1 /18 M.Chrzanowski: Strength of Materials SM2-12: Fracture Introduction to FRACTURE MECHANICS

2 /18 M.Chrzanowski: Strength of Materials SM2-12: Fracture  ekspl =R m /s < R H This fundamental design formula is valid: 1.Below load bearing capacity 2.Within elastic region but it turned out to be unsatisfactory in two situations: q=q(t), P=P(t) Because of material FATIGUE Beacuse of material CRACKING (Fatigue Mechanics) (Fracture Mechanics) When loading (and consequently – stress) is varying in time: When geometry of a structure yields stress concentration σ t Elasticity versus fracture

3 /18 M.Chrzanowski: Strength of Materials SM2-12: Fracture s FrFr FaFa FaFa FrFr Reactive force (repulsion)Active force (attraction) m > n (m  10, n  5) For s=s o soso FrFr FaFa FaFa FrFr Lennart-Jones model

4 /18 M.Chrzanowski: Strength of Materials SM2-12: Fracture s > s o FrFr FaFa FaFa FrFr FF soso F  0  =s-s o F s 0 Lennart-Jones model

5 /18 M.Chrzanowski: Strength of Materials SM2-12: Fracture F  0 F=F R  =  R  =  R  Lennart-Jones model

6 /18 M.Chrzanowski: Strength of Materials SM2-12: Fracture m=10, n =5 =(10  1000)( F R experimental ) Reasons for discrepancy: 1.Extremely simplified two-atomic model 2.Defects of crystalline structure (theory of dislocation) Lennart-Jones model

7 /18 M.Chrzanowski: Strength of Materials SM2-12: Fracture XV century XIX century X century Fatigue Observed since prehistoric times: example – technology development in shipbuilding

8 /18 M.Chrzanowski: Strength of Materials SM2-12: Fracture A ship is working alternatively as double cantilever and simply supported beam Deck under tension! Keel under compression! Keel under tension! Deck under compression! Until the middle of XX century the problem of ship cracking caused by fatigue remained unsolved. Fatigue

9 /18 M.Chrzanowski: Strength of Materials SM2-12: Fracture Transporter SS Schenctady, cracked in half when docked in the port on , Portland, OR

10 /18 M.Chrzanowski: Strength of Materials SM2-12: Fracture Fatigue appearance accelerated when steam railways were introduced in XIX century. „The Rocket”, steam locomotive built by R. Stephenson, 1829 Fatigue

11 /18 M.Chrzanowski: Strength of Materials SM2-12: Fracture M  t Wheel axis cross-section A A A A

12 /18 M.Chrzanowski: Strength of Materials SM2-12: Fracture A.Wöhler ( ) Wöhler stand for fatigue investigation Wöhler diagram for high-cycle fatigue Fatigue limit Fatigue

13 /18 M.Chrzanowski: Strength of Materials SM2-12: Fracture Stress approach q[Pa] y x σ y = 3q G.Kirsch, 1898 – band of infinite width with circular hole Cracking

14 /18 M.Chrzanowski: Strength of Materials SM2-12: Fracture Stress approach q[Pa] y x C.E.Inglis, 1913 – A band of infinite width with elliptic hole a b b  0 σ  a  b σ  3q Independent of the hole half- radius size a !!! Cracking

15 /18 M.Chrzanowski: Strength of Materials SM2-12: Fracture H.M.Westergaard, 1939, N.I.Muskhelischvili, 1943 – analysed 2D stress field at the tip of a sharp slit yy AA r  x  0 0 For Singularity! Stress intenstiy factor a Cracking For

16 /18 M.Chrzanowski: Strength of Materials SM2-12: Fracture Three typical cases (modes) can be distinguished: Stress intensity factors were calculated for different configuration of loading and specimen geometry (G.Sih) Mode I - Tearing; crack surfaces separate perpendicularly to the crack front. Mode II – In-plane shear; crack surfaces slide perpendicularly to the crack front Mode III – Out-of-plane shear; crack surfaces slide parallely to the crack front. KIKI K II K III Cracking

17 /18 M.Chrzanowski: Strength of Materials SM2-12: Fracture Design criteria are: K I < K Ic K II < K IIc K III < K IIIc where K Ic, K IIc, K IIIc are critical values of corresponding stress factors, being determined experimentally. Cracking

18 /18 M.Chrzanowski: Strength of Materials SM2-12: Fracture 2l2l q q c c What is the length of a central slit we can introduce to the structure shown without the reduction of its load bearing capacity? (no interaction assumed) For side crack For central crack If e.g. c = 2 cm 2 l  5 cm Simple example of Fracture Mechanics application 2l ≤2c or 2l  2c ?

19 /18 M.Chrzanowski: Strength of Materials SM2-12: Fracture  stop