CONSTITUTIVE EQUATIONS FOR SOLID PROPELLANTS

Cyclic Eect
In order to represent cyclic eects, i. e. the rapid decrease of stress during unloading, the large
amount ofhysteresis in load/unload cycles, and the dependence of response on the maximum strain
previously attained, we introduce another internal state variable as an argument ofd and postulate
the form
d(s i )=g(s 1 )f(s 2 ) (12)
where g is the damage function with argument s 1 = c max as introduced above. The function f
represents the eect of cyclic loading and is allowed to dier for the situations of unloading and
reloading.
A reasonable choice for the argument of function f representing the joining of the reloading
curve to the original loading curve beyond the previous maximum strain is
s 2 =
I
I max
(13)
where I max
represents the maximum I previously achieved during the loading history. During
loading s 2 is unity. Its value decreases during unloading, and increases during reloading. For
the latter we allow s 2 > 1, since we consider a given state as reloading until it rejoins the original
loading curve. In this manner we account for the fact that reloading curve joins the original loading
curve gradually.
In order to implement the eect of cyclic loading into the model we apply the following criterion
to distinguish between rst loading, on the one hand, and unloading/reloading, on the other
_c(t) < 0 unloading ) _ f = _ f u
_c(t) 0 and f(t) < 1 reloading ) _ f = _ f r
_c(t) 0 and f(t) =1 loading ) _ f =0 (14)
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