Experiments show that the failure of ductile materials can be characterized by a
rate-independent parameter, relative spacing d defined as the ratio of the distance between two voids and
the radius of voids. In this study, this experimental phenomenon is analyzed via numerical simulations
using 3-D finite element model. Considering that hydrostatic stress is a dominant factor in the evolution
of microvoid nucleation, growth and coalescence in ductile materials, numerical simulations are
performed to obtain the relationship between relative spacing d and hydrostatic stress in the ligament
between voids. Numerical results show that hydrostatic stress along matrix ligament is sensitive to the
change of the relative spacing. Further analysis shows that the failure of ductile materials can modeled by
using a criterion of the threshold of local hydrostatic stress in the ligament. Based on such a criterion, a
curve displaying the relationship between the strength of ductile material and strain rate is obtained
numerically. It is concluded that the failure criterion of ductile materials can be described by using local
hydrostatic stress and relative spacing between two voids, which is not sensitive to strain rates.
Crown Ratio and Relative Spacing Relationships for Loblolly Pine Plantations
