1. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 1/25 FRACTURE MECHANICS Energy approach

2. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 2/25 H.M.Westergaard, 1939, N.I.Muskhelischvili, 1943 – 2D analysis of the stress field around the notch tip yy AA r  x  0 0 Dla Singularity ! Stress intensity factor a

3. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 3/25 To stystemise above, three cases are distinguished: Stress intensity factors for different orientation of crack plane and loading has been calculated around 1960 (G.Sih) Mode I - Tearing; crack surfaces open perpendicular to the planes of its movement Mode II – Out-of-plane shear; crack surfaces slide perpendicular to the planes of its movement Mode III - Out-of-plane shear; crack surfaces slide paralel to the planes of its movement KIKI K II K III

4. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 4/25 Stress-based approach Safe design requires following conditions to be fulfilled: K I < K Ic K II < K IIc K III < K IIIc where K Ic, K IIc, K IIIc are critical values of responsible stress intensity factors, experimentally determined

5. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 5/25 q[Pa] y x G.V. Kolosow 1910 & C.E.Inglis, 1913 – Thin plate of unlimited width containing an elliptical hole a b b  0 σ  a  b σ  3q INDEPENDENT OF THE MAGNITUDE OF a !!! Griffith theory

6. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 6/25 With if but ! u x y for |x|<l w PSS for |x|<l w PSD l l for |x|>l The work of external load on displacement u (c=1m) then where c

7. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 7/25 The work of external load on displacement u Energy required to form a free surface of the length 2x2 l (over unit thickness of the plate)  - surface energy of unit thickness [N/m] W0W0 W d(W 0 -W)/dl=0 l kr l Energy required Energy supplied Instability point W0-WW0-W u x y l l q q

8. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 8/25 d(W 0 -W)/dl=0 For l=l kr LFM:

9. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 9/25 You will see an experiment showing how elastic energy released with the crack length growth is transferred to the region of final failure

10. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 10/25 q q

11. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 11/25 q q

12. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 12/25 q q

13. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 13/25 q q

14. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 14/25 q q

15. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 15/25 q q

16. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 16/25 q q

17. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 17/25 q q Even 5-fold reduction of cross section area does not reduce strength of the specimen!

18. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 18/25 q q 0,5 sec

19. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 19/25 q q 1 sec

20. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 20/25 q q 2 sec

21. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 21/25 q q 1 sek q q

22. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 22/25 q q 1 sek q q 2l kr 2l

23. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 23/25 STOP

24. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 24/25 FAST ANIMATION

25. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 25/25 q q

26. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 26/25 q q

27. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 27/25 q q

28. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 28/25 q q 1 sek

29. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education M.Chrzanowski: Strength of Materials 3 SM3-0 4 : Energy of fracture 29/25 q q 1 sek q q