Fracturing rate effect and creep in microplane model for dynamics

The formulation of microplane model M4 in Parts I and II is extended to rat e dependence. Two types of rate effect in the nonlinear triaxial behavior o f concrete are distinguished: (1) Rate dependence of fracturing (microcrack growth) associated with the activation energy of bond ruptures, and (2) cr eep (or viscoelasticity). Short-time linear creep (viscoelasticity) is appr oximated by a nonaging Maxwell spring-dashpot model calibrated so that its response at constant stress would be tangent to the compliance function of model B3 for a time delay characteristic of the problem at hand. An effecti ve explicit algorithm for step-by-step finite-element analysis is formulate d. The main reason that the rate dependence of fracturing must be taken int o account is to simulate the sudden reversal of postpeak strain softening i nto hardening revealed by recent tests. The main reason that short-time cre ep (viscoelasticity) must be taken into account is to simulate the rate dep endence of the initial and unloading stiffness. Good approximations of the rate effects observed in material testing are achieved. The model is suitab le for finite-element analysis of impact, blast, earthquake, and short-time loads up to several hours duration.