# Simulation of Stress Distribution near Weld Line in the Viscoelastic Melt Mold Filling Process

Authors: Binxin Yang; Jie Ouyang; Fang Wang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Simulations of interface evolution and stress distribution near weld line in the viscoelastic melt mold filling process are achieved according to the viscoelastic-Newtonian two-phase model. The finite volume methods on nonstaggered grids are used to solve the model. The level set method is used to capture the melt interface. The interface evolution of the viscoelastic melt in the mold filling process with an insert in is captured accurately and compared with the result obtained in the experiment. Numerical results show that the stress distribution is anisotropic near the weld line district and the stress distribution varies greatly at different positions of the weld line district due to the complicated flow behavior after the two streams of melt meet. The stress increases quickly near the weld line district and then decreases gradually until reaching the tail of the mold cavity. The maximum value of the stress appears at some point after the insert.


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## Body

## 1. Introduction

The plastic mold filling process produces large numbers of parts of high quality. Plastic material in the form of granules is melted until it is soft enough to be injected under pressure to fill a mold. Early simulations of mold filling process mostly used the Hele-Shaw model coupled with the finite element method, which is based on the creeping flow lubrication model [1–4]. With the development of computer hardware, 3D simulations of mold filling process have been realized by using Navier-Stokes equations and different numerical methods [5–10]. The papers mentioned above studied the mold filling process without the consideration of the interface motion. The development of the interface capturing or tracking techniques, such as volume of fluid method (VOF) and the level set method has propelled greatly the development of mold filling simulation techniques. Many papers studying mold filling process coupled with interface tracking techniques can be found [11–20]. In these papers, the viscoelastic properties of materials were ignored. However, the melt for mold filling process is often viscoelastic materials. Some papers made a study on mold filling problems with viscoelastic free surfaces [21, 22]. However, these papers studied the problem with only viscoelastic fluid phase considered and the gas phase in the cavity ignored, in which case complex boundary conditions must be properly dealt with. Yang et al. [23] proposed a model for mold filling process in which the governing equations for the viscoelastic fluid (melt phase) and the Newtonian fluid (gas phase) are successfully united into a system of generalized Navier-Stokes equations, avoiding dealing with complex boundary conditions.The viscoelastic behaviour in mold filling process has been tested in [23], in which the die swelling phenomenon and the influences of elasticity and viscosity on velocity, stresses, pressure, stretch, and the first normal-stress difference have been discussed in detail. However, it is well known that weld line is unavoidable in most products of even moderate complexity by mold filling process and influences weightily the quality of the products and the stress distribution near the weld line influences the mechanical property of the products greatly. This paper uses the viscoelastic-Newtonian two-phase flow model established in [23] and finite volume method on nonstaggered grids to study the mold filling process with an insert in the cavity and analyze the stress distribution near the weld line. The comparison between the numerical results and those obtained from experiments shows the correctness of the model and the numerical methods.


## 2. Mathematical Model (see [23])

### 2.1. Interface Capturing Equations

We use the corrected level set method proposed by Sussman et al. [27] to capture the interface. The level set and its reinitialization equations are described as follows:
(1)∂φ∂t+u·∇φ=0(2)∂φ∂tr+sign(φ0)(|∇φ|-1)=ωδε(φ)|∇φ|,φ(x,y,0)=φ0(x,y),whereφ is the interface, u=(u,v) is the velocity vector of the melt, t is the time, ω is the weight coefficient, tr is a pseudo time, and  sign(φ0) is the sign function of φ which is defined as
(3)sign(φ0)=φ0φ02+[min(Δx,Δy)]2.Here,Δx and Δy are the grid widths along x and y direction, respectively, and [min(Δx,Δy)]2 is used to avoid denominator’s dividing by zero. δε(φ) is the Dirac function defined as
(4)δε(φ)={12ε(1+cos(πφε))|φ|<ε,0otherwise.Here,ε is a small positive number about a grid width. See Sussman et al. [27] for more details.


### 2.2. Governing Equations for Flow Field

The governing equations for the flow field with the consideration of fibers are given as follows.Continuity(5)∂ρ∂t+∂u∂x+∂v∂y=0,u-momentum
(6)∂(ρu)∂t+∂(ρuu)∂x+∂(ρvu)∂y-1Re(∂2(μu)∂x2+∂2(μu)∂y2)=-∂p∂xHε(φ)+(β-1)Re(∂2(μu)∂x2+∂2(μu)∂y2)Hε(φ)+1Re∂τxx∂xHε(φ)+1Re∂τxy∂yHε(φ),v-momentum
(7)∂(ρv)∂t+∂(ρuv)∂x+∂(ρvv)∂y-1Re(∂2(μv)∂x2+∂2(μv)∂y2)=-∂p∂yHε(φ)+(β-1)Re(∂2(μv)∂x2+∂2(μv)∂y2)Hε(φ)+1Re∂τyx∂xHε(φ)+1Re∂τyy∂yHε(φ),where the Reynolds numberRe=ρlLU/μl, ρ(φ)=ξ+(1-ξ)Hε(φ), μ(φ)=η+(1-η)Hε(φ), ξ=ρg/ρl, η=μg/μl, β is the ratio of the Newtonian viscosity and the total viscosity. Since isothermal mold filling process is considered here, β is a constant. The subscripts l and g denote the liquid phase and the gas phase,  respectively, and L and U are parameters for nondimensionalization.Constitutive(8)ϖ∂ψ∂t+∇·(ϖuψ)-∇·(Λ∇ψ)=Sψ.Here, the extended Pom-Pom (XPP) constitutive equation developed by Verbeeten et al. [28] is used as the constitutive relationship. The constants and functions in (8) are defined in Table 1 [24, 25], where f(λ,τ)=2λ0b/λ0seν(λ-1)(1-1/λ)+1/λ2[1-αIτ·τ/3G02], λ=1+|Iτ|/3G0, ν=2/q, and the Weissenberg number is defined as We=λ0bU/L. Here λ is the backbone stretch used to represent the stretched degree of the polymer molecule, λ0b and λ0s denote the orientation and backbone stretch relaxation time scales of the polymer chains, respectively, G0 is the linear relaxation modulus, α is an adjustable parameter, which controls the anisotropic drag, I is the identity tensor, q is the number of arms of polymer chains and d is the strain tensor. Some hints on choosing meaningful values of the parameters in XPP model can be found in [25]; in this paper, we take β=1.0/9.0, α=0.15, q=2.0, ε=1.0/3.0.Table 1
Definition of the constants and functions in the constitutive equation [24, 25].
Equation
ϖ
ψ
Λ
S
ψ
τ
x
x normal stress
We
τ
x
x
0
2
(
1
-
β
)
∂
u
∂
x
+
2
We
τ
x
x
∂
u
∂
x
+
2
We
τ
x
y
∂
u
∂
y
-
f
(
λ
,
τ
)
τ
x
x
-
[
f
(
λ
,
τ
)
-
1
]
1
-
β
We
-
α
We
1
-
β
(
τ
x
x
2
+
τ
x
y
2
)
τ
x
y shear stress
We
τ
x
y
0
(
1
-
β
)
(
∂
v
∂
x
+
∂
u
∂
y
)
+
We
τ
x
x
∂
v
∂
x
+
We
τ
y
y
∂
u
∂
y
-
f
(
λ
,
τ
)
τ
x
y
-
α
We
1
-
β
τ
x
y
(
τ
x
x
+
τ
y
y
)
τ
y
y normal stress
We
τ
y
y
0
2
(
1
-
β
)
∂
v
∂
y
+
2
We
τ
y
y
∂
v
∂
y
+
2
We
τ
x
y
∂
v
∂
x
-
f
(
λ
,
τ
)
τ
y
y
-
[
f
(
λ
,
τ
)
-
1
]
1
-
β
We
-
α
We
1
-
β
(
τ
y
y
2
+
τ
x
y
2
)
τ
z
z stress
We
τ
z
z
0
-
f
(
λ
,
τ
)
τ
z
z
-
[
f
(
λ
,
τ
)
-
1
]
1
-
β
We
-
α
We
1
-
β
τ
z
z
2


### 2.3. Boundary Conditions

Proper boundary conditions must be posed on the solid walls of the cavity. In this paper, no-slip boundary conditions are used for the velocities, that is,u=v=0. As for the pressure boundary conditions, for the air in the cavity, we use no-slip boundary conditions, that is, p=0, while for the melt in the cavity, no-penetration boundary conditions are used, that is, ∂p/∂n=0.


## 2.1. Interface Capturing Equations

We use the corrected level set method proposed by Sussman et al. [27] to capture the interface. The level set and its reinitialization equations are described as follows:
(1)∂φ∂t+u·∇φ=0(2)∂φ∂tr+sign(φ0)(|∇φ|-1)=ωδε(φ)|∇φ|,φ(x,y,0)=φ0(x,y),whereφ is the interface, u=(u,v) is the velocity vector of the melt, t is the time, ω is the weight coefficient, tr is a pseudo time, and  sign(φ0) is the sign function of φ which is defined as
(3)sign(φ0)=φ0φ02+[min(Δx,Δy)]2.Here,Δx and Δy are the grid widths along x and y direction, respectively, and [min(Δx,Δy)]2 is used to avoid denominator’s dividing by zero. δε(φ) is the Dirac function defined as
(4)δε(φ)={12ε(1+cos(πφε))|φ|<ε,0otherwise.Here,ε is a small positive number about a grid width. See Sussman et al. [27] for more details.


## 2.2. Governing Equations for Flow Field

The governing equations for the flow field with the consideration of fibers are given as follows.Continuity(5)∂ρ∂t+∂u∂x+∂v∂y=0,u-momentum
(6)∂(ρu)∂t+∂(ρuu)∂x+∂(ρvu)∂y-1Re(∂2(μu)∂x2+∂2(μu)∂y2)=-∂p∂xHε(φ)+(β-1)Re(∂2(μu)∂x2+∂2(μu)∂y2)Hε(φ)+1Re∂τxx∂xHε(φ)+1Re∂τxy∂yHε(φ),v-momentum
(7)∂(ρv)∂t+∂(ρuv)∂x+∂(ρvv)∂y-1Re(∂2(μv)∂x2+∂2(μv)∂y2)=-∂p∂yHε(φ)+(β-1)Re(∂2(μv)∂x2+∂2(μv)∂y2)Hε(φ)+1Re∂τyx∂xHε(φ)+1Re∂τyy∂yHε(φ),where the Reynolds numberRe=ρlLU/μl, ρ(φ)=ξ+(1-ξ)Hε(φ), μ(φ)=η+(1-η)Hε(φ), ξ=ρg/ρl, η=μg/μl, β is the ratio of the Newtonian viscosity and the total viscosity. Since isothermal mold filling process is considered here, β is a constant. The subscripts l and g denote the liquid phase and the gas phase,  respectively, and L and U are parameters for nondimensionalization.Constitutive(8)ϖ∂ψ∂t+∇·(ϖuψ)-∇·(Λ∇ψ)=Sψ.Here, the extended Pom-Pom (XPP) constitutive equation developed by Verbeeten et al. [28] is used as the constitutive relationship. The constants and functions in (8) are defined in Table 1 [24, 25], where f(λ,τ)=2λ0b/λ0seν(λ-1)(1-1/λ)+1/λ2[1-αIτ·τ/3G02], λ=1+|Iτ|/3G0, ν=2/q, and the Weissenberg number is defined as We=λ0bU/L. Here λ is the backbone stretch used to represent the stretched degree of the polymer molecule, λ0b and λ0s denote the orientation and backbone stretch relaxation time scales of the polymer chains, respectively, G0 is the linear relaxation modulus, α is an adjustable parameter, which controls the anisotropic drag, I is the identity tensor, q is the number of arms of polymer chains and d is the strain tensor. Some hints on choosing meaningful values of the parameters in XPP model can be found in [25]; in this paper, we take β=1.0/9.0, α=0.15, q=2.0, ε=1.0/3.0.Table 1
Definition of the constants and functions in the constitutive equation [24, 25].
Equation
ϖ
ψ
Λ
S
ψ
τ
x
x normal stress
We
τ
x
x
0
2
(
1
-
β
)
∂
u
∂
x
+
2
We
τ
x
x
∂
u
∂
x
+
2
We
τ
x
y
∂
u
∂
y
-
f
(
λ
,
τ
)
τ
x
x
-
[
f
(
λ
,
τ
)
-
1
]
1
-
β
We
-
α
We
1
-
β
(
τ
x
x
2
+
τ
x
y
2
)
τ
x
y shear stress
We
τ
x
y
0
(
1
-
β
)
(
∂
v
∂
x
+
∂
u
∂
y
)
+
We
τ
x
x
∂
v
∂
x
+
We
τ
y
y
∂
u
∂
y
-
f
(
λ
,
τ
)
τ
x
y
-
α
We
1
-
β
τ
x
y
(
τ
x
x
+
τ
y
y
)
τ
y
y normal stress
We
τ
y
y
0
2
(
1
-
β
)
∂
v
∂
y
+
2
We
τ
y
y
∂
v
∂
y
+
2
We
τ
x
y
∂
v
∂
x
-
f
(
λ
,
τ
)
τ
y
y
-
[
f
(
λ
,
τ
)
-
1
]
1
-
β
We
-
α
We
1
-
β
(
τ
y
y
2
+
τ
x
y
2
)
τ
z
z stress
We
τ
z
z
0
-
f
(
λ
,
τ
)
τ
z
z
-
[
f
(
λ
,
τ
)
-
1
]
1
-
β
We
-
α
We
1
-
β
τ
z
z
2


## 2.3. Boundary Conditions

Proper boundary conditions must be posed on the solid walls of the cavity. In this paper, no-slip boundary conditions are used for the velocities, that is,u=v=0. As for the pressure boundary conditions, for the air in the cavity, we use no-slip boundary conditions, that is, p=0, while for the melt in the cavity, no-penetration boundary conditions are used, that is, ∂p/∂n=0.


## 3. Numerical Methods

Level set evolution equation (1) and the reinitialization equation (2) are solved by the finite difference method on a rectangular grid. The spatial derivatives are discretized by the 5th-order weighted essentially non-oscillatory (WENO) scheme [29, 30] and the temporal derivatives are discretized by the 3rd-order total variation diminishing Runge-Kutta (TVD-R-K) scheme [31].The finite volume SIMPLE methods on a nonstaggered grid are used to solve the governing equations (5)–(8).The validity of the methods has been verified in [23].


## 4. Numerical Results and Analysis

### 4.1. Mold Filling Process

Figure1 shows the sketch map of the mold with an insert in. Figure 2 gives the top view of the mold, which also represents the computational area. In this paper, we take x=16.2, y=11.6, a=3.3, b=3.4, c=4.9, d=1.8, which are the same as those in [26]. The inlet lies in the middle of the left wall of the computational area. The grid number is 200×120.Figure 1
Sketch map of the mold with an insert in.Figure 2
Computational domain of the mold.Figure3 gives the melt interface positions at different times in mold filling process.Melt positions at different time in the mold filling process.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)From Figure3 we can see that the melt is divided into two streams when it reaches the insert. The two streams move forward until they pass the insert; then they begin to move toward each other until they meet at some point behind the insert, from where the weld line begins to form, and a hole appears between the insert and the meet point of the two streams of melt. After meeting, the melt moves toward two opposite directions. One direction is moving forward and  the other direction is moving backward to fill in the hole formed between the insert and the meeting point. The hole is finally filled with melt and two streams of melt reunion into one stream and move forward. Figure 4 gives the comparison between the interface evolution after the insert and that obtained in  te experiment [26]. The well agreement shows the validity of our model and method.Comparison between the interface evolution after the insert and that obtained in experiment.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)


### 4.2. Distribution of the Stress Birefringence

In order to get the stress distribution and make a comparison with the experimental results given by the stress birefringence distribution, we use the formula in [32] to compute the numerical stress birefringence, which is given as follows:
(9)Δn=C[(τxx-τyy)2+4τxy2]1/2.Here,Δn is the stress birefringence. C is the stress-optical coefficient, which is a constant for linear stress-optical principle, and we take C=1.Figure5 shows the distribution of the stress birefringence at t=1.05 in the simulation. We can see that the stress birefringence distribution is anisotropic near the weld line district and varies greatly at different positions of the weld line district. This phenomenon is induced by the complicated flow behavior after the two streams of melt meet. Since the melt moves toward two opposite directions, the stress birefringence distribution decreases progressively along the two opposite directions. The stress distribution birefringence obtained in experiment after the product is completely produced is given in Figure 6 [26], Since only the mold filling process is considered in the simulation, while the birefringence is obtained after the cooling stage, some difference exists between the numerical and experimental results. However, both the numerical and the experimental results are qualitative agreeable.Figure 5
The distribution of the stress birefringence att=1.05 in the simulation.Figure 6
The stress distribution birefringence obtained in experiment [26] after the product is completely produced.Figure7 gives the change of the stress birefringence from the tail of the insert until the end of the cavity, which is in accordance qualitatively with the experiment in [26], that is, the stress birefringence increases quickly near the weld line district and then decreases gradually until reaching the tail of the mold cavity. The maximum value of the stress birefringence appears at some point after the insert.Figure 7
The change of the stress birefringence from the tail of the insert until the end of the cavity.


## 4.1. Mold Filling Process

Figure1 shows the sketch map of the mold with an insert in. Figure 2 gives the top view of the mold, which also represents the computational area. In this paper, we take x=16.2, y=11.6, a=3.3, b=3.4, c=4.9, d=1.8, which are the same as those in [26]. The inlet lies in the middle of the left wall of the computational area. The grid number is 200×120.Figure 1
Sketch map of the mold with an insert in.Figure 2
Computational domain of the mold.Figure3 gives the melt interface positions at different times in mold filling process.Melt positions at different time in the mold filling process.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)From Figure3 we can see that the melt is divided into two streams when it reaches the insert. The two streams move forward until they pass the insert; then they begin to move toward each other until they meet at some point behind the insert, from where the weld line begins to form, and a hole appears between the insert and the meet point of the two streams of melt. After meeting, the melt moves toward two opposite directions. One direction is moving forward and  the other direction is moving backward to fill in the hole formed between the insert and the meeting point. The hole is finally filled with melt and two streams of melt reunion into one stream and move forward. Figure 4 gives the comparison between the interface evolution after the insert and that obtained in  te experiment [26]. The well agreement shows the validity of our model and method.Comparison between the interface evolution after the insert and that obtained in experiment.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)


## 4.2. Distribution of the Stress Birefringence

In order to get the stress distribution and make a comparison with the experimental results given by the stress birefringence distribution, we use the formula in [32] to compute the numerical stress birefringence, which is given as follows:
(9)Δn=C[(τxx-τyy)2+4τxy2]1/2.Here,Δn is the stress birefringence. C is the stress-optical coefficient, which is a constant for linear stress-optical principle, and we take C=1.Figure5 shows the distribution of the stress birefringence at t=1.05 in the simulation. We can see that the stress birefringence distribution is anisotropic near the weld line district and varies greatly at different positions of the weld line district. This phenomenon is induced by the complicated flow behavior after the two streams of melt meet. Since the melt moves toward two opposite directions, the stress birefringence distribution decreases progressively along the two opposite directions. The stress distribution birefringence obtained in experiment after the product is completely produced is given in Figure 6 [26], Since only the mold filling process is considered in the simulation, while the birefringence is obtained after the cooling stage, some difference exists between the numerical and experimental results. However, both the numerical and the experimental results are qualitative agreeable.Figure 5
The distribution of the stress birefringence att=1.05 in the simulation.Figure 6
The stress distribution birefringence obtained in experiment [26] after the product is completely produced.Figure7 gives the change of the stress birefringence from the tail of the insert until the end of the cavity, which is in accordance qualitatively with the experiment in [26], that is, the stress birefringence increases quickly near the weld line district and then decreases gradually until reaching the tail of the mold cavity. The maximum value of the stress birefringence appears at some point after the insert.Figure 7
The change of the stress birefringence from the tail of the insert until the end of the cavity.


## 5. Conclusion

In this paper, simulations of interface evolution and stress distribution near weld line in the viscoelastic melt mold filling process are achieved according to the viscoelastic-Newtonian two-phase model established by Yang et al. [23]. The interface evolution of the viscoelastic melt in the mold filling process with an insert in is captured accurately. The distribution of the stress birefringence is qualitative agreeable with that of experiment. The stress increases quickly near the weld line district and then decreases gradually until reaching the tail of the mold cavity. The maximum value of the stress birefringence appears at some point after the insert.



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*Source: 856171-2013-07-15.xml*