Sung Kyu Ha, Kyo Kook Jin, Yuanchen Huang

St t d C it L b t H U i it KStructures and Composites Laboratory, Hanyang University, Korea

1

Motivation – To Achieve Better Fatigue Life Prediction of Composite Structures

Fatigue test dataaσmaxσ

Most challenging task to predict the fatigue life !

GFRP [(±45)8/07]S

t

a

mσminσ1N6N

The real fatigue loads and i t l d t h

2N

Goodman diagram for fatigue life prediction;

experimental data show…...

σ• Bi‐axial loads

Uniaxial only, Laminate dependent, Accuracy ??

GFRP [90/0/±45/0]S

aσ • Laminate dependent• Nonlinear life diagram

mσDevelop a generic methodology and tools !!

Minimize the number of testsMinimize  the number of  testsMulti‐scale approachPredict Life 

2

Multi-Scale Modeling for Structural Analysis of Composites

Multi‐scale Modeling • Life prediction• Strength PredictionF

Structures

40

60

80

100

120

T=25' C

T=70 'C

T=120 'C

Test Data

x. st

ress

[MPa

]

10000

12000

14000

16000

18000

20000

orce

F

Laminates(+cohesive)

0

20

0 1 2 3 4 5 6

Max

Time to Failure, log tf [min]

master curves (structures)0

2000

4000

6000

8000

0 0.5 1 1.5 2 2.5 3

Displacement

Fo

Ply

( )

Fiber, Matrix & Interface s‐s relations 2T)

Temperature [°C]Micromechanics : MMF, SIFT & MCT

Interface s‐s relationsσyield

linear12

-10-8-6-4-202

0 20 40 60 80 100 120 140 160 180

T<110 'Cacto

r, lo

gaT0

(T

ε

Molecular level

-16-14-12 T<110 C

T>110 'C

shift

fa

3

Static and Fatigue Strength Prediction Using MMFMi h i d l i• Micromechanics models to compute micro-stresses.

• Failure criteria and Fatigue life diagram for each constituent.Compute micro‐stresses

micromechanical models Matrix

Fiber

TΔHomogeneous Strains calculated by FEM

( )C Tσ ε α= − Δ

σ ⎫

FiberInterface

macroσ Macro  micro

Failure Criteria for each constituent

( )macro imicroT

σσ

⎫⇒⎬Δ ⎭

Macro‐stressLaminatesF(t)

Fatigue Life Diagram for each  Failure Criteria for each constituent

MatrixFiberaσ

aσS-N cure

Mean Stress Effectsaσ

R 0 7

g gconstituent

ct static

engths

life

Interface( )( ) 1imicroF σ < σ2

R= 0R= ‐0.3

R= ‐0.5R= ‐0.7

R= 0.5

1aσ

2aσpred

icstre

Pred

ict l σ1

k<1 k=1mσN

N N1N2N3 mσ

4

5

Micromechanical Model – Regular Fiber Arrays

Cross-sectional view of continuous fiber reinforced composites

6

Calculation of Micro Stresses – The Concept of Stress Amplification Factor (SAF)

σTΔ σ3

2

32

T= + Δσ Mσ A

2

M Stress amplification factor for macro stressesA Stress amplification factor for temperature increment

Pl l l tPly-level macro stressesConstituent-level micro stresses

ΔT Temperature incrementσσ

7

Calculation of Micro Stresses: Formulas in Extenso (1)

σTΔ( )kP

For a material point within the fiber( ) ( ) ( )k k k T+ Δσ M σ A( )

fP( ) ( ) ( )f f f T= + Δσ M σ A

( ) ( )1 11 11 12 13 14 1 11

2 22 21 22 23 24 2 22

0 00 0

k kM M M MM M M M

σ σ σ σσ σ σ σ

= =⎡ ⎤ ⎡ ⎤ ⎡⎢ ⎥ ⎢ ⎥ ⎢= =⎢ ⎥ ⎢ ⎥ ⎢

( )1

2

kAA

⎤ ⎡ ⎤⎥ ⎢ ⎥⎥ ⎢ ⎥2 22 21 22 23 24 2 22

3 33 31 32 33 34 3 33

4 23 41 42 43 44 4 23

5 31 55 56 5 31

0 00 0

0 0 0 0

M M M MM M M M

M M

σ σ σ σσ σ σ σσ σ σ σ

⎢ ⎥ ⎢ ⎥ ⎢⎢ ⎥ ⎢ ⎥ ⎢= =

=⎢ ⎥ ⎢ ⎥ ⎢= =⎢ ⎥ ⎢ ⎥ ⎢⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥

2

3

4

5

AT

AA

⎥ ⎢ ⎥⎥ ⎢ ⎥+ Δ⎥ ⎢ ⎥

⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥2

3

TΔ

6 12 65 66 6 120 0 0 0f f

M Mσ σ σ σ⎢ ⎥ ⎢ ⎥

= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ 6 fA

⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎦ ⎣ ⎦

F t i l i t ithi th t i

2

σTΔ( )kmP

( ) ( ) ( )k k km m m T= + Δσ M σ A

For a material point within the matrix

( ) ( )1 11 11 12 13 14 1 11

2 22 21 22 23 24 2 22

3 33 31 32 33 34 3 33

0 00 00 0

k kM M M MM M M MM M M M

σ σ σ σσ σ σ σσ σ σ σ

= =⎡ ⎤ ⎡ ⎤ ⎡⎢ ⎥ ⎢ ⎥ ⎢= =⎢ ⎥ ⎢ ⎥ ⎢⎢ ⎥ ⎢ ⎥ ⎢= =

=⎢ ⎥ ⎢ ⎥ ⎢

( )1

2

3

kAAA

T

⎤ ⎡ ⎤⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥+ Δ⎥ ⎢ ⎥3

4 23 41 42 43 44 4 23

5 31 55 56 5 31

6 12 65 66 6 12

0 00 0 0 00 0 0 0

m m

M M M MM MM M

σ σ σ σσ σ σ σσ σ σ σ

⎢ ⎥ ⎢ ⎥ ⎢= =⎢ ⎥ ⎢ ⎥ ⎢⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥

= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣

4

5

6 m

TAAA

+ Δ⎥ ⎢ ⎥⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎦ ⎣ ⎦

2

8

Calculation of Micro Stresses: Formulas in Extenso (2)

( ) ( ) ( )k k k T+ Δσ M σ A

For a point at the fiber-matrix interfacetn( )k

iP ( ) ( ) ( )i i i T= + Δσ M σ A

( ) ( ) ( )

1 11

2 22k k k

σ σσ σ

=⎡ ⎤⎢ ⎥=⎢ ⎥⎡ ⎤ ⎡ ⎤ ⎡ ⎤

( ) ( ) ( )2 2215 16 1

3 3321 22 23 24 2

4 2331 32 33 34 3

0 0 0 00 00 0

k k kx

n

t i i i

t M M At M M M M A Tt M M M M A

σ σσ σσ σ

⎢ ⎥⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥= + Δ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎢ ⎥=

tx35 31

6 12

σ σσ σ

⎢ ⎥=⎢ ⎥

=⎣ ⎦tt

21

tx Longitudinal interfacial tractiont Tangential interfacial tractiontt Tangential interfacial tractiontn Normal interfacial traction

9

Examples of Micro Stress Distribution in Unit Cells

Contour of micro stress σ2 Contour of micro stress σ2 Contour of micro stress σ2

1

Micro transverse stress due to unit macro transverse tensile stress 2 22Mσ =

2 1σ =

23

23

23

2 1σ = 2 1σ =

2

1

2

12

1

Micro in plane shear stress due to unit macro in plane shear stress MσContour of micro stress σ6 Contour of micro stress σ6 Contour of micro stress σ6

6 1σ = 6 1σ = 6 1σ =

Micro in-plane shear stress due to unit macro in-plane shear stress 6 66Mσ =

23

23

23

1 1 1

10

Micromechanical Model – Random Fiber Array

Cross-sectional view of continuous fiber reinforced composites

11

Finite Element Modeling of the Random Fiber Array• Vf = 0.6• Number of fibers: 120

Mesh generationCross-sectional view

3D view of fibers 3D view of matrix

12

Comparison between Random Array and Regular Arrays

Random fiber array (Vf = 0.6) Regular fiber arrays (Vf = 0.6)

16

18

10

12

14

(MP

a)

Sqr.4

6

8

max

. tn

set 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

qHex.Dia.

0

2

4

0 20 40 60 80 100 120

Case1Case2 2 0.1%ε = 2 0.1%ε =set 1set 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80 20 40 60 80 100 120

(a) random array (vf = 0.6) (b) regular arrays

f iber number f iber volume f raction

13

14

Constituent Failure CriteriaFiber failure criterion• Maximum longitudinal stress failure criterion

3

1 1 1f f fC Tσ− < < Tf1 Fiber longitudinal tensile strengthCf1 Fiber longitudinal compressive strength

1

3

2

• Quadratic failure criterion

6 6 66 6 6

1 1 11ij fi fj i fi

j i iF Fσ σ σ

= = =

+ =∑∑ ∑

M t i f il it i

Fij , Fi Coefficients associated with fiber strengths

Matrix failure criterion

( )21VM m m m mC T I C Tσ + − = Tm Matrix tensile strength

Cm Matrix compressive strength

Interface failure criterion

m p g

tn

2 2

1n s

n s

t tY Y

⎛ ⎞ ⎛ ⎞+ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠2 2 2

Yn Interface normal strengthYs Interface shear strength

3

txttts

15

2 2 2s x tt t t= +21

Incorporation of Kinking Model for Longitudinal Compressive Failure

3D kinking model

⎧ ( ) ( )11 22 11 2211 12

22 11 22 11

33 33

ˆ cos 2 sin 22 2

ˆ ˆˆ

ψ ψψ

ψ

ψ

σ σ σ σσ ϕ τ ϕ

σ σ σ σσ σ

⎧ + −= + +⎪

⎪= + −⎪

⎪ =⎪⎨

Kinking Macro

Micro stresses (Matrix)( ) ( )( )

( ) ( )

33 33

23 23 31

31 31

11 22

ˆ cos sinˆ cos

ˆ sin 2 cos 2

ψ ψ

ψ

ψψ

τ τ ϕ τ ϕτ τ ϕ

σ στ ϕ τ ϕ

⎪⎨ = −⎪⎪ =⎪

−⎪ = − +⎪

Macro Stresses

MMF (Matrix FC)

SAF

16

( ) ( )12 12sin 2 cos 22

τ ϕ τ ϕ= +⎪⎩ MMF (Matrix FC)

Biaxial Tests Using Cruciform Specimens

MMF failure envelope & biaxial test data

[0/±45/90]

Laminate specimen for biaxial test

Specimen held in grips of biaxial test machine

[90] Matrix failure

[0/±45/90]sfor biaxial test of biaxial test machine

800

1000 σy (MPa)Test (smooth)

Test (open hole)

MMF (smooth)

400

600

( )

MMF (open hole)

MaterialProperty

IM7/8551-7(WWFE-Ⅰ)

IM7/8552(Hexcel data)

E11 (GPa) 167 164200

400

E22 (GPa) 8.4 12

nu12 0.27 0.27*

X (MPa) 2550 2724-200

0-800 -600 -400 -200 0 200 400 600 800 1000

σx (MPa)

[±45] Matrix

X (MPa) 2550 2724

X’ (MPa) 1600 1690

Y (MPa) 72 111

Y’ (MPa) 189 189*-600

-400

[±45] Matrix failure

[90] Fiber failure

Y’ (MPa) 189 189*

S (MPa) 65 120

* Value not provided-800

17

failure

Tri-Axial Strength Prediction : WWFE-II Test Case (2)T t C (2) V i ti f th h t th f UD b / ith h d t tiStress Strain Curve : σ2 (=σ3 =σ1 ) vs. τ 12 for T300/PR319 epoxyTest Case (2): Variation of the shear strength of UD carbon/epoxy with hydrostatic pressure

σ3

τ12 (MPa) σ2

τ12

250

300EXP (90°tubes)EXP (0°tubes)EXP (0°tubes)‐Shear failure

12 ( )σ1

100

150

200Final matrixfailure

0

50

100

Initial matrixfailure

MMF prediction• Averaged SAF

‐1250 ‐1000 ‐750 ‐500 ‐250 0 250

σ2 (=σ3=σ1) (MPa)• Averaged SAF• Quadratic fiber FC• Progressive damage

18

Triaxial Test Results For Fibre Reinforced Composites: Second World‐Wide Failure Exercise Benchmark Data, M J Hinton and A S Kaddour, 2009

Tri-Axial Strength Prediction : WWFE-II Test Case (6)Test Case (6): Variation of the longitudinal strength with through-thickness stress for a UD S-glass/Epoxy2Failure Envelope : σ1 vs. σ3 (=σ2) for S-glass/Epoxy2

σ3=σ2 (MPa) Finalσ

0

250

σ (MPa)

σ3 σ2 ( )

FinalMatrix Failuredue to Kinking

MatrixFailure

σ3

‐250

0

‐3500 ‐3000 ‐2500 ‐2000 ‐1500 ‐1000 ‐500 0 500 1000 1500 2000 2500

σ1 (MPa)due to Kinking

σ1

σ2

‐500 FinalFiberF il

InitialMatrixFailure

‐750

FailureFailure

MMF prediction‐1000

FinalMatrixFailure

MMF prediction• Averaged SAF (Hex) for M55 & M66• Quadratic fiber FC• Kinking model (θ=4.5°)

‐1500

‐1250

Test Data

FinalMatrix Failuredue to Kinking

g ( )• Progressive damage

19

‐1500Triaxial Test Results For Fibre Reinforced Composites: Second World‐Wide Failure Exercise Benchmark Data, M J Hinton and A S Kaddour, 2009

Tri-Axial Strength Prediction : WWFE-II Test Case (8)Test Case (8): Variation of the axial compressive strength with through-thickness stress for [±35]s laminateFailure Envelope : σx (=σz) vs. σy for [±35]s; E-glass/MY750 epoxy

σ (MPa)200

σy (MPa)

σx=σz (MPa)

‐200

0

‐1200 ‐1000 ‐800 ‐600 ‐400 ‐200 0 200

Initial envelope

x z ( )

‐400

Final envelope

EXP

‐600

1000

‐800

‐1200

‐1000

20

Triaxial Test Results For Fibre Reinforced Composites: Second World‐Wide Failure Exercise Benchmark Data, M J Hinton and A S Kaddour, 2009

21

Key Issues to Address in Fatigue Life Prediction of Composites

• Why Micromechanics?

• Uni-axial fatigue loads: S-N curvesg

• Multi-axial fatigue loadsu gue o ds

• Mean stress effectsMean stress effects

• Random fatigue loads• Random fatigue loads• Low cycle fatigue (N<100,000) or High cycle fatigue (N>100,000) ?• Strain or Stress based ?• Crack initiation, Crack propagation, or life prediction ?

22

Procedures of MMF-Based Fatigue Life Prediction

Laminate Ply

Macro Stress Analysis

1N2N

6N1M

2M

6M xσ sσ

yσ

Loads &T t

3D beam or FEMTΔF

σ1

NM

time time

CLT

Multi‐axial Randoml d

FEM(3D beam)

Temperatures

time

Damage accumulation 

Fatigue loading

σ,maxσ eqMicro Fatigue Analysis

Multi-axial gand/or Rain‐flow counting

n∑σ

aσS-N cure

Mean Stress Effects

aσt

,σ eq a,σ eq m,minσ eq

,q

iσ,i aσ,i mσMatrix

σnτσij

Stress

di Lif

i

i

nDN

=∑aσS-N cure

R= 0R= ‐0.3

R= ‐0.5R= ‐0.7

R= 0.5

1aσ

2aσ

t,

σ eq ,σ eq aFiber

z

nσij

σxx,f

Interface

Micro mechanical modelConstant Life Diagram

Predict Life

mσN

N N1N2N3 mσTime

,σ eq m

Micro stresses

xy

23

Co sta t e ag aOf Each Constituent

Laminates Subjected to Fatigue Loadings• Laminate • Ply

( )x x tσ σ⎧ =⎪on‐axis( )N N t⎧ =⎪

⎨In‐plane

( )( )

y y

s s

tt

σ σσ σ

⎪ =⎨⎪ =⎩

on axismacro stresses

( )( )M M t⎨ =⎪⎩

ploads

2N

( )s s⎩FEM or CLT TΔ

1N6N

yσ

2M

M σ sσy

1M6M xσ

NIn‐planeloads

ijσ

M

NTime

ij

24

M t

Calculation of Micro Stresses in Constituents

( )( )

1 1 tt

σ σσ σ⎧ =⎪ =⎨

on‐axismacro

( ) ( )1 111 12 13 14

2 221 22 23 24

0 00 0

k kM M M MM M M M

σ σσ σ⎡ ⎤ ⎛ ⎞⎡ ⎤

⎜ ⎟⎢ ⎥ ⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥

Stress Amplification Factors (SAF)

( )( )

2 2

6 6

tt

σ σσ σ

=⎨⎪ =⎩

macro stresses

2 221 22 23 24

3 331 32 33 34

41 42 43 444 4

55 565 5

0 00 0

0 0 0 0

M M M MM M M M

M M

σ σσ σσ σ

⎜ ⎟⎢ ⎥ ⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥

= ⎜ ⎟⎢ ⎥ ⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎜ ⎟

MMF

65 666 60 0 0 0mm

M Mσ σ⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎜ ⎟⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎝ ⎠

TΔMatrix

MMF

2σFiber

Interfacetiσm

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

kf

km

k kf f

k km m

k kk

T

T

= + Δ

= + Δ

σ

σ

M A

M A

σ

σ1σ 6σ

12

3

Micro model

σf

Ply level macro cyclic stress micro cyclic stresses (Micro‐mechanics)

( ) ( ) ( )k ki i

ki T= + ΔM Aσt Micro model

Ply level macro cyclic stress  micro cyclic stresses (Micro mechanics)

Applied to constituent fatigue models (multi‐axial stress & mean stress effects )

25

MMF-Based Fatigue Models

( ) ( )2 2 21, 1 ,1 1 4m VM mI Iβ β βσ

σ− ± − +

= ( ) ( )22i k

Critical plane modelat the interface

Equivalent stress distribution in the matrix

, 2eq mσβ

= ( ) ( )22, ,eq i n nsign kσ σ τ σ τ= +

φ

z nστMatri

iσ

θ

φ

x

y

τMatrix

Interface

σnτσmtlinear σ

Time

τaτ

On a plane with directions ,θ φ

Fiber

Interfacen=1,m=1

n=1.5,m=1

n=1,m=2

n=3,m=3

,eq aσ

Ma im m stress model

Time

nσ,maxnσ

2

3 σf1,eq mσ

mTmC

σ

, 1eq f fσ σ=

Maximum stress modelfor fiber

1 Micro model

1fσ

aσ

R= 0R= ‐0.3

R= ‐0.5R= ‐0.7

1f

tmσNN1N2N3

26

tm

Uniaxial Fatigue Tests to Generate S-N Curves of Resin

• Uni‐axial fatigue loads • Life prediction using S‐N curve

IsotropicS

( )a ff Nσ =

σσ

aσ( )f

t

maxσ

iσ

aσNf N

minσ

27

Effects of Mean Stress on Fatigue Life

For isotropic material Life depends on both amplitude and mean stresses

σmaxσσ

constaσ =aσmσminσ

max

min

max

Stress ratio, R σσ

= t: mean stressσ

t

• Mean stress effect : Constant Life Diagram

R=∞ R= ‐2 R= ‐1 R= ‐0.5 R=0 R=0.5

: mean stress,: stress amplitude

m

a

σσ

σ N li MσMean stress effect : Constant Life Diagram

aeff

m

TT C T C

σσσ

=+ −

− − 1 1

σσσ σ

=⎛ ⎞ ⎛ ⎞− +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

aeff n m

m m

T C

•Modified Goodman •Harris (non‐linear) aσaσ

S-N cure

Nonlinear MeanStress Effects

aσ

R= ‐0.5

R= ‐0.7

2 2m ⎝ ⎠ ⎝ ⎠T C

80

100linear

n=1,m=1

n=1.5,m=1

n=1,m=2

R= ‐1.0

R= 0

R= ‐1.5R= ‐0.3

N1

S N cureR= 0

R= ‐0.3

R= 0.5

1aσ

2aσ

20

40

60 n=3,m=3 R= 0

R= 0.3

R= 0.5

mσN

NN1

N2

N3

mσ

28

0

‐150 ‐100 ‐50 0 50 100

m

Equivalent Stress Method for Multi-Axial Fatigue Loadings

For isotropic material •Determination of Equivalent Stress

σ EQ modelσ iσ,i aσ,i mσ

iσeqσ

( ) ( )2 2 21, 1, ,1 1 4β β βσ− ± − +a a vm aI I

tt

σ sσyσ

( ) ( )

( ) ( )

1, 1, ,,

2 2 21, 1, ,

2

1 1 4

σβ

β β βσσ

=

− ± − +=

a a vm aeq a

m m vm mI I

, CwhereT

β =xσ , 2

σβ

=eq m

M t ff t σ•Mean stress effects aσ

R= 0R= ‐0.3

R= ‐0.5R= ‐0.7

60

80

100linear

n=1,m=1

n=1.5,m=1

n=1,m=2

n=3,m=3

R= ‐1.0

R= 0

R= ‐1.5R= ‐0.3

aσ

mσNN1N2N3

0

20

40

,

R= 0.3

R= 0.5

mσ

29

m‐150 ‐100 ‐50 0 50 100

m

Critical Plane Model for the Fiber-Matrix Interface• Determination of normal and shear stresses at the 

interface

σ • Effective stress

( ) ( )22,σ σ τ σ τ= +eq n nsign kφ

z nστ

Effective stress

θx

y

On a plane with directions θ φ

σ eq ,σ eq a

Time

τOn a plane with directions ,θ φ

Time

,σ eq m

•Mean stressTime

nσ

•Mean stress effects

aσ

R= 0R= ‐0.3

R= ‐0.5R= ‐0.7

80

100linear

n=1,m=1

n=1.5,m=1

n=1,m=2

R= ‐1.0

R= 0

R= ‐1.5R= ‐0.3

σNN1N2N3

20

40

60 n=3,m=3 R= 0

R= 0.3

R= 0.5

30

mσ0

‐150 ‐100 ‐50 0 50 100

Cycle Counting for Random Fatigue LoadingsSi l i fl i f l i l• Simple rainflow counting for only one signal

E(N) N=4E(N) N=2E(N) N=4N=2E(N)

Time

N=2

N=3Time

N=3

N=1 N=1 N=5 N=1 N=3

• Multi‐axial rainflow counting for several signals

τ

Time

Countingsignal

(simple rainfow

aτ

nσ

Counting)

maxnσ

Time

Auxiliarysignal

,maxn

31

Cumulative Damage Law

Random Fatigue Load Case1 Case2 Case3 Case4 Case5

…..

RainflowCounting

1

,1f

nN

Number of cycle

Number of cycle to failure

…..…..

2

,2f

nN

3

,3f

nN

4

,4f

nN

5

,5f

nN

• Linear cumulative damage law (load segments with same failure modes failure

1DDamage index …..= i

inDN

2D 3D 4D 5D

Linear cumulative damage law (load segments with same failure modes failure modes, Miner’s rule)

1

k

iD D= ∑

,f iN

• Non‐linear cumulative damage law

{ }exp 1k

D D F D⎡ ⎤= −⎣ ⎦∑

1i=

ORk

mD D= ∑

32

{ }1

exp 1i i ii

D D F D=

⎡ ⎤= ⎣ ⎦∑ OR1

ii

D D=

= ∑

Fatigue Life Prediction of Off-Axis UD

50

60

er (M

Pa)

• Matrix Fatigue S‐N curves from the UD * ( ) ( ) 0.07100 19f ff N N

−= −

20

30

40

paramete

xσ15ω = °E‐glass/Epoxy (Vf =0.6)

,min 0.1xRσ

= =

3σ

2σijσ

matrix

fiber

0

10

20

TEST‐15deg

Curve fitting

Dam

age 

Time

,maxxσ

,minxσ

,maxxσ MMFxyσ

2

1σ Matrix

300TEST‐5deg

2 3 4 5 6 7

• Well predict the fatigue life of UD with other fiber angle

log Nf

200

250

TEST‐5degTEST‐10degTEST‐15degTEST‐60degPrediction

with other fiber angle

ss (M

Pa)

xσ10ω = °

xσ5ω = °

50

100

150

mum

 stre0.1R =E-glass/Epoxy (Vf =0.6)

xσ60ω = ° , m a xxσ

, m inxσ

0

50

2 3 4 5 6 7

Maxim

log Nf

Time

But off‐axis UD with small fiber angles are slightly overestimated because of matrix fatigue models.

UD

33

* Hashin Z, Rotem A. A fatigue failure criterion for fibre reinforced materials. Journal of Composite Materials 1973;7:448–64.

log Nf

Material Properties: Off-Axis UDMaterial properties and analysis options for fatigue life prediction of the off-axis UD

1. Matrix material propertiesMatrix : Epoxy (Epon826)a)

2. Fiber material propertiesFiber : E glass b)

3. Interface material propertiesFiber : E glass/EpoxyMatrix : Epoxy (Epon826)a)

Multi-axial stress : Equivalent stressMean stress effect : Modified Goodman

S-N curve : Basquin’s eqn.

Fiber : E-glass b)

Multi-axial stress : Max. stress (sxx,f)Mean stress effect : Modified Goodman

S-N curve : Basquin’s eqn.

Fiber : E-glass/EpoxyMulti-axial stress : Critical planeMean stress effect : Modified Goodman

S-N curve : Basquin’s eqn.

( ) ( )2 2 21, 1 ,

,

1 1 42

m VM meq m

I Iβ β βσσ

β− ± − +

=iσ ( ) ( )22

, ,eq i n nsign kσ σ τ σ τ= +

θ

φ

x

z

y

nστ

, ,eq f xx fσ σ=,xx fσ

2150=fT MPa

1.9 

2.0 

MPa

)

tTime

Time

nσ

τaτ

,maxnσ

On a plane with directions,θ φ

2000

2500

a)

t

2150σ = =MAXeq fT MPa

1450=f

fC MPa

80σ = =MAXeq mT MPa

y = ‐0.073x + 1.955

1.5 

1.6 

1.7 

1.8 

log sig_eq

_bar (log

, M

E d

4. Micromechanics model500

1000

1500

sig_eq

_bar ( MPa

660σ =Endurance MPa

0.136log lo 3.636gσ −= +eq fN

0 073log lo 1 955gσ −= +fN

1.4 

0  1  2  3  4  5  6  7 

l

log Nf

32.9σ =Enduranceeq MPa

0

0  1  2  3  4  5  6  7 

log Nf

660σ =eq MPaSQR(FEM) HEX(FEM) MCT

VolumeAverage

Multi-Continuum Theory

0.073log lo 1.955gσ +eq fN

a) Tao G., and Xia Z., “Biaxial Fatigue Behavior of an Epoxy Polymer with Mean Stress Effect,” International Journal of Fatigue, Vol. 31, No. 4, 2009, 678 685

34

pp. 678-685.b) Z. Hashin and A. Rotem, A Fatigue Failure Criterion for Fiber Reinforced Materials, Journal of Composite Materials 1973; 7; 448

Fatigue life prediction of the off-axis UD : off-axis angle

300

350SQRHEX

S-N curve @ θ = 15°

a)

Off-axis UD •Material : E-glass/Epoxy(Epon826)•Layup : [15°] [30°] [60°]

150

200

250 MCTTEST b)

σ x,m

ax(M

PaLayup : [15 ], [30 ], [60 ]

xσθ

R= 0.1xσ

0

50

100σ

•Micro-mechanics models (No averaged SAF)SQR(FEM) HEX(FEM) MCT

0 1 2 3 4 5 6 7fNlog

S N curve @ θ = 60°S N curve @ θ = 30°

VolumeAverage

Multi-Continuum Theory

250

300

350SQRHEXMCT 250

300

350SQRHEXMCT(M

Pa)

S-N curve @ θ = 60S-N curve @ θ = 30

(MP

a)100

150

200

250TEST b)

100

150

200

250TEST b)

σ x,m

ax(

σ x,m

ax(

0

50

100

0 1 2 3 4 5 6 7

0

50

100

0 1 2 3 4 5 6 7

35

fNlogfNlog

Fatigue Life Prediction of the Angled Laminates

600

70015deg30deg45d

S-N curve using Square model

a)

Angled Laminates

•Material : E glass/Epoxy(Epon826)

300

400

50045deg60deg

σ x,m

ax(M

Pa•Material : E-glass/Epoxy(Epon826)

•Layup : [±θ°]SA l θ 15 30 45 60 90°

0

100

200

σ•Angle, θ = 15, 30, 45, 60, 90°R= 0.25

xσxσ

θ

0 1 2 3 4 5 6 7

•Micro-mechanics models (Averaged SAF)HEX(FEM)

fNlog

S-N curve using Hexagonal modelSQR(FEM)

500

600

70015deg30deg45deg60deg(M

Pa)

g g

200

300

400

g

σ x,m

axStatic strengths

0

100

0 1 2 3 4 5 6 7Nl

prediction usingTsai‐Wu criteriain E‐glass/MY750(WWFE)

36

fNlog(WWFE)

Material Properties : UDT, BX and TXM i l i d l i i f f i lif di i f E l /E (H i )Material properties and analysis options for fatigue life prediction for E-glass/Epoxy(Hexion)

1. Matrix material properties a)

Matrix : Epoxy (Hexion)2. Fiber material properties b)

Fib E l

3. Interface material propertiesFib E l /EMatrix : Epoxy (Hexion)

Multi-axial stress : Equivalent stressMean stress effect : Modified GoodmanS-N curve : Basquin’s eq.

Fiber : E-glassMulti-axial stress : Max. stress (σf1)Mean stress effect : Modified GoodmanS-N curve : Basquin’s eq.

Fiber : E-glass/EpoxyMulti-axial stress : Critical planeMean stress effect : Modified GoodmanS-N curve : Basquin’s eq.q q

( ) ( )2 2 21, 1 ,

,

1 1 42

m VM meq m

I Iβ β βσσ

β− ± − +

=iσ , 1eq f fσ σ=1fσ

q q

( ) ( )22, ,eq i n nsign kσ σ τ σ τ= +

θ

φ

x

z

y

nστ

68=T MPa 2150=T MPa

2000

2500

)60

70

80

)

t tTime

nσ

τaτ

,maxnσ

On a plane with directions,θ φ

68σ = =MAXeq mT MPa 2150σ = =MAX

eq fT MPa

68125

==

m

m

T MPaC MPa

2150

1450

=

=f

f

T MPa

C MPa

500

1000

1500

sig_eq

_bar ( MPa

)

20

30

40

50

60

sig_eq

_bar ( MPa

)

0.13log lo 2.037gσ = +−eq fN0.227log lo 3.746gσ −= +eq fN

4. Micromechanics models

Time

E d

0

0 1 2 3 4 5 6 7

log Nf

0

10

0  1  2  3  4  5  6  7 

log Nf

242σ =Enduranceeq MPa

4. Micromechanics models18σ =Enduranceeq MPa

SQR(FEM) HEX(FEM) MCT

VolumeAverage

Multi-Continuum Theory

a) Tested by HSCL

Average

Vf = 0.42

37

) yb) Z. Hashin and A. Rotem, A Fatigue Failure Criterion for Fiber Reinforced Materials, Journal of Composite Materials 1973; 7; 448

Fatigue Life Prediction of the UDT, BX and TX

140

160SQRHEX

S-N curve @ BX[±45°]s

a)

UDT, BX and TX• Material : E-glass/Epoxy (Hexion)• Layup : UDT[90°], BX[±45°]s and TX[0°2/±45°]s

xσ

60

80

100

120 MCTTEST a)

σ x,m

ax(M

Pas 2 s

• Micromechanical modelsSQR(FEM) HEX(FEM) MCT

Multi-C ti

0

20

40

60σ

R= 0 1

VolumeAverage

Continuum Theory

• Fatigue loading a) tested by HSCL

[±45°] ply, matrix fail

0 1 2 3 4 5 6 7fNlog

R  0.1xσ

S N curve @ TX[0° /±45°]S N curve @ UDT[90°] xσ xσ

400

500

600SQRHEXMCT

60708090

SQRHEXMCTM

Pa)

S-N curve @ TX[0 2/±45 ]sS-N curve @ UDT[90°]

(MP

a)200

300

400 TEST

2030405060 TEST

σ x,m

ax(M

σ x,m

ax(

0

100

0 1 2 3 4 5 6 7

01020

0 1 2 3 4 5 6 7

[90°] ply, matrix fail LPF [0°] ply, fiber fail

38

fNlogfNlogStatic failure

Implementation of MMF-Based Fatigue Analysis into FEM

FEM ToolsImport *.mmi file;

MMFatigue ProgramSMM+ Preprocessor

for SMM+

Import  .mmi file;• Node & Element information

Import *.mms file;• Ply macro stresses

MMFatigue Program(Visual Basic)

3D BEAM

for SMM

Interfacefor 3D BEAM •Micromechanics based fatigue analysis

NASTRAN

for 3D BEAM

Interface

•Micromechanics based fatigue analysis•Multi‐axial fatigue stress•Mean stress effect (modified Goodman, Harris)

•Random fatigue loadsNASTRAN for NASTRAN

I t f

•Random fatigue loads (rainflow counting method)

•Damage accumulation (Miner’s rule)

ABAQUSInterfacefor ABAQUS

Export *.out file;• Damage index

*.mmf file•Fatigue loading•Fatigue parameters (constituents)•Stress amplification factor•Options

GUI

39

Summary and Conclusion

• The MMF fatigue life prediction of laminates, applied to real composites structures.• Fatigue loads with various mean stresses and multi-axial loads orFatigue loads with various mean stresses and multi axial loads orspectrum, random fatigue loads.

• Environmental effects (temperature and moisture) can be readily investigated.( p ) y g

• A MMF based life prediction module for commercial SMM+ is developed.• A linkage with the 3D-beam, ABAQUS and NASTRAN is under development.

• Need to generate constituent Failure and Fatigue D/B from Experiments.

Thank you for your attention.sungkha@gmail.comyuanchen.huang@gmail.comSee you in ICCM18, 2011, Jeju island, Korea.

40