Article : 8 Article : 8 Stress Field. and. Singularity Problem

Transcription

1 Aticle : 8 Aticle : 8 Stess Field and Singulaity Poblem

2 (fatigue cack gowth) Repeated load cycles cack development Time (cycles) Cack length 3

3 Weakening due to gowing cacks Cack length stess concentation ate of cack popagation stength of stuctue 4

4 Cack, a, in infinite plate unde emote etenal stess 5

5 At distance,, fom the a long cack tip: y y a a a

6 At distance,, fom the cack tip: (cont.) These equations ae valid fo a limited aea aound the cack Also: z ; plane stess z y ; plane stain 7

7 At distance,, fom the cack tip: K I f ij ij ; Whee: K I a (K I is known as mode I stess intensity facto) 8

8 Stessed Solid: Assumed stesses:, y, z, y, z, yz ; Whee in a condition of plane stess : z,= y = z, =0 and in a condition of plane stain : z = 0; hence: z = ( y + z ) 9

9 Plane poblem; equilibium: y y 0, y y y 0 Stain - displacement epessions: u y v y y u y v stess - stain elations: E E y y y y y 0

10 Assuming a function,, such that: y y, y y Satisfies the equilibium equations

11 This is obtained by substitution of stesses in tems of fom these epessions into stess - stain elations and diffeentiating twice: y y 0 o:

12 3

13 K I Z z Ug pola co-odinates with 0 z K I f ij ij z K I e i 4

14 K I y y 0 K K K I I I K I Plane Stess, Plane Stain 5 z z y

15 K I One may conclude that: At the cack tip: 0 and thus stesses become infinite. Facto K I is then a measue of cack tip gulaity Stesses ae elastic thus fo uni-aial tension K I is popotional to K I must also be popotional to the squae oot of a length The only chaacteistic length fo infinite plate is the cack size K I c a 6

16 K I Fo mode-i cack in infinite plate unde bi-aial stess with oigin of co-odinate system at cack tip: z Z z z a a Thus: K I a 7

17 K I ; uni-aial vs bi-aial infinite plate solution It is epected that the stess system paallel to the cack is not distubed by the cack Thus the solution fo the uni-aial case must be the same as fo bi-aial Hence, the facto C is equal to 8

18 u Re Z y Im E v Im Z y Re E O: K I u E K I v E Z Z 9

19 Stess components in tems of K I ae eact fo cack tip. Howeve they can also be used to a distance fom tip that is small compaed with the cack size. The geneal solution with the highe ode tems: 0 ij C f ij C f ij C 3 f a a a 3 ij... O: C n ij f ij C n f n nij 0

20 The geneal solution neglecting the highe ode tems: C f ij ij With: C K I

21 K II y y K K K II II II z y z yz 0

22 K II Fo a long Cack in infinite plate with unifom in-plane shea,, at infinity K II a 3

23 z yz K III K K III III y z y 0 4

24 epesenting the stess state in, out and on the plastic zone! Cacks with same K I have equal size plastic zones and out of the PZ the stess field is the same, this is also the case on the bounday of the PZ Thus: the stesses and stains in the PZ must be the same; it means that K I also detemines what occus inside the PZ! K I is a measue fo all stesses and stains 5

25 c cack etension occus when the stesses and stains at the cack tip each a citical value; o one can say: when K I eaches a citical value it is anticipated that K IC, the citical K I, can be seen as a mateial paamete 6

26 c (cont.) within cetain limitations, this is the case as epeiments indicate Ic c c, failue stess The above fomula is valid fo infinite plate only; 7

27 fo a plate of finite size: K I a f a w ;Whee w is the plate width 8