Steady-state heat transfer in microcracked media

Abstract

Material behaviour is often affected by the heterogeneities existing at the microscopic level. Especially the presence of cracks, voids, etc collectively known as defects, can play a major role in their overall response. Homogenization can be used to study the influence of these heterogeneities and also to estimate the effective properties of a given material. Several research works have been dedicated to determining the elastic behaviour of microcracked media. Yet, thermal properties are not investigated as much. Moreover, the question of unilateral effect (opening/closing of cracks) still remains an important issue. So, this paper aims to provide the effective thermal conductivity of 2D microcracked media with arbitrarily orientated cracks, either open or closed. With the help of Eshelby-like approach, homogenization schemes (dilute and Mori-Tanaka) and bounds (Ponte Castañeda-Willis) are developed to provide the closed-form expressions. In addition, these results are compared to numerical simulations performed based on finite element modelling.

Thermal conductivity of crack-containing media: A numerical study

“…Also included in Figure 7 are analytical solutions which are used here for the purpose of model verification. Rangasamy Mahendren 25 applied Eshelby’s equivalent inclusion method 44 and treated cracks as the limit case of thin elliptical inclusions (in the same spirit as the current numerical work). They developed closed-form expressions of effective thermal conductivity for materials containing parallel cracks with arbitrary orientations relative to the nominal heat conduction direction, based on various homogenization approaches including dilute inhomogeneities, 25 Mori & Tanaka homogenization scheme, 45 and energy-based bounds from Ponte Castaneda and Willies.…”

“…The M&T and PC&W approaches take into consideration interactions between cracks, with PC&W also accounting for certain degrees of spatial distribution of cracks. 25 These expressions in tensor notation are rewritten below, in the form of simple algebraic equations using the crack size parameter a / L and the crack orientations under the xy -coordinate system considered in the present study.…”

“…[18][19][20] The theoretical framework has found extensive applications to composite systems, and is able to deliver high-fidelity results across a wide range of geometric features of the heterogeneity encompassing fibers and porosity. [21][22][23][24] As for cracks as a specific form of defects, there have been theoretical estimates of steadystate thermal conductivity of solids with various crack configurations, such as oriented cracks at various angles in two dimensions, 25 random penny-shaped cracks, 26,27 oriented elliptical cracks, 28 randomly distributed elliptical cracks, 29 cracks of arbitrary non-flat shape, 30 uniformly oriented penny-shaped cracks, 14,27,31 and ribbonshaped cracks. 32 While the analytical expressions offer useful closed forms and shed light on the general trend, they are less capable of providing sufficient insight into how local arrangement of cracks dictates the overall behavior.…”