New Third-Order Bounds on the Effective Moduli of n-Phase Composites (original) (raw)
Related papers
Geometrical-parameter bounds on the effective moduli of composites
Journal of the Mechanics and Physics of Solids, 1995
We study bounds on the effective conductivity and elastic moduli of two-phase isotropic composites that depend on geometrical parameters that take into account up to three-point statistical information concerning the composite microstructure. We summarize existing bounds, apply a special fractional linear transformation to simplify their functional forms, and describe two approaches to improve such bounds. These approaches allow us to get new and improved geometrical-parameter bounds on the elastic moduli of two-dimensional composites. Applications of the bounds for effective-medium geometries as well as random arrays of aligned fibers in a matrix are discussed.
Variational bounds on some bulk properties of a two-phase composite material
Physical Review B, 1976
A variational principle due to Hashin and Shtrikman is used to obtain theoretical upper and lower bounds on the effective bulk dielectric constant c, (or an analogous property such as magnetic permeability, electrical or thermal conductivity, or a diffusivity} of a two-phase macroscopically homogeneous composite material from information about another similar eA'ective bulk property. For the case of a composite whose macroscopic symmetry under rotations is either isotropic or cubic, we obtain i new and rather simple pair of bounds that are usually considerably better than any of those that are presently obtainable under these conditions.
Construction of Bounds on the Effective Shear Modulus of Isotropic Multicomponent Materials
Vietnam Journal of Mechanics, 2013
In our previous paper, we constructed bounds on the effective bulk modulus of isotropic multicomponent composites using minimum energy principles and modified Hashin-Shtrikman polarization trial fields. In this paper, following the variational approach, we construct more sophisticated bounds on the effective shear modulus. Applications to particular models are presented.
Phase-interchange relations for the elastic moduli of two-phase composites
International Journal of Engineering Science, 1996
For isotropic two-phase composites, we derive phase-interchange inequalities for the bulk and shear moduli in two and three space dimensions. We find optimal microstructures that realize part of the bulb: and shear moduli bounds in two dimensions. Geometrical-parameter bounds and the translation method are used to prove the results. The phase-interchange relations are applied to composites with cavities or a perfectly rigid phase, composites near the percolation threshold, incompressible composites, composites with equal bulk or shear phase moduli, and symmetric composites..
2011
Bounds are obtained on the volume fraction in a two-dimensional body containing two elastically isotropic materials with known bulk and shear moduli. These bounds use information about the average stress and strain fields, energy, determinant of the stress, and determinant of the displacement gradient, which can be determined from measurements of the traction and displacement at the boundary. The bounds are sharp if in each phase certain displacement field components are constant. The inequalities we obtain also directly give bounds on the possible (average stress, average strain) pairs in a two-phase, two-dimensional, periodic or statistically homogeneous composite
Bounds on effective dynamic properties of elastic composites
Journal of the Mechanics and Physics of Solids, 2013
We present general, computable, improvable, and rigorous bounds for the total energy of a finite heterogeneous volume element O or a periodically distributed unit cell of an elastic composite of any known distribution of inhomogeneities of any geometry and elasticity, undergoing a harmonic motion at a fixed frequency or supporting a singlefrequency Bloch-form elastic wave of a given wavevector. These bounds are rigorously valid for any consistent boundary conditions that produce in the finite sample or in the unit cell, either a common average strain or a common average momentum. No other restrictions are imposed. We do not assume statistical homogeneity or isotropy. Our approach is based on the Hashin-Shtrikman bounds in elastostatics, which have been shown to provide strict bounds for the overall elastic moduli commonly defined (or actually measured) using uniform boundary tractions and/or linear boundary displacements; i.e., boundary data corresponding to the overall uniform stress and/or uniform strain conditions. Here we present strict bounds for the dynamic frequencydependent constitutive parameters of the composite and give explicit expressions for a direct calculation of these bounds.
Effective Elastic Moduli of Composite Materials: Reduced Parameter Dependence
Applied Mechanics Reviews, 1997
In this paper, we focus on the effective elastic constants of composite materials and pay attention to the possibility of reducing the number of independent variables. Surprisingly, this important issue has hardly been explored before. In our analysis, we rely on a new result in plane elasticity due to , and use Dundurs constants (Dundurs, 1967(Dundurs, , 1969. As an example, we consider a result for the effective elastic moduli of a composite containing a dilute concentration of perfectly-bonded circular inclusions.
Multiphase composites with extremal bulk modulus
Journal of the Mechanics and Physics of Solids, 2000
This paper is devoted to the analytical and numerical study of isotropic elastic composites made of three or more isotropic phases. The ranges of their eective bulk and shear moduli are restricted by the Hashin±Shtrikman±Walpole (HSW) bounds. For twophase composites, these bounds are attainable, that is, there exist composites with extreme bulk and shear moduli. For multiphase composites, they may or may not be attainable depending on phase moduli and volume fractions. Sucient conditions of attainability of the bounds and various previously known and new types of optimal composites are described. Most of our new results are related to the two-dimensional problem. A numerical topology optimization procedure that solves the inverse homogenization problem is adopted and used to look for two-dimensional three-phase composites with a maximal eective bulk modulus. For the combination of parameters where the HSW bound is known to be attainable, new microstructures are found numerically that possess bulk moduli close to the bound. Moreover, new types of microstructures with bulk moduli close to the bound are found numerically for the situations where the aforementioned attainability conditions are not met. Based on the numerical results, several new types of structures that possess extremal bulk modulus are suggested and studied analytically. The bulk moduli of the new structures are either equal to the HSW bound or higher than the bulk modulus of any other known composite with the same phase moduli and volume fractions. It is proved that the HSW bound is attainable in a much wider range than it was
Three-point correlation bounds on the effective bulk modulus of isotropic multicomponent materials
Vietnam Journal of Mechanics, 2012
Three-point correlation bounds based on minimum energy principles are constructed to give estimates on the effective elastic bulk modulus of disordered multi-component materials. The constructed trial fields are extensions of Hashin-Shtrikman polarization ones used in our previous approach and lead to tighter bounds. Some examples of applications are presented.