Showing papers in "Quarterly of Applied Mathematics in 1996"

On the exponential stability of linear viscoelasticity and thermoviscoelasticity

Abstract: The exponential stability of the semigroup associated with one-dimensional linear viscoelastic and thermoviscoelastic equations with several types of boundary conditions is proved for a class of kernel functions, including the weakly singular kernels.

Growth rates for blow-up solutions of nonlinear Volterra equations

Abstract: An investigation is made of the blow-up growth property of the solution to certain nonlinear Volterra integral equations which model explosive behavior in a diffusive medium. The basic results provide the asymptotic form of the blow-up solution for a large class of kernels as well as various nonlinearities based on examples from solid combustion and adiabatic shear band formation.

Linear stability analysis of reaction fronts in liquids

Abstract: The propagation of a reaction front for liquid-to-solid reaction is studied. The model includes the heat equation, an equation for the concentration of the liquid reactant, and the equations of liquid motion under the Boussinesq approximation. The linear stability of the reaction front is studied, and conditions for cellular and oscillatory instability are determined.

Bending energy of highly elastic membranes II

Abstract: The strain energy per unit area for a deformed sheet of elastic material is estimated by representing the deformation as a power series in the thickness variable. The membrane energy is the lowest-order approximation obtained in this way. Straingradient and bending energies appear in the next order of approximation. Neither the membrane energy nor the higher-order approximation satisfy the Legendre-Hadamard material stability condition if the stress is compressive in some direction, so the theories based on either of these approximations can lead to problems with no stable solution. An energy function that does satisfy the material stability conditions is obtained by omitting the strain-gradient term, provided that certain longitudinal moduli are positive. A modified form for which existence of solutions can be guaranteed is proposed.

Mapping flat cracks onto penny-shaped cracks, with application to somewhat circular tensile cracks

Abstract: Consider a three-dimensional homogeneous isotropic elastic solid containing a flat pressurized crack, fi. The problem of finding the resulting stress distribution can be reduced to a hypersingular integral equation over Q for the crack-opening displacement. Here, this equation is transformed into a similar equation over a circular region D, using a conformal mapping between Q and D. This new equation is then regularized analytically by using an appropriate expansion method (Fourier series in the azimuthal direction and series of orthogonal polynomials in the radial direction). Analytical results for regions that are approximately circular are also obtained. The method will generalize to other scalar problems and to vector problems (such as shear loading of the crack).

On stiffness maximization of variable thickness sheet with unilateral contact

Abstract: The problem of maximizing the stiffness of a linearly elastic sheet, in unilateral contact with a rigid frictionless support, is considered. The design variable is the thickness distribution, which is subject to an isoperimetric volume constraint and upper and lower bounds. The bounds may vary over the domain of the sheet, and the lower one is allowed to be zero, hence giving the possibility of obtaining topology information about an optimal design. By using saddle point theory, the existence of solutions, i.e., thickness functions and corresponding displacement states, is proved. In general, one cannot expect uniqueness of solutions, unless the lower bound is strictly positive, and the uniqueness of optimal states is shown in this case.

Single phase energy minimizers for materials with nonlocal spatial dependence

Abstract: We consider a one-dimensional body and assume that the total stored energy functional depends not only on the local strain field but also on the spatial average of the strain field over the body, weighted with an influence kernel. We investigate the problem of minimizing the total stored energy subject to given end displacements. The general existence theory for this problem is reviewed. Then, we narrow our study and concentrate on certain fundamental aspects of nonlocal spatial dependence by restricting our consideration to the case of a convex local energy and an exponential-type influence function for the nonlocal part. We find explicit solutions and show their characteristic properties as a function of the parameter that measures the extent of influence in the nonlocal kernel. We then study in detail the behavior that results when the total stored energy functional loses its coercivity. In this case, issues concerning the local and global stability of extremal fields are considered. Table of contents

Love's circular patch problems revisited: closed form solutions for transverse isotropy and shear loading

Abstract: This paper considers the title problem of uniform pressure or shear traction applied over a circular area on the surface of an elastic half space. The half space is transversely isotropic, where the planes of isotropy are parallel to the surface. A potential function method is adopted where the elastic field is written in terms of three harmonic functions. The known point force potential functions are used to find the solution for uniform pressure or shear traction over a circular area by quadrature. Using methods developed by Love (1929) and Fabrikant (1988), the elastic displacement and stress fields for normal and shear loading are evaluated in terms of closed form expressions containing complete elliptic integrals of the first, second, and third kinds. The solution for uniform normal pressure on an isotropic half space was previously found by Love (1929). The present results for transverse isotropy including shear loading are new. During the course of this research, a new relation has been discovered between different forms of the complete elliptic integral of the third kind. This has allowed the present solution to be put in a more convenient form than that used by Love. Following a limiting procedure allows the isotropic solution to be obtained. It is shown that for normal loading the present results agree with Love's solution, while the results for shear loading of an isotropic half space are also apparently new. Special consideration is also given to derive the limiting form of the elastic field on the z-axis and the surface (z = 0).

Analytic solutions of the vector Burgers' equation

Abstract: The well-known analytical solution of Burgers' equation is extended to curvilinear coordinate systems in three dimensions by a method that is much simpler and more suitable to practical applications than that previously used The results obtained are applied to incompressible flow with cylindrical symmetry, and also to the decay of an initially linearly increasing wind

Stability of a linear integro-differential equation with periodic coefficients

Abstract: Stability of a linear integro-differential equation with periodic coefficients is studied. Such an equation arises in the dynamics of thin-walled viscoelastic elements of structures under periodic compressive loading. The equation under consideration has a specific peculiarity which makes its analysis difficult: in the absence of the integral term it is only stable, but not asymptotically stable. Therefore, in order to derive stability conditions we have to introduce some specific restrictions on the behavior of the kernel of the integral operator. These restrictions are taken from the analysis of the relaxation measures for a linear viscoelastic material. We suggest an approach to the study of the stability based on the direct Lyapunov method and construct new stability functionals. Employing these techniques we derive some new sufficient stability conditions which are close enough to the necessary ones. In particular, when the integral term vanishes, our stability conditions pass into the well-known stability criterion for a linear differential equation with periodic coefficients. In the general case, the proposed stability conditions have the following mechanical meaning: a viscoelastic structure under periodic excitations is asymptotically stable if the corresponding elastic structure is stable and the material viscosity is sufficiently large. As an example, the stability problem is considered for a linear viscoelastic beam compressed by periodic-in-time forces. Explicit limitations on the material parameters are obtained which guarantee the beam stability, and the dependence of the critical relaxation rate on the material viscosity is analysed numerically for different frequencies of the periodic compressive load.

On existence of periodic orbits for the FitzHugh nerve system

Abstract: Applying bifurcation theory, we construct various phase portraits of the FitzHugh differential system and describe the set of parameters for which this system has periodic solutions.

The initial-value problem for the Kelvin-Helmholtz instabilities of high-velocity and magnetized shear layers

Abstract: The general initial-value problem for the linear Kelvin-Helmholtz instability of arbitrarily compressible velocity shear layers is considered for both the unmagnetized and magnetized cases. The time evolution of the physical quantities characterizing the layer is treated using Laplace transform techniques. Singularity analysis of the resulting equations using Fuchs-Frobenius theory yields the large-time asymptotic solutions. The instability is found to remain, within the linear theory, of the translationally convective or shear type. No onset of rotational or vortex motion, i.e., formation of \"coherent structures\" occurs.

Transient response of an impulsively loaded plastic string on a plastic foundation

Abstract: The problem of an impulsive loading of a long rigid-plastic string resting on a rigid-plastic foundation is studied. A closed form solution is obtained by disregarding the longitudinal motion and considering an arbitrarily large transversal motion. Expressions for the final shape of the string are derived in terms of the magnitude of the applied impulse. It is found that the stress and the foundation reaction force are not uniquely determined, while the shape of the string is.

General contact-angle conditions with and without kinetics

Abstract: The classical condition for the contact angle of a phase interface at a container wall is generalized to include both anisotropy and kinetics The derivation, which does not involve an assumption of local equilibrium, is based on a capillary force balance, a dissipation inequality representing the second law, and suitable constitutive assumptions

A comparison of time domain boundary conditions for acoustic waves in wave guides

Abstract: Researchers consider several types of boundary conditions in the context of time domain models for acoustic waves. Experiments with four different duct terminations (hard wall, free radiation, foam, and wedge) were carried out in a wave duct from which reflection coefficients over a wide frequency range were measured. These reflection coefficients were used to estimate parameters in the time domain boundary conditions. A comparison of the relative merits of the models in describing the data is presented. Boundary conditions which yield a good fit of the model to the experimental data were found for all duct terminations except the wedge.

Derivative of a function of a nonsymmetric second-order tensor

Abstract: Exact explicit expressions are obtained for an isotropic tensor-valued function of a nonsymmetric second-order tensor, and its derivative, without resort to eigenvector calculations. These are then used to derive explicit expressions for the material time derivative of the general strain measures in terms of the deformation rate tensor.

Time-dependent drop deformation in a rotating high viscosity fluid

Abstract: An exact solution is presented which describes the time-dependent deformation of a nearly spherical drop suspended on the rotation axis of a more dense rotating viscous fluid. The solution is demonstrated to be similar, though not identical, to that derived from the commonly invoked assumption that the external flow field is purely extensional.

On the accurate computation of the prolate spheroidal radial functions of the second kind

Abstract: The series expansion of the prolate radial functions of the second kind, expressed in terms of the spherical Neumann functions, converges very slowly when the spheroid's surface coordinate £ approaches 1 (thin spheroids). In this paper an analytical series expansion in powers of (£2 — 1) is obtained to facilitate the convergence. Then, by using the Wronskian test, it is shown that this newly developed expansion has been computed with a double precision accuracy. Introduction. The prolate spheroidal radial functions satisfy the following differential equation [1], [2]: i ty \" !) (ATM h2¥ + ^rr^) Rmn(h,Z) = 0, m = 0,l,2,..., n = m, m + 1, m + 2,... where £ is the spheroid's radial coordinate (£ > 1), and Xmn is the spheroid's eigenvalue for the given h parameter, i.e., h = kF where k = 2-k/X is the operating wavenumber and F is the semi-interfocal distance of the spheroid. According to Flammer [2] the second solution of the above differential equation, which can be expressed in terms of the associated Legendre functions of the first and second kinds, is given by {oo OO \\ dr(h)QE+r(o+ E' r=2m,2m+l r=2m+2,2m+l J (2) where denotes the summation over even or odd values of r if n — m is even or odd. Also, the expansion coefficients and follow the recursion relations given in [2] along with the spheroid's joining factor Kmh{h) [2, p. 33]. Another representation of the prolate spheroidal wave function of the second kind is given in terms of the following spherical Neumann expansion [2]: ' f^ar(h)nm+r(hO, (3) V ^ / r=0,l Received May 2, 1994. ©1996 Brown University 677 678 T. DO-NHAT and R. H. MACPHIE where aTMn are the normalized expansion coefficients, and nm+r(/i£) are the spherical Neumann functions. Equations (2) and (3) are the main interest. It is historically well known that the series in (3) converges very slowly when /i£ is small. According to Morse and Feshbach [3, p. 1506], \"the series does not converge well for h£ small, in fact it is an asymptotic series not being absolutely convergent for any finite value of /i£.\" Recently, Sinha and MacPhie [4] summed this series up to 40 terms and replaced the residual series by an integral. However, the integrand of this integral is a curve-fitting function which may not be reliable for large to or n. In this paper we focus on the series given by (2) expressed in terms of the associated Legendre functions of the first and second kinds. Here, due to the lack of the development of the QTM+r(£) function, Flammer [2] expanded the prolate spheroidal function of the second kind in powers of (£2 — 1) by using that of the first kind and the Wronskian of Rmh(h,{;) and Rmh{h,£). However, Flammer's prolate spheroidal wave expression of the second kind is cumbersome and complicated, and is limited to some lower values of m. For the above purpose we first derive the representations of the associated Legendre function of the second kind QTM+r(£) for any integer m + r (r — —2m, —2m + 1,..., m — 0,1,2,...). By using the linear hypergeometric transformation, QTM+r{£) is given in closed form for —2m < r < — 1. However, for r > 0, QTM+r(£) is explicitly expressed in terms of the associated Legendre functions of the first kind. By using these representations it will be proved that the prolate spheroidal radial function of the second kind can be expressed in terms of its first kind. Nevertheless, when £ is near to 1 an analytical series expansion in powers of (£2 — 1) is obtained and all the expansion coefficients are expressed in closed forms in terms of the dTMn and coefficients. 2. Closed form expression of <517(0 {v > —to). First, for v = —to, —to + 1,..., -1, we start with the general definition of Barnes [5, Ch. XV, p. 326] for positive integers m as follows: = sin(/z + to)tt T{u + m + l)F(l/2) (£2 l)m/2 \" sin utt 2\"+1r(is + 3/2) 1 „ /v m v m 1 3 ,_9\\ . .. ' F ( o + TT + 1) o + ~o~ + o' V + o ' £ )' l£l — (4) where F(a, 6; c; z) is the hypergeometric function with z = £~2, a — v/2 + to/2 +1,6 = v/2 + to/2 + 1/2, and c = a + b — m. By using the linear hypergeometric transformation for to = 1, 2,3,..., F(a, 6; c; z) is given [6, p. 560, Eq. 15.3.12] as follows: F(a b. a+b — mz) — r(m)r(a + 6-m)(1 z)— Y (a ~ rn)k(b m)k _ k r (a,o,a+o m,z) r(a)r(6) 1 ' k\\ (1 m)k [ ' _(_pm r(a + 6 to) y, (g)k(b)k F(a — m)T(b — m) fc! (fc + to)! V(k + 1) V(k + to + 1) + V(k + a) + (k + 6)] PROLATE SPHEROIDAL RADIAL FUNCTIONS OF THE SECOND KIND 679 for m = 1,2,, | arg(l — z)\\ < n, |1 — z\\ < 1, where ^(a:) = j-[InT(a;)] is the Digamma function. In our case F(a — m) = T{v/2 — to/2 + 1), T(6 — m) = T(v/2 — to/2 + 1/2) and, since u — —to, —to + 1,..., —1, the Gamma function T(a — to), or T(6 — to) with argument 0, —1, -2,..., tends to infinity. Hence, (5) is reduced to m—1 F(a b-a + b-m-z)= ^ [ ^ ~ ' (1 z)-m V — — )k[ ~ ,k fj _ z)k [ ' ' + ' ' r(a)r(6) 1 j ^ fc! (1 — m)fc ( j ' m = 1,2,... . (6) If we now substitute (6) into (4), then QTM(£) (i/ = m + r) is given in closed form as follows: oIC) I 1 ra-'fo, + V,„+rK)-( i) •! lTM 1). £,+1 Zj fe! (1 — m)t {\" ' (7) r — —2m, —2m + 1,..., —1. If to = 1 and r = —2, by using (7) we obtain Q—i(0 — —£(£2 ~ l)\"1^2) which agrees with that which is derived by using the recursion formula of the associated Legendre functions [6, p. 334, Eq. 8.5.3], For v = 0,1,2,..., from the definition of Hobson for the associated Legendre functions of argument greater than 1 [5, p. 325],

Inertial manifolds of incompressible, nonlinear bipolar viscous fluids

Abstract: The existence of an inertial manifold is established for the nonlinear system of equations describing the motion of a bipolar incompressible viscous fluid. In this paper we consider only the case of a spatially periodic velocity field. Existence of an inertial manifold for the model complements earlier work on the existence of compact global attractors for bipolar viscous fluids and serves to further highlight the differences between the bipolar model and the usual model based on the linear Stokes constitutive relation.

Non-classical diffusion equations related to birth-death processes with two boundaries

Abstract: A pair of forward and backward diffusion equations is considered. In the forward equation, boundary values appear in the differential equation, and in the backward equation, boundary values are related to average values of the solution in the interior of the domain. The forward equation can be regarded as a diffusion approximation to a type of birth-death process with returns to the interior, or as a heat equation in one dimension where heat flowing out from the boundaries is returned to the interior. Existence and uniqueness theorems are proved, and some properties of the associated eigenvalues and eigenfunctions are deduced. An expression for the steady-state solution is obtained. Some information on the goodness of the diffusion approximation is also obtained.

Differentiability of the scalar coefficients in two representation formulae for isotropic tensor functions in two dimensions

Abstract: here T : Sym —> Sym is a given isotropic tensor function; Sym denotes the set of symmetric second-order tensors, and I is the identity tensor. The algebraic theorem asserts that for each isotropic function T( •), there are coefficients y,, y2, and y3, which are scalar functions of the principal invariants of B, such that Eq. (1.1) is valid for each B in Sym. Moreover, it is well known that the coefficients yi are uniquely determined by Eq. (1.1) only if the eigenvalues of B are distinct. A natural question on the continuity and differentiability properties of y( arises: suppose T( •) is of class C" (n > 0); can we choose the coefficients yi so that they are of class Ck (0 < k < «)? In 1959, Serrin [4] gave a counterexample in which T( •) is differentiable but there is no way to choose the yt such as to remain continuous. He went on to prove in the same article that the yt can be chosen to be continuous if T( •) is of class C . Recently Man [5] improved Serrin's sufficient 3 2 condition of smoothness from C to C , but that is all that we currently know about the analytical aspects of the representation formula (l.l).1 Except for the analogue of (1.1) in two-dimensional space (cf. Man [5]) and a major memoir of Ball [6] on scalar-valued functions, as far as we are aware, nothing else has been published on the analytical aspects of other representation formulae for isotropic tensor functions. As an explanation for such a deplorable situation, there is the argument that the continuity and differentiability of representation formulae are ignored because they are irrelevant in many applications. We do not accept this argument. Even if we set

Webs, iteration groups, and equivalent changes in probabilities

Abstract: Yew-Kwang Ng [12] listed several \"reasonable properties\" for equivalent changes of probabilities and other proportions. He produced a family of functions satisfying all properties and asked whether there exist essentially different ones. We show that this is the case, by constructing uncountably many families of functions satisfying all properties. We show also that there are no other solutions. Our method establishes connections with webs (nets) and iteration groups. This may be of interest both in itself and for applications.