# Development of warpage and residual stresses in film insert molded parts.

INTRODUCTION

Film insert molding (FIM) technology also known as in-mold decorating (IMD) is one of the most important polymer processing methods, where surfaces of plastic parts can be decorated during injection molding by inserting a printed film into the cavity. A thin film is attached on the cavity wall and the resin is injected into the cavity in the FIM process. Film insert molding is one of the new and highly advanced methods offering particular benefits to designers, processors, and manufacturers of many plastic products because it has many advantages, for example, cost effectiveness, longtime durability, wide spectrum of formability, and creativity of custom textures when compared with the traditional injection molding. However, it is more complex because complicated phenomena usually occur, for example, asymmetric cooling of the part in the cavity, remelting of the inserted film, wrinkling of the film, serious residual stress development, and so forth. It is important to understand its complex process accurately and to identify unique problems such as thermal residual stresses and warpage developed during FIM process.

In general, residual stresses are generated in an injection molded part by many processing factors including packing stresses as well as flow-induced and thermal-induced stresses. Packing stresses resulting from the high packing pressure imposed during compression stage affect the final shrinkage of a molded product (1). Flow-induced residual stresses are usually generated by shear and extensional flows during filling stage and they lead to anisotropy of physical properties because molecular orientations in the direction parallel to the flow are different from those in the perpendicular direction. Unlike the previous two types of stresses, thermal residual stresses are generated in the FIM process because of the temperature gradient generated during solidification of polymer melts (2-4). Magnitude of the thermal residual stress depends on various molding conditions, in particular, on the surface temperature of the mold during solidification (5), (6). Although complex residual stresses are generated in the injection molded parts, thermal residual stresses are dominant for most molded polymeric parts. Hence, the model used in this study considered only thermally induced residual stresses that are developed in the film insert molded parts.

Thermal residual stresses are generated in injection molded parts due to nonuniform temperature distribution in the cavity during filling, packing, and cooling (7), (8). The difference in temperature between two mold walls causes asymmetric residual stresses. Although symmetric boundary conditions for mold wall temperature are applied to the FIM process, the thermally induced residual stress becomes asymmetric because two different materials are used in the molding process and it is difficult to control the temperature profile distribution perfectly. If the molded part is not stiff enough, it will be deformed considerably after ejection from the mold (9), (10). Warp-age is caused by distortion of the molded part after it is ejected from the mold cavity. Part warpage results from molded-in residual stresses, which are attributed to nonuniform shrinkage of the molded part. Figure 1 illustrates the solidification of a FIM part with simple geometry schematically. The heat transfer rate of the part 1 across the thickness differs from that of the part 2. Therefore, the residual stress distribution across the thickness of the FIM part is asymmetric and the bending moment generated by the residual stress distribution causes warpage of the final product.

[FIGURE 1 OMITTED]

In this study, the thermally induced residual stress is investigated and governing equations are proposed for modeling of the FIM. For complete modeling of the residual stress development, heat transfer from the mold cavity and stress equilibrium must be considered simultaneously. Solutions of the energy equation and moment equilibrium are obtained with a finite difference method (FDM) to comprehend the influence of temperature difference in the mold and the Biot number. A house numerical code is developed for calculation of the residual stress by using C/C + + programming language. To validate the calculated results, they are compared with both experimental results and numerical predictions obtained by using commercial software.

THEORETICAL BACKGROUND

The thermoelastic residual stress model used in this study is constructed based on the work of Osswald and Menges (11). In this study, two different models, that is. thermoelastic residual stress model and thermal stress-moment model are used to analyze thermally induced residual stresses and bending moment generated during the FIM process qualitatively and quantitatively. Because mold temperature distribution is asymmetric in the FIM process as shown in Fig. 1, stress distribution of the molded part is calculated by considering part 1 and 2 separately. For an FIM part with the dimension in thickness much smaller than the dimension in planar direction, the following assumptions are made for constraint conditions: Amorphous polymer resin is used for FIM. There is no strain or stress present in the stationary polymer melt and no residual stress in the film part before being inserted in the mold. As the part is thin, the major direction of heat transfer during the cooling stage is the thickness direction and heat transfer in the other directions is neglected. The polymer melt experiences solidification at the glass transition temperature [T.sub.g]. The solidified thin polymeric layer undergoes volume shrinkage and flow-induced stresses are generated by the flow strain. [[epsilon].sub.[upsilon], which can also be referred to as the viscous strain. It is assumed that the flow strain occurs under hydrostatic stress and that it is frozen in the solid layer, which means that it is a function of space alone. In the solid region, thermal strain, [[epsilon].sub.th] = [beta](T(z,t) - [T.sub.g]), is formed, where [beta]is the thermal expansion coefficient, and bending is taken into account only after the molded part is ejected from the mold. The continuity condition for temperature and heat flux is applied to the interface between the inserted film and injected resin.

[FIGURE 2 OMITTED]

Thermoelastic Residual Stress Model

According to Indenbom theory (3), (12), the total strains for the part 1 and part 2, [epsilon].sub.1] (t) and [[epsilon].sub.2] (t), are expressed as a function of time and location as below.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [[[epsilon].sub.e] is the elastic strain and I stands for the interface of solidification. To calculate the total strain in Eq. 1, a new method is required to determine the flow strain, [[epsilon].sub.[upsilon]] (z). After using Hooke's Law and assuming that Poisson's ratio and Young's modulus are constant, the stress in the frozen layer is given by

[sigma] (z, t) = E/1 - v[[epsilon](t) - [beta] (T (z, t) - [T.sub.g]) - [[epsilon].sub.[upsilon]] (z)] (2)

The equilibrium condition is satisfied by the following integration:

[[integral].sub.-L.sup.L] [sigma] (z, t) dz = 0 (3)

Combining Eqs. 2 and 3 gives the following equations.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Total strains are expressed by the following equations by some manipulation and integration.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

At the solidification front, z = [z.sub.s], the thermal and elastic strains are zero. Eq. 1 becomes [epsilon] (t) = [[epsilon].sub.[upsilon]] (z), and Eq. 5 can be written as below.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

It is assumed that the stress relaxation of the part does not occur during injection molding and flow strains are not allowed to be changed. Hence, the transient temperature field T(z,t) and the location of [z.sub.s] should be determined to obtain the flow strain profile, [[epsilon].sub.v](z). Calculation of the flow strain distribution is carried out up to the moment that the weld surface, z =I, becomes solid, at which point the flow strain profile is fully determined. When the ejected plate cools down to room temperature, [T.sub.f], the following strain distribution can be obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Thermal Stress-Moment Model

The total strain, [[epsilon].sub.tot], is related to the entire (or actual) shrinkage of the plate and expressed as a function of temperature. Young's modulus. Poisson's ratio, and the thermal expansion coefficient. Based on the classical shell theory (13), (14) and the stress distribution in the plate, the thermal moment of a beam with unit width is computed as follows.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The moment generated by the thermal stress distribution leads to warpage of the final part.

NUMERICAL PROCEDURE

Numerical analyses were conducted to calculate transient temperature field, residual stresses, and bending moments generated by the FIM process. The transient temperature distribution was predicted by using the Do Fort-Frankel method (15), (16), which is an unconditionally stable scheme. The numerical method was derived by considering the leapfrog time advancement and the second-order central spatial differencing. Thermally induced residual stresses and bending moments are calculated by using the trapezoidal rule.

Three-dimensional numerical simulation was carried out additionally for modeling of the film insert molding by using Moldflow, a commercial program for numerical simulation of injection molding. To predict the warpage of the FIM part, three-dimensional tetrahedral finite elements were generated by Hypermesh, a commercial program for construction of finite element mesh, and then they were exported to Moldflow for building of the finite element mesh as shown in Fig. 2. For the three dimensional simulation, 70102 elements were used. Moldflow simulations were carried out under three different boundary conditions of mold temperature difference for comparison with experimental results and numerical predictions obtained by the house code developed in this study.

Transient Temperature Field

Transient temperature field, T(z,t) and location of the solidification front, [z.sub.s], included in Eq. 6 are determined numerically. For polymers, a model can be expressed based on the enthalpy formation (17), (18).

[rho] [partial derivative]H/[partial derivative]t = k [[partial derivative].sup.2] T/[partial derivative]z.sup.2]

H = [c.sub.p]T + gL (9)

where L is the latent heat of the polymer, [c.sub.p] is the specific heat, k is the thermal conductivity, [rho] is the density, and g is the liquid fraction that is zero for solid and unity for liquid. Dimensionless variables are defined as follows.

z* = z/b t* = kSt/[rho][c.sub.p][b.sup.2]t T* = [T - [T.sub.s]/[T.sub.s] - [T.sub.f]

H* = StT* + g St = [c.sub.p]([T.sub.s] - [t.sub.f])/L] Bi = hb/k (10)

where b is the thickness of a FIM part and h is the convective heat transfer coefficient at the surface z = 0. By assuming no internal heat generation, the governing equation is expressed as below.

[partial derivative]H*/[partial derivative]t* = [[partial derivative].sup.2]T*/[[partial derivative][z*.sup.2] (11)

Boundary conditions are given as the following.

[partial derivative]T*/[partial derivative]z* [|.sub.z=0] = -Bi(T* + 1) and [partial derivative]T*/[partial derivative]z* [|.sub.z=b] = 0 (12)

Because a perfect contact between the film and the resin is assumed, thermal interfacial resistance is negligible and the temperature continuity at the interface yields the following boundary conditions (19).

[T*.sub.F](L - [Florin], t) = [T*.sub.R] (L - [Florin], t) (13)

where [T*.sub.F] and [T*.sub.R] are the dimensionless temperatures of the film and resin surfaces. When there is no heat generation at the interface as in the case of phase changes, the heat flux continuity at the interface is expressed as below.

[k.sub.F] [partial derivative][T*.sub.F]/[partial derivative]z = [k.sub.R] [partial derivative][T*.sub.R]/[partial derivative]z (14)

where [k.sub.F] and [k.sub.R] are the thermal conductivity of the film and resin, respectively.

Key dimensionless groups in Eq. 14 are the Stefan number (St) and the Biot numbers (Bi) of the resin and film parts. The former is the ratio of latent heat and controls the phase change. The latter is the ratio of convective heat removal to heat conduction and controls the cooling process.

The heat equation, an exact differential equation, is reduced to an approximate algebraic Eq. 15 by applying the FDM. The finite difference formulation of the heat equation is applied to any interior node that is equidistant from its two neighboring nodes.

(1 - 2[beta]) [H.sub.i.sup.n+1] = (1 - 2[beta]) [H.sub.i.sup.n-1] + 2[beta]([T.sub.i+1.sup.n] + [T.sub.i-1.sup.n])

[beta] = [DELTA]t/[DELTA][z.sup.2] (15)

where, i is a node counter on space grid with uniform step [DELTA]z, and [DELTA]t a time step. The superscript n + 1 refers to values at the new time step and for presentation convenience, the superscript "*" is omitted. In a given time step, solution of Eq. 13 provides a nodal enthalpy field which updates the temperature field as follows.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

The time at which the solidification front reaches node i. that is, when [z.sub.s] = i[DELTA]z, is given when the nodal enthalpy, [H.sub.i], is 0.5. At this moment, the nodal temperature distribution is used for [[epsilon].sub.v]([z.sub.s]). These nodal values are used to calculate the residual stress profile across the thickness.

Thermal Residual Stress and Bending Moment

To compute the thermally induced residual stresses, [sigma](z, t), calculation of the transient temperature field and integration of the total strain are needed. For the interval, [z.sub.i] [less than or equal to] z [less than or equal to] [z.sub.i+1], the trapezoidal rule yields the following integration.

[[integral].sup.i+1] [[epsilon].sub.tot]dz [approximately equal to] [[DELTA]z/2 ([[epsilon].sub.tot] ([z.sub.i], t) - [[epsilon].sub.tot] ([z.sub.i+1], t)) (17)

where [DELTA]z = [z.sub.i+1] - [z.sub.i]. The geometrical assumption of this formula is that the function [[epsilon].sub.tot] within the interval is approximated by a straight line passing through the two end points and that an area under the curve in the interval is approximated by the resulting trapezoid. For the entire resin interval [L - [Florin], -L], uniform spacing is assumed and the trapezoidal rule is used over all the sub-intervals.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

In addition, bending moments are also computed by using the trapezoidal rule as below.

[[integral].sub.-L.sup.L-[Florin]] M. zdz [approximately equal to] [DELTA]z (- 1/2 [M.sub.0]L + 1/2 [M.sub.n]L + [[n-1] summation over (i=1)] [M.sub.i] [z.sub.i]) (19)

EXPERIMENTAL

PC/ABS (polycarbonate/acrylonitrile-butadiene-styrene) alloy resin (Staroy HP-1000X supplied by the Cheil industries. Korea) was injected into a dog-bone-shape mold cavity to produce specimens, as shown in Fig. 3 using an injection molding machine (Engel 120, Engel, Austria). The PMMA/ABS laminated film (Shin-Etsu Mille-tint Film, Japan) was attached to one side of the mold wall before injection. It was properly cut by a die cutting machine before inserted into the mold. The main processing parameters are summarized in Table 1, and the FIM specimens were molded under the symmetric and asymmetric mold temperature conditions. Deflection and curvature of the FIM parts were measured after ejection by using a scanner (Image-Pro Plus, Media Cybernetics).

[FIGURE 3 OMITTED]

RESULTS AND DISCUSSION

Transient Temperature Profiles and Residual Stresses

Thermal history of the injection molded part was investigated by using a FDM to determine boundary conditions for thermal equilibrium and residual stress analyses. Under various processing conditions, transient temperature fields, residual stress profiles, and bending moments were predicted by applying the material property data given in Table 2. It was assumed that Young's modulus and Poisson's ratio are constant in the solid resin and film parts. While calculating the transient temperature field for the film and resin parts, it was assumed that the thermal expansion coefficient of the polymeric resin, the heat transfer coefficient for the steel mold, the specific gravity, and the heat capacity are constant. In addition, the main numerical simulation conditions are summarized in Table 3. Cooling time should be predetermined for solidification of the resin so that local temperature at any point should drop below the glass transition temperature of the resin at the end of the cooling step. In the condition (1). film thickness was varied in order to consider the influence of the Biot number of the film on the bending moment. Moreover, variations of the resin thickness were taken into account in the simulation.

Transient asymmetric temperature profiles are shown in Fig. 3a-d. Figure 3a shows the temperature distribution when the temperature at a node reaches the glass transition temperature, [T.sub.g] = 137[degrees]C. Several pairs of curves appear in the graphs and numbers indicate the time when the internal temperature of each node reaches the glass transition temperature. This result implies that the more the resin-side mold wall temperature, [T.sub.2], increases, the more similar rate of heat flux is obtained on both the resin and film side surfaces. Therefore, the residual stress distributions become more symmetric as the resin side mold wall temperature increases as described in Fig. 4. This behavior can be explained by the fact that symmetric temperature distribution generates symmetric thermal residual stress distribution (9), (20), (21).

[FIGURE 4 OMITTED]

Remelting of the film substrate is often caused by the heat transfer from the molten resin during filling stage. Figure 5 shows the transient temperature profile at the interface between the resin and film parts. The calculated interfacial temperature at the beginning of filling was as higher than the glass transition temperature of the film material. However, the interfacial temperature was rapidly decreased with respect to time and approached to a constant value along with solidification of the injected resin. Hence, the effect of film remelting on the temperature profile and residual stress distribution was not considered in this study. Calculated bending moment of the FIM part is shown with respect to the mold temperature difference in Fig. 6. The bending moment was linearly decreased as the mold temperature difference (|[T.sub.2] - [T.sub.1]|) and the film-side mold wall temperature ([T.sub.1]) were increased. It is important that the direction of the curvature is opposite to that obtained by the "asymmetric cooling analysis in general injection molding." It indicates that the heat transfer is slow through the surface of the film part. Variation in the cooling rate from the mold wall to its center can cause thermally induced residual stresses and the bending moment will be induced by the thermal residual stresses (9), (11), (20).

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

When cooling rates of the two surfaces are different, asymmetric thermal residual stresses can be generated due to different material shrinkage and thermal-mechanical histories. As illustrated in Fig. 7a, unbalanced cooling results in an asymmetric tension-compression stress field across the part and leads to a bending moment that causes warpage of the part. Although both upper and lower parts of the mold have the same wall temperature in the FIM process, asymmetric thermal residual stress distribution can be obtained because of the unbalanced cooling rates caused by the inserted film. On the contrary to asymmetric molding temperature conditions in Fig. 7a, the bending moment created in the FIM process is negative because of the retarded heat transfer through the mold wall as shown in Fig. 7b. In Fig. 7c, larger negative bending moment is generated and it implies that the magnitude of compressive stress near the film side surface is larger than that shown in Fig. 7b as illustrated by the bending moment results computed in Fig. 5 previously. It is explained by the fact that high cooling rate is achieved through the film side mold wall even though the inserted film disturbs the heat diffusion through the mold wall.

[FIGURE 7 OMITTED]

Bending Moments and Warpage of the FIM Parts

The Biot number is a critical parameter in understanding the warpage of the FIM part because it includes the effect of thickness and thermal properties of the part. The effect of the Biot number of the film and resin on the bending moment is shown in Fig. 8. The bending moment of the film part is decreased slowly with the increasing Biot number, whereas the resin part shows deeper decrement with respect to the number because the Biot number of the resin part induces higher negative moment. In addition, the bending moment per MTD (Mold Temperature Difference) is more sensitive to the resin because the thermal conductivity of the resin is larger than that of the film. As a result, it is clear that the Biot number of the resin part strongly influences the warpage of a whole part because the film has a small thickness when compared with the whole part (22).

[FIGURE 8 OMITTED]

Film insert injection molding was used to produce specimens and to investigate the process phenomenally by using an image processing method. The warpage of the tensile specimen produced with the PC/ABS alloy resin is displayed in Fig. 9. Although a symmetric mold wall temperature is used for processing (condition 1). asymmetric thermal residual stress distribution was generated by variation in the cooling rate from the mold wall to its center due to the inserted film on one side of the mold wall as shown in Fig. 9a. It is possible that an asymmetric stress distribution across the FIM part occurs because of the unbalanced cooling rate, and then the bending moment induced by the asymmetric stress distribution generates the warpage of the final product. As expected from asymmetric thermally induced residual stress profile, which is obtained when asymmetric mold wall temperature is applied to the FIM process, deflection of the FIM part is increased significantly when compared with specimen (a) as shown in Fig. 9b. It is concluded that the bending moment of the FIM part decreases as the mold temperature difference increases because the negative bending moment becomes larger with the increasing deflection against the normal direction of the film surface.

[FIGURE 9 OMITTED]

For comparison with the experimental results and the numerical predictions obtained from the house code developed in this study, Moldflow simulations were carried out for three different conditions for mold temperature difference. The warpage of the FIM part predicted by Moldflow simulation is illustrated in Fig. 10a-c. From the above results, it is understood that higher mold temperature difference causes bigger deflection because of asymmetrical cooling rate conditions (9), (23). The effect of asymmetrical cooling rate on the thermally induced residual stress was verified by the numerical simulation. However, as shown in Fig. 11, the numerical simulation results overestimate the warpage of the FIM parts when compared with the experimental results. Finally, the bending moments predicted by the numerical simulations are compared with the experimental results as shown in Fig. 12. The predictions calculated by the house code underestimate the bending moment of the FIM part because the numerical calculation considers only the thermal residual stresses among total residual stresses, which are pressure induced, flow induced, and thermally induced residual stresses (20). In the case of actual experiments, the bending moment of the FIM specimen is often influenced by pressure history, such as injection, packing, and holding pressure variations as well as injection conditions such as the injection speed. Although thermal residual stresses and bending moments of the FIM parts were explained by using numerical analyses and experimental methods, a further study is needed to understand the effects of the inserted film and asymmetric heat flux conditions on the residual stresses generated during the FIM process.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

CONCLUSIONS

Thermally induced residual stresses in FIM parts were predicted by numerical simulation and bending moment of the FIM part was calculated and compared with the experimental results. Numerical modeling requires solution of thermal solidification problems and calculation of the stress integration through the part thickness. Although a one-dimensional thermoelastic model was considered with the assumption of constant material properties, it could provide an insight into the FIM process. It was found that many factors contribute to the residual stress distribution during the FIM process. Thermally induced residual stress distribution depended remarkably on the Biot number and heat was removed through the surfaces rapidly, thereby resulting in high residual stresses. The asymmetrical residual stresses generated by a nonuniform cooling rate through the part thickness leads to nonuniform shrinkage of the molded part. The bending moment of the FIM part was linearly decreased and warpagc was increased with respect to the temperature difference of the mold because the thermal strain mainly affects the total strain and the material cools down and shrinks inconsistently from the mold wall to the center of the FIM part.

REFERENCES

(1.) K.M.B. Jansen and G. Titomanlio, Polym. Eng. Sci., 36, 2029 (1996).

(2.) J.R. White, Polym. Test., 165, 4 (1984).

(3.) A.I. Isayev and D.L. Crouthamel, Polym-Plast. Technol., 22, 177 (1984).

(4.) A. Siegmann, A. Buchman, and S. Kenig, Polym. Eng. Sci., 22, 561 (1982).

(5.) R.F. Eduljee, J.W. Gillespie Jr., and R.L. McCullough, J. Thermoplastic Compos. Mater., 2(4), 319 (1989).

(6.) B. Haworth, C.S. Hindle. G.J. Sandilands, and J.R. White, Plast. Rub. Proc. Appl., 2, 59 (1982).

(7.) S.K. Kim, S.W. Lee, and J.R. Youn. Korea-Aust. Rheol. J., 14(3). 107 (2002).

(8.) Y.I. Kwon, K. Chung, T.J. Kang, and J.R. Youn, Fiber Polym., 2(4). 203 (2001).

(9.) O. Denizart, M. Vincent, and J.F. Agassant. J. Mater. Sci., 30, 552 (1995).

(10.) H.H. Chiang. K. Himasekhar, N. Santhanam, and K.K. Wang, J. Eng. Mater. Tech., 37, 115 (1993).

(11.) T.A. Osswald and G. Menges, Material Science of Polymers for Engineers, Hanser Publishers, Munich (1996).

(12.) V.L. Indenbom, J. Tech. Phys., 24, 925 (1954).

(13.) R.G. Treating and W.T. Read Jr., J. Appl. Phys., 22, 130 (1951).

(14.) J.M. Gere, Mechanics of Materials. 5th ed., Nelson Thornes. Cheltenham (2001).

(15.) P. Moin, Fundamentals of Engineering Numerical Analysis, Cambridge University Press, New York (2001).

(16.) J.H. Ferziger, Numerical Methods for Engineering Application, Wiley, New York (1998).

(17.) V.R. Voller. Adv. Numer. Heat Transfer. 1, 341 (1997).

(18.) B.A. Boley and J.H. Weiner, Theory of Thermal Stresses, Dover Publications, New York (1997).

(19.) J.F. Agassant, P. Avenas, J.P. Sergent. and P.J. Carreau, Polymer Processing: Principles and Modeling, Hanser Publishers, Munich (1982).

(20.) A.I. Isayev. Injection and Compression Molding Fundamentals, Marcel Dekker, New York (1987).

(21.) E.H. Lee, T.G. Rogers, and T.C. Woo, J. Am. Ceram. Soc., 48. 480 (1965).

(22.) C. Nylund and K. Meinander. Int. J. Heat Mass Trans., 41, 428 (2005).

(23.) J.L. White, Polym. Eng. Sci., 15, 44 (1975).

Hwa Jin Oh, (1) Young Seok Song, (2) Seung Hwan Lee, (1) Jae Ryoun Youn (1)

(1) Research Institute of Advanced Materials (RIAM), Department of Materials Science and Engineering, Seoul National University, 56-1, Silllim-Dong, Gwanak-Gu, Seoul 151-742, Korea

(2) Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Correspondence to: Jae Ryoun Youn: e-mail: jaeryoun@snu.ac.kr

Contract grant sponsor: The Korea Science and Engineering Foundation [Applied Rheology Center (ARC)].

DOI 10.1002/pen.21354

Published online in Wiley InterScience (www.interscience.wiley.com).

[C] 2009 Society of Plastics Engineers

Film insert molding (FIM) technology also known as in-mold decorating (IMD) is one of the most important polymer processing methods, where surfaces of plastic parts can be decorated during injection molding by inserting a printed film into the cavity. A thin film is attached on the cavity wall and the resin is injected into the cavity in the FIM process. Film insert molding is one of the new and highly advanced methods offering particular benefits to designers, processors, and manufacturers of many plastic products because it has many advantages, for example, cost effectiveness, longtime durability, wide spectrum of formability, and creativity of custom textures when compared with the traditional injection molding. However, it is more complex because complicated phenomena usually occur, for example, asymmetric cooling of the part in the cavity, remelting of the inserted film, wrinkling of the film, serious residual stress development, and so forth. It is important to understand its complex process accurately and to identify unique problems such as thermal residual stresses and warpage developed during FIM process.

In general, residual stresses are generated in an injection molded part by many processing factors including packing stresses as well as flow-induced and thermal-induced stresses. Packing stresses resulting from the high packing pressure imposed during compression stage affect the final shrinkage of a molded product (1). Flow-induced residual stresses are usually generated by shear and extensional flows during filling stage and they lead to anisotropy of physical properties because molecular orientations in the direction parallel to the flow are different from those in the perpendicular direction. Unlike the previous two types of stresses, thermal residual stresses are generated in the FIM process because of the temperature gradient generated during solidification of polymer melts (2-4). Magnitude of the thermal residual stress depends on various molding conditions, in particular, on the surface temperature of the mold during solidification (5), (6). Although complex residual stresses are generated in the injection molded parts, thermal residual stresses are dominant for most molded polymeric parts. Hence, the model used in this study considered only thermally induced residual stresses that are developed in the film insert molded parts.

Thermal residual stresses are generated in injection molded parts due to nonuniform temperature distribution in the cavity during filling, packing, and cooling (7), (8). The difference in temperature between two mold walls causes asymmetric residual stresses. Although symmetric boundary conditions for mold wall temperature are applied to the FIM process, the thermally induced residual stress becomes asymmetric because two different materials are used in the molding process and it is difficult to control the temperature profile distribution perfectly. If the molded part is not stiff enough, it will be deformed considerably after ejection from the mold (9), (10). Warp-age is caused by distortion of the molded part after it is ejected from the mold cavity. Part warpage results from molded-in residual stresses, which are attributed to nonuniform shrinkage of the molded part. Figure 1 illustrates the solidification of a FIM part with simple geometry schematically. The heat transfer rate of the part 1 across the thickness differs from that of the part 2. Therefore, the residual stress distribution across the thickness of the FIM part is asymmetric and the bending moment generated by the residual stress distribution causes warpage of the final product.

[FIGURE 1 OMITTED]

In this study, the thermally induced residual stress is investigated and governing equations are proposed for modeling of the FIM. For complete modeling of the residual stress development, heat transfer from the mold cavity and stress equilibrium must be considered simultaneously. Solutions of the energy equation and moment equilibrium are obtained with a finite difference method (FDM) to comprehend the influence of temperature difference in the mold and the Biot number. A house numerical code is developed for calculation of the residual stress by using C/C + + programming language. To validate the calculated results, they are compared with both experimental results and numerical predictions obtained by using commercial software.

THEORETICAL BACKGROUND

The thermoelastic residual stress model used in this study is constructed based on the work of Osswald and Menges (11). In this study, two different models, that is. thermoelastic residual stress model and thermal stress-moment model are used to analyze thermally induced residual stresses and bending moment generated during the FIM process qualitatively and quantitatively. Because mold temperature distribution is asymmetric in the FIM process as shown in Fig. 1, stress distribution of the molded part is calculated by considering part 1 and 2 separately. For an FIM part with the dimension in thickness much smaller than the dimension in planar direction, the following assumptions are made for constraint conditions: Amorphous polymer resin is used for FIM. There is no strain or stress present in the stationary polymer melt and no residual stress in the film part before being inserted in the mold. As the part is thin, the major direction of heat transfer during the cooling stage is the thickness direction and heat transfer in the other directions is neglected. The polymer melt experiences solidification at the glass transition temperature [T.sub.g]. The solidified thin polymeric layer undergoes volume shrinkage and flow-induced stresses are generated by the flow strain. [[epsilon].sub.[upsilon], which can also be referred to as the viscous strain. It is assumed that the flow strain occurs under hydrostatic stress and that it is frozen in the solid layer, which means that it is a function of space alone. In the solid region, thermal strain, [[epsilon].sub.th] = [beta](T(z,t) - [T.sub.g]), is formed, where [beta]is the thermal expansion coefficient, and bending is taken into account only after the molded part is ejected from the mold. The continuity condition for temperature and heat flux is applied to the interface between the inserted film and injected resin.

[FIGURE 2 OMITTED]

Thermoelastic Residual Stress Model

According to Indenbom theory (3), (12), the total strains for the part 1 and part 2, [epsilon].sub.1] (t) and [[epsilon].sub.2] (t), are expressed as a function of time and location as below.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [[[epsilon].sub.e] is the elastic strain and I stands for the interface of solidification. To calculate the total strain in Eq. 1, a new method is required to determine the flow strain, [[epsilon].sub.[upsilon]] (z). After using Hooke's Law and assuming that Poisson's ratio and Young's modulus are constant, the stress in the frozen layer is given by

[sigma] (z, t) = E/1 - v[[epsilon](t) - [beta] (T (z, t) - [T.sub.g]) - [[epsilon].sub.[upsilon]] (z)] (2)

The equilibrium condition is satisfied by the following integration:

[[integral].sub.-L.sup.L] [sigma] (z, t) dz = 0 (3)

Combining Eqs. 2 and 3 gives the following equations.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Total strains are expressed by the following equations by some manipulation and integration.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

At the solidification front, z = [z.sub.s], the thermal and elastic strains are zero. Eq. 1 becomes [epsilon] (t) = [[epsilon].sub.[upsilon]] (z), and Eq. 5 can be written as below.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

It is assumed that the stress relaxation of the part does not occur during injection molding and flow strains are not allowed to be changed. Hence, the transient temperature field T(z,t) and the location of [z.sub.s] should be determined to obtain the flow strain profile, [[epsilon].sub.v](z). Calculation of the flow strain distribution is carried out up to the moment that the weld surface, z =I, becomes solid, at which point the flow strain profile is fully determined. When the ejected plate cools down to room temperature, [T.sub.f], the following strain distribution can be obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Thermal Stress-Moment Model

The total strain, [[epsilon].sub.tot], is related to the entire (or actual) shrinkage of the plate and expressed as a function of temperature. Young's modulus. Poisson's ratio, and the thermal expansion coefficient. Based on the classical shell theory (13), (14) and the stress distribution in the plate, the thermal moment of a beam with unit width is computed as follows.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The moment generated by the thermal stress distribution leads to warpage of the final part.

NUMERICAL PROCEDURE

Numerical analyses were conducted to calculate transient temperature field, residual stresses, and bending moments generated by the FIM process. The transient temperature distribution was predicted by using the Do Fort-Frankel method (15), (16), which is an unconditionally stable scheme. The numerical method was derived by considering the leapfrog time advancement and the second-order central spatial differencing. Thermally induced residual stresses and bending moments are calculated by using the trapezoidal rule.

Three-dimensional numerical simulation was carried out additionally for modeling of the film insert molding by using Moldflow, a commercial program for numerical simulation of injection molding. To predict the warpage of the FIM part, three-dimensional tetrahedral finite elements were generated by Hypermesh, a commercial program for construction of finite element mesh, and then they were exported to Moldflow for building of the finite element mesh as shown in Fig. 2. For the three dimensional simulation, 70102 elements were used. Moldflow simulations were carried out under three different boundary conditions of mold temperature difference for comparison with experimental results and numerical predictions obtained by the house code developed in this study.

Transient Temperature Field

Transient temperature field, T(z,t) and location of the solidification front, [z.sub.s], included in Eq. 6 are determined numerically. For polymers, a model can be expressed based on the enthalpy formation (17), (18).

[rho] [partial derivative]H/[partial derivative]t = k [[partial derivative].sup.2] T/[partial derivative]z.sup.2]

H = [c.sub.p]T + gL (9)

where L is the latent heat of the polymer, [c.sub.p] is the specific heat, k is the thermal conductivity, [rho] is the density, and g is the liquid fraction that is zero for solid and unity for liquid. Dimensionless variables are defined as follows.

z* = z/b t* = kSt/[rho][c.sub.p][b.sup.2]t T* = [T - [T.sub.s]/[T.sub.s] - [T.sub.f]

H* = StT* + g St = [c.sub.p]([T.sub.s] - [t.sub.f])/L] Bi = hb/k (10)

where b is the thickness of a FIM part and h is the convective heat transfer coefficient at the surface z = 0. By assuming no internal heat generation, the governing equation is expressed as below.

[partial derivative]H*/[partial derivative]t* = [[partial derivative].sup.2]T*/[[partial derivative][z*.sup.2] (11)

Boundary conditions are given as the following.

[partial derivative]T*/[partial derivative]z* [|.sub.z=0] = -Bi(T* + 1) and [partial derivative]T*/[partial derivative]z* [|.sub.z=b] = 0 (12)

Because a perfect contact between the film and the resin is assumed, thermal interfacial resistance is negligible and the temperature continuity at the interface yields the following boundary conditions (19).

[T*.sub.F](L - [Florin], t) = [T*.sub.R] (L - [Florin], t) (13)

where [T*.sub.F] and [T*.sub.R] are the dimensionless temperatures of the film and resin surfaces. When there is no heat generation at the interface as in the case of phase changes, the heat flux continuity at the interface is expressed as below.

[k.sub.F] [partial derivative][T*.sub.F]/[partial derivative]z = [k.sub.R] [partial derivative][T*.sub.R]/[partial derivative]z (14)

where [k.sub.F] and [k.sub.R] are the thermal conductivity of the film and resin, respectively.

Key dimensionless groups in Eq. 14 are the Stefan number (St) and the Biot numbers (Bi) of the resin and film parts. The former is the ratio of latent heat and controls the phase change. The latter is the ratio of convective heat removal to heat conduction and controls the cooling process.

The heat equation, an exact differential equation, is reduced to an approximate algebraic Eq. 15 by applying the FDM. The finite difference formulation of the heat equation is applied to any interior node that is equidistant from its two neighboring nodes.

(1 - 2[beta]) [H.sub.i.sup.n+1] = (1 - 2[beta]) [H.sub.i.sup.n-1] + 2[beta]([T.sub.i+1.sup.n] + [T.sub.i-1.sup.n])

[beta] = [DELTA]t/[DELTA][z.sup.2] (15)

where, i is a node counter on space grid with uniform step [DELTA]z, and [DELTA]t a time step. The superscript n + 1 refers to values at the new time step and for presentation convenience, the superscript "*" is omitted. In a given time step, solution of Eq. 13 provides a nodal enthalpy field which updates the temperature field as follows.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

The time at which the solidification front reaches node i. that is, when [z.sub.s] = i[DELTA]z, is given when the nodal enthalpy, [H.sub.i], is 0.5. At this moment, the nodal temperature distribution is used for [[epsilon].sub.v]([z.sub.s]). These nodal values are used to calculate the residual stress profile across the thickness.

Thermal Residual Stress and Bending Moment

To compute the thermally induced residual stresses, [sigma](z, t), calculation of the transient temperature field and integration of the total strain are needed. For the interval, [z.sub.i] [less than or equal to] z [less than or equal to] [z.sub.i+1], the trapezoidal rule yields the following integration.

[[integral].sup.i+1] [[epsilon].sub.tot]dz [approximately equal to] [[DELTA]z/2 ([[epsilon].sub.tot] ([z.sub.i], t) - [[epsilon].sub.tot] ([z.sub.i+1], t)) (17)

where [DELTA]z = [z.sub.i+1] - [z.sub.i]. The geometrical assumption of this formula is that the function [[epsilon].sub.tot] within the interval is approximated by a straight line passing through the two end points and that an area under the curve in the interval is approximated by the resulting trapezoid. For the entire resin interval [L - [Florin], -L], uniform spacing is assumed and the trapezoidal rule is used over all the sub-intervals.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

In addition, bending moments are also computed by using the trapezoidal rule as below.

[[integral].sub.-L.sup.L-[Florin]] M. zdz [approximately equal to] [DELTA]z (- 1/2 [M.sub.0]L + 1/2 [M.sub.n]L + [[n-1] summation over (i=1)] [M.sub.i] [z.sub.i]) (19)

EXPERIMENTAL

PC/ABS (polycarbonate/acrylonitrile-butadiene-styrene) alloy resin (Staroy HP-1000X supplied by the Cheil industries. Korea) was injected into a dog-bone-shape mold cavity to produce specimens, as shown in Fig. 3 using an injection molding machine (Engel 120, Engel, Austria). The PMMA/ABS laminated film (Shin-Etsu Mille-tint Film, Japan) was attached to one side of the mold wall before injection. It was properly cut by a die cutting machine before inserted into the mold. The main processing parameters are summarized in Table 1, and the FIM specimens were molded under the symmetric and asymmetric mold temperature conditions. Deflection and curvature of the FIM parts were measured after ejection by using a scanner (Image-Pro Plus, Media Cybernetics).

[FIGURE 3 OMITTED]

TABLE 1. Processing conditions for the FIM experiments. Thickness of the part (mm) Condition Cooling Mold temperature Film-side mold wall Film Resin number time difference temperature (sec) ([DELTA]T) ([T.sub.t]) ([degrees]C) ([degrees]C) (1) 30 0 43 0.5 3.5 (2) 35 20 43 0.5 3.5

RESULTS AND DISCUSSION

Transient Temperature Profiles and Residual Stresses

Thermal history of the injection molded part was investigated by using a FDM to determine boundary conditions for thermal equilibrium and residual stress analyses. Under various processing conditions, transient temperature fields, residual stress profiles, and bending moments were predicted by applying the material property data given in Table 2. It was assumed that Young's modulus and Poisson's ratio are constant in the solid resin and film parts. While calculating the transient temperature field for the film and resin parts, it was assumed that the thermal expansion coefficient of the polymeric resin, the heat transfer coefficient for the steel mold, the specific gravity, and the heat capacity are constant. In addition, the main numerical simulation conditions are summarized in Table 3. Cooling time should be predetermined for solidification of the resin so that local temperature at any point should drop below the glass transition temperature of the resin at the end of the cooling step. In the condition (1). film thickness was varied in order to consider the influence of the Biot number of the film on the bending moment. Moreover, variations of the resin thickness were taken into account in the simulation.

TABLE 2. Material properties used for numerical and experimental studies. Material Resin Film Mold properties Young's 2780 2780 -- modulus (Mpa) (a) Poisson's 0.23 0.23 -- ratio Thermal 6.7 x [10.sup.-5] -- -- expansion coefficient ([degrees] [C.sup.-1]) Heat transfer -- -- 2.5 x [10.sup.4] coefficient (W/[m.sup.2] [C.sup.-1]) Thermal 0.21 0.15 29 conductivity (W/[m.sup.2] [C.sup.-1]) Specific 1.13 1.05 7.8 gravity Heat capacity 2.13 x [10.sup.6] 2.082 x [10.sup.6] 4.6 x [10.sup.5] (J/kg [degrees]C) (a) Young's modulus and Poisson's ratio of the PC/ABS resin and PMMA/ABS film were assumed constant values. TABLE 3. Processing conditions for numerical simulation of FIM. Thickness of the part (mm) Condition Cooling Mold Film-side Film Resin number time temperature mold wall (sec) difference temperature ([degrees]C) ([T.sub.1]) ([degrees]C) (1) 15 0, 10, 20, 30, 40 30, 40, 50 0.25, 0.5, 2.5 1.0 (2) 20 0, 10, 20 30, 40, 50 0.5 3.5 (3) 30 0, 10, 20 30, 40, 50 0.5 4.5

Transient asymmetric temperature profiles are shown in Fig. 3a-d. Figure 3a shows the temperature distribution when the temperature at a node reaches the glass transition temperature, [T.sub.g] = 137[degrees]C. Several pairs of curves appear in the graphs and numbers indicate the time when the internal temperature of each node reaches the glass transition temperature. This result implies that the more the resin-side mold wall temperature, [T.sub.2], increases, the more similar rate of heat flux is obtained on both the resin and film side surfaces. Therefore, the residual stress distributions become more symmetric as the resin side mold wall temperature increases as described in Fig. 4. This behavior can be explained by the fact that symmetric temperature distribution generates symmetric thermal residual stress distribution (9), (20), (21).

[FIGURE 4 OMITTED]

Remelting of the film substrate is often caused by the heat transfer from the molten resin during filling stage. Figure 5 shows the transient temperature profile at the interface between the resin and film parts. The calculated interfacial temperature at the beginning of filling was as higher than the glass transition temperature of the film material. However, the interfacial temperature was rapidly decreased with respect to time and approached to a constant value along with solidification of the injected resin. Hence, the effect of film remelting on the temperature profile and residual stress distribution was not considered in this study. Calculated bending moment of the FIM part is shown with respect to the mold temperature difference in Fig. 6. The bending moment was linearly decreased as the mold temperature difference (|[T.sub.2] - [T.sub.1]|) and the film-side mold wall temperature ([T.sub.1]) were increased. It is important that the direction of the curvature is opposite to that obtained by the "asymmetric cooling analysis in general injection molding." It indicates that the heat transfer is slow through the surface of the film part. Variation in the cooling rate from the mold wall to its center can cause thermally induced residual stresses and the bending moment will be induced by the thermal residual stresses (9), (11), (20).

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

When cooling rates of the two surfaces are different, asymmetric thermal residual stresses can be generated due to different material shrinkage and thermal-mechanical histories. As illustrated in Fig. 7a, unbalanced cooling results in an asymmetric tension-compression stress field across the part and leads to a bending moment that causes warpage of the part. Although both upper and lower parts of the mold have the same wall temperature in the FIM process, asymmetric thermal residual stress distribution can be obtained because of the unbalanced cooling rates caused by the inserted film. On the contrary to asymmetric molding temperature conditions in Fig. 7a, the bending moment created in the FIM process is negative because of the retarded heat transfer through the mold wall as shown in Fig. 7b. In Fig. 7c, larger negative bending moment is generated and it implies that the magnitude of compressive stress near the film side surface is larger than that shown in Fig. 7b as illustrated by the bending moment results computed in Fig. 5 previously. It is explained by the fact that high cooling rate is achieved through the film side mold wall even though the inserted film disturbs the heat diffusion through the mold wall.

[FIGURE 7 OMITTED]

Bending Moments and Warpage of the FIM Parts

The Biot number is a critical parameter in understanding the warpage of the FIM part because it includes the effect of thickness and thermal properties of the part. The effect of the Biot number of the film and resin on the bending moment is shown in Fig. 8. The bending moment of the film part is decreased slowly with the increasing Biot number, whereas the resin part shows deeper decrement with respect to the number because the Biot number of the resin part induces higher negative moment. In addition, the bending moment per MTD (Mold Temperature Difference) is more sensitive to the resin because the thermal conductivity of the resin is larger than that of the film. As a result, it is clear that the Biot number of the resin part strongly influences the warpage of a whole part because the film has a small thickness when compared with the whole part (22).

[FIGURE 8 OMITTED]

Film insert injection molding was used to produce specimens and to investigate the process phenomenally by using an image processing method. The warpage of the tensile specimen produced with the PC/ABS alloy resin is displayed in Fig. 9. Although a symmetric mold wall temperature is used for processing (condition 1). asymmetric thermal residual stress distribution was generated by variation in the cooling rate from the mold wall to its center due to the inserted film on one side of the mold wall as shown in Fig. 9a. It is possible that an asymmetric stress distribution across the FIM part occurs because of the unbalanced cooling rate, and then the bending moment induced by the asymmetric stress distribution generates the warpage of the final product. As expected from asymmetric thermally induced residual stress profile, which is obtained when asymmetric mold wall temperature is applied to the FIM process, deflection of the FIM part is increased significantly when compared with specimen (a) as shown in Fig. 9b. It is concluded that the bending moment of the FIM part decreases as the mold temperature difference increases because the negative bending moment becomes larger with the increasing deflection against the normal direction of the film surface.

[FIGURE 9 OMITTED]

For comparison with the experimental results and the numerical predictions obtained from the house code developed in this study, Moldflow simulations were carried out for three different conditions for mold temperature difference. The warpage of the FIM part predicted by Moldflow simulation is illustrated in Fig. 10a-c. From the above results, it is understood that higher mold temperature difference causes bigger deflection because of asymmetrical cooling rate conditions (9), (23). The effect of asymmetrical cooling rate on the thermally induced residual stress was verified by the numerical simulation. However, as shown in Fig. 11, the numerical simulation results overestimate the warpage of the FIM parts when compared with the experimental results. Finally, the bending moments predicted by the numerical simulations are compared with the experimental results as shown in Fig. 12. The predictions calculated by the house code underestimate the bending moment of the FIM part because the numerical calculation considers only the thermal residual stresses among total residual stresses, which are pressure induced, flow induced, and thermally induced residual stresses (20). In the case of actual experiments, the bending moment of the FIM specimen is often influenced by pressure history, such as injection, packing, and holding pressure variations as well as injection conditions such as the injection speed. Although thermal residual stresses and bending moments of the FIM parts were explained by using numerical analyses and experimental methods, a further study is needed to understand the effects of the inserted film and asymmetric heat flux conditions on the residual stresses generated during the FIM process.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

CONCLUSIONS

Thermally induced residual stresses in FIM parts were predicted by numerical simulation and bending moment of the FIM part was calculated and compared with the experimental results. Numerical modeling requires solution of thermal solidification problems and calculation of the stress integration through the part thickness. Although a one-dimensional thermoelastic model was considered with the assumption of constant material properties, it could provide an insight into the FIM process. It was found that many factors contribute to the residual stress distribution during the FIM process. Thermally induced residual stress distribution depended remarkably on the Biot number and heat was removed through the surfaces rapidly, thereby resulting in high residual stresses. The asymmetrical residual stresses generated by a nonuniform cooling rate through the part thickness leads to nonuniform shrinkage of the molded part. The bending moment of the FIM part was linearly decreased and warpagc was increased with respect to the temperature difference of the mold because the thermal strain mainly affects the total strain and the material cools down and shrinks inconsistently from the mold wall to the center of the FIM part.

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Hwa Jin Oh, (1) Young Seok Song, (2) Seung Hwan Lee, (1) Jae Ryoun Youn (1)

(1) Research Institute of Advanced Materials (RIAM), Department of Materials Science and Engineering, Seoul National University, 56-1, Silllim-Dong, Gwanak-Gu, Seoul 151-742, Korea

(2) Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Correspondence to: Jae Ryoun Youn: e-mail: jaeryoun@snu.ac.kr

Contract grant sponsor: The Korea Science and Engineering Foundation [Applied Rheology Center (ARC)].

DOI 10.1002/pen.21354

Published online in Wiley InterScience (www.interscience.wiley.com).

[C] 2009 Society of Plastics Engineers

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Author: | Jin, Hwa; Song, Young Seok; Lee, Seung Hwan; Youn, Jae Ryoun |
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Publication: | Polymer Engineering and Science |

Article Type: | Technical report |

Date: | Jul 1, 2009 |

Words: | 4804 |

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