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Estimates for Material Shrinkage in Molded Parts Caused by Time-Varying Cavity Pressures.

INTRODUCTION

During many part shaping processes, in particular injection molding, plastic resins undergo very large decreases in volume during solidification, so that the final dimensions of the part can be significantly smaller than those of the cavity in which it was molded. This change in dimensions is genetically referred to as shrinkage. As an example, on demolding the length of a 1 -m long thermoplastic part can shrink by as much as I cm. Differences in shrinkage between the surfaces of two sides of a part can result in the out of plane distortion--even in a nominally flat panel. Such distortions are referred to as warpage. Another important, and mostly undesirable, consequence of molding processes is self-equilibrating residual stresses that are induced in the part as a result of nonuniform cooling during solidification. In general, shrinkage, warpage, and residual stresses are coupled and need to be considered simultaneously.

From a part design and processing standpoint, mold design requires an a priori knowledge of the dimensional changes that occur on molding. For, then, the mold dimensions can be adjusted to achieve the desired final part dimensions. The increased mold dimensions are based on shrinkage data for the material being molded--the tacit assumption being that shrinkage is a material property that is unaffected by part geometry and processing conditions. This strategy assumes that the shrinkage is uniform, so that every part dimension shrinks by the same ratio, thereby assuring a part in which each dimension is a scaled down version of the mold dimension. However, local shrinkage is far from uniform and varies across the part depending on the part geometry and processing conditions. As a result, the uniform shrinkage strategy cannot be used for obtaining parts in which all the dimensions meet specifications. Instead, only critical dimensions can be controlled, and that too by an iterative trial-and-error process. While critical dimensions may be achieved, this strategy can result in higher than necessary residual stresses. Prediction and control of shrinkage, warpage, and residual stresses are, perhaps, the most difficult generic problems associated with the use of plastics.

The solidification of molten resin involves viscoelastic relaxation, so that the final part geometry and residual stresses depend on the time-temperature and constraint history imposed on the resin. The analysis of simple model viscoelastic process problems has resulted in a better understanding of how packing pressure, gate freeze-off, constraints, and asymmetric cooling affect the part shape and residual stresses [1-5]. Such analyses tend to be quite complex.

At the other extreme are highly idealized models for the molding process, in which the cooling rates are assumed to be so slow that the melt cools homogeneously, so that there are no temperature gradients in the solidifying melt. By following the process conditions on the PVT diagram for the resin, this model can be used to understand how packing pressure affects shrinkage. However, the homogeneous temperature assumption rules out cooling rate and fill-time, or cycle time, effects. Also, the pressure is assumed to be uniform throughout the material, ruling out the possibility of any pressure gradients.

This article first establishes the phenomenology of shrinkage during molding, and then develops a simple nonviscoelastic model that accounts for temperature and pressure gradients during molding.

MOLDING EXPERIMENT

The phenomenology of shrinkage was established by measurements on edge-gated 355.6 x 177.7.8 x 3 [mm.sup.3] (14 x 7 x 0.118 in.) rectangular plaques, schematically shown in Fig. 1. Data were obtained from 25 plaques, injection molded at five different nominal processing conditions. The large dimensions of the plaques made shrinkage measurements more accurate. A generous gate, thicker than the plaque, assured that gate solidification occurred only after complete solidification of the plaque. This solidification sequence assured a continuous liquid path into the cavity during the packing phase.

A grid of 5 x 11 squares, approximately 25.4 mm (1 in.) on a side, was scribed onto the mold surfaces, resulting in this grid being imprinted on each side of the plaque. Dimensional measurements on this grid were used to determine the local, flow and cross-flow shrinkage relative to dimensions of the grid on the mold surface. The coordinates of the intersections points on grid lines on the mold-cavity surfaces and on the molded plaques were accurately determined by using an optical coordinate measuring system. Each grid block was then treated as a fournoded element of a mesh, allowing a polynomial interpolation scheme to capture continuous deformation of the plaque surface. Local flow and cross-flow shrinkages were calculated at the intersections of three lines along the flow, designated by the letters A, B and C, and five lines across the flow, indicated by the numbers 1-5.

The thermoplastic used in the molding experiments was polycarbonate (LEXAN LS2). The nominal molding conditions were a mold temperature of 93[degrees]C (200[degrees]F), a melt temperature of 316[degrees]C (600[degrees]F), an injection pressure of 117.2 MPa (17 x [10.sup.3] psi), and a hold time of 20 s. The effect of the cavity pressure on the distribution of local shrinkage was investigated for five nominal hold pressures of 27.6 (4), 41.4 (6), 55.2, 68.9 (8), and 82.7 MPa (12 x [10.sup.3] psi). The pressure was monitored with four pressure transducers: one located in the nozzle and the others in the cavity at three locations, indicated by "gate," "center," and "end" in Fig. 1. Plaques were molded in batches of five for the same processing conditions. The pressure recorded for the same nominal processing condition showed very little variation. Figure 2 shows typical pressure traces for a hold pressure of 55.2 MPa (8 x [10.sup.3] psi). The pressure traces were very similar within each batch of five plaques. Therefore, the average of five traces for the gate, center, and the cavity end was used to represent the cavity pressure history for that batch. Such representative averages were obtained for each of the five different molding conditions. The large (300 mm) length of the mold cavity and the small thickness (3 mm) resulted in a significant pressure drop along the mold, as shown in Fig. 2.

The shrinkage distribution over a plaque surface was calculated from the measured locations of the scribe line intersections on the mold surface and the corresponding scribe line imprints on the molded plaque surface. The local shrinkage was calculated at 15 locations for each plaque: five points, denoted 1-5, along each of the three lines A, B, and C, as shown in Fig. 1. For the nominal hold pressure of 55.2 MPa (8 x [10.sup.3] psi), the distributions of flow and cross-flow shrinkage are shown, respectively, in Figs. 3 and 4. The standard deviation of the measured values is indicated by error bars. The flow-direction shrinkage has a smaller variation among longitudinal lines A, B, and C. Both the flow and cross-flow shrinkages increase by about 40% from the gate to the plaque end. The shrinkage along centerline B exhibits a noticeable decrease at about x = 200 mm. This anomalous decrease in flow shrinkage was observed at this location at all hold pressures, including those not shown here. A closer examination of the plaque surfaces revealed that, in comparison to the imprints from the other two transducers, the central pressure transducer left significantly deeper imprints on all plaques. This suggests that this nonflush pressure transducer introduced an undesirable constraint resulting in the anomalous flow shrinkage in its vicinity.

Figure 5 shows the distribution of the flow and cross-flow shrinkage along the plaque length. Each point in this figure was obtained by averaging the flow and cross-flow shrinkages at the three lines A, B, and C for each point along the length of the plaque. The rationale for this averaging process being the assumption that flow and cross-flow shrinkages and shrinkages at lines A, B, and C are statistically similar and represent the same physical quantity. This assumption is justified for molding amorphous materials, such as polycarbonate, when a uniform melt flow front is achieved along the plaque during molding. Averaging flow and cross-flow shrinkage data at the three lines A, B, and C resulted in the relatively high standard deviation, especially at the plaque end, which experienced less consistent cavity pressure history during solidification. For different levels of nominal hold pressures, Figs. 6-8 show, respectively, the distribution of the flow, cross-flow, and combined shrinkages. All the traces show consistently decreasing shrinkage with increasing cavity pressure. The combined flow and cross-flow shrinkage (Fig. 8) varies from 0.44% at 82.7 MPa (12 x [10.sup.3] psi) in the gate region to 0.77% at 27.6 MPa (4 x [10.sup.3] psi) at the end of cavity. The data in Fig. 8 have very little practical value, because it is presented as a function of the plaque length and the nominal hold pressure. Both these independent variables are specific to part geometry, gate, and runner system configuration. The data in Fig. 8 clearly show that shrinkage is not a material property. A model that accounts for shrinkage during an abrupt cooling of the melt in a mold is required to make data in Fig. 8 useful from a practical standpoint. One goal of the simple solidification model presented in the next section is to clearly separate material properties that effect final part shrinkage from all other geometrical and processing parameters. Without such model, the data in Fig. 8 can only be used to calculate an average shrinkage from all data points. This average shrinkage is 0.62%, which is close to the number 0.6% that is usually reported by material manufacturers for shrinkage of polycarbonate.

SOLIDIFICATION MODEL

Post-molding shrinkage of thermoplastic parts is a manifestation of thermally induced volumetric changes in the part material caused by the rapid cooling of a hot melt after injection into a colder mold cavity. The PVT diagram, which represents the (equilibrium) equation of state for the material, can be used to estimate the volumetric shrinkage, [S.sub.V], from the specific volumes of the material at the solidification and final states as

[mathematical expression not reproducible] (1)

where [mathematical expression not reproducible] are, respectively, the specific volumes at the solidification and final states, marked by S and [E.sub.0] on the PVT diagram for polycarbonate (Fig. 9). In this figure, points [S.sub.0] and [E.sub.0] lie on the p = 0 isobar. Normally, the symbol [T.sub.g] is used for the temperature at the solid-liquid transition temperature at p = 0, which is point [S.sub.0] in Fig. 9. More generally, the glass transition temperature at pressure p is the location of the point on the liquidto-solid transition line [S.sub.0]S corresponding to this pressure, and is written as [T.sub.g] = [T.sub.g] (p). The normally defined glass transition temperature is then [T.sub.g](0), which will be abbreviated as [mathematical expression not reproducible]. Here, point S lies on the 250 MPa isobar. The double-domain modified Tait equation of state (DDTE)

[mathematical expression not reproducible] (2)

can be used to determine the specific volume for any given pressure and temperature. Subscript x indicates the solid phase when the temperature T is less than the transition temperature [T.sub.g] or the melt temperature otherwise. The transition temperature [T.sub.g] = [T.sub.g](p) is assumed to be linearly dependent on the pressure:

[T.sub.g](p) = [b.sub.5] + [b.sub.6]p (3)

The constant C = 0.0894 and the coefficients [b.sub.ix] (i = 1, 2, 3, 4), [b.sub.5], and be for polycarbonate are listed in Table 1.

The specific volumes on the solid-liquid transition line are obtained from Eq. 2 by substituting the temperature on the liquid-solid transition line given by Eq. 2, resulting in

vs ([bar.p]) = ([b.sub.1x] + [b.sub.2x] [b.sub.6] [bar.p]){1 - C ln [1 + [bar.p]/[b.sub.3x] exp {[b.sub.4x] [b.sub.6] [bar.p])} (4)

where [bar.p] denotes the pressure at the time of solidification. This equation gives the specific volume of the melt that solidified at pressure [bar.p].

Equation (4) is not unique, because parameters [b.sub.ix], i = 1, 2, 3, 4 can be taken either from the solid domain or from the liquid domain. The solidification line must be uniquely defined by the equation of state, since it is the common boundary between both domains. This uniqueness requires [b.sub.i] (solid) = [b.sub.i] (melt), i = 1, 2, 3, 4. The imposition of these conditions would eliminate the double-domain character of the Tait equation that would no longer discriminate between the solid and melt phases. As normally understood, the DDTE is continuous at the solidification point only on the p = 0 isobar. For p > 0, the isobars are discontinuous at the solidification point, that is, the specific volume undergoes a discontinuity. This discontinuity is normally overlooked because the DDTE, which is an empirical fit to data, is not considered accurate in the solidification regime where nonequilibrium effects are important. However, this regime is very important for estimating shrinkage; even a small discontinuity in the specific volume can cause large errors in the shrinkage estimates.

This discontinuity in the specific volume along the solidification line can be resolved by requiring that [v.sub.S] ([bar.p]) be uniquely defined to first order in the solidification pressure. An expansion of [v.sub.S] ([bar.p]) in [bar.p] gives

[v.sub.S]([bar.p]) = [b.sub.1x] + [[b.sub.2x] [b.sub.6] - C [b.sub.1x]/[b.sub.3x]] [bar.p] + *** (5)

The proposed modified uniqueness condition then requires that

[mathematical expression not reproducible] (6)

The first of these equations guarantees the continuity of v(0) = v(0, T) on the PVT diagram. While imposing a constraint on coefficients [b.sub.2x] and [b.sub.3x], the second equation does not overconstrain the Tait equation; it can still discriminate between the solid and melt phases. The function [v.sub.S]([bar.p]) (Eq. 4) and its linear approximation (Eq. 4) are shown in Fig. 10.

The volumetric shrinkage [S.sub.V] (Eq. 7) can be expressed as a function of the cavity pressure at solidification as

[mathematical expression not reproducible] (7)

where [beta] = C [b.sub.1]/[b.sub.3] - [b.sub.2][b.sub.6] and [v.sub.A] = [b.sub.1]. The last step in Eq. 7 involves an expansion in [beta][bar.p]--which is small compared to [mathematical expression not reproducible], even for pressures up to 100 MPa--in which only linear terms in [bar.p] are retained. The ratio [mathematical expression not reproducible] can also be approximated as [mathematical expression not reproducible], where [[alpha].sub.V] is a volumetric thermal expansion coefficient for the solid phase at ambient pressure, and [mathematical expression not reproducible] is the temperature difference between points [S.sub.0] and [E.sub.0]. Thus

[mathematical expression not reproducible] (8)

where [mathematical expression not reproducible].

If every portion of the part solidifies at the same time at the same pressure, the volumetric shrinkage [S.sub.V] will be equally distributed in all directions, so that the linear shrinkage will approximately equal one-third of the volumetric shrinkage ,[S.sub.V]. Thus, this linear shrinkage--which would be the linear shrinkage of a part that has been cooled infinitely slowly--referred to as the isotropic shrinkage [S.sub.I], is given by

[mathematical expression not reproducible] (9)

where [[alpha].sub.L] = [[alpha].sub.V]/3 is a linear coefficient of thermal expansion for the solid phase and

[mathematical expression not reproducible].

In the case of a time-varying cavity pressure, each layer of material solidifies at a different pressure. And the nonuniform temperature distribution in real parts results in a sequential solidification during which newly formed solid layers are constrained by the material that solidified earlier.

Consider an elastic layer at a distance z from the center of the cavity (Fig. 11) that is subjected to stresses [[sigma].sub.x](z, t), [[sigma].sub.y](z, t), and [[sigma].sub.z] (z, t), the corresponding strains, respectively, being [[epsilon].sub.x](z, t), [[epsilon].sub.y](z, t), and [[epsilon].sub.z] (z, t). The in-plane strains [[epsilon].sub.x] and [[epsilon].sub.y] are related to the stresses [[sigma].sub.x], [[sigma].sub.y], and [[sigma].sub.z] and the isotropic shrinkage [S.sub.I] (z, t) through

[[epsilon].sub.x] + [S.sub.I] = 1/E [[[sigma].sub.x] - v([[sigma].sub.y] + [[sigma].sub.z])] [[epsilon].sub.y] + [S.sub.I] = 1/E [[[sigma].sub.y] - v([[sigma].sub.x] + [[sigma].sub.z])] (10)

where E and v are, respectively, the Young's modulus and the Poisson's ratio of the material, which is assumed to be elastic from the instant it turns solid. Alternative viscoelastic material models are used in Refs. 1 and 2. The isotropic shrinkage is also a function of position and time.

[mathematical expression not reproducible] (11)

Note that pressure at the time of solidification [bar.p] is only a function of the layer position z.

If the layer is constrained in the same manner in the x- and y-directions, [[epsilon].sub.x] = [[epsilon].sub.y] = [[epsilon].sub.L](z, t) and [[sigma].sub.x] = [[sigma].sub.y] = [[sigma].sub.L] (z, t). Equation 10 can then be simplified to

[[epsilon].sub.L] (z, t) + [S.sub.I](z, t) = 1/E {[[sigma].sub.L] (z, t) - v[[[sigma].sub.L] (z, t) + [[sigma].sub.z]{z, t)]} (12)

While in the mold, the solidified layer is subject to the same time-dependent cavity pressure. Equilibrium in the thickness direction requires [[sigma].sub.z] = -p(t) so that [[sigma].sub.z] depends on time.

After demolding and cooling to the final, constant, and uniform temperature, [[sigma].sub.z] = 0, [[sigma].sub.L] (z) is the residual stress, and [[epsilon].sub.L](z) is the final in-plane strain in the layer. All the layers located between z = [+ or -] h/2 will have the same final strain [[epsilon].sub.L](z) = [[epsilon].sub.L], which equals the negative of the final in-plane part shrinkage [S.sub.L], so that

-[S.sub.L] + [S.sub.I][[bar.p](z)] = 1 - v/E [[sigma].sub.L](z) (13)

An integration of Eq. 13 across the wall thickness after a substitution for [S.sub.I] from Eq. 9 gives

[mathematical expression not reproducible] (14)

The right-hand side of this equation, essentially the total inplane force, must equal zero for an unconstrained, demolded part, so that

[mathematical expression not reproducible] (15)

or

[mathematical expression not reproducible] (16)

where [p.sub.eff] is the thickness averaged cavity pressure defined by

[p.sub.eff] [??] 1/h [[integral].sup.h/2.sub.-h/2] [bar.p](z) dz (17)

The parameter [p.sub.eff], referred to as the effective pressure, accounts for the effect of the time-varying cavity pressure on the in-plane shrinkage. While two different cavity-pressure histories can result in the same effective pressure, each effective pressure corresponds to a unique shrinkage.

An elimination of [S.sub.L] from Eqs. 13 and 15 gives the residual stress [[sigma].sub.L] (z) as

[[sigma].sub.L](z) = E/1 - v [gamma] [[p.sub.eff] -p(z)] (18)

While this model does give a formal expression for the residual stress (Eq. 18), this estimate is rather crude. For, consider the case of a constant cavity pressure p(t) = [p.sub.0], which gives [p.sub.eff] = [p.sub.0], so that the model predicts zero residual stresses. However, for this case, a viscoelastic material model predicts [3] substantial residual stresses that depend on the applied pressure--a result that is supported by experiments. As such, the residual stresses (Eq. 18) predicted by the simple model in this article are not reliable, and must be interpreted with care.

EXAMPLE

Figure 12 shows a typical cavity-pressure history p(t) for the cavity end region of a mold. The initial rapid increase in the cavity pressure during injection is followed by a slowly decaying pressure during the packing phase. Variations in the cavity pressure during solidification cause changes in the transition temperature [T.sub.g](t) = [b.sub.5] + [b.sub.6][bar.p](t).

The position of the melt-solid interface z = z(t) is determined from the temperature history T(z, t) by solving T(z, t) = [b.sub.5] + [b.sub.6] [bar.p](t) for z = z(t).

In this example, the temperature history was obtained from an analytical solution to the heat transfer equation using a constant thermal diffusivity of 0.135 [mm.sup.2] [s.sup.-1]. A uniform initial temperature of 300[degrees] C was assumed over the domain -h/2 [less than or equal to] z [less than or equal to] h/2, and a constant mold temperature of 80[degrees]C was imposed at time zero on the surfaces at z = [+ or -] h/2. Figure 13 shows z = z(t) for the packing-pressure history in Fig. 12. To illustrate the effect of elevated cavity pressure on the solidification front, Fig. 13 also shows the solid/melt interface for p = 0.

Figure 14 shows the cavity pressure as a function of solid/melt interface position. The effective pressure [p.sub.eff], the thickness average of [bar.p](z), is 15.2 MPa, as indicated by the dashed horizontal line.

Determining the effective pressure involves an integration of the cavity pressure from the beginning of solidification at the external surfaces z = [??] h/2 to the end of the solidification that takes place in the center plane of the part for symmetric cooling. In order to investigate the development of in-plane shrinkage during solidification, it is convenient to define the partial effective pressure

[mathematical expression not reproducible] (19)

where [t.sub.sol] is the solidification time. Figure 15 shows the time history of the partial effective pressure for the case of symmetric cooling considered in this article. The partial effective pressure in Fig. 15 shows that the cavity pressure at the end of the solidification has a larger effect on the final part shrinkage than the initial packing pressure. For t > 11.1 s, the cavity pressure has no effect on shrinkage since all the material has solidified. It can also be shown that, even for a constant cavity pressure, the dependence of the effective pressure on time (Fig. 15) is not linear. The nonlinearity comes from the nonlinear progression of the solidification front.

EFFECTIVE PRESSURE AS AN INDEPENDENT VARIABLE

The effective pressure defined by Eq. 17 is a thickness average of the pressure- and temperature-dependent solidification pressure. It combines all geometric and processing parameters into one number, which can be calculated for any point in the cavity when the local pressure history is known from a direct measurement or from a melt flow analysis. The only material property indirectly involved in the effective pressure is the material thermal diffusivity, which for most unfilled amorphous thermoplastics does not vary much. Thus, for this class of materials, the effective pressure depends entirely on nonmaterial process parameters. Consequently, the effective pressure is the best independent geometry/processing parameter for presenting shrinkage data. Figure 16 presents shrinkage data form Fig. 8 as a function of the effective pressure calculated for each point from the measured cavity pressure history. The simple solidification model, on the basis of which the effective pressure is defined, can be used to predict local shrinkage from the knowledge of the PVT diagram for the material (Table 1). The predicted shrinkage agrees well with the measured values. The model captures the material shrinkage that occurs during the abrupt cooling in the mold. Eq. 16 for predicting in-plane shrinkage indicates a linear dependence on the effective pressure, which accounts for all nonmaterial parameters. The intercept and the slope in Eq. 16 are two parameters that are of practical interest because they can be considered as two material parameters that determine material shrinkage. The intercept

[mathematical expression not reproducible] (20)

depends on the linear coefficient of thermal expansion [[alpha].sub.L] and the glass transition temperature[mathematical expression not reproducible]--both of which are material properties--and on the final temperature [mathematical expression not reproducible], which can be assumed to be the room temperature of 20[degrees]C. Coefficient b gives an upper limit for the in-plane shrinkage of the material.

The slope

[mathematical expression not reproducible] (21)

depends on the final and glass-transition specific volumes and on the coefficient [beta] = C[b.sub.1]/[b.sub.3] - [b.sub.2][b.sub.6], which can be determined from the PVT diagram for the material. The slope a determines the sensitivity of material shrinkage to the variation in the local cavity pressure.

SHRINKAGE IN THE THICKNESS DIRECTION

The thickness-direction shrinkage is normally not as important as the in-plane shrinkage. Although the thickness-direction shrinkage can vary in the [+ or -]30% range, or even more, it is rarely of a concern because it does not visibly affect part quality. When highly localized, such high shrinkage is referred to as a sink mark, which can result in the part being rejected.

The through-thickness shrinkage can be analyzed by following the approach used in the previous sections. The starting point is the third constitutive relationship that can be added to the set of two equations describing stress-strain relationships in an elastic layer at a distance z from the center of the cavity (see Eqs. 9 and 10)

[e.sub.z] + [S.sub.I] = 1/E [[[sigma].sub.z] - v([[sigma].sub.x] + [[sigma].sub.y])] (22)

where [S.sub.I] is the isotropic shrinkage given by Eq. 9.

Here again, the solidifying layers are assumed to experience the same constraint history in the x- and y-directions. That is

[[epsilon].sub.x] = [[epsilon].sub.y] = [[epsilon].sub.L](z, t) [[sigma].sub.x] = [[sigma].sub.y] = [[sigma].sub.L] (z, t) (23)

A substitution of Eq. 23 in Eq. 22 gives

[[epsilon].sub.z] + [S.sub.I] = 1/E ([[sigma].sub.z] - 2v [[sigma].sub.L]) (24)

By substituting in Eq. 24 the in-plane stress [[sigma].sub.L] from Eq. 18, and the isotropic shrinkage [S.sub.I] from Eq. 9, the through-thickness strain in the solidified layer is given by

[mathematical expression not reproducible] (25)

The through-thickness shrinkage is defined as

[S.sub.z] = - 1/h [[integral].sup.h/2.sub.-h/2] [[??].sub.z](z) dz (26)

A substitution from Eq. 25 in Eq. 26 gives

[mathematical expression not reproducible] (27)

which is identical to the expression for the in-plane shrinkage. However, experiments have shown that while the in-plane shrinkage is normally on the order of 1%, the through thickness has much larger variations, often on the order of 10%.

It will now be shown that while the simple expression in Eq. 16 captures the in-plane plane shrinkage well, the through-thickness shrinkage given by Eq. 27 is only accurate under very special conditions. Consider the material in the mold before complete solidification at t = [t.sub.sol]. The solidified polymer forms two layers, one adjacent to each mold surface, which are separated by a liquid resin layer. The in-plane and through-thickness strains are related to the stresses through the isotropic relationship.

[mathematical expression not reproducible] (28)

where

[S.sub.I](z, t) = [alpha][[T.sub.g] - T(z, t)] - [gamma][bar.p](z) (29)

While the material is still in the mold, it is assumed that [[epsilon].sub.L] (z, t) = 0. It then follows from the first equation in Eq. 28 that the in-plane stress is

[[sigma].sub.L] (z, t) = 1/1 - v [E [S.sub.I](z, t) -vp(t)] t [less than or equal to] [t.sub.sol] (30)

By substituting the above expression into the second equation in Eq. 28, the through-thickness strain is given by

[[epsilon].sub.z](z,t) = - 1 + v/1 - v [S.sub.I](z,t) - 1/E 1 - v - 2[v.sup.2]/1 - v p(t) (31)

By integrating the above expression over the current layer thickness, the change in the thickness of each solidified layer is given by

[mathematical expression not reproducible] (32)

where [z.sup.(1).sub.s](t) is the current location of the solid-liquid interface in the layer.

Consider the instant at which the two solidifying layers are about to fuse. It is assumed that the solid layers are still separated by a melt layer that is much thinner than the part thickness. This melt is under pressure

[p.sub.s] = p([t.sub.s]) (33)

and at temperature

[T.sub.s](z) = T(z, [t.sub.s]) (34)

Also [z.sub.s] = 0 in the case of symmetric cooling, the case considered in this article. The change in thickness of the solidified layers is given by Eq. 32, an integration of which after a substitution from Eq. 31 results in

[DELTA][h.sub.1] = h/2 {1 + v/1 - v [-[[alpha].sub.L]([T.sub.g] - [[bar.T].sub.s]) + [gamma][p.sub.eff]] - 1/E 1 - v - 2 [v.sup.2]/1 - v [p.sub.s]} (35)

The structure of the above expression is similar to the expressions for the in-plane and through-thickness shrinkages. However, the final temperature [mathematical expression not reproducible] in Eqs. 16 and 27 is replaced here with the average through-thickness temperature at the time of solidification

[[bar.T].sub.s] = 2/h [[integral].sup.h/2.sub.0] T(z, [t.sub.s]) dz (36)

The extra term in Eq. 35 accounts for the melt pressure at the time of through-thickness solidification. If the cavity thickness at the time of solidification is

[h.sub.s] = h + [DELTA][h.sub.s] (37)

in which [DELTA][h.sub.s], the mold cavity opening caused by high cavity pressure, depends on the compliance of the tool; it also depends on the ability of the molding machine to maintain adequate clamping force, so that this [DELTA][h.sub.s] is specific to the process.

The assumption [p.sub.s] [greater than or equal to] 0 implies that solidifying polymer is in contact with the mold surface till the instant of complete solidification. Consequently, the thickness of the solidifying part must equal the thickness of the cavity. That is

[mathematical expression not reproducible] (38)

Since the mold opening [DELTA][h.sub.s] and all the variables in Eq. 35 are independent, this condition (Eq. 38) generally cannot be satisfied without introducing an additional quantity [DELTA][h.sup.*.sub.s], which denotes an extra gap between solidifying surfaces. This gap is filled by the melt at pressure [p.sub.s] and temperature [T.sub.g]. With this correction Eq. 38 becomes

[mathematical expression not reproducible] (39)

The above equation can be used to calculate the gap thickness [DELTA][h.sup.*.sub.s]. The through-thickness shrinkage model basically consists of two symmetric resin layers solidifying under a time-dependent pressure, and a central layer that accounts for the extra melt that has to be injected into the cavity to satisfy the resin-mold contact condition. This extra material has a small effect on in-plane shrinkage since the region accounts for a small fraction of the total part thickness. It can, however, significantly affect the through-thickness shrinkage. After complete solidification, the thickness of this region can be calculated by noting that it has thickness [DELTA][h.sup.*.sub.s] given by Eq. 39, that the in-plane and through-thickness strain is zero, and that the material is under pressure [p.sub.s] and temperature [T.sub.g]. After cooling, the in-plane strains are given by Eq. 16, the through-thickness stress is zero, and the material is at the constant final temperature [mathematical expression not reproducible]. The correction to the expression in Eq. 27 for the through-thickness shrinkage is then

[mathematical expression not reproducible] (40)

When the third term in this equation is on the order of the in-plane shrinkage, the effect of the mold opening [DELTA][h.sub.s] and melt redistribution before complete solidification may cause the first two terms to be significantly larger.

CONCLUSIONS

Estimates for the in-plane shrinkage in parts molded under time-varying cavity pressures are obtained from a simple model that uses material shrinkage data from (equilibrium) PVT diagrams. This model neglects viscoelastic effects and the liquid phase is assumed to solidify with a sharp transition into an elastic solid. The in-plane shrinkage is given by a simple expression in terms of an effective pressure that captures the effects of the time-varying pressure. This effective pressure, which is the through-thickness average of the solid/melt interface pressure during solidification, combines several variables, such as the material thermal diffusivity and the part thickness, into a single parameter. The effective pressure can be used as an independent variable for presenting shrinkage data. The shrinkage model predicts a linear dependence of shrinkage on the effective pressure. Both the intercept and the slope can be considered as material properties that can be estimated from the PVT diagram for the material.

The position of the solidification front as a function of time, determined from an analytical conduction heat transfer calculation, is used to obtain the pressure at the solidification front as a function of the position for a given cavity-pressure time history. A spatial average of this pressure-position function gives the effective pressure, which is then used to determine the in-plane shrinkage from a simple, closed form expression.

While this model also gives an expression for the residual stresses, the estimates are rather crude, and must be interpreted with care. For example, contrary to well-established phenomenology, this model predicts zero residual stresses for a constant cavity pressure.

REFERENCES

[1.] W.C. Bushko and V.K. Stokes, Polym. Eng. Sci., 35, 351 (1995).

[2.] N. Santhanam, Ph.D. Thesis, Cornell University, Ithaca, NY (1992).

[3.] W.C. Bushko and V.K. Stokes, Polym. Eng. Sci., 35, 365 (1995).

[4.] W.C. Bushko and V.K. Stokes, Polym. Eng. Sci., 36, 322 (1996).

[5.] W.C. Bushko and V.K. Stokes, Polym. Eng. Sci., 36, 658 (1996).

Wit C. Bushko, ([dagger]) Vijay K. Stokes (iD [double dagger])

GE Global Research, Niskayuna, New York, 12301

Correspondence to: V. K. Stokes; e-mail: vijay.stokes@gmail.com

([dagger]) Present address: Bose Corporation, Framingham, MA 01701

([double dagger]) Present address: 2365 Jade Lane, Niskayuna, NY 12309

This paper is an extended version of results presented at the 1997 SPE ANTEC.

DOI 10.1002/pen.25163

Published online in Wiley Online Library (wileyonlinelibrary.com).

Caption: FIG. 1. Plaque geometry. All dimensions in millimeters.

Caption: FIG. 2. Pressure traces recorded for a nominal hold pressure of 55.2 MPa (8 x [10.sup.3] psi).

Caption: FIG. 3. Flow direction shrinkage along Lines A, B, and C (see Fig. 1) for a nominal hold pressure of 55.2 M (8 x [10.sup.3] psi).

Caption: FIG. 4. Cross-flow direction shrinkage along Lines A, B, and C (see Fig. 1) for a nominal hold pressure of 55.2 MPa (8 x [10.sup.3] psi).

Caption: FIG. 5. Combined flow and cross-flow direction shrinkage along the plaque length (see Fig. 1) for a nominal hold pressure of 55.2 MPa (8 x [10.sup.3] psi).

Caption: FIG. 6. Combined (Lines A, B, and C) flow-direction shrinkage along the plaque length (see Fig. 1) for five nominal hold pressure levels.

Caption: FIG. 7. Combined (Lines A, B, and C) cross-flow direction shrinkage along the plaque length (see Fig. 1) for five nominal hold pressure levels.

Caption: FIG. 8. Combined (Lines A, B, C, flow, and cross-flow) shrinkage along the plaque length (see Fig. 1) for five nominal hold pressure levels.

Caption: FIG. 9. PVT diagram for polycarbonate.

Caption: FIG. 10. Specific volume of polycarbonate at the melt-solid transition as a function of pressure.

Caption: FIG. 11. Wall cross section showing a layer located at distance z from the center plane.

Caption: FIG. 12. Cavity-end cavity pressure as a function of time for a hold pressure of 82.7 MPa (12 x [10.sup.3] psi).

Caption: FIG. 13. Cavity-end melt-solid interface position as a function of time at a hold pressure of 82.7 MPa (12 x [10.sup.3] psi).

Caption: FIG. 14. Cavity-end solidification front pressure as a function of through-thickness position for a hold pressure of 82.7 MPa (12 x [10.sup.3] psi).

Caption: FIG. 15. Cavity-end partial effective pressure as a function of time for a hold pressure of 82.7 MPa (12 x [10.sup.3] psi).

Caption: FIG. 16. Shrinkage as a function of effective pressure.
TABLE 1. DDTE constants for polycarbonate.

Constant           Solid                   Melt

[b.sub.1]          854.16                 854.16
[b.sub.2]          0.159                  0.562
[b.sub.3]          299.95                 182.88
[b.sub.4]   1.7051 x [10.sup.-3]   3.9864 x [10.sup.-3]
[b.sub.5]                      144.01
[b.sub.6]                     0.34877

Constant                  Units

[b.sub.1]         [mm.sup.3] [g.sup.-1]
[b.sub.2]   [mm.sup.3] [(g[degrees]C).sup.-1]
[b.sub.3]                  MPa
[b.sub.4]          [degrees][C.sup.-1]
[b.sub.5]               [degrees]C
[b.sub.6]        [degrees]C [MPa.sup.-1]
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Author:Bushko, Wit C.; Stokes, Vijay K.
Publication:Polymer Engineering and Science
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Geographic Code:1USA
Date:Aug 1, 2019
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