ORIGINAL ARTICLE

Rheological measurement of the nonlinear viscoelasticity of the ABSpolymer and numerical simulation of thermoforming process

Jemyung Cha1 & Hyeong Yong Song2& Kyu Hyun2

& Jeung Sang Go1

Received: 7 October 2019 /Accepted: 22 January 2020# The Author(s) 2020

AbstractThe thickness distribution of thermoformed products is greatly affected by the viscoelastic behavior of the extruded polymersheet. In this work, linear and nonlinear rheological experiments are carried out to characterize the viscoelastic properties ofacrylonitrile-butadiene-styrene sheets under thermoforming conditions including a wide range of temperatures, strains, and strainrates. First, aspects of linear viscoelasticity such as the storage modulus and loss modulus are measured by small-amplitudeoscillatory shear experiments. The discrete relaxation spectra and the Williams-Landel-Ferry parameters are obtained from theconstructed linear master curves. Then, nonlinear time-dependent extensional viscosity is measured by uniaxial extensionalexperiments. The parameters of the damping function are evaluated using an optimization method. In addition, the effect ofthe orientation of the polymer is analyzed. The uniaxial extensional stress and viscosity in the extruder direction demonstratehigher resistance against tearing and extreme thickness reduction during processing. Finally, the linear and nonlinear inputparameters for the numerical simulation are prepared. Numerical simulations are performed using the Wagner model with theobtained nonlinear viscoelasticity. The thickness distribution in thermoformed ABS sheets, obtained numerically, shows goodagreement with the experimentally obtained values.

Keywords Thermoforming .ABSpolymer .Rheologicalmeasurement .Nonlinear viscoelasticity .Orientation .Wagnermodel .

Finite element simulation

1 Introduction

1.1 Thermoforming

Thermoforming is a manufacturing process used to convertpolymer sheets into thin-walled plastic products with desiredshapes. In the thermoforming process, the polymer sheet isdeformed by the heat and pressure applied. The sheet, heatedabove the glass transition temperature, has both viscous andelastic components. The sheet undergoes rapid and large de-formation during the process, and this results in a change inviscoelasticity, which greatly affects the thickness distributionof the thermoformed product. Therefore, it is very important

to investigate the viscoelastic behavior of polymer sheets un-der thermoforming process conditions such as strain, strainrate, and temperature.

The advantages of thermoforming are low cost in molddesign, precise shape, low pressure, automation for mass pro-duction, and the ability to use a variety of applicable polymermaterials. Thermoforming applications are classified intothin-gauge and heavy gauge types, depending on the thicknessof the product. The former group includes packaging con-tainers with thicknesses of less than 1 mm, such as food,beverage, and pharmaceutical and semiconductor containers.The latter group includes products with thicknesses from 1 to10 mm and includes automotive, ship, and aircraft parts thatrequire high durability. Polymer material undergoes large andrapid deformation during the thermoforming process, and thiscan result in nonuniform thickness distribution, excessivethinning and a decrease in mechanical properties. Typically,nonuniform thickness distribution occurs more frequently inheavy gauge thermoforming, which involves deep drawingmolds with complex shapes, as compared with thin gaugethermoforming.

* Jeung Sang Gomicros@pusan.ac.kr

1 School of Mechanical Engineering, Pusan National University,Busan, South Korea

2 School of Chemical and Biomolecular Engineering, Pusan NationalUniversity, Busan 46241, South Korea

https://doi.org/10.1007/s00170-020-04979-7The International Journal of Advanced Manufacturing Technology (2020) 107:2449–2464

/Published online: 20 March 2020

The thermoforming process consists of three main steps:heating, pre-stretching, and forming, as shown in Fig. 1. Inthe heating step, shown in Fig. 1a, the polymer sheet is heatedabove the glass transition temperature (Tg) by infrared radia-tion, which causes it to lose its mechanical strength; eventually,it sags due to gravity. In the pre-stretching step shown in Fig.1b, the polymer sheet is pre-stretched to improve the uniformi-ty of thickness distribution in the thermoformed product. Theedge area of the polymer sheet is fixed by a clamping tool andvacuum pressure is applied to inflate the sheet surface like abubble. In the forming step shown in Fig. 1c, the polymer sheetundergoes extensional deformation due to contact with themold, and local cooling occurs where areas of the polymersheet are in contact with the lower temperature mold. Thisreduces the mobility of the styrene-acrylonitrile (SAN) matrix,and the extension of the polymer sheet stops. The deformationbehavior that occurs in each step of thermoforming, such assagging, expansion, extension, and cooling by the mold, af-fects the thickness distribution of the thermoformed product.

In the thermoforming process, polymer materials exhibitdeformation- and temperature-dependent nonlinear viscoelas-ticity depending on time and space. Under particular processconditions, the internal stress grows or relaxes over time. Thecharacteristics of the polymer material are affected by temper-ature. In the glassy state, the polymer material is brittle and onlysmall deformation is possible. However, since the material isductile in the rubbery state above the Tg, large deformation ispossible. In addition, stress can be increased or decreased de-pending on the strain rates applied to the material. Polymermaterials can have different mechanical properties dependingon their molecular weight and structure, and polymer sheets

produced by the extrusion process can exhibit strong anisotropydue to resulting molecular orientation. The thickness distribu-tion of thermoformed products is greatly affected by the visco-elastic characteristics of the polymer materials.

Industrial thermoforming processes are typically designedto improve quality and uniform thickness distribution, increaseproduction efficiency, and reduce material costs. These objec-tives can be affected by various parameters of thethermoforming process, such as vacuum pressure, sheet tem-perature, mold temperature, ambient temperature, convectiveheat transfer coefficient, mold speed,moldmaterial, coefficientof friction, and thermal conductivity. Trial-and-error-based ex-periments have been conducted to investigate the effects ofprocess parameters. However, that approach requires signifi-cant effort and cost, and it can be difficult to analyze the de-formation behavior of the polymer materials in detail becausethere are so many parameters. In addition, a lack of detailedtheoretical background can lead to problems, such asunoptimized process, process instability, material waste, andlow-quality products. A numerical simulation is an excellenttool for conducting in-depth analyses of the deformation ofpolymer materials and the thickness distribution ofthermoformed products. To perform numerical simulations, anonlinear viscoelastic constitutive model is required, with ma-terial properties under process conditions.

1.2 Literature review

Many previous works have been devoted to the rheological mea-surement of linear and nonlinear viscoelastic properties, as wellas numerical simulations with constitutive models. Wang et al.

(a) Heating (b) Pre-stretching

(c) Forming (mold moving and vacuuming)

Radiative energy Vacuum pressure

Contact with the mold Vacuum pressure

Fig. 1 Main three steps of thethermoforming process usingdeep drawing mold and vacuumpressure. a Heating, b Pre-stretching, c Forming (moldmoving, vacuuming)

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[1] performed uniaxial tensile experiments using aMeissner-typerheometer to measure the properties of ABS and proposed amaterial model that considers the effects of strain rate, strainhardening, and temperature sensitivity. Lau et al. [2] showed thatABS exhibited high resistance to sheet sag because of its highviscosity and high modulus of elasticity. Because of its broadermolecular weight distribution, polypropylene had a sag resis-tance lower than that of ABS. Aoki et al. [3] studied the effectof the dispersion of rubber particles in ABS polymer on dampingfunction, which indicated stress reduction withmaterial deforma-tion. It was found that the damping function became strongerwith increased rubber particle content in the ABS sample con-taining well-dispersed rubber particles. Lee et al. [4] measuredthe rheological properties of ABS, such as elongational viscosity,strain hardening, and strain softening using a Meissner-type ex-tensional rheometer and mechanical spectrometer. They reportedthat the strain-hardening behavior of the polymer material wasimportant for uniform thickness distribution. Münstedt et al. [5]showed that the strain hardening of polymer materials, such aspolyethylene and polypropylene, had a significant impact on theuniformity of deformation. The innovative thermoforming pro-cess has been recently developed to form composite sheets bymeans of polymer melt pressure during the injection phase [6].This process is an integration of the thermoforming and injectionmolding processes and can result in significant reductions oftooling, machinery, and operational cost [7].

In general, the extruded polymer sheets used inthermoforming exhibit anisotropic stresses along the machinedirection or in the transverse direction. Chan and Lee [8] re-ported that the orientation stress of an extruded polypropylenesheet depended strongly on the draw ratio and that higher melttemperature reduces the orientation effect. Stephenson et al.[9] conducted an experimental study of the sag of an extrudedpolypropylene sheet and found that the extension ratio in thetransverse direction was higher than in the machine direction.The polystyrene was found to be temperature-sensitive in thethermoforming range, and temperature effects should be in-cluded to obtain a better understanding of sheet sag.

Two different types of constitutive models can be consid-ered to perform numerical simulations of the thermoformingprocess. Hyperelastic models are commonly used because oftheir simplicity, and viscoelastic models are employed to de-scribe the deformation- and temperature-dependent nonlinearviscoelasticity of polymer materials. Koziey et al. [10] com-pared the hyperelastic Ogden model and the Kaye–Bernstein–Kearsley–Zapas (K-BKZ) viscoelastic model in a numericalsimulation of thermoforming. The Ogden model provided re-liable results for molds with shallow depths and simpleshapes, while the K-BKZmodel showed good agreement withthickness distribution using deep drawing molds with com-plex shapes. Novotný et al. [11] reported that the hyperelasticmodel was not suitable to describe the deformation of polymermaterials in thermoforming in detail and that the nonlinear

viscoelastic model was capable of accurately representingthe behavior of polymer materials. The K-BKZ model wasused to describe viscoelasticity for large deformations. Salaet al. [12] performed numerical simulations with the G’sellviscoelastic model of high impact polystyrene (HIPS)thermoforming for a refrigerator liner. A 20% discrepancybetween experimental measurements and the numerical re-sults was found, which it may be possible to improve byintegrating the heat transfer and the material behaviors.Dong et al. [13] used hyperelastic models such as theMooney-Rivlin and Ogden models to carry out numericalsimulations of bubble inflation of a polymethylmethacrylatesheet. Anisotropic material properties can have a significantimpact on simulation results when applying an isotropic ma-terial model, but there were no significant differences inPMMA ratings after shrinkage testing.

There have been several studies that have demonstrated theinfluences of various process parameters, such as heating,temperature of sheet/mold/plug, speed and shape ofmold/plug, pressure in air/vacuum, and coefficient of frictionon the thickness distribution of thermoformed products.Ayhan and Zhang [14] studied the effects of sheet tempera-ture, air pressure, and heating time for simple-shaped foodcontainers in plug-assisted thermoforming using multi-layered polymer sheets. The most important parameter affect-ing the wall thickness distribution was sheet temperature.Bhattacharyya et al. [15] used the grid strain analysis tech-nique to quantify the difference in the strain distribution dur-ing the thermoforming of composite sheets made of woodfiber and polypropylene. They reported that the forming tem-perature and the size of the clamp frame had the most signif-icant effect on thermoformability. Chen et al. [16] investigatedthe thickness distribution of thermoformed thin films using0.125 mm and 0.2 mm polycarbonate. They found that thick-ness at the sidewalls increased with increasing mold tempera-ture, preheating temperature, plug depth, and holding time butdecreased with increasing plug speed. McCool and Martin[17] studied plug-assisted thermoforming for a simple shapedcup using a 1.25-mm thick HIPS polymer sheet. Plug dis-placement, sheet temperature, plug temperature, and plugshape greatly influenced thickness distribution, but plug speedand air pressure had relatively small influence.

Collins et al. [18] studied the role of contact between thesheet and several plug materials such as syntactic foam,polyoxymethylene, and aluminum. The experimental resultsshowed that under Tg the coefficient of friction (COF) is in therange of 0.1 to 0.3, but COF increased sharply with increasingtemperature. At a thermoforming temperature above Tg, thesheet material became sticky and could not slip on the plugsurface. O’Connor et al. [19] found that in the plug-assistedthermoforming of polypropylene cups the contact friction be-tween the sheet and plugwere themost sensitive parameters tocontrol thickness distribution.Morales et al. [20] measured the

Int J Adv Manuf Technol (2020) 107:2449–2464 2451

coefficient of friction between a HIPS sheet and plugmaterialssuch as velocities of aluminum and steel in thermoforming.The COF increased with increasing sheet temperature, butthere was no significant change with plug speed. Maratheet al. [21] studied the temperature dependence of COF inplug-assisted thermoforming using 3.0 mm thick HIPS. Itwas found that the COF rapidly increased when the tempera-ture was higher than the Tg, resulting in a significant reductionin slip and increasing uniaxial stretching at the clamp.

Furthermore, many studies have highlighted the signifi-cance of the heating step in the thermoforming process, be-cause the thickness distribution is affected by thetemperature-dependent viscoelastic properties of the polymermaterials. Yousefi et al. [22] focused on accurate measurementof heat transfer parameters such as heat transfer coefficient,atmospheric temperature, emissivity, view factor, specific heat,and thermal conductivity for the 1.6-mm thick ABS sheet.Duarte and Covas [23] attempted to solve the inverse heatingproblem in thermoforming, minimize the temperature gradi-ents across the sheet thickness, and obtain uniform thicknessdistribution. Because of significant sheet deformation due tosheet sag during the heating step,Wiesche [24] used a dynamicview factor to perform radiative and conductive heat transfersimulations. Labeas et al. [25] studied the optimization of theheating step by conducting a simulation involving heat transferbetween thermoplastic carbon/polyetherimide compositesheets. Various parameters, such as the number of heating ele-ments, and location, and heat flux of infrared heating lampswere investigated for optimal heating. Giacomin et al. [26]reported that the complicated processability of thermoformingcomes from temperature gradients, either across the sheet orthrough its thickness, due to sheet sag. Ghobadnam et al. [27]found that different heating times are required depending onthe shape of the mold. Increasing the heating time was impor-tant to achieve a more uniform thickness distribution for thefemale mold. However, for male molds, the heating time had tobe reduced.

From the previous literature, it is clear that the thicknessdistribution of thermoformed products is dependent on thebehavior of the polymer materials. In addition, the investiga-tion of polymer materials should be conducted under processconditions including a wide range of temperatures, strains, andstrain rates. Moreover, numerical simulation is essential to abetter understanding of the thermoforming process via thestudy of the effects of various process parameters. Hence,rheological measurement of the nonlinear viscoelasticity ofthe polymer material should be considered.

1.3 Objectives

The objective of this study is to investigate the nonlinear vis-coelastic characteristics of the extruded polymer sheet used inheavy gauge thermoforming. Rheological measurements were

carried out under thermoforming conditions including a widerange of strains, strain rates, and temperatures. The linear andnonlinear viscoelastic properties of the extruded ABS sheetwere extensively investigated, including by the constitutivemodel, and the effects of orientation in terms of viscosity,stress, and homogeneous deformation were analyzed.Numerical simulations based on the finite element methodwere performed with the obtained nonlinear viscoelastic pa-rameters. The numerical results were validated by comparingthem with the thickness distr ibution of productsthermoformed by an industrial thermoforming machine. Thiswork provides important insight into the deformation behaviorof polymer materials that undergo large and rapid deforma-tions in thermoforming.

2 Nonlinear viscoelastic constitutive model

Generally, polymers are viscoelastic materials that dissipate aportion of the applied energy and simultaneously rememberthe deformation history. Polymer materials exhibit linear vis-coelasticity in small deformations, but gradually display non-linear viscoelastic behavior under large strain and high strainrates over a wide range of temperatures. Considering that de-formations are large and rapid in most processing operations,it is necessary to describe the nonlinear behavior of polymersaccurately [28]. To this end, many constitutive equations havebeen proposed in integral or differential forms [29]. Amongthe constitutive equations, the K-BKZ equation, developed byKaye [30] and Bernstein et al. [31], has been extensively eval-uated due to its wide applicability to polymer materials. Inparticular, it was reported that the K-BKZ model results in acredible agreement with the experimental uniaxial extensionalbehavior of ABS [4]. Thus, in this work, the K-BKZ modelwas selected to investigate the linear and nonlinear behaviorof ABS.

The K-BKZ equation is a class of equations because itspredictability depends on the choice of the kernel function.A simple and useful equation form was suggested byWagner, which is called the Wagner model [32]. TheWagner model is expressed as follows:

σ tð Þ ¼ ∫t

−∞M t−t

0; I1; I2

� �B t; t

0� �

dt0 ð1Þ

where σ is an extra molecular stress tensor, M is a kernelfunction that depends on both time and strain, B is a Fingerstrain tensor, and I1 and I2 are the first and second invariants ofthe Finger strain tensor, respectively.

Wagner proposed the kernel function as the product of atime-dependent memory function and a strain-dependentdamping function, which is called time-strain separability.

Int J Adv Manuf Technol (2020) 107:2449–24642452

M t−t0; I1; I2

� �¼ m t−t

0� �

h I1; I2ð Þ ð2Þ

wherem(t − t′) is the time-dependent memory function used toexplain the linear viscoelasticity and h(I1, I2) is the strain-dependent damping function used to describe the nonlinearviscoelasticity. The time-dependent memory function andstrain-dependent damping function should be specified in thismodel to explain the linear and nonlinear viscoelasticbehaviors.

The memory function is presented as the sum of the dis-crete relaxation modes with discrete relaxation moduli gi anddiscrete relaxation time λi,

m t−t0

� �¼ ∂G t−t0

� �∂t0

¼ ∑N

i¼1

giλi

� �exp −

t−t0

λi

� �ð3Þ

Generally, N sets of relaxation moduli are called the dis-crete relaxation spectra. The values gi and λi can be experi-mentally obtained by fitting the following equation to smallamplitude oscillatory shear (SAOS) data,

G0ωð Þ ¼ ∑

N

i¼1

giω2λ2

i

1þ ω2λ2i

;G″ ωð Þ ¼ ∑N

i¼1

giωλi

1þ ω2λ2i

ð4Þ

Various damping functions have been suggested based ontheoretical and empirical approaches [33–38]. The equationforms for damping functions are summarized well in a recentreview by Rolón-Garrido and Wagner [39]. The dampingfunction h(I1, I2) as a function of the invariants of the Fingerstrain tensor has a value between 0 and 1.

0 < h I1; I2ð Þ≤1 ð5Þ

When h(I1, I2) is one, the materials exhibit linear viscoelas-ticity. In this case, Eq. (1) reduces to the Lodge model devel-oped by Lodge [40]. As h(I1, I2) goes to zero, the nonlinearviscoelasticity of the materials becomes noticeable. Thedamping function of the Wagner model, used in this study, isexpressed as

h I1; I2ð Þ ¼ exp −αffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβI1 þ 1−βð ÞI2−3

p� �ð6Þ

where α and β are the nonlinear parameters of the dampingfunction, obtained from the uniaxial extensional experiments.The first and second invariants of the Finger strain tensor foruniaxial extensional flow are as follows:

I1 ¼ exp 2εð Þ þ 2exp −εð Þ; I2 ¼ 2exp εð Þ þ exp −2εð Þ ð7Þ

where εh is the Hencky strain, defined as the product of ex-tensional rate and time (≡ ε̇ t). By combining Eq. (1) with Eqs.(3), (6), and (7), the Wagner model equation can be solvednumerically for start-up uniaxial extensional flow.

3 Rheological measurement of materialproperties

3.1 ABS polymer

ABS is a copolymer composed of acrylonitrile, butadiene, andstyrene. It is widely used in thermoforming because it has highimpact resistance, rigidity, gloss, hardness, and temperaturestability. Furthermore, ABS has high melt elasticity becausebutadiene rubber particles of several, hundred nanometer sizesare dispersed in the SAN matrix.

The commercial ABS copolymer used was provided by LGChemistry Corporation (South Korea). The SAN matrix(Mw = 206,500 g/mol) consisted of 31.8 wt% acrylonitrileand 68.2 wt% styrene. SAN-grafted butadiene rubber phaseof 10 wt% was added to the matrix. The rubber particles havea narrow size distribution with an average diameter of 230 nm.

3.2 Glass transition temperature

Glass transition temperature (Tg) can be used to determine thethermoformable temperature range. ABS is an amorphouspolymer and, thus, does not have a clear melting point. ABSis hard and brittle in the glassy state below Tg. In the rubberystate above Tg, the mobility of the polymer chain increases,and thus ductility increases and rigidity decreases.Accordingly, in this work, the thermoforming temperaturerange was set from Tg + 30 to Tg + 60 °C.

To measure the Tg of the ABS used, a dynamic mechanicalanalysis (DMA) experiment in cantilever mode was per-formed using a DMA Q800 (TA Instruments). Specimenswere obtained by cutting the extruded ABS sheet into dimen-sions of 35 mm long, 13 mm wide, and 3.4 mm thick. Theelastic and viscous responses (E′ and E″) were measured atthree frequencies of 1, 5, and 10 Hz in a range of temperaturesfrom 30 to 150 °C, with a heating rate of 2 °C/min.

From Fig. 2, it is clear that the viscoelasticity of the ABShas temperature dependence. The E′ of the ABS did notchange significantly up to 100 °C, while E″ changed sensi-tively with increasing temperature. Tg was obtained at thepeak of the tan δΕ (≡ E″ / E′) curve. Near Tg, the polymermaterial exhibited a transition from glassy to rubbery state,and thus E′ and E″ changed dramatically. The measuredvalues of Tg at the three frequencies are summarized inTable 1.

3.3 Linear rheological measurements

At low strain and strain rates, a polymer material exhibitslinear viscoelastic behavior. However, as strain and strain ratesincrease, the material response deviates from linear viscoelas-ticity and gradually moves to nonlinear viscoelastic behavior.SAOS experiments were performed to characterize the linear

Int J Adv Manuf Technol (2020) 107:2449–2464 2453

viscoelastic properties of ABS over a wide range of tempera-tures, from 130 to 220 °C, with an angular frequencyω of 10−1

to 102 rad/s, and at small shear strain amplitude, γ0 = 0.5%,corresponding to the linear viscoelastic condition.

The linear viscoelastic storage modulus (G′) and loss mod-ulus (G″) are measured in SAOS tests. When an input strainsignal is a simple sine function, the shear stress signal underlinear oscillatory shear is expressed as follows:

σ tð Þ ¼ σ0sin ωt þ δð Þ

¼ γ0 G0ωð Þsin ωtð Þ þ G″ ωð Þcos ωtð Þ

h ið8Þ

tanδ ¼ G″ ωð ÞG

0ωð Þ ð9Þ

whereG′(ω) is the storage modulus,G″(ω) is the loss modulus,and tan δ is the ratio of G″ (ω) to G′(ω). SAOS experimentswere carried out using a rotational rheometer (ARES-G2, TAinstruments) with parallel-plate geometry and a diameter of13 mm. A circular specimen with a thickness of 3.4 mm anda diameter of 13 mm was cut from the extruded ABS sheet.Each experimental test was repeated three to five times, andone representative test was selected for the analysis. Based onthe time-temperature superposition principle, the experimentaldata at different temperatures were shifted to construct a linearmaster curve. The data were shifted along the frequency axis,and the horizontal shift factors aT were fitted with theWilliams-Landel-Ferry (WLF) equation.

log aTð Þ ¼ −C1 T−T rð ÞC2 þ T−T rð Þ ð10Þ

where T is the current temperature, Tr is the reference temper-ature, and C1 and C2 are the parameters of the WLF model.The parameters C1 = 4.67 and C2 = 105.14 K were obtained ata Tr of 170 °C. As depicted in Fig. 3, the shift factors aredescribed well by theWLF equation. The obtained linear mas-ter curve is shown in Fig. 4, with the prediction from Eq. (4).Ten sets of the discrete relaxation time λi and discrete relaxa-tion modulus gi were obtained and are summarized in Table 2.

3.4 Nonlinear rheological measurements

Linear viscoelastic behavior can be analyzed using G′(ω) andG″(ω) when the deformation is small. However, the linearviscoelastic theory is no longer valid when polymer materialsundergo large and rapid deformation. As strain increases, ma-terial response gradually departs from the linear behavior, andnonlinear behavior such as strain hardening and strain soften-ing becomes noticeable. It is therefore very important in thethermoforming process to analyze the nonlinear viscoelasticbehavior of polymer materials.

To this end, the uniaxial extensional viscosity of ABS wasmeasured using an ARES-G2 equipped with an extensionalviscosity fixture. Figure 5 shows the principle of uniaxial ex-tensional viscosity measurement. One end of the specimen isconnected to a fixed drum where the force is measured. Theother end is connected to a rotating drum pulling the speci-men. The rotating drum moves along a circular orbit aroundthe stationary drum to apply uniaxial stretch with a constantextensional rate to the specimen.

Since the polymer sheet used in thermoforming is extrud-ed, a strong orientation occurs inside the sheet with respect tothe extrusion direction, which may exhibit significant anisot-ropy. The oriented polymer sheet has a high resistance todeformation in the extrusion direction and exhibits a weakresistance in the transverse direction, which may have a largeinfluence on thermoforming. To investigate the orientation

Table 1 The frequencydependence of the glasstransition temperature ofthe ABS

Frequency [Hz] Tg [°C]

1 118.6

5 122.2

10 123.6

Fig. 2 Determination of glass transition temperature (Tg) by using DMA

Fig. 3 Storage modulus G′(ω) and the loss modulus G″(ω) of ABSpolymer used. The experimental data is represented by the color circlesand the prediction by Eq. (4) is represented by the dash-dotted lines

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effect of the extruded ABS sheet, uniaxial extensional exper-iments were performed on two specimens cut from the samesheet. The two specimens were named type A and type B.Type A shows ABS in the direction parallel to the rollers,and type B shows that it is in the transverse direction.

The test specimens were 18 mm in length, 10 mm in width,and 1.1 mm in thickness. The measurements were carried outat 0.1, 0.5, and 1.0 s−1 and at four different temperatures of160, 170, 180, and 190 °C. Uniaxial extensional viscosity ηþEt; ε̇ð Þ is expressed as follows:

ηþE t; ε˙� � ¼ σE t; ε̇ð Þ

ε̇¼ F tð Þ

l0w0e−ε˙ tε̇

ð11Þ

where σE t; ε̇ð Þ is the extensional stress in terms of time andstrain rates, F(t) is the force, and l0 and w0 are the length andwidth of the specimens, respectively. To check the accuracy ofthe extensional measurements, ηþE t; ε̇ð Þ was compared withthe linear viscoelastic behavior, calculated using λi and gifrom the SAOS experiments,

ηþE tð Þ ¼ 3 ∑N

i¼1giλi 1−e−t=λi

� �ð12Þ

Figure 6 shows the uniaxial extensional viscosity curves ofthe extruded ABS sheet for different extensional rates at var-ious temperatures. The dashed line indicates the linear visco-elastic response of ηþE t; ε̇ð Þ (Eq. 12). The filled symbol (typeA) and open symbols (type B) indicate the experimentallyobtained ηþE t; ε̇ð Þ.

In the initial stage, ηþE t; ε̇ð Þ of both types showed goodagreement with the linear viscoelastic response when theHencky strain was small. As the strain increased, ηþE t; ε̇ð Þgradually deviated from ηþE tð Þ, with significant strain harden-ing. Strain-hardening behavior became more remarkable asthe extensional rates increased at a fixed temperature.Accordingly, rapid deformation can have a positive effect onthe thickness distribution of thermoformed products, bypreventing excessive thinning. As the strain increased further,softening behavior was observed, indicating that melt failureoccurred near the maximum strain of 3.4. These behaviorsindicate that a nonlinear constitutive model is required to de-scribe the nonlinear viscoelasticity of the ABS underthermoforming conditions, such as large strain and/or highstrain rate.

As the temperature increased, it was found through analysisthat the increase in mobility of the SAN matrix influences theuniaxial extensional behavior of ABS sheets. The ηþE t; ε̇ð Þcurves at 160 °C displayed strong strain hardening behavior.However, as the temperature decreased, the strength of strainhardening also decreased. The ηþE t; ε̇ð Þ at 190 °C even showedstrain-softening behavior and almost linear viscous character-istics at 0.1 s−1. At high temperatures, low ηþE t; ε̇ð Þ and strainsoftening can lead to unstable deformation, such as a bubbleexplosion in the pre-stretching step and tearing in the formingstep.

The stress-strain curves were plotted to clearly show theeffect of orientation of the ABS sheet, as shown in Fig. 7.Within for students the experimental conditions, type A most-ly exhibited a higher stress value and earlier strain hardeningthan type B. This difference between the two types has asignificant effect on the thickness distribution of thethermoformed products. The earlier strain hardening of typeA provides high resistance to the applied deformation,preventing extreme thickness reduction or tearing due to highfrictional force. In contrast, type B has a lower resistance todeformation at εh < 3. Type B is also expected to displayenough strain hardening at εh > 4. However, considering thatthe maximum εh value during the thermoforming process isless than 4, as determined by Throne [41], a more uniformthickness distribution can be achieved when the ABS chainsare oriented toward the extension direction. Extensional vis-cosity, strain hardening, and the effects of orientation on stress

Fig. 4 Horizontal shift factor (aT) and fitted result with the WLFequation. The parameters C1 and C2 were obtained at the referencetemperature of 170 °C

Table 2 The discrete relaxation modulus (gi) and discrete relaxationtime (λi) of the ABS polymer at a reference temperature of 170 °C

i gi [Pa] λi [s]

1 5.93 × 106 4.16 × 10−6

2 4.13 × 105 8.98 × 10−5

3 1.55 × 105 8.39 × 10−4

4 1.10 × 105 6.67 × 10−3

5 1.02 × 105 4.35 × 10−2

6 9.39 × 104 2.59 × 10−1

7 6.94 × 104 1.41 × 100

8 3.61 × 104 7.22 × 100

9 1.13 × 104 4.18 × 101

10 3.42 × 103 4.40 × 102

Int J Adv Manuf Technol (2020) 107:2449–2464 2455

and strain can be very useful information in polymerprocessing.

The strain hardening behavior of ABS can be controlled bythe amount and hardness of butadiene particles. Takahashiet al. [42] investigated the effect of the hardness of butadieneparticles for a series of ABS samples that had similar particle

size and dispersion state. Notably, the strain hardening becameweaker with increasing content of soft or hard particles. Theseresults suggest that the degree of strain hardening is stronglyaffected by the particle hardness. Thus, the role of butadieneparticles on strain hardening behavior is highly important.Compared with their results, a particle content of 10 wt% is

Fig. 6 a–d Uniaxial extensional viscosity of ABS at various temperatures for different extensional rates. The linear viscoelastic response which wasobtained from SAOS experiments is shown by the dashed lines

Rotating drum

Fixed drum

Sample

Fig. 5 Schematic of theextensional viscosity fixture tomeasure the uniaxial extensionalviscosity

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adequate for thermoforming processes because it generatesenough strain hardening to prevent excess thickness thinning.

3.5 Orientation effects of extruded sheet

Strain hardening of polymer material has a positive effect onhomogeneous deformation in the polymer process, and thiswas quantitatively analyzed using the Considère criterion[43].

dFdεh

¼ 0 ð13Þ

The Considère criterion has been used to predict the failureof polymer melts and solutions. Lee et al. [4] in particular usedthe Considère criterion as an effective indicator of the nonuni-form deformation of ABS in uniaxial extension. Figure 8shows the critical strain (εc) obtained with Eq. (13) along withforce and strain curves for type A and type B, respectively.The εc values are summarized in Table 3. In all the results, εc

increased with increasing extensional rates at a lowertemperature.

The εc value represents the maximum Hencky strain atwhich samples are deformed homogeneously. Thus, it canbe regarded as a measure to determine the operating limit inpolymer processes such as thermoforming, which are impor-tant for achieving uniform thickness distribution. At 180 and190 °C, the two types displayed similar εc values (Table 3).However, with decreasing temperature, the εc value of type Ashifted to two, while type B remained less than one. Thus, typeA can undergo homogeneous deformation until higher εc,compared with type B. From these results, it can be seen thata critical strain of an extruded ABS sheet exists at whichhomogeneous deformation can be maintained using strain,strain rate, and temperature.

Taken together, the thickness and uniformity of thethermoformed products highly depend on the extensional rhe-ology of the polymer material. The different extension direc-tions with respect to chain orientation, in particular, can resultin different nonlinear responses, which leads to the final

Fig. 7 a–d Stress-strain curves of ABS for various extensional rates in the temperature range of 160 to 190 °C. Under the same conditions, the stress ofthe type A specimen is larger than that of type B specimen

Int J Adv Manuf Technol (2020) 107:2449–2464 2457

properties of the thermoformed products. As a result, it wasspeculated that uniform thickness distribution and high tearresistance can be obtained when the sheet is pulled in theextrusion direction.

4 Numerical simulation of thermoformingprocess

4.1 Parameters of damping function

The Wagner model from Section 2 was exploited to describethe uniaxial extensional behavior of the ABS. The specificform of the equation is as follows:

σE t; ε˙� � ¼ G tð Þh tð Þ e2ε

˙ t−e−ε˙ t

h i

þ ∫s

0m sð Þh sð Þ e2ε

˙ s−e−ε˙ s

h ids ð14Þ

where s = t − t′. The linear viscoelastic relaxation modulusG(t) and memory function m(s) were calculated from the dis-crete relaxation spectra (Eq. 3). Equation (6) was used for thedamping function h(t) and h(s). To determine the nonlinearparameters α and β in the model, the following objectivefunction (RSS) was optimized numerically.

Fig. 8 a–d Force-strain curve of ABS at various temperatures. The homogeneous deformation is indicated by the position of the critical strain (εc),which is calculated with the Considère criterion

Table 3 The achievable maximum strain (εc) calculated by theConsidère criterion in the uniaxial extensional deformation for type Aand type B

Extensional rates [s−1] 160 °C 170 °C 180 °C 190 °C

Type A

0.1 1.75 0.57 0.33 0.22

0.5 2.08 1.86 0.58 0.45

1.0 2.09 1.97 0.68 0.52

Type B

0.1 0.67 0.56 0.47 0.39

0.5 0.94 0.69 0.59 0.51

1.0 1.10 0.83 0.62 0.58

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RSS ¼ ∑D

i¼1Fexpi −Fmodel

i

� �2 ð15Þ

where Fexpi is the experimental data, Fmodel

i is the Wagnermodel predictions, and D is the total number of data points.

During the thermoforming process, because deformations oc-curredmainly in themachinedirectionof the extrudedABSsheet,the parametersα andβwere obtained using the typeA specimen.The reference temperature of 170 °Cwas selected to describe thetemperature-dependence, using the WLF equation. Figure 9shows the uniaxial extensional viscosity curves of experimentsand the predictions from the Wagner model using the obtainedparameters. The Wagner model prediction for ηþE t; ε̇ð Þ can beused to quantitatively understand the characteristics of the exper-imental results. At a short time, the model prediction coincidedwith the linear viscoelastic response, validating the accuracy ofthe numerical solutions. As time increased, the Wagner modelgave good predictions for strain hardening behavior, especiallythe onset time of strain hardening. In contrast to the Lodgemodel,which predicts infinite growth of the extensional viscosity(Lodge, 1956), the Wagner model allows one to obtain the max-imum viscosity value as a function of the damping function term.However, at an extensional rate of 0.1 s−1, the model predictedhigherηþE t; ε̇ð Þ, and stronger strainhardening than the experimen-tal data. Accordingly, a complete description of the experimentaldata was not clear when using the parameter sets ofα and β. Thevalidity of the parameters will be discussed in Section 4.3; theresults of numerical simulations using the Wagner model canreasonably explain the deformation of the extruded ABS sheetcompared with that in the experimental results.

4.2 Numerical setup

Three-dimensional simulations based on the finite elementmethod were performed using T-SIM (version 4.83,

Accuform), a commercial code package for thermoforming.The deformation of the polymer materials and the thicknessdistribution of the thermoformed products were numericallyinvestigated. Numerical simulations require a nonlinear visco-elastic constitutive model that can describe the relationshipbetween stress and deformation. The discrete relaxation spec-tra, the WLF equation, and the Wagner model were utilized inthe numerical simulations.

A mold for a refrigerator inner case using heavy gauge anddeep drawing thermoforming was designed using the com-mercial CAD software NX (version 8.0, Siemens). This re-frigerator case had a complex shape because it containedstructures to increase the structural rigidity of the final productand to mount various parts. Figure 10 shows the shape anddimensions of the mold. The width, depth, and height of themold were 840 mm, 740 mm, and 1620mm, respectively. Themold consisted of an upper compartment 820 mm in lengthand a lower compartment of 760 mm length. The partition ofthe narrow gap between the compartments prevents the self-contact of the polymer sheet during bubble inflation in the pre-stretching step. The frame clamps the polymer sheet duringthe entire process.

Figure 11 shows the finite element model used for the nu-merical simulation. The polymer sheet was approximated astriangular membrane elements because the thickness of thesheet was very thin compared with the length or width, andthus the velocity and stress changes in the thickness directioncould be ignored. Approximately 100,000 finite elementswere established by performing a grid dependency test; testresults showed excellent numerical convergence. Because themold was symmetrical, only half of the polymer sheet wasused for efficient calculations. Tools such as molds, partitions,and frames are considered rigid bodies that are not deformedand penetrated. The boundary conditions for the sheet were setto have zero velocity for the clamped edge and a symmetricalcondition for the half-line. The initial temperature fields for

Fig. 9 Uniaxial extensional viscosity of ABS (type A) at a referencetemperature of 170 °C for different extensional rates. The experimentaldata is shown by the filled symbols, and the prediction by the Wagnermodel is demonstrated by the solid lines

740 mm

40 mm

820 mm

760 mm

840 mmUpper compartment

Lower compartment

1620 mm

Fig. 10 The deep drawing mold consisting of an upper and lowercompartment

Int J Adv Manuf Technol (2020) 107:2449–2464 2459

the upper and lower compartments of the polymer sheet wereset at 175 °C and 185 °C, respectively. The convective heattransfer coefficient of 8 W/(m2∙K) was used. Contact betweenthe polymer sheet and the mold was considered by Coulombfriction and the coefficient of friction was assumed to be 0.2.The thermal properties of the ABS polymer, such as density,thermal conductivity, and specific heat were 1000 kg/m3,0.2 W/(m∙K), and 2000 J/(kg∙K), respectively.

Figure 12 shows the process conditions such as differentialpressure and mold position over time. From 0 to 4 s in the pre-stretching step, differential pressure was applied to inflate thepolymer sheet. The mold stayed at the initial position until t =2 s and then reached the maximum position at t = 4 s. From t =4 s to t = 8 s in the forming step, the polymer sheet wasstretched by contact with the mold; then, a vacuum was ap-plied to the sheet through fine holes in the mold to obtain thedesired shaped products.

During the process, the mold temperature can be increasedby repeated contact with the heated polymer sheet, and sowater flow channels were fabricated in the mold for cooling.The heat transfer between the sheet and the mold is expressedas follows.

q ¼ U T sheet−Tmoldð Þ ð16Þwhere q is the heat flux, U is the overall heat transfer coeffi-cient, Tsheet is the temperature of the sheet, and Tmold is thetemperature of the mold.

4.3 Numerical results

The numerical results at each step of the thermoforming pro-cess are illustrated in Fig. 13. The polymer sheet was assumedto be preheated and sagged due to gravity, as shown inFig. 13a. Next, vacuum pressure was applied to inflate thepolymer sheet like a bubble for more uniform thickness dis-tribution, as shown in Fig. 13b. This helps to distribute thematerial in contact with the rear surface of the mold into otherareas before the forming step. As plotted in Fig. 13c, theinflated sheet begins to touch the edge of the mold. The smallcontact area between the sheet and the mold causes partialslips, not complete sticking. Then, the sheet continues to ex-perience extensional deformation due to the movement of themold. Finally, the mold stops at its predesigned position, andthen vacuum pressure is applied through fine holes in the moldto fully contact the polymer sheet with the entire surface of themold. A thermoformed product with the desired shape is ob-tained; the thinnest area was found near the edge of the mold.

Figure 14 shows the thickness distribution on the symmet-ric line A-A’ of products obtained from the experiments andnumerical simulations. Thermoforming products were thenmanufactured using industrial thermoforming machines witha deep drawing mold with a complex shape. The same exper-imental and numerical conditions were used for comparison,as described in Section 4.2. Thickness distributions ofthermoformed products were measured by micrometer andcompared with results from numerical simulations. The thick-ness distribution can be analyzed in three regions: the top,bottom, and rear regions in the upper and lower compart-ments. The top and bottom regions between the frame tool

Clamped

Triangular membrane elementsof the sheet (3.4 mm)

Symmetric

1900 mm

550 mm Symmetric

Clamped

Fig. 11 The finite element modelfor the numerical simulation

FormingPre-stretching

Fig. 12 Differential pressure and mold position for thermoformingsimulations

Int J Adv Manuf Technol (2020) 107:2449–24642460

were free surfaces in contact with the mold. They were mainlyuniaxially deformed, and significant thickness reduction wasobserved because these regions continued to deform until themold stopped. In contrast, the rear region was mainly biaxiallydeformed and the thickness change was relatively small. Thehot sheet was cooled by contact with the edge of the mold sothat it became difficult to pull the material from the rear regionto the top and bottom regions. Due to a combination of

cooling and partial slips, nonuniform thickness distributionswere found in both the experimental and numerical results.Maximum deviation between experimental data (1.1 mm)and numerical results (0.77 mm) was observed at the edgebetween the top region and rear region in the upper compart-ments because temperature-dependent COF was not included.

Figure 15 provides a visualization of the deformation of theside walls: the experimental and numerical thickness distribu-tions are compared. Before deformation, a grid was drawn onthe sheet with a height of 62.5 mm and a width of 125 mm.After deformation, the height of the grid in the upper compart-ment was 208 mm in the experiments (Fig. 15a) and 170 mmin the numerical results (Fig. 15b), and the height in the lowercompartment was 214 mm in the experiments (Fig. 15a) and158 mm in the numerical results (Fig. 15b). High extensionaldeformation at the side walls was observed in both results, andthe experimental results showed a larger change than that ofthe numerical results. As a result, in the sidewalls, the exper-imental thickness was lower than the numerical results (500–1000 mm in line B-B’ and line C-C’, corresponding to theupper and lower compartments, respectively). This is due tothe limitations of the constitutive model, which cannot de-scribe anisotropy, such as the effect of the orientation.

As shown in Figs. 6, 7, and 8, the extruded polymer sheetwas sensitive to the applied deformation along a specific di-rection. However, isotropic models, such as the Wagner mod-el, used in this work, do not take this effect into account.Numerical simulations of thermoforming could be improved

(a) Heating (b) Pre-stretching

(c) Forming (mold moving and vacuuming)

Contact with the mold

Fig. 13 Main three steps of thethermoforming process usingdeep drawing mold and vacuumpressure. a Heating, b Pre-stretching, c Forming (moldmoving and vacuuming)

A A’

Top Rear Bottom Top Rear Bottom

Upper Lower

Fig. 14 Comparison between the numerical analysis and experimentalresults of the thickness distribution along the line A-A’

Int J Adv Manuf Technol (2020) 107:2449–2464 2461

by the development of anisotropic viscoelastic constitutivemodels that are capable of a prediction considering the orien-tation of the polymer chains.

The thickness distributions in the numerical simulationsshowed good qualitative agreement with the experimental re-sults, but some discrepancies remain. This can be caused bythe use of a constant value of COF in the simulation. Asshown in previous works [18–21], COF is a key parameterto control the thickness distribution in thermoforming and is afunction of temperature. Since the inflation of the polymersheet leads to a free surface in the pre-stretching step, theinfluence of the coefficient of friction on the deformation be-havior is not large. On the other hand, the extensional defor-mation of the polymer sheet can be greatly affected by contactfriction in the forming step. As Marathe et al. reported [21],COF is highly dependent on temperature and increases verysteeply above Tg. The measured COF of HIPS was low for T< 100 °C, whereas it increased considerably at and above100 °C in their work. It is expected that the behavior of sheetsticking or sliding on the surface of the thermoforming toolscan significantly change with temperature change. For exam-ple, using a high COF can increase the thickness, as the sheetsticks to the surface of the mold, and deformation stops in thatarea. Contact friction is the most important factor affecting thethickness distribution of thermoformed products. Hence, for amore accurate prediction of the thickness distribution,temperature-dependent COF of the polymer sheet should beincorporated into the numerical simulation.

Furthermore, the assumption of a uniform temperature fieldat the initial state for each compartment can cause the differencesbetween the experimental results and the simulated predictions.In the heating step, the industrial thermoforming machine useshundreds of heating elements to heat the polymer sheet, and it isdifficult to obtain a uniform temperature field of the sheet. Smalltemperature differences across the sheet surface can exist, andtemperature-dependent viscoelastic properties of the sheet canlead to error.

5 Conclusion

In this work, using a deep drawing mold with a complexshape, rheological measurements were carried out to charac-terize the linear and nonlinear viscoelasticity of the ABS usedin heavy gauge thermoforming. Moreover, the effect of theorientation of the extruded polymeric sheets was investigated.Numerical simulations using the Wagner model were per-formed to explore the thickness distribution of thethermoformed products, and the results were compared withthose of thermoformed products manufactured using industri-al thermoforming machines.

The linear and nonlinear viscoelastic properties of the ABSpolymers were studied by rheological measurements and theparameters of the constitutive model were obtained. In theSAOS experiments, the storage modulus and loss moduluswere measured over a wide temperature and frequency ranges.

(a) Experimental result

(b) Numerical result

208 mm

214 mm

C

C’

170 mm

158 mm

B

B’

Fig. 15 Comparison between the numerical analysis and experimental results of the thickness distribution along the line B-B’ and C-C’. a Experimentalresults, b numerical results

Int J Adv Manuf Technol (2020) 107:2449–24642462

A linear master curve was constructed to cover the widerrange of frequencies. The discrete relaxation spectra and theparameters of theWLF equation were extracted from the mas-ter curve. To explain the nonlinear viscoelasticity, uniaxialextensional experiments were carried out and the parametersof the damping function were obtained. The extensional prop-erties of the extruded ABS polymer sheets were found tosignificantly change depending on the orientation of the poly-mer chains. The type A specimen, which was oriented in theextrusion direction, had higher uniaxial extensional viscosityand stress than did type B, which was in the transverse direc-tion. It was also found that, in type A, high resistance to ma-terial deformation and a wide range of uniform deformationsare possible at 160 °C and 170 °C.

The nonlinear parameters α and β of the damping functionwere calculated by the optimization method. Using theWagner model, numerical simulations were performed withthe obtained nonlinear parameters, together with the visco-elastic properties. The thickness distribution of the numericalresults was found to be qualitatively consistent with the prod-ucts manufactured by an industrial thermoforming machine.Maximum difference between experiments and numerical re-sults was observed at the edge in the upper compartment.Material deformations of the side walls were visualized, andgrid deformations in the transverse direction in experimentswere higher than those in the numerical simulations. Variousreasons for the differences between the experimental and sim-ulated predictions were analyzed with regard to the coefficientof friction, the orientation of the polymeric material, and thelimitations of the constitutive model.

Funding information This research was supported by the NationalResearch Foundation of Korea (NRF) grant funded by the Korean gov-ernment (MOE) (No. NRF-2017R1A2B2006264).

Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing, adap-tation, distribution and reproduction in any medium or format, as long asyou give appropriate credit to the original author(s) and the source, pro-vide a link to the Creative Commons licence, and indicate if changes weremade. The images or other third party material in this article are includedin the article's Creative Commons licence, unless indicated otherwise in acredit line to the material. If material is not included in the article'sCreative Commons licence and your intended use is not permitted bystatutory regulation or exceeds the permitted use, you will need to obtainpermission directly from the copyright holder. To view a copy of thislicence, visit http://creativecommons.org/licenses/by/4.0/.

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