Dynamic Behavior of Concrete: Constitutive Models - Dynamic Behavior of Concrete and Seismic Engineering - page 72

2.5.2.2. Phenomenological law

The apparent strength of numerous materials depends on the loading rate used to
test it. The experimental results related to strength in a uniaxial stress state are
generally represented in a diagram of strength versus strain rate on a logarithmic
scale. For some rate ranges, strength varies more or less linearly with the strain rate
logarithm. The observation has led to empirical modeling:

E

V K H [2.22]

By analogy, models have been designed for application to concrete, with the
intention of deriving the strain rate effect from tests carried out at different rates.
The main problem with this approach, apart from the precautions to be taken when
interpreting the results of the tests, involves converting to a tensorial formulation. A
rather simplistic approach in this field could lead to models that have dubious
thermodynamic acceptability (dissipation not necessarily demonstrated outside the
radial loading paths).

2.5.3. Elasto-plasticity and criteria

Most of the models have been built on an elastoplastic theoretical framework. If
the plasticity is perfect, the plasticity criterion corresponds to the failure criterion,
but many models differentiate the elastic limit criterion from the failure criterion.
The loading surface evolves from the first to the second with the evolution of a
strain-hardening variable.

These criteria are expressed in terms of the stress state and the strain-hardening
variables:

f V , k 0 [2.23]

i

The material independence leads us to express the criterion as a function of the
three stress invariants:

- the first invariant (or pressure);

I trace 3 p [2.24]

V

1