The mechanics of interface fracture in layered composite materials: (2) cohesive interfaces

The author’s mixed-mode partition theories [1-9] for rigid interfaces are extended to non-rigid cohesive interfaces for one dimensional (1D) interface fracture. In the absence of crack tip through thickness shear forces both classical and shear deformable partition theories have identical mode I and II energy release rate (ERR) partitions which are the same as those of shear deformable partitions for a mixed mode at rigid interfaces and independent of interface cohesive laws. Consequently, the mode mixity remains constant during fracture evolution. In the case of interface fracture in the layered isotropic materials, the pure modes in 2D elasticity partition theory only depend on the ratio between the penalty stiffness to the Young’s modulus of the materials and are independent of the shape of the cohesive laws. A mixed fracture mode can be readily partitioned by using the pure modes and a constant mode mixity is shown

The mechanics of interface fracture in layered composite materials: (4) buckling driven delamination of thin layer materials

Analytical theories were developed for studying post-local buckling-driven delamination of thin layer materials under in-plane compressive stresses which can arise from externally applied mechanical loads, thermal stresses due to mismatch of coefficients of thermal expansion between the thin layer material and the thick substrates, the intercalation stresses due to electrochemical lithiation and delithiation, and etc. The development was based on three mixed mode partition theories. They are Euler beam or classical plate, Timoshenko beam or shear deformable plate [1-5] and 2D-elasticity [6-8] theories. Independent experimental tests [9] show that, in general, the analytical partitions based on the Euler beam or classical plate theory predicts the propagation behaviour very well and much better than the partitions based on the Timoshenko beam and 2D-elasticity theories

Epsilon-noncrossing partitions and cumulants in free probability

Motivated by recent work on mixtures of classical and free probabilities, we introduce and study the notion of $\epsilon$-noncrossing partitions. It is shown that the set of such partitions forms a lattice, which interpolates as a poset between the poset of partitions and the one of noncrossing partitions. Moreover, $\epsilon$-cumulants are introduced and shown to characterize the notion of $\epsilon$-independence

Logical Entropy: Introduction to Classical and Quantum Logical Information theory

Logical information theory is the quantitative version of the logic of partitions just as logical probability theory is the quantitative version of the dual Boolean logic of subsets. The resulting notion of information is about distinctions, differences and distinguishability and is formalized using the distinctions of a partition. All the definitions of simple, joint, conditional and mutual entropy of Shannon information theory are derived by a uniform transformation from the corresponding definitions at the logical level. The purpose of this paper is to give the direct generalization to quantum logical information theory that similarly focuses on the pairs of eigenstates distinguished by an observable, i.e., qudits of an observable. The fundamental theorem for quantum logical entropy and measurement establishes a direct quantitative connection between the increase in quantum logical entropy due to a projective measurement and the eigenstates that are distinguished by the measurement. Both the classical and quantum versions of logical entropy have simple interpretations as “two-draw” probabilities for distinctions. The conclusion is that quantum logical entropy is the simple and natural notion of information for quantum information theory focusing on the distinguishing of quantum states

Combinatorial Topology Of Multipartite Entangled States

With any state of a multipartite quantum system its separability polytope is associated. This is an algebro-topological object (non-trivial only for mixed states) which captures the localisation of entanglement of the state. Particular examples of separability polytopes for 3-partite systems are explicitly provided. It turns out that this characterisation of entanglement is associated with simulation of arbitrary unitary operations by 1- and 2-qubit gates. A topological description of how entanglement changes in course of such simulation is provided.Comment: 14 pages, LaTeX2e. Slightly revised version of the poster resented on the International Conference on Quantum Information, Oviedo, Spain, 13-18 July, 2002. To appear in the special issue of Journal of Modern Optic