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1

On strict pseudoconvexity

Ivanov V., I.

Journal of Applied Analysis

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2007

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Vol. 13, nr 2

183-196

EN

The present paper provides first and second-order characterizations of a radilly lower semicontinuous strictly pseudoconvex function ∫ : X → R defined on a convex set X in the real Euclidean space Rn in twerms of the lower Dini-directional derivative. In particular we obtain connections between the strictly pseudoconvex functions, nonlinear programming problem, Stampacchia variational inequality, and strict Minty variational inequality. We extend to the radially continuous functions the characterization due to Diewert, Avriel, Zang [6]. A new implication appears in our conditions. Connections with other classes of functions are also derived

2

The maximum principle in optimal control, then and now

Clarke F.

Control and Cybernetics

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2005

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Vol. 34, no 3

709-722

EN

We discuss the evolution of the Pontryagin maximum principle, focusing primarily on the hypotheses required for its validity. We proceed to describe briefly a unifying result giving rise to both classical and new versions, a recent theorem of the author giving necessary conditions for optimal control problems formulated in terms of differential inclusions. We conclude with a new application of this result for the case in which mixed constraints on the state and control are imposed in terms of equalities, inequalities, and unilateral set constraints. In order to lighten the exposition, the discussion is limited to differentiable data, thereby avoiding mention of generalized gradients or normal cones, except, in the technical section on differential inclusions.

3

Sformułowanie zagadnienia dynamiki ciała z materiału sprężysto-idealnie plastycznego z zastosowaniem metod analizy niegładkiej

Grzesikiewicz W., Wojewódzki W., Zbiciak A.

Prace Naukowe Politechniki Warszawskiej. Budownictwo

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2004

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z. 142

69--94

PL

W pracy podajemy relacje konstytutywne materiału sprężysto-idealnie plastycznego, a następnie formułujemy zagadnienie brzegowo-początkowe. Ze względu na nieróźniczkowalność funkcjonałów definiujących dyssypację, wprowadzamy pojęcie subróżniczki funkcjonału. Związki fizyczne omawianego materiału nie pozwalają na jednoznaczne określenie prędkości odkształceń plastycznych, dlatego też korzystamy z ich następstw różniczkowych. W kolejnych rozdziałach specyfikujemy zależności konstytutywne w formie następstw na przypadek gładkiej powierzchni plastyczności. W szczególności rozpatrujemy powierzchnię Hubera-Misesa--Hencky'ego. Następnie formułujemy zagadnienie płaskiego stanu naprężenia oraz podajemy przykład analizy dynamicznej tarczy prostokątnej poddanej impulsowi ciśnienia.

EN

The aim of the paper was the initial-boundary-value problem formulation for the elastic-perfectly plastic solid. The main difficulty in analysis of such material is that the dissipation potential is non-smooth. The lack of differentiability in dissipation function becomes a structural analysis problem in the study of structures with multi-valued stress-strain law. Non-smooth convex analysis methods are used to describe problems which involve non-differentiable, in the classical sense, functions or for problems which have inequalities in their definition. It is because these methods represent the natural framework for the study of plasticity phenomenon. At first we have introduced the constitutive relations for the elastic perfectly-plastic material. After the formulation of the initial-boundary-value problem it has been proved that the form of the constitutive relations given by the consistency equation and the maximum dissipation principle does not allow us to determine the velocities of the plastic strains. We then defined the succession form of the constitutive equations. We have focused on the definition of the sets of admissible velocities of the stress tensor and the plastic strain tensor. We have demonstrated the procedure of determining the detailed form of the constitutive equations in the case of the smooth plasticity surface. In particular the Huber- -Mises-Hencky yield criterion was considered. Then we have analyzed the plane stress problem. The oadditional Lagrange multipliers have been introduced in order to determine the set of constitutive equations. These multipliers can be interpreted as the plane stress reactions. In the last section we have shown the numerical example of dynamic analysis of rectangular disc loaded by an impulsive load.

4

Unilateral Contact Applications Using Fem Software

Stavroulaki M. E., Stavroulakis G. E.

International Journal of Applied Mathematics and Computer Science

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2002

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Vol. 12, no 1

115-125

EN

Nonsmooth analysis, inequality constrained optimization and variational inequalities are involved in the modelling of unilateral contact problems. The corresponding theoretical and algorithmic tools, which are part of the area known as nonsmooth mechanics, are by no means classical. In general purpose software some of these tools (perhaps in a simplified way) are currently available. Two engineering applications, a rubber-coated roller contact problem and a masonry wall, solved with MARC, are briefly presented, together with elements of the underlying theory.