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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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