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For warpage analysis, nodal displacements and curvatures for shell elements are expressed as:
where [k] is the element stiffness matrix, [Be] is the derivative operator matrix, {d} is the displacements, and {re} is the element load vector which can be evaluated by:
The use of a full three-dimensional FEM analysis can achieve accurate warpage results, however, it is cumbersome when the shape of the part is very complicated. In this paper, a two dimensional FEM method, based on shell theory, was used because most injection-molded parts have a sheet-like geometry in which the thickness is much smaller than the other dimensions of the part. Therefore, the part can be regarded as an assembly of flat elements to predict warpage. Each three-node shell element is a combination of a constant strain triangular element (CST) and a discrete Kirchhoff triangular element (DKT), as shown in Fig. 3. Thus, the warpage can be separated into plane-stretching deformation of the CST and plate-bending deformation of the DKT, and correspondingly, the element stiffness matrix to describe warpage can also be pided into the stretching-stiffness matrix and bending-stiffness matrix.
Fig. 3a–c. Deformation decomposition of shell element in the local coordinate system. a In-plane stretching element b Plate-bending element c Shell element
3 Experimental validation
To assess the usefulness of the proposed model and developed program, verification is important. The distortions obtained from the simulation model are compared to the ones from SL injection molding experiments whose data is presented in the literature [8]. A common injection molded part with the dimensions of 36×36×6 mm is considered in the experiment, as shown in Fig. 4. The thickness dimensions of the thin walls and rib are both 1.5 mm; and polypropylene was used as the injection material. The injection machine was a production level ARGURY Hydronica 320-210-750 with the following process parameters: a melt temperature of 250 ◦C; an ambient temperature of 30 ◦C; an injection pressure of 13.79 MPa; an injection time of 3 s; and a cooling time of 48 s. The SL material used, Dupont SOMOSTM 6110 resin, has the ability to resist temperatures of up to 300 ◦C temperatures. As mentioned above, thermal conductivity of the mold is a major factor that differentiates between an SL and a traditional mold. Poor heat transfer in the mold would produce a non-uniform temperature distribution, thus causing warpage that distorts the completed parts. For an SL mold, a longer cycle time would be expected. The method of using a thin shell SL mold backed with a higher thermal conductivity metal (aluminum) was selected to increase thermal conductivity of the SL mold.
Fig. 4. Experimental cavity model
Fig. 5. A comparison of the distortion variation in the X direction for different thermal conductivity; where “Experimental”, “present”, “three-step”, and “conventional” mean the results of the experimental, the presented simulation, the three-step simulation process and the conventional injection molding simulation, respectively.
Fig. 6. Comparison of the distortion variation in the Y direction for different thermal conductivities
Fig. 7. Comparison of the distortion variation in the Z direction for different thermal conductivities
Fig. 8. Comparison of the twist variation for different thermal conductivities
For this part, distortion includes the displacements in three directions and the twist (the difference in angle between two initially parallel edges). The validation results are shown in Fig. 5 to Fig. 8. These figures also include the distortion values predicted by conventional injection molding simulation and the three-step model reported in [3].
4 Conclusions
In this paper, an integrated model to accomplish the numerical simulation of injection molding into rapid-prototyped molds is established and a corresponding simulation system is developed. For verification, an experiment is also carried out with an RP fabricated SL mold.