VI2007_Chow_Damage by HC111111103511

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									  Damage and Residual Life
Prediction of Vehicle Structures

             Chi L. Chow
  Department of Mechanical Engineering
    University of Michigan-Dearborn


                                         1
           Table of Contents
• Introduction
• Theory of Damage Mechanics
• Example Projects
  i) Bumper Damage under multiple Impact
  Loading
  ii) Crashworthiness of High Strain Rate Plastics
  iii) Fatigue Damage of Strain-Rate and
  Temperature Dependent Solder Alloys
  iv) Forming Limit Diagram (FLD) of Strain Rate
  Dependent Metals

                                                     2
           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.


                                         3
Micro-Meso-Macro scales




                          4
‘Real-Life’ Materials




                        5
‘Real-Life’ Materials




                        6
Micro-defects - Inclusions




                             7
Macro-Crack Formation




                        8
Fracture/Rupture Process
   Before            Initial       Subsequent       Final
   Loading           Loading        Loading         Loading




            50

            40
   stress




            30

            20

            10

            0
                 0   0.01   0.02     0.03   0.04   0.05   0.06


                                   strain
                                                                 9
       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


                                            10
     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.

                                           11
Rupture Criterion - Conventional
For Smooth Specimen (with/without
  notches)
                                                P
• Static loading

   – Stress, strain or energy-based criteria

• Fatigue loading

   – S-N curve (Due to Wohler in 1858) for      P
     constant amplitude loading
   – Miner’s rule for variable amplitude
     loading
   – Rainfall counting method for ‘real life’
     fatigue loading
                                                    12
Rupture Criterion – Conventional
For Cracked Specimen

                                            P
•       Static loading

    –     Fracture Mechanics

                                    crack
•       Fatigue loading
                                            P
    –     Paris Law
    –     Others based on  G, J



                                                13
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



                                                 14
             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


                                                    15
   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
                                                 16
  Past Projects: Electronic Packaging

• Fatigue Damage of Strain-Rate and
  Temperature dependent Solder Alloys

• Scale Effect of Solder Joints
• Micro-structural Evolution of Solders




                                          17
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

                                                18
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


                                              19
2. Theory of Damage
     Mechanics




                      20
Definition of scalar damage variable

                n
                             
            A



                                    A0  A
                                 D
                                     A0


where   A0 = original surface area (with defects);
        A = surface area excluding defects
                                                     21
        Damage-coupled Elasticity
                                           
• True stress was replaced by       
  effective stress                       1 D

• Based on strain equivalent      E0 (1  D)
  principle

• Based on energy equivalent      E0 (1  D)  2

  principle

• Damage evolution equation     
                                D  f ( , D,......)
                                                       22
Relationship Between Scalar Damage
   Variable and Young’s Modulus

  D  1 E             or     D  1 E
              E0                           E0




  Undamaged material        Damaged Material



                                                23
Effective Young’s Modulus: an example




                                    24
      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
                            1
                  M( D ) =      I
                           1- D
I is a unit tensor .

                                                            25
              Damage Effect Tensor
For multi-axial stress state,
  T = [  1  2  3  4  5  6 ] = [  11  22  33  23  31  12 ]
  T = [ 1  2  3  4  5  6 ] = [ 11  22  33 2  23 2  31 2 12 ]

damage effect tensor is
                                           1      0    0    0
                                                                
                                           1      0    0    0
                                                                
                                      1      1    0    0    0
                            M( D ) =                            
                                     1- D  0 0 0 1-     0    0
                                                                
                                           0 0 0    0 1-     0
                                                                
                                           0 0 0
                                                    0    0 1-  
                                                                 

 where D and  are scalar damage variables
                                                                            26
      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
                           1
              W e  p   T : C01 :   p
                                   

                           2
• With damage
            1        1          1
            T : C0 :   p   T : C1 :   p
            2                    2

                                                        27
Damage-Coupled Elastic Equation
                           
                       =
                        e
                              = C 1 : 
                           
where C is effective/damage stiffness matrix
                   1 -  -    0      0      0
                                                
                   -  1 -    0      0      0
                                             0
                1  - -  1    0      0
                                                 
          C 1 = 
                E    0 0 0 2(1+ )     0      0
                                                
                   0 0 0       0 2(1+ )     0
                                                
                   0 0 0
                               0      0 2(1+ ) 
                                                 
and
                E0 (1 - D )2             0 - 2(1- 0 ) - (1 - 3 0 ) 2
      E=                             =
         1 - 4 0  + 2(1- 0 ) 2          1 - 4 0  + 2(1- 0 ) 2

                                                                            28
Damage Energy Release Rate
 The conjugates of the damage variables,
 known as damage energy release rate,
 are defined as

                      1
       YD              T : C1 : 
                D    1 D
                     1
       Y              T : A :
                    1 D



                                           29
       Formulation of [A] Matrix

                  A1       A2   A2          0               0             0
                                                                             
                  A2       A1   A2          0               0             0
                                                                             
       1          A2       A2   A1          0               0             0
A=                                                                           
   E0 (1 - D)       0      0    0 2( A1 - A2 )              0             0
                                                                             
                    0      0    0           0 2( A1 - A2 )                0
                                                                             
                    0      0    0           0               0   2( A1 - A2 ) 


A1 = 2 (1 -  0 ) - 2 0                A2 = (1 +  )(1 -  0 ) - 2  0




                                                                                  30
Isotropic Strain Hardening
                Isotropic Hardening Based
                      Yield Surfaces


                          400
                          300
                          200
                          100
 stes2




                            0
         -400    -200    -100 0        200   400
                         -200
                         -300
                         -400
                           stes1


                                                   31
Kinematic Strain Hardening
                Kinematic Hardening Based
                     Yield Surfaces


                          400


                          200
 stes2




                            0
         -400     -200          0      200   400
                         -200


                         -400
                           stes1


                                                   32
Damage-Coupled Yield Surface
                        1 
    F p (  ,D,  , R)=       eq - [ R0 + R(p)]= 0
                        1 D

where  eq is defined as


 eq    1
         2   {(1  2 ) 2  (1  3 ) 2  (2  3 ) 2  6 (42  52  62 )}


and R0 and R(p) are yield stress and strain hardening
threshold. p is overall effective plastic strain.


                                                                                   33
Damage-Coupled Plastic Equation
                         Fp
                         1   3S
              dε  
                 p
                                   
                      1  D 2 eq
                        Fp
              dp  
                        ( R)


  where S is the true stress deviator tensor,
        λ is a Lagrange multiplier.




                                                34
Damage and Yield Surfaces

                         300



                         200


                                    dmg sur.
                          100
                                    yld sur.

                           0
    -300   -200   -100          0   100        200   300

                         -100



                         -200



                         -300



                                                           35
Fatigue and Plastic Damage Surfaces

                2
                           plastic damage
                           surface

   no damage




                               1


                     fatigue damage
                     surface



                                            36
     Plastic Damage surface
The expanding plastic damage surface is
expressed in terms of the damage energy
release rates YD and Y as

    F pd
                    1/
         ( Y, B )= Ypd2 - [ B0p + B( wp ) ] = 0

                    1
              Y pd = ( Y Dp +  Y p )
                         2        2

                    2



                                                  37
Plastic Damage Evolution Equations

                           F pd     pd Y Dp
            dD p =   pd        =
                           Y Dp           /
                                     2 Y 1pd2
                           F pd     pd  Y p
            d p =   pd        =
                           Y p            /
                                      2 Y 1pd2
                         Fpd
            dwp =  pd          =  pd
                        (  B)

 where dwp is overall plastic damage increment,
        λpd is the Lagrange multiplier



                                                  38
            Fatigue Damage
Fatigue damage surface
                F fd (  , D,  ) = 0
Fatigue damage evolution equations
                                F fd
                 dD f =   fd
                                Y Df
                                F fd
                 d f =   fd
                                Y f
                 dw f =  fd
where dwf is overall fatigue damage increment,
       λfd is a Lagrange multiplier
                                                 39
            Total Damage
Total damage is the summation of fatigue
damage and plastic damage

               D  D f  Dp
                  f  p
               w  w f  wp



                                           40
   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.



                                            41
      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.




                                           42
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
                                                                43
Damage Analysis Approach

  Damage-Coupled Material Model




               FEA




           Damage Index


                                  44
3. Example Projects




                      45
   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.

                                                          46
Testing Procedure for E and 


                                                     3

                                       2
                         1
   stress




                    E0            E1            E2            E3




            0
                0             2             4             6             8    10   12
                                                     strain        , 0.01




                                                                                       47
Measured Young’s Modulus E

                         210000
  Young's modulus, MPa

                         180000



                         150000



                         120000
                                  0   0.1                 0.2   0.3
                                            true strain




                                                                      48
      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

  {2(1  0 )  1  3 0 } 2  2(1  0  2 0 )     0  0

and                E
         (1  D) 
                 2
                      {1  4 0   2(1  0 )  2 }
                   E0



                                                                     49
            Critical Damage

Critical overall damage for SAE 950 steel was
measured to be wc=0.112 and martinsite sheet
steel, wc=0.04.




                                                50
       Effective Stress-Strain Curve
                   700
                   600
true stress, MPa



                   500
                   400
                   300
                   200
                   100
                    0
                         0   0.1                 0.2   0.3
                                   true strain

                                                             51
Strain Hardening Curve of R versus p

                           300
   strain hardening, MPa


                           250
                           200
                           150
                           100
                           50
                            0
                                 0      0.1            0.2       0.3
                                     equivalent plastic strain



                                                                       52
                         Damage Hardening of
                             B versus w
damage hardening, MPa   1.5



                         1



                        0.5



                         0
                              0   0.05         0.1   0.15
                                   overall damage



                                                            53
One Bumper Model
(made of SAE 950 sheet steel)




                                54
Spring Assisted Drop Weight
        Test Fixture




                              55
       FEA and Test Results:
Maximum Impact Force under 6th impact of 13mph




                                                 56
Damage Contours
  (after six impacts)




            Maximum damage near support ends

                                               57
            Photo Micrographs
(SAE 950 beam near weld and support showing damage)
               1            2                  4
                                3

                            6   7
              5                                8




              Flange Edge           Surface




                                    Middle of Flange




                                                       58
            Photo Micrograph
(SAE 950 beam in center showing no damage)

        1             2                        4
                             3

                       6     7
        5                                      8




            Maximum deformation, but no damage




                                 Center Under Impactor




                                                         59
                       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.


                                                            60
    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

                                                            61
            SHPB for Material Tests

Split Hopkinson Pressure Bar (SHPB) for high strain-rate
testing (102~104/s)

                          i , r
                                               t



   Striker Bar
                    Input Bar                   Output Bar
                                    Specimen



  i    incident strain wave
  r    reflection strain wave
  t    transmission strain wave

                                                             62
  Measurement of Stress-Strain Curve

                 2C                   A
       (t )  
                    r (t )   σ(t)     E t (t )
                 Ls                   As
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

                                                     63
Rate-Dependent Stress-Strain Curves




                                  64
Rate-Dependent Constitutive Model

                                          m
                
                 e      p
                                      ae
                                E
E, a and m are material constants.

when the strain rate  is constant
                     
                   1      a mE
             E  ln[1  ( e  1)]
                   m      
                          


                                                 65
      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




                                              66
Damage Measurement




                     67
Three-Point Bend Impact Test




                               68
               Deformation
High-speed imaging sequence of a 2.2 m/s three-
point bend test at 22°C for material C




                                                  69
                       Load-Displacement Curves

            Impact velocity 2.2m/s, temperature 22C
                    Material A                   Material B
            0.16
                                                                             0.2
            0.14                              test
                                                                            0.18                              test
                                              simulation                                                      simulation
            0.12                                                            0.16
                                                                            0.14
             0.1
Load (KN)




                                                                            0.12




                                                                Load (KN)
            0.08                                                             0.1
            0.06                                                            0.08
                                                                            0.06
            0.04
                                                                            0.04
            0.02
                                                                            0.02
              0                                                               0
                   0   10    20       30      40           50                      0   10      20       30     40          50
                            Deflection (mm)                                                 Deflection (mm)




                                                                                                                                70
              Maximum Load


Material              Max Load (KN)
           Measured              Computed   Percent Difference
   A        0.14                    0.15            7.1
   B        0.17                    0.17             0
   C        0.19                    0.18            5.2
   D        0.24                    0.23            4.2
   E        0.24                    0.24             0
   F        0.30                    0.26            1.3
   G        0.31                    0.30            3.2




                                                                 71
           Damage Accumulation and
               Failure Analysis
Materi   Critical damage   Maximum damage    Fail or safe   Test result
 al            value            value        prediction
 A             0.63             0.49             Safe       No failure
  B            0.61             0.59           Safe but     No failure
                                               critical
  C           0.76              0.68             safe       No failure
  D           0.67              0.66           Safe but     No failure
                                               critical
  E           0.44              0.39             Safe       No failure
  F           0.41              0.24             Safe       No failure
  G           0.30              0.33        Fail at 8.5mm    failure
                                              deflection




                                                                          72
                  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

                                               73
Strain-Rate and Temperature
Dependent Fatigue Damage




                              74
              Objective


To develop a predictive Viscoplasticity
Model for fatigue damage of solder alloys
based on the theory of Damage
Mechanics




                                        75
           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.

                                                 76
Fracture/Rupture Process
  Nf = 0          Nf = 1000 Nf = 10,000 Nf = 1,000,000




             50

             40
    stress



             30

             20

             10

             0
                  0   0.01   0.02    0.03    0.04   0.05   0.06


                                    strain



                                                                  77
 Aging of Solder Alloys
at Elevated Temperature




16 hours aged specimen   25 days aged specimen
       at 100oC                 at 100oC




                                                 78
                  Effects of microstructure
                     on hysteresis loops
• Two batches of 63Sn-37Pb
Batch A (100C/16h): 16 hours at 1000C
Batch B (125C/24h): 24 hours at 1250C
• Two temperatures
250C and 800C
• Three strain rates
10-3/s, 10-4/s and 10-5/s




                                              79
         Rate and Temperature Effects
         on 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




                                          80
    Load-drop tests under TMF loading
• Batch A (100C, 16h)
ramp: 30 minutes
strain rate: 210-5/s
temperature range: 25 to 800C
strain range: 0 to 3.6%




                                        81
   Viscoplastic Constitutive Equations
• Elastic equation
                   
                
                 e
                       C1 :                        C: e
                   
• Inelastic equation
                                                                    1
                     3 SX                  3                  
          in  p in
                                                                    2
                                    J 2   (S  X)T : (S  X)
                     2 J2                   2                  
                1         Q  0   m  1      J2 
                                      p

         p in 
                    f exp       sinh  1  D  (c  c ) 
                1 D       R                       ˆ  

     X is back stress, S is deviatoric stress,
      is current diameter, 0 is initial diameter,
            
     c and c state variables
                                                                        82
                    State Variables
              
X, k , c and c

       1 
  k 
              C k 0 ( A 5 p  A 6 ) k  k :  k
               in
                               in         2 T
                                                           
       1 D                                3

   c  A1 p in  ( A2 p in  A3 )( c  c0 ) 2
                    

           0 
                   A8
                                                v x  v0
   c  A7  
   ˆ                                 
                                       A11
                                                
  v x  A9 p in  A10 v x
                                    2
                                   X  Ck 0 k
                                      3
                                                               83
Fatigue and inelastic Damage Surfaces

                 є2
                            inelastic damage
                            surface

    no damage




                                є1


                      fatigue damage
                      surface



                                               84
               Damage Evolution

Damage surface in strain space to characterize the type
of damage (inelastic damage and fatigue damage)

                   Fd  p in  pm ax  0
                                in



Damage evolution

           Y                  Yd 
                                     B1
                                                      Fd in
    
    D  -w D
                       p in   , if Fd  0 and
                                                          :  0
                                                               
           2Yd                Yh                    in
                     
                   w
                   
             Y      in Yd                                Fd in
      -w
                     
                       p        , if Fd  0 or F  0 and          :  0
                                                                   
             2Yd         Yhf
                                                d
                                                             in
                     
                     

                                                                           85
     Numerical Implementation

The damage coupled viscoplasticity model is
coded in ABAQUS through its user-defined
material subroutine UMAT.




                                         86
Creep Behavior Simulation w/o Damage

• three stress levels, namely 2000 psi, 3000 psi and 4050
  psi, at 22C under load control
                     0.4
                               4050 psi   3000 psi              2000 psi



                     0.3
       true strain




                     0.2


                                                                           w ith damage

                     0.1                                                   w ithout damage
                                                                           test



                      0
                           0                           2500                                  5000
                                                     time (s)


                                                                                                    87
       Softening and Creep Behavior
        of 63Sn-37Pb bulk material
• softening tensile behavior   • tensile creep behavior at
  at strain rate 10-4/s          applied stress 4.14 MPa




                                                         88
                                                  Fatigue Life Definition
• level of stress-range drop, e.g. 50%
• acceleration in stress-range drop
                                                      stress drop curve

                            1.10
                                                                                                       Guo: 0.3~3%
                            1.00
  Normalized Stress Range




                            0.90                                                                     about 40% difference
                                                                                 Guo's data
                            0.80
                                                                                 simulation, 0.5%
                                                                                                    in fatigue life prediction
                            0.70

                            0.60

                            0.50
                                0.00       0.50                1.00       1.50
                                       Normalized Number of Cycles N/Nf



                      Normalized stress range /max
                 against normalized number of cycles N/Nf


                                                                                                                                 89
                                      Miniature Specimen
                                                                                                                           displacement curve

• Under monotonic tension




                                                                        displacement rate, mm/s
                                                                                                  3.0E-04


                                                                                                  2.0E-04                                                   test,16h
                                                                                                                                                            test, 25d
                                                      Gage Length                                 1.0E-04                                                   average


                                                                                                  0.0E+00
                                                                                                            0            2000             4000   6000
                  Brass                                   Brass
                                                                                                                                time, s

                          118°                                      D
                                                                                                                       load-displacement, ps_a7_03


   Load Cell                                                                                      3
   (Force                         Gage
   Measurement)                   Diameter
                                                   PbSn
                                                                                                  2                                                        test,16h




                                                                        force, N
                                                                                                                                                           test, 25d
                                    Displacement                                                  1                                                        simulation

                                 For 1mm specimen,
                                 D = 1/4”
                                                                                                  0
                                 For 0.35mm specimen,                                                 0         0.2          0.4           0.6       0.8
                                 D=1/8”
                                                                                                                      displacement, mm



gage diameter: 0.354 mm                                                                                   10% lower than the test data
gage length: 2.58 mm                                                                                      at the peak load

                                                                                                                                                                        90
                               Miniature Specimen
• Under fatigue loading: 0.015 mm, 310-4 mm/s
gage diameter: 1 mm, gage length: 3 mm
                                                                              load drop curve, 10E-4/s,25C


                                                                 1.20




                                         Normalized Load Range
           Specimen Diameter                                     1.00

                                                                 0.80
                                                                                                                       test
                                                                 0.60
                                                                                                                       simulation
                                   90°                           0.40

                                                                 0.20

                                                                 0.00
   Specimen                                                         0.00   500.00         1000.00            1500.00
   Gage Length                                                                Number of Cycles


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


                                                                                                                                    91
     Damage in Miniature Specimen




                        Necking




Equivalent damage under tension   Equivalent damage distribution
      at 1.3 mm displacement      under fatigue at 1500 cycles



                                                                   92
No Observed Mesh Sensitivity

              a710, 63Sn-37Pb, 0.5E-3 mm, 6.6E-4 mm/s

                       coarse element      fine element


              20

              15
    load, N




              10

              5

              0
                   0               0.1                    0.2
                            displacement, mm



                                                                93
   Notched Specimen
  for Shear Simulation




Monotonic Tensile   Isothermal Fatigue
  error = 3%          error = 10%




                                         94
Rate Effects on Fatigue Life




                               95
         Other Application: Lap-joints

• 18 joints


      aluminum             copper




                 Solder joint




                                Loading Direction




                                                    96
                    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.


                                                    97
 Forming Limit Diagram (FLD)
of Strain Rate Dependent Metals




                                  98
  Localized Necking in Sheet Metals


                                 1

                                     2



            n                                    n=(1,0)
                




.Uni-axial to plane strain tension       b. Bi-axial tension

                                                               99
          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)

                                                100
               Isotropic Hardening Based
                     Yield Surfaces

                         400
                         300
                         200
                         100
stes2




                           0
        -400     -200   -100 0       200   400
                        -200
                        -300
                        -400
                          stes1


                                                 101
               Kinematic Hardening Based
                    Yield Surfaces

                         400


                         200
stes2




                           0
        -400      -200          0    200   400
                         -200


                         -400
                           stes1


                                                 102
       Vertex Theory
Due to Storen and Rice, 1975
                                    Uncertain plastic
                                    flow direction
 Initial yield locus
                          2
 Subsequent yield locus
                               Loading path

                               1




                                                        103
                        Anisotropic Damage
σ  M(D) : σ

         1                                                                                      
        1  D      0        0             0                    0                    0           
             1
                                                                                                 
         0          1                                                                           
                             0             0                    0                    0
                 1  D2                                                                         
                             1                                                                  
         0         0                      0                    0                    0           
                          1  D3                                                                
M (D)                                    1                                                     
         0         0        0                                  0                    0           
                                   (1  D2 )(1  D3 )                                           
                                                               1                                
         0         0        0             0
                                                         (1  D3 )(1  D1 )
                                                                                     0           
                                                                                                
         0         0        0             0                    0
                                                                                     1           
                                                                             (1  D1 )(1  D2 ) 
                                                                                                




                                                                                                     104
 Effective Strain Energy Release Rate

• Elastic strain energy of damage material
                  1 T   e 1  1 T   e 1
        W (σ, D)  σ : C : σ  σ : C : σ
          e

                  2           2

• The elastic strain energy release rate is

                  W e (σ, D)
              Y
                     D


                                              105
Damage and Plastic Yield Surfaces

                       300



                       200


                                  dmg sur.
                        100
                                  yld sur.

                         0
  -300   -200   -100          0   100        200   300

                       -100



                       -200



                       -300


                                                         106
         Damage Evolution Equations

• The damage surface is expressed as
          Fd  Yeq  [C0  C(Z )]  0
• where C0 is initial damage threshold, C(Z) is damage
  increment and Yeq is equivalent damage energy
  release rate defined as
                                 1
                     1 T
              Yeq  ( Y : N : Y) 2
                     2
• And N is the a characteristic tensor of plastic damage

                                                      107
               Equation of Yield Surface

• For isotropic strain-hardening material
          Fp (σ, p )   eq  [T0  T ( p )]  0

• eq=equivalent stress; To=initial yield limit;
• T(p)=strain-hardening increment; p=plastic strain
     Hosford 1979 yield equation
                                                                        1
                       1                  a         a
       eq                       ( R90  1  R0  2  R0 R90  1   2 )
                                                                      a a
                              1
               [ R90 (1  R0 )]
                              a


                                                                            108
                   Vertex Theory

• Storen and Rice (1975)
• Equilibrium eqns across the localized band
  under principal stress coordinates

 n1 11  n2  12  n1 11 ( g1n1  g 2 n2 )  0
                
 
 n1 12  n2  22  n2 22 ( g1n1  g 2 n2 )  0
                



                                                      109
                   Vertex Theory

• Zhu, Weinmann and Chandra (2001)
• Shear stress rate continuous across localized
  band, or
                   12  0
                     
    Rice equation is simplified to become

            1   1 ( g1n1  g 2 n2 )  0
              
           
            2   2 ( g1n1  g 2 n2 )  0
              

                                                  110
 Power Law Hardening Equation


                        eq  K           n
                                           eq


Localized necking at LHS of FLD


             (1  R0 )[ra  2  R90 (1  r )(1  r ) a  2 ] f a (r )
 1* 
       (a  1)(1  r )(1  r r )[ra  2  ( R90  R0 ra  2 )(1  r ) a  2 ]
            (a - 1)n - 1
        
          (a - 1)(1 r )

                                                                                      111
                  Localized Necking

• Localized Necking at RHS of FLD


                 (1  R0 )[ra  2  R90 (1  r ) a  2 ] f a (r )
     1* 
           (a  1)(1  r r )[ra  2  ( R90  R0 ra  2 )(1  r ) a  2 ]
                 (a - 1)n - 1
            
              (a - 1)(1 r r )



                                                                                 112
                              FLD of AL 2028
                                        0.4



                                       0.35



                                        0.3



                                       0.25
Major strain




                                        0.2



                                       0.15
                                                                        necking
                                        0.1
                                                                        safe
                                                                        theory (a=2)
                                       0.05
                                                                        theory (a=4)
                                         0
                                                                        theory (a=6)
               -0.15   -0.1    -0.05          0    0.05    0.1   0.15    0.2   0.25    0.3


                                                  Minor strain



                                                                                             113
                              FLD of AL 6111-T4
                                               0.4



                                              0.35



                                               0.3



                                              0.25
Major strain




                                               0.2



                                              0.15
                                                                             theory (a=6)
                                               0.1                           theory (a=4)
                                                                             thepry (a=2)
                                              0.05
                                                                             test

                                                0
               -0.2   -0.15    -0.1   -0.05          0   0.05   0.1   0.15      0.2    0.25   0.3


                                                 Minor strain


                                                                                                    114
      Stress-Strain of AKDG Steel
                    600


                    500


                    400
true stress (MPa)




                    300
                                                                0.0005/s
                    200                                         0.05/s
                                                                0.4/s
                    100                                         4/s
                                                                regression
                     0
                          0          0.05                 0.1                0.15
                                            true strain

                              stress-strain curves of AKDQ

                                                                                    115
       FLD of AKDQ Steel
                 1
                          necking
                0.9       safe
                          rate-independent theory
                0.8
                          rate-dependent theory
                0.7
                0.6
                0.5
                0.4
                0.3
                0.2
                0.1
                 0
-0.4     -0.2         0        0.2              0.4



                                                      116
            Rate-Dependent FLD

• Rate dependent power-hardening law
                  K 
                       mn

• Quasi-linear constitutive equation
            ( ,  )  ( ,  )
                       
• where =elastoplastic response of materials and
  =dynamic relaxation or rate-dependent power-
  hardening rule.


                                                    117
     Definitions of ( and 

( and  should satisfy the following
                         
                         
                         
   Then we have
                                 
                                        1

                   ( ,  )  ( n )    m
                                K
    and
                                       c
           ( ,  )               
                         (m  n)           n

                                           m


                                                 118
             Equations of Rate dependent FLD

   • The LHS of FLD
                         mn      3 (m  n) s ( eq ,  eq )
                    1         
                         1  r      2 1  r  r2
   • The RHS of FLD

     3r2  (m  n)(2  r ) 2   (m  n)s( eq ,  eq )
1                                                      [(2  r ) 3(1  r  r2 ) eq  3r2 ]
      2(2  r )(1  r  r2 ) 2(2  r )(1  r  r2 )



                                                                                             119
       FLD of AKDQ Steel
                 1
                          necking
                0.9       safe
                          rate-independent theory
                0.8
                          rate-dependent theory
                0.7
                0.6
                0.5
                0.4
                0.3
                0.2
                0.1
                 0
-0.4     -0.2         0        0.2              0.4



                                                      120
                 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.



                                             121

								
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