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Item type:Publication, Access status: without files , Closed-form formulas for the effective properties of random particulate nanocomposites with complete Gurtin–Murdoch model of material surfaces(2017-01-01); ; The objective of this work is to present an approach allowing for inclusion of the complete Gurtin–Murdoch material surface equations in methods leading to closed-form formulas defining effective properties of particle-reinforced nanocomposites. Considering that all previous developments of the closed-form formulas for effective properties employ only some parts of the Gurtin–Murdoch model, its complete inclusion constitutes the main focus of this work. To this end, the recently introduced new notion of the energy-equivalent inhomogeneity is generalized to precisely include all terms of the model. The crucial aspect of that generalization is the identification of the energy associated with the last term of the Gurtin–Murdoch equation, i.e., with the surface gradient of displacements. With the help of that definition, the real nanoparticle and its surface possessing its own distinct elastic properties and residual stresses are replaced by an energy-equivalent inhomogeneity with properties incorporating all surface effects. Such equivalent inhomogeneity can then be used in combination with any existing homogenization method. In this work, the method of conditional moments is used to analyze composites with randomly dispersed spherical nanoparticles. Closed-form expressions for effective moduli are derived for both bulk and shear moduli. As numerical examples, nanoporous aluminum is investigated. The normalized bulk and shear moduli of nanoporous aluminum as a function of residual stresses are analyzed and evaluated in the context of other theoretical predictions.Publicationtype: Journal Article114 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Lurie solution for spherical particle and spring layer model of interphases : its application in analysis of effective properties of compositesA new approach to the determination of equivalent inhomogeneity for spherical particles and the spring layer model of their interphases with the matrix material is developed. To validate this approach the effective properties of random composites containing spherical inhomogeneities surrounded by an interphase material of constant thickness are evaluated. The properties of equivalent inhomogeneity, incorporating only properties of the original inhomogeneity and its interphase, are determined employing a new approach based on the exact Lurie's solution for spheres. This constitutes the central aspect of the proposed approach being in contrast with some existing definitions of equivalent inhomogeneity whose properties dependent also on the properties of the matrix. With the equivalent inhomogeneity specified as proposed here, the effective properties of the material with interphases can be found using any method applicable to analysis of the materials with perfect interfaces (i.e., without interphases) and any properties of the matrix. In this work, the method of conditional moments is employed to this end. The choice of that method is motivated by the method's solid formal foundations, its potential applicability to inhomogeneities other than spheres and to anisotropic materials. The resulting effective properties of materials with randomly distributed spherical particles are presented in the closed-form and are in excellent agreement with values reported in technical literature, which are based on both formally exact and approximate methods.Publicationtype: Journal ArticleScopus© Citations 19 209 323 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Energy-equivalent inhomogeneity approach to analysis of effective properties of nanomaterials with stochastic structure(2015-05-01); ; A mathematical model based on the method of conditional moments combined with a new notion of the energy-equivalent inhomogeneity is presented and applied in the investigation of the effective properties of a material with randomly distributed nanoparticles. The surface effect is introduced via Gurtin-Murdoch equations describing the properties of the matrix/nanoparticle interface. The real system, consisting of the inhomogeneities and their surfaces possessing different properties and, possibly, residual stresses, is replaced by energy-equivalent inhomogeneities with modified bulk properties which incorporate the surface effects. The effective stiffness tensor of the material with so defined equivalent inhomogeneities is determined by the method of conditional moments. Closed-form expressions for the effective moduli of a composite consisting of a matrix and randomly distributed spherical inhomogeneities are derived for both the bulk and the shear moduli. Dependence of those moduli on the radius of nanoparticles is included in these expressions exhibiting analytically the nature of the size-dependence in nanomaterials. As numerical examples, nanoporous aluminum and nanoporous gold are investigated. The dependence of the normalized bulk and shear moduli of nanoporous aluminum (for two sets of surface properties) on the pore volume fraction (for different radii of nanopores) and on the radius of nanopores (for fixed volume fraction of nanopores) are compared to and discussed in the context of other theoretical predictions. Further, the normalized effective Young's modulus of nanoporous gold as a function of void volume fraction for various ligament radii is analyzed.Publicationtype: Journal Article89 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Influence of interfaces on effective properties of nanomaterials with stochastically distributed spherical inclusionsThe method of conditional moments is generalized to include evaluation of the effective elastic properties of particulate nanomaterials and to investigate the size effect in those materials. Determining the effective constants necessitates finding a stochastically averaged solution to the fundamental equations of linear elasticity coupled with surface/interface conditions (Gurtin-Murdoch model). To obtain such a solution the system of governing stochastic differential equations is first transformed to an equivalent system of stochastic integral equations. Using statistical averaging, the boundary-value problem is then converted to an infinite system of linear algebraic equations. A two-point approximation is considered and the stress fluctuations within the inclusions are neglected in order to obtain a finite system of algebraic equations in terms of component-average strains. Closed-form expressions are derived for the effective moduli of a composite consisting of a matrix and randomly distributed spherical inhomogeneities. As a numerical example a nanoporous material is investigated assuming a model in which the interface effects influence only the bulk modulus of the material. In that model the resulting shear modulus is the same as for the material without surface effects. Dependence of the bulk moduli on the radius of nanopores and on the pore volume fraction is analyzed. The results are compared to, and discussed in the context of other theoretical predictions.Publicationtype: Journal Article37