Fracture Mechanics experimental tests (ASTM standards and data) 2/2

Transcription

1 Task 6 - Safety Review and Licensing On the Job Training on Stress Analysis Fracture Mechanics experimental tests (ASTM standards and data) 2/2 Davide Mazzini Ciro Santus Pisa (Italy) June 15 July 14, 2015

2 Standard for Fracture Mechanics Content Fracture Toughness K Ic - Plane strain condition - ASTM standard E399 High Toughness J Ic - Limitation of the K Ic - ASTM standard E1820 Measurement of Fatigue Crack Growth Rates - Paris curve experimental determination, ASTM standard E647 2

3 Large Fracture Toughness Wide plasticity CT specimen A very large specimen would be required to test according to K Ic 3

4 Rice 1968, J - integral Definition Let us assume an alternative material with Elastic behavior, not linear (hyperelastic material) and the same Stress/ Strain curve than the actual Elastic-Plastic one For a monotonic loading the Stress/ Strain history is the same 4

5 Rice 1968, J - integral Definition J is an integral along a path. It does not depend on the path that can be arbitrary provided that it is around the tip of the crack from side to side. x is the crack direction 5

6 Rice 1968, J - integral Definition J does not depend on the path that can be arbitrary provided that it is around the tip of the crack from side to side. Zero integral for free surfaces Zero integral for a closed path * * 1 2 = 1 2 6

7 J - integral equivalences Energy Release Rate For an Elastic behavior, even not linear, the J integral equals the Energy Release Rate Potential energy: Energy release rate: same as the G (elastic) parameter, A is the crack area. U is the stored strain energy F is the work done by external forces 7

8 J - integral equivalences The Griffith problem Infinite plate with a crack, with linear elastic material, and plane stress, Griffith found: 8

9 J - integral equivalences The Griffith problem Under these conditions: 2 d a J da E Being for this problem: KI F a, F 1.0 Then it follows (that s the reason of π): J 2 K I E 9

10 J - integral equivalences General case Only if the material is linear: where: E E plane stress E E 2 1 plane strain E G 2(1 ) Shear modulus J KI KII KIII E 2G 10

11 CTOD Crack Tip Opening Displacement CMOD Crack Mouth Opening Displacement COD - Crack Opening Displacement CMOD is just the clip gauge measurement CTOD 11

12 CTOD Crack Tip Opening Displacement COD - Crack Opening Displacement From ASTM standard E

13 COD - Crack Opening Displacement CTOD definition Crack tip blunting (approx. circumferential) Alternative way to define the CTOD: 2 45 lines 13

14 CTOD under SSY hypothesis CTOD can be related to the SIF under Small Scale Yielding 14

15 CTOD can also be related to the J integral well beyond the validity limits of LEFM CTOD under large plasticity Path around the strip-yield zone ahead of a crack tip Plane stress conditions and a nonhardening material. More generally: m is a dimensionless constant that depends on the stress state and material properties 15

16 CTOD similar value under SSY hypothesis CTOD under large plasticity Plane stress: being: J 4 K YS 2 I E 2 KI 4 J E YS approximately: 4 1 J YS ( m 1) 16

17 ASTM standard E1820 J and CTOD J and CTOD are still representative even over the 80% of the fully plastic load (plastic collapse) 17

18 ASTM standard E

19 ASTM standard E1820 From J back to K J KI KII KIII E 2G where: E E plane stress E E 2 1 plane stress E G 2(1 ) Shear modulus ASTM plane strain assumption, mode I only 19

20 ASTM standard E1820 Similar to the previous standard The displacement gage 20

21 ASTM standard E1820 Similar to the previous standard Types of preparation notches 21

22 ASTM standard E1820 Similar to the previous standard The SEB specimen The CT specimen The DCT specimen 22

23 ASTM standard E1820 The CT specimens Two CT specimen types, the increased size for the pin seat is for very high tension loads 23

24 ASTM standard E1820 Displacement gage fixture details Previous standard 24

25 ASTM standard E1820 J calculation for CT specimen Elastic term Plastic term 25

26 ASTM standard E1820 J calculation for CT specimen These values are for the initial crack, just after the (fatigue) precrack 26

27 ASTM standard E1820 J-R curve determination The suggested procedure is to determine a series of J values for increasingly crack size, and then obtain a (fit) Resistance curve a a 27

28 ASTM standard E1820 J-R curve determination For each crack increment (i) there is an unload cycle to determine the crack size through the compliance J, a () i () i Load, kn Total displacement, mm 28

29 ASTM standard E1820 J-R curve determination (i) stands for the updated values for the crack increments (i=0) is just the initial crack K (i) is calculated exactly as the previous standard 29

30 ASTM standard E1820 J-R curve determination 30

31 ASTM standard E1820 J-R curve determination Compliance Load Line calculation: 31

32 ASTM standard E1820 J-R curve determination Crack size calculation (compliance method): 32

33 ASTM standard E1820 J-R curve determination Compliance correction 33

34 ASTM standard E1820 J-R curve Power law regression line a oq, B, C are the regression coefficients for fitting the points J, a () i () i Least squares fit procedure to find the regression coefficients 34

35 ASTM standard E1820 J-R curve 35

36 ASTM standard E1820 J-R curve J Ic is defined near the initiation of stable crack growth. The precise point is usually ill-defined an offset (similar to yield strength) is required 36

37 ASTM standard E1820 J-R curve Yield and Ultimateaverage Example: J Q Y kj N N MPa mm 800MPa Y 2 m m mm JQ B mm...easy to be satisfied! Conversion: E E K Ic 2 /(1 ) 220GPa E J 6630MPa mm 210MPa m Ic 37

38 ASTM standard E1820 Alternatively the δ-r curve (CTOD-R curve) CTOD instead of J 38

39 ASTM standard E1820 J-R curve J J K Q Ic Ic J E Ic J Ic is just theonset of fracture, the specimen can sustain higher values of J,indeed the curve continues with a (stable) increase of the crack size. 39

40 The Resistance curve J-R curve Rising curve after J Ic 40

41 The Resistance curve J-R curve Rising curve after J Ic Stable further propagation 41

42 The Resistance curve J-R curve P paths: Load control i P, P stablecrack growth P limit stable crack P unst 4 able crack Load control is usually less stable than displacement control, in most structures the conditions are between the extremes of load and displacement control P 4 i paths: Displacement control paths 1 5 stable crack growth 42

43 The Resistance curve J-R curve Steady State Three stages of crack growth in an infinite body 43

44 The Resistance curve R curve, single fracture toughness value K, J Unstable K Ic, J Ic K( a), J( a) Stable With a single value the curve is just asymptotic a 0 crack size a 44

45 ASTM standard E1820 Homework: Apply the ASTM E1820 procedure extracting data from the test file: Test J_Ic.xlsx 70 Then estimate J Q and verify if it can be converted into J Ic Load, kn Total displacement, mm? 45

46 The damage tolerant approach Crack (stable) propagation under fatigue The presence of a crack (actually detected or just postulated) can be tolerated if the propagation rate is reliably estimated. The size of the postulated crack is the minimum detectable of the inspection method. 46

47 Paris law (also known as the Paris-Erdogan law) Fatigue crack propagation da dn C K m K (usually K ) I N 1 N 2 N... where: K K K max min Time Cm, are parameters depending on: -thematerial -theload ratio R K K min max Paul C. Paris 47

48 Fatigue crack propagation Paris law validity range Near threshold K th Paris validity Sudden (unstable) fracture Kc K th is the threshold stressintensityfactor range below this amplitude the crack remains same size though fatigue loaded K c is the threshold stressintensityfactor range for which K max K Approaching this value thecrack turnsinto unstable propagation Ic 48

49 Fatigue crack propagation Paris law integration da dn m C K C( F a) m after assuming F does not change(at least not significantly) the a dependece can be integrated: a1 a2 N m m m/2 ( m 2)/2 ( m 2)/2 ( m 2) CF ( ) a1 a2 with m 2(usually m 2) 49

50 Fatigue crack propagation Example: a 1mm, a 1 2 5mm a b, for exampleb 100 mm F MPa m 3.25 C mm/cycle (MPa mm) m a1 a2 N m m m/2 ( m 2)/2 ( m 2)/2 ( m 2) CF ( ) a1 a cycles b 50

51 Fatigue crack propagation Example: a 1mm, a 1 2 5mm limited width, b 10 mm thecalculation can be donestepwise dividing the crack range in small steps: F i 1.38( a 1 2mm) F 1.65, F 2.1, F 2.7 ii iii iv N N N N N 1-2 i ii iii iv cycles a1 a2 b 51

52 Fatigue crack propagation Example: a 1mm, a 1 2 5mm limited width, b 10 mm Paris' law (numerical) integration MATLAB N cycles (previous result N cycles) 100 cycles (small) steps integration 52

53 AFGROW software for crack propagation calculation Fatigue crack propagation 53

54 Fatigue crack propagation ASTM E647 54

55 ASTM standard E647 Definitions Usually tests are performed at positive load ratios e.g.: R

56 ASTM standard E647 Specimen geometry For fracture toughness B = W/2, while for fatigue this requirement is less demanding. The K I values experienced by the specimen under fatigue propagation are much lower. 56

57 ASTM standard E647 Specimen geometry LEFM validity condition, plane stress which is more demanding 57

58 ASTM standard E647 Specimen notch preparation 58

59 ASTM standard E647 Delta SIF calculation Same relation to find the Stress Intensity Factor for the CT specimen Current a value? 59

60 ASTM standard E647 Compliance method for the crack size Different positions of the crack gage clip Here is the clip displacement 60

61 Crack size determination Alternative ways for accurate crack size measurement during the test - Potential drop - Fractomat (Crack Gage) 61

62 Crack size determination Potential drop Calibration procedure Example coefficients: X X a Y Y 2 62

63 Crack size determination Fractomat crack gage As the crack propagates the foil resistance increases. Having a predefined geometry the calibration equation is already given by the manufacturer Crack range 63

64 Crack size determination Possible not parallel crack propagation, especially at the beginning Application of Fractomat at the two sides a' [mm] ch1 ch2 5 Channels 1 and 2 0 Number of cycles 64

65 ASTM standard E647 Near threshold/ high propagation rate High rate The final part of the curve is usually not of interest once K thic is known 5 10 mm/cycle K da dn th : 7 10 mm/cycle Near threshold 65

66 ASTM standard E647 Near threshold Decreasing procedure 66

67 ASTM standard E647 Propagation rate calculation A large number of cycles is recommended between steps the i and i 1 (instead of just two consecutive cycles) to have a significant, still small,crack size increment 67

68 Data example Q+T steel, similar to AISI 4340 m 3.6 C mm/cycle (MPa m) m Several tests with Fractomat and Potential Drop K th 9.4 MPa m 68

69 Data example Homework: Compare the experimental data with the alternative Paris model: da m m C( K Kth ) dn POTENTIAL DROP CRACK GAGE K th m 3.6 C 9.4 MPa m mm/cycle (MPa m) m