DOCSLIB.ORG

Computational Micromechanics for Composites with Finite Boundaries

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Computational Micromechanics for Composites with Finite Boundaries TH 18 INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS COMPUTATIONAL MICROMECHANICS FOR COMPOSITES WITH FINITE BOUNDARIES S. Nomura1* and T. Pathapalli1 1Mechanical and Aerospace Engineering Department, The University of Texas at Arlington, Arlington, TX 76019-0023 USA * Corresponding author ([email protected]) Keywords: Micromechanics, elliptic inclusions, computer algebra system Abstract primarily involves determining a set of continuous permissible functions that satisfy the boundary A semi-analytical approach is presented to solve conditions and continuity conditions at the matrix- general boundary value problems (BVPs) that arise inclusion interface, expressing the BVP (governing in the analysis of composite materials. As an equation) in terms of the Sturm-Liouville (S-L) illustration, a rectangular shaped medium that system [3], subjecting the S-L problem (eigenvalue contains elliptical inclusions will be considered. problem) to the Galerkin method to obtain an Permissible functions, which satisfy the orthonormal set of eigenfunctions, and finally, homogeneous boundary conditions and the representing the mechanical/physical field as a linear continuity conditions at the matrix-inclusion combination of the evaluated eigenfunctions. In interface, are analytically derived. The order to facilitate the analytical derivation of the eigenfunctions are subsequently derived from the permissible functions in terms of the relevant permissible functions using the Galerkin method. material and geometrical parameters, a computer The mechanical and physical fields (temperature or algebra system [4, 5] has been extensively used. As elastic deformation), with an arbitrarily given source a demonstration example, a steady-state heat term, can be expressed as a linear combination of the conduction problem for a rectangular shaped eigenfunctions. The method is favorably compared medium with elliptical inclusions is presented. The with numerical results obtained from the finite results are favorably compared with those obtained element method. from the finite element method. 1 Introduction 2 Formulations The conventional micromechanics [1], pioneered by The governing differential equations for both Eshelby [2], has been widely used to analyze the elasticity and steady-state heat conduction can be microstructure of elastic solids that contain expressed as inclusions. However, the major limitation of micromechanics is that the medium is assumed to be [()] + () = 0, (1) infinitely extended (no boundary) and the geometry of the microstructure is strictly limited, thus making where is a self-adjoint differential operator, () is it inconvenient to accurately represent actual the unknown physical field (temperature or composite materials. Although purely numerical displacement) and () is the source term (heat methods such as the finite element method can be source or body force). For steady state heat employed for the analysis of composites, analytical conduction, takes the following form: or semi-analytical solutions, if available, are needlessly valuable. This paper presents a new semi- [()] = . () ∇(), (2) analytical method for heterogeneous materials that makes use of both analytical and numerical where () is the thermal conductivity and for the techniques to obtain mechanical and physical fields static elasticity case, takes the form: associated with the given BVP. The method ([()]) = ,,, (3) = [ ()] (), (9) where is the elastic modulus. For isotropic materials, it can be expressed in terms of the shear = () (). (10) modulus, , and the bulk modulus, , as: 2 The components of and are obtained using = − + + , (4) 3 equations (9) and (10). Therefore, equation (8) can be solved using a standard numerical techniques to where is the Kronecker delta. obtain the eigenvalues, , and the corresponding set of unknown coefficients, . Based on the Sturm-Liouville theory, the solution to equation (1) can be obtained if the eigenfunction, In accordance with the Sturm-Liouville theory, the (), is available. The eigenfunction is defined as solution to equation (1) can be expressed as a linear combination of eigenfunctions as [()] + () = 0, (5) ( ) ( ) where and () are the eigenvalue and = , (11) eigenfunction, respectively. An intrinsic property of the Sturm-Liouville system is that its differential where is the eigenfunction expansion coefficient operator, , is Hermitian, hence suggesting that all of the source term, (), and can be obtained as the eigenvalues are real and the eigenfunctions are mutually orthogonal as = () (). (12) () () = . (6) The most demanding task is to obtain the permissible function, () , that satisfies the The solution to equation (5) can be obtained by boundary conditions and the required continuity expressing the eigenfunctions, (), as a series of conditions at the interface. For instance, a 2-D body permissible functions as that contains an elliptical inclusion, the permissible function needs to be defined separately for each phase. For the inclusion phase, (, ) is assumed to () = (), (7) be in the polynomial form as where () is a permissible function chosen from elementary functions such as polynomials to satisfy (, ) = , (13) the homogeneous boundary conditions and the continuity conditions across the matrix-inclusion where represents the order of the polynomial and interface. The quantity, , is the coefficient of the summation ensures that the permissible function, () of the eigenfunction and is determined using the Galerkin method. By substituting equation (, ), encompasses all the polynomials up to the order. Similarly, for the matrix phase, (, ) (7) into equation (5), multiplying () on both sides and integrating them over the entire domain, is assumed to be of the form: equation (5) can be converted to the following generalized eigenvalue problem: ( ) ( ) (14) , = , , + = 0, (8) where (, ) is a function that vanishes at the where boundary (for the boundary condition of the first kind). For example, if the boundary is rectangular COMPUTATIONAL MICROMECHANICS FOR COMPOSITES WITH FINITE BOUNDARIES shaped with , , , defining the corners of the different thermal conductivities, and , as shown rectangle, then, (, ) takes the form: in Fig. 1. The governing equation is expressed as (, ) = ( − )( − )( − )( − ), (15) ∇. (, ) ∇(, ) = , (19) The homogeneous boundary condition is where (, ) is the unknown temperature field and categorically satisfied by equation (14). represents the heat source. The boundaries of the square shaped region are subjected to the boundary Equations (13) and (14) also need to satisfy the condition of the first kind ( (, ) = 0 ). The continuity conditions at the matrix-inclusion associated continuity conditions are interface. This is achieved by satisfying the following conditions: (, )| = (, )|, (20) (, ) (, ) | = | , (16) | = | , (21) | = | , (17) where the indices, 1 and 2, represent the inclusion phase and matrix phase, respectively. where and are material constants associated with the inclusion and matrix, respectively. For steady state heat conduction, the material constant would be thermal conductivity, () , and for the elastic equilibrium equation, it would be the shear modulus, , and the bulk modulus, . The term, , represents the directional derivative on the matrix-inclusion interface. The surface normal, , for an elliptical boundary is Figure 1. Elliptical Inclusions. defined as A set of continuous permissible functions are ⁄ analytically derived that satisfy the homogeneous ⎛⁄ + ⁄⎞ boundary condition and the continuity conditions = , (18) ⎜ ⁄ ⎟ indicated above. Despite the simplicity of the BVP, ⁄ + ⁄ the procedure involved in obtaining the permissible ⎝ ⎠ functions is a cumbersome one. This is facilitated where and are the semi-major and semi-minor with the aid of a computer algebra system, axis of the ellipse respectively. Mathematica [6]. For illustration purposes, one of the computer generated permissible functions is In the case of steady state heat conduction, equations shown below: (16) and (17) represent the continuity of temperature and heat flux respectively. The same equations, in (, ) = − + + − the case of elastic equilibrium, represent the continuity of displacement and traction force across − + + the matrix-inclusion interface. − + − − (22) 3 Examples + + + The Poisson type equation is considered that governs the steady state heat conduction in a square shaped − − , matrix medium consisting of two elliptical inclusions with the inclusions and matrix having 3 where is half the length of the square shaped matrix, and represent the semi-major and semi- minor axes of the elliptical inclusion, and are the thermal conductivities of the inclusion and matrix, respectively. From the above equation, it can be observed that (, ) is a function of both the geometrical and material parameters. From the permissible functions, the matrix elements of equations (9) and (10) are computed, following which an orthonormal set of eigenfunctions are obtained using equation (7). Figures 2, 3, and 4 show three arbitrary eigenfunctions for aspect ratios Figure 5. Cross-sectional profile of (, ). of 1 (circular inclusion), 2, and 5, respectively. A noticeable feature in all of the representations below Figure 5 depicts the cross sectional views of is the shape of
Load more

Recommended publications
  • Chapter 10: Composite Micromechanics
    Application of the Finite Element Method Using MARC and Mentat 10-1 Chapter 10: Composite Micromechanics 10.1 Problem Statement and Objectives Given the micromechanical geometry and the material properties of each constituent, it is possible to estimate the effective composite material properties and the micromechanical stress/strain state of a composite material. The objectives of this project are (1) to determine the effective stiffness c c ν c ν c properties E1 , E2 , 12 , 23 of a unidirectional composite material and (2) to determine the strain concentration factor in the matrix region when the composite material is subjected to a uniform transverse normal strain in the X 2 direction. NOTE TO ME 424 CLASS: Only do Part (2). 10.2 Background A composite material is often defined as a combination of two or more materials fabricated in such a way that the individual constituents (materials) can still be readily identified in the final form. If designed properly, this combination of materials yields a composite material that exhibits the best properties of each constituent as well as some advantageous properties not exhibited by the individual constituents. One example of such a material is a unidirectional fiber reinforced composite, which is often used in aerospace structures. An idealized micromechanical view of a unidirectional fiber reinforced composite material is shown in Figure 10.1. In these materials, the fibers have a very small diameter and a very high length-to-diameter ratio. This geometry yields excellent stiffness and strength characteristics in the fiber, since the crystals tend to align along the fiber axis and there are fewer internal and surface defects than in the bulk material.
    [Show full text]
  • Micromechanics As a Basis of Stochastic Finite Elements And
    Micromechanics as a basis of stochastic finite elements and differences: An overview M Ostoja-Starzewski Department of Materials Science and Mechanics Michigan State University, East Lansing, MI 48824-1226 A generalization of conventional deterministic finite element and difference methods to deal with spatial material fluctuations hinges on the problem of determination of stochastic constitutive laws. This problem is analyzed here through a paradigm of micromechanics of elastic polycrystals and matrix-inclusion composites. Passage to a sought-for random meso-continuum is based on a scale dependent window playing the role of a Representative Volume Element (RVE). It turns out that the microstructure cannot be uniquely approximated by a random field of stiffness with continuous realizations, but, rather, two random continuum fields may be introduced to bound the material response from above and from below. Since the RVE corresponds to a single finite element, or finite difference cell, not infinitely larger than the crystal size, these two random fields are to be used to bound the solution of a given boundary value problem at a given scale of resolution. The window- based random continuum formulation is also employed in analysis of rigid perfectly-plastic mate­ rials, whereby the classical method of slip-lines is generalized to a stochastic finite difference scheme. The present paper is complemented by a comparison of this methodology to other existing stochastic solution methods. 1. INTRODUCTION a locally isotropic form The necessity to account for random effects in determining a.. = X(x, co) 5..e +2u(x, co)e.. n ^ the response of a mechanical system is due, in general, to »J iJ kk 'J {Li> three different sources: random external forcing, random is adopted by simply postulating one or both elastic constants, boundary conditions, and random material parameters.
    [Show full text]
  • MICROMECHANICS of STRESS-INDUCED MARTENSITIC TRANSFORMATION in MONO- and POLYCRYSTALLINE SHAPE MEMORY ALLOYS; Ni-Ti
    Advanced Materials for the 21st Century: The 1999 Julia R. Weertman Symposium, 1999 TMS Fall Meeting in Cincinnati, OH, USA., Oct. 31-Nov.4, pp.385-396 MICROMECHANICS OF STRESS-INDUCED MARTENSITIC TRANSFORMATION IN MONO- AND POLYCRYSTALLINE SHAPE MEMORY ALLOYS; Ni-Ti Y. Liang, M. Taya and T. Mori* Department of Mechanical Engineering University of Washington Seattle, WA 98195-2600 Abstract Stress-induced martensitic transformation in single crystals and polycrystals are examined on the basis of micromechanics. A simple method to find a stress- and elastic energy-free martensite plate (combined variant), which consists of two variants, is presented. External and internal stresses preferentially produce a combined variant, to which the stresses supply the largest work upon its formation. Using the chemical energy change with temperature, the phase boundary between the parent and martensitic phases is determined in stress-temperature diagrams. The method is extended to a polycrystal, modeled as an aggregate of spherical grains. The grains constitute axisymmetric multiple fiber textures and a uniaxial load is applied to the fiber axis. The occurrence and progress of transformation are followed by examining a stress state in the grains. The stress is the sum of the external stress and internal stress. The difference in the fraction of transformation and, thus, in transformation strains between the grains causes the internal stress, which is calculated with the average field method. After a short transition stage, all the grains start to transform, and the external uniaxial stress to continue the transformation increases linearly thereafter. The external stress at the end of the transition is defined as the macroscopic yield stress due to the transformation in polycrystals.
    [Show full text]
  • Viscoelastic Behavior of Polymer-Modified Cement
    applied sciences Article Viscoelastic Behavior of Polymer-Modified Cement Pastes: Insight from Downscaling Short-Term Macroscopic Creep Tests by Means of Multiscale Modeling Luise Göbel 1,2,*,†, Markus Königsberger 2,3 ID , Andrea Osburg 1 ID and Bernhard Pichler 2 1 F.A. Finger-Institute for Building Material Engineering, Bauhaus-Universität Weimar, 99423 Weimar, Germany; [email protected] 2 Institute for Mechanics of Materials and Structures, TU Wien—Vienna University of Technology, 1040 Vienna, Austria; [email protected] (M.K.); [email protected] (B.P.) 3 BATir Department, Université libre de Bruxelles (ULB), 1050 Bruxelles, Belgium * Correspondence: [email protected]; Tel.: +49-3643-584743 † Current address: Coudraystraße 11 A, 99423 Weimar, Germany. Received: 28 February 2018; Accepted: 21 March 2018; Published: 23 March 2018 Abstract: Adding polymers to cementitious materials improves their workability and impermeability, but also increases their creep activity. In the present paper, the creep behavior of polymer-modified cement pastes is analyzed based on macroscopic creep tests and a multiscale model. The continuum micromechanics model allows for “downscaling” the results of macroscopic hourly-repeated ultra-short creep experiments to the viscoelastic behavior of micron-sized hydration products and polymer particles. This way, the increased creep activity of polymer-modified cement pastes is traced back to an isochoric power-law-type creep behavior of the polymers. The shear creep modulus of the polymers is found (i) to be two orders of magnitude smaller than that of the hydrates and (ii) to increase considerably with increasing material age. The latter result suggests that the creep activity of the polymers decreases with the self-desiccation-related decrease of the relative humidity inside the air-filled pores of cement paste.
    [Show full text]
  • Chiral Metamaterial Predicted by Granular Micromechanics: Verified With
    1 Chiral Metamaterial Predicted by Granular Micromechanics: Verified with 2 1D Example Synthesized using Additive Manufacturing 3 4 5 Anil Misra1*, Nima Nejadsadeghi 2, Michele De Angelo 1,3, Luca Placidi 4 6 7 8 1Civil, Environmental and Architectural Engineering Department, 9 University of Kansas, 1530 W. 15th Street, Learned Hall, Lawrence, KS 66045-7609. 10 2Mechanical Engineering Department, 11 University of Kansas, 1530 W. 15th Street, Learned Hall, Lawrence, KS 66045-7609. 12 3Dipartimento di Ingegneria Civile, Edile-Architettura e Ambientale, Università degli Studi 13 dellAquila, Via Giovanni Gronchi 18 - Zona industriale di Pile, 67100 LAquila, Italy 14 3International Telematic University Uninettuno, 15 C. so Vittorio Emanuele II 39, 00186 Rome, Italy 16 *corresponding author: Ph: (785) 864-1750, Fax: (785) 864-5631, Email: [email protected] 17 18 19 20 21 22 23 Continuum Mechanics and Thermodynamics 24 25 26 27 28 Abstract 29 Granular micromechanics approach (GMA) provides a predictive theory for granular material 30 behavior by connecting the grain-scale interactions to continuum models. Here we have used 31 GMA to predict the closed-form expressions for elastic constants of macro-scale chiral granular 32 metamaterial. It is shown that for macro-scale chirality, the grain-pair interactions must include 33 coupling between normal and tangential deformations. We have designed such a grain-pair 34 connection for physical realization and quantified with FE model. The verification of the 35 prediction is then performed using a physical model of 1D bead string obtained by 3D printing. 36 The behavior is also verified using a discrete model of 1D bead string.
    [Show full text]
  • Micromechanics Models for Viscoelastic Plain-Weave Composite Tape Springs
    AIAA JOURNAL Vol. 55, No. 1, January 2017 Micromechanics Models for Viscoelastic Plain-Weave Composite Tape Springs Kawai Kwok∗ Technical University of Denmark, 4000 Roskilde, Denmark and Sergio Pellegrino† California Institute of Technology, Pasadena, California 91125 DOI: 10.2514/1.J055041 The viscoelastic behavior of polymer composites decreases the deployment force and the postdeployment shape accuracy of composite deployable space structures. This paper presents a viscoelastic model for single-ply cylindrical shells (tape springs) that are deployed after being held folded for a given period of time. The model is derived from a representative unit cell of the composite material, based on the microstructure geometry. Key ingredients are the fiber volume density in the composite tows and the constitutive behavior of the fibers (assumed to be linear elastic and transversely isotropic) and of the matrix (assumed to be linear viscoelastic). Finite-element-based homogenizations at two scales are conducted to obtain the Prony series that characterize the orthotropic behavior of the composite tow, using the measured relaxation modulus of the matrix as an input. A further homogenization leads to the lamina relaxation ABD matrix. The accuracy of the proposed model is verified against the experimentally measured time- dependent compliance of single lamina in either pure tension or pure bending. Finite element simulations of single-ply tape springs based on the proposed model are compared to experimental measurements that were also obtained during
    [Show full text]
  • Variational Asymptotic Micromechanics Modeling of Composite Materials
    Utah State University DigitalCommons@USU All Graduate Theses and Dissertations Graduate Studies 12-2008 Variational Asymptotic Micromechanics Modeling of Composite Materials Tian Tang Follow this and additional works at: https://digitalcommons.usu.edu/etd Part of the Applied Mechanics Commons Recommended Citation Tang, Tian, "Variational Asymptotic Micromechanics Modeling of Composite Materials" (2008). All Graduate Theses and Dissertations. 72. https://digitalcommons.usu.edu/etd/72 This Dissertation is brought to you for free and open access by the Graduate Studies at DigitalCommons@USU. It has been accepted for inclusion in All Graduate Theses and Dissertations by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. VARIATIONAL ASYMPTOTIC MICROMECHANICS MODELING OF COMPOSITE MATERIALS by Tian Tang A dissertation submitted in partial ful¯llment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Mechanical Engineering Approved: Dr. Wenbin Yu Dr. Leijun Li Major Professor Committee Member Dr. Brent E. Stucker Dr. Thomas H. Fronk Committee Member Committee Member Dr. David E. Richardson Dr. Byron R. Burnham Committee Member Dean of Graduate Studies UTAH STATE UNIVERSITY Logan, Utah 2008 ii Copyright °c Tian Tang 2008 All Rights Reserved iii Abstract Variational Asymptotic Micromechanics Modeling of Composite Materials by Tian Tang, Doctor of Philosophy Utah State University, 2008 Major Professor: Dr. Wenbin Yu Department: Mechanical and Aerospace Engineering The issue of accurately determining the e®ective properties of composite materials has received the attention of numerous researchers in the last few decades and continues to be in the forefront of material research. Micromechanics models have been proven to be very useful tools for design and analysis of composite materials.
    [Show full text]
  • Micromechanics-Based Prediction of the Effective Properties of Piezoelectric Composite Having Interfacial Imperfections
    Micromechanics-based prediction of the effective properties of piezoelectric composite having interfacial imperfections Sangryun Leea, Jiyoung Junga, and Seunghwa Ryua,* Affiliations aDepartment of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea *Corresponding author e-mail: [email protected] Abstract We derive an analytical expression to predict the effective properties of a particulate- reinforced piezoelectric composite with interfacial imperfections using a micromechanics- based mean–field approach. We correctly derive the analytical formula of the modified Eshelby tensor, the modified concentration tensor, and the effective property equations based on the modified Mori–Tanaka method in the presence of interfacial imperfections. Our results are validated against finite element analyses (FEA) for the entire range of interfacial damage levels, from a perfect to a completely disconnected and insulated interface. For the facile evaluation of the nontrivial tensorial equations, we adopt the Mandel notation to perform tensor operations with 99 symmetric matrix operations. We apply the method to predict the effective properties of a representative piezoelectric composite consisting of PVDF and SiC reinforcements. Keywords Piezoelectric composite, Eshelby tensor, Interfacial damage, Effective modulus 1. Introduction Piezoelectricity refers to the electric charge accumulation in response to an applied mechanical loading, or, conversely, the mechanical
    [Show full text]
  • Viscoelastic Interphases in Polymer–Matrix Composites: Theoretical Models and finite-Element Analysis
    Composites Science and Technology 61 (2001) 731–748 www.elsevier.com/locate/compscitech Viscoelastic interphases in polymer–matrix composites: theoretical models and finite-element analysis F.T. Fisher, L.C. Brinson * Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA Received 21 January 2000; received in revised form 7 December 2000; accepted 18 January 2001 Abstract We investigate the mechanical property predictions for a three-phase viscoelastic (VE) composite by the use of two micro- mechanical models: the original Mori–Tanaka (MT) method and an extension of the Mori–Tanaka solution developed by Benve- niste to treat fibers with interphase regions. These micro-mechanical solutions were compared to a suitable finite-element analysis, which provided the benchmark numerical results for a periodic array of inclusions. Several case studies compare the composite moduli predicted by each of these methods, highlighting the role of the interphase. We show that the MT method, in general, provides the better micromechanical approximation of the viscoelastic behavior of the composite; however, the micromechanical methods only provide an order-of-magnitude approximation for the effective moduli. Finally, these methods were used to study the physical aging of a viscoelastic composite. The results imply that the existence of an interphase region, with viscoelastic moduli different from those of the bulk matrix, is not responsible for the difference in the shift rates, 22 and 66, describing the transverse Young’s axial shear moduli, found experimentally. # 2001 Published by Elsevier Science Ltd. All rights reserved. Keywords: Interphase; Micromechanics; Viscoelasticity; Finite element analysis; Physical aging 1. Introduction PMC with a viscoelastic interphase may be undertaken.
    [Show full text]
  • A New Definition of the Representative Volument Element in Numerical
    UNIVERSITE´ DE MONTREAL´ A NEW DEFINITION OF THE REPRESENTATIVE VOLUMENT ELEMENT IN NUMERICAL HOMOGENIZATION PROBLEMS AND ITS APPLICATION TO THE PERFORMANCE EVALUATION OF ANALYTICAL HOMOGENIZATION MODELS HADI MOUSSADDY DEPARTEMENT´ DE GENIE´ MECANIQUE´ ECOLE´ POLYTECHNIQUE DE MONTREAL´ THESE` PRESENT´ EE´ EN VUE DE L'OBTENTION DU DIPLOME^ DE PHILOSOPHIÆ DOCTOR (GENIE´ MECANIQUE)´ AVRIL 2013 © Hadi Moussaddy, 2013. UNIVERSITE´ DE MONTREAL´ ECOLE´ POLYTECHNIQUE DE MONTREAL´ Cette th`eseintitul´ee : A NEW DEFINITION OF THE REPRESENTATIVE VOLUMENT ELEMENT IN NUMERICAL HOMOGENIZATION PROBLEMS AND ITS APPLICATION TO THE PERFORMANCE EVALUATION OF ANALYTICAL HOMOGENIZATION MODELS pr´esent´ee par : MOUSSADDY Hadi en vue de l'obtention du dipl^ome de : Philosophiæ Doctor a ´et´ed^ument accept´eepar le jury d'examen constitu´ede : M. GOSSELIN Fr´ed´eric, Ph.D., pr´esident M. THERRIAULT Daniel, Ph.D., membre et directeur de recherche M. LE´VESQUE Martin, Ph.D., membre et codirecteur de recherche M. LESSARD Larry, Ph.D., membre M. BOHM¨ Helmut J., Techn. Doct., membre iii To my parents Safwan and Aisha To my wife Nour To my whole family... iv ACKNOWLEDGEMENTS I would like to thank Pr. Daniel Therriault and Pr. Martin L´evesque for their guidance throughout this project. Your insightful criticism guided me to complete a rigorous work, while gaining scientific maturity. Also, I want to thank Pr. B¨ohm, Pr. Lessard and Pr. Henry for being my committee members and Pr. Gosselin for being the Jury president. I owe special gratitude to the members of Laboratory for Multiscale Mechanics (LM2) research group. I would like to paricularly acknowledge the contributions of Maryam Pah- lavanPour by conducting parts of the analytical modeling included in this thesis, and for numerous fruitful dicussions all along this project.
    [Show full text]
  • Micromechanically Based Stochastic Finite Elements
    Finite Elements in Analysis and Design 15 (1993) 35-41 35 Elsevier FINEL 333 Micromechanically based stochastic finite elements K. Alzebdeh and M. Ostoja-Starzewski Department of Materials Science and Mechanics, Michigan State UniL'ersity, East Lansing, MI 48824-1226, USA Abstract. A stochastic finite element method for analysis of effects of spatial variability of material properties is developed with the help of a micromechanics approach. The method is illustrated by evaluating the first and second moments of the global response of a membrane with microstructure of a spatially random inclusion- matrix composite under a deterministic uniformly distributed load. It is shown that two mesoscale random continuum fields have to be introduced to bound the material properties and, in turn, the global response from above and from below. The intrinsic scale dependence of these two random fields is dictated by the choice of the finite element mesh. Introduction In elliptic boundary value problems there may be three different sources of randomness: distortion of the boundaries, fluctuation in the external force fields, and heterogeneity of the medium. All of these sources necessitate a stochastic generalization of the conventional solution methods. Such generalizations of the finite element method were attempted since the early eighties with a goal of grasping the randomness of either external force fields or medium's variability [1-6]. However, the latter aspect has been lacking a correct formulation in all the previous works, mainly due to purely hypothetical assumptions concerning the random field description of the material. That is, in the area of linear elastic structures, all these studies relied on a stochastic interpretation of the locally isotropic constitutive law o'ij = a(x, W)ekkaij + 2/x(x, w)Eij, (1) where the Lam6 constants are taken to be random fields.
    [Show full text]
  • Micromechanics Modeling of Functionally Graded Materials Containing Multiple Heterogeneities
    Vol. 26, No. 6, 392-397 (2013) DOI: http://dx.doi.org/10.7234/composres.2013.26.6.392 ISSN 2288-2103(Print), ISSN 2288-2111(Online) Paper Micromechanics Modeling of Functionally Graded Materials Containing Multiple Heterogeneities Jaesang Yu*†, Cheol-Min Yang**, Yong Chae Jung** ABSTRACT: Functionally graded materials graded continuously and discretely, and are modeled using modified Mori- Tanaka and self-consistent methods. The proposed micromechanics model accounts for multi-phase heterogeneity and arbitrary number of layers. The influence of geometries and distinct elastic material properties of each constituent and voids on the effective elastic properties of FGM is investigated. Numerical examples of different functionally graded materials are presented. The predicted elastic properties obtained from the current model agree well with experimental results from the literature. Key Words: functionally graded material (FGM), micromechanics, heterogeneity, composites 1. INTRODUCTION of three phase FGMs. Pindera et al. [10] used a computational micromechanical model, the generalized method of cells, to Due to their unique structure and multi-functionality, func- predict local stress in the fiber and matrix phases of FGMs. tionally graded materials (FGMs) are gaining attention and are Fang et al. [11], developed a micromechanics-based elasto- being proposed for many applications [1-3]. In FGMs, the dynamic model to predict the dynamic behavior of two-phase variation of material properties along a specific direction helps functionally graded materials. the material to have the desirable material properties of each Zuiker [12] reviewed the micromechanical modeling of constituent at a specific material point with structural conti- FGMs and concluded that the self-consistent method (SCM) nuity.
    [Show full text]