Conclusions - 探討缺陷石墨板之局部性質

Hardy local stress formulation was employed successfully to calculate the local stress distribution of the graphene sheet with free surface and central cracks. Based on the local stress distribution, the local properties of graphene sheet with presence of free surface and cracks were determined. Results indicated that for the graphene with free surfaces, when van der Waals force is present, the bond length at the edge is shortened; therefore the edge of the graphene sustains compressive stress. However, when van der Waals force is absent, the bond length in the whole atomistic graphene structure is remaining constant and therefore there is no stress induced on the atomistic structure. Moreover, the local stress distribution near the free surface of the graphene can be characterized successfully using Hardy stress formulation which exactly exhibits the local deformation of the microstructures within the graphene sheet.

With regard to the graphene with central cracks subjected to remote uniaxial loading, it was observed that for continuum models, both FEA and LEFM solution yield stress singularity near the crack tip. In addition, for small cracks, LEFM solution would deviate from the FEA. For discrete models, it was found that Hardy stress formulation exhibits non-local attribute near the crack tip and the maximum stress is in agreement with the non-local elasticity solution. Based on the local stress distribution, the fracture properties such as stress intensity factor and fracture toughness can be deduced directly from the maximum stress hypotheses. It is revealed that the stress intensity factors obtained from continuum models and discrete models are close to each other; therefore it can be used to predict the state of stress field near the crack tip for both continuum solids and atomistic structures. On the other hand, in the discrete models, the fracture toughness is found to be sensitive to the crack length

when the length is below 40 lattices. The smaller the crack length is, the lower the fracture toughness will be. Therefore, for the small crack length, the concept of LEFM may not be applicable to characterize the fracture behaviors of atomistic graphene structure.

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Appendix A

MATLAB code of calculating non-local stress field for crack-tip problem using Gaussian function as distribution curve

%---beginning of the code---clc;

clear;

%---Eringen's non-local

elasticity---% Reference:

% Eringen, A.C., Speziale, C.G., Kim, B.S., 1977.

% Crack-tip problem in non-local elasticity. Journal of the Mechanics and Physics of Solids 25, 339¡V355.

% Distribution curve alpha is Gaussian function

%---input parameters---ad=2.46; % graphene lattice length

L=2.46*23/2; % half crack length (L)

h=1.9; % smoothing length in Gaussian function beta=ad/h;

e=ad/(2*beta);

tou=-2; % applied loading on "crack surface" unit: GPa ndiv=1000; % output data number

tol=1e-10; % integral error tolerance x=linspace(0.0*L,2.0*L,ndiv); % interval of stress field

%---calculating non-local stress field---for i=1:ndiv

xx=x(i)/L;

f=@(K)besselj(1,K).*cos(K*xx).*((1-erf(e*K/L)).*(1-2*(e*K/L).^2)+(2*(

e*K/L)/sqrt(pi)).*exp(-(e*K/L).^2)); % eqn. (4.1.52) in the thesis

y(i)=quadgk(f,0,inf,'RelTol',tol,'AbsTol',tol,'MaxIntervalCount',8000 );

end

%---superposition of stress field---str=tou*y-tou;

xx=(x'-L)./ad; % plot data from crack tip and normalized the position

% with respect to lattice

str=str'./(-tou); % normalized the stress field with respect

% to applied loading plot(xx,str,'r')

%---end of the

code---Appendix B

MATLAB code of calculating non-local stress field for crack-tip problem using Triangular function as distribution curve

%---beginning of the code---clc;

clear;

%---Eringen's non-local

elasticity---% Reference:

% Eringen, A.C., Speziale, C.G., Kim, B.S., 1977.

% Crack-tip problem in non-local elasticity. Journal of the Mechanics and Physics of Solids 25, 339¡V355.

% Distribution curve alpha is Triangular function

%---input parameters---ad=2.46; % graphene lattice length

L=2.46*23/2; % half crack length (L) e=ad/L;

ndiv=3000; % output data number

tol=1e-10; % integral error tolerance

uplim=2*pi*L/ad; % upper limit of integral, as suggested in

%Eringen's paper

x=linspace(0.0*L,2*L,ndiv); % interval of stress field

p=2; % applied loading on "crack surface" unit: GPa

%---calculating non-local stress field---xx=x./L;

% the following is the eqn. (4.1.57) in the thesis

f=@(K)(-6/pi*((13/30*(K*e).^-1+32/15*(K*e).^-3-1/20*(K*e)).*cos((K*e) )+(19/30*(K*e).^-2-1/20).*sin((K*e))+(1/3-1/20*(K*e).^2).*sinint(K*e) -1/6*pi-32/15*(K*e).^-3+pi/40*(K*e).^2)).*besselj(1,K).*cos(K.*xx).*(

-p);

y=quadv(f,0,uplim);

%---superposition of stress field---yy=y+p;

yy=yy./p; % normalized the stress field with respect

% to applied loading

xx=x-L; % plot data from crack tip and normalized the

%position with respect to lattice xx=xx'./ad;

plot(xx,yy,'b')

%---end of the

code---Table 3.1.1 Cut-off radius corresponding to different smoothing lengths h

h=1.8 h=2.5 h=3.0

Rc (Å) 8 9 12

Table 4.2.1 Dimension of finite element model with different crack lengths

Unit: Å 2L/a=3 2L/a=41 2L/a=81

2W 73.8 1008.6 1992.6

2H 76.68 1005.36 1985.16

Table 4.2.2 Material properties of graphene sheet obtained from MD simulation Young's Modulus (GPa) Poisson's ratio

790.7 0.27

Table 4.3.1 Different widths of discrete graphene model with crack length of 5 lattices

Unit: Å L/W=0.05 L/W=0.1 L/W=0.3

2W 246 123 49.2

2H 247.08 119.28 51.12

Table 4.3.2 Dimension of discrete graphene model with different crack lengths 2L/a

3 5 7 19 21 23 41 61 81 2W 73.8 123 172.2 467.4 516.6 565.8 1008.6 1500.6 1992.6

2H 76.68 119.28 170.4 468.6 519.72 562.32 1005.36 1499.52 1985.16 Unit: Å

Table 4.3.3 Maximum local stress with different crack lengths Maximum stress (GPa)

2L/a

Non-local (Gaussian function) Hardy stress Tsai stress

3 2.705 2.588 3.29

5 3.224 3.171 4.19

7 3.678 3.672 4.94

19 5.701 5.850 8.08

21 5.967 6.141 8.49

23 6.230 6.412 8.88

41 8.198 8.503 11.84

61 9.950 10.356 14.44

81 11.431 11.923 16.64

Table 4.4.1 Stress intensity factor from Hardy, FEM and continuum mechanics KI (Pa m)x10⁴

2L/a

Hardy stress FEM KI =σ₀ πL Error

(Hardy with FEM)

3 6.44 6.99 6.81 -7.87 %

5 8.26 8.92 8.79 -7.40 %

7 9.67 10.39 10.40 -6.93 %

19 16.12 17.02 17.14 -5.29 %

21 17.02 17.88 18.02 -4.81 %

23 17.87 18.70 18.85 -4.44 %

41 24.23 25.26 25.17 -4.08 %

61 29.34 30.42 30.71 -3.55 %

81 34.04 35.28 35.38 -3.51 %

Table 4.4.2 Stress concentration factor with different crack lengths 2L/a C

Non-local (Gaussian function) Hardy stress

3 0.781 0.747

5 0.721 0.709

7 0.695 0.694

19 0.654 0.671

21 0.651 0.670

23 0.649 0.669

41 0.640 0.664

61 0.637 0.663

81 0.635 0.662

Table 4.4.3 Applied loading to achieve σc with different crack lengths σ0 (GPa)

2L/a

Non-local (Gaussian function) Hardy stress

3 0.739 0.773

5 0.620 0.631

7 0.544 0.545

19 0.351 0.342

21 0.335 0.326

23 0.321 0.312

41 0.244 0.235

61 0.201 0.193

81 0.175 0.168

Table 4.4.4 Fracture toughness with different crack lengths KIC (Pa m)x10⁴

2L/a

Non-local (Gaussian function) Hardy stress

3 2.517 2.632

5 2.726 2.773

7 2.828 2.832

19 3.006 2.930

21 3.020 2.934

23 3.029 2.938

41 3.071 2.960

61 3.086 2.965

81 3.096 2.969

ΔH=3.4 Å ΔW=2.46 Å

ΔW ΔH ΔH=3.4 Å

ΔW=2.46 Å

ΔW ΔH

Fig. 3.1.1 Dimension of the dividing plane adopted in Tsai's stress formulation

α β

x λ=1 λ=0

r

r r

^β

− = r

r r

^α

− =

m

Figure 3.1.2 Interpretation of bond function

r^α-r (Å) Localizationfunction(Å-3 )

-9 -6 -3 0 3 6 9

0.000 0.010 0.020 0.030

h=1.8 h=2.5 h=3.0

Figure 3.1.3 Localization functions with various smoothing lengths

X Y

51.65 Å

46.86 Å

Periodic B.C

Periodic B.C.

Periodic B.C

X Y

51.65 Å

46.86 Å

Periodic B.C

Periodic B.C.

Periodic B.C

` Figure 3.2.1 Continuous graphene sheet with periodic boundary conditions

X(Å) σ yy(GPa)

-20 -10 0 10 20

8.0 9.0 10.0 11.0 12.0

Hardy stress Lutsko stress Tsai stress

Figure 3.2.2 Local stress in the periodic graphene with bonded interaction

X(Å) σ yy(GPa)

-20 -10 0 10 20

8 9 10 11 12

Hardy stress Lutsko stress

Figure 3.2.3 Hardy and Lutsko stress in the periodic graphene with bonded and non-bonded interaction

X(Å) σ yy(GPa)

-20 -10 0 10 20

0.0 5.0 10.0 15.0 20.0

ΔW=4ΔW=8 ΔW=12 ΔW=18

Figure 3.2.4. Tsai stress in the periodic graphene with bonded and non-bonded interaction with different dividing planes

46.86 Å

70.65 Å

Y

X

surface surface

46.86 Å

70.65 Å

Y

X Y

X

surface surface

Figure 3.2.5 Finite graphene sheet with free surfaces in the x direction

X(Å) σ yy(GPa)

-20 -10 0 10 20

-0.0001 -5E-05 0 5E-05 0.0001

Hardy stress Lutsko stress Tsai stress

Figure 3.3.1 Local stress in the finite graphene sheet with bonded interaction

X(Å)

ΔL(Å)

-20 -10 0 10 20

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

Figure 3.3.2 Bond length of finite graphene sheet with bonded interactions at stress free state

X(Å) σ yy(GPa)

-25 -20 -15 -10 -5 0 5 10 15 20 25 0

2 4 6 8 10 12 14 16 18

Hardy stress Lutsko stress Tsai stress

(a)

X(Å) σ yy(GPa)

10 15 20 25

0 2 4 6 8 10 12 14 16 18

Hardy stress Lutsko stress Tsai stress

(b)

Figure 3.3.3 Local stress distribution in the finite graphene sheet with bonded interactions (a) global view (b) local view

X(Å) σ yy(GPa)

10 15 20 25

0 2 4 6 8 10 12 14 16 18

Hardy stress

Tsai stress (ΔW=2.46) Tsai stress (ΔW=7.38)

Figure 3.3.4 Tsai stress with different dividing planes in the finite graphene sheet with bonded interactions near surface

X(Å)

ΔL(Å)

-20 -10 0 10 20

0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 0.022

Figure 3.3.5 Bond length of the finite graphene sheet with bonded interactions at uniaxial stress state

X(Å)

ΔL(Å)

-20 -10 0 10 20

-0.010 -0.008 -0.006 -0.004 -0.002 0.000 0.002

Figure 3.4.1 Variation of bond length for the finite graphene sheet with bonded and non-bonded interactions at stress free state

X(Å)

Stress(GPa)

-35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 -0.8

-0.4 0.0 0.4

(a)

X(Å)

Stress(GPa)

10 15 20 25 30 35

-0.8 -0.4 0.0 0.4

(b)

Figure 3.4.2 Hardy stress distribution of the finite graphene sheet with bonded and non-bonded interactions at stress free state (a) global view (b) local view

X(Å) σ yy(GPa)

-30 -20 -10 0 10 20 30

-1.5 -1 -0.5 0 0.5 1

Figure 3.4.3 Lutsko stress distribution of the finite graphene sheet with bonded and non-bonded interactions at stress free state

X(Å) σ yy(GPa)

-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 -12

-8 -4 0 4 8 12

Figure 3.4.4 Tsai stress distribution of the finite graphene sheet with bonded and non-bonded interactions at stress free state

X(Å) σ yy(GPa)

-30 -20 -10 0 10 20 30

0 2 4 6 8 10 12 14 16

(a)

X(Å) σ yy(GPa)

10 15 20 25 30

0 2 4 6 8 10 12 14 16

(b)

Figure 3.4.5 Hardy stress distribution of the finte graphene sheet with bonded and non-bonded interactions at uniaxial stress state (a) global view (b) local view

X(Å)

ΔL(Å)

-20 -10 0 10 20

0.007 0.010 0.012 0.015 0.017 0.020

Figure 3.4.6 Variation of bond length for the finite graphene sheet with bonded and non-bonded interactions at stress state of 10GPa

X(Å) σ yy(GPa)

-30 -20 -10 0 10 20 30

0 2 4 6 8 10 12 14 16 18

Figure 3.4.7 Lutsko stress distribution of the finite graphene sheet with bonded and non-bonded interactions at uniaxial stress state

X(Å) σ yy(GPa)

-30 -20 -10 0 10 20 30

-12 -8 -4 0 4 8 12 16 20 24 28

Figure 3.4.8 Tsai stress distribution of the finite graphene sheet with bonded and non-bonded interactions at uniaxial stress state

r'-r (Å)

α(Å-2 )

-10 -5 0 5 10

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Gaussian (h=1.9) Triangle

Fig. 4.1.1 Distribution curves α employed in non-local elasticity

t

-l l

t

-L L

Y X Y

X Y

t

-l l

t

-L L

t

0₀

t

-l l

t

0₀

t

-L L

Y X Y

X Y

Fig. 4.1.2 Boundary conditions of the line-crack problem in non-local elasticity

Fig. 4.1.3 Superimposition of the boundary condition in the non-local elasticity

x/a

Fig. 4.1.4 Different methods of solving non-local elasticity problem

x/a σ yy/σ 0

0 1 2 3 4 5 6 7 8 9 10

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

Nonlocal-Gaussian Nonlocal-Triangle

Fig. 4.1.5 Stress distribution in the graphene with crack lengths of 3 lattices

x/a σ yy/σ 0

0 2 4 6 8 10

1 2 3 4 5

Nonlocal-Gaussian Nonlocal-Triangle

Fig. 4.1.6 Stress distribution in the graphene with crack lengths of 41 lattices

x/a σ yy/σ 0

0 2 4 6 8 10

1 2 3 4 5 6 7

Nonlocal-Gaussian Nonlocal-Triangle

Fig. 4.1.7 Stress distribution in the graphene with crack lengths of 81 lattices

X 2L

Fig. 4.2.1 Finite element model for continuum graphene sheet. Note that the dimension has different values for the three different models.

Fig. 4.2.2 Finite element mesh for the graphene with crack lengths of 41 lattices: (a) mesh for the entire model (quarter model) and (b) magnified view of the fine mesh

around the crack tip.

x/a σ yy/σ 0

0 1 2 3 4 5 6

0.5 1 1.5 2 2.5

FEM LEFM

Fig. 4.2.3 Stress distribution in the graphene with crack lengths of 3 lattices

x/a σ yy/σ 0

0 2 4 6

1 2 3 4 5 6 7

FEM LEFM

Fig. 4.2.4 Stress distribution in the graphene with crack lengths of 41 lattices

x/a σ yy/σ 0

0 2 4 6

1 2 3 4 5 6 7 8 9 10 11

FEM LEFM

Fig. 4.2.5 Stress distribution in the graphene with crack lengths of 81 lattices

Figure 4.3.1 Atomistic structure of the graphene sheet subjected to uniaxial loading.

X(Å) σ yy(GPa)

0 2 4 6 8 10

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4

L/W=0.05 L/W=0.1 L/W=0.3

Figure 4.3.2 Local stress distribution of the graphene with crack lengths of 5 lattices and different graphene widths.

x/a σ yy/σ 0

0 2 4 6 8 10

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7