Numerical model for compression molding process of hybridly laminated thermoplastic composites based on anisotropic rheology

Available online 7 December 2021

2666-6820/© 2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by- nc-nd/4.0/).

Numerical model for compression molding process of hybridly laminated thermoplastic composites based on anisotropic rheology

Sooyoung Lee, Dongwoo Shin, Gyeongchan Kim, Wooseok Ji

^*

Department of Mechanical Engineering, Ulsan National Institute of Science and Technology (UNIST), Ulsan, S. Korea

A R T I C L E I N F O Keywords:

Process simulation Compression molding Hybrid composite Suspension rheology Fiber orientation Battery pack

A B S T R A C T

In this work, a numerical model is developed for simulating the compression molding process of a hybrid composite material, alternately laminated with continuous and discontinuous fiber-reinforced layers. Although various process simulation models are already available for plastic materials embedding each type of the re- inforcements, they are incapable of simultaneously dealing with the continuity and discontinuity. Here, ther- momechanical behavior of the continuous fiber-reinforced layer and rheological behavior of the discontinuous fiber-reinforced layer are separately modeled and eventually integrated assuming perfectly bonded interfaces.

The unified process model is applied to the simulation of compression molding of a full-scale battery pack structure of an electric vehicle. A simple yet robust rheology test is utilized to measure rheological properties necessary for the numerical simulation. In the full-scale simulation, thermoforming process of the hybrid charge is successfully simulated and fiber direction changes due to the suspension flow are also predicted. It is found that the reoriented fibers significantly affect stress distributions at the final stage of the process. The process model developed in the present study can be implemented into either the Lagrangian or Eulerian framework.

1. Introduction

Composite materials are now actively employed in the automotive industry due to the ever-increasing demand for lightweight vehicles to improve fuel efficiency and reduce carbon dioxide emissions in response to the strengthening environmental regulations [1]. The powertrain electrification also strongly drives the effort for the weight reduction of automobile structures because of heavy batteries. Discontinuous fiber-reinforced plastic (DFRP) is a suitable material for commercial scale production because existing plastic molding methods with short process cycles can be utilized. Injection molding (IM) of a molten short or long fiber-reinforced thermoplastic (SFT or LFT) has been widely used for manufacturing composite products [2]. The compression molding (CM) process is applicable to thermoplastic composites such as LFT or a glass mat-reinforced thermoplastic (GMT) as well as thermosetting composites like a sheet molding compound (SMC) [3–5]. Molding DFRP materials is proven to be an efficient process to produce lightweight components for automotive applications [6].

DFRP is advantageous over continuous fiber-reinforced plastic (CFRP) for automotive parts owing to lower material and manufacturing costs as well as higher design freedom. However, unlike CFRP, DFRP is

seldom applied alone for a core structural load-bearing component because of its inherent discontinuity. The deficiency of DFRP is often overcome by hybridization with CFRP or sheet metal layers [7]. Hybrid composite products are often manufactured through an additional overmolding process; CFRP is firstly placed in (and preformed by) a mold and then a bulk compound material, such as DFRP, is molded over the continuous layer by the following compression molding step [6] [8].

The overmolding process adds load-bearing structural elements at spe- cific sites and locally strengthens the designated area. Since the primary intention of the overmolding is local reinforcement and functionaliza- tion, excessive process time may be required to hybridize a broad area.

Recently, hybrid composites that do not require the additional over- molding process are developed as intermediate materials and now commercially available.

The hybrid composite considered in the present study is also an in- termediate material, named as WLFT, that can be easily processed with CM only. WLFT is a laminate consisting of woven fabric fully impreg- nated with thermoplastic resin (WFT) and long fiber-reinforced ther- moplastic (LFT). Both WFT and LFT are glass fiber-reinforced polypropylene (GF/PP) composites. The WLFT laminate has two LFT layers, each of which covers the top and bottom of WFT. The layup

* Corresponding author.

E-mail address: wsji@unist.ac.kr (W. Ji).

Contents lists available at ScienceDirect

Composites Part C: Open Access

journal homepage: www.sciencedirect.com/journal/composites-part-c-open-access

https://doi.org/10.1016/j.jcomc.2021.100215

Received 19 July 2021; Received in revised form 25 November 2021; Accepted 4 December 2021

sequence is not a typical sandwich concept in which a stiffer material covers a bulky core material to maximize flexural resistance. Instead, the soft LFT layers are intentionally situated on the outer surfaces to facil- itate joining with other parts through welding or flowing into locally complex mold channels or pockets. In this manner, WLFT inherits not only the strong mechanical properties of WFT but also the excellent formability of LFT.

Fiber directions in WLFT significantly change during the CM process.

It is crucial to accurately predict reoriented fiber directions because fiber orientation greatly affects the mechanical performance of a final prod- uct. While the WLFT material is processed through CM, the woven layer conforms to a mold shape and the deformation determines the change in the directions of the continuous fibers. LFT can flow during the CM process and the suspension flow highly reorients the directions of the discontinuous fibers. Several models were developed in the Lagrangian framework to simulate the CM process of DFRP and thereby predict fiber reorientation [9] [10]. Some researchers [11] [12] and commercial

software packages (e.g., Moldflow and Moldex3D) deal with DFRP in the Eulerian framework. Forming of CFRP has been vigorously studied by several research groups [13-16]. However, none of them is capable of dealing with DFRP and CFRP simultaneously.

In this work, a process simulation model is developed for compres- sion molding of the WLFT laminate. Although the WFT and LFT layers are integrated into a single laminate, each layer exhibits distinctive material behavior during the CM process; the resin-rich LFT layer flows like a viscous suspension filled with chopped fibers while the fiber-rich WFT layer is deformed like an orthotropic solid material owing to its continuous fiber architecture. Therefore, the present simulation model characterizes WFT and LFT with their own constitutive models and unifies them assuming perfect bonding across the interface to predict the total behavior of WLFT. By doing so, the evolution of fiber orientation and stress states in each LFT and WFT layer can be individually taken into account. The WLFT simulation model is demonstrated for the compression molding process of the battery carrier of an electric vehicle.

As previously mentioned, the automotive electrification requires inno- vative lightweighting technology that can replace conventional heavy metallic parts to compensate the accommodation of heavy batteries. The battery carrier parts are usually designed with metal alloys to carry the heavy weight of the battery cells and withstand various types of service loads such as vibrational fatigue loading, impact loading and so on. The WLFT composite can be a good replacement for the weighty metals owing to it superior specific stiffness, strength, and high impact resis- tance. Furthermore, because each of the cover and tray can be fabricated into one piece by molding the WLFT sheet, numerous mating parts of the existing metal-based carrier can be eliminated. However, there exist no predictive process model yet for the hybrid composite to facilitate its deployment into production.

The rest of this paper is organized as follows. Section 2 describes the LFT and WFT materials with the rheological and thermomechanical testing protocols to measure their material properties needed for process simulations. Section 3 presents the theoretical backgrounds of the fiber reorientation and anisotropic flow models for the LFT layer. Section 4 deals with the full-scale process simulation of a battery carrier using the WLFT material. The conclusions of the present study are summarized in Section 4. In the appendix A1, supplemental materials about thermo- mechanical tests is provided. Appendix A2 first explains the numerical implementation of the WFT and LFT models and discusses the validity of the constitutive model of LFT through simple anisotropic flow simulations.

Fig. 1. Disk-shaped LFT specimen for compression test (Note: White dots indicate the thickness measurement points with the dial gage).

Fig. 2. (a) Experimental setup for the compression test (b) Compression test fixture.

2. Material characterization

2.1. Rheological property of LFT suspension 2.1.1. Extensional viscosity measurement

For an isotropic Newtonian fluid, the extensional viscosity is given by the triple of the shear viscosity according to the famous Trouton’s ratio.

The ratio of three is no longer valid for an inhomogeneous fluid like a fiber suspension, and thus its extensional and shear viscosities should be individually measured. In the present work, however, only the exten- sional viscosity of the LFT suspension is measured at a specific pro- cessing temperature. According to the previous studies in [10, 11, 17, 18, 19], it has been observed that the extensional viscosity mainly dominates the flow kinetics of thermosetting or thermoplastic DFRP suspension inside the mold cavity under isothermal or hot-mold condi- tions. When a homogeneous fluid such as neat resin flows inside a mold cavity, it may exhibit fountain flow behavior; the flow front becomes parabolic due to the velocity and stress gradients (Fig. 5(a) in [19]). The viscous friction at the mold surface can shear and retard the flow since this hydrodynamic friction is often dominant over the viscous contri- bution. Hence, the shearing viscosity plays an important role in shaping the flow front. On the other hand, for the inhomogeneous fiber sus- pension, the extensional viscosity is much larger than the triple of shearing viscosity due to the fiber alignment along the flow direction [20] and the viscous flow may overcome the hydrodynamic friction. In this case, thermal interaction with the mold is crucial for determining the shape of flow front [19]. When the mold temperature is relatively higher than the charge or both are in isothermal condition, a very thin resin-rich lubricating layer having a relatively low viscosity exists on the mold surface and fiber suspensions flow through the mold cavity almost without any friction [17]-[19]. Hence, the bulky core area of the fiber suspension exhibits extensional plug-flow behavior, and the flow front has a fairly flat shape with no velocity gradient (Fig. 5(c), (d) in [19]).

However, when the hot charge interacts with a relatively cold mold, heat transfer to the mold surface cause high viscosity near the mold and low viscosity at the core. As a result, the perfect slip of the core area is no longer available and the velocity gradient may exist with a shearing effect (Fig. 5(b) in [19]). The present study considers the compression molding of a thin thermoplastic charge with highly planar fiber-orientation under the isothermal condition (see Section 4). Ac- cording to [19], thin glass mat thermoplastic (GMT) shows a plug flow behavior under isothermal molding, which is comparable to the current simulation condition. Therefore, the shear effect is negligible and the extensional viscosity is sufficient for modeling the bulky extensional flow of the LFT suspension [10] [11] [19].

The extensional viscosity of the polypropylene composite reinforced with long glass fibers is measured at the CM process temperature of 200

◦C. The process temperature is set to be higher than the melting tem- perature of polypropylene (160 ^◦C). Several researchers [9] [20-23]

have implemented various test methods to measure the extensional viscosity of discontinuous fiber suspensions (either SMC or LFT) since there exists no test standard. In this work, the test method in [20] is employed because it is easy to apply and proven to be valid from various applications [21]-[23]. This test method is based on the compression of a disk-shaped LFT specimen. The specimen should be thick enough to undergo compressive deformation over sufficiently long duration.

2.1.2. Test setup and procedure

In the present study, two 5 mm-thick LFT panels (KwangSung Co.

Ltd., S. Korea) are laminated by a hot press and a water-jet cutter is utilized to fabricate the specimens with a diameter of 100 mm and an average thickness of 9.5 mm. The average thickness is obtained from the measurements at 5 points (white dots in Fig. 1) within a specimen using a dial gage. The disk-shaped specimen is then compressed at 200 ^◦C using a servo hydraulic loading frame equipped with a convection oven as shown in Fig. 2(a). The compression fixture illustrated in Fig. 2(b) is

made of AISI H13 tool steel to minimize thermal expansion. The upper and lower platens contacting with the molten specimen are covered with a polytetrafluoroethylene (PTFE) film acting as a lubricant. The oven temperature for heating up the specimens at 200 ^◦C is controlled by a thermocouple attached to the top surface of the lower disk. Thermo- couples are not directly attached to the specimen so as to avoid its disruption to rheological behavior during the compression test. In order to indirectly determine the specimen temperature, we have performed several pre-tests. A dummy specimen attached with a few thermocouples is placed on the lower platen and the temperatures of the specimen and disk are measured and calibrated. We have also measured the time for the dummy specimen at room temperature to reach the processing temperature of 200 ^◦C after entering the heated oven. It takes about 12 min for the specimen to reach the uniform temperature distribution at 200 ^◦C.

After the 12-minute heating, the upper disk starts to compress the 9.5mm-thick specimen to the thickness of 6 mm with the compression rate of 0.1 mm/s. The load cell connected to the upper disk measures the reaction force of the LFT charge and the compressive displacement is measured from the hydraulic actuator. The compressive true stress σ, true strain ε, and strain rate ϵ are calculated as: ˙

σ= 4Fh πφ²₀h0

, ϵ=ln (h0

h )

, ϵ˙=h˙

h (1)

where F is the reaction force, φ0 is the initial diameter of the specimen and h is the compression rate. h˙ 0 is the initial mold gap height between upper and lower disks, which is set to the thickness of each specimen. h is the current gap distance, which is calculated from the subtraction of the compressive displacement from h0.

2.1.3. Test results

Fig. 3 reports the stress-strain responses of three LFT specimens.

Finite element analysis (FEA) results are also displayed in Fig. 3, which will be discussed in detail in appendix A2.3. The reproducibility of the test results from the three specimens demonstrates the robustness and reliability of the presenting rheology experimentation. As can be seen in Fig. 3, all three graphs show fairly plateau phases after the initial rising.

Similar trends can be found in other researchers’ works [20-22]. In the initial rising phase, the material tends to resist against the compressive loading rather than flow. Therefore, the extensional viscosity of LFT is characterized over the plateau region where the molten charge flows

Fig. 3. Stress-strain response of LFT from the compression tests and finite element analysis.

along the mold gap. The measurement section is defined in the range of 0.2 to 1 (section between two pink dashed lines in Fig. 3) in terms of the compressive true strain. The average of the true stresses in the mea- surement window is calculated as σ_avg. Then, the extensional viscosity can be obtained from

η_E=σavg

ϵ˙ (2)

Fig. 4 shows the change of strain rates between the true strains of 0.2 and 1. Since h ˙is fixed during the rheological tests, the average strain rate is utilized to compute the extensional viscosity in Eq. (2). As a result, the extensional viscosity of LFT is measured to be 4.7 MPa•s. The mea- surement is in a similar range reported in other studies on similar discontinuous fiber-reinforced thermoplastic materials: 4 MPa•s (fiber volume fraction, VFfiber, 19% GF/PP) [24]-[25], 8 MPa•s (VFfiber 25%

GF/PP) [26], and 1.31 MPa•s (VFfiber 60% CF/PP) [11]. As previously mentioned, the shear viscosity of LFT is not within the scope of the current work since it is not as crucial as η_E for characterizing the plug-flow behavior of the fiber suspension considered here. Instead, the shear viscosity η_S is estimated based on the ratio between extensional and shear viscosity measured at various fiber volume fractions reported in [20]. In the present study, with the fiber volume fraction of 20%, η_S of LFT is determined to be one fiftieth of η_E according to [20].

2.2. Thermomechanical properties of WFT 2.2.1. Specimen preparation

The WFT material incorporates plain-weave E-glass fabric with areal weight of 290 g/m². The fiber volume fraction is measured to be 42.85%

from the matrix burn-out test per ASTM D2584. In the present WLFT

molding simulation, the WFT layer is modeled as a linear elastic mate- rial. Therefore, the orthotropic mechanical properties of WFT are measured at the process temperature of 200 ^◦C. Dog-bone shaped specimens as shown in Fig. 5 are cut from WFT panels of 1 mm thickness using a water jet. Specimens with 0-degree and 45-degree fiber orien- tations are prepared to measure the longitudinal and shear properties of the composite, respectively. Since there is no general standard for tensile testing of composite materials at elevated temperature, the specimen dimensions have been decided such that the gage width can accom- modate several textile patterns.

In the present study, two different strain measuring methods are employed for each of the two fiber orientations. For the 0^◦specimen, a simple and fast 2-point tracking technique is used to measure uniaxial strain only. Details of the 2-point tracking method is given in the appendix A1. For the 45^◦specimen, a digital image correlation (DIC) technique is utilized to measure both longitudinal and transverse strains.

The two measurement methods necessitate artificial patterns on the specimen surfaces. For the 0^◦specimen, two white dots are painted on the black specimen surface as shown in Fig. 5. A heat resistant paint (Dupli-color®, USA) is selected considering its durability at the process temperature. The paint is sprayed on the specimen masked with a perforated masking tape (see Fig. 5). On the other hand, a random speckle pattern is applied on the 45^◦specimen for DIC analysis using the same heat resistant paint.

2.2.2. Tensile test setup and procedure

Tensile tests are performed on the painted specimens using the same loading frame introduced in Section 2.1 at the extension rate of 2 mm/

min. The specimens are mounted with wedge grips that are made of vacuum heat-treated AISI H13 tool steel to minimize thermal expansion.

If any initial load is detected from the load cell due to the gripping mechanism, the crosshead position is adjusted until the unnecessary load approaches near 0 N. After the adjustment, the convection oven starts to heat up the specimen until the average temperature from the three thermocouples attached on the specimen surface (see Fig. 6) rea- ches the target temperature. The high oven temperature may develop compressive stress in the specimen due to thermal expansion while its both ends are clamped by the wedge grips. This thermal stress has been released before the tensile loading is applied, again by adjusting the crosshead until the load cell indicates near 0 Newton. After the loading is applied, load data is recorded from the load cell at the frequency of 10 Hz. A high-resolution digital single-lens reflex (DSLR) camera is utilized as shown in Fig. 6 to take sequential pictures of the specimen through the oven window in every single second. These images are processed to calculate strains. During the tensile testing, the oven temperature is controlled by the average temperature from the three locations of the specimen surface indicated in Fig. 6. As a result, the specimen temper- ature has been maintained in a fairly uniform range (200 ±1 ^◦C).

Fig. 4.Change of strain rates in the measurement range during the compres- sion test.

Fig. 5.Typical specimen configuration. Two white dots are painted for measuring uniaxial strain of the 0^◦specimen. For the 45^◦specimen, a speckle pattern is utilized for the measurement of shear strain.

2.2.3. Test results

The results of the tensile tests on the 0^◦ and 45^◦ specimens are summarized in Table 1. Two tests are performed per each measurement in Table 1. The tensile modulus E11( = E22) is calculated per ASTM D3039 based on the axial stress-axial strain curve of the 0^◦specimen.

The axial stress is defined as the load cell data divided by the initial cross-sectional area of the specimen. The axial strain of the 0^◦specimen is obtained from the 2-point tracking technique. For the shear modulus, the axial and transverse strains of the 45^◦specimen are first computed from DIC analysis. The strains and axial stress are then transformed per ASTM D3518 to obtain the shear stress-shear strain response. Table 1 also reports the mechanical properties of WFT measured at the room temperature for the sake of comparison.

3. Theoretical background

The hybrid composite considered in the present study (WLFT) is an intermediate material that do not require a typical overmolding process.

WLFT is a laminated sheet consisting of woven fabric fully impregnated with thermoplastic resin (WFT) and long fiber-reinforced thermoplastic (LFT). Since two distinct layers of WLFT exhibit different mechanical responses, different models should be integrated into a single numerical framework to simultaneously simulate the responses of the two phases during the CM process. In the present work, the LFT layers in the WLFT charge is modeled based on the rheological constitutive model. Then, the LFT model is integrated with the orthotropic linear elasticity model for WFT. In this section, for the sake of completeness of current work, we revisit the well-known models of Jeffrey [27], Tucker [28] and Fava- loro’s publications 25,29] regarding the orientation-dependent rheology of fiber suspensions.

3.1. Fiber reorientation model

When a single discontinuous fiber (or particle) is modeled as an axisymmetric rigid cylinder, its orientation can be described with a unit vector (p) defined along the longitudinal direction. Jeffery [27]

introduced the reorientation model of a single ellipsoid immersed in a viscous Newtonian fluid:

dp

dt =W⋅p+ξ[D⋅p− (p⋅D⋅p)p] (3)

where W and D are the rate tensors of rotation and deformation, respectively. ξ is the particle shape factor expressed as

ξ=a²− 1

a²+1 (4)

where a =l/d is the aspect ratio between the particle length (l) and diameter (d). For a continuous fiber, the aspect ratio is infinite and thus ξ =1. Although Jeffery’s model can describe the motion of a single fiber in the viscous fluid, it would be computationally burdensome to deal with multiple fibers simultaneously. For practical application of the reorientation model, Advani and Tucker [28] proposed the volume-averaged fiber orientation concept based on a probability dis- tribution function, Ψ. As shown in Fig. 7, when the orientation of a fiber Fig. 6. Typical experimental setup for tensile testing of WFT specimens at the process temperature.

Table 1

Mechanical properties of WFT at room and process temperatures.

Temperature E11 [GPa] E22 [GPa] G12 [GPa] ν₁₂

25 ^◦C 20 20 3.95 0.067

200 ^◦C 4 4 0.3 –

Fig. 7. Definition of the orientation vector p for a rigid discontinuous fiber.

is expressed with a polar angle θ and azimuthal angle φ in the spherical coordinate system, the probability of finding the fiber between angles [θ1,θ1 +dθ] and [φ1, φ₁ +dφ] is

P(θ1≤θ≤θ1+dθandφ₁≤φ≤φ₁+dφ) =Ψ(θ1,φ₁)sinθ1dθdφ (5) Therefore, the integration of the right hand side of Eq. (5) over the entire orientation space (surface of the unit sphere) should return unity:

∫^2π

φ=0

∫^π

θ=0

Ψ(θ,φ)sinθdθdφ=

∮_Ψ

(p)dp=1 (6)

Advani and Tucker defined a fiber orientation tensor A by integrating the dyadic product of fiber orientation vectors over the orientation space with the weighting function Ψ(p):

A=Aij=

∮

pipjΨ(p)dp (7)

If the probability is uniform in all directions, the fiber orientation tensor A in Eq. (7) can be rewritten as

A= 1 M

∑^M

α=1

pα⊗pα (8)

where M is the total number of fibers. The fiber orientation tensor A is symmetric and its trace is equal to unity.

The three diagonal components, A11, A22, and A33, conform to the directions of the 1, 2 and 3 axes of the Cartesian coordinate system in Fig. 7, respectively. The off-diagonal terms in A indicate how much the fibers are misaligned from the axes. For example, A =I/3 represents the isotropic fiber orientation state, where I is the identity matrix. By using this fiber orientation tensor, the reorientation model in Eq. (3) is modified as:

A˙ =W⋅A− A⋅W+ξ(D⋅A+A⋅D− 2A:D) (9) where A ˙ is the material time derivative of the second-order fiber orientation tensor and A is the fourth order fiber orientation tensor defined as A_ijkl =∮_pi

pjpkplΨ(p)dp. The fourth-order tensor is typically simplified through a closure approximation based on the second-order tensor A [28] [31]. In the present work, a quadratic closure approach (A=A⊗A) in [28] [30] is employed to predict the reorientation of fibers using Eq. (9).

3.2. Constitutive model for suspension with reorienting fibers

The effect of reinforcing fibers on suspension behavior has been vigorously studied for decades [32]-[35]. It is well known that fiber orientation greatly affects the rheological behavior of a suspension because flow viscosity is higher along the fiber-aligned direction. The flow of a fiber-filled suspension and its fiber orientation can be coupled through the fourth-order viscosity tensor, η. The viscosity tensor η and deformation rate tensor D yield the deviatoric stress τ of the suspension, which is mainly related to the flow:

τ=η:D (10)

For a transversely isotropic fluid with collimated fibers [28] [33], the fourth-order tensor η is fully defined with only three viscosity co- efficients: extensional viscosity in the fiber direction (η₁₁), longitudinal shear viscosity (η₁₂ =η₁₃), and transverse shear viscosity (η₂₃). The transverse extensional viscosity is given as η₂₂ =η₃₃ =4η₁₁η₂₃/(η₁₁ + η₂₃) owing to the isotropy in the 2–3 plane [25]. Practically, only the extensional and shear viscosities are sufficient to define η because the longitudinal and transverse shear viscosities are almost identical (η₁₂≅η₂₃) for typical fiber suspensions [25]. Now, by perturbing the fiber orientation of the transversely isotropic suspension, a fully aniso- tropic viscosity tensor can be obtained. In conjunction with the fiber

orientation tensors defined in the prior section, the anisotropic viscosity tensor is expressed as [28]

η_ijkl= (η_E− 3η_S)A_ijkl+ (

− 1 3η_E+η_S

)[

Aijδkl+Aklδij

]+ (1

9η_E− η_S )

δijδkl

+η_S^[δikδjl+δilδjk

]

(11) where η_E( =η₁₁) is the extensional viscosity in the fiber direction, and η_S is the shear viscosity, i.e., η₁₂ =η₁₃≅η₂₃.

Although the flow and fiber orientation can be coupled through the fourth-order viscosity tensor in Eq. (11) or other tensor-based ap- proaches [34, 35], it may lead to numerical difficulty in finding solu- tions for the tensor components of each element. Therefore, a simple scalar viscosity approach [25] [37] can be appropriate for practical applications. We resolve the issue here by implementing the informed isotropic (IISO) viscosity model proposed by Favaloro et al. [25] [29]

that couples the flow and fiber orientation through a scalar viscosity function. Using the IISO viscosity tensor, η_IISO, the deviatoric stress in Eq. (10) is now rewritten as

τ⁼2ηIISOD (12)

Equating the energy dissipation rate (τ: D) of the fully anisotropic relationship in Eq. (10) and the equivalent model in Eq. (12) yields η_IISO=D:η:D

2D:D =d:η:d (13)

where d= ̅̅̅̅̅̅̅̅^D

2D:D

√ is the deformation mode [25]. Eq. (13) is the general form of the informed isotropic viscosity. When the anisotropic viscosity tensor is defined as Eq. (11), Eq. (13) becomes

η_IISO=η_S+ (η_E− 3η_S)D:A:D

2D:D (14)

In Eq. (14), η_IISO is dependent on two viscosities (η_E and η_S), strain rate (D), and fiber orientation (A). For an isotropic fluid with the Trouton’s ratio of three (η_E =3η_S), the informed isotropic viscosity be- comes η_S, which is the lower bound. In order to fully define η_IISO of a heterogeneous suspension, both the extensional viscosity and shear viscosity need to be measured at a specific deformation rate through rheological tests as presented in Section 2.1.

The constitutive model of the LFT suspension in Eq. (12) is imple- mented in ABAQUS/Explicit through a user subroutine (VUMAT). The numerical implementation of the rheological model is explained in appendix A2.1 and the validity of the constitutive model of LFT is dis- cussed through simple anisotropic flow simulations in appendix A2.3.

4. Process simulation of a full-scale battery carrier

The presenting rheological model is employed to simulate the compression molding process for manufacturing a real-scale battery pack of an electric vehicle using the hybridly laminated WFLT material.

The battery pack is typically composed of several parts including the upper cover, lower carrier, and supporting frames. The presenting simulation focuses on the lower carrier with some insignificant struc- tural details simplified. The carrier part has a shell-type tray shape (1736 ×1240 ×73 mm³), of which geometrical features of curved surfaces and flanges are shown in Fig. 8. In the compression molding process, the upper male mold presses down the flat WLFT charge placed on the fixed lower female mold so that the WLFT sheet is formed ac- cording to the mold shape (see Fig. 9). The isothermal condition is assumed during the entire molding process to neglect the thermal interaction with the mold surface. The interaction between the WLFT charge and molds is assumed to be frictionless with hard contact to indirectly take the lubrication layer of LFT into account. While the no- slip boundary condition is often employed for fluid mechanics prob- lems without characterizing hydrodynamic friction, the perfect-slip

condition is appropriate for simulating the plug-flow behavior of the planar-oriented fiber suspension in the narrow mold gap under the isothermal condition [10, 11, 19]. The WLFT sheet consists of a 2 mm-thick WFT layer covered by two 1-mm-thick LFT layers. As can be seen in Fig. 9, the flat charge and two surface-type molds are initially in contact with each other with the initial mold gap of 74 mm, and the upper mold moves down 71 mm in the z-direction so that the final thickness of the formed charge (or under-cover part) becomes 3 mm.

In the present simulation, the surface-type upper and lower molds are modeled with Lagrangian discrete rigid type elements and the initial charge is discretized with 3D Lagrangian continuum elements (C3D8R).

Although the Eulerian approach is commonly accepted for simulating flow behavior, the flowing space (Eulerian domain) of a fluid (or ma- terial) should be additionally modeled and extremely fine meshing is required to achieve convergent and accurate solutions. According to the Abaqus guideline for the CEL approach [39], the Eulerian part should be

discretized with at least three elements along the direction of the smallest dimension. The smallest dimension of the lower housing is the sheet thickness (=3 mm), which is extremely smaller than other di- mensions (see Fig. 8). Compliance with the Abaqus instruction may result in billions of tiny elements for the discretization of the entire Eulerian domain. However, the present viscous constitutive model can be implemented into both Lagrangian (see appendix A2.2) and Eulerian (see appendix A2.3) frameworks. Furthermore, it was already shown that only FVM was sufficient in [25],[38] to simulate the molding pro- cess of discontinuous fiber reinforced composites. Therefore, the Lagrangian approach is employed here to simulate the LFT flow within a reasonable computational cost and number of elements. Total 196,752 C3D8R elements are used for the WLFT charge, and the average element size is 5 mm in plane and 1 mm in thickness.

The LFT layers of the WLFT charge is modeled with the present rheological model implemented into ABAQUS/Explicit through Fig. 8.Lower carrier of a battery pack (a) Outer dimensions (b) Section A: details on curved surfaces and flange.

Fig. 9. WLFT battery carrier model for compression molding simulation (Note: ‘t’ denotes thickness in millimeters.).

Fig. 10.Compression molding process of the battery carrier (Note: ‘h’ denotes the current mold gap distance which is 74 mm at the beginning and 3 mm at the end of molding. The upper mold is hidden and half of the model cut through xz-plane is displayed.).

Fig. 11. Fiber reorientation in the LFT layer from initial xy-planar isotropy orientation (Note: The product of the first eigenvector and eigenvalue of the fiber orientation tensor is assigned to a line segment. The color and length of a line segment indicate the degree of fiber alignment along the eigenvector direction: blue dot

~ red line).

Fig. 12. Predicted fiber orientation versus stress distributions in LFT layer (a) Fiber orientation. Note that rainbow-colored fibers in Fig. 11(f) are converted into uniform black color. (b) Contour plot of in-plane shear stress, σxy (c) Contour plot of in-plane transverse stress, σyy (d) Contour plot of in-plane longitudinal stress, σxx.

VUMAT. The rheological properties of the LFT layer measured at the process temperature 200 ^◦C are reported in Section 2.1. Since the fibers in the LFT layers cannot be oriented along the out-of-plane direction due to the geometrical constraints, the initial distribution of the long fibers are assumed to be isotopic in the xy-plane, such that

A=

⎡

⎣0.5 0 0 0 0.5 0

0 0 0

⎤

⎦ (15)

The WFT layer is modelled as an orthotropic linear elastic material, of which constitutive model is available in typical FEA software pack- ages. The thermomechanical properties of the WFT layer are listed in Table 1. The WFT and LFT layers are assumed to be perfectly bonded with each other during the compression molding. Table 2 reports the material properties of the WLFT charge used in the molding simulation.

Fig. 10 illustrates the deformation process of the WLFT charge during the compression molding simulation. The upper mold is hidden and xz- planar section (cut) view is presented in Fig. 10 to clearly visualize the deformed shape of the charge. The initially flat charge is mounted on the lower female mold and the upper male mold deforms the charge downward until the gap between the two molds reaches the target thickness of the carrier (=3 mm). During the molding process, wrinkling is observed around the charge edge as shown in Fig. 10(c) to (e). The wrinkles are flattened in the final stage of the molding process (Fig. 10 (e) and (f)) where the total thickness of the charge is reduced by 1 mm.

The final shape of the deformed charge conforms to the mold shape (or target dimension) without any discrepancy. The results are obtained without any excessively distorted elements during the entire forming process, implying the numerical stability of the presenting constitutive model based on the Lagrangian framework.

Fig. 11 illustrates reorienting fibers in the LFT layer during the compression molding process. The fiber orientation is computed at each element. Each fiber orientation is visualized through a line segment assigned with the product of the first eigenvector and eigenvalue of A at the corresponding element. The first eigenvector represents the orien- tation vector, and the eigenvalue indicates the degree of aligned fibers along the eigenvector direction (with the maximum value of 1). For example, A matrix of the xy-planar isotropy has 0.5 for the first and second eigenvalues and the corresponding eigenvectors are {1, 0, 0} and {0, 1,0}, respectively, while the third eigenvalue is zero. In the present work, since the fibers in the LFT layer can change their directions only in the xy-plane, the first eigenvalue is utilized as a magnitude of a fiber orientation vector at each element. Each vector is assigned to the line segment of the corresponding element, which is added to the result file at the end of analysis through a Python script. The line segments are visualized through the symbol plot function of the ABAQUS visualiza- tion module. The color and length of a line segment indicate the degree of average fiber alignment within an element; a red line signifies that fibers are fairly aligned in the direction of a line segment whereas a blue dot implies isotropic fiber distribution. Fig. 11(a) shows the initially isotropic fiber distribution in the xy-plane. As can be seen in Fig. 11(b) and (c), blue dots are still dominantly observed in the early stages of the

molding process. The line segments turn green in Fig. 11(d) and even- tually red in Fig. 11(f). The changes gradually appear around the charge edge, which forms the curved surface and flange area of the battery carrier.

The evolution of the fiber orientation in Fig. 11 and the charge deformation in Fig. 10 can be directly correlated. The flat charge in Fig. 10(a) is compressed and deformed into an concave-like shape until the mold gap of 38.5 mm. Detailed curvatures are partially formed from Fig. 10(d) when h =20.75 mm, which corresponds to the fiber orien- tation changes (blue to green and orange) in Fig. 11(d). During the CM process, the fiber suspension flow (or deform) along the mold surface and the long fibers would be collimated along the flow (or deformation) direction. After the mold gap reaches 38.5 mm, since the edge section of the LFT charge is gradually forming the curved surface and flange, the long fibers may be aligned along the valley of the mold surface and further along the flange.

Fig. 12 compares the predicted fiber orientation and the stress dis- tribution over the final molded part. The rainbow color of the line seg- ments in Fig. 11(f) are converted into uniform black to emphasize only the orientation, as shown in Fig. 12(a). Reoriented fibers are clearly observed at each corner highlighted with the pink dashed lines in Fig. 12 (a). The fiber alignment at the corners is directly associated with the high shear stresses displayed in Fig. 12(b). When fibers are aligned, the region should be much stiffer than other areas having randomly oriented fibers. Therefore, the corner sections in Fig. 12 exhibit higher resistance to the transverse directional expansion during the compression molding, which may act as barriers that impede the expansion of other softer areas. As can be seen in Fig. 12(c), the two vertical flange sections (highlighted with red dashed boxes) cannot freely expand vertically due to the upper and lower stiff corner sections, leading to the development of compressive stresses. Similarly, Fig. 12(d) shows compressive stresses in the horizontal flange areas. The predicted fiber orientation and molded shape can be utilized for the structural analysis of battery carrier by implementing the stiffness estimation methods for discontinuous fiber reinforced composites [40-42].

5. Conclusions

A process simulation model for compression molding of WLFT is developed in the present study. WLFT is a hybrid composite laminated with continuous (WLFT) and discontinuous (LFT) fiber-reinforced layers. WFT and LFT are characterized separately with their own constitutive models. For the LFT suspension, anisotropic flow behavior is fully coupled with the fiber reorientation model (sections 3 and appendix A2.1). The deviatoric behavior of LFT is governed by the effective scalar viscosity model, which can replace the fully anisotropic viscosity tensor with a less computational burden. The scalar viscosity model embeds the fiber orientation tensor, enabling the present model to predict the direction changes of the fibers in the whole domain caused by the suspension flow. The viscous constitutive model for LFT is veri- fied through the compression test on a single element, which also pro- vide the penalty stiffness that ensures the nearly incompressible Table 2

Simulation parameters.

Material Property Value Testing method

LFT Extensional viscosity (η_{E) 4.7 MPa}•s Section 2.1

Shear viscosity (η_S) 0.094 MPa•s (= 1

50η_E) [20]

Bulk modulus (κ) 88.7 MPa Appendix A2.2

Density (ρ_{LFT) 1.455} ×10⁹ ton/mm³

WFT Warp directional modulus (E11) 4 GPa ASTM D3039 (Section 2.2)

Weft directional modulus (E22) 4 GPa

Poisson’s ratio (ν_{12) 0.067}

Shear modulus (G12) 300 MPa ASTM D3518

Density (ρ_WFT) 1.73 ×10⁹ ton/mm³ ASTM D2584

behavior of the fiber suspension (appendix A2.2). It has also been demonstrated that the LFT model can be implemented in both Lagrangian (appendix A2.2) and Eulerian (appendix A2.3) frameworks.

The LFT model is integrated with the orthotropic linear elasticity model for WFT to simulate the WLFT molding process (Section 4). Rheological properties of LFT, necessary for the process simulation, are measured through a customized rheometer (Section 2.1) based on a compression test. The extensional viscosity of LFT is measured at the process tem- perature and the repetitive test results show excellent reproducibility.

Thermomechanical properties of WFT, also required for the simulation, are measured from tensile tests at room and process temperatures (Section 2.2). The compression molding simulation of the actual scale of the EV battery carrier is demonstrated using the Lagrangian approach. It is shown that the reorienting fiber directions during the molding process can be predicted and clearly visualized. In the initial stage of the process, the reorientation behavior of the long fibers is greatly influenced by the flow (or deformation) of the LFT charge. The reoriented fibers mutually

affect the deforming behavior and result in stress concentrated regions in the final stage.

Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This research work is supported by Kwangsung Corporation, Ltd. The authors are also grateful for the financial support through “Overseas order-linked aviation parts industry process technology development in 2019^′′of the Korea Institute for Advancement of Technology (KIAT), granted from the Ministry of Trade, Industry and Energy, Republic of Korea.

Appendix

A1. Image-based strain measuring method for thermomechanical tests

Measuring strains at elevated temperature has been a challenging issue in the field of material testing. Special strain gages are available in the heating environment, but they require a special installation technique such as heat curing, which may damage the polymeric specimen even before testing. An extensometer can be another option for measuring strains, but it requires an application hole on the furnace wall, which is extremely undesirable for heat insulation. Furthermore, multiple devices are required to simultaneously measure axial and transverse strains, which might be burdensome. In this research, two types of image-based methods are utilized to measure the strains of specimens having different fiber orientations.

Firstly, the well-established digital image correlation (DIC) method is employed. The DIC technique can measure real-time full-field data pertaining to displacement and strain [43]. However, its computation burden is quite large, and applying a speckle pattern is often cumbersome. A simple yet robust strain measurement technique is more desirable when a single axial strain value is sufficient. For this reason, a simple image-based strain measurement method named as a “2-point tracking technique” has been developed particularly for axial strain.

The core working principle of the 2-point tracking technique is tracking two dots that are painted on a specimen surface. Note that the DIC technique tracks small areas, i.e., facets, consisting of randomly-distributed numerous tiny dots whereas this method traces the central coordinates of the two dots only. The first step of this approach starts from applying two dots on the specimen surface with high contrast to background as shown in Fig. 5. The line connecting the centers of the two circles should be parallel to the strain measuring direction. Sequential images are acquired during tensile testing and they are binarized by the in-house algorithm with a user-defined greyscale threshold. The threshold is recommended to be the local minimum point of a bimodal curve from the histogram of an image. In the next step, the algorithm removes any undesirable pixel found inside the dots but filled with the color of the surrounding, using the morphological reconstruction function of Matlab, imfill. Such an artifact may be caused by a lighting condition or during the binarization process. The undesirable pixels cause an error when the algorithm computes the central coordinates of the tracing circles. In the algorithm, the central coordinates are identified through regionprops of Matlab, which finds a centroid of a mono-colored area. Finally, the axial strain is calculated from the changes in the distance between the centers of the two circular dots. The accuracy of the present strain-measuring technique is validated through tensile tests on aluminum specimens carried out per ASTM B557. The axial strain measured from the present method is in excellent agreement with the benchmark data obtained using strain gages as well as the DIC technique.

A2. Numerical implementation

A2.1. Constitutive model for LFT suspension

The constitutive model of the LFT suspension in Eq. (12) is implemented in ABAQUS/Explicit through a user subroutine (VUMAT). The modified fiber reorientation model in Eq. (9) and the orientation-dependent IISO viscosity in Eq.(14) are updated using D and W that are calculated at each time increment. VUMAT passes in deformation gradient tensors at the beginning and the end of each time increment; the change of the deformation gradients over the time increment (dt) becomes the time-derivative of the deformation gradient, F˙. Now, the velocity gradient, L, is computed from L

=FF˙ ⁻¹, where F⁻¹ is the inverse of the deformation gradient at the end of the increment. Finally, the deformation rate and rotation rate tenors are calculated from D=¹₂(L+L^T)and W =¹₂(L− L^T). The rate tensors are then used to compute the time-derivative of the fiber orientation tensor A in ˙ Eq. (9) with A at the beginning of the time increment. The fiber orientation tensor at the end of the increment is updated by A+Ad˙ t. In a similar manner, the informed isotropic viscosity, η_IISO, in Eq.(14) is updated from the fiber orientation and deformation gradient tensors at each time step.

The total stress tensor, σ, of the LFT suspension can be obtained by adding volumetric stress to the deviatoric stress, τ, defined in Eq. (12):

σ= − pI+τ (16)

where the volumetric stress that the fiber suspension endures is represented by the hydrostatic pressure, p, and the identity matrix, I. In the present work, the fiber suspension is modeled as a nearly incompressible fluid and the hydrostatic pressure enforces the near-incompressibility on material’s deformation. In this case, the hydrostatic pressure can be defined as

p= − κ(det(F) − 1) (17)

where the bulk modulus, κ, is employed as a numerical penalty stiffness to control the deformation. Sommer et al. implemented a similar viscous constitutive model in the ABAQUS implicit solver using the incompressible element formulation, in which the bulk modulus is intrinsically defined [10]. In their next work [11], the authors performed a simple tension test on a single element with fibers aligned in the loading direction. They compared the numerical results against analytical solutions with various penalty stiffness values. It was shown that the nearly incompressible model can successfully simulate full incompressibility in terms of the hydrostatic pressure when the bulk modulus is one order greater than the maximum deviatoric stress experienced by the model. Indeed, it is sufficient if the bulk modulus exceeds a certain threshold since κ is a penalty stiffness helping the material to stably deform under near-incompressibility (as like the bulk modulus is infinite in the case of full incompressibility).

A2.2. Validation of near incompressibility: Lagrangian approach

A simple compression test on a single 3D Lagrangian continuum element (C3D8R) is performed to validate the numerical model for the LFT suspension. General boundary conditions for the single-element model is illustrated in Fig. A1. The viscous properties measured through the rheo- logical tests presented in Section 2.1 are utilized to construct the IISO viscosity in Eq. (13). Because the fibers cannot be oriented along the out-of-plane direction in the WLFT plate due to the geometrical constraints, the initial fiber orientation is assumed to be random in the 1–3 plane. The corre- sponding fiber orientation tensor is given as

A=

⎡

⎣0.5 0 0

0 0 0

0 0 0.5

⎤

⎦ (18)

These material properties are also applied in the full-scale compression molding simulation in Section 4. Since the ultimate goal of the presenting research is the simulation of the CM process at the scale of actual products, the element-scale test is performed under equivalent compressive loading conditions.

In the uniaxial compressive loading with the simply supported boundary condition, the material should expand freely in two transverse directions (x and z directions in Fig. A1) due to traction-free conditions. Therefore, the total stresses in the transverse directions, σ₁₁ and σ₃₃, should be ideally zero for an incompressible material, i.e., the hydrostatic pressure, p, and deviatoric stress components, τ₁₁ and τ₃₃, should be identical. Since there may exist small discrepancy between τ₁₁ and τ₃₃ due to round-off errors, their average value is utilized to compute the “adaptive κ” at each increment step, which is required to enforce the near-incompressibility on the element deformation:

κ= − p

det(F) − 1= − (τ11+τ33)/2

det(F) − 1 (19)

In this way, at each deformed state, the bulk modulus that satisfies the incompressibility can be updated during the compression process. As shown in Fig. A2, the unit element is expanded along the two transverse directions in a near-incompressible manner during 0.5 mm compression. If fully incompressible, the element volume should be maintained even after the deformation, and the maximum displacement should be ideally 0.414 mm ( =

̅̅̅2

√ − 1), which is comparable to the results in Fig. A2. The DENSITY variable of Abaqus is tracked during element deformation and the deviation from initial density is 0.85% reduction, which can validate the quasi-incompressible behavior. The stress response of the “adaptive κ” method is presented in Fig. A3(a): the transverse stresses, σ₁₁ and σ₃₃, remains almost zero while the loading directional stress, σ₂₂, and hydrostatic pressure, p, increase monotonically as the element deforms. As can be seen in Fig. A3(b), the bulk modulus calculated from Eq. (19) shows fairly increasing trend during the compressive deformation. These results highlight the effectiveness of the “adaptive κ” approach, which actively satisfies the near-incompressibility condition during the element deformation.

Notwithstanding its effectiveness of the “adaptive κ” approach, it may not be practical to compute κ at every time increment during the process simulation at the product scale. In the full-scale simulation, the traction boundary conditions are complex and they are not identically applied to numerous elements. Instead of computing κ for each element along its deformation, a fixed representative κ value is utilized uniformly for every element in the product scale simulation. The representative κ value is determined from the average of the bulk modulus that changes with the element deformation of a simple compression model with “adaptive κ” method in Fig. A1. The validity of a fixed κ value is demonstrated through the same single-element model. The fixed bulk modulus of 88.7 MPa is calculated by averaging κ in Fig. A3(b), and the hydrostatic pressure is now calculated using Eq. (17) to enforce near-incompressibility. As shown in Fig. A4(a), the transverse stress components are almost zero while the loading directional component, σ₂₂, and the hydrostatic pressure monotonically increase. They are almost identical to the results of the “adaptive κ” case as shown in Fig. A4(b).

In appendix A2.4, it is shown that the average κ value determined from a single element model can be applicable to the compression model with multiple elements. Since bulk modulus determined from the presenting approach is sensitive to boundary conditions and material properties (e.g., strain rate, fiber orientation, density), all the analysis conditions of the single-element model should be identical or equivalent to the full-scale process simulation model. Since the hydrostatic pressure guides the element deformation, a bulk modulus determined from ill-treated condition will violate the near-incompressibility, leading to invalid element deformation. In summary, the bulk modulus of the viscous constitutive model in Eq. (17) can be determined by averaging κ values computed from the “adaptive κ” approach using a single-element model subjected to equivalent process conditions of a full-scale product.

Fig. A1.Single-element compression model for κ characterization.

Fig. A2. Results of the single-element compression model based on the “adaptive κ” approach: 0.5 mm compression along y-direction. (a) x-directional displacement (b) z-directional displacement (Note: Initial and deformed shapes of the single-element model are overlaid.).

Fig. A3. Mechanical behavior of the single element based on the “adaptive κ” approach (a) stress responses: FEA vs. analytical solution (b) change of the bulk modulus.

A2.3. Validation of anisotropic flow: Eulerian approach

The rheological model for the LFT material is validated through anisotropic flow behavior of a circular sample as illustrated in Fig. A5 Herein, the coupled Eulerian Lagrangian (CEL) approach available in ABAQUS/Explicit is employed to simulate the compression process of the cylindrical LFT charge. The CEL approach has been widely used to model the molding process of LFT or SMC materials [6,12,44]. In the presenting work, the upper and lower molds are modeled with Lagrangian discrete rigid elements while the flowing space of the initial charge (red dashed box in Fig. A5) is discretized with Eulerian elements based on the finite volume method (FVM). The initial charge is molten polypropylene reinforced with a long glass fiber. The viscous properties at 200 ^◦C reported in Section 2.1 are utilized for the validation. The interaction between the charge and molds is considered frictionless with hard contact. The lower mold is fixed and the upper mold compresses down the LFT charge along the z-direction for 10 s, and the circular charge flows over the xy-plane as shown in Fig. A5. Three samples having initially different fiber alignment are tested. Fig. A6 compares the flow front shapes of the three charges after the 10-second compression. Fig. A6(a) shows the result when the fibers are initially aligned along the x-direction (A_xx =1 and others=0). In this case, the LFT suspension flows more rapidly in the y-direction than in the x-direction due to the higher viscous resistance resulting from the fibers aligned in the x-direction. Therefore, the flow front becomes the elliptical shape with the major axis conforming to the y-axis as shown in Fig. A6(a). The similar elliptical shape is observed from the y-directional alignment case (Ayy =1, Axx =Azz = 0) in Fig. A6(b). On the other hand, the initial xy planar isotropy (A_xx = A_yy =0.5, A_zz =0) leads to the circular flow front as shown in Fig. A6(c).

These results demonstrate the capability of the presenting rheological constitutive model for simulating both isotropic and anisotropic flows taking account of the initial fiber orientation. The validation results here are also consistent with the similar simulation and experiment results of the center-gated disk reported in [25] and [45].

The CEL model in Fig. A5 is utilized to validate the present rheological model with the test data in Fig. 3. For the validation, the dimensions of the CEL model in Fig. A5 are changed to those of the actual specimen and the disk fixture in Section 2.1.2. The initial fiber orientation of the LFT charge is assumed to be xy-planar isotropy. The materials properties of the LFT material reported in Table 2 are used except the bulk modulus. Because the bulk Fig. A4. Results of the single-element compression model using a fixed bulk modulus value (88.7 MPa) determined from the average of the “adaptive κ” approach (a) stress responses of the model (b) comparison with the “adaptive κ” approach in Fig. A3(a).

Fig. A5.CEL model for compression molding of a cylindrical LFT charge.

modulus in Table 2 is obtained using the CM process condition, the representative bulk modulus for the validation model is determined based on the loading rate of the actual compression test (0.1 mm/s) in Section 2.1.2 using the approach described in appendix A2.2. The interaction between the LFT charge and molds is assumed to be frictionless with hard contact following the actual interfacial condition of the PTFE film (see Section 2.1.2). The loading directional reaction force and displacement are obtained from the rigid upper mold during the compression simulation to compute stresses and strains. As shown in Fig. 3, the FEA result is in good agreement with the three test data. The stress computed from the FE model, however, slightly decreases in the plateau region, probably because of using the fixed bulk modulus while the resistance of the specimens to compression increases after they are excessively compressed. This may also cause a numerical issue and the early termination of the FEA result. The accuracy of the FEA result would be improved if the “adaptive κ” is implemented at the cost of enormous computation time. Nevertheless, the efficient “average κ” approach effectively reproduces the experimental stress-strain curves in the region where the viscosity is defined.

A2.4. Validation of near incompressibility using multiple elements

The validation presented in appendix A2.2 is extended to multiple elements here. The main purpose of this section is to demonstrate whether the average κ value obtained from the “adaptive κ” approach on the single-element model can be applied to the multi-element model. The inputs for each element remain the same for the single-element model in appendix A2.2: the viscous properties measured through the rheological tests in Section 2.1 and the initial fiber orientation in Eq. (18) are employed. General boundary conditions for the 25-element model is illustrated in Fig. A7. A fixed bulk modulus of 88.7 MPa determined from the single-element model using the “adaptive κ” approach in appendix A2.2 is used for all the 25 elements. The total stress and hydrostatic pressure are calculated from Eq. (16) and Eq. (17), respectively. As shown in Fig. A8, the 25-element model is transversely expanded during 0.5 mm compression. If fully incompressible, the transverse extensional displacements of the whole model should be 2.07 mm (≈5 ̅̅̅

√2

− 5)to maintain the initial volume. The ideal displacement is comparable to the maximum U1 and U3 in Fig. A8, implying that the 25-element model deforms in a nearly incompressible manner.

The loading directional total stress (σ₂₂) and the hydrostatic pressure (p) of the 25 elements are individually compared in Fig. A9. The element numbers in the legend of Fig. A9 match with the element numbers marked in Fig. A8. The stress responses of all the elements are fairly consistent with each other. Although each 25 element has different boundary conditions compared to the single-element model in appendix A2.2, the global boundary conditions are all the same and, thus, the average κ value determined from the single-element model is applicable. The stress components of the 25 elements are averaged and compared in Fig. A10(a). σ₂₂ and p of each element monotonically increase while other total stress components are almost zero, similarly with the results in Fig. A4(a). As can be seen in Fig. A10(b), the axial stress and hydrostatic pressure between the multi-element and single-element models are consistent. This comparison results demonstrate that the average bulk modulus can be determined from the single-element

“adaptive κ” model and the κ value is directly applicable to the multi-element model and further to the real part-scale model.

Fig. A6. Flow front vs. initial fiber alignment (a) x-directional alignment (b) y-directional alignment (c) xy-planar isotropy (Note: Flowing process of a charge is visualized through overlaid plots with different colors. For example, the red disk indicates the initial charge and the green represents the final deformed shape.).

Fig. A7. Multi-element compression model using a fixed bulk modulus, which is determined from the average of “adaptive κ” approach implemented on the single element model.

Fig. A8. FE analysis result of the multi-element compression model: 0.5 mm compression in the y-direction. (a) Transverse expansion in the x-direction (b) Transverse expansion in the z-direction (Note: Superimpose plot of undeformed and deformed shapes).

Fig. A9. Mechanical response of each 25 elements under compressive loading (a) Loading directional axial stress, σ₂₂ (b) Hydrostatic pressure, p (Note: The numbers in the legend correspond to the element numbers marked in Fig. A8).

Fig. A10. Averaged mechanical response of the multi-element compression model (a) Average stress components of 25 elements (b) Model validation through comparison with the single-element model in Fig. A4(a).