# Trouser tear testing of thin anisotropic polymer films and laminates

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## Abstract

This research has investigated the essential work of fracture (EWF) from trouser tear test of polyethylene terephthalate (PET), low-density polyethylene (LDPE) films and their corresponding laminate using a convenient cyclic tear test method. Propagation of tear crack in these thermoplastics deflects from the initial crack path due to the material anisotropy. An improvement to a two-zone tear model for determining tear EWF was proposed for LDPE-like materials. Energy dissipation due to non-uniform bending of the trouser-legs was determined to be significant in EWF calculation of tearing and this was therefore considered in this study. To measure the tear EWF in laminates, contribution from delamination energy dissipation was accounted for.

## Keywords

Trouser tear test Essential work of fracture Flexible laminate Crack path deviation Delamination## 1 Introduction

Polymers films of low-density polyethylene (LDPE) and polyethylene terephthalate (PET) are widely used in packaging industries, and tearing is a common method of package-opening. Although there are many existing in-plane mode I studies on single layer as well as laminates of these thin polymer films (Kao-Walter 2004; Kao-Walter et al. 2006; Andreasson et al. 2014; Zhang et al. 2016), not many out-of-plane investigations could be found in the literature (Kim and Karger-Kocsis 2004; Bjerkén et al. 2006; Kao-Walter et al. 2009, 2011; Martínez et al. 2010; Andreasson et al. 2013).

The essential work of fracture (EWF) of tear in thin polymer films has increased popularity to characterize out-of-plane shear fracture toughness (Wong et al. 2003). The two-leg trouser tear test which was first used by Rivlin and Thomas (1953) with rubber, rapidly became a preferred test method for tear testing of thin sheets and films. ‘Trouser tear test’ will be referred simply as ‘tear test’ later in this article. One of the challenges is that, when highly extensible materials experience tearing, it is hard to separate the plastic work done in the legs from the plastic work done at the vicinity of the crack. The total fracture energy from a tear test can be separated into geometry dependent (non-essential work) and geometry independent (essential work) contributions to characterize EWF. For energy separation in a tear test, a two-zone model was proposed by Wong et al. (2003). The authors showed that the plastic zone width increases with progress of tearing up to a certain length (zone one) from the initial crack tip; and for any further tearing, the plastic zone width remains constant (zone two). This research was extended by Kim and Karger-Kocsis (2004), who included a third zone that divides ‘zone one’ proposed by Wong et al. (2003) into two separate zones, wherein the new ‘zone one’ considers the crack tip deformation prior to any crack propagation during a tear test. A two-zone model was selected as the foundation of the current study, and additional observations were incorporated, as discussed in Sect. 4. All the earlier studies used multiple tear tests for tear EWF calculation, whereas, the current study showed that a single cyclic tear test can be sufficient.

During a tear test, the specimen legs bends plastically close to the crack tip and along the leg width in some cases. Dissipation of energy from the work done by plastic bending and straightening of trouser-legs during tearing is non-essential work of fracture. It was considered for EWF calculation by Mai and Cotterell (1984) for metal with constant curvature of leg bending. Kim and Karger-Kocsis (2004) later reported that dissipation of energy due to plastic bending and straightening is negligible for polymers; they did not report any measurement on this. However, for PET, LDPE, and their laminate, the bending was observed to vary along the width of the specimen leg. The work of plastic dissipation from this non-uniform bending was considered as presented in Sect. 3, and its magnitude was determined to be significant to EWF as shown in Sect. 4.

Studying the tearing of thin polymer laminates become involved as the films in the laminate respond differently under load compared to individual layer; this has been investigated by several authors (Bjerkén et al. 2006; Andreasson et al. 2014; Zhang et al. 2016; Kao-Walter et al. 2006, 2009, 2011; Islam et al. 2016). Kao-Walter et al. (2011) investigated the LDPE–PET laminate under tearing and observed delamination in the interface which increased along with tear crack propagation. This study has also investigated the significance of delamination in a laminate while calculating laminate EWF.

Further, both LDPE and PET are anisotropic which resulted in deviation of the tearing crack from its initial path as it propagates. Mode mixing is another challenge. Wong et al. (2003) described mode III tearing EWF to be very similar to that of mode I and attributed this to the fact that mode III tearing at the crack tip becomes a mixture of mode I and mode III due to high local deformation (Kim and Karger-Kocsis 2004; Mai and Cotterell 1984). Bárány studied EWF of PET for mode I and mode III and found similar correlation between them (Bárány et al. 2005).

This article presents a new cyclic tear test method, proposes tear EWF calculation method for laminates and considers non-uniform bending and delamination in the trouser-legs as non-essential work of fracture. The effect of material anisotropy was also checked. The article is organized as follows: Experiments on standard tear test and its extension to cyclic tear teat is presented in Sect. 2. Section 3 describes the application of a existing plastic energy dissipation (non-essential work of fracture) theory to non-uniform bending of trouser-legs. EWF of tearing was calculated for thin LDPE, PET films and their laminate in Sect. 4. Section 5 presents some scanning electron microscope (SEM) observations of fracture surface and delamination of the tested materials. Results are discussed in Sect. 6 and the paper ends with some conclusions.

## 2 Experiments

Comparison of mechanical and geometric properties of the tested materials in MD

| \(\sigma _y\) (MPa) | \(\sigma _b\) (MPa) | \(\nu \) (–) | \(\alpha \) (–) | t (\(\upmu \hbox {m}\)) | |
---|---|---|---|---|---|---|

LDPE | 172 | 5 | 12.2 | 0.45 | 0.0058 | 25 |

PET | 1550 | 32.8 | 72 | 0.40 | 0.0364 | 50 |

Laminate | 1090 | – | – | – | – | 75 |

Cyclic trouser tear tests were utilized to determine tear EWF in single layers and in the laminate. To complement the calculation of tear EWF and to find a relation between tear and mode I fracture, additional tensile tests were performed on the continuum and center crack specimens of the same materials.

LDPE, PET and LDPE–PET laminate are anisotropic. Tear and tensile tests were performed in five different material orientations as depicted in Fig. 1a for checking material anisotropy . Angles of orientation were measured from MD; the direction 90\(^\circ \) from the MD is referred to as cross direction (CD). From the tensile test results shown in Appendix A, the elastic modulus and yield stress of LDPE and PET were observed to be isotropic. Anisotropy of ultimate stress was significant in PET but not substantial in LDPE. The Young’s modulus (*E*), Poisson’s ratio (\(\nu \)), initial yield stress (\(\sigma _b\)), work-hardening parameter (\(\alpha \)) and film thickness for LDPE and PET in MD are presented in Table 1.

### 2.1 Standard trouser tear test

Crack deviation in PET and laminate together with delamination due to tearing in degrees (small crack deviation of LDPE was disregarded)

MD | \(22.5^\circ \) | \(45^\circ \) | \(67.5^\circ \) | CD | |
---|---|---|---|---|---|

Crack angle-PET | 12 | 2–4 | 1 | \(-\) (3–4) | \(-\) 6 |

Crack angle-laminate | 7–8 | 6–7 | 1 | \(-\) (1–1) | \(-\) (4–6) |

Delaminated area \((\text {mm}^2)\) | 6.30 | 5.32 | 3.30 | 5.572 | 7.66 |

### 2.2 Cyclic trouser tear test

## 3 Bending dissipation of tear

*da*). For energy dissipation during bending (\(dU_{db}\)) at maximum curvature, the bending energy release rate can be written as,

*Case 1*: For \(0<k_0<1\), bending involves elastic loading and elastic unloading with no plasticity.

*Case 2*: For \(1< k_0<2\), bending involves elastic–plastic loading and elastic unloading, but no reverse plasticity.

*Case 3*: For \(k_0>2\), elastic–plastic loading and reverse plastic deformation are involved.

*b*) of the trouser-legs bending, the material will experience all three possible cases of plastic energy dissipation with gradual changes in curvature, from inner to outer curvature (Fig. 8).

*t*is trouser-leg’s thickness. Assuming a linear change of curvature along the width of the legs, the normalized curvature can be expressed as a function of width,

*b*(Eq. 4). The leg width can be divided into three zones as illustrated in Fig. 8.

*Case 1*: For \(b<\frac{b_{max}\left( k_{0\ max}-1\right) }{k_{0\ max}-k_{0\ min}}\) ; no plastic dissipation.

*Case 2*: For \(\frac{b_{max}\left( k_{0\ max}-1\right) }{k_{0\ max}-k_{0\ min}}<b<\frac{b_{max}\left[ k_{0\ max}-\frac{2\left( 1-\alpha \right) }{\left( 1-2\alpha \right) }\right] }{k_{0\ max}-k_{0\ min}}\)

*Case 3*: For \(b>\frac{b_{max}\left[ k_{0\ max}-\frac{2\left( 1-\alpha \right) }{\left( 1-2\alpha \right) }\right] }{k_{0\ max}-k_{0\ min}}\)

## 4 Essential work of fracture

### 4.1 Separation of plastic dissipation from bending

### 4.2 Evaluation of EWF from only Zone I

*h*) of the plastic zone (also called ligament width) increases as the crack tip progresses in a certain manner depending on the mechanical and geometric properties of the material. This is illustrated in the schematic of Fig. 9a. This plastic zone is visible in a teared LDPE specimen as wrinkles of increasing width (Fig. 9b) and as a thin white zone close to fracture surface for PET teared specimen (Fig. 9c). Zone I of a post-tear specimen is the small zone ahead of initial crack tip where plastic zone width (

*h*) increases more rapidly. Observation of post-test tear specimens (Fig. 9c) indicates that the plastic zone area (\(S_{pa}\)) for PET (for the tested thickness) was small and barely spread from the fracture surface. The subsequent SEM study made similar observations. However, LDPE plastic zone width increased faster in zone I (Fig. 9b). In zone II (Fig. 9b), the plastic zone width increased slowly and steadily with the increasing ligament length. The triangularly shaped plastic area in zone I (Fig. 9b) can be calculated as \(S_{pa}=\alpha ^\prime {l_a}^2\). The plastic area multiplier, \(\alpha ^\prime \), as presented by Wong et al. (2003) is the slope of the ligament width outer boundary with respect to pre-crack path in zone I (Fig. 9a). Therefore, Eq. 11 can be written as follows:

### 4.3 Evaluation of EWF from only zone II

*h*) (slope \(\alpha ^\prime \)), followed by zone II (length \(l_2\)), in which width of the ligament (

*h*) increases comparatively slower (slope \(\alpha ^{\prime \prime }\)). As in Fig. 9a, it is possible to extend zone II backwards to achieve zero ligament width at a distance of \(l_2^\prime \) from the beginning of zone II. Further, total work of fracture can be partitioned as zone I and zone II work of fracture. The total work of fracture in zone I (\(W_{TF-I}\)) can be experimentally quantified using Eq. 12. It is then possible to plot a relation between \({W_{TF-II}=W}_{TF}-W_{db}-W_{TF-I}\) and \(l_2=l_a-l_1\). Figure 9a implies that specific plastic work of fracture (\(w_{TP}\)) is zero at \(l_2={-l}_2^\prime \). Hence according to Eq. 14, extrapolating this curve to \(l_2={-l}_2^\prime \) (\(l_a=l_1{-l}_2^\prime \)) provides the specific essential work of fracture.

*h*),

Specific work of PET fracture calculation for different cycles using Eq. 12

PET (MD) | Ligament length (\(l_a\)) [mm] | Total work of fracture (\(W_{TF}\)) [N-mm] | Bending dissipation (\(W_{db}\)) [N-mm] | Specific total work of fracture \(w_{Te}+w_{TP}\alpha ^\prime l_a\) [N/mm] |
---|---|---|---|---|

Cycle-1 | 6.80 | 2.04 | 0.28 | 5.20 |

Cycle-2 | 11.30 | 3.66 | 0.28 | 5.49 |

Cycle-3 | 15.40 | 5.24 | 0.29 | 5.71 |

Cycle-4 | 20.00 | 6.84 | 0.28 | 5.72 |

Cycle-5 | 24.40 | 8.41 | 0.29 | 5.75 |

Specific work of LDPE fracture calculation for different cycles using Eq. 14

LDPE (MD) | Ligament length in zone II (\(l_2\))[mm] | Total work of fracture in zone II (\((W_{TF}-W_{TF-I})\)) [N-mm] | Bending dissipation (\(W_{db}\)) [N-mm] | Specific total work of fracture in zone II \(w_{Te}+w_{TP}(h+\alpha ^{\prime \prime } l_2)\) [N/mm] |
---|---|---|---|---|

Cycle-1 | 4.80 | 2.51 | 0.10 | 20.11 |

Cycle-2 | 9.50 | 5.78 | 0.10 | 23.51 |

Cycle-3 | 14.00 | 9.36 | 0.10 | 25.88 |

Cycle-4 | 18.50 | 12.99 | 0.12 | 27.22 |

Cycle-5 | 22.90 | 16.78 | 0.13 | 28.44 |

Specific work of laminate fracture calculation for different cycles

Laminate (MD) | Ligament length (\(l_a\)) [mm] | Total work of fracture (\(W_{TF}\)) [N-mm] | Bending dissipation (\(W_{db}\)) [N-mm] | Delamination dissipation \(W_{del}=w_{del}S_{del}\) [N-mm] | Specific total work of fracture \(w_{Te}+w_{TP}\alpha ^\prime l_a\) [N/mm] |
---|---|---|---|---|---|

Cycle-1 | 6.60 | 4.32 | 0.97 | 0.0179 | 6.63 |

Cycle-2 | 11.5 | 9.05 | 1.06 | 0.04 | 8.07 |

Cycle-3 | 15.50 | 14.52 | 1.18 | 0.06 | 9.62 |

Cycle-4 | 20.20 | 20.41 | 1.15 | 0.08 | 10.45 |

Cycle-5 | 24.80 | 26.64 | 1.21 | 0.15 | 11.13 |

### 4.4 EWF in laminates

### 4.5 Calculation of EWF from cyclic tear test

## 5 Scanning electron microscopy

LDPE crack surface fracture appeared highly ductile (Fig. 13c); more importantly, there was a significant reduction in thickness which indicates damage on LDPE in the laminate is due to thinning.

Further observation on all three sections according to Fig. 12b agreed that the spread of this thickness reduction increases with tear crack propagation. An increase in the area that experiences reduction in thickness with crack propagation is equivalent to an increase in ligament length width (*h*) (Fig. 9a).

Figure 12c schematically and Fig. 13a in SEM crack tip section view of a torn laminate illustrates significant local strain with thickness reduction in the LDPE layer, relatively smaller thickness reduction, and strain of the PET layer and delamination of the interface. With increasingly torn ligament, more area is delaminated, and width of delamination is increased. As a result, more LDPE is unconstrained and therefore available for thinning (Fig. 2b).

## 6 Results and discussion

Plastic dissipations in the trouser-legs during tear are regarded as non-essential work of fracture. Measuring the dissipation from the leg bending curvature renders the calculated SEWF more independent of leg width. This method of measurement can be beneficial when it is practical to minimize the number of tests by not testing for multiple leg width to omit any width effect. The formulation provided in this study can be applied to both uniform and non-uniform curvature distribution.

SEWF of PET measured by the proposed cyclic tearing was comparable with results found in the literature. The calculated SEWF for \(50\,\upmu \hbox {m}\) PET was 4.80 N/mm in this study and 6.35 N/mm for \(250\,\upmu \hbox {m}\) PET in the literature (Kim and Karger-Kocsis 2004). The current value was smaller because the dissipation of bending was regarded as non-essential work of fracture. The literature reports that smaller tear SEWF for thinner PET is expected (Kim and Karger-Kocsis 2004). LDPE was divided into two zones based on visual inspection, and the proposed zone II method bypasses any necessary calculation for near crack tip plastic dissipation. This method is applicable to materials that develop long ligaments with steadily increasing plastic zone width during tear. The authors did not find SEWF for LDPE tear reported in the literature. However, the specific essential work of fracture for blown LDPE film in mode I was reported to be approximately 16 N/mm for \(150\,\upmu \hbox {m}\) thick LDPE film. The SEWF is more anisotropic for thinner LDPE, and it ranges between 9 and 43 N/mm for \(15\,\upmu \hbox {m}\) thick LDPE (Rennert et al. 2013). Macroscopic crack tip observation of a tear test suggests that since LDPE is more ductile and flexible, it tends to shift mode, and load more in mode I; therefore, a relation can be expected with mode I SEWF. In this study, the calculated SEWF of tear for \(25\,\upmu \hbox {m}\) LDPE was 11.1 N/mm, which is comparable with the results reported earlier in the literature. The calculated SEWF of the laminate of LDPE–PET was significantly low and necessitates further investigation. Moreover, although the thickness can also exert a significant effect on SEWF value (Kim and Karger-Kocsis 2004), merely one particular thickness for each film was tested.

The tear tests were performed such that trouser-legs were separated vertically, which caused the tail of the tear specimen to hang and bend down due to gravity (Fig. 2a). This can cause the bending curvature at the bottom leg to be larger than the top one and contribute to additional tear crack deviation. Particularly in the laminate, if the more compliant LDPE side is facing upwards, delamination width increases faster with tearing relative to when the PET side faces upwards. This can affect the SEWF and worth further investigation. This effect can be negated by pulling the tear specimen sidewise such that the tail hangs vertically.

## 7 Conclusions

Trouser tear test of PET, LDPE films, and the corresponding laminate have been examined in this study in five different material orientations. Propagation of tearing in these thermoplastics demonstrated deviation from the initial and parallel crack path with a mixed mode I and mode III. This was determined to be caused by the material anisotropy, and the deviation can be related to the difference in mode I fracture toughness at different material orientations. The crack tends to deflect toward the weakest material orientation. The amount of delamination was also discovered to be influenced by the material orientation.

The proposed cyclic tear test method for SEWF measurement could produce results comparable to those reported in the literature. Energy dissipation due to non-uniform bending of the trouser-legs was demonstrated to be significant in the tearing SEWF calculation and was therefore considered in this study. Analytical expressions for the calculation of non-uniform bending energy dissipation for a bi-linear isotropic hardening material model were presented. A variation of a two-zone tear model was proposed to bypass any plastic dissipation calculation for SEWF calculation in LDPE. To measure the SEWF of laminates, delamination energy dissipation was accounted for. However, delamination appeared to expose more unconstrained LDPE that effects the laminate behaviour more than that caused by energy dissipation due to delamination. Further study is necessary to use EWF for characterization of laminate tear fracture. Also, this study particularly focused on thin LDPE, PET and their laminate. Additional studies are necessary to check the applicability of the presented methods and formulations for other polymers.

## Notes

### Acknowledgements

Open access funding provided by Blekinge Institute of Technology. This work is a part of the research activity of the Model Driven Development and Decision Support project (MD3S) at Blekinge Institute of Technology, which is funded by the Swedish Knowledge and Competence Development Foundation (KKS). Special thanks to SEM laboratory at Shanghai Polytechnic University, China for allowing us to use their facility, and to Dr. Lun Zhao from Kunming University of Science and Technology in China for helping us with the SEM study.

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