On the general form of the law of dynamics

The dynamics of Lagrangian systems is formulated with a differential geometric approach and according to a new paradigm of the calculus of variations. Discontinuities in the trajectory, non-potential force systems and linear constraints are taken into account with a coordinate-free treatment. The law of dynamics, characterizing the trajectory in a general non-linear configuration manifold, is expressed in terms of a variational principle and of differential and jump conditions. By endowing the configuration manifold with a connection, the general law is shown to be tensorial in the velocity of virtual flows and to depend on the torsion of the connection. This result provides a general expression of the EULER-LAGRANGE operator. POINCARÉ and LAGRANGE forms of the law are recovered as special cases corresponding, respectively, to the connection induced by natural and mobile reference frames. For free motions, the geodesic property of the trajectory is directly inferred by adopting the LEVI-CIVITA connection induced by the kinetic energy.     

Morawetz's method for the decay of the solution of the exterior initial-boundary value problem for the linearized equation of dynamic elasticity

In this paper, the behavior of the solution of the time-dependent linearized equation of dynamic elasticity is examined.For the homogeneous problem, it is proved that in the exterior of a star-shaped body on the surface of which the displacement field is zero, the solution decays at the rate t ^-1 as the time t tends to infinity.For the non-homogeneous problem with a harmonic forcing term, it is proved that for large times, the elastic material in the exterior of the body, tends to a harmonic motion, with the period of the external force.The convergence to the steady harmonic state solution is at the rate t ^-1/2 as t tends to infinity, and is uniform on bounded sets.     

Opportunities of the elastometry method for investigating the elastic properties of the eyeball shell

On the basis of a maximally simple mechanical model, the possibility to estimate the stiffness of the eyeball shell using the intraocular pressures measured in the presence of different weights applied to the cornea (elastometry method) is investigated. On the basis of general considerations of dimensional theory, it is shown that the pressure difference used in ophthalmology for estimating the rigidity of the eye ball and determined in the elastometry procedure depends on the ratio E/p ₀, where E is a quantity characterizing the shell stiffness and p ₀ is the intraocular pressure in the unloaded eye. An experimental procedure, which can be treated as a development of the elastometry method and makes it possible to estimate a certain parameter dependent only on the eyeball shell stiffness, is proposed. The practical realization of the method proposed needs to be checked experimentally.     

On the singularity index and the unfolding of the diffraction catastrophes

The caustics of high-frequency wave propagation may be classified using catastrophe theory. The wavefield in the vicinity of any caustic is described by the corresponding diffraction catastrophe. The singularity index, β, is a measure of the rate at which such a wavefield diverges as ω→∞ at the point where all control parameters and moduli are set equal to zero. It is shown that away from this point β also describes a balance between two different measures of the unfolding of the wavefield in each control direction, β = σ_nϱ_n. The indices σ_n and ϱ_n describe, respectively, the rate at which individual ray arrivals separate in time and decay as a function of control parameter.     

The effect of stereotomy on the shape of the thrust-line and the minimum thickness of semicircular masonry arches

More than a century ago, the Serbian engineer and astronomer Milutin Milankovitch presented a remarkable formulation for the thrust-line of arches that do not sustain tension, and by taking radial cuts and a polar coordinate system, he published for the first time the correct and complete solution for the theoretical minimum thickness, t, of a monolithic semicircular arch with radius R. This paper shows that Milankovitch’s solution, t/R = 0.1075, is not unique and that it depends on the stereotomy exercised. The adoption of vertical cuts which are associated with a cartesian coordinate system yields a neighboring thrust-line and a different, slightly higher value for the minimum thickness (t/R = 0.1095) than the value computed by Milankovitch. This result has been obtained in this paper with a geometric and a variational formulation. The Milankovitch minimum thrust-line derived with radial stereotomy and our minimum thrust-line derived with vertical stereotomy are two distinguishable, physically admissible thrust-lines which do not coincide with R. Hooke’s catenary that meets the extrados of the arch at the three extreme points. Furthermore, the paper shows that the catenary (the “hanging chain”) is not a physically admissible minimum thrust-line of the semicircular arch, although it is a neighboring line to the aforementioned physically admissible thrust-lines. The minimum thickness of a semicircular arch that is needed to accommodate the catenary curve is t/R = 0.1117—a value that is even higher than the enhanced minimum thickness t/R = 0.1095 computed in this paper after adopting a cartesian coordinate system; therefore, it works toward the safety of the arch.     

On the reliability theory of the structures

Summary The researches into the reliability theory of the structures, in hypothesis of elasto-plastic material, compels one to determine the probability function of structural resistance. As it is very difficult to find this function exactly, various authors approximate its values following different methods.The paper starts from the comparison among these ways and proposes as more efficient a simulation method based on parametric linear programming. To the latter it gives an appropriate physical interpretation that allows one to be able to control the computational process.

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Sommario Le difficoltà di determinare la distribuzione di probabilità della resistenza di strutture elasto-plastiche (costruite con materiale a resistenza aleatoria) hanno spinto molti autori a mettere a punto metodi approssimati. Partendo da un esame comparativo si propone, in questa sede, una tecnica di simulazione, che sembra dotata di particolare efficienza computazionale, basata sulla programmazione lineare parametrica. Di essa viene data anche una interpretazione fisica che consente di controllare il processo di calcolo.

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Research conducted under a grant of C.N.R. and accepted by the 15th Polish Solid Mechanics Conference. Zakopane, September 1973.     

Testing the accuracy of the overlap criterion

Here we investigate the accuracy of the overlap criterion when applied to a simple near-integrable model in both its 2D and 3D versions. To this end, we consider, respectively, two and three quartic oscillators as the unperturbed system, and couple the degrees of freedom by a cubic, non-integrable perturbation. For both systems we compute the unperturbed resonances up to order O(ε²), and model each resonance by means of the pendulum approximation in order to estimate the theoretical critical value of the perturbation parameter for a global transition to chaos. We perform several surface of sections for the bi-dimensional case to derive an empirical value to be compared to our theoretical estimation. Although both values are of the same order of magnitude, there is a significant difference between them. For the 3D case a numerical estimate is attained that we observe matches quite well the critical value resulting from theoretical means. This confirms once again that calculating resonances up to O(ε²) suffices in order the overlap criterion to work out.     

Dependence of the stability criterion of nucleate boiling on the compressibility of the vapor

The article gives the results of a study of the motion of bubbles and their deformation near the heating surface at different pressures. It was observed that, during the time of their growth, the gaseous medium in the bubbles is in a compressed state.Nomenclature R) radius of bubble - R_h) maximul radius of a deformed bubble in the horizontal plane - R_v) maximal radius of a deformed bubble in the vertical plane - ) specific weight - B) universal gas constant - ) surface-tension coefficient - p) pressure - ) edge wetting angle - g) acceleration due to gravity - V) volume - ) molecular weight - C_T) isothermal velocity of sound Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 77–81, July–August, 1971.     

On the relativistic extension of the equations of the classical gravitational potential

Summary The meaning of the vector V which characterizes the difference between the two tensor potentials of the Riemann and of the Weyl tensor is here studied. In the context of general relativity, the equations satisfied by such a vector and by the scalar potential characterizing its irrotational part, are connected to the equation proposed by Cattaneo as a relativistic extension of the Gauss-Poisson classical one. The link between the vector potential V and the standard gravitational field is thus obtained, for static universes.

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Sommario Si studia il significato del vettore V che caratterizza il divario fra i tensori potenziali dei tensori di Riemann e di Weyl. Le equazioni cui soddisfano, in relatività generale, detto vettore ed il potenziale scalare che ne individua la parte irrotazionale, vengono collegate con l'equazione relativistica proposta da Cattaneo quale estensione di quella classica di Gauss-Poisson. Si perviene così, nel caso statico, a stabilire un legame fra il potenziale vettore V ed il campo gravitazionale standard.

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This work was done in the sphere of activity of the C.N.R. Groups for mathematical research.     

On the theory of the dynamic viscometer

Summary It is shown that by using a dynamic viscometer, complex dynamic viscosities can be determined rather easily over a wide frequency range. Expansions of the exact solution which takes inertia effects into account are in the case of small gap widths almost identical for low and high frequencies. For intermediate frequencies the deviation from this limiting behavior is only a few percent.

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Zusammenfassung Die Anwendung von dynamischen Viskometern ermöglicht eine einfache Bestimmung von komplexen dynamischen Viskositäten über einen weiten Frequenzbereich. Reihenentwicklungen der exakten Lösung, welche Trägheitseffekte berücksichtigt, sind für den Fall enger Zylinder-Spalte für niedrige und hohe Frequenzen fast identisch. Aber auch bei Frequenzen im Zwischenbereich weichen die Ergebnisse nur um einige Prozent ab.

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With 1 figure     

A logrithmic bound on the location of the poles of the scattering matrix

It is shown that the complex poles z of the scattering matrix satisfy the inequality: Im z≧a+b log ¦z¦, b>0, in three instances of classical scattering in three space dimensions described by the wave equation u_{t t}?c²Δu+qu=0.

1. c and q smooth with c=1 and q=0 for ¦x¦>p, all rays going to infinity, and the energy form positive definite.
2. c=1 and q=0 outside of a convex body on which u=0.
3. c=1, q bounded and measurable, q=0 for ¦x¦>p, and the energy form not necessarily positive definite.

     

On the origin of the hole pressure

Summary One wall of a slit die apparatus contains a slot filled with the test liquid. Photographs of marked streamlines near the slot mouth show a pronounced curvature. A 0.5% aqueous solution of Separan AP 30 and molten polystyrene and polyethylene were investigated at low values of Reynolds number (Re < 1.0). In each case, streamlines are convex towards the slot mouth. This supports the hypothesis ofTanner andPipkin which attributes hole pressures in flowing viscoelastic liquids to the combined effect of holegenerated streamline curvature and shear-generated normal stress differences.

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Zusammenfassung In einer der Wände einer Schlitzdüsenvorrichtung ist eine mit Testflüssigkeit gefüllte Quernut angebracht. Es wurde die Strömung einer 0,5%-igen wässerigen Separan AP 30-Lösung, sowie einer Polystyrol- und einer Polyäthylenschmelze bei kleinen Reynoldszahlen (Re < 1.0) untersucht. In allen Fällen waren die Stromlinien konvex, also zur Schlitzmündung hin gekrümmt. Dadurch wird die Hypothese vonTanner andPipkin unterstützt, wonach die Lochdrücke in strömenden viskoelastischen Flüssigkeiten als Wirkung der durch die Scherung erzeugten Normalspannungsdifferenzen auf die gekrümmten Stromlinien erklärt werden.

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Department of Engineering Mechanics.

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Material Science Program.

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With 4 figures     

On the solution of the Chazy system of equations

We obtain a solution of the Chazy system that consists of nine nonlinear algebraic equations. This system gives a necessary condition for the class of nonlinear third-order differential equations with six singularities to be of P-type. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 92–98, January–March, 2009.     

The influence of the stabilization parameter on the convergence factor of iterative methods for the solution of the discretized Stokes problem

This paper discusses the influence of the stabilization parameter on the convergence factor of various iterative methods for the solution of the Stokes problem discretized by the so-called locally stabilized Q1-P0 finite element. Our objective is to point out optimal parameters which ensure rapid convergence. The first part of the paper is concerned with the dual formulation of the problem. It gives the theoretical precision and practical developments of our stabilized context Uzawa-type algorithm. We assert that the convergence factor of such a method is majored independently of the mesh size by a function of the stabilization parameter. Moreover, we point out that there exists an optimal value of this parameter that minimizes this upper bound. This gives a theoretical justification of pre-existing numerical results. We show that the optimal parameter can be determined a priori. This is a key point when the method has to be implemented. Finally, we base an interpretation of the iterated penalty method numerical behaviour on some theoretical results about the minimum eigenvalue of the stabilized dual operator. This algorithm involves a penalty parameter and a stabilization parameter and we discuss a strategy for choosing optimal parameters. The mixed formulation of the problem is dealt with in the second part of the paper, which proposes several preconditioned conjugate-gradient-type methods. The indefinite character of the problem makes it intrinsically hard. However, if one chooses a suitable preconditioner, this difficulty is overcome, since the preconditioned operator becomes positive definite. We study the eigenvalue spectrum of the preconditioned operator and thereby the convergence factor of the algorithm. In contrast with the two previous formulations, we show that this convergence factor is majored independently of the stabilization parameter. More precisely, we point out convergence factors comparable with those obtained for Poisson-type problems. Finally, we present a variant of the latter method which uses our so-called macroblock-type preconditioner. A comparison with the simple case of diagonal preconditioning is addressed and the improved performance of the macroblock-type preconditioner is evidenced. Various 2D numerical experiments are given to corroborate the theories presented herein.     

Effect of the wettability of the reservoir rock on the flow into a well

The effect of the wettability of the reservoir rock on the productivity of a well in a flooded reservoir is analyzed for immiscible fluid systems of the oil-water type by means of a computational experiment. The calculations take into account the presence of a drilling mud filtrate penetration zone in the neighborhood of the well.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 172–175, September–October, 1990.     

Determination of the Closure of the Set of Elasticity Functionals

We determine the closure for Mosco-convergence in L ² (,³) of the set of elasticity functionals. We prove that this closure coincides with the set of all non-negative lower-semicontinuous quadratic functionals which are objective, i.e., which vanish for rigid motions. The result is still valid if we consider only the set of isotropic elasticity functionals which have a prescribed Poisson coefficient. This shows that a very large family of materials can be reached when homogenizing a composite material with highly contrasted rigidity coefficients.     

Role of the polymer phase in the mechanics of nacre-like composites

Although strength and toughness are often mutually exclusive properties in man-made structural materials, nature is full of examples of composite materials that combine these properties in a remarkable way through sophisticated multiscale architectures. Understanding the contributions of the different constituents to the energy dissipating toughening mechanisms active in these natural materials is crucial for the development of strong artificial composites with a high resistance to fracture. Here, we systematically study the influence of the polymer properties on the mechanics of nacre-like composites containing an intermediate fraction of mineral phase (57 vol%). To this end, we infiltrate ceramic scaffolds prepared by magnetically assisted slip casting (MASC) with monomers that are subsequently cured to yield three drastically different polymers: (i) poly(lauryl methacrylate) (PLMA), a soft and weak elastomer; (ii) poly(methyl methacrylate) (PMMA), a strong, stiff and brittle thermoplastic; and (iii) polyether urethane diacrylate-co-poly(2-hydroxyethyl methacrylate) (PUA-PHEMA), a tough polymer of intermediate strength and stiffness. By combining our experimental data with finite element modeling, we find that stiffer polymers can increase the strength of the composite by reducing stress concentrations in the inorganic scaffold. Moreover, infiltrating the scaffolds with tough polymers leads to composites with high crack initiation toughness K_IC. An organic phase with a minimum strength and toughness is also required to fully activate the mechanisms programmed within the ceramic structure for a rising R-curve behavior. Our results indicate that a high modulus of toughness is a key parameter for the selection of polymers leading to strong and tough bioinspired nacre-like composites.     

An extension of the Levi-Weckesser method to the stabilization of the inverted pendulum under gravity

Sufficient conditions are given for the stability of the upper equilibrium of the mathematical pendulum (inverted pendulum) when the suspension point is vibrating vertically with high frequency. The equation of the motion is of the form $$ \ddot{\theta}-\frac{1}{l}\bigl(g+a(t)\bigr) \theta=0, $$ where l,g are constants and a is a periodic step function. M. Levi and W. Weckesser gave a simple geometrical explanation for the stability effect provided that the frequency is so high that the gravity g can be neglected. They also obtained a lower estimate for the stabilizing frequency. This method is improved and extended to the arbitrary inverted pendulum not assuming even symmetricity between the upward and downward phases in the vibration of the suspension point.     

On the solution of the sector problem

A rigorous study of the sector problem is presented by using the Mellin transform technique. The stress function is obtained as an asymptotic expansion of the complex inversion integral. The number of terms of this expansion, as well as the differentiability of the stress function, depend on the differential properties of boundary conditions on the radial edges. If these boundary conditions belong toC , this asymptotic expansion is transformed to a uniformly convergent infinite series. The coefficients of the series, which depend only on the boundary conditions along the circumferential edges, are calculated by applying a bi-orthogonality condition, or, by a technique based on the Betti formula.

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Résumé En utilisant la technique de la transformation de Mellin on présente une étude rigoureuse du problème d'un secteur. La fonction des contraintes est obtenue par un développement asymptotique de l'intégral complexe d'inversion. Le nombre de termes de ce développement, ainsi que la differentiabilité de la fonction des contraintes, dépendent de proprietés différentielles des conditions aux limites sur les bords radiaux. Si ces conditions aux limites sont dansC , ce développement asymptotique est transformé en série infinie uniformement convergente. Les coefficients de cette série, qui dépendent uniquement de conditions aux limites sur le bord circulaire, sont calculés à l'aide d'une condition de bi-orthogonalité que nous démontrons, ou, à l'aide d'une technique basée sur la formule de Betti.