2. Introduction#

2.1. Main characteristics of the model#

The model presented in this document is a nonlinear law of behavior for concrete. It is based on the theory of plasticity, it is valid for three-dimensional stress states. The modeling hypotheses adopted partly use the models developed by G. Heinfling [bib2] and J. F. Georgin [bib1] and are as follows:

  • there is a field of reversibility of constraints delimited by two criteria of the Drücker Prager type,

  • each of the criteria falls apart, the rupture domain corresponds to the maximum of the reversibility domain,

  • in compression, the work hardening is positive up to a peak, then it becomes negative,

  • under traction, the work hardening is exclusively negative,

  • the post-peak curves in both cases (traction/compression) vary with the size of the finite element (the shape of this curve is linked to negative work hardening and to the cracking energy),

  • the plastic flow is governed by a rule of normality (associated plasticity) the formulation of the work hardenings is of an isotropic type,

  • the elasticity module and the reversibility thresholds vary with temperature.

Note:

*The terminology of traction criteria and compression criteria is questionable. We will use it out of habit, being**well aware that a state of tensile stresses can lead to the activation of the so-called compression criterion. *

2.2. Why two Drücker Prager criteria#

The authors of the theses cited in reference [bib1] and [bib2] use a Drücker Prager criterion in compression and a Rankine criterion in traction. They justify these choices by physical considerations by showing that the reversibility domain thus obtained is close to experimental reality. On the other hand, they limit their models to states of two-dimensional constraints. We preferred to replace the traction criterion with a surface also of the Drücker Prager type. By this choice, certain difficulties are overcome, particularly in three-dimensional formulations. The 3D surface defining the admissible stress states with respect to traction is no longer a pyramid (3D Rankine) but a conical surface whose vertex is located on the hydrostatic axis. The trace of the « so-called traction » criterion on the deviatoric plane is no longer a triangle, but a circle. The formulation obtained is simpler. The difference between the two criteria is minimal for stress states close to plane stress states. On the other hand, for highly confined stress states, the two approaches (Rankine and Drücker Prager) are different, which is a limitation of the proposed model.