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.