SIPS2022 Volume 12 Trovalusci Intl. Symp Multiscale & Multiphysics Modelling of Complex Materials

Editors: | F. Kongoli, E. Aifantis, R. Das, V.Eremeyev, N. Fantuzzi. |

Publisher: | Flogen Star OUTREACH |

Publication Year: | 2022 |

Pages: | 116 pages |

ISBN: | 978-1-989820-56-8(CD) |

ISSN: | 2291-1227 (Metals and Materials Processing in a Clean Environment Series) |

Here we apply our general method [1] for providing rigorous asymptotic mathematical models to the case of a structure made of a thin linearly thermo-elastic adhesive layer connecting two linearly thermo-elastic bodies. The principle is to consider the geometrical and physical data as parameters and to rigorously study the asymptotic behavior of the structure when the parameters go to their natural limits. We will provide 5x5 asymptotic models depending on 5 possible relative behaviors for the stiffness and for the thermal conductivity with respect to the thickness of the adhesive layer. From the physical and mathematical points of view, thermo-elasticity is interesting in asymptotic modeling because it involves coupled transient phenomena which, at the limit, may induce a change in the nature of the constitutive equations! More precisely, we are facing a coupling between a hyperbolic equation (the motion equation) and a parabolic one (the heat equation). Our strategy is to formulate the problem in terms of a sequence of evolution equations set in Hilbert spaces of possible states with finite thermomechanical energy governed by m-dissipative operators. According to Trotter theory [1,2] it suffices to study the limit of the associated static problems. In certain cases (low stiffness and conductivity) state variables additional to the traces of the displacement and temperature of the adherents do appear in order to describe the state of the surface the layer shrinks to. By keeping these additional state variables, the structure of the limit equations for the surface remains as those of the layer. The 25 models are very different ranging from thermomechanical constraints to material thermo-elastic surfaces with constitutive equations strongly depending on the relative behaviors of the thermo-mechanical parameters with respect to the thickness of the layer. This study improves [3] and may be considered as a framework to assess the partial and formal study obtained through asymptotic expansion [4] devoted to poro-elasticity...

[2] H. F. TROTTER, Pacific J. Math., vol.8 (1958), 887-919.

[3] C. LICHT, A. OULD KHAOUA, T. WELLER, C. R. Mecanique, 342 (2015) 18-26.

[4] M. SERPILLI, Annals of Solids Structural Mechanics, 11 (2019) 1-10.