Analytical aspects in strain gradient theory for chiral Cosserat thermoelastic materials within three Green-Naghdi models

Abstract

This work is motivated by the recent interest in using strain gradient theory to model the chiral behavior of elastic materials. In this paper, we derive a linear strain gradient theory for Cosserat thermoelastic materials according to the three models (types I, II and III) of Green-Naghdi theory. Models II and III permit propagation of thermal waves at finite speeds, while model I coincides with the classical Fourier’s law. The thermal field is influenced by the displacement and the microrotation fields and by some additional parameters that describe the chiral behavior. We prove the well-posedness for the three models and the asymptotic behavior for models I and III by the semigroup theory of linear operators.

Keywords:

• Asymptotic behavior
• chiral cosserat elasticity
• Green-Naghdi
• strain gradient
• well-posedness

Acknowledgments

Part of this work was done when the first author visited the Dipartimento di Ingegneria Civile in February 2018 and the Dipartimento di Matematica in April 2018 and July 2018, Università degli Studi di Salerno. He thanks them for their hospitality.

Notes

1 This condition holds only for type-III problem and the inequality sign is a consequence of the Second Law of Thermodynamics, which requires the non-negativeness of the functional ξ (see [4]) and of our choice (45)₃

Additional information

Funding

The first author thanks both Dipartimento di Ingegneria Civile and Dipartimento di Matematica, Università degli Studi di Salerno for their financial support through FARB 2016, 2017