlinear anisotropically hardened medium based on the concept of slip. Unlike the commonly used procedure of application ofa priori known laws of hardening (variation of the surface of loading in the process of plastic deformation), we suggest a method for the experimental evaluation of a universal function of the material appearing in the constitutive equations for arbitrary complex loading processes including elastic unloading and plastic deformation in the direction opposite to the initial one. The constitutive equations are relatively simple and, hence, can easily be used for the statement and solution of boundary-value problems in the theory of plastic yield. Dnepropetrovsk State University, Dnepropetrovsk. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 33, No. 6, pp. 63–70, November–December, 1997.     

Damage of structural materials under complex low-cycle loading

On the basis of the concepts of the continuum mechanics of damage, we propose an engineering method for the analysis of the kinetics of accumulation of scattered defects in metallic structural materials under conditions of elastoplastic deformation and low-cycle fatigue. It is shown that, in the general case of complex loading for the complex stress state, it is reasonable to use the specific energy of additional stresses (with regard for the arc of plastic strains in a loading cycle) as a parameter of damage for two types of fracture (rupture and shear). __________ Translated from Problemy Prochnosti, No. 6, pp. 25–34, November–December, 2007.     

多步非稳态载荷下钢轨滚动接触应力和弹塑性变形分析

采用弹塑性有限元法,分析了多步非稳态载荷下钢轨滚动接触应力和变形。多步载荷指的是钢轨同时受到机车和车辆车轮的反复作用或多趟列车通过钢轨。通过在钢轨表面重复移动Hertz法向压力分布和切向力分布来模拟车轮的反复滚动作用。材料循环塑性本构模型采用考虑材料棘轮效应的Jiang-Sehitoglu模型。分析结果表明:在非稳态载荷作用下,钢轨接触表面产生不均匀塑性变形而形成波状表面;多步载荷对钢轨残余应力影响不大;随着机车车轮通过次数的增加,钢轨残余剪应变、表面材料位移、波深和残余累积等效塑性应变将增大,在机车车轮通过之后,随着车辆车轮通过次数的增加,前三个量将减小,而残余累积等效塑性应变继续增大,但其增大的速率变小。随着机车和车辆车轮反复滚过钢轨,钢轨残余剪应变、表面材料位移和波深变化速率即棘轮率呈衰减性。     

A simple model of elastoplasticity for nonproportional cyclic loading

On the basis of the analysis of experimental data, we formulate requirements to the constitutive relations of plasticity under the conditions of complex cyclic loading. We propose a version of constitutive relations obtained by a simple generalization of the Mazing model to the three-dimensional case and introduction of a function of cyclic hardening. We also suggest a procedure for the identification of this function. According to the results of numerical analysis, this model adequately describes the main effects of cyclic plasticity for austenitic stainless steels. Perm State Technical University, Perm, Russia. Translated from Problemy Prochnosti, No. 1, pp. 15 – 24, January – February, 1998.     

Non-coaxial version of Rowe’s stress-dilatancy relation

Non-coaxiality occurs when the directions of the principal plastic strain increments and the principal stresses deviate. Extensive experimental data have now conclusively shown that plastic flow in granular soils is non-coaxial particularly during loadings involving rotation of the principal stress directions. One way to integrate the effects of non-coaxiality is by modifying the expressions for energy dissipation and stress-dilatancy used in modeling plastic deformation of granular soils. In this regard, the paper’s main objective is to derive a non-coaxial version of Rowe’s stress-dilatancy relation, thereby making it more general and applicable to loadings involving principal stress rotation. The paper also applies Rowe’s non-coaxial stress-dilatancy equation in the determination of the effects of principal stress rotation in granular soils during simple shear loading conditions. Previous experimental data from simple shear tests on sand are used to validate the proposed non-coaxial version of Rowe’s stress-dilatancy relation.     

Mixed projection-mesh scheme of the finite-element method to solve boundary-value problems describing the non-isothermal processes of elastoplastic deformation

A mixed projection-mesh scheme for solving a boundary-value problem of thermal plasticity is formulated in a quasi-static statement when the process of non-isothermal elastoplastic deformation of a body is a sequence of equilibrium states. In this case, the stress-strain state depends on the loading history, and the process of inelastic deformation is to be observed over the whole time interval under study. The correctness and convergence of the mixed approximations for stresses, strains and displacements are investigated as applied to the solution of nonlinear boundary-value problems that describe the non-isothermal processes of active loading taking into account the initial strains dependent on the history of deformation and heating. The properties of the projecting operators are studied in detail, and on this basis, the condition that ensures the existence, uniqueness and stability of solution is formulated. The results of the analysis of special formulas of the interpolation-type numerical integration are presented, the use of which considerably simplifies the computation procedure for solving equations of the mixed method. The convergence and accuracy estimations are based on the results of the theory of the generalized boundary-value problems and methods of the functional analysis. According to the estimations obtained, the accuracy of solution of a finite-dimensional problem at the initial stages of loading should be sufficient to avoid the effect of increase of the first coefficients in the expansion of the total error on the accuracy of solution of the elastoplastic problem at the subsequent stages of loading. __________ Translated from Problemy Prochnosti, No. 3, pp. 87–117, May–June, 2007.     

Instability of thermal fracture under the conditions of constrained deformation

We study the processes of quasistatic deformation and fracture of brittle materials under the action of rapidly varying temperature fields. As a fracture criterion, we use the condition of attainment of the critical levels of stresses. The analyses of the stressed state and crack growth are performed under the assumptions that the corresponding elements of the stress field are equal to zero on the newly formed free surfaces and that the conditions of the fracture criterion are satisfied at the ends of the crack. It is shown that the process of crack propagation is unstable for the major part of modes of thermomechanical loading: as soon as the critical stresses are attained at a certain point of the body, the crack instantaneously propagates to a critical size corresponding to a new stable state. It is shown that the mechanical overloading of a specimen can substantially weaken the effect of instability of development of the fracture zone. Examples of fracture of elastic brittle bodies are presented. We also perform the numerical analyses of the processes of initiation and propagation of cracks with regard for the plasticity of the material near its heated surface. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 42, No. 6, pp. 55–60, November–December, 2006.     

A variational void coalescence model for ductile metals

We present a variational void coalescence model that includes all the essential ingredients of failure in ductile porous metals. The model is an extension of the variational void growth model by Weinberg et al. (Comput Mech 37:142–152, 2006). The extended model contains all the deformation phases in ductile porous materials, i.e. elastic deformation, plastic deformation including deviatoric and volumetric (void growth) plasticity followed by damage initiation and evolution due to void coalescence. Parametric studies have been performed to assess the model’s dependence on the different input parameters. The model is then validated against uniaxial loading experiments for different materials. We finally show the model’s ability to predict the damage mechanisms and fracture surface profile of a notched round bar under tension as observed in experiments.     

Enlargement of a circular hole in a disc of plastically compressible material

A semi-analytic solution for the elastic/plastic distribution stress and strain in a thin annular disc subject to pressure over its inner radius is presented. It is assumed that a pressure-dependent yield criterion and its associated flow rule are valid in the plastic zone. Thus, the material is plastically compressible, which is a distinguished feature of the solution. Also, in contrast to most studies on elastic/plastic deformation of thin plates and discs under plane stress conditions, the flow theory of plasticity is adopted in conjunction with a smooth yield surface. Numerical methods are only necessary to evaluate ordinary integrals and to solve simple transcendental equations. It is shown that the stress path is not proportional and, therefore, the application of deformation theories of plasticity widely used to calculate the distribution of stresses and strains in thin plates and discs is not justified.     

Construction of Constitutive Relationships of the Deformation Theory for Simple in Noll's Sense Hardening Materials with Elastoplastic Behavior

Based on the theory of simple hardening materials with elastoplastic behavior, the general constitutive relationships of the deformation theory of plasticity are mathematically strictly constructed for arbitrary continuous, piece-wise continuously differentiable deformation trajectories, any strains and symmetry types of the material properties. Two conditions under which this is possible are considered. The approaches to a strict specialization of general constitutive relationships of the deformation theory of plasticity have been developed by imposing restrictions on the material strains, deformation processes and properties. In this case, the restrictions on the properties of materials formalize the data obtained in the experimental investigations. A series of both new and known constitutive relationships have been constructed that are arranged into a hierarchy according to the level of complexity of the response to deformation. The area of applicability of the derived physical equations has been defined. Special attention has been given to the modeling of finite and infinitesimal strains of isotropic materials. __________ Translated from Problemy Prochnosti, No. 6, pp. 35 – 49, November – December, 2005.     

Measurement of fatigue-induced surface plasticity

A method is presented for measuring at a surface the localized plastic strains induced by fatigue within individual grains. The technique uses mica flakes distributed on a sample surface as reference gauges, relative to which strains in the surface can accurately be determined. An application of the method to the study of fatigue induced microplasticity in an Al 2219-T851 alloy is discussed. On an unfatigued specimen, subjected to applied stresses less than the yield strength, deformation is elastic over gauge lengths comparable with the grain size. After fully-reversed cyclic loading at a peak tensile stress of 75% of the yield strength for 20×10³ cycles, the larger grains in the alloy exhibit a residual tensile strain after a tensile loading cycle. Neighbouring smaller grains are driven into elastic compression to accommodate this tensile plastic deformation. Peak localized tensile plastic strains may exceed 0.5% at the surface. This technique will be useful in evaluating models of fatigue crack initiation and surface damage accumulation.     

Finite deformation plasticity and viscoplasticity laws exhibiting nonlinear hardening rules

  We consider two finite deformation plasticity models, which differ mainly in the evolution equation governing the response of kinematic hardening. Both models reduce to the same constitutive law in the case of small deformations. The aim of the paper is to discuss these models by calculating the predicted responses for some representative loading conditions. The numerical calculations needed are performed by using an efficient time-integration algorithm which has been developed with a view to implementation in the ABAQUS finite element code. Generally, there are some differences between the predicted responses and in particular between the second-order effects predicted by the two models. For some simple deformation processes, e.g. simple shear and simple torsion, the differences concerning second-order effects exhibit some kind of regularities, which are independent of material parameters. Also, even if boundary value problems are considered where global deformations are small, significant differences can exist between the predicted model responses according to the finite deformation and the limiting small deformation theory.     

Strain hardening of metastable austenitic steel under repeated simple loading at different temperatures

On the basis of experimental data obtained under repeated simple loading of 18-10 steel at temperatures of 77, 123 and 293 K, we discovered that the direction of maximum hardening in the space of stresses does not coincide with the direction of preloading. This effect becomes more pronounced if preloading and repeated loading are performed at different temperatures. It is shown that the maximum degree of hardening is attained after repeated loading preceded by uniaxial preloading in the direction of trajectories close to uniform biaxial tension. It is also noted that the yield surface of structurally unstable steel undergoes complicated transformations. Translated from Problemy Prochnosti, No. 6, pp. 59–65, November–December, 1997.