Dynamic Behavior of Concrete: Constitutive Models - Dynamic Behavior of Concrete and Seismic Engineering - page 65

1
12

A v

vB

V [2.11]

1

Note that this threshold only exists if A ! 0 and B 1 - 2v . Beyond that
threshold, the lateral stress equals:

1
1

B A

V V [2.12]

2 1

B 1 B

If we retain the hypothesis of an incompressible plastic flow, which in that case
corresponds to a non-associated flow:

31 KB 21 AB

V H [2.13]

1 1

3 B 1 B 3 B

In this case, we can observe that the conclusion considering the von Mises
criterion established above is no longer valid. The volume-pressure relationship
cannot be derived through a simple translation from the oedometric path strain-stress
relationship. Nevertheless, if the deviatoric behavior is well known, the deduction is
still possible with the hypothesis of an incompressible plastic flow.

Here we have been considering perfect plasticity models. Yet, whenever there is
strain hardening, the plastic flow is either dilating or contracting, and although it is
not impossible, expressing the volume-pressure relationship from the strain-stress
relationship can be quite difficult.

2.2.1.2. Creation of a shock wave

A shock wave will be created during an impact if the impact speed is sufficient
for the stress and strain levels to be located within the “positive curvature” part of
the axial stress-axial strain curve. The remark means that a shock wave is possible
in uniaxial strain, but is impossible in uniaxial stress. The stress level is located in
the upper part of Hugoniot's curve, as shown in Figure 2.8. In such conditions, a
shock wave is created with a velocity of:

R

v [2.14]

U