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1

Governing Differential Equations for the Mechanics of Undamageable Materials

Voyiadjis G. Z., Kattan P. I.

Engineering Transactions

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2014

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Vol. 62, iss. 3

241–-267

EN

In this work the mathematical foundations of the mechanics of elastic undamageable materials are presented. In particular the governing differential equations are derived for both the scalar and tensorial cases. In the isotropic case it is found that the resulting scalar differential equations are simple and easy to solve. However, in the anisotropic case the tensorial differential equations are complicated and unsolvable at this time. The current work presents the solution in the form of explicit nonlinear stress-strain relations for the simple one-dimensional case. However, the general solution of the three-dimensional case remains unattainable at the present time. Only the governing tensorial differential equations are derived for this latter case. It is to be noted that the term “undamageable” is reflected in the context of the material stiffness and not the property of indestructibility due to various loading conditions. Thus, the undamageable material reflects that no microcracks or microvoids occur as well as no plastic yielding in the material. To illustrate this concept, a last section is added on applications.

2

Stability of fractional discrete-time linear scalar systems with one delay

Busłowicz M.

Pomiary Automatyka Robotyka

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2013

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R. 17, nr 2

327-332

EN

In the paper the problems of practical stability and asymptotic stability of fractional discrete-time linear scalar systems with one constant delay are addressed. Standard and positive systems are considered. New conditions for practical stability and for asymptotic stability are established.

PL

Rozpatrzono problem stabilności liniowych skalarnych układów dyskretnych niecałkowitego rzędu z jednym opóźnieniem zmiennych stanu. Wykorzystując metodę podziału D, podano granczne warunki konieczne i wystarczające praktycznej stabilności. Bazując na tych warunkach, sformułowano proste analityczne warunki wystarczające stabilności praktycznej oraz stabilności asymptotycznej. W przypadku układów dodatnich podano proste analityczne warunki konieczne i wystarczające stabilności praktycznej oraz stabilności asymptotycznej.

3

The invariants of a pair of directions in geometry [En1] and their interpretation

Misiak A., Stasiak E., Szmuksta-Zawadzka M.

Demonstratio Mathematica

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2013

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Vol. 46, nr 2

361--371

EN

Solving a certain functional equation, we find all invariants of a pair of directions in n-dimensional pseudo-Euclidean geometry of index one [En1]. In (n-1)-dimensional space we construct a model for these directions by means of concepts characteristic of Euclidean geometry. Because it is a pseudo-orthogonal group, not orthogonal, that operates in this model, the distance between two points and the measure of an angle are not invariants. Using these changeable quantities we construct invariant quantities.