Crack reinforcement by distributed springs

An analysis is presented for a centre crack which is reinforced by linear springs over a distance l from both tips. The reinforcement introduces a characteristic length k⁻¹ where k denotes a suitably defined spring constant. Two functions F, V are defined corresponding respectively to the stress intensity factor and the maximum spring-stretch, normalized relative to their values in the absence of reinforcement, for modes I, II or III. Asymptotic expansions for these functions are derived for the limiting cases of soft springs (kl ⪡ 1) and hard springs (kl ⪢ 1). Interpolating functions constructed from these asymptotic expansions are shown to agree with the numerical solution of the governing integral equation over the intermediate range of kl, to within an error of less than 1%. It is noted that the leading term of the expansions for hard springs can be derived by physical reasoning, requiring relatively simple calculations, which can also be used for nonlinear springs.