“Antient” philosophy-science in Magna Graecia: Cassiodorus and Pythagoras and 20th c. literary travellers, Gissing and Douglas

The paper's philosophy-science time-scale stretches over a millenium, from 6th c. B.C. Pythagoras and into 6th c. A.D., Cassiodorus. Interest centres on Greek “science”, especially that of Pythagoras, in the colonies in Greater Greece — the foot of Italy — and on the writings of Capella and Boethius in the 5th c. A.D., as appreciated by Cassiodorus. His regard for the latters’ work and interest in its transmission to the universities in the Middle Ages became the Trivium and Quadrivium, the seven Liberal Arts-Sciences. The case is argued that Cassiodorus was probably the instigator of Benedictine regard for scholarship and was pivotal in facilitating the transmission of the latter “Seven Pillars of Wisdom”. Current general interest in the ancient classical scholarship of this region was stimulated by two early 20th c. literary travellers, Gissing and Douglas, whose writings, after visiting the region, we describe.     

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)].     

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.     

Kinematic hardening analysis of ratchet strain in the “pulley test”

The classical Bree problem—which represents an uniaxial model of a thin tube subjected to combined internal pressure and cyclic thermal stress across its wall—can be simulated by means of the pulley test in which a wire or strip specimen is subjected to combined steady tensile stress and cyclic bending stress. In this paper, accumulation of ratchet strain in the pulley test is investigated using a linear kinematic hardening material model from which perfect plasticity can be generated as a special case. The results of the investigation show that asymptotic ratchet strains are linearly related to the excess in mean stress σ_D above its value σ*_D at the ratchetting limit regardless of the thermal stress amplitude. Comparisons with test results on copper wire specimens—which exhibit non-linear hardening rate—confirm the qualitative validity of this simple relation. Deviations between theory and experiment are attributed to metallic cyclic creep. Further, perfect plasticity results are shown to be well predicted by a linearized lower bound estimate.     

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.     

Benjamin robins: Anonymous pamphlets and reports of 1739–1742, shortly summarized

Benjamin Robins had printed two political pamphlets (Nos 1 and 2 below) in 1739 and a third one (No. 3) which is his to the extent that it was grossly “disfigured” before being put “abroad”. A pamphlet (No. 4) not by Robins is included here because it is a short, interesting, anonymous answer, in effect, to pamphlet No. 1.In 1742, Robins was Secretary of a House of Commons Secret Enquiry into Lord Orford's conduct which produced its report in May (No. 5) and which was followed by a “leaked” one (No. 6) in June.All the pamphlets (save No. 4) and reports came out anonymously and were it not for James Wilson's biography of Robins which prefaces Wilson's collection of his Mathematical Tracts (printed in 1761), we should not know of Robins' involvement in them.Wilson refers only minimally to these documents (again, No. 4 excepted), but historians of science since 1761 seem not to have read and commented on them. For this reason, we now give below summaries of their contents. Among other things, the issues addressed reflect the turmoil of the age as described in a companion paper, W. Johnson, “Called to publick employment … a very honourable post.” To be published (1993).     

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.     

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.     

Concluding note about a university at Stamford and Francis Peck's academia tertia anglicana: The antiquarian annals of Stamford [1]

The author herewith concludes his work [Int. J. Mech. Sci.33, 675 (1991); 34, 831 (1992)] on the subject of a university at Stamford—a university which “there never was”—but an institution of which there was promise in the early 14th Century. When one did seem likely to be realized, it was suppressed by the then-King of England. This association, mostly in “halls”, was within 13 miles of Newton's birthplace, Colsterworth, Lincolnshire and so might well have been attended by him had it existed in the mid-17th Century. Reliable details of this potential centre of learning are difficult to come by, but a major source of information chanced upon by the writer has the title given above and is a large volume of several hundred pages, Academia tertia Anglicana (The Third English Academy), composed by Francis Peck in 1727. The author gives, briefly, some items from the latter work which should help those who might henceforth wish to penetrate more deeply into this subject; for the mass of readers it provides a simple though partial picture of how some European universities started early in this millenium.     

Improvements in the prediction and characterization of the thermoelastic response of glass-fibre reinforced composite products

Significant improvements in the prediction of the thermoelastic constant of SMC (sheet moulding compound) have been made as a result of the use of the iso-stress version of the Young's modulus of the polyester—limestone matrix (i.e. the “series” rule of mixtures version) in place of the iso-strain version (i.e. the “parallel” rule of mixtures version). The thermoelastic constants of GMT (glass-mat-reinforced thermoplastic) and RTM (resin-transfer-moulded material) can also be accurately predicted. The property variations in large moulded GMT panels, with nominal glass contents of 20% and 40%, have been studied using the SPATE equipment and a thermoelastic technique has been developed, based on the simple disc specimen subjected to two opposed in-plane compressive forces, for the determination of the principal material directions in these panels.     

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.     

An efficient approach to characterize pseudo-merohedral twins by precession electron diffraction: Application to the LaGaO3 perovskite

Pseudo-merohedral twins are frequently observed in crystals displaying pseudo-symmetry. In these crystals, many [u v w] zone axis electron diffraction patterns are very close and can only be distinguished from intensity considerations. On conventional diffraction patterns (selected-area electron diffraction or microdiffraction), a strong dynamical behaviour averages the diffracted intensities so that only the positions of the reflections on a pattern can be considered. On precession electron diffraction patterns, the diffracted beams display an integrated intensity and a “few-beam” or “systematic row” behaviour prevails which strongly reduces the dynamical interactions. Therefore the diffracted intensity can be taken into account. A procedure based on observation of the weak extra-reflections connected with the pseudo-symmetry is given to identify without ambiguity any zone axis. It is successfully applied to the identification and characterization of {1 2 1} reflection twins present in the LaGaO₃ perovskite.     

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.     

Tribological behaviour of duplex treated Ti–6Al–4V: combining nitrogen PSII with a DLC coating

The chemical structure and tribological behaviour of Ti–6Al–4V plasma source ion implanted with nitrogen then DLC-coated in an acetylene plus hydrogen-glow discharge (bias voltage −10 to −30 kV) were investigated. The as-modified samples have a TiN/H:DLC multilayer architecture (coating resistivity 1.6×10⁹ to 2.4×10¹¹ Ω/cm) and exhibit higher hardness, especially at low loads or plastic penetrations in the order of deposition bias voltage −10, −20 and −30 kV. At a lower contact load (1 N) and higher sliding speed (0.05 m/s), frictional properties in most cases improved, as did wear properties. At a higher contact load (5 N) and lower sliding speed (0.04 m/s), friction showed almost no improvement, and wear properties deteriorated. When the material of the counterbody was then changed from AISI 52100 to Ti–6Al–4V modified as the disc (contact load 5 N unchanged, sliding speed decreased), the friction coefficient decreased (but showed no improvement compared with the unmodified sample), while wear properties deteriorated further, and wear was changed from just the disc to both disc and ball, abrasive and adhesive dominated. Transfer films, mainly made up of wear debris transferred from the disc wear surfaces, were formed on the wear scars of the counterbodies. The deterioration of wear properties of the modified samples at the higher contact load is considered to be caused by the “thin ice” effect.     

Fireworks and firemasters of England, 1662–1856

The origin and facts of some early firework-rocket history are related mainly as applied to Restoration England. The two facets of its management, the tending to the details of compositions for rocket propulsion and to the actual shooting of rockets, are identified in the titles “Fireworkers” and “Firemasters”. A fine example of the art of the latter was the triumphal celebration of the peace of Aix-la-Chapelle, which we describe noting a small part for Benjamin Robins. The little known works of Sir Martin Beckman, Colonel H. J. Hopkey and Sir William Congreve are recalled. Rockets in firework displays, though once used on occasions of national celebration in a manner seldom remembered today, were discontinued in 1856 with the end of the Crimean War.Essentially, this paper continues some of the themes of W. Johnson, The rise and fall of early war rockets, in Int. J. Impact Engng15(4), 365 (1994); Congreve's details of the rocket system and the Artillery Museum in the Rotunda at Woolwich, London, in Int. J. Impact Engng (in press) (1994); and S. Clyens and W. Johnson, Fra Leipzig til London or from Leipzig to London: a translation from the Danish, in Int. J. Impact Engng (in press) (1994).     

New cylinder liner surfaces for low oil consumption

Anticipated emission legislation and reduced fuel consumption are the main driving forces when developing new engines. Optimization of the active surfaces in the piston system is one possible way to meet the above demands. In this study the effects of surface topography and texture direction of the ring/liner contact on oil film thickness and friction were simulated and experimentally tested. “Low wear” results from the experimental wear tests with “glide honed” smooth liner surfaces supported the “low friction” simulation results. In addition a new wear volume sensitive surface roughness parameter, R_ktot, based on the Abbot–Firestone bearing area curve was introduced.     

Analysing the vibrating conveyor

Several possible vibrating conveyor designs—the “sealskin” conveyor, the “jerk” conveyor, the conveyor with inclined motion and the conveyor with out-of-phase motion—are first discussed and their relative performances are compared. The most promising (and most widely employed) design, namely the conveyor with inclined motion, is then analysed in detail. The analysis is in all cases limited to conveyors with a horizontal track and with vertical track accelerations not exceeding g, the gravitational constant. Since the analytical performance prediction of the conveyor with sinusoidal motion is very difficult, an alternative type of motion having a triangular velocity profile is introduced. This allows analytical expressions for conveyor performance by simple methods to be derived. The results of this analysis are compared with the theoretical predictions for sinusoidal motion obtained by means of a simulation on the analogue computer. To simplify the presentation of the performance results, the design variables are reduced to two non-dimensional parameters, one for vertical acceleration and the other for horizontal sliding. The performance, too, is described by a non-dimensional parameter, so that the completely general performance maps presented for triangular and sinusoidal track velocity profiles need contain only three parameters.     

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.     

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.     

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.