等效介面層模型程式

第五章結論

附錄 7 等效介面層模型程式

clear all

%---

% Gm: matrix shear modulus

% Em: matrix young modulus

% vm: matrix ratio

% Gp: particle shear modulus

% Ep: particle young modulus

% vp: interface possion ratio

% Gi: interface shear modulus

% Ei: interface young modulus

% vi: interface possion ratio

%--- Gm=1.332;

Em=3.678;

vm=(Em/(2*Gm))-1;

Gp=37.95;

Ep=83.93;

vp=Ep/(2*Gp)-1;

v=vm;

Ei=7.1455;

vi=0.3;

Gi=Ei/(2*(1+vi));

%---

% c1: volume fraction of particle

% c2: volume fraction of interface

% c3: volume fraction of matrix

%--- c1=0.0454

c2=0.1147 c3=0.8399

% Se: Eshelby tensor

Cm=[cm11,cm12,cm13,0,0,0 cm21,cm22,cm23,0,0,0 cm31,cm32,cm33,0,0,0 0,0,0,cm44,0,0

0,0,0,0,cm55,0 0,0,0,0,0,cm66];

%---

%---particle [C]---

cp11=Ep./(1+vp).*(1-vp)./(1-2*vp);

cp12=Ep./(1+vp).*vp./(1-2*vp);

cp44=Gp;

cp22=cp11;

cp33=cp11;

cp21=cp12;

cp13=cp12;

cp31=cp12;

cp23=cp12;

cp32=cp12;

cp55=cp44;

cp66=cp44;

Cp=[cp11,cp12,cp13,0,0,0 cp21,cp22,cp23,0,0,0 cp31,cp32,cp33,0,0,0 0,0,0,cp44,0,0

0,0,0,0,cp55,0 0,0,0,0,0,cp66];

%---

%---interface [C]--- ci11=Ei/(1+vi)*(1-vi)/(1-2*vi);

ci12=Ei/(1+vi)*vi/(1-2*vi);

ci44=Gi;

ci22=ci11;

ci21=ci12;

ci13=ci12;

ci31=ci12;

ci23=ci12;

ci32=ci12;

ci55=ci44;

ci66=ci44;

Ci=[ci11,ci12,ci13,0,0,0 ci21,ci22,ci23,0,0,0 ci31,ci32,ci33,0,0,0 0,0,0,ci44,0,0 0,0,0,0,ci55,0 0,0,0,0,0,ci66];

%---

%----effective [C]--- I=eye(6);

Tp=I-Se*inv(Se+inv(Cp-Cm)*Cm);

Tpi=I-Se*(c1/(c1+c2)*inv(Se+inv(Cp-Cm)*Cm)+c2/(c2+c1)*inv((Se+inv(Ci-Cm)*C m)));

Ceff=Cm+((c1+c2)*(Ci-Cm)*Tpi+c1*(Cp-Ci)*Tp)*inv(c3*I+(c1+c2)*Tpi);

D=inv(Ceff);

elastic=1/(D(1,1));

表

1-1 A

與

B

材料的機械性質

Material A B

Young’s modulus (GPa) 3 70

Possion’s ratio 0.3 0.3

表

1-2

比較藉由有限單元法分析代表性之體積元素模型的楊氏模數

RVE 1 RVE 2

Young’s modulus (GPa) 4.56GPa 4.56GPa

表

2-1

聚亞醯胺以及

PI-SiO

^B2^B奈米複材進行分子動力學模擬前的特性

PI PI-SiO^B2^B

No. of unit in polymer chain 10 10 No. of chain in polymer model 7 7

No, of SiO^B2^B - 1

Radius of particle (Å) - 9.79

Initial density of composite (g/cc) 1.33 1.433

Forcefield Compass Compass

Constructional Temperature (K) 650 650 Initial density of matrix (g/cc) 1.33 1.324

Cubic length (Å) 36.37 37.39

表

2-2

分子動力學模擬過程

Process 1 2 3

Method NVT NPT NPT

Forcefield Compass Compass Compass

Target Temperature (K) 1500 300 1

Time step (fs) 0.5 0.5 0.5

Number of steps 40000 30000 40000

Dynamic time (ps) 20 15 20

Target pressure (GPa) - 0 0

表

2-3

聚亞醯胺以及

PI-SiO

^B2^B奈米複材進行

MD

模擬後的特性

PI PI-SiO2

Radius of particle (Å) - 8.5034

Non-bond gap (Å) - 4.3758

Volume (Å^P³^P) 48827.6 54796.1

Density (g/cc) 1.31 1.34

Volume fraction (%) - 4.7

表

2-4 PI-SiO

^B2^B奈米複材介面層的非鍵結能量

PI-SiO^B2^B

Energy(Kcal/mole) -205.471229

Energy per unit area(J/m^P²^P) -1.6116

Non-bond gap(Å) 4.3758

表

4-1

各種材料以及介面層之楊氏模數以及剪力模數

SiO^B2^B Polyimide Interface

Young’s modulus (GPa) 83.9 3.678 7.2004 Shear modulus (GPa) 37.95 1.332 2.7694

表

4-2

藉由不同方法所得

PI-SiO

^B₂^B奈米複材的楊氏模數

MD simulation

Mori-Tanaka method

FEM with two phase

Effective interface model

FEM with three phase

Young’s modulus (GPa)

4.23 4.02 4.01 4.362 4.361

表

4-3

不同的非鍵結間距對楊氏模數造成的影響

Non-bond gap (Å) Interface modulus(GPa) Model prediction (GPa)

6.5813 4.7637 4.2627 4.3875 7.2004 4.364 2.19375 14.2911 4.257 1.096875 28.5822 4.1581

表

4-4

各種模擬方法與楊氏模數的比較

Force field model Interface modulus(GPa) Model prediction (GPa)

表

4-5

比較不同粒徑的二氧化矽奈米粉體

PI-SiO2 (Large) PI-SiO2 (Small)

Particle radius (Å) 8.5034 6.5564

Non-bond gap (Å) 4.3758 4.163

Normalized non-bond energy (J/m2) -1.5754 -2.1846

Volume fraction (%) 4.47 3.87

Young’s modulus of interface (GPa) 7.2004 10.4945

Young’s modulus (GPa) (MD simulation) 4.237 4.52

Young’s modulus (GPa) (Effective

interface model) 4.352 4.5224

表

4-6 PE

單位原子結構模型的分子鏈性質

[13]

PE-I PE-II PE-III

No. of Chains in

PE 1 4 7

No. of unit in chain 493 538 527

Molecular Weight

(g/mole) 6915 7546 7392

表

4-7 PE-Bucky Ball

奈米複材的幾何以及機械性質

[13]

C60-PE C180-PE C320-PE

Particle diameter (Å) 7 12 17

Cubic length (Å) 24.49 36.43 45.6

Non-bond Gap(Å) 3.394 3.436 3.478

Volume fraction 1.22% 1.87% 2.71%

UVDW(kcal/mole) -74.9 -169.5 -274

Normalized

U^Bnon-bond^B(kcal/mole-atom) -1.2483 -0.9416 -0.8562

Young’s modulus (GPa) (MD

simulation) 2.87 2.4422 2.183

表4-8 PE 及巴克球的機械性質[12][33]

PE[12] Bucky-ball[33]

Young’s modulus(GPa) 1.79 12.7

Shear modulus(GPa) 0.65 4.7

表4-9 藉由各種方法模擬奈米複材的楊氏模數

C60-PE C180-PE C320-PE

Volume fraction 1.22% 1.87% 2.71%

MD simulation (GPa) 2.87 2.4422 2.183

Effective interface model

(GPa) 2.1124 2.0233 2.0221

表4-10 比較不同的 PI 分子鏈長度

PI-1 PI-2 PI-3 PI-4 No. of unit in

PI chain 5 10 12 35

No. of chain in

PI model 14 7 10 2

Density (g/cc) 1.33 1.31 1.31 1.31

Young’s modulus (GPa)

3.043 3.67 3.95 4.53

P P

RVE 1 RVE 2

圖1-1 考慮不同顆粒大小但體積比相同的代表性之體積元素模型(representative volume elements ,RVE) ，體積比為19.6%

圖1-2 複合材料之介面層現象[17]

圖2-1 聚亞醯胺分子結構式

圖2-2 利用 MS 軟體所建立的聚亞醯胺分子結構式

(a)

(b)

(c)

圖2-3 分子結構模型 (a) 二氧化矽奈米粉體 (b) 聚亞醯胺單位原子結構模型 (c) PI-SiO^B2^B奈米複材結構模型

Radius

D e n s it y( g /cc)

0 2 4 6 8 10 12 14 16 18

0 2 4

PI-center PI-P PI-Q PI-R

1.31 g/cc

圖2-6 聚亞醯胺單位原子結構模型之密度分佈

Radius

D e n s it y( g /cc)

0 2 4 6 8 10 12 14 16 18

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2

1.31 g/cc

圖2-7 聚亞醯胺單位原子結構模型之平均密度分佈

(Å)

Radius

D e n s it y( g /cc)

8 10 12 14 16 18

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

1.8 PI

SiO₂ 1.31g/cc

圖2-8 PI-SiO^B2^B奈米複材結構模型密度分佈示意圖

Radius

P I- P I R a di a l D is tr ib ut io n F unc ti on g( r)

0 2 4 6 8 10 12

0 1 2 3 4 5 6

PI PI-SiO₂

(Å)

(a)

(b)

圖2-10 二氧化矽晶格模型 (a)單位晶格 (b)分析的晶格模型

φ

(a) z

y

(b)

y z

x

φ θ

2-11 非鍵結間距示意圖 (a) 平面圖 (b) 計算間距示意圖

Degree

Non-bond gap: 4.3959 Å

8.32531Å Non-bond gap:4.37191Å

(Å)

Degree

R a di us

-180 -135 -90 -45 0 45 90 135 180

0 2 4 6 8 10 12 14 16 18 20

PI SiO₂

圖 2-14 繞 x 軸旋轉之非鍵結間距

(a) (b)

εT ⁽^x⁾

εC

(c)

基材內含物

圖3-1 Eshelby’s 內含物問題( Eshelby’s inclusion problem) (a).無施加應力的基材 13.0554Å

8.6959Å Non-bond gap: 4.3595 Å

(Å)

ε

C

σ

(a) 非均質內含物 (b) 均質內含物

ε

圖3-2 Eshelby’s 等值內含物問題(Eshelby’s equivalent inclusion problem)

圖3-3 施加ε^A之非均質值內含物的應力及應變關係示意圖 C

ε ε

ε = +

(

^A ^C

)

C ε ε

σ = +

ε

圖3-4 施加ε^A之均質值內含物的應力及應變關係示意圖

圖3-5 施加ε^A以及ε^C之均質值內含物的應力及應變關係示意圖

ε

C A

ε

ε = + ε

ε

(

^A ^C ^T

)

= C ε + ε − ε σ

ε

ε =

A m

f C

σ = ε

ε

~

ε + ε

f m

~ ~

ε + ε + ε

圖3-6 有限長的基材添加內含物的問題

ε

圖3-7 在有限長的複合材料的等值內含物問題

Ω

圖3-8 等值雙層內含物(equivalent double inclusion)

Ω

圖3-12 等值雙層內含物問題(equivalent double inclusion problem) (a) 非均質之雙層內含物 (b) 均質之雙層內含物

Ω Γ

C^Ω

C^Γ C^m

ε

Ω Γ

C^m

ε

1 ^ε¹^T^(x) ^ε^T²

圖3-14 在有限長的複合材料的等值雙層內含物問題

圖3-15 軸對稱 RVE 模型模擬奈米複材示意圖 (a) 六角形奈米複材模型 (b)考慮 RVE 之六角形奈米複材模型 (c)軸對稱 RVE 模型

V^BA

P P

R^B0^B

V^BB

(a) (b) (c)

2R^B0^B

P x

2R₀

R₀

圖3-16 不考慮介面層之軸對稱 RVE 模型 (a) 軸對稱 RVE 模型 (b) 有限單元法模擬之軸對稱RVE 模型

2R₀

R₀ x

圖3-17 考慮介面層之軸對稱 RVE 模型 (a) 軸對稱 RVE 模型 (b) 有限單元法

(a) (b)

圖4.1 模擬介面層示意圖

圖4.2 單軸受力之構件

F

L

E A

VDW gap Matrix

Particle Spring

Equivalent inclusion material

F

Particulate radius

圖4-5 繞 z 軸旋轉之非鍵結間距 Non-bond gap: 4.1053Å

10.7723Å Non-bond gap: 4.1673Å

6.605Å

(Å)

Degree

R a di us

-180 -135 -90 -45 0 45 90 135 180

2 4 6 8 10 12 14 16

PI SiO₂

圖4-7 繞 x 軸旋轉之非鍵結間距

No. of unit in the polymer chain

Y o ung' s m odul us (G P a )

0 10 20 30 40

2 2.5 3 3.5 4 4.5 5

4-8 聚亞醯胺單位原子結構模型之楊氏模數 10.8215Å Non-bond gap: 4.2166Å

6.6098Å

(Å)

在文檔中利用多尺度模擬探討奈米複合材料之機械性質 (頁 80-112)

第五章 結論

附錄 7 等效介面層模型程式

1-1 A

B

1-2

2-1

PI-SiO

2-2

2-3

PI-SiO

MD

2-4 PI-SiO

4-1

4-2

PI-SiO

4-3

4-4

4-5

4-6 PE

[13]

4-7 PE-Bucky Ball

[13]

Radius

D e n s it y( g /cc)

Radius

D e n s it y( g /cc)

(Å)

(Å)

Radius

D e n s it y( g /cc)

Radius

P I- P I R a di a l D is tr ib ut io n F unc ti on g( r)

(Å)

(Å)

φ

(a) z

y

(b)

y z

x

φ θ

Degree

(Å)

Degree

R a di us

(Å)

ε

C

C

σ

ε

ε ε

ε = +

(

)

C ε ε

σ = +

ε

ε

ε

ε = + ε

ε

(

)

= C ε + ε − ε σ

ε

ε

ε =

ε

ε

~

ε + ε

~ ~

ε + ε + ε

ε

ε

ε

ε

ε

第五章結論