Modeling of Thermoforming

At present there seems to have been no quantitative attempts to estimate the temperature at which a polymer is 

FIGURE 10.15 Dynamic mechanical thermal analysis (G versus temperature at an angular frequency of 1.0 rad/s) of polypropylene (PP) and PP filled with 20 wt% glass fiber determined using the torsional mode of a rheometer. 

Sometimes forced convection is used cyclically to cool the sheet surface and again the methods described in Chapter 5 can be used to estimate the heat transfer coefficient. The numerical procedure associated with Example 5.5 (p. 127) can be used for determining the heating time and temperature distribution in the sheet. 

Estimating the wall thickness distribution is extremely difficult especially for irregularly shaped parts. Numerical methods, such as the finite element method, are required. Furthermore, an appropriate nonlinear constitutive equation for the polymer is required, and this may be difficult to obtain for materials that exist in the near rubbery state at the forming temperature. Obtaining rheological data at the forming temperature is very difficult. Finally, the solution of the coupled equations of motion and the nonlinear constitutive equation requires sophisticated numerical codes, which are not readily available at this time. 

FIGURE 10.16 Relative dimensions of a container generated from a thermoformed sheet using a female mold. Before forming, the area of the section to be formed is 2 units. After forming, the surface area of the box is 8 units. 

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The plastic at its forming temperature can behave as an elastic solid, a viscoelastic fluid, or a combination of the two. Modeling of thermoforming has been done using all of these models. On a molecular scale, things are just as complex. For an amorphous material, such as polystyrene or PMMA, the forming temperature determines the chain mobility and the ease of flow. For semicrystalline materials, such... 

Karamanou, M., Warby, M.K. and Whiteman, J.R. (2006) Computational modelling of thermoforming processes in the case of finite viscoelastic materials. Computer Methods in Applied Mechanics and Engineering, 195, 5220. 

We can model the thermoforming process by examining a preheated sheet that is vacuum-formed directly onto the walls of a cavity. The softened sheet undergoes a... 

With viscoelastic models used by an increasing number of researchers, time and temperature dependence, as well as strain hardening and nonisotropic properties of the deformed parison can, in principle, be accounted for. Kouba and Vlachopoulos (97) used the K-BKZ viscoelastic constitutive equation to model both thermoforming and parison membrane stretching using two-dimensional plate elements in three-dimensional space. Debbaut et al. (98,99) performed nonisothermal simulations using the Giesekus constitutive equation. 

Molds for hand lay-up (as well as others such as spray-up, contact molding, thermoforming, and casting) are usually made of TS polyester or epoxy shell set in a cradle made of a material such as steel angle. They can usually be made in-house, on a model of the product that could be made from an inexpensive material that can be shaped or sculpted, such as plaster, balsa wood, or expanded polystyrene, sealed and coated with a release agent. 

H. DeLorenzi and H. F. Nied, Finite Element Simulation of Thermoforming and Blow Molding, in A. Isayev, ed.. Modeling of Polymer Processing, Hanser, Munich, 1991. 

The modeling of the inflation and solidification portions of blow molding and thermoforming do not introduce major new concepts beyond those incorporated in... 

Mathematical modeling can provide valuable insights into mold design and process improvement. The objective of computer simulation of thermoforming is... 

A large range of standard models, mono- or multi-stations, to thermoform sheets or films from 0.1 mm to 8 mm fed in rollers or sheets. 

Research carried out by the Queen s University of Belfast in the thermoforming of PP and PP foam is reviewed. Studies were made of the influence of crystallinity on thermoformability and the nse of chemical blowing agents in the production of foams having lower density and optimum cell stracture. Finite element modelling was nsed in the prediction of wall thicknesses and an optimised process control system was developed. BELFASUQUEEN S UNIVERSITY... 

The parison is inflated fast, within seconds or less, at a predetermined rate such that it does not burst while expanding. It is a complex process that involves expansion of a nonuniform membrane-like element. Because the extension ratio is high (above 10), it is difficult to calculate the final thickness distribution. Naturally, much of the recent theoretical research on parison stretching and inflation (as in the case with thermoforming) focuses on FEM methods and the selection of the appropriate rheological constitutive models to predict parison shape, thickness, and temperature distribution during the inflation. 

Petrie and Ito (84) used numerical methods to analyze the dynamic deformation of axisymmetric cylindrical HDPE parisons and estimate final thickness. One of the early and important contributions to parison inflation simulation came from DeLorenzi et al. (85-89), who studied thermoforming and isothermal and nonisothermal parison inflation with both two- and three-dimensional formulation, using FEM with a hyperelastic, solidlike constitutive model. Hyperelastic constitutive models (i.e., models that account for the strains that go beyond the linear elastic into the nonlinear elastic region) were also used, among others, by Charrier (90) and by Marckmann et al. (91), who developed a three-dimensional dynamic FEM procedure using a nonlinear hyperelastic Mooney-Rivlin membrane, and who also used a viscoelastic model (92). However, as was pointed out by Laroche et al. (93), hyperelastic constitutive equations do not allow for time dependence and strain-rate dependence. Thus, their assumption of quasi-static equilibrium during parison inflation, and overpredicts stresses because they cannot account for stress relaxation furthermore, the solutions are prone to numerical instabilities. Hyperelastic models like viscoplastic models do allow for strain hardening, however, which is a very important element of the actual inflation process. 

H. G. deLorenzi and H. F Nied, Blow Molding and Thermoforming of Plastics Finite Element Modeling, Compu. Struct. 26, 197-206 (1987). 

Modelling and analysis of composites thermoforming KINEMATIC APPROACH... 

Scherer, R. (1995) Thermoforming of unidirectional continuous fibre-reinforced polypropylene laminates and their modeling, in Polypropylene Structure, Blends and Composites, Vol. 3, Chap. 8, (ed. J. Karger-Kocsis), Chapman Hall, London, pp. 293-315. 

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