Determination of Key Cohesive Zone Model’s Parameters for Orthotropic Paper and Its Static Fracture Simulation

WANG Yue，WANG Yongjian，LI Lingquan

College of Engineering，Nanjing Agricultural University，Nanjing 210031，P.R.China

Abstract: Investigation of paper cutting process is vital for the design of cutting tools，but the fracture mechanism of paper cutting is still unclear. Here，we focus on the cutting process of paper，including the key parameters of cohesive zone model（CZM）for the orthotropic paper，to simulate the shear fracture process. Firstly，the material constants of the orthotropic paper are determined by longitudinal and transverse tensile test. Secondly，based on the tensile stressstrain curves，combined with damage theory and numerical simulations，the key parameters of the CZM for the orthotropic paper are obtained. Finally，a model III fracture is simulated to verify the accuracy of the model. Results show that the load-displacement curves obtained by the simulation is consistent with the test results.

Key words：orthotropic；cohesive zone model（CZM）；paper；static fracture

0 Introduction

The rapid development of logistics industry has a high requirement on the characteristics of packag‐ing paper in terms of tear resistance，tensile resis‐tance and resistance to external shock load. There‐fore，a variety of mechanical theories on paper，such as damage mechanics［1-2］and fracture mechan‐ics［3‐4］were proposed. Paper needs suitable cutting tools and cutting process to make sure the smooth‐ness of the paper’s edge for various products. How‐ever，cutting technology aimed at ensuring the edg‐es of paper are smooth mostly depend on a large number of experiments，which means the fracture mechanism of paper is still unclear.

The mechanical properties of the paper have been studied for decades. He and Zhang［5］applied the nonlinear theory of thin plate to simulate the vis‐coelastic mechanical behavior of printing paper and obtained incremental equilibrium equation. Xie［6］re‐ported a relationship between the shear modulus and elastic modulus，under the assumption of ortho‐tropic paper. Based on the updated Lagrangian de‐scription，Ma et al.［7］derived the incremental equi‐librium equations of triangular plate-shell elements for a geometrically nonlinear problem. He and Liu［8］obtained the integral form of stress and strain consti‐tutive equation of the z-direction mechanistic model for printing paper by using Laplace transform. Addi‐tionally，the Nomex paper［9］and the honeycomb pa‐per［10］were also studied，to make sure that the used packaging paper is strong enough. It is crucial to fo‐cus on paper’s fracture deeply including mechanical properties of paper.

Many types of research on fracture mechanics of metal materials can be found，but only a few on paper. In the field of crack propagation，the cohe‐sive zone model（CZM）［11-12］has been widely used in metal materials［13］and some other smart materi‐als［14］. It is believed that the maximum traction and the critical fracture energy are the key parameters for CZM［15-16］. In most literatures，these two key pa‐rameters were obtained by specific experiments［17］or numerical simulations［18］，both of which are com‐plicated and expensive. Hence，Wang and Ru［19］proposed a new method to determine the two key parameters for X80 pipeline steel，based on the damage mechanics and numerical simulations. Shi and Wang［20］studied the shape of the traction-sepa‐ration law（TSL）of the CZM，and found that the shape of TSL does not affect the global response for the quasi-static fracture. However，to the best of the author’s knowledge，it is not reported that the two key parameters were obtained without specific ex‐periments and numerical simulations for the paper fracture. To cover this gap，the current study pro‐posed an engineering method to determine the two key CZM parameters for an orthotropic paper，which can help some related researchers to obtain the best cutting process parameters and the best size of cutting tools to make sure the smoothness of the paper’s edge.

1 Mathematical Model

To be simplified，it is assumed to be ortho‐tropic in the proposed model［6］，and its constitutive equation is as follows［21］

where σij，εijand cijare the stress，the strain and the stiffness matrix coefficients. The stiffness matrix C has the following relationship with the elastic modu‐lus and Poisson’s ratio［21］.

where，Eiis the elastic modulus along the i direc‐tion，and υijand Gijare the Poisson’s ratio and shear modulus，respectively. Due to the symmetry of the stiffness matrix，there are［21］

The tensile test samples were performed along with the longitudinal and transverse directions. The size of tensile test sample is shown in Fig.1，and the thickness of the paper is 0.4 mm. The wider area of 70 mm×20 mm on both sides is the grip section，the area of 80 mm×13 mm in the middle is the ef‐fective section of the experiment，the rounded cor‐ner of 15 mm is the transition zone.

Fig.1 Size of tensile test sample (half sample)

The stress-strain curves are obtained by five tensile tests on transverse and longitudinal direc‐tions，as shown in Fig.2. After taking the average value（thick line in Fig.2），the transverse and longi‐tudinal elastic modulus can be obtained as E1=1 065.8 MPa and E2=563.22 MPa.

Fig.2 Stress-strain curves for transverse and longitudinal di‐rections

The relationship between shear modulus and elastic modulus is given by Xie［6］

Let υ12=0.34［6］，and thus G12=298.35 MPa.

A micro indentation instrument was used to find E3along the thickness direction，the needle size is 50 μm×50 μm. The stress-strain curves along the thickness for both sides of the paper were shown in Fig. 3. After taking the average of the curves（thick line in Fig.3），the elastic modulus of the rough and smooth surface were obtained：E3c=278.15 MPa，E3s=392.79 MPa，and thus the aver‐age value is E3=335.47 MPa.

Fig.3 Stress-strain curves along the thickness direction of smooth and rough surfaces

Let υ23=υ31=0.01［6］，and thus according to Eq.（4），G23=208.69 MPa，G31=251.33 MPa.The engineering material constants are listed in Ta‐ble 1. Substituting the engineering material con‐stants into Eq.（2），the stiffness matrix can be ob‐tained.

Table 1 Engineering material constants

2 Analysis of Parameters

A bilinear TSL was chosen in this paper，sub‐sequently the shape of the TSL of CZM has limited influence on the macro response of the structure［20］，as shown in Fig.4. Tmaxis the maximum traction，S0the corresponding separation where the damage was beginning，and Smaxthe maximum separation at the fracture point. The area enclosed to the TSL curve is the critical fracture energy，denoted as J，and the maximum traction Tmaxand the critical fracture energy J are the two key parameters for CZM.

Fig.4 Bilinear TSL for a CZM

2.1 Critical fracture energy

Based on the material constants obtained by transverse and longitudinal tensile tests，the critical fracture energy will be estimated by numerical simu‐lations［19］.

A two-dimensional tensile specimen model［19］is established，as shown in Fig.1. The left end was fixed and the right end was loaded at a speed of 3 mm/s. To obtain the value of J-integral，the crack tip is arranged in the middle of the specimen，with a length of 0.1 mm. There are 744 CPS8R elements in this model，and the mesh at the crack tip is shown in Fig.5.

Fig.5 Half model of a model I crack specimen

The relationship between J-integral，nominal strain and analysis step time are shown in Fig.6.When the global nominal strain given by the univer‐sal testing machine reaches the fracture strain，the corresponding J-integral value achieves as critical fracture energy. It can be seen in Fig.6，the critical fracture energies for longitudinal and transverse di‐rections are 120.8 J/m2and 104.8 J/m2，respective‐ly.

Fig.6 Critical fracture energies for transverse and longitu‐dinal directions (Model I crack)

To obtain the critical fracture energy for model III crack，a three-dimensional model with a thick‐ness of 0.4 mm and a notch of 90° at the crack tip is established，as shown in Fig.7. The left side with 5 mm under the crack is fixed but the other side is loaded at a speed of 3 mm/s. A total of 1 872 C3D20 elements were used in this model. The same as the tensile test simulation，the corresponding Jintegral value is the critical fracture energy as the to‐tal strain reaches the fracture strain.

Fig.7 Sizes and boundary conditions of the model III crack(Unit:mm)

In Fig. 8，the critical fracture energies of model III crack for longitudinal and transverse directions are 28.47 J/m2and 27.27 J/m2.

2.2 Maximum traction

According to the tensile test，the critical frac‐ture stresses here are=24.68 MPa and=15.58 MPa. Based on the principle of damage me‐chanics，a damage factor d is introduced，and the critical stress is the maximum traction，which can be expressed as［19］

Fig.8 Critical fracture energies for transverse and longitudi‐nal directions (Model III crack)

Let d = 0.3，and the maximum tractions for t he CZM are22.25 MPa.

According to the simulation results of the mod‐el III crack，the critical shear strain corresponding to the total strain at the crack tip for transverse and longitudinal directions are=0.044 and=0.068. Thus，the corresponding shear stress are=1.58 MPa and=2.42 MPa. Substituted these values into Eq.（5），the maximum tractions for model III crack can be obtained，i.e.，=2.26 MPa and=3.45 MPa.

Above all，the key parameters of CZM for models I and III cracks on the orthotropic paper are shown in Table 2. It can be seen that the maximum traction and fracture energy for the model III crack are much smaller than that of the model I crack，which is consistent with the existing literature find‐ings.

3 Discussion

To verify CZM，the fracture of the model III crack is simulated. Fig.9 shows the size of the speci‐men，with a thickness of 0.4 mm. The cohesive sur‐face（red line in Fig.9）is used to represent the direc‐tion of crack propagation. There are a total of 36 000 C3D8R and 61 506 C3D6 elements. The left10 mm length of the notch is fixed，and the right 10 mm length of the notch is loaded at a speed of 10 mm/min.

Table 2 Key parameters of CZM for model I and model III cracks

Fig.9 Simulation of the fracture for model III crack

Fig.10 shows the load-displacement curves for the simulation and experimental results. It can be seen that the simulation results agreed well with the experimental results for both longitudinal and trans‐verse directions. Once the load begins to drop，the paper has been torn. For the experiments，it is diffi‐cult to make sure that the mode III crack propagates along the planned route. Furthermore，in most cas‐es，the layers of paper will be pulled apart after the load decreased. However，the simulation test does not consider the interaction between layers of paper.Therefore，only the beginning of the model III frac‐ture was compared.

4 Conclusions

Fig.10 Comparison of load-displacement curves between re‐sults of simulations and experiments

During the paper cutting process，burr along the cutting side are mostly not considered. The main reason is that the mechanical behavior of paper cut‐ting is unclear. In this paper，the model III fracture based on CZM for the orthotropic paper was stud‐ied. The material constants for the orthotropic paper were obtained by tensile and micro indentation tests.Based on the tensile test，the maximum traction and the critical fracture energy of CZM were obtained ac‐cording to the damage mechanics and numerical sim‐ulations. The model III fracture of the orthotropic paper was simulated to verify the accuracy of CZM.By comparing the load displacement curves between the results of simulations and experiments，it is proved that the key CZM parameters estimated in this paper are reasonable and credible.