Christopher R. Cater and Xinran Xiao

Michigan State University, East Lansing, Michigan

Robert K. Goldberg and Lee W. Kohlman

Glenn Research Center, Cleveland, Ohio

Improved Subcell Model for the Prediction of

Braided Composite Response

NASA/TM—2013-217875

June 2013

AIAA–2013–1618
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Christopher R. Cater and Xinran Xiao

Michigan State University, East Lansing, Michigan

Robert K. Goldberg and Lee W. Kohlman

Glenn Research Center, Cleveland, Ohio

Improved Subcell Model for the Prediction of

Braided Composite Response

NASA/TM—2013-217875

June 2013

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

The 54th Structures, Structural Dynamics, and Materials Conference (SDM)

cosponsored by AIAA, ASME, ASCE, AHS, and ASC

Boston, Massachusetts, April 8–11, 2013

AIAA–2013–1618
Acknowledgments

The authors would like to thank the NASA Lewis Educational and Research Collaborative Internship Program for its support, as 

well as colleagues for their assistance and insight. These include Dr. Gary Roberts and Dr. Kelly Carney of NASA Glenn Research

Center (GRC). The work was funded by the VSST project of the Aviation Safety Program.

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Trade names and trademarks are used in this report for identifi cation 

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Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2013-217875 1 

Improved Subcell Model for the Prediction of 

Braided Composite Response 

Christopher R. Cater and Xinran Xiao 

Michigan State University 

East Lansing, Michigan 48823 

Robert K. Goldberg and Lee W. Kohlman 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

In this work, the modeling of triaxially braided composites was explored through a semi-analytical 

discretization. Four unique subcells, each approximated by a “mosaic” stacking of unidirectional 

composite plies, were modeled through the use of layered-shell elements within the explicit finite element 

code LS-DYNA. Two subcell discretizations were investigated: a model explicitly capturing pure matrix 

regions, and a novel model which absorbed pure matrix pockets into neighboring tow plies. The in-plane 

stiffness properties of both models, computed using bottom-up micromechanics, correlated well to 

experimental data. The absorbed matrix model, however, was found to best capture out-of-plane flexural 

properties by comparing numerical simulations of the out-of-plane displacements from single-ply tension 

tests to experimental full field data. This strong correlation of out-of-plane characteristics supports the 

current modeling approach as a viable candidate for future work involving impact simulations. 

1.0 Introduction 

Triaxially braided composites have been implemented into fan containment systems for jet engines due 

to the high-specific stiffness and strength of fiber-reinforced polymer matrix systems and the improved 

impact damage tolerance and delamination resistance of braided architectures. These types of composites, 

however, present significant disadvantages for numerical modeling and simulation. First, previous impact 

studies have shown architectural dependence on damage evolution and damage patterns during a ballistic 

impact event (Ref. 1). Consequently, an appropriate modeling strategy must account for the nonhomogeneity present at the mesoscale: the scale of the braided pattern. Second, standardized coupon testing 

used for metals and simpler composite architectures have been shown to be relatively ineffective for 

determining braided composite strength properties (Ref. 2). Thus, accurate strength properties required for 

standard finite element composite damage models are currently unavailable. Additionally, the modeling 

scheme must be computationally efficient in order to be suitable for modeling of dynamic events such as a 

fan-blade-out event. The current work presented proposes to improve on already developed semi-analytical 

methods utilizing layered-shell elements and existing composite material models within the explicit finite 

element code LS-DYNA (Ref. 3). The overall goal is to develop a modeling algorithm which can best 

capture mechanisms of deformation and damage unique to the braided reinforcement architecture. 

Analytical methods for determining the elastic properties of braided composites utilizing 

micromechanics were first developed as an extension of previous literature on woven composites. These 

typically involved first identifying the repeating unit cell, RUC, of the braided mesostructure. Ishikawa 

and Chou developed mosaic, fiber undulation and bridging models, modifying classical laminate theory to 

account for fiber undulations (Refs. 4 and 5). Naik and Shembekar (Refs. 6 and 7) extended the onedimensional mosaic model into two dimensions, whereas Yang et al. (Ref. 8) improved on accounting for 

fiber undulation in a fiber inclination model. Branch et al. (Ref. 9) extended a fiber inclination model into 

three dimensions. Redman and Douglas (Ref. 10) implemented the mosaic model using the rule of 
NASA/TM—2013-217875 2 

mixtures and classical lamination plate theory (CLPT). More advanced discretizations of the RUC were 

developed by Ivanov and Tabei (Refs. 11 to 13), splitting it into four unique subcells. Huang (Refs. 14 to 

16) developed a micromechanical model modeling individual subcells as regions of two yarns and two 

pure matrix regions and treating each as a unidirectional (UD) composite. Quek et al. (Ref. 17) developed 

an analytical model which consisted of one axial tow, two braider tows, and one matrix layer. Concentric 

cylinder modeling was used to determine the homogeneous properties of the RUC. Carey et al. (Ref. 18) 

divided the braided RUC into 13 distinct regions and utilized a volume weighted stiffness matrix to 

homogenize the properties of the RUC. It was noted in Masters et al. (Ref. 19) that simple analytical 

laminate models performed well for stiffness predictions, whereas Falzon (Ref. 20) noted that these 

representations were unable to accurately capture out-of-plane properties. 

There are three classifications of numerical methods for predicting the response of triaxially braided 

composites. The first includes those which model the RUC explicitly through finite element (FE) models 

(Refs. 21 to 23). In these approaches, the explicit models are used to determine the effective properties 

(stiffness/strength) of the RUC for use in structural simulations. A review of these types of numerical 

methods can be found in (Ref. 24). A second branch utilizes multiscale methodologies to link the explicit 

FE representation of the RUC with homogenized elements on the structural level (Ref. 25). Lastly, a 

semi-analytical method was developed (Refs. 26 and 27) based on a braiding-through-the-thickness 

method introduced by Cheng and Binienda (Ref. 28) utilizing layered shell elements, subcell 

discretizations, and the inherent CLPT implementation available in FE packages. Xiao et al. (Ref. 29) 

utilized a similar layered shell methodology with a unique subcell discretization. 

In the current work, the semi-analytical methodology is utilized and builds upon limitations in the 

previous models by Cheng and Binienda (Ref. 28) and Xiao et al. (Ref. 29). Section 2.0 will discuss the 

general methodology in the semi-analytical approach using subcell models and introduce the novel 

absorbed matrix model. Section 3.0 will cover the numerical implementation of the modeling procedure 

and Section 4.0 will verify the in-plane elastic properties of the subcell models. Finally, the efficacy of the 

absorbed matrix model will be demonstrated with novel simulations of single-ply transverse tension tests 

in Section 5.0. 

2.0 Subcell Modeling 

2.1 Previous Works 

The material system of interest is a [0/+60/–60] triaxially braided composite made of 24 k carbon 

fiber bundles in the 0° tow direction and 12 k carbon fiber bundles in the 60° braider directions. Braider 

geometries were formed by A&P Technologies and RTM molding was done by North Coast Composites 

(Refs. 1, 2, and 30). Fiber and matrix constituent sources and properties are provided in Section 3.1. A 

typical RUC, the smallest characteristic volume capturing the effective properties of this braided 

composite, is shown in Figure 1(a). The unit cell can be decomposed into 4 subcells as outlined in 

Figure 1(b). This partitioning corresponds to the existence of regions where axial (0°) and braider 

(+60°/–60°) tows are both present and regions where only the braider tows are present. Figure 1(b) only 

displays the fiber tows in the braided composite and not the pure matrix regions which exist between and 

around fiber tows. The boundary of subcells A and C are drawn to lie at the edges of axial fiber tows. The 

individual subcells can be subsequently discretized in a “mosaic” model of UD composite plies to capture 

the orientation and response of the fiber tows within the triaxial braid. Previous subcell modeling 

(Refs. 26 to 28, and 30) developed in collaboration with the NASA Glenn Research Center (GRC) and the 

University of Akron, shown in Figure 2(a), utilized subcells containing two unique UD plies: one 

containing the axial tow, and another containing the braider tow and surrounding matrix material. The 

two UD plies were assigned unique fiber volume fractions. These volume fractions were determined by 

matching global fiber volume fractions and assuming 80 percent fiber volume fraction in the 0° plies. The 

fiber volume fraction of the axial tows in this braided composite have been previously assumed to be 

80 percent (Ref. 26). 
NASA/TM—2013-217875 3 

There were two limitations to these previous models which are addressed in this work. First, the latest 

revision of the semi-analytical model homogenized each subcell through the thickness, resulting in a 

single layer for each subcell (Ref. 27). Although the model was still able to capture in-plane stiffness and 

strength properties of the braided composite, unique flexural properties of individual subcells were lost, 

such as the unique tension-twist coupling of the unsymmetric subcell laminate documented by Kohlman 

(Ref. 2). Secondly, the fiber volume fraction of the 60° UD plies in the model in Figure 2(a) were 

determined by meeting the global composite fiber volume fraction, but did not reflect the actual fiber 

volume fraction contained in the given subcells. 

The subcell model developed by Xiao et al. (Ref. 29) overcame these limitations by retaining the 

layered shell formulation for each individual subcell as well as developing a simple straight line model for 

determining ply thicknesses and UD fiber volume fractions. Xiao et al. also modeled fiber tows as unique 

UD plies, where the fiber volume fraction of each ply was identical to that of the fiber tow (~80 percent) 

and pure matrix pockets were modeled explicitly using the resin properties. Figure 2(b) displays the 

discretization, which hereon is referred to as the pure matrix model. Another significant advantage to the 
NASA/TM—2013-217875 4 

 

discretization of the pure matrix model was that the B and D subcells containing only the braider tow 

plies were modeled as symmetric [–60°/+60°/–60°] and [+60°/–60°/+60°] laminates. Figure 3 shows a 

detailed schematic highlighting the overlapping of +60° and –60° tows in subcells B and D. Although the 

orientation of the overlap differs between the subcells, they both contain equal amounts of 60° and 

-/+ 60° tow regions within the entire subcell. Although the local green/yellow triangular regions on the 

braid are locally unsymmetric, one can approximate that these two subcells are symmetric with respect to 

the subcell discretization. 

A limitation of the pure matrix model, however, is the fact that the pure matrix pockets contained 

between the UD plies do not accurately represent the true interaction of pure resin pockets with adjacent 

fiber tows. Figure 4 presents a cross-sectional micrograph of the braided composite with a red arrow 

indicating a pure matrix region adjacent to an axial fiber tow and a green arrow indicating matrix rich 

regions between axial and braider tows. From the micrograph images, more pure matrix regions are found 

lying in-plane to either the braider or axial tow rather than sandwiched between braider and axial tows as 

is modeled in the pure matrix model in Figure 2(b). Despite this deficiency, the pure matrix model is used 

as a comparative reference to a new absorbed matrix model. 

2.2 Absorbed Matrix Model 

In an attempt to combine aspects of both the previous semi-analytical models with that of the pure 

matrix model, it was decided that a suitable subcell discretization would contain the following elements: 

1. Explicit modeling of UD layers 

2. Accurate subcell fiber volume fractions 

3. Symmetric B/D subcells 

4. Better matrix representation 

The absorbed matrix model utilizes the straight line model used by Xiao et al. for determining fiber 

volumes in each subcell for the axial and braider directions. Rather than modeling pure matrix pockets as 

unique layers which are in parallel (that is, with respect to in-plane loading) to the UD fiber tow plies, the 

volume of pure matrix is “absorbed” by the surrounding UD fiber tow plies to create a homogenized 

representation of the tow/matrix interaction. This representation of the tow/matrix interaction is notably 

similar to the original subcell model in Figure 2(a), however, the fiber volume fraction of the braider plies 

in this model are now thought to best capture the amount of fibers in the respective subcells. Figure 5 

compares the fiber volume fractions between the previous semi-analytical subcell models and those of the 
NASA/TM—2013-217875 5 

pure matrix and absorbed matrix models. Both the pure and absorbed matrix models have similar volume 

fractions since they are derived based on the same straight line model approach. Figure 5 shows fiber 

volume fraction totals (blue) for subcells A and B and fiber volume fractions segregated by direction (red 

and green). It should be noted that the volume fractions shown for the axial and braider plies are given 

with respect to the volume of the subcell, thus are not ply-level fiber volume fractions discussed later. 

The geometric parameters required to use the straight line model for this braided composite system 

are given in Table 1. Both the pure matrix model and the absorbed matrix model are applied in the 

sections to follow for testing both in-plane stiffness properties and single ply coupon tests. The resulting 

subcell discretizations are presented in Tables 2 and 3, respectively. The necessary equations and figures 

for the straight line model used for computing the thicknesses in Tables 2 and 3 are presented in the 

Appendix. The thicknesses in Tables 2 and 3 are with respect to a total shell thickness of unity. It should 

be noted that the absorbed matrix model assumes that pure matrix pockets in subcells A and C are 

assimilated only into the 60° plies. Thus, the axial UD ply in subcells A and C has a fiber volume 

fraction of 80 percent for both the pure matrix and absorbed matrix models. The next section will cover 

the numerical implementation of both models within LS-DYNA. 

TABLE 1.—GEOMETRICAL PARAMETERS USED FOR 

CALCULATING THE LAMINA THICKNESSES 

(T700/PR520 SYSTEM) 

Label Description Value 

W Width of RUC (m) 0.0089 

WA Width of cell A (m) 0.004201 

WB Width of cell B (m) 0.004765 

h Ply thickness (m) 0.00056 

Vf,UD Fiber volume (tow) 0.8 

na Number of fibers in axial tow 24000 

nb Number of fibers in braider tow 12000 

da Diameter of fiber filament in axial tow (m) 0.000007 

db Diameter of fiber filament in the braider tow (m) 0.000007 

L Height of unit cell (length) (m) 0.0051 

θ Braid angle (degrees) 60 
NASA/TM—2013-217875 6 

TABLE 2.—SUBCELL A & B DISCRETIZATION FOR 

THE PURE MATRIX MODEL 

Subcell A Lay-up Angle,

degree 

Vf,

percent 

Thickness 

Braider tow –60 80 0.2334 

Matrix --- 0.02123 

Axial tow 0 80 0.49074 

Matrix --- --- 0.02123 

Braider tow 60 80 0.2334 

Subcell B Lay-up --- --- -------- 

Braider tow –60 80 0.1174 

Matrix --- --- 0.2652 

Braider tow 60 80 0.2348 

Matrix --- --- 0.2652 

Braider tow –60 80 0.1174 

TABLE 3.—SUBCELL A AND B DISCRETIZATION FOR 

THE ABSORBED MATRIX MODEL 

Subcell A Lay-up Angle,

degrees 

Vf, 

percent

Thickness 

Braider tow –60 73.0 0.255 

Axial tow 0 80.0 0.49 

Braider tow 60 73.0 0.255 

Subcell B Lay-up --- ----- ----- 

Braider tow –60 37.5 0.25 

Braider tow 60 37.5 0.50 

Braider tow –60 37.5 0.25 

3.0 Numerical Implementation 

3.1 Material Models 

The work presented here focuses on single-ply coupon tests which remained within the elastic range 

of the composite, thus all plies in both subcell models required only elastic properties. However, in 

preparation for future work to identify a top-down approach for determining ply-level strength, viable 

candidates for material models were investigated and utilized in the work. The matrix material was 

assumed to be elastic, perfectly-plastic and was implemented within LS-DYNA using material type 24, 

*MAT_PIECEWISE_LINEAR_PLASTICITY. The matrix properties were taken from Table 4. The 

efficacy of the elastic, perfectly-plastic assumption was not validated here, and it should be noted that the 

explicit modeling of the matrix was not necessary for the absorbed matrix model. Two resident material 

models of LS-DYNA were identified as candidates for modeling the unidirectional composite material: 

Mat_54, a linear-elastic composite damage model, and Mat_58, a non-linear composite model utilizing a 

continuum damage mechanics (CDM) approach. The former utilizes the Chang-Chang failure criterion 

and models elastic/brittle failure in the fiber direction and an elastic, perfectly-plastic response in the 

transverse and shear matrix directions. The latter model uses a modified-Hashin failure criterion and a 

strain-driven CDM formulation developed by Matzenmiller (Ref. 31). A purely elastic composite 

response was simulated by setting arbitrarily high strength values in Mat_58 for all directions in the 

composite damage model, thereby requiring only updated elastic properties. For these stiffness values, a 

bottom-up micromechanics approach was employed to calculate the moduli and Poisson’s ratio of the 

unidirectional layers in each of the subcells. In this way, the effective properties of the UD ply were 

determined based on the constituent properties of the fiber and matrix. The micromechanics software 

MAC/GMC developed at NASA GRC (Ref. 32) was used to compute the effective properties (elastic 

moduli, Poisson ratio) of the UD lamina based on the constituent properties provided in Table 4. It was 

hypothesized to be sufficient to use this bottom-up approach since the UD discretization of the sub-cell 

accurately captured the fiber contributions in all material axes present in the braided composite 
NASA/TM—2013-217875 7 

TABLE 4.—CONSTITUENT MATERIAL PROPERTIES 

(REFS. 26 TO 28 AND 33) 

Material Density,

g/cm3

E11, 

GPa 

E22, 

GPa 

Poisson, 

v12 

G12,

GPa 

T700 1.8 230 15 0.2 27 

PR520 1.256 4 4 0.4 1.47 

5208 1.263 3.8 3.8 0.4 1.36 

3502 1.266 3.6 3.6 0.4 1.29 

E-862 1.2 2.7 2.7 0.4 1.0 

(particularly for in-plane loading). The process of determining the UD lamina properties was repeated 

depending on the number of UD fiber volume fractions present in the discretization. For example, the 

absorbed matrix model would involve computing effective properties using MAC/GMC for 37.5, 73, and 

80 percent fiber volume fractions. The layered shell formulation within LS-DYNA would use laminate 

theory to determine the overall subcell response based upon the provided UD properties. This bottom-up 

process was used to compute the UD ply properties for four different material systems (T700 Toray fibers 

with different matrix systems: Cytec PR520, Cytec 5208, Hexcel 3502, Epon Epikote Resin 862, hereby 

referred to as E-862). 

3.2 Finite Element Model 

This section highlights the implementation of the layered subcell approach for modeling the braided 

composite within the FE code, LS-DYNA. Each of the individual subcells were modeled as unique parts 

composed of a layered, Belytschko-Tsay quadrilateral shell using the *INTEGRATION_SHELL 

keyword. Each layer was assigned a unique integration point and associated thickness according to the 

parameters provided in Table 2. Tensile coupon tests were simulated in both the axial direction (parallel 

to the axial tows) and the transverse direction (perpendicular to the axial tows). Axial coupon specimen 

geometries were 8 by 1.412 by 0.125 in. (H by W by D, w/240 elements) whereas the transverse coupon 

geometries were 12.002 by 1.4 by 0.125 in. (476 elements). These dimensions corresponded to a 6-ply 

specimen of the braided composite. Each shell element thus represented 6 plies of the braided composite 

and the total thickness of the shell was set to 0.125 in. It should be noted that in the real composite braid, 

nesting of axial fibers occurs and results in imperfect stacking. These effects are currently neglected and 

perfect layer stacking is assumed. The finite element models for the tensile tests are provided in Figure 6. 

The compression specimen geometries were 1 by 1.412 by 0.125 in. and 1.4 by 0.8714 by 0.125 in. 

(H by W by D) for the axial and transverse compression tests, respectively. The meshes for the axial and 

transverse compression tests used to compute the compressive moduli are shown by highlighted boxes in 

Figure 6. A white arrow in Figure 6, and in figures to follow, identifies the direction of the axial fiber tows. 
NASA/TM—2013-217875 8 

4.0 Coupon Test Results 

The finite element results from the simulated coupon tests were used to compute effective properties 

of the braided composite for various matrix systems. Cross-sectional forces acting at the center gage of 

the specimen are used to compute the uniaxial stresses, whereas crosshead displacements are used for 

strain calculations. The stress-strain curves are then used to compute the moduli for the axial and 

transverse directions in both tension and compression. Table 5 displays the results for the E-862 and 3502 

material systems. Shown are the predicted moduli for the pure matrix model, absorbed matrix model, 

previous subcell models (where available), and experimental test data. Table 6 presents similar data for 

the PR520 and 5208 material systems. General trends in the moduli predictions were that the axial tensile 

stiffnesses were slightly over-estimated (~2.4 percent) and the transverse tensile stiffnesses were 

generally under-predicted (~18 percent). Despite little variation in the experimental transverse moduli 

across the material systems, the simulations were sensitive to the matrix modulus. The pure matrix model 

was slightly stiffer in all cases, due to the localization of the fibers in the 80 percent UD plies and its 

inherent effect on the subcell response. Both investigated subcell models adequately captured the stiffness 

properties even when compared against previous subcell works listed for the E-862 and PR520 systems in 

Tables 5 and 6. The results for the compressive moduli varied across the material systems and between 

the transverse and axial directions. The transverse compression moduli were generally in better agreement 

(~6 percent), however, the axial compression moduli were only slightly over-predictive (~13 percent). For 

the case of compression, a similar trend in matrix moduli sensitivity was found for the transverse case, 

and an apparent softening in the absorbed matrix model in comparison to the pure matrix model was 

noticeable. Considering the purely predictive, bottom-up method of the proposed subcell models the moduli 

predictions were well within a reasonable range. Both the pure matrix and absorbed matrix models are 

investigated for their out-of-plane properties and verified against single ply coupon tests in the next section. 

TABLE 5.—COMPARISONS OF COMPUTED MODULI WITH EXPERIMENTAL DATA FOR 

THE E-862 AND HEXCEL 3502 MATERIAL SYSTEMS 

 E-862 3502 

Pure 

matrix 

Absorbed 

matrix 

Previous works,

(Ref. 30) 

Test, 

(Ref. 34) 

Pure 

matrix 

Absorbed 

matrix 

Test, 

(Ref. 30) 

E11T (GPa) 48.92 47.32 51.10 46.9 49.52 48.41 47.2 

E22T (GPa) 32.36 31.43 38.60 41.6 34.53 32.39 40.6 

E11C (GPa) 50.31 48.64 ------- 41.4 51.45 49.26 41.8 

E22C (GPa) 38.44 34.80 ------- 42.7 40.94 36.86 41.1 

TABLE 6.—COMPARISONS OF COMPUTED MODULI WITH EXPERIMENTAL DATA FOR 

THE CYTEC 5208 AND PR520 MATERIAL SYSTEMS 

 5208 PR520 

Pure matrix Absorbed 

matrix 

Test 

(Ref. 34) 

Pure 

matrix 

Absorbed

matrix 

Previous works 

(Refs. 30 and 27) 

Test 

(Ref. 34) 

E11T(GPa) 49.858 48.905 47.5 51.496 49.22 51.1/44.9 47.7 

E22T (GPa) 35.351 35.377 41.3 36.667 36.81 38.7/51.2 42.7 

E11C (GPa) 51.744 50.037 44.6 52.154 51.20 ------------ 47.6 

E22C (GPa) 41.42 39.790 38.6 42.434 41.76 ------------ 42.0 
NASA/TM—2013-217875 9 

5.0 Single Ply Coupon Tests and Simulations 

Previous works have investigated the out-of-plane displacements in single-ply coupon tests of the 

braided composite under axial and transverse tension (Ref. 2) using Digital Image Correlation (DIC) on 

the T700/E862 braided composite. The displacements were a result of the locally unsymmetric nature of 

the braided composite. The DIC images of the out-of-plane displacements are reproduced in Figures 7(a) 

and (b). In light of these results, it was conjectured that a modeling approach best capturing this phenomenon would be advantageous for two reasons. One, it would provide a means of quantifying the flexural 

property predictions of the subcell model. Second, it would suggest that the subcell model may be 

sufficient in capturing the effects of inter-ply delamination hypothesized to be caused by these out-ofplane deformations. These edge-related delaminations drastically affect the stress/strain distribution in the 

gage section of standard test coupons (Ref. 2) and accurate simulations of these phenomena may provide 

insight into obtaining valuable strength data from material characterization tests. 

5.1 Subcell Flexural Properties 

CLPT was used initially to determine the anticipated response of the two subcell discretizations (pure 

matrix and absorbed matrix models). This analysis would provide insight into the out-of-plane properties 

of the subcells and would allow for meaningful interpretation of the single ply coupon simulations. Two 

subcell characteristics were identified as potential contributors to the out-of-plane deformations and 

studied using CLPT: the tension/twist coupling of subcells A and C and the flexural modulus of subcell B 

and D. To quantify the tension twist coupling under uniaxial loading, relations (1) and (2) below were 

used- whereby assumptions of uniaxial stress and zero membrane moments provided a relation between 

membrane strain, εx or εy, and the twist curvature, kxy. The x-direction corresponds to that of the axial tows 

(0°), and the y-direction corresponds to the transverse direction perpendicular to the axial tows. 

Equation (1) represents the tension-twist coupling for uniaxial tension in the x-direction. Conversely, 

Equation (2) is used for the case of uniaxial tension in the y-direction. The quantities T*

x and T*y represent 

the linear relation between membrane strain and curvature for the cases of uniaxial loading in the axial 

and transverse directions, respectively. The quantities B16 and B26 are the only non-zero components of 

the extensional-bending coupling matrix of the subcell laminate; it is for this reason the curvatures are 

reduced to only kxy for the case of uniaxial tension. The computed results for the two discretization 

xy x Tx x D

B B

A

A

k   























 

 0 

66

26 16

11

12

0 (1) 

xy y Ty y D

B B

A

A

k   























 

 0 

66

16 26

22

12

0 (2) 

  ,for 1,2 1 12

* 3 1     i, j

d h D

E

ij ij

f

ij (3) 
NASA/TM—2013-217875 10 

TABLE 7.—TENSION TWIST COUPLING 

IN SUBCELL A 

Model Tx

*

(1/m) 

Ty

*

(1/m) 

Pure matrix –5800.64 –1216.6 

Absorbed matrix –5983.83 –1208.43 

TABLE 8.—PREDICTED FLEXURAL MODULI 

OF SUBCELL B 

Model E f

x

(GPa) 

E f

y

(GPa) 

E f

xy

(GPa) 

Pure matrix 9.48 21.37 7.91 

Absorbed matrix 6.53 21.01 10.4 

TABLE 9.—COMPARISON OF BENDING 

STIFFNESS MATRICES FOR SUBCELL B 

Pure matrix model Absorbed matrix model 

0.2324 0.2921 –0.1558 0.2966 0.1403 0.5667 

0.2921 0.9500 –0.4372 0.1403 0.2112 0.4890 

–0.1558 –0.4372 0.3201 0.5667 0.4890 0.2203 

models, shown in Table 7, highlight the minute differences between the pure and absorbed matrix models 

in terms of the quantities T*

x and T*y. The flexural moduli of subcell B for the two discretizations were 

computed using Equation (3) and the results are presented in Table 8. The flexural moduli in the x and xy

directions varied considerably between the subcell discretizations, however the flexural moduli for the 

transverse, y, direction were nearly identical for both models. This similarity in transverse flexural moduli 

was contrary to the unambiguously different D matrices (bending-stiffness matrices) computed for the 

two methodologies, shown in Table 9. 

5.2 Numerical Verification of Single Subcell Responses 

The curvatures seen in Figure 7(a) and (b) were hypothesized to be a direct consequence of the 

tension-twist coupling present in the unsymmetric subcells A and C. Thus, it was necessary to ensure the 

LS-DYNA implementation of the subcell models captured this phenomenon and to determine any mesh 

and/or element-type dependencies. Due to little expected differences between the two subcell 

discretizations in the tension/twist responses of subcells A and C, only the pure matrix model was used in 

the simulations to follow and shown in Figures 8 and 9. Figure 8(a) and (b) present the contour plots of 

out-of-plane displacement for subcell A under uniaxial tension. As can be seen from the contours, the 

displacements in the y and z directions were unrestrained. There was noticeable smoothing in the contours 
NASA/TM—2013-217875 11 

as a function of increased mesh dependency; however, the size and magnitude of the deformation were 

unchanged. The out-of-plane deformations were as expected from the laminate analysis reported 

previously with only twist curvature, kxy, present in the case of uniaxial tension. The various contour plots 

in Figure 9 for subcell C highlight the effects of edge boundary conditions on the displacements as well as 

the influence of shell element formulation. Two shell formulations were investigated, Belytschko-Tsay 

and Belytschko-Leviathan, based on recommendations in literature for modeling unbalanced and 

unsymmetric lamina in LS-DYNA (Ref. 35). In the case of free y and z displacements on the top and 

bottom surfaces of the subcell, the two element formulations produced nearly identical results. When the 

y and z displacements were fixed on the top and bottom boundaries, a reversal in twist was observed. Due 

to the complex nature of the induced curvatures, only Belytschko-Leviathan shells obtained smooth 

contour plots seen in Figure 9(d). In contrast, the Belytschko-Tsay elements had noticeable sharp, lined 

features which appear to be a function of the element size, highlighted in Figure 9(c). It should also be 
NASA/TM—2013-217875 12 

mentioned that comparing Figure 9(a) with Figure 8(a) indicates that the out-of-plane deformation of 

subcells A and C have reflective symmetry with respect to each other. Based on the performance of the 

Belytschko-Leviathan shells, it was deemed that the LS-DYNA layered shell implementation of the 

subcell discretizations were sufficient to capture the necessary tension-twist mechanisms. BelytschkoLeviathan shell elements are utilized in the single ply coupon simulations to follow. 

5.3 Single Ply Coupon Simulations 

Full coupon simulations were performed for both axial and transverse tension using similar coupon 

dimensions from the earlier tensile specimens. In this study, the thicknesses of the shells were reduced to 

that of a single ply (~0.022 in.). The coupon mesh was refined to include 4 elements per subcell, shown in 

Figure 10(b), in comparison to that of the standard coupon tests in Figure 6(a) and (b). Belytschko 

Leviathan shell elements are used to better account for the tension-twist coupling of the unsymmetric 

subcells as discussed in the previous section. Both coupon specimens are loaded to 1 percent strain for 

which the composite response in the experiments was purely elastic. Thus, the currently elastic material 

models for the UD plies in each subcell are again deemed appropriate for capturing the out-of-plane effects. 

Transverse coupon results are displayed in Figure 10(a) and (c) for the pure matrix model and can be 

compared to mesh/subcell location using Figure 10(b). The two contour plots in Figure 10(a) and (c) 

represent two alternating stable positions in the predicted solution, which tend to alternate through 

loading. Figure 10(a) contained the major curvatures and displayed similarities to the DIC results in 

Figure 7(a). Minor curvatures of opposite sign seen in Figure 10(c) also existed between the major 

curvatures that could be observed in the DIC data. Two distinct remarks could be made, qualitatively, 
NASA/TM—2013-217875 13 

from the contour results. First, edge twisting resulted from a collective effect of subcells A and C and the 

reflective symmetry of their tension-twist coupling. Consequently, significant edge displacements, 

although rooting from the twisting of subcells A and C, were manifested in subcells B and D. In 

Figure 10(c), the major curvatures occurred in the regions of subcell B, whereas the minor curvatures 

occurred within subcells D. This feature was determined to be a consequence of boundary conditions and 

the relative adjacency of either of these subcells with the restrained edge. For example, modifying the 

order of subcells so that subcell C would be the first from the constrained edge—that is, if looking left to 

right in Figure 10(b)—would move the location of the major curvatures to subcells D. This shifting of 

subcells was performed and the single ply transverse tension simulation is given in Figure 11. The 

z-displacement contours shown in Figure 11 provide an example of competing major/minor curvatures 

between subcell B and D by varying the subcell closest to the restrained edge (labeled on the figure). The 

left hand side of Figure 11 features dominant positive (red) displacement curvatures at the top of the 

model, whereas beyond the dashed transitional region marked on the figure the dominant displacement 

curvatures at the top of the model are negative (blue). Though the source of major/minor curvatures was 

found to be a function of boundary condition, the alternating stable contours between those in 

Figure 10(a) and (c) were thought to be a construct of the numerical implementation. The contours seen 

experimentally were entirely stable and matched closely to that of Figure 10(c), and thus supported this 

deduction. These results highlight the importance of the finite element model’s constraints on the out-ofplane displacement contours in the modeling approach, a phenomenon which must be acknowledged 

when comparing experimental and numerical results. 

A quantitative comparison of the z-displacements for the two subcell models is displayed in 

Figures 12(a) to (c) with the full field DIC data presented again in Figure 12(d). The results are presented 

using the stable contour shape from Figure 10(c), which are found to match closely to those seen 

experimentally. Two different element sizes are provided for the absorbed matrix model, Figure 12(a) and 

(b), to show the effect of a mesh refinement on smoothing the displacement contours. Note that all four 

contour plots match to the fringe levels shown at the right of Figure 12. As expected from the single 

subcell tests, the finer discretization, although smoothing the contours, did not alter the magnitude of the 

z-displacements. Comparing the two subcell models, the pure matrix model showed a significant 

reduction in out-of-plane deformation. Despite the pure matrix model showing qualitatively accurate 

curvatures in Figures 10 and 11, the contours were underpredicted in comparison to the experimental 

results. Based on the earlier investigations using CLT, the probable cause of the suppression of the out-ofplane displacements in the pure matrix model was the divergent properties of the bending stiffness matrix 

components between the two discretizations. The pure matrix model had a 145 percent higher flexural 

modulus in the axial fiber direction than the absorbed matrix counterpart, despite there being almost no 

differences in the tension-twist coupling of subcell A between the two models. Thus, the stiff outer UD plies 

in subcell B of the pure matrix model are acting like composite sandwich skin layers, suppressing the outof-plane deformation. The z-displacements predicted by the absorbed matrix model for both element 

discretizations in Figure 12(a) and (b) were in good agreement with the experimental results in Figure 12(d). 
NASA/TM—2013-217875 14 

The z-displacement contour plots for axial coupon simulations are shown in Figure 13. The axial 

coupon was meshed coarsely with two elements per subcell width, similar to that of the transverse case 

shown in Figure 12(a). Although there were out-of-plane displacements occurring near the gripped ends 

of the coupon, readily apparent in the absorbed matrix model—the results did not mimic the convex/ 

concave undulations throughout the body of the coupon as seen experimentally in Figure 7(b). The nature 

of the tension-twist coupling of subcells A and C permitted twist deformations only at the corners of the 

subcells due to anti-symmetry, thus almost no out-of-plane deformations were observed in the interior of 

the model. It must be mentioned that the contour plots from the LS-DYNA simulations in Figure 13(a) 

and (b) are on a scale 1/6th that of the contour plots in the experimental data in Figure 7(b). Thus, 

although these deformations are present within the single ply and may contribute to internal stresses 

within the 6-layer braid, the effects of the axial undulations are not as significant to coupon damage, 

unlike the edge twist in the transverse single ply coupons. Consequently, it was concluded that the 

mechanisms driving the concave/convex curvatures seen experimentally in the axial coupon tests arose 

from physical mechanisms other than the locally unsymmetric nature of the single-ply braided composite. 

It is thought that capturing concave/convex curvatures in the axial coupon simulations requires a 

methodology preserving fiber continuity. Due to the nature of the discretization and the resulting 

discontinuity of fiber tows, capturing curvature phenomena like those seen in Figure 7(b) is a limitation to 

the current subcell approach. 
NASA/TM—2013-217875 15 

6.0 Conclusion 

Based on the results from the single ply coupon tests, the absorbed matrix model was found to be an 

excellent candidate for future advancements in the semi-analytical approach. The adequate moduli 

prediction in both tension and compression, as well excellent agreement in out-of-plane tension-twist 

displacements, under global transverse loading, were based purely on constituent properties, thus 

highlighting the efficacy of the model to be extended to other material systems. The analysis of the 

flexural properties of the subcell discretizations was a novel contribution and provided insight into the 

mechanisms driving the out-of-plane deformations seen in the transverse coupon tests. It was found that 

the unsymmetric A and C subcell laminates were the direct cause of the unique material response. The 

excellent correlation with the experimental DIC data for the transverse coupon case supported the subcell 

discretization utilized in the absorbed matrix model, which left subcells B and D as symmetric laminates. 

The single ply coupon results also verified the assumption that the matrix was best modeled in series with 

surrounding braider tows, rather than in parallel as in the pure matrix model. The subcell model, however, 

was not able to capture the axial out-of-plane responses in the single ply simulations and was attributed to 

a limitation of the overall discretization scheme. Although very slight differences were found between the 

two subcell models for in-plane properties, significant differences in the axial flexural modulus 

contributed to varying tension-twist responses in the single ply. Lastly, the current method which preserves 

the through-thickness subcell properties should be beneficial when performing impact simulations. 
NASA/TM—2013-217875 16 

The current semi-analytical subcell approach has several advantages over that of the previous 

iterations, summarized in Section 2.0. Most important, however, is the possibility of using the current 

subcell implementation to better understand the braided coupon’s edge related damage, currently a 

limiting constraint in determining material strength parameters. In conjunction with understanding coupon 

damage mechanisms, future work will look to develop a top-down methodology in order to determine the 

necessary UD ply strengths for the absorbed matrix subcell discretization. 
NASA/TM—2013-217875 17 

Appendix—Straight Line Braided Composite Model 

The straight line model (Ref. 29) for determining the necessary UD ply thicknesses and computing 

the volume of fibers in the axial and braider directions contained in the subcells is presented here. The 

subcell widths WA and WB are measured as shown in Figure A1. Figure A1 also shows the amounts of 

+60° braider tow (dashed yellow box) approximated to exist inside of the subcell dimensions highlighted 

by the green box. 

Equations (4) to (6) are used to compute the volume of fiber contained in subcells A or B (the 

quantity of fibers in C and D will be identically equivalent). The subscripts a and b designate axial or 

braider tow and A and B subscripts designate the corresponding subcell. Superscript f represents volume 

of fibers whereas the subscript f represents volume fraction. The tow subscript represents the fiber tow, m

the matrix layer, L the length of the subcell shown in Figure A1, and h the height of the subcell through 

the thickness as shown in Figure A2. 

n L d V a f a aA 4

2   (4) 

  



 







    2 2 sin

2 2

A

b

b

b bA

f b

bA

W

n

d

n l d V (5) 

   

0.5

2

2 2

2 2 0.5

2

4 4 sin 















  

 









    h Wn

d

n l h d V B

b

b

b bB

f b

bB (6) 

k a b l A B

V WhL

V

V

f

f

kl

kl for , ; , ' ,tow

   (7) 

 

k

Vm,l 1 Vkl (8) 

Equations (7) and (8) compute the UD ply thicknesses for the pure matrix model. In both subcell 

models, the volume fractions of the three different UD plies (axial tow in A, braider tows in A, and 

braider tows in B) are computed according to Equations (9) to (11). The thickness of the axial UD in 

subcell A is assumed to be equivalent to the original pure matrix model; however, the braider UD plies in 

both A and B are modified according to Equations (12) and (13). 

Vf , aA Vf ,tow (9) 

  aA

f

bA

f bA V

V

V   1 , (10) 

W hL

V

V

B

f

bB

f , bB (11) 

VbA VbA VmA * (12) 

VbB VbB VmB * (13) 
NASA/TM—2013-217875 18 
NASA/TM—2013-217875 19 

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14. ABSTRACT

In this work, the modeling of triaxially braided composites was explored through a semi-analytical discretization. Four unique subcells, each 

approximated by a “mosaic” stacking of unidirectional composite plies, were modeled through the use of layered-shell elements within the 

explicit finite element code LS-DYNA. Two subcell discretizations were investigated: a model explicitly capturing pure matrix regions, and 

a novel model which absorbed pure matrix pockets into neighboring tow plies. The in-plane stiffness properties of both models, computed 

using bottom-up micromechanics, correlated well to experimental data. The absorbed matrix model, however, was found to best capture outof-plane flexural properties by comparing numerical simulations of the out-of-plane displacements from single-ply tension tests to 

experimental full field data. This strong correlation of out-of-plane characteristics supports the current modeling approach as a viable 

candidate for future work involving impact simulations.

15. SUBJECT TERMS

Polymer matrix composites; Braided composites; Finite element method 

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