Keywords: woven fabric, thermoplastic composites, thermoforming, finite element analysis.

An implicit finite element method is developed to analyze the thermoforming process of thermoplastic laminates reinforced with woven fabric. Since the deformation process is perform at melting temperature of thermoplastic resin, the reinforced laminates are regarded as a viscous material reinforced with the woven continuous fibres, while the elastic behaviour was consider for melted resin in recent research works to simplify the deformation analysis. Due to the high axial stiffness of the continuous fibres compared to the melted resin and the high ratio of bulk to shear viscosity at forming temperature [1,2], the kinematical constraints of fibre inextensibility and material incompressibility are considered in the formulations presented. To eliminate the accumulation of computational errors in each time increment, the kinematical constraints are defined according to a reference configuration. Unlike the quasi-static assumption used in some previous investigations to model the thermoforming process, the deformed geometry is calculated according to the current laminate properties at the end of each increment. Using an appropriate solution method, the computed deformations satisfy the kinematical constraints and tool contact conditions in an exact manner.

In the thermoforming process of thermoplastic reinforced laminates, the material position, fibre orientations and tool contact boundary conditions are time-dependent. Therefore, the total forming process is divided into a number of increments and the calculated deformed configuration for a single time step is used as an input for the next time step. To model the behaviour of reinforced thermoplastic laminates, the resin and fibres are considered as a homogenized anisotropic material and the woven fabric thermoplastic composites have been modelled as an incompressible viscous material constrained along two inextensible directions [3]. The finite element formulation presented defines a set of nonlinear equations calculating the material deformation in terms of the current laminate properties and boundary conditions. The deformed geometry is calculated using an iterative procedure in which the convergence criteria are met when the nodal displacement satisfies the momentum equation and kinematical constrains, simultaneously, while the maximum value of displacement modification is smaller than a predefine tolerance.

In the present research work, a user subroutine is developed to implement the finite element formulation to simulate thermoforming process in the Abaqus finite element code. The user subroutine is called to define the type of element, update material positions, define initial and deformed fibre orientations, calculate the Jacobean matrix, and the internal, kinematical and residual forces. Using the Abaqus code, the linear equation system defined by Jacobean matrix and residual force are solved to determine the incremental deformation in a specific increment using an iterative procedure.

The comparison with the analytical solutions and previous numerical results shows that the present method provides an accurate and efficient procedure for the thermoforming analysis of reinforced thermoplastic laminates. The finite element procedure developed is used to analyze the picture-frame experiment and the results are compared to the analytical solution and the quasi-static method. It is shown that the optimum value for the penalty number is times the viscosity of the thermoplastic resin for which the numerical results are in a good agreement with the analytical solution. Using the quasi-static assumption, the results obtained in one increment highly deviate from analytical solution and the quasi-static finite element method converges to an accurate result for a high number of increments. The reason for the inaccuracy in the quasi-static method is the assumption of fixed fibre orientations during each increment that contradicts with real conditions. Since the present nonlinear finite element procedure considers the variations of fibre orientations, it provides accurate results in a single increment. Therefore, the present numerical method reduces the required number of increments to obtain the accurate results compared to the previous numerical method using the quasi-static assumption. The hemispherical forming process is also analyzed using the present method and the results are evaluated in the regions where the woven fabric reinforced laminates have a high potential to wrinkle.

1

M. Tahaye Abadi, H.R. Daghyani , Sh. Fariborz, "Experimental Wrinkling Analysis in Stamp Forming of Continuous Fibre Reinforced Thermoplastic Sheets", International Congress in Manufacturing Engineering, Tehran, Iran, 2005.

2

S.G. Advani, T.S. Creasy and S.F. Shuler, "Rheoelogy of Long Fibre-Reinforced Composites in Sheet Forming", Composite sheet material, Composite material series, Vol. 11, editor D. Bhattacharyya, Elsevier Pub., 1997.

3

A.J.M. Spencer, "Theory of Fabric-Reinforced Viscose Fluid", Composite Part A, Vol. 31, 1311-1321, 2000. doi:10.1016/S1359-835X(00)00006-3

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