Residual Stress Development in the Injection Molding of Polymers

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Residual Stress Developmentin the Injection Molding ofPolymersA. I. Isayev a b & D. L. Crouthamel a ca Sibley School of Mechanical and AerospaceEngineering Cornell University , Ithaca, NewYork, 14853b Polymer Engineering Center, University ofAkron , Akron, Ohio, 44325c Bell Laboratories , Allentown, Pennsylvania,18103Published online: 13 Dec 2006.

To cite this article: A. I. Isayev & D. L. Crouthamel (1984) Residual StressDevelopment in the Injection Molding of Polymers, Polymer-Plastics Technologyand Engineering, 22:2, 177-232, DOI: 10.1080/03602558408070038

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P0LYM.-PLAST. TECHNOL. ENG., 22(2), 177-232 (1984)

RESIDUAL STRESS DEVELOPMENT IN THE INJECTION MOLDING OF POLYMERS

A. I. ISAYEV* and D. L. CROUTHAMELt

Sibley School of Mechanical and Aerospace Engineering Cornell University Ithaca, New York 14853

I.

11.

111.

IV.

V.

VI.

VII.

VIII.

IX.

X.

XI.

INTRODUCTION . . . . . . . . . . . . . . . . . . . . , . . . 178

RESIDUAL FLOW STRESSES . . . . . . . . . . . . . . . . . 178

THERMAL (COOLING) STRESSES. . . . . . , . . . . . . . . 1 80

RESIDUAL STRESSES IN MOLDED PARTS-REVIEW . . . . 188

EXPERIMENTAL TECHNIQUES FOR RESIDUAL STRESS MEASUREMENTS. . . . . . . . . . , . . . . . . . . . . . . . 192 A. Hole Drilling Method . . . . . . . . . . . . . . . . . . . . 192 B. Stress Relaxation Method . . . . . . . . . . . . . . . . . . 193

EXPERIMENTAL . . . . . . . . . . . . . . . . . . . . . . . . 194 A. Materials and Processing Conditions. . . . . . . . . . . . . 194 B. Layer-Removal Technique. . . . . . . . . . . . . . . . . . 195

THERMAL STRESSES IN FREELY QUENCHED STRIPS . . . 200

CREEP AND STRESS RELAXATION . . . . . . . . . . . . . 2 1 1

RESIDUAL STRESSES IN MOLDED PARTS . . . . . . . . . . 21 6

COMPARISON OF FLOW, THERMAL, AND RESIDUAL STRESSES IN MOLDED PARTS . . . . . . . . . . . . . . . . 224

CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . 228

References . . . . , . . . . . . . . . . . . . . . . . . . . . . . 230

*Present address: Polymer Engineering Center, University of Akron, Akron,

?Present address: Bell Laboratories, Allentown, Pennsylvania 18 103.

177

Ohio 44325.

Copyright @ 1984 by Marcel Dekker, Inc. 0360-2559/84/22024177$350/0

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178 ISAYEV AND CROUTHAMEL

1. INTRODUCTION

Injection molding is one of the most widely employed methods for the fabri- cating of polymer articles, being characterized by high production rates and accurately dimensioned products. The process includes the flow of polymer melt through a runner system and gates followed by injection into a cold mold, packing under high pressure, and subsequent cooling to solidification. Accordingly, during the injection-molding process the polymer undergoes simultaneous mechanical and therma! influences while in fluid, rubbery, and glassy states. Such effects introduce residual stresses and strains into the final product [ 1,2] , resulting in highly anisotropic mechanical behavior [3-91 and warpage and shrinkage [lo-131. Thus, understanding the factors governing the residual-stress development during molding is of great importance.

Residual stresses in an injection-molded plastic part can be attributed to two main sources. The first of these is due to the shear and normal stresses that develop during the nonisothermal flow of the polymer in the mold cavity during fding and packing [14-161. These stresses do not completely relax but get frozen in due to a rapid increase in relaxation time during cooling. Such stresses will subsequently be referred to as residual flow stresses. The second type of residual molding stress arises during the rapid cooling stage itself [ 17, 181 . The viscoelastic behavior of the polymer during its passage through glass transition temperature, Tg, coupled with nonequilibrium density changes accompanying the rapid, inhomogeneous cooling, resulting in the development of residual thermal (cooling) stresses. A rather complex picture of the development of the residual stress field in an injection-molded article thus arises in which the flow-induced and thermally induced stress fields are coupled. The precise nature of t h s coupling is not presently understood, and no single constitutive model seems capable of describing the full behavior of the polymer through the injection, packing, and cooling stages of the injection-molding process.

The purpose of the present investigation is to give a brief account of the techniques used for residual stress measurements and to review a literature devoted to various aspects of residual stresses related to injection molding, including our own theoretical and experimental results obtained for residual flow, cooling, and molding stresses in polymer articles.

II. RESIDUAL FLOW STRESSES

Flow stresses appear mostly as a result of the cavity fiuing stage with additional contribution apparently due to packing under high pressure. High strain rates

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RESIDUAL STRESS DEVELOPMENT 1 79

in the melt during injection introduces shear stresses and first and second normal stress differences [14-161 together with extensional stresses [ 191. Shear stresses arise in any polymer fluid (Newtonian or non-Newtonian) during injection behind a propagating melt front. The extensional stresses develop mainly at the melt front location and are caused by an extensional flow of the melt in the so-called “fountain region” where a hot melt from the core area moves toward the cold mold surface and subsequently solidifies. Moreover, the extensional stresses also develop in the region of contraction or expansion in the mold where the flow area suddenly or gradually changes. First and second normal stress differences arise as a consequence of the viscoelastic nature of polymer melts having large relaxation times. After the cavity is fdled, relaxation of all the above-mentioned stresses is hindered due to the fact that the relaxation time of the melt rapidly (exponentially) increases when the melt approaches its solidification temperature. Accordingly, the flow stresses become locked into the polymer article.

In general, residual flow stresses cannot be measured directly. However, since the flow stresses give rise to molecular orientation, it has been assumed that measurement of frozen-in birefringence is an indirect indication of the level of the residual flow stresses. It is clear that flow stresses develop when the polymer is in the fluid and rubbery states, i.e., above the melting point or glass-transition temperature, such that the linear stress-optical rule relating birefringence and stresses is usually valid [ 161 . Many investigations of frozen- in birefringence in molded articles reported in the literature [14-16,20-241 have been made with polystyrene for which the stress-optical coefficient, C, in the fluid and rubbery states is almost three orders-of-magnitude larger than in the glassy state [22]. Data obtained by Wales [20] show a remarkable correlation between gapwise-averaged frozen-in birefringence and shear stress at the wall for molded polystyrene slabs.

Few attempts have been made to theoretically predict residual flow stresses and correlate them with frozen-in birefringence [ 14, 15,241. All approaches are one-dimensional. The approach of Dietz and White [ 141 is based upon the formation of a frozen-in layer of polymer in the wall region and the use of an isothermal power-law model for flow in the core region with the subsequent nonisothermal stress relaxation simulated by means of a Maxwellian-type model. Later, Greener and Pearson [24] used the formulation of Dietz and White for the filling stage but modeled the nonisothermal stress relaxation according to the Marucci equation. On the other hand, Isayev and Hieber [ 151 have employed the Leonov constitutive equation for both the filling and cooling stages of injection molding in order to predict frozen-in flow stresses in molded slabs and runners [22]. These latter investigations were

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able to derive all stress components developing during the molding of polymers. Such results will be shown below concerning the level of residual flow stresses in molded articles. There are also other possible means of determining the residual flow stresses. For example, one can release the cooling stresses by annealing a molded article near the glass-transition temperature and then slow cooling to room temperature. The residual birefringence can then be measured and a distribution of the residual flow stresses be constructed by means of the stress-optical rule. However, such type measurements of flow stresses do not seem to have been carried out yet.

111. THERMAL (COOLING) STRESSES

This area of investigation has paralleled similar developments concerning thermal stress effects in inorganic glasses and metals. Presently, there exists an extensive literature devoted to measurements of the thermal stresses in polymeric slabs. A brief summary of the results obtained by various authors has been presented by Isayev et al. [18], Siegmann et al. [25], Haworth et al. [26], and Mills [27]. Here we would like to elaborate these findings.

tigate stresses introduced by free quenching of polymethyl methacrylate (PMMA) and polycarbonate (PC) sheets by using the layer-removal technique developed by Treuting and Read for metals [29]. A milling machine was used to cut the layers. Application of this technique to polymers will be reviewed in depth below. In all cases of quenching, the thermal stress profiles were found to be parabolic in nature, with a maximum compressive stress at the surface and a maximum tensile stress at the center. For both materials the surface residual-stress levels were seen to decrease with increased specimen thickness. Residual stresses in annealed specimens were found to be negligible. Later, Siegmann et al. conducted thermal stress investigations on quenched sheets of PMMA and polystyrene (PS) [30] and polyphenylene oxide (PPO) [25]. Stress profiles in the PMMA and PPO sheets were determined by layer removal at 20,000 rpm using a tensile cut mdling machine, and surface stresses in the PS sheets were qualitatively characterized by observing crazing behavior in n-heptane. The parabolic stress profile characteristic of thermal stresses was observed in the PMMA samples. An increase in the initial temperature before quenching from 120 to 130°C was shown to increase the level of compressive surface stresses in the PMMA. The experimental procedure was deemed satisfactory since residual stresses found in annealed samples were of a negligible level. In the later experiment with PPO [25], it was found

So and Broutman [28] seem to have been the first to experimentally inves-

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RESIDUAL STRESS DEVELOPMENT 181

that the compressive stresses at the surface decrease with increasing initial temperature. Such results cannot be explained within the framework of the thermoelastic theory to be described below. On the other hand, their experimental results are in accordance with the thermoelastic theory in showing that increasing the quenching temperature reduces the level of the compressive and tensile stresses. At constant initial temperature, To, a linear dependence has been found between the logarithm of the reduced surface compressive stress uJTg - Ta and the logarithm of the reduced temperature (To - T,)/(Tg - Ta, where u, is the surface stress and T , is the quenching temperature. The slopes of such curves have been found to decrease exponentially with increasing initial temperature. At an initial temperature To N Tg, a unique value of a,/(Tg - T A has been found which is independent of To. It is noted that the existence of dependence of reduced thermal stresses on the reduced temperature has been indicated by numerical calculations of Struik [ 171. Moreover, Struik has presented experimental results on the measurements of surface compressive strains in quenched PMMA sheets with thicknesses varying from 0.003 to 0.01 m. This has been done by measuring the distance between surface marks before and after quenchng. It has been found that the compressive strain depends on the Biot number, Bi = hb/h, where h is the heat transfer coefficient at the surface, h is the thermal conductivity, and b is the half-gap thickness of the sheet. Using the layer removal method, Mandell et al. [9] measured the thermal stress distribution in constrained quenched polysulfone sheets having a thickness of 3.2 X fect upon the fatigue. In this case, quenching was performed between two aluminum sheets both having a thickness of 3.2 X m. The initial temperature was 200°C and water bath temperature was 13°C. As for the case of free quenching, the thermal stress distribution was found to be parabolic with a surface compressive stress of -14.7 MPa and a tensile stress at the center of 7.4 MPa.

junction with the stress-optical rule to determine the thermal stress values in quenched polymer sheets. Broutman and Krishnakumar [3 11 subjected sheets of PC to various thermal treatments to introduce thermal stresses. Then surface stresses in these sheets were calculated with the stress-optical rule using birefringence measurements in conjunction with an experimentally determined fringe constant. In addition, surface stresses were calculated by measuring the surface strain resulting from annealing a sheet inscribed with a square grid pat- tern. Results from the two methods agreed to within a few percent. The birefringence profiles through the sheet thickness were seen to have unequal positive and negative areas, and thus utilization of the stress-optical rule would re-

m and have also determined their ef-

A number of authors attempted to use birefringence measurements in con-

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sult in a violation of stress equilibrium. To explain this, a stress component perpendicular to the plane of the sheet was proposed to exist, being zero at the surface and tensile in the interior.

Later, Saffell and Windle [32] investigated thermal residual stresses in quenched sheets of PMMA and PC by using birefringence measurements and the stress-optical rule. Cooling was performed by one of three methods: (1) exposure to still air at 20°C, (2) immersion in a water bath at 20°C, or (3) sandwiching between copper platens at various temperatures. The parabolic birefringence profiles, seen for all three quenching procedures, were observed to have unequal positive and negative areas. Since equilibrium dictates equal areas for the stress profiles, it was obviously impossible to calculate residual stresses from the birefringence profiles by direct application of the stress- optical rule. To alleviate this problem, Saffell and Windle introduced the concept of “residual birefringence.” In their measurements, birefringence values varied with sample thickness for pieces larger than 0.001 m in the direction perpendicular to the light path, but became constant for samples thinner than 0.001 m. This constant profile of “residual birefringence,” attributed to molecular orientation induced by flow during cooling, was subtracted by the authors from the measured birefringence profiles to yield a net profile, which was then used to calculate residual stresses by means of the stress-optical rule with a constant coefficient C. The surface compressive stresses were seen to increase with increased sheet thickness, contrary to the results of So and Broutman [28].

validity of the “residual birefringence” concept proposed by Saffell and Win- dle [32]. It has been demonstrated that the birefringence profile is independent of sample thickness, even for samples of thickness considerably smaller than the critical value of m cited in Ref. 32. As an alternative explana- tion for the differences between the birefringence and cooling stress profiles, the time dependence of the stress-optical coefficient in the glassy state as well as the transition zone from the glassy to the rubbery states has been proposed in Ref. 22. However, when the initial temperature prior to quenching is not far above the glass-transition temperature, one can still derive reasonable results for the residual stress distribution in slabs from birefringence measurements provided the gapwise birefringence distribution is symmetric. Greener and Kenyon [33] used the stress-optical rule in order to relate the frozen-in birefringence distribution in quenched PS and PMMA disks to the stress distribution as calculated from the theory of Aggarwala and Saibel [34]. For the case of PMMA disks at a critical temperature of 125”C, experimental results were found to be in good agreement with theory whereas for PS disks

Recently, evidence has been put forth [22] which casts some doubt on the

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the birefringence profile was not in accordance with the theory. In these calculations, the stress-optical coefficient has been taken to be constant and independent of time and temperature, and assumed to correspond to the solid material.

Wust and Bogue [35] also used the birefringence technique, measuring the gapwise birefringence distribution in quenched PS strips. In order to relate the experimental frozen-in birefringence distribution to the calculated thermal stress distribution based upon the theory of Lee, Rogers, and Woo [36], they used the strain-optical rule with both a temperature-dependent and temperature-independent strain-optical coefficient, Both approaches give satisfactory results even though a time-dependence of the strain-optical coefficient has been disegarded. It should be noted, however, that an initial temperature of 120°C has been used which is not far above the glass-transition temperature. In both investigations [33,35], procedure for separating molecular and residual orientation [32] has not been used. Recently, Mills [27] also used the birefringence technique for evaluating the residual stress distribution in quenched PC sheets of different thicknesses. These data have shown unbalanced areas of compressive and tensile stresses.

An extensive literature exists dealing with the theoretical prediction of thermal stresses in quenched inorganic glass. Since an effort is made later in this article to test the accuracy of several of these theories in predicting thermal stresses in polymer slabs, a review of some of the approaches is in order.

Bartenev [37,38] developed a theory which incorporates thermoelastic response of the glass together with instant freezing during cooling through Tg, introducing the assumption that the temperature gradient becomes frozen-in at Tg which separates the perfectly fluid and perfectly elastic regimes of the glass behavior. That is, above Tg the glass is plastic and plastic strains proceed instantaneously with internal stresses being absent. Below Tg, on the other hand, the glass is elastic with plastic strains being residual and internal stresses unrelaxed. Bartenev’s theory leads to the following equation for residual-stress evaluation in an infinite slab:

where y is the gapwise coordinate, u the normal stress in the xz plane, 0 the linear coefficient of thermal expansion, and E and u are Young’s modulus and Poisson’s ratio of the glassy state, respectively, with @ denoting a fictitious temperature, defined to within an arbitrary additive constant by

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where T ~ , I * ) Tg, i.e., t * b ) is the time at which the temperature passes through Tg for a given y , with T b , t ) being the actual gapwise temperature distribution at time r and Gay being the gapwise-averaged value of 4.

For glass quenched relatively slowly from temperature sufficiently above Tg, the Bartenev theory has been shown to predict closely experimentally observed residual stresses in works by Bartenev himself [37] and also by Gardon [39]. For initial temperatures close to Tg and also for more rapid cooling rates, however, the theory has been shown to yield much poorer correlation with experiment [39].

Subsequently, Indenbom [40,41] criticized the Bartenev theory on a number of counts. First, Bartenev’s hypothesis of the freezing of the fictive temperature gradient only takes into account the temperature gradient in the layer undergoing solidification, and ignores the state of the layers that have already hardened. Also, Indenbom pointed out that Bartenev’s formulation is not based on a consideration of the magnitudes of the strains on both sides of the transition zone. According to Indenbom, the total strain in the x-z plane, E ,

is uniform across the plate thickness and is composed of three components:

where Ee is the elastic strain, which is zero above Tg, eP is the plastic strain, which is time-independent below Tg, and To is the initial temperature. For the region of slab where T < Tg, th_e average elastic strain E, is zero and the corresponding average total strain E is given by

Accordingly, since E is independent of y , such that e(t) = ;(I), Eqs. (3) and (4) can be equated at y = y*(t) , where T b * , t ) I Tg and Ee@*,t) = 0, resulting in

which can be used to generate e P , given T b , f ) and assuming that eP remains independent of I for y > y* (i.e., T < Tg). Finally, when the temperature has

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uniformly reached its asymptotic value, T,, setting E = E with Ee = 0 results in

Accordingly, with 4 0 1 ) determined, the final stress can be calculated based upon the following relation from elasticity:

In particular, Indenborn’s theory has demonstrated [a, 411 a better correlation with experimental glass tempering data than that of Bartenev, particularly for the higher rates of cooling associated with forced convection.

The two previously described theories of Bartenev and Indenbom are based upon thermoelastic formulations, and thus cannot predict the procession of the stress profiles through the transient cooling period (recall Indenborn’s was time independent for T < Tg). In order for a theory to have such a capacity, a constitutive model based on viscoelasticity must be employed. Aggar- wala and Saibel [34] developed a theory for predicting residual thermal stresses in an infinite plate based on a four-parameter viscoelastic constitutive model,

A ( E ~ - e) t B ( i x - k ) = C(ux - a) +D(GX - 6) (8)

where the plate’s midplane is the x-z plane with surfaces at y = +b, and

e = K o t e (9)

where

in which ex and ox are the strain and stress, respectively, in the plane of the sheet; K is the compressibility; A, B, C, and D are the constitutive parameters; 01 is the coefficient of linear thermal expansion; and T is temperature measured

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relative to ambient. Applying appropriate equilibrium equations, the residual stress profile ox can be expressed as

2 A K + C 0 2 B K + D

[ ( A E ~ + B& - A e - B E ) exp (J d7)I

where ex is an unknown time function determined from the stress equilibrium of

$: oxdy = 0

Aggarwala and Saibel illustrated their theory by solving it for a Maxwell material with a temperature-dependent expansion coefficient, and with the viscosity coefficient assumed to be zero for T > Tg, and infinite for T < Tg. Such a formulation corresponds exactly to the inviscid fluid and perfectly elastic states assumed in Indenborn’s theory [40,41] and thus, not surprisingly, the two solutions coincide if the expansion coefficient is taken as temperature-independent. This latter result will be shown in a separate note.

residual thermal stresses based upon a time- and temperature-dependent modulus. They have treated the case of a slab with complete lateral restraint and thus zero lateral strain during cooling. Lee, Rogers, and Woo [36,43] subsequently modified the approach of Muki and Sternberg [42] for that of an infinite slab with free lateral expansion and zero lateral resultant force. The theory in Refs 36 and 43 is based upon nonisothermal linear viscoelasticity for the deviatoric stress-strain fields coupled with a purely elastic treatment of the trace components of the stress-strain fields. For an infinite slab of thickness 2b, the resulting governing equations reduce to

Muki and Sternberg [42] developed a viscoelastic theory for predicting

subject to the zero lateral resultant force constraint

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where e(f) is the isotropic elongational strain in the xz plane, x is the reduced time defined by

where K(T) is a temperature-dependent characteristic relaxation time, and R k ) is an auxiliary modulus function satisfying

where C(x) is the relaxation shear modulus. Narayanaswamy and Gardon [44] applied the theory of Ref. 36 to calcu-

lating the transient and residual stresses during quenching of inorganic glass by using a modified numerical integration scheme. The stress-optical law holds for such materials, and stresses were determined experimentally by birefringence measurements. It was shown that the model correlates well with experiment, provided the initial temperature is significantly above Tg. At initial temperature below this, it is necessary to consider the contribution of structural effects t o the residual stress field. Narayanaswamy [45] extended the theory to include these structural effects by evaluating the thermal strain using the fictive temperature concept of Tool [46]. The fictive temperature has been calculated from the real temperature and reduced time. It has been shown [45] that structural relaxation can introduce about 25% of the total residual stress. A further modification of the theory in Ref. 36 has been done by Ohlberg and Woo [47], where a temperature-dependent expansion coefficient has been in- corporated into both the fictive temperature concept and also the pseudo-temperature approach of Muki and Sternberg [42].

This review of the theoretical predictions of residual stresses in inorganic glass suggests the possibility of applying the theories to residual thermal stresses in polymers. Only a few authors have attempted to do this. In particular, Struik [17] modified the theory of Aggarwala and Saibel [34] as developed for a Maxwell material with zero viscosity above Tg and infinite viscosity below Tg by neglecting the temperature dependency of expansion coefficient. A reasonable correlation was demonstrated between the theoretically predicted and experimentally observed surface stresses for quenched sheets of PMMA. Later, use will be made of the thermoelastic theory of Indenbom [40,41] (which is the same as Struik's modification [ 171 of the theory of Aggarwala and Saibel

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1 88 ISAYEV AND CROUTHAMEL

[34]), and the viscoelastic theory of Lee, Rogers, and Woo [36]. in regard to present experiments with quenched PS and PMMA.

Two other approaches [ l l , 481 have been developed for prediction of the thermal stress distribution in quenched polymer slabs. Although the authors have referred to stress distribution in molded parts, their approaches are valid only for thermal stress prediction since the flow stresses have been neglected in their formulation. Menges et al. [ 1 1 1 used the thermoelastic formulation for shrinkage, A€, with a temperature dependent on the linear expansion coefficient a and Young’s modulus E. Accordingly, the residual stress value at the ith layer of the slab can be derived from the formula

where the shrinkage, A€, can be determined from a force balance at every instant of time according to the equation

Rigdahl [48] performed calculations for residual stresses by using a finite- element technique, assuming that the residual stresses arise from a complete hindering of thermal shrinkage during cooling. Accordingly, a formula from elasticity theory for the residual stress, o = EaAT, has been used and coupled with the computed transient temperature field obtained from the energy equation.

IV. RESIDUAL STRESSES IN MOLDED PARTS-REVIEW

The experimental determination of residual stresses in injection-molded parts has been performed in several works [9, 1 1 , 13,49441. In most of these, the layer removal method of Treuting and Read [29] has been utilized in evaluating the stress profiles. Unlike residual stresses of a purely thermal origin, stresses in a molded part are anisotropic in nature, owing to the contribution of flow stresses in the flow direction. However, this effect has not been taken into account in earlier investigations. Coxon and White [49] investigated the effect of cross-linking on the residual stresses in injection-molded high-density polyethylene. Layer removal showed the residual-stress profiles for all of the

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specimens to be compressive near the surface, with a subsurface compressive maximum, and tensile in the interior. The residual-stress profile for the cross- linked specimen was similar in shape to that of the uncross-linked specimen, with a slightly greater compressive stress maximum. Thus, the characteristic shape of the residual-stress profile was determined by the processing variables, and subsequent cross-linlung served only to slightly increase the level of compressive stresses near the surface.

In a separate investigation, Coxon and White [50] looked at the effect of aging on residual stresses in injection-molded polypropylene. Specimens were stored 8 to 10 months following molding before analysis at three different temperatures-room temperature, -40°C (deep freeze), and -196°C (liquid nitrogen). Stress analysis proceeded by use of the layer-removal method. For the sample aged in liquid nitrogen, the residual stress profile was marked by a compressive maximum at the surface and tensile stresses in the interior. Stor- age in liquid nitrogen w a s believed to reduce stress relaxation or aging to a negligible level, and thus the observed stress profile for this sample was thought to be that contained in a freshly molded part. A significant reduction in the residual stress levels for the sample aged at room temperature was observed. The surface stress was near zero, followed by a subsurface compressive maximum and a tensile interior.

glass-filled polypropylene. Stresses were measured using the layer-removal method in specimens parallel to the flow near the gate and perpendicular to the flow far from the gate. For the parallel-orientation specimens, stresses near the surface were compressive, with a maximum value some distance below the surface, with the interior being in tension. Residual stresses measured in the samples transverse to the flow direction were compressive at the surface, tensile in a region below this, and compressive again in the center of the sheet. Varying the mold temperature between 30 and 80°C was shown to have a negligible effect on the residual stress profiles.

Sandilands and White [51] investigated the effect of injection pressure and crazing on the residual stresses in molded bars of polystyrene using layer removal. Varying the pressure in the range of 37 to 143 MPa was shown to have essentially no effect on the observed residual stresses, which were parabolic in shape. The introduction of surface crazes by bending around a cylindrical form was shown to increase the levels of residual stresses in the molded bars.

ual stresses in injection-molded strips of PS. Stresses were evaluated qualitatively by drilling holes through the sheets and immersing then in n-heptane. In this manner, the magnitude and gapwise location of tensile stresses could

Hindle et al. [ 131 recently reported on residual stresses in injection-molded

Schmidt et al. [52] studied the effect of processing parameters on the resid-

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be determined by observing craze formation. Strips molded at low wall temperatures developed crazes in the center of the strip, while those subjected to high packing pressures crazed at the surface. Variations in melt temperature and injection rate produced no discernible crazing upon immersion. This method of stress analysis gives no information about the compressive stresses in the molded part, however. It was noted that no crazing occurred in any of the sheets if 21 days were allowed to elapse between molding and immersion in n-heptane, presumably due to relaxation of the residual stresses.

Russell and Beaumont [53] investigated the residual stresses in nylon-6 injection-molded bars. According to them, the stress distribution in the molding is found to be parabolic with a compressive stress at the surface and tensile stress at center. The magnitude of these stresses was shown to be inversely proportional to mold temperature. For example, the maximum compressive and tensile stresses were, respectively, 6.5 and 3.5 MPa for a mold temperature of 25°C compared to stresses of 1.2 and 0.8 MPa for mold temperature of 100°C. Apparently, the residual flow stresses were negligible for these nylon- 6 moldings. By applying the Aggarwala-Saibel theory for thermal stresses [34] t o predict the distribution of residual stresses in these moldings, it was found that the model overestimates the experimental data. The authors explained this discrepancy as a consequence of the stress relaxation in the molding, since the model assumes the absence of stress relaxation below the freezing temperature of the polymer. However, no experimental evidence of relaxation has been presented. Further, an exposure of the molding to boiling water was found to result in a reversal of the residual stress distribution with a tensile stress at the surface and compressive stress at center. I t was proposed that this effect was due to water-induced crystallization in the nylon-6.

Mandell et al. [9] studied residual stresses in injection-molded polysulfone bars, particularly concerning their effect on fatigue. Again, in molded as well as quenched bars, the residual stress distribution was found to be parabolic with the level of compressive and tensile stresses being about four times lower in the molded rather than in the quenched bars. In contrast to annealed samples, it was found that fatigue cracks in the molded and quenched samples ini- tiate in the interior due to the presence of compressive stresses a t the surface.

Siegmann et al. [54] made an extensive study of the effect of injection- molding conditions on the residual stresses in molded square slabs from PPO (Noryl). The measurements were made at different locations from the gate in a direction perpendicular to the flow direction. Complicated stress distributions were found. The main effect of melt temperature was observed in the surface layer where the compressive stresses range up to 20 MPa but decrease markedly with increasing melt temperature. In the interior, the tensile stresses

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were found to have a peak value of 5 MPa, depending slightly on melt temperature below 250°C. Above 250"C, these tensile stresses were found to decrease and even become compressive. At increasing distance from the gate the surface compressive stresses decrease and approach the level of stresses in the interior. The compressive stresses in the outer layers were found to be approximately constant up to mold temperature of 60°C. Above this mold temperature, a marked decrease in the stresses was observed. In the interior, the tensile stresses were found to decrease gradually with increasing mold temperature. However, with increasing distance from the gate, the dependence of stresses on mold temperature becomes weak. The injection rate has been shown to have a dominant effect on the residual stresses. At low injection rate, the surface stresses are tensile whereas at higher injection rate they become compressive, eventually passing through a maximum with subsequent decay. In general, this maximum decreases with increasing distance from the gate. In the interior, the dependence of residual stress on injection rate was found to be less pronounced. Concerning the influence of injection pressure, the surface residual stresses were found to first increase with increasing pressure, reaching a maximum and then decreasing. The range of injection pressure was 35-140 bar, with the value and position of the maximum depending on distance from the gate. A reversal of the surface stresses from compressive to tensile has been found near the gate at high pressures. In addition, at high pressures, a high compressive stress was detected in the core region near the gate.

fect upon residual stresses. According to Menges et al. [l 11 , increasing the duration of holding pressure on PS moldings was found to lead to increased surface compressive stresses. Moreover, the level of the holding pressure itself was found to have less influence on the residual stresses.

Based upon pure bending theory, Jacques [ 121 presented a thermoelastic analysis of warpage in injection-molded slabs due to unbalanced cooling caused by asymmetric cooling and geometry. He has found that warpage is more highly dependent upon the design condition of the asymmetric cooling fluid temperature than on differences in the metal thickness between cooling line and cavity surface.

It should be noted that in most of the previously cited work on stresses in molded parts, the stress profiles were calculated based solely on the curvature component along one axis of the specimen, with the curvature in the other direction assumed to be zero or equal. This treatment is valid only for the uniaxial or equal biaxial formulations. This is particularly puzzling for the investigation in Ref. 13, where the very obvious anisotropic biaxial nature of

The holding pressure duration has also been found to have a significatn ef-

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the stress field was detected, as evidenced by differences in the curvature profiles for samples oriented parallel and perpendicular to the flow direction. For proper analysis of stresses in the molded part, curvature profiles of specimens oriented parallel and perpendicular to the flow direction must be measured for samples taken from the same location in the mold, and the general biaxial formulation in Ref. 29 must be applied to determine the stress profiles parallel and perpendicular to the flow. Moreover, an incorrect treatment could account for the peculiar stress profiles observed in many cases.

V. EXPERIMENTAL TECHNIQUES FOR RESIDUAL STRESS MEASUREMENTS

A. Hole Drilling Method

Numerous examples of using the hole drilling method for residual stress mea- sutements appear in the literature. This method was first described by Mathar [ 551 in 1934 and later improved and used in many investigations of residual stresses in metals [56-581. In particular, drilling a hole disturbs the stress field equilibrium in the sample having residual stresses. Deformations which develop on the surface during hole drilling in the sample can be recorded using four strain gauges located around the hole in directions mutually perpendicular to each other. The small incremental change in surface strain AE for a small incremental increase in hole depth h is proportional to the magnitude of the average residual stress in the layer corresponding to this depth. A dis- ruption of the residual stresses in a biaxial planar stress field can be determined according to the following equations based upon pure elasticity:

For an equal biaxial stress field (a, = u,) such as occurs in the quenching of strip specimens, these equations can be simplified:

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where K 1 and K z are constants which can be determined from calibrating specimens with known stress fields.

It should be noted that the hole drilling method has not received much attention in the polymer literature. Ito [59] reported some experience with applying this method to polymers. The main difficulties arise from calibration and use of special equipment including attachment of strain-gauge rosettes to the plastic surface and a precise method for drilling without introducing a stress during the machining. Other limitations of this method concern the develop ment of cracking in plastic molded parts during hole drilling 1521.

B. Stress Relaxation Method

The stress relaxation method for determining residual stresses in polymers has also evolved from similar work with metals, in which it has been shown that the kinetics of creep and stress relaxation are dependent upon the level of the residual stresses. Presently, two approaches [60,61] are available for applying this method. Both of these approaches are based upon the following power-law relation between stress, a, and strain rate, I?, in uniaxial extension:

6 / E t Ban = E'

where B and n are empirical constants. For stress relakation, t = o such that

For the case in which a residual stress a is present in the plastic part, the effec- tive stress is given by aeff = u - ai. If flow has no effect on the residual stress, Eq. (24) can be rewritten as

By solving this equation and eliminating time t , the slope of the relaxation curve, da/d In t can be found as follows:

where a. is the initial stress. The method proposed by Li [60] uses the intercept of the plot - (da/d In t)

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versus u with the u-axis to give the value of ui. Therefore, a single relaxation experiment is sufficient to determine ui. Since the value of - (du/d In t ) may depend on uo (or deformation) in some cases, it may be necessary [26] t o re- peat the experiment at different values of uo and then extrapolate the dependence of uj versus uo to uo = 0. The intercept with the u-axis gives the residual stress. It is noted, however, that the methof of Li is difficult to implement when the curvature of the stress relaxation curve u versus In t is low or practically linear. In order to avoid these difficulties, Kubat and Rigdahl [61] proposed a method which is based upon the measurement of the maximum slope (- du/d In t),,, for a number of relaxation curves corresponding to various initial stresses uo (or strains). Then, by extrapolating the plot of (-du/d In t)max versus uo to a zero value of (-du/d In t),,,, the value of ui can be found as the intercept with the u,-axis.

The methods of Li and of Kubat and Rigdahl have both been applied to residual-stress measurement in plastic slabs produced by injection molding [51, 62-65] and by other processing operations [66-691. The main disadvan- tage of these methods seems to be the impossibility of measuring the residual stress distribution in the gapwise direction of plastic slabs. Therefore, the meaning of the residual stress values determined by these methods is somewhat unclear. It seems that the most suitable means for determining the gapwise residual stress distribution is the layer-removal method [29] which has been employed in the present investigation and will be described in the next section.

VI. EXPERIMENTAL

A. Materials and Processing Conditions

The material under investigation was polystyrene (PS) Styron 678U (Dow Chemical). Molding experiments with this polymer have been carried out on an 800-ton Beloit machine with a shot size of ~ 2 . 5 X lo-’ m3 using two different sized straight-gated rectangular cavities and a variety of processing conditions, as indicated in Table 1.

In addition to molding experiments, the present work has included free quenching experiments performed on annealed PS and PMMA strips having planar dimensions of 0.05 X 0.05 m and thicknesses of 0.0026 m for PS and 0.003 m for PMMA, and on annealed PMMA strips having planar dimensions of 0.108 X 0.108 m and a thickness of 0.0071 m. The PS specimens for quenching were removed from the abovementioned molded strips whereas the PMMA specimens were removed from extruded sheets (Plexiglas G, Rohm and Haas).

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TABLE 1 Cavity Dimensions and Processing Conditions of the Molded PS Strips

Mold Melt Sample temperature temperature Average velocity Cavity dimension number ("C) ("0 (mid WX2bXL(m)

1 60 244 0.82 0.051 x 2 60 210 0.56 0.00254 X

0.48

3 40 223 0.47 0.076 x 4 60 5 60

247 0.70 21 5 0.71

6 40 228 0.83 7 40 224 0.36 8 60 224 0.34

0.00381 X 0.456

To eliminate flow prehistory, these PS and PMMA samples were placed in a form and annealed according to the following procedure: 100°C for 24 h, 130 and 170°C for 2 h, followed by cooling to below Tg. Subsequently, the samples were placed in a cup filled with glycerol and the annealing procedure was re- peated, followed by quenching in a controlled-temperature water bath.

B. Layer-Removal Technique

Residual stresses in the quenched and molded strips were measured using the layer-removal technique of Treuting and Read [29]. In this analysis, thin layers of uniform thickness are removed from one surface of a rectangular specimen, thus upsetting the residual stress equilibrium existing in the piece. To reestablish equilibrium, the specimen warps to the shape of a circular arc. The gapwise residual stresses in the specimen prior to layer removal can be calculated by measuring the resultant curvature as a function of depth of material removed, using the general biaxial stress relation:

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where u, is the longitudinal stress, y = k b are the initial upper and lower surfaces of the specimen,y = y l is the new surface formed after each removal, E is the elastic modulus, u is Poisson’s ratio, and px and pz are the measured longitudinal and transverse curvatures, respectively. uZO1) can be derived by exchanging the x and z subscripts in Eq. (27).

Residual molding stresses vary with direction and position in the strip due to the influence of flow stresses. Accordingly, for this investigation, specimens were removed parallel and perpendicular to the flow direction in molded strips having thicknesses of 0.00254 and 0.00381 m, with the respective planar dimensions of these specimens being 0.05 m X 0.005 m and 0.076 m X 0.0076 m; such samples were removed near and far from the gate with positions of their planar center from the gate being at 0.025 and 0.455 m for the thin strip and at 0.038 and 0.418 m for the thick strip. A modified Tensilkut machine was used to mill layers in increments of 0.00003 m from these specimens. Re- sulting curvatures pxb) and pZO) for specimens respectively parallel and perpendicular to the flow direction were measured with an optical microscope immediately after every so many layer removals. Curvature data were fitted by polynomial regression, and gapwise stress profiles uxb) and uZb) were calculated using Eq. (27).

Thermal stresses from quenching are isotropic: 001) = u x b ) = u&) and p(y) = p, (y) = pZ@). For this case the layer removal has been performed on specimens of 0.05 and 0.072 m length, approximately 0.005 and 0.009 m width, and having, respectively, thicknesses of 0.0026 and 0.0071 m removed from quenched strips. Based on the measured curvature pb), the thermal stress profile has then been calculated from the following simplified form of Eq. (27):

Integration of the residual stress profile indicated an experimental error on the order of 5%.

Rigorously speaking, the method of Treuting and Read, employed in the present investigation, is applicable to a linear elastic material with isotropic elastic constants. It has been shown in several investigations [70,71] that Young’s modulus of oriented injection-molded polymer sheets is independent of the particular direction of the molecular alignment in the test specimen, be it parallel or perpendicular to the specimen’s longitudinal axis. Such modulus values are gapwise averaged, and some variations of modulus with gapwise posi-

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tion would be expected. In fact, these gapwise variations of modulus have been recently found [6, 72,731. For the present investigation, however, Young's modulus has been assumed to be constant, irrespective of direction in the sheet or gapwise location.

The materials studied in the residual stress analysis, PS and PMMA, most definitely do not follow a linear elastic constitutive model. In particular, a viscoelastic phenomena of both creep and relaxation have been observed for these materials.

With regard to creep, it has been observed that for a stress specimen reduced to its half thickness (i.e., at the end of the layer-removal operation), the magnitude of the curvature increases with the passage of time to some equilibrium value. Thus, the response of the specimen to the moment induced by removing layers can be separated into two components: (1) an instantaneous, elastic response, resulting in an initial level of curvature; and (2) a time-dependent, creeping response, whereby the degree of curvature grows to some final value. This behavior can be represented mathematically by the expression

where p ~ l , t) is the curvature of a machined specimen at time t after the upper surface has been. milled down to the plane y = y1 , p 0 ,0) is the curvature immediately after machining, and p 2 ('yl ,t) is the time-dependent additional term.

This creeping action has an important consequence with regard to the application of the layer-removal method of Treuting and Read [29] to a viscoelastic material. Since their formulation is based on a linear elastic material, the first term of Eq. (29), namely p1 ('yl ,O), must be used in applying the curvature-stress relationships to polymers. Fortunately, the creep phenomenon occurs very slowly, and measurement of the curvature within the first few minutes following machining is soon enough to ensure an accurate value for the time-independent term.

surement have been expressed in the literature. Thakkar and Broutman [74] feel that the curvature of a specimen should be measured within the first two minutes following machining in order t o avoid what they refer to as stress relaxation. In this instance, stress relaxation is a misnomer, as the observed time- dependent behavior is that of creep. However, regardless of the terminology used, the procedure of measuring the curvature soon after machining is proper in light of the preceding discussion. That is, the elastic response of the material is what one needs to measure in order to calculate the residual stresses.

Two opposite opinions on the subject of the time-dependent curvature mea- Dow

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Siegmann et al. [30], on the other hand, expressed the opinion in their work that three days should elapse between machining and curvature measurement in order to allow the specimen to warp to its equilibrium configuration. The curvature that they measured, then, was the total value, not just the elastic time-independent component. For this to be proper, a new formulation of the stress-curvature relationships would have to be developed, one based on long-time viscoelastic behavior as opposed to the elastic formulation of Treut- ing and Read [29]. As they give no indication that any such formulation was developed, it can be assumed that the curvature values used by those authors in their residual stress calculations were large (long time instead of instantaneous values). Such an error should have manifested itself in the results by over- predicting the residual stress magnitudes. A check of the results supports this assertion. For 1 /4 in. sheets of PMMA receiving the same thermal processing (quenching from 130 to O'C), Siegmann et al. [30], who waited 3 d following machining for each curvature measurement, show residual stresses in the surface layers which are 55% higher than those determined by So and Broutman [28], who allowed only 2 min to elapse between machining and measurement. This difference in magnitude is far greater than the estimated experimental errors of 5 to 15%. Thus, it seems likely that the higher residual stresses observed by Siegmann et al. [30] can be attributed to the fact that they used the larger equilibrium values of curvature rather than just the elastic component.

Relaxation of the residual stress levels has also been observed in the quenched sheets of PS and PMMA in the present investigation. A specimen tested immediately after quenching will indicate higher residual stresses than one tested several weeks after quenching. Thus, the experimentally determined stress profiles are dependent upon the time that has elapsed between when the stresses are introduced and when the layer-removal procedure is carried out. In order to draw any conclusions concerning the effect of processing conditions on residual stresses, it is imperative that the stresses be measured as soon as possible following processing. In addition, it would be of value to monitor the decay of the stresses with time in order to quantify the relaxation process in the particular material.

Creep and stress relaxation are discussed in more detail below, where the specific results for quenched PMMA and PS sheets are presented.

In using the layer-removal technique to determine residual-stress profiles, it is important that the removal process does not introduce additional stresses. Thus, before the residual stress investigation was developed in detail, several preliminary tests were conducted to determine the effects of the method of layer removal and the size of the test specimen on the observed results. With regard to the method of layer removal, two machining procedures were ex-

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FIG. 1. Effect of machining method on curvature profiles. Data for case PMMA machined on Bridgeport milling maching (A) and Tensilkut milling machine (a) and annealed PS (0 ) and PMMA (0) machine on Tensilkut.

amined. A vertical-spindle Bridgeport milling machine equipped with a 0.01 6- m diameter conventional end mill operating at 2,700 rpm and a modified Ten- silkut milling machine with a 0.01 3-m diameter carbide-tipped router bit operating at 20,000 rpm were compared. Curvature vs thickness data for the two removal procedures are shown in Fig. 1 for samples taken from a 0.003 m thick sheet of cast PMMA. The curvatures observed when the Bridgeport milling machne was used are approximately an order of magnitude higher than those detected when the Tensilkut machine was used. This additional deflec- tion can be attributed to tensile stresses introduced by the machining process. Thus, the Bridgeport milling machine with the cutter and spindle speed used is not an acceptable means of removing layers for residual stress analysis since the deflections caused by machining damage could mask the deflections due to the residual stresses present in the piece before machining. The Tensilkut machine, on the other hand, is seen to introduce negligible stresses, as shown by the small curvature values. Also, the machined surface produced by the Tensilkut remained transparent, while that produced by milling on the Bridgeport was matted. Also shown in Fig. 1 are curvature data for layer removal operations performed by the Tensilkut machine on samples of annealed PS and PMMA. Again, the minute curvatures are an indication that the Tensilkut introduces

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negligible machining stresses. Based on these results, all residual stress analysis was conducted with the use of the Tensilkut machine. With regard to the size of the test specimen, it has been found that provided the length/width ratio is large (>6), there is no appreciable effect of size on the observed curvature.

VII. THERMAL STRESSES IN FREELY QUENCHED STRIPS

In order to understand the complicated phenomena of residual stresses arising in injection molding, consideration will first be given to thermal stresses in polymer strips due to the abrupt cooling process alone. In particular, it is of interest to evaluate the effects of strip thickness, initial temperature, and bath temperature on thermal stresses and to compare experimental data with theoretical predictions based upon pure elasticity and linear viscoelasticity. Results are shown in Fig. 2 for the curvature versus fractional gapwise coordinate for PS samples freely quenched from 130 and 150°C to 23°C and for PMMA samples freely quenched from 130 and 170°C to 0°C. These results have been generated from fresh specimens for which the layer removal has been performed approximately 1 h after quenching. Curvature values are seen to vary mono- tonically with depth of material removed. Figure 3 shows the corresponding stress profdes generated from the curvature data in Fig. 2 by using Eq. (28). The stress distributions are parabolic in nature, with maximum compressive stress at the surface and maximum tensile stress at the center. For the quenched PS samples, surface compressive stresses of -9.8 and -8.2 MPa and centerline tensile stresses of 2.6 and 3.3 MPa can be seen for initial temperatures of 130 and 15O"C, respectively, while for the quenched PMMA specimens these stresses are significantly higher and lie in the range from -19.8 to 7.2 MPa for the strip quenched from 170°C and from -26.3 to 9.2 MPa for the strip quenched from 130°C. The parabolic profile is characteristic of thermal stresses and has been observed in both polymers [ 11,18,25-281 and inorganic glass [37-411. It can be seen for both materials that there is little effect of initial temperature on the magnitude of thermal stress. In Fig. 4, the effect of the bath temperature on thermal stress is shown. In this case, the PMMA strips of 0.003 m thckness were quenched from 170°C to either 22 or 0°C. The stress profile in Fig. 4 for a bath temperature of 22OC was generated from curvature data shown in Fig. 5 whereas the other stress curve is taken from Fig. 3. It is noted that the bath temperature has a more significant influence on the thermal stress level than the initial temperature does, with surface stresses increasing about 40% for a 22°C drop in bath temperature. Finally, the thermal stress profiles for PMMA strips of thicknesses 0.003 and 0.0071 m quenched from 130 to 0°C are shown

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RESIDUAL STRESS DEVELOPMENT

b) i3

i4

20 1

- - 'E

e P

Y

Q

3 c

L 3 0

A - 'E

z B 3

Y

Q

3 +

L

FIG. 2. Curvature profiles for 0.0026-m thick PS quenched from 1 3OoC (a) and 150°C (b) to 23°C and for 0.003-m thick PMMA quenched from 130°C (c) and 17OoC (d) to 0°C.

in Fig. 6. Corresponding curvature data used to geneme the stress profile for the 0.0071 m thickness case is shown in Fig. 7. It can be seen from Fig. 6 that the thickness increase has very little effect on the observed compressive stress but noticeable effect upon the tensile stress.

It is of interest to compare the experimentally determined thermal-stress profiles for quenched PS and PMMA specimens with those predicted by theory. In particular, use is made of the instant freezing linear thermoelastic theory of Indenbom [40] and the linear viscoelastic theory of Lee, Rogers, and Woo [ 3 6 ] , both of which have been taken from the inorganic glass-tempering literature.

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I - 170 PMMA, 2- 130 T, ~ 0 % 3- 150 PS, 4- I30 T, =23OC

-30 Y FIG. 3. Thermal-stress profiles for 0.003-m thick PMMA quenched from

170°C (Curve 1 ) and 130°C (Curve 2) to 0°C and for 0.0026-m thick PS quenched from 1 50°C (Curve 3) and 1 3OoC (Curve 4) to 23°C.

-30 L

FIG. 4. Thermal-stress profiles for 0.003-m thick PMMA quenched from 170 to 22OC (Curve 1) and 0°C (Curve 2).

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1

203

FIG. 5. Curvature profile for 0.003-m thick PMMA quenched from 170 to 22OC.

Table 2 shows the assumed properties used to calculate the theoretical stress profiles.

In particular, the temperature field required to conduct the thermal-stress calculations has been determined from transient one-dimensional conduction with a convective heat-transfer boundary condition at the surface, namely

aT a2T at ay2

a- - _-

subject to the following conditions on T b , t):

aT aT Tb,O)=TO, -(O,t)=O, -k-((b, t )=h[T(b, t ) -T,]

at at

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TABLE 2 Mechanical and Thermal Properties of Polystyrene and PMMA Used to

Calculate Theoretical Stress Profilesa ~~~

Polystyrene PMMA

Young's modulus E (N/m2) 2.11 x lo9 3.5 x lo9 Poisson's ratio u .33 .33 Density p (kg/m3) 1.15 x lo3 1.18 x lo3

Glass transition temperature Tg (OK) 373 3 73 Thermal expansion coefficient 0 (l/"K) 4.5 x 10-5 Thermal diffusivity (mZ Is) 6.13 x lo-* 1.095 x lo-'

7.0 x 10-~

Thermal conductivity k (J/s-m*OK) 1.30 X lo-' 1.9 X lo-' Heat capacity Cp (J/kg*OK) 1.844 X lo3 1.47 X lo3 Heat transfer coefficient h (J/s- mz *OK) 1.47 x lo3 1.47 x lo3

aData other than h were taken from Encyclopedia of Polymer Science and Technology, Wiley-Interscience, New York, 1970.

10

0

- Lu E \

g -10 - L

b

-20

-30

Thickness (m) 1-0.0030 2 -0.007 I

FIG. 6. Thermal-stress profiles for 0.003-m (Curve 1) and 0.0071-m (Curve 2) thick PMMA quenched from 130 t o O°C.

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l2

FIG. 7. Curvature profile for 0.0071-m thick PMMA quenched from 130 to OOC.

with T, being the ambient temperature of the bath and a being the thermal diffusivity. Equation (30) has been solved by using a finite-difference method with standard tri-diagonalization. The heat-transfer coefficient, h, has been based upon steady natural convection for a vertical plate in water at represen- tative temperatures [75] :

k h = 0.555 - (Gr*Pr)'/4

L

where k is the thermal conductivity of the water, L is the length of the plate, Gr is the Grashof number, and Pr is the Prandtl number. The time dependence of the relaxation modulus G(r) and the temperature dependence of the characteristic relaxation time K(T) for polystyrene, as required in the calculation

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T, O C 60 80 100 120 140 160 180 200 220 240

10

9 N

E \ 8

d 7 -

6 g 5 -

z

' 6 -

-I 4 -

3 -

- 6

- 4 8

2 r- k

- o x

-2 -I

- -4

- -6

v) - Y

0"

1~~~ I , 1

- 4 - 3 - 2 - 1 0 I 2 3 4 5 6 7 log x

FIG. 8. Relaxation G and auxiliary R moduli versus reduced time x and characteristic relaxation time K versus temperature T for PS $111. G ( x ) and K ( T ) from Ref. 7 6 ; R ( x ) from Eq. (18).

based upon the theory of Lee, Rogers, and Woo [36], have been taken from the work of Aklonis and Tobolsky [76] for monodisperse polystyrene $1 11, having a molecular weight M, = 2.39 X lo5 which is practically the same as that of polystyrene Styron 678 as reported by Wales [20]. Although these two polystyrene samples have different molecular mass distributions (MMD), it is well known that such differences would affect G(t) only in the fluid and rubbery state and not in the glassy state and transition zone where, as will be shown below, the value of G(t) is most important for determining the residual thermal stress. In Fig. 8 are shown the functions G(t) and K(T) according to Aklonis and Tobolsky [76] and the corresponding calculated auxiliary modulus function, R(x) , satisfying Eq. (18). In particular, calculations for the thermal stress have been performed with R(x) represented in tabulated form over the entire domain indicated in Fig. 8 and also with R ( x ) represented by the dashed curve in Fig. 8, as given by

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R ESI D UA L STR ESS D EVE LOPM ENT 207

Resulting predictions based upon these two representations for R(x) have been found to be essentially the same, thus indicating that the rubbery and fluid states, corresponding to where Eq. (33) does not represent R(x) well at all, give little contribution to the final stress field. That is, the glassy state and transition zone are mainly responsible for the thermal stresses; on the other hand, as shown in Ref. 15, the fluid and rubbery states characterize the residual flow-induced stresses in injection-molded parts. Shown in Figs. 9 and 10 are the gapwise experimental thermal stress profiles and corresponding theoretical predictions for quenched PMMA and PS strips. In particular, experimental results in Fig. 9 for the 0.003-m thick PMMA strips quenched from 170 and 130°C to 0°C (a and b, respectively) and from 170 to 22°C (c) together with 0.0071-m thick PMMA quenched from 130 to 0°C (d) are compared with theoretical predictions based upon the Indenbom theory [40]. It can be seen that the theory of Indenbom adequately predicts the entire residual stress profile for quenched PMMA. Struik [ 171 has demonstrated a similar correspondence between experiment and his thermoelastic theory for quenched PMMA, but only the surface stresses were measured. On the other hand, Fig. 10 shows experimental results for 0.0026-m thick PS strips quenched from 130 and 170°C to 23°C (a and b, respectively) together with theoretical predictions based upon the theories of Indenbom [40] and Lee, Rogers, and Woo [36]. As can be seen, both theories substantially overpredict the level of the experimentally measured thermal stresses. There are two major effects which may contribute to this observed discrepancy, namely, nonlinear behavior of the polymer during quenching in the transition zone between the rubbery and glassy state and stress relaxation after quenching. Since the layer removal was initiated approximately 1 h after quenching, there was not enough time for substantial stress relaxation. Thus, for PS it appears that the above-mentioned nonlinear behavior mainly accounts for the lower-than-predicted experimental stress values. Such a nonlinear behavior may reduce the value of the shear modulus of the polymer and, correspondingly, the residual stresses. It is well known from creep [77] and oscillatory-deformation experiments 1781 that a maximum nonlinearity effect for polymers occurs in the region of the glass transition temperature. Some indication of the possible importance of such an effect for residual thermal stresses in plastics can be found in the work of Weitsman [79].

It is interesting to note what effects the initial specimen temperature and

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-30

10

0 - cu E

5 b

\ z -10

Y

-20

-30

10

0 Nii E \

f -10 Y

b -20

-30

10

0 A

cu E 1 -10 I Y

b -20

-30

FIG. 9. Experimental thermal-stress profile (Curve 1) and prediction based upon theory of Indenbom (Curve 2) for 0.003-m thick PMMA quenched from 17OoC (a) and 130°C (b) to 0°C and from 170 to 22°C (c) and for 0.0071-m thick PMMA quenched from 130 to 0°C (d).

bath temperatures have on the stress as predicted by the linear elastic and viscoelastic theories. The effect of initial temperature on the stresses predicted by the theories of Indenbom [40] and of Lee, Rogers, and Woo [36] are shown in Figs. 1 l(a) and 1 l(b), respectively, for a 0.0026-m thick sample of PS quenched to 23OC from either 200 or 130°C. It can be seen that both theories indicate that the stresses are nearly independent of initial temperature. Indeed, the experimental data for both PMMA and PS samples evidenced only a small dependence of residual stress on initial temperature. On the other hand, both

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RESl DUAL STRESS DEVELOPMENT 209

- 3 O L

FIG. 10. Experimental thermal-stress profiles (Curve 1) and predictions based upon theories of Indenbom (Curve 2) and Lee, Rogers, and Woo (Curve 3) for 0.0026-m thick PS quenched from 130' (a) and 1 50'C (b) t o 23'C.

0 b

- -'"t 40

FIG. 11. Effect of initial temperature o n thermal-stress profiles in 0.0026- m thick PS as predicted by the theories of Indenbom (a) and Lee, Rogers, and Woo (b). Quenching temperature 23'C.

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a

I C

-4 -31

0 Y / b

0.c 30 F 60 F

b 2r 0 I 0

FIG. 12. Effect of quenching temperature on thermal-stress profiles in 0.0026-m thick PS as predicted by the theories of Indenbom (a) and Lee, Rogers, and Woo (b). Initial temperature 1 SO'C.

theories predict a significant effect due to temperature of the quenching medium, as shown in Fig. 12 in which a PS strip of 0.0026 m thickness is again used as the example with the quenchmg being from 150 to 60,30, or 0" c.

It might be noted that the heat-transfer coefficient h in Eq. (31) has been evaluated based upon an empirical Eq. (32) from heat-transfer literature [75]. Such a procedure may introduce some error in calculating the value of h which, in turn, could lead to incorrect predictions of the thermal stresses. Therefore, it is appropriate to assess the sensitivity of the present thermal- stress predictions to the value of the heat-transfer coefficient. In this regard, Fig. 13 shows results for the predicted gapwise thermal-stress distribution based upon the Indenbom theory [40] with two different values for the heat- transfer coefficient. It is seen that changes by a factor of 2 in h in either direction from the assumed value in Table 2 introduced a maximal change in stress of only about 20%.

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10

0

0 a I

b c

-10

-20

h, J/s-& OK

FIG. 13. Effect of heat-transfer coefficient on predicted thermal-stress profile according to theory of Indenbom in 0.0026-m thick PS quenched from 150 to 6OoC.

VIII. CREEP AND STRESS RELAXATION

The viscoelastic phenomena of creep and stress relaxation have been observed for quenched PS and PMMA specimens. When a specimen was released from the substrate following machining, the resulting curvature was seen to creep with time to some equilibrium value. With regard to the creeping, it is important to measure the initial elastic response of the material during bending after layer removal. It is this value of curvature that is needed to calculate the residual stresses by the method of Treuting and Read. Typical curvature vs time plots for PMMA and PS are shown in Fig. 14, where it is seen that the additional curvature due to creep decreases exponentially with time. For both the quenched PMMA and PS, the initial level of curvature was approximately 85%

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1.0 0-

o PMMA 0 PS

J.. O'r 1 I I I

0 500 1000 Time (mid

FIG. 14. Ratio of curvature at time t after machining, p(t) , to the final curvature value at long time (-3 d), p-, versus time for 0.0026-m thick PS (0) and 0.003-m thick PMMA (0) quenched from 170 C to 23°C.

of the equdibrium value that resulted after about 3 d following machining. Thus, by measuring the curvature within the first few minutes following layer removal, the experimental error attributable to this creeping phenomenon was kept to a neglibible level.

A substantial degree of relaxation of the residual thermal stresses has been observed for both PS and PMMA. For the results discussed in the previous section, the stresses were measured as soon after quenching as possible, typically within 60 to 90 min. However, when a longer period was allowed to elapse between quenching and stress measurement, the observed stress profiles were smaller in magnitude, presumably due to stress relaxation.

In Figs. 1 5(a) and 15(b) the curvature data and corresponding stress profiles for PS quenched from 150 to 23°C are shown, respectively, for elapsed times between quenching and measurement of 90 min (Curve 1) and 2 X lo4 min (Curve 2). It can be seen that the stresses have relaxed approximately 23% during the time between these measurements. Similar results have been observed for quenched PMMA sheets. Shown in Figs. 16 and 17 are the curvatures and corresponding stress profiles as functions of time elapsed after quenching for PMMA strips quenched as follows: (a) 0.003 m thickness, 170 to 0°C; (b) 0.003 m thickness, 130 to 0°C; (c) 0.0071 m thickness, 130 to 0°C; (d) 0.003

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a

W 0:

1.0 0 FRACTIONAL THICKNESS, y/b

FIG. 15. Curvature data (a) and corresponding thermal-stress profile (b) for 0.0026-m thick PS quenched from 150 to 23OC for elapsed time between quenching and measurement of 90 min (Curve 1) and 2 X lo4 min (Curve 2).

m thickness, 170 t o 22°C. Relaxation by as much as 50% of the initial stress levels can be seen in several cases for elapsed times of 3.2 X lo4 min (24 d). As shown for the case of the sheet quenched from 170 to 22°C (Fig. 17d), the majority of the stress relaxation has occurred during the first few weeks following quenching.

cessing parameters with residual stresses. To do so, stresses must always be Stress relaxation is a very important concept with regard to correlating pro-

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21 4

Surfoce Center

b

C

1'

ISAYEV AND CROUTHAMEL

FIG. 16. Curvature profiles for PMMA at various times after quenching: (a) 0.003-m thick, 170 to O°C; (b) 0.003-m thick, 130 t o O°C;(c) 0.0071-m thick, 130 to O°C; (d) 0.0003-m thick, 170 to 22OC.

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a

- 30 L I - N 10

E 2 5 5

0 b

v) -5 ?

\

- v)

iz -10

8 [L -20

- 0 3

-15

-25

- 30

C

21 5

b

d 10 -

-25

-30

- -

FIG. 17. Thermal-stress profiles for PMMA at various times after quenching under conditions in Fig. 16.

analyzed at the same time following processing for all samples. If this rule is not followed, a specimen with relatively low levels of residual stresses tested soon after processing might give the illusion of possessing greater residual stresses than a sample initially having hgher levels of stresses, but tested later, giving a longer period for relaxation to take place. Ideally, the residual stresses

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should be measured immediately after processing, but this is not feasible in practice since specimen preparation and the layer-removal procedure require a finite amount of time (-3 h).

The exact time to measure the stresses must be determined by the experi- menter. If stresses are measured as soon after processing as possible, errors are introduced since the residual stresses are relaxing during the layer-removal process, as this initial period marks the greatest rate of stress decay. On the other hand, if a long period is allowed to elapse following processing, so that the stress state is essentially static over the duration of the test, then the observed stress profiles can only imply the magnitude of the initial stresses if the relaxation process has been quantified for the material. For the measurement of the residual stresses in molded parts presented in the following section, a long time (- 1 year) was allowed to elapse between molding and stress measurement. This was done for practical purposes, since a great many strips could then be molded at one time without interruption for stress testing.

IX. RESIDUAL STRESSES IN MOLDED PARTS

Residual stresses in molded parts arise from the coupled effects of flow and thermal stresses. While the planar thermal stresses are directionally independent with equal areas of tensile and compressive stresses across the thickness in the case of symmetrically cooled slabs, the flow stresses are tensile in nature and highly dependent on the direction. Thus, residual stresses in molded parts are expected to be highly anisotropic with a complex distribution in the thickness direction. An attempt has been made below to elucidate the effects of molding processing variables on residual stresses and to qualitatively explain the nature of their gapwise distribution on the basis of knowledge available from residual thermal and flow stresses.

Curvature profiles for the samples taken from Strips 1 and 2 of Table 1 are shown in Fig. 18. In this case, melt temperature was the variable in question. Stress profiles generated from the curvature data are shown in Fig. 19, where U I I and u1 indicate the residual stresses parallel and perpendicular to the melt flow direction, respectively. Strip 1, molded with a melt temperature of 244"C, has a parabolic stress distribution everywhere, with higher compressive surface stresses near the end of the cavity. The stresses in Strip 2, molded with a melt temperature of 210"C, however, differ dramatically from those in Strip 1 , particularly in the flow direction near the gate. In this region, Strip 2 shows tensile stresses at the surface of the piece. Near the end of the mold, surface stresses are seen to be compressive, but smaller in magnitude than those found

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FIG. 18. Curvature profiles for molded PS strips 1 and 2 of Table 1. G11 and E l 1 correspond, respectively, t o samples taken near gate and near end of the cavity parallel to flow. Cl and E l correspond, respectively, to samples taken near gate and near end of the cavity perpendicular to flow.

in Strip 1. The trends observed here can be explained in a qualitative sense based on the contributions of the thermal and flow stresses to the total residual stress picture. Flow stresses have been shown [ 181 to be tensile in nature and maximum near the surface of the molded strip. Stresses of a purely thermal nature, on the other hand, are parabolic in profile with compressive surface stresses and tensile stresses in the interior. A molded part contains both of these stress sources, and the nature of stress near the surface can be regarded

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-2L

I

“E a \ z T

-I bi

- 2

FIG. 19. Residual-stress profiles parallel (UII ) and perpendicular (01) t o the flow direction for molded PS strips 1 and 2 of Table 1 ; (a, b) near the gate; (c, d) near the end.

as the result of the coupled competition of the tensile flow stress and the compressive thermal stress. For the case of a high melt temperature, the flow stresses are small, and the thermally induced stresses should predominate. For low melt temperatures, on the other hand, the flow stresses are much higher, and the possibility exists that they could predominate over the thermal compressive stresses near the surface, resulting in a net tensile stress. This behavior is believed to be responsible for the stress profiles seen in the region near the

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Y/b Y/b

FIG. 20. Curvature profiles for molded PS strips 4 and 5 of Table 1. Nota- tion is same as in Fig. 18.

gate for Strips 1 and 2 in Fig. 19. It was shown in Ref. 22 that the flow stresses are maximum near the gate, and drop off rapidly toward the end of the mold. Thus, the effect of the tensile flow stresses should be most evident near the gate, and this is precisely what is indicated in Fig. 19.

shown. Melt temperature has again been varied, but the molded strips were from the 0.00381-m thick cavity. The resulting stress profiles are shown in Fig. 21. Just as in the case of the 0.00254-m thick strips, the higher melt tem-

In Fig. 20 the curvature profiles for samples taken from Strips 4 and 5 are

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I

0 N

E \ z 5 b= -I

-2 b)

-2L -2 )r

FIG. 21. Residua- stress profiles parallel (011) and perpendia-ir (01 to the flow direction for molded PS strips 4 and 5 of Table 1 ; (a, b) near the gate; (c, d) near the end.

perature (Strip 4) resulted in parabolic stress profiles, while tensile stresses are seen at the surface in the flow direction near the gate for Strip 5, molded at a lower melt temperature.

The effect of cavity thickness can be seen by comparing the stress profiles of Strip 1 (Fig. 19) and Strip 5 (Fig. 21). Molding conditions for both were very similar. The surface compressive stresses in the thicker strip ( # 5 ) are

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Y/b Y/b

J.:F/.5 f ) 8 E I -- IE Y

Q

0 I 0 I 0

Y/b Y/b

h) 8Ell 1.5 -; .5 - - E x

0 0 0

I 0 I Y/b Y/b

FIG. 22. Curvature profiles for molded PS strips 7 ang 8 of Table 1. Nota- tion is same as in Fig. 18.

larger than those in the thinner strip (#l), indicating that either the thermal stresses (see Fig. 6) are higher or the flow stresses are smaller (see next section), or both, for the thicker sample.

The curvature data for Strips 7 and 8, molded with variable mold wall temperature, are shown in Fig. 22. The corresponding residual stress profiles are shown in Fig. 23. It can be seen in this case that a change in the wall temperature from 40 to 60°C has resulted in no appreciable change in the shape or magnitudes of the stress profiles, which are close to parabolic in shape. Thus, withm the range investigated, the mold wall temperature does not have a significant influence on the residual molding stresses.

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' r

-2 1

(u

E \ o z I

-I

I

-2

FIG. 23. Residual-stress profiles parallel (u 11) and perpendicular (01) to the flow direction for molded PS strips 7 and 8 of Table 1 ; (a, b) near the gate; (c, d) near the end.

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FIG. 24. Curvature profiles for molded PS strip 6 of Table 1. Notation is same as in Fig. 18.

In Fig. 24 the curvature data for Strip 6 is shown. Corresponding stress profiles are shown in Fig. 25, together with those of Strip 7, to show the effect of injection rate. All the profiles are close to parabolic in shape, with the surface compressive stresses found in the strip molded at higher flow rates (#6) being larger than those in the strip molded at a slower rate (#7). This trend can be explained in terms of the effect of flow rate o n flow stresses. Birefringence measurements in Refs. 18 and 22 have shown that the stresses induced by flow are reduced for larger flow rates, presumably due to the enhanced relaxation process. Thus, the lower flow stresses would allow the thermal stresses to predominate, giving rise to greater compressive surface stresses for a piece molded at higher injection rates. This is consis- tently shown in the case of the stresses found in Strips 6 and 7.

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' r

-2 L a)

I

c u o E \ z =. b' - I

-2

N O E \

0 N O E z s \

i b -I

e c

0

-2 1 1 C)

-2 d)

FIG. 25. Residual-stress profiles parallel (u I ) ) and perpendicular (ul) to the flow direction for molded PS strips 6 and 7 of Table 1 ; (a, b) near the gate; (c, d) near the end.

X. COMPARISON OF FLOW, THERMAL, AND RESIDUAL STRESSES IN MOLDED PARTS

The question which naturally arises concerns the relative contributions of flow- and thermally induced stresses to the total residual stresses in molded parts. In order to address this issue, we should first determine the level of the flow stresses in molded parts. The flow stresses can be calculated from the experi-

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mental birefringence distribution in the gapwise direction Anb) of molded parts by using the stress-optical rule Aa@) = An(~)/c, where Aob) is the gapwise distribution of the difference in principal flow stresses. A stress optical coefficient C = -4800 Brewsters for the PS melt has been taken from the shear-flow measurements of Wales [20] whereas Anb) has been measured experimentally as described in Ref. 22. Such a calculation of the flow stresses is justified since, according to investigations [ 15, 22, 801 which deal with predicting the maximum value of birefringence in injection molded parts, it is sufficient to use a constant value of C found frum the flow behavior of the given material. Moreover, it has been shown in Refs. 18 and 22 for molded PS strips that the contribution of thermal birefringence to the gapwise maximum in the total frozen-in birefringence An is very small near the gate. Thus, at least the gapwise peak value of the flow stresses can be correctly evaluated from frozen-in birefringence measurements in molded PS strips. Accordingly, the effect of melt and mold temperature on the gapwise flow stress distribution near the gate can be seen from Figs. 26 and 27, respectively. In general, increasing the melt temperature significantly decreases the frozen-in flow stresses (Fig. 26). This is mainly due to lower flow stresses during cavity fdling and also faster relaxation of the stresses after mold filling at higher temperatures. On the other hand, increasing the mold temperature slightly decreases the frozen-in flow stresses (Fig. 27). These findings are in accordance with theoretical results [ 151 which have been developed for prediction of the flow stresses in molded parts.

Finally, it is interesting to compare the residual and flow stresses in molded parts with the molded parts with the thermal stresses in quenched parts. Figure 28 shows the results of such comparisons.

for the quenched sample are on the order of lo' N/m2 and are much larger than the flow stresses or residual stresses for the molded part which are on the order of lo6 N/m2. While frozen-in flow stresses in the molded part are always positive, the residual stresses are positive in the core region and negative in the wall region. Since the residual stresses in the molded part are due to the coupled effects of flow and thermal stresses, it follows from Fig. 28 that these two components are of comparable magnitude and that both are extremely important for the injection.molding of polymers. This also means that the residual stresses in the molded part cannot be easily related to the value of shear stresses during cavity filing, as has been proposed in the com- mercially available Moldflow System [81], but seems to be a much more complicated problem. On the other hand, since the level of thermal stresses in the molded part is apparently much less than that in the quenched strip,

A main conclusion from the comparison in Fig. 28 is that thermal stresses

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I ”

Q

0 0 0.5 I .o

FIG. 26. Gapwise distribution of the frozen-in flow stresses in PS molded strip near the gate at different melt temperatures. Curves 1, 2, 3 and 4 correspond, respectively, to processing conditions Nos. 1, 2, 4 and 5 of Table 1.

this indicates that the cooling process in quenching is more severe than that in injection molding. Furthermore, it is clear that the flow and thermal stresses are apparently the main cause of undesirable warpage and dimensional instability effects observed in molded parts. Therefore, in the injection molding of polymers, especial attention should be addressed to the design of appropriate cooling systems which would result in reduced residual stresses or which would provide a symmetrical thermal-stress distribution.

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0 a I

b” 0.5 b -

0 0.5 I .o Y/b

0

FIG. 27. Gapwise distribution of the frozen-in flow stresses in PS molded strip near the gate at different mold temperatures. Curves 1 and 2 correspond, respectively, to processing conditions Nos. 7 and 8 of Table 1.

a b

-6 -6

FIG. 28. Comparison of gapwise distribution of frozen-in flow stresses (Curve l) , total residual stresses in directions parallel (Curve 2) and perpendicular (Curve 3) t o flow at cross sections near and far from the cavity en- trance for PS strip 3, and gapwise thermal stress distribution (Curve 4 in PS strip quenched from 170 to 3OoC.

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XI. CONCLUDING REMARKS

In the present paper, theoretical and experimental aspects of the residual-stress development in the injection molding of polymers have been considered, including effects of processing conditions, material, and cavity thickness. Since these residual stresses appear as coupled flow- and cooling-induced stresses, both injection molding and quenching experiments have been performed in order to elucidate the respective contributions of thermal and flow stresses in the overall residual-stress development and to make comparison with corresponding theoretical predictions. Experimental techniques suitable for the measurement of residual stresses in molded plastic parts have been critically reviewed. Among these methods are the hole-drilling, relaxation, and layer- removal methods. The latter method is simple to implement and gives the most extensive information concerning the residual-stress field. Although this method needs to be further formulated for viscoelastic materials, the approach based on pure elasticity has been used in the present work for the measurement of thermal stresses in quenched slabs and residual stresses in molded slabs. For those polymers for which the stress optical coefficient in the glassy state is much smaller than in the rubbery and fluid states, the level of flow- induced stresses can be derived from measurements of frozen-in birefringence by means of the stress-optical rule. The resulting difference between the principal frozen-in flow stresses in molded plastic slabs typically does not exceed 2 MPa and is tensile in nature. These flow stresses are due to the development of shear and normal stresses in viscoelastic polymeric melts and can be predicted by using viscoelastic constitutive equations. In particular the fluid and rubbery states of the polymeric melt mainly contribute to these flow stresses. An increase in melt temperature and cavity thickness significantly decreases such frozen-in flow stresses. In contrast, an increase in mold temperature was found to have little effect on the flow stresses. On the other hand, the level of compressive thermal stresses determined in quenched slabs by using the layer-removal technique has typically been found to be on the order 10 MPa. Calculation of the thermal stresses on the basis of the instant-freezing elastic theory of Indenbom and the viscoelastic theory by Lee, Rogers, and Woo have been performed and compared with experimental data. These comparisons show that the theories of Indenborn satisfactorily predict the experimental data for the gapwise distribution of thermal stresses in quenched PMMA slabs. However, for quenched PS slabs both theories systematically overpredict the experimentally measured stresses. This latter discrepancy is apparently due to the neglect of nonlinear polymeric behavior during quenching through the glass-transition temperature and to the phenomenon of thermal-stress relaxa-

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tion following quenching, as observed in the present investigation. In particular, it has been found that within a month the stresses might reduce to one- half their initial values. In addition, it might be noted that two representations of the relaxation modulus necessary for calculation of the thermal stresses according to the viscoelastic theory have been used. In one case, the relaxation modulus data covering the fluid, rubbery, and glassy states of polymer have been used whereas in the second case the relaxation modulus has been based upon data covering only the glassy state and the transition zone from the glassy to rubbery states. Since both representations have been found to result in the same predicted thermal stress distribution, it follows that the glassy state and the transition zone are mainly responsible for the thermal stresses. More- over, it is well known that the relaxation modulus in the glassy state and transition zone are independent of molecular weight and the molecular-weight distribution for polymers having molecular weights larger than the critical molecular weight corresponding to formation of the rubbery plateau. Thus, the thermal stresses will also be independent of these molecular characteristics of polymers and will only depend on the nature of the polymer chain.

Unlike thermal stresses, which are the same in two mutually perpendicular directions, residual stresses in molded slabs have been found to depend on orientation relative to flow direction during cavity filling and on distance from the gate. While gapwise distributions of thermal stresses are parabolic in nature with compressive and tensile stresses at the surface and interior, respectively, residual stress in molded slabs are, generally, nonparabolic. Characteristics of their gapwise distribution depend on processing conditions. Within the range considered in the present investigation, injection rate and mold temperature have been found to have only a minor effect on the level and gapwise distribution of molding stresses, while melt temperature has been found to have a significant effect. Lowering the melt temperature has been found to reduce the level of compressive stress or even result in a tensile stress at the surface of the molded part. The maximum level of residual stress is found to be typically less than 2 MPa. This value is comparable with the level of flow stresses in molded parts but is found to be much lower than the level of thermal stress in quenched samples. This means that the cooling procedure in quenching is more severe than that in injection molding. However, one should remember that molding stresses were obtained on molded parts aged over 1 year after molding. Definitely, during this period of time a residual-stress relaxation may occur in molded parts essentially similar to that observed for thermal stresses in quenched slabs. That is, one would expect somewhat larger residual stresses in fresh molded samples than that measured in the present work. However, this requires further investigation.

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Acknowledgments

This work is part of the Cornell Injection Molding Program, supported by the National Science Foundation under Grant MEA 82-00743 and by an industrial consortium. The authors wish to acknowledge Dr C. A. Hieber of Cornell Uni- versity for his helpful comments and assistance in the numerical computations.

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Residual Stress Development in the Injection Molding of Polymers