Showing papers in "Quarterly of Applied Mathematics in 1993"
TL;DR: In this article, the authors consider linearly elastic composite materials made by mixing two possibly anisotropic components and present optimal upper and lower bounds on the elastic energy quadratic form for given volume fractions and average strain.
Abstract: We consider linearly elastic composite materials made by mixing two possibly anisotropic components. Our main hypothesis is that the Hooke's laws of the two components be well-ordered. For given volume fractions and average strain, we present optimal upper and lower bounds on the elastic energy quadratic form. We also discuss bounds on sums of energies and bounds involving complementary energy rather than elastic energy. Our arguments are based primarily on the HashinShtrikman variational principle; however, we also discuss how the same results arise from the \"translation method\", making use of the analysis of Milton. Our bounds are equivalent to those established by Avelleneda and closely related to the \"trace bounds\" established by Milton and Kohn. The optimal energy bounds, however, are presented here as the extreme values of certain convex optimization problems. The optimal microgeometries are determined by the associated first-order optimality conditions. A similar treatment for mixtures of two incompressible, isotropic elastic materials has previously been given by Kohn and Lipton.
118 citations
TL;DR: In this paper, NITSCHE et al. considered the problem of boundary value maximization with respect to a free energy functional which is quadratic in the principal curvatures.
Abstract: The following investigation deals with surfaces governed by and extremal for a free energy functional which is quadratic in the principal curvatures. The associated Euler-Lagrange differential equations are derived, as are the corresponding intricate natural boundary conditions. Pertinent boundary value problems—without and with volume constraints—are formulated and discussed1 and existence proofs are provided for certain situations. The discussion opens the view onto an arena of rich mathematical problems which will also be of interest in engineering applications where the surfaces in question are utilized frequently as idealized models for the interfaces separating phases in real materials. 1. Let <5* = {x = \(u, v); (u, v) e P} be a differential geometric surface embedded in Euclidean space R3. The position vector x(w, v) = {x(u, v), y(u, v), z(u, v)) is defined over a prescribed parameter domain P—a connected set with or without given boundaries. We wish to consider this surface not so much as a geometric object but as an idealized model for the interfaces or middle surfaces occurring in real materials—open or closed lipid bilayers and surfactant films, thin elastic plates, 2 A etc. Thus, we associate with <5" a free energy O, per unit area, which incorporates surface tension but also reflects elastic properties. The latter involve the surface curvatures, and so will have the form O = 0(«rx, k2) . Here kx = 1 //?, and k2 = 1 /R2 denote the principal curvatures of S". As a consequence, the total energy of S? becomes = (*!, k2) dA . (1) Received August 6, 1991. 1991 Mathematics Subject Classification. Primary 35J35, 35J40, 35J60, 35J65, 49Q05, 49Q10, 53A10, 73C50; Secondary 31B30, 31G15, 35P15, 35Q35, 35R35, 53A05, 53C42, 76B45. 1 Some of the lengthier calculations are omitted. 2Realistically speaking, of course, the concept of sharp physical interfaces amenable to treatment with the tools of differential geometry is only an approximation, though often accurate and appropriate, to the actual narrow zones separating phases. For a discussion of the microstructure governing these zones in real materials, see, e.g. [1, 2, 11, 37, 39, 40, 63, 68]. ©1993 Brown University 363 364 JOHANNES C. C. NITSCHE (dA denotes the area element.) There may be further energy terms attributable to the boundary d5? of . Two universal structural conditions will be imposed on the integrand in (1): (1) 0(Kj , k2) is a symmetric function of its arguments. (2) if is definite in the following sense: There is a constant c > -oo , possibly negative, such that IP ) > c for every connected orientable surface -T of regularity class C2, with or without boundary. We allow the lower bound for i? to be negative to reflect the fact that the free surface energy of an interface need not be positive; however, disregarding possible extraneous information concerning the magnitude of physical constants or the limiting influence of boundary conditions in specific applications (see, e.g., Sec. 10 and (34) below), we shall insist on the universal condition C > —oo . It should be stressed that our condition of definiteness is mathematical in nature; it does not take into account the structural properties of the admissible comparison surfaces J7" on the molecular level; see footnote 2. Under mild regularity assumptions (for a polynomial , as a consequence of the fundamental theorem on symmetric polynomials), condition (1) above implies that can be written in the form 4>(Kr,, k2) = O(H, K), where H = (k{ + k,)/2 and K = k{k2 are the mean and Gaussian curvatures of 5?, respectively. Moreover, rather than working with general nonlinear integrands, it is consistent with standard engineering practice, at least as a first step, that we limit consideration to the inclusion of terms up to those of second order, so that becomes (//, K) — a0 + axH + a2H + a}K . The second condition (2) of definiteness imposes specific restrictions on the coefficients a0, , a}. In fact, an analysis similar to that given on p. 14 of [55] shows that O must have the form 0, 0 = (jc, — k2) = 4{H2 K), and O = k2 + k2 — AH2 — 2K—can be found in numerous sources from the earliest times (see, e.g., S. D. Poisson [59, pp. 221-225], [60, p. 321], S. Germain [21, pp. 12, 16], G. R. Kirchhoff [44, p. 63], F. Casorati [7, p. 109], A. E. H. Love [48, p. 130]) and has attracted renewed attention through the work of W. Helfrich [34] and fellow scientists (among others, [17, 28, 35, 37, 38, 58]; for further references also [54, Sec. 11]). The case = 1 leads to the venerable theory of minimal surfaces (see [49, 51, 54]), and the case = H2—to be sure, in all investigations published so far exclusively restricted to closed surfaces, embedded or immersed (i.e., surfaces with possible self-intersections)—has been of considerable recent interest to geometers; references can be found in [6, 70, 71, 73, 77] and [54, Sec. 11]. Generally speaking, there are today few mathematical treatments and no BOUNDARY VALUE PROBLEMS FOR VARIATIONAL INTEGRALS 365 general existence theorems for the various physical boundary value problems arising in connection with (1), (2). The "natural" (or "intrinsic", "spontaneous") curvature H0 in (2) can often be related to the initial state of the interface or to the fact that the two sides of a bilayer may be chemically different; see, e.g., [34, 17, 28], The actual numerical values of the constants a, ft , y , and H0 are of secondary importance at the moment. In many applications, a is larger by orders of magnitude than ft and y, a fact which often lets us approach the interface 5? as the perturbation of an interface governed by the energy functional ff dA based on the integrand O = 1 , i.e., a minimal surface (or a surface of constant mean curvature), depending on /? and y as control parameters. In other cases, for instance for menisci and vesicles with a size in the range of a few hundred A, the influence of a and that of /?, y become comparable. 2 3 The case a = 0, H0 = 0, p = y > 0, i.e., O = H K, is special. Here the associated energy W has the value zero for all spheres and spherical caps. Thus there may exist continua of solution surfaces for our Problem 1 below, and specific solution surfaces will have to be characterized by their area or by the volume they enclose. 2. One observation: The inclusion of terms of third order, that is, the addition of a4H} + a5HK to the expression 0(//, K) in (2) above, leads to an indefinite energy functional unless aA = a5 = 0. This can be seen as follows. For thin cylinders for which K = 0, the term /H3 dA can be made dominatingly negative so that the coefficient a4 must be zero. As for the other part, consider as comparison surface the torus = {x = (a + b cos u) cosv , y = (a + bcosu) sinw , z = ±6 sin u; 0
84 citations
73 citations
TL;DR: In this paper, the authors proved that the semigroup associated with the onedimensional thermoelastic system with Dirichlet boundary conditions is an exponen12 2 tially stable C0-semigroup of contraction on the space H 0 x L x L. The technique of the proof is completely different from the usual energy method.
Abstract: In this paper it is proved that the semigroup associated with the onedimensional thermoelastic system with Dirichlet boundary conditions is an exponen12 2 tially stable C0-semigroup of contraction on the space H0 x L x L . The technique of the proof is completely different from the usual energy method. It is shown that the exponential decay in 3 (s/) recently obtained by Revira is a consequence of our main result. An important application of our main result to the Linear-QuadraticGaussian optimal control problem is also discussed.
65 citations
Abstract: We discuss the classification of solutions to the zero-surface-tension model for Hele-Shaw flows in bounded and unbounded regions with suction and injection. We use results from the theory of univalent functions to derive estimates for certain geometric properties of the fluid region in the injection case.
64 citations
TL;DR: Within the linearized theory of heat conduction with fading memory, some restrictions on the constitutive equations are found as a direct consequence of thermodynamic principles as mentioned in this paper, which allow us to obtain existence, uniqueness, and stability results for the solution to the heat flux equation.
Abstract: Within the linearized theory of heat conduction with fading memory, some restrictions on the constitutive equations are found as a direct consequence of thermodynamic principles. Such restrictions allow us to obtain existence, uniqueness, and stability results for the solution to the heat flux equation. Both problems, which respectively occur when the instantaneous conductivity kQ is positive or vanishes, are considered.
59 citations
TL;DR: In this article, the Dirichlet initial boundary value problem in one-dimensional nonlinear thermoelasticity is studied and it is proved that if the initial data are close to the equilibrium then the problem admits a unique, global, smooth solution.
Abstract: We are mainly concerned with the Dirichlet initial boundary value problem in one-dimensional nonlinear thermoelasticity. It is proved that if the initial data are close to the equilibrium then the problem admits a unique, global, smooth solution. Moreover, as time tends to infinity, the solution is exponentially stable. As a corollary we also obtain the existence of periodic solutions for small, periodic righthand sides.
48 citations
TL;DR: In this article, the local behavior of a solution of a semilinear heat equation with singular power-like absorption near a quenching point was studied using the method of Herrero and Velazquez.
Abstract: We are interested in the local behavior, near a quenching point, of a solution of a semilinear heat equation with singular powerlike absorption. Using the method of Herrero and Velazquez, we obtain a precise description of the spatial profile of the solution in a neighborhood of a quenching point at the quenching time, under certain assumptions on the initial data.
43 citations
41 citations
TL;DR: In this paper, a potential theory for monotone multivalued operators is developed and an answer to Rockafellar's conjecture is provided, in which it is shown that the potential of the subdifferential of a convex functional coincides, to within an additive constant, with the restriction of the functional on the domain of its sub-differential map.
Abstract: The concept of cyclic monotonicity of a multivalued map has been introduced by R. T. Rockafellar with reference to the subdifferential operator of a convex functional. He observed that cyclic monotonicity could be viewed heuristically as a discrete substitute for the classical condition of conservativity, i.e., the vanishing of all the circuital integrals of a vector field. In the present paper a potential theory for monotone multivalued operators is developed, and, in this context, an answer to Rockafellar's conjecture is provided. It is first proved that the integral of a monotone multivalued map along lines and polylines can be properly defined. This result allows us to introduce the concept of conservativity of.a monotone multivalued map and to state its relation with cyclic monotonicity. Further, as a generalization of a classical result of integral calculus, it is proved that the potential of the subdifferential of a convex functional coincides, to within an additive constant, with the restriction of the functional on the domain of its subdifferential map. It is then shown that any conservative monotone graph admits a pair of proper convex potentials which meet a complementarity relation. Finally, sufficient conditions are given under which the complementary and the Fenchel's conjugate of the potential associated with a conservative maximal monotone graph do coincide.
32 citations
TL;DR: In this paper, the existence of global smooth solutions to initial boundary value problems in one-dimensional nonlinear thermoviscoelasticity is established by means of the Leray-Schauder fixed point theorem.
Abstract: Initial boundary value problems in one-dimensional nonlinear thermoviscoelasticity are considered, and the existence of global classical solutions is established by means of the Leray-Schauder fixed point theorem. Introduction. In this paper we study the existence of global smooth solutions to initial boundary value problems in one-dimensional nonlinear thermoviscoelasticity. The conservation laws of mass, momentum, and energy for one-dimensional materials with the reference density p0 = 1 are u,-vx = 0, Vt ~ °x = 0 ' / 2X «. (1.2) where subscripts indicate partial differentiations, u is the deformation gradient, v is the velocity, e denotes the internal energy, a is the stress, rj stands for the specific entropy, 6 for the temperature, and q for the heat flux. For one-dimensional, homogeneous, thermoviscoelastic materials, e, a , rj, and q are given by the constitutive relations (see [1]) e = e(u,6), a = a(u, 6, vx), ri = r)(u,6), q = q{u,6,0x), (1.3) which in order to be consistent with (1.2), must satisfy a(u, 6, 0) = \\j/u{u, 6), fj(u, 0) = -&e(u, 0), (a(u, 0,w)-d(u,9, 0))w >0, q(u, 9, g)g < 0, where y/ — e 9t] is the Helmholtz free energy function. (1.4) Received August 5, 1991 and, in revised form, October 28, 1991. 1991 Mathematics Subject Classification. Primary 35M10, 73C35, 73B30. Permanent address : Department of Mathematics, Xi'an Jiaotong University, Xi'an, Shaanxi Province, PR China. E-mail address : unm204@ibm.rhrz.uni-bonn.de. © 1993 Brown University 731
TL;DR: In this paper, the thermistor problem is modeled as a coupled system of nonlinear elliptic equations, and the existence of a weak solution is established under some special Dirichlet and Neumann boundary conditions.
Abstract: The thermistor problem is modeled as a coupled system of nonlinear elliptic equations. When the conductivity coefficient a(u) vanishes (u = temperature) one of the equations becomes degenerate; this situation is considered in the present paper. We establish the existence of a weak solution and, under some special Dirichlet and Neumann boundary conditions, analyze the structure of the set {a(u) = 0} and also prove uniqueness.
TL;DR: In this paper, the authors studied the associated problem for a general anisotropic elastic strip and derived the decay factor for the strip of monoclinic materials with the plane of symmetry at x3 = 0 and with the sides x2 = ± 1 being traction free.
Abstract: It is known that the stresses of an isotropic elastic semi-infinite strip decay exponentially at large distance x, from the end x, = 0 if the sides x2 = ±1 are traction free and the loading at xl = 0 is in self-equilibrium. We study the associated problem for a general anisotropic elastic strip. Eight different side conditions at x2 = ±1 and eight different end conditions at x, = 0 are considered. With the Stroh formalism, all these different side and end conditions are encompassed in one simple formulation. It is shown that, for certain side conditions, the loading at x, = 0 need not be in self-equilibrium. The decay factor for the strip of monoclinic materials with the plane of symmetry at x3 = 0 and with the sides x2 = ±1 being traction free is derived, and it has a remarkably simple expression. Numerical calculations of the smallest decay factor are presented.
TL;DR: In this article, a parameter identification problem is considered in the context of a linear abstract Cauchy problem with a parameter-dependent evolution operator, and conditions are investigated under which the gradient of the state with respect to a parameter possesses smoothness properties which lead to local convergence of an estimation algorithm based on quasi-linearization.
Abstract: A parameter identification problem is considered in the context of a linear abstract Cauchy problem with a parameter-dependent evolution operator. Conditions are investigated under which the gradient of the state with respect to a parameter possesses smoothness properties which lead to local convergence of an estimation algorithm based on quasi-linearization. Numerical results are presented concerning estimation of unknown parameters in delay-differential equations.
TL;DR: In this paper, the convexity-concavity character of the yield surface is found to be related to the changes in elastic stiffness that take place during plastic loading.
Abstract: An elementary proof of the normality condition is given. In addition, the convexity-concavity character of the yield surface is found to be related to the changes in elastic stiffness that take place during plastic loading.
TL;DR: In this paper, a model for the propagation of a combustion front through a periodically inhomogeneous medium is posed, and the existence of a steady state solution is proved, in which the front's velocity is periodic in time.
Abstract: A model problem for the propagation of a combustion front through a periodically inhomogeneous medium is posed. The existence of a steady state solution is proved, in which the front's velocity is periodic in time. Computer simulations are carried out. Finally, through rigorous homogenization techniques, it is shown that when the wavelength of the inhomogeneity is small, the solution may be approximated by a travelling wave solution of the corresponding problem for a medium with certain constant properties.
TL;DR: The question of global existence for solutions of reaction-diffusion systems presents fundamental difficulties in the case in which some components of the system satisfy Neumann boundary conditions while others satisfy nonhomogeneous Dirichlet boundary conditions as mentioned in this paper.
Abstract: The question of global existence for solutions of reaction-diffusion systems presents fundamental difficulties in the case in which some components of the system satisfy Neumann boundary conditions while others satisfy nonhomogeneous Dirichlet boundary conditions. We discuss particular examples for which classical solutions are known to exist globally when all components satisfy the same type of boundary condition and yet either finite-time blowup occurs or else global existence is unknown when mixed boundary condition types are imposed on the system. Some positive results are presented concerning global existence in the presence of mixed boundary conditions if certain structure requirements are placed on the system, and these results are applied to some particular chemical reaction models.
TL;DR: In this paper, the authors derived objective rates of tensors on the boundary line enclosing the surface under consideration of a two-dimensional continuum and gave a correct formulation of a 2D continuum.
Abstract: In this paper we derive objective, in the sense of surface, rates of tensors and we give a correct formulation of a two-dimensional continuum. Furthermore we present objective rates of tensors on the boundary line enclosing the surface under consideration. It can be considered as a first step to derive objective rates for generalized continua as Cosserat continua and Kirchhoff-Love type nonlinear shell theories.
TL;DR: The uniqueness of the solution of the problem has been examined by McLeod and Rajagopal [2] and by Troy et al. as mentioned in this paper, and motivated by the work of the above-mentioned authors, they reinvestigated analytically through a simple mathematical procedure.
Abstract: 1. Background. Rajagopal, Na, and Gupta have studied the flow of a viscoelastic fluid over a stretching sheet [1]. The uniqueness of the solution of the problem has been examined by McLeod and Rajagopal [2] and by Troy et al. [3]. WenDong Chang [4] recently claimed that the solution of the problem is not necessarily unique. Motivated by the work of the above-mentioned authors, here the solution of the problem is reinvestigated analytically through a simple mathematical procedure.
TL;DR: In this article, small oscillations of an elastic beam clamped radially to the interior of a rotating ring were discussed using a Ritz Galerkin procedure and the model equation was reduced to a nonlinear ordinary differential equation of second order.
Abstract: We discuss small oscillations of an elastic beam clamped radially to the interior of a rotating ring. Using a Ritz Galerkin procedure, the model equation is reduced to a nonlinear ordinary differential equation of second order
TL;DR: In this article, certain thermodynamic properties of elastic-plastic materials with workhardening are discussed and their corresponding free-energy functions are determined, and the corresponding free energy functions are derived.
Abstract: Certain thermodynamic properties of elastic-plastic materials with workhardening are discussed and their corresponding free-energy functions are determined.
TL;DR: In this article, the nonlinear dynamics of normal field instability in a ferrofluid under the influence of a uniform magnetic field was studied and an equation governing the evolution of small but finite amplitude was obtained.
Abstract: We study the nonlinear dynamics of normal field instability in a ferrofluid under the influence of a uniform magnetic field. In addition, a small normal sinusoidal magnetic field is superimposed on the system. An equation governing the evolution of small but finite amplitude is obtained. Applying the Melnikov method, it is shown that there exist transverse homoclinic orbits leading to chaotic motions.
TL;DR: In this article, the authors established an asymptotic lower bound for the minimum excitation needed to cause instability for the damped Mathieu equation, using Floquet theory and Liapunov-Schmidt.
Abstract: We establish an asymptotic lower bound for the minimum excitation needed to cause instability for the damped Mathieu equation. The methods used are Floquet theory and Liapunov-Schmidt, and we use a fact about the width of the instability interval for the undamped Mathieu equation. Our results are compared with published numerical data.
TL;DR: In this paper, a hyperelastic ball was shown to have monotonicity and a determinant gradient of deformation upper bound for storing energy in the ball, and the authors used the Web of Science Record created on 2008-12-10, modified on 2016-08-08
Abstract: Keywords: hyperelastic ball ; stored energy ; monotonicity ; determinant ; gradient of deformation ; upper bound Reference ANA-ARTICLE-1993-002View record in Web of Science Record created on 2008-12-10, modified on 2016-08-08
TL;DR: In this paper, the stability and bifurcation behavior of a nonlinear autonomous system in the vicinity of a compound critical point characterized by two pairs of pure imaginary eigenvalues of the Jacobian are investigated.
Abstract: This paper is concerned with the stability and bifurcation behaviour of a nonlinear autonomous system in the vicinity of a compound critical point characterized by two pairs of pure imaginary eigenvalues of the Jacobian. Attention is focused on the local dynamics of the system near-to-resonance. The methodology developed earlier for the bifurcation analysis into periodip and quasi-periodic motions (unification technique coupled with the intrinsic harmonic balancing) is extended to consider the stability and bifurcations of resonant cases. A set of simplified rate equations characterizing the local dynamics of the system is derived. These equations differ from those associated with nonresonant cases in that they are phase-coupled. Furthermore, the stability conditions of the phase-locked periodic bifurcation solutions are presented. All the results are expressed in explicit forms.
TL;DR: In this article, the effects of small perturbations in the constitutive laws on antiplane shear deformation fields arising in the theory of nonlinear elasticity were assessed, and the main result provided a comparison between two solutions, one of which is a solution to a simpler equation, for example Laplace's equation.
Abstract: : This paper is concerned with assessing the effects of small perturbations in the constitutive laws on antiplane shear deformation fields arising in the theory of nonlinear elasticity. The mathematical problem is governed by a second-order quasilinear partial differential equation in divergence form. Dirichlet (or Neumann) boundary-value problems on a semi- infinite strip, with nonzero data on one end only, are considered. Such problems arise in investigation of Saint-Venant end effects in elasticity theory. The main result provides a comparison between two solutions, one of which is a solution to a simpler equation, for example Laplace's equation. Three examples involving perturbations of power-law material models are used to illustrate the results.