Showing papers in "Zeitschrift für Angewandte Mathematik und Physik in 1990"

A critical review of the state of finite plasticity

Abstract: The object of this paper is to provide a critical review of the current state of plasticity in the presence of finite deformation. In view of the controversy regarding a number of fundamental issues between several existing schools of plasticity, the areas of agreement are described separately from those of disagreement. Attention is mainly focussed on the purely mechanical, rate-independent, theory of elastic-plastic materials, although closely related topics such as rate-dependent behavior, thermal effects, experimental and computational aspects, microstructural effects and crystal plasticity are also discussed and potentially fruitful directions are identified. A substantial portion of this review is devoted to the area of disagreement that covers a detailed presentation of argument(s), bothpro andcon, for all of the basic constitutive ingredients of the rate-independent theory such as the primitive notion or definition of plastic strain, the structure of the constitutive equation for the stress response, the yield function, the loading criteria and the flow and the hardening rules. The majority of current research in finite plasticity theory, as with its infinitesimal counterpart, still utilizes a (classical) stress-based approach which inherently possesses some shortcomings for the characterization of elastic-plastic materials. These and other anomalous behavior of a stress-based formulation are contrasted with the more recent strain-based formulation of finite plasticity. A number of important features and theoretical advantages of the latter formulation, along with its computational potential and experimental interpretation, are discussed separately.

Analysis of resonances in the spin-orbit problem in Celestial Mechanics: the synchronous resonance (part I)

Abstract: We study the stability of spin-orbit resonances in Celestial Mechanics, namely the exact commensurabilities between the periods of rotation and revolution of satellites or planets We introduce a mathematical model describing an approximation of the physical situation and we select a set of satellites for which such simplified model provides a good approximation

On the motion of a phase interface by surface diffusion

Abstract: Mullins, in a series of papers, developed a surface dynamics for phase interfaces whose evolution is controlled by mass diffusion within the interface. It is our purpose here to embed Mullins's theory within a general framework based on balance laws for mass and capillary forces in conjunction with a version of the second law, appropriate to a purely mechanical theory, which asserts that the rate at which the free energy increases cannot be greater than the energy inflow plus the power supplied. We develop an appropriate constitutive theory, and deduce general and approximate equations for the evolution of the interface.

Eigenvalue inclusions for second-order ordinary differential operators by a numerical homotopy method

Abstract: Inclusion intervals for the firstN eigenvalues of a second-order ordinary differential operator with boundary conditions of Sturm-Liouville or of periodic type are derived by a combination of “elementary” estimates, an appropriate numerical procedure and a homotopy algorithm.

Lie series method for vector fields and Hamiltonian perturbation theory

Abstract: We consider a rigorous Hamiltonian perturbation theory based on the transformation of the vector field of the system, realized by the Lie method. Such a perturbative technique presents some advantages over the standard one, which uses the transformation of the Hamilton functions. Indeed, the present method is simple, and furnishes quite detailed informations on the normal form. Moreover, it leads to estimates which are better and/or simpler than those of the scalar Lie methods. The perturbation method is presented with reference to two model problems, both pertaining to the realm of the well known Nekhoroshev theorem: the confining of actions for exponentially long times in a system of coupled harmonic oscillators, and an application to the so called problem of the realization of a holonomic constraint in classical mechanics.

Asymptotic estimates for Laguerre polynomials

Abstract: We give a brief summary of recent results concerning the asymptotic behaviour of the Laguerre polynomialsL () (x). First we summarize the results of a paper of Frenzen and Wong in whichn→∞ and α>−1 is fixed. Two different expansions are needed in that case, one with aJ-Bessel function and one with an Airy function as main approximant. Second, three other forms are given in which α is not necessarily fixed. These results follow from papers of Dunster and Olver, who considered the expansion of Whittaker functions. Again Bessel and Airy functions are used, and in another form the comparison function is a Hermite polynomial. A numerical verification of the new expansion in terms of the Hermite polynomial is given by comparing the zeros of the approximant with the related zeros of the Laguerre polynomial.

Analysis of resonances in the spin-orbit problem in celestial mechanics: higher order resonances and some numerical experiments (part II)

Abstract: In this work we study the stability of the non-synchronous resonances in the spin-orbit problem in Celestial Mechanics. Using a mathematical model obtained making some restrictions on the physical problem, we trap the 3∶2 resonance between invariant surfaces for realistic values of the physical and orbital parameters in the cases of the Moon and other three satellites of Saturn. We also provide some remarks on higher order resonances and on the Mercury-Sun system.

The Kramers problem: velocity slip and defect for a hard sphere gas with arbitrary accommodation

Abstract: The half-space problem of rarefied gas flow (the Kramers problem) is considered for the linearized Boltzmann equation and arbitrary gas-surface interaction. Accurate numerical results for the velocity slip coefficient and velocity defect are obtained for the rigid sphere interaction and Maxwellian boundary condition.

On the coagulation-fragmentation equation

Abstract: Two Theorems are presented on the subject of non-density conserving solutions to the coagulationfragmentation equation. The first deals with pure coagulation processes and the second with pure fragmentation. A third Theorem provides an asymptotic property for general solutions of the pure fragmentation equation. Some examples of exact solutions are given.

On the Kolodner-Coffman method for the uniqueness problem of Emden-Fowler BVP

Abstract: This paper surveys the Kolodner-Coffman method which has been applied very successfully by many authors to obtain uniqueness theorems for boundary value problems for differential equations of Emden-Fowler type. We recast some of the arguments in the language of Sturm's oscillation theory for linear second-order differential equations. The clarification provided by the new perspective enables us to improve several known results.

Existence of KAM tori in the phase-space of lattice vortex systems

Abstract: Under very mild conditions on the circulations, and for arbitrary vortex configurations, the existence of quasi-periodic solutions for a lattice vortex model is shown.

On the axisymmetric interaction of a rigid disc with a semi-infinite solid

Abstract: An analytical treatment is presented for the determination of the response of a vertically-loaded disc embedded in a semi-infinite elastic medium By means of Love's method of potential and a set of relaxed boundary conditions, the mixed boundary value problem is formulated as dual integral equations with the aid of Hankel transforms On the reduction of the dual integral equations to a Fredholm integral equation which features a closed-form kernel, solutions to the inclusion problem are computed In addition to including existing solutions for zero and infinite embedment as degenerate cases, the present analysis reveals a severe boundary-layer phenomenon which is apt to be of significance to this class of problems in general As illustrations, numerical results on the load-displacement relation, the response of the embedding medium, as well as the contact load distribution are included

Inversion of smooth mappings

Abstract: A method based on differential equations to evaluate the inverse mapping as well as to estimate the extension of its maximal star shaped domain is proposed here. The size of the domain of univalence, restricted to which the mapping is a diffeomorphism, is also estimated.

Visco-elastic boundary layer flow past a stretching plate and heat transfer

Abstract: In the present paper, we study the temperature profile in the case of a linear stretching plate for large Prandtl numberP by an asymptotic approach. Also, we evaluate the Nusselt number and the order of magnitude of Prandtl number for variable Nusselt numbers.

Hydromagnetic effects on the three-dimensional flow past a porous plate

Abstract: Hydromagnetic effects on the three-dimensional flow of an electrically conducting viscous incompressible fluid past a porous plate with periodic suction has been analysed. The uniform flow is subjected to a transversely applied magnetic field. The mathematical analysis is presented for the hydromagnetic boundary layer flow neglecting the induced magnetic field. Approximate solutions for the components of velocity field and the skin-frictions due to them are obtained and discussed with the help of a graph and tables.

Rayleigh-Taylor instability and Rayleigh-type waves on a Maxwell-fluid

Abstract: The maximum growth rate for Rayleigh-Taylor instability on a Maxwell-fluid has been found. The Rayleigh-Taylor instability mode exists for all wave-numbers and dimensionless time numbers. Further it was found that a propagating wave mode may exist at the same time. It was shown that this mode indeed was a Rayleigh-wave mode ask→∞. A cut-off range may exist for the propagating wave-mode.

Viscoelastic flow in a curved channel: a similarity solution for the Oldroyd-B fluid

Abstract: A similarity solution is used to analyse the flow of the Oldroyd fluid B, which includes the Newtonian and Maxwell fluids, in a curved channel modelled by the narrow annular region between two circular concentric cylinders of large radius. The solution is exact, including inertial forces. It is found that the non-Netonian kinematics are very similar to the Newtonian ones, although some stress components can become very large. At high Reynolds number a boundary layer is developed at the inner cylinder. The structure of this boundary layer is asymptotically analysed for the Newtonian fluid. Non-Newtonian stress boundary layers are also developed at the inner cylinder at large Reynolds numbers.

Decay of travelling waves in dissipative Poisson systems

Abstract: In many finite and infinite dimensional systemslow-dimensional behaviour is often observed. That is to say, the dynamics, observed experimentally or numerically, looks as if it can be described (approximately) with only a few essential parameters. Choosing the correct set of such ldquorobust observablesrdquo is an essential ingredient of a successful low dimensional description. This paper reports on a specific example of a more general approach that aims at describing certain (low dimensional) phenomena in (high dimensional) damped/driven equations with parameters that are essentially determined from the underlying conservative part of the equation. In particular, a Hamiltonian or a Poisson structure of the conservative part is exploited to find (characterize) families of exact solutions. These solutions are then used as the ldquobase functionsrdquo with the aid of which the solutions of the disturbed system are approximated. This approximation is accomplished using the parameters that characterize the family as variables that depend on time. In this paper, this procedure is applied to a class of systems which admit travelling waves when dissipation is ignored.

Flow due to the longitudinal and torsional oscillation of a cylinder

Abstract: An exact solution is obtained for the motion of a fluid contained in an infinite circular cylinder which is undergoing both torsional and longitudinal oscillations. An analytical expression is obtained for the viscous drag on the cylinder and the velocity is depicted graphically. Where possible, comparisons are made with corresponding results for the external problem.

Elastostatics of the orthotropic double-cantilever-beam fracture specimen

Abstract: Crack-plane stresses and the stress intensity factor were determined in an orthotropic double-cantilever-beam configuration. The DCB fracture specimen was modeled as an infinite strip containing a semi-infinite crack at its midplane. Concentrated loads acted upon the crack surfaces, whereas the strip surfaces were traction free. Constitutive equations of an orthotropic body involving four independent material constants were considered. Fourier transforms and the Wiener-Hopf technique were utilized for an analytical solution within the context of the two-dimensional, linear theory of elasticity.

On the Boltzmann system for a mixture of reacting gases

Abstract: Exact analytical solutions to the nonlinear spatially homogeneous integro-partiaS differential Boltzmann system, governing the distribution functions of the rarefied gases of a given mixture in the presence of scattering, removal and chemical reaction effects, are derived upon the hypothesis of constant collision frequencies, BGK-type scattering and creation laws, and particles possessing only translational energy. Four classes of second-order chemical reactions are investigated together with the relevant continuity system for the number densities of the participating species.

The Stokes-system in exterior domains: L p estimates for small values of a resolvent parameter

Abstract: We consider the resolvent problem for the Stokes-system in an exterior domain: $$- v \cdot \Delta u + \lambda \cdot u + abla \pi = f,divu = 0in\mathbb{R}^3 \backslash \bar \Omega ,$$ , with υe]0, ∞[, λeℂ] −∞, 0], Ω bounded domain in ℝ3, withC 2-boundary ∂Ω. In addition, Dirichlet boundary conditionsu¦∂Ω=0 are prescribed. Using the method of integral equations, we estimate solutions (u,π) inL p -norms, for small values of ¦λ¦.

Counting the number of solutions in combustion and reactive flow problems

Abstract: Computational methods and comparison theory enable, when combined, an enhanced capability for counting the number of solutions in combustion equations. Very good lower bounds for the last turning point reveal a stable high temperature “explosion branch” for very small positive exothermicity.

A generalization of thermodynamic orthogonality to random media

Abstract: Basic relations of Ziegler's theory of thermodynamic orthogonality are generalized to random media. Two basic cases of interpreting randomness in nonconservative material response are considered, and, accordingly, the mean, the average, and the effective dissipation functions are identified. It is found that when a homogeneity condition is fulfilled, these functions satisfy very simple equalities. When a quasi-homogeneity condition is satisfied, these function lead to very concise forms of effective Legendre transformations. An interpretation of the extremum principles corresponding to the derived relations is given.

Compressible flow induced by the transient motion of a wavemaker

Abstract: The effect of fluid compressibility on the evolution of the pressure distribution and free surface elevation, following the initiation of a horizontal motion of a vertical wavemaker, is analysed. This effect is significant even in a liquid (like water) when the time scale of the motion is very short (e.g. impulsive motions).

Soliton decay in a Toda chain caused by dissipation

Abstract: Consider a Toda chain with uniform friction. Starting with an initial condition that represents a soliton, we investigate its decay. The main result is that the solitary character is almost completely preserved. During the decay the wave activates other nonlinear modes. The corresponding actions appear to be bounded uniformly in time, proportional to the square of the friction coefficient. We focus upon the interaction with the same soliton but opposite direction of propagation. Comparing the numerical observations with an analytical model we conclude that the activated wave is well described by a linear equation, inhomogeneously driven by the main wave. The main wave itself decays as a nonlinear damped oscillator with one degree of freedom.

Higher order shock waves

Abstract: Shock waves and the relevant Rankine-Hugoniot conditions may be inadequate even in simple cases. As an example, the electromagnetic field generated by a point charge, whose velocity jumps from0 to a constant valueυ, not only jumps across a spherical surface expanding at light speed, but also includes aδ-distribution term. This suggests that the concept of higher order shock waves be introduced, the associated compatibility conditions being also deduced.

Spatial decay theorems for nonlinear parabolic equations in semi-infinite cylinders

Abstract: Classes of nonlinear parabolic equations in a semi-infinite cylinder are considered. The equations are of the form $$u,_{jj} + g_{ij} \left( {x,u,p,\partial ^2 u} \right)u,_i u,_j = c\left( {x,u,p} \right)\frac{{\partial u}}{{\partial t}},$$ wherep=u,ku,k and∂2u represents a general space derivative of second order. Homogeneous Dirichlet data are prescribed on the lateral sides of the cylinder for all time, along with zero initial data. At any fixed timet, the solution is assumed to be bounded throughout the cylinder, as is the corresponding symmetric matrixgij. Under these assumptions, it is proved that each solution decays pointwise exponentially to zero with distance from the face of the cylinder and the exponential decay rate depends only upon the cross-section of the cylinder, but not upon time or the bounds foru andgij. In addition, if the boundary data on the face of the cylinder satisfy certain mild smoothness conditions, one obtains a decay rate equal to the best possible rate for the Laplace equation.

Quasiperiodicity, strange non-chaotic and chaotic attractors in a forced two degrees-of-freedom system

Abstract: Strange non-chaotic, strange chaotic and quasiperiodic attractors are demonstrated to exist for a system of two non-linear coupled oscillators with almost periodic excitations. For same parameter values a transition from a strange non-chaotic to a quasiperiodic attractor is presented, whereas for other parameter values a shift from the strange chaotic attractor to a quasiperiodic one is found.

Constant vorticity Riabouchinsky flows from a variational principle

Abstract: Flows with constant vorticity regions bounded by vortex sheets are obtained by minimizing a functional which is the difference of energy in the external (irrotational) flow and the internal flow. In the zero vorticity case this reduces to the functional used by Garabedian, Lewy, and Schiffer for Riabouchinsky's problem. The discretization is done using Schwarz-Christoffel transformations for approximating polygons and FFT's to compute required Dirichlet integrals.