Algorithm for finding a solution to the inclusion 0∈F(x)

Using the steepest-descent method combined with the Armijo stepsize rule, we give an algorithm for finding a solution to the inclusion 0F(x), whereF is a set-valued map with smooth support function. As an example, we consider the special caseF(x)=g(x)+K, withK being a convex cone andg a single-valued function. The relation between the present algorithm and that given by Burke and Han is also discussed.The valuable comments and helpful suggestions of the referee are gratefully acknowledged. Sincere thanks are due to Dr. J. Burke for submitting the necessary material and to Dr. C. Lemarechal for advices and encouragement.     

Asymptotic behavior of optimal solutions to control problems for systems described by differential inclusions corresponding to partial differential equations

In the paper, we consider differential inclusions related to PDEs of parabolic type and some control problems with integral cost functionals associated to them. Given a sequence of such problems, we investigate first the asymptotic behavior of solution sets (mild solutions or more precisely selection-trajectory pairs) for differential inclusions, and we get some semicontinuity or continuity results (Kuratowski convergence of solution sets). Then, we prove the -convergence of cost functionals, related to the above Kuratowski convergence of solution sets. Finally, applying the Buttazzo-Dal Maso abstract scheme, based on the sequential -convergence, we obtain results concerning the asymptotic behavior (hence, also stability results) for optimal solutions to control problems as well as the convergence of minimal values.The authors would like to thank Professors G. Dal Maso and S. Spagnolo for helpful conversations.This work was done when the first author was visiting ICTP and ISAS in Trieste in 1990/91.     

Modeling the macroscopic behavior of two-phase nonlinear composites by infinite-rank laminates

A new approach is proposed for estimating the macroscopic behavior of two-phase nonlinear composites with random, particulate microstructures. The central idea is to model composites by sequentially laminated constructions of infinite rank whose macroscopic behavior can be determined exactly. The resulting estimates incorporate microstructural information up to the two-point correlation functions, and require the solution to a Hamilton–Jacobi equation with the inclusion concentration and the macroscopic fields playing the role of ‘time’ and ‘spatial’ variables, respectively. Because they are realizable, by construction, these estimates are guaranteed to be convex, to satisfy all pertinent bounds, to exhibit no duality gap, and to be exact to second order in the heterogeneity contrast. Sample results are provided for two- and three-dimensional power-law composites, and are compared with other homogenization estimates, as well as with numerical simulations available from the literature. The estimates are found to give physically sensible predictions for all the cases considered, even for extreme values of the nonlinearity and heterogeneity contrast. Interestingly, in the case of isotropic porous materials under hydrostatic loadings, the estimates agree exactly with standard Gurson-type models for viscoplastic porous media.     

One approach to a magnetostatic problem for bodies with inclusions in an inhomogeneous external field

A direct magnetostatic problem for magnets with a finite-size inclusion is considered in an integrodifferential form. An approach is used that, under certain conditions, reduces the problem to a single integral equation on a two-dimensional manifold-the inclusion surface. As an important illustrative example, finite formulas are derived to compute the resulting field of a magnetic half-space with a spherical cavity in an arbitrary external field.     

φ-单调型集值变分包含解的Ishikawa迭代逼近

在实自反的Banach空间中引入和研究了一类新的非线性φ-单调型集值变分包含,并证明了此类变分包含解的唯一性及其具误差项的Ishikawa迭代程序的收敛性.本文结果在多方面改进和推广了Huang和Fang等人的一些相关结果.     

Dielectric elastomer composites: small-deformation theory and applications

Abstract

We provide estimates for the effective response of Electro-Active Polymer Composites (EAPCs) consisting of aligned ellipsoidal inclusions of a stiff dielectric material which are distributed randomly in an soft elastomeric matrix with “ellipsoidal” two-point statistics. The derivation of the results for the electro-mechanical response assumes linearized deformations, but includes non-linear (quadratic) terms in the electric fields. We investigate three different physical mechanisms contributing to the macroscopic electro-mechanical response of the composite: the intrinsic effect of the particles on the Maxwell stress, the inter-particle (dipole) interactions which are accounted for by evaluating the effect of changes in the “shape” of the two-point probability functions with the deformation, and the effect of particle rotations and torques when the geometric and/or anisotropy axis of the particles are not aligned with the applied electric field. Several illustrative examples are provided to emphasize the relative importance of the different effects on the overall electrostriction of the composites. In particular, for the “compliant electrode” boundary conditions that are widely used in applications, it is shown that inter-particle interactions are synergistic with the intrinsic effect of the particles on the Maxwell stress, leading to significant enhancements in the electro-mechanical coupling of the EAPCs, especially at high particle concentrations. On the other hand, the effect of electric torques on non-aligned particles is generally deleterious for electrostriction.     

Light scattering by a spherical particle with multiple densely packed inclusions

This paper calculates light scattering by a spherical water particle containing densely packed inclusions at a visible wavelength 0.55 \mum by a combination of ray-tracing and Monte Carlo techniques. While the individual reflection and refraction events at the outer boundary of a sphere particle are considered by a ray-tracing program, the Monte Carlo routine simulates internal scattering processes. The main advantage of this method is that the shape of the particle can be arbitrary, and multiple scattering can be considered in the internal scattering processes. A dense-medium light-scattering theory based on the introduction of the static structure factor is used to calculate the phase function and asymmetry parameters for densely packed inclusions. Numerical results of the single scattering characteristics for a sphere containing multiple densely packed inclusions are given.     

平面弹性中双圆柱夹杂问题的格林函数

通过复势法并结合共形映射、解析开拓、奇点分析、Cauchy积分公式、圆环域上的Laurent级数展开等技术的联合运用,给出了含有双圆柱异相夹杂的无限大弹性基体这种三相复合系统在受到面内集中力和刃型位错作用时的级数形式的精确解答。该奇点可以作用于基体上或两夹杂内部,当一刃型位错作用于基体上时,也推导了该位错与双圆柱夹杂的交互作用能以及作用于位错上的广义。所给出的算例验证了该弹性解答的正确性并直观显示出奇点与双圆柱夹杂的相互作用。     

Antiplane Problems of Periodical Rigid Line Inclusions between Dissimilar Anisotropic Materials

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On the stability of elastic-plastic systems with hardening

This paper discusses the stability of quasi-static paths for a continuous elastic-plastic system with hardening in a one-dimensional (bar) domain. Mathematical formulations, as well as existence and uniqueness results for dynamic and quasi-static problems involving elastic-plastic systems with linear kinematic hardening are recalled in the paper. The concept of stability of quasi-static paths used here is essentially a continuity property of the system dynamic solutions relatively to the quasi-static ones, when (as in Lyapunov stability) the size of initial perturbations is decreased and the rate of application of the forces (which plays the role of the small parameter in singular perturbation problems) is also decreased to 0. The stability of the quasi-static paths of these elastic-plastic systems is the main result proved in the paper.