The present work focuses on the problem of a rigid inhomogeneity of toroidal shape embedded in an elastic matrix. Inhomogeneities of this kind occur in both natural and man-made materials. Barium titanate nanotori are used as nonvolatile memory devices, transducers, optical modulators, sensors and possible energy storage in supercapacitors. Toroidal particles represent preferred morphology of Li2O2 deposition on porous carbon electrode in lithium-oxygen batteries. Polymeric “microdonuts” are used in bioengineering; toroidal shape of nanoparticles is preferred for microwave absorption properties of BaTiO3. Toroidal particles of SiO2 may form in a Cu matrix due to internal oxidation of a Cu-Si solid-solution polycrystal. Analytical modeling of materials with such microstructure has not been well developed. In the homogenization schemes, the inhomogeneities are usually assumed to be of ellipsoidal shape. This unrealistic assumption is responsible for insufficient linkage between micromechanics and materials science applications. While for 2D non-elliptical inhomogeneities many analytical and numerical results have been obtained, only a limited number of approximate estimates are available for non-ellipsoidal 3D shapes. Asymptotic methods have been used in [1] to evaluate the contribution of a thin rigid toroidal inhomogeneity into overall stiffness. Eshelby tensor for a toroidal inclusion has been also derived by Onata. However, Eshelby tensor for non-ellipsoidal inhomogeneities is irrelevant to the problem of effective properties of a heterogeneous material. The effective conductivity of a material containing toroidal insulating inhomogeneities has been addressed in [2]. We first consider a homogeneous elastic material, with isotropic stiffness tensor C0, containing a rigid inhomogeneity of volume V(1). The contribution of the inhomogeneity to the overall stress per representative volume V (the extra stress Δσ, as compared to the homogeneous matrix) is given by the fourth-rank stiffness contribution tensor N, defined by the following relation where ε ∞ is the remotely applied strain, n is the outward unit normal to the inhomogeneity surface S. To calculate the components of N, a displacement boundary value problem has been solved for 3D elastic space containing a rigid toroidal inhomogeneity.

Effective properties of composites containing toroidal inhomogeneities / Radi, Enrico; Sevostianov, Igor; Lanzoni, Luca. - (2018). (Intervento presentato al convegno XXII GIMC e IX Riunione GMA tenutosi a Ferrara nel 13-14 Settembre 2018).

### Effective properties of composites containing toroidal inhomogeneities

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*Radi Enrico;Sevostianov Igor;Lanzoni Luca*

##### 2018-01-01

#### Abstract

The present work focuses on the problem of a rigid inhomogeneity of toroidal shape embedded in an elastic matrix. Inhomogeneities of this kind occur in both natural and man-made materials. Barium titanate nanotori are used as nonvolatile memory devices, transducers, optical modulators, sensors and possible energy storage in supercapacitors. Toroidal particles represent preferred morphology of Li2O2 deposition on porous carbon electrode in lithium-oxygen batteries. Polymeric “microdonuts” are used in bioengineering; toroidal shape of nanoparticles is preferred for microwave absorption properties of BaTiO3. Toroidal particles of SiO2 may form in a Cu matrix due to internal oxidation of a Cu-Si solid-solution polycrystal. Analytical modeling of materials with such microstructure has not been well developed. In the homogenization schemes, the inhomogeneities are usually assumed to be of ellipsoidal shape. This unrealistic assumption is responsible for insufficient linkage between micromechanics and materials science applications. While for 2D non-elliptical inhomogeneities many analytical and numerical results have been obtained, only a limited number of approximate estimates are available for non-ellipsoidal 3D shapes. Asymptotic methods have been used in [1] to evaluate the contribution of a thin rigid toroidal inhomogeneity into overall stiffness. Eshelby tensor for a toroidal inclusion has been also derived by Onata. However, Eshelby tensor for non-ellipsoidal inhomogeneities is irrelevant to the problem of effective properties of a heterogeneous material. The effective conductivity of a material containing toroidal insulating inhomogeneities has been addressed in [2]. We first consider a homogeneous elastic material, with isotropic stiffness tensor C0, containing a rigid inhomogeneity of volume V(1). The contribution of the inhomogeneity to the overall stress per representative volume V (the extra stress Δσ, as compared to the homogeneous matrix) is given by the fourth-rank stiffness contribution tensor N, defined by the following relation where ε ∞ is the remotely applied strain, n is the outward unit normal to the inhomogeneity surface S. To calculate the components of N, a displacement boundary value problem has been solved for 3D elastic space containing a rigid toroidal inhomogeneity.##### Pubblicazioni consigliate

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