Dynamic Behavior of Concrete: Constitutive Models - Dynamic Behavior of Concrete and Seismic Engineering - page 82

- Model data: decomposition of the stress tensor, within the principal axis, in a
positive part and in a negative part, to take into account the independence of
stiffness in traction and compression. Two damage variables are defined by:

V V

v E D E D

11 2 2

c

H V trace V f V

EDEDE 1 1 ED 1 ED 1

0 1 0 2 0 0 1 0 2

[2.48]

where D 1 and D 2 are the damage variables, the other parameters being characteristics
of the material.

2.6.3.3. Dubé

- Reference: [DUB 94].

- Principle: visco-damage (implicit formulation).

- Application field: cyclic loading, dynamic problem.

- Can model: stiffness loss (= deterioration under traction stress), residual
strains, stiffness recovery and rate phenomena.

- Cannot model: compaction.

- Model data: the formulation is close to La Borderie's model, but it introduces,
in the same way as Perzyna, time dependence in the evolution of the damage
variables (f being the damage threshold function)

n

§ ¢² f ·

D ¨ [2.49]

¸

m

© ¹

2.6.4. Model coupling damage and plasticity

2.6.4.1. Ulm

- Reference: [ULM 93].

- Principle: associated elastoplastic model + damage (implicit formulation).

- Application field: cyclic loadings, low-rate dynamic and monotonous loadings.

- Can model: monotonous loadings, contracting plastic behavior for high
hydrostatic pressures, dilating plastic behavior for low hydrostatic pressures.