Robert K. Goldberg

Glenn Research Center, Cleveland, Ohio

Brina J. Blinzler and Wieslaw K. Binienda

University of Akron, Akron, Ohio

Investigation of a Macromechanical Approach to 

Analyzing Triaxially-Braided Polymer Composites

NASA/TM—2010-216371

July 2010

AIAA–2010–2522
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Robert K. Goldberg

Glenn Research Center, Cleveland, Ohio

Brina J. Blinzler and Wieslaw K. Binienda

University of Akron, Akron, Ohio

Investigation of a Macromechanical Approach to 

Analyzing Triaxially-Braided Polymer Composites

NASA/TM—2010-216371

July 2010

AIAA–2010–2522

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

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Level of Review: This material has been technically reviewed by technical management. 
Investigation of a Macromechanical Approach to Analyzing 

Triaxially-Braided Polymer Composites 

Robert K. Goldberg 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Brina J. Blinzler and Wieslaw K. Binienda 

University of Akron 

Akron, Ohio 44325 

Abstract 

A macro level finite element-based model has been developed to simulate the mechanical and impact 

response of triaxially-braided polymer matrix composites. In the analytical model, the triaxial braid 

architecture is simulated by using four parallel shell elements, each of which is modeled as a laminated 

composite. The commercial transient dynamic finite element code LS-DYNA is used to conduct the 

simulations, and a continuum damage mechanics model internal to LS-DYNA is used as the material 

constitutive model. The material stiffness and strength values required for the constitutive model are 

determined based on coupon level tests on the braided composite. Simulations of quasi-static coupon tests 

of a representative braided composite are conducted. Varying the strength values that are input to the 

material model is found to have a significant influence on the effective material response predicted by the 

finite element analysis, sometimes in ways that at first glance appear non-intuitive. A parametric study 

involving the input strength parameters provides guidance on how the analysis model can be improved. 

Introduction 

The use of textile (woven and braided) polymer composites is being actively investigated for potential 

application in aerospace structures, as an alternative to metals or traditional laminated composites. For 

example, triaxially-braided composites are being utilized in fan cases for jet engines. To be certified as 

flight-worthy, a full-scale engine blade-out test must be performed to demonstrate that the fan 

containment system can contain a released fan blade and that the fan case can maintain sufficient 

structural integrity during the engine spool down. As part of the design process, the need exists to 

simulate the blade and case deformation, along with the resulting damage and failure, during a blade-out 

event through the use of commercial explicit finite element codes. For metallic case systems, a sufficient 

ability to properly simulate the blade-out event exists due to the fact that adequate material databases and 

engineering experience in correlating analysis and test results are available. However, the capability to 

simulate the impact response of composite materials is much less mature. The ultimate goal of the current 

research is to improve the blade-out simulation capability for composite fan cases to a level comparable to 

that available for metallic cases. The work presented in this paper is a step towards that goal. 

There has been a significant amount of research conducted in the analysis and modeling of textile 

composites. The majority of the efforts have concentrated on various means to determine the effective 

mechanical properties of woven materials. Among the earliest attempts to model these materials was the 

work of Chou and Ishikawa (Ref. 1). In their original mosaic model, the woven composite was 

approximated as a one-dimensional series of laminated cross-ply composites, and classical laminate 

theory in combination with iso-stress or iso-strain assumptions were applied to obtain the effective 

stiffness properties of the material. They later extended the model to account for the fiber undulations that 

are present in an actual woven material (Ref. 1). This approach was extended to two dimensions by Naik 

and Shembekar (Ref. 2), where a mixture of parallel and series assumptions were applied to obtain the 

NASA/TM—2010-216371 1 
effective properties of the material. To analyze more complicated fiber architectures, such as braided 

composites, researchers such as Pastore and Gowayed (Ref. 3) and Byun (Ref. 4) modeled the fibers as a 

series of rods at various angles, and utilized simple iso-strain assumptions to obtain the overall effective 

properties of the composite. More sophisticated analysis methods, such as those developed by Tanov and 

Tabiei (Ref. 5) and Bednarcyk and Arnold (Ref. 6), used an approach where a representative unit cell of a 

woven composite was created, and then micromechanics-based approaches were applied to compute the 

effective properties and response of the material. In the context of applying these methods within a finite 

element model, elements are created with a homogenized set of material properties, and the appropriate 

analysis method is used to generate the effective properties and response of the woven material. 

When many of these previously developed analytical methods have been used to conduct a finite 

element analysis of a structure composed of woven or braided composite materials, homogenized 

elements are used, in which the architecture of the textile material is not directly accounted for within the 

finite element model. However, for triaxially-braided composites, experimental evidence has shown 

(Ref. 7) that when these materials are subject to impact, oftentimes the damage will propagate along the 

fiber directions. To best simulate these damage patterns, the braid architecture should be directly 

simulated within the finite element model. To account for the fiber architecture in a computationally 

efficient manner, Cheng (Ref. 8) created a “Braided Through the Thickness” approach, where the braided 

composite is modeled as a series of layered shell elements, where each element is a laminated composite 

with the appropriate fiber-layup. The effective stiffness and strength material properties of the equivalent 

unidirectional composite then were included as input data for the finite element analysis. In the method 

developed by Cheng, fiber and matrix properties were combined using simple micromechanics-based 

approaches to obtain the required effective properties. The model was developed within the context of 

LS-DYNA (Ref. 9), a commercially available transient dynamic finite element code which is commonly 

used within the aerospace industry. A continuum damage mechanics based composite constitutive model 

available within LS-DYNA was used as the material model. 

Littell et al. (Ref. 10) made two major extensions to the approach developed by Cheng. First of all, in 

actual braided composites, the layers of braid often do not directly sit on top of each other, but instead 

fiber nesting takes place. In this work, the phrase “fiber nesting” refers to the in-plane shifting of fibers in 

adjacent plies by a fraction of the unit cell dimensions, not the definition that is also sometimes used in 

the context of textile composites, that of through-the-thickness shifting that also sometimes occurs. In 

their work, the discretization of the braid was adjusted to account for this phenomenon. More 

significantly, in the previously developed methods, the effective unidirectional composite properties of 

the materials utilized in a braided composite either had to be measured experimentally or computed by 

using micromechanics techniques. This approach requires an extensive additional experimental program, 

with tests on either the matrix constituent (for micromechanics approaches) or the equivalent 

unidirectional composites, besides tests on the braided composite, being required to obtain the needed 

data. These test programs can be quite complicated, with no guarantee that the experimental properties 

obtained are representative of the in-situ properties of the material within the braided composite. In the 

approach developed by Littell, et al., the equivalent unidirectional properties of the composite were 

extracted based on data obtained from coupon tests of the braided composite. In this manner, the in-situ 

properties were directly incorporated into the input for the material model, which could reduce the 

amount of testing required. Also, this approach has the potential to accurately simulate the deformation, 

damage and failure of a triaxially-braided composite in a computationally efficient manner. Alternatively, 

an approach of this type can be useful as a tool to verify more sophisticated analysis approaches. As 

described by Littell, et al. (ref 10), the stiffness and strength values obtained through simulations of quasistatic coupon tests correlated reasonably well to experimentally obtained values. However, when flat 

panel impact tests were simulated, the predicted penetration velocity correlated reasonably well to 

experimentally obtained values for some materials and the predicted damage patterns did not correlate 

well to what was observed experimentally. 

In this paper, first the “Braided Through the Thickness” analysis method developed by Cheng (Ref. 8) 

and extended by Littell, et al (Ref. 10) as applied to the analysis of triaxially-braided polymer composites 

NASA/TM—2010-216371 2 
will be summarized and explained. Next, a series of parametric studies designed to better understand how 

the material data input affects the computed response of the triaxially-braided material will be discussed. 

One of the major current goals of the research effort is to improve the ability of the analysis method to 

predict the damage patterns which result when a triaxially-braided composite is subject to impact loads. 

One approach to improving the damage predictions is to modify the strength parameters that are input into 

the material model. Since, as will be described later, the strength parameters are obtained from quasistatic coupon tests, rate dependence of the composite strengths needed for impact simulation are not 

currently accounted for within the analysis approach. Due to the nature of the constitutive model which is 

currently being utilized, the effects of varying a particular parameter on the overall simulations may not 

be straightforward. A first step in determining the sensitivity of the simulations to material model 

property input, the effect of varying the input strength parameters on the stress-strain curves and ultimate 

strength predictions obtained through simulations of quasi-static coupon tests is being quantified. Insights 

from this study can then be utilized to provide guidance on how the model input parameters can be varied 

to improve the simulation of impact events. 

Overview of Analysis Approach 

The “Braided Through the Thickness” analysis approach as developed by Cheng (Ref. 8) and Littell, 

et al (Ref. 10) has currently been designed to analyze triaxially-braided polymer composites with a 

[0°/+60°/–60°] braid architecture. Other triaxially-braided fiber architectures are currently being 

simulated using the approach, but theoretically the methodology can be applied to any textile composite 

architecture. A unit cell of a typical triaxially-braided fiber preform is shown in Figure 1(a). In the figure, 

the ±60° bias fibers are visible on the surface. Portions of the 0° axial fibers that lie below the bias fibers 

can be seen in the open spaces between the bias fibers. An assumption in the methodology is that the fiber 

bundle spacing and number of fibers per fiber bundle are adjusted such that the fiber volume fractions in 

both the axial and bias directions are equal. In actuality, this particular fiber architecture is quasi-isotropic 

in-plane. Therefore the global in-plane stiffness should be the same in all directions, but this condition is 

not actually applied within the context of the analysis model. 

To apply the analysis approach, the braided-fiber architecture is idealized. As a first step in this 

process, a schematic of the top view of the fiber architecture is shown in Figure 1(b), where the –60° bias 

fibers are green, the +60° bias fibers are red, and the 0° axial fibers are blue. As shown in the figure, the 

unit cell can be divided into four equal length parallel subcells. Next, each subcell is approximated to be a 

laminated composite composed of a stack of fiber bundles at various orientations that are determined by 

the braid architecture. Subcell A is modeled as a [+60°/0°/–60°] composite (bottom layer listed first). 

Subcell B is modeled as a [+60°/–60°] composite, subcell C is modeled as a [–60°/0°/+60°] composite, 

and subcell D is modeled as a [–60°/+60°] composite. This idealization is shown in Figure 1(c). Each 

subcell can be modeled as an individual shell element in a finite element model. A unit cell then consists 

of four shell elements. Shell elements can be used since the length and width of the composite structures 

typically considered are much greater than the thickness. A full structure can then be modeled by 

replicating this four element unit cell throughout the finite element mesh. Each subcell is assumed to be 

equal thickness with a constant fiber volume fraction. In actuality, this assumption is inaccurate. In the 

one layer model, as described here, the plies in subcell B and subcell D should have lower overall fiber 

volume fractions due to the lack of axial fibers, resulting in these subcells only having two plies while 

subcell A and subcell C have three plies. However, this assumption simplifies the process of determining 

the material properties to be input into the model using the techniques to be described below. Future 

efforts will concentrate on modifying the analysis method to remove this assumption. Note also that while 

the unit cell as a whole can be approximated to be a symmetric composite, each individual subcell is not a 

symmetric composite. The assumption is made that since the unit cell as a whole is a symmetric 

composite, any normal-bending coupling will be smeared out. As a result, the portion of the laminate 

stiffness matrix (commonly referred to as the “B” matrix) that relates the in-plane loads to out-of-plane 

curvatures is assumed to be equal to zero. 

NASA/TM—2010-216371 3 
The actual braided composites which were examined in this study had six layers of braided fibers (see 

Fig. 2). The axial (0°) fibers in each layer were aligned, but the lateral position of axial fibers in each 

layer was random. As a result, axial fibers in the various layers are not necessarily located directly over 

the top of each other, but instead are shifted in a random way relative to the axial fibers in layers above 

and below. In generating the finite element model for the full six layer composite, this behavior was 

approximated in an extreme sense for purposes of bounding the problem by shifting each layer of fibers 

by one subcell to the left in the full model. A schematic of the full finite element model is shown in 

Figure 2, where subcell A (from the original one layer model) is pink, subcell B is blue, subcell C is green 

and subcell D is yellow. The fiber shifting can be observed in this schematic. For example, in the top 

layer subcell D is the rightmost subcell, in the second layer subcell D is the second from the right subcell, 

and so forth. 

Accounting for the fiber shifting in the subcell model has other advantages for the model as currently 

formulated. First of all, since in the full model with fiber shifting each subcell has fifteen plies (layers of 

tows), the overall fiber volume fraction within each subcell is constant. Also, with this fiber layup, each 

subcell has equal number of +60° and –60° layers, which makes each subcell quasi-isotropic and 

representative of the overall composite architecture. Note though that since each subcell has an equal 

number of 0°, +60° and –60° fibers, capturing the architecture-dependent damage under impact may be 

difficult with the six-layer model as currently formulated. In future efforts, modifications to the model to 

account for this difficulty will be examined. 

NASA/TM—2010-216371 4 
Constitutive Model Overview 

Due to the desire to apply this analysis methodology to the simulation of impact problems, the model 

was implemented within the context of LS-DYNA (Ref. 9), a commercially available transient dynamic 

explicit finite element code that is widely used within the aerospace industry to conduct impact analyses. 

The constitutive model resident within LS-DYNA that was chosen for use with the analysis method is a 

continuum damage mechanics-based model for unidirectional composites, based on a plane-stress 

progressive failure theory developed by Matzenmiller, et al. (Ref. 11), that is implemented in LS–DYNA 

as MAT_58. The failure criterion implemented in the model is based on the Hashin failure criterion. For 

this material model, the required input stiffness parameters include the unidirectional-ply-level axial and 

transverse moduli, in-plane shear modulus, and the in-plane Poisson’s ratio. The unidirectional-ply-level 

strength data that is required includes the axial tensile and axial compressive failure stress and failure 

strain, the transverse tensile and transverse compressive failure stress and failure strain, and the in-plane 

shear failure stress and failure strain. The stiffness values are used by the material model to simulate the 

initial linear portion of the composite response. The ply-level strength and ultimate strain values are used 

to determine the nonlinear portion of the material response and to determine how the damage parameters 

used as internal variables within the material model evolve over the loading cycle. Furthermore, the user 

is allowed to specify a “stress limiting factor” in the longitudinal and/or transverse directions. If the value 

of the “stress limiting factor” is set to zero, the failure stress value is assumed to occur at the specified 

failure strain, and the material model then “connects the dots” (including enforcing any required material 

nonlinearity) to create the best-fit stress strain curve that accounts for the specified stiffness and strength 

values. If the “stress limiting factor” is set to one, the stress-strain curve of the composite is assumed to 

increase linearly until the maximum strength value is reached. At that point, the material is assumed to act 

as if it were a perfectly plastic material, with the strain in the specified direction increasing with no 

increase in stress until such point as the specified strain level is reached, at which point material failure is 

assumed to occur. 

The properties required by the model described above are the properties of the lamina (individual 

fiber bundles plus a portion of the surrounding resin material). A fundamental problem in the analysis of 

NASA/TM—2010-216371 5 
textile composites is that the lamina properties are not known and cannot be directly measured. Attempts 

can be made to fabricate unidirectional laminates using the same fiber tows that are used to make the 

preform. This approach is often not easy to accomplish, and the properties of the unidirectional lamina 

could be different from the properties of the fiber tows within the textile composite because of processing 

differences. For these reasons, a method was chosen that utilizes an approach based on a combination of 

classical laminate theory and concepts from composite micromechanics theory to calculate the lamina 

properties from experimental data obtained on the braided composite. Although there are also some 

limitations using this approach, the experimental requirements are simplified, and the calculated lamina 

properties represent the in-situ properties of fiber bundles within the braided composite. 

Determination of Material Properties 

The full procedure for the determination of the equivalent unidirectional material properties based on 

the results of coupon tests on the braided composite is described in detail in Littell, et al (Ref. 10). 

However, the procedure is summarized below. To back-calculate the stiffness properties, first the 

geometry of a coupon-level transverse tensile test was considered, where the axial fibers are oriented 

perpendicular to the applied load and the four subcells of the unit cell are oriented perpendicular to the 

applied load (see Fig. 2). In addition, the geometry of an axial tensile test was also examined. In this case, 

the unit cell and subcells were aligned in the direction of the applied load. Mechanics of materials types 

of assumptions were combined with the assumptions and methods of classical lamination theory to 

produce a series of equations that relate the applied global stress for the composite coupon (and thus the 

unit cell) and the subcell-level axial and transverse strains (determined from the experimental data using 

methods to be described below) to the unidirectional-ply-level stiffness properties. Note that a key 

assumption in the model development is that only in-plane loads were considered, and in-plane normalshear couplings were neglected. 

To apply these equations, the unit cell level stresses and the subcell level strains were assumed to be 

the known values, and the effective ply-level unidirectional material properties were assumed to be the 

unknown values. The results from the axial tension and transverse tension tests were applied 

independently, i.e. stress values from axial tension tests were paired with axial and transverse strain 

values from axial tension tests, and stress values and axial and transverse strain values from transverse 

tension tests were linked. To determine the unit cell level stresses, a stress value from the initial linear 

portion of the stress-strain curve was identified. Full-field strain measurements obtained during axial 

tension and transverse tension coupon-level tests (see Littell, et al (Ref. 10) for details) were examined to 

determine the average strain within regions corresponding to specific subcell locations. By utilizing fullfield strain measurements, the overall braid architecture and regions corresponding to specific subcells 

can be identified. The average strain in each subcell can be measured by tracking points near the edge of 

each subcell. Strain values from multiple unit cells and subcells were measured and averaged. A visual 

demonstration of determining the average subcell strains based on optical strain measurements is shown 

in Figure 3 (Ref. 10). Global stress (and corresponding subcell strain) values were determined from 

several locations in the linear portion of the axial and transverse stress-strain curves, with their values 

being applied to the equations. The corresponding computed property values were then averaged to arrive 

at a single value for each of the unidirectional-ply-level properties. Note that a potential source of error in 

this procedure is related to the fact that the subcell strains used in the calculations are based on surface 

strain measurements which may not be totally representative of the average strains through the thickness. 

These procedures will be modified in future work to account for the fact that in the actual composite the 

stiffness values for the individual subcells will vary since the fiber volume fraction in each subcell is not 

constant. 

NASA/TM—2010-216371 6 
The unidirectional strength values were also determined from the results of coupon tests on the 

braided material. The assumption was made that in an axial tensile test the load is primarily carried by the 

axial fibers, with the bias fibers making relatively small contributions. As a result, the axial failure stress 

and axial failure strain observed for the braided composite were assumed to be equal to the equivalent 

ply-level unidirectional-axial failure stress and axial failure strain. The strain field measured during axial 

compression tests was nearly uniform throughout the gage section with little effect of the braid 

architecture (see Littell, et al (Ref. 10)). The compressive strains in the axial fibers were therefore 

assumed to be equal to the global compressive strains in the composite. The compressive stress in the 

axial fibers was assumed to be equal to the global compressive stress. Therefore, the unidirectional 

compressive-failure stress and -failure strain were set equal to the measured global compressive-failure 

stress and -failure strain. To determine the equivalent transverse compressive-failure stress and -failure 

strain, once again the observation was made that during a transverse compression test, the compressive 

strains are relatively uniform throughout the gage section, indicating that the compressive strain (and by 

extension the compressive stress) for the zero degree layers are equal to the compressive stress and strain 

for the braided composite. Therefore, once again the equivalent unidirectional transverse compressivefailure stress and -failure strain were set equal to the transverse compressive failure stress and failure 

strain for the braided composite. 

NASA/TM—2010-216371 7 
To determine the equivalent transverse tensile-failure stress of the unidirectional composite, the 

transverse tensile test data from the braided composites could not be used directly since in a transverse 

tension test, unlike in an axial tension test, the bias fibers significantly contribute to the global response of 

the composite. Therefore, full-field strain measurements obtained during transverse tension tests were 

utilized to obtain the required information. The axial strain field in a transverse tensile specimen (for a 

T700S/PR520 triaxially-braided composite to be described later in this paper) loaded close to failure is 

shown in Figure 4(a) (Ref. 10). Global stress versus local strain is shown in Figure 4(b) for the four 

locations indicated by points in Figure 4(a). Localized damage is visible as small horizontal lines 

throughout the specimen in Figure 4(a). These local regions of high strain occur as a result of local matrix 

cracking within the fiber tows, which will hereafter be referred to as “fiber splitting”, in the subsurface 

axial-braid fiber bundles, which lie in a horizontal direction in Figure 4(a). The locations of the first four 

splits that occur are indicated by colored dots in Figure 4(a). The local strain in the regions of these fiber 

bundle splits can be measured for each time step in the test. Figure 4(b) shows the global stress versus 

local strain curves for the four split locations. The curves in Figure 4(b) show an abrupt increase in local 

strain at the locations indicated by arrows. These abrupt increases are associated with splitting of the 

NASA/TM—2010-216371 8 
subsurface axial braid fiber bundles. The global stress at which the abrupt increase in strain occurs can be 

assumed to approximate the transverse-bundle failure stress. The lowest global transverse stress at which 

fiber splitting occurred was used as the transverse failure-stress for the unidirectional composite. In a 

unidirectional composite, complete failure occurs when the transverse failure stress is reached. In a 

braided composite the fiber bundles are constrained by adjacent fiber bundles, so multiple splits occur 

within a fiber bundle before complete failure of the composite. To account for this behavior, the “stress 

limiting factor” in the transverse direction was set equal to one. This allows the material to act as if it 

were a perfectly plastic material after the onset of fiber bundle splitting. Furthermore, the transversefailure strain for the equivalent unidirectional composite was set equal to the global transverse-failure 

strain of the braided composite. 

To compute the shear-failure stress and the shear-failure strain, the assumption was made that under 

pure shear loading, the shear stress and shear strain in every layer in all of the subcells was equal to the 

global-shear stress and -shear strain. Therefore, the equivalent unidirectional shear strength and shear 

failure strain was assumed to be equal to the shear-failure stress and shear-failure strain measured for the 

composite. 

These approximations will be adjusted in future work to both more accurately account for the fact that 

each subcell has a different fiber volume fraction (resulting in different strengths in each subcell) and to 

more accurately relate the global stresses to the actual local stresses present in each composite layer in the 

analysis model. 

Simulation of Quasi-Static Tension Tests 

A series of simulations were conducted to evaluate the analysis methods described above. 

Specifically, quasi-static axial and transverse coupon-level tensile tests on a T700S/PR520 material were 

simulated. This material is composed of a two-dimensional triaxial-braided preform and a 177 °C 

(350 °F) cure epoxy resin. Specifically, TORAYCA T700S fibers (Toray Carbon Fibers America, Inc.) 

and a CYCOM PR 520 matrix (Cytec Industries, Inc.) were used. T700S is a high strength, standard 

modulus carbon fiber. PR520 is a one part toughened resin. 

A [0°/+60°/–60°] triaxial-braid architecture was used. The 0° axial fibers were 24k flattened tows 

while the ±60° bias fibers were 12k flattened tows. Although larger fiber bundles were used in the axial 

direction, the fiber bundle spacing in the axial and bias directions were adjusted to give the same fiber 

volume in the axial and bias directions. This is a nominally quasi-isotropic in-plane fiber architecture, so 

the global in-plane stiffness is expected to be the same in all directions when the region of interest 

includes several unit cells of the braid material. Composite panels were fabricated by placing six layers of 

the braided preform into a mold with the 0° fibers aligned in the same direction. Although the axial (0°) 

fibers in the various layers were aligned, the lateral position of the axial tows in the six layers was 

random. Resin was injected into the closed mold and cured according to processing conditions 

recommended by the resin manufacturer. The global fiber-volume ratio of the cured composites 

was measured using the acid digestion technique. The T700S/PR520 composite had a fiber volume of 

55.9 ± 0.18 percent. 

The finite element meshes used in the study are shown in Figure 5. The finite element model for the 

axial tension test was 20.32 cm (8.0 in.) long and 3.56 cm (1.40 in.) wide and had 369 nodes and 320 

shell elements. The model for the transverse tensile test was 30.48 cm (12 in.) long and 3.56 cm (1.40 in.) 

wide and had 522 nodes and 476 elements. The fixed end of the model was constrained in all three 

displacement and rotation directions. The loading was displacement controlled at the rate of 0.0635 cm/s 

(0.025 in./s), which was consistent with the test conditions. Each subcell was modeled as a layered-shell 

element, which allows the braid architecture to be explicitly modeled with the finite element mesh of the 

coupon. In Figure 5, the red elements represent subcell A, the blue elements represent subcell B, the green 

elements represent subcell C, and the yellow elements represent subcell D. The unit cell orientations for 

NASA/TM—2010-216371 9 
both of the modeling conditions are highlighted in Figure 5 for reference. In the axial test simulation, the 

unit cell is oriented parallel to the direction of the applied load. In the transverse test simulation, the unit 

cell is oriented perpendicular to the direction of the applied load. An important point to note is that the 

analytical techniques developed in this work will ultimately be applied to the analysis of the impact 

response of braided composites. However, since the experiments that were used to characterize the model 

were based on quasi-static tests, only these low-rate tests will be simulated in this work. Future efforts 

will involve using the model to simulate impact events. Furthermore, the capability to account for the 

strain rate dependent and high strain rate response of the material will be added to the model. 

Tensile stress versus strain curves were generated for the T700S/PR520 material in both the axial and 

transverse directions to examine the correlation of the analysis model to the test data. Even though the 

coupon level test data was used to obtain the input data applied to the model, due to the complexity of the 

material model and the ways that the various stiffness and strength parameters interact, the stiffnesses and 

strengths predicted by the analytical simulations may not necessarily correspond to the parameters 

determined from the test data. The material properties that were used for the analysis are shown in 

Table 1. Specimen failure was deemed to occur when all of the elements in a row failed. The element 

failure occurred nearly simultaneously in the simulations, leading to a fairly abrupt ultimate failure. The 

resulting stress-strain curves are shown in Figure 6 for the axial tension simulations and Figure 7 for the 

transverse tension simulations. In these figures and all remaining figures, the test data is as determined by 

Littell, et al (Ref. 10). 

NASA/TM—2010-216371 10 
TABLE 1.—MATERIAL PROPERTY VALUES USED FOR 

THE QUASI-STATIC COUPON SIMULATIONS 

Material T700 – PR520 

Material Parameter Name (LS-DYNA name) Value (unit) 

Axial Modulus (EA) 51.37 (GPa) 

Transverse Modulus (EB) 25.03 (GPa) 

In Plane Shear Modulus (GAB) 18.96 (GPa) 

In Plane Poisson Ratio (PRBA) 0.071 

Axial Tensile Failure Strain (E11T) 0.0216 

Axial Compressive Failure Strain (E11C) 0.018 

Transverse Tensile Failure Strain (E22T) 0.0168 

Transverse Compressive Failure Strain (E22C) 0.011 

In Plane Shear Failure Strain (GMS) 0.024 

Axial Tensile Stress at Failure (XT) 1044.59 (MPa) 

Axial Compressive Stress at Failure(XC) 377.09 (MPa) 

Transverse Tensile Stress at Failure (YT) 361.99 (MPa) 

Transverse Compressive Stress at Failure (YC) 344.75 (MPa) 

In Plane Shear Stress at Failure (SC) 307.31 (MPa) 

Stress Limiting Parameter for Axial Tension (SLIMT1) 0 

Stress Limiting Parameter for Transverse Tension (SLIMT2) 1 

Stress Limiting Parameter for Axial Compression (SLIMC1) 0 

Stress Limiting Parameter for Transverse Compression (SLIMC2) 0 

Stress Limiting Parameter for Shear (SLIMS) 0 

NASA/TM—2010-216371 11 
For the axial tension test, the initial portions of the simulated and experimental curves coincide, but in 

general the simulated curve is stiffer than the experimental curve. This discrepancy could be due to the 

fact that in the analysis model every subcell has 0° layers, which is probably not representative of the 

actual material architecture, and could lead to the predicted results being overly stiff under axial tension 

loading (since the 0° layers promote increased stiffness). Furthermore, results obtained by Littell, et al. 

(Ref. 10) indicated that in the actual axial tensile tests, fiber splitting occurred in the bias fibers, and there 

was also some out-of-plane damage, both of which could lead to a weaker response. The current analysis 

model, due to the shell element formulation, cannot capture any out-of-plane damage. In addition, the 

presence of the 0° layers in every subcell of the analysis model could have reduced the influence of any 

bias fiber damage and weakening in the simulations. However, the fact that the axial tensile specimen 

exhibited a near linear response until failure was captured in a general sense by the simulations. The 

simulated axial tensile strength of the specimen also was closely correlated with the experimental results. 

The ultimate strength of the transverse tensile specimen was significantly overpredicted, and the 

simulated response of the transverse tensile test was much more linear than observed in the experiments. 

A possible cause of the discrepancy in the transverse tensile results can be seen in the full-field strain 

maps taken from a transverse tensile test which are shown in Figure 4(a). Premature failure near fiber 

bundles that terminate at free edges can influence the shape of the stress-strain curve and the measured 

failure strain. A large amount of edge damage was observed in both axial and transverse tensile tests. In 

an axial tensile test the axial braid fibers lie along the specimen axis and are gripped on both ends during 

a test. Since these axial braid fibers carry most of the load, the reduction in strength caused by edge 

damage is small for the axial tensile test. The experimental axial tensile stress-strain curve shown in 

Figure 6 is therefore nearly linear until failure. The edge damage has a larger effect on the measured 

strength for the transverse tensile test because all of the axial braid fibers are perpendicular to the applied 

load. There is no continuous load path between the grips through fibers because all axial and bias fiber 

bundles terminate at a free edge. The edge damage contributes to the nonlinearity of the experimental 

transverse tensile stress-strain curve in Figure 7 and also contributes to the reduced transverse failure 

stress compared with the axial failure stress. The dark blue triangular regions near the edges of the 

specimen in Figure 4(a) are regions of low strain that are not carrying load because of edge damage. Here, 

the edge-damaged region is about 10 percent of the specimen width. Since the damaged region does not 

carry load, the stress in the undamaged region is about 10 percent higher than the stress that is calculated 

based on the initial cross-sectional area of the specimen. The experimental transverse stress-strain curve 

in Figure 7 is therefore not an accurate representation of the transverse tensile properties. Due to the 

nature of the way in which the braid architecture was discretized and simulated in the analytical model, 

NASA/TM—2010-216371 12 
with the fiber tows not being modeled in a continuous manner, these local effects could not be captured in 

the simulations as each subcell in a sense is an independent laminated composite. As a result, the 

simulated results from the transverse tension test in all likelihood were more representative of the 

response that would be observed if the response was measured at a location far away from where edge 

effects would be significant. Therefore, the fact that the transverse tension response of the material was 

nearly linear, like the axial tension response, with a transverse failure stress only slightly lower than the 

axial failure stress, is most likely actually more representative of the response of the material as utilized in 

an engineering structure. Another possible cause of the discrepancy is due to the fact that instead of the 

elastic-perfectly plastic response in the transverse direction imposed by the material model, the actual 

material response may be more continuous and generally nonlinear, which could also lead to the 

experimental results being weaker than the simulated curves. Further studies do need to be made, 

however, to explore whether the observed discrepancies are caused by limitations in the experimental 

techniques, limitations imposed by the assumptions used in the analysis method, or both. 

Parametric Study of Input Unidirectional Strength Parameters 

To devise ways to modify some of the assumptions that were applied to the original braided through 

the thickness model (such as the constant fiber volume fraction in each subcell), and to extend the model 

to account for mechanisms such as material aging and strain rate effects, the need exists to gain a further 

understanding of the sensitivity of the model predictions to variations in the input data. In particular, due 

to the complex composite architecture being utilized in the modeling approach (modeling the braid using 

a series of varying laminated composites), and the complex nature of the constitutive model being 

utilized, the sensitivity of the model simulations to the various input parameters may not be 

straightforward. As a result, a study was undertaken where several of the input unidirectional strength 

parameters were varied independently. The strength parameters were considered to be the most important 

since these values would most likely to have the most significant effect on the damage and failure of the 

composite. Repeats of the simulations of the axial tension and transverse tension tests of the braided 

composite were then conducted to quantify the effects of the parameter variation on the predicted strength 

of the braided composite. 

Six-Layer Model 

For this study, first the unidirectional axial tensile strength (and corresponding axial tensile failure 

strain) was increased 50 percent and then decreased 50 percent from the baseline values. The baseline 

“nested” six layer model shown in Figure 2 was analyzed. The results from this study are shown in 

Figure 8 (composite axial tension simulations) and Figure 9 (composite transverse tension simulations). 

Next, the unidirectional transverse tensile strength (and corresponding failure strain) was increased 

50 percent and then decreased 50 percent. The results from this study are shown in Figure 10 (axial 

tension simulations) and Figure 11 (transverse tensile simulations). Finally, the in-plane shear strength 

(and failure strain) was increased 50 percent and decreased 50 percent. The results from this study are 

shown in Figure 12 (axial tension simulations) and Figure 13 (transverse tension simulations). In these 

(and several of the following) figures, some stiffening can be seen in several of the simulated curves right 

before failure occurs. This discrepancy is a numerical artifact resulting from the fact that some 

instabilities show up in the analysis right before failure occurs. These instabilities are the result of the 

explicit integration routines used in the finite element solver. Some of the results of the parametric study 

are intuitive. For example, the fact that varying the unidirectional axial-tensile strength significantly 

affected the composite axial tensile simulations (Fig. 8) is not surprising since for an axial test the load is 

primarily carried by the axial fibers. Similarly, the fact that varying the unidirectional transverse-tensile 

strength did not significantly affect the composite axial tensile response (Fig. 10) is also not surprising 

since the composite architecture as currently simulated is dominated by the axial fibers (each subcell has a 

number of 0° layers), and thus the transverse stresses would not significantly affect the results. 

NASA/TM—2010-216371 13 
 

NASA/TM—2010-216371 14 
 

Furthermore, as seen in Figure 13, varying the unidirectional in-plane shear strength significantly 

affected the predicted composite transverse-tensile strength. These results were expected since the 

stresses in the matrix are dominant under transverse tensile loading, and the in-plane shear strength is 

primarily driven by the stresses in the matrix. 

However, some of the results obtained from the parametric study were not as intuitive. For example, 

as seen in Figure 9, decreasing the uniaxial axial-tensile strength significantly lowered the predicted 

composite transverse-tensile strength. This result is counterintuitive since one would not expect the 

longitudinal stresses to play a key role in a transverse tension test. However, these results indicate that the 

bias fibers might be failing in an axial manner. Likewise, as is seen in Figure 11, while decreasing the 

unidirectional transverse-tensile strength resulted in a lower predicted composite transverse-tensile 

strength, which would be expected (due to the fact that both are driven by matrix stresses), increasing the 

unidirectional transverse-tensile strength resulted in a minimal increase in the predicted composite 

transverse strength, which is somewhat unexpected. Furthermore, as seen in Figure 12 reducing the 

unidirectional in-plane shear strength resulted in a reduction in the predicted composite axial-tensile 

strength, which is somewhat unexpected since the composite axial tensile strength should primarily be 

driven by the axial tensile strength of the longitudinal fibers (layers). The exact reasons for these 

discrepancies are currently unclear and are under further investigation. However, the causes of these 

results are most likely due to the fact that the equations predicting the initiation and propagation of 

damage within the material constitutive model are based on interactive combinations of the stresses and 

strengths, resulting in fairly smooth damage and failure surfaces. Particularly in the bias (±60°) fibers, the 

stress state is highly multiaxial, which when combined with the damage and failure criteria most likely 

leads to nonintuitive results. 

Single-Layer Model 

To further explore the effects of varying the input strength parameters, a revised model composed of 

only one layer of the braided material, similar to that shown in Figure 1(c), was analyzed. By examining 

the single layer model, some of the complexity of the six layer model is removed, which allows for a 

simpler analysis of the effects of the input strength parameters. Furthermore, to eventually model the 

impact damage observed in actual structures, an analysis model with different fiber layups in each subcell 

will be required in order to capture the architecturally-dependent damage. However, by shifting to this 

revised analysis model, the approximation made for the six-layer model that the fiber volume ratio is the 

same in all subcells is no longer valid. The fiber volume ratio in subcells A and C should now be greater 

NASA/TM—2010-216371 15 
than the fiber volume ratio in subcells B and D, with corresponding changes in the material properties. 

Future efforts will concentrate on properly modifying the material property input to reflect this situation. 

However, for this study, the same material properties used for the baseline study will be applied, with the 

understanding that the quantitative comparison of the analysis results to the experimental values 

(particularly in the axial tension simulations) will not be as good. However, the qualitative examination of 

the effects of the input parameters on the computed results should still be valid. Another difference 

between the six-layer model and the single layer model is that in the six-layer model the layer thickness in 

all of the subcells was equal. In the single-layer model, due to the fact that subcells A and C have 3 layers 

and subcells B and D have two layers, the layer thicknesses in subcells B and D are larger than those in 

subcells A and C. The overall layout for the finite element model is the same as was shown in Figure 5. 

To explore the effects of using the single-layer model (as compared to the “nested” six-layer model), 

the axial tension test described earlier was simulated using the single-layer model with the results shown 

in Figure 14. The transverse tensile test was also simulated with the results shown in Figure 15. As can be 

seen in the figures, the single layer simulations were softer, particularly in the nonlinear region, as 

compared to the six layer simulations. The results were more pronounced with the axial tension 

simulations. To provide a better “apples to apples” comparison between the two analysis models, a sixlayer “unnested” model was created by stacking six layers of the single-layer model to create a modified 

six-layer model, as shown schematically in Figure 16. The results computed using this six-layer 

“unnested” model are shown in Figure 17 for the axial tension simulation and in Figure 18 for the 

transverse tension simulation. As can be seen in Figure 17, the computed results under axial tension 

conditions for the single layer and “unnested” six-layer model were very close, with the “unnested” six 

layer model having a slightly higher final strength. As seen in Figure 18, using the “unnested” six-layer 

model under transverse tension conditions resulted in a slightly stiffer material response as compared to 

the single-layer model, to the point where the results obtained using the “unnested” six-layer model very 

closely tracked the results from the original “nested” six-layer model, with the only major difference 

being that the “unnested” six-layer model had a somewhat lower final strength as compared to the 

“nested” model. Even though the results obtained using the single layer model are closer to the 

experimental results than the results obtained using either of the six-layer models, given the deficiencies 

identified in the transverse tension tests and the approximations in the analysis method (which as 

discussed above indicate that the results obtained using the “nested” six-layer model may be closer to 

reality), the fact that the “unnested” six layer results tracked closer to the “nested” six layer results may 

indicate that the “unnested” six layer model yields more accurate results that the single-layer model. 

 

NASA/TM—2010-216371 16 
 

To examine the reasons for the differences between the “nested” six-layer and the single-layer (and 

by extension the “unnested” six layer) models in more detail, the stress-strain response for subcells A and 

C and subcells B and D were individually computed, with the results shown in Figure 19 for the axial 

tension case and Figure 20 for the transverse tension case. In each of these figures, the curves labeled 

“LS-DYNA Simulation 1 ply” and “LS-DYNA simulation 6 ply” are the overall, effective stress-strain 

curves for the composite for the single layer and “nested” six layer cases, respectively. The curves labeled 

“Simulation 1 ply AC” are the results specifically for subcells A and C from the single layer analysis and 

the curves labeled “Simulation 1 ply BD” are the results specifically for subcells B and D from the single 

NASA/TM—2010-216371 17 
layer analysis. As can be seen in Figure 19, for the axial tension case, the results for subcells A and C 

almost identically track the results from the six-layer “nested” model. These results are reasonable since 

subcells A and C and the “nested” six-layer model both have a mixture of 0°, +60° and –60° layers. 

Conversely, the results for subcells B and D are much softer, reflecting the fact that these subcells do not 

have any 0° layers, which in an axial tension test provide the majority of the stiffness. The average of the 

two subcell sets then would result in the overall, relatively softer material response in the case of the 

single layer simulations. The fact that the response for subcells B and D flattens out indicates that the 

local transverse stresses have reached the transverse-failure stress for the single ply simulations. In 

examining the transverse tension results in Figure 20, the results for each of the subcells tracked along the 

same path as the average results for the full single-layer model. However, the results for subcells A and C 

had a final predicted strength near to that of the “nested” six-layer model, and the results for subcells B 

and D had a predicted strength that was significantly lower, leading to the average curve for the single 

layer simulations that has a strength that is lower than the “nested” six-layer results. If the same study was 

conducted using the “unnested” six layer model, based on the case shown in Figure 18, the results for 

subcells A and C would most likely track very closely to the results for the “nested” six-layer model. 

Hence, the axial 0° layers do not significantly contribute to the overall transverse stiffness, which is 

reasonable, but they do cause an increase in the transverse strength. 

The parametric study that was conducted for the “nested” six-layer model was repeated for the singlelayer model. First the unidirectional axial-tensile strength (and corresponding axial-tensile failure strain) 

was increased 50 percent and then decreased 50 percent from the baseline values. The results from this 

study are shown in Figure 21 (composite axial-tension simulations) and Figure 22 (composite transversetension simulations). Next, the unidirectional transverse-tensile strength (and corresponding failure strain) 

was increased 50 percent and then decreased 50 percent. The results from this study are shown in 

Figure 23 (axial tension simulations) and Figure 24 (transverse tensile simulations). Finally, the in-plane 

shear strength (and failure strain) was increased 50 percent and decreased 50 percent. The results from 

this study are shown in Figure 25 (axial tension simulations) and Figure 26 (transverse tension 

simulations). 

NASA/TM—2010-216371 18 
 

 

NASA/TM—2010-216371 19 
 

As can be seen in Figures 21 and 22, varying the unidirectional axial-tensile strength resulted in 

significant changes in the composite axial-tensile response but minimal changes in the composite 

transverse-tensile response. These results are both intuitive based on the fiber architecture, as the 

unidirectional axial-tensile strength should primarily effect the composite axial-tensile response. The 

reason that in the “nested” six-layer model the unidirectional axial-tensile strength affected the composite 

transverse-tensile response is still unclear. 

Examining the effects of varying the unidirectional transverse-tensile strength in Figures 23 and 24, 

varying the transverse strength affected both the composite axial-tension and transverse-tension response. 

While the fact that varying the unidirectional transverse-tensile strength would significantly affect the 

composite transverse tensile response is fairly intuitive (since the loading in all of the layers is primarily a 

transverse load), the fact that varying the unidirectional transverse-tensile strength would affect the 

composite axial-tensile response is not intuitive. However, the reason for this variation can be seen in 

Figure 27, where the effect of varying the unidirectional transverse-tensile strength on the response of the 

individual subcells is shown for the axial-tension simulations. As can be seen in Figure 27, changing the 

unidirectional transverse-tensile strength resulted in no change in the response of subcells A and C, which 

would be expected due to the axial response of the 0° layers driving this response. This also most likely 

explains why for the “nested” six-layer model changing the unidirectional transverse-tensile strength did 

not affect the composite axial-tensile response. However, changing the unidirectional transverse-tensile 

strength significantly affected the axial response of subcells B and D. Specifically, the response of 

subcells B and D became much more linear as the transverse strength was increased, and the response of 

the two subcells became nonlinear at a much lower stress as the transverse strength was reduced. This 

result is actually reasonable since even under an axial tensile load the bias fibers are under a multiaxial 

stress state, and the interaction between the stresses could be significant enough to affect how the 

constitutive model simulates the material behavior. In particular, these results indicate that by raising the 

transverse strength, the local transverse stresses now stayed below the transverse failure strength. 

Furthermore, reducing the transverse strength caused the transverse stresses in the subcells to reach the 

revised, lower transverse strength at a relatively low strain. These results were reflected in the overall 

average composite curve, as raising the unidirectional transverse-failure stress caused the overall curve to 

become more linear, and reducing the transverse-failure stress caused the overall curve to become more 

nonlinear. 

NASA/TM—2010-216371 20 
 

Examining the effects of varying the unidirectional in-plane shear strength in Figures 25 and 26, for 

both the composite axial tension and transverse tension cases varying the in-plane shear strength once 

again affected both the composite axial-tension and transverse-tension responses. As for the case of the 

transverse-tensile strength, the fact that changing the unidirectional in-plane shear strength would affect 

the composite transverse-tensile response is fairly intuitive, but the fact that lowering the in-plane shear 

strength would affect the axial-tensile response is once again not necessarily what would be expected. By 

examining the individual subcell responses for the axial tensile simulations in Figure 28, however, the 

results make sense. As can be seen in this figure, lowering the unidirectional in-plane shear strength 

caused the failure stress of all the subcells to be significantly reduced for the case of axial tensile loading. 

These results indicate that for the multiaxial stress state that results in the bias fibers in all of the subcells, 

the local shear stresses must reach the reduced shear strength, resulting in relatively early failure. 

Conclusions 

A macro-level finite element-based approach has been developed which allows for the simulation of 

the response of a triaxially-braided composite in a manner which takes into account the architecture of the 

braided material within a structure. The input material properties utilized for the analysis can be 

determined based on experimental tests conducted on the braided composite. The analysis model, which 

utilizes LS-DYNA shell elements, appears promising in its ability to simulate the axial and transverse 

deformation and failure of the braided material. The various material strength parameters that are input to 

the model have a significant effect on the overall results of the simulations, sometimes in ways that at first 

glance appear to be non-intuitive. However, detailed studies of the material response indicated that many 

of what appeared to be non-intuitive effects could be explained and made sense within the context of the 

analysis model. Future efforts will involve modifying the analysis model to allow for each of the subcells 

in the discretization to have different fiber-volume ratios, and thus different material properties, which is 

more representative of the actual composite. Furthermore, the analysis model will be adapted to permit 

the analysis of the strain rate-dependent response of the braided-polymer composites. In particular, 

simulations of impact events will be conducted. Methods to utilize the model to explore composite 

materials subject to aging, or materials with weak fiber-matrix interfaces or brittle matrices will also be 

explored. 

NASA/TM—2010-216371 21 
NASA/TM—2010-216371 22 

References 

1. Chou, T.W. and Ishikawa, T.: “Analysis and Modeling of Two-Dimensional Fabric Composites,” in 

Textile Structural Composites, Composite Material Series, Vol. 3, T.W. Chou and F.K. Ko, Eds., 

Elsevier Science Publishers B.V., Amsterdam, 1989, Ch. 7, pp. 209–277. 

2. Naik, N.K.; and Shembedkar, P.S.: “Elastic Behavior of Woven Fabric Composites: I-Lamina 

Analysis,” Journal of Composite Materials, Vol. 26, No. 15, pp. 2196–2225, 1992. 

3. Pastore, C.M. and Gowayed, Y.A.: “A Self-Consistent Fabric Geometry Model: Modifications and 

Application of a Fabric Geometry Model to Predict the Elastic properties of Textile Composites,” 

Journal of Composites Technology and Research, Vol. 16, No. 1, pp. 32–36, 1994. 

4. Byun, J.-H.: “The analytical characterization of 2-D braided textile composites,” Composites Science 

and Technology, Vol. 60, No. 5, pp. 705–716. 2000. 

5. Tanov, R. and Tabiei, A.: “Computationally Efficient Micromechanical Models for Woven Fabric 

Composite Elastic Moduli,” Journal of Applied Mechanics, Vol. 68, No. 4, pp. 553–560, 2001. 

6. Bednarcyk, B.A.; and Arnold, S.M.: “Micromechanics-Based Modeling of Woven Polymer Matrix 

Composites,” AIAA Journal, Vol. 41, No. 9, 2003. 

7. Roberts, G.D. et al. “Impact Testing and Analysis of Composites for Aircraft Engine Fan Cases,” 

NASA/TM—2002-211493, 2002. 

8. Cheng, Jingyun. Material Modeling of Strain Rate Dependent Polymer and 2D Triaxially Braided 

Composites. Ph.D. Dissertation. University of Akron. Akron, Ohio. 2006. 

9. Livermore Software Technology Corporation. LS-DYNA Keyword Manual v 971. Livermore, CA, 

2007. 

10. Littell, J.D.; Binienda, W.K.; Goldberg, R.K.; and Roberts, G.D.: “A Modeling Technique and 

Representation of Failure in the Analysis of Triaxial Braided Carbon Fiber Composites,” 

NASA/TM—2008-215245, 2008. 

11. Matzenmiller, A.; Lubliner, J.; and Taylor, R.L.: “A Constitutive Model for Anisotropic Damage in 

Fiber-Composites,” Mechanics of Materials 20, pp. 125–152, 1995. 
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Investigation of a Macromechanical Approach to Analyzing Triaxially-Braided Polymer 

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14. ABSTRACT

A macro level finite element-based model has been developed to simulate the mechanical and impact response of triaxially-braided polymer 

matrix composites. In the analytical model, the triaxial braid architecture is simulated by using four parallel shell elements, each of which is 

modeled as a laminated composite. The commercial transient dynamic finite element code LS-DYNA is used to conduct the simulations, 

and a continuum damage mechanics model internal to LS-DYNA is used as the material constitutive model. The material stiffness and 

strength values required for the constitutive model are determined based on coupon level tests on the braided composite. Simulations of 

quasi-static coupon tests of a representative braided composite are conducted. Varying the strength values that are input to the material 

model is found to have a significant influence on the effective material response predicted by the finite element analysis, sometimes in ways 

that at first glance appear non-intuitive. A parametric study involving the input strength parameters provides guidance on how the analysis 

model can be improved. 

15. SUBJECT TERMS

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

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