A micromechanics constitutive model for pure dilatant martensitic transformation of ZrO2-containing ceramics

A new micromechanics constitutive model for pure dilatant transformation plasticity of structure ceramics is proposed in this paper. Based on the thermodynamics, micromechanics and microscalet→m transformation mechanism analysis of the TZP and PSZ ZrO₂-containing ceramics, an analytic expressions of the Helmholtz and complementary free energy of the constitutive element for the case of pure dilatant transformation is derived for the first time in a self-consistent manner. By the analysis of energy dissipation in the forward and reverse transformations, the micromechanics constitutive law is derived in the framework of Hill-Rice's internal variable constitutive theory. The project is supported by the National Natural Science Foundation of China.     

A spectrum relation of almost periodic solution of second order scalar functional differential equations with piecewise constant argument

In this paper, the spectrum relation of almost periodic solution for the equation (x(t) + px(t − 1))″ = qx([t]) + f(t) is investigated. Although this has been discussed in an article, some counterexamples are constructed to show that some part of the spectrum inclusion in that article is not correct. The key point which causes such problem is found out. A new statement is formulated and proved.     

Bagley-Torvik型分数阶微分方程和包含非局部积分边值问题

In this article,we consider the Bagley-Torvik type fractional differential equation ~cD~(ν1) l(t)-a~cD~(ν2) l(t) = g(t, l(t)) and differential inclusion ~cD~(ν1) l(t)-a~cD~(ν2) l(t) ∈ G(t, l(t)),t ∈(0, 1) subjecting to l(0) = l_0,and■ ds where 1 ν_1 ≤ 2, 1 ≤ν_2 ν_1,0 ω≤ 1, χ = ν_1-ν_2 0, a, λ′are given constants. By using Leray-Schauder degree theory and fixed point theorems, we prove the existence of solutions. Our results extend the existence theorems for the classical Bagley-Torvik equation and some related models.     

Localization of elastic deformation in strongly anisotropic, porous, linear materials with periodic microstructures: Exact solutions and dilute expansions

Exact solutions are derived for the problem of a two-dimensional, infinitely anisotropic, linear-elastic medium containing a periodic lattice of voids. The matrix material possesses either one infinitely soft, or one infinitely hard loading direction, which induces localized (singular) field configurations. The effective elastic moduli are computed as functions of the porosity in each case. Their dilute expansions feature half-integer powers of the porosity, which can be correlated to the localized field patterns. Statistical characterizations of the fields, such as their first moments and their histograms are provided, with particular emphasis on the singularities of the latter. The behavior of the system near the void close-packing fraction is also investigated. The results of this work shed light on corresponding results for strongly non-linear porous media, which have been obtained recently by means of the “second-order” homogenization method, and where the dilute estimates also exhibit fractional powers of the porosity.     

Directionally differentiable multiobjective optimization involving discrete inclusions

This paper is devoted to multiobjective optimization problems involving discrete inclusions. The objective functions are assumed to be directionally differentiable and the domination structure is defined by a closed convex cone. The directional derivatives are not assumed to be linear or convex. Several concepts of optimal solutions are analyzed, and the corresponding necessary conditions are obtained as well in maximum principle form. As an application of the main results, a maximum principle is also derived for multiobjective optimization with extremalvalue fucctions involving discrete inclusions.The authors are indebted to the referee for detailed comments.The paper was written while the second author was visiting the laboratory of Prof. S. Suzuki, Department of Mechanical Engineering, Sophia University, Tokyo, Japan.     

Viability Theorems for Higher-Order Differential Inclusions

The aim of this paper is to establish both necessary and sufficient conditions for the existence of viable solutions of a higher-order differential inclusion. Higher-order tangent sets are introduced and studied to get such conditions. These tangent sets are analysed in the case of the viable set defined by equality or inequality constraints and viability conditions are rewritten in terms of these constraints.     

Optimal control of nonlinear evolution inclusions

In this paper, we study the optimal control of nonlinear evolution inclusions. First, we prove the existence of admissible trajectories and then we show that the set that they form is relatively sequentially compact and in certain cases sequentially compact in an appropriate function space. Then, with the help of a convexity hypothesis and using Cesari's approach, we solve a general Lagrange optimal control problem. After that, we drop the convexity hypothesis and pass to the relaxed system, for which we prove the existence of optimal controls, we show that it has a value equal to that of the original one, and also we prove that the original trajectories are dense in an appropriate topology to the relaxed ones. Finally, we present an example of a nonlinear parabolic optimal control that illustrates the applicability of our results.This research was supported by NSF Grant No. DMS-88-02688.     

The perturbed proximal point algorithm and some of its applications

Following the works of R. T. Rockafellar, to search for a zero of a maximal monotone operator, and of B. Lemaire, to solve convex optimization problems, we present a perturbed version of the proximal point algorithm. We apply this new algorithm to convex optimization and to variational inclusions or, more particularly, to variational inequalities.     

Nonlinear Impulse Target Problems under State Constraint: A Numerical Analysis Based on Viability Theory

We study the problem of reaching a target without leaving a prescribed set for controlled impulse dynamics. First, we provide a numerical procedure for the approximation of the set of initial conditions from which the objective can be met. Then we show that the minimum time function associated with this target problem can be approached by a sequence of value functions for suitable discrete-time systems. This can be deduced from the fact that the epigraph of the minimum time function is the set of initial conditions from which a target can be reached without leaving a constraint set for an auxiliary impulse system. In this case, the numerical procedure for the qualitative target problem can be simplified. We provide results for estimating the convergence rate of the simplified scheme.     

On singular perturbations for differential inclusions on the infinite interval

We consider a differential inclusion subject to a singular perturbation, i.e., part of the derivatives are multiplied by a small parameter >0. We show that under some stability and structural assumptions, every solution of the singularly perturbed inclusion comes close to a solution of the degenerate inclusion (obtained for =0) when tends to 0. The goal of the present paper is to provide a new result of Tikhonov type on the time interval [0,+∞[.