Structured Cohesive Zone Crack Model

1
TIME DEPENDENT FRACTURE IN VISCO-ELASTIC AND
DUCTILE SOLIDS

• 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
3
Figure 1. Structured cohesive zone crack model of
Wnuk (1972,1974). Note that of the two length
parameters ? and R the latter is time dependent
analogous to length a, which denotes the length
of the moving crack. Process zone size ? is the
material property and it remains constant during
the crack growth process. Ratio R/ ? serves as a
measure of material ductility for R/ ?gtgt1
material is ductile, while for R/ ? -gt 1,
material is brittle.
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
Fig. 2. Schematic diagram of the standard linear
solid model.
?1 E1/E2
?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 E1/E2
10
Solution of Mueller-Knauss-Schapery equation for
a Moving Crack in SLS
x a/a0 ? t/?2
11
Crack Motion in Visco-elastic Solid
x a/a0 ? ?/a0 ? t/?2
?t ?/a? a? da/dt
12
Fig. 4. Incubation and Propagation of Crack
in Linear Visco-elastic Solid
13
Critical Time / Life Time

• t1 incubation time
• t2 propagation time
• ?/a0
• n (?G/?0)2
• ?1 E1/E2

14
Fig. 3a. Logarithm of the incubation time in
units of t2 shown as a function of the loading
parameter s for two different values of the
material constant ß1 E1/E2.
15
Fig. 3b. Logarithm of the time-to-failure used
during the crack propagation phase, in units of
t2, shown as a function of the loading parameter
s for two different values of the material
constant ß1 E1/E2.
16
Fig. 5. Wnuk Model of Crack Motion in Ductile
Solid
17

• Material Parameters
• Process Zone Size ?
• Length of Cohesive Zone at Onsetof Crack Growth
  Rini

Material Ductility
Profile of the Cohesive Zone (R ltlt a)
18
Wnuks 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 gt 0, stable crack growth dR/da lt 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

26
Fig. 6. Material Resistance R-curves for Ductile
Solids of Various Ductility ?
27
Fig. 8. Stability Index as Function of Crack
Length for Various Ductility
28
Fig. 9. Current Crack Length as Function of
Time for Various Ductility
29
Fig. 10. Material Resistance R-curves for Ductile
Solids of Various Roughness
30
Fig. 11. Load as Function of Crack Length
for Various Material Roughness ?
31
Fig. 12. Stability Index as Function of Crack
Length for Various Material Roughness
32
Fig. 13. Crack Length as Function of Time
for Various Material Roughness
33
Fig. 14a. Stages of Subcritical Crack Growth
in Ductile Solids
34
Fig. 14b. Examples of Quasi-static Stable Crack
Growth in Ductile Solids
35
Fig. A1. Equilibrium and Terminal Instability
States for Various Material
Ductility
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