Dynamic Behavior of Concrete: Constitutive Models - Dynamic Behavior of Concrete and Seismic Engineering - page 76

strongly dependent. Location limiters (either explicitly or implicitly) induce the
dependence of behavior on the strain rate.

For instance, in visco-plasticity, owing to a study of wave dispersion, we can
establish a characteristic length (K is the coefficient of viscosity and c e the velocity
of elastic waves):

2 K c

E

e

[2.33]

A

c

We can then wonder if this time-dependence can be mixed up with the
rheological aspect of the behavior. As a matter of fact, both aspects are separated by
the values of the viscosities in play. In fact, the value of a characteristic length is
centimetric [PIJ 87] and therefore must be related to a specific coefficient of
viscosity.

2 4

[2.34]

A | 10 m l| K 10 Pa s .

c

Such a coefficient of viscosity value (taken as a visco-plasticity coefficient in
Perzyna's model) is without any noteworthy effect on the macroscopic behavior (for
example, the apparent strength as a function of stress/speed). If we are to apply a
coefficient of viscosity to express a rheological aspect, like the dependence of
compaction on the strain rate [GAR 98b], We will have to introduce a coefficient
about 10 8 Pa.s.

2.6. Models

An illustrative selection of models that are used in fast dynamics codes to
represent concrete behavior are presented below. This collection is obviously
restricted and does not pretend to be exhaustive. It is not a critical or comparative
study of the models. For each model, a reference is provided, affording the reader
the opportunity to find details related to each, as explained by its author. The “basic”
models, such as Prager's model, are not included.

2.6.1. Elasticity-based model

2.6.1.1. Cedolin

- Reference: [CED 77].