Finite-Element Model

Here we construct the finite-element model of Eq. (13) for the two-dimensional case (see Reddy 7)). Let each displacement increment be approximated as 

since Eq. (13) is a linearized version of Eq. (11), the error introduced into the calculation of the displacements between configurations can drift the solution away from the true solution (especially, if the load steps are large). Therefore, a correction should be made to the displacements at each load step. This can be done as follows The solution A of Eq. (15) allows us [with the aid of Eq. (3)] to compute the total displacements at time (t + At), 

Equations (18) and (19) correspond to the Newton-Raphson iteration. If the left-hand side (i.e., [ ]) is not updated during the iteration, the iterative scheme is known as the modified Newton-Raphson iteration. 

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Several types of experiments have been carried out to investigate the stress state in the head of the bolt created by the body forces. The results of the finite element model experiment can be seen in Fig. 2, and those of the optical plane model experiment are presented in Fig. 3. 

In order to describe inherited stress state of weldment the finite element modelling results are used. A series of finite element calculations were conducted to model step-by-step residual stresses as well as its redistribution due to heat treatment and operation [3]. The solutions for the reference weldment geometries are collected in the data base. If necessary (some variants of repair) the modelling is executed for this specific case. 

Finite Element Modelling of Polymeric Flow Processes... 

FINITE ELEMENT MODELLING OF POLYMERIC FLOW PROCESSES 3.1.1 The U-V-P scheme... 

FINITE ELEMENT MODELLING OF POLYMERIC FLOW PROCESSES... 

Descriptions given in Section 4 of this chapter about the imposition of boundary conditions are mainly in the context of finite element models that use elements. In models that use Hermite elements derivatives of field variable should also be included in the set of required boundai conditions. In these problems it is necessary to ensure tluit appropriate normality and tangen-tiality conditions along the boundaries of the domain are satisfied (Petera and Pittman, 1994). 

An example describing the application of this algorithm to the finite element modelling of free surface flow of a Maxwell fluid is given in Chapter 5. 

Petera, J. and Nassehi, V., 1996. Finite element modelling of free surface viscoelastic flows with particular application to rubber mixing. Int. J. Numer. Methods Fluids 23, 1117-1132. 

The majority of polymer flow processes are characterized as low Reynolds number Stokes (i.e. creeping) flow regimes. Therefore in the formulation of finite element models for polymeric flow systems the inertia terms in the equation of motion are usually neglected. In addition, highly viscous polymer flow systems are, in general, dominated by stress and pressure variations and in comparison the body forces acting upon them are small and can be safely ignored. 

The majority of polymer flow processes involve significant heat dissipation and should be regarded as nou-isothermal regimes. Therefore in the finite element modelling of polymeric flow, in conjunction with the equations of continuity... 

Keeping all of the flow regime conditions identical to the previous example, we now consider a finite element model based on treating silicon rubber as a viscoelastic fluid whose constitutive behaviour is defined by the following upper-convected Maxwell equation... 

I liis simulation provides the quantitative measures required for evaluation of the extent of deviation from a perfect viscometric flow. Specifically, the finite element model results can be used to calculate the torque corresponding to a given set of experimentally determined material parameters as... 

Finite element modelling of flow distribution in an extrusion die... 

In Figure 5.23 the finite element model predictions based on with constraint and unconstrained boundary conditions for the modulus of a glass/epoxy resin composite for various filler volume fractions are shown. 

Nassehi, V., Kinsella, M. and Mascia, 1.., 1993b. Finite element modelling of the stress distribution in polymer composites with coated fibre interlayers. J. Compos. Mater. 27, 195-214. 

Nassehi, V. and Pittman,. 1. F. T., 1989. Finite element modelling of flow distribution in an extrusion die. In Bush, A. W., Lewis, B.A. and Warren, M.D. (eds), Flow Modelling in Industrial Processes, Chapter 8, Ellis Horwood, Chichester. 

Petera, J. and Nassehi, V., 1995. Use of the finite element modelling technique for the improvement of viscometry results obtained by cone-and-plate rheometers. J. Non-Newtonian Fluid Mech. 58, 1-24. 

As mentioned in Chapter 2, the numerical solution of the systems of algebraic equations is based on the general categories of direct or iterative procedures. In the finite element modelling of polymer processing problems the most frequently used methods are the direet methods. 

Iterative solution methods are more effective for problems arising in solid mechanics and are not a common feature of the finite element modelling of polymer processes. However, under certain conditions they may provide better computer economy than direct methods. In particular, these methods have an inherent compatibility with algorithms used for parallel processing and hence are potentially more suitable for three-dimensional flow modelling. In this chapter we focus on the direct methods commonly used in flow simulation models. 

PRACTICAL ASPECTS OF FINITE ELEMENT MODELLING OF POLYMER PROCESSING... 

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