Teik-Cheng Lim

Modelling of composite sheet forming: a review

Abstract The use of composite materials in sheet forming applications is gaining popularity with the rise of consumer demands and specific mechanical properties. In addition to unidirectional (UD) fibres, the use of textile reinforcements such as woven fabric and knitted fabric has been shown to be feasible in recent years. This paper gives a survey on the modelling of composite sheet forming for both UD fibre and textile composites. Two broad approaches are reviewed here—the mapping approach and the mechanics approach. Mapping approaches for UD fibre composites, woven fabric composites and knitted fabric composites are elucidated on the basis of their fibre geometry. For the mechanics approach both the viscous fluid models and elastic solid models, as a means of describing the constitutive properties, are reviewed. Various updating methods for modelling large deformation found in sheet forming are then described. Finally, a guideline for the choice of modelling techniques for various types of fibre/fabric reinforcements and suggestions for future work are given.

Axisymmetric sheet forming of knitted fabric composite by combined stretch forming and deep drawing

Previous studies have demonstrated that sheet formability is governed by the interlacing effect of pure stretch forming and pure deep drawing. In this paper, knitted fabric reinforced thermoplastic composite sheets, which exhibit excellent stretchability and drapeability, are investigated for their formability. Initially a new parameter, X, which describes the amount of stretching relative to drawing during forming is introduced. It is shown that sheet forming processes, specified by the use of various combinations of forming conditions, can be characterized by the parameter X. Finally, the overall formability of knitted fabric composite sheet is discussed with reference to the X-factor.

Connection among classical interatomic potential functions

The four classical interatomic pair potentials of Lennard-Jones [Proc. Roy. Soc. Lond. A 106 (1924) 463], Morse [Phys. Rev. 34 (1929) 57], Rydberg [Z. Phys. 73, (1931) 376] and Buckingham [Proc. Roy. Soc. Lond. A 168 (1938) 264] have been widely adopted in molecular force fields of computational chemistry softwares and also incorporated into 2-body potential energy description in many-body solid-state physics. This paper demonstrates how mathematical methods such as infinite series expansions can be used to connect parameters across these interatomic potential functions.

Connection between parameters of the Murrell-Sorbie and Fayyazuddin potentials

A connection between the Murrell–Sorbie and the Fayyazuddin potential energy functions were established by obtaining conversion matrices that convert the formers parameters into the latter and vice versa. This was attained by rearranging the Fayyazuddin potential into another, but equivalent, form comparable to the Murrell–Sorbie function with the aid of series expansion. A list of 71 sets of Fayyazuddin diatomic parameters were generated by applying one of the conversion matrices on the Huxley–Murrell data [J. Chem. Soc. Faraday Trans. II 79, 323 (1983)]. Verification of the developed conversion matrices was performed by plotting the Murrell–Sorbie and the Fayyazuddin functions using the Huxley–Murrell data and the converted data, respectively. Potential energy plots of typical (OSi) and extreme (FO, BeS and HH) diatomic parameters exhibit very good agreement between the two potential functions considered, thereby confirming the conversion matrices’ validity. Slight discrepancies of the plotted diatomic energy curves further suggest the comparative suitability of the Fayyazuddin potential for curve-fitting harder, rather than softer, bonds.

The Relationship between Lennard-Jones (12-6) and Morse Potential Functions

The most commonly used potentials for van der Waals interactions are the Exponential-6 and the Lennard-Jones (12-6) potential. In this paper a correlation between them is described. The Morse function, which is normally applied for quantifying 2-body interactions, has been adopted in one software. This paper deals with the validity of the Morse function for non-bonded interactions by means of obtaining a relationship between the Morse and the Lennard-Jones (12-6) potential functions. An approximate and an exact mathematical relationship is demonstrated to exist between these two potentials.

Application of Maclaurin series in relating interatomic potential functions: A review

This paper provides a short review on the application of Maclaurin series in relating potential functions within the same category of interatomic interaction. The potential functions covered are those commonly adopted in computational chemistry softwares. While various mathematical approaches have been employed in generating relationships amongst potential functions, the use of Maclaurin series has been prevalent recently due to the increasing application of polynomial series-type potential functions. In the case of covalent bond-stretching, the Maclaurin series for the exponential function is used to transform the Morse potential into the polynomial series potential, and vice versa. For covalent bond-bending, Maclaurin series for the sine and cosine functions were employed to extract polynomial angle series potential from the Fourier series and harmonic cosine potential functions, and vice versa. Finally, both the exponential and the 1/(1 − x) expressions in Maclaurin series were used in obtaining the exact relationship for the repulsive terms between two potential functions.

Alignment of Buckingham Parameters to Generalized Lennard-Jones Potential Functions

Abstract The Lennard-Jones(12-6) and the Exponential-6 potential functions are commonly used in computational softwares for describing the van der Waals interaction energy. Some softwares allow switching between these two potentials under prescribed condition(s) that attempt to connect the parameter relationship between the two functions. Here we propose a technique by which the parameter relationship between both potentials is extracted by simultaneously imposing an equal force constant at the well depth’s minimum and an equal mean interatomic energy from the point of equilibrium to the point of total separation. The former imposition induces good agreement for the interatomic compression and a small change in the interatomic distance near the equilibrium while the latter enables good agreement for large interatomic separation. The excellent agreement exhibited by the plots validates the technique of combined criteria proposed herein

Simplified Model for the Influence of Inclusion Aspect Ratio on the Stiffness of Aligned Reinforced Composites

A simplified model for the Young’s Modulus of aligned composites possessing spheroidal inclusions, of any aspect ratio α, is developed herein based on a triply periodic array of parallelpiped inclusions. The Young’s Modulus of continuous unidirectional fiber composites (α→∞), particulate composites (α = 1) and periodically bilaminated composites (α = 0) are obtained as limiting cases based on the Parallel-Series and Series-Parallel schemes of analyses. A reinforcing parameter, R, of the inclusion is then identified from the limiting cases such that R ∊ [0,1]. By curve-fitting for a relationship between the reinforcement parameter and the inclusion aspect ratio for α∊ [0,∞], the Young’s Modulus for aligned short fiber composites (α > 1) and aligned flake composites (α < 1) are obtained. For verification, the present model at limiting cases (α =0, 1, ∞) were reduced to other simple models and/or compared with experimental results. For nonlimiting cases (α <1, α > 1), comparison with other theoretical models show reasonable agreement.

Relationship between the 2-body Parameters of the Biswas-Hamann and the Bauer-Maysenholder-Seeger Potential Functions

A theoretical relation between the Bauer-Maysenholder-Seeger (BMS) and the Biswas-Hamann (BH) potential function has been developed herein for the case of 2-body interaction. By equating derivatives of these two potential functions for the 2-body part, a loose form of BMS is expressed in terms of BH parameters with a scaling factor. The validity of the developed parametric relationships has been graphically demonstrated, and the suitability discussed with reference to the extent and direction of bond distortion.

Spring constant analogy for estimating stiffness of a single polyethylene molecule

The stiffness of a simple planar polymeric chain is modeled using analogies of mechanical springs arranged in series and in parallel assemblies. The stiffness of chemical bonds is resolved into two perpendicular axes defined by the longitudinal and transverse axes in the molecular plane. Using Hookes definition of spring stiffness, the molecular stiffness of polyethylene is obtained along the longitudinal and transverse directions. This paper demonstrates the use of physical analogies and mathematical approximations for obtaining an analytical form for the stiffness of a simple single-molecule.