Penrose''s volume: An investigation of the principles of Athenian Architecture

We describe the contents of the second edition of Penrose's book of 1888 about his on-site investigations, findings and conclusions concerning his researches into Athenian Architecture. The volume emphasises, “the incorrectness of the belief that the system of design in (classical) Greek architecture was absolutely rectilinear”. We review only the chapters of his book of immediate relevance to us, though referring to seven Appendices (the first by author W. W. Lloyd), for historical and mathematical background. The attention of the reader is drawn to an accompanying paper ^[2] giving the biographies of, mostly, English investigators who, principally, explored the title subject before Penrose, extending over a period of about 150 years.     

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.     

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.     

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.     

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.     

“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.     

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).     

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.     

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.     

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).     

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.     

The curious mechanical euclid of William Whewell, F.R.S. (1774–1866)

The origin of the little known and virtually unread work of well-known polymath William Whewell, The Mechanical Euclid, (1837) seems ascribable to 19th century concern with the exact teaching of Euclid. The book probably initially stimulated the interest of Isaac Todhunter and Charles Dodgson who so loudly championed Euclid a generation later. The main purpose of this Note relates to their concern with Euclid and rigour.In writing about one of Whewell's books we do not neglect to observe that this year is the 200th anniversary of his birth.^†     

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.     

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.     

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     

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.     

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.