ABSTRACT
These residual values are indicated as the ratio of residual f'cr to f'c for an identical virgin specimen. We observe values of damage induced by various trajectories in stress space. The default failure surface used in our concrete is included for reference. The conclusion to be drawn from this figure is that significant damage is only observed for stress states near the failure surface. Large compressive strains do not induce appreciable damage. Motivated by this diagram and by the residual failure surface shown in Fig. 2, we implement softening by shifting the failure surface and J1 limit towards zero along the J1 axis as a function of expansive volumetric plastic strain (see Fig. 4). Because plastic strain increments on the cap are equivoluminal or compressive, no softening occurs when the stress state is on the cap. To account for the increased ductility observed in compression relative to that observed in tension, we introduce the following pressure dependent relation for failure surface shifting,
(13) Here, 6a denotes the hydrostatic pressure (compression negative), XIO denotes the intersection of the unshifted failure surface with the pressure axis, and pcut is the pressure cutoff, d is the failure surface evolution shift parameter that is determined by the positive volumetric plastic strain,
and the confinement pressures, 6a. Hooke's Law relates the elastic stress increments to the strain increments. For the viscoplastic flow, we adopt a piecewise linear Duvaut-Lions style formulation loosely following the work of Simo et al [12]. This formulation lends itself to a clean and simple numerical implementation. Details are presented in [13].
