Kurzweil integral representation of interacting Prandtl-Ishlinskii operators
2015
We consider a system of operator equations involving play and Prandtl-Ishlinskii hysteresis operators.
This system generalizes the classical mechanical models of elastoplasticity, friction and fatigue by introducing coupling
between the operators. We show that under quite general assumptions the coupled system is equivalent
to one effective Prandtl-Ishlinskii operator or, more precisely, to
a discontinuous extension of the Prandtl-Ishlinskii operator based on the Kurzweil integral of the derivative
of the state function. This effective operator is described constructively in terms of the parameters
of the coupled system. Our result is based on a substitution formula which we prove for the Kurzweil integral of regulated functions integrated with respect to functions of bounded variation.
This formula allows us to prove the composition rule for the generalized (discontinuous) Prandtl-Ishlinskii
operators. The composition rule, which underpins the analysis of the
coupled model, then establishes that
a composition of generalized Prandtl-Ishlinskii operators is also a generalized Prandtl-Ishlinskii operator provided that a monotonicity condition is satisfied.
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