Introduction

Injection molding is the most widely used processing technology for plastics. It is an extremely versatile and flexible process for producing a wide range of simple or complex plastic components with high precision, good surface finish, and low operational costs for mass production. It consists of three stages: filling, packing, and cooling. First, raw material is heated until a homogeneous melt is obtained, which is then forced by pressure into a cavity. When filling is nearly completed, a packing pressure is applied to fill the remaining volume of the cavity and to compensate for the shrinkage caused by cooling of the material. When the gate of the cavity solidifies, no more pressure is needed and the material is allowed to cool into the desired shape. Once the part is rigid enough, the mold is opened and the part is ejected, at which time the cycle is repeated.

During these processes, the polymer melt experiences a complex deformation, and temperature and pressure histories that affect the final properties of the component. Residual stresses are originated due to the high pressure, temperature differences, and relaxation of polymer chains, which result in shrinkage and warpage of the product.

Engineering plastics are a group of thermoplastic materials that exhibit superior mechanical and thermal properties as compared to the more widely used commodity plastics. They are used in applications generally requiring exceptional properties such as stiffness, toughness, and heat and chemical resistance. To meet the technical requirements in demanding applications such as the automotive industry, different additives, fillers, and modifiers are commonly used. However, these additives also affect the behavior of the polymer melt. Opposite to commodity plastics that are relatively easy to process, engineering plastics processing is more complex, requiring higher temperatures and pressures, having a significant effect on residual stresses.

Current demands on close dimensional tolerances and high dimensional stability make it necessary to be able to predict residual stresses and warpage of the molded part. This requires a deep understanding of the mechanisms that originate residual stresses and the factors that influence them. Residual stresses have been a topic of interest in the last decades. Several reviews on the build-up of flow-induced stresses during injection molding are found in the literature [1–6]. However, most of these reviews are from three decades ago, and therefore, an update to more recent work is required.

The aim of this review is to describe the up-to-date state of knowledge regarding residual stresses in injection molded products, including not only the description of its formation mechanisms and models but also a review of the processing conditions that most affect them. The effect of residual stresses in different material properties which affect the product performance is also included, as well as a summary of the different residual stress measurement techniques. At the end, some comments on the current state of the art are included.

Types of residual stresses in injection molding products

Residual stresses mainly originate from two effects [7, 8]: the flow-induced stresses, which correspond with the orientation of the molecules and are developed during the filling and packing of the polymer into the cavity; and the thermall-induced stresses developed during the cooling stage. Generally, flow-induced stresses are an order of magnitude smaller than the thermal-induced stresses and are usually neglected [6, 7]. However, some authors [9] suggest that it is impossible to neglect them as they induce anisotropy of several properties [7, 10] because of the different frozen-in orientations of polymer molecules, which affect the long-term dimensional stability of the component. In the following sections, these two types of stresses, their main features, and build-up mechanisms and models will be described.

Flow-induced stresses

During the injection molding process, orientation [11, 12] and flow-induced stresses develop during the viscoelastic flow of the polymer in both the filling and the post-filling stage [9, 13, 14]. In the 1980s, attention was focused on the filling stage [6, 15]; however, calculation of the flow-induced stresses in the filling stage has been extended to the post-filling stage [16–18] because birefringence measurements [19–21] have shown that during packing, large orientations are also developed [22]. They develop when the polymer is in its fluid or melt state and are accompanied with alignment of the chain molecules with respect to the flow direction [23]. Because of the high solidification rates in injection molding, the complete relaxation of this flow-induced stresses is prevented and they remain locked within the solidified material as reported by Bushko and Stokes [24–27]. Analytical investigations of flow-induced stresses require an accurate calculation of the moving flow front during filling and the chain relaxation in the post-filling stage. Since rheological and geometrical nonlinearities are involved in polymer melt flow [28–31], the task becomes difficult and extremely complex, thus many times simple one-dimensional theories are used, which allow to grasp the main features of the flow.

Flow-induced stresses formation mechanism

Daly et al. [5] described the build-up or formation mechanism of flow-induced stresses during filling and packing based on Vinogradov [32] earlier qualitative study. It is suggested that during filling, polymer chains are stretched and oriented in the flow direction (Fig. 1a). During cooling, these deformations relax. The more oriented the chains, the more the relaxation they will undergo. However, relaxation of the most oriented chains (in the outer layers) is restricted by the less oriented inner layers (Fig. 1b), and thus flow-induced residual stress in the filling stage is tensile in the outer layers and compressive in the inner ones.

Fig. 1
figure 1

Flow-induced stresses during the filling stage: a stretched polymer chains and b relaxed polymer chains

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For the packing stage, three cases are presented and they are illustrated in Fig. 2. In the first case, Fig. 2a, it is assumed that the polymer melt is not sufficiently compressed and that during the cooling stage, the pressure in the core layers fall to zero when the thickness of the solidified outer layer is relatively small. Upon further cooling, the inner layers reduce their volume and compress the outer layers, yielding a compressive-skin–tensile-core distribution, usually evident as a sink mark. This profile is always observed in free quenched products [33]. Similar results were found by Sandilands and White [34] and Mlekusch [35]. For the case when the melt is over-compressed, Fig. 2b, the pressure at the inner cores remains very high even after complete cooling, and therefore, the stress distribution will be tensile-skin–compressive core, as found by Wang and Young [36]. In the third case, Fig. 2c, it is assumed that the pressure in the core layers fall to zero when the thickness of the solidified outer layer is much greater than that of the inner ones, resulting in a more complex distribution: tensile-skin—compressive-subskin—tensile core, as reported by Young [8] and Zoetelief et al. [7].

Fig. 2
figure 2

Flow-induced stresses during the packing stage subjected to a low packing pressure (under-compression), b high packing pressure (over-compression), and c high/low packing pressure profile

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Jansen et al. [37] found a similar behavior when molding plates at different packing pressures. They reported that as the packing pressure was increased, the plate started warping toward the opposite direction. This is, at low packing pressures it warped toward the hot side, as the pressure was increased, warpage was reduced, but as the packing pressure continued to increase, warpage increased toward the cold side. As residual stresses originate warpage, this change in warpage behavior must be related to a change in the residual stress distribution. Since the pressure is not uniform along the flow path during injection molding (it is higher at the gate and it decreases along the flow path), all three cases can be present in a single component, resulting in different residual stress distributions. Therefore, it is important to be able to predict or monitor the temperature and pressure history during the complete injection molding cycle, which are known as temperature and pressure profiles.

Flow-induced stresses models

The three-dimensional governing equations of the non-isothermal flow of viscoelastic fluids are [6]

  1. (1)

    Continuity equation:

    \( \frac{{\dot{\varrho }}}{\varrho } + \vec{\nabla } \cdot \vec{\upsilon } = 0 \), where ϱ represents the density, and \( \vec{\nabla } \cdot \vec{\upsilon } \) is the divergence of the velocity field.

  2. (2)

    Momentum equation:

    \( \vec{\nabla } \cdot \sigma + \varrho \vec{f} = \varrho \dot{\vec{\upsilon }} \), where σ is the Cauchy stress tensor and \( \vec{f} \) is the body force per unit mass.

  3. (3)

    Energy equation:

    \( \varrho \dot{\varepsilon } = \sigma :D - \vec{\nabla } \cdot \vec{h} + \varrho r \), where ɛ is the specific internal energy, D is the rate of strain tensor, \( \vec{h} \) the heat flux, and r is an internal heat source.

To solve these governing equations, constitutive equations for ϱ, σ, ɛ, \( \vec{h} \), and r must be given.

During the last decades, different approaches to the problem have been developed. Table 1 summarizes some of the most important studies developed in this area. It is observed that a great amount of this work has been based on Leonov viscoelastic constitutive equation [38], where irreversible thermodynamics are used for constructing the rheological equations capable of describing the behavior of polymer melts in a range of large elastic strains. It is also observed that, depending on the processing conditions, reported values of residual stresses are approximately up to 8 MPa, both in tension and compression when the packing stage is considered. However, when packing is neglected (i.e., Kabanemi et al. [39]) reported values are much lower.

Table 1 Flow-induced stresses models and simulations
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It has been noticed that most of the research work on residual stresses and warpage has been focused on amorphous polymers. However, it is clear from several studies [51–55] that the flow and thermal history experienced by the melt during injection can enhance crystallization kinetics, and thus lead to different types of crystalline structures, which is known as flow-induced crystallization. This is of great technological importance as polymer properties are to a great extent affected by morphology [56–61]. Differences in crystallinity and orientation throughout an injected product will lead to anisotropy and other changes in mechanical properties as reported by Santis et al. [62] and Pantani et al. [63]. The crystalline phase has a tighter packing than the amorphous phase and, therefore, a higher density. This densification process is accompanied by changes in mechanical properties such as elastic modulus, yield strength, elongation at break, ultimate strength, and thermal properties such as the thermal expansion coefficient. This, in turn, will have an effect on residual stress distributions. Various models [64–69] have been developed to describe the crystallization process of polymers. However, most of these studies are limited to idealized situations, in which external conditions such as temperature or cooling rate are considered as constant. In real situations, however, the polymer is cooled down at different rates and with high thermal gradients, which makes the crystallization process dependent on instantaneous conditions [70]. Several modifications of the Avrami theory [65, 71, 72] have been proposed over the years to model non-isothermal crystallization. Most of these models neglect spherulite impingement and variations in cooling rates. Empirical or experimental approaches have also been developed to calculate the main parameters of non-isothermal crystallization [73] in which crystallization kinetics are observed by differential scanning calorimetry.

For the particular case of injection molding, an exhaustive literature review of the modeling of morphology evolution during processing of semicrystalline polymers was presented by Pantani et al. [74], including a thorough analysis of the effect of different processing parameters in the morphology of the product. However, work investigating the effect of crystallinity distribution on residual stresses in plastic parts is still required.

Thermally induced stresses

Thermoplastics processing usually involves the non-uniform cooling of molten polymer, which results in the presence of thermal residual stresses in the final product. The thermoviscoelastic theory of residual stresses was initially developed for inorganic glasses [75–79], with previous work by Adams and Williamson [80] on the annealing of glass, and first applied to polymers by Struik [81] under free or unconstrained quenching conditions.

According to his work, it can be assumed that during injection molding, the outer layer of the polymer melt, in contact with the cold mold, undergoes an instantaneous step change in temperature, while the core remains hot. Heat removal is almost entirely from the outer surfaces where a solid external layer is formed. At this point, the surface layers are almost stress free as they were allowed to contract freely. The inner layers are still hot and behave as a liquid free of stresses. While cooling continues, the solidifying material further in is prevented from freely contracting by the outer solid layer. A compressive-skin/tensile-core distribution as the one shown in Fig. 2a is obtained.

These results were reported by Siegmann et al. [2], who investigated the distribution of residual stresses in quenched PPO. Compressive stresses were measured at the surface layers, while tensile stresses were measured in the inner layers. The level of residual surface stresses was found to depend on both the total temperature difference during cooling and the initial specimen temperature. Similarly, Rigdahl [82] used the finite element method to calculate the distribution of residual internal stresses in an injection molded PS specimen. By determining the temperature distributions in the plate and its variation with cooling time, the corresponding stress distribution was found. It was found that the surface layer of the plate is subject to compressive stresses, while the interior accommodates stresses of tensile type. Anisotropy and viscoelastic relaxation have been neglected as well as the effect of packing pressure.

However, during injection molding, the polymer melt conditions differ from those of free quenching. The material is constrained by the mold geometry, the holding pressure, and the adhesion between the mold and part, which change the thermal-induced residual stresses build-up. Therefore, shrinkage of the solidified layer is prevented, and a residual stress distribution as the one shown in Fig. 2c is obtained.

Thermally induced stresses formation mechanism

Zoetelief et al. [7] illustrated the development of residual thermal stresses using the schematic representation similar as the one shown in Fig. 3. This representation is analogous to the work presented by Struik [81] in which the theory of the quenching of flat glass plates is described and shown to be applicable to polymers. Similar numerical formulations were used by Jansen and Titomanlio [83]. As shown in Fig. 3, cooling is idealized in five steps and pressure varies as a function of time. T freeze is the glass transition temperature and it is assumed that the material behaves as an ideal fluid when T > T freeze, and as a linear elastic material when T < T freeze. Temperature drops and residual stresses develop as follows:

Fig. 3
figure 3

Thermally induced residual stress development in injection molded products: a t 1, b t 2, c t 3, d t 4, and e t 5. Drawn based on [7]

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t = t 0: Homogeneous temperature throughout the specimen; pressure = 0; material free of stresses.

t = t 1 : No-slip condition hinders contraction of outer layers; small tensile stress develops (Fig. 3a).

t = t 2: Holding pressure σ = −p h acts on the melt; the rigid shell is also compressed; stress levels decrease by \( \Delta \sigma = \nu p_{\text{h}} /\left( {1 - \nu } \right), \) where ν is the Poisson’s ratio (Fig. 3b).

t = t 3: Packing stage: pressure remains constant; a small layer solidifies, and its contraction is hindered, decreasing compressive stresses in it (Fig. 3c).

t = t 4: Packing stage finishes, pressure is set to zero; stress in the melt disappears, and the stress levels increase by \( \Delta \sigma \) (Fig. 3d).

t = t 5: Mold is opened, product is released from the mold; further cooling sets tensile stresses in the core, which are in equilibrium with the rest of the specimen. At the end of this interval, a tensile-skin, compressive-subskin, and tensile-core distribution are obtained, which have been reported by several authors [33, 35, 84–86] (Fig. 3e).

Thermally induced stresses models

Initially, most of the models developed to predict residual stresses or warpage of plastic parts assumed a linear thermoelastic behavior in which the conservations of momentum and energy equations are similar to those presented in “Flow-induced stresses models” Section with the Cauchy stress tensor, commonly decomposed into a hydrostatic part p, and a deviatoric part σ d is determined by [5]

$$ \sigma = pI + \sigma^{\text{d}}, $$

where I is the identity tensor,

$$ p = \int\limits_{0}^{t} {\left[ {\frac{\alpha }{\kappa }T - \frac{i}{\kappa }{\text{tr}}D} \right]{\text{d}}s ,} $$
$$ \sigma^{\text{d}} = 2\int\limits_{0}^{t} {G\left( {\xi \left( t \right),\xi (s)} \right)D\left( s \right){\text{d}}s,\;\;{\text{and}}} $$
$$ \xi \left( t \right) = \int\limits_{0}^{t} {\frac{1}{{a_{\text{T}} }}{\text{d}}s}, $$

where α is the coefficient of thermal expansion, κ the coefficient of compressibility, G(t,s) is the shear relaxation modulus, ξ is the reduced time, and a T is the shift factor of the time–temperature superposition principle.

For the free or unconstrained quenching, \( \sigma \cdot n = 0. \)

For constrained quenching, u = 0, where n is the unit vector normal to the surface and u is the displacement on the surface.

St-Jacques [87] was the first author to present work on warpage in injection molding flat parts due to unbalanced cooling conditions. He used a one-dimensional, transient heat conduction model (with constant material properties) to predict the temperature profiles in a solidifying slab and used them to estimate the thermal warpage. His simulation, using finite differences, allowed to analyze asymmetrical cooling, and results showed good agreement with experimental data. Nowadays, several approaches have been developed to study the shrinkage, warpage, or residual stresses in injection molded components due to cooling, some of which are summarized in Table 2.

Table 2 Thermal-induced stresses models and simulations
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An interesting method for predicting thermal-induced residual stresses in polymeric materials was proposed by Tropsa et al. [99], based on previous work by Williams [100]. They introduced the “residual temperature field” concept to describe the relationship between the thermal history that the material goes through during processing and the frozen-in strains. When this temperature field is applied as an actual temperature distribution, it produces thermal stresses and distortions equal to those caused by residual stresses. The derivation of thermally induced residual stresses starts with the equilibrium equations and the constitutive law for a linear thermoelastic solid. However, the analysis is extended afterward to include anelastic effects to give residual stresses. By knowing the residual temperature field (T res), the residual stress distribution σ res(z) can be calculated as

$$ \sigma_{\text{res}} \left( z \right) = \frac{{E_{\infty } }}{1 - \nu }\alpha \left[ {\bar{T}_{{_{\text{res}} }} - T_{\text{res}} \left( z \right)} \right], $$

where E ∞ is the long-term modulus, ν is the Poisson ratio, α is the thermal expansion coefficient, and \( \bar{T}_{\text{res}} \) is the average value of T res through the specimen thickness.

Effective factors of residual stress build-up during injection molding

Several studies have focused on analyzing the effects of different processing parameters, such as mold and melt temperature, packing pressure and time, and injection and cooling rate, on the residual stress distribution on injection molded parts. In most of these studies, warpage has been used as an indicator of residual stresses.

There are two basic approaches on these studies. In one hand, experimental investigations have been carried out, in which processing conditions are varied on each injection cycle, and their effect on residual stresses measured and recorded. Packing pressure, mold and melt temperature, and the design of the cooling system have been pointed out as the most significant processing conditions, with packing pressure as the most important parameter on which shrinkage, warpage, and residual stresses depend, as it can be observed in Table 3 in which results of several authors are summarized. Other processing and design factors such as mold deflection, gate dimensions, and wall thickness have also been studied [101, 102]. However, it has been found that their effect in residual stresses is less significant.

Table 3 Effects of processing conditions on shrinkage, warpage, and residual stresses
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On the other hand, with the development of computational tools, a different software has been used to simulate the flow of the polymer inside the cavities and to predict the part quality after ejection. Again, processing conditions on each simulation are changed and their effect on residual stresses recorded.

Huang and Tai [103] used the experimental design of Taguchi method to determine the effects of injection molding conditions on warpage, and the injection process was simulated using C-MOLD. They found that packing pressure has the greatest effect in warpage, followed by mold temperature, melt temperature, and packing time. However, by analyzing the interaction between factors, they found that the interaction between the mold temperature and melt temperature has in fact a greater effect on warpage than the packing pressure by itself and should not be neglected. Other authors [104–111] have used Taguchi method and injection molding simulation software as Moldflow to minimize warpage and sink marks in injection molded components while obtaining the optimum processing conditions. Gao and Wang [112] proposed a Kriging model in combination with Moldflow simulations to minimize the warpage in injection molding.

More recently, neural networks [113–116] and genetic algorithms [117] have been used to predict the quality of injection molded products and to obtain the optimum processing conditions. Similar to the experimental results, it has been reported that the packing pressure and melt temperature contribute significantly to the shrinkage, warpage, and quality of the injected parts; also, the effect of a higher temperature gradient between polymer melt and mold was overcome by the effect of a higher packing pressure [110, 117].

Although numerical methods combined with statistical tools are useful, some authors [118, 119] suggest that these results should be combined with experimental ones, mainly because factors such as available clamp force, cycle time, and mold surface finish are not considered. Other factors affecting residual stress development are the molecular weight of the polymer (with higher molecular weight polymers resulting in higher residual stresses [120]), its degree of crystallinity [121], its relaxation behavior, and geometric parameters such as thickness and other processing considerations such as the use of release agents [122].

Other effects of residual stresses on the performance of injection molded parts

It is well known that residual stresses influence the properties of injection molded products. These stresses act internally at room temperature and have the same effects on the material as externally applied stresses do [122, 137]. Their magnitude can be high enough to induce severe shape changes in the product, as well as changes in the overall material performance.

In addition to the shape distortion, the presence of residual stresses is also expected to affect the mechanical behavior of the product. Broutman et al. [138, 139] found a large increase of the notched Izod impact strength and a decrease in the ductile–brittle transition temperature of PC and other materials when tensile residual stresses were reduced. They suggested that the presence of compressive residual stresses in the surface suppressed craze initiation in advance of the notch [140]. However, in the case of PVC and ABS, the impact properties were not significantly modified by the presence of compressive stresses, leading to the conclusion that the influence of residual stresses on impact strength is only significant on those polymers whose failure initiation is highly localized. When the failure initiation is not limited to a single craze, as in rubber modified polymers, the extent of deformation is controlled by multiple crazing or by shear yielding, and the effect of the compressive residual stresses is limited [138].

Chaoui et al. [141] studied the effect of residual stresses on slow crack propagation in MDPE pipes. They found that the pipes exhibit more resistance to crack propagation in the outer surface than in the inner one, with compressive and tensile residual stresses, respectively. They concluded that the material resistance to fracture is strongly influenced by its thermal history, which determines not only the residual stresses distribution but also the morphology of the material. Guevara and Leevers [142] studied the effect of residual stresses on rapid crack propagation of polyethylene pipes. They found that the lower the residual stress, the lower the S4 (Small Scale Steady State) critical temperature. It was suggested that the additional stored strain energy prior to fracture helps to drive the crack, and that a change in crack front shape due to the release of a bending moment can change crack propagation. Similar results were found by Argyrakis [143]. Davis [144] also found a difference in rapid crack propagation of single and dual cooled pipes; however, he attributed the results mainly to differences in crystallinity.

The effect of residual stresses on the fatigue life of a polymer is also known. Hornberger and Devries [145, 146] found that compressive residual stresses enhance fatigue life, while tensile stresses usually decrease it. Compressive stresses decrease the sensitivity of the polymer to flaws; and since fatigue is dependent on the stress intensity factors at those flaws, the fatigue life is increased. Sauer et al. [147] have reported increases in fatigue life by a factor of 20 in PS with the reduction of tensile stresses. Hornberger and Devries reported a tenfold increase in the mean fatigue life of PC samples [145]; however, they also suggested that the morphology of the polymer also has an important influence on the fatigue life of the polymer.

Siegmann et al. [3] studied the effect of residual stresses on density distribution and tensile properties of quenched PPO specimens. A steep density gradient at the surface of the specimens was found. Tensile modulus and ultimate tensile stress increased significantly from the surface to the inner layers. By analyzing fracture surfaces, they also found that fracture initiation sites and thus the fracture energy are influenced by residual stresses, the latter being higher when fracture initiates at the inner layers.

Turnbull et al. [148] studied the impact of residual stress and molecular orientation on environmental stress cracking. The difference in threshold stress measured for the annealed and as-processed specimens indicated the existence of a net tensile residual stress in the very near-surface region of their specimens.

It has been found that tensile stresses accelerate the rate of photochemical degradation of polymers by accelerating molecular scission, while compressive stresses generally retard it [149–152]. Kwok et al. [153] suggested a premature ejection and quenching of injection molded components with an aim of obtaining compressive stresses at the outer layers and improving resistance against ultraviolet irradiation.

Residual stress measurement techniques

Techniques for characterizing residual stresses in plastics are basically of two types: destructive and non-destructive [5, 154, 155]. Destructive methods are based on the relaxation of strains after the removal of a specific amount of material, and the measurement of these strains to calculate the residual stresses. Non-destructive methods are mainly used to measure stresses at the surface of the specimens, such as photoelastic methods. Hughes et al. [156] also mention predictive techniques, in which computer simulations are used to predict residual stresses; however, results must be validated by experimental data collected using one of the previous techniques.

Destructive methods

Layer removal

The layer-removal technique, first applied to metal sheets, was developed by Treuting and co-author [157]. It involves the removal of successive uniform layers of material from the surface of the specimen and the measurement of the resulting curvatures as a function of specimen thickness. The measured curvature as a function of depth removed can be used to calculate the stress distribution through the thickness of the sample prior to layer removal. The technique has been the primary method used for plastics [158], but the limitation to flat sheets is a major constraint as is the inability to assess very near-surface stresses. In contrast to other methods, it provides a complete picture of the distribution of residual stresses. When the following conditions are satisfied, the accuracy of the method is limited only by the precision of the measurements [157]: the specimen is linear in pure bending for the range of curvature, the stress does not vary in the plane of the specimen but only through the thickness, and the process of removing successive layers does not disturb the stresses in the remaining material.

Coxon and White [159] examined the residual stresses in injection molded PP bars using a stress-relaxation method and the layer-removal technique. The layer-removal technique showed that the stresses near to the surface were compressive and those in the interior tensile. White et al. [160–162] examined the layer-removal technique for determining residual stress distributions for moldings with depth-varying elastic modulus. They concluded that although this is a more exact method, in the majority of cases, the unmodified Treuting and Read procedure is perfectly adequate. Hastenberg et al. [130] used the layer-removal method to determine the influence of annealing on the thermal stress distribution on flat plates of three amorphous polymers: PS, PC, and polyphenylene ether/high-impact polystyrene blend. A good reproductibility was obtained. They found that an annealing treatment significantly reduces the overall stress level, without affecting the stress pattern.

Akay and Ozden [131, 163] measured the residual stresses in injection molded ABS and PC specimen using the layer-removal technique and evaluated the effect of the curvature measurement device on the reliability of the results. The accuracy of the measurements depended on the type of device employed (coordinate machine, dial gage, and optical scanner). A peg/pegboard arrangement was found to enable accurate reproduction of the specimen curvature. As expected, a non-contact method such as an optical scanner produced the most reliable curvature measurements.

As pointed out by Denizart et al. [91], the time elapsed between the layer removal and the measurement of the curvature is critical. Some authors prefer to reduce this time to a minimum [4, 130], while others wait until the curvature reaches its maximum value [1, 164].

Although Siegmann [3] and White [158] consider that machining does not affect the residual stress distribution in a significant way when the correct cutting techniques are used, other authors suggest the opposite [91, 163]. Jansen et al. [165] used an Excimer laser as the milling tool for applying the layer-removal method. They found that some disadvantages associated with the layer-removal method were overcome: stress-relaxation effects were effectively excluded since the heating of adjacent material during milling was shown to be negligible. Moreover, an improvement of the measurement resolution was possible as with the laser technique small layers of well-controlled thickness could be removed.

Hole-drilling

The hole-drilling technique was first proposed by Mathar [166] for residual stress measurement. The technique is relatively simple and has been standardized for metallic plates as ASTM Standard E837 [167]. It is a semi-destructive residual stress measurement technique in which a rosette of strain gages is bonded on the surface of the specimen at the point where residual stresses are to be measured. Then, a hole is drilled precisely through the center, and the measured strains are used to calculate the stresses for the two principal axes in the plane of the sample [168]. Sicot et al. [169] studied the influence of two experimental parameters—the depth of each drilled increment and the influence of the relative position of the strain gages compared with the radius of the hole drilled—on the determination of residual stresses using the incremental hole-drilling method. Results showed that these parameters have a significant effect on the magnitude and stability of the residual stresses, mainly because of significant stresses relaxation. Kim et al. [170] used the incremental hole-drilling method to measure the residual stresses in injection molded PS parts. Results were compared with the ones from the layer-removal method. They found that the measured residual stresses are in fact affected by additional stresses generated during these techniques, and thus the experimental environment needs to be improved. In other work, Kim et al. [171, 172] used the finite element method for calibration of residual stresses in each increment on the hole-drilling method. Residual stress distributions obtained by both experiments and numerical methods accorded well with each other. Maxwell and Turnbull [173] made a comparative evaluation of the layer-removal method and hole-drilling techniques for measurement of residual stress in ABS samples. They found that residual stresses determined by hole drilling were not equi-biaxial and did not balance through the thickness of the specimen, and concluded that although hole-drilling technique is a more flexible technique than layer-removal technique, the results obtained are not as reliable.

Although it is a relatively simple technique, the utilization of the strain gage rosettes presents some practical disadvantages [174] such as the hole must be drilled exactly at the center of the rosette, the strains measured by the gages are average values in the range of the length of the strain gage, and, as in the layer-removal techniques, it is very difficult to identify the additional deformation which resulted from machining (hole drilling). To overcome some of these issues, Chen et al. [174, 175] and Shankar et.al. [176] propose to combine hole-drilling technique with Moiré interferometry, an optical technique that allows to obtain more accurate measurements.

Chemical probe technique

The chemical probe technique is a more speculative approach based on exposing the stressed part or product for a specific period of time to an environment of varying aggressiveness. When a polymer, for example, is immersed in a solvent, it will craze [177–179]. Reference data exist for the relationship between stress and time to crazing for different polymer–environment combinations. Observation of the crazing and the size of the cracks will indicate the level of stresses at the external surface of the part. This technique is also known as solvent crazing, and is similar to the ASTM method for determining residual stresses in ABS parts by immersion in glacial acetic acid [180]. Turnbull et al. [168] used the chemical probe technique to measure the residual stresses in annealed PC and ABS specimens. They concluded that this technique detects only very near tensile surface stresses, which is its major limitation as it lacks the ability to measure the residual stress distribution across the part thickness.

Non-destructive methods

Photoelasticity is a well-known technique for measuring the stress state in complex parts. Residual stresses result in distortion of the polymer chains and induce anisotropy of polarizability, which can be determined by birefringence measurements [168]. Although the technique is limited to transparent materials and its analysis can be complicated due to the effect of molecular orientation induced by processing, it has been successfully applied to the analysis of frozen-in orientation in injection molded samples [40, 181–184]. Wimberger-Friedl and Hendriks [17] measured birefringence in quenched PC specimens. They found that PC is very suitable for the measurement of stress-induced birefringence because of its high positive stress-optical coefficient in the melt and in the glassy state. Wiesauer et al. [185] used polarization-sensitive optical coherence tomography (PS-OCT) to determine and map the internal birefringence properties of PS samples, and obtain information about the stress state within the materials. OCT is an imaging technique capable of recording cross-sectional images of transparent and turbid structures with micrometer-scale resolution [186]. PS-OCT provides additional information on the birefringence properties of a material, as it maps the retardation between the vertical and horizontal polarization components and the orientation of the fast optical axis within the sample, leading to enhanced structural contrast.

Hauk et al. [187] presented an evaluation of different X-ray techniques used to measure residual stresses in semicrystalline polymers. Although some of its limitations are that the specimens should have a crystalline structure and that the measurement depth is limited, and research has extended to amorphous polymers. Barret and Predecki [188, 189] introduced fillers consisting of crystalline particles or powders in amorphous polymers, and then measured their lattice deformation in the injected part by diffracting X-rays at high Bragg angles. Assuming a perfect contact between the particles and the matrix, the stress state was deduced. Hughes et al. [156] used synchrotron X-rays to measure residual strains in commercial HDPE gas pipeline samples. Measurements were feasible in samples of complex geometry and although the technique is used in crystalline polymers, it is suggested that there is also applicability for low- and non-crystalline polymers via the mixing of small volumes of metal powders.

Sanchez and Hornberger [190] used holographic interferometry to monitor the physical relaxation of a plastic-molded component during heating and estimated the initial stress state. Colpo et al. [191] used an embedded Optical Fibre Bragg Grating sensor for characterizing residual strains in an epoxy block during the curing and postcuring stage. Other techniques for measuring residual stresses based on the change in material properties such as refraction of light or electrical conductivity, mainly in thermoplastic composites, are described by Parlevliet et al. [192].

The indentation method, which is almost non-destructive, can be used to measure residual stresses in plastic parts for practical applications, particularly for small or complex parts. Pak et al. [193] applied an indentation method to measure residual stresses in injection molded components. The load–displacement curve was measured for indentation at stressed and non-stressed positions. Residual stress distribution of the injection molded part was calculated by comparing the load–displacement curve results with respect to the indentation depth. Good agreement with numerical results and those measured by the hole-drilling method was found.

Comments on the state of the art

Previous sections show the great effort that has to be done to understand the mechanisms of residual stress build-up. Several models for estimating temperature history during filling, packing, and cooling have been developed, and different models such as the residual temperature field have been applied to estimate thermal residual stresses. The same has happened for flow-induced stress, although more complex situations have been found here. The viscoelastic nature of polymers and the high shear and pressure conditions to which the polymer is subjected to, result in a complex flow of the polymer inside the mold, which in turns results in a complex deformation, orientation, stretching, and relaxation of polymer chains. All these set up residual stresses, which results in part warpage and shrinkage.

Most of these models assume constant through thickness polymer properties, such as constant modulus, density, thermal properties, and orientation, among others, which is far from reality. A more detailed analysis is needed for more complex shapes where the thickness varies across the geometry, and where processing conditions generate different temperature and pressure histories, and thus different polymer structures and properties, affecting the residual stress distribution.

Despite the huge effort for estimating warpage, most of the work has focused on simple geometries such as flat plates, disks, L-shaped specimens, and rectangular boxes. Estimation of warpage of more complex parts is challenging as the polymer melt faces different restrictions during flow which changes the polymer chains' orientation. During filling and packing, restriction comes from the mold itself, while after ejection, it is the part geometry which inhibits uniform shrinkage and polymer chains relaxation. Although some authors have worked with ribbed specimens [84], other common features such as snap fits and bosses require attention, especially in parts subjected to critical loading.

Another interesting area to pursuit is simulation. Commercial software predict warpage of injection molded components based on semi-empirical data. Developing a model that can cope with complex geometries and be used on finite element analysis is needed. As mentioned previously, simulations must be accompanied by experimental data and one way to achieve it is using instrumented molds for complex geometries, in which pressure and temperature histories could be monitored and a detailed analysis of the effect of different processing conditions in residual stress built up could be done.

Finally, transformation-induced residual stresses might need special attention. Although some models consider crystallization and include its effect on changing density, modulus and other properties, the effect of spherulite formation on the volume of the part might be of interest.