Multi-scale Progressive Failure Analysis of Triaxially ...

Copyright ⓒ The Korean Society for Aeronautical & Space SciencesReceived: March 2, 2017 Revised: June 21, 2017 Accepted: June 22, 2017

436 http://ijass.org pISSN: 2093-274x eISSN: 2093-2480

PaperInt’l J. of Aeronautical & Space Sci. 18(3), 436–449 (2017)DOI: http://dx.doi.org/10.5139/IJASS.2017.18.3.436

Multi-scale Progressive Failure Analysis of Triaxially Braided Textile Composites

Tsinuel N. Geleta*Department of Civil Systems Engineering, Chungbuk National University, Chungbuk 28644, Republic of Korea

Kyeongsik Woo**School of Civil Engineering, Chungbuk National University, Chungbuk 28644, Republic of Korea

Abstract

In this paper, the damage and failure behavior of triaxially braided textile composites was studied using progressive failure

analysis. The analysis was performed at both micro and meso-scales through iterative cycles. Stress based failure criteria

were used to define the failure states at both micro- and meso-scale models. The stress-strain curve under uniaxial tensile

loading was drawn based on the load-displacement curve from the progressive failure analysis and compared to those by test

and computational results from reference for verification. Then, the detailed failure initiation and propagation was studied

using the verified model for both tensile and compression loading cases. The failure modes of each part of the model were

assessed at different stages of failure. Effect of ply stacking and number of unit cells considered were then investigated using

the resulting stress-strain curves and damage patterns. Finally, the effect of matrix plasticity was examined for the compressive

failure behavior of the same model using elastic, elastic – perfectly plastic and multi-linear elastic-plastic matrix properties.

Key words: Triaxially braided composites, Progressive failure analysis (PFA), Failure criteria, Unit cell, Stress – Strain curve,

Failure progression behavior

1. Introduction

In order to use composite materials for structural

applications in engineering fields, their mechanical properties

have to be determined beforehand. These properties include

the elastic and failure behaviors often represented by the

full stress-strain curve. This curve can be determined from

experimental tests on the material, analytical methods or

computational tools. While providing the true nature of the

material behavior, the experimental test, however, is much

more expensive and sometimes takes more time to conduct

making it difficult and uneconomical to use. The analytical

method, on the other hand, is the easiest, cheapest and

fastest; but has limited reliability since it is based on many

assumptions. This leaves computational tools midway in all

aspects, which calculate the values based on the mechanical

properties of the constituent fiber and matrix materials.

Triaxially braided textile composites are not different in

choosing the methodology for the determination of their

mechanical properties. All the three methods can be applied

to draw their stress-strain curves.

Over the years, researchers have been conducting

experiments on specimens targeting the required type of

response. Littell et al. [1] conducted a series of experiments

to examine the damage characteristics of triaxial braided

composites under tensile loading. They discussed the failure

loads and mode of failure in different parts of the braided

composite. Similarly, Ivanov et al. [2] studied the multiple

stages of damage in triaxial braids and compared the results

to FE model. Miravete et al. [3] also described analytical

meso-mechanical approach for the prediction of elastic and

failure properties valid for triaxial and 3D braided composite

This is an Open Access article distributed under the terms of the Creative Com-mons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduc-tion in any medium, provided the original work is properly cited.

* Master Student ** Professor, Corresponding author:[email protected]

437

Tsinuel N. Geleta Multi-scale Progressive Failure Analysis of Triaxially Braided Textile Composites

http://ijass.org

materials. They used a unit cell scheme in which the

geometry of both fiber and matrix has been considered. They

also conducted experimental tests for result validation.

Another less expensive alternative to predict the failure

properties of triaxially braided composites is through the

use of computational methods. Brief review of some of

these methods and a broader explanation of the element

failure method (EFM) are given by Tay et al. [4]. One of these

methods is through the use of material property degradation

method (MPDM) within continuum damage mechanics

(CDM) scheme. Blackketter et al. [5] applied this method to

model damage in a plain weave fabric reinforced composite

material. Naik [6] also used it for woven and braided

composites considering the discrete slices of the tows and

solved the effective stiffness and failure predictions using FE

technique. He used material property degradation method

based on maximum stress and strain in the constituent

materials. PFA and plastic behavior of biaxial braids was

studied by Tang et al. [7] and Goyal et al [8], respectively.

Nobeen et al. [9] also used MPDM by applying 3D-Hashin

[10, 11] and Stassi [12] failure criteria for the progressive

damage of constituent materials. Xu et al. [13] conducted

multiscale analysis between mesoscale and microscale

regimes. The constituent stresses are related to mesoscale

stresses using stress amplification factor.

Other computational technique that can be applied for

triaxial braids is cohesive zone modeling (CZM). Xie et al.

[14] applied discrete CZM to simulate static fracture in 2D

triaxially braided carbon fiber composites by inserting spring

elements between the nodes of bulk plane stress elements.

They compared their results with that of experimental tests

as well. In the paper of Li et al. [15] failure initiation and

progressive material degradation has been simulated using

MPDM while CZM is used to model the fiber tow-matrix

interface.

These methods can be used to predict either the tensile

or compressive behavior of the textile composites. Some

examples of researches on the tensile behavior of triaxial

braids are Littell et al. [1], Miravete et al. [3], Xu et al. [13],

and Li et al. [15]. All of them considered the tensile damage

behavior of triaxial braided composites when loaded in the

axial direction although they have different configurations.

Compressive failure behavior, on the other hand, has been

studied by Littell et al. [1] and Li et al. [15] in different

methods.

In comparison to unidirectional laminates and other non-

textile composites, textile composites, especially braided

textile composites are not studied very well for their damage

and failure behavior. The common way of determining

these behaviors is using experimental techniques which

are relatively expensive. Therefore, computational methods

such as the PFA provide insight into the detailed damage and

failure distribution when applied at meso- and micro levels.

This type of analysis enables the determination of damage

and failure scheme based on the individual constituent

material properties.

This study is organized in two major sections. Firstly,

the configuration, material properties and modeling of the

selected triaxial braid is discussed followed by over-view

of progressive failure analysis methodology. The types and

description of the failure criteria applied is also discussed

in this section. Then, the results of the PFA are presented by

discussing the damage progression behavior and modes of

failure. The method is then validated using tensile failure

results from references before discussions on the damage

and failure behavior. The effects of finite thickness and

stacking assumption on the damage behavior are examined.

Finally, the damage behavior is discussed in compression

loading case before closing with conclusion.

2. Analysis

2.1 Configuration

Triaxially braided textile composites have complex

interlaced structures of axial and bias tows. The tows in turn

consist of both fibers and matrix at the final form. Due to the

complicated fiber tow geometry as well as the composition of

fiber and matrix materials, the modeling of these composites

is a very formidable task.

The modeling of triaxially braided textile composites can

be done in multi-scales. Since the material is made from

interlacing of fiber bundles called tows, the modeling of these

tows and the matrix between them is regarded as mesoscale

modeling. On the other hand, since individual tows are made

from thousands of fiber strands, their mechanical behavior

can be modeled at individual filament level which is called

microscale modeling.

Triaxially braided textile composite model is characterized

by a number of geometric parameters. Since it is not possible

to perfectly model the real braid geometry, simplified

geometric models are made for the tows. The modeling scale

at these tows and matrix between them is called mesoscale

modeling. One important parameter is the braiding angle (θ)

defined as the acute angle between the axial tow and bias.

The axial tows are the straight tows running in the major

material direction while bias tows are the undulating tows

running below and above the axial tows and other bias

tows. The other important parameters are the widths (wa,

DOI: http://dx.doi.org/10.5139/IJASS.2017.18.3.436 438

Int’l J. of Aeronautical & Space Sci. 18(3), 436–449 (2017)

wb), thicknesses (ta, tb) and in-between gaps (εa, εb) of the

axial and bias tows, respectively. All of these parameters are

shown in Fig. 1.

Other parameters defining a triaxially braided composite

are the three different volume fractions. The first is the tow

volume fraction (v tow) which is the ratio of volume of tows

to the total volume of the model. The second is the tow fiber

volume fraction (vftow) defined as the ratio of the volume of

fiber in the tows and the volume of the tow. The last volume

fraction is the total fiber volume fraction (vf0) defined as the

total volume of fiber in the whole braid divided by the total

volume of the braid. It can also be defined in terms of the tow

volume fraction and tow fiber volume fraction as shown in

equation 1 below:

6

�� (1)

The geometric modeling of the tows is based on some assumptions to simplify the geometric

parameters such as cross-sectional shape and tow paths. Both the axial tow and bias tows are assumed

to have lenticular cross-sections of circular arcs. The bias tows are modelled by sweeping the

lenticular cross-section along an undulating tow path while the cross-section is kept parallel to the

bias tows running in the other direction. Then, these tows are duplicated and arranged to create the

full braid model from which the repeating unit cell can be cut out. Some geometric components of the

bias tow which was partitioned in to a number of groups for material orientation assignment. (See Fig.

3.) Detailed description of the modeling has been discussed in detail in a paper preceding this one [16].

The triaxially braided composite used in this study is a regular braid with a braiding angle of 30⁰. The width (��), thickness (��) and in-between gaps (��) of the axial tows are 2.40 mm, 0.50 mm and

2.38 mm, respectively. Similarly, the width (��), thickness (��) and in-between gaps (��) of the bias

tows were 2.20 mm, 0.50 mm and 1.94 mm, respectively. The braided unit cell is shown in Fig. 2 with

the dimensions. The tow volume fraction (��), fiber volume fraction (��) and total fiber volume

fraction (��) were 34.1%, 78.0% and 26.6%, respectively.

Fig. 2. Unit cell of the 30⁰ triaxial braid used in this study

W = 9.56 mm

t = 1.54 mm

L = 8.28 mm

. (1)

The geometric modeling of the tows is based on some

assumptions to simplify the geometric parameters such as

cross-sectional shape and tow paths. Both the axial tow and

bias tows are assumed to have lenticular cross-sections of

circular arcs. The bias tows are modelled by sweeping the

lenticular cross-section along an undulating tow path while

the cross-section is kept parallel to the bias tows running

in the other direction. Then, these tows are duplicated

and arranged to create the full braid model from which

the repeating unit cell can be cut out. Bias tow segments

were partitioned in to a number of groups for material

orientation assignment. (See Fig. 3.) Detailed description of

the modeling has been discussed in a paper preceding this

one [16].

The triaxially braided composite used in this study is a

regular braid with a braiding angle of 30⁰. The width (wa),

thickness (ta) and in-between gaps (εa) of the axial tows are

2.40 mm, 0.50 mm and 2.38 mm, respectively. Similarly, the

width (wb), thickness (tb) and in-between gaps (εb) of the bias

tows were 2.20 mm, 0.50 mm and 1.94 mm, respectively. The

braided unit cell is shown in Fig. 2 with the dimensions. The

tow volume fraction (v tow), fiber volume fraction (vftow) and

total fiber volume fraction (vf0) were 34.1%, 78.0% and 26.6%,

respectively.

2.2 Material Properties

The triaxial braid considered in this study is made from

carbon fiber and epoxy matrix constituent materials. The

elastic and failure properties of these materials are given

in Tables 1 and 2. These material properties are used at the

micro-scale computation to get the homogenized properties

for the tows. The homogenization is based on modified

classical laminate theory and micromechanics that has

been incorporated in a commercial software called MCQ-

Composites [17]. These calculations were made throughout

the progressive failure analysis process since the degradation

of material properties was made at the micro level. This

process is discussed in detail later.

The homogenized properties were assigned to the tows

in the mesoscale unit cells which needed the material

orientation assignments. The material orientation was

assigned by aligning the fiber direction to the tangent of the

tow paths while the transverse directions were perpendicular

to it. This material orientation definition is shown in Fig. 3.

5

Triaxially braided textile composite model is characterized by a number of geometric parameters.

Since it is not possible to perfectly model the real braid geometry, simplified geometric models are

made for the tows. The modeling scale at these tows and matrix between them is called mesoscale

modeling. One important parameter is the braiding angle ( ) defined as the acute angle between the

axial tow and bias. The axial tows are the straight tows running in the major material direction while

bias tows are the undulating tows running below and above the axial tows and other bias tows. The

other important parameters are the widths ( , ), thicknesses ( , ) and in-between gaps ( , )

of the axial and bias tows, respectively. All of these parameters are shown in Fig. 1.

Fig. 1. Major geometric parameters defining triaxially braided textile composite

Other parameters defining a triaxially braided composite are the three different volume fractions.

The first is the tow volume fraction ( ) which is the ratio of volume tows to the total volume of the

model. The second is the tow fiber volume fraction ( ) defined as the ratio between the volume of

fiber in the tows and the volume of the tow. The last volume fraction is the total fiber volume fraction

( ) defined as the total volume of fiber in the whole braid divided by the total volume of the braid. It

can also be defined in terms of the tow volume fraction and tow fiber volume fraction as shown in

equation 1 below.

wa εa ta

θ

εbwb

tb

θ – Braiding angle wa – Width of axial tow wb – Width of bias tow ta – Thickness of axial tow tb – Thickness of bias tow a – Gap between axial towsb – Gap between bias tows

Fig. 1. Major geometric parameters defining triaxially braided textile composite

6

�� (1)

The geometric modeling of the tows is based on some assumptions to simplify the geometric

parameters such as cross-sectional shape and tow paths. Both the axial tow and bias tows are assumed

to have lenticular cross-sections of circular arcs. The bias tows are modelled by sweeping the

lenticular cross-section along an undulating tow path while the cross-section is kept parallel to the

bias tows running in the other direction. Then, these tows are duplicated and arranged to create the

full braid model from which the repeating unit cell can be cut out. Some geometric components of the

bias tow which was partitioned in to a number of groups for material orientation assignment. (See Fig.

3.) Detailed description of the modeling has been discussed in detail in a paper preceding this one [16].

The triaxially braided composite used in this study is a regular braid with a braiding angle of 30⁰. The width (��), thickness (��) and in-between gaps (��) of the axial tows are 2.40 mm, 0.50 mm and

2.38 mm, respectively. Similarly, the width (��), thickness (��) and in-between gaps (��) of the bias

tows were 2.20 mm, 0.50 mm and 1.94 mm, respectively. The braided unit cell is shown in Fig. 2 with

the dimensions. The tow volume fraction (��), fiber volume fraction (��) and total fiber volume

fraction (��) were 34.1%, 78.0% and 26.6%, respectively.


W = 9.56 mm

t = 1.54 mm

L = 8.28 mm


Table 1. Mechanical properties of carbon fiber IM7 [18]

7

2.2. Material Properties

The triaxial braid considered in this study is made from carbon fiber and epoxy matrix constituent

materials. The elastic and failure properties of these materials has been given in Tables 1 and 2. These

material properties are used at the micro-scale computation to get the homogenized properties for the

tows. The homogenization based on modified classical laminate theory and micromechanics that has

been incorporated in a commercial software called MCQ-Composites [17]. These calculations were

made throughout the progressive failure analysis process since the degradation of material properties

was made at the micro level. This process is discussed in detail later.


Elastic properties Longitudinal modulus (GPa) 276.0 Transverse modulus (GPa) 27.6 In-plane shear modulus (GPa) 138 Transverse modulus (GPa) 7.8 Poisson’s ratio 0.3 Poisson’s ratio 0.8 Strength properties Longitudinal tensile strength (MPa) 3800 Longitudinal compressive strength (MPa) 2980

Table 2. Mechanical properties of the matrix [13]

Elastic properties Elastic modulus (GPa) 3.0 Poisson’s ratio 0.35 Strength properties Tensile strength (MPa) 65 Compressive strength (MPa) 130

These homogenized properties were assigned to the tows in the mesoscale unit cells which needed

the material orientation assignments. The material orientation was assigned by aligning the fiber

direction to the tangent of the tow paths while the transverse directions were perpendicular to it. This


7

2.2. Material Properties

The triaxial braid considered in this study is made from carbon fiber and epoxy matrix constituent

materials. The elastic and failure properties of these materials has been given in Tables 1 and 2. These

material properties are used at the micro-scale computation to get the homogenized properties for the

tows. The homogenization based on modified classical laminate theory and micromechanics that has

been incorporated in a commercial software called MCQ-Composites [17]. These calculations were

made throughout the progressive failure analysis process since the degradation of material properties

was made at the micro level. This process is discussed in detail later.


Elastic properties Longitudinal modulus (GPa) 276.0 Transverse modulus (GPa) 27.6 In-plane shear modulus (GPa) 138 Transverse modulus (GPa) 7.8 Poisson’s ratio 0.3 Poisson’s ratio 0.8 Strength properties Longitudinal tensile strength (MPa) 3800 Longitudinal compressive strength (MPa) 2980


Elastic properties Elastic modulus (GPa) 3.0 Poisson’s ratio 0.35 Strength properties Tensile strength (MPa) 65 Compressive strength (MPa) 130

These homogenized properties were assigned to the tows in the mesoscale unit cells which needed

the material orientation assignments. The material orientation was assigned by aligning the fiber

direction to the tangent of the tow paths while the transverse directions were perpendicular to it. This

439


http://ijass.org

2.3. FE Model and Periodic Boundary Condition

The finite element model for the meso-scale unit cell was

generated using 3D four node tetrahedral linear elements

(C3D4). In order to simulate the repeating nature of the unit

cells, periodic boundary conditions (PBC) were applied to

the opposite faces of the unit cell. The faces of the unit cell are

named as shown in Fig. 4. The number of linear tetrahedral

elements and nodes used for the unit cell are 42,052 and

8,476, respectively. Three different types of ply stackings have

been modeled by varying the boundary condition applied on

the top-bottom pair of faces. The first is the antisymmetric

stacking infinite number of plies where PBCs are applied in

all faces. The second one is the symmetric stacking infinite

plies where PBCs are applied at the in-plane boundary faces

while multi-point constraints (MPCs) were applied to the

z-direction degree of freedom of the out-of-plane boundary

faces. This makes the z-displacements of all nodes in the top

and bottom faces to be constants, which means the faces

stay flat after loading. The third is single ply or free-free case

where the top and the bottom faces are simply set free of any

kind of boundary condition, while applying PBCs at the in-

plane boundary faces. All of these boundary conditions are

summarized in Table 3.

2.4 Multi-scale Progressive Failure Analysis

Progressive failure analysis is a computational method

8

material orientation definition is shown in Fig. 3.

Fig. 3. Material orientation

2.3. FE Model and Periodic Boundary Condition

The finite element model for the meso-scale unit cell was generated using 3D four node tetrahedral

linear elements (C3D4). In order to simulate the repeating nature of the unit cells, periodic boundary

conditions (PBC) were applied to the opposite faces of the unit cell. The faces of the unit cell are

named as shown in Fig. 4. The number of linear tetrahedral elements and nodes used for the unit cell

are 42,052 and 8,476, respectively. Three different types of ply stacking have been modeled by

varying the boundary condition applied on the top-bottom pair of faces. The first is the antisymmetric

stacking infinite number of plies where PBCs are applied in all faces. The second one is the

symmetric stacking infinite plies where PBCs are applied at the in-plane boundary faces while multi-

point constraints (MPCs) were applied to the z-direction degree of freedom of the out-of-plane

boundary faces. This makes the z-displacements of all nodes in the top and bottom faces to be

Coordinate systems x, y, z - Global x1, y1, z1 - Tow path 1, 2, 3 - Local material axis

Fig. 3. Material orientation

9

constants, which means the faces stay flat after loading. The third is single ply or free-free case where

the top and the bottom faces are simply set free of any kind of boundary condition, while applying

PBCs at the in-plane boundary faces. All of these boundary conditions are summarized in Table 3.

Fig. 4. Matching opposite faces of meso-scale unit cell mesh

Table 3. Boundary conditions applied to unit cell for different ply stacking types

Ply stacking Top-Bottom Front-Back Left-Right

Single ply – PBC PBC

Symmetric multiple MPC PBC PBC

Antisymmetric multiple PBC PBC PBC

2.4. Multi-scale Progressive Failure Analysis

Progressive failure analysis is a computational method using finite element analysis with which the

step-by-step damage and failure of materials is simulated. This method is applied by iteratively

conducting finite element analyses on a model by progressively changing either the load or the

material properties. The overall load on the structure is systematically increased gradually with each

iteration by monitoring the amount of damage and structural stability. The damage and fracture of the

structure is represented by local degradation of material property and removal of elements,

respectively. The decision whether or not to degrade the material properties is decided based on

conditions called failure criteria. The process of PFA starts with the ordinary FEA of the model using

Left

Right

Front

Back

Top

Bottom



9

constants, which means the faces stay flat after loading. The third is single ply or free-free case where

the top and the bottom faces are simply set free of any kind of boundary condition, while applying

PBCs at the in-plane boundary faces. All of these boundary conditions are summarized in Table 3.



Ply stacking Top-Bottom Front-Back Left-Right

Single ply – PBC PBC

Symmetric multiple MPC PBC PBC

Antisymmetric multiple PBC PBC PBC

2.4. Multi-scale Progressive Failure Analysis

Progressive failure analysis is a computational method using finite element analysis with which the

step-by-step damage and failure of materials is simulated. This method is applied by iteratively

conducting finite element analyses on a model by progressively changing either the load or the

material properties. The overall load on the structure is systematically increased gradually with each

iteration by monitoring the amount of damage and structural stability. The damage and fracture of the

structure is represented by local degradation of material property and removal of elements,

respectively. The decision whether or not to degrade the material properties is decided based on

conditions called failure criteria. The process of PFA starts with the ordinary FEA of the model using

Left

Right

Front

Back

Top

Bottom



using finite element analysis with which the step-by-step

damage and failure of materials is simulated. This method

is applied by iteratively conducting finite element analyses

on a model by progressively changing either the load or

the material properties. The overall load on the structure

is systematically increased gradually with each iteration by

monitoring the amount of damage and structural stability.

The damage and fracture of the structure is represented

by local degradation of material property and removal

of elements, respectively. The decision whether or not

to degrade the material properties is decided based on

conditions called failure criteria. The process of PFA starts

with the ordinary FEA of the model using a small load. Then,

the stress and strain of every element is checked against the

failure criteria defined by every stress or strain component or

their combinations.

The complete process of the progressive failure analysis

is done in two levels of computations conjugated in to one

cycle. The lower level is the micro-scale analysis where

the local material property homogenization, micro-stress

computation and degradation takes place. The other level is

the meso-scale analysis in which generalized finite element

analysis and stability of the structure is computed. The multi-

scale analysis is discussed in detail in Section 2.4.2.

The process of PFA described here has been applied

using the commercial software GENOA [19]. The software is

capable of importing the input file from ABAQUS, another

commercial software, and apply all the required failure

criteria before running. The advantage of this software is

that it can run both micro and meso-scale analysis within

the broader PFA. It checks for the micro failures and

degrades the corresponding matrix or fiber properties before

homogenizing them for the next iteration. The microscale

analysis is done by MCQ-Composites [17]. The whole PFA

process used by these programs and the failure criteria

applied are described in the following section.

2.4.1 Failure Criteria

Failure criteria are functions in stress or strain which

identify material elements that are in “failed” state from the

“un-failed” ones. From the long list failure criteria [20, 21],

maximum stress criteria have been applied in this study.

At each individual load step, the orthotropic composite

domain such as matrix pocket, tow matrix and tow stresses

and strains are obtained from micro-stress analysis. The

first twelve failure modes are associated with the positive

and negative limits of the six local stress components in the

material direction as follows [19]:

10

a small load. Then, the stress and strain of every element is checked against the failure criteria defined

by every stress or strain component or their combinations.

The complete process of the progressive failure analysis is done in two levels of computations

conjugated in to one cycle. The lower level is the micro-scale analysis where the local material

property homogenization, micro-stress computation and degradation takes place. The other level is the

meso-scale analysis in which generalized finite element analysis and stability of the structure is

computed. The multi-scale analysis is discussed in detail in Section 2.4.2.

The process of PFA described here has been applied using the commercial software GENOA [19].

The software is capable of importing the input file from ABAQUS, another commercial software, and

apply all the required failure criteria before running. The advantage of this software is that it can run

both micro and meso-scale analysis within the broader PFA. It checks for the micro failures and

degrades the corresponding matrix or fiber properties before homogenizing them for the next iteration.

The microscale analysis is done by MCQ-Composites [17]. The whole PFA process used by these

programs and the failure criteria applied are described in the following section.

2.4.1. Failure Criteria

Failure criteria are functions in stress or strain which identify material elements that are in “failed”

state from the “un-failed” ones. From the long list failure criteria [20, 21], maximum stress criteria

have been applied in this study.

At each individual load step, the orthotropic composite domain such as matrix pocket, tow matrix

and tow stresses and strains are obtained micro-stress analysis. The first twelve failure modes are

associated with the positive and negative limits of the six local stress components in the material

direction as follows [19]:

(2a) , (2a)

11

(2b)

(2c)

(2d)

(2e)

(2f)

where S, C and T indicate the strength in the specified direction, compression mode and tension mode,

respectively. Each of the terms on the left and right of the inequality in equation 2 are defined in Table

4. These failure criteria represent the micro-scale matrix and fiber failure which in-turn defines the

individual tow failures. These criteria can also be assigned to isotropic pure matrix pockets between

tows even though the three material directions behave in the same way.

The failure criteria application in the current study is grouped in to two. The first is, the criteria for

the pure matrix elements where the criteria simplify to compression, tension and shear modes. Since

there is no fiber in the pure matrix pocket, the strength parameters of the matrix are directly used to

check the status of the matrix. All the three criteria were turned on for both damage and fracture. This

means, fracture of an element in any mode will cause its damage and eventual removal.

For the fiber tows, on the other hand, all of the failure criteria depicted in Table 4 were applied for

the damage and fracture modeling. However, the only criteria turned on to fracture the tows was the

longitudinal tensile criteria. This is because the elements of the fiber tows should not be removed but

damaged unless fiber breakage occurs. The fiber breakage means loss of longitudinal tensile strength

of the tows. The rest of the failure criteria will only cause degradation of the material properties in

that particular direction or damage.

2.4.2. PFA Process

The damage tracking process can be illustrated in terms of load – displacement curve as shown in Fig.

5. At point A of the load increment, assessment of initial composite material damage based on the

, (2b)

11

(2b)

(2c)

(2d)

(2e)

(2f)

















2.4.2. PFA Process



, (2c)

11

(2b)

(2c)

(2d)

(2e)

(2f)

















2.4.2. PFA Process



(+), (2d)

11

(2b)

(2c)

(2d)

(2e)

(2f)

















2.4.2. PFA Process



(+), (2e)

11

(2b)

(2c)

(2d)

(2e)

(2f)

















2.4.2. PFA Process



(+), (2f)

where S, C and T indicate the strength in the specified

direction, compression mode and tension mode,

respectively. Each of the terms on the left and right of the

inequality in equation 2 are defined in Table 4. These failure

criteria represent the micro-scale matrix and fiber failure

which in-turn defines the individual tow failures. These

criteria can also be assigned to isotropic pure matrix pockets

between tows even though the three material directions

behave in the same way.

The failure criteria application in the current study is

grouped in to two. The first is the criteria for the pure matrix

elements where the criteria simplify to compression, tension

and shear modes. Since there is no fiber in the pure matrix

pocket, the strength parameters of the matrix are directly

used to check the status of the matrix. All the three criteria

were turned on for both damage and fracture. This means,

fracture of an element in any mode will cause its damage and

eventual removal.

For the fiber tows, on the other hand, all of the failure

criteria depicted in Table 4 were applied for the damage

and fracture modeling. However, the only criteria turned

on to fracture the tows was the longitudinal tensile criteria.

This is because the elements of the fiber tows should not be

removed but damaged unless fiber breakage occurs. The

fiber breakage means loss of longitudinal tensile strength

of the tows. The rest of the failure criteria will only cause

degradation of the material properties in that particular

direction or damage.

2.4.2 PFA Process

The damage tracking process can be illustrated in terms

of load – displacement curve as shown in Fig. 5. At point A of

the load increment, assessment of initial composite material

damage based on the proposed failure criteria is made.

Based on the selected failure criteria, the material properties

are degraded and the FEM repeated. The applied load is

maintained at a certain value (Pa) at which equilibrium

was achieved while running a number of trials with a load

increment until the next equilibrium is attained at point B.

Whenever an equilibrium point like B is achieved, the whole

structure is checked for stability. If it is not stable, then PFA

441


http://ijass.org

process is deemed complete and the program terminates

which is represented by point N in Fig. 5. Otherwise, the

current point B will turn in to point A for the next iteration

until the structure becomes unstable.

In its simplest form, material property degradation is

done by factoring the Young’s moduli and shear modulus

with degradation factors. A simplified form used by Tan et al.

[22] is shown by equation 3 as:

12

proposed failure criteria is made. Based on the selected failure criteria, the material properties are

degraded and the FEM repeated. The applied load is maintained at a certain value ( ) at which

equilibrium was achieved while running a number of trials with a load increment until the next

equilibrium is attained at point B. Whenever an equilibrium point like B is achieved, the whole

structure is checked for stability. If it is not stable, then PFA process is deemed complete and the

program terminates which is represented by point N in Fig. 5. Otherwise, the current point B will turn

in to point A for the next iteration until the structure becomes unstable.

In its simplest form, material property degradation is done by factoring the Young’s moduli and

shear modulus with degradation factors. A simplified form used by Tan et al. [22] is shown by

equation 3 as:

(3a)

(3b)

(3c)

, (3a)

12










equation 3 as:

(3a)

(3b)

(3c)

, (3b)

12










equation 3 as:

(3a)

(3b)

(3c)

, (3c)

where Ed11, Ed

22 and Gd12 are the effective material properties of

the damaged elements while E 011, E 022 and G 012 are those of the

elements before damage. D1, D2, and D6 are the directional

damage parameters for extension and shear. Similarly,

Camanho and Matthews [23] considered four damage modes 12










equation 3 as:

(3a)

(3b)

(3c)

Fig. 5. PFA in the form of load – displacement curve [19]

Table 4. Maximum stress failure criteria [19]

13

Fig. 5. PFA in the form of load – displacement curve [19]

Table 4. Maximum stress failure criteria [19]

Mode of failure Expression Longitudinal tensile

Longitudinal compressive

Fiber crushing mode:

Delamination mode:

Micro-buckling mode:

The minimum value of , and is considered as ply longitudinal compressive strength

, ,

Transverse tensile strength

Transverse compressive strength

In-plane shear strength

Longitudinal normal shear (through-the-thickness) strength

Transverse normal shear (through-the-thickness) strength

where , and are the effective material properties of the damaged elements while ,



by using Hashin’s failure theory. For the degradation, they

considered the damage of constituent materials individually

including compression as shown by equation 4.

Matrix tensile or shear cracking:

14

�� and �� are those of the elements before damage. ��, ��, and �� are the directional damage

parameters for extension and shear. Similarly, Camanho and Matthews [23] considered four damage

modes by using Hashin’s failure theory. For the degradation, they considered the damage of

constituent materials individually including compression as shown by equation 4.


��

Fiber tensile fracture:

��

Matrix compressive or shear cracking:

��

Fiber compressive fracture:

��

where superscripts d, C and T represent degraded property, compression and tension, respectively. The

degradation factors used in this study are all taken from the default values used by GENOA software

[19]. The degradation factors used are �� ; �� ; �� and

�� .

The overall process of the progressive failure analysis has been summarized in Fig. 6 divided into

(4a)


14






��


��


��


��




�� .


, (4b)


14






��


��


��


��




�� .


(4c)


14






��


��


��


��




�� .


, (4d)

where superscripts d, C and T represent degraded property,

compression and tension, respectively. The degradation

factors used in this study are all taken from the default values

used by GENOA software [19]. The degradation factors used

are DT1 = DT

2 = 1%; DT4 = 10%; DC

1 = DC2 =20% and DC

4.

The overall process of the progressive failure analysis

has been summarized in Fig. 6 divided into two scales. At

the micro-scale, elastic property homogenization, material

property degradation and micro-stress computations are

done. This level involves the constituent matrix and fiber

properties. At this level, there is no finite element model

involved, instead modified classical laminate theory and

other micromechanics principles are used [17]. The other

level of analysis is the mesoscale where the 3D finite element

model is used to represent different parts of the unit cell. At

this level, the geometric and finite element models discussed

before are used to represent the fiber bundle tows and pure

matrix parts of the unit cell.

The failure criteria discussed before can be selectively

applied, for damage and fracture assessment. Here, a

damaged element means the one with reduced properties but

can transfer displacements and/or has considerable stiffness

and strength in other directions. Fractured element, on the

other hand, is an element that has no capability to transfer

force or displacement in any direction. Therefore, a fractured

element can be removed from the full model as described in

the PFA process. In the current case, for instance, even if tows

are damaged in their transverse direction in either tension,

compression or tension, the elements are not deemed to be

fractured as long as the tensile fracture of fibers occurs. This

is reasonable because, tows are even capable of resisting

tensile load even without matrix. However, the isotropic

matrix is analyzed with both damage and fracture turned on

for all failure criteria considered.

15

two scales. At the micro-scale, elastic property homogenization, material property degradation and

micro-stress computations are done. This level involves the constituent matrix and fiber properties. At

this level, there is no finite element model involved, instead modified classical laminate theory and

other micromechanics principles are used [17]. The other level of analysis is the mesoscale where the

3D finite element model is used to represent different parts of the unit cell. At this level, the geometric

and finite element models discussed before are used to represent the fiber bundle tows and pure

matrix parts of the unit cell.

Fig. 6. Multiscale progressive failure analysis process [19]

Start

Generate elastic properties from constituent properties

Compute generalized stresses using FEM

Compute micro-stresses for stuffer, filler and

weaver of the composite

Did local damage/ fracture occur?

Equilibrium reached, increase load

Too many nodes failed, reduce the load

and restart from the previous equilibrium

New damaged/ fractured nodes >

the limit?

Entire structure collapsed?

Print out the ultimate loads and all relative

End

Separate all elements connecting to the fractured

nodes, create additional nodes & renumber

Degrade the material properties for the damaged nodes according to the damage model

YN

N

Y

Y

N

Meso-scale Micro-scale

Fig. 6. Multiscale progressive failure analysis process [19]

,

,

,

,

,

,

443


http://ijass.org

3. Results and Discussion

The PFA analysis was performed on a number of models

of the triaxially braided textile composite. Firstly, the

methodology was validated by comparing the results

obtained to experimental and computational results from

two different references. Then, the same configuration was

used to discuss the damage mode and progress with the

use of the stress-strain curve. Next, the effect of ply stacking

and boundary conditions has been discussed. Finally, the

compressive failure behavior is discussed using the three

material behavior assumptions for matrix.

3.1 Axial Tension and Compression Failure

Before conducting validation of the method an attempt

was made to achieve a reasonable convergence of mesh. For

this single ply stacking model was used, in which PBCs were

applied in the in-plane direction and free-free boundary

condition in the out-of-plane direction. Next, the mesh

dependency of damage behavior was investigated. Fig. 7

shows the stress-strain curve for four meshes with different

element sizes. However, the variation of the curves did not

converge to a specific value. For instance, comparing the

maximum stress values, percentage variation increased from

9.7% to 16.3% as the mesh size decreased from 0.4 mm to 0.3

mm and from 0.3 mm to 0.2 mm, respectively.

This is because of the inherent behavior of continuum

damage mechanics employed in GENOA that it does not

consider the energy dissipation through the process of

damage and fracture. Therefore, it varies with the variation

of mesh no matter how fine the mesh size is [24]. Rather

than trying to find a ‘best’ mesh, a mesh size was chosen in

two steps. The first was from the convergence of the elastic

solution. Then a mesh size which resulted in the closest

stress-strain curve to the experimental test was chosen.

This is reasonable because if someone wants to apply this

model to a larger application, use of the same mesh with the

same material set will result in reasonably similar damage

behavior. Therefore, a mesh size of 0.3 mm was chosen by

comparing the stress-strain curves which produced less

than 1% difference in initial elastic modulus compared to

that of the refined reference mesh and used for the rest of

the analyses.

Figure 8 shows the PFA predicted complete stress-strain

curve for the triaxial braid unit cell model under uniaxial

tension in the axial direction. Also plotted in the figure

are the experimental result by Miravete et al. [3] and the

analysis result by Xu et al. [13] using Micro-Mechanics of

Failure (MMF) model. The model used for this comparison

had infinitely stacked antisymmetric boundary condition

applied. The comparison was made using the stress-strain

curve which showed very good match with the experimental

results. As can be seen in the figure, a better match in

shape of the curve was obtained as compared to the MMF

prediction by Xu et al. [13]. However, the predicted strength

is slightly lower than that of the experimental model. This is

because of the possible difference in fiber volume fraction

and geometric dimensions assumed from the test specimen,

in addition to the uncertainty of mesh dependency of

CDM approach employed in GENOA. Since the geometric

dimensions and fiber volume fraction were directly taken

from the work of Xu et al. [13], the difference between the

current analysis and the experimental result is within a

reasonable range.

The progress of damage was demonstrated using the same

configuration with infinite plies in antisymmetric stacking.

This progress is shown in Fig. 9 and Table 5. The percentage

values shown in the table indicate the proportion of

damaged elements facing the specified failure mode relative

17

For instance, comparing the maximum stress values, percentage variation increased from 9.7% to 16.3%

as the mesh size decreased from 0.4 mm to 0.3 mm and from 0.3 mm to 0.2 mm, respectively.

This is because of the inherent behavior of continuum damage mechanics employed in GENOA

that it does not consider the energy dissipation through the process of damage and fracture. Therefore,

it varies with the variation of mesh no matter how fine the mesh size is [24]. Rather than trying to find

a ‘best’ mesh, a mesh size was chosen in two steps. The first was from the convergence of the elastic

solution. Then a mesh size which resulted in the closest stress-strain curve to the experimental test

was chosen. This is reasonable because if someone wants to apply this model to a larger application,

use of the same mesh with the same material set will result in reasonably similar damage behavior.

Therefore, a mesh size of 0.3 mm was chosen by comparing the stress-strain curves which produced

less than 1% difference in initial elastic modulus compared to that of the refined reference mesh and

used for the rest of the analyses.

Fig. 7. Mesh dependency of damage and failure progression

Figure 8 shows the PFA predicted complete stress-strain curve for the triaxial braided unit cell

model under uniaxial tension in the axial direction. Also plotted in the figure are the experimental

0

100

200

300

400

500

0.0% 0.5% 1.0% 1.5% 2.0%

Stre

ss (M

Pa)

Strain

0.2 mm

0.25 mm

0.4 mm

0.3 mm

Fig. 7. Mesh dependency of damage and failure progression

18

result by Miravete et al. [3] and the analysis result by Xu et al. [13] using Micro-Mechanics of Failure

(MMF) model. The model used for this comparison had infinitely stacked antisymmetric boundary

condition applied. The comparison was made using the stress-strain curve which showed very good

match with the experimental results. As can be seen in the figure, a better match in shape of the curve

was obtained as compared to the MMF prediction by Xu et al. [13]. However, the predicted strength is

slightly lower than that of the experimental model. This is because of the possible difference in fiber

volume fraction and geometric dimensions assumed from the test specimen, in addition to the

uncertainty of mesh dependency of CDM approach employed in GENOA. Since the geometric

dimensions and fiber volume fraction were directly taken from the work of Xu et al. [13], the

difference between the current analysis and the experimental result is within a reasonable range.

Fig. 8. Validation of PFA result against test and computational results from references for axial tension

loading

The progress of damage was demonstrated using the same configuration with infinite plies in

antisymmetric stacking. This progress is shown in Fig. 9 and Table 5. The percentage values shown in

the table indicate the proportion of damaged elements facing the specified failure mode relative to the

total number of elements in the considered unit cell part at that particular stage of fracture. (See Table

0

100

200

300

400

500

600

0.0% 0.5% 1.0% 1.5% 2.0%

Stre

ss (M

Pa)

Strain (%)

Test [3]

Current

MMF [10]

Fig. 8. Validation of PFA result against test and computational results from references for axial tension loading



to the total number of elements in the considered unit cell

part at that particular stage of fracture. (See Table 4 for the

explanation of the damage mode symbols.) The first elastic

portion of the stress-strain curve continues up to point P1

of Fig. 9 which has a strain and stress value of about 0.69%

and 290.9 MPa, respectively. The first sign of damage was

seen in most of the bias tows and edges of the axial tows as

shown by the segment from point P1 to P2 in Table 5. Most of

the damage seen in the bias tows is the in-plane shear that

resulted from their diagonal orientation from the loading

direction. The axial tows also showed slight damage at their

edges in transverse compression mode resulting from the

Poisson’s effect. However, this has very small effect on the

stress-strain curve since the axial tows are supporting the

load in their fiber direction while the damage is occurring in

perpendicular direction. At this stage, no significant damage

was seen in the pure matrix pocket.

The next stage of damage is shown by the segment of the

stress-strain curve from P2 to P3 of Fig. 9 and Table 5. At this

20

tows running in the two different directions. However, there is almost no additional damage or failure

observed in the bias tows at this stage. Therefore, the predominant modes of failure leading to the

major failure are the starting of axial tow fiber breakage and the damage to some parts of the matrix

pocket in crushing and mode II interface separation.

From point to of Fig. 9, the material shows slight resistance until the remaining matrix

pockets fail in similar fashion as before. The axial tows also keep failing in the form of tow fiber

breakage leading to point of Fig. 9. Beyond this point, most of the fiber in the axial tow has

already been damaged and is no longer supporting the axial load. Therefore, the bias tows start to

deform excessively with some failure in the axial direction.

Fig. 9. Stress-strain curve for damage progress in infinite antisymmetric stacking triaxial braid loaded

in axial tension

0

100

200

300

400

500

0.0% 0.5% 1.0% 1.5% 2.0%

Stre

ss (M

Pa)

Strain (%)

P1(0.69%, 290.9)

P2(1.04%, 388.8)

P3(1.21%, 439.6)

P4(1.25%, 331.3)P5(1.31%, 326.2)

P6(1.34%, 184.6)

P7(1.68%, 71.8)

Fig. 9. Stress-strain curve for damage progress in infinite antisymmet-ric stacking triaxial braid loaded in axial tension

Table 5. Damage modes with progressing damage in infinite antisymmetric stacking loaded in axial tension

Table 1. Damage modes with progressing damage in infinite antisymmetric stacking loaded in axial

tension

Damage and failure modes at levels shown by points on stress-strain curve in Fig. 9

Mat

rix

Dam

aged

All

Axi

al to

ws

-2.15% -18.9% -56.4% -78.7% -3.33% -2.86%

-89.0% -3.45% -5.44%

-89.8% -7.97% -5.59% -6.60%

-90.3% -10.9% -5.72% -9.57%

Bia

s tow

s

All

Dam

aged

-10.8% -86.1%-4.72%-2.88%

-95.7% -6.97% -2.88%

-95.8% -5.60%-4.44%

-95.8% -6.00%-5.32%

-94.6% -7.31% -5.01%

-96.3% -13.2% -12.0%

No damage Damaged Fractured

(Percentage values indicate the portion of elements facing the specified mode of failure from the current number elements in the corresponding unit cell part)

445


http://ijass.org

stage the only significant damages were more transverse

compression in axial tows and complete damage of the bias

tows through in-plane shear. However, the curve does not

seem to show significant change in slope, which is because

the axial tow damage in the transverse mode has very little

effect and the bias tow damage was very little as compared to

the number of elements already damaged in this mode. Once

again, the damage to the matrix pocket is still not occurring.

In both of these stages the pure matrix pocket is not damaged

because modes of failure in the tows are internal to the tows

instead of interfacial interaction with the matrix pocket.

The peak stress or the failure strength occurs at point

P3 of Fig. 9, beyond which the major failure of the braided

structure happens. The stress and strain values at this

point are 439.6 MPa and 1.21%, respectively. From point

22

Damage progress in the case of compressive loading has been shown in Fig. 10 and Table 6. The

first sign of damage occurred in small region of the axial tow edges indicated by point with a

strain and stress values of 0.44% and 185.3 MPa. Then, further increase in the axial compression

resulted in fiber micro-crushing in the axial tows as shown by point of the stress-strain curve

(0.49% and 202.3 MPa) and pictures. At this stage, the bias tows also start to show compressive

failure in the form of fiber micro-buckling. The next stage is the major drop in strength of the whole

braided structure shown as the part from to . The sudden drop in strength is mainly because of

the compressive fiber micro buckling damage in the bias tows and compressive failure in the pure

matrix pocket. Beyond this point, the stress did not change significantly while the strain increased.

However, the amount matrix pocket damage increases with increase in the global strain level as

shown by the damage patterns at point .

Fig. 10. Stress-strain curve for damage progress in infinite antisymmetric stacking of triaxial braid

loaded in axial compression

0

50

100

150

200

250

0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2% 1.4%

Stre

ss (M

Pa)

Strain (%)

P1(0.44%, 185.3)P2(0.49%, 202.3)

P3(0.52%, 60.5) P4(1.32%, 58.3)

Fig. 10. Stress-strain curve for damage progress in infinite antisym-metric stacking of triaxial braid loaded in axial compression

Table 6. Damage modes with progressing damage in infinite antisymmetric stacking with axial compression loading

Table 2. Damage modes with progressing damage in infinite antisymmetric stacking with axial

compression loading

Damage and failure modes at levels shown by points on stress-strain curve in Fig. 10

Mat

rix

Dam

aged

All

Axi

al to

ws

-8.03%-1.89%-8.03%-6.15%

-40.5%-32.1%-40.5%-8.54%

-67.9%-34.5%-67.9%-30.2%

-71.6%-34.6%-71.6%-32.5%

Bia

s tow

s

Dam

aged

All

-0.1%-0.1%-0.1%

-0.13%-0.13%-0.13%

-13.2%-13.2%-10.1%-3.61%

-26.3%-26.3%-17.1%

No damage Damaged Fractured(Percentage values indicate the portion of elements facing the specified mode of failure from the current number elements in the corresponding unit cell part)



P3 to P4, the stress-strain curve drops suddenly due to the

fiber breakage in some parts of the axial tow as well as the

beginning of damage in the matrix pocket. The matrix pocket

started damage and failure at some parts of the confined

spaces between axial tow and bias tows as well as the thin

matrix layer covering the bias tow in the top and bottom

faces of the unit cell. The failure mode in the pure matrix is

mainly in compression and shear. This kind of matrix pocket

failure may lead to interface separation in mode II. On the

other hand, the axial tow failure in the form of fiber breakage

has started at the ends of the axial tow where it is closely

confined between the bias tows running in the two different

directions. However, there is almost no additional damage or

failure observed in the bias tows at this stage. Therefore, the

predominant modes of failure leading to the major failure

are the starting of axial tow fiber breakage and the damage

to some parts of the matrix pocket in crushing and mode II

interface separation.

From point P4 to P5 of Fig. 9, the material shows slight

resistance until the remaining matrix pockets fail in similar

fashion as before. The axial tows also keep failing in the form

of tow fiber breakage leading to point P6 of Fig. 9. Beyond

this point, most of the fiber in the axial tow has already

been damaged and is no longer supporting the axial load.

Therefore, the bias tows start to deform excessively with

some failure in the axial direction.

Damage progress in the case of compressive loading has

been shown in Fig. 10 and Table 6. The first sign of damage

occurred in small region of the axial tow edges indicated by

point P1 with a strain and stress values of 0.44% and 185.3

MPa. Then, further increase in the axial compression resulted

in fiber micro-crushing in the axial tows as shown by point P2

of the stress-strain curve (0.49% and 202.3 MPa) and pictures.

At this stage, the bias tows also start to show compressive

failure in the form of fiber micro-buckling. The next stage

is the major drop in strength of the whole braided structure

shown as the part from P2 to P3. The sudden drop in strength

is mainly because of the compressive fiber micro buckling

damage in the bias tows and compressive failure in the pure

matrix pocket. Beyond this point, the stress did not change

significantly while the strain increased. However, the amount

matrix pocket damage increases with increase in the global

strain level as shown by the damage patterns at point P4.

3.2 Effect of Ply Stacking Type on Damage Behavior

The number and type of stacking of plies was checked if it

has effect on the damage and failure behavior of the triaxial

braid. The number of ply and stacking plays an important

role when there is a significant out-of-plane displacement

up on loading. The PFA was run on three conditions of ply

stacking under the same axial tensile loading condition;

single ply, infinite symmetric and anti-symmetrically stacked

cases. All three cases had the PBCs applied at the in-plane

boundaries. For the case of infinite number of plies stacked

in antisymmetric manner, periodic boundary conditions that

the displacement pattern is repeating is applied at the out-

of-plane faces of the unit cell. For the symmetrically stacked

infinite plies, the out-of-plane displacement in the top and

bottom faces are restrained to be flat after deformation. For

the single ply, the free-free condition is applied at the out-of-

plane boundaries.

For the current braid configuration, the variation of elastic

modulus in the axial direction has been shown to have very

small variation in the previous work [16]. It was shown that

under axial tension the predominant part of the load was

transferred by the axial tow exhibiting limited amount of

deformation at the bias tows. Therefore, the elastic behavior

of the whole assembly did not vary by changing the stacking

types since the stacking type varies the amount of out-of-

plane constraint.

The variation of stress-strain curves with the three ply

stacking conditions loaded in axial tension are shown in

Fig. 11. The results show very small variation between these

three conditions. This is because as shown in [16] the axial

x-direction tensile loading results in relatively small out-of-

plane deformation in the ply. Therefore, whether the unit

cell is restrained or not, its effect on the axial strength of the

triaxial braiding is found not significant.

3.3. Single Versus Multiple Unit Cell

The repeating nature of the unit cell is generally expected

to be seen in the elastic stress distribution. In other words,

the pattern of geometric repetition in the unit cell is reflected

25

Fig. 11. Variation of failure and damage behavior with ply stacking conditions

To verify this, larger models were made having dimensions in integer multiple of the single unit cell.

Firstly, the repeating nature of stress distribution was checked at the linear region of the stress-strain

curve as shown in Fig. 12. The comparison was made between single unit cell, nine unit cells and

twenty-five unit cells arranged in square fashion. In all three cases the load applied was in x –

direction which resulted in equal values of nominal stress (�� 277 MPa) and strain (��̅ = 0.67%) at

the ply level. This point lies in the linear range of the stress-strain curves in Fig. 12. The repeating

nature of the stress can be seen very well, which proves the ability of the periodic boundary condition

to model the pattern of stress distribution.

However, repeating nature of the elastic range stress distribution does not guarantee the repeating

nature of the damage progression. Naturally, the repeating nature of unit cells goes up to the

occurrence of the first damage. Sometimes, once the initial damage has occurred, the next stage of

damage is more governed by numerical uncertainties than the geometry of the structure. (This can be

an equivalent representation of many material and structural uncertainties in the real specimen.)

Therefore, the pattern of damage post the initial one may or may not have repeating nature.

0

100

200

300

400

500

0.0% 0.5% 1.0% 1.5% 2.0%

Stre

ss (M

Pa)

Strain (%)

Antisymmetric_infiniteSymmetric_infiniteFree-free_single ply


447


http://ijass.org

in the stress distribution.

To verify this, larger models were made having dimensions

in integer multiple of the single unit cell. Firstly, the repeating

nature of stress distribution was checked at the linear region

of the stress-strain curve as shown in Fig. 12. The comparison

was made between single unit cell, nine unit cells and

twenty-five unit cells arranged in square fashion. In all three

cases the load applied was in x –direction which resulted in

equal values of nominal stress

25
















0

100

200

300

400

500

0.0% 0.5% 1.0% 1.5% 2.0%St

ress

(MPa

)Strain (%)


and strain

25
















0

100

200

300

400

500

0.0% 0.5% 1.0% 1.5% 2.0%

Stre

ss (M

Pa)

Strain (%)


at the ply level. This point lies in the linear range

of the stress-strain curves in Fig. 12. The repeating nature of

the stress can be seen very well, which proves the ability of

the periodic boundary condition to model the pattern of

stress distribution.

However, repeating nature of the elastic range stress

distribution does not guarantee the repeating nature of

the damage progression. Naturally, the repeating nature

of unit cells goes up to the occurrence of the first damage.

Sometimes, once the initial damage has occurred, the next

stage of damage is more governed by numerical uncertainties

than the geometry of the structure. (This can be an equivalent

representation of many material and structural uncertainties

in the real specimen.) Therefore, the pattern of damage post

the initial one may or may not have repeating nature.

Previously, it has been established that the current

configuration starts its damage in the form of in-plane shear

in the bias tows. In order to check the damage pattern, a single

unit cell model was compared to a nine-unit cell model. Fig.

13 shows this comparison made between these two models

at two different nominal stress levels. The first is at a nominal

stress and strain level of 270 MPa and 0.64%, respectively. At

this stage, there is very small amount of damage in the bias

tows which has very good match. This can be considered the

end of the linear stress range for the model. Then, at nominal

stress and strain level of 328 MPa and 0.80%, respectively,

the damage in the bias tows can be seen in both single unit

cell and nine-unit cell models. The damage shape begins to

lose the repeating pattern beyond this point, and the damage

cannot be guaranteed to be the same between models since

it becomes too sensitive to be stable.

3.4 Effect of Matrix Plasticity on Compression Failure

The effect of plasticity in the matrix was investigated for the

compressive failure case. Ernst et al. [25] showed that epoxy

resin behaves in brittle manner when it fails under tension

while it shows some plasticity in shear mode. Therefore,

it was thought to be needed to investigate the effect of the

plasticity of the epoxy matrix for the compressive failure.

The plasticity of the matrix was considered by taking two

different types of the assumptions based on experimental

test result taken from reference. The first is the elastic-

perfectly plastic assumption where the failure stress and

failure strain correspond to those of the experimental data.

26

Fig. 12. Repeating nature of stress patterns in single and multi-cell models under the same macro

stress and strain levels (�� 277 MPa, ��̅ �0.67%)

Previously, it has been established that the current configuration starts its damage in the form of in-

plane shear in the bias tows. In order to check the damage pattern, a single unit cell model was

compared to a nine-unit cell model. Fig. 13 shows this comparison made between these two models at

two different nominal stress levels. The first is at a nominal stress and strain level of 270 MPa and

0.64%, respectively. At this stage, there is very small amount of damage in the bias tows which has

very good match. This can be considered the end of the linear stress range for the model. Then, at

nominal stress and strain level of 328 MPa and 0.80%, respectively, the damage in the bias tows can

be seen in both single unit cell and nine-unit cell models. The damage shape begins to lose the

repeating pattern beyond this point, and the damage cannot be guaranteed to be the same between

models since it becomes too sensitive to be stable.

3.4. Effect of Matrix Plasticity on Compression Failure

The effect of plasticity in the matrix was investigated for the compressive failure case. Ernst et al.

[25] showed that epoxy resin behaves in brittle manner when it fails under tension while it shows

some plasticity in shear mode. Therefore, it was thought to be needed to investigate the effect of the


σ11(MPa)

25

15

5

x

y

1 RUC

9 RUCs

25 RUCs

Fig. 12. Repeating nature of stress patterns in single and multi-cell models under the same macro stress and strain levels (

26


















σ11(MPa)

25

15

5

x

y

1 RUC

9 RUCs

25 RUCs

=277 MPa,

26


















σ11(MPa)

25

15

5

x

y

1 RUC

9 RUCs

25 RUCs

=0.67%)

27

Fig. 13. Repeating nature of initial failure with multiple unit cells (Single ply model)

The plasticity of the matrix was considered by taking two different types of the assumptions based

on experimental test result taken from reference. The first is the elastic-perfectly plastic assumption

where the failure stress and failure strain correspond to those of the experimental data. The second

assumption is taking multilinear elastic-plastic curve by taking different points on the experimental

curve. Both kinds of assumptions are shown in Fig. 14 relative to the test data.

The same configuration mentioned in the foregoing discussions was used for the compressive

failure analysis of triaxial braiding. The infinite antisymmetric stacking case was considered for the

compressive analysis with the three assumptions of the matrix material behavior shown in Fig. 14.

The resulting stress-strain curves are shown in Fig. 15 with multi-linear elastic-plastic, perfectly

σ = 270 MPa ε = 0.64%

σ = 328 MPa, ε = 0.80%

Nine cell model Single cell model

x

y

Damaged No Damage

Fig. 13. Repeating nature of initial failure with multiple unit cells (Single ply model)



The second assumption is taking multilinear elastic-plastic

curve by taking different points on the experimental curve.

Both kinds of assumptions are shown in Fig. 14 relative to

the test data.

The same configuration mentioned in the foregoing

discussions was used for the compressive failure analysis of

triaxial braiding. The infinite antisymmetric stacking case

was considered for the compressive analysis with the three

assumptions of the matrix material behavior shown in Fig.

14. The resulting stress-strain curves are shown in Fig. 15

with multi-linear elastic-plastic, perfectly elastic and elastic-

perfectly plastic matrix material properties. In a similar loading

scenario, Li et al. [15] obtained results for perfectly plastic

matrix model with 60⁰ triaxial braid. Comparing their result

with the current stress-strain curve of the elastic-perfectly

plastic model qualitatively, it shows resemblance that the slope

reduces significantly once major damage starts to occur.

All three cases had the same initial linear portion matching

until the first sign of damage shows. After very slight change

in shape, the braid with perfectly elastic matrix model drops

suddenly. This is because the failure in the matrix does not

allow any plastic deformation. For the other two, however,

the slope of the stress-strain curve changes smoothly before

the sudden drop occurs. The change in the slope is higher

for the elastic-perfectly plastic model than the multi- linear

elastic-plastic model. However, the multilinear elastic-

plastic case reaches the maximum strength earlier than the

elastic-perfectly plastic model.

4. Conclusion

Progressive failure analysis of a triaxially braided textile

composite was successfully conducted using repeating unit

cell models. Maximum stress based failure criteria were used

based on individual components to enable directional damage

and failure of elements. Since the analyses were conducted

in multi-scale between micro and meso-scale, damage

and failure was modeled at constituent material level. The

progressive failure analysis was conducted on the FE model by

progressively degrading the material properties and eventually

removing completely failed elements based on the defined

failure criteria until the whole model becomes unstable.

The analysis result was verified against test and

computational results from references which showed very

good match. Under uniaxial tensile load, the initial damage

in the model occurred due to in-plane shear of bias tows.

However, this did not have the biggest contribution to the

major failure. The major damage that causes the sudden

drop in the stress-strain curve occurred due to local damage

of pure matrix pocket and beginning of axial tow fiber

breakage. The matrix pocket failed in shear and compression

at the region where the bias tows and axial tows overlap.

Beyond the major failure, propagation of matrix pocket

failure, axial tow breakage continues. At high level of strain,

bias tow breakage starts once the whole assembly becomes

very soft with low stress and large deformation. The same

approach was used to investigate compressive failure case of

the same configuration.

The effect of unit cell geometry and material property

variation on the damage and fracture patterns was

parametrically investigated. The variation in the ply stacking

arrangement in which free-free single ply, antisymmetric

infinite ply and symmetric infinite ply arrangements were

considered resulted in no significant difference in the stress-

strain curves between these three cases. This was because

of the limited out-of-plane deformation of the model when

it is subjected to axial tension. The repeating nature of stress

distribution and initial failure was demonstrated by comparing

single unit cell models with nine-unit cell and twenty-five

28

Fig. 14. Bi-linear and multi-linear models for plastic stress-strain curve of the epoxy matrix

Fig. 15. Stress-strain curves for triaxial braid unit cells loaded in axial compression with different

matrix materials

elastic and elastic-perfectly plastic matrix material properties. In a similar loading scenario, Li et al.

[15] obtained results for perfectly plastic matrix model with 60⁰ triaxial braid. Comparing their result

with the current stress-strain curve of the elastic-perfectly plastic model qualitatively, it shows

0

10

20

30

40

50

60

70

0% 1% 2% 3% 4%

Stre

ss (M

Pa)

Strain (%)

Multi-linear elastic-plastic [13]

Elastic - perfectly plastic

Test [13]

0

50

100

150

200

250

0.0% 0.4% 0.8% 1.2% 1.6% 2.0%

Stre

ss (M

Pa)

Strain (%)

Multi-linear elastic-plasticPerfectly elasticElastic - perfectly plastic


28


Fig. 15. Stress-strain curves for triaxial braid unit cells loaded in axial compression with different

matrix materials

elastic and elastic-perfectly plastic matrix material properties. In a similar loading scenario, Li et al.

[15] obtained results for perfectly plastic matrix model with 60⁰ triaxial braid. Comparing their result

with the current stress-strain curve of the elastic-perfectly plastic model qualitatively, it shows

0

10

20

30

40

50

60

70

0% 1% 2% 3% 4%

Stre

ss (M

Pa)

Strain (%)

Multi-linear elastic-plastic [13]

Elastic - perfectly plastic

Test [13]

0

50

100

150

200

250

0.0% 0.4% 0.8% 1.2% 1.6% 2.0%

Stre

ss (M

Pa)

Strain (%)

Multi-linear elastic-plasticPerfectly elasticElastic - perfectly plastic

Fig. 15. Stress-strain curves for triaxial braid unit cells loaded in axial compression with different matrix materials

449


http://ijass.org

unit cells model. Finally, the effect plasticity of matrix on the

compressive failure behavior of the triaxially braided composite

was investigated. The results showed that the perfectly elastic

matrix modeling resulted in early failure while the multi-linear

elastic-plastic case had highest strength value.

References

[1] Littell, J. D., Binienda, W. K., Roberts, G. D. and

Goldberg, R. K., “Characterization of Damage in Triaxial

Braided Composites Under Tensile Loading”, Journal of

Aerospace Engineering, Vol. 22, No. 3, 2009, pp. 270-279.

[2] Ivanov, D. S., Baudry, F., Van Den Broucke, B., Lomov,

S. V., Xie, H. and Verpoest, I., “Failure Analysis of Triaxial

Braided Composite”, Composites Science and Technology,

Vol. 69, No. 9, 2009, pp. 1372-1380.

[3] Miravete, A., Bielsa, J. M., Chiminelli, A., Cuartero,

J., Serrano, S., Tolosana, N. and De Villoria, R. G., “3D

Mesomechanical Analysis of Three-axial Braided Composite

Materials”, Composites Science and Technology, Vol. 66, 2006,

pp. 2954-2964.

[4] Tay, T. E., Liu, G., Tan, V. B., Sun, X. S. and Pham, D.

C., “Progressive Failure Analysis of Composites”, Journal of

Composite Materials, 2008.

DOI: 10.1177/0021998308093912

[5] Blackketter, D. M., Walrath, D. E. and Hansen, A. C.,

“Modeling Damage in a Plain Weave Fabric-Reinforced

Composite Material”, Journal of Composites Technology and

Research, Vol. 15, No. 2, 1993, pp. 136-142.

[6] Naik, R. A., “Failure Analysis of Woven and Braided

Fabric Reinforced Composites”, Journal of Composite

Materials, Vol. 29, No. 17, 1995. pp. 2334-2363.

[7] Tang, X., Whitcomb, J. D., Kelkar, A. D. and Tate, J.,

“Progressive Failure Analysis of 2 x 2 Braided Composites

Exhibiting Multiscale Heterogeneity”, Composites Science

and Technology, Vol. 66, 2006, pp. 2580-2590.

[8] Goyal, D. and Whitcomb, J. D., “Effect of Fiber

Properties on Plastic Behavior of 2 x 2 Biaxial Braided

Composites”, Composites Science and Technology, Vol. 68,

2008, pp. 969-977.

[9] Nobeen, N. S., Zhong, Y., Francis, B. A., Ji, X., Chia, E.

S., Joshi, S. C. and Chen, Z., “Constituent Materials Micro-

Damage Modeling in Predicting Progressive Failure of

Braided Fiber Composites”, Composite Structures, Vol. 145,

2016, pp. 194-202.

[10] Hashin, Z. and Rotem, A., “A Fatigue Failure Criterion

for Fiber Reinforced Materials”, Journal of Composite

Materials, Vol. 7, No. 4, 1973, pp. 448-464.

[11] Hashin, Z., “Failure Criteria for Unidirectional Fiber

Composites”, Journal of Applied Mechanics, Vol. 47, No. 2,

1980, pp. 329-334.

[12] Stassi-D'Alia, F., Limiting Conditions of Yielding

of Thick Walled Cylinders and Spherical Shells, US Army

European Research Office, Frankfurt, Germany, (9851), 1959.

[13] Xu, L., Jin, C. Z. and Ha, S. K., “Ultimate Strength

Prediction of Braided Textile Composites Using a Multi-scale

Approach”, Journal of Composite Materials, 2014.

DOI: 10.1177/0021998314521062

[14] Xie, D., Salvi, A. G., Sun, C. E., Waas, A. M. and

Caliskan, A., “Discrete Cohesive Zone Model to Simulate

Static Fracture in 2D Triaxially Braided Carbon Fiber

Composites”, Journal of Composite Materials, Vol. 40, No. 22,

2006, pp. 2025-2046.

[15] Li, X., Binienda, W. K. and Goldberg, R. K., “Finite-

Element Model for Failure Study of Two-Dimensional

Triaxially Braided Composite”, Journal of Aerospace

Engineering, Vol. 24, No. 2, 2011, pp. 170-180.

[16] Geleta, T. N., Woo, K. and Lee, B., “Prediction

of Effective Material Properties for Triaxially Braided

Textile Composite”, Submitted to: International Journal of

Aeronautical and Space Sciences.

[17] Alpha STAR Corporation, “MCQ-Composites

Theoretical Manual”, Alpha STAR Corporation, 2014.

[18] Sun, X. S., Tan, V. B. and Tay, T. E., “Micromechanics-

Based Progressive Failure Analysis of Fibre-Reinforced

Composites with Non-Iterative Element-Failure Method”,

Computers & Structures, Vol. 89, No. 11, 2011, pp. 1103-1116.

[19] Alpha STAR Corporation, “GENOA 5 Theoretical

Manual”, Alpha STAR Corporation, 2014.

[20] Echaabi, J., Trochu, F. and Gauvin, R., “Review of

Failure Criteria of Fibrous Composite Materials”, Polymer

Composites, Vol. 17, No. 6, 1996, pp. 786-798.

[21] Soden, P. D., Hinton, M. J. and Kaddour, A. S., “A

Comparison of the Predictive Capabilities of Current Failure

Theories for Composite Laminates”, Composites Science and

Technology, Vol. 58, No. 7, 1998. pp. 1225-1254.

[22] Tan, S. C. and Nuismer, R. J., “A Theory for Progressive

Matrix Cracking in Composite Laminates”, Journal of

Composite Materials, Vol. 23, No. 10, 1989, pp. 1029-1047.

[23] Camanho, P. P. and Matthews, F. L., “A Progressive

Damage Model for Mechanically Fastened Joints in

Composite Laminates”, Journal of Composite Materials, Vol.

33, No. 24, 1999, pp. 2248-2280.

[24] Brekelmans, W. A. M. and De Vree, J. H. P., “Reduction

of Mesh Sensitivity in Continuum Damage Mechanics”, Acta

Mechanica, Vol. 110, No.1, 1995, pp. 49-56.

[25] Ernst, G., Vogler, M., Hühne, C. and Rolfes,

R., “Multiscale Progressive Failure Analysis of Textile

Composites”, Composites Science and Technology, Vol. 70,

No. 1, 2010, pp. 61-72.

Multi-scale Progressive Failure Analysis of Triaxially ...

Documents