Lecture #8: Ductile Fracture (Theory & Experiments) by Dirk Mohr ETH Zurich, Department of Mechanical and Process Engineering, Chair of Computational Modeling of Materials in Manufacturing 2015 1 1 1
Ductile Fracture (continuation from previous lecture) 2 2 2
Lode angle parameter Stress triaxiality: m III s III plane I II III Normalized third stress invariant 27 2 J 3 3 Lode angle parameter 2 1 arccos( ) I II s I +1 0-1 s II 3 3 3
Lode angle parameter Lode parameter (Lode, 1926) II I L 2 I III III III s III plane I II III s II Lode angle parameter 2 1 arccos( ) L I II s I +1 0-1 1 axisymmetric tension 0 generalized shear 1 axisymmetric compression III II I III II I III II I 4 4 4
Plane stress states For isotropic materials, the stress tensor is fully characterized by three stress tensor invariants, { I1, J 2, J 3} or { I, II, III} while the stress state is characterized by the two dimensionless ratios of the invariants, e.g. { I1 / J 2, J 3 / J 2 3/ 2 } or {, } or /, / } { II I III I with I 1 3 3 J 2 and 2 3 3 1 arccos 2 3 3/ 2 J 2 J 5 5 5
Plane stress states Under plane stress conditions, one principal stress is zero. The stress state may thus be characterized by the ratio of the two nonzero principal stresses. As a result, the stress triaxiality and the Lode angle parameter are no longer independent for plane stress, i.e. we have a functional relationship [] axisymmetric tension (1) Biaxial comp. ( I 0) Tensioncompression ( II 0) generalized shear (0) Biaxial tension ( III 0) axisymmetric compression (1) 6 6 6
Unit Cell with Central Void 151-0735: Dynamic behavior of materials and structures Results from Localization Analysis Stresses on Plane of Localization t Linear Mohr-Coulomb approximation 7 7 7
Mohr-Coulomb Failure Criterion n n t According to the Mohr-Coulomb model, failure occurs along a plane of normal vector n for which the linear combination of the shear stress t and the normal stress n stresses acting on that plane reaches a critical value c 2 : max[ t n c n 1 ] c 2 with the dimensionless friction coefficient c 1. 8 8 8
Mohr-Coulomb criterion The maximization problem max[ t c n n 1 ] has an analytical solution which is given by the solution of the equality for the ordered principal stresses with the coefficients ) c ) b c I III c 1 1 c 2 1 I III 2 t max I and b n [ t max c III ] 2 I III c 2 2c 2 1 c Observe that the first term corresponds to the maximum shear stress, while the second term is the normal stress acting on the plane of maximum shear: b 2 2 1 9 9 9
Mohr-Coulomb criterion for plane stress after Bai (2008) 10 10 10
Hosford-Coulomb criterion The Mohr-Coulomb model can be seen as a linear combination of the Tresca equivalent stress and the normal stress, HF 1 2 I III c( I III ) Tresca stress As a generalization, the Tresca stress is substituted by the Hosford equivalent stress, ( I II ) a ( II which results in the so-called Hosford-Coulomb criterion: III ) a b ( I III ) 1 a ) a Hf c( ) I III b 11 11 11
HF 1 2 Hosford equivalent stress 151-0735: Dynamic behavior of materials and structures ) a a a a ( ) ( ) ( ) k I II II III I III 1 a 2 a 1 The Hosford-Coulomb stress may be considered as an interpolation between the Tresca and von Mises envelopes. The limiting cases are obtained for: a=1 (Tresca): HF a1 I III 1 a 2 a=2 (von Mises): HF a2 12 12 12
Second in-plane stress Hosford-Coulomb criterion 151-0735: Dynamic behavior of materials and structures 0.67 EMC Hosford- Coulomb Mohr- Coulomb 0.58 Yield Mises 0.33 First in-plane stress 0 13 13 13
Coordinate Transformation 151-0735: Dynamic behavior of materials and structures Principal stress space,, } { I II III Haigh-Westergaard space {,, } III s III plane I II III I II III ) f 1 ) f 2 ) f 3 s II with +1 0-1 f 2 [ ] cos (1 ) 3 6 1 I II s I f f 2 [ ] cos (3 ) 3 6 2 2 [ ] cos (1 ) 3 6 3 14 14 14
Hosford-Coulomb Ductile Fracture Model Principal stress space,, } { I II III τ + c(σ I + σ III ) = b തσ Hf Hosford- Mohr-Coulomb Coordinate transformation Haigh-Westergaard Mixed strain-stress space {,, } space,, } f [, ] f Isotropic hardening law k[ p ] f 1 k { p [, ] f f f 15 15 15
Hosford-Coulomb Ductile Fracture Model General form von Mises equivalent plastic strain to fracture [,, a, b, c] f f f Stress triaxiality Lode angle parameter 3 material parameters Detailed expressions g f HC b g 1 c [, ] HC 1 n 1 1 a 1 a 1 a f f f f f f ) a c 2 f f ) 2 I II 2 II III 2 I III I III f I 2 [ ] cos (1 ) 3 6 f II 2 [ ] cos (3 ) 3 6 2 [ ] cos (1 ) 3 6 3 16 16 16 f
Hosford-Coulomb Fracture Initiation Model - for proportional loading - 151-0735: Dynamic behavior of materials and structures f 3D View plane stress f 2D View plane stress 17 17 17
Hosford-Coulomb Ductile Fracture Model Influence of parameter b a=1.3 c=0.05 b=0.5 b=0.4 b=0.3 b=0.2 b = strain to fracture for uniaxial tension (or equi-biaxial tension) 18 18 18
Hosford-Coulomb Ductile Fracture Model Influence of parameter a c 0.1 a 2 a a 1.5 1.2 Can easily adjust the depth of the plane strain valley Compare: Mohr-Coulomb a 1 c 0.2 c 0.35 a 0.8 c 0 c 0.1 a 1 19 19 19
Hosford-Coulomb Ductile Fracture Model Influence of parameter c c=0.1 c=0.2 a=1.3 n=0.1 c=0.05 c=0 20 20 20
Application of the Hosford-Coulomb Model f f f DP780 DP590 TRIP780 CH PU SH CH PU NT20 NT6 SH CH PU SH NT20 NT6 NT20 NT6 a=1.47 b=1020.8 c=0.008 a=1.89 b=522.2 c=0.001 a=1.29 b=1371.5 c=0.096 21 21 21
Application of the Hosford-Coulomb Model DP1000 CP1000 CP1200 22 22 22
Common feature for most metals: Biaxial Tension Valley /2 /2 23 23 23
Biaxial Tension Valley 151-0735: Dynamic behavior of materials and structures f plane stress f plane stress Biaxial tension valley Biaxial tension valley is due to Lode effect! 24 24 24
Biaxial Tension Valley 151-0735: Dynamic behavior of materials and structures f Biaxial tension valley Biaxial tension valley is due to Lode effect! 25 25 25
Hosford-Coulomb Ductile Fracture Model 3D View f 2D View f plane stress plane stress heart of the model: [, ] f f 26 26 26
Damage Accumulation Define damage indicator f [, ] f D d [, ] f p D D 0 1 (initial) (fracture) Example: uniaxial tension VIDEO 27 27 27
Damage Accumulation f [, ] f Define damage indicator D d [, ] Example: uniaxial compression followed by tension f p D 0 D 1 (initial) (fracture) VIDEO 28 28 28
Damage Accumulation Define damage indicator f [, ] f D d [, ] f p D D 0 1 (initial) (fracture) Example: uniaxial compression followed by tension Non-linear loading path effect! 29 29 29
Calibration Experiments Focus on simplicity and robustness of experimental technique: All experiments can be performed in a uniaxial testing machine Strains to fracture can be directly measured on specimen surface (no FEA needed) I. Shear test II. Plate bending III. Mini-Punch 20mm 60mm 30 30 30
Plate bending 31 31 31
Punch test 32 32 32
Model Calibration SHEAR BENDING SHEAR PUNCH PUNCH BENDING Non-linearity in loading paths negligible 33 33 33
Shear Fracture Specimen Design Many different flat shear specimen designs exist for use in uniaxial loading frames 20mm but we nonetheless developed a new geometry 34 34 34
Shear Fracture Specimen Design Stress Triaxiality Major challenge: Fracture prone to initiate prematurely at nearly plane strain tension conditions near boundaries! 0.667 0.50 0.33 0.167 0.00 35 35 35
Shear Specimen Optimization Problem 36 36 36
Shear Specimen - Optimization 37 37 37
Basic geometry 151-0735: Dynamic behavior of materials and structures Smiley Shear Specimen apparent shear fracture strain: 0.74 Optimized geometry apparent shear fracture strain: 0.86 38 38 38
Typical smiley-shear experiment Average equivalent plastic strain rate: ~0.001 /s Camera resolution: 4 mm/pixel 39 39 39
Other fracture experiments 151-0735: Dynamic behavior of materials and structures Punch Butterfly shear f Notched tension Central hole tension 40 40 40
Flat Notched Tensile Specimens 41 41 41
Hybrid experimental-numerical determination of the loading history Back view: Gage section 1pix<10µm Experiment Front view: Whole specimen 1pix=50µm FEA Loading history Boundary displacement up to the onset of fracture (first surface crack) Surface strain field Location of onset of fracture: Not known experimentally Element with highest plastic strain 42 42 42 42
Approach 1. Identification of plasticity model for large strains based on multi-axial experiments on specimens with homogeneous stress and strain fields ( material test ) 2. Validation of the plasticity model for very large strains and multi-axial loading based on experiments on specimens with heterogeneous stress and strain fields ( structural test ) 3. Determination of loading path to fracture and assessment of errors 43 43 43
Strain hardening p n A ( s ) 44 44 44 44
Discretization errors Eq. plastic strain [-] Coarse 2 elements through half thickness Medium 4 elements through half thickness Fine 8 elements through half thickness Very fine 16 elements through half thickness Fine mesh gives a converged result 45 45 45 45
Notched tension: Exp. & FEA 46 46 46
Notched tension: Exp. & FEA Side view of FEA (R=20mm) Side view of FEA (R=10mm) Side view of FEA (R=6.67mm) 47 47 47
Experimental detection of the onset of fracture Crack propagation unstable in most experiments t = 617s Instant of onset of fracture: Location of onset of fracture: t = 618s appearance of the first surface crack unknown experimentally 48 48 48 48
Loading path to fracture 49 49 49 49
Summary plots 50 50 50
Tensile specimen with central hole 51 51 51
Tensile specimen w/ central hole 52 52 52
Punch experiments 53 53 53
Punch experiments 54 54 54
Reading Materials for Lecture #8 C. Roth and (2015), Ductile fracture experiments with locally proportional loading histories, Int. J. Plasticity, http://www.sciencedirect.com/science/article/pii/s0749641915001412 55 55 55