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Residual stresses in gas-assisted injection molding Abstract Residual stresses are a major issue in the mechanical and optical behavior of injection-molded parts. In this study, we analyze their development in the case of gas-assisted injection molding (GAIM) of amorphous polymers. Flow-induced residual stresses are computed within a decoupled approach, in which elastic effects are neglected in the momentum balance, assuming a generalized Newtonian material behavior. In a staggered procedure, the computed viscous flow kinematics are used to calculate normal stresses employing a compressible version of the Rolie-Poly model.For the computation of thermally and pressure-induced residual stresses, a linear thermo-viscoelastic model is used. A 3-D finite element model for GAIM is employed, which is able to capture the kinematics of the flow front and whose capabilities to predict the thickness of the residual material layer have been validated by Haagh and Van de Vosse (Int J Numer Methods Fluids 28:1355–1369, 1998). In order to establish a clear comparison, the development of residual stresses is analyzed using standard injection molding and GAIM for a test geometry.37190
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Keywords Gas-assisted injection molding •Viscoelasticity •Compressible Rolie-Poly model •Flow-induced residual stresses •Thermally and pressure-induced residual stresses • Amorphous polymers • Numerical analysis
Governing equations for the injection molding problem
The balance equations for mass, momentum, and energy are now presented and simplified with respect to the process requirements and modeling assumptions.
The related constitutive equations and boundary conditions are given and justified. The general form of the balance equations for mass, momentum, and internal energy read:
where ρ represents density and u the velocity field, σ the Cauchy stress tensor, g is the body force per unit mass, and e˙ is the rate of change of internal energy. The terms on the right-hand side of the energy equation, Eq. 3, represent the work done to deform the material, with D the rate of deformation tensor, the heat transferred by conduction, with q the heat flux, the heat transferred by radiation, r, and internal heat generation with Rc the reaction rate, and hr the reaction heat. To solve these equations, appropriate constitutive equations have to be specified for the Cauchy stress tensor, the heat flux, and an equation of state for the density and internal energy, i.e., e = e( p, T ), where p and T denote pressure and temperature, respectively, introduced in the forthcoming section. Additionally, initial and boundary conditions have to be prescribed. We now state the basic assumptions to simplify the above equations and to motivate the choice for constitutive relations.
Assumptions
The assumptions given below are quite standard for injection molding. Justification of these can be found in literature (Baaijens 1991; Douven 1991; Douven et al.1995; Haagh and Van de Vosse 1998) and are discussed in some detail further on when appropriate.
− Compressibility effects are negligible during the filling phase.
− Flow kinematics are determined by kinematic boundary conditions.
− The melt behaves according to a generalized Newtonian flow description.
− Inertial effects are negligible.
− Thermal radiation is negligible.
− No heat source is present.
− Heat generated due to compression is negligible.
− Isotropic heat conduction.
It is known that, upon processing, the thermal conductivity of polymers is increased in the flow direction and decreased in the direction normal to the flow. Such observations were reported by, e.g., Hansen and Bernier (1972) and van den Brule and O’Brien (1990) and explained on the basis that the heat conductance is much higher along covalent bonds than throughout weak secondary bonds. Furthermore, the anisotropy resulting from the frozen-in molecular orientation can differ substantially according to the molecular structure of the polymer, being favored by linear compact polymers, and for polymers with higher relaxation times and molecular weight. All this is not taken into account due to the lack of experimental data. According to the26 Rheol Acta (2010) 49:23–44 above assumptions, the governing equations are simplified, yielding during the filling phase an incompressible Stokes flow problem.