S. A. Grishanov

The Simulation of the Geometry of a Two-component Yarn Part II: Fibre Distribution in the Yarn Cross-section

This paper considers the distribution of fibres in yarns and discusses the development of a model for the simulation of fibre distributions; the concept of ‘virtual location’ within the cross-section domain, which may contain a fibre, is introduced. The distribution of fibres in real yarns is compared with computer-generated distributions. The distributions predicted by using the model demonstrate all of the properties of the real yarns considered.The change in the fibre distribution caused by compressive forces is considered, and a model to describe the behaviour of real yarns is proposed.

Modelling of Two-component Yarns Part I: The Compressibility of Yarns

This paper considers the compression properties of yarns. A model is proposed which predicts the behaviour of yarns in compression; the model takes account of the structural characteristics of yarns and the finite dimensions of fibres. A device is described which has been successfully used to measure the effects of compression forces on the dimensions of yarn cross-sections.A comparison between the theoretical behaviour of yarns and experimental results demonstrates the good predictive ability of the theory.

A Topological Study of Textile Structures. Part I: An Introduction to Topological Methods

This paper proposes a new systematic approach for the description and classification of textile structures based on topological principles. It is shown that textile structures can be considered as a specific case of knots or links and can be represented by diagrams on a torus. This enables modern methods of knot theory to be applied to the study of the topology of textiles. The basics of knot theory are briefly introduced. Some specific matters relating to the application of these methods to textiles are discussed, including enumeration of textile structures and topological invariants of doubly-periodic structures.

DOUBLY PERIODIC TEXTILE STRUCTURES

Knitted and woven textile structures are examples of doubly periodic structures in a thickened plane made out of intertwining strands of yarn. Factoring out the group of translation symmetries of such a structure gives rise to a link diagram in a thickened torus, as in [2]. Such a diagram on a standard torus in S3 is converted into a classical link by including two auxiliary components which form the cores of the complementary solid tori. The resulting link, called a kernel for the structure, is determined by a choice of generators u, v for the group of symmetries. A normalized form of the multi-variable Alexander polynomial of a kernel is used to provide polynomial invariants of the original structure which are essentially independent of the choice of generators u and v. It gives immediate information about the existence of closed curves and other topological features in the original textile structure. Because of its natural algebraic properties under coverings we can recover the polynomial for kernels based on a proper subgroup from the polynomial derived from the full symmetry group of the structure. This enables two structures to be compared at similar scales, even when one has a much smaller minimal repeating cell than the other. Examples of simple traditional structures are given, and their Alexander data polynomials are presented to illustrate the techniques and results.

RECOGNIZING TEXTILE STRUCTURES BY FINITE TYPE KNOT INVARIANTS

Typical examples of textile structures are separated by finite type invariants of knots in non-trivial (in particular, non-orientable) manifolds. A new series of such invariants is described.

Modelling the load-extension behaviour of plain-knitted fabric part I: a unit-cell approach towards knitted-fabric mechanics

The objective of the study reported in this paper was to generate new models of the mechanical behaviour of knitted fabrics in quasi-static deformation from an initially relaxed state to the extended state. The problem of extension of a knitted structure is complicated by the combination of non-linear properties derived from both the characteristics of the structure and the properties of the yarn. A model of in-plane deformation of a plain-knitted structure is proposed. To facilitate study of the mechanical properties of the proposed model, standard dimensional parameters of a fabric combined with the geometry and mechanical properties of a yarn are used. With the purpose of obtaining important yarn characteristics for the subsequent evaluation of the model, an advanced analysis of the yarn path in plain-knitted fabric is performed. An algorithm for evaluating the loop geometry from the given fabric dimensions in both course and wale directions and the yarn properties is developed. Kinematic relationships in tensioned fabric are considered. A finite-element approach to the problem of modelling knitted-fabric deformation is proposed in which the knitted structure is presented as a 2D mesh of unit cells.

Kauffman-type polynomial invariants for doubly periodic structures

A two-variable polynomial invariant of non-oriented doubly periodic structures is proposed. A possible application of this polynomial for the classification of textile structures is suggested.

A Topological Study of Textile Structures. Part II: Topological Invariants in Application to Textile Structures

This paper is the second in the series on topological classification of textile structures. The classification problem can be resolved with the aid of invariants used in knot theory for classification of knots and links. Various numerical and polynomial invariants are considered in application to textile structures. A new Kauffman-type polynomial invariant is constructed for doubly-periodic textile structures. The values of the numerical and polynomial invariants are calculated for some simplest doubly-periodic interlaced structures and for some woven and knitted textiles.

An Application of Queuing Theory to Modeling of Melange Yarns Part I: A Queuing Model of Melange Yarn Structure

A queuing model of staple fiber yarn is presented that enables the modeling and a better understanding of fiber migration in a yarn. The model provides a fine yarn structure where the migrational behavior of fibers is associated with the behavior of customers traveling across an open network of queuing systems to get services. Based on this analogy, the underlying mathematical foundation of the queuing theory is used for the modeling of yarn structure and properties. The model uses yarn technical specifications including yarn linear density and twist level, fiber linear density and length distribution, together with specific parameters such as fiber packing density distribution and migration probabilities. The model can be used for modeling a wide range of structurally different yarns; examples include marl, mottle and melange yarns, yarns with different levels of hairiness, and yarns produced by various spinning systems. The model can be used for 3D simulation of yarns in computer-aided design systems for textile design and for the prediction of mechanical properties of yarns.

Invariants of links in 3-manifolds and splitting problem of textile structures

An infinite family of invariants of multicomponent links in 3-manifolds is introduced and used to prove the non-splitting and non-equivalence of textile structures.