Locally Exact Homogenization of Unidirectional Composites With Cylindrically Orthotropic Fibers

2016 ◽  
Vol 83 (7) ◽  
Author(s):  
Guannan Wang ◽  
Marek-Jerzy Pindera
Keyword(s):  
Volume Fraction ◽  
Local Stress ◽  
Transversely Isotropic ◽  
Local Fields ◽  
Homogenization Theory ◽  
Classical Solutions ◽  
Equilibrium Equations ◽  
Fiber Volume ◽  
Unidirectional Composites ◽  
Cylindrically Orthotropic

The elasticity-based, locally exact homogenization theory for unidirectional composites with hexagonal and tetragonal symmetries and transversely isotropic phases is further extended to accommodate cylindrically orthotropic reinforcement. The theory employs Fourier series representations of the fiber and matrix displacement fields in cylindrical coordinate system that satisfy exactly equilibrium equations and continuity conditions in the interior of the unit cell. Satisfaction of periodicity conditions for the inseparable exterior problem is efficiently accomplished using previously introduced balanced variational principle which ensures rapid displacement solution convergence with relatively few harmonic terms. As demonstrated in this contribution, this also applies to cylindrically orthotropic reinforcement for which the eigenvalues depend on both the orthotropic elastic moduli and harmonic number. The solution's demonstrated stability facilitates rapid identification of cylindrical orthotropy's impact on homogenized moduli and local fields in wide ranges of fiber volume fraction and orthotropy ratios. The developed theory provides a unified approach that accounts for cylindrical orthotropy explicitly in both the homogenization process and local stress field calculations previously treated separately through a fiber replacement scheme. Comparison of the locally exact solution with classical solutions based on an idealized microstructural representation and fiber moduli replacement with equivalent transversely isotropic properties delineates their applicability and limitations.

Semianalytical compressive strength criteria for unidirectional composites

2017 ◽  
Vol 37 (4) ◽  
pp. 238-246
Author(s):  
Uri Breiman ◽  
Jacob Aboudi ◽  
Rami Haj-Ali
Keyword(s):  
Experimental Data ◽  
Compressive Strength ◽  
Volume Fraction ◽  
Unidirectional Composite ◽  
Fiber Volume ◽  
Unidirectional Composites ◽  
Analysis And Design ◽  
Composite Systems ◽  
Wide Range ◽  
Laminated Structures

The compressive strength of unidirectional composites is strongly influenced by the elastic and strength properties of the fiber and matrix phases, as well as by the local geometrical properties, such as fiber volume fraction, misalignment, and waviness. In the present investigation, two microbuckling criteria are proposed and examined against a large volume of measured data of unidirectional composites taken from the literature. The first criterion is based on the compressive strength formulation using the buckling of Timoshenko’s beam. It contains a single parameter that can be determined according to the best fit to experimental data for various types of polymeric matrix composites. The second criterion is based on buckling-wave propagation analogy using the solution of an eigenvalue problem. Both criteria provide closed-form expressions for the compressive strength of unidirectional composites. We propose modifications of the two criteria by a fitting approach, for a wide range of fiber volume fractions, applied to four classes of unidirectional composite systems. Furthermore, a normalized form of the two models is presented after calibration in order to compare their prediction against experimental data for each of the material systems. The new modified criteria are shown to give a good match to a wide range of unidirectional composite systems. They can be employed as practical compression failure criteria in the analysis and design of laminated structures.

On Boundary Condition Implementation Via Variational Principles in Elasticity-Based Homogenization

2016 ◽  
Vol 83 (10) ◽  
Author(s):  
Guannan Wang ◽  
Marek-Jerzy Pindera
Keyword(s):  
Variational Principle ◽  
Variational Principles ◽  
Transversely Isotropic ◽  
Homogenization Theory ◽  
Element Analysis ◽  
Field Representation ◽  
Unidirectional Composites ◽  
Exterior Problems ◽  
Periodic Microstructures ◽  
Periodic Materials

Convergence characteristics of the locally exact homogenization theory for periodic materials, first proposed by Drago and Pindera (2008, “A Locally-Exact Homogenization Theory for Periodic Microstructures With Isotropic Phases,” ASME J. Appl. Mech., 75(5), p. 051010) and recently generalized by Wang and Pindera (“Locally-Exact Homogenization Theory for Transversely Isotropic Unidirectional Composites,” Mech. Res. Commun. (in press); 2016, “Locally-Exact Homogenization of Unidirectional Composites With Coated or Hollow Reinforcement,” Mater. Des., 93, pp. 514–528; and 2016, “Locally Exact Homogenization of Unidirectional Composites With Cylindrically Orthotropic Fibers,” ASME J. Appl. Mech., 83(7), p. 071010), are examined vis-a-vis the manner of implementing periodic boundary conditions. The locally exact theory separates the unit cell problem into interior and exterior problems, with the separable interior problem solved exactly in cylindrical coordinates and the inseparable exterior problem tackled using a balanced variational principle. This variational principle leads to exceptionally fast and well-behaved convergence of the Fourier series coefficients in the displacement field representation of the unit cell's different phases. Herein, we compare the solution's convergence behavior based on the balanced variational principle with that based on the constrained energy-based principle originally proposed by Jirousek (1978, “Basis for Development of Large Finite Elements Locally Satisfying All Fields Equations,” Comput. Methods Appl. Mech. Eng., 14, pp. 65–92) in the context of locally exact finite-element analysis. The relevance of this comparison lies in the recently rediscovered implementation of Jirousek's constrained variational principle in the homogenization of periodic materials.

Development of a Constitutive Theory for Short Fiber Yarns: Mechanics of Staple Yarn Without Slippage Effect

1992 ◽  
Vol 62 (12) ◽  
pp. 749-765 ◽  
Author(s):  
Ning Pan
Keyword(s):  
Fiber Length ◽  
Fiber Volume Fraction ◽  
Volume Fraction ◽  
Stress Transfer ◽  
Transversely Isotropic ◽  
Short Fiber ◽  
Constitutive Theory ◽  
Fiber Volume ◽  
Shear Moduli ◽  
Yarn Twist

This article reports an attempt to develop a general constitutive theory governing the mechanical behavior of twisted short fiber structures, starting with a high twist case, so that the effect of fiber slippage during yarn extension can be ignored. A differential equation describing the stress transfer mechanism in a staple yarn is proposed by which both the distributions of fiber tension and lateral pressure along a fiber length during yarn extension are derived. Factors such as fiber dimensions and properties and the effect of the discontinuity of fiber length within the structure are all included in the theory. With certain assumptions, the relationship between the mean fiber-volume fraction and the twist level of the yarn is also established. A quantity called the cohesion factor is defined based on yarn twist and fiber properties as well as on the form of fiber arrangement in the yarn to reflect the effectiveness of fiber gripping by the yarn. By considering the yarn structure as transversely isotropic with a variable fiber-volume fraction depending on the level of twist, the tensile and shear moduli as well as the Poisson's ratios of the structures are theoretically determined. All these predicted results have been verified according to the constitutive restraints of the continuum mechanics, and the final results are also illustrated schematically.

Micromechanics of a Smart Composite Actuator Embedded With Hollow Piezoelectric Fibers

2011 ◽  
Author(s):  
Sontipee Aimmanee ◽  
Supharoek Trakarnkulchai ◽  
Pakinee Aimmanee
Keyword(s):  
Volume Fraction ◽  
Transversely Isotropic ◽  
Material System ◽  
Smart Composite ◽  
Concentric Cylinder ◽  
Fiber Volume ◽  
Reinforced Composite ◽  
Concentric Cylinders ◽  
Matrix Volume ◽  
Composite Actuator

This paper presents a development of mathematical models for predicting the effective elastic and piezoelectric properties of a Smart Composite Actuator (SCA) reinforced with transversely isotropic piezoelectric hollow fibers. The models are established based on micromechanics of representative volume element of concentric cylinders or so-called concentric cylinder model (CCM). Five elastic constants and two piezoelectric coefficients are predicted as a function of fiber volume fraction, matrix volume fraction, and their constituents’ properties in the SCA. Numerical results of a chosen material system are obtained and discussed. The models can be found useful for developing a SCA or a novel hollow fiber-reinforced composite with the desired properties.

Elastic and Plastic Response of Perforated Metal Sheets With Different Porosity Architectures

2009 ◽  
Vol 131 (3) ◽  
Author(s):  
Hamed Khatam ◽  
Linfeng Chen ◽  
Marek-Jerzy Pindera
Keyword(s):  
Elastic Moduli ◽  
Volume Fraction ◽  
Stress Transfer ◽  
Local Stress ◽  
Shear Loading ◽  
Homogenization Theory ◽  
Plastic Response ◽  
Elastic Plastic ◽  
Aluminum Sheets ◽  
Metal Sheets

The effects of porosity architecture and volume fraction on the homogenized elastic moduli and elastic-plastic response of perforated thin metal sheets are investigated under three fundamental loading modes using an efficient homogenization theory. Steel and aluminum sheets weakened by circular, hexagonal, square, and slotted holes arranged in square and hexagonal arrays subjected to inplane normal and shear loading are considered with porosity volume fractions in the range 0.1–0.6. Substantial variations are observed in the homogenized elastic moduli with porosity shape and array type. The differences are rooted in the stress transfer mechanism around traction-free porosities whose shape and distribution play major roles in altering the local stress fields and thus the homogenized response in the elastic-plastic domain. This response is characterized by four parameters that define different stages of micro- and macrolevel yielding. The variations in these parameters due to porosity architecture and loading direction provide useful data for design purposes under monotonic and cyclic loading.

Effect of Fiber Anisotropy on Thermal Stresses in Fibrous Composites

1986 ◽  
Vol 53 (4) ◽  
pp. 751-756 ◽  
Author(s):  
W. B. Avery ◽  
C. T. Herakovich
Keyword(s):  
Fiber Volume Fraction ◽  
Maximum Stress ◽  
Thermal Stresses ◽  
Volume Fraction ◽  
Transversely Isotropic ◽  
Radial Coordinate ◽  
Stress Distributions ◽  
Matrix Interface ◽  
Fiber Volume ◽  
Fiber Matrix Interface

An elasticity solution is utilized to analyze an orthotropic fiber in an isotropic matrix under uniform thermal load. The analysis reveals that stress distributions in the fiber are singular in the radial coordinate when the radial fiber stiffness (Crr) is greater than the hoop stiffness (Cθθ). Conversely, if Crr < Cθθ the maximum stress in the composite is finite and occurs at the fiber-matrix interface. In both cases the stress distributions are radically different than those predicted assuming the fiber to be transversely isotropic (Crr=Cθθ). It is also shown that fiber volume fraction greatly influences the stress distribution for transversely isotropic fibers, but has little effect on the distribution if the fibers are transversely orthotropic.

The Eshelby Tensors in a Finite Spherical Domain—Part I: Theoretical Formulations

2006 ◽  
Vol 74 (4) ◽  
pp. 770-783 ◽  
Author(s):  
Shaofan Li ◽  
Roger A. Sauer ◽  
Gang Wang
Keyword(s):  
Boundary Condition ◽  
Volume Fraction ◽  
Spherical Inclusion ◽  
Finite Size ◽  
Transversely Isotropic ◽  
Solution Procedure ◽  
Eshelby Tensor ◽  
Homogenization Theory ◽  
Fredholm Type ◽  
Spherical Domain

This work is concerned with the precise characterization of the elastic fields due to a spherical inclusion embedded within a spherical representative volume element (RVE). The RVE is considered having finite size, with either a prescribed uniform displacement or a prescribed uniform traction boundary condition. Based on symmetry and group theoretic arguments, we identify that the Eshelby tensor for a spherical inclusion admits a unique decomposition, which we coin the “radial transversely isotropic tensor.” Based on this notion, a novel solution procedure is presented to solve the resulting Fredholm type integral equations. By using this technique, exact and closed form solutions have been obtained for the elastic disturbance fields. In the solution two new tensors appear, which are termed the Dirichlet–Eshelby tensor and the Neumann–Eshelby tensor. In contrast to the classical Eshelby tensor they both are position dependent and contain information about the boundary condition of the RVE as well as the volume fraction of the inclusion. The new finite Eshelby tensors have far-reaching consequences in applications such as nanotechnology, homogenization theory of composite materials, and defects mechanics.

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