  This paper develops an analytic framework to design both stress-controlled
and displacement-controlled T-periodic loadings which make the quasistatic
evolution of a one-dimensional network of elastoplastic springs converging to a
unique periodic regime. The solution of such an evolution problem is a function
t-> (e(t),p(t)), where e_i(t) and p_i(t) are the elastic and plastic
deformations of spring i, defined on [t0,\infty) by the initial condition
(e(t0),p(t0)).
  After we rigorously convert the problem into a Moreau sweeping process with a
moving polyhedron C(t) in a vector space E of dimension d, it becomes natural
to expect (based on a result by Krejci) that the solution t->(e(t),p(t)) always
converges to a T-periodic function. The achievement of this paper is in
spotting a class of loadings where the Krejci's limit doesn't depend on the
initial condition (e(t0),p(t0)) and so all the trajectories approach the same
T-periodic regime. The proposed class of sweeping processes is the one for
which the normal vectors of any d different facets of the moving polyhedron
C(t) are linearly independent. We further link this geometric condition to
mechanical properties of the given network of springs. We discover that the
normal vectors of any d different facets of the moving polyhedron C(t) are
linearly independent, if the number of displacement-controlled loadings is two
less the number of nodes of the given network of springs and when the magnitude
of the stress-controlled loading is sufficiently large (but admissible). The
result can be viewed as an analogue of the high-gain control method for
elastoplastic systems. In continuum theory of plasticity, the respective result
is known as Frederick-Armstrong theorem.
