On frames, dual frames, and the duality principle

2015 ◽  
Vol 45 (1) ◽  
pp. 183-200 ◽  
Author(s):  
Diana T. Stoeva
Keyword(s):  
Duality Principle ◽  
Dual Frames

Analysis of Multiple Rigid-Line Inclusions for Application to Bio-Materials

2007 ◽  
Author(s):  
Pawan S. Pingle ◽  
Larissa Gorbatikh ◽  
James A. Sherwood
Keyword(s):  
Stress Intensity ◽  
Stress Intensity Factors ◽  
Crack Problem ◽  
Building Blocks ◽  
Duality Principle ◽  
Inclusion Problem ◽  
Multiple Cracks ◽  
Intensity Factors ◽  
Aspect Ratios ◽  
Rigid Line

Hard biological materials such as nacre and enamel employ strong interactions between building blocks (mineral crystals) to achieve superior mechanical properties. The interactions are especially profound if building blocks have high aspect ratios and their bulk properties differ from properties of the matrix by several orders of magnitude. In the present work, a method is proposed to study interactions between multiple rigid-line inclusions with the goal to predict stress intensity factors. Rigid-line inclusions provide a good approximation of building blocks in hard biomaterials as they possess the above properties. The approach is based on the analytical method of analysis of multiple interacting cracks (Kachanov, 1987) and the duality existing between solutions for cracks and rigid-line inclusions (Ni and Nasser, 1996). Kachanov’s method is an approximate method that focuses on physical effects produced by crack interactions on stress intensity factors and material effective elastic properties. It is based on the superposition technique and the assumption that only average tractions on individual cracks contribute to the interaction effect. The duality principle states that displacement vector field for cracks and stress vector-potential field for anticracks are each other’s dual, in the sense that solution to the crack problem with prescribed tractions provides solution to the corresponding dual inclusion problem with prescribed displacement gradients. The latter allows us to modify the method for multiple cracks (that is based on approximation of tractions) into the method for multiple rigid-line inclusions (that is based on approximation of displacement gradients). This paper presents an analytical derivation of the proposed method and is applied to the special case of two collinear inclusions.

Finding Dead-Point Positions of Planar Pin-Connected Linkages Through Graph Theoretical Duality Principle

2005 ◽  
Vol 128 (3) ◽  
pp. 599-609 ◽  
Author(s):  
O. Shai ◽  
I. Polansky
Keyword(s):  
Graph Theory ◽  
Discrete Mathematics ◽  
Duality Principle ◽  
Mathematical Foundation ◽  
The Dead

The paper brings another view on detecting the dead-point positions of an arbitrary planar pin-connected linkage by employing the duality principle of graph theory. It is first shown how the dead-point positions are derived through the interplay between the linkage and its dual determinate truss—the relation developed in the previous works by means of graph theory. At the next stage, the process is shown to be performed solely upon the linkage by employing a new variable, the dual of potential, termed face force. Since the mathematical foundation of the presented method is discrete mathematics, the paper points to possible computerization of the method.

scholarly journals Duality of Complex Systems Built from Higher-Order Elements

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Dalibor Biolek ◽  
Zdeněk Biolek ◽  
Viera Biolková
Keyword(s):  
Nonlinear Systems ◽  
Complex Systems ◽  
Higher Order ◽  
Duality Principle ◽  
Shift Type ◽  
Hardware Emulation ◽  
Classical Duality ◽  
Memristive Circuits ◽  
Circuit Elements

The duality of nonlinear systems built from higher-order two-terminal Chua’s elements and independent voltage and current sources is analyzed. Two different approaches are now being generalized for circuits with higher-order elements: the classical duality principle, hitherto restricted to circuits built from R-C-L elements, and Chua’s duality of memristive circuits. The so-called storeyed structure of fundamental elements is used as an integrating platform of both approaches. It is shown that the combination of associated flip-type and shift-type transformations of the circuit elements can generate dual networks with interesting features. The regularities of the duality can be used for modeling, hardware emulation, or synthesis of systems built from elements that are not commonly available, such as memristors, via classical dual elements.

Sign in / Sign up

Export Citation Format

Share Document