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Published byAllison Beasley Modified over 9 years ago
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Structured Cohesive Zone Crack Model Michael P Wnuk College of Engineering and Applied Science University of Wisconsin - Milwaukee
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Preliminary Propagation of Crack in Visco-elastic or Ductile Solid
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Constitutive Equations of Linear Visco-elastic Solid
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Wnuk-Knauss equation for the Incubation Phase Mueller-Knauss-Schapery equation for the Propagation Phase
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1 = E 1 /E 2 2 – relaxation time
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Creep Compliance for Standard Linear Solid
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Solution of Wnuk-Knauss Equation for Standard Linear Solid
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Range of Validity of Crack Motion Phenomenon 1 = E 1 /E 2
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Solution of Mueller-Knauss- Schapery equation for a Moving Crack in SLS x = a/a 0 = t/ 2
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Crack Motion in Visco-elastic Solid x = a/a 0 = /a 0 = t/ 2 t = /a a = da/dt
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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
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Critical Time / Life Time t 1 = incubation time t 2 = propagation time = /a 0 n = ( G / 0 ) 2 1 = E 1 /E 2
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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)
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Wnuk’s Criterion for Subcritical Crack Growth in Ductile Solids
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Governing Differential Equation
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Wnuk-Rice-Sorensen Equation for Slow Crack Growth in Ductile Solids
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Necessary Conditions Determining Nature of Crack Propagation dR/da > 0, stable crack growth dR/da < 0, catastrophic crack growth dR/da = 0, Griffith case
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Auxiliary Relations
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Terminal Instability Point =
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Rough Crack Described by Fractal Geometry Solution of Khezrzadeh, Wnuk and Yavari (2011)
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Governing Differential Equation for Stable Growth of Fractal Crack = (2-D)/2 D – fractal dimension
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* 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
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* New Law of Physics of Fracture Discovered: Ten Commandments from God and one equation from Wnuk
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