Showing papers in "Quarterly of Applied Mathematics in 1968"

Variational principles for linear coupled thermoelasticity

Abstract: Several variational principles are derived for the initial-boundary-value problem of fully coupled linear thermoelasticity for an inhomogeneous, anisotropic continuum. A consistent set of field variables is employed and a method based on the Laplace transform is used to incorporate the initial conditions explicitly into the formulation. These principles lend themselves readily to numerical solutions based on an extended Ritz method.

General solutions for plane extensible elasticae having nonlinear stress-strain laws

Abstract: A general finite deformation solution is obtained for the equilibrium of hydrostatically loaded elasticae whose underformed shapes are circular arcs. The nonlinear stress-strain relations employed give the bending moment and the axial force as derivatives of a strain energy function with respect to suitable strain measures. The representation of the solution involves arbitrary constants of integration which can accommodate any end conditions consistent with equilibrium. Examples are given. The constraint of inextensibility is examined and a perturbation procedure for small extension is developed. In an appendix, the stress-strain laws are derived by an appropriate reduction of the equations for three-dimensional hyperelasticity.

On the existence of normal mode vibrations in nonlinear systems

Abstract: Introduction. Cor tain properties of normal mode vibrations have been studied fairly extensively in recent times [1]. However, the question of their existence has largely remained open (except for the case that the normal mode vibrations are \"similar\". This is the name applied to normal mode vibrations x,(t)/Xj(t) = cu = const, (i, j = 1, 2, • • •); i.e. where the wave shapes are similar in the sense of plane geometry). The first significant advance in establishing existence theorems for cases including nonsimilar normal mode vibrations was made recently in a paper by Cooke and Struble [2] in which the motion is, as the authors say, \"near to linearized motion\". This statement implies actually two properties, one referring to the structure of the system, and the other to its motion. It implies, in fact, that the system is \"linearizable\" [1] and that the motions are small. Under these assumptions, one can readily show that the equations of motion used by Cooke and Struble can always be modelled by a straight, anchored chain of elastically coupled particles, each possessing a single translational degree of freedom in the direction of the chain. For instance, a space array of elastically coupled particles each having three translational degrees of freedom cannot, in general, be a model for their equations because that system may be nonlinearizable, owing to the so-called \"kinematic\" nonlinearities [3]. Also, strongly nonlinear systems do not fall within the compass of the work by Cooke and Struble. It is probably for this reason that these authors conclude that the general question of the existence of normal mode vibrations \"remains an unanswered... mathematical problem\". In this paper we demonstrate the existence of normal mode vibrations of elastically coupled, nonlinear systems where neither the system nor its motion need lie near the linear case. In fact, the systems admitted here need not have isolated equilibrium positions; they may be strongly nonlinear or nonlinearizable, and the nonlinearities may be \"elastic\" or \"kinematic\"; hence, the physically interesting space array of particles is admitted. The methods used in the existence proof are purely geometrical. They depend on the construction of a metric Riemann space on which one can utilize all theorems of Riemannian geometry; in this way, the existence proof is reduced to demonstrating the existence of certain extremal arcs in a Riemann space. The system. A Hamiltonian system is called simple if it is conservative, holonomic and scleronomic. We consider a simple Hamiltonian system having a finite number n of degrees of freedom. The generalized coordinates are the components g,of the vector 9 = (Qi > <72, • • • i 5»), and the velocity vector is q = (ji , q2, •• • ,

The neuron as a Lie group germ and a Lie product

Abstract: A basic Lie group derived earlier that describes the \"constancies\" of visual perception is extended to one more dimension to take into account the nonlinear flow of an afferent volley of nerve impulses through the layers of the visual cortex. One is then led, via the usual determination of the solutions of a Lagrange partial differential equation in terms of an associated Pfaffian system of ordinary differential equations, to a correspondence between neuron cell body, Lie group germ, and critical point of the system of ordinary differential equations governing the orbits. The local phase portraits of the latter bear a marked resemblance to one or the other of the neuron types defined by Sholl. Since \"brains are as different as faces\", the concept of structural stability plays an important role in analyzing the connectivity of the neural network. Finally, Lukasiewicz's theory of parentheses is used to obtain a graph-theoretic representation of the Jacobi identity, which then serves to explain the branching of neuronal processes (dendrites).

Steiner’s problem for set-terminals

Abstract: 1. Let Li , ■ ■ • , Ln be n separate lakes which are to be interconnected by a network of canals of minimum possible length. Or, let il/, , • • ■ , Mn be n metropolitan areas which are to be joined by the shortest possible network of roads. Again, let Py , • •• , P„ be n metal plates in the plane which are to be soldered together by the least possible amount of wire of fixed diameter. Assuming spatial homogeneity, we formulate an abstract problem which underlies the given examples. By a net N we shall understand a finite set of plane rectifiable arcs; the sum L(N) of their lengths is the length of N. Let n > 3 and let A = \Al , • • ■ , An\ be a plane set with exactly n components A{ ; it will be assumed throughout that each A { is compact. Any such A will be called an w-tcrminal set and its n components will be called terminals. Our problem is:

Some nonlocal results for weakly nonlinear dynamical systems

Abstract: where x is an n vector, e the same small parameter as in (1.1) and / an n vector function periodic or almost periodic in t. The results in the literature, valid for all time, for either systems (1.1) or (1.2), are concerned with the existence and stability of periodic or almost periodic solutions and usually a technique for obtaining approximations to these solutions is also given. See, for example, the books by Hale [1], Coddington and Levinson [2], and Bogoliuboff and Mitropolskiy [3]. Bogoliuboff and Mitropolskiy [4] also discuss solutions other than periodic or almost periodic solutions, for systems of the type (1.2), by comparing the solution of (1.2) with the solution of the associated "averaged equation", but their results are valid only for a finite interval of time. Our study is concerned with the solutions of (1.2), valid for all time, but which are not restricted to only periodic or almost periodic solutions. Our main result gives sufficient conditions for the solutions of (1.2) to approach stable almost periodic solutions of (1.2) as time goes to infinity. As in [4], we also compare the solutions of (1.2) with the solutions of the associated "averaged equations" but in our case asymptotically valid estimates for solutions of (1.2) are given for all time. On the basis of this main result, it is possible to give, with the aid of a theorem of LaSalle [5], results regarding the domains of attraction of almost periodic solutions of (1.2). These results regarding the domains of attraction are generalizations of the results for periodic systems given by Loud and Sethna in [6]. In [6] explicit estimates are given for the domains of attraction for second order systems. In the present study we give explicit estimates for the domains of attraction of almost periodic solutions of dynamical systems of order greater then two but in the special case of systems

Rolling contact between dissimilar viscoelastic cylinders

Abstract: This paper treats the plane problem of rolling contact between linear viscoelastic cylinders with different radii and different quantitative mechanical response. The analysis is an extension of that previously given for the simpler problem of rolling contact between two identical cylinders (or equivalently one cylinder and a rigid halfplane), for which a singular integral equation was derived connecting pressure and normal displacement in the contact region. The present problem is shown to lead to an integral equation of identical form but containing further parameters which reflect the difference in the properties of the two cylinders. A neater construction of the closed form solution of the integral equation is presented and the final formulae are expressed in terms of tabulated functions. An illustration is given for a viscoelastic model with two characteristic times.

On local stability of a finitely deformed solid subjected to follower type loads.

Abstract: : The problem of the local stability (stability in the small) of a finitely deformed solid subjected to a set of follower type surface loads is analyzed, and a necessary, and a sufficient condition for asymptotic stability is established. Certain implications of the commonly used modal analysis are also investigated, and necessary and sufficient conditions for stability are formulated. (Author)

On the concept of rate-independence

Abstract: Mechanical and thermodynamical theories of material behavior based on rate independence concept, discussing equivalence of various theories

Optimal elimination for sparse symmetric systems as a graph problem.

Abstract: The optimal (requiring the minimum number of multiplications) ordering of a sparse symmetric system of linear algebraic equations to be used with Gaussian elimination is first developed as a graph problem which is then treated using the functional equation techniques of dynamic programming. A simple algorithm is proposed as an alternative to the more lengthy procedures of dynamic programming and this algorithm is shown to be effective for systems whose graphs are \"grids\". Introduction. The motivation for this work is the fact that there exists a large class of physical problems which give rise to sparse symmetric linear systems, for which the computational effort required to obtain a solution by elimination is highly dependent upon the ordering of the equations. Here a system of n linear algebraic equations A'x' = b' (1) is called symmetric if the coefficient matrix is symmetric and sparse if A' has a large number of zero elements. In many problems dealing with structures, networks, finite difference formulations, etc., this is precisely the case. Certainly if there are no zero elements, there is no such thing as an optimum procedure in the sense in which the term is used here. The origins of this work can be traced back to Ivron [1] in the work which he calls \"Diakoptics\" and more recently to the work of Branin [2] and Roth [3]. It has been pointed out (see also [4], [5]) that to solve these sparse systems by first computing the inverse system matrix can be highly inefficient, and that Gaussian elimination, which is in fact a special case of one of Kron's techniques, is apparently the most efficient procedure, excluding special cases such as, e.g., systems which are highly symmetric (systems in which there is much repetition of elements or groups of elements). There are now digital computer programs available for the automatic analysis of many physical systems. Since the computational effort and therefore the cost is sensitive to the procedure used, it is important to proceed efficiently. In the following, Gaussian elimination is first developed as a graph problem; its functional equation is then treated using the dynamic programming techniques of Bellman [6]; and finally a simple algorithm is discussed which is a computationally attractive alternative to the dynamic programming procedures and which can be easily included in computer programs for automatic analysis. Gaussian elimination. Given a system in the form of Eq. (1), the question considered here is how to find the solution matrix x' using elimination, so that the computing time, and therefore computing cost, is as small as possible. Following von Neumann, the number of multiplications required will be counted as a measure of the computing time. *Received April 11, 1966; revised manuscript received July 21, 1967. 126 W. R, SPILLERS AND NORRIS HICKERSON [Vol. XXVI, No. 3 Consider the distinct phases, elimination and backsubstitution. Generally, a typical step in the elimination phase consists of using, e.g., the zth row of A' to remove all the nonzero terms except a'tl in the jth column by taking linear combinations of rows. Here, a restrictive form is used in which the zth row is used only to delete terms in the ith column. This implies that the system is well conditioned; it also, however, makes the problem tractable. In the backsubstitution phase, known components of the solution matrix x' are used to compute, e.g., the unknown ith component. In this restricted form, the numerical procedure is completely specified for a given system once the equations have been ordered. Further, if only the number of multiplications are to be counted, a reduced matrix A whose elements are defined to be a,, = 1 if a'a 9^ 0 ^

Convergence of Iterative Solutions to Integral Equations for Sound Radiation

Abstract: A preferred method of solving the integral equation for sound radiation problems is by some iterative method. This paper proposes and illustrates a more general iterative scheme than the classical Neumann series and derives conditions for its convergence. I Il I II I 111 1 eg Reprinted from QUARTERLY OF APPLIED MATHEIMATICS Vol. XXVI, No. 2, July 1968 CONVERGENCE OF ITERATIVE SOLUTIONS TO INTEGRAL EQUATIONS FOR SOUND RADIATION* BY GEORGE CHERTOCK, (Naval Ship Research and Development Center, Washington, D. C.)

An exact general solution for the temperature distribution and the composite radiation convection heat exchange along a constant cross-sectional area fin

Abstract: Differential equations for composite radiation and convection heat exchange along constant area fins

On the displacement boundary-value problem of linear elastodynamics

Abstract: Here, cijki are the elastic constants, x is the position vector, uk(x, t) are the components of the displacement u(x, t), p is the (positive) density of the elastic body, and time is denoted by t. The displacement u is always assumed to exist and to be real and twice continuously difTerentiable (i.e. u £ (f). The particular uniqueness problem considered here is to find necessary and sufficient, conditions on the cijkl so that the only twice continuously difTerentiable solution of (l)-(4) is identically zero. After earlier work on isotropic bodies by Neumann [1] and Gurtin and Sternberg [2], Gurtin and Toupin [3] recently have shown that if

Dispersion relations, stored energy and group velocity for anisotropic electromagnetic media

Abstract: The Hermitian and skew-Hermitian components of the susceptibility matrix of a general linear electromagnetic medium are represented as Hilbert transforms of each other. These so-called dispersion relations lead to a priori inequalities which must be satisfied by the susceptibility of a passive medium in a frequency interval in which the medium is lossless. One such inequality states that the stored energy density for a given E(«) and H(ai) is always greater than in free space. This is also verified directly from the usual gyrotropic susceptibilities of ferrites and plasmas. The group velocity for an eigenwave or mode of a structure with one, two or three independent translational symmetry vectors is shown to be, in general, an average Poynting vector divided by an average stored energy density. This formula is then combined with the above inequality for the stored energy density to show that the magnitude of the group velocity is less than c, the velocity of light in free space.

The lift and drag on a flat plate at low Reynolds number via variational methods

Abstract: The problem of determining the low Reynolds number lift and drag on a flat plate in Oseen flow is treated by the application of a variational principle due to Levine and Schwinger. This approach circumvents the difficulty which has limited previous results to drastically low values of the Reynolds number. In particular, an asymptotic form for the drag coefficient is derived, and found to be suitable for a much larger range of this parameter. The ease with which the variational principle can be applied rests heavily on the ability to reduce the pair of coupled integral equations which govern this problem to one integral equation. This equation and its adjoint are shown to have solutions which are quite simply related, which further facilitates the treatment of the variational problem.