Boundary integral equation methods in the theory of elasticity of hemitropic materials: A brief review

The theory of elasticity of hemitropic materials has recently been the object of rigorous mathematical analysis. In particular, the potential method and the theory of pseudodifferential equations have been used in studying the solvability in various function spaces of the main boundary value and transmission problems, in smooth and in Lipschitz domains. The main features and results of this boundary integral equations approach are briefly reviewed here.     

Mathematical problems of the theory of elasticity of chiral materials for Lipschitz domains

By the potential method, we investigate the Dirichlet and Neumann boundary value problems of the elasticity theory of hemitropic (chiral) materials in the case of Lipschitz domains. We study properties of the single‐ and double‐layer potentials and of certain, generated by them, boundary integral operators. These results are applied to reduce the boundary value problems to the equivalent first and the second kind integral equations and the uniqueness and existence theorems are proved in various function spaces. Copyright © 2005 John Wiley & Sons, Ltd.     

Contact problems with friction for hemitropic solids: boundary variational inequality approach

We study the interior and exterior contact problems for hemitropic elastic solids. We treat the cases when the friction effects, described by Tresca friction (given friction model), are taken into consideration either on some part of the boundary of the body or on the whole boundary. We equivalently reduce these problems to a boundary variational inequality with the help of the Steklov–Poincaré type operator. Based on our boundary variational inequality approach we prove existence and uniqueness theorems for weak solutions. We prove that the solutions continuously depend on the data of the original problem and on the friction coefficient. For the interior problem, necessary and sufficient conditions of solvability are established when friction is taken into consideration on the whole boundary.     

Thermo-poroacoustic acceleration waves in elastic materials with voids

A model for coupled elasto-acoustic waves, thermal waves, and waves associated with the voids, in a porous medium is investigated. Due to the use of lighter materials in modern buildings and noise concerns in the environment such models for thermo-poroacoustic waves are of much interest to the building industry. Analysis of such waves is also of interest in acoustic microscopy where the identification of material defects is of paramount importance to industry and medicine. We present a model for acoustic wave propagation in a porous material which also allows for propagation of a thermal wave. The thermodynamics is based on an entropy inequality of A.E. Green, F.R.S. and N. Laws and is presented for a modification of the theory of elastic materials with voids due to J.W. Nunziato and S.C. Cowin. A fully nonlinear acceleration wave analysis is initiated.     

An extended beam theory for smart materials applications part I: Extended beam models,duality theory,and finite element simulations

An extended theory for elastic and plastic beam problems is studied. By introducing new dependent and independent variables, the standard Timoshenko beam model is extended to take account of shear variation in the lateral direction. The dynamic governing equations are established via Hamilton's principle, and existence and uniqueness results for the solution of the static problem are proved. Using the theory of convex analysis, the duality theory for the extended beam model is developed. Moreover, the extended theory for rigid-perfectly plastic beams is also established. Based on the extended model, a finite-element method is proposed and numerical results are obtained indicating the usefulness of the extended theory in applications.The work of the first author was supported in part by National Science Foundation under Grant DMS9400565.     

Solvability and regularity results to boundary‐transmission problems for metallic and piezoelectric elastic materials

We investigate three‐dimensional transmission problems related to the interaction of metallic and piezoelectric ceramic bodies. We give a mathematical formulation of the physical problem when the metallic and ceramic sub‐domains are bonded along some proper parts of their boundaries. The corresponding nonclassical mixed boundary‐transmission problem is reduced by the potential method to an equivalent nonselfadjoint strongly elliptic system of pseudo‐differential equations on manifolds with boundary. We investigate the solvability of this system in different function spaces. On the basis of these results we prove uniqueness and existence theorems for the original boundary‐transmission problem. We study also the regularity of the electrical and mechanical fields near the curves where the boundary conditions change and where the interfaces intersect the exterior boundary. The electrical and mechanical fields can be decomposed into singular and more regular terms near these curves. A power of the distance from a reference point to the corresponding edge‐curves occurs in the singular terms and describes the regularity explicitly. We compute these complex‐valued exponents and demonstrate their dependence on the material parameters (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mixed impedance transmission problems for vibrating layered elastic bodies

Direct scattering problems for partially coated piecewise homogenous and inhomogeneous layered obstacles in linear elasticity lead to mixed impedance transmission problems for the steady‐state elastic oscillation equations. For a piecewise homogenous isotropic composite body, we employ the potential method and reduce the mixed impedance transmission problem to an equivalent system of boundary pseudodifferential equations. We give a detailed analysis of the corresponding pseudodifferential operators, which live on the interface between the layers and on a proper submanifold of the boundary of the composite elastic body, and establish uniqueness and existence results for the original mixed impedance transmission problem for arbitrary values of the oscillation frequency parameter; this is crucial in the study of inverse elastic scattering problems for partially coated layered obstacles. We also investigate regularity properties of solutions near the collision curves, where the different boundary conditions collide, and establish almost best Hölder smoothness results. Further, we analyze the asymptotic behavior of the stress vector near the collision curve and derive explicit formulas for the stress singularity exponents. The case of Lipschitz surfaces is briefly treated separately. In the case of a composite body containing homogeneous or inhomogeneous finite anisotropic inclusions, we develop an alternative hybrid method based on the so‐called nonlocal approach and reduce the mixed transmission problem to an equivalent functional‐variational equation with a sesquilinear form that ‘lives’ on a bounded part of the layered composite body and its boundary. We show that this sesquilinear form is coercive and that the corresponding variational equation is uniquely solvable. Copyright © 2015 John Wiley & Sons, Ltd.     

Some programming problems for teaching foundations of group theory

This paper contains some programming problems which can be suggested for students starting to learn group theory. These problems are related to important notations such as subgroup, coset, normal divisor, symmetric group, normalizer, centralizer, homomorphism and automorphism. Carefully selected problems provide a successful understanding of the basic themes of finite group theory.     

出现在化学反应器理论中的奇摄动边值问题的渐近解

研究了一类出现在化学反应器理论中的奇摄动边值问题.在适当的条件下,用合成展开法构造出该问题的形式近似式,并应用微分不等式理论证明了解的存在性及其渐近性质.     

Perturbation theory for abstract optimization problems

A general perturbation theory is given for optimization problems in locally convex, linear spaces. Neither differentiability of the constraints nor regularity of the solutions of the unperturbed problem are assumed. Without reference to a particular multiplier rule, multipliers of the unperturbed problem are defined and used for characterizing solutions of a perturbed problem. In case of differentiable constraints or finite-dimensional spaces, the results exceed those known so far.     

Non-linear higher-order boundary value problems describing thin viscous flows near edges

Two boundary value problems for non-linear higher-order ordinary differential equations are analyzed, which have been recently proposed in the modeling of steady and quasi-steady thin viscous flows over a bounded solid substrate. The first problem concerns steady states and consists of a third-order ODE for the height of the liquid; the ODE contains an unknown parameter, the flux, and the boundary conditions relate, near the edges of the substrate, the height and its second derivative to the flux itself. For this problem, (non-)existence and non-uniqueness results are proved depending on the behavior, as the flux approaches zero, of the “height-function” (the function which relates the height to the flux near the edge out of which the liquid flows). The second problem concerns quasi-steady states and consists of a fourth-order ODE for the (suitably scaled) height of the liquid; non-linear boundary conditions relate the height to the flux near the edges of the substrate. For this problem, the existence of a solution is proved for a suitable class of height-functions.     

Inverse spectral theory for Sturm-Liouville problems with finite spectrum

For any positive integer and any given distinct real numbers we construct a Sturm-Liouville problem whose spectrum is precisely the given set of numbers. Such problems are of Atkinson type in the sense that the weight function or the reciprocal of the leading coefficient is identically zero on at least one subinterval.

     

Value of information in two-team zero-sum problems

The situation in which two groups of people have conflicts of interest is considered as a two-team zero-sum game problem. Two special cases of this problem are solved to illustrate that communication among members of a team may not be worth-while and extra information need not always be desired by decision makers. In the appendix, it is shown that the optimal saddle-point solution exists and is still affine for the general problem with quadratic Gaussian performance index.Dedicated to Professor M. R. HestenesThe research reported in this paper was made possible through support extended to the Division of Engineering and Applied Physics, Harvard University, by the U.S. Office of Naval Research under the Joint Services Electronics Program, Contract No. N00014-67-A-0298-0006, and by the National Science Foundation, Grant No. GK-31511.     

Applications of Toland's duality theory to nonconvex optimization problems

Through a suitable application of Toland's duality theory to certain nonconvex and nonsmooth problems one obtain an unbounded minimization problem with Fréchet:-differentiable cost function as dual problem and one can establish a gradient projection method for the solution of these problems.     

The scattering theory of C. Wilcox in elasticity

We extend the abstract time‐dependent scattering theory of C.H. Wilcox to the case of elastic waves. Most of the results are proved with the minimal assumption that the obstacle satisfies the energy local compactness condition (ELC). This holds especially for the existence and unitarity of the wave operators. Copyright © 2002 John Wiley & Sons, Ltd.     

Uniqueness and Stability of Determining the Residual Stress by One Measurement

In this paper we prove a Hölder and Lipschitz stability estimates of determining the residual stress by a single pair of observations from a part of the lateral boundary or from the whole boundary. These estimates imply first uniqueness results for determination of residual stress from few boundary measurements.     

Some results for impulsive problems via Morse theory

We use Morse theory to study impulsive problems. First we consider asymptotically piecewise linear problems with superlinear impulses, and prove a new existence result for this class of problems using the saddle point theorem. Next we compute the critical groups at zero when the impulses are asymptotically linear near zero, in particular, we identify an important resonance set for this problem. As an application, we finally obtain a nontrivial solution for asymptotically piecewise linear problems with impulses that are asymptotically linear at zero and superlinear at infinity. Our results here are based on the simple observation that the underlying Sobolev space naturally splits into a certain finite dimensional subspace where all the impulses take place and its orthogonal complement that is free of impulsive effects.     

Boundary-value problems for shallow elastic membrane caps

Consider the general nonlinear boundary-value problem (p(t)y' (t))' = p(t)q(t) f (t, y(t), y' (t)), t 1, g(y(1), y' (1))= 0, where the function f may be singular at the point y(1)= 0 and p(1) 0. We obtain conditions which guarantee existenceof positive and bounded solutions of the above problem. As anapplication we prove existence and uniqueness of rotationallysymmetric solutions to a nonlinear boundary-value problem, representingthe elastic deformation of a spherical cap.