m. kawai institute of engineering mechanics and systems,

A New Strength Parameter and a Damage Mechanics Model for Off-Axis Fatigue of Unidirectional

Composites Under Different Stress Ratios

M. Kawai

Institute of Engineering Mechanics and Systems,

University of Tsukuba, Tsukuba 305-8573, JAPAN

Background

Objectives

Experimental Results

Modeling & Verification

Conclusions

Strength Measures

Outline

UD Lamina:

Fatigue Failure Analysis of Composites

MD Laminate:

Fiber

Matrix

Local off-axis loading of inclined plies

Matrix-Dominated Behavior

Loading Mode Dependence of Fatigue

time

Service Loading of Structural Laminates

(in general)

・ Alternating stress (Amplitude)

・ Mean stress

・ Waveshape

・ Frequency

Effects of Mean Stress on Off-Axis Fatigue Behavior of PMCs

—Experimental Data—

Unidirectional Carbon/Epoxy

(Kawai, M., Suda, H. and Koizumi, M., 2002)

Unidirectional Glass/Epoxy

(El Kadi, H. and Ellyin, F., 1994)

Fatigue Model Considering Mean Stress Effects

Mean Stress Effects on Off-Axis Fatigue Behavior of UD PMCs

for the range –1 ≤ R ≤ 1

Objectives

Fatigue Strength Measure

Stress Ratio:Stress Ratio:

time

max

min

Mean

R min

max

Effects of Mean Stress on Off-Axis Fatigue Behavior of PMCs

—Experimental Data—

Unidirectional Carbon/Epoxy

(Kawai, M., Suda, H. and Koizumi, M., 2002)


(El-Kadi, H. and Ellyin, F., 1994)

10

50 100 50

21

(unit:mm)

50 100 50

2120

(unit:mm)

= 0°

= 10 , 15 , 30 , 45 , 90°

Carbon/Epoxy (T800H/2500)

Specimens:

Material System

Comparison Between Tensile and Compressive Strengths

0

0.5

1

1.5

2

0 10 15 30 45 90

T800H/2500

TensionCompression

Fra

ctur

e st

ress

x

f , M

Pa

Fra

ctur

e st

ress

x

f , M

Pa

Fra

ctur

e st

ress

x

f , M

Pa

Fra

ctur

e st

ress

x

f , M

Pa

xU

TS /

xU

TS ,

xU

CS /

xU

TS

Off-axis angle , degree

R = - 0.3R = - 1.0

Experimental (RT)

Off-Axis Fatigue Testing

・ Load control

R = 0.5 R = 0.1

・ Frequency 　 10 Hz

・ Temperature 　 RT

・ Stress ratio R = 0.5, 0.1, –0.3 ( = 0°) R = 0.5, 0.1, –1.0 ( > 0°)

Fatigue Testing on CFRP

time

max

min

R = –0.3, –1.0

time

max

min

time

max

min

Antibuckling Guide Fixtures

Effects of Stress Ratio on Off-Axis Fatigue (CFRP)

Nf

0

50

100

150

200

100 101 102 103 104 105 106 107

Fatigue = 30 (UD) RT 10Hz L/w=5

max ,

MPa

● R = 0.5● R = 0.1● R = -1.0

0

20

40

60

80

100

100 101 102 103 104 105 106 107


Nf

max ,

MPa

● R = 0.5● R = 0.1● R = -1.0

0

20

40

60

80

100

100 101 102 103 104 105 106 107


Nf

max ,

MPa

● R = 0.5● R = 0.1● R = -1.0

Nf

0

100

200

300

400

500

100 101 102 103 104 105 106 107


max ,

MPa

● R = 0.5● R = 0.1● R = -1.0

T-T Fatigue Failure Morphology (CFRP)

R = 0.5 R = 0.1

0°

10°

15°

30°

45°

90°

Failure along fibers

T-C Fatigue Failure Morphology (CFRP)

( R = -0.3 )

0° 30°

10° 45°

15° 90°

Failure along fibersOut-of-plane shear, Microbuckling

0

50

100

150

200

250

300

350

400

100 101 102 103 104 105 106

m

ax,

MP

a

Nf

E-Glass/Epoxy (RT)

Experimental (Kadi & Ellyin, 1994)(3.3 Hz)

▲ R = 0.5○ R = 0× R = -1

= 19°

0

20

40

60

80

100

100 101 102 103 104 105 106 107

m

ax,

MP

a

Nf

E-Glass/Epoxy (RT)


▲ R = 0.5○ R = 0× R = -1

= 45°

0

20

40

60

80

100

100 101 102 103 104 105 106 107

m

ax,

MP

a

Nf

E-Glass/Epoxy (RT)


▲ R = 0.5○ R = 0× R = -1

=71°

0

20

40

60

80

100

100 101 102 103 104 105 106 107

m

ax,

MP

a

Nf

E-Glass/Epoxy (RT)


▲ R = 0.5○ R = 0× R = -1

= 90°

Effects of Stress Ratio on Off-Axis Fatigue (GFRP)

Non-Dimensional Fatigue Strength Measure

Strength Ratio:

s* max

B

max

B

Maximum fatigue stress

Static strength

where

Off-Axis S-N Relationship Using Strength Ratio

Unidirectional T800H/Epoxy (R = 0.1)

101

102

103

104

105

100 101 102 103 104 105 106 107

Nf

m

ax,

MP

a

Experimental (Kawai & Suda)(RT, f = 10 Hz, R = 0.1)

T800H/2500

○ 0°▲ 10°◇ 15°

● 30°▽ 45°■ 90°

s* max

B

10-2

10-1

100

101

102

100 101 102 103 104 105 106 107

　 ○　 0°　 ▲　 10°　 ◇　 15°　 ●　 30°　 ▽　 45°　 ■　 90°

Nf

T800H/2500 Experimental (Kawai & Suda)(RT, f = 10 Hz, R = 0.1)

m

ax/

B(e

xp)

Effect of Stress Ratio on Off-Axis S-N Relationship

Unidirectional T800H/Epoxy

10-2

10-1

100

101

102

100 101 102 103 104 105 106 107

m

ax/

B(e

xp)

Nf

T800H/2500 Experimental (Kawai & Suda)(RT, f = 10 Hz)

○ R = 0.5△ R = 0.1● R = -1

＋ R = -0.3

Modified Strength Ratio:

S* a

B m

Non-Dimensional Fatigue Strength Measure

a 12

(1 R)max

m 12

(1 R)max

where

S*

12

(1 R)s*

1 12

(1 R)s*

S*

max

B

1

S*

R 1s*

Master S-N RelationshipR = –1

Modified Strength Ratio


Off-Axis S-N Relationship Using


10-2

10-1

100

101

102

100 101 102 103 104 105 106 107

m

ax/

B(e

xp)

Nf


○ R = 0.5△ R = 0.1● R = -1

＋ R = -0.3

S* a

B m

10-2

10-1

100

101

102

100 101 102 103 104 105 106 107

S*

Nf


○ R = 0.5△ R = 0.1● R = -1

Unidirectional Glass/Epoxy (R = 0)

101

102

103

104

105

100 101 102 103 104 105 106 107

E-Glass/Epoxy Experimental (Kadi & Ellyin) (RT, f = 3.3 Hz, R = 0)

m

ax

Nf

　　　　　 ○ 0°　 ▽　 71°　　　　　 ▲ 19°　 ■ 90°　　　　　 ◇ 45° 　

10-2

10-1

100

101

102

100 101 102 103 104 105 106 107

m

ax/

B(e

xp)

Nf

E-Glass/Epoxy Experimental (Kadi & Ellyin)(RT, f = 3.3 Hz, R = 0)

　　　　　 ○ 0°　　　　　　 ▲ 19°　　　　　 ◇ 45° 　　　　　 ▽ 71°　　　　　 ■ 90°

s* max

B

Off-Axis S-N Relationship Using Strength Ratio

Off-Axis S-N Relationship Using



10-2

10-1

100

101

102

100 101 102 103 104 105 106 107

m

ax/

B(e

xp)

Nf

E-Glass/Epoxy Experimental (Kadi & Ellyin)(RT, f = 3.3 Hz)

○ R = 0.5△ R = 0● R = -1

10-2

10-1

100

101

102

100 101 102 103 104 105 106 107

S*

Nf


○ R = 0.5△ R = 0● R = -1

S* a

B m

A Unified Fatigue Strength Measure

—Experimental—

Modified Strength Ratio:

S* a

B m

Stress ratio effect

Fiber orientation effect

(for the tested range of R)

Tsai-Hill Static Failure Criterion:

11

X

2

11 22

X 2 22

Y

2

12

S

2

1

Non-Dimensional Effective Stress

Y

２

１

X

X: Longitudinal strength

Y: Transverse strength

S: Shear strength

Non-Dimensional Effective Stress:

* 11

X

2

11 22

X 2 22

Y

2

12

S

2

Theoretical Strength Ratio

Off-Axis Fatigue Loading of UD Composites

* x 1

Non-Dimensional Effective Stress

* x Static Failure Condition:

B ( pred ) 1

max

* max max

1

max

B ( pred )

Maximum Non-Dimensional Effective Stress

Off-Axis S-N Relationship Using Theoretical Strength Ratio

10-1

100

101

100 101 102 103 104 105 106 107

Nf

T800H/2500

* m

ax

R = 0.5 n = 64.8

R = -1.0 n = 11.9R = -0.3 n = 12.0R = 0.1 n = 23.1

R = 0.1

R = 0.5

R = -0.3

R = -1.0

(*

max)nN

f=1Experimental (RT, 10 Hz)

　 ● R = 0.5　 ○ R = 0.1　 ▲ R = -0.3　 × R = -1.0


s* vs N f

10-1

100

101

100 101 102 103 104 105 106 107

*m

ax

Nf


○ R = 0.5△ R = 0● R = -1

Off-Axis S-N Relationship Using Theoretical Strength Ratio


s* vs N f

Modified Non-Dimensional Effective Stress:

* a

*

1 m*

a

* 12

(1 R)max

*

m

* 12

(1 R)max

*

where

Non-Dimensional Effective Stress for Fatigue

R [ 1, 1]

*

12

(1 R) max

*

1 12

(1 R) max*

*

R 1 max

*

Theoretical Modified Strength Ratio

Master S-N RelationshipR = –1

10-2

10-1

100

101

102

100 101 102 103 104 105 106 107

*

Nf

T800H/2500 Experimental (Kawai & Suda)(RT, 10 Hz)

○ R = 0.5△ R = 0.1● R = -1

＋ R = -0.3

Off-Axis S-N Relationship Using Theoretical Modified Strength Ratio


S* vs N f

Off-Axis S-N Relationship Using Theoretical Modified Strength Ratio

10-2

10-1

100

101

102

100 101 102 103 104 105 106 107

*

Nf


○ R = 0.5△ R = 0● R = -1


S* vs N f

Damage Mechanics Modeling of Composite Fatigue

ddN

Kn 11

k

: Fatigue strength parameter

Fatigue Damage Growth Law:

N f 1

(k 1)Kn

Fatigue Life Equation:

1

N f 1

N f 1

n

Off-Axis Fatigue Model

ddN

K * n* 11

k

N f 1

* n*

*

*-Based Fatigue Damage Model:

Master S-N Relationship:


Master S-N Relationship

10-2

10-1

100

101

102

100 101 102 103 104 105 106 107

S*

Nf


○ R = 0.5△ R = 0.1● R = -1

10-2

10-1

100

101

102

100 101 102 103 104 105 106 107

s* =

m

ax/

B(e

xp

),

*m

ax

Nf


○ R = 0.5△ R = 0.1● R = -1

N f 1

* n

N f 1

* n*

max 2*

( ) (1 R) (1 R)*

max

* 2*

(1 R) (1 R)*

Transformation of Master S-N Relationship

101

102

103

104

105

100

101

102

103

104

105

106

107

m

ax, M

Pa

Nf

T800H/2500 Experimental (Kawai & Suda)(RT, f = 10 Hz, R = -1)

○ 0° (R = -0.3)

▲ 10°◇ 15°　

Predicted

● 30°▽ 45°■ 90°　

101

102

103

104

105

100

101

102

103

104

105

106

107

m

ax, M

Pa

Nf


○ 0°▲ 10° ◇ 15°　　

Predicted

● 30°▽ 45°■ 90°　

101

102

103

104

105

100

101

102

103

104

105

106

107

m

ax, M

Pa

Nf


Predicted

○ 0°▲ 10° ◇ 15°　　

● 30°▽ 45°■ 90°　

Comparisons With Experimental Results



Master S-N Relationship

10-2

10-1

100

101

102

100 101 102 103 104 105 106 107

S*

Nf


○ R = 0.5△ R = 0● R = -1

10-2

10-1

100

101

102

100 101 102 103 104 105 106 107

Sm

ax =

m

ax/

B(e

xp

),

*m

ax

Nf


○ R = 0.5△ R = 0● R = -1

N f 1

* n

N f 1

* n*

100

101

102

103

104

105

100

101

102

103

104

105

106

107

m

ax, M

Pa

Nf

E-Glass/Epoxy Experimental (Kadi & Ellyin)(RT, f = 3.3 Hz, R= -1)

　　　　　 ○ 0° ▽　 71°

　　　　　 ▲ 19° ■ 90°

　　　　　 ◇ 45° 　

Predicted

101

102

103

104

105

100 101 102 103 104 105 106 107

m

ax, M

Pa

Nf

E-Glass/Epoxy Experimental (Kadi & Ellyin)(RT, f = 3.3 Hz, R= 0)　　　　　 ○ 0° ▽　 71°

　　　　　 ▲ 19° ■ 90°　　　　　 ◇ 45° 　

Predicted10

1

102

103

104

105

100

101

102

103

104

105

106

107

m

ax, M

Pa

Nf

E-Glass/Epoxy Experimental (Kadi & Ellyin)(RT, f = 3.3 Hz, R = 0.5)

　　　　　 ○ 0° ▽　 71°

　　　　　 ▲ 19° ■ 90°

　　　　　 ◇ 45° 　

Predicted


Comparisons With Experimental Results

Constant Fatigue Life Diagram (CFLD)

N f 1

* n* ( R 1)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

R = -1.0

a/B

(ex

p)

R = 0.5

R = 0.1

m/

B(exp)

Experimental (RT, 10Hz)

　　 ● Nf=104

　　 ○ Nf=105

　　 × Nf=106

T800H/2500 = 45

a

* 1N f

1

n*

1 m

*

A non-dimensional strength measure * that considers the mean stress as well as fiber orientation effects on the off-axis fatigue behavior of unidirectional polymer matrix composites was proposed.

Validity of the fatigue model based on the non-dimensional strength measure * was evaluated by comparing with experimental results.

Conclusions

For ” ,

Using the modified strength ratio S*, we can substantially remove the fiber orientation as well as stress ratio dependence of the off-axis fatigue data to obtain an experimental master S-N relationship.

A general expression * of the modified fatigue strength ratio is obtained as a natural extension of the non-dimensional effective stress based on the Tsai-Hill static failure criterion.

A fatigue damage mechanics model that considers the fiber orientation as well as stress ratio effects is formulated using the modified non-dimensional effective stress *.

The proposed fatigue model can adequately describe the off-axis S-N relationships of unidirectional glass/epoxy and carbon/epoxy laminates under constant-amplitude cyclic loading with non-negative mean stresses.

ConclusionsFor ” ,

Summary Chart

Experimental Theoretical

s* max

B

S* a

B m

max

*

*

Application to Fatigue

Metals UD-PMCs

Basquin

(1910)

Awerbuch-Hahn

(1981)

Landgraf

(1970)?

(* )n*N f 1

(R [ 1,1])

Thank you for your kind attention !

m. kawai institute of engineering mechanics and systems,

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