Replacement relations for thermal conductivities of heterogeneous materials having different matrices

doi:10.1016/j.mechmat.2018.03.003

Mechanics of Materials

Volume 121, June 2018, Pages 50-56

https://doi.org/10.1016/j.mechmat.2018.03.003 Get rights and content

Highlights

•
Replacement relations that link the overall thermal or electrical conductivities of heterogeneous materials having the same microstructure and inhomogeneities, but different matrix materials are derived. Relations of this kind have never been proposed in literature before.
•
These replacement relations can be used to determine conductivities of the small particles in matrix composites which cannot be evaluated by other methods. The corresponding equation is derived.
•
The relations are verified by comparison with FEA calculations and with experimental data available in literature.

Abstract

We derive explicit closed form relation between thermal (or electrical) conductivities of two matrix composite materials having the same microstructure and inhomogeneities, but different matrices. The relation is verified numerically and by comparison with experimental data available in literature. We also discuss a possible application of this replacement relation which allows one to evaluate conductive properties of particles in a matrix composite – the problem of crucial importance in geophysics, mechanics of nanocomposites, biomechanics, etc.

Introduction

In the present paper, we develop replacement relations for thermal (or electrical) conductivities of heterogeneous materials having the same microstructures formed by isolated inhomogeneities of the same properties, while the matrix materials are different. The theoretical derivation is validated by comparison with numerical and experimental data.

Usually, the replacement relations link overall properties of heterogeneous materials that have the same matrix material and microstructure while properties of the inclusions are different. First relations of this kind have been proposed by Gassmann (1951) in the context of the effect of saturation on seismic properties of rock in geomechanics (see Mavko et al. (2009) and Jaeger et al. (2007) for application of these relation). He proved that the bulk modulus of fully saturated rock in terms of the elastic properties of dry rock as $K_{e f f} = K_{e f f}^{d r y} + \frac{K_{0} {(1 - K_{e f f}^{d r y} / K_{0})}^{2}}{1 - ϕ - K_{e f f}^{d r y} / K_{0} + ϕ K_{0} / K_{1}}; μ_{e f f} = μ_{e f f}^{d r y}$ where subscripts “0”, “1”, and “eff” denote elastic constants of the matrix material, material filling the pores, and effective properties of a composite (material with saturated pores), respectively; ϕ is volume fraction of the inhomogeneities (porosity for the material with unfilled pores); $K_{e f f}^{d r y}$ and $μ_{e f f}^{d r y}$ are bulk and shear moduli of the porous material of the same morphology. Eq. (1.1) is known in geomechanics as Gassmann equation. Brown and Korringa (1975) generalized this equation to the case of anisotropic materials. Sevostianov and Kachanov (2007) derived relations that link the contributions of inhomogeneities having the same shape but different elastic constants, to the overall elastic properties. The relations are exact for ellipsoids and may be used as approximations for certain non-ellipsoidal shapes – the authors showed that relations can be used with satisfactory accuracy for a cube and for various 2-D shapes. Saxena and Mavko, 2014, Saxena and Mavko, 2015) discussed the impactful applications of Gassmann equation in geophysics, they highlighted the importance of these relation in evaluating the effective properties of a heterogeneous material from those of a porous material and the benefices of calculating effective properties using Eshelby tensor. Saxena and Mavko (2014), also derived replacement relations analytically (they use term “substitution relations”) under assumption that the strains and stresses inside the inhomogeneities are uniform (overall properties and properties of the constituents are isotropic). Note, however, that the assumption used in the derivation is equivalent to the statement that the inhomogeneities are ellipsoids subjected to the uniform external field (Eshelby, 1957, Lubarda and Markenscoff, 1998, Rodin, 1996) and do not interact with each other, due to that relations of Saxena and Mavko (2014) follow from the replacement relations of Sevostianov and Kachanov (2007).

In the context of thermal or electrical conductivity problem, the first result that can be interpreted as a replacement relation, has been derived by Keller (1964). He proved that, for a double-periodic 2-D two-phase material with isotropic constituents with conductivities k_A and k_B the following relation connects two principal effective conductivities $k_{11}^{e f f}$ and $k_{22}^{e f f}$ $k_{11}^{e f f} (k_{A}, k_{B}) k_{22}^{e f f} (k_{B}, k_{A}) = k_{A} k_{B}$ where the first argument denotes conductivity of the first phase (matrix in the case of the matrix composite) and the second - conductivity of the second phase (inhomogeneities in the case of the matrix composite). Schulgasser (1992) showed that this result is valid for any two phase material, not necessary periodic. In another work, Schulgasser (1976) showed that this result cannot be extended to 3-D materials. Zimmerman (1996) calls this result Keller–Schulgasser theorem. In particular, for isotropic two phase material (1.2) after some algebra yields $k_{e f f} (α k_{A}, k_{A}) k_{e f f} (k_{A} / α, k_{A}) = k_{A}^{2}$ or $k_{e f f} (k_{A}, α k_{A}) k_{e f f} (k_{A}, k_{A} / α) = k_{A}^{2}$ that represent certain replacement relations for either matrix material or material of inhomogeneities. In particular, (1.4) means that conductivities of a 2-D material with non-conductive and superconductive inhomogeneities, normalized to the conductivity of the matrix, are inverse to each other.

The problem of the replacement relation for materials with the same microstructures and matrix, but different inhomogeneities first attracted attention of experimentalists in the context of geomechanical applications. Schärli and Rybach (1984) compared air saturated and water saturated low-porosity granitic rocks and reported that thermal conductivity of water-saturated samples is 30% higher than “dry” conductivities. Zimmerman (1989) proposed the theoretical background for such observations. He evaluated thermal conductivity of air- and water- saturated rock using Fricke (1924) formula for electrical conductivity of a material containing randomly oriented spheroidal inhomogeneities. In particular, he proposed a methodology to evaluate thermal conductivity of water saturated rock from the dry rock measurements and validated the prediction on experimental data for sedimentary and granitic rock data. He stated that the procedure requires inversion of the equations and, while the required equations can be inverted in closed form, the final results can be obtained only numerically. Recently, Chen et al. (2017) derived replacement relations for conductivity, adopting approach of Sevostianov and Kachanov (2007). They showed, in particular, that the replacement relations can be applied with good accuracy for materials containing non-ellipsoidal inhomogeneities of convex shape while the error is significant for concave shapes. For this case, they suggested a modification involving an extra shape factor and experimentally verified their approach measuring thermal conductivity of sandstone saturated with water and kerosene. They also derived exact replacement inequalities based on Hashin–Shtrikman bounds (Hashin and Shtrikman, 1962).

Note that all the available results on replacement relations (excepting (1.3)) connect the effective properties (elastic, thermal, electric) of the materials having the same matrix and the same microstructure, while the inhomogeneities have different properties. To the best of our knowledge, three-dimensional replacement relations have never been derived for the heterogeneous materials that have different matrices. We do it in the present paper in the context of thermal conductivity. We focus on the case when properties of all the constituents as well as overall properties of the composites are isotropic.

Section snippets

Background material: property contribution tensors

Replacement relations for a single inhomogeneity in the context of elastic properties have been derived by Sevostianov and Kachanov (2007) using the concept of property contribution tensors (Horii and Nemat-Nasser, 1983). For the thermal (or electrical) conductivity problem, replacement relation of this kind was proposed by Chen et al. (2017). Our approach is also based on the property contribution tensors and, for reader convenience, we briefly outline this concept below.

For the thermal (or

Replacement relations for composites with different matrix materials

We focus now on the case when both matrix and inhomogeneity are isotropic and the inhomogeneity has ellipsoidal shape with semi-axes a₁, a₂, and a₃. In this case, tensor Q defined by (2.3) has the following form (see, for example, Fricke (1924)) $Q = k_{0} [(1 - I_{1}) e_{1} e_{1} + (1 - I_{2}) e_{2} e_{2} + (1 - I_{3}) e_{3} e_{3}] \equiv k_{0} \tilde{Q}$ where $I_{i} = \frac{a_{1} a_{2} a_{3}}{2} \int_{0}^{\infty} \frac{d s}{(a_{i}^{2} + x) Δ (s)}; Δ (s) = \sqrt{(a_{1}^{2} + s) (a_{2}^{2} + s) (a_{3}^{2} + s)} .$

Tensor $\tilde{Q}$ is completely determined by the shape of the inhomogeneity and is independent of the material properties. Similar statement can be made about

Numerical simulations

In this section, we provide numerical procedure for verification of replacement relation (3.7) using finite element analysis (FEA) of computer generated microstructures containing randomly distributed spherical inhomogeneities.

We start with microstructure generation using a custom script based on modified collective rearrangement method (Drach et al., 2016, Trofimov et al., 2017) modified by Trofimov et al. (2018). The procedure is valid for both regular and irregular shapes and results in

Numerical and experimental verification

In this section, we present results of verification of the replacement relations (3.8) using numerical and experimental results. For this goal, to avoid dependence on multiple parameters, we introduce two scaling parameters for (3.8) as follows $x \equiv \frac{1}{(r_{1} / r_{A}) - 1} - \frac{1}{(r_{1} / r_{B}) - 1} and$ $y \equiv φ [\frac{1}{(r_{e f f}^{A} / r_{A}) - 1} - \frac{1}{(r_{e f f}^{B} / r_{B}) - 1}] .$

The replacement relation (3.8) predicts that $y = x$ . Plotting y versus x for different numerical and experimental data, we may compare them on the same plot where the replacement relation gives the

Conclusions

We derived the replacement relations that link the overall thermal or electrical conductivities of heterogeneous materials having the same microstructure and inhomogeneities, but different matrix materials. Relations of this kind have never been proposed in literature before. The relations are verified by comparison with FEA calculations and with experimental data available in literature. The agreement is very good in the entire interval of variation of the volume fractions of the

Acknowledgment

Financial support from NASA Cooperative Agreement NNX15AK41A to New Mexico State University through NM Space Grant Consortium is gratefully acknowledged.

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Research paperReplacement relations for thermal conductivities of heterogeneous materials having different matrices

Highlights

Abstract

Introduction

Section snippets

Background material: property contribution tensors

Replacement relations for composites with different matrix materials

Numerical simulations

Numerical and experimental verification

Conclusions

Acknowledgment

Int. J. Solids Struct.

Int. J. Rock Mech. Min. Sci.

Int. J. Solids Struct.

J. Mech. Phys. Solids

Int. J. Solids Struct.

J. Mech. Phys. Solids

Tectonophysics

Int. Commun. Heat Mass Transf.

Int. J. Eng. Sci.

J. Mech. Phys. Solids

Int. J. Solids Struct.

J. Mech. Phys. Solids

Chem. Phys.

J. Pet. Sci. Eng.

Binary mixtures of water and glycerol - thermal conductivity of liquids

Ind. Eng. Chem.

On the dependence of the elastic properties of a porous rock on the compressibility of the pore fluid

Geophysics

Research paper
Replacement relations for thermal conductivities of heterogeneous materials having different matrices