Efficient Multiscale Modeling Framework For Triaxially .

Table 1. Microstructural ParametersParameterDetermination methodVfV f 0 , V f θw0 , w θ , θ, t 0 , t θa0 , a θASTM D3171Assume 80% for polymer matrix compositesOptical microscope measurementShape function or approximationDownloaded from ascelibrary.org by Arizona State Univ on 07/10/19. Copyright ASCE. For personal use only; all rights reserved.w0 θ¼a0 V1f ðV f 0 þV f θm Þðcos1 θÞðt 0 þ 2t θ ÞFig. 4. Mesoscale RUC of a fiber tow w θð7ÞMesoscaleAt this point, the architecture is constrained in terms of severalmicrostructure parameters, but the tow cross-sectional area is stillvaguely defined. Therefore an assumption is made regarding thecross-sectional area of the tows. For simplicity of integration withthe generalized method of cells, the present analysis considers towsto be of a rectangular cross section of width w and thickness t. Thisyields a final equation providing the braided tow spacing expressedas follows in Eq. (8):w0 θ¼ðw0 t0 Þ V1f ðV f 0 þV f θm Þðcos1 θÞðt 0 þ 2t θ Þ w θð8ÞThe approximation of tows to be rectangular is an acceptableassumption in this case as the overall fiber volume fraction andlocal fiber volume fractions are accurately represented. Becausethe analysis portion of this paper only considers the effective elasticmoduli and first failure modes, the representation of the tows withrectangular cross section provides acceptable results. If further detail into nonlinear damage progression and failure is to be considered, a more refined cross section may be necessary. The axial towspacing is not arbitrary, as mentioned previously, and can be determined using Eqs. (3c) and (8), expressed as follows:1w00 ¼ðw þ w0 θ Þ w0 ð9Þcos θ θLast, the undulation angle of the braided tows can be defined interms of the thickness and the braided tow spacing. t 0 þ t θφ ¼ arctanð10Þw0 θsin θThe resulting characterization of the TriBC architecture is dependent on 10 microstructural parameters. Table 1 lists the parameterswith suggestions as to how they may be determined. Typically, in anidealized problem, it is assumed that the tow fiber volume fractionsV f 0 and V f θ are equivalent and uniform. If the cross-sectional areasare assumed to be rectangular, this results in only seven microstructural parameters being needed. The two parameters defining towspacing are constrained based on the parameters listed in Table 1.The mesoscale is used to represent discrete sections of fiber towsand matrix-rich regions. The fiber tows are assumed able to be represented by a doubly periodic RUC of dimensions h by l, consistingof constituents from the microscale. An example of such an RUCdiscretized for GMC is shown in Fig. 4, where the inner regiondenotes the fiber and the outer region is the matrix. The RUC isdiscretized in such a manner that it is composed of nβ nγ rectangular subcells, with each subcell having dimensions hβ lγ . Fromthis point forward, superscripts with lowercase Greek letters denotea specific subcell at the microscale, superscripts with uppercaseGreek letters denote a specific subcell at the macroscale, and nosuperscript denotes macroscale properties. Fiber tow packingand volume fraction typically govern the architecture of theRUC but must be in accordance with the previously describedRUC microstructural parameters. The resulting stress in the fibertow can be determined from the GMC homogenization process.In GMC, the current stress and current tangent moduli of a particular fiber tow at a point are determined through a volume averagingintegral over the repeating unit cell, as represented by a summationin Eqs. (11) and (12), producing the first homogenization in themultiscale modeling framework. In these equations, σ denotesthe Cauchy true stress, A denotes the strain concentration matrix,and C denotes the stiffness matrix (see Paley 1992 for details).The microscale subcell stresses and tangent moduli needed to complete the summation are determined through the applied constitutive models for each constituent based on their current strain state.Because the matrix-rich regions are monolithically represented bythe matrix constituent, the microscale response is propogated directly through to the mesoscale without need for homogenization. X ABΓnγn1 β XσABΓ ¼σβγ hβ lγð11Þhl β¼1 γ¼1C ABΓ ¼ X ABΓnγn1 β XC βγ Aβγ hβ lγhl β¼1 γ¼1ð12ÞMacroscaleHomogenizationMicroscaleThe present homogenization methodology represents the TriBCstarting with its constituent materials, i.e., the fiber and matrix. Thisis the only length scale where constitutive models are applied.Stress states and tangent moduli for larger length scales are determined through the GMC homogenization procedure (Paley 1992).The results presented here are based on simple linear elastic constitutive models to demonstrate the capabilities of the multiscalemodeling methodology, although advanced constitutive modelswill be implemented in the future.The RUC for the TriBC is modeled at the macroscale. For thisstudy, two unique types of triply periodic repeating unit cells, ofsize D H L and matching the dimensions given by the microstructural parameters, were discretized into nA nB nΓ parallelepiped subcells, with each subcell having dimensions d A hB lΓ .A simplified RUC was developed for the purpose of reducing computational effort and for eventual implementation in finite-elementanalysis and optimization algorithms. A refined RUC has also beendeveloped for material characterization purposes and providesmore detailed local stress/strain states (see Fig. 5).The simplified model discretizes the TriBC RUC into four architecturally governed sections through the width and four sections164 / JOURNAL OF AEROSPACE ENGINEERING ASCE / APRIL 2011J. Aerosp. Eng., 2011, 24(2): 162-169

h1 ¼ h4 ¼ t θð13b Þh2 ¼ h3 ¼ t0 ð13c Þl1 ¼ l3 ¼ w0 ð13d ÞDownloaded from ascelibrary.org by Arizona State Univ on 07/10/19. Copyright ASCE. For personal use only; all rights reserved.l2 ¼ l4 ¼ w0 1ðw þ w0 θ Þcos θ θð13e ÞWith the exception of subcells containing axial tows, all subcellshave an effective volume fraction that encompasses the resin-richsections that are not directly represented. As can be seen fromEq. (8), as the fiber volume fraction increases, w0 θ decreases, alongwith the volume of resin rich sections. This indicates that at highvolume fractions the simplified model is less of an approximation.The subcell effective volume fractions are determined by enforcingthe correct overall volume fraction as well as the correct volumefraction per column and per row of subcells. The overall fiber volume fraction in the simplified RUC is matched analytically to thatof the true RUC, with the final expression shown in Eq. (14). w0 θt 0 t θγ1V f ¼ V f 0 Vþ 1 t 0 þ t θðw θ þ w0 θ Þ f θ t 0 þ t θ γ1 þ γ2 w0 θγ þ γ2γγ2þ 1Vf 1 1 Vγ2γ2ðw θ þ w0 θ Þ f θ γ1 þ γ2ð14ÞFig. 5. (a) TriBC RUC; (b) simplified macroscale RUC isometric view;and (c) refined RUC top viewthrough the thickness, as shown in Fig. 5(b). The subcells throughthe width are separated into sections containing axial tows (Γ ¼ 1and Γ ¼ 3) and those that do not (Γ ¼ 2 and Γ ¼ 4). The subcellsthrough the width are separated into sections of either axial orbraided tows. For subcells with A ¼ 1 and A ¼ 4, the propertiesfrom the mesoscale are rotated by an angle θ to represent the braid;subcells Γ ¼ 2 and Γ ¼ 4 are rotated once more by angle φ torepresent the undulation. The end result is that each subcell contains only a single tow material, either axial or braided with correctorientation. No pure matrix regions are explicitly represented. Tomaintain a true representation of the previously described microstructural parameters, the geometric dimensions and fiber volumefraction of each subcell are analytically determined and expressedin terms of the microstructural parameters in Eq. (13).d1 ¼1ðw þ w0 θ Þsin θ θð13a ÞThe expressions for the fiber volume fractions of each subcellare given in Table 2 and have been determined through the use ofEqs. (1)–(10). The effective homogenized properties for each subcell are carried through the microscale using the tow (see Fig. 4)RUC representing that subcell’s respective volume fraction. It isimportant to note that undulation is represented in the Γ2 andΓ4 subcell columns through rotation of the effective microscaleproperties of the braided tow by angle φ.A refined RUC that further discretizes the TriBC RUC intosubcells that explicitly include the resin-rich regions is shown inFig. 5(c). Note that a top view of the RUC is shown in this case.This RUC also directly takes into consideration the braided towspacing, which was only effectively represented in the simplifiedmodel. The through-thickness discretization is identical to that ofthe simplified model, separating the axial and braided tows. The inplane discretization also follows a similar methodology to the simplified RUC, where subcell dimensions are governed by architectural parameters. This is in sharp contrast to a typical finite-elementmesh, in which element shape and dimensions lack physical meaning. In Fig. 5(c), key dimensions are shown and are defined inTable 3. These dimensions are derived from the microstructuralparameters mentioned previously. The refined TriBC RUC containsa total of 572 subcells.In a manner similar to the homogenization used at the mesoscale, the effective stresses and tangent moduli are determinedthrough a homogenizing volume averaging integral over themacroscale RUC. The homogenization is broken into two stepsas a way of introducing normal/shear coupling that is not presentin GMC. This method, as implemented within GMC, was firstTable 2. Simplified RUC Effective Fiber Volume Fraction by Subcell [see Fig. 5(b)]Subcell fA; B; ΓgFiber volume fractionf2; 1; 1g, f2; 1; 3g, f3; 1; 1g, f3; 1; 3gf1; 1; 1g, f1; 1; 3g, f4; 1; 1g, f4; 1; 3gf1 4; 1; 2g, f1 4; 1; 4gV f 0 ½1 w0 θ ðw θ þ w0 θ Þ V f θ½ðl1 þ l2 Þ l2 V f ðl1 l2 Þ½1 w0 θ ðw θ þ w0 θ Þ V f θJOURNAL OF AEROSPACE ENGINEERING ASCE / APRIL 2011 / 165J. Aerosp. Eng., 2011, 24(2): 162-169

Table 3. Refined RUC Subcell Dimension ParametersParameterValueð1 2 sin θÞðw θ w0 θ Þw0 θ sin θð1 2 cos θÞðw θ w0 θ Þw0 θ 2 cos θ0ð1 2Þf½ðw θ þ w θ Þ 2 cos θ w0 gðw0 θ 4 cos θÞ ð1 2Þ½ðw θ 2 cos θÞ w0 Downloaded from ascelibrary.org by Arizona State Univ on 07/10/19. Copyright ASCE. For personal use only; all rights reserved.x1x2x3x4x5x6presented by Bednarcyk (Bednarcyk 2000) and showed significantimprovement over a single-step homogenization. The RUC is firsthomogenized through the thickness and then in-plane. It is important to note that the stresses and stiffness are rotated into thecoordinate system of the macroscale from the local microscale.This occurs for the braided tows that are either in a rotated θor ( θ, φ) state. The in-plane (first step) stresses are determinedby homogenizing the mesoscale stress in the through thicknessdirection.A1XσABΓ d Ad A¼1nσBΓ ¼ð15ÞThe macroscale stresses (second step) are determined fromhomogenizing the in-plane stresses from Eq. (15).B XΓ1XσBΓ hB lΓhl B¼1 Γ¼1nσ¼nð16ÞThis same process is repeated to determine the effective stiffnessmatrix for each step, as represented in Eqs. (17) and (18).A1XAC ABΓ AABΓ1st step dd A¼1nC BΓ ¼B XΓ1XB ΓC BΓ ABΓ2nd step h lhl B¼1 Γ¼1nC¼ð17Þnð18ÞThe mesoscale stresses and stiffness appearing in Eq. (15) and(16) are determined by Eqs. (11) and (12), which in turn rely on thecurrent state of the microscale constituents for that particular mesoscale RUC. The finalized stiffness matrix at the macroscale can bewritten in terms of the microscale and mesoscale terms as shown inEq. (19). From the stiffness matrix the appropriate elastic constantscan be determined.C¼ ABΓ nγnnB XnΓnA X1X1X1 β XAC βγ Aβγ hβ lγAABΓd1st stephl B¼1 Γ¼1 d A¼1 hl β¼1 γ¼1 B ΓABΓ2nd step h lFig. 6. Example of mesoscale localization under transverse loadingLocalizationMesoscaleBecause GMC allows the determination of local fields using concentration matrices (matrices relating global stress and strains tolocal stresses and strains, denoted as A), it is possible to determinethe local stress and strain fields based on the global macroscalestress. In order to achieve this localization from the macroscaleto the mesoscale, the strain concentration matrix needs to be determined, but this is already computed during the homogenizationprocess. The two-step homogenization process used at the macroscale is, essentially, performed in reverse. Using the global strainsdetermined from the macroscale analysis and the strain concentration matrix for the second step (from Eq. (18)), the local in-planestrains (homogenized through the thickness) are determined.εBΓ ¼ ABΓ2nd step εð20ÞThe strains through the thickness can then be determined fromthe strains of Eq. (20) and the concentration matrix from the macroscale RUC [Eq. (17)].BΓεABΓ ¼ AABΓ1st step εð21Þð19ÞAt the macroscale, the effective homogenized tangent modulican be readily used to determine global stress and deformationbased on globally applied loads.The mesoscale steps are finally determined through the effectivestiffness matrix of Eq. (17).σABΓ ¼ C ABΓ εABΓð22ÞTable 4. Elastic Properties of T700/E862 and T700/PR520 Material SystemT700E862PR520Axial modulus (GPa)Transverse modulus (GPa)Axial Poisson’s ratioTransverse Poisson’s ratioShear modulus 3166 / JOURNAL OF AEROSPACE ENGINEERING ASCE / APRIL 2011J. Aerosp. Eng., 2011, 24(2): 162-169

Downloaded from ascelibrary.org by Arizona State Univ on 07/10/19. Copyright ASCE. For personal use only; all rights reserved.Fig. 6 demonstrates the mesoscale localization for the case of arefined TriBC RUC loaded transversely at the macroscale. Thelocalized mesoscale strains are contour plotted. The high strainin the pure matrix regions is captured well with the localizationtechnique.microscale RUC, the local strains for the constituents are determined and shown in Eq. (23).MicroscaleEq. (24) shows the relationship between the microscale strainsand stresses and the mesoscale strains.Localization from the mesoscale to the microscale requires the useof strain concentration matrices determined during mesoscalehomogenization (Eqs. (11) and (12)). This localization, like theprevious length scale, uses the local strains from the higher level(mesoscale) as global strains. The strains through the thicknessdetermined at the final step of the mesoscale localization serveas the global strains for the microscale localization. Using the strainconcentration matrix and these global strains, along with theεβγABΓ ¼ AβγABΓ εABΓσβγABΓ ¼ C βγABΓ εβγABΓð23Þð24ÞEq. (25) shows the final relationship between the microscalestresses and global applied strains.ABΓσβγABΓ ¼ C βγABΓ AβγABΓ AABΓ2nd step A1st step εð25ÞFig. 7. Elastic properties varied with braid angle and fiber volume fraction: (a) transverse modulus versus braid angle; (b) axial modulus versus braidangle; (c) shear modulus versus braid angle; (d) transverse modulus versus volume fraction; (e) axial modulus versus volume fraction; and (f) shearmodulus versus volume fractionJOURNAL OF AEROSPACE ENGINEERING ASCE / APRIL 2011 / 167J. Aerosp. Eng., 2011, 24(2): 162-169

Downloaded from ascelibrary.org by Arizona State Univ on 07/10/19. Copyright ASCE. For personal use only; all rights reserved.Fig. 8. Digital image correlation of TriBC specimen subject to transverse load and corresponding stress/strainOnce the local constituent stresses are determined, failure ordamage could be applied locally and then homogenized back tothe macroscale in the context of a simulation involving incrementally applied loading.60 braid, the volume fractions are equal transversally and axially,so CLT works well directly. For other brand angles this is not thecase and must be accounted for.Transverse TensionResults and DiscussionElastic ModuliA simple application of this multiscale methodology is the prediction of effective elastic properties at the macroscale. To validate thismethodology, predictive results are compared to experimental andfinite-element data for a T700/E862 material system with 56% fibervolume fraction. Using the multiscale modeling methodology, theelastic properties for the macroscale RUC are predicted for avarying volume fraction and braid angle and compared with limitedexperimental and finite-element results (Littell 2008). For the mesoscale RUC, the generalized method of cells RUC, which is a 2 2RUC, was applied. The linear elastic constituent properties used forthe analysis are presented in Table 4. Results are also compared toclassical lamination theory for reference.The predicted transverse Young’s, in-plane shear, and axialYoung’s moduli of the 56% triaxially braided T700/E862composite, as a function of braid angle and fiber volume fraction,are plotted in Fig. 7. The two RUCs bounded the predicted modulifor both axial and transverse moduli. The simplified RUC resultedin slightly stiffer properties, whereas the refined RUC presentedslightly compliant properties. The coarse and refined models onceagain bounded the shear modulus results. Digital image correlationshowed nonuniform state of shear strain in the gauge section. Dueto the inherently large RUC size of the TriBC, there was difficultyin measuring the macroscale properties. Future tubular test specimens are therefore planned. Results show good correlation to boththe experimental and finite-element results. In most cases, the simplified RUC and CLT followed similar trends. The CLT results arein good agreement with the multiscale GMC results for the transverse modulus for variation of braid angle, but the shear and axialmoduli differ greatly. This is due in part to the limitations of usingCLT directly without modification. When using CLT, the thicknesses of equivalent laminates must be varied to enforce that theoverall fiber volume fraction in each direction is enforced. For aAn additional application of the multiscale modeling methodologyis to predict the failure strengths or the onset of damage in TriBCs.The described multiscale GMC methodology is well-suited forsuch an application because the local fiber and matrix stress fieldsthroughout the composite can be calculated, as described earlier.As such, the onset of failure under transverse tensile loadingwas predicted for a 56% T700/PR520 TriBC. For this analysis,two assumptions about the failure modes were made. First, itwas assumed that failure occurred in only the matrix materialand was determined by the maximum stress criterion. Second,the axial tows (oriented at 90 to the applied transverse tensile loading) were assumed to fail first. These assumptions were made basedon experimental results for a T700/PR520 material system. Thesimplified RUC at the macroscale [see Fig. 5(b)] was used topredict the failure onset strength. A nominal failure strength of82 MPa, which was provided by the resin vendor, was used forthe PR520, along with the elastic properties given in Table 4. Firstmatrix failure within the axial tows was predicted to occur at338 MPa compared to the experimentally measured failure initiation stress of 344 MPa. Fig. 8 presents a digital image correlation ofa TriBC specimen loaded transversely until failure. The high strainregions denote the transverse tow failure. The corresponding stressstrain plot shows a distinct initial failure correlating to first towsplitting.ConclusionThe multiscale modeling methodology using the generalizedmethod of cells

Efficient Multiscale Modeling Framework for Triaxially Braided Composites using Generalized Method of Cells Kuang C. Liu1; Aditi Chattopadhyay2; Brett Bednarcyk3; and Steven M. Arnold4 Abstract: In this paper, a framework for a three-scale analysis, beginning at the co