Automated Computational Mechanics Laboratory | - Effects of ......An integrated computational framework is presented for the automated modeling and simulation of the failure response

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Computational MechanicsSolids, Fluids, Structures, Fluid-Structure Interactions, Biomechanics,Micromechanics, Multiscale Mechanics,Materials, Constitutive Modeling,Nonlinear Mechanics, Aerodynamics ISSN 0178-7675 Comput MechDOI 10.1007/s00466-018-1634-1

Effects of shape and misalignment of fiberson the failure response of carbon fiberreinforced polymers

Hossein Ahmadian, Ming Yang, AnandNagarajan & Soheil Soghrati

Computational Mechanicshttps://doi.org/10.1007/s00466-018-1634-1

ORIG INAL PAPER

Effects of shape andmisalignment of fibers on the failure response ofcarbon fiber reinforced polymers

Hossein Ahmadian1 ·Ming Yang2 · Anand Nagarajan2 · Soheil Soghrati3

Received: 13 May 2018 / Accepted: 25 August 2018© Springer-Verlag GmbH Germany, part of Springer Nature 2018

AbstractAn integrated computational framework is presented for the automated modeling and simulation of the failure responseof carbon fiber reinforced polymers (CFRPs) with arbitrary-shaped, randomly-misaligned, embedded fibers. The proposedapproach relies on a new packing/relocation-based reconstruction algorithm to synthesize realistic 3D representative volumeelements (RVEs) of CFRP. A non-iterative mesh generation algorithm is then employed to create high-quality finite elementmodels of each RVE. The failure response of CFRP is simulated using ductile and cohesive-contact damage models for theepoxy matrix and along fiber-matrix interfaces, respectively. In addition to studying the impact of fiber misalignments, thiscomputational framework is employed to investigate the effect of cross-sectional geometry of fibers (circular versus ovalshaped) on the strength, ductility, and toughness of CFRP subject to tensile and compressive loads applied transverse to thefibers direction.

Keywords Fiber reinforced composite · Misalignment · Cross-sectional geometry · Damage · Finite element

1 Introduction

There has been a significant growth in the application ofcarbon fiber reinforced polymers (CFRPs) in the aerospaceand automotive industry within the past few decades [1].The industrial application of CFRPs was pioneered by theaerospace sector for manufacturing structural components ofthe fuselage. Compared to high-strength aluminum and steelalloys, CFRPs provide unique advantages such as a higherfatigue life, better corrosion resistance, and high strength-to-weight ratio [2,3], which are very attractive for the design ofaerospace structures. More recently, the increasing demandfor manufacturing lightweight vehicles that meet strict fuel

B Soheil [email protected]

1 Department of Integrated Systems Engineering, The OhioState University, 1971 Neil Avenue, Columbus, OH 43210,USA

2 Department of Mechanical and Aerospace Engineering, TheOhio State University, 201 W 19th Avenue, Columbus,OH 43210, USA

3 Department of Mechanical and Aerospace Engineering,Department of Materials Science and Engineering, The OhioState University, 201 W 19th Avenue, Columbus, OH 43210,USA

economy and safety regulations has promoted the applica-tion of CFRPs in the automotive industry [4]. However,the high cost of CFRPs compared to conventional metallicalloys is still a major barrier toward their widespread use inthis industry. To reduce this cost, one solution is to replaceaerospace-grade carbon fibers, which have circular-shapedcross-sections and a low diameter variability, with cheaperfibers that oftendonot attain suchproperties.Anothermethodto reduce the manufacturing cost is to accelerate the man-ufacturing process, which could cause small-angle fibermisalignments in each ply. In this article, we investigatethe effects of cross-sectional geometry (circular versus oval)and misalignments between fibers on the failure responseof CFRPs subject to tensile and compressive loads appliedtransverse to the fiber direction.

Predicting the mechanical behavior (strength, toughness,etc.) of CFRPs requires the ability to create realistic mod-els of the composite microstructure, which is characterizedby the volume fraction, shape, size distribution, and spatialarrangement of fibers. However, creating multiple realisticrepeating unit cells (RUCs) of a CFRP microstructure andvirtually modifying their geometrical features (e.g., volumefraction and spatial arrangement of fibers) throughout thedesign process could be a challenging task. One can buildsuchmodels directly based on imaging data such as Scanning

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Computational Mechanics

ElectronMicroscope (SEM) andmicro-Computed Tomogra-phy (micro-CT) images [5–7]. However, the expensive andlaborious nature of the image preparation, as well as chal-lenge posed by identifying interfaces between fibers thatare in close proximity could prohibit the direct image-to-model transformation process [8]. Even if such challengesare resolved, this approach only yields a single microstruc-tural model of an existing CFRP, which is insufficient for usein the computational design or for quantifying the uncertaintyof the mechanical behavior due to microstructural variations.

Alternatively, an appropriate microstructure reconstruc-tion algorithm can be implemented to virtually createthe microstructural model of a CFRP [9]. For exam-ple, a descriptor-based technique [10,11], together withan optimization algorithm such as the Genetic Algorithm(GA) [12,13], can be employed to replicate desired statisticaldescriptors (particles/fibers shape, volume fraction, spatialarrangement, etc.) in the reconstructedmicrostructure. In cor-relation function-based reconstruction algorithms [14,15],techniques such as the Voronoi tessellation [16,17], Ran-dom Sequential Adsorption (RSA) [18,19], and randomfield-based method [20,21] can be utilized to synthesizethe microstructure. An iterative stochastic method such asthe Monte-Carlo [22], pixel switching [23,24], or the mass-spring mutation operator [25,26] is then applied to thesynthesized microstructure to replicate target correlationfunctions.

Simulating the failure response of composite materialsusing the finite element method (FEM) also requires the con-struction of an appropriate conforming mesh to discretizereconstructed RUC. Several sophisticated algorithms suchas the Delaunay triangulation [27], Octree-based methods[28,29], and the advancing front [30,31] have been developedto build high-quality conforming meshes with proper ele-ment aspect ratios and a negligible geometric discretizationerror. However, the iterative phase used in such algorithmsto improve the quality of elements could be computationallydemanding andmay even fail to converge formodelingmate-rials with intricate 3Dmicrostructures. In order to avoid suchchallenges, one can implement enriched FE methods suchas the eXtended/Generalized FEM (X/GFEM) [32–34] andthe hierarchical interface-enriched FEM (HIFEM) [35,36]that allow using a nonconforming mesh for discretizing thedomain. However, the successful implementation of suchmethods requires resolving several new challenges, such asimposing Dirichlet boundary conditions, ill-conditioning ofthe stiffness matrix, and most importantly accurate recov-ery of the gradient field (stresses) along material interfaces[35,37,38]. Note that the latter has a crucial impact on thefidelity of simulations for predicting the initiation and evo-lution of damage in a CFRP microstructure.

In addition to reconstructing themicrostructure and gener-ating a conformingmesh, simulating the failure response of a

CFRP requires using appropriate constitutive damage mod-els for the matrix and fiber phases, and along their interfaces.It has been shown that when macroscopic loads are appliedtransverse to the fibers direction, fiber-matrix debonding isone of the main damage mechanisms governing the mechan-ical behavior [39,40]. Under transverse compression andshear loads, failure is dominated by the formation of shearbands in the matrix [41,42]. For loadings applied longitu-dinally along the fibers direction, one must also take intoaccount the fracture of fibers and the kink-band formationas potential damage mechanisms [43]. Several computa-tional studies have been carried out to characterize the failureresponse of CFRPs subject to transverse and longitudinalloadings with respect to the fibers direction [44–48], someof which rely on 3D FE models [49–52].

In this manuscript, we implement an automated computa-tional framework for modeling CFRPs with arbitrary-shapedembedded fibers to investigate effects of cross-sectionalgeometry and small-angle misalignments between fibers onthe failure response under tensile and compressive loadsapplied in the transverse fibers direction. This is achievedby further expanding reconstruction and meshing algorithmsused in [9] for modeling CFRPs with embedded circular-shapedfibers, aswell as improving damagemodels to captureplastic deformation of the matrix and fiber-matrix contactunder compression. A literature survey shows that only ahandful of prior computational and experimental studieshave attempted at characterizing the mechanical behaviorof CFRPs with non-circular shaped fiber cross-sections.It has been shown that carbon fiber reinforced cementi-tious composites with C-shaped fiber cross sections provideimproved mechanical properties compared to those rein-forced with circular-shaped fibers [53,54]. Experimentaltesting has shown that reinforcing the polymer matrix withkidney shaped carbon fibers could improve the interfacialshear strength of the composite [55]. According to [56],carbon fibers with triangular-shaped cross-sections increasethe flexural strength of CFRP. In a more recent study, asignificant increase in damping properties of CFRPs hasbeen reported with four-lobed star shaped carbon fibers [57].The computational study presented in [58] shows that undertransverse compression, composites reinforced with lobularfibers yield a higher strength compared to those with embed-ded circular-shaped fibers. Pathan et al. [59] numericallystudied the mechanical behavior of CFRPs with concave-shaped embedded fibers (e.g., star-shaped) and concludedthat such fibers provide enhanced stiffness but a lower damp-ing effect. In another study, Yang et al. [60] reported slightlyimproved transverse stiffness and strength when using fiberswith triangular-shaped cross-sections compared to circular-shaped fibers.

Another important microstructural feature of CFRPs thathas not been rigorously studied in the past is the impact of

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fiber misalignment on the failure response [61,62]. Majorityof computational studies mentioned in paragraphs above relyon 2D models, which can only predict the failure responseof an ideal CFRP with perfectly aligned fibers subject totransverse loads. However, depending on the manufacturingprocess, some of the embedded fibers might be misalignedwith the principal longitudinal fiber direction [63]. Signifi-cant misalignment of fibers (e.g., fiber kinking) could havea notable impact on the mechanical behavior of a CFRP,including its longitudinal tensile modulus [64,65], longitudi-nal compressive strength [66,67], and transverse mechanicalproperties [68,69]. It has been shown that misaligned fibersin particular deteriorate the compressive strength of CFRPlaminae [70,71]. Under compression, misaligned fibers aresubject to shear deformation, which lead to the acceleratedfailure of the surrounding matrix, followed by the formationof kink-bands [72]. Predicting the compressive strength ofsuchCFRPshas been the subject of several analytical [73–75]and numerical [76,77] studies. Gutkin et al. [78] presented a2D micromechanical model to investigate the effect of uni-form misalignments of fibers on the failure response subjectto in-plane shear and longitudinal compression. By conduct-ing high-fidelitymicromechanical simulations, Bai et al. [79]showed that the shear stress developed along fibers-matrixinterfaces is the main cause of the formation of kink bands.More recently, a coupled plasticity-damage model has beenimplemented in [80] to investigate the effect of fiber kinkingon the longitudinal compressive strength of a CFRP. How-ever, none of the previous works have studied the impact ofrandom, small angular misalignments between fibers on themechanical behavior of CFRPs.

In the remainder of this manuscript, we first presentthe governing equations for modeling the micromechan-ical behavior of CFRP in Sect. 2, together with a briefdescription of continuum and cohesive-contact damagemod-els used for simulating its failure response under transversetension and compression. A microstructure reconstructionalgorithm is employed in Sect. 3 for synthesizing 3D peri-odic RUCs of CFRPs with arbitrary-shaped fibers, which canalso take into account misalignments between fibers. Thisalgorithm is composed of two main phases, starting withthe virtual packing of fibers in RUC and then implement-ing a relocation-based optimization phase to simulate theirtarget spatial arrangement. In Sect. 3, we also show the appli-cation of a non-iterative mesh generation algorithm namedConforming to Interface Structured Adaptive Mesh Refine-ment (CISAMR) [81,82] for creating high-quality FEmodelsof each RUC. A thorough study is presented in Sect. 4 onthe effects of cross-sectional geometry and misalignments offibers on the failure response of CFRP subject to transversetensile and compressive loads. Final concluding remarks aresummarized in Sect. 5.

There are several novel aspects pertaining the algorithmsused and the study conducted in this manuscript, whichworth clarifying before proceeding further. The computa-tional framework used for creating 3D FE models of CFRPmicrostructures relies on automated microstructure recon-struction and mesh generation algorithms, where comparedto previous works by the authors [83], new aspects areintroduced in the former to allow incorporating random mis-alignments between fibers. The meshing algorithm is alsoenhanced to enable modeling cohesive debonding and con-tact along fiber-matrix interfaces during damage simulations.The main novelty of this article, however, pertains to thestudies presented in Sect. 4 to quantify effects of the cross-sectional morphology and misalignments between fibers onthe failure response of CFRP. Regarding the impact of cross-sectional shapes of fibers, this article is one of the very fewstudies that considers realistic shape and size distribution offibers in 3D reconstructed FE models. Further, to the best ofthe authors’ knowledge, the current work is the first studyaiming at quantifying the impact of random misalignmentsbetween fibers on the failure response of CFRPs.

2 Problem formulation

2.1 Micromechanical analysis

Consider a CFRP panel described in the macroscopic coor-dinate system xM. Also, assume that open domain Θ withboundary Λ and outward unit normal vector nm representsan RUC of this composite material in the microscopic coor-dinate system xm. A first order asymptotic expansion ofthe displacement field u(xM, xm) can be used to decomposethat into macroscopic uM(xM) and microscopic um(xM, xm)fields as

u(xM, xm) = uM(xM)+ ξum(xM, xm), (1)

where the asymptotic scaling parameter ξ ≪ 1 is the ratioof the microscopic to macroscopic length scales. The equi-librium equation at the microscale can then be expressed as

∇C : (εM + εm) = 0 in Θ, (2)

where C is the elasticity tensor, and εM and εm are macro-scopic and microscopic strain tensors, respectively.

We employ the Hill–Mandel micro-homogeneity princi-ple [84] to link the macroscopic potential energy densityΦM = 1

2εM and the microscopic energy density Φm in Θ

as

infuM

ΦM(εM) = infεM

infum

1|Θ|

∫

ΘΦm(εM + εm) dΘ, (3)

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where |Θ| is the measure of Θ . Also, Φm is given by

Φm = 12(εM + εm) : σm, (4)

where σm is the microscopic Cauchy stress tensor. Note thatthe macroscopic stress σM corresponding to Θ is given by[85]

σM = 1|Θ|

∫

Θσm dΘ. (5)

2.2 Boundary conditions

Boundary conditions (BCs) imposed on Θ must satisfy thestrain averaging theorem expressed as

εM(xM) = 1|Θ|

∫

Θεm(xm) dΘ. (6)

Several types of BCs can satisfy this condition, includ-ing applied tractions or prescribed displacements evaluatedbased on macroscopic stresses and strains. However, suchchoices often cause unrealistic damage localizations neardomain boundaries and require using a largermicrostructuralmodel as RVE, which in turn leads to a higher computationalcost [86,87]. To avoid this issue, one can impose periodic BC(PBC) along parallel faces Λi and Λ j of RUC as [9]

um|Λi = um|Λ j . (7)

The mechanical loading is then applied as a macroscopicstress or strain to every point in Θ . Despite such advantages,applying PBC to an RUC would only be feasible if it hasperiodic geometrical features. In this work, we will imple-ment amicrostructure reconstruction algorithm (described inSect. 3) to virtually create periodic microstructural models ofCFRPs with perfectly aligned fibers. However, incorporatingmisaligned fibers in the virtual microstructure results in thelack of periodicity for RUC faces perpendicular to the prin-cipal longitudinal fibers direction; hence it would no longerbe feasible to impose PBC on these faces. Thereby, we willlater study appropriate boundary conditions for these facesand use larger RUCs to compensate for potential boundaryeffects in numerical simulations.

2.3 Continuum damagemodel

While the polymer matrix of CFRP shows a brittle failureresponse in tension, it undergoes a significant amount of plas-tic deformation before the propagation of damage subject tocompressive loads. In order to simulate this behavior, theyield criterion in the damage model must be a function of thestress triaxiality η = − p

q , where p is the hydrostatic stress

and q is the effective von-Mises stress. In this work, this isachieved by implementing a phenomenological ductile dam-age model [88,89]. Using the strain equivalence hypothesis,the strain response of the damaged material can be evalu-ated by replacing the actual microscopic stress σm with aneffective stress σ eff as [90]

σ eff =σm

1 − w, (8)

where the scalar parameter w yields the magnitude of dam-age, ranging from 0 (undamaged) to 1 (fully damaged). Theyield surface can be expressed as

f (σ ) = q − σy(εpeq), (9)

where εpeq is the equivalent plastic strain tensor and σy(ε

peq)

is the yield function.The onset of damage in the matrix occurs when the equiv-

alent plastic strain reaches a threshold value εp0 , which is a

function of the stress triaxiality and equivalent plastic strainrate ε

p0 given by

εp0 = ε

p0 (η, ε

p0 ). (10)

A thorough discussion on the dependence of the equivalentplastic strain threshold on the stress triaxiality is presentedin [88]. The damage initiates and subsequently grows when

∫dε p

εp0 (η, ε

p0 )

= 1, (11)

which monotonically increases with the plastic deformationuntil it reaches the initiation state. The propagation of damagein the matrix (i.e., growth of ω from 0 to 1) will then lead tothe mechanical degradation of the load bearing capacity.

In order to avoid severe ill-conditioning and challengesassociated with modeling contact along fiber-matrix inter-faces, in FE simulations presented hereafter, an element isremoved from the mesh when ω reaches the critical valueof 1. Also, mesh dependency effects are alleviated by intro-ducing the length scale L for each element into the damagemodel.In this non-local regularization approach, the soften-ing response is characterized using a stress–displacementrelationship [91], in which L is used to evaluate the frac-ture energy G f as

G f =∫ ε

peq

εp0

Lσydεpeq =

∫ u p

0σydu p, (12)

where u p is the fracture energy conjugate of the yield stressafter the initiation of damage, the rate of which is given by

u p = L ε p, (13)

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For the calibration of the model with experimental data (c f .Sect. 4), it is assumed that the damage evolution parameterexponentially varies with the effective plastic displacement.More information regarding the implementation and calibra-tion of this continuum ductile damage model is provided in[88,92]

2.4 Cohesive damagemodel

The interfacial debonding between carbon fibers and thepolymer matrix is simulated using a cohesive zone model(CZM), together with a surface-based contact model to pre-vent interpenetration between these phases [93,94]. In CZM,the effective interfacial separation along material interfacesis defined as

δm =√

δ2s + δ2t + ⟨δn⟩2, (14)

where subscripts s and t refer to orthogonal in-place shearcomponents of the contact separation vector δ, while n indi-cates its normal component. The cohesive behavior alongfiber-matrix interfaces is modeled using a linear traction-separation law given by

⎧⎨

⎩

tstttn

⎫⎬

⎭ =

⎡

⎣Ks 0 00 Kt 00 0 Kn

⎤

⎦

⎧⎨

⎩

δsδtδn

⎫⎬

⎭ , (15)

where the left-hand-side is the surface traction vector t, whichis the product of the cohesive stiffness tensorK and the con-tact separation vector δ shown in the right-hand-side. Theinterfacial damage initiates when

max{ ⟨tn⟩

t0n,tst0s,ttt0t

}= 1, (16)

where t0s and t0t are in-plane shear strength values and t

0n is the

normal strength. Note that the Macaulay brackets ⟨⟩ appliedto the latter indicate that a compressive normal traction alongthe interface does not affect the cohesive damage response.Instead, a surface-based contact model is implemented toavoid the interpretation of surfaces subject to a compressivetraction.

Once the damage initiation criterion given in (16) is satis-fied, the damage propagates along the cohesive surface. Theevolution of damage leads to the degradation of the cohe-sive stiffness, which is characterized by a damage variableD ranging from 0 (intact interfacial bonding) to 1 (cohesivefailure) as

t = (1 − D)Kδ, (17)

Adopting a bilinear cohesive law, the damage evolution alongthe fiber-matrix interface can be evaluated as

D = δfm(δmax − δ0m)

δmax(δfm − δ0m)

, (18)

where δ0m and δfm are effective separation values corre-

sponding to the damage initiation and complete failure,respectively. Also, δmax is the maximum effective separationalong the interface during the loading history.

3 Microstructure reconstruction andmeshing

Several algorithms, including the CVT-GA-based algo-rithm introduced in [9], have been used for reconstruct-ing CFRP microstructures reinforced with fibers that havecircular-shaped cross-sections. Such algorithms can prop-erly simulate morphological and statistical features of themicrostructure by replicating the desired volume fraction,size distribution, and spatial arrangement of fibers. However,for CFRPs with non-circular shaped fibers cross-sections(e.g., oval-shaped fibers shown in Fig. 1b), additional con-siderations would be necessary to virtually reconstruct themicrostructural model. Further, studying the impact of thefiber misalignments on the mechanical behavior of a CFRPrequires incorporating this microstructural feature in recon-structed RUCs. In this section, we introduce a modifiedversion of the packing/relocation-based algorithm presentedin [83] for synthesizing RUCs reinforced with arbitrary-shaped, misaligned fibers.

3.1 Shape library and statistical descriptors

Before reconstructing microstructures of CFRPs reinforcedwith circular and oval shaped fibers, morphologies of a rep-resentative subset of fibers are extracted from correspondingSEM images and stored in a shape library. Note that dueto the presence of misaligned fibers, the SEM image visu-alizes oblique cross-sections of such fibers, as they are notperpendicular to the 2D plane captured in the image. Hence,the projection of a misaligned fiber with a circular-shapedcross sectionwould be an ellipse on this plane, which enablesidentifying such fibers. However, for fibers with non-circularshaped cross-sections (e.g., oval-shaped fibers in Fig. 1c), itwould practically be impossible to distinguish misalignedfrom aligned fibers in the SEM image. However, for theCFRP systemwith randomlymisaligned fibers studied in thiswork, the maximummisalignment angle is max(θmis) < 10◦

and thereby the difference between the actual cross-sectionand its projection on the plane perpendicular to the principal

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4 µm

(a)

4 µm

(b)

5 µmInterface BBox

Enlarged BBox

(c)

Fig. 1 SEM images of carbon fibers with a circular-shaped and b oval-shaped cross sections; c SEM image of a CFRP with oval-shaped fibers(V f ≈ 50%), together with enlarged and interface BBoxes representingthe cross-sectional geometry of one of the fibers

fibers direction is negligible. For example, for an oval-shapedfiber with θmis = 10◦, the difference between projected andactual dimensions of the cross-section is ≈ 1.5%. Thus,we directly use the aligned and misaligned cross-sectionalgeometries obtained after the segmentation of SEM imagesto build the shape library.

In addition to the shape library, a number of statisticaldescriptors must also be extracted from the imaging data andused in the reconstruction algorithm to synthesize realisticmicrostructuralmodels of aCFRP. In order to characterize thevariation in the area A of fiber cross-sections, we implementa normal distribution function fad(A) given by

fad(A) =1√

2π Sadexp

(

− (r − Nad)2

2S2ad

)

, (19)

where Nad is themean value and Sad is the standard deviation.Another set of key statistical descriptor is associated with thespatial arrangement of fibers in the CFRP microstructure,which distinguishes resin-rich areas from those with denselypacked fibers that have a considerable impact on the failureresponse. In this work, we use a two-point correlation func-tion to statistically quantify the spatial arrangement of fibers

as

S(xi , x j ) = ⟨I (xi )I (x j )⟩, (20)

where xi and x j are two arbitrary points in the microstruc-ture, while the angular brackets denote a linear expectationoperator. Here, the indicator function I (x) is defined as

I (x) ={1 x ⊂ Vi ,

0 otherwise,(21)

The sampling template algorithm described in [95] isemployed to extract the target two-point spatial correla-tion function of fibers from SEM images. This function,together with the volume fraction and size (area) distribu-tion of fibers and coupled with their actual cross-sectionalgeometries stored in the shape library, fully characterize theCFRP microstructure.

3.2 Reconstruction of microstructural models

Theproposed algorithm for reconstructingCFRPmicrostruc-tures relies on three main phases: (i) virtual packing of fibersto creates an initial, periodic, 2D microstructural model withthe desired volume fraction, size distribution, and cross-sectional geometries of fibers; (ii) an optimization phaseto adjust locations of fibers in order to replicate the targettwo-point spatial correlation function; (iii) extruding fibersto generate a 3D virtual microstructure, while taking intoaccount random, small (θmis < 10◦) misalignments betweenfibers.

• Virtual packing This process begins by representing thecross-sectional geometry of each fiber using two typesof Bounding Boxes (BBoxes), as shown in Fig. 1c. Inthis approach, an enlarged BBox is constructed for eachfiber by scaling up dimensions of its actual BBox by10%. A set of interface BBoxes are also created along theperiphery of each fiber, which provides a more realisticbut low resolution (pixelated) approximation of its cross-sectional geometry.

In order to add a new fiber to the microstructure, we firstselect its cross-sectional geometry from the shape library.Note that this selection is not random and made based onthe normal distribution function fad(A) associated with thearea distribution of fibers. The fiber can be embedded ata randomly-assigned location in the 2D microstructure ifit overlaps with none of the existing fibers. The processof checking overlaps between new and existing fibers isdepicted in Fig. 2a. As a computationally inexpensive pre-screening phase, intersections between enlarged BBoxes offibers are checked to identify existing fibers that are in the

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(c)(b)(a)

Newfiber1

23

4

5

67

Fig. 2 a Adding a new fiber to the CFRP microstructure by checkingintersections between its enlarged and interface BBoxes with those ofpre-existing fibers; b initial RVE reconstructed using this algorithm,

where arrows represent directions of subsequent fiber relocations per-formed to replicate the target two-point spatial correlation function; cfinal RVE

vicinity of the new fiber (e.g., fibers 4 and 7 in Fig. 2a).We then only need to check intersections between interfaceBBoxes of these fibers with those of the new fiber to deter-mine whether they overlap. If no overlap is detected, the fibercan be embedded in the microstructure and otherwise a newrandom location must be assigned to the incoming new fiberand restart the entire process.

The algorithm described above is recursively continueduntil reaching the target volume fraction (here, 50%). Dur-ing the packing process, the periodicity of the microstructureis automatically satisfied by creating an identical copy ofeach new fiber intersecting with one of the edges of RVE onthe opposing edge and checking their overlaps with existingfibers simultaneously. Figure 2b illustrates a periodic RVEgenerated using this algorithm for a CFRP reinforced withoval-shaped fibers. Note that while realistic shapes, volumefraction, and size distributionoffibers canbe accurately repli-cated in this microstructural model, the packing algorithmhas no control on the spatial arrangement of fibers.

• Relocation-based optimization Next, virtually packedfibers fully confined within RVE boundaries (i .e., not cutby its edges) are visited in a random order and relocatedto replicate the target two-point spatial correlation func-tion. In this sequential optimization algorithm, a fiber isallowed to relocate if: (i) it does not overlap with sur-rounding fibers; (ii) L2-norm of the error EL2 betweentarget and simulated two-point correlation functions isreduced. This error function is evaluated as

EL2 = ∥S(xi , x j ) − S(xi , x j )∥L2 , (22)

where S(xi , x j ) is the two-point correlation function ofsynthesized RVE. The process of randomly visiting and relo-

cation fibers is recursively continued until EL2 becomessmaller than a threshold value to build the final microstruc-tural model that replicates the target two-point correlationfunction (cf. Fig. 2c). Note that throughout this relocationprocess, the volume fraction and size distribution of fibersthat are already replicated by the packing algorithm remainintact. Also, by avoiding the relocation of fibers intersectingwith RVE boundaries, the final microstructure maintains itsperiodicity.

The most important aspect of this optimization algorithmis to avoid overlaps between fibers during the relocation pro-cess. In order to satisfy this criterion in a computationallyefficient manner, at each visit, a fiber is only allowed tomove horizontally or vertically to edges of its enlarged BBox(cf. Fig. 2a). Because the fiber is still confined within thisenlarged BBox after this relocation, we only need to checkpotential overlaps with neighboring fibers whose enlargedBBoxes intersect with it. This check can be efficiently madeusing secondary BBoxes of fibers, which in fact would be anextension of the algorithm described previously for the vir-tual packing of fibers. After identifying allowable directionsfor relocating a fiber (i .e., no overlap with adjacent fibers), itis moved in the direction that yields the least updated EL2 . Ifrelocating the fiber in all directions leads to its overlap withone of the neighboring fibers or an increase in the updatedEL2 , it maintains its location.

Figure 3 illustrates two reconstructed RVEs reinforcedwith circular and oval shaped fibers (l = 80 µm, V f = 50%)using the proposed packing-relocation algorithm. The figurealso provides a comparison between target (extracted fromSEM images) and simulated two-point correlation functionscorresponding to each RVE, showing a perfect agreementbetween them. It must be noted that l = 80 µm is adopted

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TargetOptimized

Pro

babi

lity

Den

sity

Distance (µm)0 20 40 60 80

0.18

0.34

0.5

(a)

TargetOptimized

Distance (µm)0 20 40 60 80

0.18

0.34

0.5

Pro

babi

lity

dens

ity

(b)

Fig. 3 Virtually reconstructed periodic RVEs and corresponding targetand optimized two-point correlation functions for CFRPs reinforcedwith a circular and b oval shaped fibers

as the RVE size based on a study similar to that presented in[9] to ensure that the failure response of the microstructuralmodel is independent of its characteristic length scales.

• Extrusion to 3D with misalignment Finally, the 2DRVE reconstructed using the packing-relocation algo-rithm is extruded to build a 3D microstructural modelwith randomly misaligned fibers. This feature is clearlyobservable in Fig. 4, which shows the micro-CT imageof a cross-ply CFRP with embedded oval-shaped fibers.An enlarged perspective view of fibers in the middle plyis illustrated in the inset of this figure, showing that somefibers form a small misalignment angle (θmis < 10◦)with the principal longitudinal fibers direction. Here itis worthwhile to reemphasize the importance of using areconstruction algorithm to build RVE rather than directsegmentation of micro-CT data by showing the modelcreated via the latter approach in Fig. 4b. Despite thehigh resolution of the imaging data (voxel size: 700µm) and applying several advanced image processingsteps (noise filtration, smoothing, etc.), the complexityof microstructure and close proximity of fibers prohibits

proper identification of their interfaces with the sur-rounding matrix. In particular, note that the full lengthof some fibers are not recognized and some adjacentfibers are modeled as single fibers with an unrealisticallyhigh diameter after the image segmentation in Fig. 4b.Clearly, the inability to capture thin layers of the matrixbetween adjacent fibers in the resulting microstructuralmodel deteriorates the fidelity of subsequent FE simula-tions aiming at predicting the failure response of CFRP.

In order to synthesize a 3Dmicrostructurewithmisalignedfibers, we first reutilize the BBox-based relocation algorithmto replicate the target two-point correlation function in 2DRVE, which determines the spatial arrangement of extrudedfibers on the opposing face (cf. Fig. 5a). In this approach, anumber of misaligned fibers (shown in orange) are relocatedto simulate the variation in their locations on the opposingface. After checking the feasibility of moving a fiber in all4 possible directions (i.e., no intersection with neighboringfibers), it is relocated by distance dreloc such that correspond-ing max(θmis) < 10◦, i.e., dreloc ≤ lz tan(θmis), where lz isthe RVE length in the longitudinal fibers direction. Note thatin this algorithm, a fibermight be revisitedmultiple times andmoved in different directions to achieve the desiredmisalign-ment angles (here, determined based on a normal distributionfunction).

Next, the 2D cross-sectional geometry of each fiber isextruded toward its (potentially relocated) position on theopposing face to reconstruct a 3D fiber. In order to facili-tate this process, we parameterize the morphology of fibercross-sections in terms of Non-Uniform Rational B-Splines(NURBS)C(u). A NURBS curve utilizes B-spline functionsof parametric coordinate u to interpolate a given geometry ona set of control points with physical coordinates xi . Thereby,as shown in Fig. 5b, the 3D shape of a fiber can be fullycharacterized using control points of NURBS curves corre-sponding to its cross-sections on two parallel faces of RVEperpendicular to the principal longitudinal fibers direction.For this purpose, the control points parameterizing cross-sectional geometries on the front and back faces, where thelatter is obtained by an orthogonal transformation (transla-tionalmapping) of the former, is extruded in the direction of aline connecting them to one another. Figure 5c illustrates thexy view of the resulting 3DRVE for the spatial arrangementsshown in Fig. 5a (lz = 35 µm). Note that this microstructureis still periodic along faces parallel to the principal longi-tudinal fibers direction, as identical misaligned angles areassigned to fibers intersecting these edges.

3.3 Non-iterative mesh generation

In order to simulate the micromechanical behavior, virtuallyreconstructed periodic RVEs of CFRPmust be converted into

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25 µm100 µm0◦ ply

90◦ ply

0◦ ply

(a)

25 µm

(b)

Fig. 4 aMicro-CT image of a cross-ply CFRP microstructure, together with an enlarged perspective view of the middle laminate showing the fibermisalignment; b inappropriate microstructural model directly reconstructed from the imaging data after image processing and segmentation

NURBScontrol point

(a) (b) (c)x

y

z

Fig. 5 a Relocated fibers (shown in light color) in the 2D RUC depicted in Fig. 2c to simulate their misalignment in the 3D microstructural model;b 3D fiber generated by mapping the NURBS control points in the extrusion direction and c x − y plane view of the final 3D RUC with misalignedfibers

appropriate FE models, which requires the construction of aperiodic conforming mesh with small element aspect ratios.Preferably, this mesh must be adaptively refined along fiber-matrix interfaces to more accurately simulate the interfacialdebonding and damage nucleation in the surrounding matrixwithout a significant increase in the number of degrees offreedom (DOFs). In this work, we employ a new mesh gen-eration algorithm named conforming to interface structuredadaptive mesh refinement (CISAMR) [81,82] to create suchFEmodels. Thismeshing algorithmcannon-iteratively trans-form a structured mesh composed of tetrahedral elementsinto a high-quality conforming mesh. While CISAMR caneasily handle problems with complex geometries and adap-tively refine the mesh in the vicinity of material interfaces,it also ensures that aspect ratio of resulting elements do notexceed 5.

For 3D problems, the CISAMR algorithm is composedof four main phases [82]: (i) h-adaptive refinement of back-ground elements along material interfaces; (ii) r -adaptivityof nodes of nonconforming elements by relocating someof them to the material interface; (iii) face-swapping toeliminate a small number of cap and sliver shaped ele-ments with poor aspect ratios, which may emerge duringthe r -adaptivity process; (iv) sub-tetrahedralization of theremaining elements intersecting with material interfaces togenerate the conforming mesh. Figure 6 illustrates a smallportion of the conforming mesh generated using CISAMRfor the 80µm× 80µm× 35µm RVE with embedded oval-shaped misaligned fibers (cf. Fig. 5c). An 80 × 80 × 35initial background mesh is used for discretizing the domain,resulting in a total number of 1.59 million elements (4.63million DOFs) after implementing CISAMR. The inset of

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xy

z

z

xy

Cohesive interface

Carbon fiber

Epoxy matrix

(a) (b)

Fig. 6 a Small portion of the conforming mesh generated using CISAMR for 3D RVE with misaligned fibers shown in Fig. 5c; b enlarged viewof tetrahedral elements discretizing fiber and matrix phases, as well as cohesive elements used for simulating the interfacial debonding

Fig. 7 Summary of the automated modeling process used for creatingFE models of CFRP RVEs

Fig. 6 shows proper aspect ratios of conforming elementsgenerated by this algorithm, together with the cohesive ele-ments added along fiber-matrix interfaces to simulate theinterfacial damage. It must be noted that the mesh size usedhere is selected after a convergence study through compar-

σy

(MPa)

εy

Comp. Cal.Comp. Exp.Ten. Cal.Ten. Exp.

0 0.1 0.2 0.30

20

40

60

80

100

120

140

Fig. 8 Comparison between the experimental data and calibrated FEsimulated macroscopic stress–strain responses of the epoxy matrix

ison with a finer mesh (160 × 160 × 50 initial grid, 4.05million elements, 9.45 million DOFs). Both meshes yieldidentical macroscopic stress–strain responses and fairly sim-ilar damage patterns after the FE simulation, but obviouslythe computational cost associated with the current mesh is asmall fraction of that of the fiber mesh. Different phases ofthe automated modeling framework described in this sectionfor creating 3D FE models of CFRP RVEs are summarizedin Fig. 7.

4 Results and discussions

In this section, we study effects of the cross-sectional geom-etry of fibers and their misalignment on the failure response

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Table 1 Yield function used inthe continuum ductile damagemodel for the epoxy matrix

σy (MPa) 29.0 37.0 52.2 84.8 95.3 96.3 97.8

εpeq 0.0 4.95e−4 1.50e−3 1.01e−2 2.27e−2 2.72e−2 1.42e−1

of CFRP under transverse tensile and compressive loads. Inthe FE simulations presented next, the epoxy matrix has anelastic modulus of Em = 3 GPa and a Poisson’s ratio of tν = 0.35. Elastic properties of circular-shaped carbon fibersare Ez = 214 GPa, Ex = Ey = 23 GPa, νxz = νyz = 0.28,and νxy = 0.445 [9]. Note that while small differences wereexperimentally observed between mechanical properties ofoval and circular shaped fibers (< 8%), in order to isolatethe impact of their shapes on the micromechanical behav-ior, same material properties are considered for both types offibers. Note that since only loadings in the transverse fibersdirection are analyzed in this work, material properties offibers do not have a notable impact on resulting simulations,as main damage mechanisms in such cases are fiber-matrixdebonding and crack propagation in the matrix.

In order to simulate the initiation and evolution of damagein the matrix, the parameters used in the continuum plasticdamagemodel are calibratedwith experimental data obtainedfrom tension and compression tests on pure epoxy specimens[96,97]. A comparison between calibrated FE simulationsand corresponding experimental results is depicted in Fig. 8,which shows an excellent agreement for both case scenar-ios. The resulting plastic yield function σy(ε

peq) is presented

in Table 1, together with thresholds εpl0 = 0.033 and 0.13

for equivalent plastic strains before the damage initiationunder tensile and compressive loads, respectively. Also, theeffective plastic displacement u pl allowed before the com-plete failure are calibrated to be 0.01 µm (tension) and 0.1µm (compression). Material properties used in the surface-based cohesive damage model for simulating the debondingalong fiber-matrix interfaces include the cohesive stiffnessKn = Ks = Kt = 4.54 GPa, cohesive strength t0n = t0s =t0t = 33.5MPa, and the effective separation after the damageinitiation δ

fm−δ0m = 0.5µm.Since the bonding epoxy/carbon

interfaces is mainly governed by hydrogen bonds [98], theseparameters are evaluated through calibration of the model byexperimental peel tests and molecular dynamics simulationspresented in [99,100].

4.1 Effect of cross-sectional shape of fibers

Figure 9 illustrates the macroscopic stress–strain responsesof RVEs reinforced with aligned circular and oval shapedfibers subject to macroscopic tensile and compressive strainsapplied along the y axis, i.e., transverse to the fibers direction.As shown in this figure, under a tensile loading, using fiberswith circular cross-sections only leads to a slightly higher

0 0.02 0.040

40

80

120

160

εy

σy

(MPa)

CircularOval

2.1%

4.15%

Tension

Compression

Fig. 9 Comparison between macroscopic stress-strain responses ofRVEs with different fiber shapes subject to tensile and compressiveloads

strength (≈ 2.1%). This can be attributed to higher stress con-centrations in regions with higher curvature along interfacesbetween oval-shaped fibers and the polymer matrix, whichcould accelerate the damage nucleation. Considering themuch better economic picture formanufacturingCFRPswithembedded oval-shaped fibers compared to circular fibers, a2.1% reduction in the tensile strength can be regarded as neg-ligible. Further, one can argue that this reduction is linked tothe size distribution of fibers and not merely their morpholo-gies, as oval-shaped fibers studied in this work have a largeraverage cross-sectional area than circular-shaped fibers (bothCFRPs have identical volume fractions).

Figure 10 shows the initiation and evolution of damage inRVEs, showing that in bothCFRPs the spatial arrangement offibers has amore pronounced impact compared to their shapeand size distributions on the failure response. As shown inthis figure, in both cases the damage initiates in thin layers ofmatrix between two fibers that are in close proximity, whichis due to higher stress concentrations in such regions (cf.damage patterns at εy = 0.017). This damage is triggeredby deboning along fiber-matrix interfaces and then propa-gates into the matrix perpendicular to the loading direction,whichwhich corresponds to sudden drops in themacroscopicstress–strain response (cf. Fig. 9). Note that the damage pathdoes not remain perpendicular to the loading direction andinstead forms a jagged surface as it deviates after interactingwith embedded fibers. Thereby, although shape and size offibers affect their spatial arrangement in RVE, this study sug-gests that this arrangement and in particular relative distances

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εy = 0.017

ω1

0

x

y

z εy = 0.022 εy = 0.035

(a)

εy = 0.017

ω1

0

zx

y

εy = 0.022 εy = 0.035

(b)

Fig. 10 Damage initiation and evolution in RVEs reinforced with a circular-shaped and b oval-shaped fibers subject to a transverse tensile load

between fibers (under tension, distances in the direction of xaxis) has the most crucial impact on the failure response ofCFRP.

The macroscopic stress–strain responses of RVEs undertransverse compression in Fig. 9 shows that using oval-shaped fibers as the reinforcement leads to a more notabledeterioration of the strength and toughness. Comparedto CFRP with embedded circular fibers, the strength isdecreased by 4.2% and themacroscopic strain correspondingto the peak macroscopic stress is also decreased by 11.6%.Further, the absorbed energy before failure inRVE reinforcedwith oval shaped fibers is ≈ 7% is lower than that withembedded circular-shaped fibers. Note that for both typesof fibers, CFRP shows a considerably higher strength andductility subject to a compressive load compered to a tensileload (cf. Fig. 9), which is due to the ductile behavior of theepoxy matrix under this loading condition.

Analyzing damage patterns at three different stages of thecompressive loading in Fig. 11 could better elucidate dif-ferences between mechanical behaviors of RVEs reinforcedwith circular and oval shaped fibers. Similar to RVEs subject

to a macroscopic transverse tensile strain, at initial stages ofcompression (εy = 0.2), the damage nucleation is more evi-dent in regions that fibers have a relatively small distance,but in this case in the direction of the y axis. Unlike the caseof tensile loading, a comparison of RVEs reinforced withcircular and oval shaped fibers under compression shows amore severe damage growth in the latter. A more meticulouslook at this RVE reveals that, across multiple regions thatfibers are nearly touching one another, the damage nucle-ation is more severe near oval-shaped fibers whose majoraxes (longer diameter) are aligned with the loading direction.This is because the region along the interface of such fiberswith the highest curvature is punching into the matrix undercompression, which leads to higher stress concentrations andthereby accelerated damage initiation compared to circularfibers. Also, the localization of damage near embedded fiberswith limited propagationwithin thematrix explains the strainhardening behavior observed in macroscopic stress–strainresponse of both RVEs (cf. Fig. 9).

Figure 11 also illustrates damage patterns in both RVEs atεy = 0.32 and 0.35, which correspond to partially evolved

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εy = 0.02x

z

y

ω1

0

εy = 0.032 εy = 0.05

(a)

εy = 0.05x

z

y

ω1

0

εy = 0.032εy = 0.02

(b)

Fig. 11 Damage initiation and evolution in RVEs reinforced with a circular-shaped and b oval-shaped fibers subject to a transverse compressiveload

and fully developed fracture surfaces, respectively. For thelatter, the FE model can properly capture fiber-matrix con-tact and fibers collision, which limits the formation of shearbands (specially evident in RVE reinforced with oval-shapedfibers). Note that the damage pattern after failure in RVEreinforced with oval-shaped fibers clearly shows both thecoalescence of shear bands and crushingmodes of compositefailure (Fig. 11b), while only local shear banding is observedin RVE with circular fibers (Fig. 11). Despite a small deteri-oration of mechanical performance under transverse tensileloading, the much lower cost of oval-shaped fibers comparedto circular fibers could still justify their application for thedesign of CFRPs.

4.2 Effect of fiber misalignments

The study presented in Sect. 4.1 revealed that for both circularand oval shaped fibers, the spatial arrangement of fibers hasthe most prominent effect on the failure response of CFRP.

In particular, it was shown that the close proximity of fibersis more decisive than their shapes/sizes in the nucleation ofdamage. Thereby, it would also be important to quantify theimpact of variation in the spatial arrangement of fibers dueto small angular misalignments formed between them duringthe manufacturing process. As noted previously, for CFRPsstudied in thismanuscript, themaximummisalignment angleof fibers with their principal longitudinal direction is lessthan 10◦. RVEs of such CFRPs are reconstructed using thepacking-relocation-based algorithm described in Sect. 3 andtransformed into high-fidelity FE models using CISAMR.We preserve the periodicity of each RVE along faces paral-lel to the principal fibers direction to allow imposing PBC.For the non-periodic faces perpendicular to this direction,displacement DOFs are constrained, which was found to bea better choice compared to a traction-free BC in terms ofpreventing unrealistic damage nucleation along these faces.

Figure 12 illustrates macroscopic stress–strain responsesof CFRPs with circular and oval shaped misaligned fibers

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0 0.02 0.04 0.060

40

80

120

160

εy

σy

(MPa)

AlignedMisalgined

(b)

0 0.02 0.04 0.060

40

80

120

160

εy

σy

(MPa)

(a)

AlignedMisalgined

Tension

Compression

Tension

Compression

10.5%1.8%

3.3%2.6%

Fig. 12 Comparison between macroscopic stress–strain responses of RVEs reinforced with aligned and misaligned fibers of both shapes, undermacroscopic tensile and compressive loadings. a Circular fibers and b oval fibers

(a) (b)

Fig. 13 Damage patterns in RVEs reinforced with misaligned fiberssubjected to a macroscopic transverse tensile load, showing frustum-shaped damage zones formed due to the fibers misalignment. a Circularfibers and b oval fibers

subject to transverse tensile and compressive loads, togetherwith their comparison with failure responses of RVEs rein-forced with perfectly aligned fibers. When subject to a trans-verse tensile load, 3.3% and 2.8% reductions are observed instrengths of RVEs reinforced with circular and oval shapedfibers, respectively, due to fiber misalignments. Both RVEsalso experience a small reduction in the toughness (< 3%).At first glance, onemight expect an apposite trend, as some ofcarbon fibers in such RVEs are slightly aligned with the load-ing direction, which could potentially improve mechanicalproperties of CFRP due to their significantly higher strengthand stiffness compared to the epoxy matrix.

The negative impact of small angular misalignments offibers on the mechanical behavior of CFRP is attributed tothe variation in their spatial arrangement in the principal lon-gitudinal fibers direction, e.g., on the front and back facesof RVE (the former is identical to RVE with aligned fibers).

As a result, in addition to regions on the front face that couldserve as damage nucleation sites due to the close proximity offibers, new damage nucleation sites could form on the backface, as relative distances between some fibers are reduceddue to misalignments. This leads to frustum-shaped damagepatterns between two adjacent fibers, starting from the faceat which they are closer to one another and then propagat-ing to the opposing face, as shown in Fig. 13. A comparisonbetween xy views of damage patterns in RVEs with alignedand misaligned fibers is provided in Fig. 14.

This study shows that small misalignments between fiberslead to insignificant reductions in the strength and tough-ness of CFRP under a transverse tensile loading. It is worthmentioning that in an earlier study [9], the authors showedthat using a 2D RVE to simulate the failure response underthe plane strain assumption yields an identical macroscopicstress–strain response and damage pattern as those of a 3Dsimulation for a CFRP with perfectly aligned fibers. Clearly,the computational cost and difficulty associated with themesh generation process and the FE simulation phase areconsiderably less for the former, although it is not capable ofmodeling the impact of fiber misalignments. Thereby, froma practical computational design perspective, it would stillbe preferable to utilize 2D RVEs when CFRP is subject to atransverse tensile load provided that the uncertainty associ-ated with small angular misalignments of fibers is taken intoaccount in interpreting results.

Similar to RVEs subject to transverse tension, Fig. 12ashows that the impact of fibers misalignment on the deteri-oration of macroscopic stress–strain response of CFRP withembedded circular-shapedfibers isminimalwhen subject to atransverse compressive load,with only 1.8%and2.4% reduc-

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z = 0µm z = 35µm

(b)(a)

x

y

ω1

0

Aligned fibers z

Fig. 14 Damage patterns after failure in RVEs reinforced with oval-shaped fibers subject to a transverse tensile load: a aligned fibers; bmisalignedfibers (shown in dark grey)—front and back faces

tions in the strength and toughness, respectively. However, asshown in Fig. 12b, the RVE reinforced with misaligned oval-shaped fibers yields a considerably different failure responsecompared to its counterpart with aligned fibers. In this case,small angular misalignments between fibers leads to a 10.5%drop in the strength and a similar decrease in the toughness.Given the notable difference between this response comparedto that of CFRP with embedded circular fibers under com-pression, as well as failure responses of both RVEs subjectto a transverse tensile load, it is worthwhile to further inves-tigate the impact of BCs on the fidelity of simulations. Notethat in order to evaluate the failure response of a CFRP withperfectly aligned fibers, it would be feasible to impose PBCon all faces of RVE. On the other hand, the presence of mis-aligned fibers RVE necessitates constraining displacementDOFs on faces perpendicular the principal fibers direction,as the lack of geometric periodicity prohibits imposing PBC.

In order to elucidate the impact ofBC,macroscopic stress–strain responses of RVEs reinforced with aligned circularand oval shaped fibers, simulated using both PBC (on allfaces) and a combination of constrained displacement DOF(on faces perpendicular to the longitudinal fibers direction)and PBC (on lateral faces) are depicted in Fig. 15. Note thatthe latter replicates BCs applied to RVEs with embeddedmisaligned fibers. As shown in the figure, the impact of dif-ferent choices of BC on the failure response of each RVEis negligible, with less than 0.4% difference in predictedstrengths for both circular and oval shaped fibers. Figure 16aillustrates xy views of simulated damage patterns in RVEsreinforced with oval-shaped fibers using both BCs. The dif-ferences observed in these damage patterns are emanatingfrom unrealistic stress concentrations on faces perpendicu-lar to the fibers direction in the absence of PBC. However,Fig. 15 indicates that the RVE size employed for approxi-

0 0.02 0.04 0.060

40

80

120

160

εy

σy

(MPa)

Oval (periodic)

Circular (periodic)Circular (nonperiodic)

Oval (nonperiodic)

Fig. 15 Macroscopic stress–strain responses of RVEs reinforced withcircular and oval-shaped fibers, each with and without PBC on facesperpendicular to the principal fibers direction

mating homogenized stress–strain curves is sufficiently largeand can mitigate this impact to obtain nearly identical failureresponses for both choices of BC. This study shed light onthe significant difference between failure response of RVEsreinforced with aligned and misaligned oval-shaped fibers inFig. 12b is not an artifact of BCs and instead emanates fromdifferences in their microstructural features.

Figure 16b illustrates xy views of damage patterns onfront and back faces of RVE reinforcedwithmisaligned oval-shaped fibers. Although both damage patterns look identical,they are considerably different than that of RVE withaligned fibers simulated using the same BC (cf. Fig. 16a—nonperiodic). The 3D view of damage patterns on faces ofRVE reinforced with misaligned oval-shaped fibers at two

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z = 0µm z = 35µm

ω1

0

(a) (b)x

y

Periodic Nonperiodic z

Fig. 16 Damage patterns after failure in RVEs reinforced with oval-shaped fibers subject to a transverse compressive load: a aligned fibers; bmisaligned fibers (shown in dark grey)—front and back faces

εy = 0.05

ω1

0

xz

y

(b)(a)εy = 0.032

Fig. 17 a Damage evolution and b 3D view of damage pattern after failure in RVE reinforced with misaligned, oval-shaped fibers (shown in darkgrey) subject to a macroscopic transverse compressive strain

different stages of loading, as well as within its microstruc-ture, are depicted in Fig. 17. The damage pattern at εy =0.032 shows that it is mainly concentrated near the centroidof RVE, where the matrix is fully damaged and fibers arecrushed into one another. Note that at this loading stage,multiple damage zones are developed along lateral faces ofRVE parallel to the yz-plane due to misalignments betweenfibers, which are not even observed in the fully damaged stateof RVEwith aligned fibers (cf. Fig. 16). These damage zoneslater merge with the main damaged regions near the RVE’scentroid at εy = 0.05, as shown in Fig. 17, which leads to anaccelerated failure compared to RVE reinforced with alignedfibers.

5 Conclusion

A comprehensive numerical study was presented to quantifythe effects of cross-sectional geometry and random smallangular misalignments of fibers on the failure response ofCFRPs. This study was made possible via an automated

computational framework for creating high-fidelity FE mod-els of the composite, involving the virtual reconstruction ofits realistic RVEs, followed by their automated transforma-tion into 3D conforming meshes. The latter is accomplishedusing a new non-iterative mesh generation algorithm namedCISAMR, which is capable of transforming a simple struc-tured mesh composed of tetrahedral elements into a high-quality conforming mesh with adaptively refined elementsalong material interfaces. The microstructure reconstruc-tion phase was carried out by first virtually packing 2Dcross-sectional geometries of fibers in RVE, then using anoptimization phase to replicate their target spatial arrange-ment, and finally extruding them to build the 3Dmodel whiletaking into account potential misalignments between fibers.The failure response of each RVE were simulated using con-tinuum ductile and cohesive-contact damage models for theepoxy matrix and fiber-matrix interfaces, respectively. Mainoutcomes of this study are summarized below:

• Effect of fibers cross-sectional shape Under both tensileand compressive loads applied transverse to the fibers

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direction, RVEs reinforced with oval-shaped fibers yieldslightly lower strength and toughness (< 3% in tensionand < 5% in compression) than those with embeddedcircular-shaped fibers. More than the shape of cross-section, the spatial arrangement of fibers is a determiningfactor on the CFRP’s failure response, as damage ofteninitiates in regions where fibers are in close proximity.

• Effect of fiber misalignments small angular misalign-ments (θmis < 10◦) between oval-shaped fibers andthe principal longitudinal fibers direction could lead toa notable decrease in the strength and toughness undertransverse compression (> 10%). For both CFRPs rein-forced with circular and oval shaped fibers subject toa transverse tensile strain, as well as the former undertransverse compression, the deterioration of mechanicalproperties due to fibermisalignmentwas found to bemin-imal (< 4%).

Acknowledgements This work has been supported by the Air ForceOffice of Scientific Research (AFOSR) under Grant Number FA9550-17-1-0350 and the Ohio State University Simulation Innovation andModeling Center (SIMCenter) through support from Honda R&DAmericas, Inc. The authors also acknowledge the allocation of com-puting time from the Ohio Supercomputer Center (OSC).

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