Incorporation of density relaxation in the analysis of residual stresses in molded parts.
1. INTRODUCTIONPrediction of the residual stresses and post-molding deformation is one of the most challenging tasks in injection molding. Several attempts have been made to estimate the residual stresses and shrinkage/warpage in injection molded parts. A review of some of the significant attempts is presented by Bushko and Stokes (1, 2) and Santhanam (3). Most of these attempts have incorporated a simplistic material model. St. Jacques (4), Tamma et al. (5), and Maruyama et al. (6) applied an elastic material-behavior model to predict the residual stresses and warpage in injection-molded parts of simple geometry. Using the elastic model, Jansen (7) studied the residual-stress-formation mechanisms for free quenching (extrusion) and constrained quenching with varying applied pressure (injection molding). He showed that the driving force for the stress formation during an extrusion process is of a thermal nature. In the case of injection molding, he found that the driving force is the pressure.
Using an isotropic, thermo-rheologically simple viscoelastic material whose dilatational behavior is elastic, Crouthamel and Isayev (8) compared the experimental stresses due to free quenching of amorphous polymers with theoretical predictions. Santhanam (3) adopted an analogous model to study the residual-stress development and post-molding deformation in a strip cavity. The analysis was also extended to thin three-dimensional molded parts, making use of a local treatment. This analysis shows that the equilibrium stress variation across the thickness of a thin injection-molded strip exhibits a high surface tensile value changing to a compressive peak value close to the surface with the core region experiencing a flattened tensile peak. This stress profile was found (3) to be relatively insensitive to the molding conditions such as the mold-wall temperature, melt temperature, flow rate, holding pressure and axial location. In general, the results agreed qualitatively with the measured stress distribution. However, the experimental stress results demonstrated a much lower value for the surface tensile-stress, a parabolic tensile-stress distribution in the core and a greater sensitivity to molding conditions. In an attempt to improve the predicted stress profile, Santhanam implemented a partially constrained condition at the surface of the mold. This moderated the flattened tensile stress in the core, rendering a parabolic stress distribution. However, it did not reduce the high value of the surface tensile stress nor did it increase the sensitivity of the stress results to molding conditions. Bushko and Stokes (1, 2) utilized the problem of solidification of a molten polymeric layer between cooled parallel plates to study the mechanics of part shrinkage and warpage and the buildup of residual stresses in the injection molding process. An orthotropic thermo-rheologically simple viscoelastic model was assumed (1). The model allows molten material to be added to fill up the space created by the compression caused by the packing pressure during solidification such that packing pressure effects can be assessed. Bushko and Stokes applied the model to conduct a parametric study (2) on the effects of the mold temperature, melt temperature, part thickness and packing pressure on the in-plane and through-thickness shrinkage and on the residual stresses. The packing pressure was shown to have a dramatic effect on part shrinkage but smaller effect on residual stresses. Mold and melt temperatures were shown to have a much smaller effect. The parametric study, however, was limited to the case of isotropic, elastic material-behavior in dilatation with constant thermal properties, i.e., the effect of density relaxation was ignored.
The objective of this paper is to investigate the effect of density relaxation on the residual-stress profile developed upon injection molding a simple one-dimensional cavity. To account for the density relaxation, viscoelastic constitutive equations are adopted for the material-behavior of the polymer in both the deviatoric and dilatational domains. Various combinations of the relaxation function (sum-of-exponential function and b-function) and the shift factor (WLF and Gibbs equations) are proposed. In order to check the validity of the developed constitutive equations and the proposed functions, they are applied to predict the isobaric density relaxation following a sudden quench from above the glass-transition temperature and upon constant rates of cooling of an amorphous polymer (polystyrene). Using the sum-of-exponential function (Prony series) and the modified Gibbs equation, the proposed constitutive equations are applied to predict the residual-stress profile in a strip cavity. Though some comparison of the predicted density relaxation with experimental results are presented, a qualitative investigation of the effect of incorporating the density relaxation on the predicted residual stress in a one-dimensional post-molded part is the focus of this paper.
2. THE CONSTITUTIVE EQUATIONS
2.1. Shear Stress and Dilatation Strain
The polymer is assumed to be an amorphous, isotropic, and thermo-rheologically and piezo-rheologically simple viscoelastic material in both the deviatoric and dilatational domains; i.e.,
[S.sub.ij] = [integral of] 2[Mu](t - [Xi]) [Delta][e.sub.iy]/[Delta][Xi] d[Xi] between limits t and 0 (1)
[[Epsilon].sub.v] = [integral of] J(t - [Xi]) [Delta][[Sigma].sub.m]/[Delta][Xi] d[Xi] between limits t and 0 + [integral of] [Alpha](t - [Xi]) [Delta]T/[Delta][Xi] d[Xi] between limits t and 0 (2)
where T is the absolute temperature, [S.sub.ij] is the deviatoric stress tensor, [e.sub.ij] is the deviatoric strain tensor, [[Epsilon].sub.v] is the dilatational strain, [[Sigma].sub.m] is the mean stress, [Mu](t) is the sheer-relaxation modulus, J(t) is the dilatational-creep compliance, and [Alpha](t) is the volumetric "thermal-expansion" modulus.
In addition,
[Mu](t) = [[Mu].sub.1] - ([[Mu].sub.1] - [[Mu].sub.g])[M.sub.1](t) (3)
J(t) = [[Kappa].sub.1] - ([[Kappa].sub.1] - [[Kappa].sub.g])[M.sub.2](t) (4)
[Alpha](t) = [[Alpha].sub.1] - ([[Alpha].sub.1] - [[Alpha].sub.g])[M.sub.2](t) (5)
where [[Mu].sub.1], [[Kappa].sub.1], and [[Alpha].sub.1] are the shear modulus, compressibility, and volumetric thermal-expansion coefficient for the liquid state, respectively. Similarly, [[Mu].sub.g], [[Kappa].sub.g], and [[Alpha].sub.g] have the same definitions but for the glassy state. [M.sub.1](t) and [M.sub.2](t) are the shear-relaxation and the dilatational-creep functions, respectively. They assume values between 0 and 1 (0 [less than or equal to] M(t) [less than or equal to] 1), with M = 0 in the liquid state and M = 1 in the glassy state. Two functions are suggested for the M(t)'s:
1 - The sum-of-exponential function
M(t) = [summation over i][w.sub.i]Exp{- [integral of] dt[prime]/[[Tau].sub.i] between limits t and 0} (6a)
2 - The b-function (Williams-Watts equation)
M(t) = Exp{-[([integral of] dt[prime]/[[Tau].sub.c] between limits t and 0).sup.b]} (6b)
where [w.sub.i] and [[Tau].sub.i] are the weighting parameter and the relaxation or retardation time of mode i, [[Tau].sub.c] is the critical time and b is a parameter (0 [less than] b [less than] 1). The b-function involves only two material parameters but is nonlinear and can lead to some numerical difficulties. Note that, for the sake of brevity, M(t) in the above-mentioned analysis represents either the shear-relaxation function, [M.sub.1](t), or the dilatational-creep function, [M.sub.2](t). Consequently, it is understood that there are two sets of ([w.sub.i], [[Tau].sub.i], [[Tau].sub.c], and b), namely for the shear relaxation [([w.sub.i], [[Tau].sub.i], [[Tau].sub.c], and b).sub.s] and for the dilatational creep [([w.sub.i], [[Tau].sub.i], [[Tau].sub.c], and b).sub.D]. The integral [integral of] dt[prime]/[Tau] between limits t and 0, for both [[Tau].sub.i] and [[Tau].sub.c], is introduced to accommodate the variation of [Tau] with time during general non-isothermal, non-isobaric processes.
The compressibilities ([[Kappa].sub.1] and [[Kappa].sub.g]) and the thermal-expansion coefficients ([[Alpha].sub.1] and [[Alpha].sub.g]) are assumed to be functions of the hydrostatic pressure and the temperature. Utilizing the Tait equation (9), v(T, p) = [v.sub.0](T){1 - C ln(1 + p/B(T))}, we can get
[Alpha](p, T) = [C.sub.2]/[v.sub.0](T) - [CC.sub.4] [v.sub.0](T)/v(p/p + B(T))
and
[Kappa](p, T) = [v.sub.0](T)/v(C/p + B(T))
where [v.sub.0](T) = [C.sub.1] + [C.sub.2]T, and B(T) = [C.sub.3]Exp{-[C.sub.4]T}.
In the above equations, C is the empirical universal Tait constant (C = 0.0894) and the [C.sub.i]'s are material parameters. Notice that the subscripts g and 1 are omitted for the sake of brevity; therefore, it is understood that there are two sets of [C.sub.i]'s (for the glassy and the liquid states). It should be pointed out that, as will be stated later, the variation of the pressure and temperature with time are assumed known at all material points (layers) for the entire injection molding process. Consequently, the values of the compressibility and thermal-expansion coefficient for both the melt and glassy states are considered known throughout the entire process as well.
2.2. The Relaxation and Retardation Times
The relaxation/retardation times, [[Tau].sub.i]'s, and the critical time, [[Tau].sub.c], for both the deviatoric and dilatational domains, are assumed to be functions of the temperature, mean stress and the structural state. For a simple thermo-rheological/piezo-rheological material, we have
[Tau] = a(T, p, [Beta])[[Tau].sub.r] (7)
where [Tau] stands for either [[Tau].sub.i] or [[Tau].sub.c] in any state, [[Tau].sub.r] is the value of [Tau] in a reference state and a(T, p, [Beta]) is the shift factor (function) where [Beta] is a state variable that represents the structural state of the material. Since the molecular motions are probed by different means in the shear (deviatoric) and dilatational modes, the respective distributions of the relaxation times are expected to be different. However, Espinoza and Aklonis (10) have argued that since the underlying molecular modes for both distributions are the same ones, they should move in an identical manner. Consequently, the same shift factor can be assumed for both the deviatoric and dilatational modes.
Equations 6 and 7 imply that the shear-relaxation function as well as the dilatational-creep function obtained under isothermal, isobaric and "isostructural" (same [Beta]) conditions has the same shape at every thermo-mechanical state (T, p, [Beta]) when plotted on a logarithmic time scale. Also, the shear-relaxation functions as well as the dilatational-retardation functions may be brought into coincidence by a parallel shift along the logarithmic time scale. The amount of this shift is log{a(T, p, [Beta])}. For general thermo-mechanical processes, we assume that shifting under non-isothermal, non-isobaric and non-isostructural conditions will be valid instantaneously. This assumption renders Eqs 6 and 7 applicable for any thermomechanical process. It may be worthwhile mentioning that upon substituting Eqs 6 into 7, alternative expressions for M(t) can be obtained:
M(t) = [summation over i] [w.sub.i] Exp{-[Zeta]/[[Tau].sub.i,r]} for the sum-of-exponential function, and
M(t) = Exp {-[Zeta]/[[[Tau].sub.c,r]).sup.b]} for the b-function,
where [Zeta] is known as the reduced time (11, 12), [Zeta] = [integral of] dt[prime]/a(T, p, [Beta]) between limits t and 0
In particular, two shift factors are investigated:
1. The modified Williams-Landel-Ferry (WLF) equation (13), which is based on the fractional-free volume theory;
a = Exp{Q/[f.sub.[infinity]] - Q/[f.sub.r]} Exp{-Q[Delta]/[f.sub.[infinity]]([f.sub.[infinity]] + [Delta])} (8a)
where [f.sub.r] is the fractional free volume at the reference state, [f.sub.[infinity]] is the fractional free volume at the equilibrium state, [f.sub.[infinity]] = [f.sub.r] + ([[Alpha].sub.1] - [[Alpha].sub.g])(T - [T.sub.r]), Q is a material constant and [Delta] is a measure of the fractional free volume deviation from equilibrium, [Delta] = (v - [v.sub.[infinity]])/[v.sub.[infinity]], where v is the specific volume.
2. The modified Gibbs function (14),
a = Exp{Q/T/1 - [T.sub.2]/[T.sub.f] + [C.sub.0][[Sigma].sub.m] - Q/[T.sub.r] - [T.sub.2]} (8b)
where Q and [C.sub.0] are material constants, [T.sub.2] is the ground-state temperature at which the configurational entropy extrapolates to zero, [T.sub.r] is the temperature at the reference state, and [T.sub.f] is the fictive temperature, which represents the structural state of the material ([T.sub.f] [equivalent to] [Beta]). The fictive temperature can be expressed as (15)
[T.sub.f] = [T.sub.0] + [integral of] {1 - [M.sub.2](t - [Xi])} [Delta]T/[Delta][Xi] d[Xi] between limits t and 0 (9)
It is noted that the WLF equation has been successfully used before to predict the shear creep (16) and relaxation (17) behavior of amorphous polymers during physical aging. However, Matsuoka (18) showed that the Gibbs equation is more satisfactory in some cases. Consequently, both equations are investigated in the present density-relaxation analysis.
3. DENSITY-RELATION ANALYSIS
Since the objective of this study is to investigate the effect of density relaxation on the development of residual stresses in post-molded parts, a preliminary investigation of the volume-relaxation prediction of the proposed material-behavior model is first presented. Volume relaxation following a sudden quench from the equilibrium melt to a temperature below the glass transition, [T.sub.g], and due to constant cooling rate through [T.sub.g], will be studied for different combinations of the relaxation function, M(t), and the shift factor, a(T, p, [Beta]). Comparisons with corresponding available experimental data will be presented for some commercial-grade polystyrenes in order to assess the best combination of M(t) and a(T, p, [Beta]).
Predictions and corresponding experimental results for volume relaxation following a sudden quench are shown in Fig. 1 whereas results for constant cooling rate are shown in Fig. 2. The experimental results in both cases are from Greiner and Schwarzl (19). Three combinations have been used for the analytical results: the b-function with the modified Gibbs (BFN/GIB), the sum-of-exponential function with the modified Gibbs (EXP/GIB), and the sum-of-exponential function with the modified WLF equation (EXP/WLF). The material properties used are given in Table 1. Tait's constants have been obtained from Hieber (20), whereas the best-fit model parameters are the optimum parameters that minimize the root-mean-square error between the respective model and corresponding experimental results for the sudden quench shown in Fig. 1. In general, good agreement between the experimental and the analytical results is obtained, with the EXP/WLF combination being the least accurate (Table 1b). For constant rates of cooling [ILLUSTRATION FOR FIGURE 2 OMITTED], all three models predict much faster rates of relaxation, particularly at higher rates of cooling. For the volume relaxation under a constant rate of cooling at different pressures, all combinations perform well. In particular, Fig. 3 displays predictions based upon the EXP/GIB combination as applied to data from (21). The corresponding optimum parameters are presented in Table 1c. Though the BFN/GIB combination, in general, performed slightly better than the EXP/GIB combination, the latter is selected for the residual-stress analysis presented in the next section in order to avoid possible numerical complication associated with the nonlinearity of the b-function.
[TABULAR DATA FOR TABLE 1 OMITTED]
4. RESIDUAL-STRESS ANALYSIS
4.1. Assumptions
For the determination of the evolution of the residual stresses in post-injection molded thin parts, the following assumptions are considered:
1. The melt pressure and temperature history are available.
2. The stresses and the strains in the polymer are sufficiently small that the linear constitutive equations described in Section 2 are applicable.
3. The shear-relaxation and dilatational-creep functions are described by the sum-of-exponential function, and the shift factor obeys the modified Gibbs function.
4. A one-dimensional scenario; i.e., there are no variations in the directions orthogonal to the thickness direction (1 and 2 directions in [ILLUSTRATION FOR FIGURE 4 OMITTED]).
5. The melt pressure is constant across the thickness (lubrication theory).
6. The shear stresses in the material are negligible.
4.2. Outline of the Numerical Technique
Based on the above-mentioned assumptions, Eqs 1 and 2 become
[Mathematical Expression Omitted] (10)
[Mathematical Expression Omitted] (11)
where [Sigma] ([Sigma] = [[Sigma].sub.11] = [[Sigma].sub.22]) and [Epsilon]([Epsilon] = [[Epsilon].sub.11] = [[Epsilon].sub.22]) are the lateral stress and strain, respectively. At a given location along the cavity, if one knows the temperature distribution and the pressure history of the material points across the thickness, Eqs 10 and 11 can be solved simultaneously at each material point (layer) over the time history in a step-by-step manner. During this time the polymer goes from the melt state, through the transition state, to the glassy state. It is interesting to note that the constitutive equations, Eqs 10 and 11, together with the complementary equations, Eqs 3-9, are applicable throughout all three stages.
When the material point is in the melt (liquid) state, the relaxation and retardation times are negligible ([[Tau].sub.imax] [much less than] 1), and consequently we have:
* From Eq 6a, [M.sub.1] [approximately equal to] [M.sub.2] [approximately equal to] 0.
* From Eqs 3-5, [Mu] [approximately equal to] [[Mu].sub.1], [approximately equal to] 0, J [approximately equal to] [[Kappa].sub.1], and [Alpha] [approximately equal to] [[Alpha].sub.1].
* From Eqs 10 and 11, [Sigma] [approximately equal to] -p and [[Epsilon].sub.v] [approximately equal to] [[Kappa].sub.1]([p.sub.0] - p) + [[Alpha].sub.1](T - [T.sub.0]).
As the polymer cools, the shift factor, a, increases and consequently the relaxation and retardation times, [[Tau].sub.i]'s, increase. When a reaches a critical value, [a.sub.c], the process of vitrification starts, at which time the constrained-end condition is imposed. This constrained boundary condition implies that the lateral deformation of the polymer, while still in the mold, is restricted. The assumption is relevant in the case of injection molding large, complex structures in which the lateral contractions and/or expansions of the cooling polymer are restricted by abrupt geometric changes. The melt pressure also pins the vitrified polymer against the wall, further restricting the lateral motion (3). During the vitrification process, Eqs 10 and 11 are solved numerically in a step-by-step manner. At each time step, [t.sup.j], the stress at each layer is evaluated as follows:
* A mean stress value, [Mathematical Expression Omitted], is assumed.
* Knowing the time increment, [Delta]t = [t.sup.j] - [t.sup.j-1], the current value of the pressure, [p.sup.j], and the temperature, [T.sup.j]; the fictive temperature, [Mathematical Expression Omitted], the shift factor, [a.sup.j], the relaxation times, [Mathematical Expression Omitted]'s, and the retardation times, [Mathematical Expression Omitted]'s, are evaluated using Eqs 7-9 in conjunction with Eq 6.
* The stress, [[Sigma].sup.j], and the deviatoric strain, [Mathematical Expression Omitted], are computed using Eqs 10 and 11 in conjunction with Eqs 3-5.
* A new value of [Mathematical Expression Omitted] is obtained, [[Sigma].sub.m] = (1/3)(2[Sigma] - p), and the above steps are repeated until convergence is achieved.
Upon ejection of the part, the constrained-end boundary condition is removed and the lateral force per unit width, F, vanishes; i.e., F = [Sigma] [[Sigma].sup.(l)] [Delta][h.sup.(l)] [approximately equal to] 0, where [[Sigma].sup.(l)] and [Delta][h.sup.(l)] are the lateral stress and thickness of layer "l", respectively. In addition, we assume that the cross-sectional plane, across the thickness, remains a plane; i.e., [Delta][[Epsilon].sup.(l)] = [Delta][Epsilon] = constant. Consequently, at each time step, the numerical computation of the stress proceeds as follows:
* An incremental strain [Delta][[Epsilon].sup.j] is assumed.
* The value of the mean stress at each layer, [Mathematical Expression Omitted], as well as [{[[Sigma].sup.(l)]}.sup.j] is computed using the algorithm described in the pre-ejection vitrification process.
* The value of the axial force, [F.sup.j], is calculated.
* Based on the computed value of [F.sup.j], a new value of [Delta][[Epsilon].sup.j] is evaluated and the procedure is repeated until [F.sup.j] vanishes (becomes very small).
It should be mentioned that numerical solution of the convolution integrals in Eqs 9-11 is carried out using the algorithm described by Scherer (12), and that the melt temperature and pressure histories are determined using a special code developed under the Cornell Injection Molding Program (22).
4.3. Boundary Condition Assumptions
It is important to emphasize that the current analysis assumes the pressure and temperature histories are known for the entire injection molding process including filling, packing and post ejection, and that upon vitrification of a material point (layer) the fixed-end boundary condition is imposed until demolding. These assumptions are valid only under certain conditions and may lead to contradictions if applied indiscriminately.
During packing, the pressure can be controlled only when the molten resin is connected to the barrel through a continuous molten core. That is, when the part is partially solidified with the core layers in the molten state and the outer layers in the solidified state. For each of these outer layers, taking into consideration that the lateral strain, [Epsilon], is fixed (equal to the strain at vitrification) and [[Sigma].sub.m] = (2[Sigma] - p)/3, Eqs 10 and 11 can be solved simultaneously for the unknowns [Sigma] and [[Epsilon].sub.v]. This implies that the through-thickness strains and consequently shrinkage of the solidifying layers are fully determined ([[Epsilon].sub.z] = [[Epsilon].sub.v] - 2[Epsilon]). The through-thickness shrinkage is compensated for by the molten material added to the molten core from the barrel during packing. However, when the core vitrifies and all layers are in the solidified state (part is completely solidified), the total through-thickness strain as well as [[Epsilon].sub.v] becomes constrained and assumes a fixed value. Consequently, the pressure can no longer be prescribed but rather solved for, using Eqs 10 and 11, till it vanishes (1, 2). Prescribing the pressure during that time (from core vitrification till p = 0) would cause a contradiction: an over-constrained problem. In our current analysis, the pressure and temperature histories are obtained from a special (PACK1D) mold-filling simulation program (22), which considers the core-solidification effect but not the viscoelastic and volume-relaxation effects. Therefore, it is expected that the above-mentioned contradiction, though still existing, is alleviated in the current analysis. A complete remedy of this contradiction would be accomplished by integrating the current analysis into the mold-filling simulation program. This task is the target of future work.
Furthermore, when the pressure drops to zero during packing and before demolding, the through-thickness-strain constraint is released and Eqs 10 and 11 can be solved once more for [Sigma] and [[Epsilon].sub.v]. However, in the absence of the pinning pressure and for a simple strip cavity, it becomes possible for the solidified layers to move freely in the in-plane direction upon further cooling. Therefore, it may be more appropriate to apply the free-end lateral boundary-condition assumption during the period from p = 0 till demolding. This assumption is not considered in this paper but is currently being investigated and will be reported in a future work. The preliminary results of the investigation indicate that applying the free-end boundary condition instead of the fixed-end one, during the period from p = 0 till demolding, promotes the development of the tensile parabolic residual-stress profile in the core region.
4.4. Results and Discussion
Numerical predictions of the evolution of the thermal stresses in a rectangle cavity (10 x 2 x 0.2 cm) have been evaluated. The effect of molding conditions and material parameters have also been investigated, with illustrative results presented in Figs. 6-12. For purposes of comparison, we have chosen the reference case to be: flow rate = 20 [cm.sup.3]/sec, melt temperature = 230 [degrees] C, wall temperature = 60 [degrees] C, holding pressure = 11.55 Mpa (1732 psi), ejection time = 7.6 sec, and an axial location, x = L/2, where L is the axial length of the cavity. The material properties used are given in Table 2. These material properties are representative of commercial-grade polystyrenes (3). The relaxation and retardation spectra of the shear modulus and the dilatational compliance are shown in Fig. 5, evaluated at a reference temperature of [T.sub.r] = 112 [degrees] C. The material [TABULAR DATA FOR TABLE 2 OMITTED] constants of the shear-relaxation spectrum have been obtained from (3) and the dilatational-spectrum constants are the same as for the PS N7000 6-parameter model of Greiner and Schwarzl (17). It should be pointed out that the results and analysis presented here are strictly qualitative, though an effort has been made to use material properties that are as representative as possible.
Figure 6 displays the predicted evolution of stress across the thickness, at different instants during the molding cycle, for the reference case. During the early stage of molding, the stress profile assumes two distinct regions: 1) an exterior region where the stress is tensile with a maximum at the surface, and 2) an interior region where the temperature is hot and the polymer is in the melt state such that the corresponding stress profile is flat with a negligible compressive value. As time progresses and cooling continues, the exterior region expands inward at the expense of the interior region. A plateau behavior, however, emerges in an intermediate region. This is because the layers closer to the surface vitrify at a higher pressure than the layers further inward. The frozen-in pressure tends to counteract the shrinkage due to the fall in temperature, rendering a local drop in the tensile stress and consequently a local peak in the inward neighboring layer (3). During post-ejection, the free-end condition is imposed and the residual stresses are equilibrated, engendering a quasi-flat tensile stress profile in the interior region with a concave compressive-stress behavior closer to the wall and a maximum tensile stress at the surface. Since the interior region is still hot upon ejection [ILLUSTRATION FOR FIGURE 7 OMITTED], the residual stresses continue to develop in this region in a manner similar to the unconstrained-quench case, owing to the free-end condition, producing a parabolic tensile-stress profile with a maximum stress at the center.
The temperature, shear modulus and shift factor profiles at ejection time (t = 7.6 sec), for the reference case, are displayed in Fig. 7. It clearly depicts the two regions: the relatively stiff/cold exterior region and the soft/hot interior region. This Figure also suggests that the interior region, within which the residual stresses continue to develop after ejection, can be defined in terms of the value of the shift factor, such as a [less than] 100 (a [less than] 40 dB).
The effect of ejection time, axial location and average dilatational retardation time on the "equilibrated stress" profile is demonstrated in Figs. 8, 9, and 11, respectively, where the thick-solid curve represents the reference case. Figure 8 demonstrates that delaying ejection, in general, alleviates the severity of the residual stresses, especially in the core region. As the ejection time increases, the polymer is allowed more time for the temperature to drop and the shift factor to increase, causing a diminishing of the soft/hot interior region and consequently an inhibition of the development of the residual stresses in the core region after ejection. The axial location of the material point, in general, does not significantly affect the "equilibrated stress" profile [ILLUSTRATION FOR FIGURE 9 OMITTED]. Near the inlet, however, the concave compressive stress of the exterior region expands inward and the parabolic tensile stress profile of the interior region becomes more pronounced. Near the inlet, the holding pressure is sustained for a longer period of time than further away from it, as shown in Fig. 10, causing the frozen-in pressure to be distributed over a wider region across the thickness and thereby producing a wider concave compressive zone. Also, because of the continuing replenishment of the melt from the nearby inlet, the Interior region remains hotter and softer longer, rendering a more pronounced parabolic tensile stress in the core.
Increasing the average dilatational retardation time, [[Tau].sub.D], i.e., shifting the retardation spectrum towards the right in Fig. 5, generally reduces the level of the residual stresses and tends to eliminate the high surface tensile stress [ILLUSTRATION FOR FIGURE 11 OMITTED]. Also, increasing [[Tau].sub.D] tends to expand the parabolic tensile-stress region in the core at the expense of the concave compressive-stress region nearer the surface. The relationships among [[Tau].sub.D], [T.sub.f] and a explain the behavior demonstrated in Fig. 11. The fictive temperature, [T.sub.f], representative of the structural configuration, depends primarily on the dilatational relaxation time, [[Tau].sub.D], as can be seen from Eqs 6 and 9. As [[Tau].sub.D] increases, the drop in the value of [T.sub.f] with decreasing temperature gets smaller (structural orientation becomes more cumbersome), producing a smaller shift factor, a. Figure 12 shows the shift-factor profile at the ejection time for the average retardation times presented in Fig. 11. Clearly, as [[Tau].sub.D] increases the shift factor, a, across the thickness generally drops and the soft/hot interior region expands at the expense of the exterior region, engendering a flattening down and expansion of the parabolic residual stresses in the core. For [[Tau].sub.D] = 2.756 sec, the soft/hot region almost covers the entire cross section and consequently the residual stresses evolve in a manner similar to the free-quench case. The results in Fig. 11 demonstrate the importance of incorporating the viscoelastic dilatational material-behavior (density relaxation) on the development of the thermal residual stresses in injection molded parts.
5. CONCLUSIONS
Using thermo-rheologically/piezo-rheologically simple viscoelastic constitutive equations for the material-behavior of a generic polystyrene in both the deviatoric and dilatational domains, the development of the thermal residual stresses in a one-dimensional injection-molded part has been studied. The study shows that the predicted stress distribution, generally, agrees with established measured stress profiles. The predicted profile exhibits two regions:
* A parabolic tensile stress profile in the core (interior region), due to the post-ejection free-end vitrification.
* A concave compressive stress profile near the surface (exterior region), due to the frozen-in pressure at vitrification.
The study also indicates that the relative shapes and magnitudes of the two regions of the stress profile, as experienced experimentally, depend on certain material properties and molding conditions such as ejection time, axial location and average retardation time. The sensitivity of the stress profile to the retardation time suggests that density relaxation (viscoelastic material-behavior in dilatation) may play an important role in the development of the residual-stress distribution in injection-molded parts.
ACKNOWLEDGMENT
This work has been supported by the Cornell Injection Molding Program (CIMP).
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| Author: | Ghoneim, H.; Hieber, C.A. |
|---|---|
| Publication: | Polymer Engineering and Science |
| Date: | Jan 1, 1997 |
| Words: | 5471 |
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