Meso-scale compaction simulation of multi-layer 2D textile reinforcements: A Kirchhoff-based large-strain non-linear elastic constitutive tow model

Abstract

This paper presents a non-linear, large strain constitutive approach to modelling the mechanics of fibrous tows. The isotropic and linear St. Venant hyperelastic model was modified to extend its applicability to a transversely isotropic continuum. From this model, a set of large-strain material parameters which resemble the linear-elastic moduli was derived. The non-linear constitutive response of fibrous tows was accounted for through a modelling scheme that incorporates the instantaneous microstructural arrangement of the tows. Tow samples were isolated from the studied textile roll and subjected to a compression study utilising a custom-made tow compression rig, from which a set of material functions that fully defined the constitutive model were generated. The developed model was then implemented as a material subroutine and utilised in a mesoscopic compaction simulation of a multi-layer 2D woven reinforcement. It was found that the constitutive model performed very well in predicting the shape changes and stresses in deforming tows.

Introduction

Liquid Composite Moulding (LCM) encompasses a family of closed-mould composite manufacturing techniques with a general methodology that can be divided into preforming of a dry textile reinforcement, consolidation of the preform to the designed fibre volume fraction, injection and subsequent cure of a low viscosity liquid thermoset resin, and ejection of the cured composite part [1]. During the consolidation step, a textile preform is subjected to a combination of deformation modes, with transverse compaction being the most significant [2]. This through-thickness loading significantly alters the preform geometry, spatial placements, and overall fibre volume fraction, which ultimately influences its injectability and cured properties. To anticipate these changes, studies to realistically replicate the textile compaction step are of importance. The predicted compacted textile geometries can be used as the inputs to simulations that predict the preform permeability [3], [4] and the homogenised mechanical properties of the cured composite system [5], [6].

When considering interlaced fibrous reinforcements (e.g. woven, braided, etc.), a transverse textile compaction study can be executed at different length scales, owing to the fabrics’ hierarchical nature: micro (fibre-level), meso (tow-level), and macro (part-level). At the mesoscopic level, each tow is approximated as a transversely isotropic continuum [7], and the interaction between adjacent tows is treated as a type of contact problem. This length scale provides a balance between detailed representation of textile geometry and computational efficiency. An issue which arises when adopting this length scale is the need to define a constitutive model which can accurately describe the tow mechanics. Over the past decades, various tow models have been proposed for mesoscopic consolidation studies. Very early works by Kawabata et al. [8], [9] in the early 1970s modelled the response of woven fabrics under various in-plane loadings by treating each tow as a chain of two-line elements with linear elastic behaviour and no bending resistance. In this way, the tows were effectively treated as a system of trusses with frictionless joints. The elastic beam theory later emerged as a popular means to represent the tow mechanics. Various analytical and semi-empirical equations to describe fabric compression have been derived [10], [11] based on this theory. Further developments in the application of elasticity theory led to the textile modelling software, WiseTex [12], [13], which determines the stable configuration of a textile unit cell by finding the solution to a minimum bending energy problem through an iterative procedure.

With the advancement of computing power, Finite Element Analysis (FEA) has become the primary tool to perform mesoscopic consolidation studies. This tool opens up the possibility to implement more advanced 3D constitutive tow models. The simple linear-elastic model has traditionally been used in various FEA-based consolidation simulations [14], [15], [16]. Through a set of constant elastic moduli (i.e. Young’s moduli, Poisson’s ratios, and shear moduli), the mechanics of a linear-elastic material is fully defined. These moduli have a well-defined physical significance; thus, their experimental characterisation is reasonably straight-forward. However, in real textile consolidation processes, the tows undergo significant displacements and rotations which violate the core assumptions of linear elasticity. To address these limitations, some works [7], [17], [18] have adopted a hypoelastic model, a form of constitutive model which relates the objective Cauchy stress rate tensor with the rate of deformation tensor. Despite its widespread implementation, this model is not consistent in the sense that a zero net work is not guaranteed when a continuum is subjected to a closed-loop loading path. A hyperelastic model, on the other hand, is free of these limitations. This model assumes the existence of a strain energy function, which is a measure of the potential energy stored at a material point due to elastic deformations. The stress-strain relationship of a hyperelastic material is entirely governed by its strain energy function, and therefore, it is imperative to formulate the appropriate mathematical expression of this energy function, which, however, is not readily apparent. For this reason, the treatment of a tow as a hyperelastic continuum is not yet commonplace.

Some examples of the implementation of hyperelastic tow models can be seen in [19], [20]. Charmetant et al. [20] decomposed the tow deformation into four separate modes: elongation in the fibre direction, compaction in the transverse section, shear in the transverse section, and shear along the fibre direction, each assigned a unique (elementary) strain energy function. The mathematical forms of the strain energy functions associated with the elongation (polynomial) and transverse compaction (power-law) were found from fitting experimental data, whereas the forms related to the remaining modes were assumed to be quadratic with the associated constants quantified via an inverse method. The model was based on the assumptions that the four deformation modes were independent and that the strain energy was a linear combination of the elementary strain energy functions. The developed constitutive law was implemented in a mesoscopic in-plane shearing simulation of a single-layer plain-weave unit cell where a good comparison with experimental data was reported.

Recognising the simplicity of linear-elasticity and the suitability of hyperelastic theory for large deformation problems, this paper proposes a simple approach to large-strain constitutive modelling that utilises a set of material moduli with physical significance resembling that of the well-known linear-elastic moduli. The constitutive model accounts for the generally assumed transversely isotropic nature of fibrous tows, and is able to describe the shape changes and stresses of deforming tows within a transversely compacted textile assembly. First, the development of the constitutive model is described. The St. Venant hyperelastic model [21], [22], which is a simple extension of the isotropic linear-elastic model to the large deformation regime, is extended for application to transversely isotropic hyperelastic materials, according to the methodology presented by Bonet et al. [23]. A constant material elastic tangent stiffness tensor is derived from this model’s strain energy function, which resembles that of the linear elasticity theory. From this stiffness tensor, quantities analogous to the familiar linear-elastic moduli are derived, which naturally couple the different deformation modes of the tow. Second, a comprehensive custom-designed tow compaction procedure to experimentally quantify the constitutive model parameters is presented. Finally, a truly tow-scale comparison method to validate the tow model is presented: the constitutive model is implemented in a mesoscopic compaction simulation in which the unconsolidated textile FEA geometry is derived directly from a µCT-scan of 4-layer plain weave sample. From the same sample, µCT-scans corresponding to different compaction levels are available and are compared to the compacted models.