stress analysis and serr calculation in fiber/matrix interfacial crack

8/14/2019 Stress Analysis and SERR Calculation in Fiber/Matrix Interfacial Crack

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INT J APPL ELECTR PHYS & ROBOT, Vol. 1 No. 1 July (2013) pp. 59

ISSN: 2203-0069

International Journal of Applied

Electronics in Physics & Robotics

Research Article

Stress Analysis and SERR Calculation in Fiber/MatrixInterfacial Crack

Ahamd Ghanbari1,2,, Arash Rahmani2, Ramin Soleimani3, Farshid Tabatabaie3

(1) School of Engineering Emerging Technologies, Mechatronic Laboratory. University of Tabriz, Tabriz, Iran.

(2) Department of Mechanical Engineering, University of Tabriz, Tabriz, Iran.

(3) Department of Engineering, Ajabshir Branch, Islamic Azad University, Ajabshir, Iran.

Copyright 2013 Australian International Academic Centre, Australia doi:10.7575/aiac.ijaepr.v.1n.1p.5

Article history:

Received 10 April 2013Reviewed 18 April 2013Revised 21 April 2013

Accepted 23 April 2013

Published 6 July 2013

Abstract. Stress analysis method was used to establish a theoretical model to findthe energy release rate forinitiation of an interfacial crack and progressive debonding with friction at debonded interface. For this pro-

pose, using stress equilibrium equations, boundary and continuity conditions and minimum complementaryenergy principle, we defined an expression for energy release rate, G, for a single fibre embedded in a concen-tric cylindrical matrix, to explore the fibre/matrix interfacial fracture properties. We determine the criticalcrack length by interfacial debonding criterion. Also, Numerical calculation results for fibre-reinforced com-

posite, SiC/LAS, were compared with experimental data witch obtained by other methods.

Keywords: energy release rate, fiberreinforced composite, interfacial crack, Stress analysis.

1 Introduction

For both theoretical analysis and experimental stud-

ies to develop a successful fiberreinforced composite, theproperties of the fiber/matrix interface have been identi-

fied as a key factor. The most important of these proper-

ties is the constraint between the fiber and matrix that

is related to slipping and debonding, which is associated

with the work of fracture for composite failure. Therefore,

many researches have used single fiber composites model

to study and explore the initiation and progressing of in-

terfacial fiber/matrix debonding. By testing fiber pull

out,Hampe et al. pointed out that a debonded interface

may appear before the interfacial shear stress reaches

the shear strength, which shows that the shear strength

based criterion is invalid to some extent [1]. Honda and

Kagawa showed that, according to the energybased in-terfacial debonding criterion, the interface crack grows

when the energy release rate G exceeds the interfacial

debonding toughness [2]. By applying the shearlag mod-

els and the Lam method respectively,HsuehandOchiai

et al. obtained solutions for the energy release rate and

the bridging law [3,4]. However, they neglected the shear

stress and strain energy in the fiber, the interfacial ra-

dial stress, the variation of axial stress in the matrix

with radial positions, and the Poissons effect. When the

axial stress in the matrix is substituted by an equiv-

alent axial stress concentrating on an effective radius,

Chiangfurther derived an expression for the energy re-

lease rate including the axial strain energy in the fiber,

Corresponding author: A. Ghanbari: +98 (0)914 116 0796: [email protected]

Fig. 1. A composite cylindrical model of length L having aninterracial crack of length a

and the axial and shear strain energy in the matrix [5].

Rauchs and Withers obtained numerical solutions of the

energy release rate by using the finite element method [6].

However, oversimplifications resulted in serious errors.

Damage growth by debonding in a single fibre metal ma-trix composite is investigated byPapakaliatakis and Kar-

alekas [7]. They Elastoplasticity and strain energy den-

sity criterion for this propose. Kushch et al. are ap-

plied numerical simulation of progressive debonding in

fiber reinforced composite under transverse loading[8].

Johnson et al. studied the role of matrix cracks and fi-

bre/matrix debonding on the stress transfer between fibre

and matrix in a single fibre fragmentation test [9].

In this paper we extract an expression for strain en-

ergy release rate (SERR) for a crack propagation analy-

sis by using stress equilibrium equations, boundary and

continuity conditions and minimum complementary en-

ergy principle. For this propose, as shown in Fig. 1a sin-gle fiber embedded in a concentric cylindrical matrix in

a cylindrical coordinate (r,z,) is considered. In Fig.1 aspecimen with an interfacial crack with a crack length of
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IJAEPR 1(1):59, 2013

ais divided into two regions [10]:

1. Region I is the region within the interfacial crack.

2. Region II is the region with an intact region.

Our first step is to find stresses in each region. Af-

ter that, we should calculate total strain energies in the

specimen. Then, according to strain energy release rate

definition we extract an expression for SERR.

2 Stress Analysis

2.1 Region I

Consider the equilibrium of the axial force acting on

element of length in the fiber of region I, leads to the fol-

lowing equation.

dfr2f

=s

rf

.d z

df

dz =

2

rfs (1)

Considering the boundary conditions as bellow.

(z=L)= Ef/E (2)m (z=L)= (Em/E) (3)

f(z=0)=/Vf

(4)

m (z= 0)= 0 (5)

and total axial stresses satisfaction

Vff+Vmm = (6)

Solving (1) and using (2) to (6), the fiber and matrix

stresses in the cracked iterfacial length, region I, (0za)become

f,I(z)=

Vf2s

rf

z (7)

m,I(z)=

Vf

Vm

2s

rf

z (8)

wherea denote the crack length, Vm, Vfshow matrix andfiber volume fraction ands,m,f andare interfacialshear stress, matrix and fiber and far field axial stresses.

In the cylindrical coordinate, the equilibrium equations

for a 3D axisymmetric problem are given by

r

r +

rz

z +

r

r = 0 (9)

z

z +

rz

r +

rz

r = 0 (10)

Solving(9) and (10)and using boundary conditions as

below.

f

r = rf,z=m

r= rf,z

=s

(11)

m (r = rm,z)=0 (12)

Shear stresses in region I are calculated.

f,I(r,z)=r

rf

s (13)

m,I(r,z)=Vf

r2m r

2

Vm rf r s (14)

2.2 Region II

For calculation the stresses in this region we introduce

stress function as

Fk =Hk (z)Ik (r) (15)

where k = f, m represent the fiber and matrix, respec-

tively. The stress solutions satisfying (9)and (10)at re-

gion II are expressed as

k (z)=2Fk

r2 +

1

r

Fk

r

=

2Ik (r)

r2 +

1

r

Ik (r)

r

Hk

(16)

k (r,z)=2Fk

rz

=Ik

r .Hk

z

(17)

k ()=k (r)

=2Fk

z2

=Ik (r) .2Hk

z2

(18)

Therefore, using stress boundary conditions as below.f,I I(z= a)=

Vf

2sa

rf

m,I I(z= a)=2Vfsa

Vm rf

(19)

f,I I(z=L)=

Ef

E

m,I I(z=L)=Em

E

(20)

f

r= rf=m

r = rf

(21)

f

r = rf,z=m

rf,z

=i (z)

(22)

m (r, rm)=m (r= rm,z)

=0 (23)

where rf, rm and i(z) denote fiber and matrix radiusand interfacial shear stress in region II, respectively.

By solving(16) and substituting(19) the functions If,Im and Hm are expressed as

If =1

4

Vf

2sa

rf

r2+Af (24)

Im (r)= Am

r2+BmLn (r)+Cm (25)

Hm =

4VmAm

1+

2sa

rfHf

(26)

where Af, Am, Bm and Cm are constant coefficients

which by using(20) to(23) are calculated as

Af =1

4

Vf

2s a

rf

Vf

Vm

r2f r

2m2r

2mLn

rm

rf

+ r2f

(27)

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International Journal of Applied Electronics in Physics & Robotics

Am =Vf

4Vm

Vf

2sa

rf

(28)

Bm =

Vf

Vm

r2m

2Ln rm

Vf

2s a

rf

(29)

Cm =V

fr2

m4Vm

Vf

2s arf

(30)

By substituting(27)to(30) into(24) and(25) and using

(16)to (18), the stresses in region II are written as

f,I I(z)=

VmEm

VfE

2s a

rf

(z)+

Ef

E (31)

m,I I(z)=

Vfsa

Vm rf

Em

E

(z)+

Em

E (32)

i (z)=

2VmEm

VfE

2sa

rf(z) (33)

f,I I(r,z)=r

2rf

VmEm

VfE

2s a

rf

(z) (34)

m,I I(r,z)=Vf

r2m r

2

2Vm rf r

VmEm

VfE

2s a

rf

(z)

(35)

f,I I(r)=f,I I()

=

r2 r2

f V

f

r2

fr2

m

12Ln

rf

rm

Vm

2

4r2f

VmEm

VfE

2sa

rf

(z)

(36)

f,I I(r)=f,I I()

=

r2 r2m

12Ln

rfrm

2

4r2f

2Vfs a

Vm

rf

Em

E (z)

(37)

where

(z)=e(za)/rf (38)

and

=E

EfVm (1vm)(39)

Andis a nondimensional parameter, given byBudian-sky and Cui [11] as

=1

2V2m 2LnVf+Vm

3Vf

(40)

In above equations E, Em and Ef are Youngs modu-lus of composite, matrix and fiber, respectively, and vmdenotes the Poissons ratio for matrix.

3 Strain Energy Release Rate

The total strain energy in the specimen is

=UI+UI I+UW (41)

where UIand UI Idenote the total strain energy in region

I and II, respectively. UW is the work done by externalforces. Therefore, according to crack propagation concept

and Minimum complementary energy principle, the to-

tal energy release rate associated with growth of crack in

Fig.1is

G =

A

=1

2rf

a

G =1

2rf

UI

a +

UI I

a +

UW

a

(42)

Then, we should calculate UI, UI Iand UWseparately.

Ui =Uf,i +Um,i i =I,I I (43)

In (43), Uf,i andUm,i are total strain energy for fiber

and matrix respectively, in region I and II with respect

to suffix i and according to strain energy definition is ex-pressed as

Uf,i =

20

rf0

yx

f,i (z)f(z)

2

+f,i (r)f(r)

2 +

f,i ()f()

2

+f,i (r,z)f(r,z) r dz dr d

(44)

Um,i =2

0

rm0

yx

m,i (z)m (z)2

+m,i (r)m (r)

2 +

m,i ()m ()

2

+m,i (r,z)m (r,z) r d z dr d

(45)

In(44) and (45), if i =I then x = 0 and y =a; and ifi=I Ithen x = aand y=L.

By using stresses which are calculated in the previous

section and substituting (7), (8), (13) and (14) into (44)

and substituting(31)to(37) into(44), we find expressions

for total strain energy in each region as below

UI= 1a3+2a2+3a (46)

wherea is crack length and

1 =22s

3

1

rfEf+

V2f

V2mEm

(47)

2 =rfs

VfEf(48)

3 =V2

f2s

4V2mGm r2f

4r4

mLn

rm

rf 3r2

m+ r4

f4r2

f r2

m

+

r2f

2

2

Ef+

2s

2Gf

(49)

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IJAEPR 1(1):59, 2013

and

UI I=1a3+2a

2+3a (50)

where

1 =

3

6E2s

VmEmEf

22s

Ef rf

2V2f2s

V2mEm

(51)

2 =

2

2V2frf

1+vm2s

V2mEm

+

2

6Er f

2s

VmEmEf

rf2s

VfE f

2Er fs

VmEmEf

(52)

3 =rfs

Vf

V2m2 (1vm)

4VfE +

3

Ef

4rf

Ef

+

r2f2

VfEf

VmEm

2VfE +

Ef

E

1

Vf+

Vf

2

(53)

Also, for calculatingUWwe have

UW= UFUP (54)

UFis the work done by friction stress s andUP is thework done by tensile stress acting on fiber which are pre-

sented as [10]

UF= 2rf

a0

s v (z)d z (55)

where v (z) is relative axial displacement between thefiber and the matrix which was given byChiang[5].

Then, by solving(51) we find

UF= 1a3+2a

2+3a (56)

where

1 =4

3

E2s

VmEmEf

(57)

2 =

rfs

VfEf

4Er fs

VmEmEf

(58)

3 =

2r2

fs

VfEf

(59)

And the work is done by friction stress s is

Up =r2

f

Vfv (0) (60)

Up = 1a2+2a+3 (61)

where

1 =rfs

VfEf(62)

2 =

VfEf+

2s

Ef+

E

r2f

Vf (63)

3 =

rfVmEm+LVfEf

V2f

EfE

r2f

2 (64)

By Combining (42), (46), (50), (56)and (61), an expres-

sion for the energy release rate is obtained as

Up =1a2+2a+3 (65)

where

1 =E2s

rfVmEfEm(66)

2 =V2

f (1vm)

2s

V2mEm+

3E2s

VmEmEf

2VfEf+ EVmEmEf

s

(67)

3 =V2m(1+vm)s

8V2f

E

3s

2VfEf

+rfVmEm

2

4V2f

E fE

r2f2s

8Gf

V2f2s

8V2mGm r3f

4r4mLn

rm

rf3r2m+ r

4f4r

2f r

2m

(68)

4 Results

We Show that the energy release rate is a secondorderfunction of the cracked length a when the material andgeometry parameters are known. When an interfacial

crack growth criterion G i is introduced, the critical

crack length can be determined by

G i 1a22a+3i 0 (69)

ac1,2 =2

2

241 (3i)

21(70)

Only the smaller cof the two roots of (70)is physicallymeaningful.

Fig. 2 shows Distributions of the energy release rate viathe normalized cracked length which is extracted with an

exact solution and good agreement between the methods

is presented in this paper with the others.

As we can see in Fig.2the curves G relative to normal-ized crack length have the first decreasing and then re

increasing tendency. The reincreasing part is physically

meaningless because interfacial debonding appears only

at G i and stops after the condition G =i is satisfied.The effect of friction, between fiber and matrix in region I

is considered in this paper as important factor for calcu-

lation of strain energy release rate that is illustrated in

Fig.3.

Fig. 2. Distributions of the energy release rate via the normal-ized cracked length

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International Journal of Applied Electronics in Physics & Robotics

Fig. 3. The effect of friction between fiber and matrix in region II

According to Fig.3,increasing friction stresss resultsin a smaller G at the same a/rf , improving the interfa-

cial debonding toughness i. The curvesG

a/rf

tend

to be smooth when the friction stress s decreases and

approaches the minimum values=

0, where the energyrelease rate reaches the maximum value G = 6.02

Jm2

for the decreasing parts of curves.

5 Conclusion

For calculating strain energy release rate, we divided

the single fiber embedded in a concentric cylindrical ma-

trix into two regions. Region I is cracked and region II is

intact region. Using stress equilibrium equations, bound-

ary and continuity conditions for each region we calculate

stresses separately and then according to minimum com-

plementary energy principle, we define strain energy re-

lease rate as second order function of crack length.

The following conclusions are obtained

1. The presented method in this paper is feasible to de-

termine critical debond length in fiber/matrix inter-

facial crack.

2. Numerical calculation results for fiberreinforced

composite, SiC/LAS, have good agreement with ex-

perimental data which obtained by other methods.

3. The curves G relative to normalized crack lengthhave the first decreasing and then re-increasing ten-

dency. The reincreasing part is physically mean-

ingless because interfacial debonding appears only

atG >i and stops after the condition G =i is sat-

isfied.

4. When the interfacial friction between fiber and ma-

trix at region I increase, the curves G relative to the

normalized crack length have sharp slope and when

we have complete debonding in interface of fiber and

matrix,G is independent from crack length.

5. The shear effects in the fiber and matrix and Pois-

sons effect neglected by the shearlag models be-come more remarkable with the increase of friction

stresss for suppressing the interface failure.

REFERENCES

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[2] K. Honda and Y. Kagawa, Debonding criterion in thepushout process of fiber-reinforced ceramics, Acta materi-alia, vol. 44, no. 8, pp. 32673277, 1996.

[3] C. H. Hsueh, Crack-wake interfacial debonding criteriafor fiber-reinforced ceramic composites, Acta materialia,vol. 44, no. 6, pp. 22112216, 1996.

[4] S. Ochiai, M. Hojo, and T. Inoue, Shear-lag simulation of

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[8] V. I. Kushch, S. V. Shmegera, P. Brndsted, and L. Mish-naevsky Jr., Numerical simulation of progressive debond-ing in fiber reinforced composite under transverse loading,

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[9] A. C. Johnson, S. A. Hayes, and F. R. Jones, The role of ma-trix cracks and fibre/matrix debonding on the stress trans-fer between fibre and matrix in a single fibre fragmentationtest, Composites Part A: Applied Science and Manufactur-ing, vol. 43, no. 1, pp. 6572, 2012.

[10] P. F. Liu, W. M. Tao, and Y. M. Guo, Properties of frictionalbridging in fiber pull-out for fiber-reinforced composites,

Journal of Zhejiang University Science, vol. 6, no. 1, pp. 816, 2005.

[11] B. Budiansky and Y. L. Cui, Toughening of ceramics byshort aligned fibers, Mechanics of materials, vol. 21, no. 2,pp. 139146, 1995.

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stress analysis and serr calculation in fiber/matrix interfacial crack

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