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Flow-induced stresses
The total Cauchy stress tensor is the sum of an elastic and a viscous part: σ = σe+ σv. The elastic part is split into a hydrostatic part and a deviatoric part:σ = − pI + τ + σv . For a proper description of the polymer rheology, i.e., accurate residual stress predictions, a discrete number of relaxation times and their corresponding moduli are required; see Bird et al. (1987), Macosko (1994). Using a multi-mode description, the deviatoric part of the elastic extra stress tensor is given by:
in which Gj is the relaxation modulus of a specific mode, and Be the elastic Finger or conformation tensor.
The thermo-linear viscoelastic model is only solved for T ≤ Tg . Above the glass transition temperature, the residual stresses are isotropic and equal to minus the pressure in the melt. The computation of thermally and pressure-induced stresses can be substantially simplified by considering a series of assumptions commonly employed:
1. The material is assumed to stick to the mold for as long the pressure in the symmetry line remains positive.
2. Continuity of stress and strain at the solid melt interface.
3. The normal stress σ22, see Fig. 1, is constant across the part thickness and equals minus the pressure in the melt as long as the temperature at the symmetry line in the mold, T∗ , is larger than the glass transition temperature Tg.
. 4. In a coordinate system with the 22 direction perpendicular to the filling direction11 , the shear straincomponent ε12 is disregarded.
5. As long as pressure remains above zero, at the symmetry line, the only non-zero strain componentis ε22.
6. Solidification takes place when the no-flow temperature (Tg) is reached.
7. Mold elasticity is disregarded.
8. Frozen-in or flow-induced stresses can be neglected.
A detailed discussion on these assumptions is given by Baaijens (1991); we do not want to repeat all this in this paper except for the last two assumptions. By disregarding the mold, elasticity errors are introduced in the pressure history inside the mold cavity. This will, of course, have an effect on the development of the pressure-induced stresses. Baaijens (1991) showed that introducing mold elasticity slows the decay of the pressure inside the mold cavity, causing some changes in the final profile of the thermally and pressureinduced stresses. Regarding assumption 8, it was shown in Baaijens (1991) and Zoetelief et al. (1996) that the order of magnitude of flow-induced stresses is about 10平方 lower than the thermally and pressure-induced stresses. Moreover, the addition of flow-induced stresses to the total residual stresses is still a matter of some debate. Zoetelief et al. (1996) used the removal layer technique to measure residual stresses parallel and perpendicular to flow direction. They found a difference of less than 20%, suggesting that the influence of flow-induced stresses is small.
Injection molding case
To evaluate the development of flow-induced stresses in GAIM parts, we depart from an earlier study,Baaijens (1991), in which residual stresses were computed for injection-molded PC plates. Baaijens used a Hele–Shaw formulation to predict the flow kinematics and used the Leonov model to compute viscoelastic flow-induced stresses. In our study, we apply a fully 3-D model to the same injection problem, i.e., geometry, material, and processing conditions, to study the development of flow-induced stresses when using conventional or GAIM. Small differences are obviously expected when using a fully 3-D-based approach without the Hele–Shaw assumptions, which are: constant pressure across the mold thickness, negligible velocity in the thickness direction, small velocity gradients, and negligible thermal conduction parallel to the mid plane. The main difference, however, is that we capture the fountain flow, and the melt stretched at the flow front region will contribute to the final flow-induced stress profiles. To compare with the results of Baaijens, we have only to investigate a 2-D problem, of a cross section of the original 3-D geometry of 80 × 2 mm (length,height); see Fig. 1. The gate is located at the channel entrance. In Baaijens (1991), flow-induced stresses were computed in an injection-molded PC plate. Next, we study the effect of GAIM on flow-induced stresses, using the same material, processing conditions, and geometry. Again, we approximate the original 3-D geometry to a 2-D problem by taking a cross-section along the channel’s length; see Fig. 1. The gate occupies the total height of the channel.