a constant-thickness plate specimen of width w

Question

A constant-thickness plate specimen (of width, w, and thickness, t) containing a crack has a compliance, C, given by:

C = A₁ + A₂ α³

Where a is the crack length, and A₁ and A₂ are constants.

The specimen is loaded by a cyclic load which varies from 0 (zero) to P (i.e, the stress ratio, R = 0) and the crack propagates in Mode I.

B and m are curve-fitting constants and ΔK_I is the amplitude of the opening-mode stress intensity factor.

The fracture toughness is K_IC , Young’s modulus is E, the yield strength is Sy and the ultimate tensile strength is S_ut.

If the crack growth rate per cycle, da/dN_f, is given empirically by the power law;

da/dN_f = B (ΔK_I ) ^m

Determine an expression for the number of cycles to failure, N_f if the initial crack size is a₀. State any assumptions you make.