Xuetao Li and Wieslaw K. Binienda

The University of Akron, Akron, Ohio

Robert K. Goldberg 

Glenn Research Center, Cleveland, Ohio

Finite Element Model for Failure Study of 

Two-Dimensional Triaxially Braided Composite

NASA/TM—2010-216372

July 2010
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Xuetao Li and Wieslaw K. Binienda

The University of Akron, Akron, Ohio

Robert K. Goldberg 

Glenn Research Center, Cleveland, Ohio

Finite Element Model for Failure Study of 

Two-Dimensional Triaxially Braided Composite

NASA/TM—2010-216372

July 2010

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Acknowledgments

The fi rst two authors acknowledge the support of the National Aeronautics and Space Administration and associated Glenn 

Research Center, grant number NNX07AV60A and technical monitor Mr. Dale Hopkins. Dr. Justin Littell is acknowledged for 

performing the experimental tests described in this report. This work was funded by the Aging Aircraft and Durability Project of 

the Aviation Safety Program.

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only. Their usage does not constitute an offi cial endorsement, 

either expressed or implied, by the National Aeronautics and 

Space Administration.

Level of Review: This material has been technically reviewed by technical management. 
Finite Element Model for Failure Study of 

Two-Dimensional Triaxially Braided Composite 

Xuetao Li and Wieslaw K. Binienda 

The University of Akron 

Akron, Ohio 44325 

Robert K. Goldberg 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

A new three-dimensional finite element model of two-dimensional triaxially braided composites is 

presented in this paper. This meso-scale modeling technique is used to examine and predict the 

deformation and damage observed in tests of straight sided specimens. A unit cell based approach is used 

to take into account the braiding architecture as well as the mechanical properties of the fiber tows, the 

matrix and the fiber tow-matrix interface. A 0°/±60° braiding configuration has been investigated by 

conducting static finite element analyses. Failure initiation and progressive degradation has been 

simulated in the fiber tows by use of the Hashin failure criteria and a damage evolution law. The fiber 

tow-matrix interface was modeled by using a cohesive zone approach to capture any fiber-matrix 

debonding. By comparing the analytical results to those obtained experimentally, the applicability of the 

developed model was assessed and the failure process was investigated. 

1.0 Introduction 

Carbon fiber/polymer composites have been heavily utilized in recent years due to their excellent 

mechanical properties, such as high strength and stiffness, with a corresponding low weight. These 

properties make them particularly popular in aerospace applications. Classically, laminated composites 

are used when in-plane properties are of primary importance. However, laminated composites generally 

have poor through thickness properties which will cause delamination under low level of load. To avoid 

these difficulties, textile architectures, such as braided composites, have been developed to improve the 

through-thickness behavior. 

In particular, in two-dimensionally triaxially braided composites, three sets of fiber tows with 

different orientations are intertwined to form a single layer of 0°/±θ° material. Bias tows alternatively 

undulate over and under each other while 0° tows (also called axial tows) are straight and define the axial 

direction of the composite. In particular, braided composites with a bias fiber orientation angle of ±60° 

fiber architecture are often utilized due to the nonlinear quasi isotropic nature of the material. 

Furthermore, it has been reported that braided composites with a 60° braiding angle, as shown in 

Figure 1, can resist crack initiation and propagation as well as formation of delaminations during impact 

(Roberts et al., 2003). In this work, when a flat panel was impacted by a soft projectile, the damage area 

on the braided panel was observed to be arrested and only slightly larger than the initial impact area, 

compared with a damage area on aluminum plates which extended to the panel boundaries as a result of 

dynamic crack propagation. A state-of-the-art review of the application of two dimensional braided 

composites in structural components has been conducted by Ayranci and Carey (2008). 

NASA/TM—2010-216372 1 
Due to the complexity and uncertainty in the architecture of fabric reinforcement, analyses of this 

type of composite are challenging. There has been some work conducted on the prediction of mechanical 

properties as well as the prediction of damage and delamination in textile composites. A limited amount 

of work has been done on triaxially braided composites. However, some efforts have been conducted in 

analyzing woven composites and other types of braided composites. Many of the models have focused on 

the macroscopic level, leading to homogenized constitutive material models (Iannucci, 2006), (Castejon, 

Miravete and Cuartero, 2001), (Zeng, 2004). There also have been some efforts focused on meso-scale 

modeling of three-dimensional braided composites by Fang (Fang 2009) and triaxially braided 

composites by Ivanov (Ivanov et al., 2009). In these analyses, the effects of the fiber matrix interface 

were not included. 

Littell (Littell, 2008, Littell, et al., 2009), conducted an extensive experimental and analytical study of 

the mechanical and impact response of 0°/±60° triaxially braided polymer matrix composites. In this 

study, features of the effective macroscopic response were tightly linked to mechanisms occurring on the 

local level. Furthermore, a modeling approach was developed in which the braid architecture was 

explicitly accounted for within the analytical model, and features of the local response were heavily 

utilized in characterizing the input parameters of the computational model. One important experimental 

result from this study is that for different composite material systems (i.e., different matrix materials were 

used) where the fiber volume ratio and braiding architecture were the same and the elastic stiffness 

properties of the fiber and matrix (even though the matrix material used in each system was different) 

were nominally the same, the resulting static and impact strengths of the resulting braided composites 

significantly varied. One possible cause of this variation was hypothesized to be related to the fiber towmatrix interface in the various composites. Some of the composite systems were assumed to have a strong 

fiber/matrix interfacial bond, where other composites were assumed to have a weak interfacial bond. For 

example, as shown in Figure 2, for a particular braided composite system during a transverse tension test 

the fiber/matrix interfaces opened up during the course of the testing, indicating a weak interface. 

Furthermore, in the studies conducted by Littell, other local failure mechanisms such as “fiber splitting” 

and matrix cracking (both within and outside of the fiber tows) were found to be significant. 

To gain further insight into the features of the material response identified by Littell, in this study a 

meso-scale finite element model of a quasi-isotropic triaxially braided composite will be constructed and 

analyzed. The goal of this study will be to gain insight into and quantify the local damage propagation 

and failure of these materials. In the finite element model, the braiding architecture of the fiber tows will 

be modeled in detail. To account for the damage mechanisms within the composite, progressive damage 

algorithms will be applied within the fiber tows. Furthermore, a cohesive zone model will be used to 

model the fiber tow-matrix interface to capture the effects of a weak interface. In Section 2.0 of this 

paper, the meso-scale finite element model will be described in detail. In Section 3.0 the methods used to 

simulate the material properties and response of each of the three constituent phases (fiber tow, matrix 

and interface) will be described in detail. In Section 4.0, the ability of the meso-scale model to simulate 

the effective deformation and strength response of triaxially braided composites will be examined. In 

Section 5.0, the developed model will be applied to analyze the local initiation and progression of damage 

for a triaxially braided material. 

NASA/TM—2010-216372 2 
2.0 Meso-Scale Unit Cell Model of Triaxially Braided Composite 

For this work, the primary focus was on modeling a unit cell of a 0°/±60° triaxially braided 

composite. A unit cell is the smallest portion of a composite whose behavior is representative of the 

overall behavior of the composite. For braided composites, the unit cell size depends on the dimensions of 

the fiber tows and the magnitude of the bias angle (Chou and Ko, 1989). A mathematical formulation of 

methods of determining the unit cell dimensions has been proposed which accounts for the tow width, 

fiber undulation and braiding angle (Li, 2009). For the materials studied here, the axial fiber tows consist 

of 24,000 fibers, the bias fiber tows have 12,000 fibers, and the global fiber volume ratio is approximately 

56 percent. Based on scanning electron microscope measurements, the average unit cell size was 

determined to be 0.706 in. (17.93 mm) (w) by 0.2 in. (5.08 mm) (l) by 0.021 in. (0.53 mm) (t). As can be 

seen in Figure 2, the actual cross-sectional shape and alignment of the fiber tows is variable. However, as 

a first approximation to simplify the model generation process, an average cross-sectional shape and area 

was determined for the fiber tows and a lenticular shape was assumed. Furthermore, a regular spacing for 

the fiber tows was also assumed. Future efforts can involve examining the effects of the fiber architecture 

on the model predictions. As will be discussed shortly, the average unit cell model used for this work is 

shown in Figure 3. Figure 3(d) shows a comparison between the actual cross-section of the material and 

the idealized cross-section generated in the model. 

For this study, the composites which were examined all have the same fiber system, Toray’s T700s, 

but are infused with different polymer resin systems: Cytek Industries PR520 and Epon’s 862. The Toray 

fiber is a high strength intermediate modulus fiber with elastic brittle failure behavior. Cytec’s PR520 is a 

toughened thermo-set resin. Studies by Littell, et al. (Littell, 2009) have indicated that the T700s/PR520 

composite most likely has a very strong fiber matrix interface. Epon’s 862 resin is a low viscosity and 

high flow thermo-set resin. The T700s/E-862 composite has been found to have an intermediate interface 

cohesion strength. Tables 1 and 2 list the mechanical properties of fiber and each resin, as reported by the 

manufacturer. 

TABLE 1.—MECHANICAL PROPERTY OF TORAY’S T700S 

Density 

(g/cm3

)

Tensile strength 

SFT 

(MPa) 

Longitudinal modulus 

EF11 

(GPa) 

Transverse modulus 

EF22 

(GPa) 

In-plane shear modulus 

GF12 

(GPa) 

1.8 4900 230 15 24 

TABLE 2.—MECHANICAL PROPERTY OF MATRIX 

Density 

(g/cm3

) 

Tensile strength 

SMT 

(MPa) 

Young’s modulus 

EM 

(GPa) 

Cytec PR520 1.256 82 4.0 

Epon 862 1.200 61 2.7 

NASA/TM—2010-216372 3 
Figure 3 illustrates the finite element model for the braided composite unit cell. The model utilizes 

six-noded pentahedron elements. In Figure 3(a), the elements representing the matrix material are not 

included so as to clearly show the braid architecture. In the isometric view of the unit cell shown in 

Figure 3(b), the tow cross-sectional shape and bias tow undulation can be observed. Based on information 

obtained from micrographs such as those shown in Figure 2, the interfaces between fiber tows were 

modeled by using cohesive elements around axial fiber tows and between bias fiber tows, as is shown in 

Figure 3(c). As discussed earlier, Figure 3(d) shows a comparison between the idealized and actual fiber 

architectures. 

By modeling the braided unit cell in this manner, the tow undulation and cross-sectional shape were 

controllable, and a high global fiber volume fraction could be achieved. Furthermore, the tow to tow 

interface was able to be modeled using cohesive zone elements in a fairly straightforward manner to be 

described shortly, which will allow for more flexibility in simulating failure modes. Finite element 

models of braided composites with other braiding angles can be generated by adjusting the shape of the 

pentahedron elements. Finally, the number of elements used in the model was relatively small, which 

results in the model being reasonably computationally efficient. 

NASA/TM—2010-216372 4 
3.0 Material Property Characterization 

3.1 Phase I: Fiber Tows 

The resin transfer molding (RTM) technique is widely used during resin impregnation of fiber tows, 

in such a way that consistency in fiber volume fractions and hence mechanical properties may be 

achieved. Therefore, in this paper, the material properties are assumed to be uniform within all fiber tows. 

The fiber tows are further assumed to be unidirectional transversely isotropic material under plane stress 

conditions. The fiber tows are assumed to behave linear elastically before damage initiation. As will be 

discussed later, the onset of damage and its subsequent evolution can be modeled by using a combination 

of a failure criteria (for example that developed by Hashin (Hashin, 1973)) and an anisotropic damage 

evolution law (for example that developed by Matzenmiller (Matzenmiller, 1995)). Plane stress 

continuum shell elements with a three-dimensional geometry are used to model the fiber tows. 

Due to the presence of matrix only regions in the composite, the fiber volume fraction of the composite 

C Vf is lower than the fiber volume fractions within the fiber tows tow Vf . The tow fiber volume fraction 

tow Vf is calculated based on the geometry of the unit cell model by using Equation (1), 

Unitcell

tow C

f f tows

Volume V V

Volume

= × (1) 

where VolumeUnitcell and Volumetows are the volume of the unit cell and the volume of the fiber tows in the 

unit cell, respectively. tow Vf is calculated to be about 80 percent, which will be further used to calculate 

the effective mechanical properties of the fiber tows based on a micromechanical model. 

To compute the effective stiffness properties of the fiber tows, the Generalized Method of Cells 

(GMC) developed by Aboudi (Aboudi, 1991) was utilized. In this micromechanical model, the assumed 

representative volume element (RVE) used a four subcell model, with one subcell for the fiber and three 

subcells for the matrix. In this model, traction and displacement continuity conditions are applied over the 

unit cell and combined with the constituent material properties to obtain effective stiffness values. The 

strength parameters were much more difficult to determine since they may be related to local defects, 

local variations and fiber matrix interface bond quality. The simplified micromechanics formulations, 

Equation (2), included in Chamis’s Model (Chamis, 1984) were used to obtain a first approximation of 

the required strength values. 

1

11

(1 ) M

T FT F F

F

E F SV V

E

⎡ ⎤

= +− ⎢ ⎥ ⎣ ⎦

 (2.1) 

(1) 1

11

(1 ) M

C FC F F

F

E F SV V

E

⎡ ⎤

= +− ⎢ ⎥ ⎣ ⎦

 (2.2.1) 

(2) 1

12

1 1

M

C

M

F

F

G F

G V

G

= ⎛ ⎞ − − ⎜ ⎟ ⎝ ⎠

 (2.2.2) 

(3) 12

1

12

13 1 M

C

F M

G G F S S MS MC G G

⎛ ⎞ = β α− + ⎜ ⎟ +

α ⋅ ⎝ ⎠

 (2.2.3) 

NASA/TM—2010-216372 5 
( ) 22 22

2

22 22

1 F

T F

M F F

E E F V SMT E E VE

=β − − (2.3) 

 ( ) 22 22

2

22 22

1 F

C F

M F F

E E F V SMC E E VE =β − − (2.4) 

 ( ) 12 12

12 12

1 F

S F

M F F

G G F V MS G G VG =β − − S (2.5) 

where F1T, F1C, F2T, F2C and Fs are the computed composite longitudinal tension and compression, transverse 

tension and compression, and shear strengths, respectively, Vf is the fiber volume ratio, EM and GM define the 

matrix Young’s modulus and shear modulus, EF11, EF22 and GF12 define the fiber longitudinal, transverse and 

shear moduli; E22 and G12 define the effective composite transverse and shear moduli (computed based on 

GMC), SFT and SFC are the fiber tensile and compressive strengths (assumed to be same), and SMT, SMC and SMS

are the matrix tensile, compressive and shear strengths (determined by Littell (2008)). Additionally, two 

coefficients are calculated as 

4 VF

π

α = ⋅ and 

22

22

1

1

1 1

M

M

F F

F

E

E E V

E

⎛ ⎞ ⎜ ⎟

β= α−

α − ⎡ ⎛ ⎞⎤ ⎢ − − ⎜ ⎟⎥ ⎝ ⎠ ⎣ ⎝ ⎠⎦

. 

To help in determining the accuracy and applicability of these values, later in this paper a sensitivity 

study which examined the sensitivity of the composite global behavior to the values of the fiber tow 

strength parameters will be described. The effective stiffness values determined for the fiber tows (for 

both of the composite systems examined in this study) are shown in Table 3, and the effective strength 

values for the fiber tows are given in Table 4. The failure mechanisms of unidirectional composites under 

longitudinal compression are complicated. Chamis (Chamis, 1984) hypothesized that there are three 

primary compressive failure modes, and thus three compressive failure strengths that need to be 

determined: 1) fiber rupture under compression, computed using Equation (2.2.1); 2) buckling or kinking, 

computed using Equation (2.2.2); and 3) fiber matrix debonding or matrix cracking, computed using 

Equation (2.2.3). Strong interfacial bonding between the fiber and matrix has been hypothesized to lead to 

compressive failure due to fiber rupture (Madhukar, 1992), while weak bonding leads to failure due to 

debonding or matrix cracking. The failure stresses due to each of these failure modes can be predicted by 

using the Chamis equations shown above. The fiber tow longitudinal compression strengths shown in 

Table 4 were first approximated by averaging the values predicted using Equations (2.2.1), (2.2.2) and 

(2.2.3) for each of the three failure modes, and then further refined by correlating the values using the 

results of experimental coupon tests (using a process to be described later). The values of F1C listed in 

Table 4, for the two different composite systems examined in this study, were determined using this 

procedure. 

TABLE 3.—FIBER TOW MODULUS 

E1 

(GPa) 

E2 

(GPa) 

G12 

(GPa) 

v12 G23 

(GPa) 

T700s/E862 184.5 10.13 6.95 0.296 2.77 

T700s/PR520 184.8 11.63 9.16 0.296 3.39 

TABLE 4.—FIBER TOW STRENGTHS 

F1T 

(MPa) 

F1C 

(MPa) 

F2T 

(MPa) 

F2C 

(MPa) 

FS 

(MPa) 

T700s/E862 3931.5 1250 61 152 61 

T700s/PR520 3937.2 1500 82 205 82 

Note: Subscript ‘1’ is for fiber longitudinal direction, ‘2’ for transverse direction, and ‘T’ for tension, ‘C’ for compression, ‘S’ for shear. 

NASA/TM—2010-216372 6 
3.1.1 Failure Criteria 

For the elements which compose the fiber tows, developing methods to predict the initiation and 

propagation of damage are required. An existing ABAQUS fiber-reinforced composite material model 

was used in this paper to predict the damage and failure of the fiber tows. The details of this ABAQUS 

material model are summarized below. Complete details concerning the material model can be found in 

the ABAQUS user documentation (ABAQUS, 2005). To predict the initiation of damage, the plane stress 

two-dimensional Hashin failure criterion (Hashin, 1973) was utilized. As listed in Table 5, in the Hashin 

criterion four different failure modes are considered, which correspond to tensile or compressive failure 

of the fiber and matrix. The material strength parameters used in the equations listed in the table were 

computed by the equations of Chamis’s Model shown in Equation (2). The material degradation is 

assumed to begin once one of the Hashin failure criteria functions identified in Table 5 reaches a value 

greater than or equal to one. 

TABLE 5.—FAILURE CRITERIA FOR FIBER TOWS 

Tensile fiber mode 11 σ > 0

2 2

11 12

1

t

f

T S

e F F

⎛ ⎞ σ σ⎛ ⎞

= +α ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

Compressive fiber mode 11 σ < 0

2

11

1

c

f

C

e F

⎛ ⎞ σ = ⎜ ⎟ ⎝ ⎠

Tensile matrix mode 22 σ > 0

2 2

22 12

2

t

m

T S

e F F

⎛ ⎞ σ σ⎛ ⎞

= + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

Compressive matrix mode 22 σ < 0

2 2

22 2 22 12

2

1

2 2

c C

m

SS C

F e F F FF

⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ σ σ =+ − + ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎢ ⎥ ⎣ ⎦

2

S

σ ⎞

⎟

⎠

When value of ‘em,f’ equals to ‘1’ triggers the damage initiation. 

3.1.2 Progressive Degradation Model 

After the initiation of damage, the progression of damage is characterized by material stiffness 

degradation, leading to material failure (Matzenmiller, 1995). Imposing stiffness degradation over a range 

of increasing strain, representing softening and fracture energy dissipation, as opposed to an abrupt failure 

of the elements, has been found to be a more reasonable approach (Hufner, 2009), since the 

implementation of a degradation model is designed to more accurately simulate the process of failure and 

to improve the convergence of the numerical analysis by the implicit finite element solver. 

An anisotropic damage evolution law is used to take into account the four different failure modes 

(fiber- or matrix-driven tensile or compressive failure). It can model damage evolution for elastic brittle 

materials with anisotropic behavior, such as fiber-reinforced materials. Four damage variables, t

f d , cf d , 

and , describe damage for each failure mode based on energy dissipation. The stiffness matrix for 

the damaged material is computed as follows 

t dm c dm

1 21 1

12 2 2

(1 ) (1 )(1 ) 0

1 (1 )(1 ) (1 ) 0

0 0(1

f f m

d fm m

s

d E d d vE

C d d vE d E

D

d GD

⎡ ⎤ − −−

⎢ ⎥ =−− −

− ⎣ ⎦ )

 (3) 

where D =− − − 1 (1 )(1 ) d d vv f m 21 12 , and df, dm and ds are the internal variables characterizing fiber, 

matrix and shear damage, respectively. The values of these variables can be computed using the 

expressions shown below, based on the fact that there are assumed to be four failure modes (fiber tensile 

NASA/TM—2010-216372 7 
and compressive failure, matrix tensile and compressive failure) possible for the composite. Once damage 

occurs for at least one mode, it will affect the criteria for damage initiation of other modes. 

⎪⎩

⎪

⎨

⎧

c

f

t

f

f d

d

d 11

11

0

0

σ >

σ < 

(4.1) 

22

22

0

0

t

m

m c

m

d

d

d

⎪

⎧ σ > = ⎨

⎪ σ < ⎩

 (4.2)

1 (1 )(1 )(1 )(1 ) t ct c

s f fm d dddd =− − − − − m (4.3) 

The damage variable for a particular mode is computed by using the following expression: 

( )

( )

0

0

f

eq eq eq

f

eq eq eq

d

δ δ −δ =

δ δ −δ (5) 

eq , defined as the equivalent displacement, is calculated by the characteristic length of the element and 

related strain component (ABAQUS, 2005). Next, the material stiffness matrix, Equation (3), is expressed 

as a stress-displacement relation, shown schematically in Figure 4 (ABAQUS, 2005), to alleviate mesh 

dependency during the energy dissipation process. 

δ

0 δeq is the initial equivalent displacement at which the 

damage initiation criterion for that mode was met. The values of 0 δeq depend on the elastic stiffness and 

strength. f δeq is the equivalent displacement at which the material is completely damaged. The value of 

f δeq for each mode depends on the respective fracture toughness Gc. 

Due to an absence of experimental data, the values for the fracture energies for the corresponding four 

failure modes were obtained from literature (Alfano, 2001) for similar materials, and were used as a first 

approximation for the current numerical analysis. 

TABLE 6.—FIBER TOW FRACTURE ENERGY 

Mode I 

GLt 

(J/mm2) 

Mode II 

GLc 

(J/mm2) 

Mode III 

GTt 

(J/mm2) 

Mode IV 

GTc 

(J/mm2) 

12.5 12.5 1.0 1.0 

NASA/TM—2010-216372 8 
3.2 Phase II: Matrix 

For this study, the bulk properties of the neat resin were assumed to represent the in-situ properties of 

the matrix in the composite, which is a common assumption in micromechanics analysis. For this study, 

to simplify the analysis a simple elastic-perfectly plastic model was used to simulate the matrix behavior. 

Furthermore, since only simulations of static coupon tests were conducted for this study, the strain rate 

dependence of the matrix material that exists in reality was neglected. Future studies will examine the 

effects of incorporating strain rate dependence into the material analysis. Representative stress-strain 

curves obtained by Littell (Littell, 2008) for the E862 material used in this study are shown in Figure 5. 

The low strain rate curve was used for this analysis. As shown in the figure, the stress-strain response was 

approximated to be an elastic-perfectly plastic response (see the red dotted line in the figure), with the 

yield stress set equal to the material tensile strength. Once this stress value was reached, the material was 

assumed to behave as a perfectly plastic material until the strain reached the failure strain. 

3.3 Phase III: Tow Interface 

Interfacial debonding between fiber tows and between fiber tows and regions of pure resin (local tow 

delamination) has been identified as a potential key damage mode in braided composites (Littell, 2008). 

The high stress gradients occurring near geometric discontinuities promote initiation of the debonding, 

which may further cause significant loss of structure integrity. Considering that in some respects 

debonding can be considered to be a local delamination, three potential methods for simulating this 

delamination have been found in the literature. The first method is to include delamination as one type of 

failure mode in the failure criterion, like the three-dimensional failure criterion adopted in MAT162 in the 

LS-DYNA material library (Xiao et al., 2007). This method is computationally efficient, but the interface 

behavior is not independent and not modeled explicitly. The second approach is based on the direct use of 

fracture mechanics, such as the virtual crack closure technique (Raju, 1987), and the J-integral (Rice, 

1968). The drawback of this method is that the initial position of the crack needs to be known and the 

computational burden increases significantly for three-dimensional problems. The third method involves 

an indirect use of fracture mechanics, combined with a strength-based failure criterion and damage 

evolution procedure. This method is utilized within the cohesive element formulation available within 

ABAQUS (ABAQUS, 2005), which was utilized for this study, and is summarized below. This technique 

initiated from Hilleborg’s cohesive zone model (Hilleborg, 1976), and was further applied in conjunction 

with interface elements (cohesive element) by Camanho (Camanho, 2002). By using this method, no 

NASA/TM—2010-216372 9 
information on an initial crack needs to be known, and the onset of crack initiation can be predicted 

within a preset cohesive zone, which is considered to be a potential crack propagation path. This method 

is particularly suitable to problems such as braided composites, where the fiber tow interface is relatively 

weak when compared with the adjoining material. 

Zero thickness cohesive elements were utilized in this study to simulate the fiber tow interface. It is 

helpful to consider cohesive elements as top and bottom surfaces that are initially well bonded. When 

they deform together with surrounding elements, there is no membrane stiffness. As illustrated in Figure 6 

(ABAQUS, 2005), the constitutive equation of the cohesive element, also called the cohesive law, is 

established in terms of three components of tractions and separations between two surfaces, one through 

thickness component, and two transverse shear components. Each component represents a corresponding 

fracture mode, such as opening, sliding and shearing modes. In this sense, tractions are considered to be 

interfacial stresses. 

It has been shown that a cohesive law can be related to a theory of fracture if the area under the 

traction-separation curve is equal to the corresponding fracture toughness, for a cohesive element with 

unit mid-surface area (Rice, 1968). The effects of the shape of the traction separation curve have been 

investigated by Alfano (Alfano, 2006). For behavior representing a pure fracture mode, a bilinear 

traction-displacement law represented in Figure 6 is used. A very high initial stiffness, the so called 

penalty stiffness, is used to hold the top and bottom surfaces together. After the traction reaches a peak 

value, which is considered to be the interface strength, the traction decreases with increasing separation. 

At the moment the traction is equal to zero, debonding between the top surface and the bottom surface 

occurs. For completeness, the unloading behavior is also defined such that a linear curve unloads towards 

the origin with a degraded stiffness. The area under the bilinear curves is the respective fracture 

toughness. Two critical separations in the cohesive constitutive relationship could be obtained once the 

interface parameters are known. 

0 c

i

i

i

t

K

δ = (6.1) 

NASA/TM—2010-216372 10 
f 2 i

i c

i

G

t

δ = c

 (6.2)

where is the interfacial strength, c

it Ki is the penalty stiffness and is the interface fracture toughness. Gi

For the triaxially braided composite being investigated in this study, the three interfacial strengths are 

assumed to be equal to the respective fiber tow transverse strengths shown in Table 4. A very large value 

was used for the penalty stiffness K (106

) to keep the top and bottom faces together in the elastic region. 

The fracture toughness can be measured by a single-mode fracture test carried out on simple specimens, 

such as a double cantilever beam test (DCB) for the opening mode, an end-notched flexure test (ENF) for 

the sliding mode and a splitted DCB test (SDCB) for tearing mode. Due to lack of the relevant 

experimental data for this material, literature values in Table 7 obtained by Alfano (Alfano 2006), for a 

composite composed of a carbon fiber impregnated by an epoxy resin, were used for this study. 

TABLE 7.—INTERFACE FRACTURE TOUGHNESS (J/mm2) 

Mode I 

Gn

Mode II 

Gs

Mode III 

Gt

0.268 1.45 1.45 

In a realistic loading situation, the fracture propagation is most likely to occur under mixed mode 

conditions. Therefore, an element formulation which accounts for mixed mode behavior is required, see 

Figure 7 (ABAQUS, 2005). 

The crack is initiated when the value of the quadratic interaction equation of the ratio of interfacial 

stresses to interfacial strengths reaches ‘1’, see Equation (7). 

2 22

1 nst

ccc

nst

ttt

ttt

⎛ ⎞⎛⎞⎛⎞ ⎜ ⎟⎜⎟⎜⎟ + + =

⎝ ⎠⎝⎠⎝⎠

 (7) 

where, tn, ts, and tt are stress components corresponding to opening, shearing and tearing modes; the 

critical stress on the denominator are indicated by superscript ‘c’. 

NASA/TM—2010-216372 11 
Damage evolution is defined based on fracture energy. Linear softening behavior is utilized. The 

dependency of fracture energy on mixed fracture modes is expressed by the widely used BK formulation 

(Benzeggagh and Kenane, 1996), which gives an analytical formula shown in Equation (8). 

( )

C C

CC CC s t n sn CCC

nst

G G GG GG

GGG

η ⎧ + ⎫

=+ − ⎨ ⎬ ⎩ ⎭ + + (8) 

where, Gn, Gs and Gt are the work done by tractions and their conjugate relative displacements 

corresponding to opening, shearing and tearing modes respectively. The power, η, is a material parameter, 

here selected to 1.45 for a carbon fiber composite (Benzeggagh and Kenane, 1996). 

4.0 Specimen Test for 60°/0°/–60° Braided Composite 

Static tensile tests on 0°/±60° braided composite specimens for both material systems considered in 

this study were performed by Littell (Littell, 2008) using an MTS axial-torsional test machine under 

displacement control. Straight sided specimens 12 in. long by 1.409 in. wide with six braided layers were 

tested. Axial tensile tests were conducted using specimens with the 0° fibers aligned along the applied 

load direction. Transverse tensile tests were conducted using specimens where the 0° fibers were 

perpendicular to the direction of the applied load, as shown in Figure 8. 

In the numerical simulation of the straight sided specimen tests, in this study the actual loading and 

boundary conditions of the mid-span of the test specimen was approximated. A finite element model was 

created with three unit cells aligned along the 0° fiber tow direction, shown as Figures 9(b) and 10(b). 

Symmetric boundary conditions were applied to generate a plate size of 1.2 in by 1.409 in. with two 

layers in the thickness direction, shown as Figures 9(a) and 10(a). The model dimensions are comparable 

to the middle part of the test specimen, shown as the shadow region in Figure 8. The free edge was also 

explicitly simulated as the part of the actual test conditions. To obtain the global stress strain curves 

shown later in this paper, the forces and displacements along the loading surface were utilized. The 

stresses are calculated by dividing the force over the surface area, and strain by dividing the surface 

displacement by the original length. Normally, in a unit cell type of finite element analysis a full set of 

periodic boundary conditions are applied to simulate the fact that the unit cell represents a material point 

in an overall continuum. However, in the current simulation, a modified symmetry set of boundary 

conditions, meant to represent a portion of the actual test specimen, were applied. The fact that the actual 

test specimen was simulated is significant for two important reasons. First, since the unit cell size of the 

braided composite is very large compared to the dimensions of the test coupon, the unit cell cannot be 

considered to be a material point in a continuum, rather the unit cell is almost a substructure of the 

structure of the composite coupon, thus requiring a more accurate modeling of the test geometry. 

Furthermore, as discussed by Littell et al. (Littell et al., 2009), there are deficiencies in the transverse 

tensile tests conducted using straight sided coupons simulated in this work. Since a number of the bias 

fibers in the gage section are ungripped due to the braiding geometry and specimen design, the stressstrain curves and ultimate strengths obtained in the transverse tensile test may not be truly representative 

of the actual composite response due to the edge effects (such as low strain regions) present in the tests. 

Specifically, the stress-strain curves obtained in the transverse tensile test may be too soft and the 

predicted strength may be too low. This point will be important later in this paper as several of the 

material properties were correlated based on the results of the experimental transverse tensile data. While 

valid for the particular test geometry presented, the results may not be truly indicative of the actual 

composite properties and results. 

NASA/TM—2010-216372 12 
NASA/TM—2010-216372 13 
To investigate the effects of the interface on the effective response of the model, three different 

interfacial conditions were simulated; perfect, moderate and weak interfaces. The moderate and weak

interfaces were designed to investigate the effects of a non-perfect bond, and are differentiated by the 

interfacial strengths of the cohesive elements. The strength of the weak interface was assumed to be equal 

to the transverse strength of the fiber tows, as computed in Table 4. Since both the interface strengths and 

the fiber tow transverse strengths are matrix dominated, they would be of the same order-of-magnitude, 

thus they were initially assumed to be the same for the case of a weak interface. The interfacial strength of 

the model with a “moderate” interface was set to be double the value used for the weak interface model, 

to investigate the effects of varying interfacial strength. In the perfect interface model, meant to simulate 

the case of a perfect bond, there were no cohesive elements or interface zone applied; instead node 

sharing was used on the contacting surfaces of the fiber tows. 

The goal of the model correlation was to match the experimental effective stress-strain curves and the 

sequence of damage. The study was separated into two parts. In the first part, model correlation was 

conducted on the T700s/E862 composite, which experimental evidence indicated (Littell, 2008) has a 

moderate interface. By simulating the axial and transverse tensile response of this material, the interface 

effect was also investigated by comparing the results obtained by applying all three of the previously 

discussed interfacial conditions. In the second part of the study, the axial and transverse tensile response 

of the T700s/PR520 system, which experimental evidence indicated had a strong interfacial bond (Littell 

2008), was simulated by using the “perfect” interface model. 

4.1 Model Correlation of the T700s/E862 Composite 

The model correlations for the T700s/E862 composite are shown in Figures 11 to 14. Figure 11 

shows the experimental results and computed values from an axial tension test, Figure 12 shows the 

results from a transverse tension test, Figure 13 shows the results from an axial compression test and 

Figure 14 shows the results from a transverse compression test. For all of the conditions shown in 

Figures 11 to 14, the linear portions of the simulated curves correlate very well with the experimental 

results. 

For the axial tension tests shown in Figure 11 and the axial compression tests shown in Figure 13, the 

nonlinear portion of the computed stress-strain curves and the computed ultimate strength correlated well 

with the experimental results, indicating that the damage and failure model applied for the fiber tows was 

reasonably accurate. Further discussions on the simulated nonlinearity in the stress-strain curves will be 

given in the next section. The extended unloading observed in the computed stress-strain curves is most 

likely a numerical artifact, as the experimental specimens most likely had a more brittle fiber failure, 

while the computed curves had a more gradual unloading to promote numerical stability. Varying the 

interfacial conditions also had a negligible effect on the computed results, which is reasonable since under 

axial loading conditions the 0° fibers should take the majority of the load, and interfacial effects should 

not be significant when 0° fibers are loaded axially. 

For the transverse tension tests shown in Figure 12 and the transverse compression tests shown in 

Figure 14, the nonlinearity in the stress-strain curves observed in the experimental results was once again 

captured reasonably well in the simulations. In these tests, varying the strength of the interface had a 

significant effect on the predicted ultimate strength of the composite, with lower interfacial strengths 

resulting in a weaker ultimate strength of the material. These results are reasonable since under a 

transverse loading condition both the axial and bias fibers carry significant portions of the load transverse 

to the fiber axis, which would result in the interface playing a significant role in the material response, 

with a stronger interface being able to carry more load. The results computed using the “moderate” 

interface appeared to correlate the best with the experimental values, which is reasonable since this 

material has been identified as having a moderately strong interface through experimental investigations 

(Littell, 2008). Overall, the developed model appears to have the capability to capture the effects of 

varying interfacial strength on the global stress-strain response of a braided composite. 

NASA/TM—2010-216372 14 
NASA/TM—2010-216372 15 
4.2 Model Correlation on T700s/PR520 Composite System 

To verify the capability of the finite element model to simulate the effective response of triaxially 

braided composites without regard to interface effects, the effective response of the T700s/PR520 system, 

which is known to have a strong fiber tow interface (Littell, 2008), was computed. Since this system has a 

strong interfacial bond, only the “perfect” interface condition described above was applied for these 

simulations. Experimental and computed effective stress-strain curves from axial tension and transverse 

tension tests are shown in Figure 15, and results from axial compression and transverse compression tests 

are shown in Figure 16. 

Overall, the correlation between the experimental and analytical results is reasonably good, indicating 

that the modeling approach is reasonably accurate, and that the “perfect” interface bond captures the 

expected response of the material. The ultimate strengths under transverse loading are somewhat 

overpredicted. The axial tensile strength is somewhat underpredicted, but this result could be due to the 

fact that the method used to compute the fiber tow axial longitudinal strength based on a mechanics of 

materials approach (Chamis, 1984) is not completely accurate (Kaw, 2005). The fact that the nonlinear 

portion of the axial tension curves was not predicted as well for this material could indicate that there 

were some out-of-plane delaminations present in the material that were not captured in the simulations. 

This feature will be examined in more detail in future work. 

For both of the materials discussed in this section, several model parameters, such as fiber tow 

stiffness and strength, interface strength and toughness, were correlated based on the results of the four 

specimen tests. Firstly, the elastic properties were correlated based on the stiffness of the four testing 

curves; the tow strengths were correlated based both on tests conducted with a material with a perfect 

interface and indirectly by correlating based on axial tension and compression tests of a material with an 

imperfect interface. From the transverse tension tests, the interface plays a major role in these tests so the 

tow strengths could not be correlated just from these tests. However, the interface strengths were 

correlated based on the transverse tests. From the numerical parameter studies, the interface toughness 

was found to mostly affect the post-peak part of the stress-strain curves. More detailed parameter 

sensitivity analysis will be addressed in future work. 

NASA/TM—2010-216372 16 
An important point to note is that, as discussed above, there are deficiencies in the transverse tension 

test that result in the corresponding curves most likely not being representative of the actual material 

response. Therefore, the correlated model parameters are correct for the test geometry as given, but are 

most likely not representative of how the material would respond when used within a large structure 

where edge effects are not as significant. In actual structures, the bias tows will most likely fully 

contribute to the material response. However, the procedure as shown in this paper demonstrates how test 

data could be used to correlate the various model parameters. Furthermore, the general trends of how the 

interfacial strength affects the simulated results are valid. 

5.0 Local Failure Investigation 

A primary motivation for utilizing a high-fidelity finite element approach for analyzing the response 

of composites is to gain insight into the local mechanisms which cause damage and failure in the material. 

In the experimental program conducted by Littell (Littell, 2008), full-field strain maps on the surface of 

the triaxially braided composite specimens were generated by use of an optical strain measurement 

system. An overlay technique was then applied to correlate the full-field strain results to the geometry of 

the braided composite. In this manner, the relative location (axial fiber, bias fiber) of the damage in the 

composite could be quantified. 

NASA/TM—2010-216372 17 
The numerical model developed in this study, due to its explicit modeling of the composite 

architecture, has the capacity to capture damage mechanisms such as fiber tow failure, matrix cracking 

and interfacial failure. Since the individual failure variable corresponding to a certain failure mode can be 

reported for each element, with the assistance of the post-processing function of ABAQUS, contour plots 

can be generated as field information for all elements at each computational step for different global stress 

or strain levels. To correlate the identified failure modes to particular damage mechanisms, the contour 

plots relating to failure patterns are overlaid on the undamaged mesh (Fig. 9(b) and Fig. 10(b)). 

For all the contour plots in Figures 17 to 22, each plot represents one particular failure mode, which is 

indicated under the plot; the darkness of the color defines the status of damage, which corresponds to the 

value of the particular damage variable ranging from 0 to 1, shown in the left picture in Figure 16. A 

value of ‘0’ means damage has not occurred yet, while a value of ‘1’ means the element is totally 

damaged. Meanwhile, the global strain and stress values at which the contour plots were generated are 

shown. 

For this study, the T700s/E862 material was investigated. The “moderate” interface was utilized in 

the numerical simulations since this interfacial condition resulted in the best correlation with the 

experimental results. Furthermore, by investigating a system with an imperfect interface the capability of 

the model to capture interfacial failure events could be quantified. Axial tension, axial compression, 

transverse tension and transverse compression tests were simulated. 

5.1 Axial Tension Test 

In the simulations of the axial tension tests shown in Figure 17, at a global strain level of 0.6 percent, 

failure in the bias fiber tows, indicated by the Hashin criteria to be due to matrix failure (due to large 

transverse tensile and shear stresses), was observed. These results correlate to observations made in the 

experiments, Figure 18(b), in which high strains in the bias fibers were identified at the corresponding 

global strain level (representing local matrix cracking within the fiber tows). These results appear to 

confirm that the high strains in the bias fiber identified experimentally are due to fiber splitting (local 

matrix cracking) within the fiber tows. As seen in the experimental axial tensile stress-strain curve shown 

in Figure 11, at this global strain level the stress-strain curve became nonlinear, which appears to indicate 

and confirm that the nonlinearity observed in the experimental global stress strain curve (and predicted 

numerically) is due to fiber splitting within the bias fiber tows. At a global strain level of 1.9 percent, 

fiber tow failure due to tensile fiber failure, as well as failure of the interface elements was identified. As 

observed in Figure 11, in the experimental axial tensile test the composite failure was identified as 

occurring at a global strain level of 1.9 percent. The results obtained here indicate that the ultimate failure 

of the composite is a result of failure of the axial fibers followed by a corresponding failure in the 

interface, which was also reflected in the computed effective composite response reaching its ultimate 

strength at this global strain level. These results are reasonable since for an axial tension test one would 

expect the ultimate failure of the composite to be governed by the failure of the axial fibers. 

NASA/TM—2010-216372 18 
5.2 Axial Compression Test 

In the detailed damage contour plots obtained in the simulations of the axial compression tests shown 

in Figure 19, at a global strain level of 0.5 percent significant levels of damage in the axial fiber tows due 

to tensile matrix failure and significant levels of damage in the bias fiber tows due to compressive matrix 

failure were identified. By comparison to the global stress-strain curve shown in Figure 13, at this global 

stress and strain level the effective curve became nonlinear. These results indicate that nonlinearity in the 

effective stress-strain curve is a result of the identified damage in the fiber tows. At a global strain level of 

1.0 percent, compressive fiber failure in the axial fiber tows, along with slight amounts of damage in the 

interfacial elements, was identified. Since the strain level of 1.0 percent corresponded to the strain level at 

which the ultimate compressive strength occurred in the effective stress-strain curve, these results indicate 

that the ultimate failure in an axial compression test is due to compressive failure in the axial fiber tows. 

5.3 Transverse Tension Test 

In the simulation of the transverse tension test, multiple damage modes were identified as seen in 

Figure 20. Direct correlations between the damage modes identified in these simulations and the global 

stress-strain curves could not be made due to identified deficiencies in the transverse tensile test 

methodology, which result in complexities in the detail of the obtained results. First, at a relatively low 

strain level 0.4 percent, damage due to tensile matrix cracking within the axial fiber tows was observed. 

At approximately the same global strain level, as seen in Figure 21(b), optical strain measurements 

obtained experimentally identified high strain regions in the axial fiber tows, which could correlate to 

damage occurring in the axial fiber tows. At a slightly higher strain level, 0.6 percent, some regions of the 

axial fiber tows in the simulations exhibited damage due to compressive fiber failure. At a global strain 

NASA/TM—2010-216372 19 
level of 1.2 percent, failure occurred in the simulations, which according to the damage contour plots was 

due to a combination of tensile matrix cracking in the all of the fiber tows, which would be expected in a 

transverse tensile test, and interfacial failure, which would be expected in a material with a weaker 

interfacial bond. 

5.4 Transverse Compression Test 

In the simulations of a transverse compression test shown in Figure 22, damage due to matrix 

compressive failure in the fiber tows was identified as occurring first, followed by compressive failure in 

the fiber tows at the point of ultimate failure. Some interfacial failure was identified as occurring along 

the edges of the specimen. 

NASA/TM—2010-216372 20 
From the detailed analysis of failure process for each test, together with the section of this report 

describing the correlation of the global stress-strain curves, the results indicate that the presented model 

gives a good local failure prediction and can be used to explain, and correlate with, experimental 

measurements. 

6.0 Conclusions 

A finite element based approach has been developed for examining the behavior of triaxially braided 

polymer matrix composites. The architecture of the braided composite unit cell has been explicitly 

modeled. A transversely isotropic constitutive, damage and failure model has been utilized for the fiber 

tows. A cohesive element approach has been applied to model the imperfect interfaces between fiber tows 

and between the fiber tows and regions of pure matrix. 

The global stress-strain response of two representative triaxially braided composite systems has been 

simulated, with reasonably good correlations obtained with experimentally obtained results. The effects of 

varying the strength of the interface on the simulations of the global response have been identified. By 

examining detailed contour plots of the simulated damage and failure within the composite, the 

progression of damage and failure within the composite has been identified, and in many cases 

correlations between the local damage mechanisms and details of the effective stress-strain response 

could be made. 

By utilizing a modeling approach of this type, numerical experiments could be performed on a 

braided composite system for situations where experimental data is not available, such as tests at high 

strain rates. Furthermore, by studying the details of the local damage and failure within the finite element 

model, significant insight could be gained into the details of the material response, which could be 

extremely useful in understanding the behavior of the material and developing more approximate 

numerical models of the material behavior. 

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Finite Element Model for Failure Study of Two-Dimensional Triaxially Braided Composite 

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

A new three-dimensional finite element model of two-dimensional triaxially braided composites is presented in this paper. This meso-scale 

modeling technique is used to examine and predict the deformation and damage observed in tests of straight sided specimens. A unit cell 

based approach is used to take into account the braiding architecture as well as the mechanical properties of the fiber tows, the matrix and 

the fiber tow-matrix interface. A 0°/±60° braiding configuration has been investigated by conducting static finite element analyses. Failure 

initiation and progressive degradation has been simulated in the fiber tows by use of the Hashin failure criteria and a damage evolution law. 

The fiber tow-matrix interface was modeled by using a cohesive zone approach to capture any fiber-matrix debonding. By comparing the 

analytical results to those obtained experimentally, the applicability of the developed model was assessed and the failure process was 

investigated. 

15. SUBJECT TERMS

Polymer matrix composites; Braided composites; Finite element method; Micromechanics 

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF

 ABSTRACT

UU 

18. NUMBER

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 PAGES

28 

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email:help@sti.nasa.gov) 

a. REPORT

U 

b. ABSTRACT

U 

c. THIS 

PAGE

U 

19b. TELEPHONE NUMBER (include area code)

443-757-5802 

Standard Form 298 (Rev. 8-98)

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