The failure properties of short fiber reinforced compositeswith polymer matrix with consideration of the fiber/matrix­debonding

a a* b b bK. Zhu, S. Schmauder' , R. Stößel, S. Predak , G. Busse, Wolfgang Lutz

a Staatliche Materialprüfungsanstalt (MP A) Universität Stuttgart, Pfaffenwaldring 32, D-70569Stuttgart, Germany

b Institut für Kunststoffprüung und KunststoftKunde (ISK) Universität Stuttgart, Pfaffenwaldring 32,D-70569 Stuttgart, Germany

The statistical strength of short fiber reinforced composites such as polymer matrix compo­sites (PMCs) with different fiber volume fractions is investigated in this work using com­bined cell models and Weibull statistical approach. based on the combined cell models [5],statistical combined cell models with consideration of the fiberlmatrix-debonding have beendeveloped in this work. The global mechanical behaviour of PMCs reinforced with randomlyoriented short fibers (e.g. polymer matrix composite with 20vol.% or 30vol.% glass fibers) iscaIculated based on the rule of mixture. In all cases, predictions of the behaviour by thestatistical combined cell model are compared with experiments and they show goodagreement.

KEY WORDS: Short fiber reinforced composites, failure and Weibulliaw

1. Introduction

Polymer matrix composites (PMCs) are frequently reinforced with strong continuous or short fibers.In the case of short fiber reinforced PMCs, not only random arrangements of fibers but also theirrandom orientations are observed and their mechanical properties are highly dependent on their com­position, the matrix, as weil as the type and volume fraction of reinforcernents. The complexity ofsuch affecting parameters makes a complete theoretical description of the behaviour and the failure

properties of short fiber reinforced composites with metal and polymer matrix nearly impossible. Inthis respect, a micromechanical analysis of the local composite failure process opens a possibility topredict the microscopic failure property of composites [1-3].

In this work, basis ofthe work [4], cornbined cell models [5] are applied to describe the overall flowbehaviour of composites reinforced with randomly oriented short fibers (polymer matrix with 20vol.% or 30 vol.% glass-fibers) with good and amiss adhesive strength (Fig. 1). The failure ofsuchcomposites with different fiber volume fractions is investigated using statistical combined cellmodels based on Dong's combined cell models [5] and Weibull statistical approach [6, 7]. The sta­tistical combined cell model has been adopted, which originally was developed for micromecha­nically considering fiber fracture in composites. It allows to caIculate separately two types of unitcells, i.e. unit cells without damage, or with damage. In all cases, the predictions by the statisticalcombined cell models will be compared to the experiments. Micromechanical models shown in this

work can be applied to describe most technical relevant composites varying from the simpleinc1usion type and interpenetrating microstructures to functionally graded materials.

2. Numerical modeling

2. J. Combined cell models (CCM)

The plastic flow in the matrix surrounding a single fiber is analysed for different external loadingdirections using unit cells involving two 2D models and two 3D models (Fig. 2), which represent thegeometrie description of an arbitrarily oriented fiber in three principal (perpendicular) cross-sections

in the Iocal co-ordinate system, from which local stress-strain curves (JA3(c,e), (JA2(c,e), (JB(c,e),

(Je (c, e) have been obtained as a function ofthe fiber orientation e. By averaging the stress-strain

* Corresponding author. Fax: 0049-711-685-2635

E-mail: Siegfried.Schmauder@mpa.uni-stuttgart.de.Prof..Ph.D.

curves of different cells weighted according to their volume relationship, the overall flow behaviour

Cf(E,e) of the composite has been estimated.

The overall flow behaviour of composites with arbitrary ShOli fiber orientations (3D randomorientation: j(e)= 1 or preferred orientations, e.g. planar random orientation: f(e) = sin e) can becalculated by an appropriate integration over all fiber orientations:

n/2 / n/2a(E) = j[aA(E,e)+ 2aBC(E,e)]r(e)de 3 jf(e)de (1)with

aA (E,e) = aA3 (E,e )cos e + aA2 (E,e )sin ecos e +sin e

aBC(E,e)= aB(E,e)sin8+aC(E,e)COSesine+cose }

(2)

2.2 Statistical description of the mechanical behaviour without consideration of theadhesion effect between jibers and matrix

For modelling the damage behaviour for short fiber reinforced composites, a damage rule shall bedefined in accordance with experimental results. Fiber failure is the principal source of damage ifthere occurs no debonding between fiber and matrix. Usually, the tensile strength of brittle materialscan be weil described by Weibull's law as

p,(cr')= [-exp[ -(:: r) (3)with

"0F : Stress in an unbroken fiber,

PF(CfF): Fracture probability ofthe brittle phase at fiber stress crF,

a~: Characteristic stress,

mF: Weibull modulus in case offiber failure

and mF and ()~ are shape and scale parameters of Weibull's law.

Therefore, statistical combined cell models (SCCM) have been adopted, which originally were

developed based on combined cell models for fiber fracture in composites (Fig. 3). It separatelyallows to calculate two types of unit cells, i.e. unit cells with unbroken, or with broken fibers. Foreach unit cell, the fracture probability and local mechanical behaviour shall be separately calculated.

For the unit cell A3 in the statistical combined cell models, Weibull's law can be rewritten as

with

Cf:A: stress in an unbroken fiber in the unit cell A3

The mechanical behaviour for unit cell A3 is combined by the mixing rule as,

-A3(- e)_[1_pA3(-F\'.-A3(- e) pA3(-F).-A3(- e)a E,- F a JJ a UF E, + F a a DF E,

(4)

(5)

with

-A3(- e)aUF E, :

-A3(- e)aDF E, :

Stress in an undamaged unit cell A3 with fiber orientation 8,

Stress in a damaged unit cell A3 with fiber orientation e.

Also, we can obtain the mechanical behaviour of composites for the unit cell A2, unit cell Band unitcell C by analogy with Eq. 4 and Eq. 5. The damage behaviourof composites can be calculated,according to Eq. 1 and Eq. 2.

Fig. 4 and Fig. 5 show the stress-strain curves of short fiber reinforced polymer matrix compositeswith different fiber volume fractions (20% and 30% glass-fibers) in dependence on global strain

using SCCM. In Fig. 4 the Weibull modulus mF is varied from 8 to infinite for u~= 450 MPa and in

Fig. 5 the characteristic stress O"~ of fibers was varied from 500 MPa to infinite for mF = 12 (20%fibers) or mF= 8 (30% fibers). It is found that an increase in fiber volume fraction from 20% to 30%results in an increase in the stress-strain curve. With decreasing Weibull modulus and decreasing

characteristic stress O"~ of fibers the stress-strain curve of the composite decreases. The curve formF = (f) (Fig. 4) and O"~ = (f) (Fig. 5) means there is hardly any damage in the fibers. The same resultcan be obtained by applying the CCM (Fig. 2). With the SCCM the deviation between numerical andexperimental results is found to be smaller than that with the CCM.

In comparison with numerical and experimental results good agreement is found when mF =12 and

O"~ = 450 MPa for the polymer matrix composite with 20% short glass-fibers and when mF = 8 and

O"~ = 450 MPa for a composite with 30% short glass fibers.

2.3 Statistical description of the mechanical behaviour with consideration of theadhesion e.fJectbetween fibers and matrix

The quality of the fiber-matrix adhesion affects the strength and toughness properties of compositessignificantly. It is defined through the structure and the properties of the boundary layer betweenfibers and matrix and through the adhesive and the cohesive bonding forces at the phase boundarysurfaces. In this work the investigation of the fiber/matrix adhesion is visualized only with inclusionofthe damage in the boundary layer between fibers and matrix.

The portion of the damage in the boundary layer between fibers and matrix between fibers andmatrix can be described according to Weibull's law. The statistical model, which was originallydeveloped for fiber fracture can, therefore, be used for both types of unit cells: those withundamaged and those with damaged boundary layers between fibers and matrix (Fig. 6).The Weibull's law can be written for the unit cell A2 in SCCM as

p;'(crG,e)= 1- Cx{- ( cr~:i,e T) (6)with

O"~~(E,e):

p~2(cyG,er

,.,.G.Va'

Stress in the boundary layer in an undamaged unit cell A2 with fiber orientation e,

Damage probability in the boundary layer,

Characteristic stress for the boundary layer,

Weibull modulus for the boundary layer.

The mechanical behaviour ofthe unit cell A2 with consideration ofthe damage in the boundary layerbetween fibers and matrix can be described with the mixture rule as follows

0" A 2 (8, e) = [1 - P ~ 2 (0" G , e)]. 0" ~~ (8, e) + P ~ 2 (0" G , e). 0" ~~ (8, e) (7)

with

0"~~ (8, e): Stress in the undamaged boundary layer in the unit cell A2 with fiber orientation e.

The mechanical behaviour derived from the unit cells A3, Band C with consideration ofthe damagebetween fibers and matrix follows in an analogous manner.

For the numerical investigation with consideration of the fiber-matrix adhesion effect, fibers werearranged in the tensile specimen perpendicularly to the loading direction. According to theobservation of experiment, it was ascertained that in this case there was damage at the boundarylayer between fibers and matrix, but no fiber fractures took place. When using the CCM, we have inthis case e = 0°, so that a description of the model through the model part A2 is sufficient (Fig. 6).As a first application of the SCCM, the parameters in equations (6) - (7) can be calculated by acomparison between the computation and the experiment.

The Weibull moduli and the characteristic stresses are varied in eq. (6). The results are depicted inFig. 7. From this figure it can be recognized that at different characteristic stresses and at differentWeibull moduli the global stress at strains under 0,5% follows the experimental curve. At risingWeibull modulus (Fig. 7a) und at increasing characteristic stresses (Fig. 7b) the global stressincreases. Through a comparison between the simulation and the experiment, the fitting Weibull

modulus and the convenient characteristic stress were derived as mG = 5 and (j"~ = 450 MPa.

2.4 Statistical description of the mechanical behaviour with consideration of fiberfractures and damage at the phase boundary layer

With consideration of fiber failure and damage at the phase boundary layer in the unit cell A3, themechanical behaviour can be obtained according to the mixture rule as follows

(JA3(E,8):= [1- p;3( (JG,8) - p~3((JG,8)]. (J~~(E,8)+ p;3( (JG,8). (J~;(E,8)+ p~3((JG,8)· (J~~(E,8) (9)with

(J~~(E,8): Stress in an undamaged unit cell A3 with fiber orientation 8.

Therein pt3 (erG, e) and cr~~(e, 8) can be obtained from the equations in section 2.2 and p~3 (erG, 8) andcr~b(E,e)from section 2.3.

The parameters of Weibull's law for short fiber reinforced thermoplastics with different fibervolume fractions, which are won from the simulation are recapitulated in Tab. 1. Through theinsertion ofthe parameters ofWeibuIl's law in equation (9), respectively in equations (1) and (2), thestress-strain curve of short fiber reinforced thermoplastics with amiss fiber-matrix adhesion can bedescribed.

The calculated stress-strain diagrams of short fiber reinforced thermoplastics with good and amissfiber-matrix adhesion with consideration of the statistical fiber fractions and damages in theboundary layer are compared with the experimental results in Fig. 8. The simulation reproduces weIlthe experimentally obtained curves when the global strain is smaller than 1.5%. At strains largerthan 1.5%, the simulation results deviate fram the experimental ones. This deviation permits toguess that crack propagation in this area plays an important rale.

3 Conclusions

The statistical strength of short fiber reinforced polymer matrix composites PMCs with differentvolume fractions is investigated in this work. In comparison with numerical and experimental resultsgood agreement has been obtained. Some major results of the present work are mentioned in thefollowing:(1) Combined cell models (CCM) can be only used to simulate the linear mechanical behaviour of

short fiber reinforced composites.(2) Statistical combined cell models (SCCM) can be used to predict the failure properties of short

fiber reinforced composites with consideration of fiber fractures and/or damage in the boundarylayer at the interface.

(3) With the developed SCCM it is possible to simulate the mechanical behaviourof short fiberreinforced composites with consideration ofthe damage between matrix and fiber.

(4) From numerical and experimental results the Weibull parameters were obtained for the studied

materials, i.e. a) with consideration of fiber fractures: mF =12, (J~ = 450 MPa for the PMC with20vol.% short glass fibers and mF = 8, (J~ = 450 MPa for the PMC with 30vol.% short glassfibers; b) with consideration of damage in the boundary layer between fiber and matrix: mG =5,

(J~ = 450 MPa for the PMC with 20vol.% and 30vol.% short glass fibers.

Acknowledgments

The authors gratefully acknowledge the financial support by the Deutsche Forschungsgemein­schaft (DFG) via the Sonderforschungsbereich 381 'Charakterisierung des Schädigungsverlaufes inFaserverbundwerkstoffen mittels zerstärungsfreier Prüfung' (Characterization of damage evolutionin fiber reinforced composites using nondestructive testing methods).

References

[1] J. Bohse, K. Kroh: Journal ofMaterials Science, 1992, 27,298.[2] Guo-Zheng Kang, Qing Gao: Composites Part A., 2002, 33, 657.[3] K. Derrien, D. Baptiste, D. Guedra-Degeorgesb: International Journal of Plasticity, 1999, 15,667.[4] K. Zhu, S. Schmauder: Computational Materials Science (accepted) (2002).[5] M. Dong, S. Schmauder, T. Bidlingmaier, A. Wanner: Computational Materials Science, 1997,9,121.[6] Characterization of failure propagation in fiber reinforced materials using non-destructive methods, Cooperated

Research Center, Report 1994-1997 University Stuttgart ( 1997)[7] Characterization of failure propagation in fiber reinforced materials using non-destructive methods, Cooperated

Research Center, Report 1997-1999 University Stuttga.J1 ( 2000)[8] ABAQUS Version 5.8, Hibbitt, Karlsson & Sorensen

Table list:

Table 1: The parameters ofWeibull's law for different fiber volume fractions

parameters ofWeibull's law with 20vol.% short fibers with 30vol.% short fibers

cr~ (MPa) 450450

mF

128

cr~(MPa)

450450

mü

55

Figures list:

a) b)

Fig. 1: micrograph of a polymer matrix with 20 vol% glass-fibers witha) good adhesive strength, and b) amiss adhesive strength.

.~~i'"

[IJ ~lIJ 00f~' Ql' Z

P Y

,.. : disinS

,c::v,,',. P

d :..•. ----- ..... ' 'LsIIl8 ~ .'1.•

: d/cosQ

A 'I---..J )'x Bx C

PlaneA

Plane BPlane{J UOA3(J UD"(J UDe(J UDß

{7

{7{7

JD 20 (plßne

20 (plane3D

[JJ ~[l] [!]Lco{[Q] + ~

[Q]}/Sins[Q]}/COSS

""'(E.S) "A2(E,S)

,,8(E,S)""(E,S) {J FOAJ(J FOAlcr FOC{J FDB

Fig.2: Schematic description of cell modelsand corresponding cross-sections of asingle oriented fiber in three ortho­

gonal planes.

Fig. 3: Schematic description ofthe statisticalcombined cell models (SCCM) withfracture of brittle fibers in short fiber

strengthened composites.

3.0

IExperiment I

2.52.01.5

0"0= 450 MPa

1.00.5

80

90

20

70

10

C? 600-

650'"'" 40

~ti 30

~xperiment I

~\si~\\~

\ ImF~ 81

2

Strain [%]

cro~ 450 MPa

Strain [%]

a) with 20vol.% fibers b) with 30vol.% fibers

Comparison of stress-strain curves for different Weibull moduli mF and (Ja = 450 MPa forthe composite: a) with 20vol.% fibers, and b) with 30vol.% fibers.

80

7060~50

0-;:Lo~~,g 30Vl

2°l

I#10

0

0

Fig.4:

80

70

60

g:50;;E~40:;i0)

.b 30'"

20

10

(Ja = 00

mF=12

2

Strain [%]

90

80

70

60

~ 40e:

c/) 30

20

10

0.5 1.0 1.5 2.0

Strain [%]

2.5

IExperiment I

3.0

a) with 20vol.% fibers b) with 30vol.% fibersFig. 5: Comparison of stress-strain diagrams for different characteristic stresses (Ja with different

Weibull moduli: a) mF= 12 (20vol.% fibers), and b) mF = 8 (30vol.% fibers)

the boundary layer

[IJ ~[iJ 00P\1. C Cl

OOGA"J aDG GDG GDS

Fig.6: Schematic description ofthe statistical combined cell models with damagein the boundary layer between fibers and matrix.

40

30

'"'"6 20

~~eC;;

2

Strain[%]

cr~ = 450 MPa

40

30

;;-

'"6 20~ '"10

2

Strain [%]

~ ~Fig. 7: Comparison ofthe stress strain curves with the experimental results: a) with different

Weibull moduli at cr g = 450 MPa, and b) with different characteristic stresses ag and aWeibull modulus ofmG = 5

3.02.51.0 1.5 2.0

Strain [%]

IExp: amiss adhesion 1 /;

lwith broken fiber and

amage in boundary layer

0.5

60

!undamaged I&90 iwith broken fiber I ...

80] IExp good adhesion ~;.:::::: ..:::.:::=-70 /;:

~;:::-;/.

/~;;:;

2

Strain [%]

amiss adhesion Ilwith broken fiber and

amage in boundary layer

fmdamaged I ~

lwith broken fiber I

70~ IExp.: good adhesion ~.:;;:~

80

60

20

10

~ 50~~ 40"~ 30

a) with 20vol.% fibers b) with 30vol.% fibers

Fig.8: Comparison of the ca\culated stress-strain curves offiber reinforced composites withgood/amiss fiber-matrix adhesion with the experimental results under tensile loading