Showing papers in "Quarterly of Applied Mathematics in 1973"

Elastic waves in rotating media

Abstract: Plane harmonic waves in a rotating elastic medium are considered. The inclusion of centripetal and Coriolis accelerations in the equations of motion with respect to a rotating frame of reference leads to the result that the medium behaves as if it were dispersive and anisotropic. The general techniques of treating anistropic media are used with some necessary modifications. Results concerning slowness surfaces, energy flux and mode shapes are derived. These concepts are applied in a discussion of the behavior of harmonic waves at a free surface. Introduction. In this paper plane wave propagation in a linear, homogeneous, isotropic elastic medium will be considered, with the assumption that the entire elastic medium is rotating with a uniform angular velocity. If the coordinate system is taken as fixed in the rotating medium, this introduces additional terms in the equations of motion: a centripetal and a Coriolis acceleration. We consider small-amplitude waves propagating in the medium and exclude any discussion of the time-independent stresses and displacements that are caused by centrifugal forces and other possible body forces. In the following section the equations governing plane-wave solutions in an infinite rotating medium are formulated and it is shown that there are three real slowness surfaces, each corresponding to the root of a cubic characteristic equation. It is shown that the phase speed in all cases depends on the ratio of the wave frequency to the rotational frequency, thus making it clear that the rotation causes the material to be dispersive. The actual slowness surfaces are given for various values of Poisson's ratio and the frequency ratio. In the next section the energy flux for plane waves is discussed and it is proved, for any admissible plane wave, to be perpendicular to the slowness surface at the point indicated by the slowness vector (essentially the wave number vector) of the wave. Actual displacements that occur are discussed qualitatively in the subsequent section. It is seen that, in general, the various modes are neither shear nor compressional, but combinations of both. All exceptional cases of pure shear or pure compressional modes are discussed. In the last section free surface phenomena are discussed. To describe the reflection of plane waves from a plane free surface, use is made of the slowness surfaces. Much qualitative information on types of reflected waves can be brought out even without the use of the explicit expressions for the slowness vector, such as under what circumstances one or two of the reflected waves will be surface waves, i.e. have a complex * Received December 30, 1971; revised version received March 28, 1972. 116 MICHAEL SCHOENBERG AND DAN CENSOR slowness vector. As an extension of this, a solution which consists only of surface waves is discussed. This can be thought of as a generalized Rayleigh wave. The analysis involved in the derivations is roughly similar to that used for wave propagation in anisotropic media by Synge [1], [2] and Musgrave [3], among others. A rotating medium can be thought of as a type of transverse isotropic medium; i.e., all directions orthogonal to some direction, in this case the axis of rotation, are equivalent. However, there are basic differences between rotational and material anisotropy. For a rotating medium, substitution of a plane wave solution into the equations of motion leads to a homogeneous set of linear equations for which the matrix of the coefficients is Hermitian, instead of symmetric, as is the case for non-rotating media. Hence the eigenvectors, i.e. the displacement vectors, even for real slowness vectors, are complex instead of real, implying that the particle trajectories are elliptical. Further, as mentioned above, the solutions for a rotating medium are frequency-dependent. In addition, there is no easily perceived eigenvector that can be used to find one root of the cubic characteristic equation, thus leaving only a quadratic equation to be solved, as for a nonrotating transverse isotropic medium (see [1, p. 331]). The standard index notation is used throughout, e.g. for the position vector x = Xi&i = x1e1 + x2e2 + x3es . The cap will always denote unit vectors. Use is made both of vector and indicial notation. The rotating elastic medium. Consider an infinite homogeneous, isotropic, linear elastic medium characterized by a density p, a shear modulus and a bulk modulus. These quantities determine, in the usual fashion, a shear wave speed C, and a pressure wave speed Cv . The medium is rotating uniformly with respect to an inertial frame, and the constant rotation vector in an Xi , x2 , rectangular Cartesian frame rotating with the medium is ii = S!w. The unit vector w will denote the direction of the axis of rotation (according to the right-hand rule) throughout. The displacement equation of motion in such a rotating frame has two terms that do not appear in the non-rotating situation. As we are looking for time-varying dynamic solutions, the time-independent part of the centripetal acceleration X (£} X x), as well as all body forces, will be neglected. Thus our dynamic displacement u is actually measured from a steady-state deformed position, the deformation of which, however, is assumed small. The equation which governs the dynamic displacement u is (iCl Cl)V(V-u) + C2,V2u = u + a X (n X u) + 2£2 X u, (1) where the dot denotes time differentiation. The term Q X (H X u) is the additional centripetal acceleration due to the time-varying motion only, and the term 2£i X u is the Coriolis acceleration. All other terms are as usual for a linear elastic medium under the assumptions of smajl strains and displacements. We look for plane wave solutions of the form u = (R U exp iw ^ -')\" = (R U exp ~ 1 (2) where (R denotes the real part, U is a constant vector, in general complex, a; is the angular frequency, C is the phase speed, n is a unit vector in the direction of propagation, and S is the non-dimensional slowness vector with amplitude S = CJC in the direction of fi. ELASTIC WAVES IN ROTATING MEDIA 117 Substitution of (2) into (1) gives (1 0)(S-U)S + /3S2U = U (1/oj2)£1 X (Q X U) + (2i/u)Q X U, /3 = (C./C,,)2, or, by making use of Si = S2w and the vector identity w X (w X U) = (wU)w — U (i /s)(s-u)s + /3S2u = (i + r2)u r2w-Uw + 2irw x u, r = a/«. (4) This vector equation stands for three scalar equations on the three components U, , and they can be written as Mull, = [(1 + T2) 5,,— T3wtu— 2iTeiikwk — (1 JS)sis/ — fisksk 8u]Ui = 0, (5) where Sif is the Kronecker delta and eijk is the permutation tensor. Note that, because of the Coriolis term, the matrix Mu is no longer symmetric but Hermitian. The necessary and sufficient condition for the existence of non-trivial eigenvectors, [/,•, is det Mu = 0. In general, there may be real and complex vectors, , which satisfy the condition det M a = 0. However, complex

Triangular, nine-degrees-of-freedom, ⁰ plate bending element of quadratic accuracy

Abstract: Introduction. The triangular plate bending element with the three nodal values w, wx and wv at the vertices has a particular appeal—its simplicity. But a C1 polynomial interpolation scheme defined over the whole element does not exist. To overcome this Clough and Tocher [1] resorted to subdividing the element into three subelements with the transverse displacement w interpolated individually [2] over each subtriangle so as to maintain a C1 continuity both in the interior of this complex element and on its boundaries. Bazeley et al. [3] interpolated w by some rational functions to obtain the desired variation of w and its derivatives along the sides of the element for assuring a C1 continuity of displacements. They considered also the use of elements which violate the continuity requirements (non-conforming) and for which the variational principle on minimum total potential energy does not hold any more. These elements may, nevertheless, sometimes produce a valid, stable difference scheme and converge to a useful solution. But their excessive flexibility and precarious convergence did not endear them to engineers. Severn, Taylor and Dungar [4, 5] and Allman [6] used a mixed variational principle [7] for generating a compact hybrid [8] finite element. Stricklin et al. [9] (and also others) made a more radical approach to the generation of plate bending elements in general and the nine-degrees-of-freedom triangular element in particular. They started from the basic equations of elasticity rather than from a ready plate theory and did away with the Kirchhoff assumption (the shear) except at the vertices and along the sides of the element (this is the \"discrete Kirchhoff assumption\" [10]). Since the plate is now considered a three-dimensional solid, the in-plane displacements are introduced independently of the transverse displacements and the continuity requirement for them is only C°, as in three-dimensional elasticity. Assuming that the shear energy is at any rate negligible in thin plates, Stricklin neglected it altogether. Removing the Krichhoff assumption from the finite-element analysis of plates and starting with the basic three-dimensional elasticity would seem the most natural approach to the generation of bending elements, particularily since the thin plate is obtained as a limiting case from a three-dimensional solid. However, without the a priori assumptions of Kirchhoff, and with the shear energy retained, the global stiffness matrix becomes violently ill-conditioned [11] as the thickness t of the structure is reduced. The difficulties, then, in constructing thin-plate bending elements directly from three-dimensional elasticity are of a numerical or computational nature. The decline in the conditioning of the matrix may cause, in the computational stage of the solution, grave numerical (round-off) errors, or for a computer with insufficient significant digits the matrix may even be numerically singular.

Refraction of acoustic duct waveguide modes by exhaust jets.

Abstract: The refraction of acoustic duct waveguide modes emitted from the open end of a semiinfinite rectangular duct by a jet-like exhaust flow is studied theoretically. The problem is formulated as a Wiener-Hopf problem and is ultimately solved by an approximate method due to Carrier and Koiter. Continuity of transverse acoustic particle displacement and of acoustic pressure is assumed at the jet/still-air interface. The solution exhibits several features of the acoustics of moving media such as a source convection effect, zones of relative silence, and simple refraction. Plots of far-field directivity patterns are presented for several cases and show refraction effects to be important even at modest exhaust Mach numbers of order 0.3. Only subsonic exhaust Mach numbers are considered.

Concentration-dependent diffusion. II. Singular problems

Abstract: when the diffusion coefficient D is a function of the concentration c. Only positive D were considered although we remarked that there are problems of physical interest approximated by D which vanish. Ames [2, Sec. 1.2] discusses a variety of physical situations leading to such problems, as do [3-6]. The ordinary differential equation resulting from the use of the similarity variable t) = x/\\/t, mi)+ !! = »■ «

Unsymmetric wrinkling of circular plates

Abstract: The branching of un,symmetric equilibrium states from axisymmetric equilibrium states for clamped circular plates subjected to a uniform edge thrust and a uniform lateral pressure is analyzed in this paper. The branching process is called wrinkling and the loads at which branching occurs are called wrinkling loads. The nonlinear von Karman plate theory is employed. The wrinkling loads are determined by solving numerically the eigenvalue problem obtained by linearizing about a symmetric equilibrium state. The post-wrinkling behavior is studied by a perturbation expansion in the neighborhood of the wrinkling loads.

Divergence instability of multiple-parameter circulatory systems

Abstract: The free vibrations and stability of a discrete linear non-dissipative system subjected to the combined effect of several independent circulatory and conservative loads are studied. Attention is restricted to systems which exhibit only divergence-type instability and are incapable of flutter. Two theorems concerning the basic properties of the characteristic surjaces (loading-frequency relationship) and the stability boundary are established. One of the several other theorems insures that the stability boundary of the corresponding conservative system will always provide a lower bound for the actual stability boundary of the non-conservative system. This particular result is valid for all types of divergence-instability problems, including those which can exhibit flutter after divergence. The theorems established are quite general in the sense that they are valid for all systems in the class considered, without further analytical conditions. Practical implications of the general results are briefly discussed.

On balance of forces for mixtures

Abstract: The usual theory of mixtures assumes the existence of a partial stress tensor; this leads to a questionable relation of symmetry of forces. We show how a reinterpretation of the concept of partial stress tensor avoids this problem. The usual balance equations hold, but the distinction between the new stress tensor and the old leads to a different interpretation of boundary data. 1. A paradox in the theory of mixtures. Consider a mixture M, and suppose that M occupies a fixed region B of Euclidean space (at a given time). Given two subregions A and C of B, let fa/s(A, C) denote the force exerted by constituent /3 in C on constituent a in A. When a 5^/3, fap(A, C) is defined for all A and C, even when they overlap, since it is meaningful to talk about, e.g., fap(A, A), the force exerted by constituent (i in A on a in A. On the other hand, faa(A, C) is defined only when A and C are separate.1 We assume that the force system obeys the following law of action and reaction:

Energy-like Liapunov functionals for linear elastic systems on a Hilbert space.

Abstract: An approach is presented for generating energy-like functionals for linear elastic dynamic systems on a Hilbert space. The objective is to obtain a family of functionals which may be used for stability analysis of the equilibrium, i.e., Liapunov functionals. Although the energy functional, when one exists, is always a member of this family, the family is shown to exist even when an energy functional does not. Several discrete and distributed-parameter examples are presented, as are certain specific techniques for utilizing this approach.

An asymptotically efficient estimate in time series analysis

Abstract: 1. A problem in linear regression. It is well known that often a theoretically best statistical estimate may be of limited value in actual practice. The computation required may be excessive, for instance, or more critically, a priori information may be needed which is unavailable to the investigator. As a result, there has been considerable interest traditionally in the formulation of other estimates which are close to optimal and at the same time computationally feasible and less dependent on intimate knowledge of the particular phenomenon being studied. The purpose of this work is to make such an addition to an already rich branch of statistics—time series analysis. Let us consider the simplest example of linear regression in time series. We have a discrete time process

On the existence of solutions to optimization problems with eigenvalue constraints

Abstract: The optimum tapering of Bernoulli-Euler beams, i.e. the shape for which a given total mass yields the highest possible value of the first fundamental frequency of harmonic transverse small oscillations, is determined. The question of the existence of a solution to the optimization problem is considered. It is shown that, irrespective of the relationship between the flexural rigidity and linear mass density of the cantilever beam, the necessary conditions for optimality lead to a contradiction. This result is in partial disagreement with that obtained by earlier investigators. By imposing additional constraints on the optimization variable, a numerical solution for the case of the cantilever beam is obtained, using the formulation of the maximum principle of Pontryagin. Introduction. Optimization of elasto-mechanic systems in which the behavioral constraint is a deformation bound has become a standard design procedure. However, optimization with a frequency constraint is a more difficult problem. The object of optimization in such problems is to establish, from among all designs of given style and specified total mass, the one for which the lowest natural mode is a maximum. This design is at the same time the minimum-volume (or -mass) structure for a specified fundamental natural frequency. The proper vehicle for modelling the above optimization problem is the calculus of variations. The theory for the extremum problem of Bolza, which is the classical mathematical tool, was developed by Bliss, McShane, Hestenes and others. On the non-classical side two principles have evolved and since become popular, namely the Maximum Principle of Pontryagin [1] which will be used here and the Principle of Optimality of Bellman. These represent the necessary extremum conditions obtainable by the use of first derivatives. This paper is motivated by two publications, one by Brach [2] and the other by Karihaloo and Niordson [3]. Brach has raised the questions of the existence of the solution of the optimization problem for a cantilever beam (and free-free beams) for the restricted case where the flexural rigidity and the mass distribution along the span of the beam are linearly related. Although the distinction between the simply supported case and the cantilever/free-free cases is not very clearly brought out, it is clear that the existence * Received April 19, 1972. This research was carried out with the support of the National Research Council of Canada, grant No. A3728. The author is also grateful to the NRC of Canada for a PIER fellowship during 1969-1971. The author acknowledges with pleasure helpful discussions with Prof. John Roorda of the Solid Mechanics Division, University of Waterloo, who guided his postgraduate research work.

Oscillation and nonoscillation of even-order nonlinear delay-differential equations

Abstract: for their oscillatory and nonoscillatory nature. In Eqs. (1) and (2) y(\"(x) = (d%/dx')y{x), i = 1, 2, • • • , 2n; yr(x) = y(x — t(x))] dy/dx and d2y/dx2 will also be denoted by y' and y\" respectively. Throughout this paper it will be assumed that p(x), j(x), t{x) are continuous real-valued functions on the real line (—°°, °°); j(x), p(x) and t(x), in addition, are nonnegative, r(x) is bounded and f(x), p(x) eventually become positive to the right of the origin. In regard to the function g we assume the following: (i) g : R2 —* R is continuous, R being the real line, (ii) g(\\x, \\y) — X2a+1^(x, y) for all real \\ ^ 0 and some integer q > 1, (iii) sgn g(x, y) = sgn x, (iv) g(x, y) —* as x, y —* <=° ; g is increasing in both arguments monotonically. Eq. (I) is called oscillatory if every nontrivial solution y(t) £ [<0 , 00) has arbitrarily large zeros; i.e., for every such solution y(t), if y(t,) = 0 then there exists <2 > a > 0. A similar definition holds for eq. (2). All solutions of (1) and (2) considered henceforth are continuous and nontrivial, existing on some halfline [t0 , 0° ). 2. Nonoscillation of Eq. (1). We will need the following lemmas. Lemma 1. (Staikos and Petsoulas [8, p. 697].) If y(t) > 0, y'(t) > 0 and y\"(t) < 0 for large t, then lim,_.„ (yT(t)/y(t)) = I. Lemma 2. Suppose J\" t2\"~1p(t) dt — Let y(t) be a nonoscillatory solution of Eq. (2) such that y(t) <0 for large t. Then

Nonlinear effect of initial stress on crack propagation between similar and dissimilar orthotropic media

Abstract: The theory of crack propagation in orthotopic media is developed by applying the theory of incremental deformations in the vicinity of a state of initial stress. This is carried out in the context of a new approach to analytical methods and a physical analysis which takes into account plastic deformation under prestress. The state of initial stress is triaxial along the directions of elastic symmetry, and the crack is parallel to these directions. An additional shear component for the initial stress is also taken into account and general conditions are derived for crack propagation, including the case of fluid injection into the crack. The analysis is first carried out for an homogeneous medium. The nonlinear influence of the initial stress appears in two ways: first, through a fundamental purely elastic effect related to the occurrence of surface instability, and second, through the influence of the initial stress on plastic behavior. The particular cases of an isotropic elastic medium with finite initial strain and an orthotropic incompressible medium are discussed. The analysis is extended to a crack between dissimilar orthotropic media with initial stress. The method of analysis leads to a number of simplifications and brings out new properties of the solutions for this type of problem. For incompressible media without initial stress, the typical oscillatory behavior disappears. Uniqueness of the solutions is also derived.

On uniqueness, stability and pointwise estimates in the Cauchy problem for coupled elliptic equations

Abstract: Results of the kind cited in the title are obtained for the improperly posed Cauchy problem for a system of two coupled elliptic partial differential equations. We assume one stabilizing condition is imposed on one of the dependent variables. The results follow from an a priori inequality which is derived as a consequence of the logarithmic convexity of a suitable functional.

Interaction of a plane compressional elastic wave with a rigid spheroidal inclusion

Abstract: In this paper we have considered the problem of diffraction of a plane compressional harmonic elastic wave by a rigid spheroidal inclusion embedded in a homogeneous isotropic medium. For simplicity we have confined our attention to the axisymmetric case when the incident wave propagates along the axis of symmetry of the spheroid. The inclusion is assumed to be movable. Since the exact solution to this problem is not obtainable analytically we have used a boundary perturbation technique that is applicable at any finite frequency. We have derived exact analytical expression for the amplitude of oscillation of the inclusion correct to first order in a shape correction factor e. It is shown that the low-frequency expansion of the amplitude agrees with the expansion derived by other means correct to first order in frequency. We have also given a high-frequency expansion of the amplitude. Furthermore, we have derived the asymptotic expansion of the field in the illuminated zone and have shown that these are compatible with those obtained by an application of Keller's ray theory.

Asymptotic solutions for shells with general boundary curves

Abstract: The influence of arbitrary edge loads on the stresses and deformations of thin, elastic shells with general boundaries is studied by means of asymptotic expansions of a general tensor equation. Expansions are made in terms of an exponential or an Airy function and a series in powers of a small-thickness parameter. Most of the steps in the procedure are effected by using the dyadic form of the tensors. Solutions are obtained that are valid in the large, with no restrictions on the loading or on the boundary geometry. Results indicate that the behavior of shells with arbitrary boundary geometry can be quite different from that in the ordinary case. Specific results show the presence of an interior caustic which is the envelope of the characteristics of an eiconal equation. The exponential expansion becomes singular at the caustic, which would generally be expected to be a local region of stress concentration. The results have a close identity with asymptotic solutions obtained in geometric optics. Following some new techniques used for solution of the reduced wave equation, a solution of the shell equation is obtained using an asymptotic expansion in terms of Airy functions which provides a solution that is uniformly valid in the neighborhood of the caustic. Introduction. The primary purpose of this study is to investigate the influence of general edge loads on the stresses and deformations of thin elastic shells. The shell equations were cast into a single, compact dyadic form by Steele [1] that will be used here without development. Some details of that derivation and additional algebraic details of equations developed in this paper are given in [2], The type of solution sought is a generalization of the well-known solution, with a decaying behavior from the shell boundary into the shell interior, commonly referred to as an edge-effect solution. Analytic solutions available in the literature are applicable only to shells with restricted boundaries and loading conditions, or are valid only close to the boundary. Certainly the most widely used edge-effect solution is that for the axisymmetricallyloaded circular cylinder with a boundary on a circumferential line of curvature, and that result may be obtained as a special case of our general solution. Perhaps the best known solution for a shell with an arbitrary boundary is that of Goldenveizer [3]. However, that solution involves a basic assumption that restricts the region of applicability of the solution to a small boundary layer near the shell edge. * Received October 10, 1971. This study, performed at Stanford University, was supported by the McDonnell Douglas Corporation and by the National Science Foundation under Grant GK-2701.

Dual extremum principles relating to cooling fins

Abstract: Under consideration is a differential equation (pu')' = qu of the SturmLiouville type where the function q(x) > 0 is given. The problem is to find a function p{x) > 0 in 0 < z < 6, a constant b and a solution u(x) of the corresponding differential equation such that the energy functional /„ [p(u')2 + qu] dx is maximized when p{x) is subject to the constraint J„ p\" dx < K\" and u is subject to the boundary conditions u = 1 at x = 0 and p(du/dx) — 0 at x = b. Here K > 0 and p > 1 are constants. A key relation \\du/dx\\ = Apc\"_1)/2, where X is a positive constant, is found. This criterion leads to explicit solution of the problem. A further consequence of this criterion together with a pair of dual extremum principles is a \"duality inequality\" giving sharp upper and lower estimates of the maximum value of the energy functional. This study is a sequel to previous studies on the optimization of cooling fins by Diffin and McLain [1, 2], In those papers p = 1. However, the latter deals with fins on convex cylinders and thus the Sturm-Liouville equation studied is a partial differential equation.

A steering problem

Abstract: It' two circular gear-wheels of different sizes engage, this may be regarded as a mechanism which generates a linear relationship between the angles turned through, the ratio of the angles being constant. In this paper it is shown that, if the circular gear wheels are replaced by suitably shaped oval wheels (or cams) which engage without slipping, it is possible to generate an arbitrary functional relationship between the angles turned through. It is further shown how this mechanism might be used in the steering of a four-wheeled vehicle with theoretically perfect satisfaction of the condition that, at any instant, the vehicle as a whole has a unique centre of rotation, so that no side-slip of the wheels occurs. The angle through which the plane of the inner front wheel is turned might be as great as a right angle, this limit being much greater than that attainable with the usual Ackermann linkage. 1. Mechanism to generate an arbitrary functional relationship between two angles. Consider two oval wheels (or cams) which lie in a plane and can turn about fixed points, A and B (Fig. 1). Initially the cams are in contact and a rolling contact is maintained by means of fine teeth on the edges of the cams, or in some other way. If the initial position is as in Fig. 1, no such motion is possible. For in rolling contact the particles

On symmetric-tensor-valued isotropic functions of two symmetric tensors

Abstract: + (h7 + A8/10)A2 + (hg + /i10/io)(AB + BA) + (hn + ^12710)B2 + (hi3 + A14/i0)(A2B + BA2) + (A1S + A16/10)(AB2 + B2A) + A17(A2B2 + B2A2) (1.1) where h0 , ■ • • , hir are polynomials in the isotropic invariants Ii , • ■ • , Ia defined by 11 , • • • , 79 = tr A, tr B, tr A2, tr AB, tr B2, tr A3, tr A2B, tr AB2, tr B3 (1.2) and where I10 = tr A2B2. (1.3) It has been shown by Rivlin [1] that any symmetric-tensor-valued isotropic polynomial function of the symmetric tensors A and B is expressible as T(A, B) = 7oI + 7iA + 72B + 73A2 + 74(AB + BA) + 75B2 + 76(A2B + BA2) + 7t(AB2 + B2A) + 7s(A2B2 + B2A2) (1.4) where the yk are polynomials in the isotropic invariants Iy , • • ■ , Iw defined by (1.2) and (1.3). There are a number of redundant terms in the expression (1.4). In Sec. 2 we outline the procedures employed to generate the matrix identities which enable us to eliminate these redundant terms and thus to proceed from the expression (1.4) for T(A, B) to that defined by (1.1), • • • , (1.3). In Sec. 3 we show that there are no redundant terms in the expression for T(A, B) given by (1.1) and hence no further simplification of the expression for T(A, B) is possible.