Analysis Of Fiber Clustering In Composite Materials Using .
NASA/TM—2015-218838Analysis of Fiber Clustering in Composite MaterialsUsing High-Fidelity Multiscale MicromechanicsBrett A. BednarcykGlenn Research Center, Cleveland, OhioJacob AboudiTel Aviv University, Ramat Aviv, IsraelSteven M. ArnoldGlenn Research Center, Cleveland, OhioJuly 2015
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NASA/TM—2015-218838Analysis of Fiber Clustering in Composite MaterialsUsing High-Fidelity Multiscale MicromechanicsBrett A. BednarcykGlenn Research Center, Cleveland, OhioJacob AboudiTel Aviv University, Ramat Aviv, IsraelSteven M. ArnoldGlenn Research Center, Cleveland, OhioNational Aeronautics andSpace AdministrationGlenn Research CenterCleveland, Ohio 44135July 2015
AcknowledgmentsThe authors gratefully acknowledge Drs. Bradley A. Lerch and Pappu L.N. Murthy of NASA Glenn Research Centerwho provided the composite micrographs and the image pixel analysis results used herein.Trade names and trademarks are used in this report for identificationonly. Their usage does not constitute an official endorsement,either expressed or implied, by the National Aeronautics andSpace Administration.Level of Review: This material has been technically reviewed by technical management.Available fromNASA STI ProgramMail Stop 148NASA Langley Research CenterHampton, VA 23681-2199National Technical Information Service5285 Port Royal RoadSpringfield, VA 22161703-605-6000This report is available in electronic form at http://www.sti.nasa.gov/ and http://ntrs.nasa.gov/
Analysis of Fiber Clustering in Composite Materials UsingHigh-Fidelity Multiscale MicromechanicsBrett A. BednarcykNational Aeronautics and Space AdministrationGlenn Research CenterCleveland, Ohio 44135Jacob AboudiTel Aviv UniversityRamat Aviv 69978, IsraelSteven M. ArnoldNational Aeronautics and Space AdministrationGlenn Research CenterCleveland, Ohio 44135AbstractA new multiscale micromechanical approach is developed for the prediction of the behavior of fiberreinforced composites in presence of fiber clustering. The developed method is based on a coupled twoscale implementation of the High-Fidelity Generalized Method of Cells theory, wherein both the local andglobal scales are represented using this micromechanical method. Concentration tensors and effectiveconstitutive equations are established on both scales and linked to establish the required coupling, thusproviding the local fields throughout the composite as well as the global properties and effectivenonlinear response. Two nondimensional parameters, in conjunction with actual composite micrographs,are used to characterize the clustering of fibers in the composite. Based on the predicted local fields,initial yield and damage envelopes are generated for various clustering parameters for a polymer matrixcomposite with both carbon and glass fibers. Nonlinear epoxy matrix behavior is also considered, withresults in the form of effective nonlinear response curves, with varying fiber clustering and for two sets ofnonlinear matrix parameters.IntroductionClustering in composite materials refers to the aggregation of the constituents such that the compositedoes not exhibit uniform microstructural distribution. Composite microstructures are sensitive to thespecific manufacturing processes utilized, and because the processing cannot be perfectly controlled,some degree of clustering will always be present. Clustering is more prevalent in composites with lowfiber or inclusion volume fractions as there is more volume available in which one constituent may besegregated. In fact, in nanocomposites, whose volume fractions are typically very low, clustering is adominant phenomenon, which strongly influences the properties and performance of the compositematerial (c.f., Shaffer and Windle (1999), Vigolo et al. (2000), Shi et al. (2004)).Several investigators have presented methods to characterize the degree of clustering in compositematerials (c.f., Guild and Summerscales (1993), Pyrz (1994), Scanlon et. al. (2003), Al-Ostaz et al.(2007), Vaughan and McCarthy (2010), Wilding and Fulwood (2011), Zangenberg et al. (2013)). In termsof modeling the effects of fiber clustering in composites, most investigations have been finite elementbased, as this approach can explicitly account for the composite microstructure. Examples includeNASA/TM—2015-2188381
Sorensen and Talreja (1993), Ghosh et al. (1997), Yang et al. (1997, 2000), Zeman and Sejnoha (2001),Wongsto and Li (2005), Melro et al. (2008), Abhilash et al. (2011), and Romanov et al. (2013).Additionally, the Generalized Method of Cells semi-analytical micromechanics theory has been employedto model random fiber distributions in continuous composites with a moving-window technique (Baxterand Graham (2000), Graham-Brady et al. (2003), Baxter et al. (2005), Acton and Graham-Brady (2010)).In the present investigation, the triply-periodic High-Fidelity Generalized Method of Cells (HFGMC)(Aboudi et al., 2013) is first enhanced to enable coupled multiscale analysis, wherein both the local andglobal scales are synergistically linked. The HFGMC method determines the strain (or stress)concentration tensors, which are used to establish the macroscopic constitutive equations of thecomposite, and also to provide the local stress and strain fields throughout the composite. This enables theprediction of not only effective composite properties, but also the effective nonlinear response (due tolocal inelasticity and damage) in response to full multiaxial loading, as well as generation of initial yieldand damage envelopes. The present multiscale HFGMC implementation considers an arbitrary number oflocal repeating unit cells (RUCs), which consist of the composite constituent materials (e.g., fiber andmatrix). These local RUCs are then arranged in a global RUC, whose behavior represents that of themultiscale composite material. The scales are explicitly coupled so that any local nonlinearity (which canbe predicted using the local fields) is homogenized and passed to the global scale analysis. The examplespresented here utilize a two scale analysis, but the methodology could be generalized to admit an arbitrarynumber of scales, as has been done in the case of the Multiscale Generalized Method of Cells (Liu et al.,2011, Liu and Arnold, 2013; Bednarcyk et al., 2015).Herein, the multiscale HFGMC approach is employed to examine the effects of fiber clustering incontinuous fiber reinforced composites. A key aspect of the approach taken involves segregating thecomposite microstructure into two zones, A and B, see Figure 1. Each zone consists of a compositematerial with its own fiber volume fraction, thus, by varying the fiber volume fractions and thearrangement/size of zones A and B, fiber clustering can be represented. For example, if the fiber volumefraction in zone A is higher than that in zone B, the fiber-rich zone A represents a clustered region.Conversely, if the fiber volume fraction in zone B is higher than that in zone A, zone A will be a matrixrich region as compared to zone B. It should be noted that the present method admits an arbitrary numberof zones. The alternative single scale HFGMC or finite element approach to modeling fiber clusteringwould involve full discretization of the actual composite microstructure. Obviously, this implies the useof a very dense mesh with great potential for meshing issues in the local fiber-matrix regions. In contrast,the proposed multiscale HFGMC methodology captures the effects of fiber clustering through local scalefiber volume fraction variations. Thus coarse meshes can be used on both scales, avoiding these meshingconcerns while balancing efficiency and fidelity. Further, while mean field approaches (e.g., Mori-Tanakamethod) can predict reliable effective properties, the predicted mean fields do not capture the variations ofthe stress fields in the composite constituents, even in the case of uniform fiber distribution. Utilization ofmean fields to predict initial yield envelopes was shown to be problematic via comparison to finiteelement micromechanics analyses by Pindera and Aboudi (1988) in the case of uniform fiber distribution.In the presence of fiber clustering, the matrix stress field variations are clearly important, thus motivatingthe use of a higher fidelity micromechanical approach such as HFGMC. It should be noted, however, thatlower fidelity methods with better computational efficiency, such as the Generalized Method of Cells orthe Mori-Tanaka Method, could be used to generate the results like those presented herein. The impact onthese results of the better local fields given by HFGMC versus other methods has not been investigated.NASA/TM—2015-2188382
Figure 1.—Schematic drawing of a compositedivided into two zones, A and B, which havedifferent fiber volume fractions.In the present investigation, actual micrographs from polymer matrix composites are used todetermine these two zones within the composite RUC. The fiber clustering is characterized by twonondimensional parameters, one representing the volume fraction of zone A within the composite, and theother representing the fraction of the fibers that are located in zone A. Studies are performed by varyingthese parameters and observing their impact on the predicted initial global-scale yield and damageenvelopes, as well as the nonlinear global stress-strain response. It should be noted that the availability ofthe concentration tensors in HFGMC enables straightforward and efficient generation of these initialenvelopes. In contrast, the use of the standard finite element method would require repeated application ofmany loading combinations to trace out these surfaces.The remainder of this paper is organized as follows. First, the multiscale HFGMC theory is presentedfor the generally triply-periodic case. Due to the possible nonlinearity of the constituent materials, anincremental (tangential) formulation is employed. The local and global concentration tensors andconstitutive equations, as well as the coupling between them, are established. Next the methodology fordetermination of the initial yield and damage envelopes is described, followed by the method used tomodel fiber clustering. The Results and Discussion section details the processing of the compositemicrographs to determine the analyzed composite RUCs. Results in the form of predicted effectiveproperties, initial yield and damage envelopes, and macroscopic nonlinear stress-strain curves arepresented for carbon/epoxy, glass/epoxy, and two different nonlinear representations of the epoxy matrix.The Conclusion section summarizes the paper and offers future research directions.Multiscale High-Fidelity Generalized Method of CellsIn order to model the clustering of fibers in a composite material, the High-Fidelity GeneralizedMethod of Cells (HFGMC) micromechanical model has been implemented in a two-scale framework.These two scales are referred to as global and local. The global scale represents a repeating unit cell of aperiodic composite material, whose constituents are themselves periodic composite materials. Thus, thelocal scale represents the RUCs present within the global scale constituents, see Figure 2.NASA/TM—2015-2188383
(a)(b)(c)Figure 2.—(a) A multiphase composite with triply-periodic microstructure defined with respect to global coordinates(X1, X2, X3). (b) Its repeating unit cell (RUC) is represented with respect to the coordinates (Y1, Y2, Y3) and isdivided into NΑ, NΒ, and NΓ global-scale subcells, in the Y1, Y2, and Y3 directions, respectively. (c) The materialswithin the global-scale subcells (ΑΒΓ) are represented by local-scale RUCs, defined with respect to localcoordinates (y1, y2, y3), and is divided into Nα, Nβ, and Nγ local-scale subcells.Global Scale AnalysisThe HFGMC theory, which has been fully described by Aboudi et al. (2013), considers a compositematerial with triply-periodic microstructure, Figure 2(a), wherein periodicity conditions are enforced inall three Cartesian coordinate directions. The global repeating unit cell (RUC), Figure 2(b), defined withrespect to local coordinates (Y1, Y2, Y3), is divided into NΑ, NΒ, and NΓ global subcells in the Y1, Y2, and Y3directions, respectively. Each global subcell is labeled by the indices (ΑΒΓ) with Α 1 , ,NΑ,Β 1, ,NΒ and Γ 1, ,NΓ, and may contain a distinct homogeneous material or a composite material.The dimensions of the RUC are D, H, and L, whereas the dimensions of global subcell (ΑΒΓ) in the Y1,Y2, and Y3 directions are denoted by DΑ, HΒ, and LΓ, respectively. A coordinate system ( Y2( Α) , Y2(Β) , Y3(Γ ) )is introduced in each subcell whose origin is located at its center. The global subcell nonlinear elasticconstitutive equation of the anisotropic material is given in an incremental form by,NASA/TM—2015-2188384
( ΑΒΓ ) ( ΑΒΓ ) σij( ΑΒΓ ) Cijkl ε kl(1)(ΑΒΓ ))where σ ij(ΑΒΓ ) , ε (ΑΒΓ, and Cijklare the components of the stress increment, strain increment, andklinstantaneous (tangent) stiffness tensors of global subcell (ΑΒΓ), respectively.The basic assumption in HFGMC is that the increment of the displacement vector U i(ΑΒΓ ) in eachglobal subcell is represented as a second-order expansion in terms of its coordinates ( Y1( Α) , Y2(Β) , Y3(Γ ) ) , asfollows,ΑΒΓ )( ΑΒΓ )(Γ)( ΑΒΓ )( Β))( Α) U i( ΑΒΓ ) εij X j Wi((ΑΒΓ Wi((100) Y2 Wi ( 010) Y3 Wi ( 001)000) Y1D21 3Y1( Α ) 2 Α2 422 ) 1 (Β) 2 H Β ( ΑΒΓ ) 1 ( Γ ) 2 LΓ Wi((ΑΒΓ 33 YWY200)i ( 020) 2 24 2 34 ) Wi((ΑΒΓ002) (2))where εij are the applied (external) average strain increments, and the unknown terms Wi((ΑΒΓlmn ) mustbe determined from the fulfillment of the equilibrium conditions, the periodic boundary conditions, andthe interfacial continuity conditions of displacements and tractions between global subcells. The periodicboundary conditions ensure that the increments of displacement and traction at opposite surfaces of theglobal RUC are identical. A principal ingredient in the HFGMC micromechanical analysis is that all theseconditions are imposed in the average (integral) sense.As a result of the imposition of these conditions, a linear system of algebraic equations is obtained,which can be represented in the following form:K V F(3)where the matrix K contains information on the global geometry and instantaneous properties of thematerials (or composites) within the individual subcells (ΑΒΓ), and the displacement vector increment,) V, contains the unknown displacement coefficients Wi((ΑΒΓlmn ) , which appear on the right-hand side ofEquation (2). The vector ΔF contains information on the applied average strain increments εij . Thesolution of Equation (3) enables the establishment of the following localization relation which expressesthe average strain increments εij(ΑΒΓ ) in the global subcell (ΑΒΓ) to the externally applied averagestrain increments εij in the form,( ΑΒΓ ) εij( ΑΒΓ ) Aijkl ε kl(4)(ΑΒΓ )where Aijklare the instantaneous global strain concentration tensor components, of the subcell (ΑΒΓ).The final form of the effective incremental constitutive law of the multiphase composite, whichrelates the average stress increments σij and strain increments εkl , is established as follows:NASA/TM—2015-2188385
ε σij Cijklkl(5) are components of the instantaneous effective global stiffness tensor, which areIn this equation Cijklgiven by, CijklNNNΑΒΓ1( ΑΒΓ ) ( ΑΒΓ )DΑ H Β LΓ CijpqA pqklDHL Α 1 Β 1 Γ 1 (6)(ΑΒΓ )Next, the components of the global instantaneous stress concentration tensor, Bijkl, which relate theaverage stress increments in the global subcell, σij(ΑΒΓ ) , to the average (global) stress increments, σij ,are determined. By combining Equations (1) and (4), the subcell stresses are given by,( ΑΒΓ ) ( ΑΒΓ ) σij( ΑΒΓ ) CijpqA pqkl ε kl(7)Then, using Equation (5), one obtains,( ΑΒΓ ) σij( ΑΒΓ ) Bijkl σkl(8)( ΑΒΓ )( ΑΒΓ ) ( ΑΒΓ ) Bijkl CijpqA pqrs S rskl(9)where, and S rsklare components of the effective instantaneous global compliance tensor, which are thecomponents of the inverse of the instantaneous global effective stiffness tensor, C*, see Equation (6).Local Scale AnalysisAs stated above and shown in Figure 2, the global subcells can be occupied by periodic compositematerials, whose responses are determined by the micromechanical analysis of local RUCs. Therefore theglobal scale analysis subcell quantities are the RUC quantities in the local scale analysis. As such, theglobal subcell incremental constitutive equation (1) serves as the RUC constitutive equation for the localscale. The local subcell incremental nonlinear elastic constitutive equation is given by,( ΑΒΓ ,αβγ ) σ ij( ΑΒΓ,αβγ ) Cijkl ε (klΑΒΓ,αβγ )(10)where the indices (ΑΒΓ) indicate the global subcell and the indices (αβγ) indicate the local subcell. As inthe global scale analysis, the increment of the displacement vector ui( ΑΒΓ,αβγ ) in each local subcell isexpanded into quadratic forms in terms the coordinates ( y1( ΑΒΓ,α ) , y 2( ΑΒΓ,β) , y3( ΑΒΓ, γ ) ) centered in thelocal subcell, as follows,NASA/TM—2015-2188386
,αβγ ),αβγ ),αβγ ) ui( ΑΒΓ ,αβγ ) εij( ΑΒΓ ) Y j( p ) wi((ΑΒΓ y1( ΑΒΓ,α ) wi((ΑΒΓ y2( ΑΒΓ,β) wi((ΑΒΓ000)100)010) ( ΑΒΓ,α ) 2 d (2ΑΒΓ,α ) ( ΑΒΓ,αβγ ),αβγ ) 1 w y3( ΑΒΓ,γ ) wi((ΑΒΓ 3y001) i ( 200)2 14 1 3 y2( ΑΒΓ,β) 2 2 h(2ΑΒΓ,β)4 ( ΑΒΓ,αβγ ) 1 ( ΑΒΓ,γ ) 2 w 3 y3 i (020)2 l(2ΑΒΓ,γ )4(11) ( ΑΒΓ,αβγ ) w i (002) where p Α,Β,Γ for j 1,2,3, respectively, and d(ΑΒΓ,α), h(ΑΒΓ,β), and l(ΑΒΓ,γ)are the dimensions of the local,
Analysis of Fiber Clustering in Composite Materials Using High-Fidelity Multiscale Micromechanics Brett A. Bednarcyk National Aeronautics and Space Administration Glenn Research Center Cleveland, Ohio 44135 Jacob Aboudi Tel Aviv University Ramat Aviv 69978, Israel Steven M. Arnold National Aeronautics and Space Administration Glenn Research Center
Caiado, J., Maharaj, E. A., and D’Urso, P. (2015) Time series clustering. In: Handbook of cluster analysis. Chapman and Hall/CRC. Andrés M. Alonso Time series clustering. Introduction Time series clustering by features Model based time series clustering Time series clustering by dependence Introduction to clustering
properties of fiber composites [1]. A number of tests involving specimens with a single fiber have been developed, such as single fiber pull-out tests, single fiber fragmentation tests and fiber push-out tests [2-4]. Yet it still remains a challenge to characterize the mechanical properties of the fiber/matrix interface for several reasons.
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