Effective Medium Approach to Matrix-Inclusion Type Composite Materials

1987 ◽  
Vol 54 (4) ◽  
pp. 880-883 ◽  
Author(s):  
S. Nomura
Keyword(s):  
Displacement Field ◽  
Mass Density ◽  
Spherical Inclusion ◽  
Homogeneous Medium ◽  
Dynamic Effect ◽  
Effective Medium ◽  
Effective Stiffness ◽  
Inclusion Type ◽  
The Matrix ◽  
Homogenous Medium

This paper addresses a problem of finding the effective medium that exhibits the same overall response as a given composite material reinfored by unidirectional short-fibers (chopped fibers). The expression for the displacement field in composites is obtained by converting the equilibrium equation into an integro-differential equation using the quasi-static Green’s function for a homogeneous medium. The “effective medium” is chosen that the ensemble averaged displacement field for the composite is equal to that of an equivalent homogenous medium. The “effective stiffness” and the “effective mass density” are defined as those properties of the effective medium. This is a first preliminary attempt to analyze the elasto-dynamic effect of matrix-inclusion type of composites. The obtained result for the effective stiffness is new and is not symmetrical with the interchange of the matrix phase and the fiber phase, unlike previous models. The result is also favorably compared with experimental data for spherical-inclusion reinforced composites.

Effective Medium of Unidirectional Short-Fiber Composites by a Self-Consistent Method

1987 ◽  
Vol 109 (1) ◽  
pp. 64-66
Author(s):  
Seiichi Nomura
Keyword(s):  
Fiber Composite ◽  
Homogeneous Medium ◽  
Fiber Composites ◽  
Short Fiber ◽  
Effective Medium ◽  
Matrix Phase ◽  
Effective Stiffness ◽  
Self Consistent ◽  
The Matrix ◽  
Consistent Method

A new self-consistent method is proposed to calculate the effective stiffness of unidirectional short-fiber composites where each transversely-isotropic short-fibers is embedded in an infinite homogeneous matrix phase. The equilibrium equation for the elastic field in short-fiber composite materials is converted into an integro-differential equation using the Green’s function for a homogeneous medium. The “effective medium” is chosen in such a way that the ensemble averaged strain field for the composite is equal to that of the homogeneous medium that exhibits the same overall response as the composite. The “effective stiffness” and the “effective mass density” are defined as those properties of the effective medium. The obtained expression for the effective stiffness is new and is not symmetrical with the matrix phase and the fiber phase, thus, reflecting the matrix role more properly than previous works which gave symmetrical results. The result is also favorably compared with experimental data.

Ballistic impact response of carbon/epoxy tubes with variable nanosilica content

2017 ◽  
Vol 52 (12) ◽  
pp. 1589-1604 ◽  
Author(s):  
Aniruddh Vashisth ◽  
Charles E Bakis ◽  
Charles R Ruggeri ◽  
Todd C Henry ◽  
Gary D Roberts
Keyword(s):  
Fracture Toughness ◽  
Shear Strength ◽  
Glass Transition ◽  
Transition Temperature ◽  
Impact Damage ◽  
Mass Density ◽  
Impact Response ◽  
Residual Shear Strength ◽  
Damage Area ◽  
The Matrix

Laminated fiber reinforced polymer composites are known for high specific strength and stiffness in the plane of lamination, yet relatively low out-of-plane impact damage tolerance due to matrix dominated interlaminar mechanical properties. A number of factors including the toughness of the matrix can influence the response of composites to impact. The objective of the current investigation is to evaluate the ballistic impact response of carbon/epoxy tubes with variable amounts of nanosilica particles added to the matrix as a toughening agent. Mass density, elastic modulus, glass transition temperature and Mode I fracture toughness of the matrix materials were measured. Tubes manufactured with these matrix materials were ballistically impacted using a round steel projectile aimed at normal incidence across the major diameter. After impact, the tubes were nondestructively inspected and subjected to mechanical tests to determine the residual shear strength in torsion. Increasing concentrations of nanosilica monotonically increased the modulus and fracture toughness of the matrix materials. Tubes with nanosilica had smaller impact damage area, higher residual shear strength, and higher energy absorbed per unit damage area versus control materials with no nanosilica. Overall, the addition of nanosilica improved the impact damage resistance and tolerance of carbon/epoxy tubes loaded in torsion, with minimal adverse effects on mass density and glass transition temperature.

A Spherical Inclusion with Inhomogeneous Interface in Conduction

2003 ◽  
Vol 19 (1) ◽  
pp. 1-8
Author(s):  
T. Chen ◽  
C. H. Hsieh ◽  
P. C. Chuang
Keyword(s):  
Infinite Number ◽  
Effective Conductivity ◽  
Spherical Inclusion ◽  
Cone Angle ◽  
Imperfect Interface ◽  
Infinite Matrix ◽  
Algebraic Equations ◽  
Legendre Series ◽  
Linear Set ◽  
The Matrix

ABSTRACTA series solution is presented for a spherical inclusion embedded in an infinite matrix under a remotely applied uniform intensity. Particularly, the interface between the inclusion and the matrix is considered to be inhomegeneously bonded. We examine the axisymmetric case in which the interface parameter varies with the cone angle θ. Two kinds of imperfect interfaces are considered: an imperfect interface which models a thin interphase of low conductivity and an imperfect interface which models a thin interphase of high conductivity. We show that, by expanding the solutions of terms of Legendre polynomials, the field solution is governed by a linear set of algebraic equations with an infinite number of unknowns. The key step of the formulation relies on algebraic identities between coefficients of products of Legendre series. Some numerical illustrations are presented to show the correctness of the presented procedures. Further, solutions of the boundary-value problem are employed to estimate the effective conductivity tensor of a composite consisting of dispersions of spherical inclusions with equal size. The effective conductivity solely depends on one particular constant among an infinite number of unknowns.

On the Dynamic Growth of a Spherical Inclusion With Dilatational Transformation Strain in Infinite Elastic Domain

1999 ◽  
Vol 66 (4) ◽  
pp. 879-884 ◽  
Author(s):  
B. Wang ◽  
Q. Sun ◽  
Z. Xiao
Keyword(s):  
Hydrostatic Stress ◽  
Spherical Inclusion ◽  
Reduction Rate ◽  
Dynamic Effect ◽  
Propagation Speed ◽  
Transformation Strain ◽  
Elastic Fields ◽  
Initiation And Propagation ◽  
Dynamic Growth ◽  
Basic Ideas

In this paper, the dynamic effect was incorporated into the initiation and propagation process of a transformation inclusion. Based on the time-varying propagation equation of a spherical transformation inclusion with pure dilatational eigenstrain, the dynamic elastic fields both inside and outside the inclusion were derived explicitly, and it is found that when the transformation region expands at a constant speed, the strain field inside the inclusion is time-independent and uniform for uniform eigenstrain. Following the basic ideas of crack propagation problems in dynamic fracture mechanics, the reduction rate of the mechanical part of the free energy accompanying the growth of the transformation inclusion was derived as the driving force for the move of the interface. Then the equation to determine the propagation speed was established. It is found that there exists a steady speed for the growth of the transformation inclusion when time is approaching infinity. Finally the relation between the steady speed and the applied hydrostatic stress was derived explicitly.

scholarly journals Translation of a Coated Rigid Spherical Inclusion in an Elastic Matrix: Exact Solution, and Implications for Mechanobiology

2019 ◽  
Vol 86 (5) ◽  
Author(s):  
Xin Chen ◽  
Moxiao Li ◽  
Shaobao Liu ◽  
Fusheng Liu ◽  
Guy M. Genin ◽  
...  
Keyword(s):  
Mechanical Properties ◽  
Exact Solution ◽  
Analytical Solution ◽  
Spherical Inclusion ◽  
Matrix Material ◽  
Elastic Matrix ◽  
Protein Ligands ◽  
First Case ◽  
The Matrix ◽  
Matrix Modulus

The displacement of relatively rigid beads within a relatively compliant, elastic matrix can be used to measure the mechanical properties of the matrix. For example, in mechanobiological studies, magnetic or reflective beads can be displaced with a known external force to estimate the matrix modulus. Although such beads are generally rigid compared to the matrix, the material surrounding the beads typically differs from the matrix in one or two ways. The first case, as is common in mechanobiological experimentation, is the situation in which the bead must be coated with materials such as protein ligands that enable adhesion to the matrix. These layers typically differ in stiffness relative to the matrix material. The second case, common for uncoated beads, is the situation in which the beads disrupt the structure of the hydrogel or polymer, leading to a region of enhanced or reduced stiffness in the neighborhood of the bead. To address both cases, we developed the first analytical solution of the problem of translation of a coated, rigid spherical inclusion displaced within an isotropic elastic matrix by a remotely applied force. The solution is applicable to cases of arbitrary coating stiffness and size of the coating. We conclude by discussing applications of the solution to mechanobiology.

A Versatile Micromechanical Model for Estimating the Effective Properties of Isotropic Composite Materials

2009 ◽  
Vol 614 ◽  
pp. 255-260
Author(s):  
Qi Chang He ◽  
H. Le Quang
Keyword(s):  
Volume Fraction ◽  
Effective Properties ◽  
Micromechanical Model ◽  
Characteristic Parameter ◽  
Effective Medium ◽  
Inclusion Type ◽  
Composite Sphere ◽  
Host Matrix ◽  
Shear Moduli ◽  
Isotropic Composite

This work is concerned with a versatile and efficient model for estimating the effective moduli of isotropic composites consisting of isotropic phases whose microstructure may be of matrix-inclusion type, disordered or intermediate. This extended version of generalized self-consistent model (GSCM) is built by inserting a composite sphere embedded in an infinite unknown effective medium has the core made of the unknown effective medium and coated by the constituent phases. The volume fraction of the constituent phases in this composite sphere is the characteristic parameter of the relevant microstructure. By imposing the an energy equivalency condition, the equations thus obtained to estimate the effective bulk and shear moduli involve the microstructural parameter which turns out to be capable of describing in some sense how far a microstructure is from the host matrix/inclusion morphology

Vertical propagation of low-frequency waves in finely layered media

Geophysics ◽  
2006 ◽  
Vol 71 (3) ◽  
pp. T87-T94 ◽  
Author(s):  
Alexey Stovas ◽  
Børge Arntsen
Keyword(s):  
Reflection Coefficient ◽  
Low Frequency ◽  
Effective Medium ◽  
Frequency Limit ◽  
Data Matching ◽  
Anisotropy Effect ◽  
Vertical Propagation ◽  
The Matrix ◽  
Backus Averaging ◽  
Time Average

Multiple scattering in finely layered sediments is important for interpreting stratigraphic data, matching well-log data with seismic data, and seismic modeling. Two methods have been used to treat this problem in seismic applications: the O’Doherty-Anstey approximation and Backus averaging. The O’Doherty-Anstey approximation describes the stratigraphic-filtering effects, while Backus averaging defines the elastic properties for an effective medium from the stack of the layers. It is very important to know when the layered medium can be considered as an effective medium. In this paper, we only investigate vertical propagation. Therefore, no anisotropy effect is taken into consideration. Using the matrix-propagator method, we derive equations for transmission and reflection responses from the stack of horizontal layers. From the transmission response, we compute the phase velocity and compare the zero-frequency limit with the effective-medium velocity from Backus averaging. We also investigate how the transition from time-average medium to effective medium depends on contrast; i.e., strength of the reflection-coefficient series. Using numerical examples, we show that a transition zone exists between the effective medium (low-frequency limit) and the time-average medium (high-frequency limit), and that the width of this zone depends on the strength of the reflection-coefficient series.

scholarly journals Equivalent-inclusion approach for the conductivity of isotropic matrix composites with anisotropic inclusions

2016 ◽  
Vol 38 (4) ◽  
pp. 239-248 ◽  
Author(s):  
Do Quoc Hoang ◽  
Pham Duc Chinh ◽  
Tran Anh Binh
Keyword(s):  
Finite Elements ◽  
Effective Conductivity ◽  
Spherical Inclusion ◽  
Effective Medium ◽  
Matrix Composites ◽  
Isotropic Matrix ◽  
Equivalent Inclusion

Many effective medium approximations for effective conductivity are elaborated for matrix composites made from isotropic continuous matrix and isotropic inclusions associated with simple shapes such as circles or spheres, ... In this paper, we focus specially on the effective conductivity of the isotropic composites containing the disorderly oriented anisotropic inclusions. We aim to replace those inhomogeneities by simple equivalent circular (spherical) isotropic inclusions with modified conductivities. Available simple approximations for the equivalent circular (spherical)-inclusion media then can be used to estimate the effective conductivity of the original composite. The equivalent-inclusion approach agrees well with numerical extended finite elements results.

Transformation of the earthquake displacement field for spherical earth—Static case

1971 ◽  
Vol 61 (4) ◽  
pp. 861-874
Author(s):  
Hans R. Wason ◽  
Sarva Jit Singh
Keyword(s):  
Displacement Field ◽  
Spherical Surface ◽  
Homogeneous Medium ◽  
Addition Theorem ◽  
Polar Coordinate System ◽  
Earth Model ◽  
Static Case ◽  
Spherical Earth ◽  
Polar Coordinate ◽  
Internal Sources

abstract Explicit expressions for the static displacement field for a Volterra dislocation and a center of compression in an infinite homogeneous medium are obtained. Using an addition theorem, the field is transformed to a polar coordinate system with origin at the center of the Earth. Expressions for the discontinuity in the motion stress vector across the concentric spherical surface through the source are then obtained. These results can be used in studying the deformation of a multilayered spherical earth model induced by internal sources by the Thomson-Haskell matrix method which has so far been mostly applied to dynamic problems.

Characterization of Nanoporous Low-k Thin Films by Contrast Match SANS

2003 ◽  
Vol 766 ◽  
Author(s):  
Ronald C. Hedden ◽  
Barry J. Bauer ◽  
Hae-Jeong Lee
Keyword(s):  
Thin Films ◽  
Pore Size ◽  
Mass Density ◽  
Solvent Mixture ◽  
Wall Material ◽  
Ion Scattering ◽  
Low Dielectric ◽  
The Matrix ◽  
Low K

AbstractSmall-angle neutron scattering (SANS) contrast variation is used to characterize matrix properties and pore size in nanoporous low-dielectric constant (low-k) thin films. Using a vapor adsorption technique, SANS measurements are used to identify a “contrast match” solvent mixture containing the hydrogen– and deuterium-containing versions of a probe solvent. The contrast match solvent is subsequently used to conduct SANS porosimetry experiments. With information from specular X-ray reflectivity and ion scattering, the technique is useful for estimating the mass density of the matrix (wall) material and the pore size distribution. To illustrate the technique, a porous methylsilsesquioxane (MSQ) spin-on dielectric is characterized.

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