Showing papers in "Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik in 2002"
TL;DR: In this article, the equilibrium and stability equations of rectangular functionally graded plates (FGPs) are determined using the variational approach, where the material properties vary with the power product form of thickness coordinate variable z.
Abstract: In the present article, equilibrium and stability equations of rectangular functionally graded plates (FGPs) are determined using the variational approach. Derivation of equations are based on the classical plate theory. It is assumed that the material properties vary with the power product form of thickness coordinate variable z. Equilibrium and stability equations for FGPs are the same as the equations for homogeneous plates. The equilibrium and stability equations are employed to study the buckling behaviour of functionally graded plates with all edges simply supported and subjected to in-plane loading conditions. By equating power law index to zero, predicted relation is reduced to the buckling equation of homogeneous plates which is available in the literature.
248 citations
TL;DR: In this article, it was shown that loss of hyperbolicity, that is the occurrence of complex eigenvalues of the Jacobian of the first order system, can be viewed as an instability criterion for arbitrary polydisperse suspensions, and that for tridisperse mixtures this criterion can be evaluated by a convenient calculation of a discriminant.
Abstract: The one-dimensional kinematical sedimentation theory for suspensions of small spheres of equal size and density is generalized to polydisperse suspensions and several space dimensions. The resulting mathematical model, obtained by introducing constitutive assumptions and performing a dimensional analysis, is a system of first-order conservation laws for the concentrations of the solids species coupled to a variant of the Stokes system for incompressible flow describing the mixture. Various flux density vectors for the first-order system have been proposed in the literature. Some of them cause the first-order system of conservation laws to be non-hyperbolic, or to be of mixed hyperbolic-elliptic type in the bidisperse case. The criterion for ellipticity is equivalent to a well-known instability criterion predicting phenomena like blobs and viscous fingering in bidisperse sedimentation. We show that loss of hyperbolicity, that is the occurrence of complex eigenvalues of the Jacobian of the first-order system, can be viewed as an instability criterion for arbitrary polydisperse suspensions, and that for tridisperse mixtures this criterion can be evaluated by a convenient calculation of a discriminant. We determine instability regions (or alternatively prove stability) for three different choices of the flux vector of the first-order system of conservation laws. Consequently, mixed or non-hyperbolic, rather than hyperbolic, systems of conservation laws are the appropriate general mathematical framework for polydisperse sedimentation. The stability analysis examines a first-order system of conservation laws, but its predictions are applicable to the full multidimensional system of model equations. The findings are consistent with experimental evidence and are appropriately embedded into the current state of knowledge of non-hyperbolic systems of conservation laws.
62 citations
TL;DR: In this article, the authors studied the dynamics of a viscoelastic rod with fractional derivative type of dissipation under time dependent loading and derived the moment curvature relation for the rod from the generalized Zener model of a standard linear solid.
Abstract: We study dynamics of a viscoelastic rod with fractional derivative type of dissipation under time dependent loading. First the moment curvature relation for the rod from the generalized Zener model of a standard linear solid is derived. Then, we show that the dynamics of the problem is governed by a single linear differential equation with fractional derivative. The existence and uniqueness of solutions for this equation is studied in detail. The form of the solution for the case of periodic compressive force is also obtained.
60 citations
TL;DR: In this paper, it was shown that micro effects are scaling effects with respect to models developed for macro systems, and that all systems as long as they can be described within a continuum theory should be treated uniquely and in a nondimensional form.
Abstract: In almost all studies on flow and heat transfer published so far deviations from what is known in macro systems occur. Often special micro effects are proposed to explain these unexpected results. From our point of view, however, this approach is misleading. Instead, all systems - micro and macro - as long as they can be described within a continuum theory should be treated uniquely and in a nondimensional form. Then it turns out that micro effects are scaling effects with respect to models developed for macro systems.
46 citations
TL;DR: In this article, the longitudinal permeability of a spatially periodic rectangular array of circular cylinders is analyzed in terms of the radius of the cylinders and the aspect ratio of the unit cell.
Abstract: We study the longitudinal permeability of a spatially periodic rectangular array of circular cylinders, when a Newtonian fluid is flowing at low Reynolds number along the cylinders. The longitudinal component of the velocity obeys a Poisson equation which is transformed into a functional equation. This equation can be solved by the method of successive approximations. The major advantage of this technique is that the permeability of the array can be expressed analytically in terms of the radius of the cylinders and of the aspect ratio of the unit cell.
45 citations
TL;DR: The present article focuses on the use of difference methods in order to approximate the solutions of stochastic partial differential equations of Ito-type, in particular hyperbolic equations.
Abstract: The present article focuses on the use of difference methods in order to approximate the solutions of stochastic partial differential equations of Ito-type, in particular hyperbolic equations. The main notions of deterministic difference methods, i.e. convergence, consistency, and stability, are developped for the stochastic case. It is shown that the proposed stochastic difference schemes for several partial differential equations have these properties.
38 citations
TL;DR: In this article, the authors discuss Lyapunov stability of the trivial solution of linear differential-algebraic equations and propose numerical parameters characterizing the property of a regular matrix pencil λA -B to have all finite eigenvalues in the open left half-plane.
Abstract: This paper discusses Lyapunov stability of the trivial solution of linear differential-algebraic equations. As a criterion for the asymptotic stability we propose numerical parameters characterizing the property of a regular matrix pencil λA - B to have all finite eigenvalues in the open left half-plane. Numerical aspects for computing these parameters are discussed.
30 citations
TL;DR: In this article, the authors derived the equations driving the flow of an incompressible fluid through a porous deformable medium and recast them in a material frame of reference fixed on the solid skeleton.
Abstract: The equations driving the flow of an incompressible fluid through a porous deformable medium are derived in the framework of the mixture theory. This mechanical system is described as a binary saturated mixture of incompressible components. The mathematical problem is characterized by the presence of two moving boundaries, the material boundaries of the solid and the fluid, respectively. The boundary and interface conditions to be supplied to ensure the well-posedness of the initial boundary value problem are inspired by typical processes in the manufacturing of composite materials. They are discussed in their connections with the nature of the partial stress tensors. Then the equations are conveniently recast in a material frame of reference fixed on the solid skeleton. By a proper choice of the characteristic magnitudes of the problem at hand, the equations are rewritten in non-dimensional form and the conditions which enable neglecting the inertial terms are discussed. The second part of the paper is devoted to the study of one-dimensional infiltration by the inertia-neglected model. It is shown that when the flow is driven through an elastic matrix by a constant rate liquid inflow at the border some exact results can be obtained.
25 citations
TL;DR: In this paper, a conformal mapping of the Zhukovskii domain (halfplane) onto the hodograph domain (lune) is used to solve the problem of a phreatic surface flow from a soil channel, whose equipotential contour satisfies the condition of constant entrance gradient.
Abstract: Soil slopes satisfying the condition of a constant exit gradient (constant Darcian velocity) and the seepage face (isobaricity) condition are found by complex analysis. For an empty drainage trench, soil is infinitely deep and the flow domain is bounded by two branches of a phreatic surface and the trench contour. The problem is solved by conformal mapping of the Zhukovskii domain (half-plane) onto the hodograph domain (lune). The ultimately stable seepage face and the inflow rate are determined as functions of a specified exit gradient. With decreasing of the gradient the trench flattens. If the gradient is 1, the depth-width aspect ratio of the trench reaches 0.21. Similar conformal mapping of a lune in the hodograph plane on a half-strip in the complex potential plane is used to solve the problem of a phreatic surface flow from a soil channel, whose equipotential contour satisfies the condition of constant entrance gradient. For a dam slope, the flow domain is underlain by an impermeable bottom and the hodograph is a circular triangle with a unit-gradient circle as the slope image. The Polubarinova-Kochina method of analytical differential equations is modified to reconstruct the complex coordinate and the complex potential as a function of an auxiliary variable. The resulting slope of unit exit gradient has a depth-width ratio of 1.22.
24 citations
TL;DR: In this paper, the wave function expansion method was used to solve the problem of a circular piezoelectric inclusion embedded in an infinite PEM matrix subjected to horizontally polarized shear waves and a steady-state inplane electrical load.
Abstract: The dynamic theory of antiplane piezoelectricity is applied to solve the problem of a circular piezoelectric inclusion embedded in an infinite piezoelectric matrix subjected to horizontally polarized shear waves and a steady-state inplane electrical load. The problem is formulated by means of the wave function expansion method. Numerical calculations are carried out for the dynamic stress and electric field concentrations, and the results are presented in graphical form.
22 citations
TL;DR: In this article, the problem of modifying the spectrum of gyroscopic systems by applying external forces is considered, and the analysis of the problem leads to a certain inverse integro-differential eigenvalue problem in the distributed parameter case, and to a non-symmetric restricted rank modification in the finite-dimensional case.
Abstract: The problem of modifying the spectrum of gyroscopic systems by applying external forces is considered. The analysis of the problem leads to a certain inverse integro-differential eigenvalue problem in the distributed parameter case, and to a non-symmetric restricted rank modification in the finite-dimensional case. The explicit solution presented may be applied to stabilize structures and systems, where a small part of the spectrum is required to be assigned and the rest of the spectrum is to remain unchanged. The required forces are determined by using a partial knowledge of the eigenvalues which are intended to be changed, and their associated eigenfunctions (eigenvectors).
TL;DR: In this paper, the authors discuss similarities and differences of two candidate approaches connected with discrete time decision processes and with uncertainties of probabilistic nature, and discuss the importance of choosing or building a suitable model taking into account the nature of the real-life problem, character of input data, availability of software, and computer technology.
Abstract: When solving a dynamic decision problem under uncertainty it is essential to choose or to build a suitable model taking into account the nature of the real-life problem, character of input data, availability of software, and computer technology. The purpose of this paper is to discuss similarities and differences of two candidate approaches connected with discrete time decision processes and with uncertainties of probabilistic nature.
TL;DR: In this paper, a non-parallel extension of the Gaussian asymptotic representation of the two dimensional laminar incompressible far wake past a symmetrical body is presented.
Abstract: A non parallel extension of the Gaussian asymptotic representation of the two dimensional laminar incompressible far wake past a symmetrical body is presented. Under the one and only condition that the middle and far field be governed by the thin shear layer theory that keeps the complete non linearity of the equation of motion, we determined a solution in terms of an infinite power series of the streamwise space variable with fractional negative exponents. The general n-th order term has been analytically established. The behaviour of these expansions inserted into the Navier-Stokes equations was analyzed to verify the consistency of the approximation in the intermediate region of the wake. At the third order the correction due to pressure variations identically vanishes while the contribution of the longitudinal diffusion is still two-three order of magnitude smaller than that of the transversal diffusion, depending on the Reynolds number.
TL;DR: In this article, the authors established conditions for the existence of an optimal solution in (P) involving (£, p, θ)-quasiconvex/pseudoconvex analytic functions.
Abstract: (P) minimize Re'f(z, z) + (z H Az) 1/2 ]/Re[g(z, z) - (z H Bz)1/2] subject to h(z, z) ∈ S ⊂ C m , z ∈ C m , where f, g: C 2n → C and h: C 2n → C m are analytic functions, A, B ∈ C m×n are positive semidefinite Hermitian matrices, S is a polyhedral cone in C m . In this paper, we establish conditions for the existence of an optimal solution in (P) involving (£, p, θ)-quasiconvex/-pseudoconvex analytic functions. Based on the sufficient optimality theorem, we construct a duality model and then establish weak/strong, and strict converse duality theorems.
TL;DR: In this paper, the mathematical modeling of polymer crystallization processes in a bounded region under heat transfer conditions, i.e. in time-dependent temperature fields, is devoted to the mathematical modelling of polymer crystal processes.
Abstract: This paper is devoted to the mathematical modelling of polymer crystallization processes in a bounded region under heat transfer conditions, i.e. in time-dependent temperature fields. A stochastic model in a general setup is developed based on the theory of marked point processes, and a hybrid model on a macroscopic scale is derived using expected values. The stochastic modelling part is supplemented by a detailed discussion of the relevant parameters and their dependence upon temperature. In the special case of one-dimensional crystallization a system of partial differential equations describing the evolution of temperature and of the degree of crystallinity is derived.
TL;DR: In this article, the authors present new results regarding a solution to such a game in randomized strategies, and an example related to the best choice problem with non-trivial randomized strategies from the subclass of stopping games with random priorities is given.
Abstract: The paper reviews recent results on Dynkin games and presents new results regarding a solution to such a game in randomized strategies. An example related to the best choice problem with non-trivial randomized strategies from the subclass of stopping games with random priorities is given. Asymptotic results for large N are presented.
TL;DR: In this paper, the effects of continuous spinline cooling and surface tension on the stability of stationary nonisothermal fiber spinning flow was studied and an appropriate extension of the one-dimensional Matovich-Pearson thin filament equations of viscous liquids was derived.
Abstract: In contrast to isothermal fiber spinning, nonisothermal fiber spinning is a remarkably stable process. In this note, we shall study the effects of continuous spinline cooling and surface tension on the stability of stationary nonisothermal fiber spinning flow. We systematically derive an appropriate extension of the one-dimensional Matovich-Pearson thin filament equations of viscous liquids. This model will account for the physical chemistry of the fiber surface and the temperature dependence of the material parameters. We employ semigroup theory to discuss the linear stability of stationary solutions. To this end, we prove the spectral determinacy of the associated semigroup. This approach is made viable by a reformulation of the governing equations to avoid a moving flow domain. Finally, we shall give some computational results to study the onset of surface tension instabilities and their suppression by cooling.
TL;DR: This work presents some sufficient conditions under which the interval hull (or some bounds) of the solution set of a parametrised interval linear system coincides with the interval Hull of the non-parametric intervallinear system corresponding to the parametric one.
Abstract: Consider linear systems whose matrix and right-hand side vector depend affine-linearly on parameters varying within prescribed bounds. We present some sufficient conditions under which the interval hull (or some bounds) of the solution set of a parametrised interval linear system coincides with the interval hull (or bounds) of the non-parametric interval linear system corresponding to the parametric one.
TL;DR: In this article, the authors consider a general discrete structural optimization problem including unilateral constraints arising from, for example, nonpenetration conditions in contact mechanics or noncompression conditions for elastic ropes, and prove that the global optimal designs and equilibrium states converge to the correct ones as the lower bound converges to zero.
Abstract: We consider a general discrete structural optimization problem including unilateral constraints arising from, for example, non-penetration conditions in contact mechanics or non-compression conditions for elastic ropes. The loads applied (and, in principle, also other data such as the initial distances to the supports), are allowed to be stochastic, which we handle through a discretization of the probability space. The existence of optimal solutions to the resulting problem is established, as well as the continuity properties of the equilibrium displacements and forces with respect to the lower bounds on the design variables. The latter feature is important in topology optimization, in which one includes the possibility of vanishing structural parts by setting design variable values to zero. In design optimization computations, one usually replaces the zero lower design bound by a strictly positive number, hence rewriting the problem into a sizing form. For several such perturbations, we prove that the global optimal designs and equilibrium states converge to the correct ones as the lower bound converges to zero.
TL;DR: In this paper, it is shown that decohesive carrying capacity may also occur at a certain point inside the disk, and qualitative analysis of this phenomenon is then illustrated by an analytical-numerical example.
Abstract: Die Trennungstragfahigkeit (DCC) wurde von Szuwalski und Życzkowski als Belastungsparameter p entsprechend unendlich wachsender Normalverzerrung im ideal elastisch-plastischen Korper definiert. In den bisherigen Arbeiten uber DCC von kreissymmetrischen Scheiben konstanter oder veranderlicher Wandstarke ist die unendlich grose Verzerrung er immer am Rand der Scheibe aufgetreten. In dieser Arbeit zeigen wir, dass DCC auch einem gewissen Punkt innerhalb der Scheibe entsprechen kann. Die qualitative Analyse dieser Erscheinung wird mit einem analytisch-numerischen Beispiel illustriert. Das Problem der Scheibe der gleichmasigen Trennungstragfahigkeit wird auch eingehend untersucht.
Decohesive carrying capacity (DCC) was defined by Szuwalski and Życzkowski as the loading parameter p corresponding to infinite increase of a normal strain in perfectly elastic-plastic body. Several papers have been devoted to DCC of circularly symmetric disks of constant or variable thickness and infinitely large normal strains er occured always at the boundary of the disk. In the present paper it is shown that DCC may also occur at a certain point inside the disk. Qualitative analysis of this phenomenon is then illustrated by an analytical-numerical example. The problem of the disk of uniform decohesive carrying capacity is also discussed in detail.
TL;DR: In this paper, nonexplosion of solutions to semilinear stochastic parabolic equations with non-Lipschitz drift and diffusion terms is proved by means of an infinite-dimensional version of the Khasminskii test.
Abstract: Nonexplosion of solutions to semilinear stochastic parabolic equations with non-Lipschitz drift and diffusion terms is proved by means of an infinite-dimensional version of Khasminskii test. The results are applied to stochastic reaction-diffusion equations with polynomial coefficients.
TL;DR: In this paper, the authors proposed a theoretical model for a predator-prey interaction where the prey is a stage-structured population with varying maturity times for the immature stage, and the interaction terms between immatures and adults are interpreted as cooperation.
Abstract: We propose a theoretical model for a predator-prey interaction where the prey is a stage-structured population with varying maturity times for the immature stage. The interaction terms between immatures and adults are interpreted as cooperation. The predator consumes both the young and the adult of the prey and this prey population is more prone to predation at higher densities. The existence and stability of the equilibrium set are discussed. Using maturity time as a bifurcation parameter it is shown that Hopf bifurcation could occur.
TL;DR: In this article, the existence criteria for multiple (at least three) positive solutions to the three-point boundary value problem (g(u'))' + a(t) f(u) = 0, u(0) = 1, u (η) = u(1), where g(v):= |v| p-2 v, p > 1, and η E (0,1) is prescribed.
Abstract: We establish existence criteria for multiple (at least three) positive solutions to the three-point boundary value problem (g(u'))' + a(t) f(u) = 0, u(0) = 0, u(η) = u(1), where g(v):= |v| p-2 v, p > 1, and η E (0,1) is prescribed. The results improve and extend earlier results due to WANG JUNYU and ZHENG DA WEI [1] by an application of the Leggett- Williams fixed point theorem in a cone.
TL;DR: A matrix-free iterative method that is inspired by a Newton type coupling of the subsystems but aims at efficiently controlled linear convergence is presented and it is proved its convergence and proposed a control mechanism to optimize its efficiency.
Abstract: Complex technical systems are often assembled from well-studied subsystems. Here, we elaborate on the obvious idea of coupling existing subsystem solvers to solve a coupled system. More specifically, we present a matrix-free iterative method that is inspired by a Newton type coupling of the subsystems but aims at efficiently controlled linear convergence. We prove its convergence and propose a control mechanism to optimize its efficiency. We illustrate the method's properties with the help of a numerical example.
Note: (As yet) we only deal with the stationary case; mathematically speaking, we are looking for roots of systems of nonlinear equations.
TL;DR: In this paper, the behavior of bounded band-limited functions in the neighborhood of an extremum is characterized and an upper bound on the peak magnitude is derived provided that the function is bounded on oversampling sets.
Abstract: In this paper the behaviour of bounded, bandlimited functions in the neighborhood of an extremum is characterized. In particular, an upper bound on the peak magnitude is derived provided that the function is bounded on oversampling sets. Moreover, it is shown that in this situation an extremal function exists. The question of bounding of band-limited functions naturally arises in interpolation theory and is also of fundamental importance in communication engineering.
TL;DR: In this article, the dependence of the contact angle on the velocity of a liquid at rest in a container to the boundary wall was investigated. But the dependence was not shown to be independent of the velocity.
Abstract: A liquid at rest in a container will show a contact angle at the wall depending on material properties. If the liquid, or the boundary walls, are moving with constant speed, this angle will change with velocity. We perform numerical experiments for a two-dimensional model free boundary value problem that has been proved to be well-posed and show that the dependence of the contact angle on the velocity is qualitatively correct.
TL;DR: In this paper, an impulse response in a viscoelastic rod experimental data are compared with a calculation using fractional derivatives for modelling viscous damping for the material under consideration (teflon).
Abstract: For an impulse response in a viscoelastic rod experimental data are compared with a calculation using fractional derivatives for modelling viscoelastic damping. For the material under consideration (teflon) the coincidence is excellent. The calculation is done for an ideal impulse as well as for a realistic exciting force; it displays also the numerical and theoretical advantages of the residue calculus.
TL;DR: In this article, the authors transform discontinuous systems to special forms being natural for the corresponding problem, and prove averaging theorems for these forms of dynamical systems, illustrated with examples of vibration-induced displacement.
Abstract: Discontinuous systems are usual in mechanics. Main examples are systems with dry friction and systems with collisions. Peculiarities of these systems make them very difficult for conventional asymptotic analysis developed for continuous systems, i.e. requiring from the right-hand sides of the equations of motions a definite level of continuity. The main idea of this paper is to transform discontinuous systems to special forms being natural for the corresponding problem. Averaging theorems are formulated and proved for these forms of dynamical systems. The approach is illustrated with examples of vibration-induced displacement.
TL;DR: In this paper, the general and special solutions of two ordinary second-order differential equations, which are coupled, are derived, are modified Bessel functions of the first and second kind, and powers of the variable.
Abstract: This problem describes the motion of a micropolar suspension between two coaxial cylinders. The two unknown functions are the velocity and the velocity of microrotation of the micropolar theory. The general and special solution of two ordinary second-order differential equations, which are coupled, are derived. These solutions are modified Bessel functions of the first and second kind, and powers of the variable.
TL;DR: In this article, two creep behavior models for thin-walled structural elements are presented, and a detailed discussion of the historical development and the introduction of some important references are presented.
Abstract: In addition to the elastic behavior, engineering materials at elevated temperatures tend to creep or, with other words, a spontaneous and a time-dependent material response can be observed. Due to the increasing safety requirements for power plants, aircraft components, equipment for chemical processes, etc. the time-dependent material behavior should be taken into account in the design process. Since many structural elements working under creep conditions can be classified as thin-walled structures the analysis is connected with the following three items: the choice of a suitable material behavior model, the choice of an adequate structural analysis model, and the choice of a suitable numerical solution technique. All three items are interlinked. For example, the choice of the structural analysis model (beam, plate, shell, etc.) has a significant influence on the numerical effort. On the other hand, the accuracy of the calculations is influenced by the material behavior model taking into account more or less effects. Creep mechanics is a branch of solid mechanics with a history of more than 100 years. After a brief discussion of the historical development and the introduction of some important references two creep behavior models are presented. For the first one the material behavior is not influenced by the kind of stress state while in the second case significant differences in dependence on the kind of the stress state can be observed. A typical example of such behavior is the different behavior in tension and compression. This behavior can be observed, for example, for the tertiary creep characterized by the damage evolution under tensile conditions. If we establish compression conditions, the creep deformations are changing but the damage state is partly frozen (no nucleation of new voids, for instance). The last part is devoted to the structural models. For thin-walled structures all simplifications of the analysis equations are founded on the assumptions with respect to the thinness of the structure. In this case, for instance, one can introduce some hypotheses for the stresses, strains and/or displacements in the thickness direction, and with the help of these hypotheses governing equations, reduced with respect to the dimension, can be established (instead of a system of three-dimensional equations we have a system of two- or one-dimensional equations). On the correctness and accuracy of such an approach will be reported.