In this section we will discuss the development of a numerical simulation method for the in-jection molding process as it was described in the previous section. Our focus is on enabling
optimization and inverse design methods. This goal forces us to consider certain physical phe-nomena, while other aspects of the material behavior can be neglected in favor of a more effi-cient simulation. In our simulation method, we consider isotropic materials only. Therefore, considerable adjustments would have to be made to the formulation, if composite materials, such as fiber-reinforced polymers, were to be used.
Despite the fact that injection molding is a very popular manufacturing process, and polymers in general are in widespread use, their physical behavior, especially where solidification is concerned, has still not been completely understood (see Sections 2.1.5 and 2.1.4) and fully reliable numerical simulation methods, e.g., for viscoelastic flow, are sometimes lacking (see Chapter 5).
Numerical simulation methods specifically for injection molding have been in development for a while, though. An early example is the work by Chiang et al. in 1991, described in [98].
They use a generalized Newtonian law (see Section 2.2.3) for a shear rate dependent viscosity.
Their simulation only encompasses the filling stages of the process and does not provide residual stresses in the material. In the same year, Baaijens and Douven [14] described a more elaborate approach, specifically for the simulation of injection molding with amorphous materials. They compare a generalized Newtonian law with a fully viscoelastic model (see Section 2.2.4) and describe how to obtain flow-induced residual stresses for both approaches.
Baaijens built upon this in [78] to additionally incorporate thermally induced residual stresses.
An example of an injection molding simulation with crystallization kinetics is the work by Hieber [82]. He applies the Nakamura equation to model crystallization. Solidification is mod-eled through a crystallinity-dependent viscosity, but viscoelasticity is not considered. More recently, Spekowius et al. [99] have applied multi-scale simulation techniques to increase the accuracy of their crystallization models.
For an extensive overview of modeling methods for injection molding, we refer to the work of Zheng et al. [13], or, for a focus on the filling simulation, Kennedy and Zheng [15].
Our simulation method is kept very general when compared to some previously described methods. The reason for this is the already mentioned focus on the inverse design methods.
However, our method is designed in a manner that allows the combination with some more specific models, e.g., for crystallization.
As we have mentioned before, models for certain physical phenomena are absolutely neces-sary for the application of the inverse design method proposed in this document. Most im-portantly, we need models that provide the inhomogeneous temperature and stress or strain distributions at ejection time. This means that heat conduction, and also elastic material be-havior — or, more precisely, viscoelasticity —, has to be modeled even before the material is completely solid. Obviously, models for the solidification itself are required, to even allow the build-up of stresses. On the other hand, certain details, such as the sensibility of the material parameters to temperature, do not affect the applicability of the inverse design method. In the interest of enabling a reliable simulation, however, we have made some provisions for such temperature-dependencies in our formulations.
We have given an overview of the relevant physical phenomena in the injection molding pro-cess in Figure 6.2. To focus now on the phenomena mentioned in the previous paragraph, we
propose a simplification of the overall process. This approximation of the process is depicted in Figure 6.3.
Filling Packing Cooling Ejection
Cavity filling Air discharge
Cooling Shrinkage Solidification
Viscosity
Elasticity
Solidification Shrinkage & Warpage
Figure 6.3:Overview of an approximation of the injection molding process that is suitable for numerical sim-ulation. Depicted are the physical phenomena that we allow during certain stages, in the top half of the figure, and the material behavior we account for, in the bottom half.
One important simplification that we make is that we assume solidification to be complete at the time of ejection. This means that viscous material behavior is not considered afterwards.
Additionally, we do not simulate the filling stage before the packing stage. Instead, we assume that cooling, shrinkage, and solidification are negligible before the cavity is filled with melt.
This allows us to ignore the air discharge from the cavity, which would add a third gaseous phase to this two-phase simulation.
As a result of these simplifications, it makes sense to consider multiple different simulation parts. Figure 6.4, shows a sketch of the input and output quantities of the entire injection molding simulation. Given a cavity shape and a melt temperature, the simulation calculates the shape, homogeneous temperature, and residual stresses in the cooled down molding.
Cavity Shape
Injection Molding Simulation
temperatureshape
stress temperatureshape
Molding Shape
Figure 6.4: The full injection molding simulation takes a cavity shape and boundary temperature as inputs calculates a molding shape, along with temperature and residual stress fields.
If we split the simulation at the ejection time we obtain one simulation part for solidification
and another for shrinkage and warpage. As we have indicated in Figure 6.3, both of them contain cooling and shrinkage effects. The separation is sketched in Figure 6.5. In addition to the input and output quantities mentioned before, we now have some intermediate quantities that are transferred from one simulation part to the other. These are the shape, temperature, and strain distribution in the material at the time of ejection.
Cavity Shape
Figure 6.5:We split the injection molding simulation into a solidification and a shrinkage and warpage simu-lation. We make this split at the point in the process when the molding is ejected and keep the molding shape fixed until this point.
6.2.1 Solidification Simulation
As shown in Figure 6.3, the relevant physical phenomena for the solidification simulation are cooling, shrinkage, and, of course, solidification. Concerning the material behavior, both viscous and elastic behavior occurs. For the latter, we also need to consider relaxation and, therefore, viscoelasticity.
The formulation of time-dependent compressible Eulerian thermoviscoelasticity in Chapter 5 fits these requirements. We can now describe the input and output quantities of this simu-lation more precisely. As depicted in Figure 6.6, we have to provide the physical domain Ω along with an initial temperature 𝜃. The simulation will not affect the shape, but will pro-duce a temperature 𝜃 and a conformation tensor 𝐊 at ejection time. The latter is required to calculate the strain at this point in time (see Section 5.10).
Cavity Shape
Solidification Simulation
temperature 𝜃shape Ω
conformation tensor 𝐊temperature 𝜃shape Ω
Ejection Shape
Figure 6.6:The solidification simulation takes a cavity shape and boundary temperature as input. It calculates the temperature and stress fields at ejection time. The shape is left unchanged.
As mentioned in the introduction of Chapter 5, the formulation does not include movable boundaries. This is the reason why the output shape of this simulation is identical to the input shape. This makes sense during the packing stage of injection molding, but during the cooling stage, when the gate has solidified, we have to expect the molding to partially loose contact with the mold walls. If these effects are to be considered, the formulation can be enhanced using a free-surface boundary approach as described in, e.g., [77].
6.2.2 Shrinkage and Warpage Simulation
The second part of the separated injection molding simulation includes cooling and shrinkage effects, but no solidification or viscous material behavior (see Figure 6.3). This means that we
are dealing with purely elastic material behavior. Since we are additionally only interested in the equilibrium result of the simulation, a stationary formulation is sufficient for this