Scale transition of a higher order plasticity model – A consistent homogenization theory from meso to macro

Standard plasticity models cannot capture the microstructural size effect associated with grain sizes, as well as structural size effects induced by external boundaries and overall gradients. Many higher-order plasticity models introduce a length scale parameter to resolve the latter limitation – microstructural influences are not explicitly account for. This paper adopts two distinct length scales in the formulation, i.e. an intrinsic length scale (l) governing micro-processes such as dislocation pile-up at internal boundaries, as well as the characteristic grain size (L), and aims to unravel the interaction between these two length scales and the characteristic specimen size (H) at the macro level. At the meso-scale, we adopt the strain gradient plasticity model developed in Gurtin (2004) [Gurtin, M.E., 2004. A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin. J. Mech. Phys. Solids 52, 2545–2568] which accounts for the direct influence of grain boundaries. Through a novel homogenization theory, the plasticity model is translated consistently from meso to macro. The two length scale parameters (l and L) manifest themselves naturally at the macro scale, hence capturing both types of size effects in an average sense. The resulting (macro) higher-order model is thermodynamically consistent to the meso model, and has the same structure as a micromorphic continuum. Finally, we consider a bending example for the two limiting cases – microhard and microfree conditions at grain boundaries – and illustrate the excellent match between the meso and homogenized solutions.     

Extended Thermodynamics of dense gases with many moments – The macroscopic approach

To understand the non-equilibrium phenomena beyond the assumption of local equilibrium in rarefied polyatomic gases, a new version of extended thermodynamics has been recently introduced. It proposes two hierarchies of balance equations; the first of these is the usual one, while the second is an “energy block”. In this framework, we investigate here the model with an arbitrary but fixed number of moments and obtain its closure up to every order with respect to equilibrium. This is done in the context of the macroscopic approach. This is not a trivial aspect; in fact, if we impose the conditions up to a given order with respect to equilibrium, there is always the risk to obtain restrictions on the same expressions by imposing the equations at an higher order! An article has been published to prove that this eventuality does not occur. Similarly, if we impose the conditions for a model with a given number of moments, there is always the risk to obtain restrictions on the same expressions by imposing the equations for the model with an higher number of moments! Here this risk is confirmed, but controlled.     

A consistent equilibrium in a cross-section of an elastic–plastic beam

A phenomenon of inequality of equilibrium and constitutive internal forces in a cross-section of elastic–plastic beams is common to many finite element formulations. It is here discussed in a rate-independent, elastic–plastic beam context, and a possible treatment is presented. The starting point of our discussion is Reissners finite-strain beam theory, and its finite element implementation. The questions of the consistency of interpolations for displacements and rotations, and the related locking phenomena are fully avoided by considering the rotation function of the centroid axis of a beam as the only unknown function of the problem. Approximate equilibrium equations are derived by the use of the distribution theory in conjunction with the collocation method. The novelty of our formulation is an inclusion of a balance function that measures the error between the equilibrium and constitutive bending moments in a cross-section. An advantage of the present approach is that the locations, where the balance of equilibrium and constitutive moments should be satisfied, can be prescribed in advance. In order to minimize the error, explicit analytical expressions are used for the constitutive forces; for a rectangular cross-section and bilinear constitutive law, they are given in Appendix A. The comparison between the results of the two finite element formulations, the one using consistent, and the other inconsistent equilibrium in a cross-section, is shown for a cantilever beam subjected to a point load. The problem of high curvature gradients in a plastified region is also discussed and solved by using an adapted collocation method, in which the coordinate system is transformed such to follow high gradients of curvature.     

Onset of Rayleigh-Bénard convection in gases - classical and extended thermodynamics

This paper deals with the determination of the critical parameters that induce instability in the Rayleigh-Bénard arrangement, viz. a gas confined between two rigid plates and heated from below. In order to compute the critical values of the parameters, classical thermodynamics with the Navier-Stokes and Fourier constitutive relations and extended thermodynamics with thirteen moments are used. Linear stability analysis is performed. The results of both theories are illustrated and compared.Received: 26 August 2003, Accepted: 1 September 2003, Published online: 5 December 2003     

On the thermodynamics of the Swift–Hohenberg theory

We present the microbalance including the microforces, the first- and second-order microstresses for the Swift–Hohenberg equation concomitantly with their constitutive equations, which are consistent with the free-energy imbalance. We provide an explicit form for the microstress structure for a free-energy functional endowed with second-order spatial derivatives. Additionally, we generalize the Swift–Hohenberg theory via a proper constitutive process. Finally, we present one highly resolved three-dimensional numerical simulation to demonstrate the particular form of the resulting microstresses and their interactions in the evolution of the Swift–Hohenberg equation.     

Theory of Extended Solutions¶for Fast-Diffusion Equations¶in Optimal Classes of Data.¶Radiation from Singularities

This paper is devoted to constructing a general theory of nonnegative solutions for the equation called “the fast-diffusion equation” in the literature. We consider the Cauchy problem taking initial data in the set ?⁺ of all nonnegative Borel measures, which forces us to work with singular solutions which are not locally bounded, not even locally integrable. A satisfactory theory can be formulated in this generality in the range 1 > m > m _c = max {(N? 2)/N,0}, in which the limits of classical solutions are also continuous in ? ^N as extended functions with values in ?₊∪{∞}. We introduce a precise class of extended continuous solutions ? _c and prove (i) that the initial-value problem is well posed in this class, (ii) that every solution u(x,t) in ? _c has an initial trace in ?⁺, and (iii) that the solutions in ? _c are limits of classical solutions. Our results settle the well-posedness of two other related problems. On the one hand, they solve the initial-and-boundary-value problem in ?× (0,∞) in the class of large solutions which take the value u=∞ on the lateral boundary x∈??, t>0. Well-posedness is established for this problem for m _c < m > 1 when ? is any open subset of ? ^N and the restriction of the initial data to ? is any locally finite nonnegative measure in ?. On the other hand, by using the special solutions which have the separate-variables form, our results apply to the elliptic problem Δf=f ^q posed in any open set ?. For 1 > q > N/(N? 2)₊ this problem is well posed in the class of large solutions which tend to infinity on the boundary in a strong sense. As is well known, initial data with such a generality are not allowed for m≧ 1. On the other hand, the present theory fails in several aspects in the subcritical range 0> m≦m _c , where the limits of smooth solutions need not be extended-continuously.     

On a Generalized Nonlinear K–ε Model and the Use of Extended Thermodynamics in Turbulence

A resent extension of the nonlinear K–ε model is critically discussed from a basic theoretical standpoint. While it was said in the paper that this model was formulated to incorporate relaxation effects, it will be shown that the model is incapable of describing one of the most basic such turbulent flows as is obvious but is described for clarity. It will be shown in detail that this generalized nonlinear K–ε model yields erroneous results for the Reynolds stress tensor when the mean strains are set to zero in a turbulent flow – the return-to-isotropy problem which is one of the most elementary relaxational turbulent flows. It is clear that K–ε type models cannot describe relaxation effects. While their general formalism can describe relaxation effects, the nonlinear K–ε model – which the paper is centered on – cannot. The deviatoric part of the Reynolds stress tensor is predicted to be zero when it actually only gradually relaxes to zero. Since this model was formulated by using the extended thermodynamics, it too will be critically assessed. It will be argued that there is an unsubstantial physical basis for the use of extended thermodynamics in turbulence. The role of Material Frame-Indifference and the implications for future research in turbulence modeling are also discussed. Received 19 February 1998 and accepted 23 October 1998     

A systematic derivation of a consistent set of “Boussinesq equations”

The often used “Boussinesq equations” for the determination of the coupled flow and temperature field in natural convection are systematically deduced by an asymptotic approach. With the nondimensional temperature difference that drives the flow, ?, as a perturbation parameter the leading order equations are identified as the appropriate equations, named “asymptotic Boussinesq equations”. These equations appear as the distinguished limit $\varepsilon\rightarrow0The often used “Boussinesq equations” for the determination of the coupled flow and temperature field in natural convection are systematically deduced by an asymptotic approach. With the nondimensional temperature difference that drives the flow, ɛ, as a perturbation parameter the leading order equations are identified as the appropriate equations, named “asymptotic Boussinesq equations”. These equations appear as the distinguished limit e?0\varepsilon\rightarrow0 and Ec? 0{Ec}\rightarrow 0 with Ec/e = const.{Ec}/\varepsilon =const. The equations are compared to “Boussinesq equations” of other studies and used to calculate Nusselt numbers in laminar and turbulent flows in infinite vertical channels as an example and for the justification of the asymptotic approach.     

Three-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD

Based on the successive iteration in the Taylor series expansion method,a three-point explicit compact difference scheme with arbitrary order of accuracy is derived in this paper.Numerical characteristics of the scheme are studied by the Fourier analysis. Unlike the conventional compact difference schemes which need to solve the equation to obtain the unknown derivatives in each node,the proposed scheme is explicit and can achieve arbitrary order of accuracy in space.Application examples for the convection- diffusion problem with a sharp front gradient and the typical lid-driven cavity flow are given.It is found that the proposed compact scheme is not only simple to implement and economical to use,but also is effective to simulate the convection-dominated problem and obtain high-order accurate solution in coarse grid systems.     

Fhree-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD

Based on the successive iteration in the Taylor series expansion method, a three-point explicit compact difference scheme with arbitrary order of accuracy is derived in this paper. Numerical characteristics of the scheme are studied by the Fourier analysisl Unlike the conventional compact difference schemes which need to solve the equation to obtain the unknown derivatives in each node, the proposed scheme is explicit and can achieve arbitrary order of accuracy in space. Application examples for the convectiondiffusion problem with a sharp front gradient and the typical lid-driven cavity flow are given. It is found that the proposed compact scheme is not only simple to implement and economical to use, but also is effective to simulate the convection-dominated problem and obtain high-order accurate solution in coarse grid systems.     

Stabilization of projection-based reduced order models of the Navier–Stokes

A new method of stabilizing low-order, proper orthogonal decomposition based reduced-order models of the Navier?CStokes equations is proposed. Unlike traditional approaches, this method does not rely on empirical turbulence modeling or modification of the Navier?CStokes equations. It provides spatial basis functions different from the usual proper orthogonal decomposition basis function in that, in addition to optimally representing the solution, the new proposed basis functions also provide stable reduced-order models. The proposed approach is illustrated with two test cases: two-dimensional flow inside a square lid-driven cavity and a two-dimensional mixing layer.     

FSI research in pipeline systems – A review of the literature

This paper provides a broad overview of the literature pertaining to the dynamic analysis of fluid-filled pipe systems considering fluid–structure interaction (FSI). Various types of models and simulation algorithms of different levels of sophistication are compared and their application range discussed. The effects of fluid parameters, structural properties, fluid–structure couplings and boundary conditions on the inherent and dynamic character of pipes conveying fluid are comprehensively compared and contrasted.     

Describing function based methods for predicting chaos in a class of fractional order differential equations

This paper deals with two different methods for predicting chaotic dynamics in fractional order differential equations. These methods, which have been previously proposed for detecting chaos in classical integer order systems, are based on using the describing function method. One of these methods is constructed based on Genesio–Tesi conjecture for existence of chaos, and another method is introduced based on Hirai conjecture about occurrence of chaos in a nonlinear system. These methods are restated to use in predicting chaos in a fractional order differential equation of the order between 2 and 3. Numerical simulation results are presented to show the ability of these methods to detect chaos in two fractional order differential equations with quadratic and cubic nonlinearities.     

Localized bulging in an inflated cylindrical tube of arbitrary thickness – the effect of bending stiffness

We study localized bulging of a cylindrical hyperelastic tube of arbitrary thickness when it is subjected to the combined action of inflation and axial extension. It is shown that with the internal pressure P and resultant axial force F viewed as functions of the azimuthal stretch on the inner surface and the axial stretch, the bifurcation condition for the initiation of a localized bulge is that the Jacobian of the vector function $(P,F)$ should vanish. This is established using the dynamical systems theory by first computing the eigenvalues of a certain eigenvalue problem governing incremental deformations, and then deriving the bifurcation condition explicitly. The bifurcation condition is valid for all loading conditions, and in the special case of fixed resultant axial force it gives the expected result that the initiation pressure for localized bulging is precisely the maximum pressure in uniform inflation. It is shown that even if localized bulging cannot take place when the axial force is fixed, it is still possible if the axial stretch is fixed instead. The explicit bifurcation condition also provides a means to quantify precisely the effect of bending stiffness on the initiation pressure. It is shown that the (approximate) membrane theory gives good predictions for the initiation pressure, with a relative error less than 5%, for thickness/radius ratios up to 0.67. A two-term asymptotic bifurcation condition for localized bulging that incorporates the effect of bending stiffness is proposed, and is shown to be capable of giving extremely accurate predictions for the initiation pressure for thickness/radius ratios up to as large as 1.2.     

Development of fractional order capacitors based on electrolyte processes

In recent years, significant research in the field of electrochemistry was developed. The performance of electrical devices, depending on the processes of the electrolytes, was described and the physical origin of each parameter was established. However, the influence of the irregularity of the electrodes was not a subject of study and only recently this problem became relevant in the viewpoint of fractional calculus. This paper describes an electrolytic process in the perspective of fractional order capacitors. In this line of thought, are developed several experiments for measuring the electrical impedance of the devices. The results are analyzed through the frequency response, revealing capacitances of fractional order that can constitute an alternative to the classical integer order elements. Fractional order electric circuits are used to model and study the performance of the electrolyte processes.     

On the evolutions of the discontinuities of orders ⩾ m in solutions for quasi-linear systems of PDEs of order m,or for the linearizations of these systems

Summary One considers a system L[u]=0 of PDEs, quasi-linear (according to [1]) and of order m, which possesses a bicharacteristic line , as it happens in the hyperbolic case. For v=0, , –m (>0) let u(^v) be a discontinuity wave of order m+v that solves the system above and whose discontinuity hypersurface includes . The corresponding transport equations along are considered. Furthermore some interesting cases are pointed out, in which these equations turn out to be mutually equivalent in a suitable sense. Some theorems are stated to compare the transport equations for the discontinuities of the above kinds, that are connected with the systems d^hL[u]/dt^h=0 (h=0, , –m) and/or the linearization of the system L[u]=0 around any regular solution of it.

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Sommario Si considera un sistema L[u]=0 di equazioni alle derivate parziali, quasi lineare (secondo [1]) e di ordine m, il quale sia dotato di qualche bicaratteristica , come accade nel caso iperbolico. Per v=0, , –m(>0) sia u(^v) un'onda di discontinuità di ordine m+v risolvente il detto sistema e avente ipersuperficie di discontinuità contenente Si considerano le relative equazioni di trasporto lungo e si determinano casi interessanti in cui queste equazioni sono mutuamente equivalenti in senso opportuno. Si stabiliscono teoremi di confronto per il trasporto delle discontinuità del tipo suddetto, relative ai sistemi d^hL[u]/dt^h=0 (h=0, , –m) e/o alla linearizazione del sistema L[u]=0 attorno a qualche sua soluzione regolare.

     

Table of Contents of Volume 36 – 2001

Volume Contents

Table of Contents of Volume 36 – 2001