byMasayuki Nakada and Yasushi Miyano

Materials System Research Laboratory, Kanazawa Institute of Technology

Hongneng CaiSchool of Materials Science and Engineering, Xi'an Jiaotong University

Advanced Accelerated Testing Methodology for Long-term Life Prediction of CFRP

The 15th Composites Durability Workshop (CDW-15), October 17 to 20, 2010, Kanazawa Institute of Technology

October 18, 2010Sakai Memorial Hall, Kanazawa Institute of Technology

Japan

2

The most important hypothesis used in our accelerated testing for the prediction of long-term life of CFRP is the time-temperature superposition principle (TTSP).

Developed mainly for polymeric based materials, elevated temperature states are used to accelerate the mechanisms of mechanical degradation which occur under loads over very long time.

The method has been widely employed to characterize nondestructive properties, and has shown good success in characterizing failureproperties recently.

The degree of acceleration per increment of elevated temperature is found through the use of the time-temperature superposition hypothesis, along with procedures of properties testing.

Basis of accelerated testing methodology (ATM)

3

TTSPStatic Creep Fatigue As shown on the table, the strength

of PAN-based CFRP meets the time-temperature superposition principle (TTSP) regardless the structural configuration and loading style. TTSP also holds for static, creep and fatigue strengths for the adhesive and bolted joints of CFRP and GFRP. These facts were confirmed experimentally.

Experimental verification of time-temperature superposition principle for CFRP strength

4

Objectives

The advanced accelerated testing methodology (ATM-2) for the long-term life prediction of CFRP laminates exposed to an actual loading having general stress and temperature history is proposed.

Three conditions as the basis of ATM-2 were introduced with the scientific bases. The long-term fatigue strength of CFRP laminates under an actual loading is formulated based on the three conditions.

The creep compliance and time-temperature shift factors of matrix resin, which perform an important role for time and temperature dependence of long-term life of CFRP laminates, are also formulated based on the modified time-temperature superposition principle.

Finally, the applicability of ATM-2 is confirmed by predicting the long-term static and fatigue strengths in the typical four directions of unidirectional CFRP laminates.

5

Advanced accelerated testing methodology for long-term fatigue life prediction of CFRP

CBAf0f fff ⋅⋅⋅= σσ

Condition A: The failure probability is independent of temperature and load histories.

Condition B: The time and temperature dependence of strength of CFRP is controlled by the viscoelasticity of matrix resin. Therefore, the time-temperature superposition principle for the viscoelasticity of matrix resin holds for the strength of CFRP.

Condition C: The strength degradation of CFRP holds the linear cumulative damage law as the cumulative damage under cyclic loading.

The long-term life concerning with the strength of CFRP can be shown by the following equation based on three conditions of A, B and C.

σfo: The static strength at room temperature determined by types of fiber and weave, volume fraction, load direction and others

fA : The failure probability determined by types of fiber and weave, volume fraction, load direction and others

fB : The viscoelasticity of matrix resin determined by the histories of temperature and loadfC: The cumulative damage determined by the load amplitude and number of cycles to failure

or CBAf0f logloglogloglog fff +++= σσ

6

Ref. Richard Christensen and Yasushi Miyano, “Stress intensity controlled kinetic crack growth and stress history dependent life prediction with statistical variability”, International Journal of Fracture (2006), 137, 77-87

Time and temperature dependent static and creep strengths can be shown by the following equation of Weibull distribution.

σfo: Scale parameter α: Shape parameter

Pf: Failure probability

The shape parameter does not change with time and temperature.

Condition AVariability of tensile strength of unidirectional CFRP

Weibull distribution of static and creep strengths

Condition A : Failure probability to be independent of temperature and load histories

Static (R.T.)

Static (150oC)Creep (R.T.)

Creep (150oC)

( ) ( )[ ]ασ

σ 1

ff0

ffA 1ln PPf −−==

7

The relationship between the longitudinal tensile (LT) and flexural (LB) static strengths and transverse flexural (TB) static strength of unidirectional CFRP and the inverse of compliance of matrix resin (1/D*) are uniquely determined and these slopes are constant. It is cleared that the time and temperature dependence of LT, LC and TB static strengths is controlled by the viscoelasticity of matrix resin.

The variation of strength of CFRP is shown by the following equation.

nr: Parameter determined by failure mechanismDc: Creep compliance of matrix resin

Condition BThe viscoelastic compliance D* of matrix resin is determined by temperature and load histories.

Condition B :Strength of CFRP controlled by viscoelasticity of matrix resin

Sta

tic s

treng

th σ

(t,T)

[MP

a]

( )( )

( )( )

r

00c

0

00f

0fB ,'/1

,'*/1,','

n

TtDTtD

TtTtf

==

σσ

( ) ( )( )

( ) ( )

( )0

'

0 0c

0

00 ,'

'd'd'd,''

,',','*

Tt

TtD

TtTtTtD

t

σ

τττστ

σε ∫ −

==

8

Yes

Yes

Yes

Time-temperature superposition principle

Failure of matrix resinMatrix crackTransverse tension

Viscoelasticity of matrix resinMicrobuckling of fiber

Longitudinal compression

Viscoelasticity of matrix resinCumulative damage of fiber

Longitudinal tension

Controller of time and temperature dependence

Failure mechanismLoad direction

The applicability of time-temperature superposition principle for the static strength to three kinds of load direction of unidirectional CFRP is theoretically confirmed.

Longitudinal tension Longitudinal compression Transverse tension

Theoretical verification of time-temperature superposition principle for CFRP strength

9

Condition C :Strength degradation by cumulative damage under cyclic load

Condition C

where N0 = 1/2

The slope depends on the stress amplitude determined by maximum stress and stress ratio and is independent of time, temperature and frequency.

S-N curve of polymer compositesAccumulation index of damage kD

n1, n2 : Number of cycles at constant load cycle “1,2 ”

Nf1, Nf2 : Number of cycles to failure at constant load cycle “1,2”

R1, R2 : σ1min / σ1max , σ2min/σ2max

12

2

1

1 <+=ff

D Nn

Nnk

( )( )0f

ffC N

Nfσσ

=

( )( )

( ) ( )

−−

⋅⋅

−−= D

0

ff

0f

ff 1loglog2

1log kNNnR

NN

σσ

( ) ( )1

fi12

fi

0f0

i

fi

iD <

== ∑∑

− nRNN

nNnk

σσ

10

Formulation of strength of CFRP based on ATM-2

The long-term fatigue strength exposed to the actual loading where the temperature and load change with time can be shown by the following equation based on the conditions of A, B and C.

The first term of left part shows the scale parameter for the strength at the reference temperature To, the reduced reference time to’, the number of cycles to failure Nf = ½ and the stress ratio R = 1.The second term shows Weibull distribution as the function of failure probability Pf. (Condition A)The third term shows the variation by the viscoelastic compliance of matrix resin which depend on temperature and load histories. The viscoelatic compliance can be shown by the following equation. (Condition B)

( ) ( ) ( )[ ] ( )( )

( )( )

−

⋅⋅−

−

−−−+=

D

ff

00c

0rf00f00 1

2log2

1,','*log1lnlog1,'log,,,,',log

kNnR

TtDTtDnPTtRNTtP fff α

σσ

( ) ( )( )

( ) ( )

( )0

'

0 0c

0

00 ,'

'd'd'd,''

,',','*

Tt

TtD

TtTtTtD

t

σ

τττστ

σε ∫ −

==where, Dc shows the creep compliance of matrix resin and σ(τ’) shows the stress history.

t’ in the above equation is the reduced time at the reference temperature T0 and can be shown by the following equation.

( )( )∫=t

T Tat

0 0

d'τ

τ where, aTo shows the time-temperature shift factor of matrix resin and T(τ) shows the temperature history.

11

Formulation of strength of CFRP based on ATM-2

11 fi

iD <=∑

=

n

i Nnk

Procedure for determination of material parameters for formulationFirst step: Determination of aTo and Dc of matrix resinSecond step: Determination of σfo, α, nr and nf of CFRP

The fourth term shows the degradation by the cumulative damage under cyclic load. The Nf and R in this term show the number of cycles to failure and the stress ratio at the final step, respectively. The kD shows the Accumulation index of damage. (Condition C)

where, the Nfiand ni show the number of cycles to failure and the number of load cycles at each step of actual loading, respectively.

Condition A: Failure probability to be independent of temperature and load historiesCondition B: Strength of CFRP controlled by viscoelasticity of matrix resin Condition C: Strength degradation by cumulative damage under cyclic load

( ) ( ) ( )[ ] ( )( )

( )( )

−

⋅⋅−

−

−−−+=

D

ff

00c

0rf00f00 1

2log2

1,','*log1lnlog1,'log,,,,',log

kNnR

TtDTtDnPTtRNTtP fff α

σσ

12

Determination procedure of material parameters for formulation

Master curve of creep strength

Master curve of fatigue strength at zero stress ratio

Time-temperature shift factor

（Accelerating rate）

Matrix resin CFRPCondition A: Failure probability to be independent of temperature and load histories

Condition B: Strength of CFRP controlled by viscoelasticity of matrix resin

Condition C: Strength degradation by cumulative damage under cyclic load

Dc

aTo

σf0 nrα

nf

First step: Determination of aTo and Dc of matrix resinSecond step: Determination of σfo, α, nr and nf of CFRP

Master curve of creep compliance

Viscoelastic tests at various times and

temperatures

Static tests at a constant strain rate

and various temperatures

Master curve of static strength

Fatigue tests at a constant frequency, zero stress ratio and various temperatures

13

T0 25 [oC]Tg 162 [oC]

∆H1 101 [kJ/mol]

∆H2 760 [kJ/mol]

b0 1.13E-02 [-]

b1 -9.85E-04 [-]

b2 2.43E-05 [-]

b3 -2.23E-07 [-]

b4 6.98E-10 [-]

Time-temperature and temperature shift factors:

G: gas constant ∆H: activation energy Tg: glass transition temp.

Measuring of the storage modulus for the transverse direction of unidirectional CFRP (MR60H/1053)

( ) ( )

( )( )TTTTG

HTTG

H

TTTTG

HTa gT

−−

−

∆+

−

∆+

−

−

∆=

gg

1

0g

1

0

1

H111303.2

11303.2

H11303.2

log0

( ) ( ) ( ) ( ) ( )[ ] ( )

( ) ( ) ( ) ( ) ( )( )TTTT

bTTbTTbTTbTTb

TTbTTbTTbTTbTTbTb gT

−−

++−+−+−+−+

−+−+−+−+−=

gg

00g12

0g23

0g34

0g4

0012

023

034

04

H1log

Hlog0

14

Dc(t’0,T0) 0.347 [1/GPa]

t’o 1 [min]

t’g 4.23 x 1010 [min]

mg 0.0116 [-]

mr 0.2876 [-]

Creep compliance of matrix resin: c m( ) 1/ ( )D t E t′ ′=

Back-calculation of Emusing the rule of mixture :

Em: storage modulus of matrix rein Vf ,Vm: volume fractions of fiber and matrix

ET, EfT : storage moduli in the transverse direction of CFRP and carbon fiber

Rule of mixture

Formulation of creep compliance

Dc: creep complianceTo: reference temperaturet’: reduced time at To

t’o: reference reduced time at To

t’g: glassy reduced time at To

where

Creep compliance of matrix resin(MR60H1053)

+

+=

rg

g000c,0c '

'''log),'(loglog

mm

tt

ttTtDD

f

m*y

fTT

*y

*ym

516.0,1111VVV

EEV

VE=

−

+=

15

X

ISO 527（JIS K7073）

Fitting parameters

σｆo [MPa] 2924

nr 0.40

nf 0.07

αs 25.7

αf 7.7

Measuring of the tensile strength for the longitudinaldirection of unidirectional CFRP (MR60H/1053)

( )

( )

( )[ ]

( )( )

( )( )

−

⋅⋅−

−

−

−−+

=

D

ff

00c

0r

f

00f0

0

12log

21

,','*log

1lnlog1,'log

,,,,',log

kNnR

TtDTtDn

P

Tt

RNTtP fff

α

σ

σ

16

X’

ISO 14125(JIS K7074)

(With cushion)Fitting parameters

σｆo [MPa] 2399

nr 0.58

nf 0.04

αs 40.0

αf 17.2

Measuring of the compressive strength for the longitudinaldirection of unidirectional CFRP (MR60H/1053)

( )

( )

( )[ ]

( )( )

( )( )

−

⋅⋅−

−

−

−−+

=

D

ff

00c

0r

f

00f0

0

12log

21

,','*log

1lnlog1,'log

,,,,',log

kNnR

TtDTtDn

P

Tt

RNTtP fff

α

σ

σ

17

Y

ISO 14125(JIS K7074) Fitting parameters

σｆo [MPa] 124

nr 3.05

nf 0.08

αs 12.2

αf 4.4

Measuring of the tensile strength for the transverse direction of unidirectional CFRP (MR60H/1053)

( )

( )

( )[ ]

( )( )

( )( )

−

⋅⋅−

−

−

−−+

=

D

ff

00c

0r

f

00f0

0

12log

21

,','*log

1lnlog1,'log

,,,,',log

kNnR

TtDTtDn

P

Tt

RNTtP fff

α

σ

σ

18

Y’

Fitting parameters

σｆo [MPa] 215

nr 3.21

nf 0.05

αs 13.4

αf 7.6

Measuring of the compressive strength for the transverse direction of unidirectional CFRP (MR60H/1053)

( )

( )

( )[ ]

( )( )

( )( )

−

⋅⋅−

−

−

−−+

=

D

ff

00c

0r

f

00f0

0

12log

21

,','*log

1lnlog1,'log

,,,,',log

kNnR

TtDTtDn

P

Tt

RNTtP fff

α

σ

σ

19

Conclusions

Thank you for your attention!

The advanced accelerated testing methodology (ATM-2) for the long-term life prediction of CFRP laminates exposed to an actual loading having general stress and temperature history was proposed.

Three conditions as the basis of ATM-2 were introduced with the scientific bases. The long-term fatigue strength of CFRP laminates under an actual loading was formulated based on the three conditions.

The creep compliance and time-temperature shift factors of matrix resin, which perform an important role for time and temperature dependence of long-term life of CFRP laminates, were also formulated based on the modified time-temperature superposition principle.

Finally, the applicability of ATM-2 was confirmed by predicting the long-term static and fatigue strengths in the typical four directions of unidirectional CFRP laminates.