9.7. Broad-Range Size Effect Law and Its Dirichlet Series Expansion

The simple size effect law in (9.40) for the basic Case 1 is normally adequate for a size range up to about 1:20, which suffices for most structural engineering applications. The generalized size effect law (9.43) has been shown to give excellent approximation of the numerical results obtained by Hillerborg with the cohesive crack model for the size range of 1:250 (Ba ant 1985b). However, only the first term of the large-size power series expansion of that law in 1/ D is correct. The following broad-range size effect law for Case 1, which is a generalization of formula (9.46), is capable of approximating the large-size asymptotic behavior up to order n + 2 in 1/ D;

(9.110)

which has the asymptotic expansion:

(9.111)

with the notations:

(9.112)

Here D ₁ , D ₂ , ....D _N, ? ₁, ? ₂, ... ? _N are constants; D ₁ , D ₂ ,...D _N are positive and may be assumed to form an increasing sequence. It is evident that this formula also preserves the finiteness of lim ? ₀ for D ? 0.

From the foregoing expansion one can further obtain:

(9.113)

in which

(9.114)

Case 1 is obviously characterized by

(9.115)

Formula (9.110) has been deliberately structured so that the expression for D/E' be the well-known Dirichlet series (called also the Prony series). This series is the real counterpart of the Fourier...