TIME DEPENDENT FRACTURE IN VISCOELASTIC AND DUCTILE SOLIDS

*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

Preliminary Propagation of Crack in Visco-elastic or Ductile Solid

Constitutive Equations of Linear Visco-elastic Solid

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

* 1 = E 1/E 2 2 – relaxation time

Creep Compliance for Standard Linear Solid

Solution of Wnuk-Knauss Equation for Standard Linear Solid

Range of Validity of Crack Motion Phenomenon 1 = E 1/E 2

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

Crack Motion in Visco-elastic Solid x = a/a 0 = /a 0 t = /a = t/ 2 a = da/dt

n=8. 16 t 1=1. 26τ2 1. 5 n=6. 25 t 1=0. 744τ2 1. 0 NONDIMENTIONAL CRACK LENGTH, x=a/ao * n=4 t 1=0. 375τ2 0. 5 NONDIMENSIONAL TIME IN UNITS OF (τ2) n=6. 25 t 2=0. 720τ2/δ 6 5 n=8. 16 t 2=1. 232τ2/δ n=4 t 2=0. 277τ2/δ 4 3 2 1 0 0. 5 1. 0 NONDIMENSIONAL TIME IN UNITS OF (τ2/δ) 1. 5

Critical Time / Life Time t 1 = incubation time t 2 = propagation time = /a 0 n = ( G/ 0)2 1 = E 1/E 2

Material Parameters: • Process Zone Size • Length of Cohesive Zone at Onset of Crack Growth Rini Material Ductility Profile of the Cohesive Zone (R << a)

Wnuk’s Criterion for Subcritical Crack Growth in Ductile Solids

Governing Differential Equation

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

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

Auxiliary Relations

Terminal Instability Point =

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

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

* More research is needed in the nano range of fracture and deformation example: fatigue due to short cracks