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 [email protected]@gmail.comSee you in ICCM18, 2011, Jeju island, Korea.
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