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.     

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.     

Rotor dynamic instability analysis on hybrid air journal bearings

Hybrid air journal bearings with multi-array of 1, 2, 3, 4, or 5-row orifice feedings are analyzed for the problem of rotor dynamic instability. The bearing stiffness and damping coefficients are calculated numerically to determine threshold rotor mass under various operating conditions. The hybrid porous air journal bearings are also analyzed for comparison to investigate the similarities in dynamic characteristics between the multi-array of orifice feeding bearings and the porous bearings. The results show that the porous bearing is more stable than the orifice feeding bearing at lower rotation speeds (Λ<0.1) or at higher rotation speeds (Λ>1) with lower feeding parameters (λ_P<10⁻⁸). The 5-row orifice feeding bearing is more stable than the porous bearing at moderate speeds (0.3<Λ<0.6) with lower feeding parameters (λ₀<10⁻⁴).     

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.     

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)].     

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.     

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.     

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.     

Cohesive and continuum damage models applied to fracture characterization of bonded joints

In this work, two different methods for simulating damage propagation are presented and applied to fracture characterization of bonded joints in pure modes I and II. The cohesive damage model is based on a special developed interface finite element including a linear softening damage process. In the continuum damage model the softening process is performed by including a characteristic length associated with a given Gauss point. The models were applied to the simulation of “double cantilever beam” (DCB) and “end notched flexure” (ENF) tests used to obtain the critical strain release rates in mode I and II of bonded joints. In mode I it was observed, under certain conditions, a good agreement between the results obtained by the two models with the reference value of critical strain energy release rate in mode I (G_Ic), which is an inputted parameter. However, in mode II some discrepancies on the obtained G_IIc values were observed between the two models. These inaccuracies can be explained by the simplifying assumptions inherent to the cohesive model. Better results were achieved considering the crack equivalent concept.     

Frictionally excited thermoelastic instability in viscoelastic and poroelastic media

Friction materials commonly used in sliding applications, such as clutches and brakes, can be poroelastic and exhibit a viscoelastic behaviour. To the author's knowledge, there are no comprehensive analysis of the influence of poroelastic and viscoelastic material properties on the onset of the phenomenon of frictionally excited thermoelastic instability in sliding systems. This issue is here analysed in some details. Firstly, a linear standard model for the friction material is adopted, introducing an effective complex dynamic modulus E=|E|e^jδ and individuating three independent parameters, E₁, E₂/E₁ and c₂/E₁, that fully describe its viscoelastic behaviour. Subsequently, a similarity between viscoelastic and poroelastic formulation is presented and the three independent parameters introduced are related to the viscosity of the fluid μ_f, the permeability k_p and elastic properties M, α_B of the porous material.The linear elastic formulation proposed by Decuzzi et al. (ASME J. Tribiol. 2001;123:865) has been modified in order to take account of the new constitutive model and the variation of the critical sliding speed with the wave parameter, and viscoelastic/poroelastic properties of the material are examined.It has been found that the susceptibility towards thermoelastic instability increases by increasing both the elastic E₂/E₁ and viscoelastic c₂/E₁ parameters, or by increasing the Biot modulus M and effective stress coefficient α_B, the viscosity μ_f of the fluid, and by reducing the permeability k_p of the porous skeleton. It has been shown that for porous friction materials employed in wet clutches which are weakly viscoelastic, the neglect of its poroelastic behaviour leads to an overestimation of the critical speed smaller than 10%. However, much larger variations are predicted for elastomeric and porous materials with more pronounced viscoelastic behaviour.     

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.     

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.     

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.     

Stability of a rigid rotor in turbulent hybrid porous journal bearings

Stability characteristics of hybrid porous journal bearings with a turbulent fluid film have been investigated theoretically following Constantinescu's turbulent lubrication theory. The stability curves have been drawn for different Re, eccentricity ratios, slenderness ratios and bearing speed parameters. In the absence of any experimental data, laminar flow results obtained by this analysis have been compared and found to be in excellent agreement with the previous results. It is observed that turbulence deteriorates the stability of the rotor and for better performance the value of the bearing feeding parameter, β, should be kept small.     

Plastic flow localization in mechanism-based strain gradient plasticity

The theory of mechanism-based strain gradient (MSG) plasticity is used to study plastic flow localization in ductile materials. Unlike classical plasticity, the thickness of the shear band in MSG plasticity can be determined analytically from a bifurcation analysis, and the shear band thickness is directly proportional to the intrinsic material length, (μ/σ_Y)²b associated with strain gradients, where μ is the shear modulus, σ_Y is the yield stress, and b is the Burgers vector. The shear band thickness also depends on the softening behavior of the material. The analytical solution of the shear strain rate yields that the maximum shear strain rate inside the shear band is two orders of magnitude higher than that outside, which is a clear indication of plastic flow localization. The limitation of the present model is also discussed.     

Modeling mechanical response of intumescent mat material at room temperature

Intumescent mat material is widely used to support ceramic substrates in catalytic converters and behaves very much like hyper-foam material under compressive loading. Experiments show that compressive loading curves depend on the ram speed and the number of cycles. The unloading curves show different slopes and paths that depend less on the ram speed and number of cycles. The slopes of the unloading curves decrease as the plastic strain increases; this is referred to as “softening” in this study. The effects of rate, softening, and plastic deformation must be considered to model the mechanical response of intumescent mat material. Finite deformation theory is applied with a multiplicative decomposition of the deformation gradient tensor. The developed theory is implemented as an implicit finite element algorithm in ABAQUS^TM/STANDARD. The necessary material parameters are extracted from experiments. Numerical simulations show good agreement with experiments.     

Viscoelastic flow between two infinite parallel porous plates,one plate oscillating and the other plate in uniform motion

The problem of viscoelastic incompressible flow between two infinite parallel porous plates, one oscillating and the other in uniform motion, was studied and an iteration method was used to solve the fluid dynamical equations. The solution obtained is valid for small values of the elastic parameter S. The effects of the flow parameters S and σ (the Reynolds number) on the velocity distribution and on the shearing stress at the plates are presented graphically.     

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.     

Scaling in combined plastic flow and fracture

The scaling laws are given for bodies undergoing simultaneous plastic flow and crack propagation, deformations which can be adequately described by rigid-plastic fracture mechanics. The laws depend on (i) a material-dependent term given by the ratio of plastic work done/volume ∫σ dε to the material fracture toughness R for the given pattern of deformation, as well as on (ii) a geometrical term given by the ratio, in the reference model structure, of the volume of material plastically deformed, V, to the crack area, A. The two contributing factors are combined in a single non-dimensional parameter . Energy scaling in prototype (p) and model (m) follows

Full-size image (<1K)

, which is the true form of λ^x-type empirical relations < 3; for fracture alone, V = 0, so ξ = 0 and x = 2; for plastic flow without fracture A = 0, so ξ = ∞ and x = 3. Associated scaling relationships for loads, stresses and displacements between model and prototype are also given. All scaling relations are functions of ξ, which is an arbitrary parameter, since it depends on the particular size chosen for the reference model structure. That is, the magnitude of the ratio of scaled quantities (and value of x in λ^x empirical relations) depends on the absolute size of the model. The reason for this curious state of affairs is found in the coordinate geometry of the linear plot of ( ) vs ( ) which is central to rigid-plastic fracture mechanics. The new scaling laws agree well with a wide range of quasi-static and dynamic experimental data on scaled bodies. They help to explain hitherto anomalous behaviour in the impact of scaled structures.     

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.