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AN EMBEDDED-TOW MODEL FOR CRUSH SIMULATION OF
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6/01 cJCIFiC/DateDuepGS-p 15

 

AN EMBEDDED-TOW MODEL FOR CRUSH SIMULATION OF TEXTILE
COMPOSITE TUBE

By

CHEE-KUANG KOK

AN ABSTRACT OF A THESIS

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

MASTER OF SCIENCE

Engineering Mechanics in Mechanical Engineering

2001

Professor Ronald C. Averill

ABSTRACT

AN EMBEDDED-TOW MODEL FOR CRUSH SIMULATION OF TEXTILE
COMPOSITE TUBE

By

Chee-Kuang Kok

An embedded-tow model was developed to simulate the response of a
circular triaxialIy-braided textile composite crush tube. The embedded-tow model
was conceived from preliminary simulations of crush tube response using its
parent model, the simpliﬁed discrete-tow model. Unlike the discrete-tow model,
which represents the ﬁber tows and resin as beam and shell ﬁnite elements
respectively, the embedded-tow model incorporates the ﬁber tows and resin into
shell ﬁnite elements. Progressive failure of individual tow and resin is accounted
for separately. The composite micro-constituent contribution to the overall
composite macro-mechanical properties is represented using the micro-
mechanical approach similar to the one adopted in the discrete-tow model. With
a FORTRAN algorithm to model the composite non-linear material properties and
the use of ABAQUS/EXPLICIT software code for simulation in the present study,
the embedded-tow model proves its ability to predict the crush tube behavior
under compression with adequate computational efﬁciency. The potential of the
model to predict general triaxiaIIy-braided compositestructure response is yet to

be studied.

Copyright by
CHEE-KUANG KOK
2001

To my parents, sister and brother,
and my church family

ACKNOWLEDGMENTS

This work was partially sponsored by the Department of Energy and the
Automotive Composites Consortium under USAMP/DOE Co-operative
agreement DE — FC05 — 950R22363. I would like to express my gratitude for the
ﬁnancial support. I would also like to thank Dr. Averill for his timely advice,
insightful knowledge and great patience that went into this work. I thank my
former colleague, Craig Carrier, for helping me along in the early stages of this
work. Last but not least, I owe this work to all who have given me tremendous

moral support.

TABLE OF CONTENTS

LIST OF TABLES .................................................................................. vii
LIST OF FIGURES ............................................................................... viii
CHAPTER 1 INTRODUCTION .................................................................. 1
1.1 Introduction ............................................................................. 1

1.2 Literature Review.......................-. ............................................ 3

1.3 The Present Study .................................................................. 13
CHAPTER 2 PRELIMINARY MODELS ..................................................... 14
2.1 Introduction ........................................................................... 14

2.2 The Checkerboard Model ......................................................... 15

2.2.1 Description of the Model ............................................... 15

2.2.2 Numerical Results Using The Checkerboard Model ............ 17

2.3 The Cross-box Model .............................................................. 18

2.3.1 Description of the Model .............................................. 18

2.3.2 Numerical Results Using The Cross-box Model ................. 19

CHAPTER 3 THE EMBEDDED-TOW MODEL ............................................ 24
3.1 Introduction ........................................................................... 24

3.2 Description of the Embedded-tow Model ...................................... 25

3.2.1 Estimating Strains of Individual Components....................26
3.2.2 Progressive Failure Schemes of

Individual Components ................................................. 27

3.2.2.1 Progressive Failure Algorithm ............................. 28

3.2.3 Description of the Analysis ............................................ 29

CHAPTER 4 EMBEDDED-TOW MODEL VERIFICATION AND

CRUSH TUBE SIMULATION NUMERICAL RESULTS ............... 35

4.1 Introduction ........................................................................... 35

4.2 Material Properties .................................................................. 36

4.3 Veriﬁcation of the Embedded-tow Model ...................................... 37

4.3.1 Simple Bending On A Strip ........................................... 37

4.3.2 Simple Tension and Compression On A Unit Cell .............. 38

4.4 Effect of Loading Rate and Mass-scaling .................................... 39

4.5 Crush Tube Numerical Simulation Results .................................. 41
4.5.1 Crush Tube Load Versus Displacement Curve and

Damage Mechanism ................................................... 41

4.5.2 Parametric Studies ..................................................... 43

4.5.2.1 Effects of Chamfer Modeling .............................. 43

4.5.2.2 Effects of Friction ............................................. 44

vi

 

CHAPTER 5 CONCLUSION ................................................................... 54

5.1 Conclusions ........................................................................... 54
5.2 Future Work ........................................................................... 54
BIBLIOGRAPHY .................................................................................. 56

vii

LIST OF TABLES

Table 4.1 Computed macro-mechanical properties of the O°l+45°/-45°
triaxialIy-braided composite material of the 80k/12k crush tube ........ 46

Table 4.2 Mechanical properties of the composite constituents ...................... 46

viii

LIST OF FIGURES

Figure 2.1 A summary of the modeling procedure the isolation of a unit
cell of a triaxiaIIy-braided composite material to the ﬁnal

discrete-tow model ................................................................. 20

Figure 2.2 Damaging scheme of individual shell and beam

elements in the discrete-tow model ............................................ 20

Figure 2.3 Crush tube model by the checkerboard model .............................. 21

Figure 2.4 Comparison between simulation results from checkerboard
model to experimental results of load versus displacement

of the crush tube .................................................................... 21

Figure 2.5 Crush tube modeled by the cross-box model. (a) A magniﬁed
view of the cross-box regions. (b) The whole crush tube,

with cross-box region on the left ................................................ 22

Figure 2.6 Comparison between simulation results from cross-box
model to experimental results of load versus displacement

of the crush tube .................................................................... 23

Figure 3.1 (a) Half of a repetitive volume element; (b) Axial tows,
braider tows and resin represented as material stacks
before the application of classical laminate theory;

(0) The overall anisotropic material block, as in a unit cell ............... 31
Figure 3.2 (a) Unit cell before deformation; (b) Unit cell after deformation. ........31
Figure 3.3 A stress-strain curve representing progressive failure scheme

of each individual constituent in a unit cell .................................. 32
Figure 3.4 A summary of the progressive failure algorithm ............................ 33
Figure 3.5 (a) Force applied at the platen, which moves towards the

initiator, as in the experiment. (b) Initiator moves towards the

tube in current simulation ........................................................ 34
Figure 4.1 A simple bending analysis on a strip of shell elements. Two

bending forces, each of P/2 in magnitude, were applied into

the plane ............................................................................. 47

Figure 4.2 Comparison between the simulated and the theoretical

out-of-plane displacements of the end of bending strip .................. 47
Figure 4.3 Comparison between the simulated and the theoretical
bending stresses of the end of bending strip ............................... 48
Figure 4.4 Uniaxial tensile and compressive test on a unit cell ....................... 48
Figure 4.5 Stress-strain curve of a unit cell subjected to tensile force .............. 49
Figure 4.6 Stress-strain curve of a unit cell subjected to compressive force. . ....49
Figure 4.7 The effects of loading rate on the simulated strip
out-of-plane displacement ....................................................... 50
Figure 4.8 The effects of different mass scaling on the simulated
strip out-of—plane displacement ................................................ 50
Figure 4.9 Simulated and experimental load versus displacement
curve .................................................................................. 51
Figure 4.10 Simulated and experimental load versus displacement
curve ................................................................................. 51
Figure 4.11 Modeling the chamfer. (a) The chamfer as in the tube
specimen (b) Chamfer modeled with one shell element
lengthwise (c) Chamfer modeled with three shell
elements lengthwise ............................................................ 52
Figure 4.12 Effects of chamfer modeling on the load versus
displacement curve .............................................................. 52
Figure 4.13 Effects of Coulomb friction on the load versus
displacement curve .............................................................. 53

 

Chapter 1

INTRODUCTION

1.1 INTRODUCTION

Composite laminates have been used in the aerospace industry for many
years, and efforts have been made to introduce their use in the automotive
industry. Traditional composite laminates are known for their high in-plane
stiffness and low weight. However, they do not consider out-of-plane stiffness
and strength. To achieve better out-of-plane mechanical properties, woven textile
composites were introduced as an alternative to traditional composites. With ﬁber
bundles woven to form a cloth preform before resin impregnation, textile
composites not only possess high in-plane stiffness and strength, but also helps
improve out-of-plane strength. They are less likely to delaminate and thus are

capable of carrying transverse loads.

However, unlike the traditional composite laminates, which can be
analyzed by classical laminate theory, reliable yet computationally efﬁcient
formulations to relate the mechanical properties of textile composites have not
been ﬁnalized. These mechanical properties include elastic constants and
strengths to their micro-mechanics properties such as ﬁber architecture, ﬁber
volume fraction and laminate conﬁgurations. Over the years, researchers have

tried different approaches to model the behavior of textile composites and have

conducted numerous experiments, ranging from simple tensile tests to crush

composite tubes, whose application is of interest in the present work.

An automotive crush tube is a structure that is part of the front rail of a
vehicle, which is designed to absorb crash energy during a collision and thus
prevent car passengers from high impact injuries. Traditionally manufactured
using metal, textile composites are now thought of as a potential alternative due

to their high speciﬁc impact resistance and energy absorption.

The goal of the present research is to simulate, using a ﬁnite element
analysis, the response of a triaxiaIIy-braided textile-composite-based automotive
crush tube subjected to axial compression. First, a simple yet computationally
efﬁcient strength and stiffness prediction model named the simpliﬁed discrete tow
model was selected to model the entire crush tube. This model identiﬁes a
repetitive unit in the triaxially—braided composite material known as a unit cell,
and represents the ﬁber bundles and the resin in the unit cell as beam elements
and shell elements respectively. A commercial ﬁnite element software, ABAQUS,
was then employed to simulate the crush tube behavior, along with the use of a

FORTRAN algorithm to model the material non-linearity.

A brief literature review concerning the various textile composite stiffness
and strength models, together with related experiments performed on textile-

composite crush tube are presented as follows.

 

1.2 LITERATURE REVIEW

The attractiveness of textile composites lies in their in-plane mechanical
properties being comparable to their traditional laminates counterparts, their
improved out-of-plane strength and impact resistance, their ability to conform to
irregular shapes, and their relative manufacturability. Yet, all these desirable
features come with a price. Due to their micro-structural complexity, textile

composite materials are difﬁcult to analyze, let alone textile composite structures.

Since the size of tow and unit cell are usually within the same order as the
important geometrical features of textile composite base structures, the
microstructures of textile composite are too complicated and computationally
costly to be modeled explicitly, yet too distinct to be “smeared out" using a
homogenization scheme. In fact, this is the primary difﬁculty in the design of an
optimum textile composite architecture. However, researchers have sought after
different approaches, varying in the degree of simpliﬁcations, to come up with
approximate solutions to the problem at hand. Many solutions on how to predict
textile composite stiffness have been presented, but not as many have been
extended to predict the material strength. Efforts to incorporate these models
into a textile composite structure to simulate the structural response are even

scantier.

Most researchers have approached the problem of textile composite
stiffness and strength modeling by ﬁrst identifying a repetitive region in the textile
composite, or a unit cell, that can be duplicated into an entire composite. This
unit cell approach usually involves a type of volume averaging from the onset,
and is based on strength of material theory, classical lamination theory or a

homogenized ﬁnite element technique.

Early researchers lshikawa and Chou [1 2.3] developed three models to
model plain weave, twill weave, four harness satin and eight harness satin
composites. Their mosaic model regarded the textile composite as an
assemblage of cross-ply laminates. By assuming iso-strain and iso-stress
conditions, upper bound and lower bounds of composite stiffness were obtained.
One-dimensional approximations were applied to strips of warp and ﬁll strands,
thus neglecting tow undulation. In their second model, the modiﬁed ﬁber
undulation model [3], efforts were made to include tow continuity and undulation
by applying classical laminate theory to each inﬁnitesimal piece of strand-wise
strips, with further assumption of successive failure of strands transverse to
applied load. The knee phenomenon that characterizes fabric behavior after
initial strand failure was studied using this model. lshikawa and Chou had, in fact,
adopted a progressive failure strategy in this model. The strands transverse to
loading were ﬁrst failed at their highest strain region, found to be at the center of
undulation. The failed region moduli were reduced signiﬁcantly, allowing damage

to propagate until the whole strand had failed. lshikawa and Chou also

introduced a third model [3], namely a bridge model to analyze satin weaves.
This model contained a weak interlaced region among four surrounding stronger
bridge regions to transfer loads. Progressive failure was again adopted to study

knee phenomenon in satin weave composite using the bridge model.

Yang et al. [4] presented a ﬁber inclination model based on a modiﬁed
classical laminated plate theory that regarded the unit cell as an assemblage of
four inclined unidirectional laminae of the same thickness and the same ﬁber
volume fraction. The unit cell, constructed by ﬁber bundles oriented in the
diagonal directions, assumed the shape of a parallelepiped with its size
determined by the weaving and braiding parameters. Though this approach could
be extended to predict strength, it has a major drawback of ignoring ﬁber yarns

interaction at the interlocking position and bending.

Whitcomb [5] conducted a three dimensional ﬁnite element analysis of
plain weave composites to study the effect of tow waviness on the elastic
constants and internal Strain distribution of the composite. 20-noded iso-
parametric hexahedrons were used to model the unit cell. It was found that
increasing tow waviness ratio increases normal and shear strain concentration
under uniaxial loading. He, Woo and Gundapaneni [6] also developed a new type
of ﬁnite element called macro ﬁnite element to account for the microstructure in
the textile composite, rather than using the traditional ﬁnite element. This

quadrilateral macro element contains sub-elements for modeling the tow, and

adopts a single ﬁeld displacement approximation. Two-dimensional analyses
performed suggested that the macro elements were capable in predicting global
response but not local stresses or strains within the element. Whitcomb and W00
[7] then used this macro-element, together with a global/local strategy to reduce
computational effort of textile composite analysis. In this method, a relatively
crude global mesh was used to obtain the overall response of the structure.
Reﬁned local meshes were then used in the regions of interest where rapid
stress changes may occur. Solutions were found accurate at regions remote from
global/local boundaries. Whitcomb and W00 [8] later developed a multi-ﬁeld
macro-element and used it to study the effect of in-plane tow shift in a laminate
consisting of two layers. In their analysis, a reduced sub-structuring technique
was employed to obtain the macro-element stiffness matrix, and an enhanced
direct stiffness method was used to assist the numerical analysis. Whitcomb and
Srirengan [9, 10] also studied the sensitivity of predicted progressive failure to
the quadrature order (the number of Gauss points) of elements used, mesh
reﬁnement, waviness ratio and choice of material degradation scheme of a plain

weave composite subjected uniaxial loading.

The work of Naik and his colleagues included modeling of plain weave
and triaxially-braided composite lamina elastic constants and strength as well as
a laminate properties and behavior. In their earlier researches, Naik and Ganesh
[11] developed a slice array model and an element array model that discretized a

unit cell into slices and further divided each slice into elements to predict the on-

axis elastic properties of plain-weave composites. These slices and elements
were then reassembled by imposing iso-stress and iso—strain conditions. Later in
a series of publications, Naik and Ganesh [12,13,14] explained their geometry
modeling of a plain weave lamina and laminate, then proceeded to develop an
analytical model to predict the stress-strain history up to ultimate failure of 2D
orthogonal plain weave fabric laminates under on-axis uniaxial static tensile
loading, considering all the intermediate stagesof failure. The stress analysis
was conducted using the method of cells. Naik and Ganesh divided a unit cell
into slices and subcells. To account for the nonlinearities of the .strand and the
matrix due to strand undulations and material progressive failure, iterations were
performed on a unit cell by considering their stiffness convergence on the subcell
level. To predict failure and model strength, ﬁnal subcell stresses and average
slice strain were compared to the permissible stresses and strains, with the failed
region stiffness reduced accordingly. Using the same approach but different
boundary conditions on the unit cell, Naik and Ganesh [15, 16] presented a

failure analysis on plain weave composites under in-plane shear loading.

Naik et al. [17] employed a different approach in analyzing triaxially-
braided composites. Here, transversely isotropic yarns were discretized along
their path and interstitial matrix was represented as isotropic material slices. The
constituent material mechanical properties were transformed to their
corresponding values in global coordinates. A volume averaging technique was

adopted to predict the composite stiffness. Naik [18, 19] then investigated

strength and failure mechanisms on a two-dimensional triaxially—braided
composite, incorporating the effect of non-linear shear response of the
constituent materials. A yarn bending model and a crack yarn model were

introduced to study the details of tow failure.

Other researchers such as Dasgupta et al. [20] employed a
homogenization scheme based on a realistic three-dimensional numerical
simulation of simple plain-weave unit cell using the ﬁnite element method to
investigate the interactions between micro-damage mechanisms and the
macroscopic behavior of the composite. Similar to many other investigations,
volume averaging of stress and strain ﬁelds was used to predict the overall
composite properties. Takano et al. [21] also adopted a homogenization scheme
but unlike Dasgupta et al. [20], macro-micro coupling between microscopic
mechanical properties and the macroscopic structure was solved using four
levels of hierarchy. Sankar and Marrey [22] presented an approach called
selective averaging method (SAM) similar to that of Naik and Ganesh [16] to
predict the thermo-elastic constants of textile composite materials. The unit cell
was divided into slices and into elements, with the elastic constants of elements
being averaged selectively for iso-strain and iso-stress conditions. Again, the
elastic constants on the micro-mechanics level were related to those on the
macro-mechanics level by volume averaging of stiffness or compliance matrices.
Pastore and Yasser [23] performed a modiﬁcation of Fabric Geometry Model [23]

that related ﬁber architecture and material properties of textile reinforced

composites to global stiffness through micro-mechanics and stiffness averaging

techniques.

The on-going battle in textile composite modeling seems to be the need
for accuracy and the demand of computational efﬁciency. Glaessgen et al. [24]
had gone so far as to discretize the detailed architecture of a plain-weave textile
to allow internal details of unit cell to be examined. On the other hand, Dadkhah
et al. [25] suggested that any model more complicated than the simple model
they proposed, which was based on classical laminate theory, might not be
justiﬁed due to the inevitable variations in the textile architecture. Dahkhah’s
modiﬁed laminate model seemed capable in predicting the compressive strength
of a two-dimensional braided composite, as veriﬁed by the use of kink band
model utilizing experimental information. In fact, West and Adams [26] found that
the axial yarn crimping on a two-dimensional triaxially-braided composite could
reduce its compressive strength by as much as 30%. Masters et al. [27] also
suggested that minor changes in the braid geometry led to disproportionate
strength variations, as their empirical data showed. Experiments performed by
Masters and lfju [28] demonstrated signiﬁcant effect of textile architecture on

mechanical properties of the composite.

All the effort presented thus far demonstrated the capability of predicting
elastic moduli and strength of lamina and laminate. However, most of them are

only suitable for a small-scale analysis, as the modeling of a composite structure

by discretely modeling the tow and resin will be extremely computationally costly.
Recently, Carrier and Averill [29] developed a novel approach that discretely
model the tow and resin using beam and shell elements in the unit cell while
considering the tow undulation and essential geometrical features. While losing
accuracy due to simpliﬁcation, this computationally efﬁcient approach seem
promising in approximating the response of a crush tube, and will thus be

adopted in the current analysis.

With regard to material progressive failure modeling, a few common
approaches are found in the literature. Hamelin and Bigaud [30] developed a
multi-scale energy approach that minimized the strain energy and
complementary energy to predict the failure behavior of textile composite of
different textile architectures. In this approach, three levels of size hierarchy from
micro to meso and then macro were utilized. The stresses/strains of one
hierarchy was then related from one to the other through the localization
matrices. A failure criterion was chosen and progressive failure performed by
changing the constitutive matrix of the mesa-element using the selective RC (row
and column) method. Tan and Nuismer [31] also presented a theory for
progressive matrix cracking in traditional composite laminates. The appealing
feature of this model is that, unlike other theories that require some empirical
data on the laminate, only the basic material properties of constituent materials
are needed to model progressive matrix cracking. However, the theory has yet to

be extended for textile composite application. Blacketter et al. [32] adopted an

10

empirical stiffness reduction scheme and applied it to each failed Gauss point of
a FEM-discretized textile composite material element to simulate different types
of failure. Iterations were then performed to recalculate the stiffness matrix due to

occurrence of material failure.

Literature on elastic constants and strength modeling are abundant, and a
good summary of some of the work mentioned can be found in Tan et al. [33].
However, not as much analytical work has been performed on modeling the
response of the composite tube subjected to axial compression. Mamalis et al
.[34] developed an energy model to quantify the energy absorption of crush tubes
with different geometries. Based on their experimental observation of fracture
mechanisms of a tube axially crushed against a platen, they attributed the
absorbed energy to that dissipated during crack propagation, frond axial splitting,
frond bending and friction between exposed surfaces. Castejon et al. [35]
performed a ﬁnite element simulation using ABAQUS/Explicit and DYNA3D to
predict the mechanical behavior, the mean crash load and speciﬁc crash energy
of an absorber, which has a more complex geometry compared to a simple crush
tube. The absorber is analyzed using a biphase material of E-glass ﬁber and
polyester resin instead of a textile composite material. Three crushing modes,
namely local buckling, lamina bending and transverse shearing were identiﬁed
and simulated. Hamada and Nakatani [36] presented three different ﬁnite
element models to simulate the response of a simple axially-loaded textile

composite crush tubes after initial crack using plain strain elements to represent

11

tube wall, fronds and debris wedge, and truss elements to represent the contact
region between fronds and wedge. However, it was not clear as to which model

compared best to reality.

Masters and Minguet [37] conducted experimental investigation on the
elastic moduli and strength of two-dimensional triaxially-braided textile
composites. The study included the effects of primary braid architecture, namely
the braid angle, axial tow content and yarn size and secondary architecture,
namely the yarn crimp angle and yarn spacing, on the mechanical performance
of the composite. Chiu et al. [38] conducted tube-crushing experiments on six
different non-hybrid and hybrid types of 2D triaxially braided composite tubes
containing Kevlar and carbon ﬁbers to study their crush failure modes and
speciﬁc energy-absorbing capabilities under quasi-static axial compression. The
inﬂuence of ﬁber type and the hybrid structure on the crush failure mode, the
speciﬁc energy-absorption and the load/displacement response were examined.
Mamalis et al. [34] and Hull [39] performed thorough investigations on the
fracture mechanisms of an axially-loaded crush tube and related the mechanism
with the load-displacement curve. While Mamalis et al. believed that friction was
the most important parameter affecting the energy absorption of the crush tube,
as afﬁrmed by Hamada and Nakatani’s [23] ﬁnite element analysis, Hull pointed
out that the speciﬁc crushing stress was the most useful and distinctive

parameter to relate the performance of different materials and component

12

geometries. Their work revealed the inter-dependency of material, geometrical

and testing parameters and their difﬁcult-to-quantify complexity.

1.3 THE PRESENT STUDY

In the present study, a ﬁnite element model is developed to efﬁciently
simulate a two-dimensional triaxially-braided composite crush tube response
subjected to uniaxial compression. The embedded-tow model derived from the
simpliﬁed discrete-tow model [29] is adopted for the prediction of the composite
crush tube behavior. Failure of individual elements, namely tows and matrix, with
incremental compressive load represents the overall progressive failure of the
composite tube. The tows and matrix are not explicitly modeled, rather
collectively represented in shell elements. The commercial ﬁnite element code,
ABAQUS/Explicit, is used to simulate the crush tube load—displacement behavior.
Some preliminary models have been investigated before the current model is
developed, and they are presented in Chapter 2. The tube modeling procedures
are discussed in Chapter 3, and veriﬁcation of the current model is presented in
Chapter 4. The ﬁnal chapter, Chapter 5, concludes the research by summing up
important observations and possible extension of the ideas developed in this

investigation to related application. Future areas of study are also recommended.

13

Chapter 2

Preliminary Models

2.1 INTRODUCTION

The problem of simulating the textile composite crush tube behavior
depends mostly on developing a proper model. Prior to the ﬁnal model, two
preliminary models were developed. These models, namely the checkerboard

model and the cross-box model, are the major theme in this chapter.

The checkerboard board model and the cross-box model are both inspired
from the simpliﬁed discrete-tow model developed by Carrier and Averill [29]. The
discrete-tow model suggests the modeling of triaxially-braided composite
material using shell elements for the matrix material and beam elements for the
axial tows and braider tows. The main advantage of this model, as its name
suggests, is the capability to capture the behavior of individual tow elements in
the composite material, thus enabling simulation of tow scissoring and tow

sliding.

In the discrete-tow model, the braider-tow beams are connected to the
matrix shell at its diagonal ends. The axial tow, while being slightly shifted to one
side of the unit cell, could be represented as two beam elements attached to

edge of the shell. The modeling procedure, starting from the isolation of a unit

14

cell from a triaxially-braided composite material to the ﬁnal discrete-tow model

are brieﬂy summarized in Figure 2.1.

Each constituent element in the discrete-tow model follows its own failure
scheme. The damaging mechanism is such that the element behaves elastically
up to a peak load after which the element stiffness is greatly reduced until it
reaches a post-damage region, as can be seen in Figure 2.2. In the post-
damaged region, the element stiffness is so low that the failed material can be
considered incapable of transferring load. The peak point before damaging is
determined using different failure criteria. The tows are assumed to follow the
maximum strain failure criterion, whereas the matrix is assumed to obey the

maximum stress failure criterion.

2.2 THE CHECKERBOARD MODEL

2.2.1 DESCRIPTION OF THE MODEL

The checkerboard model is employed to explicitly model the crush tube.
The word explicit here means that the crush tube constituents, namely the tows
and the matrix material, are accounted for in the model as closely as possible
with regard to their physical arrangement and dimensions. However, the adoption
of the discrete-tow model mandated the use of two separate axial tows in a unit
cell, which is only an approximation of the physical reality. Nevertheless, this

approximation is desirable in the view that the anticipated response of the

15

symmetric model, resulting from the simpliﬁcation, to symmetric boundary
conditions will also be symmetric. This is more likely to match the crush tube
behavior subjected to a compressive force from a circular initiator, since the
crush tube is composed of the same textile material throughout, and the circular
initiator should impose nearly uniform loading on the end of the crushing region.
Any discrepancy from this expectation can only be attributed to imperfect
specimen, uncontrollable testing parameters, or stress waves in the tube due to
dynamic effects. Figure 2.3 shows a crush tube modeled using the checkerboard

model.

The checkerboard model is able to capture the scissoring of braider tows
and the sliding of the braider tows within the unit cell, but not the sliding with tows
of neighboring cells. In addition, the sliding of axial tow with the braider tow is not

represented since the tows are pinned at the nodes of the shell elements.

The axial tows are modeled as l-beams whereas the braider tows are
modeled as rectangular beams. The use of l-beams and rectangular beams is an
attempt to closely model the physical reality. Since the crush tube to be
simulated consists of two layers of textile fabric, the axial tows are laying mostly
at the top and bottom edges, which can be approximated with I-beams with very
thin, thus negligible web sections. Since the braider tows are spread out
uniformly in the fabric, they can be approximated as rectangular beams. The

areas of the beams should be the same as the total area of the tows, as

16

computed using the approach presented in the work of Carrier and Averill [29].
The dimensions of the beams are found by performing ﬁnite element bending
simulation on a strip of discrete-tow elements complete with beams and shells,
and then comparing them to the results obtained from a strip of homogenized

shell elements.

It is important to note that beams representing different tows should not be
joined at the same node, since the moment of beam elements of different tows
do not interfere with that of the neighboring tows. Beams of the same tow,
however, should be joined at the same node. In fact, each tow should have its
own node sets, and at the locations where a tow meet with its neighbors, only the
displacements of the tow are tied to those of the neighboring tow of the same
locations. In short, beams representing different tows should be pinned, not

rigidly joined.

2.2.2 NUMERICAL RESULTS USING THE CHECKERBOARD MODEL

Computer simulation using the ABAQUS/lmplicit ﬁnite element code was
performed on the crush tube using the checkerboard model. The force-
'displacement curve of the experimental curve is compared to that of the
simulation in Figure 2.4. These experimental results were obtained from [40]. The
analysis diverged after a certain point, notably at the moment when severe
damage had occurred on some elements. Owing to the presence of braider tows

on alternate shell elements, the locations of damaging elements did not exhibit

17

symmetry. This was mostly due to early failure of sites unsupported by braider
beams and was not what is to be expected physically, as the real tube should
deform more uniformly. The divergence of the analysis was believed to have
been caused by huge energy release when the axial and braider beams fail,
resulting in numerical problems. Due to these issues, this model was deemed

unsuitable for modeling tube behavior.

2.3 THE CROSS-BOX MODEL

2.3.1 DESCRIPTION OF THE MODEL

Due to the unsymmetrical deformation and the divergence of the analysis
of the checkerboard model, the cross-box model was developed. Again, this
model adopts the unit cell modeling from the discrete-tow model, but this time the
physical arrangement of tows in the real tube specimen is neglected. In fact, the
whole tube is deemed to consist of unit cells with matrix, braider tows and axial
tows. To account for the discrepancy with the physical reality, the original volume
fraction of matrix, axial tows and braider tows are maintained throughout the
tube. Though this is more of an implicit modeling of the microstructure, it is

expected to yield symmetrical deformation behavior.

This model can be a more reﬁned model compared to the checkerboard

model, but the increase of elements in reﬁnement may reduce the efﬁciency of

computation. Since the crush tube may not need to be totally crushed, the

18

uncrushed region may for the current analysis be represented with elastic shell
elements with homogenized properties as those obtained from the discrete-tow
model analysis. Figure 2.5 shows a crush tube modeled using the cross-box
model. The cross-box region is magniﬁed to assist visualization of the small
elements. In modeling the beams, the same geometrical argument with regard to
bending stiffness as applied to the checkerboard model is applied to the present

model.

2.3.2 NUMERICAL RESULTS USING THE CROSS-BOX MODEL

Computer simulation using the ABAQUS/lmplicit code was performed on
the tube of cross-box model. The force-displacement curve obtained in the
experiment [40] is compared to that of the simulation in Figure 2.6. Again, the
analysis diverged after a certain point, notably at the moment when severe
damage had occurred. Numerical problems were again suspected to be caused

by high-energy release of the failing elements.

It is noted that the cross-box model actually over-predicts the stiffness of
the composite crush tube. This discrepancy is not exhibited in the checkerboard
model. It was suspected that the overall bending behavior of the tube might not

be represented well using the cross-box model.

19

 

 

 

 

 

 

 

 

 

 

 

 

 

Original Triaxial Braid Discrete-tow model

Figure 2.1 A summary of the modeling procedure from the isolation of a unit cell

of a triaxially-braided composite material to the ﬁnal discrete-tow model.

Stress
A

Pre-damaged Damaging Post-damaged region
region region

’//_.

 

 

> Strain

 

 

Figure 2.2 Damaging scheme of individual shell and beam elements in the

discrete-tow model.

20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.3 Crush tube model by the checkerboard model.

Graph of Load versus Displacement

 

 

 

 

 

 

 

 

 

l
l
l
‘ 2500 ,-
‘ ’f". ~-~~~-‘ ”‘—""
‘ 2000 - r—i‘i ﬁge 7* ié’ﬂw—v’ 7'7 7__,
i I". m.#
I § 1500 7777' if? giAkirT —Simulation l ,_ i
1 E 1000 — I """ Experiment i i
‘ 0
j " 500 . -7? l
l L l

0 7 ’7’ f 7 _

0 0.1 0.2 0.3 0.4 0 5

Figure 2.4 Comparison between simulation results from checkerboard model to

experimental results of load versus displacement of the crush tube.

21

 

)

b

(

Figure 2.5 Crush tube modeled by the cross-box model. (a) A magniﬁed view of

the cross-box regions. (b) The whole crush tube, with cross-box region on the

left.

22

 

 

 

 

 

 

 

 

IT* T T" T T T T T T TI
l Graph of Load versus Displacement l
l
I 2500 v i
2000 ———.~—_* 0.45;: "-_-:~.u.,...__;,’.::"______m_
I 3 150°“ 1:29.;Jaaﬂ *— l
I F 1000 - --;_-_Extfriﬂrn_ent__ .L__

8
l " 500 4’ —~»ﬁ —em——
I o- ”MMLLHL—LL-
' o 3 0.4 0 5
l .500?

Displacement (in)

 

 

Figure 2.6 Comparison between simulation results from cross-box model to

experimental results of load versus displacement of the crush tube.

23

Chapter 3

The Embedded-Tow Model

3.1 INTRODUCTION

Chapter 2 presented two modeling approaches, namely the checkerboard
model and the cross-box model. These two models suffered from inaccurate

predictions and divergence during the simulation analysis.

In this chapter, our ﬁnal model—the embedded-tow model, will be
introduced. This model is also inspired from the simpliﬁed discrete-tow model
[29], and from the experience Obtained after working on the two previous models.
The embedded-tow model, as its name implies, implicitly represents the axial and
braider tows in the shell elements. In other words, all three components in a unit
- cell—axial and braider tows, and matrix—are embedded in a shell element.
Obviously, this model forfeits the beneﬁts of explicitly modeling tow-scissoring
and tow-sliding, since tows are not modeled discretely in the model. However,
this model will be more computationally efﬁcient, and will retain the capability to
capture the failure of each tow discretely, providing that certain simplifying

assumptions are reasonable from the engineering standpoint.

24

3.2 DESCRIPTION OF THE EMBEDDED-TOW MODEL

A quick review of the simpliﬁed discrete-tow model [29] will provide a
better understanding of the current model. Details on computing the material

stiffness matrix of the simpliﬁed discrete-tow model are also given in [29].

As shown in the discrete-tow model and again in Figure 3.1, one half of
the repeating volume element of the triaxially-braided textile composite is
sufﬁcient for modeling the macro-mechanical properties of the overall textile
composite. This repeating volume element can be further divided into two
regions. One region contains the axial tow, as in the left region in Figure 3.1(b),
and the other region does not. The stiffness matrix of each region is ﬁrst
computed using the classical laminate theory. The stiffness matrix of the overall

composite is then computed by iso-stress and iso-strain assumptions.

The current approach adopts the same idea with respect to the
computation of the material stiffness matrix. The overall composite macro-
mechanical properties are assumed in each unit cell in our embedded-tow model.
The embedded-tow model also retains the discrete-tow model capability of

modeling tow undulation.

Due to the problem of divergence in the previous analyses using
ABAQUS/IMPLICIT, the current model will be implemented within

ABAQUS/EXPLICIT. Even though ABAQUS/EXPLICIT is more commonly used

25

for dynamic analysis, with proper mass scaling and loading rate, quasi-static

analysis can be performed with sufﬁcient accuracy.

3.2.1 ESTIMATING STRAINS OF INDIVIDUAL COMPONENTS

In this model, the individual strains of the axial tows, the braider tows and
the matrix are ﬁrst predicted with certain simplifying assumptions. These
simplifying assumptions originate from the work of Mauget et al [41]. Since each
unit cell will be represented by a reduced-integration shell element with a single
in-plane integration point (ﬁve section points through thickness), and by intuition
the deformation of the shell elements will be similar to that of an elementary box
as described in [40], the work of these researchers can be adopted in the current
model. Consequently, the idea presented by Mauget et al. can be used to
describe shell deformation behavior of the current study with reasonable
accuracy only before failure. Shell deformation after failure is hard to predict, but

is not of major concern in our present study.

Figure 3.2 illustrates the presumed deformation of a shell element, or a
unit cell, before failure. Using the inﬁnitesimal strain assumption, the strain of
both the axial tows in a unit cell can therefore be estimated to be equal to the
shell strain along the loading direction, or its axial strain. The strain of the braider
tows can be computed from the shell axial strain, transverse strain and shear

strain using the strain transformation as stated in equation (3.1).

26

a +8 8 -8 7 .
£45: -L2—3’— + -O—2&COS(7Z') + 331nm) (31)

845 is the strain of the braider tow, so and 890 are the strains of the shell element in
the loading and transverse directions, and y is the shear strain of the shell
element. It is important to point out that owing to the abovementioned
assumptions and the use of one-integration-point element, the two axial tows in
the unit cell will fail simultaneously. Similarly, the two braider-tows in the unit cell
will fail at the same time. This approach is not general, but it provides an efﬁcient

method for modeling the crush tube of interest here.

3.2.2 PROGRESSIVE FAILURE SCHEMES OF INDIVIDUAL COMPONENTS

The discrete-tow model adopts the progressive failure scheme shown in
Figure 2.2 for its constituents—the tows and the matrix, both in tension and
compression. The current model will also account for failure of each constituent,
but this time employing a different failure scheme for tensile and compressive

loadings.

In tensile loading, the same progressive failure scheme as proposed in the
discrete-tow model is employed. However, in compressive loading, the
progressive failure scheme consists of only two regions instead of three, namely
the pre-damaged and post-damaged regions, as shown in Figure 3.3. This

modiﬁcation is needed because constituents failing under compression are still

27

able to carry load. Post-damage load will compact the materials, instead of
pulling them apart as in tensile loading. The compacted material can transfer

additional load.

3.2.2.1 PROGRESSIVE FAILURE ALGORITHM

Since the textile composite material of the crush tube is non-linear in
behavior, a FORTRAN algorithm has been written to simulate its behavior. This
algorithm is incorporated into ABAQUS/EXPLICIT as a user subroutine called
VUMAT. Both ABAQUS/IMPLICIT and ABAQUS/EXPLICIT provide its users the
utility to include user materials, although the user-program interface of the

software codes is different.

An incremental displacement approach is employed in the current quasi-
static contact analysis. Description of the analysis is provided in section 3.2.3. As
the initiator moves incrementally towards the crush tube, the resulting
incremental crushing load is computed. The contact between the initiator and the
crush tube produces incremental strains and stresses in the tows and matrix,
which in the current analysis are embedded into the shell elements. These
constituent materials fail progressively by their individual failure criteria,
according to the aforementioned failure scheme. Similar to the analyses
described in Chapter 2, the tows are assumed to obey a maximum strain failure

criterion and the matrix a maximum stress failure criterion.

28

When a material point fails, its stiffness is recomputed following the
progressive failure scheme. The analysis then proceeds to redistribute the load
around the failed site and continues with a new displacement increment. The

algorithm is brieﬂy summarized in Figure 3.4.

3.2.3 DESCRIPTION OF THE ANALYSIS

A quasi-static contact analysis is performed to simulate the crush tube
behavior under compression. Figure 3.5 (a) shows a crush tube subjected to
compressive force through a moving platen. Figure 3.5 (b) shows the equivalent
model employed in the current analysis. An initiator was used in the experiment
and in the simulation to prevent the onset of buckling and the occurrence of
sudden peak force. It is noted that in the simulation, the initiator is moved
incrementally towards the crush tube and the tube is ﬁxed at its location. In the
experiment, however, the initiator remains ﬁxed while the crush tube was moved

towards the initiator.

The initiator in the current simulation is modeled using an analytical rigid
surface. This rigid surface is associated to a single node in all its movements and
resulting forces. This provides the convenience of outputting the resulting force

and displacement of the crush tube through a single node.

The initiator was positioned to establish immediate contact with the crush

tube at the beginning of the simulation. The curvature of the initiator is 5/16 inch.

29

Since immediate contact was assumed, the initiator diameter is the diameter of
the tube less its thickness. The tube diameter is 2.47 inches and its thickness is
0.075 inch. The crush tube has a chamfer at the crush end. The crush tube is
made of O°/+45°/-45° triaxially-braided composite material with 80k Fortaﬁl #556
axial ﬁbers, 12k Graﬁl 34-700 braider ﬁbers and Ashland Hetron 922 epoxy vinyl
ester as its matrix. The experimental density of the material is 1.4204E-4

lbf*szlin4.

The size of unit cells used for the mesh of the tube is 0.075X0.075X0.075

cubic inch. This unit cell size was adopted assuming that the length of the

chamfer is the same as that of one unit cell.

30

 

 

//

(a) (b) (C)

 

 

 

 

 

 

 

 

 

Figure 3.1 (a) Half of a repetitive volume element; (b) Axial tows, braider tows
and resin represented as material stacks before the application of classical

laminate theory; (0) The overall anisotropic material block, as in a unit cell.

/

 

 

 

 

 

\

(a) (b)

 

Figure 3.2 (a) Unit cell before deformation; (b) Unit cell after deformation.

31

Stress . .
Compressive region + Tensrle reglon

Pre-damaged

Post-damaged

//
b

 

 

Strain
Damaging

Post-damaged

 

 

’/

Pre-damaged

 

 

Figure 3.3 A stress-strain curve representing progressive failure scheme of each

individual constituent in a unit cell.

32

 

Information from input ﬁle Strain values passed from ABAQUS/EXPLICIT

 

 

 

 

i

Only once ,
For each 1ncrement

 

 

VUMAT

I

Compute strain for axial and braider tows

 

 

 

Compute principal strain for matrix

I

Determine failure of each constituent by
applying the appropriate failure criteria

I

No Yes
Recalculate

————’l Stiffness

 

 

 

 

 

 

 

 

 

 

 

I

Calculate overall stress, pass control back to ABAQUS/EXPLICIT

I

 

 

 

 

 

 

 

 

 

 

 

 

Incremental Displacement

 

 

 

Figure 3.4 A summary of the progressive failure algorithm.

33

 

Applied force applied to the platen in contact with tube

 

 

 

 

    
 

................................
.............................
................
...........................
............
........................
...................
..........................
.........
oooooooooooooooooooooooooooooo
oooooooooooooooooooooo
ooooooooooooooooooooooooooooooooooooooooooooooo
aaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
ooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooo
F. d d

Fixed displacement
boundary conditions at the
end of tube

 

‘ Crush tube

 

 

‘..IIIIIIIIIII..... I

  

Initiator represented by
analytical rigid surface

3

 

 

 

Figure 3.5 (a) Force applied at the platen, which moves towards the initiator, as

in the experiment. (b) Initiator moves towards the tube in current simulation.

34

Chapter 4

Embedded-Tow Model Verification and Crush Tube

Simulation Numerical Results

4.1 INTRODUCTION

Chapter 3 described the development of the embedded-tow model and
how it could be used to model the crush tube behavior. This chapter attempts to
verify the use of the embedded-tow model by demonstrating its accuracy in a
simple bending situation whose analytical results can be obtained. The modeling
of the progressive failure scheme is also veriﬁed by applying simple uniaxial

tension and compression on a unit cell.

Since ABAQUS/EXPLICIT was used to simulate the force-displacement
behavior of the tube, whose behavior is quasi-static, the effects of loading rate
and mass scaling were also studied by performing simple bending analyses.
Finally, an appropriate loading rate and mass scaling were chosen to simulate
the crush tube under uniaxial compression. The appropriateness of the chosen
loading rate and mass scaling can best be assured by detecting the simulated
tube behavior for dynamic effects. The result was then plotted against the

experimental crush tube behavior.

35

4.2 MATERIAL PROPERTIES

The elastic macro-mechanical properties of a unit cell used in all the
analyses in this chapter are the same as that computed for the of 0°/+45°/-45°
triaxially-braided composite material of the crush tube, and are summarized in
Table 4.1. These values were computed from the mechanical properties of the
composite constituents as listed Table 4.2 using the micro-mechanics approach
adopted by Carrier and Averill [29]. The transverse shear stiffness of the
composite is assumed to be the same as its in-plane shear stiffness in the
absence of measured and simulated data, though this may give rise to stiffer

analytical results as compared to the theoretical ones.

It is important to note that these values are only the elastic properties of
the composite. Upon damage, the composite will behave non-linearly according
to the progressive failure scheme outlined in Chapter 3 and thus the tabulated
stiffness values will no longer apply. The constituent tows are assumed to be
transversely isotropic and the matrix to be isotropic. The failure stress of matrix

and the failure strain of axial and braider strains are also listed in Table 4.2.

36

4.3 VERIFICATION OF THE EMBEDDED-TOW MODEL

Three simple loading situations have been designed to verify the
embedded-tow model behavior in the elastic and the past failure range. To
simulate the model behavior in the elastic range, a simple bending analysis was
performed on a strip of shell elements. To study the inelastic behavior of the
model, simple uniaxial tension and compression tests were conducted on a unit
cell, or a shell element. The simulated results were then compared with the

theoretical results to verify the model usefulness.

4.3.1 SIMPLE BENDING ON A STRIP

A row of 54 0.75 in X 0.75 in X 0.75 in shell elements were connected
end-to-end to form a strip whose length is 4.05 inches, that is about the same as
the length of the 4-inch tube. A concentrated load was applied at one end, with
the other end ﬁxed as shown in Figure 4.1. The load was applied in a ramp
fashion. The loading rate employed was small enough to ensure minimal
dynamic effects. Any dynamic effects induced, as can only be observed from the
strip response after the simulation, will disqualify the analysis from being quasi-
static. The axial stress of the element at the ﬁxed end and the out-of-plane
displacement of the loaded end were obtained and compared to the theoretical
results available from the theory of elementary strength of materials. According to

this theory,

37

PL3

6 = ——
4.1
3Enyy < I
PL(t/ 2)
0' x = _I— (4.2)

yy

where 8 is the out-of—plane deﬂection, P is the force applied at one end, IEx is the
Young’s Modulus along the strip direction, lW is the second moment of area along
the y direction as consistent with Figure 4.1, and t is the thickness of the strip. At
a loading rate of 0.2 lb/s, the simulated and theoretical displacements are
compared in Figure 4.2. Note that the two lines lie closely on top of one another,
indicating the accuracy of the embedded-tow model. Similarly, the simulated
bending axial stresses of the element at the ﬁxed end also agree closely with the

theoretical values, as shown in Figure 4.3.

4.3.2 SIMPLE TENSION AND COMPRESSION ON A UNIT CELL

A 0.075 in X 0.075 in X 0.075 in shell element was subjected to uniaxial
tensile and compressive stresses as depicted in Figure 4.4. The unit cell was
stretched or compressed till damaged. Its tensile and compressive non-linear
behavior is plotted in Figure 4.5 and Figure 4.6, respectively. The damage of the
constituents was obvious as seen in the drop of the stress carried by the

element. In tension, the axial tows in the unit cell were ﬁrst observed to have

38

been damaged, followed by the matrix and ﬁnally the braider tows. In
compression, however, the axial tow damage was followed by almost
simultaneous damage of braider tows and matrix. The damaging behavior of the
two different progressive failure schemes adopted for tensile and compressive
loadings was clearly shown in these two ﬁgures. Note that the stress carried by
the element after compressive failure remains unchanged, while tensile failure

will render the element to carry less stress.

4.4. EFFECTS OF LOADING RATE AND MASS-SCALING

An inappropriate loading rate may induce dynamic effects in the computer
analysis. Since the current analysis is quasi-static in nature, dynamic response is
not desirable. This dynamic response can only be minimized by employing a low
loading rate in the simulation. However, a low loading rate usually means greater
computational efforts, and may render the analysis impractical if its computation
time is too long. There is a trade-off between accuracy and computational effort.
To accurately simulate the crush tube behavior in a signiﬁcantly less amount of

time, the use of mass scaling seems inevitable.

Mass scaling has the effect of speeding up the analysis. This is done by
artiﬁcially increasing the material density by a factor of f so that the number of
increments, n, of the analysis may be reduced to n/f. Basically, mass scaling
reduces the ratio of the event time to the time for wave propagation across an

element while leaving the event time ﬁxed, which allows rate-dependent behavior

39

to be included in the analysis. Mass scaling has exactly the same effect on inertia
forces as speeding up the time of analysis [42]. If abused, however, it induces
errors in the simulation results. Therefore effects of loading rate and mass
scaling ﬁrst need to be studied. This investigation has been done by performing

the strip bending simulation as described in Section 4.3.1.

First, the effects of different loading rates were examined. Using three
different loading rates of 0.2 lb/s, 1.0 lb/s and 30.0 lb/s, the out-of-plane
displacements of the strip subjected to bending were plotted against the
theoretical values in Figure 4.7. It is evident from this ﬁgure that the displacement
for the highest loading case deviates very much from those of the other two
cases with lower loading rates, whose results match closely with the theoretical

predictions. Therefore, it is more accurate to use a low loading rate.

A high loading rate has been shown to yield erroneous results. On the
other hand, mass scaling may be shown to produce similar undesirable dynamic
response if abused. Using different mass scale factors, namely a multiplication
factor of 100 and 1000 times this original mass, the same simple bending
simulations were repeated. Results are presented in Figure 4.8, where the
displacements of the ﬁrst two cases, one without mass scaling and the other with
a factor of 100, were closer to the predicted theoretical displacement as

compared to the one with the highest mass scaling.

40

Figure 4.8 seems to suggest the existence of a threshold value for mass
scaling. For the current problem deﬁnition, a mass scaling of 100 times the
original mass is adopted to speed up the simulation of the crush tube, since it
has been shown to work well in the case of simple bending. The appropriateness
of the loading rate, however, can only be decided upon the completion of the
simulation, whose result should demonstrate sufﬁcient indication on whether or
not the dynamic effects of the simulated behavior is critical. Dynamically induced
crushing behavior usually includes sudden and non-symmetrical failure of strips
of elements. This failure mechanism corresponds to the tube axial splitting, which
in a quasi-static experiment, is only likely to happen gradually and symmetrically
around the circular tube. A loading rate of 0.025 in/s was employed in the present

study.

4.5. CRUSH TUBE NUMERICAL SIMULATION RESULTS

4.5.1 CRUSH TUBE FORCE VERSUS DISPLACEMENT CURVE AND
DAMAGE MECHANISM

Apart from loading rate and mass scaling, there are two other important
parameters in the crush tube analysis, namely the friction between the tube
material and the surface of the initiator, and the way the 45° chamfer of the tube
is modeled. The effect of friction and chamfer modeling are discussed in more
details in the following section, “Parametric Studies”. The crushing force versus
displacement curve for a tube with the chamfer modeled with three shell

elements, Coulomb friction of 0.3 between the tube material and the surface of

41

initiator, and a loading rate of 0.025 in/s is shown in Figure 4.9. The experimental

results used throughout this chapter were obtained from [40].

It is apparent that the simulated tube stiffness is higher than that of the
experiment. Before any explanation is attempted, it is evident from the
experimental curve that the stiffness of the real tube gradually increases initially,
as opposed to the almost linear stiffness of our simulated tube before the peak
load. This might be caused by the looseness of ﬁt between the crush tube and
the initiator in the experiment. This inevitable physical reality might have caused
the increasing initial stiffness as shown in the experimental curve. The parametric
studies on the effect of chamfer modeling and friction also reveal that these
parameters do affect the initial stiffness, and the peak stress of the simulated

response.

The analysis terminated prematurely because of bad element distortion.
The analysis could be restarted after deleting highly distorted elements by
identifying a state variable for each element in the model as deletion ﬂag at the
very ﬁrst step of the simulation. However, this was not done in the current

studies.

Figure 4.10 is a replica of Figure 4.9 except that important points of failure

of constituent elements in the textile composite tube are marked on the curve

42

with the corresponding curve region zoomed. The damage mechanism

associated with these points are summarized as follows:

A: Axial tows in the chamfer begin damaging in compression

B: Matrix in the chamfer begins damaging in tension

C: Matrix in the chamfer is completely damaged in tension

D: Matrix in the chamfer begins damaging in compression

E (Peak point): No axial or braider tensile failure

F: Braider tows in chamfer start damaging and failing in compression

G: Axial and braider tows in chamfer begins damaging in tension

H: No shell elements damaged in a strip. A strip refers to shell elements

connected along the axial direction of the tube.

I: Strips of material start damaging, indicating dynamic response

The results show that the sharp drop between E and G is due to the
tensile damage of both the axial and braider tows, which could be caused by high

hoop stress at the crushing end.

4.5.2 PARAMETRIC STUDIES

4.5.2.1 EFFECTS OF CHAMFER MODELING

Since crush will initiate at the chamfer because of its weak pointed end

and its relative thinness compared to the rest of the tube, the initial response of

43

the tube is dependent on how the chamfer in modeled. The current study has
investigated the effects of modeling the chamfer with different number of shell
elements lengthwise. Three different models were adopted. The ﬁrst model has
one shell element lengthwise for the chamfer, the second has 3 shell elements,
and the third has 6 shell elements. The shell elements were not offset to the
edge of the tube as in the real physical tube, rather they were centered along the
mid-plane of the tube material, as shown in Figure 4.9. The results in Figure
4.10 show that the initial stiffness of the tube reduces as more elements are used
to model the chamfer. This is intuitive as a more reﬁned mesh in the chamfer
allows the pointed end and its vicinity to be represented more accurately and
thus, damage is allowed to occur at individually smaller elements. Also, it is
noted that modeling chamfer with fewer elements lengthwise will increase the

peak load for the same reason as mentioned.

4.5.2.2 EFFECTS OF FRICTION

The Coulomb friction has been used in the current analyses to model the
friction between the tube material and the surface of the initiator. Two different
friction values of 0.1 and 0.3 were used to investigate the difference in the
crushing response. The results were shown on Figure 4.13. Again, the analysis
with a Coulomb friction of 0.1 terminated because of highly distorted elements.
However, Figure 4.13 clearly shows that a reduction in Coulomb friction results in
a reduced initial tube stiffness and a reduced peak force. This is intuitive, since

friction contributes to crush energy, and is suspected by Mamalis et al. [34] to

44

account for 40% to 50% of the total crush energy. The points marked on the

curve correspond to material failure as listed below:

A: Matrix in chamfer damaged in tension and compression
Axial tows damaged in compression

B: Braider tows start damaging in tension and failing in compression
Matrix starts to fail in tension

C: Axial tows partly damaged and partly failed in tension

D: Strips of braider tows and matrix fail in tension, indicating dynamic response

Again, the sharp drop from A to C is most likely due to the tensile damage

of the axial and braider tows by high hoop stress at the crushing end.

45

Table 4.1 Computed macro-mechanical properties of the 0°/+45°/-45° triaxially-braided
composite material of the 80k/12k crush tube.
' Macro-mechanical Properties Values
Young’s Modulus in the loading direction, E)( 10.23 Msi
Y0ung’s Modulus in the transverse direction, I3y 1.6727 Msi

In-plane Poisson’s ratio, vxy 0.573

In-plane Shear Modulus, ny 1.262 Msi

Table 4.2 Mechanical properties of the composite constituents.

Constituent Macro-mechanical PrOperties Values
Tows I Young’s Modulus in the loading direction, Ex 619 Msi

Young’s Modulus in the transverse direction, By 1.2 Msi

In-plane Poisson’s ratio, vx’y 0.3
In-plane Shear Modulus, Q, 0.65 Msi
Tensile Failure Strain 0.01
Compressive Failure Strain 4.05267E-2
Matrix Young’s Modulus 0.46 Msi
In-plane’Poisson’s ratio, vxy 0.35
In-plane Shear Modulus, ny 0.17037 Msi
Tensile and Compressive Failure Stress 5.4E-3 Msi

46

V x
P/2

Y

 

 

 

 

 

 

 

 

 

 

 

 

 

&\\\\\V

54 shell elements lengthwise P/2

Figure 4.1 A simple bending analysis on a strip of shell elements. Two bending forces,

each of P/2 in magnitude, were applied into the plane.

 

 

I
I Force versus Displacement
I
I

 

I
I
0.12

 

 

 

 

displacement (in)

I {—SimulatedFI
I S 0.03 ,—TheoreticaII
I
9 I
I '2 0.04 : I
I
I I
I 0 . . I
I 0 0.02 0.04 0.06 0.08 0.1 t
I I
l __ I

 

Figure 4.2 Comparison between the simulated and the theoretical out-of-plane

displacements of the end of bending strip.

47

T Force versus Stress

0.12

g 0.08

43

I'.’

‘9 0.04
o

 

 

 

 

i_ Simulated I
5"” Theoretical I

 

 

 

T T l l r

1000 2000 3000 4000 5000 6000

stress (psi)

 

__ L-_L d._ ____ 1

 

Figure 4.3 Comparison between the simulated and the theoretical bending

stresses of the end of bending strip.

t\\\\V

P/2

 

 

 

 

P P/2
’///////A

Figure 4.4 Uniaxial tensile and compressive test on a unit cell.

48

 

Stress vs. Strain

 

1 .20E+05

1.00E+05 - r e A—-- — — ~ ___ A-w—ﬂm—w *-

8.00E+04 —4— -7-.. L- L A .

 

6.00E+04 ~—-——— -~----— —-— _ ____ L

Stress (psi)

4.00904 -_ 4 —v ”,L_________,_ —._ 4 ‘ V 44—4 L _

 

2.00E+04 4———— A _ _. -__. _-_____

 

 

 

0.00E+00 I I . T
0.00E+00 1.00E-02 2.00E-02 3.00E-02 4.00E-02 5.00E-02

Strain (in/in)

 

 

 

Figure 4.5 Stress-strain curve of a unit cell subjected to tensile force.

 

ﬁw'

Stress versus Strain
Strain (in/In)

n 09E+09——
l I I V.

 

 

I
I
I -8.00E-02 -6.00E-02 -4.00E-02 -2.00E-02 0.00E+00
I

I — Compressive

 

 

Stress (psi)

 

 

 

 

 

g»
E

 

 

Figure 4.6 Stress-strain curve of a unit cell subjected to compressive force.

49

 

 

 

 

 

 

 

 

 

 

Force versus Displacement
0.16
0.14 77 7 .— _...4_4_ 7 7.... 77 7 7.77 7.74 77 7
0.12 ’ ‘— 77 r ‘ ,7 .7777 7. 777777 VTY—TTOTT .7_77___.,
g 0'1 ITTT TT IT TTTTT TT TT TT’ T‘T"TT:' 1:, ‘—’-_;—a.' i—-—————— .7
0 0.08 7 77 7 7_ 7_7_7 ’ —— 6413—4777777_7_77_7 7
'5 006 -. 7- 7 7 7 7 -7- 3%T7 777I— ‘0-2'9/3 TIL
T 0I04 ,.-"_'_ C ' j—1lb/s I
' :7" TT I T TT'T T' TJ- - - 30lb/s
0'02 VTT I TT —TTT ITT TITTT T—IITT I -I~~—*--Theoretica|Valuesj
0 r , . "T' W ' .
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
I displacement (in)
I7 7 7. .777 7 7 77 , 7 7__7 77

 

Figure 4.7 The effects of loading rate on the simulated strip out-of-plane

displacement

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
displacement (in)

 

 

 

77 7 77 77 77 7777--

t Force versus Displacement

l 0.16

I 0.14~ ————— — 7, .7 W- W—— —— «W W 7 7

I 0.12 ~ W 777 7 ~ - £73” -- 77777
g 0.1 7 ~- ~—— —- ~ . W
3 0.08 I ------ 7, — T - . -E- -Original mass I
" I
«9 0.06 7_ 7 77100 times original mass I

0'04 T T'V’TT ' “- - - 1000 times original massi

I 002 777/ TTII 77... Theoretical values I

I 0 ' l T I I T I l II

I

I

I - _

 

 

Figure 4.8 The effects of different mass scaling on the simulated strip out-of-

plane displacement.

50

 

Load (lb)

 

Load versus Displacement

 

3500

3000

2500 a

2000 7_77777 —%

1500 _

1000 7 ,_
500 7 -

 

 

  

 

_— _TT ‘ _ ”—7’ l T'TT— _ TTI—TT—TT TT

0 I .
600100 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

 

   

.4

 

Experiment I
__7_I

 

 

 

 

Displacement (in)

 

Figure 4.9 Simulated and experimental load versus displacement curves.

 

Load (lb)

 

Load versus Displacement

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3500

3000 _7, f- T
1'"

2500. If "T

2000. 7//-”T‘f“"“‘-~:——-

1500 - ‘“"‘“““7W
— -Simulation

1000-4* , I

Experiment I
500 x Points of failure I
0 1 " - T I
-50634 0 0.05 0.10 0.15 0.20 A 0.25

 

 

 

 

 

 

Displacement (in)

 

Figure 4.10 Simulated and experimental load versus displacement curves.

51

 

 

 

 

 

 

 

 

 

 

>/ “If ”If

(a) (b) (C) (d)
Figure 4.11 Modeling the chamfer. (a) The chamfer as in the tube specimen (b)
Chamfer modeled with one shell element lengthwise (c) Chamfer modeled with
three shell elements lengthwise (d) Chamfer modeled with six shell elements

lengthwise.

 

Load versus Displacement
Effect of Friction

 

4500 I77. 7_ _. I
Experiment I

 

 

 

40001-I

3500 7 '— "3-elementchamfer ,7 3 ~

300° T ' " '1-elementchamfer HTﬁ——TTTTTT
I ' l

2500 TI 6-elementchamfer I /

l .7 7.— .

2000 «— —- 7 .7

 

 

 

 

 

 

 

 

Load (lb)

1500 +——— ——— -—~—-—-- v
1ooo~ ,_7_ 77 / — 7
500« ~ , ,

t) 0.05 0.1 0.15 0.2 o. 25

 

 

 

 

 

 

 

~500

Displacement (in)

 

 

 

Figure 4.12 Effects of chamfer modeling on the load versus displacement curve.

52

 

Load versus Displacement
Effect of Friction

 

3500
mme‘+H_-7_777777777777777I

2500 —

2000_. ,“ ..//”TTTT"T“\——~ ._7__77

 

 

 

Load (lb)
3
O
O

 

 

 

1000 - Experiment
- -Friction = 0.3
500 7 Friction = 0.1 7
X Points of failure

 

 

 

 

 

0 ‘1" TTT l T T

0 0.05 0.1 0.15 0.2 0.25 0 3

 

 

 

Displacement (in)

 

 

 

Figure 4.13 Effects of Coulomb friction on the load versus displacement curve.

53

Chapter 5

Conclusions

5.1 CONCLUSIONS

A method to efficiently model the crush simulation of triaxially-braided
textile composite tubes was developed. The embedded-tow model has the ability
to simulate the crush response of a circular tube, as demonstrated in the current
study. The general trend of the numerical results agrees with that of the
experiments. The use of this model may be extended for simulation of other
general structures made of triaxially-braided textile composite. However, care
has to be taken by considering the assumptions that go into the modeling of the
crush tube. Though the crush tube model needs improvement in the modeling of
chamfer to increase its accuracy in predicting the initial stiffness response, the

embedded-tow model proves to be a promising tool for crushing analysis.

5.2 FUTURE WORK

To complete the analyses of the current studies, a state variable for each
individual element has to be set apart as a deletion ﬂag at the very beginning of
the analyses. The embedded-tow model, though with much simpliﬁcation, is still

computationally consuming and may need further modiﬁcation. This model has

54

also forfeited its parent model ability to represent the sliding of tows in the
composite during crush by incorporating tows into shell elements. In addition, the
current approach has neglected delamination in the composite material, which in
reality occurs when the tube is crushed. Tow sliding and delamination are
important parameters that affect the crush energy of the composite tube. Work
has to be done to incorporate the effects of tow-sliding and delamination in this

model.

55

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59

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