Filament stretching equation in Lagrangian coordinates

The general integral of the very simple equation ^21/n/() was found to describe the cross sectional area of filaments of isothermal power law fluids while in transient stretching where is time and is the initial location of fluid molecules at time = 0 given as the distance from a reference point fixed in space. Any such stretching transient given as a solution of the above equation is physically realizable subject to the restrictions > 0 and/ < 0.     

An asymptotic treatment of the transient development of axisymmetric surface waves

A linearized theory is developed for the derivation of an asymptotic solution of the initial value problem of axisymmetric surface waves in an infinitely deep fluid produced by an arbitrary oscillating pressure distribution. An asymptotic treatment of the problem is presented in detail to obtain the solution for the surface elevation for sufficiently large time. Finally, the main prediction of this analysis for some particular pressure distributions of physical interest is exhibited.Nomenclature R, , Y cylindrical polar coordinates - frequency - g acceleration due to gravity - density of fluid - T time - (R, Y; T) velocity potential - E(R, T) vertical surface elevation - P(R, T) applied surface pressure - r, y nondimensional cylindrical polar coordinates, - p(r, t) nondimensional surface pressure - t nondimensional time, T - (r, y; t) nondimensional velocity potential, - (r, t) nondimensional vertical surface elevation, - (k) Hankel transform of a function p(r) with respect to r - I ₁ transient wave integral - I ₂ steady state wave integral     

Modell zur Beschreibung des Fließens wandgleitender Substanzen durch Düsen

Zusammenfassung Zur Analyse des Fließens einer direkt an der Düsenwand gleitendenOstwald-deWaele-Flüssigkeit (Potenzgesetz) wird ein Modell entwickelt, das die rheologischen Vorgänge tribologisch, d. h. analog derCoulombschen Reibung fester Körper beschreibt.Es zeigt sich, daß in der Düse zwei Bereiche zu unterscheiden sind: ein Haftbereich in der Nähe des Düseneinlaufs und ein am Düsenaustritt liegender Gleitbereich. Die Länge des Gleitbereichs, der Verlauf des Drucks und der Schubspannung längs der Düse sowie die Änderung des Geschwindigkeitsprofils im Gleitbereich werden ermittelt.Überschreitet die Wandschubspannung einen kritischen Betrag, so entsteht am Düsenende ein labiler Bereich, in dem der Betrag der Wandschubspannung sprunghaft auf einen kleineren Wert sinken kann. Der von verschiedenen Autoren gefundene Sprung in der Fließkurve bestimmter Polymerschmelzen kann damit grundsätzlich erklärt werden.

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Summary Starting from theCoulomb Friction Law for solids, a theoretical model is developed for the pressure flow of a viscous power-law fluid with slip at the wall.It is shown that two flow regions exist in the die: a first region at the upstream part of the die, where the fluid sticks to the wall; and a second region at the downstream part of the die, where the fluid slips at the wall. The length of the slip region, the development of pressure and shear stress along the die as well as the change of the velocity distribution are given for the slip region.For shear stresses above a critical value, an instability region is found at the exit of the die. In this region, a sudden decrease of shear stress can occur. This seems to explain the discontinuity in the flow curve reported by several investigators.

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F Querschnittsfläche der Kapillaren - Volumendurchsatz - K _R Reibkraft - L Düsenlänge - m Stoffwert (Fließexponent) - N Normalkraft - p hydrostatischer Druck - p _L Druck am Düsenende - p ₁ Druck an der Übergangsstelle Haften-Gleiten - p ₀ Druck vor der Düse - p _0H Druck vor der Düse im Falle des Wandhaftens - r Radius - R Düsenradius - v _g Gleitgeschwindigkeit - v _z Strömungsgeschwindigkeit inz-Richtung - z Koordinate in Strömungsrichtung - z ₁ Längskoordinate der Übergangsstelle Haften-Gleiten - Schergeschwindigkeit - Stoffwert - Viskosität - µ Gleitkoeffizient - µ _H Haftkoeffizient - Dichte - dimensionsloser Radiusr/R - _rz Schubspannung in der Flüssigkeit - _rz ^(R) Wandschubspannung in der Flüssigkeit - ₀ Stoffwert - _wg Wandschubspannung im Falle des Gleitens - _wH Haftschubspannung an der Wand Auszugsweise vorgetragen auf der Jahrestagung der Deutschen Rheologen in Berlin vom 28.–30. April 1975.Mit 10 Abbildungen     

Structural Health Monitoring Through Chaotic Interrogation

The field of vibration based structural health monitoring involves extracting a feature which robustly quantifies damage induced changes to the structure in the presence of ambient variation, that is, changes in ambient temperature, varying moisture levels, etc. In this paper, we present an attractor-based feature derived from the field of nonlinear time-series analysis. Emphasis is placed on the use of chaos for the purposes of system interrogation. The structure is excited with the output of a chaotic oscillator providing a deterministic (low-dimensional) input. Use is made of the Kaplan–Yorke conjecture in order to tune the Lyapunov exponents of the driving signal so that varying degrees of damage in the structure will alter the state space properties of the response attractor. The average local attractor variance ratio (ALAVR) is suggested as one possible means of quantifying the state space changes. Finite element results are presented for a thin aluminum cantilever beam subject to increasing damage, as specified by weld line separation, at the clamped end. Comparisons of the ALAVR to two modal features are evaluated through the use of a performance metric.     

Asymptotic analysis of equilibrium states for rotating turbulent flows

The equilibrium states of homogeneous turbulence simultaneously subjected to a mean velocity gradient and a rotation are examined by using asymptotic analysis. The present work is concerned with the asymptotic behavior of quantities such as the turbulent kinetic energy and its dissipation rate associated with the fixed point (/kS)=0, whereS is the shear rate. The classical form of the model transport equation for (Hanjalic and Launder, 1972) is used. The present analysis shows that, asymptotically, the turbulent kinetic energy (a) undergoes a power-law decay with time for (P/)<1, (b) is independent of time for (P/)=1, (c) undergoes a power-law growth with time for 1<(P/)<(C ₂–1), and (d) is represented by an exponential law versus time for (P/)=(C ₂–1)/(C ₁–1) and (/kS)>0 whereP is the production rate. For the commonly used second-order models the equilibrium solutions forP/,II, andIII (whereII andIII are respectively the second and third invariants of the anisotropy tensor) depend on the rotation number when (P/kS)=(/kS)=0. The variation of (P/kS) andII versusR given by the second-order model of Yakhot and Orzag are compared with results of Rapid Distortion Theory corrected for decay (Townsend, 1970).     

Heteroclinic orbits in singular systems: A unifying approach

We consider singularly perturbed systems , such that=f(, _o, 0). _o ^m , has a heteroclinic orbitu(t). We construct a bifurcation functionG(, ) such that the singular system has a heteroclinic orbit if and only ifG(, )=0 has a solution=(). We also apply this result to recover some theorems that have been proved using different approaches.     

Geiger-counter operation without dead-time

Summary The dead time of a Geiger counter can be completely overcome by operating the counter with a pulsed voltage. The theory of the method is developed for two modes of measurement and simple formulae are derived. Even with=100s, accurate counting is possible when the rate of arrival of particles at the counter is as high as 10⁶/s.The author wishes to acknowledge the award of a Turner and Newall Fellowship by the University of London.     

Existence of L 1-Connections Between Equilibria of a Semilinear Parabolic Equation

We give a necessary and sufficient condition for the existence of L ¹-connections between equilibria of a semilinear parabolic equation. By an L ¹-connection from an equilibrium ^– to an equilibrium ⁺ we mean a function u(, t) which is a classical solution on the interval (–, T) for some T and blows up at t = T but continues to exist in the space L ¹ for t [T, ) and satisfies u(, t) ^± (in a suitable sense) as t ±. The main tool in our analysis is the zero number.     

Heat transfer and equilibrium temperature of a sphere in a supersonic rarefied gas flow

Some results are presented of experimental studies of the equilibrium temperature and heat transfer of a sphere in a supersonic rarefied air flow.The notations D sphere diameter - u, , T,,l, freestream parameters (u is velocity, density, T the thermodynamic temperature,l the molecular mean free path, the viscosity coefficient, the thermal conductivity) - T₀ temperature of the adiabatically stagnated stream - T_e mean equilibrium temperature of the sphere - T_w surface temperature of the cold sphere (T_we) - mean heat transfer coefficient - _e air thermal conductivity at the temperature T_e - P Prandtl number - M Mach number     

Flow through a helically coiled annulus

In this paper the flow is studied of an incompressible viscous fluid through a helically coiled annulus, the torsion of its centre line taken into account. It has been shown that the torsion affects the secondary flow and contributes to the azimuthal component of velocity around the centre line. The symmetry of the secondary flow streamlines in the absence of torsion, is destroyed in its presence. Some stream lines penetrate from the upper half to the lower half, and if is further increased, a complete circulation around the centre line is obtained at low values of for all Reynolds numbers for which the analysis of this paper is valid, being the ratio of the torsion of the centre line to its curvature.Nomenclature A =constant - a outer radius of the annulus - b unit binormal vector to C - C helical centre line of the pipe - D rL - g 1000 - K Dean number=Re² - L 1+r sin - M (L ²+ ² r ²)^1/2 - n unit normal vector to C - P, P pressure and nondimensional pressure - p ₀, p pressures of O(1) and O() - Re Reynolds number=aW ₀/ - (r, , s), (r, , s) coordinates and nondimensional coordinates - nonorthogonal unit vectors along the coordinate directions - r ₀ radius of the projection of C - t unit tangent vector to C - V _r, V , V _s velocity components along the nonorthogonal directions - V_r, V, V _s nondimensional velocity components along - W ₀ average velocity in a straight annulus Greek symbols , curvature and nondimensional curvature of C - U, V, W lowest order terms for small in the velocity components along the orthogonal directions t - _r, , _s first approximations to V _r , V, V _s for small - =/=/ - kinematic viscosity - density of the fluid - , torsion and nondimensional torsion of C - , stream function and nondimensional stream function - nondimensional streamfunction for U, V - a inner radius of the annulus After this paper was accepted for publication, a paper entitled On the low-Reynolds number flow in a helical pipe, by C.Y. Wang, has appeared in J. Fluid. Mech., Vol 108, 1981, pp. 185–194. The results in Wangs paper are particular cases of this paper for =0, and are also contained in [9].     

Methods of Small and Large δ in the Nonlinear Dynamics – A Comparative Analysis

New asymptotic approaches for dynamical systems containing a power nonlinear term x ⁿ are proposed and analyzed. Two natural limiting cases are studied: n 1 + , 1 and n . In the firstcase, the 'small method' (SM)is used and its applicability for dynamical problems with the nonlinearterm sin as well as the usefulness of the SMfor the problem with small denominators are outlined. For n , a new asymptotic approach is proposed(conditionally we call it the 'large method' –LM). Error estimations lead to the followingconclusion: the LM may be used, even for smalln, whereas the SM has a narrow application area. Both of the discussed approaches overlap all values ofthe parameter n.     

On the localness of the spectral energy transfer in turbulence

By utilizing available experimental data for net energy transfer spectra for homogeneous turbulence, contributions P(, ) to the energy transfer at a wavenumber from various other wavenumbers are calculated. This is done by fitting a truncated power-exponential series in and to the experimental data for the net energy transfer T(), and using known properties of P(, ). Although the contributions P(, ) obtained by using this procedure are not unique, the results obtained by using various assumptions do not differ significantly. It seems clear from the results that for a region where the energy entering a wavenumber band dominates that leaving, much of the energy entering the band comes from wavenumbers which are about an order of magnitude smaller. That is, the energy transfer is rather nonlocal. This result is not significantly dependent on Reynolds number (for turbulence Reynolds numbers based on microscale from 3 to 800). For lower wavenumbers, where more energy leaves than enters a wavenumber band, the energy transfer into the band is more local, but much of the energy then leaves at distant wavenumbers.     

Concentration dependence of the critical viscoelastic properties of gelatin at the gel point

Gelatin gel properties have been studied through the evolution of the storage [G()] and the loss [G()] moduli during gelation or melting near the gel point at several concentrations. The linear viscoelastic properties at the percolation threshold follow a power-law G()G() and correspond to the behavior described by a rheological constitutive equation known as the Gel Equation. The critical point is characterized by the relation: tan = G/G = cst = tan ( · /2) and it may be precisely located using the variations of tan versus the gelation or melting parameter (time or temperature) at several frequencies. The effect of concentration and of time-temperature gel history on its variations has been studied. On gelation, critical temperatures at each concentration were extrapolated to infinite gel times. On melting, critical temperatures were determined by heating step by step after a controlled period of aging. Phase diagrams [T = f(C)] were obtained for gelation and melting and the corresponding enthalpies were calculated using the Ferry-Eldridge relation. A detailed study of the variations of A with concentration and with gel history was carried out. The values of which were generally in the 0.60–0.72 range but could be as low as 0.20–0.30 in some experimental conditions, were compared with published and theoretical values.     

Determination of permeability tensors for two-phase flow in homogeneous porous media: Theory

In this paper we continue previous studies of the closure problem for two-phase flow in homogeneous porous media, and we show how the closure problem can be transformed to a pair of Stokes-like boundary-value problems in terms of pressures that have units of length and velocities that have units of length squared. These are essentially geometrical boundary value problems that are used to calculate the four permeability tensors that appear in the volume averaged Stokes' equations. To determine the geometry associated with the closure problem, one needs to solve the physical problem; however, the closure problem can be solved using the same algorithm used to solve the physical problem, thus the entire procedure can be accomplished with a single numerical code.Nomenclature a a vector that maps V onto , m^-1. - A a tensor that maps V onto . - A area of the - interface contained within the macroscopic region, m². - A area of the -phase entrances and exits contained within the macroscopic region, m². - A area of the - interface contained within the averaging volume, m². - A area of the -phase entrances and exits contained within the averaging volume, m². - Bo Bond number (= (=(–)g²/). - Ca capillary number (= v/). - g gravitational acceleration, m/s². - H mean curvature, m^-1. - I unit tensor. - permeability tensor for the -phase, m². - viscous drag tensor that maps V onto V. - ^* dominant permeability tensor that maps onto v , m². - ^* coupling permeability tensor that maps onto v , m². - characteristic length scale for the -phase, m. - l characteristic length scale representing both and , m. - L characteristic length scale for volume averaged quantities, m. - n unit normal vector directed from the -phase toward the -phase. - n unit normal vector representing both n and n . - n unit normal vector representing both n and n . - P pressure in the -phase, N/m². - p superficial average pressure in the -phase, N/m². - p intrinsic average pressure in the -phase, N/m². - p –p , spatial deviation pressure for the -phase, N/m². - r ₀ radius of the averaging volume, m. - r position vector, m. - t time, s. - v fluid velocity in the -phase, m/s. - v superficial average velocity in the -phase, m/s. - v intrinsic average velocity in the -phase, m/s. - v –v , spatial deviation velocity in the -phase, m/s. - V volume of the -phase contained within the averaging volmue, m³. - averaging volume, m³. Greek Symbols V /, volume fraction of the -phase. - viscosity of the -phase, Ns/m². - density of the -phase, kg/m³. - surface tension, N/m. - (v +v ^T ), viscous stress tensor for the -phase, N/m².     

Parameter determination for thermodynamical models of the pseudoelastic behaviour of shape memory alloy by resistivity measurements and infrared thermography

Two thermodynamical models of pseudoelastic behaviour of shape memory alloys have been formulated. The first corresponds to the ideal reversible case. The second takes into account the hysteresis loop characteristic of this shape memory alloys.Two totally independent techniques are used during a loading-unloading tensile test to determine the whole set of model parameters, namely resistivity and infrared thermography measurements. In the ideal case, there is no difficulty in identifying parameters.Infrared thermography measurements are well adapted for observing the phase transformation thermal effects.Notations 1 austenite 2 martensite - ₍₎ Macroscopic infinitesimal strain tensor of phase - ₍₂₎ ^f Traceless strain tensor associated with the formation of martensite phase - Macroscopic infiniesimal strain tensor - Macroscopic infinitesimal strain tensor deviator - _f Trace - Equivalent strain - ^pe Macroscopic pseudoelastic strain tensor - x Distortion due to parent (austenite =1)product (martensite =2) phase transformation (traceless symmetric second order tensor) - M Total mass of a system - M() Total mass of phase - V Total volume of a system - V() Total volume of phase - z=M(2)/M Weight fraction of martensite - 1-z=M(1)/M Weight fraction of austenite - u ₀ ^* () Specific internal energy of phase (=1,2) - s ₀ ^* () Specific internal entropy of phase - Specific configurational energy - Specific configurational entropy - ₀ ^f (T) Driving force for temperature-induced martensitic transformation at stress free state ( ₀ ^f T) = T ^*–Ts ^*) - Kirchhoff stress tensor - Kirchhoff stress tensor deviator - Equivalent stress - Cauchy stress tensor - Mass density - K Bulk moduli (K ₀=K) - L Elastic moduli tensor (order 4) - E Young modulus - Energetic shear (₀ = ) - Poisson coefficient - M _s ^o (M _F ^o ) Martensite start (finish) temperature at stress free state - A _s ^o (A _F ^o ) Austenite start (finish) temperature at stress free state - C _v Specific heat at constant volume - k Conductivity - Pseudoelastic strain obtained in tensile test after complete phase transformation (AM) (unidimensional test) - ₀ Thermal expansion tensor - r Resistivity - 1MPa 10⁶ N/m ² - ⁽⁾ Specific free energy of phase - _n Specific free energy at non equilibrium (R model) - _n ^eq Specific free energy at equilibrium (R model) - _n ^v Volumic part of ^eq - Specific free energy at non equilibrium (R _L model) - _conf Specific coherency energy (R _L model) - _c Specific free energy at constrained equilibria (R _L model) - _it (T) Coherency term (R _L model)     

Macro-Scale Dynamic Effects in Homogeneous and Heterogeneous Porous Media

It is known that the classical capillary pressure-saturation relationship may be deficient under non-equilibrium conditions when large saturation changes may occur. An extended relationship has been proposed in the literature which correlates the rate of change of saturation to the difference between the phase pressures and the equilibrium capillary pressure. This linear relationship contains a damping coefficient, \tau, that may be a function of saturation. The extended relationship is examined at the macro-scale through simulations using the two-phase simulator MUFTE-UG. In these simulations, it is assumed that the traditional equilibrium relationship between the water saturation and the difference in fluid pressures holds locally. Steady-state and dynamic numerical experiments are performed where a non-wetting phase displaces a wetting phase in homogeneous and heterogeneous domains with varying boundary conditions, domain size, and soil parameters. From these simulations the damping coefficient can be identified as a (non-linear) function of the water saturation. It is shown that the value of increases with an increased domain size and/or with decreased intrinsic permeability. Also, the value of for a domain with a spatially correlated random distribution of intrinsic permeability is compared to a homogeneous domain with equivalent permeability; they are shown to be almost equal.     

StationÄrer und instationÄrer Vorgang der Filmkondensation auf der ebenen Platte,dem senkrechten Kegel,dem horizontalen Rohr und der Kugel

Zusammenfassung Die vorliegende Arbeit untersucht die Filmkondensation auf verschiedenen KörperoberflÄchen. Dabei wird sowohl der instationÄre Anlaufvorgang als auch der stationÄre Proze\ betrachtet. Die Ergebnisse für die Schichtdicke des abflie\enden Kondensates werden eingehend diskutiert. Ist die Schichtdicke als Funktion des Ortes und der Zeit bekannt, ist die Berechnung des kondensierenden bzw. abflie\enden Volumenstromes, sowie die Berechnung des lokalen bzw. für die Praxis bedeutungsvolleren globalen WÄrmeübergangs möglich.

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Steady and unsteady process of film condensation on a flat plate, a vertical coin, a horizontal pipe and a sphere

This paper investigates film condensation on different surfaces of geometric bodies. In this connection the unsteady starting process and the steady process are considered. The results for the thickness of layer of the flowing-off condensate are discussed detailed. If the thickness of layer is given as a function of time and location the computation of the condensing, respective flowing-off volume stream and the computation of the local, respective global heat transfer is possible.

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Bezeichnungen C Konstante - R Rohr- bzw. Kugelradius [m] - T Temperatur [K] - kondensierender Volumenstrom pro LÄngeneinheit [m² s^–1] - abflie\ender Volumenstrom pro LÄngeneinheit [m² s^–1] - kondensierender Volumenstrom [m³ s^–1] - abflie\ender Volumenstrom [m³ s^–1] - a Kegelachse - c spez. WÄrme der kondensierenden Flüssigkeit [J kg^–1 K^–1] - e ErzeugendenlÄnge des Kegels, an der die Randbedingung vorgeschrieben ist [m] - g Erdbeschleunigung [m s^–2] - l Platten- bzw. KegellÄnge [m] - p Druck [Nm^–2] - q WÄrmestromdichte [J m^–2 s^–1] - r VerdampfungswÄrme der Flüssigkeit [J kg^–1] - t Zeit [s] - u örtliche Geschwindigkeit des Fluids [m s^–1] - x, y kartesische Ortskoordinaten - r, Zylinder bzw. Kugelkoordinaten - WÄrmeübergangszahl [J m^–2 s^–1] - Neigungswinkel der Platte - öffnungswinkel des Kegels - Schichtdicke der kondensierten Flüssigkeit [m] - WÄrmeleitzahl der kondensierten Flüssigkeit [J m^–1 s^–1] - Dichte der kondensierten Flüssigkeit [kg m^–3] - OberflÄchenspannung der kondensierten Flüssigkeit [Nm^–1] - Schubspannung in der kondensierten Flüssigkeit [Nm^–2] - v kinematische ZÄhigkeit [m² s^–1] - dynamische ZÄhigkeit [kg m^–1 s^–1] - Winkelkoordinate (Rohr, Kugel), bei der eine Randbe-dingung vorgeschieben ist Indizes g gasförmige Phase - m mittlere - s SÄttigungszustand des gasförmigen Mediums - w auf die OberflÄche der Wand (Platte, Kegel, Rohr,Kugel) bezogen - 0 Ursprung der jeweiligen Störungsausbreitung Dimensionslose Kennzahlen Nu Nu\elt-Zahl - Pr Prandtl-Zahl - Re Reynolds-Zahl Kurzfassung der bei Prof. Dr. W. Schneider, Institut für Strömungslehre und WÄrmeübertragung TU Wien, angefertigten Diplomarbeit     

Application of the Chapman-Enskog method to the case of a binary two-temperature gas mixture

An interesting property of the flows of a binary mixture of neutral gases for which the molecular mass ratio =m/M1 is that within the limits of the applicability of continuum mechanics the components of the mixture may have different temperatures. The process of establishing the Maxwellian equilibrium state in such a mixture divides into several stages, which are characterized by relaxation times _i which differ in order of magnitude. First the state of the light component reaches equilibrium, then the heavy component, after which equilibrium between the components is established [1]. In the simplest case the relaxation times differ from one another by a factor of ^*.Here the mixture component temperature difference relaxation time _T /, where is the relaxation time for the light component. If 1, 1, so that _T ~1, then for the characteristic hydrodynamic time scale t~1 the relative temperature difference will be of order unity. In the absence of strong external force fields the component velocity difference is negligibly small, since its relaxation time _vt1.In the case of a fully ionized plasma the Chapman-Enskog method is quite easily extended to the case of the two-temperature mixture [3], since the Landau collision integral is used, which decomposes directly with respect to . In the Boltzmann cross collision integral, the quantity appears in the formulas relating the velocities before and after collision, which hinders the decomposition of this integral with respect to , which is necessary for calculating the relaxation terms in the equations for temperatures differing from zero in the Euler approximation [4] (the transport coefficients are calculated considerably more simply, since for their determination it is sufficient to account for only the first (Lorentzian [5]) terms of the decomposition of the cross collision integrals with respect to ). This led to the use in [4] for obtaining the equations of the considered continuum mixture of a specially constructed model kinetic equation (of the Bhatnagar-Krook type) which has an undetermined degree of accuracy.In the following we use the Boltzmann equations to obtain the equations of motion of a two-temperature binary gas mixture in an approximation analogous to that of Navier-Stokes (for convenience we shall term this approximation the Navier-Stokes approximation) to determine the transport coefficients and the relaxation terms of the equations for the temperatures. The equations in the Burnett approximation, and so on, may be obtained similarly, although this derivation is not useful in practice.     

Saint-Venant's principle in nonlinear plane elasticity with sufficiently small strains

A homogeneous, isotropic cylinder in an equilibrium state of plane strain, whose cross-section is a rectangle R : [0 < y ₁ < 2L; 0 < y ₂ < h] with h/L 1, is considered. There are no body forces and the long sides are stress free. At y ₁ = 0 and y ₁ = 2L, there are arbitrary loadings, each statically equivalent to a uniformly distributed tensile or compressive stress c. Within the theory of nonlinear elasticity and with the strains and strain gradients assumed to be sufficiently small (but with no such assumptions on the displacement gradients), it is proved that if (,=1,2) represents the Cauchy stress tensor and the Kronecker delta, then |–c₁₁| decays exponentially to zero in R with distance from the nearer end, and the decay constant depends only upon the material but is independent of L.     

Symplectic approach to the theory of quantized fields. II

We prove that the set D of vector fields on the configuration space B of a field whose 1-parameter groups locally associated are groups of fibre-preserving transformations of B that leave invariant that field in the sense of variational theory, is a Lie algebra with respect to ordinary addition, multiplication by real numbers and Lie brackets. We see that this Lie algebra structure can be carried over to the corresponding set of Noether invariants, which then becomes a Lie algebra in a natural way.Further, we define the n-form of Poincaré-Cartan of a field, and we use it to generalize the Lie algebras D and in a reasonable way. The algebras D and are subalgebras of the new Lie algebras D and introduced. A main result in this connection is the following: the differential d of the n-form of Poincaré-Cartan is –(d+f), where (, d+f) are the field equations on the vertical bundle B.The symplectic manifold of solutions associated with a field is introduced in a formal way and the former Lie algebras D, , D, are interpreted on this manifold. In imitation of the case of analytical dynamics, the main results in this direction are: a) Every vector field of the Lie algebra D defines, in a canonical way, a vector field on the manifold of solutions such that its polar 1-form with respect to the symplectic metric ₂ is the differential of its corresponding Noether invariant, and b) the Lie bracket [, ] of two Noether invariants , is the Noether invariant given by ₂(D, D), where D, D are the vector fields on the manifold of solutions defined, in the sense a), by two infinitesimal generators of , , respectively. This will allow us to regard the Lie algebra as the analogous object in field theory to the Poisson algebra of analytic dynamics.We apply the general formalism to the relativistic theory of non-linear scalar fields, and we compare our results with the formalism developed by I. Segal for this case.