Deformation analysis of plastic strain rate hardening material using stress function under plane strain condition

The two-dimensional plane strain equation of plastic flow in accordance with the Levy-Mises constitutive relation is expressed in terms of stress functions of complex variables. Expressions for the stress, strain rate and velocity are derived for plastic flow in a non linear viscous medium assuming the stress function in the form of both the summation and product of conjugate stress functions. The plastic states are derived, also using a mixed mode solution expressed in terms of non-seperable, independent conjugate complex variables.The analysis of the block indentation associated with a nonlinear viscous (strain rate hardening) material under plane strain condition using the product form solution is performed. The effect of the variation in the strain-rate hardening exponent on the contact stress is investigated. The predicted behavior of the vertical component of the contact stress suggests the possibility of the development of a specially instrumented plane strain block indentation tests, for the rapid determination of the strain-rate sensitivity of the real material. The vertical contact stress and strain-rate obtained from the product of the complex conjugate stress function are compared with those obtained from the summation form of the complex conjugate stream function.     

Application of spherical indentation mechanics to reversible and irreversible contact between rough surfaces

A probabilistic model for the deformation mechanics of the interface between randomly rough metal surfaces, which is geometrically and mechanically more realistic than previous models, is derived and numerically evaluated. The model is based on the premise that the contact of two nominally plane engineering surfaces is in general equivalent to loading their sum against a smooth plane. The “sum surface” is assumed to be Gaussian and isotropic; thus the height and curvature of its peaks are correlated random variables. By using an idealized peak shape which is paraboloidal only at its vertex, the surface height distribution of the population of peaks is also made Gaussian. The upper load limit of the model is estimated, beyond which the microcontacts can no longer be assumed to be geometrically discrete and mechanically independent. Each microcontact is assumed to grow by mutual spherical indentation, thus enabling the entire deformation range, from Hertzian elastic to fully plastic, to be described by previously determined empirical functions. A two-stage linear/power law stress-strain curve is assumed. By summing over all microcontacts the total contact area and load are obtained as functions of the separation of the mean surfaces. For elastic deformation both normal loading and sliding friction are treated. For plastic flow two cases are considered: (1) identical materials; (2) one surface remains elastic. The mechanics of unloading are also investigated. After enough microcontacts have yielded to cause significant deviation from totally elastic behavior, the contact mechanics depend principally on the strain hardening exponent, the ratio of the yield stress to the Young's modulus, and the measurable ratio of the mean peak radius to the r.m.s. height of the sum surface.     

On frictional effects at inelastic contact between spherical bodies

Normal inelastic contact between spherical bodies is examined theoretically and numerically. The analysis is focused on viscoplastic material behaviour. In particular the effect of Coulomb friction is analysed in some detail, both regarding global and field variables. It is shown that the solution to the problem of contact between two deformable spherical bodies is provided by the solution of the fundamental problem of indentation of a viscoplastic half-space by a rigid sphere. The indentation analysis is based on self-similarity and cumulative superposition of intermediate flat die solutions as outlined in detail in a previous study by Storåkers et al. (International Journal of Solids and Structures 1997;34:3061–83). The results show that frictional effects, when global properties such as the contact area and the mean contact pressure are at issue, will only be of importance at close to perfectly plastic material behaviour. Even in such circumstances the difference between values given by the solutions for frictionless and for full adhesive contact is no more than approximately 10%. Accordingly, it can be concluded that frictional effects are essentially negligible, when, for example, material characterization of viscoplastic solids by Brinell indentation is of interest. The situation is, however, quite different when field variables are at issue. In this case, stress and strain fields can be substantially influenced by friction with possible implications for features such as crack initiation and crack growth.     

A quantitative model of the blow-forming of spherical surfaces in superplastic sheet metal

The blow-forming of a flat superplastic metal sheet to a spherical surface is analysed with a mathematical model based on the classical theory of liquid bubbles. The model enables a quantitative account to be given of the effect of the strain-rate sensitivity, flow stress and temperature of the metal on the rate of blow-forming and the replication of detail. Formulae are derived which enable the superplastic flow stress and strain-rate sensitivity to be calculated from simple blow-forming tests.     

A numerical method for calculating temperature distributions in metal working processes

A numerical (finite element) method for calculating temperatures in metal working which largely overcomes the limitations of previous methods is described. Its application is demonstrated using an experimentally obtained flow field (streamlines) for plane strain extrusion to determine the values of strain, strain-rate and velocity needed in the temperature calculations. Account is taken of the dependence of the work material's flow stress on temperature (and strain and strain-rate) and although, to reduce computing time, the thermal properties are assumed constant it should be possible to also allow these to be dependent on temperature.     

Non-isothermal plane plastic flow of a thin layer compressed by flat rigid dies

An analysis of the plane rigid—plastic flow of a thin layer compressed by flat rigid dies coupled with the heat transfer in the die-layer system is considered. The numerical model of the thin layer compression includes non-homogeneous plastic dissipation rate, heat flux of the contact friction and thermal resistance of the die-layer interface boundary. The rigid—viscoplastic relation of the effective yield stress with accumulated plastic strain, effective strain rate and temperature is assumed for the layer. The effects of the contact friction, thermal resistance and the die velocity on the hot compression of the thin steel workpiece are investigated.     

硬度影响压痕弹塑性行为的有限元分析

以具有不同硬度40Cr为研究对象,根据集中载荷下的接触模型和赫兹理论,计算了压痕接触半径和压痕附近弹性区域的表面局部接触应力,并采用有限元法,分析硬度物理量对压痕弹塑性行为、局部接触应力、卸载后保留在内部的残余应力的影响,探讨压痕参数、压痕接触应力、残余应力与硬度之间的关系以及载荷增加时它们的发展.结果表明,相同载荷下塑性隆起量、压痕接触半径、压痕量和塑性区范围随着硬度值的提高而减小,弹性回弹量、最大接触应力和残余应力随硬度提高而增加;压痕周围处接触应力和残余应力、其分布范围和塑性区域随载荷的增加而增加.     

小范围屈服条件下复合材料裂纹尖端塑性区分析

基于复合材料力学,推导Tsai-Hill强度准则在平面应力和平面应变条件下的一般表达式,得到了小范围屈服条件下,含中心裂纹无限大板Ⅰ型裂纹、Ⅱ型裂纹和Ⅰ/Ⅱ复合型裂纹尖端塑性区的解析解。针对不同裂纹倾角及泊松比 和,对裂尖塑性区进行了计算和分析。结果表明平面应变条件下塑性区范围小于平面应力条件下塑性区范围,参数、和 对复合材料裂尖塑性区范围和形状有明显的影响,不同的参数值得到的塑性区结果差别很大。另外,该解既适用于各向异性复合材料,也适用于各向同性材料。     

Elastic–plastic deformations of rotating variable thickness annular disks with free, pressurized and radially constrained boundary conditions

Analytical solutions for the elastic–plastic stress distribution in rotating variable thickness annular disks are obtained under plane stress assumption. The analysis is based on Tresca's yield criterion, its associated flow rule and linear strain hardening material behavior. The thickness of the disk is assumed to vary in parabolic form in radial direction which leads to hypergeometric differential equations for the solution. It is shown that, depending on the boundary conditions used, the plastic core may contain one, two or three different plastic regions governed by different mathematical forms of the yield criterion. The expansion of these plastic regions with increasing angular velocity is obtained together with the distributions of stress, displacement and plastic strain. It is also shown mathematically that in the limiting case the variable thickness disk solution reduces to the solution of rotating uniform thickness disk.     

Elastic-plastic stress distribution in a rotating hyperbolic disk with rigid inclusion

The elastic-plastic stress distribution of a rotating hyperbolic annular disk mounted on a circular rigid shaft is investigated in this work. The exact solution presented is based on the usual assumptions of plane stress, Tresca's yield condition, its associated flow rule and linear strain hardening material behaviour. According to the present analysis the plastic core consists of three parts with different forms of the yield condition. The theory presented is illustrated by numerical results. The results show that the plastic flow and maximum stresses are influenced by the thickness parameter.     

Surface asperity deformation under bulk plastic straining conditions

The paper is concerned theoretically and experimentally with the flattening of asperities by tools in metal-forming processes, a topic important to lubrication in metal forming. It specifically studies the plane strain compression process. It proposes kinematically admissible velocity fields for the crushing of asperities on a plastically flowing foundation and by adjusting independent variables of the fields predicts the actual flows by an energy minimization method. The theoretical predictions are compared with plane strain compression tests on aluminium samples with surfaces abraded in the direction of bulk plastic flow (the condition that the modelling describes). Both theory and experiment predict that asperities are crushed at the start of bulk plastic compression, to generate a real area of contact a certain fraction of the tool area depending on the contact friction and aspect ratio of the flow field, between 0.75 and 0.95 according to theory and between 0.6 and 0.8 according to experiment; but that with further bulk compression these contact fractions do not change. Despite these quantitative differences, it is felt that the method developed in this paper could usefully be extended to study asperity behaviour in other forming processes such as drawing or rolling.     

叶轮机械三元流动流函数方程组——S1和S2流面的统一数学方程组

随着计算机和计算技术的发展,直接求解叶轮机械三元流动成为新发展的一个问题。本文提出了直接求解叶轮机械三元流动的一个方法。在本文中引入了三元流动流函数。对吴仲华叶轮机械三元流动通用理论中的S1和S2相对流面理论作了统一的数学表达,进一步阐明了流面的意义,对流面理论中连续方程所引入的流片厚度τ作了一个严格的数学推导。所得的结果同流面理论中的结果,和其物理意义都是一致的。这样所得的流动图案是最一般的情况。而我们以往常常使用的任意回转面流动、圆柱面流动、轴对称流动等都是其中某些简化的特例。由所引入的三元流动流函数,从叶轮机械基本方程组,推导得出了包括S1和S2两族流面上流动的统一流函数方程组。由此可以明确看到S1和S2两族流面相互迭代计算的意义和途径。进一步讨论了方程组正问题求解的边界条件和求解方法,从而提出了直接求解三元流动的一种新的方法。在这方法是同时直接求解三元空间内计算节点上ψ1和ψ2两个流函数值。这两个ψ1和ψ2是通过两个联立的流函数方程组迭代求解而得的。解得两个流函数值后,从而再解出所有气动参数。这方法不同于其它直接解速度分量形式的三元流动方程。后者是求解三元空间内计算节点上三个或三个以上未知数。因此看来这个新方法有可能比其他直接解法更容易在计算机上求得完整的三元流动解。     

Analysis of noisy data from flow measurements

An improved technique is described for graduation and analysis of data from quasi-steady state isochoric plane flow, where the data represent a mesh defined by streamlines and isochrones, equispaced in the stream function and time variable, respectively. The graduation technique is a combination of adjustment of the areas of the mesh elements, and spline smoothing of the isochrones and streamlines, with the degree of smoothing of the latter controlled by tolerance matrices derived from the data and an adjustable scalar parameter. The analysis comprises computation of the velocity and rate of deformation from the spline representations of the smoothed streamlines, and the effective strain as a line integral of the maximum shear strain rate. The technique is applied to data sets representing plane plastic flow of metals in extrusion, machining, etc., and the effect of varying the smoothing parameter is investigated. With omission of the area adjustment, the technique is also largely applicable to data for compressible flow or flow with axial symmetry.     

The analytical solution of some boundary value problems in plane plastic strain

An analytical method of treating plane plastic flow problems, developed by the author in an earlier paper, is considered here for deriving mathematically exact solutions to several problems of practical interest. The basic theory is discussed in terms of the first boundary value problem, in which two intersecting sliplines are given, and applied to the singular fields defined on the convex sides of a circular arc and a logarithmic spiral. This is followed by the analysis of a boundary value problem involving a stress-free boundary associated with one of the singular fields. An exact solution is also presented for the slipline field corresponding to the plane strain compression of a block between a pair of perfectly rough parallel platens. The manner in which the basic formulation is modified by a reversal of the sign of the curvature is illustrated by considering the field defined by a logarithmic spiral on its concave side.     

Development of plastic zones and residual stress in elasto-plastic contact problems with stress singularities in elastic range

Stress singularities appear in such elastic contact problems as a plane indenter compressing a semi-infinite body, or the axisymmetric case of a turbine disc shrunk onto a shaft. The stress is infinite at the edge of the corner of the indenter and the disc as well as in neighbouring areas of the semi-infinite body and the shaft.In this paper the complex investigation of the distributions of plastic zones, deformations, pressure and residual stress has been presented for plane indenters of different shapes being in elasto-plastic contact with semi-infinite bodies. Different strain-hardening characteristics of the material, different yield criteria and states of stress/strain have been considered.The aim of the investigation is to find an answer to the question, what happens to the stress singularity, which appears in the elastic solution? The considered problem is of interest from the cognitive point of view and is of real importance in engineering practice.     

Fracture prediction of thin plates under hemi-spherical punch with calibration and experimental verification

The response of thin clamped plates subjected to static punch indentation is investigated experimentally, analytically and numerically to determine the onset of fracture. The accumulated equivalent plastic strain with stress triaxiality as a weighing function is introduced as ductile fracture criterion in the finite-element simulation and analytical prediction. The fracture criterion was calibrated by finite-element simulations of uniaxial tensile tests. Based on the calibration, and calculated distributions and histories of stress and strain, the critical location, and penetration to fracture were predicted within 5–10% accuracy for three punch radii.The plots of force–penetration and locations of fracture initiation in the static punch indentation tests were compared with finite-element simulations and analytical approximations showing good agreement. The transverse deflection profiles of the plates at the point of fracture obtained numerically were shown to agree well with the closed-form solution derived by taking into account a variable stress ratio and varying stress triaxiality. The strain distribution along the plate radius is influenced by the friction between the interfaces of punch and plate. By changing the friction coefficient, the fracture-forming limit diagram was constructed numerically. The present procedure can replace the time-consuming experimental technique in which the strain path is controlled by changing the radius of a cut off.     

Viscoplastic analysis of plane-strain rolling using different friction models

Plane-strain rolling is analyzed numerically. The nonlinearities arising from the large plastic deformations, the strain-rate dependence of the stress and the influence of the roll velocity are described by using a finite-strain elastic-viscoplastic material model. The stress-strain variation with temperature is included by approximating the isothermal behaviour at a given temperature and strain rate. The combined contact and frictional problem at the interface between the rolls and the plate is modelled by using elastic springs with high stiffnesses and by assuming two different friction models. The influence of different rolling and material parameters on the variation of the frictional stress and the normal stress along the contact length is shown for one of the friction models. Furthermore, the difference between the results obtained by the two different friction models is emphasized.     

Effect of surface etching on the lubricated sliding wear of an eutectic aluminium–silicon alloy

The paper presents a study of the formation of wear grooves on near-eutectic aluminium–silicon alloy flats, by sliding a steel ball. The formation of the grooves are tracked on etched and unetched flats as functions of normal load and sliding distance. The groove is initially formed by plastic flow, and then expanded by micro-abrasion as the ball continues to slide on the groove. Etching causes surface hardening of the alloy, but, more importantly creates a surface topology that reduces the peak contact pressure, which discourages further plastic flow in the subsurface. This effect is rationalised using an existing contact mechanical model of indentation of rough surfaces.     

Material characterization of Inconel 718 from free bulging test at high temperature

Macroscopic superplastic behavior of metallic or non metallic materials is usually represented by the strain-rate sensitivity, and it can be determined by tensile tests in uniaxial stress state and bulging tests in multi axial stress state, which is the actual hot forming process. And macroscopic behavior of Non-SPF grade materials could be described in a similar way as that of superplastic materials, including strain hardening, cavity and so on. In this study, the material characterization of non-SPF grade Inconel 718 has been carried out to determine the material parameters for flow stress throughout free bulging test under constant temperature. The measured height of bulged plate during the test was used for estimation of strain-rate sensitivity, strain-hardening index and cavity volume fraction with the help of numerical analysis. The bulged height obtained from the simulation showed good agreement with the experimental findings. The effects of strain-hardening and cavity volume fraction factor for flow stress were also compared.     

On a complete solution for frictionless non-steady state extrusion in plane strain

The slip-line field for frictionless non-steady state extrusion can be constructed by a suitable combination of the slip-line fields for plane strain indentation by smooth and rough dies. The stress state at the vertex of the rigid regions within the slip-line field solution is checked in accordance with Hill's criterion. A statically admissible stress field is constructed by the use of Bishop's method in which some lines of stress discontinuity are introduced. A series method for determining the line of stress discontinuity is proposed by combining Ewing's series method and the method of undetermined coefficients in a power series solution of the differential equation. Some examples of the complete solution are given.