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Jump conditions in stress relaxation and creep experiments of Burgers type fluids: a study in the application of Colombeau algebra of generalized functions

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Abstract

We discuss stress relaxation and creep experiments of fluids that are generalizations of the classical model due to Burgers by allowing the material moduli such as the viscosities and relaxation and retardation times to depend on the stress. The physical problem, which is cast within the context of one dimension, leads to an ordinary differential equation that involves nonlinear terms like product of a function with a jump discontinuity and the derivative of a function with a jump discontinuity. As the equations are nonlinear, standard techniques that are used to study problems concerning linear viscoelastic fluids such as Laplace transforms and the theory of distributions are not applicable. We find it necessary to seek the solution in a more general setting. We discuss the mathematical and physical issues concerning the jump discontinuities and nonlinearity of the governing equation, and we show that the solution to the governing equation can be found in the sense of the generalized functions introduced by Colombeau. In the framework of Colombeau algebra we, under certain assumptions, derive jump conditions that shall be used in stress relaxation and creep experiments of fluids of the Burgers type. We conclude the paper with a discussion of the physical relevance of these assumptions.

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, Praha 8–Karlín, CZ, 186 75, Czech Republic

    Vít Průša

  2. Department of Mechanical Engineering, Texas A&M University, 3123 TAMU, College Station, TX, 77843-3123, United States of America

    K. R. Rajagopal

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  1. Vít Průša
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Correspondence to K. R. Rajagopal.

Additional information

Vít Průša thanks the Nečas Center for Mathematical Modeling (project LC06052 financed by the MŠMT of the Czech republic) for its support.

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Průša, V., Rajagopal, K.R. Jump conditions in stress relaxation and creep experiments of Burgers type fluids: a study in the application of Colombeau algebra of generalized functions. Z. Angew. Math. Phys. 62, 707–740 (2011). https://doi.org/10.1007/s00033-010-0109-9

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