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; 0C LIBRARY
Michigan State
University ‘

 

 

This is to certify that the
dissertation entitled

Testing and Simulation of Composite Laminates

Under Impact Loading

presented by

Xinglai Dang

has been accepted towards fulﬁllment
of the requirements for

Ph.D. degree in Engineering Mechanics

yﬂée 22;;

Major profes§r

Date 5 2?“ O

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6/01 cJCl RC/DateDue.p65-p. 1 5

 

 

 

TESTING AND SIMULATION OF COMPOSITE LANIINATES UNDER
IMPACT LOADING

By

Xinglai Dang

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Materials Science and Mechanics

2000

 

ABSTRACT

TESTING AND SIMULATION OF COMPOSITE LAMINATES UNDER IMPACT
LOADING

By

Xinglai Dang

Owing to their high stiffness-to-weight and high strength-to-weight ratios, ﬁber-
reinforced polymer-matrix composite laminates are excellent materials for high-
performance structures. However, their properties in the thickness direction are very
poor as they are weakly bonded by polymeric matrices through laminate interfaces.
Accordingly, when a composite laminate is subjected to impact loading, high
interlaminar stresses along with the low interlaminar strengths could easily result in
interlaminar damage such as delamination. This thesis investigated the response of
composite laminates under low-velocity impact and presented numerical techniques for
impact simulation. To begin with, instrumented drop-weight impacts ranging from
subperforation to perforation levels were introduced to composite laminates having
various dimensions and thicknesses. Damaged composite laminates were then subjected
to compression-after-impact tests for evaluations of residual properties. Experimental
results revealed that perforation was an important damage milestone since impact
parameters such as peak force, contact duration, maximum deﬂection and energy
absorption, and residual properties such as compressive stifﬁiess, strength and energy
absorption all reached critical levels as perforation took place. It was also found that

thickness played a more important role than in-plane dimensions in perforation process.

 

 

In order to understand more about the relationship between laminate thickness and
perforation resistance and to present an economical method to improve perforation
resistance, thick laminated composite plates and their assembled counterparts were
investigated and compared. An energy proﬁle correlating the impact energy and
absorbed energy at all energy levels for each type of composite plates investigated was
established and found to be able to address the relationship between energy and
damage. Experimental results concluded that increasing thickness was more efﬁcient
than improving assembling stiffness in raising perforation resistance. As a ﬁrst step to
simulate composite response to impact loading, LS-DYNA3D was used for numerical
analysis. However, due to its inability to describe interlaminar stresses, no delamination
simulation could be achieved. As delamination played a very important role in damage
process, a computational scheme capable of identifying interlaminar stresses and
considering both numerical accuracy and computational efﬁciency was required for
impact simulation. Accounting for interlaminar shear stress continuity and having
degrees of freedom independent of layer number, a laminate theory named Generalized
Zigzag Theory was formulated into a ﬁnite element subroutine and integrated into
ABAQUS code. Due to the uses of Truesdell rate of Cauchy stress and rate of
deformation tensors, the computational scheme was able to present reasonable
interlaminar shear stresses via an updated Lagragian algorithm. Combining the
calculated interlaminar stresses with a delamination failure criterion, the computer
program was able to predict the response of composite laminates up to the onset of
delamination. Further computational simulation involving all damage modes should be

considered in future studies.

To My Family

 

ACKNOWLEDGMENTS

From my heart, deep gratitude and appreciation goes to my major advisor, Dr.
Dahsin Liu, for his constant encouragement and invaluable guidance. His contribution to
the achievements of this work is signiﬁcant.

Special thanks also extend to my dissertation committee members, Dr. T. Pence,
Dr. Zhenfang Zhou, and Dr. H.Y. Tsai, for their academic support and encouragement
through my Ph. D. program.

I am very grateful to my parents, Decang Dang and J unlan Fan, for their
understanding, support and love. They worked very hard to support me all over past
years.

Finally, I'd like to thank my wife, Xiaomei Xi, and my son, Edward Dang.
Without their help and love, the completion of this dissertation is impossible. For this

reason, this dissertation is dedicated to them.

 

TABLE OF CONTENTS

List of Tables. .................................................................................... viii

List of Figures ....................................................................................... ix

Chapter 1

INTRODUTION ..................................................................................... 1
1.1 Literature review ........................................................................ l
1.2 Objectives ................................................................................ 7
1.3 Organization of this thesis ............................................................. 7
1.4 References ................................................................................ 8

Chapter 2

SIZE EFFECTS ON IMPACT RESPONSE OF COMPOSITE LAMINATES. . . . . . . . ....16
2.1 Introduction ............................................................................ 17
2.2 Experimental methods ................................................................ 18
2.3 Experimental results .................................................................. 25
2.4 Discussions ............................................................................. 48
2.5 Conclusions ............................................................................. 54
2.6 References .............................................................................. 55

Chapter 3

IMPACT PERFORATION RESISTANCE OF LAMINATED AND ASSEMBLED

COMPOSITE PLATES ............................................................................ 59
3.1 Introduction ............................................................................. 60
3.2 Experimental methods ................................................................ 62
3.3 Results and discussions ............................................................... 68
3.4 Conclusions ............................................................................. 86
3.5 References .............................................................................. 88

Chapter 4

TESTING AND SIMULATION OF LAMINATED COMPOSITES SUBJECTED TO
IMPACT LOADING ...................................................................... 90
4. 1 .Introduction ............................................................................. 91
4.2 Instrumented Impact Testing ......................................................... 94
4.3 Computational Technique ............................................................. 94
4.4 Experimental Results .................................................................. 97
4.5 Computational Results ............................................................... 106
4.6 Conclusion ............................................................................ 107
4.7 References ............................................................................. 111

Chapter 5

SIMULATION OF COMPOSITE LAMINATES UNDER LOW-VELOCITY IMPACT

WITH CONTINUUM-BASED FINITE ELEMENT METHOD .......................... 113
5.1 Introduction ........................................................................... 113

vi

 

5 .2 Experimental Technique ............................................................. 1 15
5.3 Computational Scheme .............................................................. 1 18
5.4 Results and Discussions ............................................................ 130
5.5 Conclusions ............................................................................ 142
5.6 References ............................................................................. 143
Chapter 6
CONCLUSIONS AND RECOMMENDATIONS ............................................ 145
6.1 Conclusions ............................................................................ 145
6.2 Recommendations ..................................................................... 150
Appendices .......................................................................................... 1 52
Appendix A .......................................................................................... 153
Appendix B .......................................................................................... 165
Appendix C .......................................................................................... 176

vii

 

*4- "Am“ . . -

LIST OF TABLES

Table 2.1. Summary of peak forces and impact energy levels at penetration for

composite laminates of various dimensions and thicknesses subjected

to impact and indentation loads ...................................................... 45
Table 2.2. Ratios of specimen dimensions, thickness, and impactor diameter .............. 50

Table 3.1: Impact bending stiffnesses, penetration thresholds and perforation
thresholds of one-laminate and two-laminate composite plates .................. 78

Table 4.1 Experimental results of impact parameters ........................................... 96

viii

LIST OF FIGURES

Figure 2.1 - Schematic diagram of impact testing machine .................................... 19
Figure 2.2 - Schematic diagram of NASA compression after impact testing ﬁxture ...... 24
Figure 2.3 - Force histories at three impact velocities .......................................... 27
Figure 2.4 - Relations between peak force and impact energy for composite laminates
with various dimensions and thicknesses .......................................... 28
Figure 2.5 - Relations between contact duration and impact energy for composite
laminates with various dimensions and thicknesses ............................. 29
Figure 2.6 - Relations between absorbed energy and impact energy for composite
laminates with various dimensions and thicknesses ............................. 30
Figure 2.7 - Relations between residual compressive stiffness and impact energy for
composite laminates with various dimensions and thicknesses ............... 33
Figure 2.8 - Relations between residual compressive maximum force and impact energy
for composite laminates with various dimensions and thicknesses. . . . . . . . ....34

Figure 2.9 - Relations between residual compressive absorbed energy and impact energy

for composite laminates with various dimensions and thicknesses. . . . . . .35
Figure 2.10 - Relations between impact characteristics and laminate thickness ............ 42
Figure 2.11 - Typical load-displacement relation from indentation testing ................. 45

Figure 2.12 - Relations between mechanical properties degradation and hole size for

composite laminates with open holes ............................................ 46
Figure 3. l - Schematic diagram of impact testing machine ................................... 64
Figure 3. 2 - Schematic diagrams of some riveting and stitching patterns .................. 66
Figure 3. 3 - F orce-deﬂection curves of eighteen 1F composite plates ...................... 70

Figure 3. 4 - The whole enegy proﬁle of 1F case based on force-deﬂection curves given

in Figure 3.3 .......................................................................... 71
Figure 3. 5 - Relation between energy thresholds and impact bending stiffness of various

two-laminate composit plates ...................................................... 80
Figure 3. 6 — The whole energy proﬁles of three laminated composite plates .............. 81
Figure 3. 7 - The whole energy proﬁles of four assembled composite plates ............... 82

ix

 

Figure 3. 8 - The relation between perforation threshold and thickness for both laminated

and assembled composite plates ................................................... 85
Figure 4.1 - Schematic diagram of impact testing setup ...................................... 93
Figure 4.2 - Force-displacement relations for laminates with various thicknesses. . . . . ....98
Figure 4.3 - Force-displacement relations for laminates with various ﬁber angles ........ 99

Figure 4.4 - F orce-displacement relations for laminates subjected to various impact
Velocities ............................................................................ 100
Figure 4.5 - Comparison of force history between ref. [4.7] and LS-DYNA3D .......... 101
Figure 4.6 - Comparison of displacement history between ref. [4.7] and
LS-DYNA3D ........................................................................ 102
Figure 4.7 - Force history of 15-layer laminate with ﬁxed boundary condition and
subjected to 1m/s impact ........................................................... 103
Figure 4.8 - Displacement history of 15-layer laminate with ﬁxed boundary condition and
subjected to lm/s impact ........................................................... 104
Figure 4.9 — Velocity history of lS-layer laminate with ﬁxed boundary condition and
subjected to lm/s impact .......................................................... 105
Figure 4.10 - Force histories of 9-layer laminate subjected to lm/s impact and simulated
by various failure criteria ........................................................ 108
Figure 4.11 - Force histories of 9-layer laminate subjected to 2.8m/s impact and
simulated by various failure criteria ............................................ 109
Figure 4.12 - Force histories of 9-layer laminate subjected to 3.5m/s impact and

simulated by various failure criteria ............................................ 110

Figure 5.1 - Schematic diagram of impact testing setup ..................................... 116
Figure 5.2(a) - Force history of 15-layer laminate with ﬁxed boundary

condition and subjected to 1m/s impact ...................................... 132
Figure 5.2(b) - Displacement history of 15-1ayer laminate with ﬁxed boundary

condition and subjected to lm/s impact ...................................... 133
Figure 5.3(a) - Force histories of 9-layer laminate subjected to lm/s impact and simulated

by various failure criteria ....................................................... 134

Figure 5.3(b) - Force histories of 9-layer laminate subjected to 2.8m/s impact and

simulated by various failure criteria ........................................... 135
Figure 5.4 - Force Vs. Displacement for [05/905/05] at V0=1.12 m/s ..................... 136
Figure 5.5 - Force vs. displacement for different impact velocities on [05/905/05]. . 137
Figure 5.6 - Force vs. displacement for different laminate thicknesses .................... 138
Figure 5.7 - Force vs. displacement for different stacking sequences ...................... 139
Figure 5.8 - Force vs. displacement for different ﬁber angles ............................... 10

xi

 

Chapter 1

INTRODUCTION

1.1 Literature review

Because of their well-known advantages compared to traditional materials,
laminated composite materials are extensively used in industry. Superior strength-to-
weight ratio, high speciﬁc modulus, tailored ability for a speciﬁc application, and
environmental stability of laminated composites is very attractive to the aerospace
industry and other industrial applications. However, under impact loading, which is
induced by impacts during manufacture, normal operations, or maintenance in the life of
the structure, laminated composites are very susceptible. Even at low velocity impact, the
impact-induced damage such as matrix cracking, ﬁber breakage and delamination, can be
very signiﬁcant, and usually invisible to the naked eye. The impact-induced damage can
cause serious structural degradation. Therefore, impact is an important subject in
laminated composite analysis and very critical to industry application.

In order to develop improved composite materials and composite structural design
method for the industry, it is essential to obtain a basic understanding of the impact
phenomenon, damage mechanism, residual properties after impact, size effects, and
assembly effects. Further development of computational simulation methods for

laminated composite structures accounting for impact is a necessary step.

For this purpose, many experimental and analytical studies were performed and
can be found in the review articles by Abrate [1.1-1.2]. In impact dynamics, the main
concern is the characteristics of impact response, which include impact force history,
impact displacement history, impact duration and impact energy absorption, etc.
Determining the contact force history has been done experimentally (Delfosse [1.11],
inhashi [1.24], and Chang and Sun [l.14]). Various mathematical models such as
spring-mass models (Alderson [1.12], Teti [1.23], and Bucinell [113]), energy balance
model (Moon [1.18], Poe [1.19-1.2l], Chuancho [1.16]), and beam, plate and shell
models (Christoforou and Swanson [1.15], and Gu and Legacy [1.17], and Prasad et al
[122]) have been developed. Since impact induced damage was not accounted for in
these models, generally, the practical effectiveness is limited. Tan and Sun [1.5] veriﬁed
that an indentation law based on a quasi-static test could be used to investigate low-
velocity impact. This indentation law established a mathematical relationship between the
contact force and the indentation for elastic contact. But, to determine the indentation
parameters for a set of composite material and impactor, an indentation test was required.
Although the indentation law could be integrated into computational scheme for various
studies (Wang [1.58], Lin [1.59], Chen [1.6], Sun [1.7], and Lee [160]), a new
characterization of indentation parameters was needed each time the material or geometry
of composite or impactor was altered. As a consequence, for investigating impact
response of laminated composites, a computational technique free of experiment-
dependent parameters might prove to be more efﬁcient.

In some previous studies, Liu et a1. [1.3-1.4] concluded that matrix cracking and

delamination were the major damage modes in laminated composites subjected to low-

velocity impact. A correlation between bending stiffness mismatch and delamination size
was established. This relation successfully used for phenomenological explanations of
delamination size, location, and orientation in impacted composite laminates. But, to
establish an accurate and efﬁcient quantitative simulation, further understanding based on
experimental testing and analysis is needed.

In practical application, in order to reduce the expense associated with impact
testing of large prototype structures, small coupons are usually used in laboratories to
characterize impact response. Then, the results are used for practical structure design.
However, even made of identical materials, small coupons and practical large structures
do not always behave the same way. This difference due to size change is referred to as
size effects [1.30]. Some investigators [1.25-1.29] have concluded that size effects should
be carefully examined in material characterizations and structural design.

Scaling laws are generally used to relate small coupon impact tests to practical
large structural impacts. Based on dimensional analysis, Monrton [1.31] scaled both
composite laminates and impact loading and veriﬁed the feasibility of using analytical
scaling laws for predicting undamaged behavior for carbon-epoxy laminated beams. With
a different approach, Qian et al. [1.32] also veriﬁed that analytical scaling laws could
accurately describe the undamaged response to impact. Sankar [1.33] presented semi-
empirical formulae for predicting impact characteristics such as peak contact force,
contact duration, and peak strain on back surface. But no damage effects were considered
in his studies. Therefore, in order to understand the size effects completely for laminated
composites under impact loading, a whole-range investigation is needed ranging from

elastic impact, subperforation impact, to perforation impact.

The size effects for laminated composites consist of in-plane dimensional effects
and thickness effects. It is believed that, because of the population of the thin laminates,
the in—plane dimensional effect gained more attention than thickness effect. As
technologies of composite manufacturing advance, more and more thick-section
laminated composites are used in non-aerospace industries. For example, thick-section
laminated composites have been proved to be feasible designs for submarine hull and
armored vehicle bodies. This implies that thickness effects may play an important role to
size effects in laminated composites subjected to impact loading. Accordingly, a further
study of thickness effects is necessary.

Impact analysis of laminated composite structures includes impact resistance
analysis and impact tolerance analysis. Impact resistance is concerned with understanding
how impact damage is initiated, developed and is helpful to ﬁnding methods to prevent or
resist the impact damage, while damage tolerance is concerned with the effect of impact
damage on the mechanical properties of laminated composite structures. Demuts [1.61]
described Air Force draft requirements for damage tolerance under low-velocity impact.
The minimum design strength after impact is speciﬁed. Many experimental studies [1 .36—
l.38] were performed to determine the residual strength of laminated composites under
tension, compression, and bending.

Since under in-plane compression, most serious strength reductions are shown,
the Compression After Impact (CAI) test [1.39] is mostly used to determine the residual
strength of impact damaged composite laminates. Therefore, to characterize the residual

properties of impacted composite laminates, CAI test always is the ﬁrst choice.

As mentioned before, more and more thick-section composites are used for
heavy-duty structures. Studies on thick laminates have gained much attention. These
studies concluded that thick composite laminates behave quite differently from their thin
counterparts, and thickness - as opposed to in-plane dimensions - has much greater
inﬂuence on impact resistance of laminated composites under impact loading [1.45-1.47].

However as composite laminates become thicker, the manufacturing cost for
high-quality composite laminates may become unaffordable. Because, in order to achieve
uniform curing and thus uniform properties through the thickness of thick composite
laminates, expensive microwave curing process is required (Wei [150]). Therefore, to
meet the design requirement for high quality and reduce the manufacturing costs,
assembled composite plates, which are organized by assembling multiple thin composite
laminates together [l.51-1.52], may be considered as alternatives for thick laminated
composite plates. A further study regarding the impact resistance and impact tolerance of
assembled composite laminates is needed.

Experimental studies on composite laminates subjected to impact loading always
come ﬁrst rather than analytical or numerical simulations, because successful simulation
must be based on the correct understanding and accumulated knowledge of experimental
studies. In order to aid industrial applications with laminated composite structures that
account for impact effects, computational simulation or prediction is a helpful tool for
engineering design.

As indicated by Abrate [1.1, 1.2], the development of models for predicting
damage and residual properties for composite laminates under impact loading is

becoming imperative. Since delamination represents a major component of damage at

low-velocity impact, the prediction and simulation of delamination during impact should
be the primary step. As many investigators concluded, the initiation of delamination
always follows the matrix crack close to ply interfaces (Liu [1.57], Wang and Chang
[1.66], and Choi Downs [1.67], etc), and the transverse stress distribution at ply interfaces
determines the onset and development of delamination.

But difﬁculties associated with prediction and simulation of delamination are due
to the complexity of transverse stress distribution in the contact area. Therefore, an
accurate stress distribution on the ply interface is essential for successful prediction of
delamination.

Finite Element method is proved to be a valuable CAD tool for engineering
application. A shell element for static analysis of laminated composites under large
deformation was developed by Lee [1.65]. This element based on the Generalized Zigzag
Theory [1.62] was demonstrated being an accurate and efﬁcient element for static
analysis of laminated composites. The interlaminar stress predicted from this element is
veriﬁed to closely agree with exact solutions. Apparently, a ﬁirther development on this
element for dynamics analysis and impact analysis is logical for prediction and
simulation of laminated composites under impact loading.

Linear Shear Slip Theory [1.63-1.64] was proved efﬁcient for simulation of
delamination. With the combination of Generalized Zigzag Theory and Linear Shear Slip
Theory, a new shell element for prediction and simulation of delamination of laminated

composites subjected impact loading may be proved as a valuable tool.

1.2 Objectives

Accordingly, the objectives of this dissertation are listed below:

1. The ﬁrst objective is to perform a whole-range investigation ranging from
elastic impact, subperforation impact, to perforation impact, to examine in—plane
dimension effects and thickness effects, and to determine residual strength after impact
with CAI test. This investigation will lead to further understanding of the response of
composite laminates under low-velocity impact and to development of an accurate and
efﬁcient quantitative simulation.

2. The second objective is to investigate the impact resistance of assembled
composite plates, as alternatives to thick laminated composites, and to examine various
joining techniques. This investigation will beneﬁt practical engineering applications with
thick-section composite laminates.

3. The third objective is to evaluate available computational tools for
analysis and simulation of composite laminates under low-velocity impact. By comparing
computational results with test results, the computational tool can be examined.

4. Based on the knowledge from the experimental studies and the evaluation
of computer simulation, to further improve the prediction and simulation of delamination
of composite laminates under impact loading, the last objective is to develop a shell
element for impact analysis of laminated composites.

1.3 Organization of this thesis

With the above objectives, this dissertation is divided into the following chapters.

Chapter 2 [1.49] starts with the ﬁrst objective, experimental studies regarding the size

effects are presented. Delamination effects on material property-degradation of impact

laminates are well recognized. A detailed discussion, and a simple explanation based on
the bending rigidity is presented to experimental results. Thickness effects to impact
resistance and tolerance is greatly signiﬁcant to in-plane dimension effects. As an effort
to meet design requirements and reduce manufacturing cost, Chapter 3 [1.69] describes
various joining techniques including mechanical riveting, adhesive bonding, stitching and
their combination for assembling multiple thin composite laminates. For evaluation of
computational tools, a series of impact testing regarding thickness effect, ﬁber angle
effect, and velocity effect is described in Chapter 4 [1.68], and comparison with computer
simulation is made. LS-DYNA3D, a commonly used computer code for impact analysis,
is evaluated. Based on Lee’s static shell element, Chapter 5 presents a continuous effort
for the development of a shell element for impact analysis, delamination prediction, and
impact simulation of laminated composites. Also, some numerical studies against
experimental results are given. Finally, in the last chapter, conclusions and

recommendations are presented.

1.4 References

l.l. Abrate, 3., “Impact on Laminated Composite Materials,” Applied Mechanics
Review, 44(4), 155-190, 1991.

1.2. Abrate, S., “Impact on Laminated Composites: Recent Advances,” Applied
Mechanics Review, 47(11), 517-544, 1994.

1.3. Liu, D. and Malvern, L.E., “Matrix Cracking in Impacted Glass/Epoxy Plates,”

Journal of Composite Materials, 21(7), 594-609, 1987.

1.4. Liu, Dahsin, “Impact-induced Delamination - A View of Material Property
Mismatching,” J. Composite Materials, 22 (7), 674-691, 1988.

1.5. Tan, TM. and Sun, C.T., “Use of Statical Indentation Laws in the Impact Analysis
of Laminated Composite Plates,” J. Applied Mechanics, 52, 6—12, 1985.

1.6. Chen, J .K. and Sun, C.T., “Dynamic Large Deﬂection Response of Composite
Laminates Subjected to Impact,” Composite Structures, 4, 59-73, 1985.

1.7. Sun, CT. and Chen, J .K., “On the Impact of Initially Stressed Composite
Laminates,” J. Composite Materials, 19(6), 490-504, 1985.

1.8. Wu, E. and Yen, C., “The Contact Behavior Plates and Rigid Spheres,” J. Applied
Mechanics, 61, 60-66, 1994.

1.9. Tsai, SW. and Wu, E.M., "A General Theory of Strength for Anisotropic
Materials,” Journal of Composite Materials, 5(1), 58-80, 1971.

1.10. Chang, F.-K. and Chang, K.-Y., “Post-Failure Analysis of Bolted Composite Joints
in Tension or Shear-Out Mode Failure,” Journal of Composite Materials, 21(5), 1987.
1.11. Delfosse D, Pageau G, Bennettt A, and Pousatrip A, "Instrumented impact testing at
high velocities," J Composite Tech Res 15(1), 38-45, 1993

1.12. Alderson KL and Evans KE, "Dynamic Analysis of ﬁlament wound pipes
undergoing low velocity transverse impact," Composites Sci and Tech 45, 17-22, 1992
1.13. Bucinell RB, Nuismer RJ, and Koury JL, "Response of composite plates to quasi-
static impact events, ASTM STP 1110,TK O'Brien, ed, 528-549, 1991

1.14. Chang C and Sun CT, "Determining transverse impact force on a composite

laminate by signal deconvolution," Exp Mech, 414-419, Dec., 1989

1.15. Christoforou AP and Swanson SR, "Analysis of impact response in composite
plates," Int J Solids Struct 2792), 161-170, 1991

1.16. Chuanchao Z and Kaida Z, "the investigation of the behavior of laminates subject
to low velocity impact," Proc 8th Int Conf on Composite Mater. (ICCM/ 8), Honlulu, 1991
1.17. Guy TA and Lagace PA, "Compressive residual strength of graphite-epoxy
laminates after impact," Proc of 9th DoD/NASA/FAA Conf on Fibrous Composites in
Struct Des, DOT/FAA/CT-95-25, 253-274, 1992

1.18. Moon D and Shively JH, "Towards a uniﬁed method of causing impact damage in
thick laminated composite," Proc 35th Int SAMPE Symp and Exhibition, Anaheim CA,
April 2-5, 35, 1466-1478, 1990

1.19. Poe CC, "Summary of a study to determine low-velocity impact damage and
residual tension strength for a thick graphite/epoxy motor case," Proc 17th Congress
ICAS, Stockholm, Sweden, Sept 9-14, 1, 994-1004, 1990

1.20. Poe CC, "Relevance of impactor shape to nonvisible damage and residual tensile
strength off a thick praphite/epoxy laminate," ASTM STP 1110, TK O'Brien, ed, 1991
1.21. Poe CC, "simulated impact damage in thick graphite/epoxy laminate using
spherical indenters,” J Reinf Plastics and Composites 10(3), 293-307, 1991

1.22. Prasad CB, Ambur DR, and Stames JH, "Response of laminated composite plates
to low—speed impact by airgun propelled and dropped weight impactors," 34th AIAA
strut, Structural Dyn and Mat Conf, La Jolla CA, Pt2, 887-900, 1993

1.23. Teti R, Langella F, Crivelli Visconti I, and Caprino G, "Impact response of carbon
cloth reinforced composites, Proc 5th Int Conf Composite Mat, ICCM-V, San Diego CA,

July l9—Aug. 1, 373-381, 1985

10

1.24. inhashi S, "An intelligent method to determine the mechanical properties of
composites under impact loading," Composite Struc 23, 149-163, 1993

1.25. Workshop on scaling effects in composite materials and structures, NASA Langley
Research Center, Hampton, VA, November 15-16, 1993, Report No. 94-6, Institute for
Mechanics and Materials, University of California, San Diego, CA.

1.26. Carpinteri, A., ed., Size-scale effects in the failure mechanisms of materials and
structures, E & FN Spon, 1994.

1.27. SE. Batdorf, Note on composite size effect, J. Composite Technology and
Research, 45, 35— 37, 1989.

1.28. I. Shahid, H.-T. Sun, and F .-K. Chang, Predicting scaling effect on the notched
strength of prepreg and ﬁber tow-placed laminated composites, J. Composite Materials,
29(8), 1063-1095, 1995.

1.29. GB. Harris and DE. Morris, Effects of laminate thickness and specimen
conﬁguration on the fracture of laminated composites, Composite Materials Testing and
Design, ASTM STP 893, 177-195, 1986.

1.30. Y. D. S. Rajapakse, ed., Mechanics of Thick Composites, AMD-Vol. 162, The
American Society of Mechanical Engineers, 1993.

1.31. J. Morton, Scaling of impact-loaded carbon-ﬁber composites, AIAA Journal, 26(8),
989-994, 1988.

1.32. Y. Qian, S.R. Swanson, R.J. Nuismer, and RB. Bucinell, An experimental study of
scaling rules for impact damage in ﬁber composites, J. Composite Materials, 24(5), 1990.
1.33. B.V. Sankar, Scaling of low-velocity impact for symmetric composite laminates, J.

Reinforced Plastics and Composites, 11, 297-305, 1992.

11

1.34. J .M. Whitney and R.J. Nuismer, Stress fracture criteria for laminated composites
containing stress concentrations, J. Composite Materials, 8(3), 253-265, 1974.

1.35. J. Awerbuch and M. Madhukar, Notched strength of composite laminates
predictions and experiments - a review, J. Reinforced Plastics and Composites, 4, 1985.
1.36. W.J. Cantwell, P.T. Curtis, and J. Morton, Low velocity impact damage tolerance in
CFRP laminates containing woven and non-woven layers, Composites, 14, 1983.

1.37. E.G. Guynn and T.K. O’Brien, The inﬂuence of lay-up and thickness on composite
impact damage and compression strength, proc. 26th AIAA SDM Conference, Orlando,
Florida, 187- 196, 1985.

1.38. A.A. Baker, R. Jones, and R.J. Callinan, Damage tolerance of graphite/epoxy
composites, Composite Structures, 4, 15-44, 1985.

1.39. Standard Tests for Toughened Resin Composites, NASA RP 1092, 1983.

1.40. Liu, Dahsin, Impact-induced delamination - a view of material property
mismatching, J. Composite Materials, 22(7), 674-691, 1988.

1.41. E. Wu, E.-Z. Tsai, and Y.-C. Chen, Penetration into glass epoxy composite
laminates,”J. Composite Materials, 28, 1783-1802, 1994.

1.42. C.T. Sun and S.V. Potti, High velocity impact and penetration of composite
laminates, proc. 9th Int. Conference on Composite Materials, 4, 157-165, 1993.

1.43. W. Goldsmity, C.K.H. Dharan and H. Chang, Quasi-static and ballistic perforation
of carbon ﬁber laminates, Int. J. Solid Structures, 32, 89-103, 1995.

1.44. S.W.R. Lee and CT. Sun, Dynamic penetration of graphite/epoxy laminates

impacted by a blunt-ended projectile, Composite Science and Technology, 49, 1993.

12

1.45. Y. D. S. Rajapakse, ed., Mechanics of Thick Composites, AMD-Vol. 162, The
American Society of Mechanical Engineers, 1993.

1.46. J. Morton, “Scaling of impact-loaded carbon-ﬁber composites,” AIAA Journal,
26(8), 989-994, 1988.

1.47. Y. Qian, S.R. Swanson, R.J. Nuismer, and RB. Bucinell, “An experimental study
of scaling rules for impact damage in ﬁber composites,” J. Composite Mat, 24(5), 1990.
1.48. B.V. Sankar, “Scaling of low-velocity impact for symmetric composite laminates,”
J. Reinforced Plastics and Composites, 11, 297-305, 1992.

1.49. D. Liu, B. B. Raju, and X. Dang, “Size effects on impact response of composite
laminates,” International Journal of Impact Engineering, Vol. 21, No. 10, 1998

1.50. J. Wei, M.C. Hawley, J. DeLong, and M. DeMeuse, "Comparison of microwave
and thermal cure of epoxy resins," Polymer Engineering and Science, 33(17), 1993.

1.51. C. Lee and D. Liu, ”Tensile strength of stitching joint in woven glass fabrics,” J.
Engineering Materials and Technology, 112 (2), 125-130, 1990.

1.52. D. Liu, “Delamination resistance in stitched and unstitched composite plates
subjected to impact loading,” Reinforced Plastics and Composites, 9 (1), 59-69, 1990.
1.53. E. Wu, E.-Z. Tsai, and Y.-C. Chen, “Penetration into glass epoxy composite
laminates,” J. Composite Materials, 28, 1783-1802, 1994.

1.54. C.T. Sun and S.V. Potti, “High velocity impact and penetration of composite
laminates,” proc. 9th Int. Conference on Composite Materials, 4, 157-165, 1993.

1.55. W. Goldsmith, C.K.H. Dharan and H. Chang, “Quasi-static and ballistic perforation

of carbon ﬁber laminates,” Int. J. Solid Structures, 32, 89-103, 1995.

13

1.56. S.W.R. Lee and CT. Sun, “Dynamic penetration of graphite/epoxy laminates
impacted by a blunt-ended projectile”, Composite Science and Technology, 49, 1993.
1.57. D. Liu, “Impact-induced delamination - a view of material property mismatching,”
J. Composite Materials, 22(7), 674-691, 1988.

1.58. Wang CY and yew CH, "Impact damage in composite laminates," Comput Struct
37(6), 967-982, 1990

1.59. Lin HI and Lee YJ, "use of statical indentation laws in the impact analysis of
composite laminated plates and shells," J Appl Mech 57(3), 787-789, 1990

1.60. Lee LJ, Pai CK, and Shiau LC, "Contact behaviours of laminated composite thin
shells and a rigid ball," Proc 3rd European Conf on Composite Mat, Bordeaux, France,
March 0-23, 315-320, 1989

1.61. Demuts E, "Damage tolerance of composites," Proc Am Soc for Composites, 4th
Tech Conf, VPI, Blacksburg VA, Oct. 3-5, 425-433, 1990

1.62. Dahsin liu and Xiaoyu Li, "An overall view of laminate theories based on
displacement hypothesis", Journal of Composite Materials, 30( 14), 1996

1.63. Dahsin Liu, Lan Xu, and Xianqiang Lu, "Stress analysis of imperfect composite
laminates with an interlaminar bonding theory," International Journal for Numerical
Methods in Engineering, 37, 2819-2839, 1994

1.64. Xianqiang Lu and Dahsin liu, "Assessment of interlayer shear slip theory for
delamination modeling," Composite Materials: Fatigue and Fracture, 4th Vol, ASTM
STP 1156, 218-235, 1993

1.65. Chienhom Lee, "A continuum-based shell element for laminated composites

under large deformation", Ph. D dissertation, Michigan State University, 1998

14

1.66. Wang HS and Chang FK, "Curvature effect on low-velocity impact damage in
laminated composites," Proc 8th Int Conf on Composite Mat (ICCM/8), Honolulu, July
15-19, 28.K.1-11,1991

1.67. Choi HY, Downs RJ, and Chang FK, "A new aproach toward understanding
damage mechanisms and mechanics of laminated composites due to low-velocity impact:
Part I - Experiments," J Composite Mat 25, 992-1011, 1991

1.68. Liu,D. and Dang, X.,"Testing and simulation of laminated composites subjected
to impact loading," Composite materials: fatigue and Fracture, 7th Vol., ASTM STP
1330, pp.273-284, 1998

1.69. Dahsin Liu, Basavaraju B. Raju, and Xinglai Dang, "Impact perforation resistance

of laminated and assembled composite plates," Int J Impact Engng, to appear

15

Chapter 2
SIZE EFFECTS ON IMPACT RESPONSE OF COMPOSITE LAMINATES

SUMMARY

Delamination was known to be one of the most important damage modes in composite
laminates subjected to impact loading. In an effort to ﬁirther understand the impact
response of composite laminates, various degrees of impact ranging from subperforation
to perforation were introduced to glass/epoxy laminates through an instrumented drop-
weight impactor. In addition, composite laminates of various in-plane dimensions and
thicknesses were examined for in-plane dimensional and thickness effects, respectively.
Experimental results showed that in-plane dimensional effect was not as signiﬁcant as
thickness effect. The impacted composite laminates were then subjected to compression
after impact (CAI) tests for characterizations of residual mechanical properties.
Experimental results showed that perforation was the most important damage stage in
composite laminates subjected to impact loading since impact characteristics (peak force,
contact duration, and absorbed energy) and mechanical properties degradation (residual
compressive maximum force and residual compressive absorbed energy) of composite
laminates became stable once perforation took place. However, it was also found that
delamination played a very important role in the characterizations of mechanical
properties degradation. Since the impact response of composite laminates is due to plate
bending to some extent, bending analysis was used to explain the greater inﬂuence of
thickness effect to in-plane dimensional effect. It was also found that bending analysis

was feasible for interpretation of delamination in mechanical properties degradation.

16

2.1 Introduction

Small coupons are usually used in laboratories for material characterizations.
Results from small coupon tests are then used for large structural designs. However,
small coupons do not always behave the same way as the large structures made of
identical material. The difference of behaviors due to size change is usually called size
effects. Some investigations regarding the performance of composite materials and
structures at different sizes have been reported [2.1-2.5]. It has been concluded that size
effects should be carefully examined in material characterizations and structural designs.

The study of size effects on impact-loaded composite laminates were focused on
scaling laws and parameters. Morton [2.7] scaled both composite laminates and impact
loading and veriﬁed the feasibility of using analytical scaling laws for predicting
undamaged behavior. He also found that smaller specimens were always stronger than
larger ones. In an effort to understand the scaling laws governing impact-loaded
composite laminates, Qian, Swanson, Nuismer, and Bucinell [2.8] also veriﬁed that
analytical scaling laws could accurately describe the undamaged response to impact. In
addition, they concluded that the damage resulting from impact involved many
complicated factors and the delamination size was consistent with the size effect. Aiming
at simplifying the design procedures for sealing impact-loaded composite laminates,
Sankar [2.9] presented semi-empirical formulae for predicting impact characteristics such
as peak force, contact duration, and peak strain on, back surface. No response beyond

initial damage was investigated in his study.

17

In an effort to understand the size effects on the response of composite laminates
subjected to impact loading, a whole-range investigation ranging from elastic impact,
subperforation impact, to perforation impact, was performed. Both in-plane dimensional
effect and thickness effect were included. In order to assess the degree of damage and to
characterize the residual mechanical properties of impacted composite laminates,
compression after impact (CAI) tests were also applied to damaged composite laminates.
2.2 Experimental Methods
1. Impact Testing

In this study, impact tests were performed on a DYNATUP GRC 8250 impact
testing machine as shown in Figure 2.1. The impactor consists of three components: a
dropping crosshead, an impactor rod, and an impactor nose. The steel impactor rod has a
diameter of 12.5 mm (0.5 in) and is attached to the dropping crosshead. A force
transducer having a force capacity of 22.24 kN (5000 lbs) was mounted at the tip of the
impactor rod and encapsulated by a hemispherical nose. The total mass of the impactor
was 11.9 kg (26.06 lbs). For an impact velocity up to 4 m/s (13 ft/ 3), the impactor was
released from a chosen height up to 0.8 m (2.6 ﬂ) and dropped freely along the loading
frame. However, for an impact velocity higher than 4 m/s and up to 8 m/s, the impactor
was raised to the highest point, i.e. 0.8 m, and a pneumatic unit located at the top of

loading frame was used to provide an additional force to increase the impact velocity.

18

 

PNEUMATIC UNIT

LOADING FRAME

DROPPING HEAD

  
 

TIME TRIGGER

12.5 mm IMPACTOR ROD

HEM IS PHERICAL
IMPACI‘OR NOSE

STEEL HOLDER
SPECIMEN
STEEL HOLDER

SPECIMEN FRAME

SENSORS

SOLID BASE

CONCRETE FLOOR

Figure 2.1. Schematic diagram of impact testing machine

Cross-ply laminates made of 3M glass-epoxy composite were investigated in this
study. Two nominal thicknesses with averages of 2.24 mm and 6.69 mm were used for
studying thickness effect. The former had a stacking sequence of [02/902/02/...]13 while
the latter [03/903/03/...]51. These two types of stacking sequences could be viewed as
thickness scaling of mixed mode since it combined a sublaminate mode which increased
the laminar number from 9 to 17 and a layer- level mode which changed the layer
thickness from 2-layer to 3-layer per each lamina. In this study, the 2.24 mm specimens
were called thin laminates while the 6.69 mm specimens were considered thick laminates.

In impact tests, composite specimens with dimensions of 250 mm x 175 mm were
placed between two steel plates. Each steel plate had a square opening in the center.
There were three different sizes of square openings for studying in-plane dimensional
effect, 125 mm, 84 mm, and 42 mm, resulting in three eﬂective impact zones with
dimensions of 125 mm x 125 mm, 84 mm x 84 mm, and 42 mm x 42 mm, respectively.
Although all specimens had the same dimensions, i.e. 250 mm x 175 mm, they were
named large, intermediate, and small laminates according to the three sizes of effective
impact zone. In the impact tests, each set of specimen and steel holders was bolted at four
corners to the specimen frame which was ﬁxed to a concrete ﬂoor as shown in Figure 2.1.

As the impactor dropped and approached a composite specimen, it triggered two
time sensors right before impact took place. The initial impact velocity could be
calculated from the time interval required for the trigger to travel between the two sensors
and the distance between them. Once impact began, the contact forces at many
consecutive instants were detected by the force transducer. The force history was

recorded in a computer. The maximum contact force was termed peak force while the

20

overall time duration of contact, contact duration. The corresponding velocity history of
the impactor could then be calculated from integrating the force history (after being
divided by the mass of impactor, i.e. 11.9 kg) and with the use of initial impact velocity.
Similarly, the corresponding displacement history of the impactor could be calculated
from integrating the velocity history.

Based on the force and displacement histories of the impactor, the energy history,
which represented the history of energy transferred from the impactor to composite,
could be calculated. In this study, the absorbed energy was termed as the amount of
energy transferred from the impactor to composite at the end of an impact event while the
impact energy was the kinetic energy of the impactor right before impact took place. The
peak force, contact duration, and absorbed energy, along with the histories of force,
displacement, and energy, were found to be the important characteristics of composite

laminates subjected to impact loading.

2. Damage Inspections

The macroscopic damage modes of impacted composite laminates includes
indentation, surface cracking, delamination, and perforation while the microscopic
damage includes ﬁber breakage, matrix cracking, ﬁber-matrix debonding, etc. Among the
macroscopic damage modes, perforation is the most apparent one when composite
laminates experience perforation. However, delamination is also an important damage
mode due to the fact that the residual mechanical properties of impacted composite
laminates are strongly dependent on delamination areas and their locations at laminate

interfaces. It is a time-consuming job, if not impossible, to identify delamination areas

21

and locations. Accordingly, in quantifying impact-induced damage, it is more realistic to
use an “equivalent damage size” to account for total damage than to count all details of
macroscopic and, microscopic damage. The “equivalent damage size” is usually obtained
from curve ﬁtting and used as damage index for invisible damage. For example, Whitney
and Nuismer [2.10] presented so-called characteristic lengths in their failure criteria for
circular hole and line crack studies. Many other studies could be found in a review paper
by Awerbuch and Madhukar [2.11].

In an effort to quantify the damage size of impacted composite laminates and its
role in mechanical properties degradation, both ultrasonic C-scan and high-intensity light
were used in this study. Besides perforation, delamination was identiﬁed as another
apparent damage mode. However, because of difﬁculties involved in determining
delamination area and location at each laminate interface, the determination of overall
delamination area in this study was based on projecting delamination areas at all
interfaces onto a single plane, i.e. enveloping the delamination areas at all interfaces. The
projected delamination area and the size of perforation were then compared to determine
their roles in mechanical properties degradation in composite laminates subjected to

impact loading.

3. Compression After Impact (CAI) Testing

Since perforation and delamination were the dominant damage modes in impacted
composite laminates and they strongly affected the compression performance of
composite laminates, compression after impact (CAI) was commonly used in

characterizing the residual mechanical properties of impacted composite laminates, e.g.

22

Ref. [2.12-2.14]. In this study, the NASA’s compression after impact test [2.15] ﬁxture
shown in Fig. 2 was used for determining the residual compressive stiffness and residual
compressive maximum force of impacted composite laminates. In addition, the residual
compressive absorbed energy was also determined.

In performing CAI tests, impacted composite laminates were cut into 250 mm x
125 mm. The specimens were snug-ﬁtted in the CA1 ﬁxture by knife edges along the two
longitudinal sides as depicted in Figure 2.2. The specimens were further clamped at the
top and bottom ends. Gaps of about 10 mm were left between the clamping ends and the
top and bottom ends of knife edges, allowing the specimens to shorten during
compression tests. A crosshead speed of 3.81 mm/min was chosen in compression tests.
The CAI tests worked well for most thick composite laminates (6.69 mm) except for a
couple of cases in which local damage due to crushing of laminate at clamping ends (top
or bottom) took place when the composite laminates had either very small or no impact-
induced damage. In order to avoid the local crushing damage, especially for those with
small or no impact damage, end tabs were bonded to composite specimens. For thin
specimens (2.24 mm), extra long end tabs which covered ahnost the entire length span of
the composite specimens except for the area with impact-induced damage were used.
However, the majority of thin specimens still experienced local crushing damage. It was
determined that thin composite specimens of 2.24 mm were not suitable for use with the

existing NASA’s CAI test ﬁxture.

23

COMPRESSIVE
* LOADING

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TOP END __
CLAMPER +OOOOOOO
I 10 mm
125 mmBETWEEN GAP FOR
A WEDGES SHORTENING
O O
SPECIMEN
THICKNESS 0 O
JUSTIFIER > _
O O
A. BOLT
O O‘
O O
COMPOSITE >
SPECIMEN O O
_ BASIC
O O “ FIXTUREFOR
SPECIMENS OF
0 O 250 mm X 125 mm
10mm
7— GAP FOR
C IT 1 SHORTENING
BOTTOM END
CLAMPER >0 O O O O O

 

 

 

 

Figure 2.2. Schematic diagram of NASA compression after impact testing ﬁxture

24

2.3 Experimental Results
1. Force History

Figure 2.3 Shows the force histories of thick composite laminates (125 mm x 125
mm x 6.69 mm) at three different impact velocity levels. At low velocity (1.06 m/S), the
impact damage in the composite laminate is insigniﬁcant and the composite laminate
responds elastically since the loading and unloading curves are ahnost symmetric with
respect to the peak force. AS impact velocity increases (3.80 m/s), the unloading curve
extends to the right, indicating that the composite laminate becomes softer as damage,
such as delamination, becomes more Signiﬁcant. When the impact velocity is high (5.46
m/S), the primary section of contact duration is dramatically reduced, Signifying the
occurrence of perforation in composite laminate. The secondary section of contact

duration is due to friction between the impactor and composite laminate after perforation.

2. Characteristics of Impact

By examining the force and energy histories, it was concluded that peak force,
contact duration, and absorbed energy were the most important characteristics of
composite laminates subjected to impact loading. Figures 2.4-2.6 Show the impact
characteristics of both thick (6.69 mm) and thin (2.24 mm) laminates. The solid circles
and open circles represent the characteristics for thick and thin composite laminates with
effective impact zones of 125 mm x 125 mm, respectively. In addition, the dashed lines
represent smooth curves of the solid circles while the solid lines represent smooth curves
of the Open circles in the diagrams. Figure 2.4 reveals that the peak forces increase as the

impact energy increases. However, the value becomes relatively stable for thick

25

laminates and reaches a constant value for thin laminates. The turning points from
nonlinear transition curves to stable or constant values are called critical points while the
corresponding impact energy levels are called the critical energy levels. Similar results
can also be seen from Fig. 6 for absorbed energy. In Figure 2.5, the contact duration is
presented as a function of the impact energy.

For both thick and thin laminates, the contact duration increases rapidly as the
impact energy increases. They reach individual peak points and sharply drop afterwards.
The impact energy levels correspond to the peak points are also termed the critical energy

levels.

3. Perforation Thresholds

AS mentioned above that perforation is the most apparent damage mode in
composite laminates subjected to impact loading. Hence, the perforation threshold is an
important parameter in characterizing the response of composite laminates subjected to
impact loading [2.17-2.20]. Since the peak force, contact duration, and absorbed energy
all reach critical points when perforation takes place and the impact energy levels to
cause perforation match with the corresponding critical energy levels, the perforation
threshold of composite laminates can be identiﬁed through the following four methods:

peak force, contact duration, absorbed energy and equal energy.

26

FORCE (1000 lbs)

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

 

peak force

q S
.. 5%,:
: ig ._.__. 1.06m/s (6.64J)
’3 g: -—--' 3.80m/s (85.6J)
,2“ 1A.; ----- 5.46m/s (176.3J)
‘ t“.
g a

2' in":
.= :32”... . .
" '~'=.~~\.'

I 3“ f.
p

 

0 a ‘;‘:\!\ﬁ\ ’\/‘
vv W ‘
0.01,- . x
0.0

2'.0 410A 6.0 8.0 10.0
TIME (msec)
l" - __ — contact duration "" " " "

F iure 2.3. Force histories at three impact velocity

27

 

12.0

L-— .M. _ _,

 

 

 

 

 

Tn . . ' . vat-91
I O - , E , - — ’ ’6
18- , ---"‘" «
0‘ ‘ """" .- .
4*". ‘ o
16*- , O O . .
I. c

14» r --- 0 125mm ~
2 ,0 A 84mm 6.69mm
5 I 42mm
gm” ‘1' ‘
1?. ‘ -—- 0125mm
510- l A 84m]2.24
.2 , D 142mm
8 o.
x 8" I ‘
“l I
O
a. I

I
6 ‘1 '
I
A o

4 U G 0 JV‘ #0 ..

2 ..

0' 1 1 i I l 4 1

O 50 100 150 200 250 300 350 400
Impact Energy (J)

Figure 2.4. Relations between peak force and impact energy for composite

laminates with various dimensions and thicknesses

28

30 I H I I I I

 

A “ " . 125m
A 84mm 6.69m
25 "- I 42mm "'

——-— 0125m
A 84mm 2.24m

U
20 42mm .-

Contact Duration (ms)

 

 

 

 

O 50 100 150 200 250 300 350 400
Impact Energy (J)

Figure 2.5. Relations between contact duration and impact energy for composite

laminates with various dimensions and thicknesses

29

 

 

 

 

 

 

20° 1 a I II 1 I 1
A /
.‘l ‘
180 b' [I "l
/' .
I _, ..
.- ‘ ______
.. .l __ _. _. — -— a- . .—
160 jl , f 0- - — 6
.’ /.
140 ~ I’ ‘
I
3 .r ,’ --- 0 125mm .
512°” / , A 84mm 6.69mmr
/' /
E ’3‘, I l42mm
b- / '1
g 100 f, __ 125mm
g I, / A 84mm 2.24m
g m r— . ‘ a 42mm "‘
/
. _..—.._ 450 line
/ A
60 ,.. / .-
G O O
I l 1 l l
150 200 250 300 350 400
Impact Energy (J)

Figure 2.6. Relations between absorbed energy and impact energy for composite

laminates with various dimensions and thicknesses

3O

3.1 Peak Force Method

In Figure 2.4, the peak forces of thick composite laminates reach a relatively
stable level, changing from a nonlinear transition curve to a straight line, around 16.2 kN
when the impact energy is about 106 J (78 ft-lb), i.e. the critical energy level. This critical
energy level was found to be Slightly lower than the level to cause perforation by
examining the impacted Specimens. For thin composite laminates, the critical energy
level is identiﬁed as 30 J (22 ft-lb) and is associated with a plateau of peak force of 4.2
kN. By examining the impacted specimens, it was found that this critical energy level
was Slightly higher than the energy level that caused perforation in thin composite
laminates. The plateau seems to indicate that there was a maximum contact force that a
thin composite laminate can sustain when it was subjected to impact loading and the

maximum contact force was what required to perforate the thin composite laminate.

3.2 Contact Duration Method

The second method to identify the perforation threshold was based on the contact
duration. Figure 2.5 shows the contact durations for thick and thin laminates at various
impact energy levels. For thick composite laminates, the critical energy level is around
120 J (88.5 ft-lb) while it is around 21.5 J (16 ft-lb) for thin composite laminates. The
former was very much the impact energy level to cause perforation since some thick
specimens were perforated and some were not when subjected to this impact energy
level. The latter was found to be slightly lower than that caused perforation since no thin

specimens were perforated under this impact energy level.

31

 

3.3 Absorbed Energy Method

Shown in Figure 2.6, the absorbed energy approaches a relatively stable level
around 150 J for thick laminates when impact energy level reaches 190 J (140 ft-lb), i.e.
the critical energy level. This impact energy level was much higher than that obtained
from the peak force analysis as given in section 3.1 and was conﬁrmed to greatly exceed
the impact energy level to cause perforation. It should be pointed out, however, that there
were relatively few data points located between 100 J and 200 J. It was believed that
insufﬁcient data points within the range were responsible for errors in generating a
smooth curve, and hence the inaccurate estimate. The estimate of perforation threshold
for 2.24 mm laminates is around 35 J (25.8 ft-lb) when the absorbed energy reaches a

constant level of 28.5 J. It was also higher than experimental observations.

3.4 Equal Energy Method

The fourth technique to identify the perforation threshold is based on comparison
between impact energy and absorbed energy. It was found that composite laminates
experienced perforation when these two energy levels became very close. In other words,
perforation seemed to take place when the kinetic energy of the impactor was almost
completely transferred to the composite laminate. Results based on this argument can be
seen in Figure 2.6. The 45° line which represents equality between impact energy and
absorbed energy goes through solid circles and an open circle, giving the perforation
threshold of 120 J (88.5 ft-lb) for thick laminates and 26 J (19.2 ﬁ-lb) for thin laminates.
These two results were found to best match with experimental observations for

perforation threshold among the four methods.

32

2000 1 1

 

 

Residual Compressive Stiffness (MPa)

 

 

 

800 - s
600 r- -
—" 0 125mm
-- - A 84m impact
400 - "" I 42mm _
0 125m
200 __ A 84m indent
D 42m 7
o l A l l l l 1
0 50 100 150 200 250 300 350 400

Impact Energy (J)

Figure 2.7. Relations between residual compressive stiffness and impact energy

for composite laminates with various dimensions and thicknesses

33

 

180

160

140

120

100

 

Residual Compressive Maximum Force (kN)

 

 

 

60 - ‘
40L —- . 125mm _
--.. A 84mm] impact
' ‘ -' I 42mm
20 ” 0125mm ‘
A 84m indent
[1 42m
0 J l 4 1 1 l
0 50 100 150 200 250 300 350 400

Impact Energy (J)

Figure 2.8. Relations between residual compressive maximum force and impact

energy for composite laminates with various dimensions and thicknesses

34

 

 

 

 

 

 

600 T I I ﬂ I I I

A i — 0 125mm

3 500 --- A 84mm] impact...
a; -'- I l02mm

L
.. “c’ - 0125mm

LIJ A 84mm]indent
'8 400 U loZImn _
.o

L-

o

U)

2

Q 300 T
.2

(I)

(I)

as

h

o.

E 200 -
o

O

B A-_Q_-__-__-_-__--_--_A
3

j 1

:3 100.— . . .1
0)

CL‘

0 l I 1 1 J l l
0 so 100 150 200 250 300 350 400
Impact Energy (J)

Figure 2.9. Relations between residual compressive absorbed energy and impact

energy for composite laminates with various dimensions and thicknesses

35

 

 

 

4. Residual Compressive Properties

In addition to nondestructive investigations, an effective way to characterize the
degree of impact-induced damage is to quantify the residual properties of composite
laminates which have been subjected to impact. It has been reported by many researchers
that compression after impact is an effective test for this purpose due to the fact that
delamination is an important damage mode in impacted composite laminates and
compressive properties of composite laminates are very sensitive to the Size and location
of delamination.

Figures 2.7-2.9 Show the residual compressive stifﬁress, residual compressive
maximum force, and residual compressive absorbed energy for thick composite laminates
based on CAI tests. The residual compressive st'iﬁ‘ness represents the slope of a force-
displacement curve obtained from CAI test; the residual compressive maximum force
represents the force to cause buckling, i.e. the peak force of the force-displacement
relation; while the residual compressive absorbed energy can be calculated from the area
under the force-displacement curve. The residual compressive stifﬁress decreases
gradually as the impact energy increases. However, both the residual compressive
maximum force and residual compressive absorbed energy drop rapidly from their initial
values and become constants when the impact energy levels exceed individual critical
levels. The critical energy levels were also identiﬁed to be closely related to the
perforation threshold. This result indicates that a maximum mechanical properties
degradation of composite laminates takes place at perforation. Once perforation takes
place, some residual compressive properties of composite laminates cannot be further

degraded. In other words, as far as the residual compressive maximum force and residual

36

compressive absorbed energy are concerned, perforation seems to be the most important
damage stage in composite laminates subjected to impact loading.

AS impact energy increases, the reduction of residual compressive stiffness is not
as dramatic as those of residual compressive maximum force and residual compressive
absorbed energy. This is believed to be related to the fact that there is no delamination-
induced local buckling involved in the measurement of compressive stiffness. In other
words, local buckling plays a very important role in the reductions of residual
compressive maximum force and residual compressive absorbed energy. Consequently,
the residual compressive maximum force and the residual compressive absorbed energy
are better than the residual compressive stiffness in presenting mechanical properties
degradation of composite laminates subjected to impact loading. For convenience of
discussions, the mechanical properties degradation will be referred to degradations of
residual compressive maximum force and residual compressive absorbed energy
hereafter.

As mentioned above, perforation took place when the residual compressive
maximum force and residual compressive absorbed energy became constants as the
impact energy approached critical values, changing from nonlinear curves to constants.
The impact energy levels corresponding to the constant values of residual compressive
maximum force and residual compressive absorbed energy, i.e. critical energy levels, for
thick-large (125 mm x 125 mm x 6.69 mm) composite laminates are 115 J and 135 J,
respectively. These two values are close to the impact energy level for perforation
threshold, i.e. 120 J, as given in section 3.4. Accordingly, besides the aforementioned

four methods, the studies of residual compressive maximum force and residual

37

 

 

compressive absorbed energy present two additional options for estimating the
perforation threshold.

The compression after impact testing was also performed for impacted thin
composite laminates. Local crushing damage close to the top and bottom clamping ends
occurred in many tests. Extra-long end tabs were used to reinforce the specimens’ ends to
prevent local crushing damage from happening. Unfortunately, among the very few
specimens which had no local crushing damage, only a couple of them Showed strong
interaction between impact-induced damage and compression-induced damage. Hence, it
was concluded that thin (2.24 mm) specimens are not suitable for CAI testing using

NASA’S test ﬁxture.

5. In-Plane Dimensional Effects

In addition to effective impact zone of 125 mm x 125 mm, other effective impact
zones of 84 mm x 84 mm and 42 mm x 42 mm, i.e. intermediate and small Specimens,
were investigated for in-plane dimensional effects. Experimental results regarding the
peak force are also shown in Figure 2.4. The solid triangles are for thick-intermediate (84
mm x 84 mm x 6.69 mm) laminates while the solid squares are for thick-small (42 mm x
42 mm x 6.69 mm) laminates. Although it can be concluded that the smaller the size of
composite laminate the higher the peak force, the difference of peak force for the three
effective impact zones is not as signiﬁcant as that for thickness difference. In fact, the
result that smaller specimens has higher peak forces coincides with the fact that the

smaller the laminate dimensions the stiffer the composite laminate. Similar results are

38

also found for thin (2.24 mm) composite laminates, represented by open triangles and
Open squares.

The effects of dimensions on the contact duration is also shown in Figure 2.5.
Although stiffer specimens are expected to have shorter contact durations than the softer
counterparts, the difference shown in Figure 2.5 is not conclusive. Figure 2.6 gives the
experimental results for the absorbed energy based on the three effective impact zones. It
can be seen that the smaller the laminate size the higher the absorbed energy for thick
composite laminates. Apparently, damage in smaller laminates is more serious than that
in larger laminates because the former have higher stiffnesses than the latter. However,
different results are found in the study for thin composite laminates.

In addition to the peak force, contact duration, and absorbed energy, the effects of
dimensions on residual compressive properties can also be found in Figures 2.7-2.9.
Although there are three effective impact zones, all the laminates have identical
dimensions, i.e. 250 mm x 125 mm, for CAI tests. However, the residual compressive
stiffness of thick-intermediate and thick-small composite laminates are smaller than that
of thick-large composite laminates. The differences for the residual compressive
maximum force and the residual compressive absorbed energy of the three effective
impact zones are not as high as that of the residual compressive stiffness. In addition, the
estimates of perforation thresholds for laminates with different effective impact zones

seem to be close.

39

 

 

 

6. Thickness Effects

As mentioned above, the impact response of composite laminates changes
signiﬁcantly as the laminate thickness increases. The ratio of peak force between the
thick-large and thin-large composite larrrinates is 4.01 (16.6 kN versus 4.14 kN). The
ratio of absorbed energy is 4.62 (120 J versus 26 J). Apparently, these two numbers are
higher than the thickness ratio, which is around 3 (6.69 mm versus 2.24 mm), exhibiting
a nonlinear proportion with respect to thickness or the existence of Size effect due to
thickness.

In an effort to understand the thickness effect, two additional tests were
performed. The ﬁrst test examined two thin laminates (2.24 mm each). They were put
together without an adhesive bonding between them. Together they were called a double-
layer system and were subjected to an impact force. The second test was similar to the
ﬁrst one but had three thin laminates, namely a triple-layer system. Since 26 J was found
to be the perforation threshold for a thin single-layer, double and triple energy levels
were applied to the double- and triple-layer systems, respectively. The double-layer and
triple-layer systems were found to be close to perforation. Experimental results for the
three systems along with a thick (125 mm x 125 mm x 6.69 mm) laminate subjected to
impact energy of 80 J are given in Figure 2.10.

Excellent linear relations were found among the single-, double-, and triple-layer
systems. However, the results from the thick composite laminate are not near the linear
regressions. Comparing to the triple-layer system, the peak force of the thick composite
laminate is considerably higher while the contact duration is considerably lower. These

results are believed to be attributed to higher stiffness of the thick composite laminate as

40

opposed to the triple-layer system. In fact, the bending stiffness of the thick composite
laminate is about nine times higher than the triple-layer system.

The investigations for multiple-layer systems provides some fundamental insights
into thickness effect. As shown in Figure 2.10, the impact energy levels for the triple-
layer system and the thick composite laminate are about the same, i.e. around 78 J. This
impact energy level is not too far away from perforating the triple-layer system, based on
experimental observation, but is certainly much lower than the 120 J required to perforate
the thick composite laminate. It is believed that the difference is attributed to the fact that
the bending rigidity of the thick composite laminate is much higher than that of the triple-
layer system, i.e. 9:1. This result concludes the superiority of impact resistance of a
single-layer thick composite laminate to a multiple-layer system, which has the same

thickness as the Single-layer thick composite laminate.

41

 

   

 

 

 

 

25 T j N I ‘00
C s
3
e “-
z 20 ” ‘\ B “ 80
.9 x ./
F4 \x -/
é ‘A / .
:3 \ / .
o ‘\ /
£- )1
2 is — /' ‘\ 4 .0
2: \g
8 ./ /.'
.15 ‘ e
”.2; 4 s
V 10 ” ‘ 0 33
m z
5.2) LL}
0 ./
rs. / 4
ﬂ CY ~‘ _. 0 peak force — _
u: , . thin
a, 5 -—— A contact duration multiple
’- 6 ...— C] impact energy la er 20
-..- O absorbed energy “ y
0 peak force a ,
A contact duration tihlclke "
I impact energy :8“:
0 Q absorbed energy J y r
l 1 l O
0 2.24 4.48 6.72 8.96

THICKNESS (mm)

Figure 2.10. Relations between impact characteristics and laminate thicknesses

42

7. Strain Rate Effects

A commonly raised issue in investigating impact dynamics is regarding the
difference and similarity Of the roles of impact velocity and impact energy. More
speciﬁcally, will composite laminates behave the same under constant impact energy but
different impact velocities? Since the impactor mass and diameter are kept constants in
the present investigation, impact energy is proportional to the second power of impact
velocity. Accordingly, the aforementioned issue cannot be addressed. However, it should
be recognized that, with the use of impact energy, both the peak force and absorbed
energy increase linearly and reach stable values, and the residual compressive maximum
force and the residual compressive absorbed energy decrease linearly and become
constants when perforation takes place. Thus. the advantage of using impact energy to
gain Simple, linear relationships is clear.

In an effort to ﬁirther understand the effect of impact velocity, or the strain rate
effect, quasi-static indentation tests were performed. In this study, thick-large composite
laminates were held by ﬁxtures similar to those used in impact tests. The specimen-
holders were then ﬁxed to the bottom grip of an Instron hydraulic testing machine and
loaded by a steel indenter held by the top grip. The indenter had a diameter of 12.5 mm
and a nose of hemispherical Shape. The indentation was performed at a crosshead speed
of 3.81 mm/min. The force history and displacement history were recorded by the load
cell and the movement of hydraulic actuator, respectively.

Figure 2.11 gives a typical load-displacement relation of the quasi-static
indentation. It starts with a linear relation until a local maximum was reached. It then

drops to a local minimum due to delamination. The overall maximum load was

43

eventually achieved before perforation takes place. A descending curve follows due to
the frictional force between the indenter and the composite laminate. Experimental results
regarding the residual mechanical properties are also shown in Figs. 7-9 for comparisons
with dynamic studies. The quasi-static results, which are represented by the half-open-
and-half-solid symbols, are lower than dynamic impact results in many cases. This seems
to explain the fact that composite laminates have higher stiffnesses and strengths at

higher strain rates.

44

 

 

Load kN
°.’
1/

 

 

 

I 1 m

it ' 4 e - is
Displacement mm

OJ

Figure 2.11. Typical load-displacement relation from indentation testing

Table 2.1. Summary of peak forces and impact energy levels at penetration for
composite laminates of various dimensions and thicknesses subjected to impact and

indentation loads

 

 

 

 

Dimensions
Large Intermediate Small
125mmx125 mm 84mmx84 mm 42mmx42mm
Thicknesses impact indent impact indent impact indent
Thick Peak force 16.6 8.6 17.5 8.7 19.8 8.9
(kN)
6.69 mm Impact energy 120 51 112 44 130 52
(1)
Thin Peak force 4.14 1.56 4.21 1.44 4.22 1.85
(kN)
2.24 mm Impact energy 26.0 10.9 25.2 11.9 21.7 9.1

(J)

 

45

 

.1
.1

 

§
0
CI 0
l

 

A

§§

STIFFNESS (MPa)

 

C)

 

 

PEAK FORCE (kN)

 

 

ABSORBED ENERGY (J)

 

 

 

 

0 L ‘ I 1
° 5 10 15 20 25 30 35
Hole Diameter (mm)

Figure 2.12. Relations between mechanical properties degradation and hole size

for composite laminates with open hole

46

Although the indentation tests provide some fundamental knowledge on strain
rate effect, the tests cannot be used to substitute impact tests. AS shown in Figures 2.7-
2.9, the level of indentation energy cannot be further raised once perforation takes place
while the level of impact energy can always be increased by increasing impact velocity.
Accordingly, the important feature that residual mechanical properties drop rapidly then
approach constant levels as obtained from impact tests cannot be reproduced from
indentation tests. Table 2.1 summarizes the peak forces and impact energy levels to cause
perforation for composite laminates with various dimensions and thicknesses. It can be
seen that there is a signiﬁcant difference between impact and indentation responses, not

to mention the difference in damage mode.

8. Open Hole Tests

Another study to understand the effects of damage mode on the compressive
response of impacted composite laminates was based on open hole tests. In this study,
tests were performed for thick-large composite laminates with holes of diameters ranging
from 11.33 mm (29/64 in) to 31.25 m (5/4 in). Figure 2.12 gives all three mechanical
properties. When comparing Figure 2.12 with Figure 2.7, the difference of residual
compressive stiffness between the ones with impact-induced damage and those with open
holes is small. This is due to the fact that the residual compressive stifﬁress is only
affected by the size of through-the-thickness opening, either impact-induced perforation
or artiﬁcial open hole. However, both the residual compressive maximum force and
residual compressive absorbed energy of composite laminates with a 12.5 mm hole,

shown in Figure 2.12, are much higher than those with impact-induced perforation

47

(which has a perforation opening of approximately 12.5 mm and large delamination
areas), given in Figurs 2.8 and 2.9. This result indicates that delamination also plays a
very important role in the compressive response of impacted composite laminates.
Another result which can be obtained from Figure 2.12 is the so-called equivalent
damage size. From Figures 2.8 and 2.9, the residual compressive maximum force and
residual compressive absorbed energy reach 100 kN and 120 J, respectively, when
perforation takes place. By extending the smooth lines in Figure 2.12, the hole Sizes
corresponding to 100 kN and 120 J, i.e. equivalent damage Sizes, can be obtained. They
are, however, much larger than the size of the impactor (12.5 mm) and even the specimen
width (i.e. 125 m) if linear regressions in Figure 2.12 are used. This result again
indicates the very important role of delamination in mechanical properties degradation

although it is not as apparent as perforation, sometimes even invisible to naked eyes.

2.4 Discussions
1. Size Effects

In this study, composite laminates of three effective impact zones and two
thicknesses were investigated. However, the distinctions between the large and small, and
the thick and thin laminates need to be further deﬁned. According to the Classical Plate
Theory, the deﬁnitions of thin and thick plates are tied to the ratio of in-plane dimension
to thickness, l/ h . Table 2.2 gives the ratios of l/ h for all composite laminates
investigated in this study. Since a ratio of 20 is usually considered as the minimum

requirement for being qualiﬁed as a thin plate, 6.69 mm laminates are considered as thick

48

 

plates. Although the thick-large (125 mm x 125 mm x 6.69 mm) Specimens and the thin-

small (42 mm x 42 mm x 2.24 mm) specimens have almost identical l/ h ratio, i.e. 18.7

and 18.8, respectively, experimental results between them were quite different. This
indicates that the impactor diameter is also an important parameter in impact study.

The ratios of specimen in-plane dimension to impactor diameter, l/d , and
specimen thickness to impactor diameter, h/d , are also Shown in Table 2.2 along with
the ratio of in-plane dimension to thickness l/ h . It can be seen from the ratios given in
Table 2.2 that l/ h displays the equal important roles of in-plane dimensional and
thickness effects, l/ d shows the sole important role of in-plane dimensional effect, while
h/d indicates the sole important role of thickness effect. Since experimental results Show

that thickness effect is much more Signiﬁcant than in-plane dimensional effect,

h/ d seems to be the most important ratio among the three ratios presented. Since the

impactor diameter is kept constant in this study, the thickness of composite laminates is

then the most important parameter in impact response.

49

 

Table 2.2. Ratios of specimen dimensions, thickness, and impactor diameter

 

Dimensions 1

 

 

l/II 125 mm 84 mm 42 mm h/d
6.69 mm 18.7 12.6 6.3 0.54
Thickness h
2.24 mm 55.8 37.5 18.8 0.18
[M 10 6.7 3.4

 

I is specimen dimension.
It is Specimen thickness.
d is impactor diameter, 12.5 mm.

50

2. Comparison between In-Plane Dimensional and Thickness Effects

The response of composite laminates to impact loading, to some extent, resembles
plate bending which is governed by bending rigidity. The bending rigidity is deﬁned as
where is Young’s modulus and is the second moment of area which can be expressed as
where is laminate dimensions and is thickness. In fact, the bending rigidity has been
successfully used in interpreting the potential of delamination in composite laminates
subjected to impact loading [2.16]. The defmition of the second moment of area Shows
that it is proportional to the third power of thickness while it is only the ﬁrst power of in-
plane dimension. Since the impact response of composite laminates as mentioned above
is more strongly affected by thickness than by in-plane dimension, the bending rigidity,
which is capable of discriminating between thickness and in-plane dimension, seems to
qualify itself as an important element, if not the element, of an analytical model for
perforation analysis.

The feasibility of utilizing bending rigidity in impact response analysis is also
well supported by experimental results. AS mentioned above, excellent linear relation of
impact response exists when the impacted composite laminate is changed from Single-
layer to double- layer, and to triple-layer system. However, it is also found that the ratios
of peak force and absorbed energy between thick and thin composite laminates are higher
than the thickness ratio, exhibiting a nonlinear proportion with respect to thickness. Both
of the results seem to provide a solid foundation of rationalizing the use of bending

rigidity for perforation analysis.

51

 

3. Damage Modes

Both the peak force and absorbed energy of impacted composite laminates reach
relatively stable levels once perforation takes place. In addition, the residual compressive
maximum force and residual compressive absorbed energy drop to constant values when
perforation takes place. All these seem to imply that the peak force and absorbed energy
should also become constant instead of just stabilized when perforation is reached.
Hence, more tests are required to ﬁrrther verify this hypothesis. However, whether the
impact response and mechanical properties degradation become constant or stabilized,
the most Signiﬁcant fact is that they both reach apparent turning points when perforation
takes place. Accordingly, as far as material response is concerned, perforation is the most
important damage stage in composite laminates subjected to impact loading.

As mentioned above, the damage modes in perforated composite laminates
includes macroscopic damage, such as indentation, surface cracking, delamination, and
perforation, and microscopic damage, such as ﬁber breakage, matrix cracking, ﬁber-
matrix debonding, etc. All individual damage modes play important roles in impact
response of composite laminates. However, based on open hole study, it can be
concluded that perforation alone causes a small portion of mechanical properties
degradation. It is delamination which is responsible for local buckling and hence
signiﬁcant mechanical properties degradation.

Perforation is easy to identify. It is about the size of impactor. However,
delamination area and location are very difﬁcult to measure, if not impossible. Since

delamination plays a very important role in impact response and mechanical properties

52

degradation, many nondestructive techniques have been developed to identify
delamination. In addition, it has been concluded that delamination cannot be represented
by an equivalent hole size. It then is important to consider the true geometry of

delamination, i.e. debonding between laminae, in delamination modeling.

4. Characteristics of Perforation and Delamination

The residual compressive stiffness is not strongly affected by delamination
because local buckling does not occur in the early stage of a CAI test. In fact, it is
dependent on a damage area combining the impactor and its surrounding area with
through—the-thickness damage. Therefore, it is concluded that the modeling of
compressive stifﬁress needs to be tied to the modeling of the through-the-thickness
perforation zone. And a linear relation between the perforation opening and residual
stiffness should be established.

The modeling of delamination can be approached from plate bending analysis. As
a composite laminate bends, high interlaminar shear stresses are formed, resulting in
delamination due to low interlaminar strengths. When the delaminated composite is
subjected to uniaxial compression, local bending—buckling can take place in individual
delaminated layers. Since bending rigidity decreases with the third power of thickness,
as composites delaminates, both compressive maximum force and compressive absorbed

energy degrade rapidly as delamination increases.

53

2.5 Conclusions

The following conclusions can be drawn from the investigations:

1. Once perforation takes place, both impact characteristics, such as peak force of
impact, impact-contact duration and, absorbed energy during impact, and mechanical
properties degradation, such as residual compressive maximum force and residual
compressive absorbed energy, reach turning points. Accordingly, these ﬁve parameters
can be used to identify the perforation thresholds of composite laminates. Since these
parameters are important elements of material response, perforation can be concluded as
the most important damage stage, as far as material response is concerned, in composite
laminates subjected to impact loading,

2. The study of Size effects on impact response of composite laminates Should be
divided into two categories: in-plane dimensional effect and thickness effect. Among the
ratios based on Specimen in-plane dimensions, specimen thickness and impactor
diameter, the ratio of specimen thickness to impactor diameter seems to best match with
the experimental results. Since the impactor diameter is kept constant in this study, the
thickness of composite laminates becomes the most important parameter in impact
response. Hence, thickness effect is much more signiﬁcant than in-plane dimensional
effect.

3. In rationalize the superiority of thickness effect to in-plane dimensional effect
on impact response, bending ri gidity should be considered as an important element for
perforation analysis since it is proportional to the third power of thickness while only the

ﬁrst power of in- plane dimension. Its capability of discriminating between thickness and

54

in-plane dimension seems to be consistent with the experimental results that thickness
effect is more Signiﬁcant than in- plane dimensional effect.

4. Although perforation is the most important damage stage, as far as material
response is concerned, in composite laminates subjected to impact loading, perforation
alone causes a small portion of mechanical properties degradation. It is delamination
which also plays an important role in impact energy absorption and mechanical properties
degradation since delamination has been identiﬁed as the other primary damage mode in
impacted composite laminates.

5. The utilization of bending rigidity for perforation analysis can also be extended
to delamination analysis. In fact bending rigidity has been successfully used in a previous
study for predicting the potential of delamination of composite laminates subjected to
impact loading. Its capability in interpreting the mechanical properties degradation is well
supported by the experimental results that both compressive maximum force and
compressive absorbed energy degrade rapidly when delamination exists while

compressive stifﬁress does not.

ACKNOWLEDGEMENTS

This investigation was supported by the Independent Laboratory in—house
Research (ILIR) program at the US. Army - Tank Automotive and Armaments
Command’s (TACOM) Research, Development & Engineering Center (TARDEC). The

authors wish to express their sincere thanks for this ﬁnancial support.

55

2.6 References

2.1. Workshop on scaling effects in composite materials and structures, NASA Langley

Research Center, Hampton, VA, November 15-16, 1993, Report No. 94-6, Institute for

Mechanics and Materials, University of California, San Diego, CA.

2.2. Carpinteri, A., ed., Size-scale effects in the failure mechanisms of materials and

structures, E & FN Spon, 1994.

2.3. SE. Batdorf, Note on composite size effect, J. Composite Technology and Research,

45, 35- 37, 1989.

2.4. I. Shahid, H.-T. Sun, and F.-K. Chang, Predicting scaling effect on the notched

strength of prepreg and ﬁber tow-placed laminated composites, J. Composite Materials,

29(8), 1063-1095, 1995.

2.5. GB. Harris and DE. Morris, Effects of laminate thickness and specimen

conﬁguration on the fracture of laminated composites, Composite Materials Testing and
Design, ASTM STP 893, 177-195, 1986.

2.6. Y. D. S. Rajapakse, ed., Mechanics of Thick Composites, AMD-Vol. 162, The
American Society of Mechanical Engineers, 1993.

2.7. J. Morton, Scaling of impact-loaded carbon—ﬁber composites, AIAA Journal, 26(8),
989-994, 1988.

2.8. Y. Qian, S.R. Swanson, R.J. Nuismer, and RB. Bucinell, An experimental study of

Scaling rules for impact damage in ﬁber composites, J. Composite Materials, 24(5), 559-

570, 1990.

56

2.9. B.V. Sankar, Scaling of low-velocity impact for symmetric composite laminates, J.

Reinforced Plastics and Composites, 11, 297-305, 1992.

2.10. J.M. Whitney and R.J. Nuismer, Stress fracture criteria for laminated composites

containing stress concentrations, J. Composite Materials, 8(3), 253-265, 1974.

2.11. J. Awerbuch and M. Madhukar, Notched strength of composite laminates

predictions and experiments - a review, J. Reinforced Plastics and Composites, 4, 3—159,

1985.

2.12. W.J. Cantwell, P.T. Curtis, and J. Morton, Low velocity impact damage tolerance in
CFRP laminates containing woven and non-woven layers, Composites, 14, 301-305,
1983.

2.13. E.G. Guynn and T.K. O’Brien, The inﬂuence of lay-up and thickness on composite
impact damage and compression strength, proc. 26th AIAA SDM Conference, Orlando,
Florida, 187- 196, 1985.

2.14. A.A. Baker, R. Jones, and R.J. Callinan, Damage tolerance of graphite/epoxy
composites, Composite Structures, 4, 15-44, 1985.

2.15. Standard Tests for Toughened Resin Composites, NASA RP 1092, 1983.

2.16. Liu, Dahsin, Impact-induced delamination - a view of material property
mismatching, J. Composite Materials, 22(7), 674-691, 1988.

2-17. E. Wu, E.-Z. Tsai, and Y.-C. Chen, Penetration into glass epoxy composite
laminates,” J. Composite Materials, 28, 1783-1802, 1994.

2.18. C.T. Sun and S.V. Potti, High velocity impact and penetration of composite

1aInitiates, proc. 9th Int. Conference on Composite Materials, 4, 157-165, 1993.

57

2.19. W. Goldsmity, C.K.H. Dharan and H. Chang, Quasi-static and ballistic perforation
of carbon ﬁber laminates, Int. J. Solid Structures, 32, 89-103, 1995.
2.20. S.W.R. Lee and CT. Sun, Dynamic penetration of graphite/epoxy laminates

impacted by a blunt-ended projectile, Composite Science and Technology, 49, 369—380,

1993.

58

 

Chapter 3
IMPACT PERFORATION RESISTANCE OF

LAMINATED AND AS SEMBLED COlVIPOSITE PLATES

ABSTRACT
A previous study on impact response of composite laminates concluded that impact
perforation was the most important damage stage in composite laminates subjected to
impact loading since impact characteristics and mechanical properties degradation of
composite laminates reached critical levels once perforation took place. It was also found
that thickness had greater inﬂuence on impact perforation resistance than in-plane
dimensions. However, as the composite laminates became very thick, the manufacturing
cost for obtaining high-quality composite laminates could become unaffordable. In an
effort to meet designing requirements and to reduce manufacturing costs, assembled
composite plates, which were organized by assembling multiple thin composite laminates
together, were considered as alternatives for thick laminated composite plates. Various
joining techniques including mechanical riveting, adhesive bonding, stitching joining and
their combinations were used in assembling two- and four-laminate plates. Experimental
results revealed that epoxy bonding outperformed other joining techniques. Although
800d bonding resulted in higher impact bending stiffness and subsequently higher
perforation threshold, increasing the laminate thickness or the number of laminates was

found to be more efﬁcient in raising perforation threshold than improving the joining

Stiffness. As a major ﬁnding of the study, the assembled composite plates were found to

59

have perforation thresholds similar to the laminated counterpart. Hence, the former co

be used to replace the latter, at least, as far as perforation threshold was concerned.

3.1 Introduction
Due to their high stiffness-to-weight and high strength-tO-weight ratios, ﬁl
reinforced polymer-matrix composites are ideal materials for high-perfonna
structures. They are usually used in thin-laminate form. AS composite technolog
advance, more and more thick-section composites are used for heavy-duty structures.
example, the applications of thick composite laminates for submarine hull and arrno
vehicle bodies have been proved to be feasible. The advancement from thin laminate:
thick laminates, however, is not trivial. It falls into a study categorized as scaling efi
[3.1-3.2] and requires modiﬁcations and improvements in almost every aspect
composite technologies such as laminate theory, contact algorithm, manufactur
technique, etc. Since thick composite laminates behave quite differently from their t
counterparts, the investigations on thick laminates have gain very much attention [3.3].
Studies regarding the scaling effects on composite response to impact loading h.
been reported [3.4-3.6]. A similar study has been presented by the authors and t1
associate [3.7]. They have concluded in their study that perforation is the most import
damage stage in composite laminates subjected to impact loading. The reason is t
inlpact characteristics (such as peak force, contact duration and absorbed energy) :
mechanical properties degradation (such as residual compressive maximum force 1

residual compressive absorbed energy) of composite laminates reach critical values W]

60

perforation takes place. They have also veriﬁed that thickness, as opposed to in-p
dimensions, has much greater inﬂuence on perforation resistance.

However, as composite laminates become thick, the manufacturing cost for h
quality composite laminates may become unaffordable. For example, in order to ach
uniform curing and thus uniform properties through the thickness of thick compi
laminates, expensive microwave curing process is required [3.8]. In an effort to mee
design requirements for high quality and to reduce the manufacturing costs, assem
composite plates, which are organized by assembling multiple thin composite lamir
together, are considered as alternatives for thick laminated composite plates in this stt

In assembling thin composite laminates together, three fundamental joi
techniques including adhesive bonding, mechanical fastening, and stitching joining
performed in this study. In addition, combined methods based on these three technir
are also explored. Both adhesive bonding and mechanical fastening have been
discussed in literature. Being virtually the combination of adhesive bonding
mechanical fastening, stitching has also been found to be feasible for composite joi
[3.9-3.10] and reinforcement [3.11]. Since it offers a relatively uniform load transfer
in structural components involved, its application to assembling thin laminates is ale

interest in this study.

Accordingly, the Objectives of this study are (l) to investigiate the joining stifl
and perforation threshold of assembled composite plates based on various joi
teChniques, (2) to identify an efﬁcient way of assembling thin composite laminate:
achieving high perforation threshold and (3) to explore the feasibility of replacing

Costly thick laminated composite plates by assembled composite plates.

61

3.2 Experimental Methods
1. Thin Composite Laminates
Composite laminates made of glass ﬁbers and an epoxy matrix were investigated

this study. The glass ﬁbers were of E-glass type whereas the epoxy matrix was of I
1002 resin. The composite laminates were of cross-ply type and had a stacking sequel
of [0/90/0/...]13. The nominal ﬁber volume fraction of the composite laminates was ab
53% and the averaged thickness was 3.2 mm. In this study, these composite lamina
were assembled together to form two-laminate composite plates by using various join
techniques.

2. Impact Testing

In order to characterize the perforation resistance of assembled composite plates

DYNATUP GRC 8200 machine was used for impact testing. A schematic diagram of ‘
impact testing machine was given in Figure 3.1. According to the diagram, the impac
consisted of three components: a dropping crosshead, an impactor rod, and an impac
nose. The steel impactor rod had a diameter of 12.5 mm and was attached to the droppi
crosshead. A force transducer having a capacity of 22.24 kN was mounted on the ff!
end of the impactor rod and encapsulated by a hemispherical nose. The impactor was
at a dropping height of 0.91 m to give a constant impact velocity at 4.22 m/s for m
tests. The total mass of the impactor, however, ranged from 10.35 kg to 17.74 kg I
adding various deadweights to the crosshead), resulting in impact energy from 92 J
15 8 J. For impact energy higher than 158 J, another similar impact testing machine wit
Pneumatic unit was used. The pneumatic unit was able to provide an additional force

increase the impact velocity up to 8 W5.

62

In each impact test, a composite specimen with dimensions of 125 mm by 100
was placed between two steel plate holders, namely the top holder and the bottom h:
as shown in Figure 1. Each holder had an opening of 100 mm by 75 mm in the or
The top holder was removable while the bottom one was attached to the frame 0
impact testing machine which was ﬁxed on a solid foundation. The specimen and th
steel holder were then C-clamped at four corners to the bottom steel holder.
composite specimen thus had a ﬁxed boundary condition. In impact testing, the imp
contacted the center of composite specimens, resulting in so-called central impact.

In most impact tests, the crosshead was released from the preset height, and dro
freely according to the gravitational force. However, for impact tests with energy hi
than 158 J, it was driven by both gravitational force and pneumatic force. As the imp;
dropped and approached the composite specimen, its time trigger passed through a
sensor right before contact-impact occurred. The initial impact velocity was
calculated from the distance between two edges on the time trigger and the time int:
they passes through the sensor. Once impact began, the contact forces at r
consecutive instants were detected by the force transducer attached to the impactor.
force history was recorded in a computer. The corresponding velocity history 0:
impactor could then be calculated from integrating the force history (after being div
by the mass of the impactor) and using the initial impact velocity. Subsequently

Corresponding displacement history of the impactor could be calculated from integr:

the velocity history.

63

 

 

 

 

 

 

 

 

(‘lnmn
. r

time

SCHSOI‘

 

 

 

 

 

 

 

 

 

guiding
rail

dropping
crosshead

impactor
rod

impactor
nose

top holder
specimen

bottom
holder

solid
foundation

Figure 3. l - Schematic diagram of impact testing machine

64

Based on the force and displacement histories, the force-displacement relation and
energy history of the impactor could be established. Assuming the impactor was perfe
rigid and the energy loss on the contact-irnpact interface between the impactor and
specimen was negligible, the force-displacement relation of the impactor could
considered as the force-deﬂection curve of the composite specimen. And the kin
energy of the impactor right before contaCt-impact took place, i.e. the impact enei
would be the energy transferred to the composite specimen. However, depending on
impact energy level and the type of specimen investigated, either a partial or the t

amount of impact energy could be absorbed by the composite Specimen in forms

damage, heat and others.

3 . Joining Techniques

In investigating assembled composite plates, the aforementioned 3.2mm-t1
glass/epoxy laminates were used as the building block. Many two-laminate compo
plates were created. Each was formed by assembling two laminates together. Vari

joining techniques such as adhesive bonding, mechanical fastening, stitching joining,

their combinations were investigated.

65

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

O O - - -------------- - — o o
50 mm . .
9 q - - -------------- - - o o
l 50 mm : ' 2R3
‘ I (3-by-3 Riverting)
2R2 E4 =
(2-by-2 Riveting) I 100 mm
‘ __________
' . ' . 75 mm
o o o o ":2- """ " ' g ’ o
5 mm . .
o o o o _ _________ o . o
I I I I . : :
V __________ . .
l I
2R4 '25 mm
(4-by-4 Riveting) 2RC
(Circular Riveting)
50 mm
fm—hi _
50 mm
283 285

(3-by-3 Stitching)

(5-by—5 Stitching)

Figure 3. 2 - Schematic diagrams of some riveting and stitching patt

66

In adhesive bonding, the bonding surfaces of composite laminates were roug
with emery papers to promote later mechanical interlocking. They were then cleane
acetone before being coated with either a Scotch double-sided tape or a two-part
temperature curing epoxy. The former was manufactured by 3M while the latter
brand name EnviroTex Lite. These two types of adhesive bonding represented di:
degrees of bonding rigidity and strength. For convenience of discussion, the taper
laminate composite plates were called 2T (2 for two-laminate and T for taped) wh
epoxy-bonded two-laminate plates 2B (B for bonded).

In mechanical fastening, square riveting patterns with various densities, such as
2, 3-by-3, and 4-by-4 per each 50mm-by-50mm area were created. Figure 2 Show
details of the riveting patterns. In performing riveting, holes were prepared by
drilling bit before 4x580 steel rivets (1/8" in diameter and 1/4" in grip) were pushec
using a riveting gun. In addition, a circular riveting pattern consisting of eight
uniformly located on the circumference of a 25mm-diameter circle in the cen
composite laminates was also employed in the study. These four riveting patterns
designated as 2R2, 2R3, 2R4 and 2RC. The ﬁrst number of the designations, i.e. 2,
represented for two-laminate plates. The second letter R stood for riveting while tl
number or letter gave riveting density or shape. That was, 2R2 was for 2-by-2, 2R
for 3-by-3, 2R4 was for 4-by-4, and 2RC was for circular riveting. The circular ri
2RC was also included in Figure 2.
Similar to riveting, several stitching patterns were also used in assembling
Immate composite plates. A 28 gauge steel wire was used as the stitching thread w:

Stitching holes prepared by a drilling bit of 1/16” in diameter. For each four

67

 

forming a square unit, six stitching lines, two longitudinal, two hor
diagonal as shown in Figure 3.2 were performed on each side of the c
The stitching joints were prepared by pulling the stitching thread thror
holes as tightly as possible by hand. Stitching densities of 3-by-3, 5-by-:
each 50mm-by-50mm area were investigated. They were denoted as 2S
respectively, where 2 was for two-laminate plates, S was for stitchi
number represented for stitching densities. Figure 3.2 also showed the :
0f2S3 and 2S5.

In addition to the individual joining techniques presented abor
composite plates were also assembled. by using combined joining tech
plate was assembled by both epoxy bonding (B for bonded) and a 3-by-Z
(R for riveting) and a 2BS3 assembled plate combined both epoxy bond
stitching pattern (S for stitching). The investigations of these cc

techniques were part of the study in search of higher joining rigidity and

3.3 Results and Discussions

1 . Force-Deﬂection Curves

Before studying assembled composite plates, the composite lamina
as the building block and mentioned earlier were investigated. These
designated as 1F in the study since they were of one-laminate and had :
Condition during impact tests. The force-deﬂection curves of eighteei

S Ubj ected to various levels of impact energy were shown in Figure 3.3.

ascending section of each force-deﬂection curve was termed the impact

68

 

 

 

due to its representation of the stifﬁress of composite laminates under impact-induced
bending in the beginning of impact process. All the force-deﬂection curves seemed to
ascend similarly, indicating similar impact bending stifﬁress. They then reached
individual maximum levels. According to Figure 3.3, the maximum forces increased as
the levels of impact energy increased. When the impact energy was high enough, the

maximum forces seemed to have a Similar value. This value was termed the peak force of

the composite laminates under the speciﬁc central impact.

69

force

 

7.0000

      

(RN)
3 5000

 

 

: : it}: :1] <-
Ito) .. ............ . 1",“: .' ‘..-...... ' I I - .
a section" 4’)": it g i i i
0 <- descending --------
,' ‘t. seetion
,, .. \-----5 i
“ ’ / s i 5
° : » «p ’ "‘gi' -_,., l 15-18
g - ....... 12...-.. WM-
6 1 .._,__ .,._
3 14 a open curves
3 Sclosed curves '
:4 .0000 1.5000 — 4.0000 6.5000 g 9.0000 11.5000 14.0000 16.5000 19.0000 21.5000 24.1

deflection (mm) '

Figure 3. 3 - F orce-deﬂection curves of eighteen 1F composite plates

70

 

O‘l
O

perforation_> ._l_6 1s

9

>‘ e ual-ener /.Ml “—-

9 4O ” clintervalgy134—penetraio

g '11

a: 30 _, ‘b

1:

3 7/

'5 20 - //

g .7 0 experiment

“3 10 ‘ // —equaI-energy
y’ —-Ieast-squares

 

l I I l l

 

o
b
\

010 20 3O 40 50 E
impact energy (J)

Figure 3. 4 - The whole enegy proﬁle of IF case based on force-deﬂection our
in Figure 3.3

71

In each subperforation impact, the force-deﬂection curve rised, reached a :
level and returned back to the origin. It formed a close curve representing the i
impacting onto the composite laminate and rebounding from the composite lam
area enveloped by the closed curve was the absorbed energy of the composite
under the Speciﬁc impact. Apparently, as the impact energy increased, the envel
increased, so did the absorbed energy. If the impact energy continued to
perforation then took place. Once perforation occurred, the force-deﬂection cur
no longer be a closed curve. The area bounded by the open force-deﬂection curi
deﬂection axis was then the energy absorbed by the perforated composite lamin:
also interesting to point out that the impact events which had sufﬁciently hi1
energy to reach the peak force seemed to share partial descending sections toge
result, regardless of the rebounding sections, all the impact events seemed 1

master force-deﬂection curve.

2. Whole Energy Proﬁle

AS mentioned earlier, as far as residual pTOperties were concemed, p
seemed to cause the ultimate damage in composite laminates subjected to impac
Once a composite laminate was perforated, any excess impact energy would b«
as kinetic energy in the impactor except that an insigniﬁcant amount would be .
into additional damage. Hence, perforation threshold was an important par:
characterizing the impact response of composite laminates [3.12-3.15]. Since pi
contact duration and absorbed energy all reached critical levels when perfora

Place [3.7], the perforation threshold of composite laminates could be identiﬁer

72

criteria based on the critical levels of peak force, contact duration, and absorbed energy.
In addition, Since impact energy should be completely absorbed by the composite
laminates when perforation took place, the perforation threshold of composite laminates
could be identiﬁed based on the equality between impact energy and absorbed energy
[3.7].

Among the four criteria to identify the perforation threshold, the one based on equal
energy between impacting and absorption was considered to be the most convenient and
accurate technique [3.7]. Hence, a comparison between the impact energy and absorbed
energy was required for judging the perforation threshold of composite plates. Results of
eighteen 1F (one-laminate with ﬁxed boundaries) Specimens were given in Figure 3.4. In
addition to the experimental data points, a least-squares ﬁtting curve for the data points
with impact energy up to the penetration threshold and a least-squares line for the data
points with impact energy beyond the perforation threshold were also identiﬁed. Shown
in Figure 3.4, the data points were represented by solid circles while the the least-squares
ﬁttings by dashed lines. Apparently, as the impact energy increased, the absorbed energy
also increased.

In addition to the raw data points and least-squares lines, a line representing the
equality between impact energy and absorbed energy was also added to Figure 4. It was
called the equal-energy line. AS could be seen from the diagram, the data points were
quite lower than the equal-energy line when the impact energy was low. As the impact
energy increased, the data points became closer to the equal-energy line. Eventually the
data points and the equal-energy line merged together. They then remained roughly the

same for an interval until the data points became smaller than the equal-energy line again.

73

The interval within that the data points overlapped with the equal-energy line was c
the equal-energy interval. The discrepancy between the data points and the equal-er
line within the equal-energy interval as shown in Figure 3.4 was believed to be cause
errors due to the numerical integrations mentioned earlier.

The equal-energy interval was bounded by two points. The point of lower bound
named penetration threshold, indicating the onset of penetration. When penetration
place, the impactor got stuck in the composite plate. Since very limited rebounding
allowed in the impactor, the impact energy was almost completely absorbed by
composite plate in forms of damage. AS the penetration proceeded (the impactor ml
deeper into the composite plate), it required more energy for the impactor to I:
through the composite plate and to overcome the friction between the impactor ant
composite. Eventually perforation of the composite plate would be achieved. (
perforation occurred, any excess impact energy would be retained in the impactor in :
of kinetic energy. And the absorbed energy would be smaller than the impact en
again. Thus, there was an upper bound for the equal-energy line. The point of u
bound was called perforation threshold, indicating the completion of perforation pI‘OI
Given in Figure 3.4, the penetration threshold was about 38 J and the perfor:
threshold was around 45.5 J for the 1F case. The difference between the penetr:
threshold and the perforation threshold, i.e. the equal-energy interval, was believed 1
dependent on the material type, the laminate thickness and the joining technique 0
composite plate.

Figure 3.4 was called the whole energy proﬁle of the IF (one-laminate with 1

boudnaries) case since it included the overall enregy exchange between the impactor

74

the composite plates. The diagram was derived directly from the force-deﬂection cuves
given in Figure 3.3. By closely comparing Figure 3.3 with Figure 3.4, it could be found
that the force-deﬂection curves up to Specimen No.11 were of closed type. Specimens
No.12 to 14 were located in the transition zone between closed curves and open curves.
They were close to the penetration threshold. Once perforation took place, approximately

around Specimen No. 15, the curves changed from closed type to open type.

3. Penetration Thresholds and Perforation Thresholds

The penetration and perforation thresholds of all assembled composite plates were
identiﬁed from corresponding whole energy proﬁles and were given in Table 3.1. The
2R2 (2-by-2 riveting) case, however, had an indentical value for both penetration
threshold and perforation threshold because there was no intesection between its least-
squares curve and the equal-energy line. A Similar result also occurred in the 2R4 (4-by-4
riveting) case. These results were possibly due to experiemental discrepancy. In addition,
it should be pointed out that only one test was performed for 289 (9-by-9 stitiching) case
due to the very much time required for the Specimen preparation.

Based on Table 3.1, the 2R4 (4-by-4 riveting) case had the highest perforation
threshold and penetration threshold. By examining the damage of impacted plates, it was
found that the four rivets closest to the center of each plate, i.e. closest to the impactor,
Were always seriously distorted by the impactor during the impact process. It was
believed that the distortions of steel rivets resulted in additional energy abosorption
during impact. A Special riveting pattern, namely 2RC (circular riveting), was then

designed to avoid the problem and to verify the effects of high-density riveting on

75

perforation threshold. The 2RC specimens had higher riveting density than the 21
specimens though the former was more locally concentrated and the latter globa
distributed. The penetration threshold and the perforation threshold for the 2RC wr
105.9 J and 123.8 J, respectively, while they were both 138.9 J for the 2R4 ca
Accordingly, the 2R4 case was not used for comparison in the remaining studies.

Among the two-laminate assembled composite plates, 2B (two-laminate bonden

2RC (circular riveting), 285 (5-by-5 stitching), 2BR3 (bonded and 3-by-3 riveting) a
2BS3 (bonded and 3-by—3 stitching) had the perforation threshold close to 123 J while
others were between 108 J and 115 J. It was believed that the adhesive bonding in the 2
2BR3 and 2BS3 cases contributed high rigidity and strength to the specimens, resulting
high perforation thresholds. The high riveting density of 2RC and high stitching dens:
of 2S5 also made similar contribution to individual specimens. In fact, the perforati
threshold increased as the riveting density increased from 2R2 (2-by-2 riveting) to 21
(3-by-3 riveting), and to 2RC. Similarly, the perforation threshold increased as t
stitching density increased from 283 (3-by-3 stitching) to 285. Due to the single test I
the 289 (9-by-9 stitching) case, the result was inconclusive and omitted from furtl
discussion.

Although high density of riveting and stitching seemed to make positive contributi-
to perforation threshold, it should be noted that their preparations also required ext
effort in both time and labor. Besides, it was also possible that they introduced ext
damage, i.e. holes, to the assembled composite plates. By examining the penetratir
tl'rresholds of 2B (two-laminate bonded), 2BR3 (bonded and 3-by-3 riveting) and 2B:

(bonding and 3-by-3 stitching) cases, the 2B case with penetration threshold of 120.5

76

 

outperformed the remaining two cases which had 105.2 J and 112.3 J, respectively.
Accordingly, the experimental results seemed to indicate that pure epoxy bonding was
sufﬁciently efﬁcient for joining thin composite laminates. The additional riveting and
stitching might cause more damage to composite laminates than effectively join them.
The penetration threshods of all other cases were between 100.8 J and 108.9 J and were
much lower than the 2B case. This result further conﬁrmed the superiority of the joining
efﬁciency of epoxy bonding in assembling thin (3.2 mm) composite laminates. In
addition, it should be noted that the equal-energy intervals for all two-laminate composite

plates seemed to be very similar.

77

Table 3.1: Impact bending stifﬁresses, penetration tresholds and perforation

 

thresholds of one-laminate and two-laminate composite plates

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Specimen Type Bending Penetration Perforation
Stiffness Threshold Threshold
(N/mm) (J) (J)
1F: one-laminate Fixed 933.0 38.0 45.5
2F: two-laminate Fixed 1831.7 100.8 114.7
2T: double-Sided Taped 1740.4 104.7 11 1.1
2B: epoxy-Bonded 3370.9 120.8 123.0
2R2: 2x2 Riveting 1752.6 108.9 108.9
2R3: 3x3 Riveting 1762.3 105.1 115.8
2R4: 4x4 Riveting 1674.6 138.9 138.9
2RC: Circular Riveting 1694.5 105.9 123.8
283: 3x3 Stitching 1670.0 103.5 115.0
285: 5x5 Stitching 1663.0 108.2 120.0
x 289: 9x9 Stitching 1559.1“ 108.5“ 1085*
2BR3: Bonded & Riveted 3538.5 105.2 121.2
2883: Bonded & Stitched 2977.8 112.3 120.8

 

 

 

* based on only one test

78

u. I ”III!!!“ .__

4. Impact Bending Stiffness

Bending stiffness had been found to be an important parameter to delamination
resistance [3.16] and perforation resistance [3.7]. In investigating the effects of joining
technique on perforation resistance, the bending stiffnesses, which were the slopes of the
force-deﬂection curves, of various assembled composite plates were identiﬁed and also
listed in Table 3.1. Results of the two—laminate plates could be essentially divided into
two groups. The bending stiffnesses of 2B (two-laminate bonded), 2BR3 (bonded and 3-
by-3 riveting) and 2BS3 (bonded and 3-by-3 stitching) cases were from 3000 N/mm to
3500 N/mm. These values were about two times those of the remaining two-laminate
cases. Since the impact bending stiffness changed with the type of assembled composite
plates, it in fact was associated with the joining stiffness dut to the corresponding joining

technique. Hence, the impact bending stiffness could be considered as an index of joining

stiffness up to some extent.

79

 

 

160
140
120
100
80
60
40
20

energy thresholds (J)

 

 

 

O penetration
(one-laminate)

Cl perforation (one-
laminate)

penetration
(two-laminate)

I perforation (two-
laminate)

— least-squares

 

 

I l l

1000 2000 3000 4000
bending stiffness (N/mm)

Figure 3. 5 - Relation between energy thresholds and impact bending stiffness of

various two-laminate composit plates

80

 

180
0224mm

A320mm D
D666mm
120 “ -equal-energy

A

0'1

0 .
r

absorbed energy (J)
03 CO
O O

00
O
1

g a
459.38:
0 30 60 90 120 150 180
impact energy (J)

 

 

 

O
l

l l I I

Figure 3. 6. The whole energy proﬁles of three laminated composite plates

81

200

01
O

absorbed energy (J)

 

150-

100 -

l

 

o 1F-2.24 (one-

laminate ﬁxed)
A 2F-2.24 (two-

laminate ﬁxed [3
A 28-224 (two-

laminate bonded

c1 4F2.24 (four-
laminate ﬁxed

equal energy D
A CI
‘ E]
A
8A
‘A Ci
El

 

I l j

50 100 150
impact energy (J)

 

200

Figure 3. 7. The whole energy proﬁles of four assembled composite plates

82

 

The perforation thresholds and bending stiffnesses of two-laminate plates were put
together in Figure 3.5. Apparently, there was a correlation between them. That was, the
higher the bending stiffness of an assembled composite plate, the higher the penetration
and perforation thresholds. However, when compared with those of the one-laminate
case, i.e. IF, the increase in penetration and perforation thresholds seemed to be more
signiﬁcant due to thickness increase than due to bending stiffness increase. In fact, by
adding one more laminate to the one-laminate plates to become two-laminate plates, both
energy thresholds and bending stiffenss were ahnost doubled. However, an increase in
bending stiffness based on any joining technique did not seem to improve the penetration

and perforation thresholds Signiﬁcantly.

5. Laminated Plates versus Assembled Plates

In verifying the feasibility of using assembled composite plates to replace laminated
composite plates, a study comparing the impact bending stiffnesses and perforation
thresholds of these two groups was necessary. In this study, besides the IF (3.2 mm
thick) case, two other types of composite laminate were investigated. These two types of
composite laminate had cross-ply laminations and thicknesses of 2.24 mm and 6.66 mm.
They were designated as 1F-2.24 and 1F --6.66. Their whole energy proﬁles were given in
Figure 3.6 along with the 1F -3.2 case. Apparently, all of them had Similar trends. And the
equal-energy interval increased from 5 J for lF-2.24 to 7.5 J for 1F-3.2, and to 40 J for
1F -6.66. This result conﬁrmed that the equal-energy interval, or the penetration process,

was dependent on the thickness of composite plates.

83

 

 

In order to further conﬁrm that thickness, instead of bending stiffness, played a r
efﬁcient role in improving perforation resistance, an investigation on thickness ef
was performed. The composite laminates used in this study were of 1F-2.24. They
material properties identical to those of 1F-3.2 specimens. However, their stac
sequence was [0/90/0/...]9 and nominal thickness was 2.24 mm. In addition to the
2.24 case, assembled composite plates such as 2F-2.24 (two-laminate with 1
boundaries), 2B-2.24 (two-laminate bonded) and 4F -2.24 (four-laminate with 1
boundaries) cases were created. Their whole energy proﬁles were presented in Figure
for comparison. Although the difference of impact bending stiffensses between 2F-
and 2B-2.24 was as much as that between 2F and 2B (both based on 3.2-mm lamin.
cases, their whole energy proﬁles were very similar. However, the whole energy pro
of lF-2.24, 2F-2.24 and 4F-2.24 were quite different. Apparently, the thicker
assembled composite plate, the higher the capability of energy absorption. Accordir
Figure 3.7, the penetration thresholds for lF-2.24, 2F-2.24, 2B-2.24 and 4F-2.24 .
about 22 J, 63 J, 62 J and 157 J, respectively, and the perforation thresholds for I
were about 27 J, 73 J, 73 J and 182 J, respectively. The equal-energy intervals, i.e
difference between the penetration thresholds and the perforation thresholds, for the
cases were thus 5 J, 10 J, 11 J and 25 J. This result combined with those of earlier str'
provided evidences that the equal-energy interval was dependent on the thicknes

composite plates.

84

 

200 -- +-

180 ' - laminated
160 1 o assembled
140 _ _ least-squares M... .
120 A 28-3.2
100 —4

 

 
 
   

O 2r-3.2

perforation threshold (J)

20 _ lF-2.24

 

 

 

O 2 4 6 8 10
thickness (mm)

Figure 3. 8 - The relation between perforation threshold and thickness for both laminated

and assembled composite plates

85

 

l -“_-HA__-_._.L

The perforation thresholds of both assembled cases, including lF-2.24, 2F-2.24, 2B-
2.24, 2F-3.2, 2B-3.2, 4F-2.24, and laminated cases, including 1F-2.24, lF-3.2 and 1F-
6.66, were also summarized in Figure 3.8 for comparison. Apparently, the assembled
cases were very similar to the laminated cases because they were all closely located along
a least-squares line. It then could be concluded that the perforation threshold of a
composite plate was dependent on thickness only, regardless of being laminated or
assembled. In other words, the assembled composite plates could be used as alternatives
for thick laminated composite plates, at least, as far as the perforation threshold was

concerned.

3.4 Conclusions
The following conclusions can be drawn from above investigations:

1. Various joining techniques such as mechanical riveting, adhesive bonding,
stiching joining and their combinations were used in assembling two-laminate and four-
laminate composite plates. Pure epoxy bonding was found to be the most efﬁcient joining
technique in assembling the composite laminates together since it gave the highest
bending stiffness and perforation threshold.

2. Among the assembled two-laminate composite plates, the perforation
threshold increased as the bending stiffness increased. However, the increase of
perforation threshold based on the improvement of bending (joining) stiffness was

limited. A more efﬁcient way to signiﬁcantly increase the perforation threshold was to

86

 

increase the thickness of composite laminates or to use assembled multi-larninate
composite plates.

3. A technique named whole energy proﬁle was presented for characterizing
impact-perforation resistance. When penetration took place, the absorbed energy was
approximately equal to the impact energy. When perforation occurred, the absorbed
energy was again smaller than the impact energy. Experimental results revealed that the
equal-energy interval, which was the difference between the penetration threshold and
perforation threshold, increased as the thickness of composite plates increased.

4. Based on the studies, it was found that assembled composite plates were very
comparative with laminated composite plates in both bending stiffenss and perforation
threshold. This result veriﬁed the advantage of using assembled composite plates over
thick laminated composite plates Since the cost of making thick laminated composite

plates with high quality increased Signiﬁcantly as the thickness increased.

ACKNOWLEDGEMENTS
This investigation was supported by both Michigan Research Excellence Fund and the

US. Army TACOM-TARDEC ILIR program. The authors wish to express their sincere

thanks for the ﬁnancial supports.

87

3.5 References

3. 1 . Workshop on scaling ejfects in composite materials and structures, NASA Langley
Research Center, Hampton, VA, November 15-16, 1993, Report No. 94-6, Institute for
Mechanics and Materials, University of California, San Diego, CA.

3 .2. A. Carpinteri, ed., Size-scale ejfects in the failure mechanisms of materials and
structures, E & FN Spon, 1994.

3.3. Y. D. S. Rajapakse, ed., Mechanics of Thick Composites, AMD-Vol. 162, The
American Society of Mechanical Engineers, 1993.

3 .4. J. Morton, “Scaling of impact-loaded carbon-ﬁber composites,” AIAA Journal, 26(8),
989-994, 1988.

3.5. Y. Qian, 8.R. Swanson, R.J. Nuismer, and RB. Bucinell, “An experimental study of
scaling rules for impact damage in ﬁber composites,” J. Composite Materials, 24(5), 559-
5 70, 1990.

3 .6. B.V. Sankar, “Scaling of low-velocity impact for symmetric composite laminates,” J.
Reinforced Plastics and Composites, 11, 297-305, 1992.

3 .7. D. Liu, B. B. Raju, and X. Dang, “Size effects on impact response of composite
laminates,” International Journal of Impact Engineering, Vol. 21, No. 10 pp837—854,

1 998
3 -8. J. Wei, M.C. Hawley, J. DeLong, and M. DeMeuse, "Comparison of microwave and

thermal cure of epoxy resins," Polymer Engineering and Science, 33(17), 1132-1140,

1 993.

88

3.9. C. Lee and D. Liu, ”Tensile strength of stitching joint in woven glass fabrics,” J.
Engineering Materials and Technology, 112 (2), 125-130, 1990.
3.10. D. Liu, ”Photoelastic study on composite stitching,” Experimental Techniques, 14
(1), 25—27, 1990.
3 . 1 1. D . Liu, “Delamination resistance in stitched and unstitched composite plates
subjected to impact loading,” Reinforced Plastics and Composites, 9 (1), 59-69, 1990.
3 . 12. E. Wu, E.-Z. Tsai, and Y.-C. Chen, “Penetration into glass epoxy composite
1aIninates,” J. Composite Materials, 28, 1783-1802, 1994.
3- 13. C.T. Sun and S.V. Potti, “High velocity impact and penetration of composite
laminates,” proc. 9th Int. Conference on Composite Materials, 4, 157-165, 1993.
3.14. W. Goldsmith, C.K.H. Dharan and H. Chang, “Quasi-static and ballistic perforation
Of Carbon ﬁber laminates,” Int. J. Solid Structures, 32, 89-103, 1995.
3.15 . S.W.R. Lee and CT. Sun, “Dynamic penetration of graphite/epoxy laminates
iI'IIIDacted by a blunt-ended projectile”, Composite Science and Technology, 49, 369-380,
1993 .
3. 1 6 . D. Liu, “Impact-induced delamination - a view of material property mismatching,”

J. Composite Materials, 22(7), 674-691, 1988.

89

 

 

 

 

Chapter 4
TESTING AND SIMULATION OF LAMINATED COMPOSITES SUBJECTED TO
IMPACT LOADING

ABSTRACT

Because composite laminates are very susceptible to impact loading even at low
velocity, low-velocity impact is an important subject in laminated composite analysis.
The impact-induced damage is usually invisible to naked eyes and can cause serious
structural degradation. Many low-velocity impact tests were performed in some previous
studies; however, most of them were phenomenological analysis. In an effort to further
understand the responses of composite laminates under low-velocity impact and to
develop an accurate and efﬁcient quantitative simulation in the future, this study was
aimed at performing some instrumented impact tests and computer Simulations. A
commonly used computer code - LS-DYNA3D - was evaluated in this study and the
results were valuable for future deVelopment of a new computer code. In the study, a
low-velocity impact event investigated by Sun and Chen with an indentation law and
veriﬁed by experiments was used to justify the ﬁnite element model and contact
parameters. Once the computational scheme was established, it was used for a broader
investigation consisting of composite laminates with various thicknesses, ﬁber angles, ‘
and impact velocities. Computational results revealed that the peak contact force and
maximum deﬂection were strongly affected by the thickness of composite laminates
while the ﬁber angles investigated seemed to play a less Signiﬁcant role. In addition, it

was concluded that because delamination modeling was not included in the L8-

90

DYNA3D, the computer code needed to be modiﬁed if it was to be used for accurate

impact simulations.

4.1 Introduction

Laminated composites are very susceptible to impact loading. Composite damage
such as ﬁber breakage, matrix cracking, ﬁber—matrix debonding, and delamination can
take place in composite laminates even when they are subjected to impact forces at low
velocity. These damage modes usually cannot be detected by naked eyes. However, their
effects on composite structural degradation are always very Signiﬁcant.

Many impact tests were performed and can be found in the review articles by
Abrate [41,42]. In some previous studies [43,44], Liu et al also concluded that matrix
cracking and delamination were the major damage modes in composite laminates
subjected to low-velocity impact. A correlation between bending stiffness mismatch and
delamination size was established. The relation was successfully used for
phenomenological explanations of delamination size, location, and orientation in
impacted composite laminates. In an effort to further understand the response of
composite laminates under low-velocity impact and to establish an accurate and efﬁcient
quantitative simulation, the objectives of the present study are to perform some
instrumented impact tests and to simulate the impact responses with a commonly used
computer code - LS- DYNA3D. The results from this study are believed to be very
valuable for future improvement and/or development of computer modeling and
simulation for impact analysis.

In modeling the composite response under low—velocity impact, Tan and Sun [4.5]

veriﬁed that an indentation law based on a quasi-static test could be used to investigate

91

 

 

low-velocity impact. For a set of composite material and impactor, an indentation test
was required to identify the corresponding indentation parameters. Once the indentation
law was established, it could be integrated into a computational scheme for various
studies [4.6, 4.7]. In their study [4.8], Wu and Yen also investigated the relationship
between impact force and laminate indention. The numerical method they used was
derived from three-dimensional anisotropic elasticity. Effects of material and geometrical
parameters on force-indentation relation were also examined in their study. Their results
seemed to agree well with Sun and Chen [4.7].

Because [of its strong dependence on parameters obtained from experiments, the
approach based on the indentation law usually gave good prediction of composite
responses under low- velocity impact. However, it should be pointed out that a new
characterization for indentation parameters was needed each time the material or
geometry of the composite or impactor was altered. AS a consequence, for investigating
the effects of material type, lamination, ﬁber orientation, and thickness on impact
response of composite laminates, a computational technique free of experiment-
dependent parameters might prove to be more efﬁcient. In search of a more efﬁcient
technique for studying impact responses of various composite laminates and impactors,
the present study examined an existing computational scheme, namely LS-DYNA3D.
The effects of laminate thickness, ﬁber angle, and impact velocity on composite response
were of primary concerns along with the feasibility of using LS-DYNA3D for impact

analysis.

92

 

 

 

 

 

 
 
  

TIME
TRIGGER DEAD WEIGHT
IlVIPACI‘ OR
TRANSDUCER
SPECIMEN
TIMER
SPECIMEN HOLDER
FIXED AT 4 CORNERS

 

 

Figure 4.1 Schematic diagram of impact testing setup

93

4.2 Instrumented Impact Testing

The impact tests in this study were performed on a DYNATUP impact testing
machine. The impactor was a steel rod which had a diameter of 12.5mm and was attached
to the dropping head as shown in Figure 4.1. A force transducer having a maximum force
limit of 13,350N was mounted at the tip of the rod and encapsulated by a spherical head.
Specimens with dimensions of 125mm x 100mm were clamped by two steel holders from
both sides with a rectangular opening of 100mm x 75mm in the center. The specimens
and the specimen holders were then C-clamped to the bottom of the impact testing frame
which was nearly ﬁxed to the ground with many dead weights.

As the impactor dropped and approached a specimen, it triggered the timer and
started the recording clock. The force history detected by the force transducer was then
recorded in a computer. The corresponding displacement of the transducer or the
deformation of the specimen could be calculated through double integrations. The force
history, the deformation history, and the energy history were found to be the important
characteristics of individual composite laminates under impact. In addition, the maximum
peak force, the maximum deﬂection, the contact duration, and the energy absorption

capability were also found to be important parameters for impact analysis.

4.3 Computational Technique

A dynamic ﬁnite element code named LS-DYNA3D was used in this study for

impact simulations. In performing the ﬁnite element analysis, ﬁnite element models for

94

 

both composite laminates and a semi-spherical impactor were built. The composite
material was AS4/3501-6 and had the following elastic constants:

E11: 120 GPa, E22: E33= 7.9 GPa, G12: G23= G31: 5.5 GPa, v12: v13 = v23 = 0.3
The composite laminates had dimensions of 100mm x 75mm. The thickness of individual
ply was considered to be 0.25mm. The solid element was used in the ﬁnite element
models because it gave better results than the shell elements. Full models were built for
all specimens, resulting in a mesh of 40 x 40 x 15 for a lS-ply laminate. The laminate
models were then ﬁxed around four edges and subjected to a spherical impactor with a
diameter of 12.5mm and a mass of 24g at the center of the laminates. The impact velocity
ranged from 1 m/s to Sm/s. These ﬁnite element models were used throughout the
investigations of this study unless otherwise mentioned.

Instead of an indentation law to identify the contact force and indentation, the LS-
DYNA3D used a contact algorithm to calculate the contact-impact response.
Accordingly, besides an initial impact velocity, a contact parameter named penalty
coefﬁcient needed to be determined through comparisons with experimental results and
then assigned in the ﬁnite element simulations. The LS-DYNA3D gave results of force
history and deformation history. Other information such as peak force, contact duration,
maximum deﬂection, and energy absorption engaged in the impact events could be drawn
from the histories. Since this study was focused on the comparison between the
composite performance and computer simulation under low-velocity impact, the failure

process available in the LS-DYNA3D was also imposed in the feasibility studies.

95

Table 4.1 Experimental results of impact parameters

 

 

 

Impact Contact Peak Maximum Energy
Types of Effect Velocity Duration, Force. Deﬂection, Absorption,
Stacking Sequences m/ 5 ms kN m J
Thickness:
[03/ 903/03/903/031 0.98 4.65 3.34 0.89
[03/903/03/903/03/903/03] 0.98 4.00 4.41 0.84
Fiber Angle:
[03 l9O3/03/903/03] 0.98 4.65 3.34 1.47 0.89
[303l—303/303l—303/303] 0.97 4.55 3.33 1.52 0.81
[453 /—45,l45,l-45,/45,] 0.97 4.30 3.50 1.27 0.81
[60,l-603l603l—603/6031 0.97 4.35 3.46 1.52 0.81
Velocity:
[03/903l03] 0.99 8.15 1.78 2.79 1.04
[03/903/03] 1.59 8.43 2.95 4.32 2.59
[03/903/03] 2.77 10.05 3.61 8.38 14.25
[03/903/03] ‘ 3.40 6.25 3.70 13.46 25.76
[03/903 [0,] 3.95 4.70 3.34 12.70 23.15
[03/903 l03] 4.49 3.20 3 .32 l 1.94 24.81

96

 

4.4 Experimental Results

The following three types of studies were performed: thickness effect, ﬁber angle
effect, and velocity effect. Table 4.1 shows the details of the stacking sequence and the
impact parameters of primary concerns while Figures 4.2, 4.3, and. 4.4 are the force-
displacement relations for the thickness effect, ﬁber angle effect, and velocity effect,
respectively. The following statements can be summarized from these results:

A. (Thickness Effect) The laminate stiffness and strength increase as the laminate
thickness increases, resulting in the increase of peak contact force and decreases of
contact duration, maximum deﬂection, and energy absorption.

B. (Fiber Angle Effect) Due to the alternative lay-up and near-square geometry,
ﬁber angle does not play a signiﬁcant role in impact response for composite laminates
with the same thickness.

C. (Velocity Effect) For thin composite laminates under subperforation impact,
contact duration, peak contact force, maximum deﬂection, and energy absorption all
increase as the impact velocity increases.

D. (Velocity Effect) Once penetration takes place, both contact duration and peak
contact force decrease while maximum deﬂection remains about the same level as the
impact velocity increases.

E. (Velocity Effect) There is a signiﬁcant jump in energy absorption due to
penetration. The energy absorption remains about the same level even when the impact

velocity increases.

97

 

 

 

 

4450 I 1 U I I I I I I U U
.35 21-layer
3560 ‘ .. o.
' ' 34 lS-layer
A 2 r . a. ‘.. .0.
a 670 _ . . .' 3
m ' . .‘ .'
Q ~ ' . .
m a O . o
o t ' ‘
LL ' ‘ '
1780 ” . . . o.
,. . . o .9 9'laYCf
890 ‘b . .0 .0 :0 ........O ........00 d
.— :.. 0‘": can"...
0 1.213;}... 1'21; 1 1 I J 1 1

0.5 . 1:0 1:5 2:0 {.5
DISPLACEMENT (mm)

Figure 4.2. Force-displacement relations for laminates with various thicknesses

98

 

 

 

3560.. . 1
. ,.-"'?I? I,
>- 0. o 0“:..'. 0 q
. . C. '0’ C....
2670 .. . '. '. ' '3':
:2“. .1‘1‘
"' . ‘ . o .0. ... ‘
E? 1780" . :. ° - -. «
O __ . o . . . . .0
u‘ o ‘ 0 o .0 ..
r- ‘ ‘ . N e. ' J
890 "’ 0 . .0 '0
I a .
O 9 9"-
r- 0 t . .:;' '1
h t ' u .0 '
0 :3 11 :3chu‘ . g 1 1 1 1 1 1 1

 

 

0.5 110 1.5
DISPLACEMENT (mm)

Figure 4.3. Force-displacement relations for laminates with various ﬁber angles

99

 

 

35601va'""u

 

 

 

 

r- /r ' ' ‘ .1
2670 i" a “ \
7 3.40 ‘
:2: ' ”
8 1780
‘6‘ 1
(L p ‘\ '1
\
4.49 h
0‘ d
890" 2.77 \
\
\
i
“x i ‘ a
O . .ﬁﬁ ‘ 1 w 5 j g 1 3 1 ‘1'???
5.0- 10.0 '

DISPLACEMENT (mm) *

Figure 4.4 - Force-displacement relations for laminates subjected to various impact
velocities

100

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.2 1....1 ..- -....
_ Elf: __ _. _
10 -— — . R-F’Fim
. y —4 Ié-DIFNABD
\
0.8<— —
"‘ A
{g 06 . A! f\
g ,_m _¢\_ 71 _
0, __ . /\ /
* .V
- __ j ‘
0.2 / \
o l.
0.1 0.2 0.3 0.4 0.5

TIME (ms)

Figure 4.5 - Comparison of force history between ref. [4.7] and LS—DYNA3D

101

 

1.0

 

 

 

 

 

 

 

 

 

0 8 l" . 124512.17]
- ‘i I “I
/ -— LS-DYNABD
0.6 _ 4
/ ..\\
0.4 /‘ \

 

 

 

DISPLACEMENT (mm)

 

0.2 / - \

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5 1-0 . . 1.5
TIME (ms)

Figure 4.6 - Comparison of displacement history between ref. [4.7] and LS-DYNA3D

102

* EXPERIMENT
MPUTATION

2670

1780

FORCE (N)

890

2.0 4.0 6.0 8.0
TIME (ms)

 

Figure 4.7 - Force history of 15-layer laminate with ﬁxed boundary condition and
subjected to lm/s impact

103

1.5.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

J 3.. k
L/ '-.\ * WWW
’/ ﬂ _ COMEU’I‘ATION
-1, w 1
E ' i 3\
E
B
E 7/ 11A
/ \
c: IV] X
I 11
l \l" /\ ‘GA
0 2.0 4.0 6.0 8.0

TIME (ms)

Figure 4.8 - Displacement history of 15-1ayer laminate with ﬁxed boundary condition and
subjected to lm/s impact

104

 

 

 

14.—I l

 

 

\( at: EXPERIMENT
. t 4 +1 1
0 6 r— COMPUTATION

 

 

WV"

 

 

 

 

VELOCITY (m/s)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

k

2.0 4.0 6.0 8.0
TIME (ms)

 

 

Figure 4.9 - Velocity history of 15-layer laminate with ﬁxed boundary condition and
subjected to 1m/s impact

105

 

4.5 Computational Results
1. Justiﬁcation

Before investigating the effects of material properties and geometrical parameters
on impact response of composite laminates, the ﬁnite element models and the associated
contact parameters were justiﬁed by comparing the LS-DYNA3D results with those
obtained from the approach based on indentation law by Sun and Chen [4.7]. In the
justiﬁcation, the stacking sequence and the impactor of the ﬁnite element simulations
were identical to those deﬁned in Reference [4.7] but different from those used in later
investigations. It was found that a 40 x 40 mesh was able to give good agreement with
Sun and Chen’s. Figures 4.5 and 4.6 show the force history and deﬂection history,
respectively. The solid line represents the results from LS-DYNA3D while the enhanced

dots are those from Reference [4.1].

2. Summary

Similar studies regarding the thickness effect, ﬁber angle effect, and velocity effect
were also performed by the LS-DYNA3D simulation. Figures 4.7, 4.8, and 4.9 show the
comparisons between the experimental results and the ﬁnite element simulations for the
lS-layer laminate, i.e. [03/903/03/903/03], under 1m/s impact. The experimental results are
designated by asterisk symbol while the ﬁnite element simulations the solid lines. The
notations A and B for the solid lines represent for the results for the impactor and the
specimen, respectively. In addition to the impact velocity of lm/s, impact velocities of
2.8m/s and 3.5 m/s were also performed and the results based on various failure criteria

and associated progressive damage models, such as Tsai- Wu failure criterion [4.9],

106

Chang-Chang failure criterion [4.10], MTL54 (partially Chang-Chang failure criterion)
[4.11], and pure elastic assumption are given in Figures 4.10, 4.11, and 4.12.

Based on the computational results, the following statements can be summarized:

A. (Thickness Effect) Based on comparisons of force history, deﬂection history, and
velocity history, the computational results for 9-layer laminates agree well with the
experimental counterparts if ﬁxed boundary conditions is imposed in the simulations.
However, simply- supported boundary conditions seems to give better results for 15-1ayer
and 21 -layer simulations.

B. (Fiber Angle Effect) Simply-supported boundary condition also give good results
for angle-ply laminates which consist of 15-layers.

C. (Velocity Effect) At subperforation impact, the lower the velocity, the better the
simulation.

D. (Velocity Effect) Both Tsai-Wu and MTL54 (partially Chang-Chang failure

criterion) seem to give reasonable simulation before penetration takes place.

4.6 Conclusion

The LS-DYNA3D gives accurate predictions for the response of impacted
composites before delamination takes place. In order to closely simulate the progress of
impact-induced damage, a new type of ﬁnite element and a new failure criterion which

account for interlaminar stresses are required.

107

 

   
  

 

 

 

2670 I I I I I I I I I I I I I I
graphite/epoxy
..[ 03 I 903 / 03 ] elastic(fixed) '
1780 ~ ~
A (simple) ,
a
In
D ,.
ad
0
u.
890 I»
. “a. ,
I"...
\ 0“.-
O l l l 3 l l l 5 l
4.0 6.0 8.0

TIME (ms)

Figure 4.10 - Force histories of 9—layer laminate subjected to lm/s impact and simulated

by various failure criteria

108

 

    
  

 

 

 

,
* elastic(fixed) «
r .
_ graphite/epoxy
3900 [03/903/03] ‘
" -l
g
8 .
g4) elastic(simple) ‘
‘h 4450 ~ .
MTL54 Tsai-Wu ‘
P \ test .
O 1 1 1 g 1 1‘ 1 1
5.0 10.0
TIME (ms)

Figure 4.11 — Force histories of 9-layer laminate subjected to 2.8m/s impact and

simulated by various failure criteria

109

 

 

 
  
   
 
     
     

 

 

 

 

_ elastic(fixed) ‘
13350 ~ graphite/epoxy
” [03/903/03] ~
g 8900 r .
g elastic(simple)
I1.
4450 - q
. MTL54
n... test
0 ‘ ‘ . n.
4-0 8.0
TIME (ms)

Figure 4.12 - Force histories of 9-layer laminate subjected to 3.5m/s impact and

simulated by various failure criteria

110

ACKNOWLEDGMENT
The authors wish to express their sincere thanks to Ford Motors Company for
ﬁnancial support through the University Research Initiation Program. Partial support

from the US. Army TARDEC (Warren, Michigan) is also greatly appreciated.

4.7 References

4.1. Abrate, 8., “Impact on Laminated Composite Materials,” Applied Mechanics
Review, 44(4), 155-190, 1991.

4.2. Abrate, 8., “Impact on Laminated Composites: Recent Advances,” Applied
Mechanics Review, 47(11), 517-544, 1994.

4.3. Liu, D. and Malvem, L.E., “Matrix Cracking in Impacted Glass/Epoxy Plates,”
Journal of Composite Materials, 21(7), 594-609, 1987.

4.4. Liu, Dahsin, “Impact-induced Delamination - A View of Material Property
Mismatching,” J. Composite Materials, 22 (7), 674-691, 1988.

4.5. Tan, TM. and Sun, C.T., “Use of Statical Indentation Laws in the Impact Analysis
of Laminated Composite Plates,” J. Applied Mechanics, 52, 6-12, 1985.

4.6. Chen, J .K. and Sun, C.T., “Dynamic Large Deﬂection Response of Composite
Laminates Subjected to Impact,” Composite Structures, 4, 59-73, 1985.

4.7. (Sun, CT. and Chen, J .K., “On the Impact of Initially Stressed Composite
Laminates,” J. Composite Materials, 19(6), 490—504, 1985.

4.8. Wu, E. and Yen, C., “The Contact Behavior Plates and Rigid Spheres,” J. Applied

Mechanics, 61, 60-66, 1994.

111

4.9. Tsai, SW. and Wu, E.M., A General Theory of Strength for Anisotropic Materials,”
Journal of Composite Materials, 5(1), 58-80, 1971.

4.10. Chang, F.—K. and Chang, K.-Y., “Post-Failure Analysis of Bolted Composite Joints
in Tension or Shear-Out Mode Failure,” Journal of Composite Materials, 21(5), 809-833,

1987.

4.11. LS-DYNA3D User’s Manual.

112

Chapter 5
SIMULATION OF COMPOSITE LAMINATES UNDER LOW—VELOCITY IMPACT
WITH CONTINUUM-BASED F INITE ELEMENT METHOD

ABSTRACT

Delamination played the primary role of damage process in composite laminates under
low-velocity impact. In order to simulate impact process, a computational scheme
capable of identifying interlaminar stresses and presenting delamination failure was
imperative. Because of the complexity of the impact analysis and delamination
simulation, both numerical accuracy and computational efﬁciency should be considered
in the development of a computational scheme. Considering interlaminar shear stress
continuity and having degrees of freedom independent of layer number, a laminate theory
named the Generalized Zigzag Theory was formulated into a ﬁnite element subroutine
and integrated into a commercial package called ABAQUS. Due to the uses of the
Truesdell rate of Cauchy stress and the rate of deformation tensors for large deformation
analysis, the computational scheme was able to present reasonable interlaminar shear
stresses via an updated Lagragian algorithm. Combining the calculated interlaminar shear
stresses and a delamination failure criterion, the computational scheme adequately
simulated the response of composite laminates under impact loading when the impact

velocity is low.
5.1 Introduction

Owing to their high stiffhess-to-weight and high strength—to-weight ratios, ﬁber-

reinforced polymer-matrix composite laminates are excellent materials for high-

113

 

 

performance structures. Their properties in the thickness direction, however, are poor as
they are assembled by weak polymeric matrices through laminate interfaces.
Accordingly, when a composite laminate is subjected to impact loading, high
interlaminar stresses, due to concentrated impact loading and sudden material change
across laminate interfaces, combined with low interlaminar strengths could easily cause '
interlaminar damage such as delamination in the composite laminate. Although the
impact-induced damage could be very complex, experimental investigations [5 .1] had
shown that delamination was the primary damage mode if the impact velocity was
relatively low.

As a ﬁrst step to simulate impact response of composite laminates, LSDYNA3D was
used in a previous study [5 .2]. However, due to its inability to calculate interlaminar
stresses, no delamination simulation could be achieved in the study. The objective of this
study was to improve the impact simulation, at least for low-velocity impact. Hence, a
computational scheme capable of identifying interlaminar stresses and presenting
delamination damage was imperative. Because of the complexity process of impact
analysis delamination simulation, both numerical accuracy and computational efﬁciency
should be considered in the development of a computational scheme. Considering
interlaminar shear stress continuity and having degrees of freedom independent of layer
number, a laminate theory named the Generalized Zigzag Theory [5.13] was chosen and
formulated into a ﬁnite element subroutine. It was then integrated into a commercial
package called ABAQUS for impact simulation. The development of the computational
scheme was closely parallel to that of Lee [5.4], Experimental investigations was also

performed to verify the computational results.

114

 

 

5.2 Experimental Investigation

In order to characterize the response of composite laminated subjected to impact
loading, a DYNATUP GRC 8250 machine was used for impact testing. A schematic
diagram of the impact testing machine was given in Figure 1. According to the diagram,
the impactor consisted of three components: a dropping crosshead, an impactor rod, and
an impactor nose. The steel impactor rod had a diameter of 12.5 mm and was attached to
the dropping crosshead. A force transducer having a capacity of 22.24 kN was mounted
on the front end of the impactor rod and encapsulated by a hemispherical nose. The total
mass of the impactor was 11.9 kg. The impactor was set at a dropping height up to 0.8m
to give an impact velocity at 4 m/s. 2

In impact tests for glass/epoxy laminates, specimens with dimensions of 150mm x
150mm was placed between two steel plate holders, namely the top holder and the
bottom holder as shown in Figure 5.1. Each holder had an opening of 125mm x 125mm
in the center. In imapct tests for graphite/epoxy laminates, the specimen dimensions were
of 150mm x 100mm and the holder opening was 100mm x 75mm. The top holder was
removable while the bottom one was attached to the frame of the impact testing machine
which was ﬁxed on a solid foundation. The specimen and the top steel holder were then
C-clamped at four comers to the bottom steel holder. The composite specimen thus had a

ﬁxed boundary condition.

115

PNEIHVIATIC UNIT

LOADING FRAME

  
 

DROPPING HEAD
TIME TRIGGER

12.5 mm IMPACTOR ROD

HEM ISPHERICAL
IMPACTOR NOSE

STEEL HOLDER
SPECIMEN
STEEL HOLDER

SPECIMEN FRAME

SENSORS

SOLID BASE

CONCRETE HOOR

Figure 5.1 — Schematic diagram ofimpact testing setup

116

 

 

 

——“—.‘_—.u-——

In each impact test, the crosshead was released from a preset height, and dropped
freely according to the gravitational force. The impactor contacted the center of the
composite specimen, resulting in so-called central impact.

As the impactor dropped and approached the composite specimen, its time trigger
passed through a time sensor right before contact-impact occurred. The initial impact
velocity was then calculated from the distance between the two edges of the time trigger
and the time interval they passes through the sensor. Once impact began, the contact
forces at many consecutive instants were detected by the force transducer attached to the
impactor. The force history was recorded in a computer. The corresponding velocity
history of the impactor could then be calculated from integrating the force history (after
being divided by the mass of the impactor) and using the initial impact velocity.
Subsequently, the corresponding displacement history of the impactor could be calculated
from integrating the velocity history.

Based on the force and displacement histories, the force-displacement relation could
be established. Assuming the impactor was perfectly rigid and the energy loss on the
contact-impact interface between the impactor and the specimen was negligible, the
force-displacement curve of the impactor could be considered as the force-deﬂection
curve of the composite specimen.

In order to gain an overall view of the feasiblity of a computational scheme in impact
simulation, several composites was investigated. They included draphite.epoxy as well as
glass/epoxy. A composite laminate with a stacking sequence of [05/905/05] was taken as
the basic case. Other stacking sequences concerning different laminations such as

[03/903/03/903/03] and [0/90/0/...]15, different ﬁber orientations such as [0]15, [05/155/05],

117

 

[05/305/05] and [05/455/05], and different thicknesses such as [07/907/07] and [09/909/09]
were also tested. In addition, various impact velocities were applied to the basic laminate
to identify the feasible range of impact simulation.
5.3 Computational Scheme

To begin with, a variational approach is performed to convert the differential
governing equations, i.e. the equations of motion, into variational equations. Because this
computional scheme was implemented through a user subroutine in ABAQUS, the
Hilber-Hughes-Taylor scheme combined with the Newmark formulae was provided for
numerical time integration, and the Newton iteration for nonlinear analysis was used in
every timeincrement. In order to construct the Jacobian matrix and the residual vector for
the Newton iteration, a consistent linearization technique based on Taylor's expansion
was used to transform the nonlinear variational equations into linear approximate
equations. Subsequently, the Truesdell rate of Cauchy stress and the rate of deformation
tensor were introduced into the linearized equations. With the uses of the Generalized
Zigzag Theory [5.13] and bilinear shape functions, a 4-node shell element was
established in the ﬁnite element formulation. Hence, from above mentioned time
integration scheme and Newton iteration method, nonlinear solutions could be obtained
systematically from many small increments. With the Zigzag J acobian for the geometric
description of deformed element, the computational scheme was of the updated
Lagrangian type.

A similar procedure had been used by Lee [5.4] in the development of a

computational scheme for composite laminates subjected to static loading. The

118

development given below was aimed at dynamic analysis and was closely parallel to

Lee's procedure. In fact, many details could be found in Reference [5.4].
1. Equations of Motion

Consider that a displacement vector u,- satisﬁes the equations of motion

0w + bi = put,"

throughout a body which has a volume V over a time interval and is subjected to the

following conditions:
A. displacement (essential) boundary condition

uizgi(xi9t) on 1ﬁg

B. traction (natural) boundary condition

hi(xi,t) = 03.112]. on 1“,,
C. initial conditions
ui(x,,0) = uf’(Xl.)
“1.: (x1. ,0) = V? (Xi)

where Pg U 1“,, = S, Pg n F, = 0, and S is the surface ofthe body.

In addition, the variables appear in the above equations are deﬁned as follows:

X,- is the vector of initial (or material) coordinate,

x,- is the vector of current (or spatial) coordinate and x,- = X,- + u,;
0",}. (x,,t) is Cauchy (true) stress,

bi(x,-,t) is the body force vector of unit volume,

,0 (x,,t) is the mass density,

n,~ is the unit normal vector of the surface with speciﬁed traction.

119

(1)

(2)

(3)

(4a)

(4b)

 

The indicies i and j denote Cartesian coordinates relative to a ﬁxed reference frame and
they range from 1 to 3.
2. Variational Formulation

The variational form of the equations of motion, i.e. Eq. (1), can be written as follows
IVéili(0-ij,j+bi—mi,n)dV:O (5)
where ﬁx, is an arbitrary weight function and must satisfy the homogeneous form of the

displacement boundary condition, i.e.

éiiiz0 onF (6)

g

Performing integration by parts for Eq. (5), it leads to

 

JValibidV+ jrhéuihidr' jV 2:111 UUdVi-Fde, = J‘Va’lila’lthdV (7)

It should be noted that the fourth term on the right-handed side of Eq. (7) is the virtual
work done by tractions on all delaminated interfaces. It does not exist until delamination
takes place. Since the tractions on the two surfaces created by each delamination have
same magnitude but opposite direction and the weight function is discontinuous across
each delaminated interface, F de, can be written as

n 1

FM = iﬂ 1f (6141" —&Jl_k+1)dr (8)

k=l
where n is the total number of layers and t,- is the traction vector on the kth interface

between the kth layer and the k+1th layer and is deﬁned as t, =0',.jnj on 1“,, (k is

omitted). The domain of area integration Fk stands for the kth interface. Eq. (8) vanishes

when there is no delamination because the weight function should be continuous across

interfaces.

, 120

-.—.._.L4.—..___._._..._,__._.__.

 

Emu-ammu- . .

3. Laminate Theory

Since delamination plays a very important role in damage process, a computational
scheme capable of identifying interlaminar stresses and accounting for both numerical
accuracy and computational efﬁciency is imperative for impact simulation. Considering
interlaminar shear stress continuity and having degrees of freedom independent of layer
number, a laminate theory named the Generalized Zigzag Theory [5 .3] is chosen for the
description of displacement increments [5.4].

According to the Generalized Zigzag Theory, the components of displacement

increments of a particle P at time t to time t + At have the following forms:

Auk(x,y,z) = Auo (x,y) + (Slk + lez + A122 + 3123)Au1(x,y)+

(S; +R§z+ A222 +3223)—a—Aw0(x,y)
5y
Avk(x,y,z)=AvO(x,y)+(Plk 'l'OIkZ'i'CIZ2 +D1Z3)Avl(x’y)+
k k 2 3 a
(P2 +022+C22 +Dzz )5Awo(x,y)

Awk (x, y, z) = AWO (x, y) (9)
where x, yand z are Cartesian coordinates of a particle P at time t, k is layer order,
counting from the bottom to the top.

The independent variables are Auo (x, y) , Av0 (x, y), Aw0 (x, y) , Aul (x, y) and
Avl (x, y). The coefﬁcients are associated with material properties, ﬁber orientations and

thicknesses. They are established from imposing displacement continuity conditions and
interlaminar shear stress continuity conditions on laminate interfaces, and zero shear

stress conditions on laminate surfaces.

121

 

In order to follow engineering conventions for ﬁnite element formulation, the
independent variables are transformed into engineering variables (see Appendix C).

Hence, Auo (x, y), AvO (x, y) and AwO (x, y) are translational increments at a point on the
mid-surface of composite laminates while Aul (x, y) and Avl (x, y) are rotational
increments of a mid-surface point about x — and y — axis, respectively.

4. Large Deformation

In the development of a computational scheme, the rate of deformation tensor is used
in kinematic relations for large deformation analysis, not only because of its linearity and
simplicity to be incorporated into the Generalized Zigzag Theory but also because of its
frame indifference. The rate of deformation tensor is deﬁned as the symmetric part of

velocity gradient, i.e.

1
D.. : 5 (v1.11. + V131); vUJ) (10)

U

It vanishes when a continuum undergoes a rigid body motion.

In describing kinetic relations, both frame indifference (also called objectivity)
condition and energy conjugate condition between stress and strains components must be
satisﬁed. Thus, the Truesdell rate of Cauchy stress [5.4] given below is used in

conjunction with the aforementioned rate of deformation tensor in the analysis:
V0"; =0'UAuH —0'ijui’k +A0',j —0',,Auj,, (11)
where A03). is the Cauchy stress rate. In addition, the Truesdell kinetic relation has a form

of
V0"; =Clljk,Au(k‘,) (12)

in which

122

 

1
Czj'kz =Cijkl + Jijé‘kl —'2‘(0'ik§j1 + 011511 + 031517: + 0175.7.) (13)

and C W is the stifﬁiess matrix of the generalized Hooke's law. By combining Eqs. (1 l)

and (12), the Cauchy stress rate becomes
A0,]. = Cgk,Au(k,,) — ayAuk’k + kaAuM + aﬂAuj’, (14)

This equation can be used for updating Cauchy stress.
5. Finite Element Formulation

Among the commercial ﬁnite element packages, ABAQUS has been commonly used
in engineering analysis. It is especially useful to researches due to its capability of
integrating with user's subroutines. The present study is aimed at presenting a subroutine
for composite laminates subjected to impact loading. The computational scheme is to be
imposed into the ABAQUS for impact simulation.

In ABAQUS/Standard, the implicit dynamic analysis is realized through the Hilber-
Hughnes-Taylor scheme. Accordingto this scheme, the solution at time t + At is not only
dependent on the condition at time t , but also is dependent on that at time t + At. Hence,
the Newton iteration method is used to solve the nonlinear equations.

In the Newton method, the formulation of Jacobian matrix is essential though
sophisticated. It is an expensive numerical process and not always reliable. Hence, a
closed form is preferred in the formulation of the Jacobian matrix. In order to do so, a
consistent linearization technique is introduced into the variational equations given
earlier. Details of the consistent linearization technique and its application to the

variational equations can be found in Appendix A.

123

 

 

Performing the consistent linearization for the variational equations and using the

Newton iteration method for establishing the Hilbert-Hughnes—Taylor scheme, the
following linear system is obtained:

t+At

+aG|, (15)

 

— M(Aii) — (1 + a)[Ks (Au) — Kde, (Au)] = MW, (ml ”A, — (1 + a)G
where

M(Aiz,)= 1,061.,p0Au,,,dV0
KS(Au)=KmatI +ngom

marl _ t

geom = P
K Iyéixid. aijui,kdv

n—l

Kdel (Au) = I: I1(er‘k1Au(II-,l) + JLAui,k)ns(éi1;" — &im+l))da
=1 '

G=F -—F

ext int

+Fdel

Fen: jV&‘ibidV+ jrhéuihida
Fint: IV&"1'j,j0-ijdv
FM = i L tik (@171 —&4i(k+l))da
1:1 '

Miner! = jV 6341101412de

They can also be found in Appendix A and B. It should be mentioned that the left side of

Eq.(15) is referred as Jacobian matrix and the right side of the Eq.(15) is termed as

124

 

———

residual vector (or residual). In the residual part, the ﬁrst term is the inertial force and the
rest part is named as "static" residual.
5.1. Finite Element Discretization

As mentioned earlier, the incremental displacements at any material point of a
composite laminate can be represented by ﬁve independent variables (three translational
increments and two rotational increments with respect to the mid-plane of the composite
laminate). Hence, a two-dimensional quadrilateral element is chosen and associated
bilinear shape functions are used for interpolation of incremental displacements within

each element, i.e.,

Auo =;N.<:.rz>mu.).
Au, =;N.(:.27><Au.).
Avo = Z N. (4’, mm. ).
Av,=;Na(§Ji)(AV1)a “9

In the above equations, Na is the bilinear shape function at the a"’ node of the element
whereas 9‘ and 77 are natural coordinates. And, (Au0)a, (Aul)a, (AI/O)“ and (AV), are

increments of nodal displacement variables for the a"’ node. Because the out-of-plane

displacement increment AwO is also dependent on 63AM) (x, y) and gAwO (x, y), the
x

Hermite cubic shape function should be used for interpolation of the transverse

displacement increment to ensure C ' continuity, i.e.,

125

Aw. -= 21H... (4% 27)(Aw.>. + H... (4‘, xxx—65:— Aw. >. + H... (5, 27)<%Aw.).1 (17)

With the above interpolation descriptions based on the nodal displacement variables, the
displacement increments and strain increments can be exercised [5.4].

The description of geometry is implemented with the Zigzag Jacobian [5.4]. In the
deformed element at a current time increment, the segments of the zigzag line through
the laminate thickness are assumed linear within each layer. Based on this assumption,
the Jacobian matrix for geometry description is formulated and referred as the Zigzag
Jacobian [5 .4].

5.2. Hilber-Hughess-Taylor Scheme
In order to understand the Hilber—Hughes-Taylor scheme, the individual terms in Eq.
(15) should be carefully addressed.
A. contribution of inertial force to Jacobian Matrix
According to Eq. (B.3.5) of Appendix B,

l

M(Aii) = ﬂAt2

 

jVO a1:700 A“idVo °

By using Eq. (5.29)-(5.31) in Ref. [5.4], it yields

 

M(Aii)=ﬁ [VG/yowukAu" +§VkAVk +§kaw")dV0
1 r
= 511 MA u 18
mt. { } { l ( >

where M is the mass matrix and is deﬁned as
M: jVo/O0[({XU0}+{XU1}Z+ {XU2}ZZ+ {XU3}Z3)T(1XU0}+{XU1}Z+ {XU2}ZZ+

{XU3}23)+({XV0}+{XV.}Z+ {XVZ 122+ {XV3}23)T

126

 

 

({XVO}+lXV.}z+ {XI/2122+ {XI/3120+ {W01T1W0}1dV. (19)
details of the matrices appeared in above equations can found in Eqs.(5.29)-(5.3 l) of Ref.
[5.4].

B. stiﬂizess matrix
A similar procedure used by Lee [5.4] has been followed for formulation of stiffness
matrix, (Eq.(5 .63) for material stifﬁress matrix and Eq.(5.64) for geometric stiffness

matrix in Ref.[5.4]). It is expressed below:
[KS]: [Kmatl]+ [ngomJ (20)
where

mat] _ t
K — (yand. CUk,Au(k,,)dv

geom__ v
K — boil“. O'ijuMdv

C. Contribution of delamination to Jacobian matrix

This term does not exist before delamination takes place. Once delamination occurs,
this term becomes imperative as delamination affects the impact response of composite
laminates. A more rigorous derivation of this term requires an additional interfacial
theory for modeling laminate interfaces and could result in a very complicated
incremental displacement ﬁeld. In order to avoid the potential complexity, an interfacial
layer technique and a stress re-calculation technique are introduced to delamination
simulation process to replace the delamination related terms in the formulation procedure.
With these two techniques, the incremental displacement description, i.e. Eq.(9), still

holds.

127

 

 

 

Therefore, in this formulation procedure, the delamination related terms are neglected,
i.e.

K..(Au)=0 (21)
D. contribution of inertial force to residual

This term can be easily determined from its deﬁnition and Eq. (1 9), i.e.
M(a):1.,ew.p.ii.dV.=6{u}”M{ii} (22)
E. static residual at the end of current time increment
As mentioned earlier, if delamination has no contribution to residual vector, it yields

16 l = 1F... ,l— {12.} (23)

 

where

Fm: j Voixibidv+ j nah/1,6162

F..= [Vow adv

if.) U
F. static residual at the end of previous time increment
Similarly to above, but for the end of previous time increment, it gives

{6er .l—{F‘m

.l (24)

 

 

6. Delamination Analysis
6.1. Quadratic Stress Criterion

Since the interlaminar shear stress is predicted from the Generalized Zigzag Theory is
accurate [5.4], the prediction of delamination initiation based on U102 is implemented
with a mechanics of materials approach. A quadratic stress criterion [5.5] is used to judge
the initiation of in delamination in laminate interfaces. The interlaminar normal stress is

neglected in the Generalized Zigzag Theory for simplicity of formulation as well as the

128

 

 

fact that it does not contribute the delamination initiation. Hence, the quadratic stress

criterion implemented in U102 is

_ 2 _ 2
o' 0'
13 + 23 = 1
S. 52
where 313 and 323 are the averaged shear stresses on the concerned interface of an

element whereas S1 and S 2 are the interlaminar shear strengths for 1-3 and 2-3 planes,

respectively. Once the above quadratic stress criterion is satisﬁed, the delamination is
assumed to occur interface of the element.
6.2. Interfacial Layer Technique

The calculation of the interlaminar shear stresses on a concerned interface of an
element is performed via the assumption of an interfacial layer on the interface. The
interfacial layer is artiﬁcially inserted into the laminate interface with thickness about
one—tenth of the adjacent laminae. The integration points for numerical analysis are
imposed in the interfacial layer. The interlaminar shear stresses on all integration points
within the interfacial layer are calculated at every time increment. The calculation of the
averaged interlaminar shear stresses can be carried out the condition of the laminate
interface can be investigated with use of the quadratic stress criterion.

6.3. Stress Re-calculation.

At any time increment, if every interface of an element is delaminated after checking
the averaged stresses against the quadratic stress criterion, the element is then considered
as organized by many sub-elements with the total sub-element number equal to the
number. Each sub-element has the same thickness as the original composite layer though

it has free surfaces on both top and bottom surfaces. However, all the sub-elements

129

 

 

 

separated from the original element share the same four nodal displacements. A process
for stress re-calculation then should be carried out at this time increment. The re-
calculation process is divided into two steps. In the ﬁrst step, the stress tensor at every
integration point in the original element is set to zero. In the second step, the stress tensor
is then re-calculated from the total displacements at the current time for all integration

points in every sub—element.

5.4 Results and Discussions
1. Comparisons with LS-DYNA3D

In order to verify the ﬁnite element scheme presented in the study, i.e. code U102,
computer simulations based on the scheme and two commercial packages, i.e. LS-
DYNA3D and ABAQUS, were performed for comparison. The commercial packages had
been used by some researchers for laminate analysis although they were based on solid
elements and did not account for the continuity of interlaminar stresses on laminate
interfaces. Figure 5.2(a) showed the force histories of a [03/903/03/903/03] graphite/epoxy
laminate under 1 m/s impact [5.2]. The mass of the hemispherical impactor was 24g and
the diameter was 12.5mm. Apparently, the results from both U102 and LS-DYNA3D
agreed very well with those from the experiment. The material response shown in Figure
5.2(b), i.e. the displacement histories, also agreed well among the three groups.

When a thinner composite laminate such as [03/903/03] was investigated, U102 gave a
better comparison with the experiment than LS-DYNA3D, as shown in Figure 5.3(a). It
was believed that delamination was an important damage mode under the impact

condition and the ability of U102 to calculate interlaminar shear stresses and to model

130

 

—‘—

delamination was responsible for the result. However, if the impact velocity was
increased ﬁom 1 m/s to 2.8m/s, the discrepancy became greater, as shown in Figure
5.3(b). Apparently, delamination failure criterion alone was not enough to account for the
damage of the composite laminate under the impact condition. Failure criteria accounting
for other damage modes such as matrix crack and ﬁber breakage should be included in
the simulation.
2. Effect of Velocity and Thickness

The comparison between U102 and ABAQUS (based on S4R shell element) was
presented in Figure 5.4 for a glass/epoxy laminate with a stacking sequence of
[05/905/05]. The impactor had identical geometry and mass as that used in the LS-
DYNA3D investigations while the impact velocity was set as 1.12m/s. As mentioned
earlier, in low-velocity impact, delamination was the primary damage mode. Since U102
accounted for interlaminar shear stresses, it was able to present simulate delamination up
to some extent, resulting in a closer comparison with the experiment than ABAQUS.
However, as the impact velocity increased, the force-displacement relation deviated
further from the corresponding experimental result. Four different impact velocities were

presented in Figure 5.5.

131

 

 

* EXPERIMENT
MPUTATION

2670
. o U102

1780

FORCE (N)

890

 

2.0 1 4.0 6.0 8.0
TIME (ms)

Figure 5.2(a) — Force history of lS-layer laminate with ﬁxed boundary

condition and subjected to lm/s impact

132

 

1.5

I“
O

DISPLACEMENT (mm)

.9
U!

Figure 5.2(b) — Displacement history of lS-layer laminate with ﬁxed boundary

 

 

 

 

 

 

 

 

 

. PEA
* / —\ Edwin'rimbN
[J] A 0 [H02

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

M“

 

”EA

 

 

 

 

 

2.0 4.0
TIME (ms)

condition and subjected to lm/s impact

133

6.0

 

 

 

    
   

 

 

 

2670 I I I I I I I I I I I I I I
graphite/epoxy
*- [ 03 I 903/ 03 ] elastic(fixed) ..
t‘ a
1780 4- .
E . ‘ (simple) .
a _ X.
g ' U102
11,, L 'S.. _ *
\\ .3.
890 “' 9... .1
_ I .
1 “w...
1‘
. a
0 5 l 11—
6.0 8 0

Figure 5.3(a) - Force histories of 9—layer laminate subjected to lm/s impact and simulated

by various failure criteria

134

 

 

 

  
  
     

 

 

 

~ elastic(fixed) *
r .

_ graphite/epoxy
890° [03/903/031 ‘

8 _
8 .
904 ” elastic(simple) ‘
”4 4450 _
Tsai-Wu ‘
test .
O v ‘ 1
5.0 10.0
TIME (ms)

Figure 5.3(b) - Force histories of 9—layer laminate subjected to 2.8m/s impact and

simulated by various failure criteria

135

 

 

  
 

 

 

 

 

5 .
0 Experiment
5 -—U102
—S4R (ABAQUS)

A 4 ~
2
E
“13
8 3- °

2 -

1 A it“, O

I If”:
0 K . . .
0 2 3 4 5 6

Displacement (mm)

Figure 5.4 - Force Vs. Displacement for [05/905/05] at V0=1.12 m/s

136

 

0 Experiment

—'U102

Force (KN)
.h

(A)
1

 

 

 

 

T TF

0 2 4 6 8 10

 

  

 

 

 

 

12
Displacement (mm)
(a) V0=1.12 m/s
7
0 Experiment
6‘ —U102
5 . I
0
A44
3:
d)
8 O
.2 3.
2 a
1 .
0 l I I I 1
0 2 4 6 8 10 12

Displacement (mm)

(c) V0=1.74 m/s

Force (KN)

Force (KN)

 

7

6 _ 0 Experiment
——U102

5 q

 

 

 

 

0 2 4 6 8

10

 

 

 

 

 

1 2

Displacement (mm)

(b) V0=l .40 m/s
7

—— U102
6 — .
0 Experiment
0 O 12

. 4 6 8 1
Displacement (mm)

(d) V0=1.96 m/s

Figure 5.5 - Force vs. displacement for differenr impact velocities on [05/905/0:

137

 

 

 

6 _ 0 Experiment

——U102

Force (KN)

 

 

 

 

0 2 4 6 8 1O

 

Force (KN)

 

 

 

 

12
Displacement (mm)
(a) [05/905/05] V0=l .40 m/s
0
O
0 Experiment
-— U102
0 2 4 6 8 1O 12

Displacement (mm)

(c) [09/909/09] V0=1.71m/s

Force (KN)

 

    

 

0 Experiment

—U102

 

 

2 4 6 8 1 0
Displacement (mm)

(b) [07/907/07] V0=1.94 m/s

Figure 5.6 - Force vs. displacement for different laminate thicknesses

138

12

 

 

0 Experiment

—U102

Force (KN)

N
1

 

 

0 Experiment

—U102

Force (KN)

 

 

 

 

0 2 4 6
Displacement (mm)

(a) [05/905/05] V0=1.40 m/s

 

 

8 0 2 4 6
Displacement (mm)

(b) [03/903/03...]15
V0=1.3lm/s

 

0 Experiment

Force (KN)

 

 

0 2 4 6
Displacement (mm)

(c) [0/90/0...]15 V0=1.30 m/s

 

Figure 5.7 - Force vs. displacement for different stacking sequences

139

 

 

  
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

—U102
0 Experiment 0 Experiment
4 ‘ -—U102 4 ‘
O
A 3 " z-x
9 a E
a a
1.. . 1..
6‘3 2 - ° 8
O
O
1 _
0 I I I I I I
0 2 4 6 8 0 2 4 6
Displacement (mm) Displacement (mm)
(a) [015] V0=l.21 m/s (b) [05/ 155/05] V0=l.29 m/s
5 5
0 Experiment 0 Experiment
4 i —U102 4 ‘ —U102
O
O
O
2 3 - A3 -
E 0 é,
8 8
[Jo-4 2 , L2 2 a
1 - 1 _
O . . . 0 . ' .
0 2. 4 6 3 0 2 4 6
Displacement (mm) Displacement (mm)
(c) [05/305/051 V0=1-28 m/s (d) [05/455/05] V0=1.28 m/s)

Figure 5.8 - Force vs. displacement for different ﬁber angles

140

 

An equivalent investigation to the velocity effect was the thickness effect. In addition
to the 15-layer laminate, i.e. [05/905/05], 21-layer and 27—layer laminates with stacking
sequences of [07/907/07] and [09/909/09], respectively, were also investigated. Although
the impact velocities were not scaled proportional to the laminate thickness, the results
(see Figure 5.6) showed that the simulation result of the lS-layer case was higher than the
corresponding experimental result. That of the 21-layer case seemed to agree very well
with the experimental results while that of the 27-layer case were lower.

3. Effect of Lamination

Simulation results based on U102 for cross-ply laminates with same number of layers
but different stacking sequences were investigated. They were [05/905/05],
[03/903/03/903/03] and [0/90/0/. . .]15, namely 3-lamina, 5-lamina and lS-lamina. Figure 5.7
depicted the results. Although the impact velocities were not quite identical, the
simulation result compared to the experimental result seemed to be higher for the 3-
lamina case, well overlapped for the 5-lamina case whereas lower for the lS-lamina case.

Another comparison concerning effect of lamination was for composite laminates
with similar type of lamination but different ﬁber orientations. The laminates involved in
the study were [0]15, [05/ 155/05], [05/305/05] and [05/455/05]. Apparently, they were all of
3-lamina cases. However, the middle lamina had ﬁber orientation ranging from 0°, 15°,
30° to 45°. Based on the results given in Figures 5.8, the simulation seemed to become
worse as the ﬁber orientation of the middle lamina became smaller. It is believed that
bending-twisting coupling became more signiﬁcant as the ﬁber orientation of the middle

lamina became smaller, resulting in damage modes other than delamination.

141

5.5 Conclusions

1. The Generalized Zigzag Theory was formulated into a ﬁnite element subroutine
(named U102) and integrated into ABAQUS code. Due to the uses of the Truesdell rate
of Cauchy stress and the rate of deformation tensors, the computational scheme was able
to predict reasonable interlaminar shear stresses Via an updated Lagragian algorithm for
composite laminates under large deformation due to impact loading.

2. A interfacial layer technique was presented for the calculation of the shear
stresses on the laminate interfaces. With the interfacial layer, the integration points on
the laminate interfaces could be speciﬁed and the stress tensor at these points could be
calculated.

3. A delamination failure criterion, namely Quadratic Stress Criterion was
integrated into the presented element U102 for the judging of delamination on the
laminate interfaces of an element.

4. A stress re-distribution technique was presented to simulate the delamination of
an element. Because of the stress re-distribution, the energy dissipation caused by
delamination is simulated and a degradated transverse stiffness was estimated

5. Combining the calculated interlaminar stresses with a delamination failure
criterion, and a stress re-distribution technique, the presented shell element U102 was
able to simulate the response of composite laminates under impact loading. Experimental
results seemed to agree with computational studies when the impact velocity was

relatively low.

142

 

’m—__
‘MWﬁ‘l‘q" 1.1“ _ )

 

5.6 References

5.1 D. Liu, B. B. Raju, and X. Dang, “Size effects on impact response of composite
laminates,” International Journal of Impact Engineering, Vol. 21, No. 10, 1998

5.2 Liu,D. and Dang, X.,"Testing and simulation of laminated composites subjected to
impact loading," Composite materials: fatigue and Fracture, 7th Vol., ASTM ST P
1330, pp.273-284, 1998

5.3 Dahsin liu and Xiaoyu Li, "An overall view of laminate theories based on
displacement hypothesis", Journal of Composite Materials, 30(14), 1996

5.4 Chienhom Lee, "A continuum-based shell element for laminated composites under
large deformation", Ph. D dissertation, Michigan State University, 1998

5.5 John C. Brewer and Paul A. Lagace, "Quadratic stress criterion for initiation of
delamination", Journal of Composite Materials, Vol. 22-December, 1988

5.6 ABAQUS/Standard Example Problems Manual, Vol. I and 11, Version 5.5, Hibbitt,
Karlsson & Sorensen, Inc., Pawtucket, RI, 1996

5.7 ABAQUS/Standard User's Manual, Vol. I, II and 111, Version 5.8, Hibbitt, Karlsson
& Sorensen, Inc, Pawtucket, RI., 2000

5.8 ABAQUS/Standard Theory Manual, Version 5.8, Hibbitt, Karlsson & Sorensen,
Inc., Pawtucket, RI, 2000

5.9 ABAQUS/Standard Veriﬁcation Manual, Version 5.8, Hibbitt, Karlsson &
Sorensen, Inc., Pawtucket, RI, 2000

5.10 Bathe, K. J., "ﬁnite element procedure", Pretice-Hall, Englewod Cliffs, New Jersey

143

5.11 Dahsin Liu, Lan Xu, and Xianqiang Lu, "Stress analysis of imperfect composite
laminates with an interlaminar bonding theory," International Journal for Numerical
Methods in Engineering, 37, 2819-2839, 1994

5.12 Xianqiang Lu and Dahsin liu, "Assessment of interlayer shear slip theory for
delamination modeling," Composite Materials: Fatigue and Fracture, 4th Vol,
ASTM STP 1156, 218-235, 1993

5.13 X. Li and D. Liu, "A Zigzag Theory for composite laminates", AIAA Journal, 33(6),

1163-1165, 1995

144

 

Chapter 6

CONCLUSIONS AND RECOMMENDATIONS

6.1 Conclusions

Owing to their high stiffness-to-weight and high strength-to-weight ratios, ﬁber-
reinforced polymer-matrix composite laminates are excellent materials for high-
performance structures. However, their properties in the thickness direction are very poor
as they are weakly bonded by polymeric matrices through laminate interfaces.
Accordingly, when a composite laminate is subjected to impact loading, high
interlaminar stresses along with the low interlaminar strengths could easily result in
interlaminar damage such as delamination. This thesis investigated the response of
composite laminates under low-velocity impact and presented numerical techniques for
impact simulation.

To begin with, instrumented drop-weight impacts ranging from subperforation to
perforation levels were introduced to composite laminates having various dimensions and
thicknesses. Damaged composite laminates were then subjected to compression-after-
impact tests for evaluations of residual properties. In order to understand more about the
relationship between laminate thickness and perforation resistance, and to present an
economical method to improve perforation resistance, thick laminated composite plates
and their assembled counterparts were investigated and compared. An energy proﬁle was
established correlating the impact energy and absorbed energy at all energy levels for
each type of composite plate investigated. This proﬁle addressed the relationship between

energy and damage. As a ﬁrst step to simulate composite response to impact loading, LS-

145

 

DYNA3D was used for numerical analysis. However, due to this software's inability to
describe interlaminar stresses, no delamination simulation could be achieved. As
delamination played a very important role in damage process, a computational scheme
capable of identifying interlaminar stresses and considering both numerical accuracy and
computational efﬁciency was required for impact simulation. A laminate theory named
Generalized Zigzag Theory was formulated into a ﬁnite element subroutine and
integrated into ABAQUS code accounting for interlaminar shear stress continuity and
having degrees of freedom independent of layer number. Due to the uses of the Truesdell
rate of Cauchy stress and the rate of deformation tensors, the computational scheme was
able to predict reasonable interlaminar shear stresses via an updated Lagragian algorithm.
Combining the calculated interlaminar stresses with a delamination failure criterion, the
computer program was able to predict the response of composite laminates up to the
onset of delamination

The following conclusions can be drawn from the four investigations:
A. For size eﬂects

1. Once perforation takes place, both impact characteristics(such as peak force of
impact, impact-contact duration and absorbed energy during impact) and mechanical
properties degradation(such as residual compressive maximum force and residual
compressive absorbed energy) reach turning points. Accordingly, these ﬁve parameters
can be used to identify the perforation thresholds of composite laminates. Since these
parameters are important elements of material response, perforation is the most important
damage stage, as far as material response is concerned, in composite laminates subjected

to impact loading,

146

 

T

2. The study of size effects on impact response of composite laminates should be
divided into two categories: iii-plane dimensional effect and thickness effect. Among the
ratios based on specimen in-plane dimensions, specimen thickness and impactor
diameter, the ratio of specimen thickness to impactor diameter seems to best agree with
the experimental results. Since the impactor diameter is kept constant in this study, the
thickness of composite laminates becomes the most important parameter in impact
response. Hence, the thickness effect is much more signiﬁcant than the in-plane
dimensional effect.

3. To rationalize the superiority of thickness effect to in-plane dimensional effect
on impact response, bending rigidity should be considered as an important element for
perforation analysis since it is proportional to the third power of thickness while only to
the ﬁrst power of in—plane dimension. Its capability of discriminating between thickness
and in-plane dimension seems to be consistent with the experimental results that the
thickness effect is more signiﬁcant than the in-plane dimensional effect.

4. Although perforation is the most important damage stage, as far as material
response is concerned, in composite laminates subjected to impact loading, perforation
alone causes a small portion of mechanical properties degradation. It is delamination
which also plays an important role in impact energy absorption and mechanical properties
degradation since delamination has been identiﬁed as the other primary damage mode in
impacted composite laminates.

5. The utilization of bending rigidity for perforation analysis can also be extended
to delamination analysis. In fact bending rigidity has been successfully used in a previous

study for predicting the potential for delamination of composite laminates subjected to

147

 

‘—

impact loading. The use of bending rigidity in interpreting the mechanical properties'
degradation is well supported by the experimental results that both compressive
maximum force and compressive absorbed energy degrade rapidly when delamination
exists while compressive stiffness does not.

B. For assembly eﬂects

6. Various joining techniques such as mechanical riveting, adhesive bonding,
stiching joining and their combinations were used in assembling two-laminate and four-
laminate composite plates. Pure epoxy bonding proved to be the most efﬁcient joining
technique in assembling the composite laminates together since it gave the highest
bending stiffness and perforation threshold.

7. Among the assembled two-laminate composite plates, the perforation threshold
increased as the bending stiffness increased. However, the increase in perforation
threshold based on the improvement of bending (joining) stifﬁress was limited. A more
efﬁcient way to signiﬁcantly increase the perforation threshold was to increase the
thickness of composite laminates or to use assembled multi-laminate composite plates.

8. A technique named whole energy proﬁle was presented for characterizing
impact-perforation resistance. When penetration took place, the absorbed energy was
approximately equal to the impact energy. When perforation occurred, the absorbed
energy was again smaller than the impact energy. Experimental results revealed that the
equal-energy interval, which was the difference between the penetration threshold and
perforation threshold, increased as the thickness of composite plates increased.

9. Assembled composite plates were very comparable to laminated composite

plates in both bending stiffenss and perforation threshold. This result veriﬁed the

 

148

 

advantage of using assembled composite plates over thick laminated composite plates
since the cost of making thick laminated composite plates with high quality increased
signiﬁcantly as the thickness increased.

C. For evaluation of LS-DYNA3D

10. The LS-DYNA3D gives accurate predictions for the response of impacted
composites before delamination takes place. In order to closely simulate the progress of
impact-induced damage, a new type of ﬁnite element and a new failure criterion, which
account for interlaminar stresses, are required.

D. For improvement of impact simulation

11. The Generalized Zigzag Theory was formulated into a ﬁnite element
subroutine (named U102) and integrated into ABAQUS code. Due to the uses of the
Truesdell rate of Cauchy stress and the rate of deformation tensors, the computational
scheme was able to predict reasonable interlaminar shear stresses via an updated
Lagragian algorithm for composite laminates under large deformation due to impact
loading.

12. A interfacial layer technique was presented for the calculation of the shear
stresses on the laminate interfaces. With the interfacial layer, the integration points 011
the laminate interfaces could be speciﬁed and the stress tensor at these points could be
calculated.

13. A delamination failure criterion, namely Quadratic Stress Criterion was
integrated into the presented element U102 for the judging of delamination on the

laminate interfaces of an element.

149

 

 

 

14. A stress re-distribution technique was presented to simulate the delamination
of an element. Because of the stress re-distribution, the energy dissipation caused by
delamination is simulated and a degradated transverse stiffness was estimated

15. Combining the calculated interlaminar stresses with a delamination failure
criterion, and a stress re-distribution technique, the presented shell element U102 was
able to simulate the response of composite laminates under impact loading. Experimental
results seemed to agree with computational studies when the impact velocity was

relatively low.

6.2 Recommendations

1. To identify the residual properties of laminated composites after impact, the
tension and bending test can be introduced into experimental studies to specify the ﬁber
breakage and matrix crack effects on the degradation of laminated composite material
properties.

2. A bilinear element can be developed based on C 0 continuous for transverse
deformation. This can signiﬁcantly reduce number of nodal variables. So the degree of
freedom of every element can be reduced signiﬁcantly.

3. The stress update based on the Truesdell rate of Cauchy stress is good for
small incremental strain. This implies that the time increment still must be small to
ensure the convergence in the nonlinear analysis. Since every strain increment can be
treated as ﬁnite strain for large deformation analysis, the integration of the deformation
rate through a time increment may improve the accuracy of strain calculation and

accelerate the convergence speed.

150

 

4. Since the transverse normal stress is not well predicted from the Generalized
Zigzag Theory, the recover technique may be used to recover the transverse normal
stress. And this stress can be integrated into the delamination failure criteria and
considered as normal pressure to determine the ﬁiction between delaminated plies.

5. Further studies should include all failure models when laminated composite
are subjected to impact loading. The matrix crack and ﬁber breakage determine the
perforation threshold. Therefore it is imperative to incorporate the all failure modes into
impact simulation.

6. An independent program may be developed to stay alone with ABAQUS.
This program can even use explicit time integration methods to improve convergence
characteristics.

7. Delamination can be easily integrated into static analysis.

8. A contact algorithm may be introduced into this element for impact
simulation. This will help to develop an independent program for laminated composite

structural analysis.

151

APPENDICES

152

 

#—

APPENDIX A
CONSISTENT LINEARIZATION AND APPLICATION TO VARATIONAL

EQUATIONS

A.1 Consistent Linearization

Let's suppose
x" = X + u(x,t)v (A.l)
where V denotes the V -th iteration. X is the vector of initial (or material) coordinate. x is

the vector of current (or spatial) coordinate and x = X + u, u is the displacement vector.

For the (v+1)-th iteration, we has
x‘v“) = x" + Au (A.2)
Now, consider a nonlinear function deﬁned by equation
f (x"*”) = 0 (A3)
The left side of equation (A.3) can be expanded in Taylor series at the V -th iteration

point x" ,

f(x(v+1))= f(xv) + (aifuv + 8AM) + R(x") (A4)
6‘

6:0
The veriﬁcation of the 2nd term in the right side of equation (A4) is given below.

Using Taylor expansion to function f (x + Ax), it gives

f(x + Ax) = f(x) + [63 f(x))Ax + R (A.5)
x

The 2nd term in the right side of equation (16) can be transformed into,

6 6
[5; f (x + 6496)] = (51(0)), Ax (A6)

6:0 :0

153

 

Then, we can ﬁnish our veriﬁcation by replacing the dummy variable y with x.

6 6
-f(y)] Ax =[— f 06)] Ax (A-7)
[6y £=0 ax 5:0
Now, we can go back to (A4). By neglecting the R term in (A4), we write (A.3) as

f(x”) = —[—6—f(xv + 5AM) (A8)
65

6:0

01'

f (xv) = -L(f) (A-9)

where L is a operator, and deﬁned by the following operation,

L( 13%; f(x” + 5mg) (A10)

5:0
and this operation is referred as Consistent Linearization.

The following examples with the use of the operator L will beneﬁt us in later
derivations:
( 1) Deformation Gradient and its transpose

By deﬁnition, we have

 

62 .
L F.. = x.‘ +£Au. A.ll
(y) [anj(, ,)]'0 ( )
Then, directly we get
L(F )= —§—Au (A 12)
'j 6X. i '

Similarly, we can get

154

L(FT )= iAu

nm m
I!

where the superscript T denotes the transpose of the matrix.

(2) The determent of J acobian matrix

a a ,
L(J) -_- [Edet[—aZ—(Xt '1’ 811110)] :0

by expanding det[5§:—(x," + aAu,)] according its deﬁnition, then doing the

I

From deﬁnition,

differentiation, we can obtain
L(J) = JvAui,j
(3). Inverse of F and its transpose
Let's consider
FF“” =1
then, applying the operator, we have
L(F)F“” + FL(F"”) = 0

Now using equation (A.12), we get

6
L(FJ—n) = *Ei—I)(a—X‘A“1]EJM

1
Please note, where superscript (— I ) represents the inverse of a matrix.

Similarly, by using equation (A.13), we have

6
L161“) = -F.l‘“[—8X ijF-t“

I}
k

155

(A.13)

(A.14)

(A.15)

(A.16)

(A17)

(A18)

(A.19)

where (- T ) denotes the inverse of a transposed matrix.
A.2 Linearization of the Variational Equations

We can rewrite equation (7) as

F... - F... + F... — M1... = 0 (A20)
01'
F... (,1...) — F... 06””) + F... 04“") — M... (x.,..‘“”) = 0 (A21)
where
Fm = [ V2.22.» + I .h étlihida (A22)
F,,,, = j .61.“. Daydv (A23)

1
“(1.1) = 50%,; + u“)

Miner! = j V 4341,0141...“

Fdel : ":3. [.1 tik (&‘f _ di:/(+0)!“
1:1 '

Applying Consistent Linearization on equation (23) will lead to

" L(F (xv) + Fdel (xv) ~ Miner! (xi,ttv)

ext

)+L(F

int

)_— L(Fdel) + L(Minert):Fext(xv) _ Pint

(A24)

A detailed discussion of every term in equation (A24) is given below.

156

(I) The derivation of the 2nd term in left side of equation (A24)

Recall that

F...= [.44 -2.

1.1 U

01'

6
Pint: IV0[a—Xk_&li]kajO-ij‘ldV0

Hence, we have

L(F...)= [flair

k

 

at,.][L(E;"°)a;JV + r;§">L(a,,.)JV + F,;.">o-;.L(J)]ctV0 (A25)

By deﬁning

L(oyj )= A0,].
and using equation (A. 14), (A. l 8) and

J VdVO = dv

we can reach

 

L(F...)= [[ a; 441— F.€;’°[a%dw.]ﬁj'”a. +ij.‘”Ao;; +F.§-"'A05Au.]dv
k I

(A26)

Because of the following relations,

a é’li FkirIl) : ﬁlim
ax, '

 

 

i,j

° 61., ijf” :61.
ax,

157

 

171,]

15” Ft") z a”
5X, m 1]

equation (A26) becomes

L(F :1.) 6a,]. (UAo' +0" jA—ukk ogkAujk)dv (A27)

int)

But please note that the increments of Cauchy stress tensor still need to be further
speciﬁed. Otherwise, equation (A27) is useless.
As we well known, the relationship between the symmetric Second Piola—Kirchhoff

(2nd PK) Stress Tensor S 0' (which is deﬁned with respect to initial reference
conﬁguration X ) and the Cauchy Stress Tensor 0'11 is

=JF,,E ”UHF? 1) (A28)
By using Linearization operator to equation (A28), we have

L(S.)= L(J>(F,,£ 00.117”) +J”L(F.t” :.(F.-t”)”

(A29)
+J(F W) L< 0.117 a”) +J”(4t”)"az.(F.-t”)

When we chose the instantaneous motion conﬁguration x at the end of V—th iteration

(instantaneous motion state) as the reference conﬁguration X , we have,

J" :1
(pg-1)) = 6,. (A30)
L(J) Auk k
L(Ef”) )= —Au,.,j (A31)
So, ifwe deﬁne
L(Sij) = A033.

158

 

 

 

 

 

 

 

namely the Truesdell rate of Cauchy stress, and substitute above equations into equation
(A29), it gives
A033. = oﬁAuk‘k — oy‘éAuM + A03]. — oﬁAuN (A32)

Hence, form V -th to (v+l)-th iteration, the increments of Cauchy stress tensor are

A0,]. = A0"; — o'yvAuM + ogAui‘k + ofjAujJ (A33)
This equation can be used to update the Cauchy stress tensor. The Truesdell rate of
Cauchy stress is the increment of 2nd P.K. stress tensor with respect to the reference
conﬁguration at time t. It can be determined experimentally with Total Lagrange
formula, in which the engineering stress increments are determined by constitutive

equations, and then transformed into 2nd P.K stress tensor with respect to present

conﬁguration. Therefore the Truesdell rate of Cauchy stress can be further speciﬁed as
A0; = Cgk,Au(k,,) (A34)
Inserting (A34) into (A33), we obtain
A0,]. = Cgk,Au(k,,) — onguM + JLAuM + oijAujJ (A35)
Let’s consider equation (A27) again and deﬁne
A05.” = A6,, + 05mm -— 0,; Au j, (A36)
Hence, equation (A27) becomes to
L(Em ) = [ VéuUAojybdv (A37)
On the other hand, by substituting (A35) into (A36), it gives
A0,?” = C3,,Au(k,,) + O'ZJAuM + oyjAuN — 0,;AuM

OT

syb _ t v

159

Therefore, the following formula are obtained:

int

L(F )=]V&1U C5,,Au(k,,)dv+ 1.61“. oéAuhkdv

OI'
L(F... ) = K ’"‘”’ + K g... (A39)
where
K""’” = [ V011,]. C3,,Au(k,,)dv (A40)
KW = [ V611”. 0,3211.de (A41)

(2) The derivation of the 4”' term of the left side of equation (A24)
In equation (A23), the integration can be expressed as with respect to the initial

conﬁguration X .

MM, = j .0 654ipoude0 (A42)
Hence, we have

L(M... )= 1.41.-p.44. )dV. (A43)
01‘

L(M,,,,, )= j ,0 éuipoAudeo (A44)

(3) The derivation of the ﬁrst term of the left side of equation (A24)

From equation (A20), and

bidv = ,oobipdVO

160

 

 

(where the superscript p represents the prescribed body force vector acted on an unit
mass), we have

Fa, = j .0 61., ,oobipa'VO + I ,h 61,/ma (A45)
When the Consistent Linearization operation is performed on Fm ,

L(F )= L([Voé‘uipobipdVo + [rhea/z da)

er!
the ﬁrst term of the right side will vanish. Then, we obtain
L(F“): L(j r[1614,72 da) (A46)
Apparently, this term is important to contact problem, but, for non-contact problem,

such as prescribed constant boundary, this term can be neglected. Since contact problem

is not our concern in this dissertation, this term is neglected in following derivation.

(4) The derivation of the 3-rd term of the left side of the equation (A24)

In equation, t, = can]. on Th, n is the unit normal vector of k -th layer surface glued
together with, or delaminated from, (k +l)-th layer. Therefore, the inﬁnitesimal surface
element on a surface of the deformed conﬁguration is

a, = njda
Now, we introduce Nanson’s Formula, which is
njda = JFj:"T)nfdA
where the superscript 0 represents at time t = 0. (- T ) represents the inverse of a

transposed matrix.

With Nanson’s Formula, equation (8) can be rewritten as

161

F_,,,, =2[ 0",]. JF.‘ T>n °(011f—0’uf"“’)dA
k:]

I

where again, subscript 0 denotes at time t = 0 , corresponding to the undeforrned

conﬁguration.

Based on above preparation, we can perform Consistent Linearization on F d ,,

L(Fd,,) = Z] l. L(o'UJFjis'T’ )n: (614‘ — 61.)“) )dA
1:1 ,
or, in detail
L(F,,,)= :(L [(L (0315:”) + a,L(J)FJ.§-“ + a,JL(Fjg'T>))n:(§uf — 614“”)dA
By using equation (A.15), (A.19), and deﬁnition
L(0'0. )= A0",j
we obtained

L(Fdel) =

n—l

Ik£AoijJFj:‘T) + 0",.jJVAum’ij‘SJ) —0'g.JFj(,"T)[a: Au )F‘ ”)713 (61" —&¢1"+”)dA
k=l ' r

I 1

Substituting
_ 6
Ffr “(KAI/ll) = Au“.
into previous equation, it leads to

(F.1d): :2:[( A0371“), T) + 0'.. Au", ij‘s T) —0'U. Au F‘ T017412." —0ufk+”)dA

By exchanging the position of dummy 1 and j in the 3-rd term of the right side in above

equation, and using Nanson’s Formula again, we have

162

 

-_—

n—l

L(Fdel) = Z L (A0}; + O—ijAumm " O-ijAuiJ 154k _ ﬁlibuya
k=l '

Comparing this equation with equation (A.27), and doing the same derivation

procedure as done from (A.27) to (A38), we ﬁnally get
71-]
L(Fdel) = Z L (Clijk/Au(k,l) + ail-Aura )“s (d‘im - dJ.i"'+l))da (A-46*)
k=l '

But, we'd like to mention that the computational implement of (A.46*) is somehow
difﬁcult, since the difference of the virtual displacement across the delaminated interfaces
of laminates is impossible for an element. That is the delaminated element has the same
4-node as pre-delamination, therefore no any degrees of freedom can be introduced into
the element to simulate the delamination. For this reason, the implementation of
delamination simulation was based on a thin ﬁlm technique and stress re-distribution

technique. They both were discussed in Chapter 5.

(5) The ﬁnal form of Consistent Linearization of Variational Equation

Based on above derivation and discussion, the Linearized Variational Equation is
M(Aii,.) + K(Au) — Kde,(Au) = Fm (xv) — Em (xv) + Fde, (xv) — M061?) (A.47)

where the following deﬁnitions are used

 

Mom, ) = j ,0 éiipOAui,“dV0 (A.48)

K(Au) = K"“’” + KW (A.49)

Km" = j Val“. C5,,Auwdv (A50)

KW = j Vat“. aéAuhkdv (A.51)

Kde, (Au) = I: L (C5,,Aumn + agAum S (ow — awn; (A-50*)
2, ,

163

 

 

Fm: j Véuibidv+ j nag/qua (A52)

Fm: j VéiJUJO'ijdv (A53)

Fde, = "i L if (a; _ alga” )da (A.53*)
k=l ’

Miner! = J‘Va’lilouiﬁdv (A54)

164

APPENDIX B

NONLINEAR SOLUTION PROCEDURE

Within this APPENDIX, the numerical methods used by ABAQUS/Standard to solve
nonlinear equations and nonlinear ordinary differential equations are brieﬂy introduced.
2.1 Nonlinear Solution Methods

For a static problem, a set of nonlinear algebra equations (equilibrium equations)
needs to be solved with numerical methods. The equilibrium equations, symbolically,
have a form of

FN(uM)=O (B.1.1)
where, N indicates the Nth equations that conjugate to the Nth variable in the problem
and M is the M-th variable. So, solving Eq. (B.1.1) is our basic problem
A numerical technique, named as Newton’s method, for solving the nonlinear

equilibrium equations is generally used in ABAQUS/Standard, which is stated as follows.

Assume that, in an increment of a step, after iteration i, an approximation uf’ , to the

M
i+l ’

solution has been obtained. If C termed as (i+1)th correction, is the difference from

this solution to the exact solution of Eq. (B. 1 .1), then,

Fwy +0“

1+1) =0 (B.1.2)
Taking expansion with Taylor series, neglecting all terms but linear terms, and

denoting

KNP _ aFN
' au”

 

(uf’) (B.1.3)

165

Then gives

K,NPC.P =—F,.” (13.1.4)

1+1

where K ,NP is the J acobian matrix and F,” is the residual. Residual is the value of left

side of Eq. (B.1.1) for approximation u ,M .

After solving Eq. (B. l .4), the next approximation to the solution is then

uM =uli" +C

i+l

M
{+1

And the iteration continues.

The measurement of the convergence of Newton’s method is based on ensuring that
all entries of current residual and correction are sufﬁciently small. If the convergence is
reached, then the ﬁrst iteration of the next increment begins after updating the loading
or/and boundaries to the next increment.

In Newton’s method, usually, it is important but difﬁcult to formulate the J acobian matrix
in close form or to calculate it numerically — an expensive (and not always reliable)
process.

To formulate the J acobian matrix in close form, the consistent linearization is done in
APPENDIX A, although the linearization is completed before ﬁnite element
discretization. The similarity between Eq. (A47) in APPENDIX A and above Eq. (B.1.4)
is easy to see by comparing two sides Of them since the left side of both equations is
linear change induced by current correction, and the other side of them is the residual of
the last approximation. Therefore, after discrezation, Eq. (A.47) will result in an exact

form of Eq. (B.1.4), directly in the corresponding side.

166

 

 

 

 

Hence, in our ﬁnite element formulation procedure, the J acobian matrix will be
constructed directly from left side of Eq. (A47) in APPENDIX A, and the residual from
the other side, by ﬁnite element discretization.

2.2 Implicit Dynamic Analysis

For a time-dependent problem, particularly, for severely nonlinear dynamic analysis,
a direct-time integration method, using central differencial operator with implicit
schemes, is provides by ABAQUS/Standard. While explicit schemes obtain dynamic the
solution at time t + At based entirely on available the solution at time t, implicit schemes
obtain dynamic solution at the time t + At based not only on the solution at time t, but
also on the same solution at time t + At. Hence, nonlinear equations must be solved by
using Newton’s method for every time increment. Recall Eq. (B. 1 . 1), since we are
dealing with dynamic problem now, we can rewrite Eq. (B.1.1) as

FN(uM)=—MNMu'M +GN(uM)=O (B.2.1)

Where we separate the dynamic equilibrium into two parts, d’Alembert force part,
—M ”Ma“ , which is directly depends on the double time derivatives, and the rest part,

GN (uM ), which is the total force at degree of freedom N, excluding d’Alembert force,

and often referred to as the “static residual.”

By comparing Eq. (A.20) with Eq. (8.2.1), it is easy to see that, —M is the

inert

d’Alembert force, and Far —— F . + F de, is the “static residual.”

met
In ABAQUS/Satndard, the Hilber-Hughness-Taylor scheme is used for time
integration of the dynamic problem. According to this scheme, at time t + At, the overall

dynamic equilibrium equation, Eq. (B.2.l), is replaced by

167

 

 

FN t+At :_MNMuM t+At +(1+a)GN

—aG” ,=0 (13.2.2)

 

 

 

t+At

This scheme is completed by the Newmark formulae for displacement and velocity

 

 

 

 

 

 

integration:
1
u ”A, :14, +Atul, +At2[(§—ﬂ)u’ ,+,6ii )+ij (B23)
and
a HA, = 54, + At((1—7)u' 5741..) (B.2.4)
with
ﬂ—la—ar
4 a
_ l _ a
7 2
and
—lSaSO

3
Where on, B, and y are the (Newmark) parameters of the integration scheme.

Therefore, Eq. (B.2.2) — (B.2.4) are the equations must be solved for time t + At if we
assume that the approximate solutions at time t are already achieved. Apparently, they are
nonlinear equations, since the solution at time t + At is not only based on solution at time
t, but also on the same solution at time t+ At. The Newton’s method again is used to
solve the nonlinear equations, with including the Jacobian contributions from effects that
depend on time derivatives.

Now, assume that, at time t+ At, after i—th iteration, approximations,

M

ui t+At a

to the solution have been obtained. Let Cf, ”A, be the

 

-M "M
“i and “i t+At’

168

 

difference between u,” ”A, and the exact solution to Eq. (B.2.2). After the same

 

procedure from Eq. (B.1.2 ) to Eq. (B.1.4), we reach the same form of linear system of

 

 

 

 

 

 

 

 

 

 

 

 

 

 

equation:
KiNPIHAt CiM lt+At = 'EN t+At (B25)
Where, the residual vector is
EN t+At : ' MNM uiM t+At + “+00 GIN t+At _anN t (B"26)
and the J acobian matrix is,
N N - P N ..
kiNP t+At 26i— t+At + £(i) t+Ar + 957(ﬂ)lt+m
6uP au” du 6L7 du
Considering following relations derived from Eq. (B.2.3) — (B.2.4),
(A) __z_
du t+At ﬂAt ’
and
("3) ——’
du ”N ﬂAtz ’
and relations derived from Eq. (B.2.2),
6F N 6GN
a? t+At : (1+ a) Ell—p— t+At
6F N
Suﬁ? t+At =0
6F N
art—N” AA, = ‘M NP,
the J acobian matrix becomes
K NP — 1 6GN M NP 1 B 2 7
i z+Ar_(+a)5;p_t+Ar— ﬂAtZ (..)

169

Once Eq. (B.2.5) is solved, the next approximations to the solution are then

determined by

M
i+l

 

=u:’|,+cM

i+l

 

u

(+A! I+At ’

M
i+1

 

 

u

HA!)

1
_ M -M .. .
t+At—u lr+Atu |1+At2[(§_ﬂ)uMiz+ﬂftN+jl

t+At )’

- M
ui+l

 

 

= W. +At((1— 7)iiNl,+ﬂi.-’fl
and the iteration continues.
2.3 Consistent Linearization Applied in Hilber-Hughess-Taylor Scheme

Follow the same procedure described as above for implicit dynamic analysis, but

develop a detailed, direct implicit dynamic integration procedure according to

APPENDIX A. First, rewrite Eq. (A20) as

F(u) = —Mmm (ii) + G(u) (B.3.1)
where,
M , (ii ) = M inert
and
G(u) = Fm — Fint + Fde, (B.3.2)

Then, for the Hilber-Hughess-Taylor scheme, at time t + At, replace Eq. (B.3.1) by

F — aG| = 0 (B33)

I

 

 

: _M inert (u)

 

”A, + (l + a)G

HA! HA!

With Eq. (B.2.3) and Eq. (B.2.4), a complete form of Hilber-Hughess-Tayor scheme

for implicit dynamic integration is obtained.

170

 

 

Accordingly, corresponding to Eq. (B.2.5) to Eq. (B.2.7), the following equations are
given, with superscript vto present the V-th iteration, instead of subscript i, which is
used in section 2.2 and 2.3:
=_FV

K" C"

 

 

t+At t+At t+At 9

 

 

 

 

 

 

 

 

 

or L[F HA,] = -F V ”A, , (consistent linearization form, since K V ”NC” ”A, = L[F my],
see APPENDIX A),

F" ”A. =—M.-....(ii") A... +(1+a)GV A... ‘aGV r’

LlF . ....l = — LlM... (.2) l + (1 + a>LlG l=

 

 

 

]+ (1+ a)L[Fert — Pint + Fdel] t+At

inert t+ At

—L[M

 

 

By using the results obtained in Eq. (A.47), the linear system can be further
determined as

+ aG|,
(13.3.4)

— M<Aii> — <1 + a>[I<. (Au) — K... mm] = M.. (ii)

 

 

”A, —(1+a)G

t+At

and the deﬁnition of above terms can be found in Eq. (A.48)-(A.54).
Here, a further discussion of the ﬁrst term in the left-hand side of Eq. (B.3.4) is needed.
Recall that

dii
(E)

1
HA! zit—2’

 

therefore

.. .. a .. (iii a .. dii
_ M(Au) = _ L[Minert(u)]=_ JV a<§1iWi)E‘—AUCIV=_ Jro 5J(&4ip0ui)EAudV

171

 

l 1 1
-— Iro&‘ipo(W)Au'dV_‘ﬁ J‘VO&Ni/00AuidV_—E2— [Va’li/OAuidV

l

(B.3.5)
Apparently, Eq. (B.3.4) will result in the linear system of dynamic equilibrium

equations when ﬁnite element discretization applied,

 

+aG|

 

(+A! t

1 ..
[— M IBM — (1 + a)(1<, — Km )JAU _ MU — (1 + a)G

where AU is the correction vector of the nodal unknowns of the whole model; U and

U are the “displacement” vector, and “acceleration” vector of the nodal unknowns of
whole model, respectively.
2.4 Newmark Algorithm

In the Hilber-Hughess-Taylor scheme, parameter, or, is used to provide numerical
damping, when — g S a <0. The numerical damping with this form is controllable, slowly

growing at low frequencies, with more rapid growth in damping at high frequencies. But
when 0t=0, the Hilber-Hughess-Taylor operator is the Newmark operator (Trapezoidal
rule, at [3:1/4, y=l/2), hence, there is no numerical damping. Put oc=0 into Eq. (8.3.6), the

Newmark Algorithm can be readily introduced as following a time-approximate recursion

algorithm:

Predictor:
off}, = U, + (AtU, + £13.20, ) (3.4.1)
U533, = U, + éAtU, (3.4.2)

172

 

 

U”) :0 (3.4.3)

t+At

 

Solver:
K‘”)AU,V++A‘, = Rx; (3.4.4)
Where
V V 1 V V
K( ) = M( )ﬂ_At7+K“(’) A... _K§.1)..A. (3.4-5)
Rf:l.= —M"’)U.‘:Z. +6.1"). (13.4.6)
Corrector:
U fit.” = U311. + AUXX.” (B-4.7)
. . 2
US.” =Ufrz. +EAUXX.” (13.4.8)

.. .. 4
U(V+1) :U(V) +

t+At (+At At 2

 

Auffg,” (3.4.9)

The Newmark Algorithm gives a complete direct implicit integration procedure for
implicit dynamic analysis. But, for a clear understanding of this procedure, the following
points are discussed:

(1) At time t=0, initial ”acceleration” is generally unavailable. Therefore,
ABAQUS/Standard provides an option to estimate initial “accelerations” with mass
matrix and initial residual. The same way is used for the calculation of accelerations
after impact. If this option is not activated, the zero initial “accelerations” will be
assumed; this may cause extremely expensive iterations to reach convergence for the
ﬁrst increment when large loads are applied suddenly.

(2) In the Newmark Algorithm, there is no numerical or artiﬁcial damping, so energy is

conserved. Again, when sudden loading is applied, such as an impact, high

173

 

frequencies will be induced, resulting in convergence difﬁculty. Under such cases,
very small time increments are needed, and the computation cost will be high. To
avoid these disadvantages, an option is available in ABAQUS/Standard: that is to
introduce material damping to depress the high frequency noise, which require a
structural damping matrix to be introduced to the model, but the accuracy of the
damping is not easy to obtain.

(3) The accuracy of the solution is controlled by half-step residual tolerance in implicit
dynamic integration. Convergence check at the end of each time increment, which
ensure any component of the residual vector and correction vector sufﬁciently small,
just ensures equilibrium at these time points but does not say anything about the
quality of equilibrium at intermediate time points. Based on the assumption that the
accelerations vary linearly over the time interval, the magnitude of the large entry of
residual at half time increment provides a measure of the transient solution. But, the
calculation of the half-step residual, the storage of more intermediate results, increase
the cost. A relative cost-effective half-step residual tolerance will be important (see
ABAQUS Theory Manual)

(4) The correction and prediction of vectors are easily carried out with the Newmark
Algorithm, but ﬁnite element rotation is not a vector although angular velocity and
angular acceleration are vectors. Therefore, the correction and prediction of ﬁnite
element rotation is not so simple as given in above algorithm. Since the correction
and prediction of ﬁnite rotation is carried out by ABAQUS automatically, we do not

pay too much attention to this problem. And in impact simulation, the time increment

174

needs to be small, so the rotation increment will be small too, an inﬁnitesimal rotation

can be approximated as a vector.

175

APPENDIX C

IDENTIFICATION OF NODAL VARIABLES

The degrees of freedom at a node in user element U101 described in [5.4], do not
follow the ABAQUS conventions—they are neither nodal displacements along, nor nodal
rotation angular about any axis of local coordinate system, or global coordinate system.
For convenience of use in an ABAQUS model including native element and user
element, the nodal variables of a user element had better follow the ABAQUS
conventions. In order to do so, this APPENDIX is given.

Suppose the mid-plane (at 2:0) of laminate is located in the mp-th layer, then the

incremental displacement ﬁeld at the mid-plane can be written as

Au '"P (x. y.0) = Au. (x. y) + S Au. (x. y) + S.” ,3 Aw. (x. y) (C.1)
x

Avm” (x, y,0) = AvO (x, y) + H” Aul (x, y) + P,” 5AM) (x, y) (C.2)

Awm” (x, y,0) = AwO (x, y) (C.3)

after substituting 2:0 into Eq. (4.30)-(4.32) in [5.4].
From the spin tensor W, we know that, at any point P(x,y,z), the incremental rotations

about x- and y—axis, respectively, are

A‘IJ,=AW23=l iAV-EAW ,
' 2 62 6y

A‘I’ =AW13 =—1—[—a—Au ——a-Aw).
y 2 62 6x

176

By using Eq. (5.11)-(5.24) of [5.4] and then substitute 2:0, above two equilibrium

equations give the mid-plane incremental rotations about x- and y-axis,

A‘Pf'p = 1(0,” Av, + 02"” 2— AwO — _6_ AWO] ,
2 5y 5y
and
ALl’y’W’ = 1(R,””’ Au, + R,” _6_ AwO — i AwO] ,
2 6x 6x

respectively. And further from these two equations, we obtain

 

 

Av, = ,1” [2A‘I’ﬁ'p +(1—02""’)3Aw0] (O4)
0, p 6y

A - 1 2A‘P""’ 1 R""’ 6 A C 5

“1 ‘lep[ y +( _ 2 )5; W0] ( -)

In order to solve AuO , Av0 from Eq. (C.1)-(C.2), substitute Eq. (C.4)-(C.5) into them,

Au '"p (x, y,0) =AuO (x, y) + S,” +[2A‘I’f’p + (1 — pr)—?—Aw0 ]+ S?” 3AwO (x, y)
R, p 6x 6x

Av””’ (x, y,0) = Ave ()6. y) + lep —01,,,,, [2qusz + (1 — 0;” ) 2:; AwO ]+ P,” (Ti—Aw, (x, y)

I
Then further we have

[2AM + (1 — 112;”)3Aw0 ] — 5;": 3A).), (C.6)
y 6x 6x

 

Auomp (X, y,0) =Au(x, y,0) _ Slmp Rmp
l

AvO mp (x, y,0) =Av(x, y,0) - P,’"” ——1—[2A‘I’;"” + (1 —— 03"”)2 Aw0 ] — P2"'p 2~Aw0 (C.7)
O’Np w 6y

1

Recall Eq. (4.30)-(4.32)[5.4],

177

Auk(x,y,z)=Auo(x,y)+(S,k +R,kz+ 24,22 +B,z3)Au,(x,y)+
k k 2 3 6
(S2 +R22+Azz +Bzz )EAWO(x,y)
Avk(anaZ)=AVo(an)+(Rk +01kZ+C122 +DIZ3)AV1(x9y)+
k k 2 3 a
(P2 +022+sz +D22 )gAwo(x,y)

Awk (x, y, z) = Aw0 (x, y)

By substituting Eq. (C3) and Eq. (C.4)-(C.7) into Eq. (4.30)-(4.32) in [5.4], we
complete our transformation with following ﬁnal increment displacement ﬁeld,

Au" (x. y. z) = Au” (x. M) — Sr” +,12A~P;W + (1— 12:63AM.) — S.” 3 Aw.
R, 6x 6x

1 m m a
+<St +sz+4.22 +3.23) R—..12.u,p Huang/3...].

(S: + Ré‘z + A222 +B,z’)a£Aw0(x,y)
x

=Au'"” (x,y,0) + (SN + R52 + A,,z2 + B,,z3)A\P)"np+

ll
(Sf, +R;‘,.2+A2,.z2 +B,,z’)§;Aw0(x,y) (C8)

1 6 6
k _ mp __ mp mp _ mp _ mp
Av (x, y, z) — Av (x, y,0) P, —0,""’ [2A‘l’x + (1 02 )_6y Awo] P2 —6y Aw0

+(P," + ofz + C,z2 + 0,23) —1—[2A‘I’f”’ + (1 — og'P)3Aw0]+
Olmp 6y

(sz +052+sz2 +Dzz3)aﬁy-Awo(x,y)

=Av’"”(x, ,O)+ P"+0"z+C 22+D z’)A‘P"”’+
y It It 1t 1! x

178

(P.’: + 0.12 + (3.22 + 0,23%... (x. y) (C9)

LMM(Ly)=Aw”KL)4D (C10)

where, following notations are used (subscript t represent the transformed coefﬁcients):

 

 

 

 

 

2 2 2 2
SI]: = mp (Slk — Slmp)’ let : mp Rik ’ A1]: = mp Alk ’ Bl]: : mP Blk
R1 R1 R1 R1
1
5;, = S; + (S." — 5.”) ,, <1 — Rs”) — S.” .
R1
Rk=Rk+RhJ—a—R”)
2: 2 1 Rmp 2
1
1 ”I 1 "I
A2r=A2+A1W(l—R2P) thsz'i'Biﬁp—(l—Rzp)
1 1
Pf: i,<P.*—P.”). 0.’:=—2,;0.". C.’:=—2,,,,,—C.".
01 01 01
D1,: : 2mp le
Q
l
P.’: = P: + (P.* — P.”) (1 — 0.”) — P.
01
1 m
0:, =0: + 0: 5.0-0;)
1
1 m 1 m
C21=C2+C16m7(1_C2p) D2r=D2+D1W(1_02P)

l 1

Therefore, according to Eq.(C.7)-(C.10), the independent unknowns are three mid-

plane displacements, two mid-plane rotations, and §Awo,%Awo. And the new
x

incremental displacement ﬁeld, after transformation, has same form of original ones. But

179

 

 

please note that the new coefﬁcients, deﬁned in above equations, are calculated from
original coefﬁcients. Hence in ﬁnite element formulation, nodal variables are mid—plane

engineering variables.

180