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1
Damage and Residual Life Prediction of Vehicle Structures
Chi L. ChowDepartment of Mechanical Engineering
University of Michigan-Dearborn
2
Table of Contents
• Introduction• Theory of Damage Mechanics• Example Projects
i) Bumper Damage under multiple Impact Loadingii) Crashworthiness of High Strain Rate Plasticsiii) Fatigue Damage of Strain-Rate and Temperature Dependent Solder Alloysiv) Forming Limit Diagram (FLD) of Strain Rate Dependent Metals
3
1. Introduction
Damages in a vehicle structure are caused by material degradation due to initiation, growth and coalescence of mirco-cracks/voids in a ‘real-life’ material element from monotonic, cyclic/fatigue, thermo-mechanical loading or dynamic/explosive impact loading.
4
Micro-Meso-Macro scales
5
‘Real-Life’ Materials
6
‘Real-Life’ Materials
7
Micro-defects - Inclusions
8
Macro-Crack Formation
9
Fracture/Rupture Process
0
10
20
30
40
50
0 0.01 0.02 0.03 0.04 0.05 0.06
strain
stre
ss
Before Initial Subsequent FinalLoading Loading Loading Loading
10
Damage Mechanics
The theory of damage mechanics takes into account the process of material degradation due to the initiation, growth and coalescence of micro-cracks/voids in a ‘real-life’ material element under monotonic or cyclic or impact or thermo-mechanical loading
11
Fracture/Rupture Criteria
A valid material failure criterion must therefore take into account the process of progressive material degradation/damage under either static or dynamic/fatigue loading. Unfortunately, all conventional failure criteria including fracture mechanics ignore the process and thus unrealistic and unreliable.
12
Rupture Criterion - Conventional
For Smooth Specimen (with/without notches)
• Static loading
– Stress, strain or energy-based criteria
• Fatigue loading
– S-N curve (Due to Wohler in 1858) for constant amplitude loading
– Miner’s rule for variable amplitude loading
– Rainfall counting method for ‘real life’ fatigue loading
P
P
13
Rupture Criterion – Conventional
For Cracked Specimen
• Static loading
– Fracture Mechanics
• Fatigue loading
– Paris Law
– Others based on G, J
P
P
crack
14
Rupture Criterion – Damage Mechanics
A material element is failed when cumulative damage reaches its critical value.
Unified criterion for different conditions:
– Smooth and cracked specimen– Static and fatigue loading– Crack initiation and propagation
15
Major Advantages
• Ability to quantify material damage and predict residual life after impact loading
• Capable of providing a unified rupture criteria for macro-crack initiation or propagation for either brittle and ductile fracture. This includes fatigue damage, localized necking, multi-phase composite failure, etc
16
Past Project: Crash Mechanics
• Bumper under Multiple Impact Loading
• Crashworthiness of High Strain Rate Plastics
• Head Impact Mechanics
• Design of Seat Impact
• Knee Bolster Design Optimization under Impact
17
Past Projects: Electronic Packaging
• Fatigue Damage of Strain-Rate and Temperature dependent Solder Alloys
• Scale Effect of Solder Joints
• Micro-structural Evolution of Solders
18
Past Projects: Sheet Metal Formability
• Forming Limit Diagram (FLD) for Strain Rate Dependent Metals
• Formability of Tailor-Welded Blanks of Aluminum, Steels and Titanium
• FLD of Warm and Hot Forming
• FLD of Multiple Stamping Processes
• Warm and Hot Magnesium Tube Hydroforming
19
Past Projects: Fatigue and Fracture
• Failure Analysis of Rubber-like Materials
• Thermo-mechanical Fatigue of Engine Block Cracking
• Fractures in Composite Structures
• Fracture in Aluminum Weld Components
• Mechanics of Fracture in Tires
20
2. Theory of Damage Mechanics
21
Definition of scalar damage variable
where A0 = original surface area (with defects);
A = surface area excluding defects
n
A
0
0
A
AAD
22
Damage-coupled Elasticity
• True stress was replaced by effective stress
• Based on strain equivalent
principle
• Based on energy equivalent principle
• Damage evolution equation
)1(0 DE
20 )1( DE
D
1
,......),( DfD
23
Relationship Between Scalar Damage Variable and Young’s Modulus
Undamaged material Damaged Material
01 E
ED 0
1 EED or
24
Effective Young’s Modulus: an example
25
Tensor Damage Variable
Relationship between Effective stress and Cauchy Stress
= ( ) : M D
where M(D) is damage effect tensor. For isotropic damage,D becomes a scale damage variable. M(D) then becomes I
D-1
1 = )( DM
I is a unit tensor .
26
Damage Effect TensorFor multi-axial stress state,
-100000
0-10000
00-1000
0001
0001
0001
D-1
1 = )(
DM
where D and are scalar damage variables
= [ ] = [ ]
= [ ] = [ 2 2 2 ]
T1 2 3 4 5 6 11 22 33 23 31 12
T1 2 3 4 5 6 11 22 33 23 31 12
damage effect tensor is
27
Free Energy Equivalence The free energy for a damaged material may be expressed in a form similar to that for a material without damage except that all stresses are replaced by their corresponding effective stresses.
• Without damage
• With damage
pT
pT ::
2
1::
2
1 110 CC
pT
peW ::
2
1 1
0C
28
Damage-Coupled Elastic Equation
)+2(100000
0)+2(10000
00)+2(1000
0001--
000-1-
000--1
E
1 =
1C
where C is effective/damage stiffness matrix
: = = e 1
C
200
2000
200
0
)-2(1+4-1
)3-(1-)-2(1- =
)-2(1+4-1
)D-(1E = E
2
and
29
Damage Energy Release Rate
The conjugates of the damage variables, known as damage energy release rate, are defined as
::1
1
::1
1 1
A
C
T
TD
DY
DDY
30
Formulation of [A] Matrix
0000
0
2-)-)(1+(1 = A 2-)-(12 = A
)A-A2(00000
0)A-A2(0000
00)A-A2(000
000AAA
000AAA
000AAA
D)-(1E
1 =
21
21
21
21
122
212
221
A
31
Isotropic Strain Hardening
Isotropic Hardening Based Yield Surfaces
-400
-300
-200
-100
0
100
200
300
400
-400 -200 0 200 400
stes1
ste
s2
32
Kinematic Strain HardeningKinematic Hardening Based
Yield Surfaces
-400
-200
0
200
400
-400 -200 0 200 400
stes1
ste
s2
33
Damage-Coupled Yield Surface
0 =R(p)] + R[ - D
= R),,(F 0eqp
1
1,D
where is defined aseq
eq 12 1 2
21 3
22 3
242
52
626{( ) ( ) ( ) ( )}
0Rand and R(p) are yield stress and strain hardening threshold. p is overall effective plastic strain.
34
Damage-Coupled Plastic Equation
eq
pp
D
Fd
2
3
1
1 Sε
)( R
Fdp p
where SS is the true stress deviator tensor,λ is a Lagrange multiplier.
35
Damage and Yield Surfaces
-300
-200
-100
0
100
200
300
-300 -200 -100 0 100 200 300
dmg sur.
yld sur.
36
Fatigue and Plastic Damage Surfaces
1
2 plastic damage surface
fatigue damage surface
no damage
37
Plastic Damage surface
The expanding plastic damage surface is expressed in terms of the damage energy release rates and as
0 =] )wB(+B [ - Y = ) B, (F p0ppdpd2/1Y
) Y + Y ( 2
1 = Y 2
p2Dppd
DY Y
38
Plastic Damage Evolution Equations
pdpd
pdp
pd
ppd
p
pdpdp
pd
Dppd
Dp
pdpdp
B)(
Fdw
Y2Y
=Y
F=d
Y2Y
=Y
F=dD
=
=
2/1
2/1
where dwp is overall plastic damage increment, λpd is the Lagrange multiplier
39
Fatigue Damage
fdf
f
fdfdf
Df
fdfdf
dw
Y
F=d
Y
F=dD
=
where dwf is overall fatigue damage increment,λfd is a Lagrange multiplier
Fatigue damage surface
Fatigue damage evolution equations
0 = ) D, (F fd ,
40
Total Damage
pf
pf
pf
www
DDD
Total damage is the summation of fatigue damage and plastic damage
41
Damage Failure Criterion
A material element is said to have rupture when the total cumulative overall damage (w) has reached the measured critical value (wc) of the material under investigation.
42
Finite Element Analysis
The damage coupled material model has been implemented in ABAQUS and LS-DYNA through UMAT and user specified material subroutine respectively. It has also been programmed in FCRASH of Ford.
43
UMAT: Variables to be defined
• Jacobian Matrix of the material model
It must be defined accurately if rapid convergence is to be achieved. However, an incorrect definition only affects the convergence rate. The results (if obtained) are unaffected.
• Stress tensor• Elastic and plastic strain tensor • Damage variables defined as the solution-
dependent state variables
44
Damage Analysis Approach
Damage-Coupled Material Model
FEA
Damage Index
45
3. Example Projects
46
Bumper Damage under Multiple Impact Loading
Objectives
• To evaluate crashworthiness of bumpers under multiple low speed impact.
• To quantify accumulative damages in two bumpers, one made of SAE 950 and another, martensitic sheet steel.
• To predict overall damages sustained in the bumpers
and their potential sites of failure using FCRASH programmed with the damage model and then compare the simulation results with those of drop-weight testing.
47
Testing Procedure for E and
0
0 2 4 6 8 10 12
strain
stre
ss
E0 E1 E2 E3
1
2
3
, 0.01
48
Measured Young’s Modulus E
120000
150000
180000
210000
0 0.1 0.2 0.3
true strain
Yo
un
g's
mo
du
lus,
MP
a
49
Damage Variables D and µ
From the equations of E and in an earlier slide, we can evaluate D and μ with measured data (E0, E, 0, ) as
and
0)21(2}31)1(2{ 0002
00
})1(241{)1( 200
0
2 E
ED
50
Critical Damage
Critical overall damage for SAE 950 steel was
measured to be wc=0.112 and martinsite sheet
steel, wc=0.04.
51
Effective Stress-Strain Curve
0
100
200
300
400
500
600
700
0 0.1 0.2 0.3
true strain
tru
e st
ress
, M
Pa
52
Strain Hardening Curve of R versus p
0
50
100
150
200
250
300
0 0.1 0.2 0.3
equivalent plastic strain
stra
in h
ard
enin
g,
MP
a
53
Damage Hardening of B versus w
0
0.5
1
1.5
0 0.05 0.1 0.15
overall damage
dam
age
har
den
ing
, M
Pa
54
One Bumper Model
(made of SAE 950 sheet steel)
55
Spring Assisted Drop WeightTest Fixture
56
FEA and Test Results:Maximum Impact Force under 6th impact of 13mph
57
Damage Contours(after six impacts)
Maximum damage near support ends
58
Photo Micrographs(SAE 950 beam near weld and support showing damage)
1
65
432
87
Flange Edge Surface
Middle of Flange
59
Photo Micrograph(SAE 950 beam in center showing no damage)
1
65
432
87
Center Under Impactor
Maximum deformation, but no damage
60
Summary• A method of crashworthiness analysis based on damage
mechanics has been developed to quantify the degree of damage in a vehicle bumper after multiple low speed impacts.
• It is verified that the failure criterion developed based on the overall damage is superior to the conventional concept of strain.
• Satisfactory numerical results on the peak value of contact forces are obtained and compared well with the test results.
• A damaged-coupled FCRASH can be used to quantify bumper damages due to multiple impacts and to improve vehicle safety.
61
Crashworthiness of High Strain Rate Plastics
Objectives
• Measurement of material behavior at different strain rates (up to 103/s)
• Development of a rate-dependent constitutive model
• Characterization of damage accumulation and development of a damage failure criterion
• LSDYNA FEA analysis and testing of three-point bend impact for damage model validation
62
SHPB for Material Tests
Split Hopkinson Pressure Bar (SHPB) for high strain-rate testing (102~104/s)
i incident strain wave
r reflection strain wave
t transmission strain wave
Striker BarInput Bar Output Bar
i , r t
Specimen
63
Measurement of Stress-Strain Curve
C: speed of stress wave
Ls: length of specimen
A: cross-section area
of incident bar
As: cross-section area
of specimen
E: Young’s modulus
of incident bar
)( )(2
)( tEA
Aσ(t)t
L
Ct t
sr
s
-1500
-1000
-500
0
500
1000
1500
0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006
tim e, s
In cid en t
Tran s m itter
64
Rate-Dependent Stress-Strain CurvesMaterial B, strain rate effect, 25C
0
10
20
30
40
50
60
0 0.05 0.1 0.15
true strain
3.5E+3/s1.0E+3/s6.0E+2/s4.0E+2/s1.0E-2/s1.0E-3/s1.0E-4/s
Materia l A, s tra in ra te effect, 25C
0
10
20
30
40
50
60
0 0.05 0.1 0.15
tru e s tra in
3.5E+3/s
1.0E+3/s
6.0E+2/s
4.0E+2/s
1.0E-2/s
1.0E-3/s
1.0E-4/s
Material C, strain rate effect, 25C
0
10
20
30
40
50
60
70
0 0.05 0.1 0.15
true strain
3.5E+3/s1.0E+3/s6.0E+2/s4.0E+2/s1.0E-2/s1.0E-3/s1.0E-4/s
Material D, strain rate effect, 25C
0
10
20
30
40
50
60
70
80
0 0.05 0.1 0.15
true strain
3.5E+3/s1.0E+3/s6.0E+2/s4.0E+2/s1.0E-2/s1.0E-3/s1.0E-4/s
65
Rate-Dependent Constitutive Model
E, a and m are material constants.
when the strain rate is constant
mpe aeE
)]1(1ln[1
mEe
a
mE
66
Damage Failure Criterion
A material element is said to have ruptured when the total cumulative damage D reaches the critical value Dc of the material
67
Damage MeasurementMaterial A, modulus change under different deformation
0
2
4
6
8
10
0 0.002 0.004 0.006 0.008 0.01 0.012
true strain
00.10.20.30.40.5
Material A, modulus change
0
100
200
300
400
500
600
700
800
0 0.1 0.2 0.3 0.4 0.5 0.6
true strain
Material B, modulus change under different deformation
0
2
4
6
8
10
0 0.002 0.004 0.006 0.008 0.01 0.012
true strain
00.10.160.30.4
Material B, modulus change
0
100
200
300
400
500
600
700
800
900
0 0.1 0.2 0.3 0.4 0.5
true strain
68
Three-Point Bend Impact Test
69
DeformationHigh-speed imaging sequence of a 2.2 m/s three-point bend test at 22°C for material C
70
Load-Displacement Curves
Impact velocity 2.2m/s, temperature 22C Material A Material B
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 10 20 30 40 50
Deflection (mm)
Lo
ad
(K
N)
test
simulation
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 10 20 30 40 50
Deflection (mm)
Lo
ad
(K
N)
test
simulation
71
Maximum Load
Max Load (KN)MaterialMeasured Computed Percent Difference
A 0.14 0.15 7.1B 0.17 0.17 0C 0.19 0.18 5.2D 0.24 0.23 4.2E 0.24 0.24 0F 0.30 0.26 1.3G 0.31 0.30 3.2
72
Damage Accumulation andFailure Analysis
Material
Critical damagevalue
Maximum damagevalue
Fail or safeprediction
Test result
A 0.63 0.49 Safe No failureB 0.61 0.59 Safe but
criticalNo failure
C 0.76 0.68 safe No failureD 0.67 0.66 Safe but
criticalNo failure
E 0.44 0.39 Safe No failureF 0.41 0.24 Safe No failureG 0.30 0.33 Fail at 8.5mm
deflectionfailure
73
Summary
• All seven polymer materials are rate-dependent
• SHPB can be used to measure stress-strain curves under high strain rates
• The damage model can be used to characterize damage behavior of polymeric components under impact loading
• A failure criterion based on overall damage accumulation has been found to be satisfactory in failure analysis
74
Strain-Rate and Temperature Dependent Fatigue Damage
75
To develop a predictive Viscoplasticity Model for fatigue damage of solder alloys based on the theory of Damage Mechanics
Objective
76
Mechanics of Fatigue
• Metal fatigue in a material element is caused by either cyclic mechanical or thermo-mechanical loading.
• Progressive material degradation or damage leads to eventual fatigue crack initiation and propagation
• The process of fatigue damage must be included in any fatigue analysis to produce a consistent and reliable prediction.
77
Fracture/Rupture Process
0
10
20
30
40
50
0 0.01 0.02 0.03 0.04 0.05 0.06
strain
stre
ss
Nf = 0 Nf = 1000 Nf = 10,000 Nf = 1,000,000
78
25 days aged specimenat 100oC
16 hours aged specimenat 100oC
Aging of Solder Alloysat Elevated Temperature
79
Effects of microstructureon hysteresis loops
• Two batches of 63Sn-37PbBatch A (100C/16h): 16 hours at 1000C
Batch B (125C/24h): 24 hours at 1250C
• Two temperatures250C and 800C
• Three strain rates10-3/s, 10-4/s and 10-5/s
63Sn-37Pb, 1.0E-3/s, 1%, 25C
-50
-40-30
-20
-100
10
20
3040
50
0 0.002 0.004 0.006 0.008 0.01
strain
125C, 24h100C, 16h
63Sn-37Pb, 1.0E-4/s, 1%, 25C
-40
-30
-20
-10
0
10
20
30
40
0 0.002 0.004 0.006 0.008 0.01
strain
125C, 24h100C, 16h
63Sn-37Pb, 1.0E-5/s, 1%, 25C
-30
-20
-10
0
10
20
30
0 0.002 0.004 0.006 0.008 0.01
strain
125C, 24h100C, 16h
80
Rate and Temperature Effectson Isothermal hysteresis loops
• 63Sn-37Pb solder• Three temperatures
250C, 800C and 1000C • Three strain rates
10-3/s, 10-4/s and 10-5/s
63Sn-37Pb, 1%, 25C
-40
-30
-20
-10
0
10
20
30
40
0 0.002 0.004 0.006 0.008 0.01
strain
1.0E-3/s1.0E-4/s1.0E-5/s
63Sn-37Pb, 80C
-30
-20
-10
0
10
20
30
0 0.002 0.004 0.006 0.008 0.01
strain
1.0E-3/s1.0E-4/s1.0E-5/s
63Sn-37Pb, 100C
-20
-15
-10
-5
0
5
10
15
20
0 0.002 0.004 0.006 0.008 0.01
strain
1.0E-3/s1.0E-4/s1.0E-5/s
81
Load-drop tests under TMF loading
• Batch A (100C, 16h)ramp: 30 minutes
strain rate: 210-5/s
temperature range: 25 to 800C
strain range: 0 to 3.6%
63Sn-37Pb, 25-80C, 30 min.,3.6%, 2.0E-5/s, 0D
-20
-15
-10
-5
0
5
10
15
20
0 0.01 0.02 0.03 0.04
strain
63Sn-37Pb, 25-80C, 30 min.,3.6%, 2.0E-5/s, 0D
-20
-15
-10
-5
0
5
10
15
20
0 10000 20000 30000 40000 50000 60000 70000 80000
time, s
82
Viscoplastic Constitutive Equations
• Elastic equation
• Inelastic equation
X is back stress, S is deviatoric stress, is current diameter, 0 is initial diameter, c and state variables
ee : :1 CC
22
3
Jp inin XS
2
1
2 )(:)(2
3
XSXS TJ
)ˆ(1
1sinhexp
1
1 20
cc
J
DR
Qf
Dp m
pin
c
83
State Variables
X, k , c and
k0kC3
2X
c
20321 ))(( ccApApAc inin
8
07ˆ
A
Ac
0x
11
vv A
x109x vv ApA in
kT
kk6in
50kin
k :3
2)ApA(C
D1
1
84
Fatigue and inelastic Damage Surfaces
є1
є2 inelastic damage surface
fatigue damage surface
no damage
85
Damage Evolution
Damage surface in strain space to characterize the type of damage (inelastic damage and fatigue damage)
Damage evolution
d
d
D
Y
Yw
Y
YwD
2-
2
-
0:F
and 0or F 0 Fif ,
0:F
and 0 Fif ,
ddd
dd
1
inin
hf
din
inin
B
h
din
Y
Yp
Y
Yp
w
0max inind ppF
86
Numerical Implementation
The damage coupled viscoplasticity model is coded in ABAQUS through its user-defined material subroutine UMAT.
87
Creep Behavior Simulation w/o Damage
• three stress levels, namely 2000 psi, 3000 psi and 4050 psi, at 22C under load control
0
0.1
0.2
0.3
0.4
0 2500 5000
time (s)
tru
e s
trai
n
w ith damage
w ithout damage
test
4050 psi 3000 psi 2000 psi
88
Softening and Creep Behaviorof 63Sn-37Pb bulk material
• softening tensile behavior at strain rate 10-4/s
• tensile creep behavior at applied stress 4.14 MPa
63Sn-37Pb, strain rate 1.0E-4/s
0
5
10
15
20
25
0 0.1 0.2 0.3 0.4
true strain
test, 25 Ccal., 25 Ctest, 75 Ccal., 75 Ctest, 100 Ccal. 100 C
63Sn-37Pb, tensile creep at 4.14 MPa
0
0.05
0.1
0.15
0.2
0 5000 10000 15000 20000
time, sec
test, 25 Ccal., 25 Ctest, 75 Ccal., 75 Ctest, 100 Ccal. 100 C
89
Fatigue Life Definition• level of stress-range drop, e.g. 50%• acceleration in stress-range drop
Guo: 0.3~3%
about 40% difference in fatigue life
prediction
Normalized stress range /max against normalized number of cycles N/Nf
stress drop curve
0.50
0.60
0.70
0.80
0.90
1.00
1.10
0.00 0.50 1.00 1.50
Normalized Number of Cycles N/Nf
Nor
mal
ized
Str
ess
Ran
ge
Guo's data
simulation, 0.5%
90
Miniature Specimen • Under monotonic tension
gage diameter: 0.354 mm 10% lower than the test datagage length: 2.58 mm at the peak load
Load Cell(ForceMeasurement)
118°
Gage Length
GageDiameter
PbSn
BrassBrass
D
Displacement
For 1mm specimen,D = 1/4”For 0.35mm specimen, D=1/8”
displacement curve
0.0E+00
1.0E-04
2.0E-04
3.0E-04
0 2000 4000 6000
time, s
disp
lace
men
t ra
te,
mm
/s
test,16h
test, 25d
average
load-displacement, ps_a7_03
0
1
2
3
0 0.2 0.4 0.6 0.8
displacement, mm
forc
e, N
test,16h
test, 25d
simulation
91
Miniature Specimen
• Under fatigue loading: 0.015 mm, 310-4 mm/sgage diameter: 1 mm, gage length: 3 mm
load-drop-acceleration failure criterion (~20% drop): about 1000 cycles
50% load-drop failure criterion: the predicted fatigue life is about 8% lower than the testing life of 1349 cycles
SpecimenGage Length
Specimen Diameter
90°
load drop curve, 10E-4/s,25C
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0.00 500.00 1000.00 1500.00
Number of Cycles
Nor
mal
ized
Loa
d R
ange
test
simulation
92
Damage in Miniature Specimen
Equivalent damage under tension Equivalent damage distributionat 1.3 mm displacement under fatigue at 1500 cycles
Necking
93
No Observed Mesh Sensitivity
a710, 63Sn-37Pb, 0.5E-3 mm, 6.6E-4 mm/s
0
5
10
15
20
0 0.1 0.2
displacement, mm
load
, Ncoarse element fine element
94
Notched Specimenfor Shear Simulation
Monotonic Tensile
error = 3%
Isothermal Fatigue
error = 10%
95
Rate Effects on Fatigue Life
63Sn-37Pb, 0 to 1% mechanical strain, 25C
300
600
900
1200
1500
1.E-05 1.E-04 1.E-03 1.E-02 1.E-01
strain rate, 1/s
Guo's datawith rate effectwithout rate effect
96
Other Application: Lap-joints
• 18 joints
copperaluminum
Solder joint
Loading Direction
97
Summary
• For miniature specimen and lap-joints, the predicted maximum load with bulk material data is about 10% lower than the testing result.
• The load drop curve based on the nonlinear damage accumulation equation can be used to more accurately determine fatigue life for different applications.
• The damage distribution in a specimen can be used to determine the possible failure location.
• The fatigue life prediction can be improved with the introduction of two back stresses.
98
Forming Limit Diagram (FLD) of Strain Rate Dependent Metals
99
Localized Necking in Sheet Metals
.Uni-axial to plane strain tension b. Bi-axial tension
n
1
2
n=(1,0)
100
Localized Necking Criterion
• Maximum Stress• Maximum Strain• Maximum Strain Energy• MK Method (Arbitrary Imperfection Factor)• Thickness Criterion• Critical Accumulative Damage• Acoustic Tensor Theory (Hill, 1952)• Vertex Theory (Storen and Rice, 1975)
101
Isotropic Hardening Based Yield Surfaces
-400
-300
-200
-100
0
100
200
300
400
-400 -200 0 200 400
stes1
stes
2
102
Kinematic Hardening Based Yield Surfaces
-400
-200
0
200
400
-400 -200 0 200 400
stes1
stes
2
103
Initial yield locus
Subsequent yield locusLoading path
Uncertain plastic flow direction
Vertex Theory Due to Storen and Rice, 1975
104
Anisotropic Damage
)1)(1(
100000
0)1)(1(
10000
00)1)(1(
1000
0001
100
00001
10
000001
1
)(
21
13
32
3
2
1
DD
DD
DD
D
D
D
DM
σDMσ :)(
105
Effective Strain Energy Release Rate
• Elastic strain energy of damage material
• The elastic strain energy release rate is
σCσσCσDσ ::2
1::
2
1),(
11 eTeTeW
D
DσY
),(eW
106
Damage and Plastic Yield Surfaces
-300
-200
-100
0
100
200
300
-300 -200 -100 0 100 200 300
dmg sur.
yld sur.
107
Damage Evolution Equations
• The damage surface is expressed as
• where C0 is initial damage threshold, C(Z) is damage increment and Yeq is equivalent damage energy
release rate defined as
• And N is the a characteristic tensor of plastic damage
0)]([ 0 ZCCYF eqd
2
1
):2
1( Y:NYT
eqY
108
Equation of Yield Surface
• For isotropic strain-hardening material
eq=equivalent stress; To=initial yield limit;
• T(p)=strain-hardening increment; p=plastic strain
aaaa
a
eq RRRR
RR
1
21900201901
090
)(
)]1([
1
0)]([),( 0 pTTpF eqp σ
Hosford 1979 yield equation
109
Vertex Theory
• Storen and Rice (1975)
• Equilibrium eqns across the localized band under principal stress coordinates
0)(
0)(
2211222222121
2211111122111
ngngnnn
ngngnnn
110
Vertex Theory
• Zhu, Weinmann and Chandra (2001)
• Shear stress rate continuous across localized band, or
012
0)(
0)(
221122
221111
ngng
ngng
Rice equation is simplified to become
111
Power Law Hardening Equation
neqeq K
)1)(1-(
1-1)-(
])1)(()[1)(1)(1(
)(])1)(1()[1(22
0902
290
20*
1
ra
na
rrRRrrrra
rfrrRrRaaa
aaa
Localized necking at LHS of FLD
112
Localized Necking
• Localized Necking at RHS of FLD
)1)(1-(
1-1)-(
])1)(()[1)(1(
)(])1()[1(22
0902
290
20*
1
rra
na
rrRRrrra
rfrRrRaaa
aaa
113
FLD of AL 2028
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
Minor strain
Majo
r str
ain
necking
safe
theory (a=2)
theory (a=4)
theory (a=6)
114
FLD of AL 6111-T4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
Minor strain
Maj
or
stra
in
theory (a=6)
theory (a=4)
thepry (a=2)
test
115
Stress-Strain of AKDG Steel
stress-strain curves of AKDQ
0
100
200
300
400
500
600
0 0.05 0.1 0.15true strain
tru
e st
ress
(M
Pa)
0.0005/s
0.05/s
0.4/s
4/s
regression
116
FLD of AKDQ Steel
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.4 -0.2 0 0.2 0.4
neckingsaferate-independent theoryrate-dependent theory
117
Rate-Dependent FLD
• Rate dependent power-hardening law
• Quasi-linear constitutive equation
• where =elastoplastic response of materials and =dynamic relaxation or rate-dependent power-hardening rule.
),(),(
mnK
118
Definitions of ( and
( and should satisfy the following
Then we have
mnK
1
)(),(
m
n
c
nm
)(
),(
and
119
Equations of Rate dependent FLD
• The LHS of FLD
• The RHS of FLD
2112
),()(3
1
rr
snm
r
nm eqeq
]3)1(3)2[()1)(2(2
),()(
)1)(2(2
)2)((3 2222
22
1
rrrrrrr
snm
rrr
rnmreq
eqeq
120
FLD of AKDQ Steel
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.4 -0.2 0 0.2 0.4
neckingsaferate-independent theoryrate-dependent theory
121
Conclusions
• A modified Vertex theory taking into account the rate-dependent power-hardening law is developed
• The modified theory is applied to a typical rate-sensitive AKDQ steel and the predicted FLD agrees well with those measured.