Moment–curvature measurement in thin sheet—part II: yielding and kinking in aged steel sheet

The measurement of the moment–curvature characteristic for thin sheet metal in the elastic–plastic range provides important information which is not readily obtained from the tensile test. Steel sheet which is roll-formed to products such as roofing typically contains a residual stress distribution and is subject to strain ageing; these conditions can be identified from bend tests. The accurate determination of the moment–curvature diagram is not easy as the limiting elastic curvature is large and the magnitude of the moment is small.Part I of this paper describes equipment and data processing which have been developed for this purpose and in Part II, results obtained in sheet which has been progressively aged and stress-relieved are presented.     

Thermomechanical analysis of elastoplastic medium in sliding contact with fractal surface

The effects of mechanical and thermal surface loadings on deformation of elastic–plastic semi-infinite medium were analyzed simultaneously by using the finite element method. Rigid rough surface of a magnetic head and smooth surface of an elastic–plastic hard disk were chosen to perform a comprehensive thermo-elastic–plastic contact analysis at the head–disk interface (HDI). A two-dimensional finite element model of a rigid rough surface characterized by fractal geometry sliding over an elastic–plastic medium was then developed. The evolution of deformation in the semi-infinite medium due to thermomechanical surface loading is interpreted in terms of temperature, von Mises equivalent stress, and equivalent plastic strain. In addition to this, the effects of friction coefficient, sliding, and interference distance on deformation behavior were also analyzed. It is shown that frictional heating increases not only the contact area but also the contact pressure and stresses.     

Plastic deformation and contact area of an elastic–plastic contact of ellipsoid bodies after unloading

This paper presents theoretical and experimental results of the residual or plastic deformation and the plastic contact area of an elastic–plastic contact of ellipsoid bodies after unloading. There are three regime responses of the deformation and contact area: elastic, elastic–plastic and fully plastic. Experimental investigation is presented in order to validate the proposed model. A new technique is introduced to measure the plastic deformation and plastic contact area. Very good correlation is found between the theoretical prediction and the experimental results.     

Dynamic asymmetrical instability of elastic–plastic beams

Dynamic instability of elastic–plastic beam is investigated by employing a three-degree-of-freedom (3-DoF) beam model. Especially, asymmetrical instability induced by symmetrical load is discussed. The asymmetrical instability is considered as a second-order buckling mode. Four types of perturbations, i.e., geometrical misalignment, material property mismatch, unsymmetry of applied load and disturbance of boundary conditions, are introduced to activate the asymmetrical responses. The asymmetrical response is characterized by a modal participation factor α₂ which corresponds to an asymmetrical mode shape. Phase plane trajectories and Poincaré map are used to illustrate the chaotic characteristics of the beam response. Results show that if the perturbations are small enough, the perturbation type has negligible influence on the critical load for the occurrence of the asymmetrical instability, which implies that the asymmetrical instability is an intrinsic feature of the beam system. However, with the increase of the magnitude of the perturbations, the influence of the asymmetrical vibration is expanded to a large extension of loading parameter.     

Numerical study of the flange earring of deep-drawing sheets with stronger anisotropy

A Barlat–Lian anisotropy yield function is introduced into a quasi-flow corner theory of elastic–plastic finite deformation and the elastic–plastic large deformation finite element formulation based on the principle of virtual velocity and the discrete Kirchhoff triangle plate shell element model. The focus of the present researches is on the numerical simulation of the flange earring of deep-drawing process of circular sheets with stronger anisotropy, based on which, the schemes for controlling the flange earring are proposed.     

Numerical simulation of the localization behavior of hydrostatic-stress-sensitive metals

The present paper deals with the numerical simulation of the elastic–plastic deformation and localization behavior of solids which are plastically dilatant and sensitive to hydrostatic stresses. The model is based on a generalized macroscopic theory taking into account macroscopic as well as microscopic experimental data obtained from tests with iron-based metals. It shows that hydrostatic components may have a significant effect on the onset of localization and the associated deformation modes, and that they generally lead to a notable decrease in ductility. The continuum formulation relies on a generalized I₁–J₂–J₃ yield criterion to describe the effect of the hydrostatic stress on the plastic flow properties of metals. In contrast to classical theories of metal plasticity, the evolution of the plastic part of the strain rate tensor is determined by a non-associated flow rule based on a plastic potential function which is expressed in terms of stress invariants and kinematic parameters. Numerical simulations of the elastic–plastic deformation behavior of hydrostatic-stress-sensitive metals show the physical effects of the model parameters and also demonstrate the efficiency of the formulation. Their results are in excellent agreement with available experimental data. A variety of large-strain elastic–plastic problems involving pronounced localizations is presented, and the influence of various model parameters on the deformation and localization behavior of hydrostatic-stress-sensitive metals is discussed.     

Scaling of pulse loaded mild steel plates with different edge restraint

This paper reports the results of scaled tests on mild steel square plates with different edge restraint subjected to uniformly distributed triangular pulse pressure loading producing large inelastic deformations without tearing or rupture. The loading is representative of overpressure loading arising from a confined hydrocarbon explosion. A scale factor of 0.5 was examined in this series of tests on 16 plates, eight 1 m by 1 m × 2 mm thick and eight 0.5 m × 0.5 m × 1 mm thick. Both static and dynamic test results are presented. A quasi-static rigid–plastic and elastic–plastic analysis of a fully restrained square plate is also presented and compared with the test results. The transient response of the plates under dynamic loading show some divergence from the laws of geometrically similar scaling while the permanent deformations in both the static and dynamic tests show good compliance with the laws of similitude, within the accuracy expected in such tests.     

Anomalous dynamic elastic-plastic response of a galerkin beam model

A study of the anomalous motion of an elastic—plastic beam under short pulse loading is presented. The geometric nonlinearity due to axial end constraints is taken into account. We apply the Galerkin method to the governing partial differential equation of the transverse motion to obtain a general model of n degrees of freedom (nDoF). The results of elastic—plastic deformation analysis and dynamic response for the 2DoF model of a pin-ended beam are presented. The regular and irregular motions of the 2DoF model for the pin-ended beam are examined by various methods including time history, phase diagram, Lyapunov characteristic exponent and power spectral density.     

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.     

Development of a Wittrick–Williams algorithm for the spectral element model of elastic–piezoelectric two-layer active beams

In this paper, a Wittrick–Williams algorithm is developed for the elastic–piezoelectric two-layer active beams. The exact dynamic stiffness matrix (or spectral element matrix) is used for the development. This algorithm may help calculate all the required natural frequencies, which lie below any chosen frequency, without the possibility of missing any due to close grouping or due to the sign change of the determinant of spectral element matrix via infinity instead of via zero. The uniform and partially patched active beams are considered as the illustrative examples to confirm the present algorithm.     

An elastic–plastic stress analysis steel-reinforced thermoplastic composite cantilever beam

In the present study an analytical elastic–plastic stress analysis is carried out for a low-density homogeneous polyethylene thermoplastic cantilever beam reinforced by steel fibers. The beam is loaded by a constant single force at its free end. The expansion of the region and the residual stress component of σ_x are determined for 0°, 30°, 45°, 60° and 90° orientation angles. Yielding begins for 0° and 90° orientation angles at the upper and lower surfaces of the beam at the same distances from the free end. However, it starts first at the upper surface for 30° and 45° orientation angles. The elastic–plastic analysis is carried out for both the plastic region which spreads only at the upper surface and the plastic region which spreads at the upper and lower surfaces together. The residual stress components of σ_x and τ_xy are also determined. The intensity of the residual stress component is maximum at the upper and lower surfaces of the beam, but the residual stress component of τ_xy is maximum on or around the x-axis. The beam can be strengthened by using the residual stresses. The distance between the plastically collapsed point and the free end is calculated for the same load in the beam for 0°, 30°, 45°, 60° and 90° orientation angles.     

Analysis of viscosity interaction on the misaligned conical–cylindrical bearing

Viscosity is an essential property in hydrodynamic lubrication. In general, the lubricant is not considered to have uniform viscosity within a given bearing. The viscosity of the lubricant is affected by both pressure and temperature. The viscosity of the lubricant increases with pressure and for most lubricants, this effect is much larger than that of temperature or shear when the pressure is significantly above than the atmospheric pressure. This study analyzes the thermal effect of conical–cylindrical bearing performance parameters via the viscosity–pressure–temperature relationships of lubricants. The results reveal that pressure increases both the film viscosity and temperature as well.     

Predictive model to estimate the stress–strain curves of bulk metals using nanoindentation

Many studies have shown that finite element modeling (FEM) can be used to fit experimental load–displacement data from nanoindentation tests. Most of the experimental data are obtained with sharp indenters. Compared to the spherical case, sharp tips do not directly allow the behavior of tested materials to be deduced because these produce a nominally-constant plastic strain impression. The aim of this work is to construct with FEM an equivalent stress–strain response of a material from a nanoindentation test, done with a pyramidal indenter. The procedure is based on two equations which link the parameters extracted from the experimental load–displacement curve with material parameters, such as Young's modulus E, yield stress Y₀ and tangent modulus E_T. We have already tested successfully the relations on well-known pure metallic surfaces. However, the load–displacement curve obtained using conical or pyramidal indenters cannot uniquely determine the stress–strain relationship of the indented material. The non-uniqueness of the solution is due to the existence of a characteristic point (ε_c, σ_c); for a given elastic modulus, all bilinear stress–strain curves that exhibit the same true stress σ_c at the specific true strain ε_C lead to the same loading and unloading indentation curve. We show that the true strain ε_c is constant for all tested materials (Fe, Zn, Cu, Ni), with an average value of 4.7% for a conical indenter with a half-included angle θ=70.3°. The ratio σ_c/ε_c is directly related to the elastic modulus of the indented material and the tip geometry.     

Investigation of sharp contact at rigid–plastic conditions

Sharp contact problems are examined theoretically and numerically. The analysis is focused on elastic–plastic material behaviour and in particular the case when the local plastic zone arising at contact is so large that elastic effects on the mean contact pressure will be small or negligible. It is shown that, save for the particular case of a rigid–plastic power-law material, at such conditions, there is no single representative value on the uniaxial stress-strain curve that can be used in order to evaluate the global parameters at contact. However, the present numerical results indicate that good accuracy predictions for the mean contact pressure can be achieved when this variable is described by two parameters corresponding to the stress levels at, approximately, 2 and 35% plastic strain. Regarding the size of the contact area, it is shown that this quantity is very sensitive to elastic effects and any general correlation with material properties is complicated at best. The numerical analysis is performed by using the finite element method and the theoretical as well as the numerical results are compared with relevant experimental ones taken from the literature. From a practical point of view, the presented results are directly applicable to material characterization or measurements of residual mechanical fields by sharp indentation tests, but also for situations such as contact in gears or in electronic devices.     

Some experiments on the micro-indentation of digital audio tape

The durability of digital audio tape is a function of the elastic and plastic properties of the magnetic layer and substrate. In this work a micro-indentation method has been used to estimate these properties. The loading part of the compliance curve (load against indentor penetration) is analysed to obtain magnetic layer hardness. The unloading part is used to determine the reduced elastic modulus. Indentations at various depths have been recorded and an extrapolation technique used to predict layer properties. Rate dependence of the magnetic layer has been studied through constant loading rate tests. Results are presented isochronally, to indicate any viscous elastic–plastic non-linearities.     

Elastic–plastic beam-on-foundation under quasi-static loading

An elastic–plastic discrete spring model is developed to represent the mechanical behavior of an elastic–plastic beam-on-foundation (BoF) system, and an analytical procedure of analyzing the BoF under general quasi-static loading is formulated. The paper describes a detailed numerical simulation and analysis for the case of a BoF subjected to a concentrated force at the mid-span, and various plastic collapse mechanisms of BoFs are identified. Two peculiar phenomena, i.e. the migration of plastic hinge in the beam and the successive propagation of plastic zone in the foundation, are demonstrated. It is found that any elastic, perfectly plastic BoF system can be characterized merely by three non-dimensional parameters, but the limit state of a rigid, perfectly plastic BoF is determined by a single non-dimensional parameter only. The non-dimensional relative rigidity of BoF and the ratio of the maximum elastic deformation energies dissipated in the beam and foundation both play important roles in governing the deformation scenario of an elastic–plastic BoF system.     

Collapse mechanisms of sandwich beams with composite faces and a foam core, loaded in three-point bending. Part I: analytical models and minimum weight design

Analytical predictions are made for the three-point bending collapse strength of sandwich beams with composite faces and polymer foam cores. Failure is by the competing modes of face sheet microbuckling, plastic shear of the core, and face sheet indentation beneath the loading rollers. Particular attention is paid to the development of an indentation model for elastic faces and an elastic–plastic core. Failure mechanism maps have been constructed to reveal the operative collapse mode as a function of geometry of sandwich beam, and minimum weight designs have been obtained as a function of an appropriate structural load index. It is shown that the optimal designs for composite–polymer foam sandwich beams are of comparable weight to sandwich beams with metallic faces and a metallic foam core.     

Ductile damage micromodeling by particles’ debonding in metal matrix composites

The elastic–plastic behaviour of particle-reinforced metal matrix composites undergoing ductile damage is modelled using a two-level micro-structural approach. The considered heterogeneous material is a polycrystal containing intra-crystalline elastic particles. Ductile damage is initiated by the matrix/particle interface debonding and the subsequent voids growth with plastic straining of the crystalline matrix. Homogenization techniques are used twice: first at mesoscale to derive the equivalent grain behaviour and then to obtain the macroscopic behaviour of the material. Plastic deformation of the crystalline matrix is due to crystallographic gliding on geometrically well-defined slip systems. The associative plastic flow rule and the hardening law are described on the slip system level. The evolution of micro-voids volume fraction is related to the plastic strain. The elastic–plastic stress–strain response of particle composite is investigated. Predictions of the proposed model are compared to experimental data to illustrate the capability of the suggested method to represent material behaviour. Furthermore, specific aspects such as the stress triaxiality and yield surfaces are discussed.     

Erosion–corrosion degradation mechanisms of Fe–Cr–C and WC–Fe–Cr–C PTA overlays in concentrated slurries

In this investigation the microstructure and erosion–corrosion behaviour of a Fe–Cr–C overlay (FeCrC–matrix) produced by plasma transferred arc welding (PTA) and its metal matrix composite (FeCrC–MMC) were assessed. The FeCrC–MMC was obtained by the addition of 65 wt.% of tungsten carbide (WC). The erosion–corrosion tests (ECTs) were carried out using a submerged impinging jet (SIJ); after the ECTs the surface of the overlays was analysed to identify the damage mechanisms. Two different temperatures (20 and 65 °C) and sand concentrations (10 and 50 g/l) were used in a solution of 1000 ppm of Cl⁻ and a pH value of 8.5; the conditions were chosen to be representative of the recycling water in the tailings line in the oilsands industry. The FeCrC–matrix showed a dendritic structure and a high concentration of carbides in the interdendritic zone. The addition of the WC reinforcing phase promoted the formation of W-rich intermetallic phases, increased the microhardness values of the matrix phase of the FeCrC–MMC overlay and dramatically improved its erosion–corrosion performance as expected. For the FeCrC–matrix overlay the main erosion–corrosion degradation mechanisms were severe plastic deformation and the formation and removal of material flakes due to consecutive impacts. At 65 °C the dendritic zone was severely corroded in the area of low impact frequency. The FeCrC–MMC showed greater attack of the matrix phase compared to the WC grains; at high sand concentration the WC grains were severely fractured and flattened. The anodic polarisation analysis showed active corrosion behaviour of the FeCrC–MMC at both temperatures and sand concentrations; however the temperature dramatically increased the corrosion process of the surface studied under erosion–corrosion conditions. The paper assesses the degradation mechanisms of both FeCrC–matrix and FeCrC–MMC with the aim of understanding what aspects of MMCs must be adapted for optimum erosion–corrosion resistance.