Viscoelastic Solids of Exponential Type. I. Minimal Representations and Controllability

Certain results about state–space representation, minimality and controllability of a linear viscoelastic solid element of exponential type (VSET) are presented. In particular, we prove that VSET can be viewed as materials with finite memory. This is a first part of a plan which will be continued in a next paper, by studying free energies, stability and attractors in viscoelasticity of exponential type.     

Injection moulding of thermoplastics. II. Freezing-off in cavities

The injection moulding of thermoplastics involves, during mould filling, flow of a hot molten polymer into a mould network, the walls of which are so cold that the polymer freezes on them. During the constant pressure drop part of the filling stage, but not during the preceding constant flow-rate part, freezing-off, that is premature blockage of the mould network by frozen polymer, is possible. A semi-quantitative analysis of such freezing-off in a mould cavity is presented here. The length-scales and time-scales of all the relevant physical processes occurring during freezing-off are identified and a criterion is obtained which enables the occurrence of freezing-off to be predicted, at least crudely. This criterion is shown to be corroborated by experimental data for the filling of a spiral mould cavity by three different polymers under a range of different operating conditions.     

The transverse and torsional vibrations of A.C. motors in the starting process

According to the electromagnetic field theory, a set of differential equations is derived for coupling of the transient process for an electrical machine and transverse and torsional vibrations for the rotor by a unified method. A non-dimensional constant coefficient state equation is obtained by transformation. Computation is done. Rules of the change are obtained for transverse and torsional vibration, unilateral magnetic pull, electrical torque, and so on in the starting process. The theoretical results are shown to be in good agreement with the experimental results. The mathematical model verified by the experiment is used to analyse the effect of electromagnetic and mechanical parameters on electromagnetic force and vibration.     

Numerical modeling of multiphase first-contact miscible flows. Part 1. Analytical Riemann solver

In this series of two papers, we present a front-tracking method for the numerical simulation of first-contact miscible gas injection processes. The method is developed for constructing very accurate (or even exact) solutions to one-dimensional initial-boundary-value problems in the form of a set of evolving discontinuities. The evolution of the discontinuities is given by analytical solutions to Riemann problems. In this paper, we present the mathematical model of the problem and the complete Riemann solver, that is, the analytical solution to the one-dimensional problem with piecewise constant initial data separated by a single discontinuity, for any left and right states. The Riemann solver presented here is the building block for the front-tracking/streamline method described and applied in the second paper.     

Hydrodynamics with quadratic pressure. 2. Examples

Exact solutions of Euler equations that describe the motion of an ideal incompressible fluid with quadratic pressure are studied. The solutions are described by explicit formulas and can be physically interpreted. The dynamics of a spherical fluid volume is studied for specified initial velocity fields. It is shown that under certain initial conditions, the spherical volume can evolve into a torusshaped body, thereby changing the connectivity of the region occupied by the fluid.     

Comments on “Vibration suppression for high-speed railway bridges using tuned mass dampers” [J.F. Wang,C.C. Lin,B.L. Chen, 2003. Int. J. Solids Struct. 40(2) 465–491]

In this comment, the discusser makes some remarks on the paper “Vibration suppression for high-speed railway bridges using tuned mass dampers” by Wang, J.F., Lin, C.C., Chen, B.L., published in the International Journal of Solids and Structures, 2003, Vol. 40, No. 2, pp. 465–491. First, the formulation of H(t, t_k) on p. 470 is questionable. Second, for a moving suspension mass model, the interaction force between moving mass and bridge is incorrectly given. Third, for a moving mass model for the train and without the installation of PTMD (passive tuned mass damper), the equation of motion of the bridge is incorrect. Lastly, for the train load model, which consists of one-half of a train car, one bogie, two wheel sets, spring and dashpot between bogie and half of a train car, and spring and dashpot between bogie and each wheel set, the authors did not put forward the formulation of interaction force between wheel set and bridge, but the discusser does.     

Hydrodynamics with quadratic pressure. 1. General results

A wide class of solutions of Euler equations with quadratic pressure are described. In Lagrangian coordinates, these solutions linearize exactly momentum equations and are characterized by special initial data: the Jacobian matrix of the initial velocity field has constant algebraic invariants. The equations are integrated using the method of separation of the time and Lagrangian coordinates. Time evolution is defined by elliptic functions. The solutions have a poletype singularity at a finite time. A representation for the velocity vortex is given.     

A static cone penetrometer for ocean sediments: T.J. Hirst, A.F. Richards and A.L. InderbitzenASTM Spec. Techn. Publ. No. STP 501, 69–80 (1972)

A static cone penetrometer for deep ocean exploration has been developed and successfully tested in soft sea floor sediments at an ocean depth in excess of 1200 m. The penetration resistance of a standard 60-deg cone is detected by appropriately insulated strain gauges whose output may be monitored remotely from the sensing unit. Initial deployment of the device was aboard the submersible Deep Quest in the San Diego Trough off the coast of southern California. Resistances as low as 7 N (1·6 lb) were measured near the surface of the sediment, generally increasing with depth. At the maximum depth of penetration of 1.1 m (restricted by the limited negative buoyancy of the submersible) tip resistance reached 80 N (18 lb). By means of a conventional bearing capacity analysis, the cone resistance was converted to in situ strength and compared to vane shear measurements taken in the immediate vicinity of the penetration tests. Reasonable agreement between vane shear and penetration strengths were obtained for an assumed bearing capacity factor N_c = 9.     

The quasi-stationary structure of radiating shock waves. I. The one-temperature fluid

We calculate the quasi-stationary structure of a radiating shock wave propagating through a spherically symmetric shell of cold gas by solving the time-dependent equations of radiation hydrodynamics on an implicit adaptive grid. We show that this code successfully resolves the shock wave in both the subcritical and supercritical cases and, for the first time, we have reproduced all the expected features – including the optically thin temperature spike at a supercritical shock front – without invoking analytic jump conditions at the discontinuity. We solve the full moment equations for the radiation flux and energy density, but the shock wave structure can also be reproduced if the radiation flux is assumed to be proportional to the gradient of the energy density (the diffusion approximation), as long as the radiation energy density is determined by the appropriate radiative transfer moment equation. We find that Zel'dovich and Raizer's (1967) analytic solution for the shock wave structure accurately describes a subcritical shock but it underestimates the gas temperature, pressure, and the radiation flux in the gas ahead of a supercritical shock. We argue that this discrepancy is a consequence of neglecting terms which are second order in the minimum inverse shock compression ratio [, where is the adiabatic index] and the inaccurate treatment of radiative transfer near the discontinuity. In addition, we verify that the maximum temperature of the gas immediately behind the shock is given by , where is the gas temperature far behind the shock. Received 21 September 1998/ Accepted 2 February 1999     

Transition waves in bistable structures. I. Delocalization of damage

We consider chains of dimensionless masses connected by breakable bistable links. A non-monotonic piecewise linear constitutive relation for each link consists of two stable branches separated by a gap of zero resistance. Mechanically, this model can be envisioned as a ”twin-element” structure which consists of two links (rods or strands) of different lengths joined by the ends. The longer link does not resist to the loading until the shorter link breaks. We call this construction the waiting link structure. We show that the chain of such strongly non-linear elements has an increased in-the-large stability under extension in comparison with a conventional chain, and can absorb a large amount of energy. This is achieved by two reasons. One is an increase of dissipation in the form of high-frequency waves transferring the mechanical energy to heat; this is a manifestation of the inner instabilities of the bonds. The other is delocalization of the damage of the chain. The increased stability is a consequence of the distribution of a partial damage over a large volume of the body instead of its localization, as in the case of a single neck formation in a conventional chain. We optimize parameters of the structure in order to improve its resistance to a slow loading and show that it can be increased significantly by delocalizing a damage process. In particular, we show that the dissipation is a function of the gap between the stable branches and find an optimal gap corresponding to maximum energy consumption under quasi-static extension. The results of numerical simulations of the dynamic behavior of bistable chains show that these chains can withstand without breaking the force which is several times larger than the force sustained by a conventional chain. The formulation and results are also related to the modelling of compressive destruction of a porous material or a frame construction which can be described by a two-branched diagram with a large gap between the branches. We also consider an extension of the model to multi-link chain that could imitate plastic behavior of material.     

Secondary buckling of a flat plate under uniaxial compression: Part 2. Analysis of clamped plate by F.E.M. and comparison with experiments

In Part 1, a theoretical analysis was used to study the secondary buckling of a simply supported plate. In Part 2, a clamped plate is analysed by the finite element method. The stability criterion for a non-linear post-buckling equilibrium state is evaluated by the sign of the determinant of the stiffness matrix. It should be noted that the secondary buckling loads of clamped plates are unexpectedly smaller than those of simply supported plates and are only one and a half times the primary buckling loads. In previous analyses, only a quarter segment of the plate was considered by assuming a stable equilibrium state with symmetrical mode. However, instability can also be predicted by considering the unsymmetrical mode over the whole plate. Results of experimental analysis of secondary instability of clamped square plates under uniaxial compression agreed with the numerical results.     

Nonlinear Dynamics and Stability of a Two D.O.F. Elastic/Elasto-Plastic Model System

The local and global nonlinear dynamics of a two-degree-of-freedom model system is studied. The undeflected model consists of an inverted T formed by three rigid bars, with the tips of the two horizontal bars supported on springs. The springs exhibit an elasto-plastic response, including the Bauschinger effect. The vertical rigid bar is subjected to a conservative (dead) or non-conservative (follower) force having static and periodic components. First, the method of multiple scales is used for the analysis of the local dynamics of the system with elastic springs. The attention is focused at modal interaction phenomena in weak excitation at primary resonance and in hard sub-harmonic excitation. Three different asymptotic expansions are utilised to get a structural response for typical ranges of excitation parameters. Numerical integration of the governing equations is then performed to validate results of asymptotic analysis in each case. A full global nonlinear dynamics analysis of the elasto-plastic system is performed to reveal the role of plastic deformations in the stability of this system. Static 'force-displacement' curves are plotted and the role of plastic deformations in the destabilisation of the system is discussed. Large-amplitude non-linear oscillations of the elasto-plastic system are studied, including the influence of material hardening and of static and sinusoidal components of the applied force. A practical method is proposed for the study of a non-conservative elasto-plastic system as a non-conservative elastic system with an 'equivalent' viscous damping. This revised version was published online in July 2006 with corrections to the Cover Date.     

On Modal Interaction,Stability and Nonlinear Dynamics of a Model Two D.O.F. Mechanical System Performing Snap-Through Motion

Dynamics of a simple two degrees of freedom (d.o.f.) mechanical system is considered, to illustrate the phenomena of modal interaction. The system has a natural symmetry of shape and is subjected to symmetric loading. Two stable equilibrium configurations are separated by an unstable one, so that the model system can perform cross-well oscillations. Nonlinear statics and dynamics are considered, with the emphasis on detecting conditions for instability of symmetric configurations and analysis of bi-modal non-symmetric motions. Nonlinear local dynamics is analyzed by multiple scales method. Direct numerical integration of original equations of motions is carried out to validate analysis of modulation equations. In global dynamics (analysis of cross-well oscillations) Lyapunov exponents are used to estimate qualitatively a type of motion exhibited by the mechanical system. Modal interactions are demonstrated both in the local dynamics and for snap-through oscillations, including chaotic motions. This mechanical system may be looked upon as a lumped parameters model of continuous elastic structures (spherical segments, cylindrical panels, buckled plates, etc.). Analyses performed in the paper qualitatively describe complicated phenomena in local and global dynamics of original structures.     

Experimental methodology for the tappet characterization of timing system in I.C.E.

The aim of present work is the containment of the inertia forces, the stiffness components optimization and the fit tolerances of valve train in internal combustion engines (I.C.E.) 4T. The proposed methodology allows, through the development of a test machine, the evaluation of axial stiffness of tappet depending on eccentricity of the cam tappet contact, performing a functional analysis that simulate the behaviour of the system in operational condition, even if, some adjustment of tolerances of the fit between tappet and his guide, occurred. The dynamic study of the valve train, through modern computer codes, is performed by connecting lumped masses, springs and dampers that characterize each element. In numerical models the tappet is represented as constituted by the tappet and by the hydraulic element. Each of these elements is characterized by stiffness and mass. The structural rigidity of the tappet has, in fact, important effects on the dynamic behaviour of the entire valve train. The test machine makes possible the choice of the dimensional and geometrical tolerances of the fit between tappet and his guide; allows furthermore the evaluation of errors occurred during construction and integration phase. In addition, the test machine is also suitable for reverse engineering applications, makes it possible to automatically draw the cam profile in polar coordinates.     

Exact Relations for Effective Tensors of Polycrystals. I. Necessary Conditions

The set of all effective moduli of a polycrystal usually has a nonempty interior. When it does not, we say that there is an exact relation for effective moduli. This can indeed happen as evidenced by recent results [4, 10, 12] on polycrystals. In this paper we describe a general method for finding such relations for effective moduli of laminates. The method is applicable to any physical setting that can be put into the Hilbert space framework developed by Milton[13]. The idea is to use the W-function of Milton[13] that transforms a lamination formula into a convex combination. The method reduces the problem of finding exact relations to a problem from representation theory of SO(d)(d= 2 or 3) corresponding to a particular physical setting. When this last problem is solved, there is a finite amount of calculation required to be done in order to answer the question completely. At present, each candidate relation has to be examined separately in order to confirm the stability under homogenization. We apply our general theory to the settings of conductivity and two‐dimensional elasticity. (Accepted April 4, 1997)     

Analysis of pipe flow transition. Part II. Energy transfer

The present article describes the results from a study of nonlinear mechanisms at work during the process of transition to turbulence in pipe flows. Using an accurate hybrid finite-difference code for the simulation of unsteady incompressible pipe flow, we have performed a direct numerical simulation designed to model experiments performed by Han, Tumin and Wygnanski [12]. Based on these numerical data, we have conducted a meticulous investigation of the dynamic interactions of the structures and flow modes that can be observed during this process. Based on this study, we can paint a detailed picture of the dynamical interactions of flow structures during both the linear and nonlinear stages of pipe flow transition. While this picture does have some similarities to earlier proposed mechanisms, we find that even for the simple cases considered here the structure of the pertinent interactions is much richer than suggested by these earlier models.     

A non-linear viscoelastic model with structure-dependent relaxation times: III. comparison with l.d. polyethylene creep and recoil data

The equations of a model previously proposed by the authors to describe the non-linear viscoelastic behaviour of polymer melts are elaborated here to predict creep and recoil responses.The comparison with existing data for both shear and elongational flow shows good agreement and supports the assumption of a material which is destructured by a prolonged application of high deformation rates.     

Run-up and spin-up in a viscoelastic fluid. I.

Summary An incompressible viscoelastic fluid is contained between parallel rigid plates which at some instant of time are subjected to a translational velocity which is then held constant. The dependence on time of the resulting flow field in the fluid is calculated. Similar calculations are carried out when the fluid is contained in an infinitely long circular cylinder which is set in longitudinal motion or in rotation.It is seen that for a certain broad class of viscoelastic fluids the transfer of momentum, or angular momentum, from the boundary to the interior of the fluid takes place by a mechanism which is essentially diffusive in character. For another broad class of fluids, of which the Maxwellian fluid is a special case, the transfer of momentum results from the reflection back and forth of a velocity shock wave. These reflections take place at the boundaries in the case of runup between parallel plates and at the boundary and axis in the case of run-up, or spin-up, in a circular cylinder.With 4 figures     

In memoriam Professor A. G. Gaydon F.R.S. (1911-2004)

I, a starting post-graduate in 1958, first met Dick Gaydon in the basement area of the Chemical Engineering Department of Imperial College where the nonchemical engineers were exiled. It was sometime later, when I suggested to my employers, the Central Electricity Research Laboratories, that they should produce sufficiently handsome funding for me to build and operate a shock tube. They agreed, but on condition that Dick become a consultant and keep an eye on my progress. For some years my colleagues, Bob Stirling and Sammy Sloan, and I received monthly visits.Some will have already read obituaries and additional tributes in the daily press and combustion journals. There, all have paid full regard to his outstanding contributions as one of the 20th centurys supreme spectroscopists, particularly in his accurate determination of the dissociation energy of nitrogen, with its implications for the correct forecasting of the strengths of blasts from explosions of atomic weapons. Whilst his great love in science concerned experimental spectroscopy, he devoted much effort to the development of various techniques to excite radiation. Principal amongst these were flames produced by novel types of burner and gases heated and compressed in shock tubes.Many will be familiar with his six books, some written in collaboration with colleagues such as Pearce on spectroscopy, Wolfhard on flames and Hurle on shock tubes. That on flames, together with the text by Lewis and von Elbe, formed the principal wells of information from the 1960s to the 1990s, with the former running to three completely revised editions over a couple of decades. Indubitably, the Gaydon and Hurle monograph will be most familiar to subscribers to Shock Waves. It was one of a number on shock waves published in the 1960s, e.g., Soloukhin, Bradley and Tonnies and Green. My feeling is that the Gaydon and Hurle monograph answered more experimental problems than did the others.Whilst continuing to be somewhat isolated from the chemical engineering interests of the department, he had over his career at Imperial College a wide range of physicists and chemists, both from this country and abroad, occupying post-graduate and post-doctoral positions, many of whom went on to highly distinguished scientific careers. All in all they contributed some 150 scientific papers of high merit to the most eminent of journals.Dick had many friends and collaborators worldwide. As with most of the Western World, he was highly disturbed by the USSR invasion of Hungary and, although much troubled by his conscience, gave up correspondence with long established friends amongst Russian scientists. It was some time before he rehabilitated these.He always regarded himself as an experimentalist. As the most modest of men, he would never have regarded himself as a brilliant one, though all his colleagues did. He was elected to the Royal Society, possibly in more harmonious times, in 1953. Dick was also the kindest of men. I can still recall his questioning which attempted to avoid causing hurt to presenters of contentious papers: so politely were these framed that the authors frequently remained completely unaware of the dismantling of the foundations of their case. I have often wondered how they felt when the penny eventually dropped.He had an abiding passion for butterflies and moths and travelled the world taking photographs of them. No easy task for somebody who lost the sight of one eye through an explosion of peroxide of diethyl ether, when purifying the ether itself, early in his career. When travel proved too much for him a pet dog took their place in his affections.Published online: 18 February 2005[/PUBLISHED]