Showing papers in "Quarterly of Applied Mathematics in 1982"

A mathematical model for linear elastic systems with structural damping

Abstract: : From empirical studies it is known that the natural modes of vibration of elastic systems have damping rates which are roughly proportional to the frequency of vibration. A number of ad hoc models exhibiting behavior of this type have been proposed in the engineering literature but they are not true dynamical systems nor are they very useful for numerical computations. In this paper we present a model of the form with B, A positive, unbounded, self-adjoint operators on a Hilbert space X, exhibiting the damping behavior just described, which is known as structural damping. Finite dimensional analogs suitable for computation of approximate solutions are also noted. (Author)

Extensions of a property of the heat equation to linear thermoelasticity and other theories

Abstract: is a decreasing function of t for 0 < t < T. It should be noted that (P) holds whatever initial values the temperature may take on f = 0. The adjective 'decreasing' is to be understood in the wide sense as meaning 'monotone nonincreasing'. The condition 9(x, t) e C2(R) can easily be replaced by one that is weaker, but to insist upon the weakest smoothness hypotheses for our theorems would overburden them and obscure their main point and instead we shall habitually assume more then is really necessary. Property (P) was formulated and proved first by Polya and Szego [1]. It is in fact a straightforward deduction from the maximum principle but a different method of proof, based upon convexity arguments, was discovered by Bellman [2] and it turns out that it is Bellman's method which is the more suited to proving the extensions we have in mind. The heat equation describes the conduction of heat with considerable success. From the point of view of continuum mechanics, though, it rests upon highly restrictive assumptions, and it is interesting to ask if (P), or some suitably modified form of (P), continues to hold in other theories which reflect more nearly the behavior of real bodies. The object of this paper is to show that modified forms of (P) remain true within both the quasi-static and the dynamic theories of coupled linear thermoelasticity—these theories remove the rigidity requirement—and also within certain theories which replace the parabolic heat equation by a linear hyperbolic equation and thereby ensure that temperature disturbances propagate at finite speed.

A test for stability of linear differential delay equations

Abstract: The changes in the stability of a system of linear differential delay equations resulting from the delay are studied by analyzing the associated eigenvalues of the characteristic equation. A specific contour is mapped by the characteristic equation into the complex plane to give an easy test for stability from an application of the argument principle. When the real part of an eigenvalue is positive, the contour gives bounds on the imaginary part which are important in certain applications to nonlinear problems.

On transversely isotropic functions of vectors, symmetric second-order tensors and skew-symmetric second-order tensors

Abstract: Q(0) corresponds to a rotation about the x3 axis. R, and R3 correspond to reflections in planes perpendicular to the xt axis and the x3 axis respectively. D2 corresponds to a rotation through 180 degrees about the x2 axis. In this paper, we determine integrity bases for polynomial functions F(A1;..., AN, Vt, ...,VM, Wl5 ...,WP) of N three-dimensional second-order symmetric tensors \\p = ||/1P|| (p = 1, ..., N), M three-dimensional vectors V, = Vf (q = 1,..., M) and P three-dimensional second-order skew-symmetric tensors Wr = || W'jW (r = 1, ..., P) which are invariant under any given group chosen from Tu ..., T5. Adkins [1,2] has considered the problem of determining integrity bases for functions F( A1( ..., AN, V1; ..., VM) which are invariant under the group and for functions F(A1; ..., \\N, Yu ..., \\M) which are invariant under the group T2. Long and Mclntire [3] have considered the problem of determining an integrity basis for functions F(A1;..., AN, \\u ..., \\M, Wx,..., WP) which are invariant under the group T4. The results obtained here for this case differ from those given in [3].

The Stefan problem with a convective boundary condition

Abstract: We study the one-phase Stefan problem on a semi-infinite strip x> or =0, with the convective boundary condition -KT/sub x/(0,t) = h(T/sub L/--T(0,t)). Points of intrest include: a) behavior of the surface temperature T(0,t); b) asymptotic behavior as h..-->..infinity; c) uniqueness, and d) bounds on the phase change front and total system energy.

On the steady-state propagation of an anti-plane shear crack in an infinite general linearly viscoelastic body

Abstract: The steady-state propagation of a semi-infinite anti-plane shear crack is considered for a general infinite homogeneous and isotropic linearly viscoelastic body. Inertial terms are retained and the only restrictions placed on the shear modulus are that it be positive, continuous, decreasing and convex. For a given integrable distribution of shearing tractions travelling with the crack, a simple closed-form solution is obtained for the stress intensity factor and for the entire stress field ahead of and in the plane of the advancing crack. As was observed previously for the standard linear solid, the separate considerations of two distinct cases, defined by parameters c and c*, arises naturally in the analysis. Specifically, c and c* denote the elastic shear wave speeds corresponding to zero and infinite time, and the two cases are (1) 0 < v < c* and (2) c* < v < c, where v is the speed of propagation of the crack. For case (1) it is shown that the stress field is the same as in the corresponding elastic problem and is hence independent of v and all material properties, whereas, for case (2) the stress field depends on both v and material properties. This dependence is shown to be of a very elementary form even for a general viscoelastic shear modulus.

A Hopf bifurcation in single-loop positive-feedback systems

Abstract: This paper gives sufficient conditions for a Hopf bifurcation in a fivedimensional system of ordinary differential equations which provides a model for positive feedback in biochemical control circuits. These conditions only depend on the feedback function and its first and second derivative. The conditions are used to exhibit Hopf bifurcations for the Griffith equations and the Tyson-Othmer equations.

Asymptotic expansion of the Hankel transform with explicit remainder terms

Abstract: be the K-transform off In recent years various techniques have been developed to obtain explicit expressions for the remainder in the asymptotic expansions of functions defined by (1.1) as x —> oo. A survey of such techniques is given by Wong [17]. It is well known that under some reasonable assumptions on / and K, the Parseval relation for the Mellin transform provides a powerful tool for obtaining the asymptotic expansion of F(x). However, until recently the potential of this technique for obtaining an explicit expression for the remainder had been largely overlooked. If M[K, s] is the Mellin transform of K evaluated at s, M\_f 1 — s] is the Mellin transform of/evaluated at 1 — s, and the integrals defining these transforms converge in a strip containing the line Re s = c then, formally, by the Parseval relation,

On static similarity deformations for isotropic materials

Abstract: For static deformations of isotropic materials satisfying the principle of material indifference, the governing equations remain invariant under arbitrary orthogonal rotations of the material and spatial coordinate systems. Moreover, the basic equations are also invariant under the same change of length scale for both coordinate systems. The general functional form of the similarity deformation corresponding to these invariances is deduced. Although these invariances involve seven arbitrary constants, it is shown that by an appropriate selection of coordinates only three arbitrary constants are involved in an essential way.

On the Steady-State Continuous Casting Stefan Problem with Non-Linear Cooling.

Abstract: A steady-state, one-phase, Stefan problem corresponding to the solidification process of an ingot of pure metal by continuous casting with nonlinear lateral cooling is considered via the weak formulation introduced by H. Brezis et al . [C. R. Acad. Sci. Paris Ser. A 287 (1978), no. 9, 711--714] for the dam problem. Two existence results are obtained, for a general nonlinear flux and for a maximal monotone flux. Comparison results and the regularity of the free boundary are discussed. A uniqueness theorem is given for the monotone case.

Thermodynamic principle of virtual dissipation and the dynamics of physical-chemical fluid mixtures including radiation pressure

Abstract: Equations of motion are obtained for a viscous fluid mixture including thermal and intermolecular diffusion as well as chemical reactions and radiation pressure. They are derived by applying the thermodynamic principle of virtual dissipation. The method also incorporates a new approach to the chemical thermodynamics of open systems which leads to new concepts and formulas for the heat of reaction and the affinity. They are simpler and more general than classical values. Instead of chemical potentials, new “convective potentials” are used which involve physical properties restricted to the system. They do not require extrapolations to absolute zero or the use of undetermined constants. No statistical theory is involved. A noncalorimetric evaluation of the heat of mixing is obtained from the concept of injection pressure of each substance in the mixture. Field equations are derived and a coupling between viscous stress gradients and diffusion is brought out. The convective potentials lead to a new evaluation of the thermodynamic functions of mixtures as well as a new generalized formulation of the Gibbs-Duhem theorem. Translational invariance of the dissipation is discussed and related to total momentum balance. Lagrangian equations of motion with generalized collective coordinates are derived directly from the variational principle and should provide a powerful approach to problems of stellar dynamics with radiation pressure.

On a second-order boundary value problem arising in combustion theory

Abstract: We obtain existence and uniqueness results for the boundary-value problem y\" = x2 — y2, y(x) ~ +x as x —> + oo. Our main result shows that there are precisely two solutionsy+(x) > — |x| andy_(x) < — | x |. Only the latter is of physical interest in the problem in combustion theory from which the equation arises.

Inverse scattering for an exterior Dirichlet problem

Abstract: Scattering due to a metallic cylinder which is in the field of a wire carrying a periodic current is considered. The location and shape of the cylinder is obtained with a far field measurement in between the wire and the cylinder. The same analysis is applicable in acoustics in the situation that the cylinder is a soft wall body and the wire is a line source. The associated direct problem in this situation is an exterior Dirichlet problem for the Helmholtz equation in two dimensions. An improved low frequency estimate for the solution of this problem using integral equation methods is presented. The far field measurements are related to the solutions of boundary integral equations in the low frequency situation. These solutions are expressed in terms of mapping function which maps the exterior of the unknown curve onto the exterior of a unit disk. The coefficients of the Laurent expansion of the conformal transformations are related to the far field coefficients. The first far field coefficient leads to the calculation of the distance between the source and the cylinder.

Some exact solutions describing unsteady plane gas flows with shocks

Abstract: A new class of exact solutions of plane gasdynamic equations is found which describes piston-driven shocks into non-uniform media. The governing equations of these flows are taken in the coordinate system used earlier by Ustinov, and their similarity form is determined by the method of infinitesimal transformations. The solutions give shocks with velocities which either decay or grown in a finite or infinite time depending on the density distribution in the ambient medium, although their strength remains constant. The results of the present study are related to earlier investigations describing the propagation of shocks of constant strength into non-uniform media.

Lower bounds for eigenvalues of Sturm-Liouville problems with discontinuous coefficients: integral equation methods

Abstract: This paper is concerned with application of the theory of Fredholm integral equations to Sturm-Liouville problems with discontinuous coefficients. Such problems occur naturally in many areas of application involving mechanics of heterogeneous media. Due to the nonsmoothness of the coefficients, the eigenvalue spectrum may exhibit severe irregularities. Lower bounds for eigenvalues are obtained which reflect this behavior. Numerical results are presented for example problems previously treated using other methods.

Dynamic instability of a homogenous deformation of a thin elastic bar

Abstract: A linear stability analysis of a homogeneous deformation at constant strain-rate of a thin elastic bar is used to show that the deformation is unstable with respect to small perturbations in the case when the stress-strain relation is concave with a single maximum.

Stability and bifurcation in a modulated Burgers system

Abstract: The stability of the null state for a nonlinear Burgers system is examined. The results include (i) an energy estimate for global stability for states involving arbitrary modulation in time, and (ii) an analysis of the bifurcation from the null state for slow modulations. For the slow modulations it is determined that the amplitude A(tau) of the bifurcated disturbance velocity satisfies a Landau-type equation with time-dependent growth rate theta(tau). Particular attention is given to periodic and quasiperiodic modulations of the system, which lead to analogous behavior in theta(tau). For each of these oscillatory-type modulations, it is found that A/sup 2/(tau) has the same long-time mean value as the unmodulated case, implying no alteration of the final mean kinetic energy. Applications to various fluid-dynamical phenomena are discussed.

Limiting behavior of a class of nonlinear reaction diffusion equations

Abstract: in a smooth bounded domain S with zero lateral data u = 0 on t > 0 x dS and sufficiently small initial data u(0, x) = M{x) for x e S. Here g(0) = gf'(0) = 0, and hence g has no linear contribution. We show in Theorem 1 that if M is sufficiently small, thenMeA'—> C*(j) as t —> oo in H'oiS) for some constant C* (positive, negative, or zero) where —A, assumed negative, is the principal eigenvalue to the operator