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A dynamic integro-differential operator of variable order is suggested for a more adequate description of processes, which involve state dependent measures of elastic and inelastic material features. For any negative constant order this operator coincides with the well-known operator of fractional integration. The suggested operator is especially effective in cases with strong dependence of the behavior of the material on its present state—i.e., with pronounced nonlinearity. Its efficiency is demonstrated for cases of viscoelastic and elastoplastic spherical indentation into such materials (aluminum, vinyl) and into an elastic material (steel) used as a reference. Peculiarities in the behavior of the order function are observed in these applications, demonstrating the “physicality” of this function which characterizes the material state. Mathematical generalization of the fractional-order integration-differentiation in the sense of variability of the operator order, as well as definitions and techniques, are discussed. [S0021-8936(00)02102-4]

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Organization of product tests in the microelectronic and optical-electronic industries by the impact method is considered. Two mathematical models: a circular plate under dynamic loading with predetermined initial conditions, and contact interaction of a falling mass and a barrier, are examined. The deflection function, curvature, and acceleration are determined. As the series for these characteristics may prove divergent, the problem is to secure sufficiently reliable results. The method of impact mechanics permits determination of the duration and force of the impact, estimation of the total energy of each mode in the expansion of the deflection function and preclusion of divergent series when determining the plate acceleration and curvature during the vibration process. In parallel, viscoelasticity is simulated with the aid of a fractional-differentiation operator, certain features of which are discussed. Representation of this operator with time-dependent order by ones with constant orders is considered. Three alternative approaches for determination of the eigenfrequency and damping decrement of a vibration process are examined. In particular, a method for calculating these characteristics under conditions of time-dependence and servo control of the order function is proposed.