Regularities of the change in the coercive force under biaxial asymmetric deformation of steel 3

The change in the coercive force under biaxial asymmetric (tension and compression in mutually perpendicular directions) cyclic deformation of cross-shaped steel 3 specimens in the elastic region of deformations was studied. Specimens were deformed beforehand under biaxial asymmetric loading to various degrees of plastic deformation. It was demonstrated that the elastic-deformation dependences of the coercive force measured along the tension and compression directions are qualitatively similar to those under uniaxial tension or compression. It was also shown that, under cyclic elastic loading, these dependences are reversible for well-annealed steel and have a hysteresis that expands with increasing degree of plastic deformation for plastically deformed steel. The possible causes of the hysteresis in the dependence of the coercive force on the elastic cyclic deformations under biaxial loading are discussed. It was supposed that the hysteresis of the coercive force was caused by the appearance of free (not bound in carbide phases) carbon atoms playing the role of interstitial impurity atoms for the α-iron lattice in plastically deformed carbon steels. The possibility of estimating the stressed-strained state of steel under biaxial loading using a magnetic method was discussed.     

Post-collapse characteristics of ductile circular honeycombs under in-plane compression

The post-collapse behaviour of a circular honeycomb material under in-plane compression is analysed in order to estimate the effect of the structural topology on the material strength. A structural approach using the limit analysis and the concept of an equivalent structure is employed to describe the large plastic deformations during post-collapse process. Based on some published experimental results (International Journal of Solids Structures 36 (1999) 4367–996) and our numerical simulations, certain deformation patterns are constructed depending on the direction of loading, and the corresponding post-collapse load-carrying capacities during large deformation until densification of cells are presented.The present analysis shows that the post-collapse stress associated with an equi-biaxial compression is not excessively larger than the corresponding uniaxial stresses, in contrast to those of hexagonal honeycombs in response to biaxial loading. This behaviour is attributed to the different deformation mechanism as the curvature of a cell wall invokes bending without stretching.The influence of the size of the connecting segment between two neighbouring cells is studied, showing that the “shape” of the limit surface varies significantly depending on this connection.     

Fatigue resistance of SMA-martensite bars subjected to flexural bending

The fatigue resistance of Ni–Ti shape memory alloy (SMA)-martensite bars in bending subjected to large deformation cycles has been experimentally evaluated. Firstly, fatigue tests under constant displacement amplitude have been carried out, at two different frequencies of loading. Then, the cumulative fatigue damage and fatigue life prediction of specimens loaded under variable load conditions have been investigated. Finally, the effect of low-cycle-fatigue (LCF) and plastic deformations on subsequent high-cycle-fatigue (HCF) has been studied.The experimental results point out that the frequency of loading (i.e. temperature) significantly affects the fatigue life of the specimens. The damage accumulation process seems to follow the “Miner” linear damage theory for low-to-high (L–H) loading sequences, while the same does not hold for high-to-low (H–L) loading sequences. Surprisingly, a small fraction of LCF life consumption seems to enhance the subsequent HCF limits.     

Collision between a ring and a beam

With car–parapet collision accidents in mind, a normal collision between a free-flying half ring and a simply supported beam with/without axial constraints is studied, in which an elastic–plastic half ring with an attached mass and the elastic–plastic beam are taken as the simplest models of a car and a parapet, respectively. Particular attention is paid to the energy partitioning between the two structures and the evolution of the contact regions during collision. A mass–spring finite difference (MS–FD) model is employed whilst the large deflection and axial stretching/compression are incorporated. The numerical results show that the less stiff (i.e. softer) structure will dissipate more energy and the contact regions will move away from the initial contact points. With the increase of the relative thickness of the beam to the ring, the final deformation of the half ring will transform from a “U” shape to a “W” shape.     

Method of movable lattice particles

A new simulation technique for modeling elastoplastic deformation and friction processes based on the dynamics of a system of “lattice particles” is proposed. In usual simulation methods like molecular dynamics, only interactions compatible to the symmetries of space (invariant with respect to translations and rotations) are used. In the proposed method, the interaction potentials depend both on the relative position of particles and the orientation of their relative radius vector with respect to prescribed “lattice directions”. We show that in spite of this relation with the “external space”, the system behaves, in linear approximation, as an isotropic elastic medium invariant to both the translations and rotations of the medium as a whole. The coupling with the external space occurs to be a surface effect, which either does not play an important role (if the motions of the boundaries are prescribed) or can be handled properly by introducing fictive compensating surface forces. Introduction of forces depending on the orientation of the local surroundings of a particle makes it possible to describe elastic media with arbitrary elastic properties by using only interactions between the next neighbours. The system of lattice particles shows better stability properties and allows one to describe large plastic deformations, avoiding problems of “packaging” typical for many particle methods.     

Reference stresses and temperatures for cylinders and spheres under internal pressure with a steady heat flow in the radial direction

The paper examines the creep behavior of thick cylinders and spheres subjected to internal pressure and a negative temperature gradient in the radial direction. It is found that at stationary state the rate of radial displacement of the vessel wall is simply proportional to the material creep behavior associated with a single stress and temperature. Such “reference stresses” and “reference temperatures” are defined for spheres and cylinders of varying wall thicknesses. These reference stresses and reference temperatures are valid for any creep problem where the material behavior may be characterized by a function of the form exp (γT)σ^m. The extension of these results to variable pressure and temperature loading cases is discussed.     

Buckling phenomena related to rolling and levelling of sheet metal

The paper deals with analytical and numerical considerations of buckling phenomena in thin plates or strips under in-plane loads which typically appear during rolling and levelling, i.e. straightening by stretching, of sheet metal. Buckling due to self-equilibrating residual stresses, caused by the rolling process, in eventual conjunction with global tensile stresses (denoted as “rolling buckling”) as well as buckling during the levelling process (denoted as “stretching buckling” or “towel buckling”) are considered. Analytical estimates are derived and compared against results of numerical simulations and field observations. Mode jumping by varying the global strip tension is explained on the basis of the derived analytical solutions. It is shown how from the waves, i.e. height and length, observed on the strip sliding over or lying on a rigid plane one can provide information about the distribution of the differences in the plastic strains over the width of the strip which lead to the buckled configuration. And, vice versa, knowledge of the plastic strain distribution can be used for estimating the expected wave heights representing a measure for the geometrical quality of the rolled product. The influence of the dead weight of the strip on the post-buckling pattern is also discussed on the basis of non-linear analyses.     

Damage evolution in creep bulging of thin sheet metal

The creeping motion of thin sheet metal, damaged by artificial cavities is observed in bulging tests and simulated ‘semi’-analytically. The sheet metal satisfies Norton’s Law for secondary creep and is subjected to a bi-directional stretch. The stretch is produced by creep bulging through elliptical dies with the virtue of sustaining nearly uniform background stress ratio for each aspect ratio of the die axes. In order to reach large deformations with significant shape evolution of the cavities, the tests were conducted at superplastic conditions. The sheet is double layered (only one layer is cavitated) made of Tin–Lead (50–50 Pb–Sn). The measured damage growth is compared to an approximate simulation. The simulation of the damage evolution, throughout its time history, makes repeated use of a so-called “Green-function solution” for the motion of a single isolated cavity in an infinite viscoplastic continuum. The solution is modified from Muskhelishvili’s elastic solution by replacing the elastic shear modulus by a “viscous-like” variable (“plastic shear modulus”) which depends (non-linearly) on the evolved average strain-rate. Similarly, the stresses in the ligaments between cavities were averaged to approximate the local stress concentrations. Due emphasis is given to the rotation of each elliptical cavity, beside its expansion (contraction) and elongation.     

Strain-rate and inertia effects in the collapse of two types of energy-absorbing structure

The dynamic plastic collapse of energy-absorbing structures is more difficult to understand than the corresponding quasi-static collapse, on account of two effects which may be described as the “strain-rate factor” and the “inertia factor” respectively. The first of these is a material property whereby the yield stress is raised, while the second can affect the collapse mode, etc. It has recently been discovered [6,7]that structures whose load-deflection curve falls sharply after an initial “peak” are much more “velocity sensitive” than structures whose load-deflection curve is “flat-topped” (Fig. 1a); that is, when a given amount of energy is delivered by a moving mass, the final deflection depends more strongly on the impact velocity. In this paper we investigate strain-rate and inertia effects in these two types of structure by means of some simple experiments performed in a “drop hammer” testing machine, together with some simple analysis which enables us to give a satisfactory account of the experimental observations. The work is motivated partly by difficulties which occur in small-scale model testing of energy-absorbing structures, on account of the fact that the “strain-rate” and “inertia” factors not only scale differently in general, but also affect the two distinct types of structure differently.     

Inelastic buckling of columns : The effect of imperfections

In the design of columns of mild steel (idealized as an elastic-perfectly plastic material) it is usual to take account of the effect of possible initial crookedness by means of a “Perry” formula. In contrast, the design of columns of aluminium alloys (and other materials which cannot reasonably be idealized as perfectly plastic) is usually made by means of the “tangent modulus” formula, which is strictly relevant only to initially perfect columns. The paper proposes a way of supplementing this formula for initially imperfect columns, and a simple graphical procedure is devised to generate a family of “column curves” for different degrees of imperfection.It turns out that although the “column curve” based on the tangent-modulus formula is sensitive to the precise shape of the rising stress-strain curve, the curves for the imperfect columns are insensitive to this shape, except for stocky columns. This suggests, paradoxically, a possible design approach using a Perry formula for columns made of aluminium alloys.     

Application of magnetic and electromagnetic-acoustic methods for assessing plastic deformations under cyclic loading of annealed intermediate-carbon steel

The influence of cyclic loading of annealed steel 45 during low-cycle fatigue on changes in its magnetic characteristics, in particular, the coercive force and the residual magnetic induction, for the major and minor magnetic-hysteresis loops and on changes in the propagation velocity of a longitudinal acoustic wave is studied. The sensitivity of the considered physical characteristics to the value of plastic deformation stored under cyclic loading in the region of both large and small strains is determined. The residual mechanical properties displayed after cyclic loading are determined, and the steady-state correlations between the coercive force measured on minor hysteresis loops in weak fields (the Rayleigh region) and the remanent clongation are obtained. The possibility of monitoring the stored plastic deformation and assessing the residual life of a material during its cyclic loading from the values of its magnetic parameters is shown.     

Upper bounds on plastic strains for elastic-perfectly plastic solids subjected to variable loads

The paper considers shakedown analysis problems for elastic-perfectly plastic solids subjected to quasi-static loads which vary arbitrarily within a given domain. It gives a general inequality which is able to generate Melan's theorem for shakedown, as well as bounds on plastic strains at any point of the solid. These bounds can be made the most stringent by solving a “perturbed” shakedown problem in “finite” or “holonomic” terms. The results presented in this paper are a generalization of those given in a previous paper by the present author[10].     

Prediction of cutting forces and machining error in ball end milling of curved surfaces -II experimental verification

This paper presents the results of a series of experiments performed to examine the validity of a theoretical model for evaluation of cutting forces and machining error in ball end milling of curved surfaces. The experiments are carried out at various cutting conditions, for both contouring and ramping of convex and concave surfaces. A high precision machining center is used in the cutting tests. In contouring, the machining error is measured with an electric micrometer, while in ramping it is measured on a 3-coordinate measuring machine. The results show that in contouring, the cutting force component that influences the machining error decreases with an increase in milling position angle, while in ramping, the two force components that influence the machining error are hardly affected by the milling position angle. Moreover, in contouring, high machining accuracy is achieved in “Up cross-feed, Up cut” and “Down cross-feed, Down cut” modes, while in ramping, high machining accuracy is achieved in “Left cross-feed, Downward cut” and “Right cross-feed, Upward cut” modes. The theoretical and experimental results show reasonably good agreement.     

Modelling the evolution of residual stresses during tensile testing of elastoplastic wires subjected to a previous bending operation

During coiling operations high residual stresses are frequently developed in steel wire. In this paper the stress distribution in wires during coiling, unwinding and subsequent tensile testing is modelled for numerous bending degrees, assuming perfect Voce plastic deformation and linear elastic behaviour. The influence of such residual stresses on the observed tensile test data can be deduced. It is shown that coiling with spool radii as used today industrially can lead to measurement of wire properties deviating significantly from the “true” properties of a properly coiled wire. Also, a method is proposed to deduce the original flow behaviour of coiled samples from tensile test curves, hence filtering the effect of the residual stresses.     

A surface flow line model of a scratching tip: apparent and true local friction coefficients

The apparent friction coefficient is the ratio between the tangential force and the normal load applied to a moving scratching tip. It includes a so-called “true local” friction coefficient, which is the scission at the interface between the tip and the surface being scratched, and a “geometrical” friction coefficient, which is the plough effect due to the wave front created ahead of the moving tip and depends on the shape of the tip. Like in any mechanical test, three basic types of behaviour of the material at the interface are observed: purely elastic, elastic–plastic and fully plastic. As is usual in polymers, the material behaviour is time and temperature dependent and may exhibit strain hardening. A surface flow line model is developed here to deduce the geometrical and true friction coefficients at the interface between a moving scratching tip and a surface from the apparent friction coefficient. Using this model, several situations may be simulated to predict the influence of the geometry of the tip on the scratch resistance of the material.     

Lateral compression of tubes and tube-systems with side constraints

The quasi-static diametral compression of a tube constrained so that its horizontal diameter cannot increase is considered both theoretically and experimentally. The relationship between the behaviour of a single tube and a nest of tubes is examined experimentally and the role of “closed” system in energy absorption is discussed.     

Coercive force of ferromagnetic steels under the biaxial symmetrical tension of a material

Patterns of coercive force variations of Ct.3 and X70 ferromagnetic steels under biaxial symmetrical tension within both elastic and plastic ranges of deformation were studied using X-shaped specimens. It was shown that the coercive force of isotropic polycrystalline materials increased during plastic deformation under biaxial symmetrical tension and was proportional to applied stresses (loads). Plastic deformation of materials with high initial coercive-force anisotropy along the principal directions (X70 steel) leads to an abrupt decrease in anisotropy and subsequent alternation of its sign. The patterns of a material’s coercive force behavior allow the coercive force to be used for estimating the stress-strain state (from both an increase in the coercive force and its variations during loading as compared to that of the initial material) of articles made from the studied steels under biaxial symmetrical tensile deformation.     

Geometry of “developable cones”

When a thin disc is supported on the rim of a bowl, and its centre is pushed down by a finger, it adopts a characteristic conformation, known as a “developable cone”, and sketched in Fig. 1(a): the main, broadly conical, shape can only form if about one-quarter of the disc buckles upwards. There is a curved intersection between the two parts, which takes the form of a crescent-shaped “crease” near its apex, but with the flanking regions less tightly deformed. The “developable cone” is a recurring motif in a wide range of physical situations—crumpling, buckling, draping—and its mechanics provides a key to understand the phenomena, whether the disc deforms in the elastic or the plastic range. The task of this paper is to study only geometrical features of the “developable cone”. The first step is to replace the actual crease (Fig. 1(a)) by an idealised “sharp” crease (Fig. 1(b)). The second step is to study the apparently “large-rotation” problem of kinematics by means of an adaptation of the classical “yield-line” pattern of folding, but with a crucial added constraint that springs from Gauss's analysis of inextensional deformation. We illustrate the method via a graded sequence of examples, and we close with a discussion.     

Possibilities of magnetic inspection of plastic deformations preceding failures of low-carbon steels constructions

The possibility of magnetic inspection of plastic deformations preceding the failure of strained steel constructions was studied by locally magnetizing them with an attachable electromagnet. Field dependences of the differential magnetic permeability of a plate made of steel 09Γ2 on the applied and residual stresses were determined. Critical fields of 90° and 180° domain-wall motion at different degrees of deformation were calculated using a model taking into account the contribution of these domain walls to magnetization reversal processes. Dependences of the coercive force and residual magnetization of the plate on the applied and residual stresses, which were measured using a SIMTEST portable magnetic measuring system, are reported.     

Effect of tensile plastic deformations on the residual magnetization and initial permeability of low-carbon steels

The behavior of residual magnetization M _r , initial magnetic permeability μ, and coercive force H _c of steels subjected to plastic extension both in the loaded state and in the unloaded state during “slow” and “fast” loading is explained from a unified point of view. It is shown that, upon unloading, the appearance of residual compressive stresses along the direction of the applied force in a considerable portion of grains leads to an abrupt decrease in M _r and μ and to an equally sharp increase in H _c . A slow decrease in M _r and μ and an equally slow increase in H _c are observed in the loaded state with an increase in ε; this effect is caused by an increase in the dislocation density. The values of M _r , μ, and H _c in the transition region of ε ≤ ε_cr obtained under slow loading differ considerably from those obtained under fast loading. In loaded and unloaded states, the dependences of M _r , μ, and H _c on ε are identical at ε ≥ ε_cr. The results obtained will be important when the magnetic parameters analyzed in this study are used for nondestructive testing of steel structures that have plastic deformations.