Begell House
International Journal for Multiscale Computational Engineering
International Journal for Multiscale Computational Engineering
1543-1649
2
1
2004
Multiscale Mechanics of Nonlocal Effects in Microheterogeneous Materials
We consider a linearly thermoelastic composite medium, which consists of a homogeneous matrix containing either deterministic (periodic and non-periodic) or random (statistically homogeneous and inhomogeneous, so-called graded) field of inclusions. For functionally graded materials when the concentration of the inclusions is a function of the coordinates, the micromechanical approach is based on the generalization of the "multiparticle effective field" method, previously proposed for statistically homogeneous random structure composites by the author (see for references and details Buryachenko, Appl. Mech. Reviews 2001, 54, 1-47). Both the Fourier transform method and iteration method are analyzed. The nonlocal integral and differential effective operators of elastic effective properties are estimated. The nonlocal dependencies of the effective elastic moduli as well as of conditional averages of the strains in the components on the concentration of the inclusions in a certain neighborhood of point considered are detected; the scale effect is discovered. The proposed theory provides the bridging of length scales which is a paramount factor in understanding and controlling material microinhomogeneity at the microscale and interpreting them at the macroscale. The combined coupled concept of introducing both the integral and differential operator linking microscale and macroscale enables one to address two issues simultaneously.
Valeriy A.
Buryachenko
Civil Engineering Department, University of Akron, Akron, Ohio 44325-3901, USA and Micromechanics and Composites LLC, 2520 Hingham Lane, Dayton, Ohio 45459, USA
14
Green's Function and Eshelby's Fields in Couple-Stress Elasticity
Conventional micromechanical schemes for estimating effective properties of composite materials in the matrix-inclusion type have no dependence upon absolute sizes of inclusions. However, there has been more and more experimental evidence that severe strain-gradient may result in remarkable size effects to mechanical behavior of materials. The strain field of an unbounded isotropic homogeneous elastic body containing a spherical inclusion subject to a uniform farfield stress may have very sharp strain-gradient within a surrounding matrix region of the inclusion, whenever the inclusion size would be very small. Consequently, the strain field variation in the whole matrix region of a composite with highly concentrated very small inclusions would be violent. Therefore, it is necessary to develop a micromechanical scheme in which the matrix phase is treated as a nonconventional material, and both the inclusion phases and the composite itself as an effective medium are treated as conventional materials. Such a scheme has been reported, with interesting applications. This scheme is based on the results of Green's functions and Eshelby's fields in couple-stress elastic theory. A thorough derivation of these results is given in the present paper. The main reason for choosing the couple-stress theory among various nonconventional theories of elasticity is that it contains the least number of material constants, in order to establish a simplest possible micromechanical scheme for taking account of absolute sizes.
Quanshui
Zheng
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
Z.-H.
Zhao
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
12
Estimation of Effective Elastic Properties of Random Structure Composites for Arbitrary Inclusion Shape and Anisotropy of Components Using Finite Element Analysis
We consider a linearly thermoelastic composite medium of arbitrary anisotropic constituents, which consists of a homogeneous matrix containing a statistically homogeneous random set of inclusions of any shape, orientation, and inhomogeneous micro structure. We use the main hypothesis of many micromechanical methods, according to which each inclusion is located inside a homogeneous so-called "effective field," accompanied by the quasi-crystalline approximation describing the inclusion interactions. We estimate effective elastic properties of composites and statistical averages of stresses, which are in general inhomogeneous in the inclusions. The proposed analytical—numerical method is efficient from a computational standpoint and is based on the use of the finite element analysis implemented for the one-particle problem in the infinite-homogeneous matrix with forthcoming incorporation of the stress concentrator tensors found in the known analytical homogenization scheme of micromechanics described above. The method is presented for both two- and three-dimensional problems, but the numerical examples are carried out just for plane strain and plane-stress problems.
Valeriy A.
Buryachenko
Civil Engineering Department, University of Akron, Akron, Ohio 44325-3901, USA and Micromechanics and Composites LLC, 2520 Hingham Lane, Dayton, Ohio 45459, USA
G. P.
Tandon
University of Dayton Research Institute, 300 College Park, Dayton, OH 45469-0168, USA
17
A Seventh-Order Accurate and Stable Algorithm for the Computation of Stress Inside Cracked Rectangular Domains
A seventh-order accurate and extremely stable algorithm for the rapid computation of stress fields inside cracked rectangular domains is presented. The algorithm is seventh-order accurate since it incorporates basis functions, taking the asymptotic shape of the stress fields close to crack tips and corners into account at least up to order six. The algorithm is stable since it is based on a Fredholm integral equation of the second kind. The particular form of the integral equation represents the solution as the limit of a function which is analytic inside the domain. This allows for an efficient implementation. In an example, involving 112 discretization points on an elastic square with a center crack, values of normalized stress intensity factors and T-stress with a relative error of 10−6 are computed in seconds on a workstation. More points reduce the relative error down to 10−15, where it saturates in double precision arithmetic. A large-scale setup with up to 1024 cracks in an elastic square is also studied, using up to 740,000 discretization points. The algorithm is intended as a basic building block in general-purpose solvers for fracture mechanics. It can also be used as a substitute for benchmark tables.
Johan
Helsing
Numerical Analysis, Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 LUND, Sweden
Anders
Jonsson
Department of Solid Mechanics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
22
Exact Relations for the Effective Properties of Nonlinearly Elastic Inhomogeneous Materials
This study is concerned with the effective behavior of nonlinearly elastic materials, which are locally inhomogeneous in one, two, or three directions and whose prototypes are layered, fiber reinforced, matrix-inclusion composites or polycrystals. A systematic method based on the implicit function theorem is proposed to find conditions for the existence of locally uniform strain fields and to exactly determine the overall stress response of such a material to a macroscopic strain associated with a locally uniform strain field. General exact connections are established between the effective elastic tangent moduli evaluated at each macroscopic strain inducing a locally uniform strain field. These results are applied to a cubic polycrystal whose elastic constitutive relation is the most general one, and to power-law fiber-reinforced composites. In particular, it is proven that the overall nonlinear elastic stress response of a cubic polycrystal to an isotropic strain is identical to that of a cubic monocrystal. This conclusion constitutes a nonlinear extension of a well-known result of Hill (1952).
Qi-Chang
He
Southwest Jiaotong University, School of Mechanical Engineering, Chengdu 610031, China; Université Paris-Est, Laboratoire de Modélisation et Simulation Multi Echelle, MSME UMR 8208 CNRS, 5 Boulevard Descartes, 77454 Marne-la-Vallée, France
B.
Bary
Laboratoire de Mécanique, Université de Marne-la-Vallee, 19 rue A. Nobel, F-77420 Champs sur Marne, France
14
3D Micromechanical Modeling of Particulate Composite Materials with Imperfect Interface
In this article, we study the effective elastic and plastic properties of particle-reinforced composite materials with imperfect interface. The imperfect interface is described by a spring-type model assuming that the traction continuity remains intact, while the displacement experiences a jump proportional to the interfacial traction. Both the tangential and normal displacement discontinuities at the interfaces are considered and the effective elastic moduli are obtained using a generalized version of dilute model, self-consistent model, Mori—Tanaka model, and differential model. A comparison between the effective elastic moduli obtained by these different models is made, and the determination of an effective plastic behavior of particulate composite materials, by means of the Mori—Tanaka model and secant moduli method, is also discussed. Finally, we examine the influence of interfacial compliance on the macroscopic averaged mechanical properties of composite materials.
Z.
Zhong
Key Laboratory of Solid Mechanics of MEC, School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, P. R. China
X. B.
Yu
Key Laboratory of Solid Mechanics of MEC, School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, P. R. China
S. A.
Meguid
Engineering Mechanics and Design Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada M5S 3G8
16
Construction of the Fiber-Matrix Interfacial Failure Envelope in a Polymer Matrix Composite
Previous research efforts have used the single-fiber cruciform test to measure the tensile normal strength of a fiber/matrix interface while eliminating the influence of free-edge stresses that are present in transverse testing of conventional straight-sided specimens. In this work, the cruciform specimen was modified to characterize the fiber/matrix interface strength under combined transverse and shear loading. Initiation and growth of interface debonds were detected optically by observation of variations in the intensity of light reflected from the surface of the fiber during loading. Test data reduction was accomplished with a 3-D finite element model of the angled cruciform sample. Using the measured value of applied stress at debond initiation, and the calculated stress concentration factors at the fiber/matrix interface, a mixed-mode failure envelope was constructed in the normal-shear stress space, and a quadratic failure criteria was proposed. Finally, a brief discussion has been included of how this interfacial strength data may be used in ysis to predict bulk characteristics of a composite laminate.
G. P.
Tandon
University of Dayton Research Institute, 300 College Park, Dayton, OH 45469-0168, USA
Ran Y.
Kim
University of Dayton Research Institute, 300 College Park, Dayton, OH 45469-0168, USA
Vernon T.
Bechel
Air Force Research Laboratory, Wright-Patterson AFB, OH 45433, USA
13
Modeling High-Frequency Acoustic Velocities in Patchy and Partially Saturated Porous Rock using Differential Effective Medium Theory
Differential effective medium (DEM) theory is applied here to the problem of modeling physical properties of poroelastic media that are partially saturated with liquid. Typical fluid saturants are air and water, or gas and oil. If the liquid and gas saturants are homogeneously mixed, then we say the medium is partially saturated. If the liquid and gas saturants are very poorly mixed, so each constituent occupies separate, but contiguous, regions of the porous medium, we say the medium has patchy saturation. Some examples are presented to show that a reasonable approach to modeling the effects of patchy saturation at high frequencies (200 kHz and above) is produced by treating the medium as if it were an homogenized mixture of gas-saturated and liquid-saturated parts that are homogeneously mixed together. Estimates of the properties for partial saturation are obtained using differential effective medium theory. The results for patchy saturation differ dramatically from those predicted by Gassmann's equations for homogeneous mixing of the fluids in individual pores. In particular, the shear modulus depends on the elastic properties of the fluid constituents, unlike the quasi-static behavior predicted by Gassmann.
James G.
Berryman
Lawrence Livermore National Laboratory, University of California, P.O. Box 808 L-200, Livermore, CA 94551-9900
17
Asymptotic Homogenization Models for Smart Composite Plates with Rapidly Varying Thickness: Part I—Theory
Asymptotic homogenization models for smart composite plates with rapidly varying thickness and periodically arranged actuators are derived. The effective elastic, actuation, thermal expansion, and hygroscopic expansion coefficients are obtained. The actuation coefficients characterize the intrinsic transducer nature of active smart materials that can be used to induce strains and stresses in a coordinated fashion. Examples of such actuators employed with smart composite material systems are derived from piezoelectric, magnetostrictive, and some other materials. It is shown that the original problem for the regularly non-homogeneous smart composite plate with rapidly oscillating thickness reduces to a system of eight simpler types of problem. It is precisely these "unit-cell" problems that enable the determination of the aforementioned effective coefficients and subsequently the strain and stress fields. In the limiting case of a thin elastic plate of uniform thickness the derived model is shown to converge to the familiar classical plate model. In Part II of this work, the theory is illustrated by means of examples pertaining to a thin smart laminated plate of uniform thickness and a wafer-type smart composite plate reinforced with smart ribs oriented along the tangential directions of the plate.
A. L.
Kalamkarov
Mechanical Engineering Department, Dalhousie University, Halifax, Nova Scotia, B3J 2X4, Canada
A. V.
Georgiades
Mechanical Engineering Department, Dalhousie University, Halifax, Nova Scotia, B3J 2X4, Canada
16
Asymptotic Homogenization Models for Smart Composite Plates with Rapidly Varying Thickness: Part II—Applications
Asymptotic homogenization models for smart composite plates with rapidly varying thickness and periodically arranged actuators were derived in Part I of this work. These models were subsequently used to determine general expressions for effective elastic, actuation, thermal expansion, and hygroscopic expansion coefficients. The present article applies the theory to determine the effective properties of constant thickness laminates composed of monoclinic materials or orthotropic materials not referred to their principal coordinate system. These effective properties can then be used to calculate strains and stresses induced in the laminates by external loads, hygrothermal effects, or electric fields. Further examples illustrate the determination of the effective properties of wafer-type smart composite plates reinforced with smart ribs or stiffeners oriented along the tangential directions of the plate. For generality, it is assumed that the ribs and the base plate are made of different orthotropic materials.
A. L.
Kalamkarov
Mechanical Engineering Department, Dalhousie University, Halifax, Nova Scotia, B3J 2X4, Canada
A. V.
Georgiades
Mechanical Engineering Department, Dalhousie University, Halifax, Nova Scotia, B3J 2X4, Canada
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