Mesoscale bounds in viscoelasticity of random composites (original) (raw)
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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.
Nonlinear viscoelastic analysis of statistically homogeneous random composites
International Journal for Multiscale Computational Engineering, 2004
Owing to the high computational cost in the analysis of large composite structures through a multi-scale or hierarchical modeling an efficient treatment of complex material systems at individual scales is of paramount importance. Limiting the attention to the level of constituents the present paper offers a prosperous modeling strategy for the predictions of nonlinear viscoelastic response of fibrous graphite-epoxy composite systems with possibly random distribution of fibers within a transverse plane section of the composite aggregate. If such a material can be marked as statistically homogeneous and the mechanisms driving the material response fall within the category of first-order homogenization scheme the variational principles of Hashin and Shtrikman emerge as an appealing option in the solution of uncoupled micro-macro computational homogenization. The material statistics up to two-point probability function that are used to describe the morphology of such a microstructure can be then directly incorporated into variational formulations to provide bounds on the effective material response of the assumed composite medium. In the present formulation the Hashin-Shtrikman variational principles are further extended to account for the presence of various transformation fields defined as local eigenstrain or eigenstress distributions in the phases. The evolution of such eigenfields is examined here within the framework of nonlinear viscoelastic behavior of polymeric matrix conveniently described by the Leonov model. Fully implicit integration scheme is implemented to enhance the stability and efficiency of the underlying numerical analysis. In this regard the Fourier transform is called when solving the resulting integral equations, which permits an arbitrary choice of the reference medium so that often encountered anisotropy of individual phases creates no obstacles in the solution procedure. Attention is also paid to the evaluation of the required material statistics. It is shown that replacing the actual microstructure of real world composites with a corresponding digitized image renders a computationally promising numerical approach for their derivation. Apart from application to nonlinear viscoelastic problem the use of the present modeling strategy is further promoted by a good agreement of estimated effective thermoelastic properties with predictions obtained from periodic unit cell models.
Linear viscoelastic analysis with random material properties
Probabilistic Engineering Mechanics, 1991
Analytical studies are presented which extend the elastic-viscoelastic analogies to stochastic processes caused by random linear viscoelastic material properties. Separation of variable as well as integral transform correspondence principles are formulated and discussed in detail. The statistical differential equation of the moment characteristic functional is derived, but rather than solving the highly complex functional equation, solutions are formulated in terms of first and second order statistical properties. Both Gaussian and beta distributions are considered for the probability density distributions of creep and relaxation functions and their effectiveness is evaluated. In order to illustrate the developed general theory, specific examples of beam bending and pressurized hollow cylinders are solved. The influence of various parameters contributing to the creep and relaxation correlation f/mctions is evaluated and the relationship between deterministic and stochastic bounds is also investigated. It is shown that deterministic bounds based on material data spread are unrealistic in the presence of random viscoelastic properties, since the do not correctly predict the limits of this stochastic process.
Thermal-Stress Concentration Near Inclusions In Viscoelastic Random Composites
Journal of Engineering Mathematics, 2008
The determination of thermal-stress concentrations near inclusions in viscoelastic random composites is concerned with the prediction of the overall response of random nonlinear viscoelastic multi-component media. The continuum considered here is assumed to be subjected to a finite deformation. First Piola's stress tensor and deformation gradient are used as conjugate field variables in a fixed reference state. A nonlinear problem is investigated in a second-order approximation theory when the gradient deformation terms higher than second order are neglected. A convex potential function in a thermo-elastic problem and time functionals in a viscoelastic one are used to construct overall constitutive relations. The technique of surface operators developed by R. Hill and others is used to determine stress concentrations near inclusions for nonlinear matrix creep.
Peridynamic Micromechanics of Random Structure Composites
Springer eBooks, 2012
In this chapter, we consider the solution methods of the GIE of peridynamic micromechanics. These methods are based on extraction from the material properties a constituent of the matrix properties. Effective moduli are expressed through the average local interface polarization tensor over the surface of the extended inclusion phase rather than over an entire space. Any spatial derivatives of displacement fields are not required. The basic hypotheses of locally elastic micromechanics are generalized to their peridynamic counterparts. In particular, in the generalized effective field method (EFM) proposed, the classical effective field hypothesis is relaxed, and the hypothesis of the ellipsoidal symmetry of the random structure of CMs is not used. One demonstrates some similarity and difference with respect to other methods (the dilute approximation and Mori-Tanaka approach) of micromechanics of peridynamic CMs. Estimation of macroscopic effective response of heterogeneous media with random structures in an averaged (or homogenized) meaning in terms of the mechanical and geometrical properties of constituents is a central focus of micromechanics denoted as micro-to-macro modeling. The general results establishing the links between the effective properties and the corresponding mechanical and transformation influence functions were inspired by Hill [640] for locally elastic composites. Some basic representations analogous to the mentioned above were generalized in Chap. 17 to the thermoperidynamics of CMs. The displacement field estimations in the constituents, in turn, are based on a substitution into the one or another micromechanical scheme of a solution (called basic problem) for one inclusion inside the infinite matrix subjected to some effective field. So, for locally elastic random structure CMs, a number of micromechanical models inspired by Eshelby [449] (see Chap. 3) were proposed in the literature for describing the thermoelastic behavior of composites with ellipsoidal inclusions (see Chaps. 8-12). Numerical solutions for the basic problem are considered in Sect. 18.1, whereas the different micromechanical models are generalized to their peridynamic counterparts in Sects. 18.2-18.4.
Scale-dependent homogenization of random composites as micropolar continua
European Journal of Mechanics - A/Solids, 2015
A multitude of composite materials ranging from polycrystals to rocks, concrete, and masonry overwhelmingly display random morphologies. While it is known that a Cosserat (micropolar) medium model of such materials is superior to a Cauchy model, the size of the Representative Volume Element (RVE) of the effective homogeneous Cosserat continuum has so far been unknown. Moreover, the determination of RVE properties has always been based on the periodic cell concept. This study presents a homogenization procedure for disordered Cosserat-type materials without assuming any spatial periodicity of the microstructures. The setting is one of linear elasticity of statistically homogeneous and ergodic two-phase (matrix-inclusion) random microstructures. The homogenization is carried out according to a generalized Hill-Mandel type condition applied on mesoscales, accounting for non-symmetric strain and stress as well as couple-stress and curvature tensors. In the setting of a two-dimensional elastic medium made of a base matrix and a random distribution of disk-shaped inclusions of given density, using Dirichlet-type and Neumann-type loadings, two hierarchies of scale-dependent bounds on classical and micropolar elastic moduli are obtained. The characteristic length scales of approximating micropolar continua are then determined. Two material cases of inclusions, either stiffer or softer than the matrix, are studied and it is found that, independent of the contrast in moduli, the RVE size for the bending micropolar moduli is smaller than that obtained for the classical moduli. The results point to the need of accounting for: spatial randomness of the medium, the presence of inclusions intersecting the edges of test windows, and the importance of additional degrees of freedom of the Cosserat continuum.
Micromechanically based stochastic finite elements: length scales and anisotropy
Probabilistic Engineering Mechanics, 1996
The present stochastic finite element (SFE) study amplifies a recently developed micromechanically based approach in which two estimates (upper and lower) of the finite element stiffness matrix and of the global response need first to be calculated. These two estimates correspond, respectively, to the principles of stationary potential and complementary energy on which the SFE is based. Both estimates of the stiffness matrix are anisotropic and tend to converge towards one another only in the infinite scale limit; this points to the fact that an approximating meso-scale continuum random field is neither unique nor isotropic. The SFE methodology based on this approach is implemented in a Monte Carlo sense for a conductivity (equivalently, out-of-plane elasticity) problem of a matrix-inclusion composite under mixed boundary conditions. Two versions are developed: in one an exact calculation of all the elements' stiffness matrices from the microstructure over the entire finite element mesh is carried out, while in the second one a second-order statistical characterization of the mesoscale continuum random field is used to generate these matrices.
Journal of Engineering Materials and Technology, 2007
A material composed of a mixture of distinct homogeneous media can be considered as a homogeneous one at a sufficiently large observation scale. In this work, the problem of the elastic mixture characterization is solved in the case of linear random mixtures, that is, materials for which the various components are isotropic, linear, and mixed together as an ensemble of particles having completely random shapes and positions. The proposed solution of this problem has been obtained in terms of the elastic properties of each constituent and of the stoichiometric coefficients. In other words, we have explicitly given the features of the micro-macro transition for a random mixture of elastic material. This result, in a large number of limiting cases, reduces to various analytical expressions that appear in earlier literature. Moreover, some comparisons with the similar problem concerning the electric characterization of random mixtures have been drawn. The specific analysis of porous ran...
Random Residual Stresses in Elasticity Homogeneous Medium with Inclusions of Noncanonical Shape
International Journal for Multiscale Computational Engineering, 2012
We consider a linearly elastic composite medium, which consists of a homogeneous matrix containing a statistically inhomogeneous random set of noncanonical inclusions. The elastic properties of the matrix and the inclusions are the same, but the stress-free strains are different. The new general volume integral equation (VIE) is proposed. These equations are obtained by a centering procedure without any auxiliary assumptions such as the effective field hypothesis implicitly exploited in the known centering methods. The results of this abandonment are quantitatively estimated for some modeled composite with homogeneous fibers of nonellipsoidal shape. New effects are detected that are impossible within the framework of a classical background of micromechanics.