Showing papers in "Archive for Rational Mechanics and Analysis in 1972"
TL;DR: In this paper, a general theory of dissipative dynamical systems is presented, where dissipativeness is defined in terms of an inequality involving the storage function and the supply function, which is bounded from below by the available storage and from above by the required supply.
Abstract: The first part of this two-part paper presents a general theory of dissipative dynamical systems. The mathematical model used is a state space model and dissipativeness is defined in terms of an inequality involving the storage function and the supply function. It is shown that the storage function satisfies an a priori inequality: it is bounded from below by the available storage and from above by the required supply. The available storage is the amount of internal storage which may be recovered from the system and the required supply is the amount of supply which has to be delivered to the system in order to transfer it from the state of minimum storage to a given state. These functions are themselves possible storage functions, i.e., they satisfy the dissipation inequality. Moreover, since the class of possible storage functions forms a convex set, there is thus a continuum of possible storage functions ranging from its lower bound, the available storage, to its upper bound, the required supply. The paper then considers interconnected systems. It is shown that dissipative systems which are interconnected via a neutral interconnection constraint define a new dissipative dynamical system and that the sum of the storage functions of the individual subsystems is a storage function for the interconnected system. The stability of dissipative systems is then investigated and it is shown that a point in the state space where the storage function attains a local minimum defines a stable equilibrium and that the storage function is a Lyapunov function for this equilibrium. These results are then applied to several examples. These concepts and results will be applied to linear dynamical systems with quadratic supply rates in the second part of this paper.
3,124 citations
TL;DR: The principal result of this work shows that there exists a simply identifiable class of kinetic expressions, including the familiar detailed balanced kinetics as a proper subclass, which ensure consistency with the extended thermodynamic conditions.
Abstract: The familiar idea of mass action kinetics is extended to embrace situations more general than chemically reacting mixtures in closed vessels. Thus, for example, many reaction regions connected by convective or diffusive mass transport, such as the cellular aggregates of biological tissue, are drawn into a common mathematical scheme. The ideas of chemical thermodynamics, such as the algebraic nature of the equilibrium conditions and the decreasing property of the free energy, are also generalized in a natural way, and it is then possible to identify classes of generalized kinetic expressions which ensure consistency with the extended thermodynamic conditions. The principal result of this work shows that there exists a simply identifiable class of kinetic expressions, including the familiar detailed balanced kinetics as a proper subclass, which ensure consistency with the extended thermodynamic conditions. For kinetics of this class, which we call complex balanced kinetics, exotic behavior such as bistability and oscillation is precluded, so the domain of search for kinetic expressions with this type of behavior, which is of considerable biological interest, is greatly narrowed. It is also shown that the ideas of complex balancing and of detailed balancing are closely related to symmetry under time reversal.
1,143 citations
TL;DR: The theory of dissipative systems in the context of finite dimensional stationary linear systems with quadratic supply rates has been studied in this paper, where a necessary and sufficient frequency domain condition for dissipativeness is derived.
Abstract: This paper presents the theory of dissipative systems in the context of finite dimensional stationary linear systems with quadratic supply rates. A necessary and sufficient frequency domain condition for dissipativeness is derived. This is followed by the evaluation of the available storage and the required supply and of a time-domain criterion for dissipativeness involving certain matrix inequalities. The quadratic storage functions and the dissipation functions are then examined. The discussion then turns to reciprocal systems and it is shown that external reciprocity and dissipativeness imply the existence of a state space realization which is also internally reciprocal and dissipative. The paper proceeds with an examination of reversible systems and of relaxation systems. In particular, it is shown how a unique internal storage function may be defined for relaxation systems. These results are applied to the synthesis of electrical networks and the theory of linear viscoelastic materials.
1,061 citations
Esso1
692 citations
TL;DR: In this article, the Lagrange multipliers are used to derive restrictions from the entropy principle for all kinds of different materials, and the method proposed here is intended to facilitate the solution of this problem.
Abstract: In thermodynamics, whenever a material is subject to investigation of its properties, the entropy principle is used to impose restrictions on the constitutive relations for the material. There is not, so far, any simple method by which restrictions from the entropy principle can be derived for all kinds of different materials. Indeed, until very recently there was not even an entropy principle appropriate to all bodies. MULLER [1] [2] [3] has proposed such a general entropy principle, and here I suggest a simple method for its exploitation: the method of Lagrange multipliers. The entropy principle requires the constitutive relations to be such that every solution of the thermodynamic field equations satisfy the entropy inequality. To satisfy this requirement is not easy, since usually complicated calculations ensue. The method proposed here is intended to facilitate the solution of this problem. It can be described roughly as follows:
644 citations
TL;DR: In this article, ESHELBY deduced a surface-integral representation for the force on an elastic singularity or inhomogeneity, which gives rise to a conservation law for regular elastostatic fields appropriate to homogeneous but not necessarily isotropic solids in the presence of infinitesimal deformations.
Abstract: Several years ago ESHELBY [1] (1956), in a paper devoted to the continuum theory of lattice defects, deduced a surface-integral representation for the "force on an elastic singularity or inhomogeneity", which-in the absence of such
defects-gives rise to a conservation law for regular elastostatic fields appropriate to homogeneous but not necessarily isotropic solids in the presence of infinitesimal deformations. Morevoer, ESHIELBY noted that his result, when suitably interpreted, remains strictly valid for finite deformations of elastic solids.
583 citations
504 citations
462 citations
442 citations
TL;DR: In this article, a necessary and sufficient condition for complex balancing in mass-action systems is derived, which excludes the occurence of equilibria in certain composition regions of general kinetic systems.
Abstract: In a recent publication (Horn & Jackson [1]) it was shown that complex balancing together with mass action type rate laws ensures certain stability properties of a kinetic system, thereby precluding sustained oscillations, bistability and other types of irregular dynamics. In this paper a necessary condition for complex balancing in general kinetics and necessary and sufficient conditions for complex balancing in mass action systems are derived. A theorem is stated which excludes the occurence of equilibria in certain composition regions of general kinetic systems. For mass action systems it is shown that it is sometimes true that the algebraic structure of the reactions suffices to ensure complex balancing, while in other cases complex balancing occurs only if certain relations between the rate constants are satisfied. The number of these relations, called the deficiency of the mass action system is determined by the algebraic structure of the set of reactions underlying that system.
397 citations
TL;DR: In this article, a different approach from usual to the formulation of conservation laws and an entropy production inequality was adopted for a single phase continuum and for a mixture of any number of constituents.
Abstract: Within the scope of classical continuum thermodynamics, we elaborate on the basic concepts and adopt a different approach from usual to the formulation of conservation laws and an entropy production inequality, both for a single phase continuum and for a mixture of any number of constituents. These conservation laws and the entropy inequality can be regarded as applicable to both local and nonlocal problems. In the case of a single phase continuum and for a simple material which is homogeneous in its reference configuration, under fairly mild smoothness assumptions, we prove that all the conservation laws reduce to the usual classical ones and the entropy production inequality reduces to the Clausius-Duhem inequality. Some attention is given to possible redundancies in the basic concepts, as well as to alternative forms of the energy equation and the entropy inequality. The latter is particularly significant in regard to different but equivalent formulations of mixture theory.
TL;DR: The theory of simple materials was formulated by as discussed by the authors in 1958 in reference [N2] and has served as a foundation for a large part of the research in continuum physics since 1958, and considering that the concepts and even the notations of [N 2] are now used routinely and without reference in textbooks on continuum mechanics.
Abstract: The original theory of simple materials was formulated by me in 1958 in reference [N2]. I proposed this theory in an attempt to unify and clarify the confusing variety of theories of mechanical behavior of materials that had been proposed in the literature up to that time. One can perhaps say that the attempt was moderately successful, considering that this first theory of simple materials has served as a foundation for a large part of the research in continuum physics since 1958, and considering that the concepts and even the notations of [N2] are now used routinely and without reference in textbooks on continuum mechanics.
TL;DR: In this article, the authors examine conditions under which a function g(x,t,...,t ) provides, via composition, a 1 m mapping between Sobolev spaces, and examine conditions which ensure that for every system of functions u_,...,u e W _ (Q) 1 m 1, q (where W, q is the class of L functions with L summable i,q q q q strong first derivatives on the domain Q, c: R )S the composite function v given by v(x) = g (x,un(
Abstract: The present paper is concerned with the circumstances under which a function g(x,t ,...,t ) provides, via composition, a ^ 1 m mapping between Sobolev spaces. That is, we examine conditions which ensure that for every system of functions u_,...,u e W _ (Q) 1 m 1, q (where W, (Q) is the class of L functions with L summable i,q q q strong first derivatives on the domain Q, c: R )S the composite function v given by v(x) = g(x,un(x) , . . . ,u (x) ) belongs to Wn _(£})> with preassigned 1 R locally absolutely continuous and ueW (H) , JL _L -L} 1 "I r\r+ lOC one has v(x) = g(u(x)) eW, ,(fl) if and only if g' (u(x) ) Vu(x) eL. (0) ~ ~ 1,1 ~ ~ l
TL;DR: In this article, the authors give a complete answer to the latter problem: Vn--g~(z,z)>cB(z) with equality if and only if f2 is conformally equivalent to the unit disc less a (possible) closed set of inner capacity zero.
Abstract: Let K(z, z), cp(z) and c~(z) be the values of the Bergman kernel, the capacity, and the analytic capacity, on an open Riemann surface 12 (with respect to a fixed local parameter z). The following problem was raised by SARIO & OIKAWA [9]: Find a relation between the magnitudes of the quantities ]/~---g~(z, z), c a (z) and cg (z). As to ]/n--K-~(z, z) and ce (z), HEMAL obtained an answer for finite Riemann surfaces f2, namely that ] / ~ > c B ( z ) if f2 is not simply connected. 1 In the present paper, we shall give the following complete answer to the latter problem: Vn--g~(z,z)>cB(z) with equality if and only if either (i) 12e0o or (ii) f2 is conformally equivalent to the unit disc less a (possible) closed set of inner capacity zero. Concerning the problem for ]/n--K~(z, z) and ca(z), we are led to conjecture that ]/n--g-(~,(z,-z))> ca(z); this will be verified for doubly-connected regions in w 4. 6 2 By proving a new identity ~ log c a (z)= n K(z, z), we show that the conjecture
TL;DR: In this article, TEMAM et al. show that there exists a unique solution of the problem of finding the optimal solution of a variation of the original problem, i.e., (1) Sgradu(gradv-gradu)dx> Sf(v-u)-dx for tout v~K. Probl~me 1.
Abstract: Soit fl ~ JR. Nun ouvert born6 de fronti~re 0t2 r6guli~re et soit K = {v aHo t (fl); [ grad v ] =< 1 p.p. sur fl} ={veC(~); v=0 sur ~fl et [v(x)-v(y)] Sf(v-u)dx pour tout v~K. Probl~me 1. Peut-on trouver une fonction 2 telle que (2) 2 >_-0 p.p. sur (3) 2(1-]gradu])=O p.p. sur f2 (4)-Au-~=I-~x~ \\ Ox~] =f au sens de ~'(f2). Notons qu'inversement, si u~K et s'il existe 2 v6rifiant (2), (3), (4), alors u est la solution de (1). En effet, on a d'apr~s (4) grad u (grad v-grad u) d x = ~ f(v-u) d x-~ 2 grad u (grad v-grad u) d x. f~ f/ f2 Or si v~K on a 2 (I grad u 12 _ grad u. grad v) > 2 (1 grad u 12 _ I grad u [ I grad v l) >2 [grad u [ (I grad u I-1) =0, d'o6 l'on d6duit (1). Lorsque N= 2 et f est constant sur I2 (ce cas particulier est important car il correspond au probl~me de la torsion 61asto-plastique d'une barre cylindrique; Cf. TING [9] et LANCHON [3]) l'existence de 2 est assur~e. 1 Je remercie R. TEMAM qui a attir6 mon attention sur ce probl~me; pour d'autres exemples o/1 intervielment la dualit6 et les multiplicateurs de Lagrange cf. TE~t [7]. I1 semble toutefois que les techniques abstraites d'analyse fonctionnelle soient ici insuffisantes.
TL;DR: In this paper, the authors present new quadratic inequalities, an improved coefficient estimate, and alternate proofs for a few important theorems for one-to-one analytic functions on the unit disc.
Abstract: An extensive literature [6] concerning one-to-one, analytic functions on the unit disc has developed. The bounds on the coefficients of the power series expansions about the origin are of particular interest. This paper presents some new quadratic inequalities, an improved coefficient estimate, and alternate proofs for a few important theorems. Although the motivation and theorems of the paper are closely tied to previous work, the central results depend logically only on Theorem 1.1 and proofs presented here.
TL;DR: In this article, the authors studied the mixed problems of (1.1) in a cylindrical domain with respect to the initial condition R'x R under the assumption that the plane is smooth.
Abstract: A brief survey of equations similar to (1.1) was made in [30], where mixed problems for pseudo-parabolic equations had been studied. We refer to the references in [30] for earlier results on equations of this type. Much progress on mixed problems in a cylindrical domain has recently been made [19, 27-31]. The objective of this work is to study the solutions of (1.1) in the whole space R ' x R under the initial condition
TL;DR: Gurtin et al. as discussed by the authors studied the notion of invariance under time-reversal and established conditions that are both necessary and sufficient for the infinitesmal entropy production to display such invariance.
Abstract: TIME-REVERSAL AND SYMMETRY IN THE THERMODYNAMICS OF MATERIALS WITH MEMORY by Morton E. Gurtin We here study, within the framework of the thermodynamics of materials with memory, the notion of invariance under timereversal. In particular, we establish conditions that are both necessary and sufficient for the infinitesmal entropy production to display such invariance.