“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.     

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.     

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.     

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.     

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).     

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.     

Deformation processing of metal powders: Part I—Cold isostatic pressing

Analytical and numerical techniques are used to analyze in detail the stage I Cold Isostatic Pressing (CIPing) of metal powders. Plastic yielding is considered to be the only densification mechanism and the constitutive model developed recently by Fleck, Kuhn and McMeeking [(J. Mech. Phys. Solids40, 1139 (1992)] is used to describe the elastoplastic behavior of the metal powder. It is shown that, for the case of powder consolidation in a long cylindrical tube, a change in length is not possible, unless it is accompanied by some distortion (“shape change”) of the CIPed specimen. The numerical implementation of the constitutive elastoplastic equations in a finite element program is discussed. The finite element method is used to analyze the CIPing of titanium powder. The predictions of the finite element solution agree well with available experimental data.     

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.     

Collapse behaviour of aluminium plates

Theoretical predictions of the collapse behaviour of thin rectangular aluminium plates are presented and discussed. The plates are simply supported around all four edges, and are subjected to uniaxial in-plane compression. The method of analysis developed by the author is described elsewhere. The material behaviour of aluminium is characterised by the Ramberg-Osgood stress-strain formula. Two methods of using this formula are discussed—one based on the usual percentage proof stress, and the other on the stress at which the plastic strain is a stated proportion of the elastic strain. It is shown why the latter method is more suited to the normalisation of results for collapse analyses. The results presented here give an indication of the effects on plate strength of plate slenderness, initial out-of-flatness, and n, the “knee” factor of the material stress-strain curve.     

Coating thickness effects on initial wear of nitrogen-doped amorphous carbon in nano-scale sliding contact: Part I—in situ examination

This paper, the first of a two-part series, presents the empirical data obtained from in situ examination on the generation of wear particles on carbon nitride coatings by a spherical diamond counter-face during repeated sliding contacts. In particular, the effect of coating thickness, varying from 1 to 500 nm, on the generation of wear particles was examined.Based on the in situ examination, the shape transition maps for generated wear particles were obtained for carbon nitride coatings of various thickness. The results show that the critical number of friction cycles, Nc, for the transition from “no observable wear particles” to “wear particle generation” generally increased with increasing coating thickness. It was noted that up to 20 friction cycles, the maximum Hertzian contact pressure, P_max, for “no observable wear particles” regime can be increased from 1.39Y to 1.53Y if silicon was coated with carbon nitride coating thicker than 10 nm, where Y is defined as the yield strength of silicon.     

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.     

Encounters between Robins, and Euler and the Bernoullis; artillery and related subjects

Euler (1707–1783) was a close friend of the Bernoulli family, and John Bernoulli I as a result of not winning a French Academy prize offered for essays on the subject of Impact, published his effort—only to have it examined and worsted by criticism from a then young, unknown 20 year old, Benjamin Robins (1707–1751). This encounter probably led Robins to his work in ballistics, the ballistic pendulum for measuring musket ball speed and the writing of his New Principles of Gunnery, ultimately published in 1742.Euler (born in the same year as Robins), and presumably knowing of the latter scientific clash, seemingly became interested in artillery and the trajectories of projectiles. His first work is dated 1727, three months after Robins' critique of Bernoulli, but it was not published until many years afterwards. Also, through association with the St Petersburg Academy, by means of which Peter the Great intended to build up the Russian Navy and commerce, Euler likely developed his deep interest in naval science, tides, fire, meteorology and artillery. Robins' book was devoured by Euler and in 1745 he brought out his own book, in German, on The New Principles of Artillery, which contained a translation of Robins' book with sections containing “Anmerkung” (Remarks) which brought it up to several times the length of Robins' own book. Fortunately, the latter was translated from German into English by Hugh Brown and appeared in 1777.Almost nothing has been written by latter-day authors concerning the interaction of Robins and Euler, and somewhat the Bernoullis, apart from that of Truesdell and Scherrer. The latter's editorial and prefatory work on Robins is in German, as is all of Euler's writing in German, French or Latin. Truesdell's writings of interest to us are in English but are mostly confined to matters of fluid dynamics.The immediate aim of this paper is to make some of the details of the aforesaid interaction between Robins, Euler and the Bernoullis better known and to try to improve understanding of it, also casting light on some of the scientific questions of the mid-18th century.     

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.     

Voltaire's contribution to the spread of newtonianism—I. letters from england: Les lettres philosophiques

Part I: In early middle life Voltaire had a considerable interest in science, provoked by a visit to London and by learning at close range of the work of men like Newton and Locke, which had developed in a more scientific direction than that of the Cartesianism then current and almost universal in France. With courage, and to the chagrin of his own countrymen, Voltaire advocated the empirically based work or Natural Philosophy of Newton. He published a thin volume known as the Letters from England, 1733 or Les Lettres Philosophiques in 1734, which brought him into conflict with the officialdom of France and then exile. We describe and discuss the contents and background of this very influential book and entertain some comments made about it, then and since. For us, the philosopher John Locke here comes to play a larger role in this subject than he normally assumes in discussions of Newtonianism. We endeavour to show too the roles played by Jacques Rohault and Samuel Clarke.Part II: Resting on Part I we discuss Voltaire's larger volume of 1738, on Elements of Newton's Philosophy. Mme Emilie du Châtelet (or Chastelet) was enormously influential in the writing of it and indeed is often credited as having been truly a co-author. She made a translation of Newton's Principia which was posthumously and openly published for her in 1759, by Clairaut, 10 years after her death. Mme du Châtelet and Voltaire were greatly responsible for spreading Newton's views in France; wrote Miller in [1(a)], “... he conquered the mind of the Continent by popularising Newton”. We review the translation of the Elements, its reception, and again describe the background of the period. Our subject is a little-known story that might well have been an entertaining and timely lecture for students of mechanics three years ago on the occasion of the tercentenary of the Principia [1(b)].     

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.     

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).     

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.     

Some observations on the CLNA model in fretting fatigue

Using the Atzori–Lazzarin criterion, the author has recently proposed a unified model for Fretting Fatigue denominated Crack-Like Notch Analogue—CLNA model, considering only two possible behaviours: either “crack-like” or “large blunt notch”. In a general FF condition, the former condition is treated with a single contact problem corresponding to the MIT Crack Analogue (CA) improved in some details also by the author. The latter, with a simple peak stress condition, i.e. a simple Notch Analogue model, simply stating that below the fatigue limit, infinite life is predicted for any size of contact. In the typical condition of constant normal load and in phase oscillating tangential and bulk loads, both limiting conditions are immediately written, and the CLNA model permits to collapse the effect of the contact loads on a single closed form equation (differently from many other models which do not permit this flexibility). For not too large contact areas (“crack-like” contact) no dependence at all on geometry is predicted, but only on 3 load factors (bulk stress, tangential load and average pressure) and size of the contact. Only in the “large blunt notch” region occurring typically only at very large sizes of contact does size-effect disappear, but the dependence on all other factors including geometry remains. The model compares favourably with some experimental results in the literature. In this paper, some aspects of the CLNA model are further elucidated.     

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.