Closure Approximations

The first problem mentioned above is referred to as the closure problem. Basically, we need to approximate the fourth order tensor ai j klas a function of the second order tensor ai j. This

is not a trivial problem and has lead to several proposals. Zheng et al. [424] have recently reviewed several available closures in the context of injection molding. We will mention only four here:

1. the linear closure 2. the quadratic closure 3. the hybrid closure 4. the orthotropic closure

It should be noted that closure approximations may take different forms in planar and 3D flows.

5.8.3.1 Linear Closure

The linear closure is described by Hand [145] and has the 3D form:

a_{i j kl}≈ a_{i j kl}^{l i n} = −1 35

¡δi jδkl+ δi kδj l+ δi lδj k¢ +1

7¡ai jδkl+ ai lδj k+ aklδi j+ aj lδi k+ aj kδi l¢ . (5.80) For planar orientation, the expression is similar but the coefficient 1/35 is changed to 1/24 and the coefficient 1/7 is changed to 1/6. For random orientation, the linear closure is exact but may be unstable when the fibers become more oriented [3].

5.8.3.2 Quadratic Closure

The quadratic closure has been employed by Doi [80]. It has the form:

a_{i j kl}≈ a_{i j kl}^quad= ai ja_kl. (5.81)

This is exact when the fibers are perfectly aligned but can have poor behavior in transient flows or when the interaction coefficient is nonzero [3].

5.8.3.3 Hybrid Closure

Advani and Tucker [3] proposed a mixture of the linear and quadratic closures. It takes the form:

a_{i j kl}≈ a_{i j kl}^{h ybr i d}= (1 − f ) a^{l i n}_{i j kl}+ f a^quad_{i j kl} , (5.82) where

f = 1 − 27det(ai j) for 3D orientation , (5.83)

and

f = 1 − 4det(ai j) for planar orientation . (5.84)

By construction, the hybrid approximation is exact when fibers are random or perfectly aligned. However, it does tend to accelerate orientation in transient shearing flows.

5.8 Fiber Orientation 83

5.8.3.4 Orthotropic Closure

In this closure scheme, introduced by Cintra and Tucker [62], the fourth-order tensor is ap-proximated as a polynomial function of the principal values of the second-order orientation tensor. The coefficients of this function are obtained by fitting to the exact solution of the orientation distribution function for some simple flows (shear, elongation, or simple combi-nations of both). For further details, see Cintra and Tucker [62].

5.8.3.5 The Interaction Coefficient

The interaction coefficient presents its own problem in that there is still a lack of fundamental theory that could dictate how to determine its value, either theoretically or experimentally.

An approach was presented by Bay [29] based on experimental values of a11in steady simple shear flows for different concentrations. These experimental results were compared to the numerically calculated a₁₁using the value of C_I that best agreed with the experimental data.

The polymer matrices used in the experiments were nylon, polycarbonate, and polybutylene terephthalate. The C_I values were then plotted againstφar to give the following empirical relationship:

C_I= 0.0184 exp(−0.7184φar) . (5.85)

In Bay’s experiments, the measured a11value increases as the fiber volume fraction increases, therefore the empirical equation shows C_I decreasing with increasingφar. This is opposed to the trend observed by Folgar and Tucker [122] for nylon fibers in silicon oil. However, Fol-gar and Tucker’s data were measured in the semi-concentrated regime, while Bay’s data were measured in the concentrated regime. Bay [29] and Tucker and Advani [370] conjectured that there could be a changeover in the fiber-fiber interaction from “disturbances” to “caging” at a certain concentration (aroundφar= 1).

It also needs to be mentioned that in Bay’s work, the hybrid closure was used to model the data. With this closure, the values of C_Iwere around 0.01 to match the experimental a₁₁ val-ues. However, if one uses the orthotropic closure, the data are matched with C_I = O(0.001).

Therefore, it is important to notice that, although the measured values of a₁₁are independent of any closure scheme, the values of C_I, and hence the coefficients of the empirical Equation 5.85, depend on the closure approximation used as noted by Zheng et al. [424].

Frequently this difficulty is unknown to users of simulation software. The key point is that any attempt to determine the interaction coefficient must be in keeping with the closure approxi-mation used in the simulation software.

Whereas Bay [29] used experimental data to determine a relationship between CI andφar, Phan-Thien et al. [289] performed a direct simulation of fiber motions in shear. They used a boundary element method to calculate the fiber orientation and, when the suspension was in equilibrium, calculated the value of CI. They proposed that

C_I= 0.03£1 − exp(−0.224φar)¤ . (5.86)

The results of the direct simulation and the empirical Equation 5.86 are comparable with the data of Folgar and Tucker [122] as shown in Figure 5.6.

Figure 5.6 A comparison of the simulated CI for a_r= 10, 16.9, 20, 30, and 31.9 [289] with experimental data of Folgar and Tucker [122] (reproduced from Phan-Thien et al. [289] with permission from Elsevier)

We complete this section by noting that there is no reason to make the interaction coefficient isotropic, that is, scalar in value. Indeed, Phan-Thien et al. [289] assumed the interaction co-efficient was a tensor and the results shown in Figure 5.6 used one third of the trace of the interaction coefficient tensor for the value of CI.

We discuss other short comings and improvements of the Folgar-Tucker model in Part II.

5.9 Shrinkage and Warpage

We have seen in the previous sections that there are many approximations made to the gov-erning equations to make them more tractable due to our lack of understanding of material properties and, to a lesser extent, compute time. The latter, though not really a hurdle in aca-demic work, has a big bearing on commercial codes. So far we have dealt with prediction of pressure, temperature, and velocities in injection molding. From this information the skilled user can determine the number of gates, their location, location of weldlines, and the size of the injection molding machine required to make the part. Using heuristic ideas and results from simulation, users can try to minimize stresses in moldings and enhance quality [31, 333].

Nevertheless, the big prize in molding simulation is the accurate prediction of shrinkage and warpage. The reader should note that there are several time scales involved. The first is post-injection, where components may be assembled shortly after molding. A longer time scale is introduced when post-molding operations, such as painting, which involves exposure to ele-vated temperatures, are considered and yet a longer time scale is the performance of the part during its lifetime. So shrinkage and warpage are time dependent. While advances in poly-mers have reduced the time scale problem, many commodity polypoly-mers of great commercial importance do require detailed consideration of their performance over time.

5.9 Shrinkage and Warpage 85

No codes, commercial or academic, exist today with the ability to consider time effects on molded products. This requires some explanation. By consideration of time effects, we mean calculating the performance of the molded part after ejection from the mold. While we can cal-culate shrinkage and warpage at ejection, a real molding may be subjected to a temperature-time history after ejection and indeed, during its life. To account for this, we need to calculate properties after ejection and consider the effects of environment including mechanical inter-actions with other parts. Time dependence remains a major challenge for polymer science.

We do not seek to blame simulation codes for their current deficiency. Instead we try to give an idea of the simplifications currently used that can create error. In Part II we give some ini-tial suggestions to improve the science behind shrinkage and warpage prediction. For now though, we consider what is done in practice.