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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 : C1 : 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 : C1 : 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 22C 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: 210-5/s temperature range: 25 to 800C strain range: 0 to 3.6% 81 Viscoplastic Constitutive Equations • Elastic equation e C1 : C: e • Inelastic equation 1 3 SX 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 22C 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, 310-4 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 )[ra 2 R90 (1 r )(1 r ) a 2 ] f a (r ) 1* (a 1)(1 r )(1 r r )[ra 2 ( R90 R0 ra 2 )(1 r ) a 2 ] (a - 1)n - 1 (a - 1)(1 r ) 111 Localized Necking • Localized Necking at RHS of FLD (1 R0 )[ra 2 R90 (1 r ) a 2 ] f a (r ) 1* (a 1)(1 r r )[ra 2 ( R90 R0 ra 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 mn 3 (m n) s ( eq , eq ) 1 1 r 2 1 r r2 • The RHS of FLD 3r2 (m n)(2 r ) 2 (m n)s( eq , eq ) 1 [(2 r ) 3(1 r r2 ) eq 3r2 ] 2(2 r )(1 r r2 ) 2(2 r )(1 r r2 ) 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