100 Impact Analysis of Fiber-Reinforced

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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

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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




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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.




8/23

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




9/23


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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.




11/23

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].




12/23

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)




13/23

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.




15/23

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.




16/23

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.




17/23

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.




18/23

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.




19/23

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.




20/23

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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100 Impact Analysis of Fiber-Reinforced