Analysis of Crack Branching Morphology in a Disk of Brittle Material under Axisymmetric Tension by Using Branching Dimension

Abstract

In order to quantify the crack branching morphology of brittle materials under axisymmetric tension, the “branching dimension” was defined in the present paper, and the relation between the branching dimension and fracture mechanics parameters was introduced.
The branching dimension D_b is defined as: D_b=d(logL_i)/d(loga_i), where L_i is the total crack length after i-th crack branching has occurred and a_i is the crack length which corresponds to the distance from crack nucleation point to crack tip. When the nominal stress σ_i varies with increasing crack length a_i according to the relation, σ_i/σ₀=(a_i/a₀)^m, the branching dimension D_b can be expressed by:
D_b=2+2m=2+2·d{log(σ_i)}/d{log(a_i)}
where σ₀ and a₀ are the stress and the crack length at crack initiation. Moreover, D_b can be expressed by the change of the number of cracks, n, with increasing crack length, a_i:
D_b=1+d{log(n)}/d{log(a_i)}
The branching dimension can well characterize the branching patterns obtained from the flexural test of Pyrex glass, ceramic and high strength steel specimens.