# Geometrical and Mechanical Modeling of Polymeric Multi-Ply Yarns
## 1 Introduction  
Twisted polymeric multi-ply yarns, also called cords, are increasingly used as reinforcements embedded in rubber compounds to form the composite layers that make up the structure of pneumatic tires [1]. Different polymeric cord materials as well as cord constructions can be employed as tire reinforcements, and many experimental investigations have been conducted over the years to explore the mechanical behavior of the various cord types. In [2], the viscoelastic properties of rayon, nylon and PET cords are studied under different temperature and humidity conditions, and considering different geometric structures (number of plies, twist levels) of the reinforcements. Tests on nylon/PET hybrid cords [3] and on aramid/nylon hybrid cords [4] have been also carried out and discussed. Experimental studies on the tensile behavior of aramid and polyamide cords are reported in [5]. Cyclic tests on nylon66 cords are described in detail in [6]. Test results on rayon greige and dipped cords are described later on in this work; they have recently been obtained within an experimental campaign that complements the one reported in [7, 8]. As for the mechanical modeling of cords, intensive research has been conducted on this topic for many years. Various approaches with different levels of sophistication have been proposed. The cord modeling presents many difficulties related to the manifestation of geometrical and material non-linearities, and to the lack of precise information about the initial geometrical arrangements of the fibers. For this reason, in some of the approachesproposed for the modeling of cord-rubber composites, cords are represented in a simplified way without considering their actual filament-based twisted structure. This is exemplified for instance in [9] and in [10] where 3D FE analyses of fiber-reinforced composites are developed, with cords represented by rebar layers containing Gauss points at which the cord material, assumed in a uniaxial stress state, follows a hyperelastic law. Other more complex approaches for the simulation of composites in finite strains have been developed more recently, in which the twisted fiber geometry of the reinforcements is explicitly considered together with advanced constitutive modeling for both the elastomer matrix and the cord material. A representative study of this type is [11] in which a preliminary FE simulation is performed to obtain a realistic initial geometry of the cord filaments and an elasto-plastic cord behavior is assumed. In [12], a flax-fiber/epoxy composite is analyzed and an anisotropic model for fibers is developed. In [13], the present authors, in the framework of the thermodynamics of irreversible processes [14], developed a new transversely isotropic viscoelastic-viscoplastic constitutive law intended to model fibrous materials. The model was applied to describe the mechanical behavior of single-ply yarns, replacing their real geometry with an equivalent three-dimensional continuum of cylindrical shape. The material direction of transverse isotropy was fixed considering the filaments arranged as concentric helices, with a local tangent unit vector \(\mathbf{a}\). The non-linearity of the uniaxial tensile force-strain response of the yarn was then attributed to the combined effect of twist and of the viscoplastic behavior of the filament material. This model, at difference from the hyperelastic constitutive laws (e.g., the well known [PERSON]'s and [PERSON]'s ones [15; 16]), allows prediction of the experimental cyclic behavior with permanent strains as well as the effect of twist. The purpose of the present work is to develop a model able to predict the complex inelastic behavior of multi-ply yarns accounting for the fiber orientation. To this end, we extend the above approach to multi-ply yarns by developing a proper strategy to define an equivalent virtual fiber orientation at each point of the cord, to fix the material direction of transverse isotropy.