Dynamic Behavior of Concrete: Constitutive Models - Dynamic Behavior of Concrete and Seismic Engineering - page 66

R is the slope of the straight line linking the material's representative points
before and after the shock. We notice that there may be two different situations; if
the shock is not too violent (Figure 2.8(i)), the shock wave will be preceded by an
elastic precursor, as the shock front velocity is lower than that of the elastic waves.
If the shock is very violent (Figure 2.8(ii)), its velocity can be higher than that of the
elastic waves, and there is no longer a precursor.

V 1

(ii)

(i)

H 1

Figure 2.8. Strain-stress relationship and stress levels involving a shock wave,
with or without precursor (i and ii respectively)

2.2.2. Impact and shock polar curve

2.2.2.1. Conservation equations

A shock wave is a non-stationary speed discontinuity that is necessarily
associated with a stress discontinuity. D is the propagation speed of the discontinuity
surface. On a specific point of the discontinuity surface (of normal line n ), the mass
and momentum conservation equations take on a particular shape (the brackets
represent the jump of the considered value when passing the discontinuity surface):

Mass conservation a U vDn b 0 [2.15]

Momentum conservation ab ab V n U VVDn [2.16]

It is interesting to write these equations in the case of a one-dimensional medium
that is at rest after the shock (Figure 2.9). The equations are: