Influence of end-conditions during tube hydroforming of aluminum extrusions

Tube hydroforming experiments were conducted to develop the forming limit diagram of AA6082-T4 by utilizing three types of end-conditions: (i) “free-end”, (ii) “pinched-end” or “fixed-end” and (iii) “forced-end”. It was found that “free-end” hydroforming gives the lowest forming limits followed by “pinched-end” and “forced-end” hydroforming. It was noticed that the tube failure occurs within 5° to the extrusion weld in the “free-end” experiments, within 7° in the “pinched-end” condition and extended up to 10° in the “forced-end” hydroforming experiments. Finite element simulations were carried out to capture the effects of the weld geometry, weld mechanical properties and the end-conditions of the extruded tube on the maximum induced stress and location of the maximum von Mises stress. It was found that the anisotropy of the weld material and the end-condition used during hydroforming experiments have the largest influence on the failure location with respect to the weld center.     

Limit strain predictions for strain-rate sensitive anisotropic sheets

The combined effects of material strain-rate sensitivity and anisotropy on necking or “limit” strain predictions are examined for thin sheets with transversely isotropic properties. Various rate dependent constitutive laws based on flow theory and deformation theory of plasticity are considered.The strong effect of material strain-rate sensitivity in increasing the amount of straining prior to localized necking is first emphasized. We then discuss the joint influence of rate dependence and anisotropy on the theoretical limit strains and forming limit curves. Both strain-rate sensitivity and the local shape of the anisotropic yield surface are shown to significantly affect the predicted limit strains.A necking-band bifurcation analysis is also carried out to reveal in an explicit manner the remarkable sensitivity of overall forming limit diagram shapes to the parameters in the anisotropic yield function.     

On the stability of nonlinear circumferential waves in a spinning rectilinearly orthotropic disk

A system of two coupled nonlinear differential equations of the second and fourth order, respectively, governing motions of a rotating membrane-like disk, made of a rectilinearly orthotropic material, is analyzed after assuming a particular form for the transverse deflections. The characteristic equation connecting the speed of propagation and the amplitude of circumferential waves is derived and discussed with respect to three classes of anisotropy. By imposing temporal perturbations on the deflections, two ordinary coupled differential equations with variable coefficients, for the perturbation prameters, are obtained. The equations become decoupled by means of a “rotational” transformation of the dependent variables. The closed form solution of the equations is found. Conditions for wave stability involving the coefficients of anisotropy are determined and illustrated graphically for the three anisotropic classes considered.     

Analysis of plane and rough contacts, subject to a shearing force

We analyse the surface traction conditions induced in plane contact between two bodies whose surfaces are rough. It is assumed that the roughness may be idealised by a surface of regularly spaced cylindrical “bumps”, and the overall geometry may be in the form of a cylinder, flat ended punch or wedge. The stick-slip regime experienced by each individual asperity contact is found, and hence it is shown how the applied shearing force produces concentrated regions of surface damage. Conditions for crack initiation are then discussed, and compared with equivalent results found for nominally smooth contacts.     

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.     

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.     

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.     

The ricochet of spheres and cylinders from the surface of water

A theory of ricochet is proposed which permits the effect of projectile spin to be accounted for. This effect is not explained by previous theories. The critical angles for ricochet for a sphere and for a spinning cylinder are calculated, and the theory of the spinning cylinder is applied to the Barnes Wallis “bouncing bomb” developed during World War II.     

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.     

The free vibrations of a thin circular finite rotating cylinder

The natural frequencies of a finite circular thin cylinder are obtained by employing an exponential matrix expansion of the so-called “fundamental matrix”. It is shown that the method is general enough and able to handle any system of linear differential equations of constant coefficients together with arbitrary boundary conditions. Results are given for rotating cylinders with clamped and free edges. The vibration frequencies of a stationary finite cylinder, previously obtained by other methods of solution, are used as a check on the present method with the special case of zero spinning velocity.     

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.     

An experimental study of the burst strength of rotating disks

Details are given of a spinning rig for burst tests on metal disks in vacuo at speeds in excess of 100,000 rpm. The permanent strain distributions and the instability and fracture conditions are observed in the spinning of disks of vacuum remelted alloy steel. It is shown for hollow disks of uniform thickness that at instability two “necks” form at either side of the bore and the bore becomes oval with the direction of the minimum diameter passing through the two “necked” regions.Good correlation between theoretical and experimental strain distributions at instability are obtained provided the ratio of outside to inside radius of the disk is greater than 10. The theory is based on a rigid-plastic material and it is thought that better correlation for radius ratios less than 10 would be achieved if elastic strains were taken into account in the theory.     

MPM/MD handshaking method for multiscale simulation and its application to high energy cluster impacts

We propose a new multiscale simulation method which seamlessly combines the conventional molecular dynamics (MD) with the continuum mechanics formulated under the material point method (MPM). In MPM, modified interpolation shape functions are adopted to reduce artificial forces on the hierarchical background grids. The multiscale method is validated using the examples of step-like wave and wave packet propagations within a bar. The method is applicable to several kinds of potentials including the Lennard–Jones, EAM and a bonding-angle related potential for silicon. Examples of high energy Cu–Cu and Si–Si cluster impacts are presented. The evolution of displaced atoms is found to depend on the underlying lattice structures. For the case of Cu–Cu cluster impacts, stacking faults play an important role. The displaced atoms, visualized in the method of “local crystalline order”, propagate in an anisotropic manner. This implies the anisotropy in energy transformation process through multi-interactions among cluster and surface atoms.     

Prediction of forming limits using Hosford's modified yield criterion

A new generalized anisotropic yield criterion has been suggested by Hosford. This criterion has been used in prediction of forming limits by Graf and Hosford using M-K analysis for a material with normal anisotropy. In this paper the effect of planar anisotropy has been incorporated and the procedure for calculation of forming limits has been modified using the approach suggested by Parmar and Mellor. The results are compared with those of Graf and Hosford in the case of planar isotropy to validate the model and procedure. The results are also compared with the predictions of the M-K model using Hill's old and new (after Parmar and Mellor) yield criteria as well as experimental values obtained under conditions of both in-plane and punch stretching.It is observed that the effect of planar anisotropy is negligible while the predictions are strongly dependent on exponent “a” (exponent in yield criterion). Predictions with a = 5, 6 or 8 match the experimental results much better than predictions using the M-K model with Hill's old criterion, or even the new criterion used by Parmar and Mellor.     

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.     

Paradoxical buckling behaviour of a thin cylindrical shell under axial compression

The widely accepted theory of buckling of thin cylindrical shells under axial compressive loading emphasises the sensitivity of the buckling load to the presence of initial imperfections. These imperfections are conventionally taken to be minor geometric perturbations of a shell which is initially stress-free. The original aim of the present study was to investigate the effect on the buckling load of imperfections in the form of local initial stress, which are probably more typical of practice than purely geometric ones. Experiments were performed on a vertical “melinex” cylinder of diameter 0.9 m and height 0.7 m, with radius/thickness ratio 1800. The upper and lower edges of the cylinder were clamped to end discs by means of circumferential belts — an arrangement that allowed states of self-stress to be introduced to the shell readily by means of local “uplift” at the base. The upper disc was made sufficiently heavy to buckle the shell, and it was supported by a vertical central rod under screw control. Many buckling tests were performed. Surprisingly, the buckling loads were generally at the upper end of the range of fractions of the classical buckling load that have been found in many previous experimental studies. Even when the local uplift at the base caused a local “dimple” to be formed before the shell was loaded, the buckling load was relatively high. A surface-scanning apparatus allowed the geometric form of the shell to be monitored, and the progress of such a dimple to be followed; and it was found that a dimple generally grew in size and migrated in a stable fashion up the shell as the load increased, until a point was reached when unstable buckling occurred. These unexpected and paradoxical features of the behaviour of the experimental shell may be attributed to the particular boundary conditions of the shell, which provide in effect statically determinate support conditions. This study raises some new issues in the field of shell buckling, both for the understanding of buckling phenomena and for the rational design of shells by engineers against buckling.     

Analysis and computation of the oil film thickness between the piston ring and cylinder liner of an internal combustion engine

A study of the essential features of piston rings in the cylinder liner of an internal combustion engine reveals that the lubrication problem posed by it is basically that of a slider bearing. According to steady-flow-hydrodynamics, viz. the oil film thickness becomes zero at the dead centre positions as the velocity, U = 0. In practice, however, such a phenomenon cannot be supported by consideration of the wear rates of pistion rings and cylinder liners. This can be explained by including the “squeeze” action term in the hydrodynamic theory, viz. .This article introduces the equations of the above theory along with the viscosity variation over the piston stroke length; the piston ring profile is assumed as a double parabola with a central straight portion.The results of this analysis as applied to internal combustion engines are presented and compared with other earlier analysis.     

Flexure of orthotropic rotating disks

The influence of off-set in mounting the blades on the disk, termed “eccentricity”, and the centrifugal stiffening on the stresses and deflection induced in laterally loaded orthotropic disks of variable thickness is analysed. The analysis is based on a series solution for the differential equation governing the deflection of the disk. Numerical results showing the effects of anisotropy, eccentricity and rotation on the stresses and deflection of the disk are presented graphically. It is shown that the stress due to radial bending moment reduces significantly with the increase in the degree of anisotropy.     

An experimental study of the in-plane stretching of sheet metal

Experimental results are presented of the limit strains for the in-plane stretching of steel, aluminium and brass sheets. The development of surface strains is studied in incremental tests and attention is drawn to “crazing” of the surface of sheet specimens and to “banding” in aluminium specimens. Despite the wide difference in the mechanical properties of the three materials the values for the limit strain are remarkably similar. For steel and aluminium the limit strains are greater than the predicted instability strains, but for brass the limit strains are considerably less than the predicted instability strains. The wide scatter in the experimental limit strains perhaps indicates that failure is largely influenced by random effects.