Closed-form solution for wedge cutting force through thin metal sheets

A simple kinematic model is developed which describes the main features of the process of the cutting of a plate by a rigid wedge. It is assumed in this model that the plate material curls up into two inclined cylinders as the wedge advances into the plate. This results in membrane stretching up to fracture of the material near the wedge tip, while the “flaps” in the wake of the cut undergo cylindrical bending. Self-consistent, single-term formulas for the indentation force and the energy absorption are arrived at by relating the “far-field” and “near-tip” deformation events through a single geometric parameter, the instantaneous rolling radius. Further analysis of this solution reveals a weak dependence on the wedge angle and a strong dependence on friction coefficient. The final equation for the approximate cutting force over a range of wedge semiangles 10° ≤ θ ≤ 30° and friction coefficients 0.1 ≤ μ ≤ 0.4 is: F = 3.28σ₀(δ_t)^0.2l^0.4t^1.6μ^0.4, which is identical in form and characteristics to the empirical results recently reported by Lu and Calladine [Int. J. Mech. Sci.32, 295–313 (1990)].This analysis is believed to resolve a controversy recently developed in the literature over the interpretation of plate cutting experiments.     

Erosion of α-Fe by spherical glass particles

Polycrystalline α-Fe has been eroded at 30° and 90° with glass spheres of average diameters 70 μm and 200 μm in the velocity range 61–122 m s⁻¹. Detailed studies of the influence of the impact variables on the erosion rate as well as scanning electron microscopy studies of the eroded surfaces have been performed. It was observed that “breaking” waves developed on erosion at 30° and hills and valleys at 90°. Several different material loss processes that operate at various positions within waves, hills and valleys have been identified. It was clear that most material loss processes involved extensive localized shear and required the surface to become “conditioned” by a specific number of impacts before material loss began.     

Stress and failure probability analysis for a transversely isotropic, brittle, elastic cylinder subjected to internal pressure and axisymmetric thermal loading

Procedures are developed for the determination of the stresses in and thence the probability of failure of a transversely isotropic cylinder made of a brittle material and loaded by an internal pressure and an axisymmetric radial temperature gradient. As examples of the application of the procedures a cylinder is analysed first with isotropic material properties, then with various degrees of anisotropy including both the “fibrous” and “laminar” types. The treatment is non-dimensional; results are presented graphically in the form of failure probability “contours”. For the dimensions and materials considered it is shown that the probability of failure is affected only slightly by the fibrous form of anisotropy but markedly by the laminar form when the thermal loading predominates.     

Buckling of thin cylindrical shells: an attempt to resolve a paradox

The classical theory of buckling of axially loaded thin cylindrical shells predicts that the buckling stress is directly proportional to the thickness t, other things being equal. But empirical data show clearly that the buckling stress is actually proportional to t^1.5, other things being equal. As is well known, there is wide scatter in the buckling-stress data, going from one half to twice the mean value for a given ratio R/t. Current theories of shell buckling explain the low buckling stress—in comparison with the classical—and the experimental scatter in terms of “imperfection-sensitive”, non-linear behaviour. But those theories always take the classical analysis of an ideal, perfect shell as their point of reference.Our present principal aim is to explain the observed t^1.5 law. So far as we know, no previous attack has been made on this particular aspect of thin-shell buckling. Our work is thus breaking new ground, and we shall deliberately avoid taking the classical analysis as our starting point.We first point out that experiments on self-weight buckling of open-topped cylindrical shells agree well with the mean experimental data mentioned above; and then we associate those results with a well-defined post-buckling “plateau” in load/deflection space, that is revealed by finite-element studies. This plateau is linked with the appearance of a characteristic “dimple” of a mainly inextensional character in the deformed shell wall. A somewhat similar post-buckling dimple is also found by quite separate finite-element studies when a thin cylindrical shell is loaded axially at an edge by a localised force; and it turns out that such a dimple grows under a more-or-less constant force that is proportional to t^2.5, other things being equal.This 2.5-power law can be explained by analogy with the inversion of a thin spherical shell by an inward-directed force. Thus, the deformation of such a shell is generally inextensional except for a narrow “knuckle” or boundary layer in which the combined local elastic energy of bending and stretching is proportional to t^2.5, other things being equal. Similarly, the modes of deformation in the post-buckling dimples in a cylindrical shell are practically independent of thickness, except in the highly deformed boundary-layer regions which separate the inextensionally distorted portions of the shell. These ideas lead in turn to an explanation of the t^1.5 law for the post-buckling stress of open-topped cylindrical shells loaded by their own weight.We attribute the absence of experimental scatter in the self-weight buckling of open-topped cylindrical shells to the statical determinacy of the situation, which allows a post-buckling dimple to grow at a well-defined “plateau load”. Conversely, the large experimental scatter in tests on cylinders with closed ends may be attributed to the lack of statical determinacy there.Our paper contains several arguments that are not mathematically water-tight, in contrast to many reports in the field of mechanics of structures. We plead that the problem which we have tackled is so difficult that the only way forward is one of “over-simplification”. We hope that our work will be judged not with respect to its absence of mathematical precision, but by the light which it sheds upon the problem under investigation.     

Asymptotic analysis of extrusion of porous metals

Plane strain extrusion of fully dense and porous metals is analysed using asymptotic techniques. The extrusion die is assumed to taper gradually down the extrusion axis. The asymptotic expansions are based on a small parameter ε which is defined as the ratio of the total reduction of the original cross-section to the length of the reduction region. Coulomb's law is used to model the frictional forces that develop along the metal-die interface and the coefficient of friction is assumed to be of order ε. Analytical solutions for the first two terms in the expansions are obtained. In the case of the fully dense metals, it is shown that the leading order [O(1)] solution involves “slab flow.” It is also shown that the next term in the expansion of the solution is O(ε²), and this provides a theoretical justification for the use of the so-called “slab methods” of analysis for dies of moderate slope. An asymptotic analysis of the extrusion of porous metals with dilute concentration of voids is also carried out. Gurson's plasticity model is used to describe the constitutive behavior of the material. The leading order solution is the same as that of the fully dense material and the effects of porosity enter as an O(ε) correction. In order to verify the asymptotic solutions developed, detailed finite element calculations are carried out for both the fully dense and the porous material. The asymptotic solutions agree well with the results of the finite element calculations.     

A polar model for synovial fluid with reference to human joints

An analysis of “Boosted Lubrication” between two approaching solids, one of which is porous, is presented with reference to normally loaded living human joints. Micropolar fluid has been considered to represent the synovial fluid in the fluid film region between the approaching surfaces and the flow of viscous fluid in the porous matrix due to filtration through the porous material. Such a situation analysed in two regions separately using the slip flow model introduced by Beavers and Joseph. The effect of concentration, shape and size of the micro molecules on the bearing characteristics is discussed. The results are in accordance with those of Dowson et al.⁸     

An experimental and theoretical investigation into the creep properties of a simple structure of 316 stainless steel

The paper describes a set of creep experiments conducted on 316 austenitic stainless steel in a coupled pair of uniaxial creep testing machines which simulate the performance of a two bar structure subjected to constant mechanical and cyclic thermal loading in the temperature range of 550–600°C. The experimental results are compared with theoretical predictions using a non-interactive viscous creep/plasticity model and a model which includes recovery effects. In addition, an upper bound solution based upon the “rapid cycle” creep solution is calculated. The comparison shows that the recovery model is in close agreement with experiment and that the rapid cycle solution can provide a simple conservative estimate of the average creep rate.Several of the tests specimens exhibited strain instabilities during the initial cycles of temperature when sudden increases in inelastic strain of up to 0.2% occurred. As the material had not shown such effects in standard isothermal constant stress tests, the effects of simultaneous variation of stress and temperature which occurred in the two bar tests seems to have contributed to the occurrence of these strain instabilities.     

The influence of complex surface geometry on contact damage in curved brittle coatings

The effects of complex geometry on contact damage in bi-layer systems composed of curved brittle coating layers on compliant polymeric substrate is investigated. Previous studies of this problem utilise relatively simple flat or singly curved (domed) model structures. It is not known the extent to which conclusions driven from such observations may extended to more complex (practical) geometry. Glass plates of 1000 μm thick are used as representative of the brittle coating layer, and epoxy filler under layer as representative of under layer support. A series of doubly curved specimens (having curvatures of 4 and 8 mm) are produced to allow investigation of the influence of complex curvature on the evolution of damage. The specimens are tested by indentation with spheres of 4 mm radius loaded along the convex axis of symmetry. For comparison, some specimens loaded parallel to the axis of symmetry but off-centre. The study explores the influence of supporting geometries on the conditions to initiate and propagate subsurface “radial” cracks, which are believed to be responsible for catastrophic failure of brittle-coating-based structures in certain applications, such as dental crowns. It is demonstrated that critical loads for initiation of radial cracks and the subsequent crack propagation are insensitive to complex geometry, so that simple monotonic indentation “axis and/or off-axis loading” with minimum geometrical complication “flat, simply curved” remains an appropriate route to study the evolution of radial cracks in practical brittle coating structures.     

Elastic contact with a “donut”-shaped asperity

The laser-textured surfaces used for the touchdown area of computer hard-disks are sometimes covered with asperities consisting of a crater surrounded by a raised rim; contact with the read-head takes place over the rim of the crater, colloquially referred to as a “donut”. In order to analyse the load/compliance relation or the stiction to be expected in contact of hard disks, a number of authors have proposed load/compliance relations for contact between such a single doughnut and a plane, usually as simple modifications of the Hertz line contact equations. In this note simple, asymptotically correct, relations for a ring asperity are derived and verified by direct solutions. In particular, the relation between elastic deflection and load is approximately δ=(W/π²RE^*)[ln(16R/b)+0.5)].     

Assessments of the kinetic and dynamic loads sustained by stationary tools during high rate plastic forming

A systematic method for evaluating the kinetic and dynamic loads sustained by stationary tools (as opposed to moving tools for which methods already exist) during high rate plastic forming is examined and exemplified by examples. It is essentially based on the momentum theorem for continua for incompressible flow, utilizing kinematically admissible velocity fields. In steady state forming processes (such as rolling, wire drawing, etc.), the difference between the active load (imposed or calculated a priori) and the reactive load, is formulated rigorously, whereas for non-steady processes (forging, impact extrusion, etc.) the formulation gives merely an approximation to the dynamic effects on the tools. The resulting velocity-dependent reactions on the tools are given in terms of two nondimensional numbers, namely, the “kinetic head” (u₀²/σ₀) (called the Euler Number) and the “dynamic head” (ú₀L/σ₀), which includes the machine speed (u₀), machine acceleration ( ), material density , yield strength ₀ and a characteristic dimension of the product, L. The same two non-dimensional heads emerged previously from energy-balance consideration in Ref. [1], while approximating dynamic loads on moving tools, hence a consistency is demonstrated. These heads are unavoidably multiplied by geometrical functions, which typify the specific process under consideration and may amplify (or diminish) the intensity of the dynamic effects. The present work is focussed on quantifying, by the above method, the inherent difference between the reactive load sustained by the non-moving tool (say, a die) and the acting load carried by the moving tool (piston, ram, etc.) In particular cases of very slow processes, these loads are equal by static equilibrium. In some practical processes (like rolling) their difference appears to be relatively small, whereas in others (like impact extrusion) it appears extremely large.     

Strengthening caused by power law creep deformation of dispersed particles in an elastic matrix

The interaction energy is approximated between an edge dislocation and a particle deformable by power law creep in an elastic matrix. The stress required to overcome the interaction energy barrier is found to be greater than the Orowan stress, and the dislocation bulges to escape the particle. If the ratio of the shear modulus of the matrix to the viscosity of the particle (μt^m/σ₀) is large, the stress required to climb over the particle is larger than the Orowan stress and the dilocation bulges before it climbs. It is concluded that even if the particle is soft enough to exhibit creep, the strengthening of alloys can be achieved by an Orowan mechanism. The critical resolved shear stress (CRSS) of Cu-B₂O₃, obtained experimentally by Onaka et al. [11], agrees closely with that obtained in our analysis. This supports our analysis that the strength of Cu-B₂O₃ alloy at high temperature may be accounted for by the Orowan mechanism and the attraction between a dislocation and viscous particles. The energy and the force to overcome the energy barrier increases significantly with decrease of m, the strain rate exponent associated with the power law creep particle. It is found through analysis that for m < 1.0 and for certain values of μt^m/σ₀ > 1, the particle repulses the dislocation, while for m = 1.0 and for all values of μt^m/σ₀ > 1, the particle attracts the dislocation, which is the expected interaction between an elastic particle and a dislocation in an elastic matrix.     

Free vibration analyses of simply supported beams carrying multiple point masses and spring-mass systems with mass of each helical spring considered

In the conventional finite element method (FEM), the dynamic characteristics of a longitudinally vibrating rod with mass density ρ_r, Young's modulus E_r, cross-sectional area A_r and total length ℓ_r are considered to be the same as those of a helical spring with stiffness constant k_r=A_rE_r/ℓ_r and total mass m_r=ρ_rA_rℓ_r. For a lumped-mass model, the mass matrix of a rod element is a 2×2 diagonal one with each of its non-zero coefficients to be equal to one half of the total rod mass (i.e., 0.5m_r). Furthermore, the dynamic characteristics of a rod on the basis of last “lumped-mass” model have been found to be very close to those on the basis of “consistent-mass” model. Thus, one can easily take into account of the inertial effect of a helical spring using a massless one with “one half of its total mass”, respectively, concentrated at its two ends (in Method 2) instead of modeling it by an elastic rod with uniform mass per unit length (in Method 1). When one more spring-mass system is attached to the beam, the total number of unknown constants increases “one” in Method 2 and “two” in Method 1, thus, Method 2 will reduce more effort than Method 1 for studying the dynamic behaviors of a beam carrying a number of spring-mass systems with mass of each helical spring considered. In this paper, the formulations of Methods 1 and 2 are presented first and then the numerical examples are illustrated to confirm the reliability of the presented theory and the developed computer programs. Finally, the effect concerning mass of each helical spring of the spring-mass systems is studied.     

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.     

A universal slip-line model with non-unique solutions for machining with curled chip formation and a restricted contact tool

A universal slip-line model and the corresponding hodograph for two-dimensional machining which can account for chip curl and chip back-flow when machining with a restricted contact tool are presented in this paper. Six major slip-line models previously developed for machining are briefly reviewed. It is shown that all the six models are special cases of the universal slip-line model presented in this paper. Dewhurst and Collins's matrix technique for numerically solving slip-line problems is employed in the mathematical modeling of the universal slip-line field. A key equation is given to determine the shape of the initial slip-line. A non-unique solution for machining processes when using restricted contact tools is obtained. The influence of four major input parameters, i.e. (a) hydrostatic pressure (P_A) at a point on the intersection line of the shear plane and the work surface to be machined; (b) ratio of the frictional shear stress on the tool rake face to the material shear yield stress (τ/k); (c) ratio of the undeformed chip thickness to the length of the tool land (t₁/h); and (d) tool primary rake angle (γ₁), upon five major output parameters, i.e. (a) four slip-line field angles (θ, η₁, η₂, ψ); (b) non-dimensionalized cutting forces (F_c/kt₁w and F_t/kt₁w); (c) chip thickness (t₂); (d) chip up-curl radius (R_u); and (e) chip back-flow angle (η_b), is theoretically established. The issue of the “built-up-edge” produced under certain conditions in machining processes is also studied. It is hoped that the research work of this paper will help in the understanding of the nature and the basic characteristics of machining processes.     

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.     

Molecular dynamics simulation of deformation behavior in amorphous polymer: nucleation of chain entanglements and network structure under uniaxial tension

In order to clarify the mechanical behavior of molecular chains in amorphous polymers, a molecular dynamics simulation is conducted on a nanoscopic specimen of amorphous polyethylene under uniaxial tension. The specimen involves 3542 random coil molecular chains composed of 500–1500 methylene monomers with about two million methylene groups. The stress–strain curve shows a linear elastic relationship at the initial stage of _zz0.03 at . Then the material “yields” by elongating without stress increase up to the strain of 1.5, where strain hardening appears. Careful investigation of changes in dihedral angle and morphology of all molecular chains reveals that the gauche→trans transition takes place during yielding, generating a new network-like structure composed of entangled molecular clusters and oriented chains bridging them. The strain hardening is due to the directional orientation and stretching of molecular chains between entanglements in the nucleated structure.     

Principally on sheet metal forming defects as described in the eleventh biennial congress of the International Deep Drawing Research Group (IDDRG)

The major defects encountered in sheet metal forming operations are listed and some appropriate references given. The most common defects that arise in press-shop situations as described in the recent congress of the IDDRG are briefly reviewed.Defect—“Want or absence of something necessary for completeness or perfection”.Failure—“Omission to perform or want of success”.From Webster's Dictionary of English.     

Measurement of springback

Springback, the elastically-driven change of shape of a part after forming, has been measured under carefully-controlled laboratory conditions corresponding to those found in press-forming operations. Constitutive equations emphasizing low-strain behavior were generated for three automotive body alloys: drawing-quality silicon-killed steel; high-strength low-alloy steel; and 6022-T4 aluminum. Strip draw-bend tests were then conducted using a range of die radii (3<R/t<17), friction coefficients (0<μ<0.20), and controlled tensile forces (0.5<F_b/F_y<1.5). Springback angles and curvatures were measured for bend and bend–unbend areas of the specimen, the latter corresponding to the “sidewall curl” region, which dominates the geometric change and the dependence on process variables. Friction coefficient and R/t (die-radius-to-sheet-thickness) greater than 5 have modest but measurable effects over the ranges tested. As expected, strip tension dominates the springback sensitivity, with higher forces reducing springback. For 6022-T4, springback is dramatically reduced as the tensile stress approaches the yield stress, corresponding to the appearance of a persistent anticlastic curvature. The presence of this curvature, orthogonal to the principal curvature, violates the simple two-dimensional models of springback reported in the literature. The measured springback angles and curvatures are reported both in graphical summary and tabular form for use in assessing analytical models of springback.     

Hydro-rim deep-drawing processes of hardening and rate-sensitive materials

The conventional deep drawing process is limited to a certain limit drawing ratio (LDR) beyond which rupture will ensue. An asymptotic solution of the complete governing equations of this process indicates that this relatively low LDR results from the steep build-up of radial tensile stress with maximum value at the die lip. This tensile stress is significantly enhanced by interfacial friction along the die/flange and by high speed of the operating load and thus holds responsible for premature ruptures. The intention of this work is to examine the possibilities of relaxing the above limitation, aiming towards a process with an ‘unlimited drawing ratio’. The ideas which may lead to this goal are:(a) exerting an external fluid pressure on the outside rim of the blank (“Hydro-rim process”) to reduce radial stress and to decrease, in parallel, the interfacial friction,(b) increasing the blank temperature to a level at which the material is more rate sensitive, and thus less prone to early failure. The benefits of these ideas are examined via parametric analysis of the solution and with experiments in deep drawing processes.A clear outcome from the solution is that if changes in the material properties (strain hardening, strain-rate sensitivity, yield stress, etc.) can be controlled, say, by controlling the temperature and/or the operating speed, the process can reach higher drawing ratios with substantially less assisted fluid pressure.     

Pressure between elastic bodies having a slender area of contact and arbitrary profiles

A numerical method is presented for calculating the pressure distribution and contact area shape between two elastic bodies of arbitrary profile which make contact over a slender contact area, i.e. where the relative curvature of the two profiles is much smaller in the longitudinal direction than in the transverse. The pressure distribution is assumed to be piecewise-linear in the longitudinal direction and semi-elliptical in the transverse. No a priori relationship is assumed between the shape of the contact area and the longitudinal variation in pressure; they are found simultaneously from dual integral equations for the compatibility of (a) the normal displacement and (b) the transverse curvature along the longitudinal axis of the contact zone.In cases where the profiles of the contacting bodies are smooth and continuous up to, and beyond, the ends of the contact area, the method gives a very reliable measure of the contact pressure distribution. Where discontinuities in profile are present, at roller ends for example, stress singularities are to be expected and like any numerical method, only approximate values of the stress concentration can be found. In the cases studied, the concentration of pressure associated with a “sharp” edge of contact is found to be very local.The method has been applied to both cylindrical and variously “crowned” rollers, also to a ball “over-riding” the edge of a closely conforming groove.