Dynamic Behavior of Concrete: Constitutive Models - Dynamic Behavior of Concrete and Seismic Engineering - page 62

material). The strain rate notion is interesting if it can be associated with a volume
element where it is almost uniform. The higher the rate, the smaller the
corresponding volume. Therefore, we have to note that in continuum mechanics, a
very high strain rate can be incompatible with a given representative elementary
volume, especially for a material like concrete.

The notion of strain rate becomes meaningless if we consider a shock wave
(finite variation of the strain amplitude for an infinitesimal period of time).

2.2. Fast dynamics applied to concrete

This chapter presents the fundamental concepts that are useful to study a
structure subject to dynamic loading. Many studies and applications have been made
for metals, but here, the emphasis is placed on the specificities of applications to
concrete.

2.2.1. Impacts and waves

2.2.1.1. Uniaxial strain state
In a uniaxial strain state, which can correspond to certain impact conditions, the
stress and strain tensors have the following forms:

§ H

1 00
000
000

· § V 00 ·

1

¨ ¸ ¨ ¸

H

[2.2]

V 0 V 0

¨ ¸ ¨ ¸

2

¨ ¸ ¨ ¸

00 V

© ¹ © ¹ 2

To study such a strain state, the most commonly used hypothesis is that the
material can be described by means of perfect elasto-plastic behavior and that the
plastic flow is incompressible. Thus, the partition between elastic and plastic strains
is made:

e p

e p

HHH HH 0 , 3 i

[2.3]

i i

and:

p

p p

[2.4] trace H 0 H 2 H 0

1 2