by Yasushi Miyano and Masayuki Nakada

Materials System Research Laboratory

Kanazawa Institute of Technology

March 29, 2013

Accelerated Testing Methodology for Long-Term Durability of CFRP Laminates and Structures

Next Generation Transport Aircraft Workshop, March 29,2013 Mary Gates Hall Room 271 in University of Washington

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2

Development of high reliable structures

Data collection by accelerated testing

Durability design

Necessity of accelerated testing

The accelerated testing methodology (ATM) should be established for the prediction of long-term life of CFRP laminates and structures.

Examples: Centrifuge Generator Flywheel Aircraft Marine Wind turbine Automobile etc.

Our developed ATM is introduced in this presentation.

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Role of matrix resin on CFRP

Formation of interface

Cure and thermal shrinkage

Mol

ding

O

pera

ting

Resin Cure accelerator Hardener

Mixing

Liquid resin

Impregnation

Curing

CFRP

Carbon Fiber

Viscoelasticity Load, Temperature, Moisture

Heat, Pressure

Chemical aging Temperature, Moisture, Oxygen, Ultraviolet rays,

Physical aging Temperature These behaviors having an influence on durability of CFRP are generated in matrix resin and interface. The most important behavior is the viscoelasticity of matrix resin. Carbon fibers are perfectly stable during operating.

4

Viscoelastic behavior of matrix resin Creep compliance Dc

Spring Elastic modulus

E

Dashpot Viscosity

η

Solid

Liquid

1

εE

σ0

σ0

Stress σ

Strain ε

Dc

Time t

Constant stressσ0

εE

εE

εη(t)

ε (t)=εE+εη(t)

σ0

σ0

(= )

(= )

Dc(t)= = + σ0 η t

E 1

1/η

E 1

η t

Maxwell model

εη(t)

εη(t)

ε (t)

0

0

0

Time t

Time t

5

Dc(t,T)= = + ε(t,T) σ0 η(T)

t E 1

σ0

σ0

E

η(T)

log Dc(t,T)= log ( + ) η(T)

t E 1

log Dc(t,T)

log t

T0 T1 T2

η(T2)<η(T1)<η(T0)

t0 t1 t2

Time-temperature shift factor aTo(T): Horizontal shift amount log aTo(T)

T0 T1 T2 T

0 log aTo(Ti)= log = log (i=1, 2) t’ ti

η(Ti) η(To)

E 1

T0

T1

T2 1/η(T1)

1/η(T2)

T2>T1>T0

0

Dc(t,T)

Time-temperature superposition principle

Maxwell model

t

1/η(T0)

E 1 log

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Static

Fatigue

Creep Time-temperature shift factor (Accelerating rate):

Strengths of FRP

Expansion of time-temperature superposition principle to static, creep and fatigue strengths of CFRP

Deformation of matrix resin

( ) si cii fi0 i

0 s0 c0 f0

, i 1,2Tt tt ta T

t t t t= = = = = ⋅ ⋅ ⋅

The same time-temperature superposition principle for the deformation of matrix resin holds for the strengths of CFRP. Therefore, the long term strengths of CFRP can be predicted from the measured short term strengths of CFRP at elevated temperatures and the time-temperature shift factor aT0(T) for the deformation of matrix resin.

Time Compression by Elevating Temperature!

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Master curves based on time-temperature superposition principle

Master curve of creep compliance Dc(t,T) and time-temperature shift factor aT0(T) for matrix resin

Master curve of static strength for CFRP is obtained by using the time-temperature shift factor aT0(T) for matrix resin. Master curves of creep and fatigue strengths are also obtained by using the aT0(T).

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TTSP Static 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. These facts were confirmed experimentally.

Verification of time-temperature superposition principle (TTSP) for various FRP strengths

Y. Miyano, M. Nakada and H. Cai, “Formulation of Long-term Creep and Fatigue Strengths of Polymer Composites Based on Accelerated Testing Methodology”, J. Composite Materials (2008), 42, 1897-1923.

General and rigorous accelerated testing methodology (ATM)

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 perfectly controlled by the viscoelasticity of matrix resin.

Condition C: The slope of S-N curve depends on the stress amplitude determined by maximum stress and stress ratio and is independent of time, temperature and frequency.

The long-term life concerning with the strength of CFRP can be shown by the following equation based on the above three conditions of A, B and C for CFRP strength adding to the most important condition that the same time-temperature superposition principle (TTSP) for the deformation of matrix resin holds for CFRP strength.

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

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

fB : Time-temperature dependent strength determined by viscoelasticity of matrix resin

fC: Strength degradation determined by the stress amplitude and number of cycles to failure

or CBAf0f logloglogloglog fff +++= σσ

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Formulation of long-term life of CFRP based on ATM

σfo: Static strength at reference time determined by types of fiber and weave, volume fraction, load direction and others

fA : Scatter of strength as a function of failure probability Pf determined by types of fiber and weave, volume fraction, load direction and others

fB : Time-temperature dependent strength determined by viscoelasticity of matrix resin D* fC: Strength degradation determined by the load amplitude Δσ and number of cycles to failure Nf

CBAf0f logloglogloglog fff +++= σσ

f (Frequency) : constant

log fA(Pf)

log fB(D*)

log fC(Δσ, Nf) lo

g σ

f(t’,T

0)

σf0

log t’

T = T0

t0’ Nf=1/2

f = f’

t’=1/(2f’)

First step: The master curves of static and fatigue strengths are constructed by using the measured data based the time-temperature superposition principle (TTSP) for the deformation of matrix resin. Second step: The mater curves of static and fatigue strengths determined experimentally are formulated by the following equation.

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Details of formulation

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 failure time to’ under the static load when the constant stress is applied. 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 the stress and temperature histories. The viscoelatic compliance can be shown by the following equation. (Condition B)

( ) ( ) ( ) ( )( )

( )0 f0 f0 0 0 f r f

c 0 0 0

* ', 11log , ', , , , log ' , log ln 1 log log' , 2f f f

D t T R NP t T N R t T P n nD t T N

σ σα

⎡ ⎤ − ⎡ ⎤⎡ ⎤= + − − − − ⋅ ⋅⎢ ⎥ ⎢ ⎥⎣ ⎦

⎢ ⎥ ⎣ ⎦⎣ ⎦

( ) ( )( )

( ) ( )

( )0

'

0 0c

0

00 ,'

'd'd'd,''

,',','*

Tt

TtD

TtTtTtD

t

σ

τττσ

τ

σε ∫ −

==

where, Dc shows the creep compliance of matrix resin and σ(τ’) shows the stress history. aTo shows the time-temperature shift factor of matrix resin and T(t) shows the temperature history.

( )( )∫=t

T Tat

0 0

d'τ

τ

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, respectively. The N0 = ½ shows the static load. (Condition C)

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Formulation of creep compliance

Dc: creep compliance To: reference temperature t’: reduced time at To

t’o: reference reduced time at To

t’g: glassy reduced time at To

where

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

rg

g000c,0c '

'''log),'(loglog

mm

tt

ttTtDD

Creep compliance of matrix resin of T800S/3900-2B

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Time-temperature and temperature shift factors

( )

( )

1o g

o

1 2g

g o g

1 1log H( )2.303

1 1 1 1 1 H( )2.303 2.303

THa T T TG T T

H H T TG T T G T T

⎛ ⎞Δ= − −⎜ ⎟

⎝ ⎠

⎡ ⎤⎛ ⎞ ⎛ ⎞Δ Δ⎢ ⎥+ − + − − −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

G: gas constant ΔH: activation energy Tg: glass transition temp.

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

TTbTTTTbTbi

ii

i

iiT −−⎥

⎦

⎤⎢⎣

⎡+−+−⎥

⎦

⎤⎢⎣

⎡−= ∑∑

==g

g4

00gg

4

00 H1logHlog

0

Tensile and compressive strengths for the longitudinal direction of unidirectional CFRP (T800S/3900-2B)

X

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X’

Tensile strength X: The static strength decreases scarcely with increasing time and temperature, and the fatigue strength decreases remarkably with increasing the number of cycles to failure. Compressive strength X’: The static strength decreases remarkably with increasing time and temperature, and the fatigue strength decreases scarecely with increasing the number of cycles to failure.

Y

Tensile and compressive strengths for the transverse direction of unidirectional CFRP (T800S/3900-2B)

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Y’

Tensile and compressive strengths Y and Y’ : The static strength decreases remarkably with increasing time and temperature, and the fatigue strength decreases also remarkably with increasing the number of cycles to failure.

MMF/ATM method

Prediction of long-term life of CFRP structures

MMF/ATM method

CFRP are treated as a heterogeneous body consisted of carbon fibers and matrix resin. The failure criterion for carbon fibers and matrix resin are applied for the judge of failure of CFRP.

Prediction of failure spot, failure layer, failure mode and failure load for CFRP structures

MMF (Micromechanics of Failure)

TTSP The mechanical behavior at low

temperature and long term is equal to that at high temperature and short term.

ATM (Accelerated Testing Methodology)

The long-term life of CFRP laminates are mainly controlled by the viscoelasticity of matr ix resin for which the t ime-temperature superposition principle (TTSP) holds.

Prediction of long-term life of CFRP laminates under temperature and load history

Carbon fiber

Matrix resin

MMF/ATM method as the durability analysis of CFRP structures based ATM is proposed. 15

Static and fatigue strengths

Unidirectional CFRP

Carbon fiber

Resin

UD CFRP layer CFRP laminates Structure

10m 10mm 10µm

MMF &

ATM

Cm : Matrix Compressive

Flow of structural analysis

Tf : Fiber Tensile

Cf : Fiber Compressive

Tm : Matrix Tensile

Tf Cf Tm Cm MMF/ATM parameters

Y’ : Transverse compressive

X : Longitudinal tensile

X’ : Longitudinal compressive

Y : Transverse tensile

Static and fatigue strengths

X X’ Y Y’

Procedure of MMF/ATM method

Equation for judgment

Strengths at time t:

, , , σ= = = =T

Ik k k kT C T C

VMf f m mt c 1

f f m mf f m m

σ −σC T C

m1I

mvmσ : Von Misses stress

in matrix resin

Stresses at time t:

: Maximum tensile stress in carbon fiber

: Maximum compressive stress in carbon fiber : First stress invariant in matrix resin

Tf Cf Tm Cm

σtf

σcf

[ ]Tf Cf Tm Cm=max , , ,k k k k k

Where, k:Failure index (k<1 : No failure , k=1 : Initial failure)

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Failure index distribution map under static OHC loading (T=25 oC, V =0.1mm/min, σs =375MPa)

kTf (fiber tensile failure)

kCf (fiber compressive failure)

kCm (matrix compressive failure)

kTm (matrix tensile failure)

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Stacking sequence:

[45/0/-45/90]2S

Failure index distribution map under cyclic OHC loading (T=80 oC, f =2Hz, σf =242MPa, N =5x103)

kTf (fiber tensile failure)

kCf (fiber compressive failure)

kCm (matrix compressive failure)

kTm (matrix tensile failure)

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Stacking sequence:

[45/0/-45/90]2S

σ =301MPa (0.96σs) σf =242MPa, N =1x105

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Observation of initial failure under static and cyclic OHC loadings

The initial failure at the static test is fiber compressive failure in 0o layer at the corner of open hole, and the initial failure at the fatigue test is matrix compressive failure in +-45o layer. These results agree well with predicted ones.

T =80 oC, f =2Hz 45o 0o -45o

-45o 0o 45o 45o 0o -45o

T =25 oC, V =0.1mm/min Static test Fatigue test

Conclusion and acknowledgement

:

The authors thank the Office of Naval Research for supporting this project through an ONR awards numbered to N62909-12-1-7024 with Dr. Yapa Rajapakse as the ONR Program Officer. The authors thank Professor Richard Christensen, Stanford University as the consultant of this project. 20

The general and rigorous accelerated testing methodology (ATM) for the prediction of long-term life time of CFRP laminates and structures exposed to an actual loading having general load and temperature history was proposed by the following steps.

1. Time-temperature superposition principle (TTSP) was introduced as the most important condition for ATM.

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

3. The applicability of ATM was confirmed by predicting the long-term static and fatigue strengths for various directions of unidirectional CFRP laminates.

4. MMF/ATM method combined with our developed ATM and the micromechanics of failure (MMF) was proposed for the fatigue life prediction of the structures made of CFRP laminates and was applied to OHC loadings to the quasi-isotropic CFRP laminates.