Showing papers in "Quarterly of Applied Mathematics in 1981"

Evolution problems for a class of dissipative materials

Abstract: This work examines the problems of dynamic and quasi-static evolution for a large class of dissipative materials, including viscoplastic, viscoelastic, and elastic perfectly plastic materials. We show that when the potential of dissipation is regular, the displacement solution is regular; however, in the case of perfect plasticity, where the potential is irregular, the solution can be discontinuous. A suitable framework is used in order to account for these discontinuities. Existence theorems are stated, and the boundary conditions are discussed. The evolution equations encountered in this work are strongly nonlinear but with a monotone time-dependent nonlinearity. A direct method of resolution is proposed, since the known results do not apply in this case.

On some epidemic models

Abstract: The qualitative behavior of the solution x of the equation x(f) = /c|p(f) J A(t s)x(s) ds j|/(f) + | a(t s)x(s) ds j, t > 0 is studied. This equation arises in the study of the spread of an infectious disease that does not induce permanent immunity.

Wrinkling in finite plane-stress theory

Abstract: A general and complete formulation is given for the wrinkling phenomenon in the context of finite plane-stress theory. The planar portion of the true three-dimensional displacement field, called the pseudo-displacement field, is used as a basis for the necessary kinematic analysis. It is assumed that the principal directions associated with the pseudodeformation field are the same as those associated with the true stress field. The true stress field is governed by equilibrium and the assumption that one of the principal stresses vanishes, and hence is statically determinate. The difference between the pseudo-strain and the true strain calculated from the true stress is a new tensor, called the wrinkle-strain tensor, and serves as a measure of the wrinkliness of the surface.

Vibrations of a pendulum consisting of a bob suspended from a wire: the method of integral equations

Abstract: The vibrations of a vertical pendulum consisting of a bob suspended from a wire are studied by the method of integral equations and the composite method, respectively. The composite method combines the minimum principles and the method of integral equations. This problem consists of the fourth-order differential equation and the boundary conditions dependent on the eigenvalue parameter. Lower bounds are established for the lowest natural frequencies by both methods. Numerical results are presented. Both theoretical and computational efficiencies are illustrated and the method of integral equations is stressed.

The potential distribution in a constricted cylinder: an exact solution

Abstract: An exact solution to the Laplace equation is derived for the distribution of electric potential within a long cylinder carrying a circular constriction along its axis. The expression obtained for the potential distribution is reduced to a form which may be readily evaluated and is highly accurate for a ratio of constriction radius to cylinder radius approaching unity. Exact expressions both for the electric current density within the constriction and for the spreading resistance (i.e., the increase in resistance of the cylinder due to constriction of current flow lines) are also obtained.

Finite-amplitude surface waves in electrohydrodynamics

Abstract: The stability of weakly nonlinear waves on the surface of a fluid layer in the presence of an applied electric field is investigated by using the derivative expansion method. A nonlinear Schrodinger equation for the complex amplitude of quasi-monochromatic traveling wave is derived. The wave train of constant amplitude is unstable against modulation. The equation governing the amplitude modulation of the standing wave is also obtained which yields the nonlinear cut-off wave number.

Stable small-amplitude solutions in reaction-diffusion systems

Abstract: Bifurcation and perturbation techniques are used to construct smallamplitude periodic wave-trains for general systems of reaction and diffusion. All solutions are characterized by the amplitude a and the wavenumber k. For scalar diffusion, k ~ a, while for certain types of nonscalar diffusion, k is bounded away from zero as a\\0. For certain ranges of a and k, linear stability of waves is demonstrated.

The axisymmetric branching behavior of complete spherical shells

Abstract: : The purpose of the paper is to describe the axisymmetric branching behavior of complete spherical shells subjected to either external pressure or a centrally directed (dead) loading. By means of an asymptotic integration technique (based on the smallness of the ratio of the shell thickness to the shell radius) applied directly to a differential equation formulation, the authors are able to continue the solution branches from the immediate neighborhood of the branch points, where the solution has the functional form predicted by the linear buckling theory, to the region where the solution consists of either one or two dimples with the remainder of the shell remaining nearly spherical. The latter deflection states are the ones usually observed in experiments. The authors also deduce a number of results for the branching of solutions from the multiple eigenvalues. Finally, a boundary layer analysis is performed to determine the possible form of large deflection states. (Author)

Large deformations of a heavy cantilever

Abstract: A cantilever of uniform cross-section and density is held at an angle a at one end. The shape of the cantilever depends heavily on a and a nondimensional parameter K which represents the relative importance of density and length so that of flexural rigidity. Perturbations on the elastica equations for small and large K show good agreement with exact numerical integration. It is found that whenever K reaches a critical value, bifurcations of the solutions occur. This nonuniqueness can be observed by the flipping phenomena as a is increased. Introduction. A cantilever deformed by its own weight is of interest both practically due to its engineering significance, and theoretically due to its inherent nonlinearity. If the cantilever is thin enough, its deformed shape can be described by the elastica theory. Using this approximation and small deflections, Euler first investigated the stability of a vertical cantilever (column) under its own weight [1]. Euler's stability problem was later corrected by Greenhill [2] who obtained the minimum unstable height for a column of given density and rigidity. The large deformations of a heavy elastica was first numerically integrated by Bickley [3] who, as we shall see later, found only one of the solutions of the originally horizontal cantilever. This paper is a study in depth of the heavy elastica cantilever. The physical problem is illustrated in Fig. 1, where a cantilever of uniform density and cross-section is anchored on one end. We ask, what is the relation between the end angle a and the end torque M'? Both Greenhill's and Bickley's results will be special cases of this study. Formulation. Let us consider a small segment of an elastica with total length L and density p (Fig. 1). A moment balance gives m — p{L — s')sin 6 ds' = m + dm (1) where m is the local moment, s' is the arc length from the origin and 6 is the local angle of inclination. According to Euler, the local moment is proportional to the curvature dO (2) * Received June 3,1980.

On the eigentheory of operators which exhibit a slow variation

Abstract: A class of linear operators which exhibit slow variation is considered. If the kernel of the operator is K(x — y, \\e(x + y)), e the parameter of slowness, then its Wigner transform is defined to be K(p, q) = J K(u, q)exp(-iup) du. The eigenvalues of such operators are shown to follow an area rule: if the curve Xn = K(p, q) contains the area s/(A) = (2n + 1 )ne then Xn is an eigenvalue. Forms for the corresponding eigenfunctions are also obtained. Classical WKB theory is shown to be a special case and other examples are given.

On a problem in the dynamic theory of cracks

Abstract: We show that it is possible, for a certain case of a traveling crack problem, to obtain an explicit solution, in the entire region of interest, in terms of elementary functions. This affords a simple way of constructing level stress curves in the entire region, in contrast with the general case when simple expressions are obtainable at best along a particular axis.

On the local form of the second law of thermodynamics in continuum mechanics

Abstract: The Clausius-Duhem inequality a(r, t) > 0, a widely adopted axiom in continuum mechanics, leads to the conclusion that for many materials the entropy s cannot depend on gradients like the temperature gradient g and the velocity gradient e. But this is at variance with the received view (since Gibbs) that entropy is a function of thermodynamic state, however detailed that state description may be. Gradients, and even higher derivatives of macroscopic variables, may be included as state variables (although only on macroscopic time scales shorter than or comparable with their natural relaxation times), and the fundamental property of entropy is its convexity—the more detailed the specification of state, the smaller is the corresponding value of entropy. The entropy of a perfect monatomic gas is evaluated via the maximum principle, on the assumption that g and e are state coordinates, and it is found that s does depend on g • g and e : e, contrary to statements appearing in many textbooks on continuum mechanics. The source of the error in these works is shown to lie in applying 0 to relations involving second derivatives. The correct form of the Clausius-Duhem inequality contains only first-order derivatives; that is, it must be confined to linear constitutive relations. In this form the inequality is a consequence of the convexity of s, which is a somewhat more general manifestation of the second law.

Normal compression waves scattering at a flat annular crack in an infinite elastic solid.

Abstract: The Hankel transform is used to obtain a complete solution for the dynamic stresses and displacements around a flat annular surface of a crack embedded in an infinite elastic solid, which is excited by normal compression waves. The singular stresses near the crack tips are obtained in closed elementary forms, while the magnitude of these stresses, governed by the dynamic stress-intensity factors, is calculated numerically from a singular integral equation of the first kind. The variations of the dynamic stress-intensity factors with the normalized frequency for the ratio of the inner radius to the outer one and Poisson's ratio are shown graphically.

On the vibrations of a heterogeneous string

Abstract: It is shown that the vibrations of a heterogeneous string can be represented by an infinite series, each term of which is the result of applying a linear integral operator to a function of position and time furnished by the initial data. The method applies also to plane waves of compression or shear in a heterogeneous elastic solid for which the elastic constants and density are functions of only one coordinate and the waves move in the direction of that coordinate. 1. The partial differential equation. Consider the partial differential equation \"« s2uxx = 0; (1.1) here s is a positive function of x only, s(x). Let x range from — oo to + oo. The characteristics are given by dx/dt = ±s, which equations define two congruences of curves in the (x, f)-plane. The function s(x) represents the local speed of propagation, and the theory applies to all physical systems essentially involving only one space-dimension for which this speed is known. Thus (1) For the transverse vibrations of a string, s2 = T/k where T is the (constant) tension and A the mass per unit length at the position x. (2) For the compressional vibrations of an isotropic elastic solid in which the density and elastic constants are functions of x only (we might call it laminated), s2 = (A + 2n)/p. (3) For the transverse vibrations of such a laminated solid, s2 = n/p. Other applications may occur to the reader; the essential condition is that (1.1) is applicable, with s(x) a given positive function. 2. Straightening the characteristics. The first step is to straighten the characteristics of (1.1) for geometrical representation in a space-time plane. To do this we define y = [ dz/s{z), (2.1) Jo giving y as a function of x. Inverting, we get x as a single-valued function of y, and this gives s(x) = c(y); (2.2) we may call c(y) the transformed local speed. Changing the independent variable from x to y, we transform (1.1) into u,t Uyy + (c'/c)uy = o, (2.3) * Received February 22, 1980. (3.2) J. L. SYNGE 293 where the prime means d/dy. In a space-time plane in which y and t are Cartesian coordinates, the characteristics are now straight lines inclined to the axes at 45°. Strictly speaking, we should not use the same symbol u in (1.1) and (2.3), because the function u(x, t) of (1.1) is not the same function as the u(y, t) of (2.3); but this is not likely to cause confusion if we remember that u may be regarded as a physical quantity expressible in terms of (x, f) or (y, t). 3. Changing the dependent variable. Let us see what happens to Eq. (2.3) if we change the dependent variable u to v by the transformation u(y, t) = v(y, t)4>(y), (3.1) where the function 4> is at present unspecified. Then u„=vtt + v(f>', uyy = vyy(f) + 2vy4>' + vtfr\", and when these are substituted in (2.3) we get, on division by 4\\ vtt Vyy = 2/ct)y + 2hv, 2k = 2^'/(f) c'/c, 2h = \"/(j> (<£'/ )(c'/c). Choose (p to make k = 0: (j) = c1/2, u = vc1/2. (3.3) Then 2h = ~\\c\"/c + |(c'/c)2 = -/ + y2, (3.4) where 7 = ic'/c. (3.5) Our partial differential equation now reads v„ vyy = 2hv, (3.6) the characteristics are straight and, instead of the partial derivative uy as in (2.3), we now have the function v itself. 4. The integral equation and its solution. If the right-hand side of (3.6) were a given function of (y, t), we would have the equation for a uniform string acted on by external force and we would know how to proceed. This suggests a similar procedure, leading to an integral equation for v. In Fig. 1, P is a general point (y, t), chosen with t positive for convenience, and A, B are the points where the characteristics through P meet the y-axis. Integrating (3.6) over the triangle PAB, we get 0=|| (vyy vtt + 2hv) dy dt = (vy dt + v, dy) + 2 |'|' hv dy dt, (4.1)

Errata: “Small-amplitude waves on the surface of a layer of a viscous liquid”

Abstract: We study the initial-value problem posed by the small-amplitude waves on the surface of a layer of a viscous fluid of infinite lateral extent. The problem of the motion of the interface is reduced to an integro-differential equation which is solved by means of the Laplace transform. Explicit numerical results for illustrative cases are presented.

The parabolic umbilic catastrophe and its application in the theory of elastic stability

Abstract: The implications of the parabolic umbilic catastrophe in the theory of elastic stability are investigated. In particular, the influence of terms in the potential energy which are deemed necessary for a complete analysis and the isolation of primary critical surfaces are considered. The results are demonstrated for the example of the buckling and initial post-buckling of a spherical shell under the influence of a constant as well as a spatially variable pressure. Introduction. Catastrophe theory has been hailed as the most important mathematical discovery in decades. It has also been termed the Emperor with no clothes! This paper makes no effort to address either of the above comments; rather, it is concerned with the application of a specific catastrophe to problems in the theory of elastic stability. Thus, the intended contribution is the evaluation of certain of the implications of catastrophe theory in the context of the theory of elastic stability. Catastrophe theory [1] and the theory of elastic stability are extremely similar and it is fair to say that the latter is a special case of the former. This relationship has led to a number of papers [2, 3] which provided comparisons between actual physical problems and catastrophe theory as well as a description of areas in which catastrophe theory may be applied. It has become apparent that catastrophe theory has certain features which are of interest in the theory of elastic stability; however, it is equally obvious that the theory of elastic stability is not a trivial application of catastrophe theory. In fact, catastrophe theory does not even address some of the most difficult aspects of the prebuckling and buckling solutions for a given problem [4, 5]. Furthermore, the physical implications involved in a loss of stability play a predominant role in the analysis of physical systems. This has been demonstrated in [6] where it is shown that the least critical surface for a problem is not necessarily related to the initial loss of stability. The above points notwithstanding, it appears that the contribution which catastrophe theory has to offer is in the realm of classification for complex systems and of the determination of the correct number of loads, imperfections, etc. (control parameters) which should be involved in a stability analysis. This paper investigates the class of elastic stability problems which are described by a potential energy expression that can be reduced to the parabolic umbilic form. This particular problem is of interest for two reasons: first, the potential energy expression used is * Received December 20, 1979; revised version received August 17, 1980. 202 DAVID HUI AND JORN S. HANSEN apparently inconsistent from a perturbation point of view and, second, catastrophe theory dictates that this problem requires two independent load-type parameters in order to achieve a proper unfolding. Neither of these features would be considered necessary in a stability analysis and it is therefore appropriate to investigate their influence. Previous investigations of the critical surfaces of the parabolic umbilic model have been examined by Godwin [8] and Brocker and Lander [9], but they were not oriented towards the theory of elastic stability. In particular, the aim of this paper is to analyze this model from a structural stability point of view while taking advantage of the classification scheme provided by Catastrophe theory. The first portion of the paper is devoted to the evaluation and analysis of the critical surfaces of the parabolic umbilic catastrophe. In particular, two forms are identified and termed the parabolic umbilic types one and two. The particular critical surfaces which are of relevance in a stability analysis have been isolated and extensive parameter studies have been undertaken. The above results are then demonstrated for the two-mode buckling problem of a shallow section of a spherical shell. The influence of the additional parameters which are implied by catastrophe theory are also evaluated and are demonstrated to be significant. The parabolic umbilic catastrophe. The parabolic umbilic catastrophe arises in the analysis of systems which have two coincident least eigenvalues and for which cubic terms in the expansion of the potential energy function about the critical state are in general non-vanishing. That is, if the expansion of the potential energy V, about the ideal critical state, takes the form V= Ax 3 + Bx2y + Cxy2 + Dy3 then there are a number of possibilities which arise. If the cubic equation V = 0 has one real root and a pair of complex conjugate roots, then V leads to the hyperbolic umbilic catastrophe. If there are three real and unequal roots, then V is classed as an elliptic umbilic. These particular forms have been considered previously in [2, 6], In addition, there are two singular cases which occur for the three-real-root situation. These are the parabolic umbilic, when there are two equal roots, and the symbolic umbilic, when there are three equal roots. As may be appreciated from the root structure, the parabolic and symbolic umbilics are the non-trivial transitions which exist between the hyperbolic and elliptic umbilics. The problem of interest in the present investigation is the parabolic umbilic model which is described in standard form as V= + x4 + xy2 + Ljx2 + L2 y2 — £iX e2y (1) where Lu L2, et, e2 are the control parameters and x, y are the behavior variables. In a typical elastic stability analysis Lj and L2 would be related to some applied loads while and e2 would be related to the amplitudes of certain geometric imperfections. This is, of course, not necessary as Lu L2, ey, e2 may represent loads, imperfections, material parameters, dimensions and so on. The behavior parameters x, y are related to the amplitudes of the critical modes of the problem. As mentioned previously, the quartic term +x4 and the independence of and L2 are not the norm in elastic stability analyses and are therefore of particular interest. These factors have been included by catastrophe theorists to provide a stable jet and a complete unfolding of the catastrophe, respectively. It should also be emphasized that the plus or minus possibility for the quartic term is extremely important THE PARABOLIC UMBILIC CATASTROPHE 203 as the change in sign leads to quite different results. In the present paper, the cases with the plus or minus sign are referred to as the parabolic umbilic type one or type two, respectively. Critical sets. The critical sets are defined by the criterion that the first and second variations of the potential energy vanish simultaneously. This yields the equilibrium equations + 4x3 + y2 + 2 LjX = (2) 2y(x + L2) = e2, (3) while critical states of equilibrium occur when the solutions of (2) and (3) also satisfy {±6x2 + Lj)(x + L2) = y2. (4) The critical sets are the surfaces defined by the relationship between Lt, L2, et, s2 when x and y are eliminated from (2), (3) and (4). This elimination is not a trivial matter and it does not appear that an explicit relationship can be obtained in closed form. In the present circumstance it was accomplished numerically by first eliminating y from (2) and (3) by way of Eq. (4). This operation yields £i = ± 10x3 ± 6L2x2 + 3LjX + LlL2 (5) and 82 = ±2(x + L2)n/(± 6x2 + Lt)(x + L2). (6) In the solution for the critical surface it is required that this surface correspond to real values of x, y, £,, e2, Ll and L2. Further, it may be noted that for real values ofe1; Lx and L2 Eq. (5) has at least one real solution for x. In addition, for real values ofe2, Eq. (6) implies that (x + L2)(± 6x2 + Lt) must be positive. Also, from Eq. (3) it follows that if x and e2 are real then y is real. Therefore, the existence of real critical states is dependent only on the condition that the discriminant in Eq. (6) be positive. In practical stability problems Lj and L2 would represent load parameters which are generally treated as the unknowns and which are functions of the imperfection parameters ex and e2. There will be real solutions for Lj and L2 if the system is capable of buckling. Thus the ideal computation technique is to determine and L2 given et ande2. This method, however, leads to excess complexity and therefore the following procedure was adopted in this study. Numerical values are provided fore1; Lt andL2 and the corresponding value of e2 was to be evaluated. Thus, eu L, and L2 were substituted into Eq. (5) which then yielded either one or three real roots for x. The real root(s) for x were then substituted into Eq. (6) in order to determine the appropriate value of e2. Of course only real values of e2 are acceptable and it can be seen that there may be one or three real values of e2 corresponding to each set of£l5 Lu L2. It is clear that a graphical presentation of the critical surfaces is difficult and they are four-dimensional. Thus the figures are presented as projections of the general surfaces onto a two-dimensional plane. This is usually accomplished by introducing the relation L2 = hcLi where k is a constant which is assigned a series of values. The critical surfaces are then evaluated on the L2 versus e2 plane for different values of £x and a series of figures are presented for each value of k. 204 DAVID HUI AND JORN S. HANSEN A particularly interesting aspect of this class of problem is the existence of more than one critical surface for a given combination of el5 e2. All of the critical surfaces so obtained are of interest; however, they are not all of physical relevance, as it is only the first critical load encountered on a particular load-deflection path which specifies the buckling load. Furthermore, it may not be the least critical load on a given load deflection path which is predominant and it is often difficult to assess which is the predominant critical load. It should also be pointed out that the critical sets do not allow this interpretation and it is onl

A free boundary problem arising in a kinematic wave model of channel flow with infiltration

Abstract: Water flowing down a dry channel and infiltrating into the channel bed constitutes a free boundary problem. The free boundary is the time history of the water edge or front. In this paper we discuss a kinematic wave model of the problem. The problem is formulated in Sec. 1 and the results summarized in Sec. 2. In Sees. 3 and 4 the mathematical details are carried out, and in Sec. 5 a model using the continuity and momentum equations of hydraulics is discussed. 1. Formulation of the problem. The problem of irrigation has been studied for many years; we list, in the references, some of the more recent papers. We are dealing, essentially, with water flowing down a channel and infiltrating into the channel bed. The time history x = s(t) of the front of the water, i.e. the interface between the covered and uncovered part of the channel, is a free boundary which has to be determined along with the velocity u(x, t). This problem has been treated at various levels of mathematical complexity. We choose here a kinematic wave model that has been discussed in [1, 3, 7, 8, 9, 11]. Let x be the distance along a channel of uniform cross-section, u(x, t) the velocity, h(x, t) the depth, q(x, t) the lateral inflow rate, and f(x, t) the infiltration rate. The latter two are in volume per unit area per unit time. We can interpret q as rainfall. Further, let S be the slope of the channel, assumed constant, and Sf the friction slope. For Sf we take the Chezy formula Sf = vT/Ch, m and C positive constants. The continuity and momentum equations are [10, Chapter 11] h, + (uh)x = q — f u, + uux + ghx = g(S Sf) qu/h. (1) Conditions under which various terms in the momentum equation can be omitted have been discussed in the literature; we refer here to [4, 5, 6], The kinematic wave model is obtained by omitting all but the terms S — Ss. Thus we get u = ah1/m, v. = (SC)llm. (2) We assume now that q = 0 and that/is a function of t only. Then, writing n = 1 + m~x, we get, from (1) and (2), h, + (ah% = (3) * Received June 27, 1979; revised version received May 8, 1980. This research has been supported by the National Science Foundation through grant ENG78-25637.

The heterogeneous string: coupled helices in Hilbert space

Abstract: 1. We solve the initial-value problem for the partial-differential equation \"« s(x)2\"** = 0, (11) given u(x, 0) =f(x), u,(x, 0) = g(x); (1.2) in matters of notation, we follow [1], We take as the phase-space for the problem the space of initial data having finite energy, with inner product derived from the energy quadratic form. As in [1], we first straighten the characteristics of (1.1) with the transformation

A note on the stability of traveling-wave solutions to a class of reaction-diffusion systems

Abstract: Many classes of reaction-diffusion systems have been shown to have traveling-wave solutions. For a class of such systems for which a comparison theorem can be used, we establish a wave stability result which roughly states that these wave solutions are asymptotically stable to perturbations which lie in some weighted Lp-space if their speeds are sufficiently large. We then apply this result to some excitable systems, namely a model of the Belousov-Zhabotinskii reaction, a substrate-inhibition biochemical model, and a class of models recently studied by Fife.

A method of solution for an ordinary differential equation containing symbolic functions

Abstract: in which L is an ordinary linear differential operator and S is the Dirac delta function. 3', S\", ..., (5

On the convergence of solutions of Volterra equations to almost-periodic functions

Abstract: The rate of convergence of solutions of a certain Volterra integral equation and a system of two Volterra equations to almost-periodic limit functions is studied. The equations considered arise from some diffusion problems with nonlinear and almostperiodic boundary conditions.