A 3-dimensional singular kernel problem in viscoelasticity: An existence result

Sandra Carillo


Materials with memory, namely those materials whose mechanical and/or thermodynamical behavior depends on time not only via the present time, but also through its past history, are considered. Specifically, a three dimensional viscoelastic body is studied. Its mechanical behavior is described via an integro-differential equation, whose kernel represents the relaxation modulus, characteristic of the viscoelastic material under investigation. According to the classical model, to guarantee the thermodynamical compatibility of the model itself, such a kernel satisfies regularity conditions which include the integrability of its time derivative. To adapt the model to a wider class of materials, this condition is relaxed; that is, conversely to what is generally assumed, no integrability condition is imposed on the time derivative of the relaxation modulus. Hence, the case of a relaxation modulus which is unbounded at the initial time t = 0, is considered, that is a singular kernel integro-differential equation, is studied. In this framework, the existence of a weak solution is proved in the case of a three dimensional singular kernel initial boundary value problem.


materials with memory; viscoelasticity; unbounded relaxation modulus; singular kernel integro-differential equations.

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DOI: https://doi.org/10.1478/AAPP.97S1A3

Copyright (c) 2019 Sandra Carillo

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