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_ Received: March 2, 2006 Foundation item: Supported by National Natural Science Foundation Council of the Peoples Republic of China (20490224) Biography: XI Guo-dong(1978-), male, born in Kunshan city, Jiangsu province, Ph.D. student, main research field is CAE of injection molding of plastic. xiguodong 1 Introduction Injection molding is a flexible production technique for the manufacturing of polymer product. A typical injection molding process consists of four stages: filling, packing, cooling and ejection of product. During these processes, residual stress is produced due to high pressure and cooling, which induce the warpage and shrinkage of product. To make the product with high precision, it is important to simulate the pressure and thermal effect induced residual stress in the processes. Residual stress of injection molding has received more and more attention recently. Many researchers15 calculated the residual stress and shrinkage in the process by using different models. In general, the study of residual stress is focused on the theoretical analysis with an isolated model. It does not take into account the specifics of injection molding and correlate with the packing and cooling analysis. In this study, distribution and history of residual stress in plaque-like geometry is simulated based on linear thermoviscoelastic model, where the effects of pressure, thermal history and stress relaxation are jointly investigated. The numerical calculation model is built by finite difference method in time duration with layered discretization along thickness and under the specified boundary conditions. The developed simulation system is used to calculate the residual stress of typical injection molded parts. 2 Model Problem CADDM Vol.16 No.1 (2006) 17 Numerical Simulation of Residual Stress in Injection Molding XI Guo-dong, ZHOU Hua-min, LI De-qun (State Key Laboratory of Mold residual stress; numerical simulation; thermoviscoelastic model CLC number: TQ 320.66；O 345 Document code: A Article ID:1003-4951(2006)01-0017-07 2.1 Linear Thermoviscoelastic Model Temperature and pressure induced stresses arise from inhomogeneous cooling of the part in combination with the hydrostatic pressure. For cooling, the relaxation time increases and becomes comparable to the filling and packing time; therefore accurate prediction of thermal stress would require viscoelastic constitutive equations. But within the limit of infinitesimal strains variation the behavior of viscoelastic material is well described by the theory of linear viscoelasticity. Most injection mould products are very thin, we can take the flow direction as 1-dimensional and the local thickness direction as the 3-direction and build the local coordinates. The total stress tensor ij can be split into a hydrostatic pressure h P and an extra stress part ij 6 ijhij IP+=(1) d) )( ()( 1 t tt ttGP thm t h = (2) d )()()( d 2 t t ttGt ij t ij = (3) Where 1 G is bulk relaxation function, 2 G is shear relaxation function, th is thermal strain, m is spherical strain tensor, d ij extra strain tensor and )(t is the material time. 2.2 Basic Assumptions and Boundary Conditions For simplifying the calculation of thermally and pressure induced stress in the injection molding process, the following assumptions are made: (1) We take the flow direction as the 1-direction and the local thickness direction as the 3-direction and build the local coordinates; (2) While in the mold, the material is locally constrained in the in-plane directions by three dimensional features of the part (bosses, ribs and other walls), so the in-plane strain is assumed to be equal to 0; (3) The normal stress 33 is constant in the thickness direction of the product and equals the reverse of cavity pressure; (4) As long as the cavity pressure is non-zero (0 33 ), the material sticks to the mold walls; (5) Mold elasticity is neglected and the warpage of wokpiece in the mold is not taken into account. 3 Results and Discussion To study the evolution and distribution of stress, simulation is performed for a fan-gated specimen (300752.5 mm) as shown in Fig.1. A minimum complexity approach is proposed to emphasize the process physics, which would result in a better understanding of the transition that takes place during the molding. The material used was ABS (Novodur P2X of Bayer). The material parameters and the main processing parameters are listed in Table 1 and Table 2, respectively. Table 1 Material properties of ABS Material parameters Units ABS Material Brand Novodur P2X Producer Bayer Solidification temperature K 368 Thermal expansion coefficient 1/K 810-5 Youngs modulus MPa 2240 Shear modulus MPa 805 Poissons ratio 0.392 18 XI Guo-dong, ZHOU Hua-min, LI De-qun Table 2 Processing conditions for injection molding simulation Processing parameter Units Value Injection time s 1.2 Packing time s 6.3 Cooling time s 21.7 Melt temperature K 503 Mold temperature K 323 Packing pressure MPa 60 150mm Flow direction P1 Central position 300mm 75mm 2.5mm Fig.1 Rectangular strip mold 3.1 Evolution of Cavity Pressure According to the Basic Assumptions (3), normal stress equals the negative of the packing pressure in simulation. Therefore, the evolution of cavity pressure shows the variation of normal stress. Fig.2 shows the calculated pressure history at different locations along the flowpath, including the pressure profiles near the gate, quarter to the gate, halfway down to the end, quarter to the end and near the end of the flowpath. During filling, the pressure increases to overcome the increasing resistance to flow. At the end of the filling stage, the maximum pressure (63.42MPa) is near the gate and the minimum pressure (15.62MPa) is near the end of flow. There is an obvious difference between these two locations. When the packing stage begins, cavity pressure at all locations increases and is close to the setting packing pressure 60 MPa quickly. Moreover, the pressure distribution tends to be uniform. At the packing stage, material is forced into the cavity to compensate for the shrinkage that occurs during cooling. The cavity pressure mainly holds constant at first, then decays slowly as the frozen layer grows toward the core region. The location near the end is harder to compensate for shrinkage, where the cavity pressure drops more quickly. The cavity pressure tends to be nonuniform in the flowpath. At the end of packing, the pressure near the gate is 50.99MPa, while the pressure is just 36.75MPa near the end of flowpath. Then, cavity pressure all over Numerical Simulation of Residual Stress in Injection Molding 19 the part drops at the same speed as the continuous cooling. After cavity pressure drops to zero, the part is detached from the mold in the thickness direction and shrinks freely. The normal stress keeps zero. Fig.2 Calculated pressure history for specimen at different locations along the flowpath 3.2 Evolution of Residual Stress We call the stress in the flow plane “in-plane stress”. To the transversely isotropic material, x-direction stress equals y-direction stress. Fig.3 and Fig.4 show the in-plane stress distribution in the thickness direction and its evolution in the central position P1 plotted in Fig.1. Fig.3 shows the whole evolution of in-plane stress. Fig.4 shows stress gapwise distribution at five typical instants. They are: at the beginning of filling stage; at the end of filling stage; at the end of packing stage; when the pressure of the central position drops to zero; just before ejection. During filling, the melt material is forced into the mold. When the melt contacts the cold mold surface, a very thin layer solidifies quickly at the mold surface, in which the stress is induced as a result of material cooling. Below the surface, the stress distribution has a sharp decline. As shown in Fig.4, the stress remains uniform over a central region in which the material is essentially in a rubbery state with low elastic modulus and equals the negative of the packing pressure. As the packing stage starts, in-plane stress continues to decrease and quickly reaches the lowest value as the quick increasing of cavity pressure. During packing, packing pressure forces the melt material into the mold to compensate for the volumetric shrinkage that occurs during cooling. In-plane stress in the core region mainly holds constant at first, and then slowly decays as the cavity pressure. Moreover, the stress evolution corresponds to the evolution of the cavity pressure in Fig.2. The stress in the constrained solidified layers increases as a result of temperature decreasing. At the end of packing, the shape of the distribution changes and shows that relatively high stress has been built up in the surface layer, followed by a sharp decline in the region more inward. Stress in central region still remains uniform. Then, no more packing pressure is applied to the part and the cavity pressure drops quickly, which causes the compressive stress begins to decrease throughout the product. Just before ejection, the residual stress distribution consists of three distinct regions made up of two skins and a core. The stress exhibits a high surface tensile value and changes to a compressive peak value close to the surface, with the core region experiencing a parabolic peak. 20 XI Guo-dong, ZHOU Hua-min, LI De-qun Fig.3 In-plate stress gapwise distribution evolution Fig.4 In-plane stress distribution in the thickness direction at some typical instants 3.3 In-plane Residual Stress Distribution in Flowpath In the injection molding, the pressure history and temperature history at different locations are not the same, which causes the variation of the in-plane stress distribution. Fig.5 shows the predicted final in-plane stress gapwise distribution along the flowpath after ejection. The shape of the Numerical Simulation of Residual Stress in Injection Molding 21 Fig.5 Stress gapwise distribution along the flowpath after ejection stress gapwise distribution is almost the same along the flowpath. However, the central tensile stress near the gate is a little lower than the stress at the locations away from the gate. 4 Experimental Verification Measurements of residual stress are frequently carried out with the layer removal method. This method is relatively easy to apply to flat specimens and gives information about the stress distribution in the gapwise direction. Zoetelief5 successfully applied this method to polymers and investigated experimentally for specimens of amorphous polymer ABS. As shown in Fig.6, the calculated stress profiles are compared with the experimental values Fig.6 Calculated and measured gapwise stress distribution 22 XI Guo-dong, ZHOU Hua-min, LI De-qun in x direction and y direction. The experimental stress results are obtained from the Zoeteliefs paper. In the calculation, material and process conditions are the same to the Zoeteliefs experiment. There is a qualitative agreement between the measured and the calculated results. Moreover, to the amorphous polymer, the calculated stress in the in-plane direction is the same, since no anisotropy is taken into account. 5 Conclusion In the present study, we simulated the evolution and distribution of the cavity pressure and residual stress along the flowpath. Calculations with a viscoelastic mode are compared with experimental results obtained with the layer removal method for ABS specimens. Calculations show in-plane stress distribution in the thickness direction varies greatly in the injection molding process. At the beginning, there is considerable higher stress in the skin region, while the stress drops quickly below the surface and remains uniform over a central region. Just before ejection, the stress exhibits a high tensile value in surface and changes to a compressive peak value close to the surface, with the core region experiencing a parabolic tensile peak. During the filling and packing process, the shape of the gapwise stress distribution is the almost the same along the flowpath. At the beginning, there is a significant variation of the stress value along flowpath and then the stress tends to