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Journal of Composite Materials
DOI: 10.1177/00219983070821772007; 41; 2985Journal of Composite Materials
Ki Ho Joo, Hansun Ryou, Kwansoo Chung and Tae Jin KangConstitutive Law
Impact Analysis of Fiber-reinforced Composites Based on Elasto-plastic
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Impact Analysis of Fiber-reinforcedComposites Based on Elasto-plastic
Constitutive Law
KI HO JOO, HANSUNRYOU, KWANSOOCHUNG AND TAE JIN KANG*
School of Materials Science and Engineering, Seoul National University
Shillim-dong, Gwanak-gu, Seoul 151-744, Korea
ABSTRACT: Impact analysis of 3D braided composites and laminated plane wovenfabric composites was performed using the elasto-plastic constitutive law to describethe nonlinear, anisotropic and asymmetric properties of fiber-reinforced composites.As for the yield criterion, the modified DruckerPrager yield criterion was utilized torepresent the anisotropic and asymmetric properties, while the anisotropic hardeningwas described based on the kinematic hardening with the anisotropic evolution ofthe back-stress. Experiments to obtain the material parameters of the proposedconstitutive law were also carried out based on uni-axial tension and compressiontests for fiber-reinforced composites. Then, the proposed constitutive law wasimplemented into a finite element code and was verified by comparing the finiteelement simulation of the impact tests with experiments. In addition, impact
performance of the braided composites and laminated composites was alsocompared with each other.
KEY WORDS: modified DruckerPrager yield criterion, elasto-plastic constitutivelaw, anisotropic back-stress evolution rule, 3D braided composites, low velocityimpact.
INTRODUCTION
LAMINATED COMPOSITES HAVE been used in aircraft, high performance automobiles,
sporting goods industries, and civil infrastructure because of their good in-plane
properties and ease of handling. However, the use of the laminated composites in many
engineering fields has been restricted by their poor impact resistance and low through-
thickness. To overcome the drawbacks of the laminated composites, three-dimensional
braided composites have been recently developed [14]. 3D braided composites have the
ability of producing complex near-net-shape performs and this can reduce the
*Author to whom correspondence should be addressed. E-mail: [email protected] 17, 912 and 16 appear in color online: http://jcm.sagepub.com
Journal ofCOMPOSITE MATERIALS, Vol. 41, No. 25/2007 2985
0021-9983/07/25 298522 $10.00/0 DOI: 10.1177/0021998307082177 SAGE Publications 2007
Los Angeles, London, New Delhi and Singapore
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manufacturing cost. The 3D braided composites also have higher delamination resistance
and damage tolerance for impact because of the through-thickness reinforcement.
In order to describe the mechanical behavior of fiber-reinforced composites, two
approaches are most common: micromechanical and continuum approaches. The
micromechanical approach is usually used in the homogenization technique to predict
the macroscopic properties of heterogeneous materials, considering the properties of
individual constituents and their geometry. However, the micromechanical analysis is
limited to a unit-cell scale, since large computational time is required. Therefore, the
continuum approach is more common, especially for structural analysis involving
numerous unit-cells as is the case of this study. In the continuum approach, the composite
is considered as a homogeneous material which has a uniform average property, ignoring
differences in the individual properties of constituents.
Fiber-reinforced composites show the nonlinear behavior in stressstrain curves because
of the evolution of defects such as matrix cracking, delamination, and fiber breakage. To
describe the nonlinear behavior of composite materials, the plasticity theory was used in
this work. Because of the micro-structural changes during the deformation, the damage
theory has been widely used to describe the nonlinear behavior of fiber-reinforced
composites [19,20]. However, the plasticity theory has also become popular for its
simplicity [2123]. In plasticity, the nonlinear behavior can be described only based on
macroscopic uni-axial tension/compression test results without microscopic information.
In this article, the impact performance has been evaluated based on the elasto-plastic
constitutive law to describe the nonlinear behavior of laminated composites and 3D
braided composites. In addition, fiber-reinforced composites show strong directional
difference (anisotropy) and also the different constitutive behavior between tension and
compression (asymmetry) [7,8]. Anisotropy and asymmetry of yield criterion have been
achieved by combining linear and quadratic terms of stress components [9,10] or lineartransformation of the stress deviator [11,12].
The experimental procedure to obtain the material parameters of the proposed elasto-
plastic constitutive law is also presented for laminated composites and 3D braided
composites, utilizing the measured tensile and compressive stressstrain curves along
various material directions.
Many test methods have been developed to measure the shear properties of composite
materials. These test methods include the Iosipescu shear, 45 degree flexure, torsional tube,
picture-frame panel, 10 degree off-axis shear, and 45 degree tensile tests. Among those
tests, the Iosipescu shear and the 45 degree tensile tests are widely used because of the ease
of specimen fabrication and low cost. In this article, those two tests are carried out tomeasure the shear properties and their performances are compared to each other using the
impact tests.
In this article, a plane-stress constitutive law was utilized to describe the
nonlinearity and anisotropic/asymmetric properties. For the anisotropy of the fiber-
reinforced composites, the initial anisotropic yielding as well as the anisotropic hardening
has been considered in this work. As for the initial anisotropic yielding (and also for
asymmetry), the modified DruckerPrager yield criterion has been used [10]. To account
for the anisotropic hardening, the anisotropic back stress evolution rules based on the
kinematic hardening law have been utilized [13]. The constitutive law has been
incorporated into the commercial dynamic finite element code ABAQUS/Explicit usingthe user subroutine VUMAT [14] and impact tests have been performed to verify the
2986 K.H. JOO ET AL.
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simulation results. The impact properties of the braided composites are compared with
plain woven laminated composites.
THEORETICAL BACKGROUND
Asymmetric Orthotropic Elasticity
The orthotropic elastic stressstrain relation under the plane stress condition can be
expressed as:
r CeT or Cee 1
or, in the matrix form,
x
y
xy
0B@
1CA
Ex
1xy
yEx
1xy0
yEx
1xy
Ey
1xy0
0 0 G
2666664
3777775
T or C
"ex
"ey
2"exy
0B@
1CA 2
where, Ex, Ey, vx, vy and G are Youngs moduli and Poissons ratios in the axial and
transverse directions, and the shear modulus respectively. Note that the subscripts x and
y stand for the two orthogonal directions of the fiber-reinforced composites respectively,
while the subscripts T and C mean the material properties for the tension andcompression behavior, respectively.
Modified DruckerPrager Yield Criterion
The modified DruckerPrager yield criterion was developed to describe the anisotropy
and asymmetry of composite materials in the plastic deformation [10]:
p_2
x
22_
x_
y 2
22
_2
y
32
33
_2
xy
1=2 q_
x _
y iso 0 3
while:
r_
r a 4
where a is the back stress, riso is the size of the yield surface, p, q, b22, b33 and k are
material constants characterizing the anisotropic and asymmetric behavior. The modified
DruckerPrager yield criterion can describe different values of tensile yield stresses in two
directions (anisotropy) and different values of tensile and compressive yield stresses
(asymmetry). Also, the shear yield stress can be given independently. The five materialparameters therefore can be determined from two tensile yield stresses Tx ,
Ty,
Impact Analysis of Fiber-reinforced Composites 2987
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two compressive yield stresses Cx, C
y in the axial and transverse directions, and the shear
yield stress Yxy or the tensile yield stress along the 45 degree direction Y1.
Hardening Rules
In the isotropic-kinematic hardening law, the effective quantities are defined considering
the following modified plastic work equivalence principle, i.e.:
dwiso r a dep isod " 5
where dep is the plastic strain increment and d " is the conjugate effective plastic strain
increment. Here, iso is obtained from the effective stress of the initial state (or the
isotropic hardening case, which is relevant to the relationship, d " rdep) by replacing
rwith r a. Note that the effective plastic strain increment surface is stationary even for
the isotropic-kinematic hardening case.
For the back stress increment, the Chaboche type back stress evolution rule [15] was
used and in order to account for the directional difference of the back stress evolution for
the highly anisotropic materials such as fiber-reinforced composites, the anisotropic back
stress evolution rule [13] was proposed as:
da !1r a
isod " !2 ad " 6
where !1 and !2 are the fourth-order tensors representing the anisotropic hardening
behavior. For the plane stress condition, the evolution rule can be written in the matrix
form:
dx
dy
dxy
264
375
g11 g12 g13
g21 g22 g23
g31 g32 g33
264
375
nx
ny
nxy
264
375
h11 h12 h13
h21 h22 h23
h31 h32 h33
264
375
x
y
xy
264
375
0B@
1CAd " 7
where n r a= iso is the normal direction on the yield surface, gij and hij are the
components of!1 and !2 respectively.
For the uni-axial tension test in the x direction, the following differential equation is
obtained from Equation (7):
dx g11nx h11xd ": 8
From the associate flow rule, the in-plane components of the plastic strain increment are
given as
depd "m 9
where m @ =@r_:
Equation (8) gives the following stressstrain relation, with Equation (9):
x nx iso g11nxh11
1eh11"x=mx
10
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wheremxis thex component ofm. Then,g11and h11were obtained from the curve fitting
of the uni-axial tensile stressstrain curve in thexdirection. The other two components for
each matrix in Equation (7) are determined from the parallelism of a and r. In this case,
g21 h21g31 h31 0. For cases of simple tension in the y direction and pure shear tests,
similar formulations are possible [11]:
y ny isog22ny
h221eh22"y=my
11
xy nxy isog33nxy
h331eh33"xy=mxy
: 12
If the uni-axial tension test in the 45 degree direction is performed instead of the pure shear
test, Equation (12) is replaced as:
xy 1
2 nxy iso
g33nxy
h331eh33"1=m1 13
where m1 is the 1 component ofm in the 12 coordinate system.
Note that the pure kinematic hardening is considered here; i.e., iso remains constant
during the plastic deformation. Also, the associate flow rule was used for the in-plane
components of the plastic strain increment while the nontrivial out-of-plane component of
plastic strain increment is given from the incompressibility; i.e.:
d"pz d"px d"py: 14
Numerical Implementation Procedures
The developed elastic-plastic constitutive law was implemented into the general
purpose finite element program ABAQUS/Explicit using the material user subroutine
VUMAT [14]. To update the stress increment which involves solving a nonlinear
equation, the NewtonRaphson method based on the incremental deformation theory was
utilized [16].
Based on the constitutive law developed here, the stress update scheme is outlinedusing the predictor-corrector method based on the NewtonRaphson method.
The updated stress is initially assumed to be elastic for a given discrete strain
increment e. Therefore:
rTn1 rn C e 15
where the superscript T stands for a trial state and the subscript denotes the process time
step. Also, the trial plastic quantities are preserved as the previous values:
"Tn1 "n and aTn1 an: 16
Impact Analysis of Fiber-reinforced Composites 2989
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If the following yield condition is satisfied with the trial values for a prescribed elastic
tolerance Tole for each active surface:
rTn1 aTn1 iso5Tol
e 17
the process at the step n 1 is considered elastic. If the above condition on yielding is
violated (4Tole), the step is considered elasto-plastic and the trial elastic stress state is
taken as an initial value for the solution of the plastic corrector problem until the yield
condition is satisfied during the iteration.
MATERIALS AND CHARACTERIZATION
Material Preparation
The preform of the 3D braided fiber reinforced composites was fabricated using
the 3D-circular braiding machine with 2014 carriers and 104 pistons and by 4-step cycle
movements [13,14] as shown in Figure 1(a). The preform of the laminated plane woven
fabric composites which best fit the volume fraction and thickness of the braided
composites, was made by laminating seven layers of the plane glass-fiber woven fabrics
with the same directional alignment as shown in Figure 1(b). For the composite preform
fabricated, RTM (Resin Transfer Molding) process was performed using the epoxy resin
as matrix. After 10 h injection of the epoxy resin into the RTM cast and curing in the oven
at 1308C for 120 min, the 3D braided fiber reinforced composites and the laminated plane
woven fabric composites were prepared. The basic mechanical properties of the glass fiber
and the epoxy resin used in this study are shown in Table 1. Because the fiber volume
(b)
(a)
2 1
45
y
x
Figure 1. Preforms of (a) 3D braided composites, and (b) laminated plane woven composites.
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fraction is one of the most important factors to determine the mechanical properties of
composite materials, the fiber volume fraction was measured by the ASTM D2584
method. The fiber volume fraction of the glass fiber was measured as 56% after
combustion in the furnace at 6008C for 3h.
Elastic Properties
The tensile and compressive tests of braided composites and laminated composites were
carried out by the standard procedures, ASTM D3039-76 and ASTM D3410-87
using the 10-ton tensile and compression test machine Instron 8516 system. The measured
true stressstrain curves of tension and compression tests are shown in Figures 24 for
braided composites and Figures 5 and 6 for laminated composites. To account for the
directional differences, the tensile and compressive tests were performed in x and y
directions. These figures show that the linear regions exist in the early strain ranges but
slight nonlinear behavior is also shown beyond the linear regions. Note that because the
laminated composites have the same structure for the x and y directions as shown
in Figure 1, tension and compression have the same material properties between the two
directions (Table 2). Beyond the measurable strain regions for the compression tests, the
specimens were buckled due to the thin specimen geometry. Youngs moduli oftension and compression of braided composites and laminated composites are listed
0.0 0.2 0.4 0.6 0.8 1.00
50
100
150
200
Initial linear lineYield point
Axial tension
Transverse tension
Truestr
ess(MPa)
True strain(102)
Figure 2. Tension test results of the braided composites.
Table 1. The basic mechanical properties of the glass fiber and the epoxy resin.
Property Fiber Epoxy
Density (g/cm3) 2.46 1.12
Youngs Modulus (GPa) 86.9 2.13
Poissons ratio 0.2 0.37
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Material Parameters for Yield Criterion
In order to account for the anisotropic and asymmetric properties of composite
materials using the modified DruckerPrager yield criterion, the five material parameters
were determined from two tensile yield stresses Tx , T
y, two compressive yield stresses
Cx, C
y in the xand y directions. These yield stresses were determined as the values which
deviate from the linearity in the measured stressstrain curves, as marked in Figures 26
and listed in Table 4. The calculated parameters for Equation (3) are also shown in Table 5and the tri-component yield criteria surfaces are shown in Figure 8.
0.0 0.3 0.6 0.9 1.2 1.5
Truestress(MPa
)
0
50
100
150
200
250
300
Tensile test
Compressive test
Initial linear line
Yield point
True strain (102)
Figure 5. Tension and compression test results of the laminated composites.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Truestress(MPa)
0
20
40
60
80
45Tension test
Pure shear test
Initial linear line
Yield point
True strain (102)
Figure 6. Tension at 45 degree and pure shear test results of the laminated composites.
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Material Parameters for Hardening Laws
In order to represent the anisotropic hardening behavior using the anisotropic
kinematic hardening law shown in Equations (1)(3), the three stressstrain curves
were considered after the initiation of plastic deformation: the x, y direction tension and
the pure shear test curves. The detailed procedure to determine the anisotropic hardening
parameters have already been discussed earlier and the results are listed in Tables 6 and 7.
Using the material parameters obtained from the measured test data, true stresstrue
strain curves were re-calculated using the proposed constitutive equation as shown in
Figures 911, which confirm good agreement with the measured data. Note that the
parameters in Equations (10)(13) were obtained from the tensile hardening curves.
However, the compressive behavior also shows good agreements with the experiments as
shown in Figures 10 and 11.
IMPACT TESTS AND SIMULATIONS
Impact tests were performed using the impact testing system ITR-2000. The tool
dimensions are 150 mm (length) 150 mm (width) for the rectangular die where thepneumatic clamp pressure of 500 kPa is applied to hold the specimen firmly,
Table 2. The number of tested specimens for each measurement.
Laminate Braid
Tension 6 Axial 6
Transverse 6
Compression 6 Axial 6Transverse 6
Shear
458 direction 5 458 direction 5
Pure shear 5 Pure shear 5
Impact
19 J 8 19 J 8
31 J 8 31 J 8
Table 3. The elastic constants of braided composites and laminated composites.
Braided Laminated Explanation
ETx (GPa) 46.81 24.55 Tensile modulus in x-direction
ETy (GPa) 19.74 24.55 Tensile modulus in y-direction
ECx (GPa) 37.74 22.61 Compressive modulus in x-direction
ECy (GPa) 13.10 22.61 Compressive modulus in y-direction
G (GPa) 11.10 5.30 Shear modulus (experiment)
13.85 2.02 Shear modulus (calculation)
x 0.18a 0.17b Poissons ratio
a Referred value [1].b
Referred value [18].
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P
Center line
Strain gagesSpecimen
38mm
12mm 20mm
76mm
4mm
90
Thumbscrew
(a)
(b)
Figure 7. Iosipescu shear test: (a) Schematic view of the test and (b) specimen.
Table 4. Measured yield stresses of braided composites and laminated composites.
Braided Laminated Explanation
Tx (GPa) 71.8 140.8 Tensile yield stress in x-direction
Ty (GPa) 16.6 140.8 Tensile yield stress in y-direction
Cx (GPa) 130.2 80.8 Compressive yield stress in x-direction
Cy (GPa) 21.4 80.8 Compressive yield stress in y-direction
Y1 (GPa) 30.8 19.9 Tensile yield stress in 458-direction
Yxy (GPa) 20.3 9.8 Shear yield stress
Table 5. The material constants of the modified DruckerPrager yield criterionof braided composites and laminated composites.
Braided Laminated
p 0.776 1.37
q 0.224 0.37
b22 4.945 1.0
k 2.146 1.0
b33 2.632 (shear test) 6.05 (shear test)
1.354 (45 degree tension) 6.24 (45 degree tension)
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2 1 0 1
0.4
0.2
0.0
0.2
(a)
(b)
Contours of every 0.1 interval of
1.0 0.5 0.0 0.5 1.0 1.5 2.0
1.0
0.5
0.0
0.5
1.0
1.5
2.0
syy/s
sxy
/s
Contours of every 0.1 interval of sxy
/s
syy
/s
sxx
/s
sxx
/s
Figure 8. Predicted yield surfaces of (a) braided composites and (b) laminated composites.
Table 6. The parameters for the kinematic hardening braided composites andlaminated composites.
Braided Laminated
Shear 458 tension Shear 458 tension
g11 (MPa) 110,437.6 141,787
g22 (MPa) 784.05 141,787
g13 (MPa) 0 31 785.8 0 225 088
g23 (MPa) 0 77 867.75 0 225 088
g33 (MPa) 36 800.04 78 651.8 70 900.0 366 875
h11 1066.0 903.1
h22 15.75 903.1
h13 0 699.7 0 814.5
h23 0 350.55 0 814.5
h33 179.9 366.3 403.5 1718.6
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Table 7. The parameters for the isotropic hardening braided compositesand laminated composites.
Braided Laminated
0
(MPa) 71.8 140.8
a (MPa) 103.6 157.0
b 1066 903.1
0.0 0.2 0.4 0.6 0.8 1.0
Truestress(MPa)
0
50
100
150
200
Experiment
CalculationTension in x
Tension in y
Tension in 45
True strain (102)
Figure 9. Tension hardening curves of braided composites.
0.0 0.2 0.4 0.6 0.8 1.0
Truestress(MPa)
0
50
100
150
200
Experiment
Calculation
Compression in x
Compression in y
Shear
True strain (102)
Figure 10. Compression and shear hardening curves of braided composites.
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130 mm (length) 130 mm (width) for the specimen, 6.25 mm for the impactor radius,
35 mm for the hole radius of specimen holder, and 3 mm for the thickness of specimen. In
the test, the specimen is initially clamped between the die and the specimen holder and the
impact test is conducted using a rigid body impactor with a mass of 6.5 kg. The impact
tests were carried out for two impact velocities, 2.4 and 3.1 m/s, which correspond to the
impact energy of 19 and 31 J, respectively. The impact test setup is shown in Figure 12.
Piezo-electric force transducer and optical displacement sensor are used to acquire load
and displacement values.
The loaddeflection curves obtained from impact tests of braided and laminatedcomposites are compared in Figure 13. Note that the impact instrument used for this study
0.0 0.5 1.0 1.5 2.0
Stress(MPa)
0
50
100
150
200
250
300
Experiment
Calculation
Tension in 45
Compression in x
Tension in x
Shear
Strain (102)
Figure 11. Hardening curves of laminated composites.
Impactor
Specimen holder
Specimen
Die
Figure 12. Schematic view of the impact test setup.
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does not provide reasonable results after peaks so that only the data before peaks are
considered. The figure shows that the peak load values of the braided composites are lower
and the deflections of the impactor tip at the peaks are larger than those of laminated
composites. In addition, Figure 14 shows that it takes longer for braided composites to
reach their peaks. Taking these things into consideration, it can be concluded that braided
composites greatly deform with lower impact loads in a longer period of time. The areas
below the curves are similar to the energy absorbed until the maximum peaks. This
behavior is clearly seen in Figure 15 which shows the peak loads depending on deflection
and time, where eight representative peak values for each test condition are projected to
the time-deflection plane.
The projected images taken with an equally distributed backlight are shown in Figure 16
where the images have the same specimen size and resolution and damaged parts aredarker. The anisotropic properties of braided composites are reflected well in the figures.
Displacement (mm)
0 2 4 6 8
Load(N)
0
2000
4000
6000
8000
Laminate, 19J
Braid, 19J
Laminate, 31J
Braid, 31J
Figure 13. Typical loaddisplacement curves for impact tests.
Time (ms)
0 1 2 3 4 5
Load(N)
0
2000
4000
6000
8000
Laminate, 19J
Braid, 19J
Laminate, 31J
Braid, 31J
Figure 14. Typical loadtime curves for impact tests.
Impact Analysis of Fiber-reinforced Composites 2999
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3000
4000
5000
6000
8000
7000
3.03.3
3.6
3.9
4.24.54.8
2345
678
Braid - 19J
Laminate - 19J
Braid - 31J
Laminate - 31J
Load(N)
Deflection(mm)
Tim
e(
ms)
Figure 15. Peak load values of impact tests depending on deflection and time.
(a) (b)
(c) (d)
Figure 16. Images of impacted specimens for braided of impact energy (a) 19 J, (b) 31 J and laminated
of impact energy (c) 19 J, (d) 31 J.
3000 K.H. JOO ET AL.
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Damaged areas of braided composites are larger and the damage propagation profiles
show stronger anisotropy than laminated composites. On the other hand, the areas for
laminated composites are smaller and relatively more confined to the regions around the
impacted points showing less anisotropy and negligible xy directional dependence. Note
that the directional dependence is described in the x and y directions, not in the direction
of the yarn path. The localized propagation along the yarn is not considered in the model.
The dominant propagation in the axial x-direction of braided composites can be
explained as follows. The bending rigidity in the transverse direction is lower than therigidity in the axial direction because the orientation angle of the yarns in braided structure
is closer to the axial direction where the braiding angle for the specimen in this research
was about 308 to the axial direction. Results of three-point bending tests for braided
composites are shown in Figure 17 [2]. The bending deformation is more likely to happen
in the less resistant direction resulting in a large deflection finally causing dominant
propagation in that direction. For the case of plain wovenyarn, the rigidities in both
directions are even so that damage does not propagate in any dominant direction but
rather are bounded in the surroundings of the impacted point.
For finite element formulations, three-dimensional rigid body elements (R3D4) were
used for the rigid body tools such as the specimen holder, die and impactor, while four
nodes reduced shell elements (S4R) with five integration points in through-thickness were
utilized for the specimen. As for the boundary conditions, the die and the specimen holder
grip the specimen and all degrees of freedom of the die and the specimen holder are fixed.
In Figure 18, finite element meshes before and after impact deformation are shown.
The friction between the impactor and the specimen was ignored.
In Figures 19 and 20, the results using asymmetric orthotropic elastic constitutive law
(Case I) are compared with the experimental results. The comparison confirms that Case I
significantly overestimates the stiffness, since it cannot account for the softness in the
nonlinear region of the hardening behavior.
Figures 19 and 20 also show the comparison of the results using the elasto-plastic
constitutive law with three different yield criteria under the isotropic hardening condition:Mises yield criterion (Case II), original DruckerPrager yield criterion (Case III) and
Displacement (mm)
0 2 4 6 8 10 12 14
Load(N
)
0
100
200
300
400
500
600
Axial
Transverse
Figure 17. Loaddisplacement curves of three-point bending tests.
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Figure 18. Finite element meshes (a) before and (b) after the impact.
Time (ms)
0 1 2 3 4 5
Force
(N)
0
2000
4000
6000
8000
10000
12000ExperimentCase ICase II
Case IIICase IV
Figure 19. Timeforce curves for impact tests and simulations using different yield criteria with isotropic
hardening, braided, impact energy 19 J.
3002 K.H. JOO ET AL.
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modified DruckerPrager yield criterion developed in this research (Case IV). The
following Voce type hardening law was utilized:
iso 0 a1expb " 18
where 0 is the initial yield stress in the reference direction and a and b are material
parameters, which were obtained from the tensile behavior in the axial direction as shown
in Table 7. Note that the results of Case II and Case III overestimated the stiffness since
the Mises yield criterion does not consider both the anisotropy and asymmetry and the
original DruckerPrager yield criterion only accounts for the asymmetry. The result of
Case IV is the best among all so far, since the anisotropy in the initial yielding and
asymmetry are included. However, the result of Case IV still overestimated. This is because
prediction of the nonlinear range in the transverse and shear directions is not good enough
even for Case IV since the anisotropic hardening is not accounted for. The stressstrain
behavior in the x direction, as shown in Figures 911 shows much higher hardening
behavior. Therefore, the behavior in the transverse and shear directions was considered as
a higher hardening rate for Case IV.
The modified DruckerPrager yield criterion was used for the following comparisons.
The results are compared for the proposed kinematic hardening (Case V) and the isotropic
hardening (Case IV) in Figures 21 and 22. Note that, as mentioned earlier, the 45 degree
tensile test can be replaced with the pure shear test to account for the shear properties [17].
Therefore, the results using the 45 degree tensile test (Case VI) are also considered in this
comparison. Figures 21 and 22 show that the results of Case VI are virtually the same as
the results of Case V since the anisotropic hardening is properly accounted. The results
also show that Cases V and VI show good agreements with the experiments since Cases V
and VI properly consider the hardening differences as well as asymmetry, while Case IV
does not account for the transverse and shear hardening. The data of impact energy 31 Jare also considered and showed similar characteristics to the data of 19 J.
Time (ms)
0 1 2 3 4 5
Force(N)
0
2000
4000
6000
8000
10000
12000
ExperimentCase ICase IICase IIICase IV
Figure 20. Timeforce curves for impact tests and simulations using different yield criteria with the isotropichardening, laminated, impact energy 19 J.
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CONCLUSIONS
Impact analysis of 3D braided composites and laminated plane woven fabric composites
was performed using the elasto-plastic constitutive law to describe the anisotropic and
asymmetric properties. Modified DruckerPrager yield criterion was utilized to represent
the anisotropic and asymmetric properties, while the anisotropic hardening was described
based on the kinematic hardening with the anisotropic evolution of the back-stress. The
model showed good agreement with the experimental data while the other models not
considering asymmetry or anisotropic hardening overestimated the stiffness. The impactproperties of the braided and laminated composites were compared and shared that braided
Time (ms)
0 1 2 3 4 5 6
Load(N)
0
1000
2000
3000
4000
5000
Experiment
Case IV
Case V
Case VI
Figure 21. Timeforce curves for impact tests and simulations using the isotropic hardening and anisotropic
kinematic hardening laws, braided, impact energy 19 J.
Time (ms)
0 1 2 3 4 5 6
Loa
d(N)
0
1000
2000
3000
4000
5000
6000
Experiment
Case IV
Case V
Case VI
Figure 22. Timeforce curves for impact tests and simulations using the isotropic hardening and anisotropic
kinematic hardening laws, laminated, impact energy 19 J.
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composites absorb impact energy with a lower load, through a longer period of time with
larger deformation as propagating the stress for the larger area than the laminated
composites. The constitutive model developed in this research was found to be valid and that
further studies on nonlinear mechanical analysis were ongoing based on the model.
ACKNOWLEDGMENTS
The authors of this paper would like to thank the Korea Science and Engineering
Foundation (KOSEF) for sponsoring this research through the SRC/ERC Program of
MOST/KOSEF (R11-2005-065).
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