Phân tích dạng phá hủy hỗn hợp (Mixed Mode) của vật liệu Graphene đơn lớp bằng phương pháp phần tử hữu hạn

PHÂN TÍCH DẠNG PHÁ HỦY HỖN HỢP (MIXED MODE) CỦA VẬT LIỆU

GRAPHENE ĐƠN LỚP BẰNG PHƯƠNG PHÁP PHẦN TỬ HỮU HẠN

ANALYSIS MIXED-MODE FRACTURE TOUGHNESS (MODE I/II) OF

A SINGLE LAYER GRAPHENE SHEET BY FINITE ELEMENT

METHODS SIMULATION

Nguyen Duc Tai ¹, Nguyen Minh Ky ²

¹Cao Thang Technical College, Vietnam.

^1,2Ho Chi Minh City University of Technology and Education, Vietnam.

Received 11/10/2017, Peer reviewed 24/10/2017, Accepted for publication 15/11/2017.

ABSTRACT

The mode I and mode II fracture behavior of a single-layer graphene sheet (SLGS)

containing a center crack was characterized with the results of an finite element method (FEM)

and an analytical model. Finite element method based micromechanical analysis is used to

understand the fracture behavior of functionally SLGS. The in-plane shear fracture of a cellular

material was analyzed theoretically for deriving the K_IC, K_IICof SLGS, and FEM results were

obtained. Crack is created by removing certain number of cells pertaining to a crack length. The

inputs to the model are the geometric parameters required to model the repeating unit cell and

tensile strength of the strut. Mixed-mode fracture of armchair direction graphene sheet was

studied for various angle mode I/II ratios. The effect of mixed-mode loading on the Arcan

specimen was investigated, all the systems considered in this study mimic real service conditions.

Applying rotation boundary conditions does not inﬂuence the calculated fracture toughness

signiﬁcantly. The mixed-mode fracture criterion was determined, and the obtained fracture

envelope was in good agreement with that of another study.

Keywords: fracture of graphene; mixed mode of fracture toughness; stress intensity factor

graphene; fracture toughness of graphene; micromechanic of graphene;

TÓM TẮT

Dạng phá hủy thứ nhất và thứ hai của tấm graphene đơn lớp với vết nứt ở giữa tấm là mô

hình điển hình với những kết quả đã được nghiên cứu bởi mô hình phân tích cơ học và phần tử

hữu hạn. Phương pháp phần tử hữu hạn dựa trên phân tích vi mô được sử dụng để hiểu rõ ràng

hành vi phá hủy của graphene đơn lớp. Bằng phương pháp phân tích lý thuyết cũng như phương

pháp phần tử hữu hạn, người ta đã tìm ra hệ số cường độ tới hạn của dạng phá hủy thứ nhất K_IC

và dạng hai K_IIC. Vết nứt được tạo ra bằng cách loại bỏ số lượng các bond theo chiều dài vết nứt.

Các thông số đầu vào của mô hình là những thông số hình học, thuộc tính của vật liệu graphene

cần thiết để tính toán mô phỏng và xác định cường độ kéo của bond ngay đầu vết nứt. Dạng phá

hủy hỗn hợp của dạng I và II theo hướng armchair đã được nghiên cứu với các góc pha khác

nhau. Mô hình thực hiện dạng phá hủy hỗn hợp này được thiết lập với hướng tải và điều kiện

biên dựa vào mô hình của Arcan. Áp dụng điều kiện biên quay không ảnh hưởng đến kết quả tính

toán. Giá trị hệ số cường độ ứng suất ở từng góc pha đã được xác định và được so sánh đánh

giá với các kết quả nghiên cứu trước.

Từ khóa: Sự phá hủy vật liệu graphene; phá hủy hỗn hợp vật liệu graphene đơn lớp; hệ số

cường độ ứng suất graphene; cường độ phá hủy tới hạn graphene; mô hình cơ học vi mô

graphene

widely used in lightweight structures [2] and

proposed a bending model of cell walls to

analyze mechanical properties of cellular

materials. They found that mechanical

properties of cellular materials are related to

cell geometry and material properties of

solid cell-wall materials. They have made an

1. INTRODUCTION

A fundamental understanding of the

fracture mechanisms in graphene is not only

scientifically interesting, but also practically

important for preventing or controlling

fracture in graphene [1]. Honeycombs are

analytical estimate for the fracture toughness

of elastic–brittle hexagonal honeycombs.

of each strut from which the critical skin

stress of the first unbroken cell edge ahead

of the crack tip can be calculated. Failure

occurs when the critical skin stress reaches

the modulus of rupture of the cell wall,

which is assumed to be constant at this

point in the study [8]. The thickness (푑) and

length ( 푙 ) of a graphene sheet vary

depending on the study, and different values

have been reported by Duplock [9], Zhou

[10], Tu and Ou-Yang [11] and Pantano

[12], Kudin [13], Goupalov [14], Reddy

[15]. We used AMBER force constants [16],

where the thickness of the C–C bond is

0.84Å and Poisson’s ratio is 0.186 and

Young’s modulus is 1.05TPa for an

equilibrium length l of 1.38 Å.

Although the above studies have

contributed to the understanding of fracture

of graphene, the important issue of

mixed-mode fracture, i.e. fracture of

graphene under combined tensile and shear

loading conditions, still needs to be

addressed.

Recently, Zhang et al [3] and models on

the fracture toughness of shear initiated

computational study on this important issue.

They reported stress intensity factors and

direction of crack propagation at room

temperature for different orientation of

loading conditions. Datta et al [4] performed

classical

molecular

dynamics (MD)

simulations on a graphene sheet with crack

like flaw and investigated the complex

mixed-mode fracture behavior of grapnene.

However, the effect of the mixed-mode

fracture of graphene has not been performed

by FEM simulation. In this study, the effect

of crystal orientation armchair directions on

the mixed mode fracture behavior of

graphene sheet needs to be studied. We

investigate the effect of applying rotations as

boundary conditions in addition to

prescribed

translational

displacement

boundary conditions on the predicted

fracture toughness. This method has been

used by Prasanna Thiyagasundaram [5] and

S. Choi [6]. We directly apply

displacements based on the K-field on the

boundary of the micromechanical model. A

modified version of the Arcan specimen is

made for the mixed-mode fracture.

Mixed-mode fracture analysis using FEM

simulations was conducted to determine the

fracture toughness of SLGS for various

angles. A commercial software ANSYS is

used for FEM calculations and comparison

with previous results.

Fig.1. The geometry for the finite

element analysis of SLGS having a central

crack with a length of 2c.

The results of the analysis [17] indicate

that if the crack length is higher than 10

cells the fracture toughness is constant. Only

one crack length is used in present work with

c= 25 cells. Modulus of rupture, _fsof SLGS

is used to [18], so that the SLGS is linear

elastic to fracture and that it has constant

modulus of rupture, _fsof 27.89 MPa is

calculated as follows. The micro model uses

beam elements to model the struts and to

determine the maximum stresses at crack tip

and contained approximately 273,000

elements. From the failure criterion for the

strut material the fracture toughness of the

graphene sheet is estimated. A known

uniform tensile stress, , is applied to the

SLGS. This generates an internal stress in the

cell wall ahead of the crack tip of:

2. MODELS AND METHODOLOGIES

For the finite element analysis, a grid of

100x60cells was employed (Fig 1) where

W(width) and H(high). Here, ANSYS 14.5

[7] was utilized to model the struts with the

armchair model. The stress distribution

along the outermost layer of cells is

uniform and the stress field far away from

the crack tip is undisturbed. The analysis

gives the forces and moments at both ends

퐾

퐼

 = 푡푎푛⁻¹(_퐾)

(4)

푦

퐼

푃

퐴

퐼퐼

 = 푀

+

=

and an effective SIF, 퐾_푒푓푓, at the initial

crack length is evaluated as

= _{퐵푒푛푑푖푛푔 푠푡푟푒푠푠}+ _{퐴푥푖푎푙 푡푟푒푠푠}(1)

where _{퐵푒푛푑푖푛푔 푠푡푟푒푠푠}and _{퐴푥푖푎푙 푡푟푒푠푠}

are found directly from the FEM analysis.

Fracture occurs when the internal stress in

the cell wall ahead of the crack tip reaches

the modulus of rupture of the cell wall, _fs, at

an applied stress of,  = _푓푠. Mode-I and

Mode-II stress intensity factors [17,18] were

evaluated as

2

( )²

)

퐾_푒푓푓= √ 퐾_퐼+ 퐾_퐼퐼

(

(5)

Therefore the far-field behavior of

pristine graphene is assumed to be well

represented by the solution since the stress

singularity decreases apart from the crack

tip. The geometry and boundary conditions

are shown in Fig 3. The average value of

mode I fracture toughness (K_IC) obtained

from the [17,18] with FEM simulations with

2

 푑

√

퐾_퐼퐶= _푓푠

(2)

(3)

c= 25cells at  = 90^ois 2.63 MPa 푚 and

√

4

3 푙(2 3푙+푑)

√

also mode II fracture toughness (K_IIC)

obtained from this chapter by FEM

3

  푑

√

퐾_퐼퐼퐶= _푓푠

2

12 2 푙

푙

√

simulations at  = 0^ois 1.68MPa 푚 .

√

where, _푓푠 and _푓푠are the tensile and

shear stress values at the instant of first bond

rupture.

The fracture toughness of SLGS was

determined from FEM with the modified

version of the Arcan specimen [19, 20] under

different mixed-mode loading conditions. In

this section, a modified version of the Arcan

specimen is made for the mixed-mode

fracture of SLGS, which allows mode-I,

mode-II, and almost any combination of

mode-I and mode-II loading with the same

specimen configuration.

Fig.3. Finite element model of a typical

Arcan specimen with 45^o.

The model must change with change in

maximum normal stress direction. It can be

noted from Fig.4-5 that the crack length

direction is almost insensitive to the change

in loading angle (from 15°, 30⁰, 45° for

armchair and from 45°, 60°, 75⁰for zigzag

sheet), or change in maximum normal stress

direction.

Fig.2. Model of the mixed-mode

loading.

Phase angle  (mixed mode parameter

or equivalent crack angle) is defined as

(b)

Fig.5. (a) FEM with bending stress

results (b) View of detail at crack tip.

(a)

Critical stress intensity factors values

were derived from the maximum stress at

crack tip. K_Iand K_IIwere calculated using

FEM simulations. K_Iincreases and K_II

decreases as the mode I loading contribution,

i.e. as  increases from 0^oto 90^o. It is seen

that for loading angles  ≥ 45^o, the mode-I

contribution is greater than that of mode-II

and the opening mode fracture becomes

dominant. For loading angles  ≤ 45^othere

is an opposite trend and the shearing mode

fracture becomes dominant.

3. RESULTS AND DISCUSSION

We investigate closed form solutions for

the fracture properties of SLGS using

mechanical cellular solids. The analysis of

this field for different loading angle has

shown which length must be employed in

the fracture toughness evaluation. In order to

verify the accuracy of the modeling

procedure and the formulation developed in

this study a finite element model of a

graphene sheet is constructed in ANSYS

commercial package [7]. The numerical

simulations presented are encouraging (Fig.

4-5). Researches, however, are still seeking

a more efficient computational modeling for

predicting fracture toughness of SLGS.

(b)

Fig. 4. (a) FEM with axial stress results (b)

View of detail at crack tip.

It is well known that the fracture energy

under mode I loading is different from that

under mode II loading in SLGS. For the

situation under mixed-mode loading, there is

an interaction between the two fracture

modes and it is necessary to find a

relationship between mode I and mode II

stress intensity factors. It is generally

(a)

proposed in the form as

1.2

1.0

0.8

0.6

0.4

0.2

0.0

Present Work

Bin Zhang [3]

Datta [4]

푚

)

푛

퐾

퐼

(

+ (_퐾^퐾^퐼퐼) = 1

(6)

퐾

퐼퐶

퐼퐼퐶

The fracture toughness of SLGS was

determined from FEM with a modified

version of the Arcan specimen for various

phase angles under different mixed-mode

loading conditions. A failure criterion was

developed by plotting the average fracture

toughness data on a normalized volume

diagram (Fig.6). It can be seen that an equal

Fracture zone

Safe zone

elliptical equation (i.e m=2, n=2 and 퐾_퐼퐶

≠

퐾_퐼퐼퐶is suitable for characterizing the

mixed-mode fracture criterion of Bin Zhang

[3] and [4].

0.0

0.2

0.4

0.6

0.8

1.0

K_I/K_IC

1.2

Fig.7. Normalized prediction of the

mixed mode fracture criteria of SLGS.

PresentWork

1.0

Bin Zhang [3]

The curves in Fig.7 are the mixed-mode

(I/II) fracture curves of the Arcan specimen

for different phase angle. To use these

curves in theory cases, first the phase angle

for each stress intensity factors should be

calculated for the phase angle for each

components under the applied load. These

factors are then normalized as K_II/K_ICand

K_I/K_ICand the corresponding point is

determined in Fig 7. A ﬁnite element based

method has been used to calculate the

fracture toughness of graphene. Increase in

K_Ican be attributed to the increase in normal

stress component at the crack tip with the

loading angle from 0⁰to 45⁰and similar

previous results. In contrast, the K_IIvalues

decrease with loading angle from a peak

value at > 45° to = 90°. Also, the K_II

values for graphene sheets loaded in

amrchair edges are lower compared to

previous results for all the loading cases.

With this results show that the Mode-II

fracture toughness converges to discrete

models are not applied and to when the

rotations are applied. The mixed mode

fracture criteria are determined and fracture

surfaces obtained at different loading angles

for pristine graphene are discussed.

Datta [4]

0.8

0.6

0.4

0.2

0.0

0.2

0.4

K_I/ K_IC

0.6

0.8

1.0

Fig.6. Normalized mixed-mode fracture

envelope and predictions of the mixed-mode

fracture criteria with a crack length of

25cells.

Since the Arcan specimen developed by

Arcan [19] has very similar geometry and

boundary conditions to the specimen used in

the investigation, results were expected to be

very close. For the purpose of comparison,

in Fig 6, the finite element results together

with those calculated MD simulations by

Bin Zhang [3], Datta [4] and a homogeneous

Arcan specimen are presented. The mixed

mode FEM results are shown in Fig. 7 in a

K_II/K_ICversus K_I/K_ICdiagram and also

shown in this figure is the MD simulations

prediction of a well-known fracture criterion

in [3] and [4].

4. CONCLUSIONS

A FEM model for mixed mode (mode I/

mode II) SLGS has been implemented. One

advantage of such approach is that it’s easy

implementation and considerably lowers

computational efforts when compared with

MD techniques. The expression for mixed

mode fracture toughness fracture criterion

for SLGS and foams are derived. It is found

that mixed mode fracture toughnesses of

SLGS depend on their cell geometry and the

modulus of rupture of materials. When the

variation of modulus of rupture is taken into

account, the finite element method study

indicates that the existing model, based on a

continuum approach, is satisfactory for

graphene sheet with semi-crack length

greater than 10 times the cell size.

dimensions and in situ properties of the

graphene cell walls, the finite element

results seem to be reasonable. Therefore, it

is suggested that the SIF is an appropriate

parameter, which can be employed in the

atomistic model and the micromechanical

model for describing the fracture of

covalently bonded graphene sheet. Our

results suggest that the mixed mode fracture

toughness of SLGS is independent from the

crack length and predicted toughness of

graphene shown in Fig 6 and Fig 7. The

simulation of fracture strength is in good

agreement with that predicted by the

energy-based quantized fracture mechanics.

For the sake of comparison, the

analytical finite element model with the

same geometric parameters and material

properties as the atomistic graphene sheet

was constructed, and the corresponding

stress intensity factors were calculated from

the crack closure method. Results indicated

that the stress intensity factors obtained

from the micromechanical model exhibit

good agreement with those derived from

discrete atomistic model. Comparing the

normalized result for fracture toughness

given in FEM with the MD results, one can

note that the difference is about 15% with

Bin Zhang and 5% with Datta. Considering

the idealizations made in the model and

variations in the unit-cell shape and

Generally, the crack propagation in

mixed mode I/II can be predicted by the

Arcan specimen and the propagation of a

mixed-mode crack in a brittle fracture

graphene is investigated. The brittle fracture

of graphene prefers along armchair edges

concerning with its lower toughness and

applied complex mechanical stresses in

static rupture. The comparison between Bin

Zhang [3], Datta [4] and our numerical

results shows that the numerical results are

in good agreement with the MD simulations

results.

Acknowledgements:

This work was supported by the 2017 Research and Development Fund of Ho Chi Minh

City University of Technology and Education.

REFERENCES

[1] T. Zhu, J. Li, S. Ogata and S. Yip. Mechanics of ultra-strength materials. Materials Research

Society Bulletin. 34, pp. 167-172, 2009.

[2] L.J. Gibson, M.F. Ashby. Cellular Solids: Structure and Properties. Cambridge University

Press. 2nd ed, 1999.

[3] Bin Zhang, Lanjv Mei, and Haifeng Xiao. Nanofracture in graphene under complex

mechanical stresses. Applied Physics Letters. 121915, 101, 2012.

[4] Dibakar Datta, Siva P.V. Nadimpalli, Yinfeng Li, Vivek B. Shenoy. Effect of crack length and

orientation on the mixed-mode fracture behavior of graphene. Extreme Mechanics Letters.

2015.

[5] Prasanna Thiyagasundaram, Junqiang Wang, Bhavani V. Sankar, Nagaraj K. Arakere.

Fracture toughness of foams with tetrakaidecahedral unit cells using finite element based

micromechanics. Engineering Fracture Mechanics. 78, pp. 1277–1288, 2011.

[6] S. Choi and B. V. Sankar. Fracture Toughness of Carbon Foam. Journal of Composite

Materials. 37, 2101, 2003.

[7] ANSYS user’smanual, 2012.

[8] Pei. QX, Zhang. YW, Shenoy. VB. A molecular dynamics study of the mechanical properties

of hydrogen functionalized graphene. CARBON 48. 3, pp. 898-904, 2010.

[9] Duplock E J, Scheffler M and Lindan P J D. Hallmark of Perfect Graphene. Physical Review

Letters. 92(22), pp.225502-225505, 2004.

[10]Zhou X, Zhou J and Ou-Yang Z. Strain energy and Young’s modulus of single-wall carbon

nanotubes calculated from electronic energy-band theory. Physical Review B. 62, pp.

13692-13696. 2000.

[11]Tu Z and Ou-Yang. Single-walled and multiwalled carbon nanotubes viewed as elastic tubes

with the effective Young’s moduli dependent on layer number. Physical Review B. 65,

233407, 2002.

[12]Pantano A, Parks D M and Boyce MC. Mechanics of Deformation of Single- and Multi-Wall

Carbon Nanotubes. Mech. Physical Solids. 52, pp. 789-821, 2004.

[13]Konstantin N Kudin, Gustavo E Scuseria and Boris I Yakobson. C₂F, BN, and C nanoshell

elasticity from ab initio computations. Physical Review B. 65, 235406, 2001.

[14]Goupalov S V. Continuum model for long-wavelength phonons in two-dimensional graphite

and carbon nanotubes. Physical Review B. 71, 085420, 2005.

[15]Reddy C D, Rajendran S and Liew K M. Equilibrium configuration and continuum elastic

properties of finite sized graphene. Nanotechnology. 17, pp. 864-870, 2006.

[16]F Scarpa, S Adhikari and A Srikantha Phani. Effective elastic mechanical properties of single

layer graphene sheets. Nanotechnology. 20, 065709 (11pp), 2009.

[17]Minh-Ky, Nguyen. Fracture analysis of single layer graphene sheets. Journal of Technical

Education Science. No.35A, 2016,

[18]Minh-Ky, Nguyen. Micromechanics Study for Fracture Behavior of Single Layer Graphene

Sheet. Ulsan of University, Thesis PhD, 2013.

[19]Banks-Sills L, Arcan M, Bortman Y. A mixed mode fracture specimen for mode II dominant

deformation. Engng Fract Mech. 20(1), pp. 145–157, 1984.

[20]Banks-Sills L, Arcan M. A compact mode II fracture specimen, fracture mechnics, ASTM

STP. 905, vol. 17, pp. 347–363, 1986.

Corresponding author:

Nguyen Minh Ky, PhD

Ho Chi Minh City University of Technology and Education, Vietnam

Email: [email protected]n

BÀI BÁO KHOA HỌC

THỰC HIỆN CÔNG BỐ THEO QUY CHẾ ĐÀO TẠO THẠC SỸ

Bài báo khoa học của học viên

có xác nhận và đề xuất cho đăng của Giảng viên hướng dẫn

Bản quyền tác phẩm đã được bảo hộ bởi Luật xuất bản và Luật Sở hữu trí tuệ Việt Nam.

Nghiêm cấm mọi hình thức xuất bản, sao chụp, phát tán nội dung khi chưa có sự đồng ý

của tác giả và Trường Đại học Sư phạm Kỹ thuật TP. Hồ Chí Minh.

ĐỂ CÓ BÀI BÁO KHOA HỌC TỐT, CẦN CHUNG TAY BẢO VỆ TÁC QUYỀN!

Thực hiện theo MTCL & KHTHMTCL Năm học 2018-2019 của Thư viện Trường Đại học

Sư phạm Kỹ thuật Tp. Hồ Chí Minh.