Lecture #8: Ductile Fracture (Theory & Experiments)

Transcription

1 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

2 Ductile Fracture (continuation from previous lecture) 2 2 2

3 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 s II 3 3 3

4 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 axisymmetric tension 0 generalized shear 1 axisymmetric compression III II I III II I III II I 4 4 4

5 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 J 2 and arccos 2 3 3/ 2 J 2 J 5 5 5

6 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

7 Unit Cell with Central Void : Dynamic behavior of materials and structures Results from Localization Analysis Stresses on Plane of Localization t Linear Mohr-Coulomb approximation 7 7 7

8 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

9 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

10 Mohr-Coulomb criterion for plane stress after Bai (2008)

11 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

12 HF 1 2 Hosford equivalent stress : 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 a

13 Second in-plane stress Hosford-Coulomb criterion : Dynamic behavior of materials and structures 0.67 EMC Hosford- Coulomb Mohr- Coulomb 0.58 Yield Mises 0.33 First in-plane stress

14 Coordinate Transformation : 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 f 2 [ ] cos (1 ) I II s I f f 2 [ ] cos (3 ) [ ] cos (1 )

15 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

16 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 ) [ ] cos (1 ) f

17 Hosford-Coulomb Fracture Initiation Model - for proportional loading : Dynamic behavior of materials and structures f 3D View plane stress f 2D View plane stress

18 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)

19 Hosford-Coulomb Ductile Fracture Model Influence of parameter a c 0.1 a 2 a a 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

20 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=

21 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= c=0.008 a=1.89 b=522.2 c=0.001 a=1.29 b= c=

22 Application of the Hosford-Coulomb Model DP1000 CP1000 CP

23 Common feature for most metals: Biaxial Tension Valley /2 /

24 Biaxial Tension Valley : Dynamic behavior of materials and structures f plane stress f plane stress Biaxial tension valley Biaxial tension valley is due to Lode effect!

25 Biaxial Tension Valley : Dynamic behavior of materials and structures f Biaxial tension valley Biaxial tension valley is due to Lode effect!

26 Hosford-Coulomb Ductile Fracture Model 3D View f 2D View f plane stress plane stress heart of the model: [, ] f f

27 Damage Accumulation Define damage indicator f [, ] f D d [, ] f p D D 0 1 (initial) (fracture) Example: uniaxial tension VIDEO

28 Damage Accumulation f [, ] f Define damage indicator D d [, ] Example: uniaxial compression followed by tension f p D 0 D 1 (initial) (fracture) VIDEO

29 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!

30 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

31 Plate bending

32 Punch test

33 Model Calibration SHEAR BENDING SHEAR PUNCH PUNCH BENDING Non-linearity in loading paths negligible

34 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

35 Shear Fracture Specimen Design Stress Triaxiality Major challenge: Fracture prone to initiate prematurely at nearly plane strain tension conditions near boundaries!

36 Shear Specimen Optimization Problem

37 Shear Specimen - Optimization

38 Basic geometry : Dynamic behavior of materials and structures Smiley Shear Specimen apparent shear fracture strain: 0.74 Optimized geometry apparent shear fracture strain:

39 Typical smiley-shear experiment Average equivalent plastic strain rate: ~0.001 /s Camera resolution: 4 mm/pixel

40 Other fracture experiments : Dynamic behavior of materials and structures Punch Butterfly shear f Notched tension Central hole tension

41 Flat Notched Tensile Specimens

42 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

43 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

44 Strain hardening p n A ( s )

45 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

46 Notched tension: Exp. & FEA

47 Notched tension: Exp. & FEA Side view of FEA (R=20mm) Side view of FEA (R=10mm) Side view of FEA (R=6.67mm)

48 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

49 Loading path to fracture

50 Summary plots

51 Tensile specimen with central hole

52 Tensile specimen w/ central hole

53 Punch experiments

54 Punch experiments

55 Reading Materials for Lecture #8 C. Roth and (2015), Ductile fracture experiments with locally proportional loading histories, Int. J. Plasticity,