Effective elastic moduli of periodic granular composite with transversely isotropic phases

Abstract

We find a rigorous solution describing the macroscopically uniform stress state of a periodic granular composite with transversely isotropic phases. The structure of the composite is modeled by a cube containing a finite number of arbitrarily arranged and oriented, transversely isotropic spherical inclusions. This provides the model with a flexible means of describing the microstructure. Applying periodic vector solutions and local expansion formulas reduces the initial boundary-value problem to a system of linear algebraic equations. By averaging the solution over the unit cell, we derived exact finite expressions for the components of the effective stiffness tensor. The numerical data presented help to evaluate the efficiency of the method and the limits of applicability of available approximate theories.Keywords: periodic granular composite, transversely isotropic elastic phases, periodic structural model, spherical inclusion, effective stiffness tensor 1. Introduction. The best known approaches to predicting the macroscopic properties of structurally inhomogeneous materials are, probably, the stochastic approach based on methods of statistical mechanics [5][6][7][8] and the regularization method [1,2,4,9,12]. The latter consists in modeling a real composite by a periodic structure and solving boundary-value problems, followed by averaging of local fields. Both approaches are well developed for periodic granular composites with isotropic phases; for a review on the subject see [2,4,5]. The statistical theory of deformation and damage of composites with anisotropic phases is developed in [6-8]. All well-known applications of the regularization method are restricted to fibrous materials alone [1,4].In the present paper, we will find a rigorous, asymptotically exact solution describing the elastic equilibrium of a periodic composite with spherical inclusions, assuming transverse isotropy of the phases. The unit cell is a cube containing a finite number of arbitrarily arranged and oriented, transversely isotropic spherical inclusions. This provides the model with a general and flexible means of describing the microstructure [2]. To derive the solution, we will employ the earlier results for a transversely isotropic body with one [10, 11] and several [3] inclusions and a fast-summation method for triple series [12].2. Composite Model. The generalized periodic structural model of a granular composite [2] is an infinite matrix containing periodically arranged spherical inclusions. This model has a unit cell in the form of a cube with side length a containing N spherical inclusions of radius R q centered at points Q q N q , , ,... , = 12 . This cell, when repeated in three mutually perpendicular directions, produces the entire structure of the composite. Both the matrix and inclusions are assumed elastic and

Fundamental solutions of equilibrium equations for gradient model describing the deformation of two‐component elastic composites

“…This is because of the widespread use of composites in various industries and necessity of developing reliable methods for their design and strength prediction. Such investigations are based on the various mathematical models that are conducted at different structural levels [1][2][3][4][5][6][7].…”

Gradient model in the problem of stress-concentration around a circular hole in two-component stochastic composites

“…Wide application of composite materials in various fields of modern engineering and construction calls for comprehensive analysis of problems associated with the selection of components and structure of materials, modeling and prediction of their mechanical behavior and properties, and calculation of the load-bearing capacity of structures. Such studies are based on certain mathematical models that allow for specific features of materials and describe the behavior of loads at the macro, micro, and nano-levels [1,3,5,10,11,16,17,19,20]. One of the first and the most developed and widely used continuum theories in the mechanics of composites is the theory of effective moduli, which deals with dynamic and kinematic parameters averaged over elementary macrovolumes and macroareas.…”

“…5) where c n is the volume fraction of the nth component (n = 1, 2); angular brackets denote statistical averaging within an elementary volume over the component indicated by the superscript; and á ¢ ñ = á ñ -the difference of the component-average body forces.The composite-average strains and body forces can be calculated from the relations…”