# Numerical Approach for the Assessment of Micro-Textured Walls Effects on Rubber Injection Moulding

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Rubber Injection Moulding: Mathematical Model

#### 2.1.1. Alpha Equation

#### 2.1.2. Mass-Conservation Equation

#### 2.1.3. Momentum-Conservation Equation

#### 2.1.4. Rubber Cure Kinetics

#### 2.1.5. Energy-Conservation Equation

#### 2.2. Wall Texture: An Effective Viscosity Modelling Approach

#### 2.3. Numerical Approach

`rubberFoam`, has been developed to solve the comprehensive mathematical model described in Section 2.1 and Section 2.2. It is based on the standard solver

`compressibleInterFoam`for two compressible, non-isothermal immiscible fluids using a VOF (volume of fluid) phase-fraction-based interface capturing approach [10].

`rubberFoam`stands thus for an evolved version of the

`compressibleInterFoam`, being the major modifications aimed at accounting for the particular non-Newtonian behaviour of rubbers (Equation (20)) and the curing chemical reaction (Equation (24)). A new structure of classes has been developed to this end, being the most relevant ones indicated below.

`mixture(U, phi, omega)`), where the thermo-physical properties of the main-phase (of a new type

`rhoThermoNonNewtonian`) are not only dependent on the operating pressure and temperature, but also on the velocity field (

`U`) and an auxiliary scalar field (

`omega`); whereas the properties of the secondary-phase (of standard type

`rhoThermo`) remain only dependent on the operating pressure and temperature.

`rhoThermoNonNewtonian`) to describe the full thermo-physical behaviour of rubbers, including the effects of the strain rate ($\mathtt{sr}=f\left(\mathtt{U}\right)$) and the cure degree ($\mathtt{omega}=\omega $) on the computation of the transport coefficients.

`nonNewtonianRubberTransport`has been implemented to describe the rubber viscosity acording to the “Reactive Viscosity Model”, Equation (20).

#### 2.3.1. Texture Wall Law: ROM

`rubberFoam`) by means of a reduced order model (ROM), aimed at describing the behaviour of a complex system using simple mathematical expressions without losing relevant information.

`constant`folder of the OpenFOAM case. The

`rubberFoam`solver reads it and calculates the viscous stress tensor at each time-step and for each face of the textured wall boundaries (modeled as smooth walls), given the resulting ${U}_{P}$, ${y}_{P}$ and T at the corresponding cell near the wall face.

#### 2.4. Experimental Rubber Characterization

^{−1}up to 750 s

^{−1}shear rate, the resulting data are used to fit the parameters of the “Reactive Viscosity Model” (Equation (20) reported in Table 1); and (ii) Moving Die Rheometer (MDR) tests at four different temperatures (160, 170, 180 and 190 °C) to estimate the value of the cure reaction-kinetics parameters (Equation (24) reported in Table 2). RCR and MDR raw data are attached as Supplementary Materials. Besides, the thermal conductivity (0.413 W/(m K)) and the specific heat capacity (0.86 J/(g K)) are obtained from hot disk and a DSC experiments respectively.

## 3. Results

#### 3.1. Rubber Flow Near the Textured Wall

#### 3.2. Rubber Injection in Textured-Wall Moulds

`rubberFoam`, is applied to simulate rubber injection in textured moulds in two different cases. First, as proof of concept, the rubber filling of a cylindrical straight pipe is considered (Section 3.2.1). Next, the methodology is applied to the simulation of the rubber injection on an industrial mould for the manufacturing of D-ring seals (Section 3.2.2).

#### 3.2.1. Test Case: Straight Pipe

`rubberFoam`is tested on a simple geometry, such as the case of a straight pipe, to analyze and quantify the influence of the micro textured wall on the filling process, specifically on the injection pressure. To this end, the following numerical study has been performed.

`rubberFoam`: (1) with textured walls; and (2) with smooth walls, henceforth named textured and smooth respectively. The results are shown in Figure 7 and Figure 8.

#### 3.2.2. Real Case: D-Ring Mould

`rubberFoam`of two cases: (1) standard mould where all walls are smooth; and (2) textured mould, where the internal face is textured according to the pattern shown in Figure 2. Based on industrial experience, a filling time of 1 s is specified, and the average injection velocity is calculated accordingly and set as inlet boundary condition. A constant and uniform pressure field (1 bar) is defined at the outlets. Injection temperature is set to 450 K. Table 6 summarises the most relevant boundary conditions. Please note that the solver will automatically activate the texture wall law for viscosity at those boundaries whose name starts with

`wallTexture`. Thus, the only difference between smooth and textured cases is that the internal ring wall is called

`wall`and

`wallTexture`respectively.

## 4. Discussion

## 5. Conclusions

`rubberFoam`was developed to calculate the rubber flow during rubber injection on textured moulds, where the effective viscosity at the wall corresponding to the textured surface is introduced by means of a reduced order model. The solver is then tested in both an academic case (cylindrical straight pipe) and an industrial case (D-ring seal mould). At any case, the textured walls do not induce any negative effect on the manufacturing process, as the injection pressure remains equal or even decreases around 2% with respect to the reference case (no textured walls). This behaviour is due to the non-Newtonian nature of the rubber, as the viscosity near the wall decreases due to the higher shear rates induced by the wall texture. The results of this study support the current trend towards mould walls texturization to improve the pieces mechanical properties, without compromising (or even enhancing) the manufacturing process or production costs.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. ROM_Output_file.txt

`rubberFoam`) reads this file and calls a function (

`evalROM`), available at the ROM builder library [28,29], which interprets the data and evaluates Equation (32) to predict the viscous shear tensor at the textured wall.

*********************************************************************** |

*** ROM summary |

*** Number of terms: 5 |

*** Number of dimensions: 3 |

*** Dimension 1 discretization: 2 2 2 2 2 |

*** Dimension 2 discretization: 8 8 8 8 8 |

*** Dimension 3 discretization: 12 12 12 12 12 |

*********************************************************************** |

ROM Data: |

Term 1 |

Alpha: 412848 |

Dimension 1 |

450 0.853266 473 0.521476 |

Dimension 2 |

3.5 × 10${}^{-5}$ 0.536588 4.39583 × 10${}^{-5}$ 0.475352 5.29167 × 10${}^{-5}$ 0.435076 9.77083 × 10${}^{-5}$ 0.381214 0.000106667 0.131042 0.000124583 0.299927 |

0.000133542 0.0146306 0.00025 0.210134 |

Dimension 3 |

0.00136384 0.0608335 0.00824975 0.132693 0.0151356 0.170953 0.0220216 0.203596 0.0289075 0.229498 0.0357934 0.252525 0.0426793 0.276441 |

0.0495652 0.294166 0.0564511 0.31351 0.063337 0.327378 0.0908806 0.391973 0.166625 0.51915 |

Term 2 |

Alpha: 2667.23 |

Dimension 1 |

450 −0.943357 473 0.33178 |

Dimension 2 |

3.5 × 10${}^{-5}$ 0.0138087 4.39583 × 10${}^{-5}$ −0.354879 5.29167 × 10${}^{-5}$ −0.890803 9.77083 × 10${}^{-5}$ 0.0510481 0.000106667 0.0175478 0.000124583 |

−0.114979 0.000133542 −0.00560872 0.00025 0.253328 |

Dimension 3 |

0.00136384 −0.168722 0.00824975 0.816757 0.0151356 −0.286778 0.0220216 −0.0292196 0.0289075 −0.131955 0.0357934 −0.138186 0.0426793 |

0.330298 0.0495652 −0.0797557 0.0564511 0.160481 0.063337 −0.093528 0.0908806 0.18641 0.166625 0.011386 |

Term 3 |

Alpha: 1645.94 |

Dimension 1 |

450 0.798857 473 0.601521 |

Dimension 2 |

3.5 × 10${}^{-5}$ −0.25005 4.39583 × 10${}^{-5}$ 0.157966 5.29167 × 10${}^{-5}$ 0.0714464 9.77083 × 10${}^{-5}$ 0.868813 0.000106667 0.298654 0.000124583 |

0.217874 0.000133542 0.010628 0.00025 0.125718 |

Dimension 3 |

0.00136384 0.623631 0.00824975 0.0585777 0.0151356 −0.631778 0.0220216 −0.255556 0.0289075 −0.00579108 0.0357934 −0.0640496 0.0426793 |

0.0493176 0.0495652 −0.0170427 0.0564511 0.252543 0.063337 −0.111981 0.0908806 0.110066 0.166625 0.218881 |

Term 4 |

Alpha: 2189.93 |

Dimension 1 |

450 −0.652909 473 −0.757436 |

Dimension 2 |

3.5 × 10${}^{-5}$ 0.0795748 4.39583 × 10${}^{-5}$ 0.824329 5.29167 × 10${}^{-5}$ −0.537909 9.77083 × 10${}^{-5}$ 0.0884352 0.000106667 0.0303996 0.000124583 |

0.0814518 0.000133542 0.00397326 0.00025 −0.097 |

Dimension 3 |

0.00136384 −0.00741717 0.00824975 0.183312 0.0151356 0.137525 0.0220216 −0.756504 0.0289075 0.535539 0.0357934 −0.174232 0.0426793 |

0.0858151 0.0495652 −0.0884338 0.0564511 −0.0504824 0.063337 0.0901165 0.0908806 0.177468 0.166625 −0.0249437 |

Term 5 |

Alpha: 1087.76 |

Dimension 1 |

450 −0.353853 473 0.935301 |

Dimension 2 |

3.5 × 10${}^{-5}$ −0.345931 4.39583 × 10${}^{-5}$ −0.171289 5.29167 × 10${}^{-5}$ 0.914857 9.77083 × 10${}^{-5}$ −0.0957467 0.000106667 −0.0329129 0.000124583 |

0.00318454 0.000133542 0.000155344 0.00025 0.0613852 |

Dimension 3 |

0.00136384 0.139599 0.00824975 0.33411 0.0151356 −0.215309 0.0220216 0.0686149 0.0289075 0.00205378 0.0357934 −0.0832903 0.0426793 |

0.0700626 0.0495652 −0.451532 0.0564511 −0.679474 0.063337 0.363554 0.0908806 0.0879399 0.166625 0.022273 |

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**Figure 2.**Wall texture—Dimples (D = 100 μm, h = 30 μm, d = 100 μm): (

**a**) profile view; (

**b**) plan view.

**Figure 3.**Representative domain near the wall: (

**a**) geometry of the base (200 μm × 200 μm) with a dimple (D = 100 μm; h = 30 μm); (

**b**) mesh close to the dimple at the base.

**Figure 4.**Flow on a representative domain near the wall, contours of viscosity, shear rate and pressure on the domain base (@ T = 450K & Δp = 1000 Pa): (

**a**) smooth wall; (

**b**) textured wall.

**Figure 5.**Flow on a representative domain near the wall, results: (

**a**) Δp vs wall shear stress; (

**b**) ${U}_{P}$ vs wall shear stress; and (

**c**) ${y}_{P}$ vs wall shear stress.

**Figure 8.**Results of the filling of a straight pipe (smooth vs textured wall): (

**a**) influence of inlet velocity; (

**b**) influence of diameter; (

**c**) influence of length.

**Figure 9.**Mould cavity of a D-ring seal: (

**a**) geometry; (

**b**) mesh; and (

**c**) mesh detail close to exit vents.

**Figure 10.**D-ring mould filling: Rubber front at different times, namely: (

**a**) 0.05 s (

**b**) 0.2 s (

**c**) 0.4 s (

**d**) 1 s.

Parameter | Value |
---|---|

B | 0.250839 |

${T}_{b}$ | 9947.615 |

${\tau}^{*}$ | 0.027699 |

n | 0.461665 |

${\omega}_{g}$ | 0.796923 |

${C}_{1}$ | 4.291385 |

${C}_{2}$ | −4.13821 |

Parameter | Value |
---|---|

${A}_{1}$ | 3.77 × ${10}^{11}$ |

${E}_{1}$ | 13,350 |

${A}_{2}$ | 8.72 × ${10}^{13}$ |

${E}_{2}$ | 13,962 |

m | 1.22 |

n | 1.36 |

${B}_{1}$ | 1.05 × ${10}^{-7}$ |

${B}_{2}$ | 9124.8 |

T [K] | Δp [Pa] |
---|---|

450 | 500 |

1000 | |

1500 | |

2000 | |

473 | 500 |

1000 | |

1500 | |

2000 |

Patch | Type | Relevant Boundary Definition |
---|---|---|

Inlet (−x) | mappedPatch | Pressure: fixedValue Velocity: mappedField |

Outlet (+x) | patch | Pressure: fixedValue |

Lateral wall (−y) | symmetry | Symmetry |

Lateral wall (+y) | symmetry | Symmetry |

Upper wall (+z) | wall | Velocity: non-texturized case |

Lower wall (−z) | wall | Velocity: No slip (0 0 0) |

Cases | ${\mathit{U}}_{\mathit{inlet}}$ [m/s] | D [mm] | L [mm] |
---|---|---|---|

1 | 0.05 | 1 | 20 |

2 | 0.08 | ||

3 | 0.1 | ||

4 | 0.1 | 0.7 | 20 |

5 | 1 | ||

6 | 1.5 | ||

7 | 0.1 | 1 | 20 |

8 | 50 | ||

9 | 100 |

Patch | Type | Relevant Boundary Definition |
---|---|---|

inlet | patch | Velocity: flowRateInletVelocity |

outlet_1, outlet_2 | patch | Pressure: fixedValue |

symmetry | symmetry | symmetry |

wall | wall | Velocity: noSlip |

wallTexture | wall | Velocity: noSlip |

Viscosity: Texture wall law |

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## Share and Cite

**MDPI and ACS Style**

García-Camprubí, M.; Alfaro-Isac, C.; Hernández-Gascón, B.; Valdés, J.R.; Izquierdo, S.
Numerical Approach for the Assessment of Micro-Textured Walls Effects on Rubber Injection Moulding. *Polymers* **2021**, *13*, 1739.
https://doi.org/10.3390/polym13111739

**AMA Style**

García-Camprubí M, Alfaro-Isac C, Hernández-Gascón B, Valdés JR, Izquierdo S.
Numerical Approach for the Assessment of Micro-Textured Walls Effects on Rubber Injection Moulding. *Polymers*. 2021; 13(11):1739.
https://doi.org/10.3390/polym13111739

**Chicago/Turabian Style**

García-Camprubí, María, Carmen Alfaro-Isac, Belén Hernández-Gascón, José Ramón Valdés, and Salvador Izquierdo.
2021. "Numerical Approach for the Assessment of Micro-Textured Walls Effects on Rubber Injection Moulding" *Polymers* 13, no. 11: 1739.
https://doi.org/10.3390/polym13111739