Abstract
An analytical model is developed for the conductivity (diffusivity, permeability, etc.) of a material that contains a dispersion of spherical inclusions, each surrounded by an inhomogeneous interphase zone in which the conductivity varies radially according to a power law. The method of Frobenius series is used to obtain an exact solution for the problem of a single such inclusion in an infinite matrix. Two versions of the solution are developed, one of which is more computationally convenient for interphase zones that are less conductive than the pure matrix, and vice versa. Maxwell’s homogenization method is then used to estimate the effective macroscopic conductivity of the medium. The developed model is used to analyze some data from the literature on the ionic diffusivity of concrete. Use of the model in an inverse mode permits the estimation of the local diffusivity variation within the interphase, and in particular at the interface with the inclusion.
Highlights
Most modeling of the behavior of composite materials is carried out under the assumption that the “matrix” and “inclusions” are both homogenous, and that there is a clearly defined interface between them
A binding agent is sometimes applied to the fibers in a polymer composite, so as to promote adhesion between the fiber and the matrix (Drzal, Rich, Koenig, & Lloyd, 1983). This binding agent diffuses into the matrix during the curing process, leading to a gradient in resin concentration, which in turn may lead to gradients in physical properties within the so-called “interphase zone”
If kif/km > 46.2, the conductivity enhancement of the interfacial transition zone (ITZ) overshadows the effect of the non-conductive inclusions, causing keff to exceed km, and to increase as the volume fraction of inclusions increases. (This critical value of kif/km = 46.2 varies with β, and increases as β increases, since larger β values correspond to thinner interphases, and a thin interphase would need to be more conductive to overshadow the effect of the non-conductive inclusion.) For very high values of kif/km, the effect of the thin super-conducting ITZ renders the nonconductive inclusions irrelevant, and the normalized conductivity approaches the classical Maxwell result for super-conductive spherical inclusions in a homogeneous matrix, (1 + 2c)/(1 − c)
Summary
Most modeling of the behavior of composite materials is carried out under the assumption that the “matrix” and “inclusions” are both homogenous, and that there is a clearly defined interface between them. Zimmerman / International Journal of Engineering Science 98 (2016) 51–59 located in a homogeneous matrix, and subjected to uniform far-field stress This model of an inclusion composed of thin concentric layers, each having its own set of physical properties, has been widely used since for both the calculation of effective elastic (i.e., Hervé, 2002) and conductive (i.e., Caré & Hervé, 2004) properties. The case in which the conductivity and the thickness of the interphase each go to zero, while their ratio approaches a finite value, leads to an interface condition in which the temperature undergoes a jump discontinuity This type of model has been used in conductivity problems by, for example, Cheng and Torquato (1997), Benveniste and Miloh (1999), Hashin (2001), and Benveniste (2012), among others. The developed model will be used to analyze some data from the literature, on the ionic diffusivity of concrete
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