1. Structured Cohesive Zone Crack Model Michael P Wnuk College of Engineering and Applied Science University of Wisconsin - Milwaukee

2. Preliminary Propagation of Crack in Visco-elastic or Ductile Solid

4. Constitutive Equations of Linear Visco-elastic Solid

5. Wnuk-Knauss equation for the Incubation Phase Mueller-Knauss-Schapery equation for the Propagation Phase

6.  1 = E 1 /E 2  2 – relaxation time

7. Creep Compliance for Standard Linear Solid

8. Solution of Wnuk-Knauss Equation for Standard Linear Solid

9. Range of Validity of Crack Motion Phenomenon  1 = E 1 /E 2

10. Solution of Mueller-Knauss- Schapery equation for a Moving Crack in SLS x = a/a 0  = t/  2

11. Crack Motion in Visco-elastic Solid x = a/a 0  =  /a 0  = t/  2  t =  /a a = da/dt

12. NONDIMENTIONAL CRACK LENGTH, x=a/a o n=4 t 1 =0.375τ 2 1 n=4 t 2 =0.277τ 2 /δ n=8.16 t 2 =1.232τ 2 /δ n=6.25 t 2 =0.720τ 2 /δ n=6.25 t 1 =0.744τ 2 NONDIMENSIONAL TIME IN UNITS OF (τ 2 ) 1.51. 0 0.5 0 1. 0 1.5 2 3 4 5 6 NONDIMENSIONAL TIME IN UNITS OF (τ 2 /δ) n=8.16 t 1 =1.26τ 2

13. Critical Time / Life Time t 1 = incubation time t 2 = propagation time  =  /a 0 n = (  G /  0 ) 2  1 = E 1 /E 2

17. Material Parameters: Process Zone Size  Length of Cohesive Zone at Onset of Crack Growth R ini Material Ductility Profile of the Cohesive Zone (R << a)

18. Wnuk’s Criterion for Subcritical Crack Growth in Ductile Solids

19. Governing Differential Equation

20. Wnuk-Rice-Sorensen Equation for Slow Crack Growth in Ductile Solids

21. Necessary Conditions Determining Nature of Crack Propagation dR/da > 0, stable crack growth dR/da < 0, catastrophic crack growth dR/da = 0, Griffith case

22. Auxiliary Relations

23. Terminal Instability Point =

24. Rough Crack Described by Fractal Geometry Solution of Khezrzadeh, Wnuk and Yavari (2011)

25. Governing Differential Equation for Stable Growth of Fractal Crack  = (2-D)/2 D – fractal dimension

36. * New mathematical tools are needed to describe fracture process at the nano-scale range * More research is needed in the nano range of fracture and deformation example: fatigue due to short cracks

37. * New Law of Physics of Fracture Discovered: Ten Commandments from God and one equation from Wnuk