BEM analysis of the single fiber fragmentation test

LICENTIATE T H E S I S

Luleå University of Technology

Department of Applied Physics and Mechanical Engineering Division of Polymer Engineering

2007:52

BEM Analysis of the Single Fiber Fragmentation Test

Enrique Graciani

BEM analysis of the

single fiber fragmentation test.

by

Enrique Graciani

Division of Polymer Engineering

Department of Applied Physics and Mechanical Engineering Luleå University of Technology

S-971 87 Luleå, Sweden

December 2007

To all the friends I’ve found in Sweden

...thanks for everything

Abstract

A Boundary Element analysis of micromechanical elastic fields in the single-fiber fragmentation test is presented in this thesis. The work carried out is roughly divided in two main tasks: the development of the BE code and the numerical simulation of the single-fiber fragmentation test.

The numerical study is primarily concerned with the analysis of the initiation and growth of a debond crack along the fiber-matrix interface in the single fiber fragmentation test, although different configurations in which the crack propagates through the matrix have also been considered.

The asymptotic behavior of the near-tip singular elastic solutions in the fiber cracks, the interface cracks and the matrix cracks are studied.

Additionally, asymptotic behavior of the Energy Release Rate for a wide range of debond lengths is analyzed. Firstly, the numerical analysis is performed in the framework of the two linear elastic models of interface cracks, open model and frictionless contact model, and a discussion of their adequacy based on the numerical results presented is given. Finally, a frictional contact model is employed to elucidate the influence that the friction between the debond crack faces may have in the near-tip singular elastic solutions and crack propagation.

Therefore, a Boundary Element code has been developed which allows

the elastic analysis of axially symmetric bodies to be carried out, permitting

the definition of multiple solids bonded or in contact, taking into account

the residual stresses developed during the curing of the samples and

allowing non conforming meshes to be used in the interfaces and contact

zones. Moreover, a novel extremely accurate integration technique has

been developed to allow the near-tip singular elastic solutions to be

precisely obtained.

Preface

This thesis is a product of the graduate studies that I have carried out both in the Group of Elasticity and Strength of Materials of the University of Seville and the Division of Polymer Engineering of the Luleå University of Technology.

First of all, I wish to thank Professor Janis Varna for giving me the opportunity of carrying out these studies and for all the help and advice provided during this period. This gratitude is also extended to Professor Lars Berglund, Professor Federico París and Dr. Vladislav Mantiþ who’s supervising have been highly instructive.

I whish to express my gratitude also to all the members of both research groups for their friendship and support, that has made the work much easy and enjoyable. My relationship with the Division of Polymer Engineering started with my stays in Luleå during the years 1996-1997, and I must make a special mention to Bobs, Kristofer, Anders, Johan, Fredrik, Mats, Lennart, Kristina, Christian, Conny, Marta-Lena… and all the people I met in those visits for being so kind and for making my stay so warm despite of the outdoor temperatures.

I’m glad to see that collaboration and contact have been maintained during these years and I hope that this thesis is not the result of this collaboration but a starting point for a longer relationship.

Seville, October 2007.

Enrique Graciani

Contents

Preface ...i

Contents...iii

1. Introduction ... 1

1.1. Micromechanical tests employed for interface characterization... 2

1.2. Description of the Single Fiber Fragmentation Test ... 3

1.3. Theoretical aspects of the Single Fiber Fragmentation Test ... 4

1.4. Summary and Objectives... 6

2. Description of the Boundary Elements formulation ... 7

2.1. Axisymmetric BEM formulation of the elastic problem... 8

2.2. Weak boundary formulation of the coupling conditions... 10

2.3. Non-linear solution of the final system of equations ... 11

3. BEM analysis of the Single Fiber Fragmentation Test ... 12

3.1 Analysis of the distinct failure mechanisms of the SFFT sample. 12 3.2 Analysis of the debond growth in the SFFT sample ... 16

3.3 Accuracy of the numerical results ... 19

3.4 Effect of friction on the stress state in the SFFT sample ... 21

4. Concluding remarks and future work... 21

References ... 23 Annex 1:

Graciani E, Mantiþ V, París F and Blázquez A. “Weak formulation of axi- symmetric frictionless contact problems with boundary elements.

Application to interface cracks”. Computers and Structures 83:836-855, 2005.

Annex 2:

Graciani E, Mantiþ V, París F and Varna J. “Single fiber fragmentation test.

A BEM analysis”. Collection of Technical Papers -

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 2:988-997, Norfolk (Virginia), United States, 2003.

Annex 3:

Graciani E, Mantiþ V, París F and Varna J. “A BEM analysis of the single fibre fragmentation test. Influence of curing temperature in crack

propagation”. Proceedings of the 11th European Conference on

Composite Materials (CD, File: A030.pdf), Rodas, Greece, 2004.

Annex 4:

Mantiþ V, Graciani E, París F and Varna J. “An Axisymmetric Boundary Element Analysis of Interface Cracks in Fiber Reinforced Composites”, in Advances in Boundary Element Techniques VI. Eds: AP Selvadurai, CL Tan ana MH Aliabadi, 21-26, Montreal, Canada, 2005.

Annex 5: Erratum

1. Introduction

Among the wide range of fibrous composite materials that are used nowadays, Fiber Reinforced Composites are one of the most extended, especially for building primary structures subjected to high loads.

A typical composite of this kind is formed by a continuous matrix (usually constituted by a polymeric resin) and a fibrous reinforcement (mainly carbon or glass fibers) distributed within it. Therefore, the resulting material combines a high stiffness and stress with a light weight, with fiber reinforcement being responsible for the high mechanical properties and the matrix providing consistency to the material.

Fiber reinforced composite materials are usually designed to work in the direction of the fibers, being its failure properties in this case mainly controlled by the fiber strength properties. However, the existence of differently oriented plies inside the laminates and/or the presence of impact loads which produce stresses transferred along many directions within the laminate, are responsible for failure mechanisms in which the failure properties of the matrix and the interface play a fundamental role.

Many experimental techniques have been developed to study the failure

mechanisms of the interface. In most of them, the sample is constituted by

a limited amount of fibers surrounded by polymeric matrix. These tests are

generally known as micromechanical tests due to the small size of the

samples.

1.1. Micromechanical tests employed for interface characterization

Among the micromechanical tests in which the sample consists of only one fiber to which the load is directly applied, the following can be cited (see Figure 1): pull-out test, push-out test, microbond test and microindentation test.

(a) (b) (c) (d)

Figure 1: Micromechanical tests for studying interfacial failure properties:

(a) pull-out, (b) push-out, (c) microbond, (c) microindentation.

Other widespread tests, in which the load is applied to the sample, are the single fiber fragmentation test and the transverse single fiber test, depicted in Figure 2.

(a) (b)

Figure 2: (a) single fiber fragmentation test, (b) transverse single fiber test.

The main reason for developing such micromechanical tests is to be

able to obtain a clear failure of the interface within the sample to gain in

understanding on interface failure mechanisms and properties.

All the abovementioned micromechanical tests present a common disadvantage associated to the small size of the samples, which makes it extremely difficult to obtain accurate measurements of the elastic behavior (displacements, deformations or stresses) within the samples during the performance of the tests. To overcome these difficulties, analytical or numerical solutions are employed during data reduction to obtain quantitative results of interfacial failure properties. Therefore, the validity of the results is clearly influenced by the assumptions made to obtain the elastic solution employed for the data reduction.

Single fiber fragmentation test has been chosen as the object of the analysis for being one of the most frequently used test for characterizing the interface of composite materials. In the following sections, the test procedure and the different approaches employed to estimate interfacial failure properties from the results of the experiments are briefly summarized.

1.2. Description of the Single Fiber Fragmentation Test

Since it was introduced by Kelly and Tyson

, single fiber

fragmentation test (SFFT) has been extensively used for fiber-matrix

interface characterization. Fragmentation samples consist of a sufficiently

long fiber embedded in a resin matrix, subjected to tensile load. This tensile

load, applied as a uniform strain at the ends of the sample, is transferred to

the fiber through shear stresses at the fiber-matrix interface. As maximum

allowable strain is much lower in the fiber than in the matrix, and the fiber

strength is characterized by a two-parameter Weibull distribution, an

increase in the applied strain, after reaching a certain value, results in

successive breaking of the fiber.

If the failure properties of the interface and the matrix are high, fragmentation continues until a critical fragment length is reached where the fragments are not long enough to transfer through the interface an axial load sufficient to cause subsequent fiber breakages. This fact is usually referred to as saturation. In this case, in which the only appreciable failure in the sample takes place in the fiber, the final average length is mainly dependent on fiber failure properties.

Nevertheless, in most cases the above described saturation is not reached during the test because of the appearance of a series of cracks which arise from the ends of fiber fragments and grow through the matrix or the interface. There are three different crack paths observed in the experiments (see, for example, Ohsawa et al.

and Sjögren et al.

): a penny-shape crack and/or a bi-conical crack growing through the matrix, or a bi-cylindrical crack, growing along the interface, which is often referred to as debond crack. The appearance of these cracks implies a relaxation of the axial stress in the fiber, which prevents the appearance of new fiber breakages, and therefore the final average length of the fragments depends on the interface or matrix failure properties.

1.3. Theoretical aspects of the Single Fiber Fragmentation Test

There are distinct approaches for characterizing interface failure properties from the results of the single fiber fragmentation test.

Differences arise in the use of analytical, experimental or numerical

techniques for estimating the stress state in the sample, and also in the use

of failure criteria based on the interface strength or failure criteria based on

fracture mechanics.

The interface strength approach relies on the initial shear lag stress distribution model proposed by Cox

. The correlation between the interface strength and the fiber strength was initially obtained by a simple equilibrium of the final fragment assuming constant interface stresses. It has been corrected afterwards to take into account several features like more realistic shear stress distribution on the interface, thermal residual stresses or statistical distribution of fragment lengths (see, for example, Ohsawa et al.

, Whitney and Drzal

, Netravali et al.

, Henstenburg and Phoenix

, Lacroix et al.

, and Tripathi and Jones

). Interesting reviews in the early developments of this approach have been carried out, among others, by Verpoest et al.

, Herrera-Franco and Drzal

, and Tripathi and Jones

.

According to the shear lag model the shear stress at the interface is finite and hence the interface failure takes place when the maximum value equals to the interface strength. This criterion can be expressed as a certain relationship between the interface strength and the fiber strength and length at saturation.

Nevertheless, as it is well known, the exact solution of stresses at the interface contains shear singular stresses in the vicinity of the crack tips.

Therefore, the use of a strength criterion is highly questionable, being the measurements carried out in the single fiber fragmentation test difficultly applicable to situations in which the interface is subjected to a different stress state.

For this reason, the energy approach, either based on energy balance or on fracture mechanics, is considered more appropriate for the characterization of the interface failure in presence of singular stress states (see, for example, the works of Nairn

, Varna et al.

, Nairn and Liu

and Wu et al.

). This approach, based on energy considerations,

considers that crack propagation would take place when the total energy released per unit length during the propagation (i.e., the energy release rate) equals the work needed to create the new unit crack surfaces. As the propagation takes place under pure mode II, failure of the interface is characterized by the interface fracture toughness, which is the critical value of the mode II energy release rate (usually denoted as G ).

Finally, the works carried out by Galiotis

and Huang and Young

, measuring fiber strain by means of laser Raman spectroscopy, and the phase-stepping photoelasticity analysis made by Zhao et al.

can be cited as an example of the few existing experimental techniques which are able to provide direct stress or strain measurements during the execution of the test.

1.4. Summary and Objectives

The main objective of the present study is to show the ability of a self- developed tool based on the Boundary Element Method to carry out a numerical simulation of the single fiber fragmentation test, and to obtain an accurate solution of the displacement and stresses within the sample, which could help in the understanding of the mechanisms of initiation and growth of debond cracks along the fiber-matrix interface.

The methods employed for developing the numerical tool and the details of the analysis are reported in Annexes 1-4. As some of the graphs included in the original publications contained misprints, corrected versions of them can be found in Annex 5.

In Annex 1, the axisymmetric BEM formulation and the weak

boundary formulation of the coupling conditions developed for the

numerical analysis are explained in detail. The results obtained in the

numerical analysis of the single fiber fragmentation test are presented in Annexes 2-4.

In the following the main and novel aspects of the numerical formulation and the stress analysis carried out are highlighted.

2. Description of the Boundary Elements formulation

Numerical analysis has been carried out by the collocation Boundary Element Method (BEM) for elastic problems with axial symmetry (the fundamentals of BEM can be found, among others, in the books of Bakr

, Baláš et al.

and París and Cañas

). Coupling conditions along the bonded part and contact conditions along the debonded part of the fiber- matrix interface have been imposed using a weak boundary formulation, analogous to the one developed in Blázquez et al.

for plane elastic problems, which allows non-conforming discretizations to be used along the interface. This approach has been chosen for the numerical analysis of the single fiber fragmentation test since, in this particular case, it offers clear advantages if compared with other alternative numerical methods (like, for example, the Finite Element Method).

Firstly, as the elastic solution of the single fiber fragmentation test presents axial symmetry with respect to the fiber axis, only the radial section of the model has to be analyzed. As the BEM requires a mesh of the boundary of the domain under study, only radial section of the boundaries and interfaces of the sample need to be meshed. Therefore, a 1D BEM mesh will be sufficient for the analysis, being very easy to refine the mesh in the vicinity of the crack tips.

Secondly, as it is well known, crack propagation can be analyzed

employing the near-tip elastic solution along crack faces and ahead of the

crack tip. Since the debond crack in which the present analysis is focused grows along the fiber-matrix interface, the displacements and tractions needed for the study of crack propagation are the actual primary unknowns in the nodes of the BEM mesh. Therefore, no postprocessing of the numerical solution is needed, which results in a high accuracy of the results in the zone of interest.

Finally, the use of a BEM approach along with a weak boundary formulation of the coupling conditions, allows the use of non conforming meshes in the interfaces to be employed. As the aspect ratio of the fiber is very high (since it is much longer than its radius) a fine mesh is required to obtain accurate results within the fiber. The use of a non conforming mesh along the fiber-matrix interface yields an important reduction on the number of elements needed for the boundary mesh in the matrix, where larger elements can be employed since sample radius is much larger than fiber radius.

2.1. Axisymmetric BEM formulation of the elastic problem

The axisymmetric collocation BEM formulation of the elastic problem

is based on the application of the axisymmetric Boundary Integral Equation (BIE) of the displacements for a point at the boundary ( * ) of the domain ( : ) under study.

For a collocation point outside the symmetry axis this BIE can be written as:

³ ^



D

ED

' u U O ' t T ' u d *

C

( x ) ( x ) [

( x , y ) ( y ) ( y )

( x , y ) ( y )] (1)

where x  * is a fixed collocation point and y  * is the integration point

which moves along the boundary. ' u

( y ) , with D r, z , represent the

global components of the displacements in cylindrical coordinates, while

) ( y

' t

, with J n, s , are respectively the normal and tangential components of the tractions associated to a plane tangent to the boundary at

y . ) O

( y is a rotation matrix relating global ( r, ) and intrinsic ( z n, ) s coordinates. ) C

( x are the components of the free term, which depend upon the local geometry of the boundary at x . Finally, U

( x , y ) and

) ,

*

( y

ED

x

T are, respectively, the components of the axisymmetric fundamental solution and their associated tractions along the boundary. As

) ,

*

( x y

U

and T

( x , y ) are singular functions when y o , the boundary x integral in (1) only exists in the sense of Cauchy Principal Value.

The main difficulty in the development of the axisymmetric BEM formulation is associated to the fact that U

( x , y ) and T

( x , y ) are long non-explicit singular expressions written in terms of the complete elliptic integrals of the first and second kind. Series expansions of U

( x , y ) and

) ,

*

( x y

T

in the vicinity of the collocation point are presented in Annex 1, in which all non regular terms are identified. The novel analytical treatment of these non regular terms yields a high accuracy in the results.

For a collocation point xˆ at the symmetry axis, ' u

( x ˆ ) 0 , and the BIE of the axial component of the displacements can be written as:

³

'

^

'

*

' u U O t T u d

C ˆ

( x ˆ )

( x ˆ ) [ ˆ

( x ˆ , y ) ( y ) ( y ) ˆ

( x ˆ , y ) ( y )] (2)

where U ˆ

( x ˆ , y ) and T ˆ

( x ˆ , y ) can be obtained taking the limit of )

,

*

( y

D

x

U

and T

( x , y ) when the collocation point x tends to the

symmetry axis. The free term C ˆ

( x ˆ ) is usually evaluated indirectly using

numerical techniques. A novel analytical expression of C ˆ

( x ˆ ) is presented in Annex 1, and its use has resulted in an increased accuracy of the results.

As described in Annex 1, the collocation BEM employed for the numerical analysis is based in the application of equations (1) and (2) in a series of nodes in the boundary of the domain under study. Boundary integrals are approximated by the sum of the integrals along linear elements defined between the nodes. Displacements and tractions within the elements are approximated by linear interpolation of their nodal values.

Therefore, primary unknowns of the collocation equations are the components of the displacements and tractions at the nodes placed along the boundary of the solids under study.

2.2. Weak boundary formulation of the coupling conditions

Coupling of the collocation equations for different solids is carried out employing a novel weak boundary formulation of the equilibrium and compatibility equation at the interfaces and contact zones.

Let us consider domains :

and :

which are in contact along a part of their boundaries denoted by *

and *

. Equilibrium, that is

0 ) ( )

( y  t y t

'

' , is guaranteed by the fulfilment of the following weak equation extended to *

:

0 )

( )]

( ) (

[ 

³

B C r B A

B

t u y d

*

t

\ D D

D

' ' *

' y y y (3)

for any field of compatible displacements ' u

( y ) .

Analogously, compatibility of the displacements along the contact

region, that is ' į

( y ) ' u

( y )  ' u

( y ) with ' į

( y ) being the relative

displacements between opposite points of the contacting solids, is

guaranteed by the fulfillment of the following weak equation extended to

A

*

^:

0 )]

( )

( ) ( [ )

(  

³

A C r A B

A

A

u u y d

*

t

D

' ' ' G *

' y y y y (4)

for any tractions field in equilibrium ' t

( y ) ^.

As detailed in Annex 1, employing the same boundary elements described in the previous section, with linear interpolations of the unknowns, to discretize equations (3) and (4) yields a system of equations which allow the coupling of the collocation equations corresponding to domains :

and :

to be made, since the primary unknowns of the equilibrium and contact equations (displacements and tractions along the contacting boundaries) are in fact a subset of the primary unknowns of the collocation equations (displacements and tractions along the whole boundaries).

Relative displacements, ' į

( y ) , appearing in equation (4) are, actually, the relative opening and sliding along the contact zone. Therefore, frictional conditions have been easily introduced in the formulation.

2.3. Non-linear solution of the final system of equations

It has to be emphasized that the final system of equations is constituted

by the collocation equations described in section 2.1, the equilibrium and

compatibility equations described in section 2.2 and the boundary

conditions along the external boundaries, the interfaces and the contact

zones. Accordingly, the unknowns are the components of the displacements

and tractions along the whole boundaries and the opening and sliding along

the contact zones.

Since in many situations the sizes of the contact zones (and the adhesion or sliding regions inside them) are not known a priori and usually depend upon the applied load, the corresponding contact boundary conditions are expressed by inequalities which make the problem to become non linear. The final solution of the system of equations is achieved following an incremental scheme which assures that all contact conditions are fulfilled during the whole load history. For this reason, in some cases, iterative stages are needed until a correct definition of the sizes of the contact zones is found which allow the solution procedure to be continued.

3. BEM analysis of the Single Fiber Fragmentation Test

In Annexes 2-4 of this thesis, the results obtained in the numerical analysis of the single fiber fragmentation test (SFFT) are presented. These results are reviewed in the following sections, in which the corresponding papers are summarized in turn.

3.1 Analysis of the distinct failure mechanisms of the SFFT sample

Comparison of the different kinds of failure of the SFFT sample is carried out in Annex 2, in terms of the energy release rate (ERR).

Therefore, results of the computed ERR associated to the growth of a crack

in the different orientations described (i.e., a penny-shape matrix crack, a

bi-conical matrix crack or a bi-cylindrical interface crack) are presented for

a range of crack lengths and fragment sizes.

In this preliminary analysis, the effect of the residual stresses and the friction between crack faces in closed cracks is neglected. Analysis of the matrix cracks is carried out using a classical fracture mechanics approach, while the interface crack is studied using two distinct approaches: the open crack model introduced by Williams

(in which traction free crack faces are prescribed) and the frictionless contact model developed by Comninou

(in which frictionless contact conditions are imposed along crack faces). ERR is evaluated in all cases with an appropriate formulation of the virtual crack closure technique (VCCT). Material properties employed are described in Annex 2 and correspond to a typical E-glass fiber with an epoxy matrix.

Due to the large aspect ratio of the fiber no interaction between the cracks appearing in the sample is assumed. Therefore, in view of the existing symmetries, the model consists in the radial section of half a fragment, with L being half of the fragment length and

r being the fiber

radius. Analyzing the axial stress along the fiber axis, after successive fragmentation, four different fragment lengths are chosen for the analysis with L

10 r

, 20 r

, 40 r

, 80 r

. Although saturation is expected to occur well above L

40 r

, results of longer fragments have not been included since they are coincident with results corresponding to L

40 r

and

f

f

r

L 80 .

Results of the ERR, corresponding to the penny-shape matrix crack, are plotted versus crack size ( a ) in Figure 3 for the selected fragment lengths.

As load is applied as a uniform strain ( H

1 % ) at the ends of the sample,

ERR obtained for small crack sizes ( a  2 r

) decreases with increasing

crack length. On the contrary, for longer cracks ( a ! 4 r

), ERR increases

with increasing crack length. This means a stable growth of the crack at the

beginning followed by a final unstable growth, like can be observed in the experiments.

0 10 20 30 40 50 60 70

0 2 4 6 8

Crack size (a/rf)

] 80

[f f

IL r

G ] 40

[f f

IL r

G

] 10

[f f

IL r

G ] 20

[f f

IL r

G ] 80

[f f

IL r

G ] 40

[f f

IL r

G

] 10

[f f

IL r

G ] 20

[f f

IL r

G

Energy release rate [GII,J/m2]

Figure 3: ERR in the penny-shape matrix crack ( H

1 % ^): 0 . 2 r

 a  8 r

,

f f

f

L r

r 80

10   .

As can be seen, the assumption of no interaction between the different cracks appearing in the sample is only valid when L

t 40 r

, as no length dependence is obtained in the ERR solution for these cases. As will be shown in the next section, there is a low possibility of finding fragments with L

 40 r

and the subsequent analyses will be focused in fragment lengths above this limit.

For a conical crack, an initial straight growth oriented forming a certain angle D respect to the fiber axis is assumed. Results of the ERR are plotted versus crack orientation in Figure 4(a) for a crack with a r

and

f

f

r

L 40 . As can be seen, cracks growing close to the interface ( D  15 º ) are closed and propagation take place under pure fracture mode II. On the contrary, for D ! 15 º an open crack growing in mixed mode is obtained.

Evolution of the ERR with crack length ( a ) is shown in Figure 4(b) for

the particular case in which D 20 º . As can be seen, ERR decreases

steadily with crack length. Curves like the ones shown in Figure 4, along

with the corresponding toughness of the matrix and the elastic solution in the vicinity of the crack tip can be used to predict the path followed by the conical crack.

0 2 4 6 8 10 12 14

0 15 30 45 60 75 90

Crack orientation (D,degrees) GI

G GII GI

G GII

Energy release rate [GII,J/m2]

0 5 10 15 20 25 30 35

0 1 2 3 4

Crack size (a/rf)

] 80 [Lf rf G

] 40 [Lf rf G

] 10 [Lf rf G

] 20 [Lf rf G

] 80 [Lf rf G

] 40 [Lf rf G

] 10 [Lf rf G

] 20 [Lf rf G

Energy release rate [GII,J/m2]

(a) (b)

Figure 4: ERR in the bi-conical matrix crack ( H

1 % ^{): (a)} a r

,

f

f

r

L 40 , º 0 º  D  90 ; (b) 0 . 2 r

 a  4 r

, 10 r

 L

 80 r

, º D 20 .

As no quantitative information about the interface failure properties can be obtained when the previously described matrix failures occurs in the sample, the focus of this study has been set in the analysis of the following case, in which a bi-cylindrical debond crack grows along fiber-matrix interface.

In this case, ERR has been evaluated using two different approaches. In

Figure 5(a), results of the total ERR obtained with the open model is

plotted as a function of the debonded length ( a ), while Figure 5(b) shows

the mode II ERR corresponding to the frictionless contact model. It has to

be emphasized that, although the asymptotic solution in both cases is

clearly different, as described in Annex 2, results of the ERR are almost

coincident in both approaches.

0 5 10 15 20 25 30 35

0 2 4 6 8

Crack size (a/r_f)

] 80 [Lf rf G

] 40 [Lf rf G

] 10 [Lf rf G

] 20 [Lf rf G

] 80 [Lf rf G

] 40 [Lf rf G

] 10 [Lf rf G

] 20 [Lf rf G

Energy release rate [GII,J/m2]

0 5 10 15 20 25 30

0 2 4 6 8

Crack size (a/r_f)

] 80

[f f

IIL r

G ] 40

[f f

IIL r

G

] 10

[f f

IIL r

G ] 20

[f f

IIL r

G ] 80

[f f

IIL r

G ] 40

[f f

IIL r

G

] 10

[f f

IIL r

G ] 20

[f f

IIL r

G

Energy release rate [GII,J/m2]

(a) (b)

Figure 5: ERR in the debond crack ( H

1 % ) with 0 . 2 r

 a  8 r

,

f f

f

L r

r 80

10   : (a) open crack model; (b) contact crack model.

As can be seen, ERR is decreasing steeply for short debond lengths and tends to a constant (at least for L

t 40 r

). Therefore, crack growth will be stable in a first stage and will become unstable when the plateau value of the ERR is reached.

Although not included here for the sake of conciseness, the effect of crack propagation on the axial stress along the fiber axis is also presented for all configurations, showing that, in all cases, the crack growth results in a relaxation of axial stresses along the fiber.

3.2 Analysis of the debond growth in the SFFT sample

In Annex 3, the debond crack propagation is studied with a greater depth and taking into consideration the effect of residual stresses due to the cooling stage after solidification in the curing of the sample using the frictionless contact approach.

As, both during the cooling stage and during the loading stage, the

matrix radial shrinkage is larger than the fiber radial shrinkage, the debond

crack is closed along its whole length. Moreover, if friction between crack

faces is neglected, the solution can be obtained linearly combining two load

cases: a first case in which only the decrease of temperature is applied, and a second case in which only the mechanical load is applied.

Firstly, the fragmentation of the sample is studied. In Figure 6(a) the axial stress along the fiber axis of a fragment with L

80 r

, corresponding to a 1% deformation of the sample ( H

1 % ), is shown in conjunction with the solution obtained after successively splitting of the original and the subsequent fragments by its middle point until L

10 r

is reached.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0 10 20 30 40 50 60 70 80

Axial stress in the fibre [GPa]

Axial position, z/rf

] 80 [ ) , 0

)(

0 , (

f f

zz^H z L r

V

] 40 [ ) , 0

)(

0 , (

f f

zz^H z L r

V

] 20 [ ) , 0

)(

0 , (

f f

zz^H z L r

V

] 10 [ ) , 0

)(

0 , (

f f

zz^H z L r

V

] 80 [ ) , 0

)(

0 , (

f f

zz^H z L r

V

] 40 [ ) , 0

)(

0 , (

f f

zz^H z L r

V

] 20 [ ) , 0

)(

0 , (

f f

zz^H z L r

V

] 10 [ ) , 0

)(

0 , (

f f

zz^H z L r

V

% H 1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0 20 40 60 80 100 120

Maximunaxial stress in thefiber[GPa]

Fragment size, Lf/rf

% H 1

zz^,0)(0,z)máx

V(H zz^,0)(0,z)máx

V(H

(a) (b)

Figure 6: (a) axial stress along the fiber during fragmentation; (b) maximum axial stress in the fragment.

As can be seen, when L

t 40 r

the axial stress in the fragment reaches

a plateau value in the middle of the fragment, in which the stress is equal to

the nominal stress corresponding to an infinite length fragment. On the

contrary, shorter fragments do not reach this nominal stress, and therefore

are subjected to lower stresses than the ones that caused the previous

breakages. Therefore, the possibility of finding fragments with L

 40 r

is very low and the analysis will be focused in fragment lengths above this

limit.

The same conclusion can be obtained from Figure 6(b) where the maximum axial stress in the fragment is plotted against the fragment length. As can be seen, for L

t 40 r

the maximum axial stress becomes constant. Therefore, it can be concluded that, in absence of interface or matrix failure, fragments having L

t 40 r

have an equal transition zone in which the axial load in the fiber rises from cero to the nominal stress, and after this zone an uniform solution is obtained which is almost equal to the one corresponding to the intact fiber.

To study the debond crack propagation, a series of configurations in which the debond length is varied in the range 0 . 2 r

 a  8 r

have been analyzed. The ERR obtained from the solution of these models, corresponding to a 1% deformation of the sample ( H

1 % ), are shown in Figure 7(a).

0 2 4 6 8 10 12

0 1 2 3 4 5 6 7 8

] 80

)[

, (

f f a

II L r

G^H ] 40

)[

, (

f f a

II L r

G^H ] 80

)[

, (

f f a

II L r

G^H ] 40

)[

, (

f f a

II L r

G^H

% H 1

Energy release rate [J/m2]

Crack size, a/rf

0 1 2 3 4 5 6 7 8 9

0 0.5 1 1.5 2 2.5 3

] 0 , 80

) [

,

( G L r T K

G IIc f f

a

II^H '

] 0 , 40

) [

,

( G L r T K

G IIc f f

a

II^H '

] 80 , 80

) [

,

( G L r T K

G IIc f f

a

II^H '

] 80 , 40

) [

,

( G L r T K

G IIc f f

a

II^H '

] 0 , 80

) [

,

( G L r T K

G IIc f f

a

II^H '

] 0 , 40

) [

,

( G L r T K

G IIc f f

a

II^H '

] 80 , 80

) [

,

( G L r T K

G IIc f f

a

II^H '

] 80 , 40

) [

,

( G L r T K

G IIc f f

a

II^H '

Crack size, a/rf

Applied strain, H [%]

J/m2 IIc 50 G

(a) (b)

Figure 7: (a) ERR during debond crack propagation; (b) average strain needed to cause debond crack propagation.

As can be seen, comparing Figure 7(a) with Figure 5(b), the residual

stresses due to temperature diminish the ERR, and therefore are in

opposition to crack propagation.

As the crack is closed and propagation occurs under pure mode II, a simple propagation criterion G

G

has been employed for the analysis, where H is the averaged applied strain, a is the debond length and G

is the interface fracture toughness. As detailed in Annex 3, the solution of displacement and stresses depend linearly upon the applied strain H and, therefore, solution of G

is a second order polynomial in

H whose coefficients can be obtained from the solution of the two load cases described at the beginning of this section.

Solving equation G

G

, assuming a value of G

50 J/m

yields the results shown in Figure 7(b), where the average strain needed to propagate a crack with a certain length is shown for two different cases:

considering and neglecting the residual stresses of the curing stage.

As can be seen, a slow crack growth is obtained in an initial stage, followed by increasingly faster crack propagation at higher loads.

Furthermore, these results clearly show that the presence of residual stresses delays crack propagation, as mentioned before.

Obtaining a series of curves like the one shown in Figure 7(b), with a parametric variation of G

, would permit the calculation of the actual interface fracture toughness by best fit of experimental debond propagation measurements. However, prior to the comparison with test results, the effect of friction between crack faces on debond propagation needs to be elucidated.

3.3 Accuracy of the numerical results

In order to show the accuracy of the results, a comparison of several

aspects of the numerical solution of the BEM models with existing

analytical or semianalytical results obtained from the literature is shown in Annex 4.

Firstly, the behavior of the BEM solution in the vicinity of the fiber crack terminating at the interface is compared with the behavior predicted by the semianalytical singularity analysis of multimaterial corners presented by Barroso et al.

. According to this analysis, when U o 0 ^, with U being the distance to the crack tip, the following asymptotic behavior should be obtained: u

O ( U

) and V

O ( U

) , with

z r, , E

D ^and O ^{0 . 187} . Numerical evaluation of the singularity exponent O carried out from the BEM near tip solution yields values in the range

198 . 0 190

.

0  O  .

Secondly, the behavior of the BEM solution in the vicinity of the debond crack is compared with the asymptotical behavior corresponding to an interface crack. In particular, using the frictionless contact model, a singularity exponent in the range 0 . 501  O  0 . 511 is evaluated from the near tip stresses obtained with BEM, which is in very good agreement with the theoretical result O 0 . 5 predicted by Comninou

.

Finally, according to the theoretical analysis presented by Leguillon and Sanchez-Palencia

, limit behavior of the ERR when a o 0 , with a being the debond crack length, should be in accordance with the previously mentioned singularity exponent of the solution corresponding to the fiber crack terminating at the interface. Using the frictionless contact model, the BEM solution yields an asymptotic behavior ^{G O} ( U

) which is again in a very good agreement with the expected theoretical behavior:

) ( )

( U

^O U

O

G .

3.4 Effect of friction on the stress state in the SFFT sample

The influence of friction between fiber and matrix along the debond crack faces on the stress field at the interface can be observed in Annex 4, where the results of the frictionless contact model are compared with those obtained considering a friction coefficient P 0 . 5 . Apart from the obvious appearance of a singular tangential stress at the interface, numerical results show that friction has a visible influence on the stress values at both sides of the crack tip, decreasing the normal contact stresses and the tangential stresses ahead of the crack tip.

To quantify the influence of friction on the interface crack growth, an extension of the previous study of ERR behavior for frictionless contact would be required. However, the value of G obtained using Irwin virtual crack closure technique for an infinitesimal virtual crack step has no finite (non zero) limit, due to the singularity exponent, which is different from the standard 0.5 value

. Additionally, energy dissipation due to friction during crack propagation should be taken into account together with possible history dependence. These issues are under research and will be addressed in forthcoming works.

4. Concluding remarks and future work

A Boundary Elements code for the numerical analysis of

micromechanical elastic fields in the single-fiber fragmentation test has

been developed, which allows the definition of multiple solids bonded or in

contact to be carried out. The code takes advantage of the axial symmetry,

permits the use of non conforming meshes in the interfaces and contact

zones and takes into account the residual stresses developed during the

curing of the samples.

Preliminary stress analysis of the single fiber fragmentation test have been carried out, neglecting the effect of friction along the debond crack faces, yielding accurate results of stresses and relative displacements. The use of these numerical results permits the evaluation of the energy release rate associated to debond propagation to be accomplished using a local approach in which only the elastic solution in the vicinity of the debond crack tip is employed.

The ability of the numerical tool to obtain accurate elastic solution within the fragmentation sample including the effect of friction along the debond crack faces has also been demonstrated. However, evaluation of energy release rate associated to debond propagation in presence of friction from the obtained numerical results is not an straight-forward task, due to the inherent non-linearity associated to the dissipation of energy due to friction.

Additionally, to be able to broaden the scope of the numerical

simulation of the single fiber fragmentation test, further enhancement of the

Boundary Elements code need to be carried out: firstly, including cohesive

conditions in the formulation of the coupling equations along the interfaces

would allow the analysis of crack propagation to be carried out in a more

efficient manner (and also taking to account the actual load history

followed by the sample, which is a key aspect if friction is considered in

the analysis) and, secondly, the fundamental solution corresponding to

axisymmetric elasticity problems with transversely isotropic elastic

material behavior should be implemented to analyze samples with carbon

fibers.

References

[1] Kelly A, Tyson WR (1965). “Tensile properties of fiber-reinforced metals: copper/tungsten and copper/molybdenum” Journal of the Mechanics and Physics of Solids, 13:329-350.

[2] Ohsawa T, Nakayama A, Miwa M, Hasegawa A (1978).

“Temperature-dependence of critical fiber length for glass fiber- reinforced thermosetting resins”. Journal of Applied Polymer Science, 22:3203-3212,.

[3] Sjögren A, Joffe R, Berglund L, Mäder E (1999). “Effects of fiber coating (size) on properties of glass fiber vinyl ester composites”.

Composites Part A, 30:1009-1015.

[4] Cox HL (1952). “The elasticity and strength of paper and other fibrous materials”. British Journal of Applied Physics, 3: 72-79.

[5] Whitney JM, Drzal LT (1987). “Axisymmetric Stress Distribution Around An Isolated Fiber Fragment”. ASTM Special Technical Publication, 179-196.

[6] Netravali AN, Henstenburg RB, Phoenix SL, Schwartz P (1989).

“Interfacial shear-strength studies using the single-filament- composite test. 1. Experiments on graphite fibers in epoxy”. Polymer Composites, 10:226-241.

[7] Henstenburg RB, Phoenix SL (1989). “Interfacial shear-strength studies using the single-filament-composite test. 2. A probability model and Monte-Carlo simulation”. Polymer Composites, 10:389- 408.

[8] Lacroix T, Tilmans B, Keunings R, Desaeger M, Verpoest I (1992). “Modeling of critical fiber length and interfacial debonding in the fragmentation testing of polymer composites”. Composites Science and Technology, 43:379-387.

[9] Lacroix T, Tilmans B, Keunings R, Desaeger M, Verpoest I (1995). “A new data reduction scheme for the fragmentation testing of polymer composites”. Journal of Materials Science, 30:683-692.

[10] Tripathi D, Jones FR (1997). “Measurement of the load-bearing

capability of the fibre/matrix interface by single-fibre

fragmentation”. Composites Science And Technology, 57:925-935.