Design and Simulation Analysis of Lightweight HDPE Milk Bottle

Abstract

A lightweight design is proposed for a 2800 ml HDPE milk bottle. In designing the bottle, a three-dimensional elastic-plastic dynamic finite element model is first constructed to explore the effect of the bottle thickness on the critical load and stress distribution. The validity of the proposed model is confirmed by comparing the numerical results for the load-displacement response of the bottle with the experimental results. A parametric analysis is then performed to examine the effects of the body thickness and bottom thickness on the critical load and weight of the HDPE bottle. The results show that the critical load increases with an increasing body thickness, but is insensitive to the bottom thickness. Finally, a lightweight design is proposed in which the structural shape of the original bottle is retained, but the thickness is varied in different regions of the body in order to reduce the weight. It is shown that the redesigned bottle achieves the same maximum top-load 315 N, but reduces the weight by 21.4% compared to the reference design obtained in the parametric analysis.

1. Introduction

High density polyethylene (HDPE) is one of the most important polymer materials currently available. With its good mechanical properties, ease of processing and low cost, HDPE is commonly used for the fabrication of milk or oil containers. However, these containers are often subjected to large externally-applied loads during delivery and handling. As a result, they are required to be physically strong and robust. However, to improve their recyclability, they should also have a light weight and be easily crushed. Consequently, the need exists for an improved bottle design, in which the weight under columnar crush conditions is reduced, while the original strength and stiffness are retained.

In general, experimental studies tend to be time consuming and expensive. Consequently, numerical methods, such as finite element modeling (FEM), are commonly preferred. Abbes et al. ¹ conducted a finite element analysis (FEA) investigation into the top-load strength of aged polypropylene bottles and showed that the results were in good agreement with the experimental findings. Hu et al. ² calculated the buckling load and stress distribution in a PET bottle using an explicit nonlinear finite element (FE) model. The analysis results were used to design an improved bottle with a lower weight and an increased buckling load. Masood and KeshavaMurthy ³ used an FEM approach to design a collapsible PET water fountain bottle. It was shown that compared to a traditional PC bottle, the new bottle had an improved collapsibility and a lighter weight, but retained the strength required to hold 15 L of water.

Miranda et al. ⁴ optimized the design of a 500 ml PET bottle by means of FE simulations and parametric computer-aided design software. The validity of the optimized design was demonstrated by fabricating and testing an experimental prototype. The results showed that the optimized design reduced the weight of the original bottle by approximately 21%. Jarod and Panos ⁵ used an FE model to examine the structural stress distribution in a 600 ml PET bottle. The results indicated that the maximum von Mises stress for the optimized design bottle occurred on the bottle bottom. Demirel and Daver ⁶ examined the effects of the dimensions and geometry of a standard PET bottle on its strength and thermal stability. It was found that the magnitude of the internal stress within the bottle could be reduced through a careful design of the base geometry. Thusneyapan and Suvanjumrat ⁷ performed simulated drop tests of PET bottles with various structural shapes and designs. The results showed that bottles with a smooth continuous surface and no edges at the wall have an improved sturdiness in resisting the impact force generated during the drop test. The work done by Keawjaroen and Suvanjumrat ⁸ was to design one liter HDPE bottle and analyzed the top load resistance of the bottle. The results indicate that the maximum vertical deformation by the top load of 250 N for the one liter bottle obtained from the finite element simulation and experimental measurement were 2.09 mm and 2.4 mm, respectively. The study of Suvanjumrat and Chaichanasiri ⁹ used the finite element model (FEM) to simulate the creep behavior of HDPE lubricant oil bottles and compared with the experimental results to validate the FEM. It was shown that the FEM simulation results agreed with the experimental results.

Although various studies have addressed the lightweight design of PET bottles, the literature contains only limited information regarding the lightweight design of HDPE milk bottles. Consequently, the present study commences by constructing a three-dimensional FE model of a 2800 ml HDPE milk bottle. The model is verified by comparing the simulated load-displacement response and deformation of the bottle with the experimental results. The model is then used to examine the effects of the body thickness and bottom thickness of the bottle, respectively, on the maximum sustainable top-load and bottle weight. Finally, a lightweight bottle design is proposed in which the body thickness is adjusted in different regions of the bottle in order to reduce the weight while maintaining the same physical strength. It is shown that the lightweight design achieves the same maximum top-load as the original bottle, but offers a weight saving of approximately 21.4%.

2. Finite Element Model

Figure 1 illustrates the 2800 ml HDPE milk bottle considered in the present study. As shown, the bottle has dimensions of 150.2 mm × 131.4 mm × 260.2 mm. In addition, the neck has a height of 13.7 mm and an external diameter of 34.3 mm. The commercial finite element package ABAQUS is used to perform the numerical simulation. Figure 2 shows the three-dimensional (3D) elastic-plastic dynamic finite element (FE) model constructed in the present study. The bottle was meshed using four-node shell elements (S4), while the upper and lower plates were constructed using eight-node 3D brick elements (C3D8). The interfaces between the upper and lower plates and the bottle rim and bottle base, respectively, were modeled using contact elements. The strength and stress distribution within the bottle were investigated by means of simulated load-displacement tests. In performing the simulations, the upper and lower plates were treated as rigid bodies and the mechanical properties of the bottle were assigned the values shown in Table 1 . The boundary conditions were set as shown in Figure 3 . In simulation, the speed of the upper plate in the vertical (i.e., downward) direction was given as 50 mm/min, while all the other degrees of freedom of the plate were fixed. In addition, the lower plate was assumed to remain stationary throughout the simulation.

Figure 1

Geometric dimensions of considered 2800 ml HDPE milk bottle (unit: mm)

Figure 2

Finite element model of HDPE milk bottle

Figure 3

Boundary conditions of finite element model

Table 1

Mechanical properties of HDPE bottle used in FE simulations

Young's modulus (MPa)	273.35
Poisson's ratio	0.419
Yield Strength (MPa)	26.13
Tensile Strength (MPa)	27.72
Density (kg/m3)	630

Preliminary simulations were performed to establish an appropriate mesh size for the convergence test of the FE model. Figure 4 presents the results obtained for the maximum critical load when using meshing schemes with 2000∼7000 elements, respectively. It is seen that the critical load of the bottle (i.e., the top-load at which the bottle collapses) converges to an approximately constant value as the number of elements increases beyond 4000. Thus, to achieve a reasonable tradeoff between the numerical accuracy and the computational cost, the simulations were performed using an FE mesh consisting of 4317 elements and 4334 nodes.

Figure 4

Convergence test results for FE model

3. Verification of Finite Element Model

The validity of the dynamic FE model was evaluated by comparing the results obtained in a simulated load-displacement test with those observed experimentally. Figure 5 presents a photograph of the experimental setup consisting of the HDPE milk bottle, an upper plate, a lower plate, a material testing machine (EZ Test-500N, Shimadzu, maximum load 500 N), and a computer system. Figure 6 compares the numerical and experimental results obtained for the load-displacement response of the bottle given an upper plate displacement speed of 50 mm/min. It is observed that a good qualitative agreement exists between the two sets of results. From inspection, the numerical and experimental values of the critical load are equal to 314.8 N and 314.9 N, respectively. The corresponding displacement values are 11.5 mm and 11.6 mm, respectively. The difference between the two sets of results is therefore equal to just 0.1 N (load) and 0.1 mm (displacement).

Figure 5

Experimental setup for HDPE milk bottle load-displacement test

Figure 6

Experimental and simulation results for force-displacement response of HDPE milk bottle

Figure 7 shows the simulation and experimental results for the milk bottom deformation under the maximum critical load. Again, a good general agreement is observed between the two sets of results. The basic validity of the numerical model is thus confirmed. Moreover, the results (both numerical and experimental) suggest a target critical top-load of 315 N for the bottle redesign process.

Figure 7

Experimental and simulation results for HDPE milk bottle deformation (a) experiment (b) finite element simulation

4. Parametric Analysis of Hdpe Bottle

A parametric analysis was performed to investigate the effect of the body thickness t _a and bottom thickness t _b on the critical top-load of the bottle. In performing the analysis, the neck thickness was set as a constant 2 mm (see Figure 1 ), while the body thickness was assigned values of t _a =1.1 mm, 1.2 mm and 1.3 mm, and the bottom thickness was set as t _b =1.1 mm, 1.2 mm and 1.3 mm. Figure 8 shows the simulated force-displacement response of the bottle given the three different body thicknesses and a constant bottom thickness of t _b =1.1 mm. The results show that the maximum critical load P _cr is equal to 266 N, 329.7 N and 396.4 N for body thicknesses of t _a =1.1, 1.2 and 1.3 mm, respectively. In other words, the maximum sustainable load increases with an increasing body thickness. It also can be seen that the slope (stiffness) increases obviously with an increasing body thickness as the displacement is greater than 8 mm, as shown in Figure 8 . The three bottles have calculated weights of 84.3, 90.4 and 96.6 g, respectively. Thus, as expected, the weight also increases with an increasing body thickness. However, while the weight increases by 14.5% as the body thickness is increased from 1.1 m to 1.3 mm, the critical load increases by almost 50%.

Figure 8

Force-displacement response of HDPE milk bottle given different body thicknesses

In other words, the body thickness of the bottle has a significant effect on the maximum sustainable load, P _cr . Figures 9(a) and 9(b) show the simulated force-displacement response of the bottle with a constant body thickness of t _a =1.1 mm and t _a =1.3 mm, respectively, for three different bottom thicknesses t _b =1.1, 1.2 and 1.3 mm.

Figure 9

Force-displacement response of HDPE milk bottle given different bottom thicknesses

As shown in Figure 9(a) , the critical loads of the three bottles are equal to 266 N, 268.9 N and 277.9 N, respectively. In other words, the maximum critical load increases by just 4.4% as the bottom thickness is increased. The critical loads obtained for t _b =1.1, 1.2 and 1.3 mm are 396.4 N, 393.8 N and 390.2 N, respectively, as shown in Figure 9(b) . Thus, it is inferred that the critical load is essentially insensitive to the bottom thickness. However, it can be observed that the slope (stiffness) increases with an increasing bottom thickness as the displacement is greater than 5 mm and then reaches a stationary value as shown in Figure 9(b) . This is the reason that the critical load is insensitive to the bottom thickness.

Table 2 summarizes the parametric analysis results for the effects of the body thickness and bottom thickness on the critical load, displacement and weight of the HDPE bottle. It is seen that for all considered values of the bottom thickness, the target critical load 315 N is satisfied using a wall thickness of 1.2 mm or 1.3 mm. Furthermore, the bottle with a body thickness of 1.3 mm and a bottom thickness of 1.1 mm achieves the highest critical load 396.4 N of all the bottles. Thus, this bottle with a calculated weight of 96.6 g was taken as a reference for the subsequent lightweight design analysis process.

Table 2

Effects of body thickness and bottom thickness on critical load, displacement and weight of HDPE bottle

t a (mm)	1.1	1.2	1.3	1.1	1.2	1.3	1.1	1.2	1.3
t b (mm)	1.1	1.1	1.1	1.2	1.2	1.2	1.3	1.3	1.3
P cr (N)	266.0	329.7	396.4	268.9	325.5	393.8	277.9	323.6	390.2
Displacement (mm)	8.9	9.6	10.5	8.4	9.4	10.1	6.4	8.7	9.8
Weight (g)	84.3	90.4	96.6	85.6	91.7	97.9	86.3	93.0	99.2

5. Lightweight Analysis of Hdpe Bottle

A lightweight analysis was performed to reduce the weight of the HDPE bottle while maintaining the critical load target of 315 N. As shown in Figure 10 , the rim and bottom regions of the bottle were assigned constant thickness values of t=2 mm and t _b =1.1 mm, respectively. A topology optimization process was then performed using commercial HyperWorks software to determine the distribution of the body thickness which minimized the bottle weight while still satisfying the critical load target. In lightweight analysis, the bottom of the HDPE bottle was fixed. In optimization analysis, the objective function was taken as minimum weight of the bottle. The design region is the body thickness. The corresponding results are shown in Figure 11 . From a production perspective, it is infeasible to produce a bottle with multiple thickness values in different regions of the body. Thus, the topology analysis results were used to construct the bottle design shown in Figure 12 (referred to hereafter as Design-1), in which the body thickness was assigned just two different values, i.e., t _a =1.3 mm in the gray regions and t _a =0.85 mm in the white regions. FE simulations showed the bottle to have a critical load of 340 N and a weight of 78.4 g. Figure 13 shows the von-Mises stress distribution within the bottle under critical load conditions. The maximum von-Mises stress is found to be 25.22 MPa, which is lower than the yield stress of HDPE (i.e., 26.13 MPa, see Table 1 ).

Figure 10

Geometrical model used for lightweight analysis

Figure 11

Body thickness distribution obtained from topology analysis for Design-1 HDPE bottle

Figure 12

Wall thickness distribution for Design-1 HDPE bottle

Figure 13

von-Mises stress distribution for Design-1 HDPE bottle

The maximum load of the Design-1 bottle 340 N is significantly higher than the target critical load of 315 N. Therefore, the thickness of the white regions in the upper area of the Design-1 bottle was reduced from 0.85 mm to 0.75 mm in order to achieve a further weight saving (see the gray regions of the upper area in Figure 14 ). The weight of the resulting bottle (referred to hereafter as Design-2) was found to be 75.9 g. Figure 15 shows the simulation results obtained for the load-displacement curves of the Design-1, Design-2 and reference bottle. As shown, the Design-2 bottle has a critical load of 315.2 N, and therefore satisfies the design target. Moreover, compared with the reference bottle (96.6 g), the Design-2 bottle achieves a weight saving of 21.4%.

Figure 14

Bottle thickness distribution of Design-2 HDPE bottle

Figure 15

Load-displacement curves of three different bottle designs

6. Conclusions

This study has performed 3D numerical simulations to design and optimize a 2800 ml HDPE milk bottle. The feasibility of the proposed FEM model has been confirmed by comparing the numerical results for the load-displacement response of the bottle under columnar crush conditions with the experimental results. Further simulations have then been performed to explore the effects of the body and bottom thickness of the bottle on the critical top-load and to develop an optimized lightweight design. The numerical results support the following main conclusions:

1. 1.

   The proposed elastic-plastic dynamic finite element model provides an accurate prediction of the critical top-load and deformation of the HDPE milk bottle.

2. 2.

   The critical top-load of the bottle increases significantly (∼50%) with an increasing body thickness, but increases only slightly (∼4.4%) with an increasing bottom thickness.

3. 3.

   The Design-2 HDPE milk bottle, in which the body thickness is varied in the range of 0.75∼1.3 mm, achieves the same maximum top-load 315 N, but has a weight saving of 21.4% compared to the reference bottle obtained in the parametric analysis.