Where G1 is the relaxation shear modulus of the material. The dilatational stresses can be related to the strain as follows:

Where K is the relaxation bulk modulus of the material, and the definition of α and Θ is:

If α(t) = α0, applying Eq. 27 to Eq. 29 results in:

Similarly, applying Eq. 31 to Eq. 28 and eliminating strain εxx(z, t) results in:

Employing a Laplace transform to Eq. 32, the auxiliary modulus R(ξ) is given by:

Using the above constitutive equation (Eq. 33) and simplified forms of the stresses and strains in the mold, the formulation of the residual stress of the injection molded part during the cooling stage is obtain by:

Equation 34 can be solved through the application of trapezoidal quadrature. Due to the rapid initial change in the material time, a quasi-numerical procedure is employed for evaluating the integral item. The auxiliary modulus is evaluated numerically by the trapezoidal rule.

For warpage analysis, nodal displacements and curvatures for shell elements are expressed as:

where [k] is the element stiffness matrix, [Be] is the derivative operator matrix, {d} is the displacements, and {re} is the element load vector which can be evaluated by:

The use of a full three-dimensional FEM analysis can achieve accurate warpage results, however, it is cumbersome when the shape of the part is very complicated. In this paper, a twodimensional FEM method, based on shell theory, was used because most injection-molded parts have a sheet-like geometry in which the thickness is much smaller than the other dimensions of the part. Therefore, the part can be regarded as an assembly of flat elements to predict warpage. Each three-node shell element is a combination of a constant strain triangular element (CST) and a discrete Kirchhoff triangular element (DKT), as shown in Fig. 3. Thus, the warpage can be separated into plane-stretching deformation of the CST and plate-bending deformation of the DKT, and correspondingly, the element stiffness matrix to describe warpage can also be pided into the stretching-stiffness matrix and bending-stiffness matrix.

Fig. 3a–c. Deformation decomposition of shell element in the local coordinate system. a In-plane stretching element b Plate-bending element c Shell element

3 Experimental validation

To assess the usefulness of the proposed model and developed program, verification is important. The distortions obtained from the simulation model are compared to the ones from SL injection molding experiments whose data is presented in the literature [8]. A common injection molded part with the dimensions of 36×36×6 mm is considered in the experiment, as shown in Fig. 4. The thickness dimensions of the thin walls and rib are both 1.5 mm; and polypropylene was used as the injection material. The injection machine was a production level ARGURY Hydronica 320-210-750 with the following process parameters: a melt temperature of 250 ◦C; an ambient temperature of 30 ◦C; an injection pressure of 13.79 MPa; an injection time of 3 s; and a cooling time of 48 s. The SL material used, Dupont SOMOSTM 6110 resin, has the ability to resist temperatures of up to 300 ◦C temperatures. As mentioned above, thermal conductivity of the mold is a major factor that differentiates between an SL and a traditional mold. Poor heat transfer in the mold would produce a non-uniform temperature distribution, thus causing warpage that distorts the completed parts. For an SL mold, a longer cycle time would be expected. The method of using a thin shell SL mold backed with a higher thermal conductivity metal (aluminum) was selected to increase thermal conductivity of the SL mold.

Fig. 4. Experimental cavity model

Fig. 5. A comparison of the distortion variation in the X direction for different thermal conductivity; where “Experimental”, “present”, “three-step”, and “conventional” mean the results of the experimental, the presented simulation, the three-step simulation process and the conventional injection molding simulation, respectively.