INSTABILTY & ACTIVATION ENERGY

The occurrence of maxima in the Gc or Kc with respect to velocity of propagation of cracks v is of considerable importance. In general they lead to rapid acceleration of propagation and fracture. The heat generated in the crack tip zone is not dissipated at the higher speeds and hence heat will accumulate. For a fixed COD (critical crack opening displacement), δc criterion, the thermal softening will occur. This will lead to rapid fall in Kc with increasing v. The increasing crack length tends to increase Kc and thus a maximum of the Gc or Kc will occur before falling. The maximum temperature generated in a line zone of the thickness b, by an energy dissipation of Gc is related by a proportionality to the difference in temperature ∆T =(T – To).

∆T α Gc/[√(πρCk t)] (A)

Here ρ = density and C = specific heat, k = thermal conductivity.For very small b, being in the order of δc the critical crack opening displacement (COD),

v α c/t α Kc2/t, Gc α Kc

2, (for c refer to COD, Fig).

Then

∆T α Kc√v (B)

Differentiating (B) wrt v

1

dT/dv α Kc√v(1/Kc)dKc/dv+1/2v]

= ∆T [(1/Kc) dKc /dv + 1/2v] (C)

dT/dv α (T – To)[ (1/Kc) dKc /dv + 1/2v] (D)

Temperature dependence of Kc is expressed via an activation energy, H, so that:

Kc α vn exp[nH(1/T -1/To)/R] (E)

Taking log of (E) and integrating

(1/ Kc)dKc/ dv = n [1/v – H dT/RT2 dv] (F)

For maximum Kc representing instability,

dKc/dv = 0, if n= 0 or [X] =0.

From 2nd condition

dT/ dv α R T2/Hv (G)

On substitution into (C) or (D)

∆T α 2R To2/H (H)

From (B) & (H), we have at instability.

Kc2 vc α To

4 (I)A critical time for the isothermal-adiabatic transition may be found from the expression (A)

tc = (H/2R)2 Gc2/( πρCkTo

4) (J)

2