Arbitrary Braid Research Articles

RUC-based multi-scale model for braid-reinforced polymers: Application to coil springs

As fiber- and braid-reinforced polymer components exhibit a vast number of additional design parameters compared to components made of monolithic materials, simulation methods are indispensable in the design and optimization process, accompanying or substituting experimental approaches. In respect thereof, we present a multi-scale approach comprising three observation scales: (i) the yarn scale considering fibers embedded in polymer, (ii) the braid scale capturing interwoven yarns, and (iii) the component scale. Information from one scale is transferred to the next higher scale by means of effective properties determined by finite element analysis of repetitive unit cell (RUC) models, subsequently applied for the determination of elastic properties on the yarn and braid scale. The depicted procedure allows the consideration of different types of braids (balanced and unbalanced regular braids) exhibiting arbitrary braid angles. Therefrom obtained effective properties enter the numerical model of the respective structural component. Finally, the multi-scale model is applied to the analysis of braid-reinforced polymer coil springs highlighting the potential of the proposed numerical approach. The obtained simulation results show excellent agreement with the respective experimental measurements and provide – additionally – insight into the stress distribution/loading within the coil as well as the sensitivity of design parameters with respect to the spring performance.

A Garside-theoretic analysis of the Burau representations

We establish relations between both the classical and the dual Garside structures of the braid group and the Burau representation. Using the classical structure, we formulate a non-vanishing criterion for the Burau representation of the 4-strand braid group. In the dual context, it is shown that the Burau representation for arbitrary braid index is injective when restricted to the set of simply-nested braids.

Prediction of Elastic Constants of Carbon Fabric/Polyurethane Composites

The purpose of this work is to study a new composite material consisting of polyurethane (PU) resin and carbon fiber fabric. This PU resin is superior in impact, viscosity, low curing temperature, and short curing time. If this resin is combined with fiber fabric by vacuum assisted resin transfer method, the fabrication time will be short. Since it is a braided composite, it’s important to have a model to predict the elastic constants for different braid angels. To predict the elastic constants including Young’s modulus, shear modulus, and Poisson’s ratio, a finite element model is established. In this model a braided layer is treated as two uni-directional layers. Then, the elastic constants of this composite with different braid angels are estimated. After that, the composites with different braid angels are fabricated and tested to obtain the elastic constants, and the comparison with the finite element results is made. The results indicate that the agreement is very good for the Young’s modulus. For the Poisson’s ratio, the difference between the prediction and the measurement is reasonable. From the comparison, it can be concluded that the finite element model is good. Then, this model is used to predict all in-plane elastic constants for arbitrary braid angles.

On Cohen braids

For a general connected surface M and an arbitrary braid α from the surface braid group Bn−1(M), we study the system of equations d1β = … = dnβ = α, where the operation di is the removal of the ith strand. We prove that for M ≠ S2 and M ≠ ℝP2, this system of equations has a solution β ∈ Bn(M) if and only if d1α = … = dn−1α. We call the set of braids satisfying the last system of equations Cohen braids. We study Cohen braids and prove that they form a subgroup. We also construct a set of generators for the group of Cohen braids. In the cases of the sphere and the projective plane we give some examples for a small number of strands.

Pseudocharacters of braid groups and prime links

Pseudocharacters of groups have recently found an application in the theory of classical knots and links in R 3 . More precisely, there is a relationship between pseudocharacters of Artin's braid groups and the properties of links repre- sented by braids. In the paper, this relationship is investigated and the notion of kernel pseudocharacters of braid groups is introduced. It is proved that if a kernel pseudocharacter φ and a braid β satisfy |φ(β)| >C φ ,w hereCφ is the defect of φ ,t henβ represents a prime link (i.e., a link that is noncomposite, nonsplit, and nontrivial). Furthermore, the space of braid group pseudocharacters is studied and a way is described to obtain nontrivial kernel pseudocharacters from an arbitrary braid group pseudocharacter that is not a homomorphism. This makes it possible to employ an arbitrary nontrivial braid group pseudocharacter for the recognition of prime knots and links.

Criteria of prime links in terms of pseudo-characters

Pseudo-characters of groups have recently found applications in the theory of classical knots and links in ℝ3. More precisely, there is a connection between pseudo-characters of Artin’s braid groups and properties of links represented by braids. In the present work, this connection is investigated and the notion of kernel pseudo-characters of braid groups is introduced. It is proved that a kernel pseudo-character ϕ and a braid β satisfy Ιϕ(β)І > C ϕ, where C ϕ is the defect of ϕ, then β represents a prime link (i.e., a link that is noncomposite, nonsplit, and nontrivial). Furthermore, the space of braid group pseudo-characters is studied and a way to obtain nontrivial kernel pseudo-characters from an arbitrary braid group pseudo-character that is not a homomorphisrn is described. This allows one to use an arbitrary nontrivial braid group pseudo-character for recognition of prime knots and links. Bibliography: 17 titles.

On presentations of generalizations of braids with few generators

In his initial paper on braids, E. Artin gave a presentation with two generators for an arbitrary braid group. We give analogs of Artin’s presentation for various generalizations of braids.

On Wick algebras with braid relations

The general theory of deformed commutation relations for creation and annihilation operators of particles equipped with arbitrary braid statistics (Ψ-statistics) is studied in terms of Wick algebras. The construction and examples of such algebras are given.

On braid statistics and noncommutative calculus

The particle system equipped with an arbitrary braid statistics is considered. The noncommutative differential calculus is used in order to describe the second quantization of a given system.