Asymptotic analysis of heat transfer in composite materials with nonlinear thermal properties

Abstract We study heat transfer through a composite with periodic microstructure. The thermal conductivity of the constituents is assumed to be temperature-dependent, and it is modeled as a polynomial in terms of the temperature. The thermal resistance between the constituents is taken to be nonlinear. In order to determine the effective thermal properties of the material, we apply the asymptotic homogenization method. We discuss different approaches to determine these effective properties for the different volume fractions of the inclusions. For high volume fractions of the inclusion, we apply the lubrication theory. In the case of low volume fractions of the inclusions, we apply the three-phase model. Comparing some special cases of our results to existing ones in the literature shows a good accuracy

Influence of geometric and physical nonlinearities on the internal resonances of a finite continuous rod with a microstructure

In this work, nonlinear longitudinal vibrations of a finite composite rod are studied including geometric and physical nonlinearities. An original boundary value problem for a heterogeneous rod yielded by the macroscopic approximation obtained earlier by the higher-order asymptotic homogenization method is used. The effects of internal resonances and modes coupling are predicted, validated and analyzed. The defined novel continuous problem governed by PDEs is solved using space-discretization and the method of multiple time scales. We are aimed at understanding and analyzing how the presence of the microstructure influences the processes of mode interaction. It is shown that, depending on a scaling relation between the amplitude of the vibrations and the size of the unit cell, different scenarios of the modes coupling can be realized. Additionally to the asymptotic solution, numerical simulation of the modes coupling is performed by means of the Runge-Kutta fourth-order method. The obtained numerical and analytical results demonstrate good qualitative agreement

Internal Resonances And Modes Interactions In Non-linear Vibrations Of Viscoelastic Heterogeneous Solids

The aim of the paper is to study how viscous damping influences mode coupling in non-linear vibrations of microstructured solids. As an illustrative example, natural longitudinal vibrations of a layered heterogeneous medium are considered. The macroscopic dynamic equation is obtained by asymptotic homogenisation. The input continuous problem is analysed using a spatial discretisation procedure. An asymptotic solution is developed by the method of multiple time scales and the fourth-order Runge-Kutta method is employed for numerical simulations. Internal resonances and energy transfers between the vibrating modes are predicted and analysed. The conditions for possible truncation of the original infinite system are discussed. The obtained numerical and analytical results are in good agreement

Shear wave propagation in layered composites with degraded matrices at locations of imperfect bonding

Biodegradable polymers find an increasing number of applications in different fields of engineering and medicine due to their environmental-friendly degradation. The process of degradation of biodegradable polymer constituents and the bonding quality between the constituents in composites can be identified by the analysis of the phononic band structure. The present article considers a layered composite, in which the matrix degradation is modeled by a multitude of layers with decreasing values of their mechanical properties. Bonding between the inclusion and the degrading matrix is taken into account by a linear elastic bonding model in the first case and by a viscoelastic model in the second case