Optimal lower bounds on the hydrostatic stress amplication inside random two-phase elastic composites (original) (raw)
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Journal of the Mechanics and Physics of Solids, 2005
Composites made from two linear isotropic elastic materials are subjected to a uniform hydrostatic stress. It is assumed that only the volume fraction of each elastic material is known. Lower bounds on all r th moments of the hydrostatic stress field inside each phase are obtained for r ≥ 2. A lower bound on the maximum value of the hydrostatic stress field is also obtained. These bounds are given by explicit formulas depending on the volume fractions of the constituent materials and their elastic moduli. All of these bounds are shown to be the best possible as they are attained by the hydrostatic stress field inside the Hashin-Shtrikman coated sphere assemblage. The bounds provide a new opportunity for the assessment of load transfer between macroscopic and microscopic scales for statistically defined microstructures.
Mechanics of Materials, 2006
Composites made from two linear isotropic elastic materials are considered. It is assumed that only the volume fraction of each elastic material is known. The composite is subjected to a uniform hydrostatic strain. For this case lower bounds on all r th moments of the dilatational strain field inside each phase are obtained for r ≥ 2. A lower bound on the maximum value of the dilatational strain field is also obtained. These bounds are given in terms of the volume fractions of the component materials. All of these bounds are shown to be the best possible as they are attained by the dilatational strain field inside the Hashin-Shtrikman coated sphere assemblage. The bounds provide a new opportunity for the assessment of the local dilatational strain in terms of a statistical description of the microstructure.
Optimal Lower Bounds on Local Stress inside Random Media
Siam Journal on Applied Mathematics, 2009
A methodology is presented for bounding all higher moments of the local hydrostatic stress field inside random two phase linear thermoelastic media undergoing macroscopic thermomechanical loading. The method also provides a lower bound on the maximum local stress. Explicit formulae for the optimal lower bounds are found that are expressed in terms of the applied macroscopic thermal and mechanical loading, coefficients of thermal expansion, elastic properties, and volume fractions. These bounds provide a means to measure load transfer across length scales relating the excursions of the local fields to the applied loads and the thermal stresses inside each phase. These bounds are shown to be the best possible in that they are attained by the Hashin-Shtrikman coated sphere assemblage.
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
Acta Mechanica, 2008
A micromechanical analytical framework is presented to predict effective elastic moduli of threephase composites containing many randomly dispersed and pairwisely interacting spherical particles. Specifically, the two inhomogeneity phases feature distinct elastic properties. A higher-order structure is proposed based on the probabilistic spatial distribution of spherical particles, the pairwise particle interactions, and the ensemble-volume homogenization method. Two non-equivalent formulations are considered in detail to derive effective elastic moduli with heterogeneous inclusions. As a special case, the effective shear modulus for an incompressible matrix containing randomly dispersed and identical rigid spheres is derived. It is demonstrated that a significant improvement in the singular problem and accuracy is achieved by employing the proposed methodology. Comparisons among our theoretical predictions, available experimental data, and other analytical predictions are rendered. Moreover, numerical examples are implemented to illustrate the potential of the present method.
Scale and boundary conditions effects in elastic properties of random composites
Acta Mechanica, 2001
We study elastic anti-plane responses of unidirectional fiber-matrix composites. The fibers are of circular cylinder shape, aligned in the axial direction, and arranged randomly, with no overlap, in the transverse plane. We assume that both fibers and matrix are linear elastic and isotropic. In particular, we focus on the effects of scale of observation and boundary conditions on the overall anti-plane (axial shear) elastic moduli. We conduct this analysis numerically, using a two-dimensional square spring network, at the mesoscale level. More specifically, we consider finite "windows of observation", which we increase in size. We subject these regions to several different boundary conditions: displacement-controlled, traction-controlled, periodic, and mixed (combination of any of the first three) to evaluate the mesoscale moduli. The first two boundary conditions give us scale-dependent bounds on the anti-plane elastic moduli. For each boundary condition case we consider many realizations of the random composite to obtain statistics. In this parametric study we cover a very wide range of stiffness ratios ranging from composites with very soft inclusions (approximating holes) to those with very stiff inclusions (approaching rigid fibers), all at several volume fractions.
Elastic property of multiphase composites with random microstructures
Journal of Computational Physics, 2009
We propose a computational method with no ad hoc empirical parameters to determine the elastic properties of multiphase composites of complex geometries by numerically solving the stress-strain relationships in heterogeneous materials. First the random microstructure of the multiphase composites is reproduced in our model by the random generation-growth method. Then a high-efficiency lattice Boltzmann method is employed to solve the governing equation on the multiphase microstructures. After validated against a few standard solutions for simple geometries, the present method is used to predict the effective elastic properties of real multiphase composites. The comparisons between the predictions and the existing experimental data have shown that the effects of pores/ voids in composites are not negligible despite their seemingly tiny amounts. Ignorance of such effects will lead to over-predictions of the effective elastic properties compared with the experimental measurements. When the pores are taken into account and treated as a separate phase, the predicted Young's modulus, shear modulus and Poisson's ratio agree well with the available experimental data. The present method provides an alternative tool for analysis, design and optimization of multiphase composite materials. Published by Elsevier Inc.
Bounds on the distribution of extreme values for the stress in composite materials
Journal of the Mechanics and Physics of Solids, 2004
Suitable macroscopic quantities beyond e ective elastic properties are used to assess the distribution of stress within a composite. The composite is composed of N anisotropic linearly elastic materials and the length scale of the microstructure relative to the loading is denoted by. The stress distribution function inside the composite (t) gives the volume of the set where the norm of the stress exceeds the value t. The analysis focuses on the case when 0 ¡ 1. A rigorous upper bound on lim →0 (t) is found. The bound is given in terms of a macroscopic quantity called the macro stress modulation function. It is used to provide a rigorous assessment of the volume of over stressed regions near stress concentrators generated by reentrant corners or by an abrupt change of boundary loading.
Stress concentration in incompressible multicomponent materials
International Applied Mechanics, 2000
A method for determining the effective elastic constants and the factors of stress concentration in microstructural elements is proposed for nonlinear incompressible multicomponent composite materials randomly reinforced with spheroidal inclusions with an arbitrary ratio of the longitudinal and lateral dimensions. Use is made of the Mori-Tanaka scheme that has, as a first approximation, the result of calculation of the elastic
Zeitschrift für angewandte Mathematik und Physik, 2006
We consider a transversal loading of a linearly elastic isotropic media containing the identical isotropic aligned circular fibers at non-dilute concentration c. By the use of solution obtained by the Kolosov-Muskhelishvili complex potential method for two interacting circles subjected to three different applied stresses at infinity, and exact integral representations for both the stress and strain distributions in a microinhomogeneous medium, one estimates the effective moduli of the composite accurately to order c 2 .