∇σ F (35)
and where the relation between the stress and strain is (assuming small deformations):
(
ε ε α T)
σFC
σ= − 0− ∆ + (36)
(
T2
1 u u
ε= ∇ +∇
)
(37)where C is the forth-order constitutive stiffness tensor related to the elastic properties of the material, ε is the infinitesimal elastic strain tensor, ε0 is the initial strain tensor due to volumetric shrinkage, α is a tensor describing the linear thermal expansion, σF is initial stress components induced by flow and u is the displacement vector. The data from the residual stress and temperature profile in the molded part serve as initial conditions for the Eq.35-37.
The displacements of the nodes are obtained by a using a finite element method and finally, the displacements are used to calculate the shrinkage, warpage and sink marks of the part.
3.3 Literature survey
The shrinkage and warpage theory described in the previous subchapters is mainly influenced by textbooks in the area of injection molding but much research, which is not reported in these books, has been carried out in this area. This chapter will focus on literature discussing shrinkage and warpage related to injection molding.
Corner deformation is an important phenomenon to take in account when the warpage of e.g.
a box-like product is evaluated. Figure 14 shows the result due to asymmetric cooling of a corner product but another important contributor to this phenomenon is the spring forward effect. This effect was first described for thin compression molding of sheet molding compounds (SMC). Due to long fiber length versus part thickness ratio, most fibers in SMC parts are oriented in the planar direction. The result is a higher thermal expansion coefficient in the thickness direction than compared with the in-plane direction and when the part cools down after polymer curing, the angle of a corner will decrease. Ammar et.al. [19] showed that the spring forward effect is generally the major source of corner deformation for injection-molded parts. They derived a spring forward indicator based on the shrinkage of a corner geometry where the in-plane shrinkage was assumed to be much lower than the shrinkage in the thickness direction due to geometry and friction between the polymer and mold wall which constrains the part during cooling and polymer solidification. Experiments using PP and variations in the mold temperature, packing pressure, and packing time, together with finite element analysis, showed that the thermal effect alone could not explain the angle deformation of the corner. As an example, predicted angle deformations due to asymmetric cooling using a viscoelastic model with isotropic expansion coefficients showed good correlation with measured variations of the angle deformation due to asymmetric cooling from the experiments. As the variations were accurately described, this was not the case for
the absolute value of the angel deformation and a difference of 3° compared with the experiments was shown and the conclusion was that this difference could be explained by the spring forward effect. The authors also used MOLDFLOW where only thermal effects could be modeled and the simulations confirmed the low influence of the asymmetric thermal effect in a corner on the warpage predictions.
Jansen et.al. [20] studied the effect of asymmetric cooling on the warpage of injection-molded products and especially, the effect of mold temperature difference and packing pressure on warpage for unfilled amorphous materials. They showed that the warpage increased linearly with applied temperature difference between the mold halves but more interesting, that a plate geometry curved towards the cold side of the mold when a high holding pressure was applied.
At lower pressures, the plate curved towards the hot side which is generally known and accepted. Similarly, the study of a corner geometry with different radii showed that the angle deformation was dependent on the temperature difference as well as the packing pressure and an increased radius also implied an increased temperature sensitivity. They also compared the warpage predictions of the injection-molding software C-MOLD with their suggested model and the experiments and the model succeeded in predicting the direction of the warpage but the absolute values were only about half of the measured values. C-MOLD predicted that the warpage would always be directed towards the hotter side of the mold whereas the measurements showed that this was only the case for the lower packing pressures.
The influence of thermal parameters on the shrinkage and warpage has been discussed by many researches, and many of them conclude that warpage predictions are sensitive to small variations of these parameters. Denizart et.al. [21] reported a considerable effect on the warpage prediction of a polystyrene disc when changing the no-flow temperature by 10°. This variation was more important than anisotropic and heterogeneous values used in the three-dimensional thermo-elastic temperature-dependent model. Jansen et.al. [20] suggested that comparable effects are to be expected from variations in thermal conductivity.
Jansen et.al. [22] carried out another study where the effect of processing conditions on shrinkage for seven common thermoplastic polymers was investigated. The holding pressure was the key parameter, and the effect of melt temperature was slightly less important. The effects of injection speed and mold temperature on the warpage were smaller and differ for each material in the study. A thermo-elastic model was developed and their model accurately described the measured shrinkage for all experimental conditions for the amorphous materials. For semi-crystalline materials the correspondence was poor and shrinkages were overpredicted.
Cavity deformation is also a contributor to final part shape and size. Delauney et.al. [23]
presented a method where local mold rigidities were determined based on the knowledge of cavity pressure history. The method together with experimental results showed that mold deformation analysis can improve the volumetric shrinkage estimation and they also showed that the mold deformation mainly influenced the cavity thickness during holding. Wu and Huang [24] estimated the cavity pressure, temperature distribution and clamping force by a filling program and these were then applied as boundary conditions in a mold-deformation analysis. Molding experiments were carried out for a wedge-shaped part and the structural analysis was verified by strain gauge measurement on the mold. The measured warpage of the parts were smaller than the simulated warpage and the difference were considered to be due to cavity deformation. The study showed that the simulated cavity
deformation was close to this deviation and if this effect is taken into account, the simulation result was improved significantly.
The gradient of oriented molecules which cause the skin to be in compression while the core is in tension (ch. 2.2.2) is not always the case for injection-molded products. This type of stress profiles is observed in free-quenched products and the model describing this phenomenon is not sensitive to holding pressure settings. Sometimes the stress distribution is inverse, i.e. the skin is in tension and the core is in compression. Another model has been proposed where the stress distribution is dependent on the pressure. The idea is that layers frozen-in at elevated pressures will expand when released from the mold. Jansen et.al. [25]
used a model where the free-quench model was assumed at low holding pressures and the pressure-induced model was assumed at high holding pressures. Measuring the residual stress is difficult and the authors used the principle of layer removal but instead of using mechanical milling, excimer laser milling was used. One of the big advantages using this method is that difficulties with excessive heating and stress relaxation during the milling are avoided.
Most of the pvT data are measured under rather slow cooling or heating rates, which is not the case for ordinary process conditions in polymer processing where high cooling rates are reached. This is a drawback with the Tait equation and will affect the prediction of the actual volumetric shrinkage and hence warpage. Chang et.al. [26] proposed a model where a new variable was introduced to the Tait model in order to include the cooling rate effect. The new variable was based on the dependence of the glass transition temperature on cooling rate and the study showed that the proposed model accurately represented volumetric behaviors of amorphous polymers. Chang and Hsieh [27] used this model and showed that for the cooling dynamics of a plastic lens, a better prediction of the volumetric shrinkage near the mold wall was achieved due to the formation of frozen layers which was taken into consideration in the cooling rate dependent Tait model.