- Email: [email protected]
Evaluation of interface parameters in push-out and pull-out tests
Evaluation of interface parameters in push-out and pull-out tests M. KUNTZ K.-H. SCHLAPSCHI, B. MEIER and G. GRATHWOHL (University of Karlsruhe, Germany) Received 27 September 1993 Push-out and pull-out tests with single-fibre model composites are performed to evaluate the frictional parameters of the interface. Analysis of the experimental data according to models based on the assumption of an ideal cylindrically shaped fibre does not satisfy all observed phenomena. Thus, a model is proposed to include the effects of roughness interaction during fibre sliding. The predictions of the model fit the experimental results very well. Key words: single-fibre composites; interface parameters; friction behaviour; fibre sliding; push-out tests; pull-out tests; rough surface model The mechanical behaviour of optimized ceramic fibre/ ceramic matrix composites (CMCS) is controlled by their interfacial properties. There is evidence that attractive mechanical performance of CMCS depends on welladjusted bond strength and sliding resistance of the fibre/ matrix interface 1,2. The bond strength is usually described by the interface toughness, Fi. It has been shown that interface debonding, which is necessary for noncatastrophic failure, depends on low values of 1-"i. When debonding occurs, composite failure is dictated by frictional sliding at the fibre/matrix interface during progressive debonding and pull-out. All essential characteristics of CMfS such as R-curve behaviour and high fracture energy are directly correlated to the interfacial
main objective is the analysis of the friction behaviour during fibre sliding in both loading cases.
friction stress.
For the pull-out test, the specimen configuration used is
Advanced models3-6 for fibre debonding and pull-out quantify the sliding resistance with two parameters: the
where cry is the interface stress relief due to Poisson's contraction of the fibre.
sketched in Fig. 1. During loading, a crack is initiated in the matrix at the circumferential notch. Thus, the classical phenomena of initial and progressive debonding, crack opening, fibre failure and pull-out are all simulated. Specimen length was varied between 8 and 60 mm, the crosshead speed was set to 2 ~m s l. The push-out tests were performed using specimens with a thickness of 200-3500 ~tm. Fig. 2 shows the principle of this technique. The diamond indenter was positioned by means of a video camera with a macro objective, axial displacement of the indenter was achieved by a piezotranslator. Experimental details were described previously7.
To correlate composite failure to the interface parameters, it is necessary first to develop highly accurate micromechanical measurement methods and second to verify the models with adequate test series. This work is dedicated to the latter case. Single-fibre model composites have been tested in pull-out and push-out tests. The
Investigation of the surface roughness was carried out using a Topometrix TMX 2000 atomic force microscope (AFM).The fibres were mounted on sample discs with thermoplastic adhesive and imaged using an Si3N4 standard cantilever. The effect of curvature of the fibres was eliminated by subtracting a fitted second-order polynomial from the data.
coefficient of friction p and an interfacial clamping stress oc, which acts perpendicular to the interface and is caused by thermal expansion mismatch or roughness interaction between fibre and matrix. The sliding resistance r is described as: r = p" (o-~ - o'v)
(1)
EXPERIMENTAL The monofilament specimens were produced from SiC fibres (SCS-6, Textron Specialty Materials, USA) and an aluminosilicate glass matrix (Supremax, Schott Glaswerke Mainz, Germany). Material data are given in Table 1. To obtain well-defined stress conditions, the specimens were treated by an annealing process above the glass transition temperature; thus only the residual stresses due to thermal expansion mismatch will remain in the composite.
0010-4361/94/07/0476-06 © 1994 Butterworth-Heinemann ktd 476 COMPOSITES.VOLUME 25. NUMBER 7. 1994
T a b l e 1. M a t e r i a l
Data
Fibre Manufacturer Trade name Chemistry Manufacturing technique Diameter, ~b (lLtm) Young's modulus, E (GPa) Poisson's ratio, v Thermal expansion coefficient, ~(10 6K 1)
Matrix
Textron Specialty Materials, USA Schott Glaswerke Mainz, Germany SCS-6 Supremax C-core, ~-SiC Si02-AI203-B20z Chemical vapour deposition Molten glass pultrusion 137 +. 1 5500 413 87 0.19 0.24 4.3
4.61
F
,::::::: ::5::::
',',',',ii',
S°,ema,,° o,, e siog,e-fib e pus,-out test
[
[
!!!!i!!i!
In Fig. 3(b), the load/displacement relation of a thick push-out specimen is shown. As expected, the critical debonding load is much higher than for the thin specimen. After the load drop, the state of pure friction is reached immediately without a second maximum.
i!!ii!iii! !!i!i!!~ i!iii:~ ~
~
develops linearly until a critical value is reached. Then a sharp load drop is observed to a lower value. Thereafter, the force increases again and reaches a second maximum. Then the force declines relatively quickly until it reaches a soft edge, which might be related to the beginning of real frictional sliding. The curve is then characterized by a low stable slope.
J
F Fig. 1 Schematicof the single-fibre pull-out test
RESULTS Load/displacement behaviour For both push-out and pull-out tests, typical phenomena can be distinguished depending on the specimen geometry. Fig. 3(a) shows the loading behaviour of a thin push-out specimen. After an initial run-in, the force
The behaviour in the pull-out test is different for short specimens, which show complete fibre pull-out, and long specimens, in which fibre failure occurs during progressive debonding. Fig. 4(a) shows the pull-out curve of a short specimen. Progressive debonding occurs only in a very short range immediately before the maximum load is reached. Then a sharp load drop occurs and the fibre is pulled out, also characterized by a low slope according to the reduction of the embedded fibre length. For long specimens (Fig. 4(b)), the first load drop indicates matrix cracking. Then, a highly extended region of progressive debonding can be observed. The maximum load indicates the strength of the embedded fibre (3500 MPa). This strength was also measured in single-fibre tensile tests with fibres that were embedded in the matrix and pulled out. As the fibre strength does not scatter very much, failure occurs within the matrix crack; thus, no final pull-out is observed.
COMPOSITES . NUMBER 7 . 1994
477
4
25
Debonding 3
20~
/I
Pure friction /
~ 15 "0
2
q lo 1
5
0
0 lO
a
20
Indenter
30
40
so
Displacement~m]
I
I
I
I
200
400
600
800
40
1,000
.13i-~nlanement [/Jm] _._~..__
a
60
Debonding
so
30
~
Z
ion
40
Z "0 30
'10 20
o,
20 10
10 0
0 ~ 0
b
I
,
10
I 20
I
30
,
I
I
i
40
50
60
,
0
70
Indenter Displacement [pro]
=
i
200
300
400
Displacement [/Jm]
b Fig. 4
Fig. 3 Typical load/displacement behaviour of push-out tests: (a) thin specimen (800 I~m); (b) thick specimen (31 70 ~,m)
I
100
Typical load/displacement behaviour of pull-out tests: (a)
short specimen (8 ram); (b) long specimen (40 ram)
where The curvature indicates a plateau of the debonding force, at which debonding progresses without further increase in load. This plateau is reached when the normal stress relief o-, eliminates the interfacial clamping stress. In the case of Fig. 4(b), the plateau cannot be reached because of premature fibre failure. From the curvature, however, the plateau can be estimated to reach a value of about 70-80 N. The debonding force plateau P* is a very important result of the pull-out test and will be discussed below, Evaluation of results It is the objective of this study to assess the influence of roughness interaction between fibre and matrix during pull-out. In order to demonstrate the necessity of such a treatment, the results are first calculated according to models based on the assumption of perfect cylinders. It will be shown that the push-out and pull-out phenomena cannot be explained by these solutions. Then it will be shown that a frictional theory based upon tribological fundamentals fits the experimental data very well. The specimen configuration in this study allows the assumption of an infinite matrix radius. Thus, the basic equations given by Kerans and Parthasarathy 5 can be applied. For push-out, the maximum frictional force Pq with respect to the specimen thickness t is given by: e q = P* (e ~' -
478
1)
COMPOSITES.NUMBER7
(2)
1994
p, _
zrr~ k o'c
t - 2pk rf Em vr k = Er(1 + Vm) -'t- Em(1 - V r ) f and m denote fibre and matrix, E and v are the Young's modulus and the coefficient of Poisson contraction, respectively. The fibre radius is indicated by rr. k is the proportionality factor of the normal stress relief due to fibre axial stress o'z; i.e., or, = k o'z. The push-out results of 30 tests and the corresponding fitting curve are illustrated in Fig. 5. The evaluation according to Equation (2) reveals a clamping stress o-c = - 75 MPa and a coefficient of friction p = 0.12. The pull-out results are evaluated by a completely different algorithm to that of the push-out tests. Concerning push-out, a sample of tests and a two-parameter regression are needed to obtain the interfacial data. Using pullout test results, the frictional parameters o'c and p are obtained independently. The clamping stress o'c is directly correlated to the debonding plateau P*, i.e., when the stress relief due to Poisson contraction o'v equals the clamping stress. From the anticipated plateau P* = 70 N, the clamping stress is immediately obtained to a value of at least 120 MPa. With this value, the coefficient of friction/~ is directly obtained from the pull-out force:
•
20
c = 75 MPa
ze 15
~p = 0.12
i . / /
E o '+"
/ °
;/
r,~2(u/s)
LI
=
.,,
(X
o ~ . .
asperity distribution and E' is defined as: 1 _I--v2+I--Vm
."_-.~' ~ ", co°
IX. 5
.".""
E'
°o~,•
S
0
Ef
2
~ m
It is now the idea of the new model that the solution of
,
,
,
2,000 3.000 Specimen thickness [pm]
1,000
0
Fig. 5
°
u/s) 5/2e-x2/2dx
t¢is the number of asperities per unit area,/3 the average asperity radius, s the Gaussian standard deviation of the
o
10
oc-
-
4,000
Frictional forces in push-out tests and two-parameter fit
curve accordingto Equation (2) PrT = e*(l - e -)a)
(3)
where l is the embedded fibre length. Evaluation of the complete pull-out data with o-~ = - 120 MPa indicates a coefficient of friction of/1 = 0.065. As can be seen, there are significant differences between the push-out and pull-out results. Additionally, the values for the coefficient of friction seem to be very low compared with literature data for frictional properties of carbon. Thus, it should be discussed whether the assumption of perfect cylinders is permissible. The following section is concerned with the effects of roughness interaction,
Kerans and Parthasarathy5, Equation (4), can still be applied with a separation u determined according to the solution of Greenwood and Tripp 9, Equation (5). Thus the roughness-induced interfacial normal stress is newly determined taking into account the actual data of the real surfaces. It is initially assumed that the asperities of fibre and matrix are intimately connected and caused by the low viscosity of the matrix during processing (Fig. 6). Then the asperity profiles of fibre and matrix have the same amplitude. A reference line to describe the separation of fibre and matrix has to be defined, which is set for convenience to the centre of each profile. Thus, the initial separation between fibre and matrix is zero. The initial state is the only one in which no roughness contribution to the interfacial clamping stress exists. During fibre sliding, there is arbitrary contact between the surfaces, which leads to the separation u. We have now two equations, (4) and (5), for the two unknowns u and an. The problem can be solved iteratively. In Fig. 7, the an/U
a)
M O D E L L I N G OF R O U G H N E S S INTERACTION
I
It has repeatedly been suggested that roughness interaction contributes to the interfacial clamping stress. Kerans and Parthasarathy5 assumed that the separation of fibre and matrix during fibre sliding equals the roughness amplitude. They applied solutions of ideal axisymmetric elastic separation of cylindrical bodies to quantify this model. The roughness-induced interfacial normal stress due to a fibre-matrix gap u was calculated usingS:
...... j
b) ~
,~' ~-i U A
......
..........
'
!
Fig. 6 Roughnessinteraction: (a) initial state, fibre and matrix rou0 nes 0ro,,,e are,n,, ate, connecte0 nose0 rat,on
an = Er(1 + Vm)-r Em(l
--
vr)
,a ,
trary state after fibre displacement, elastic contact of asperities, separation u > 0
On the other hand, it is reasonable to assume that the total separation of fibre and matrix due to roughness interaction is lower than the roughness amplitude because of the statistical distribution of the profile asperities and the larger elastic deformation of the asperities in contact. Greenwood and Tripp 9 developed a model to calculate the approach of two rough surfaces with respect to the nominal contact pressure. Their model is based on the classical theory of elastic contact (Hertzian pressure) and an appropriate statistical description of the rough surface. The approach of two rough surfaces is directly correlated to the nominal pressure o-, as:
0~
m
~" o_ -5o ~-lO0 ~ -15o ~ / ~ -20o -~ l=
separat,on
~;~hoaScU~ace
~
-250
a.(u) = 8~-(tcfls)KFh,2(u/s)
Z -3oo
0.02 0.04 0.06 0.08 0.1 0.12 0.14 separation u / approach u [pm]
where K
(5)
2X/2 (/otiS)E, ./S_
3
Fig. 7 Relationship between interfacial normal stress cro and separation u according to Equations (5) and (6)
COMPOSITES. NUMBER 7. 1994 479
relation according to Equations (4) and (5) is illustrated. It should be noted that in the following ty, means the total interface normal stress including residual clamping stresses, roughness interaction and the effect of Poisson contraction.
"-~ 0 n ~ -20
In order to apply the model, the effects of shear stresses, abrasion and plastic deformation during fibre sliding have to be neglected. The first assumption seems to be reasonable, since the calculated shear stresses are low compared with the normal stresses and do not affect the Hertzian solutions very much. The second and third assumptions are supported by a topography analysis after pull-out, which revealed no significant differences to the as-received state.
-~'~
Fig. 8 shows an AFM image of the fibre topography. The roughness parameters obtained from the AFM analysis are fl = 492 I.tm, t¢ = 14 400 per mm 2 and s = 0.040 lam. With these data, it is interesting to compare the predictions of the perfect cylinder model with this new approach, in the following named 'rough surface model'. To expand the rough surface model to fibre pull-out analysis, the elastic solution is rewritten in terms of u = f(tyn). Additionally, the radial prestrain due to residual radial stresses O'r0 and the radial strain due to Poisson contraction are also regarded. Then, the equivalent to Equation (4) with respect to these extensions is expressed as: -rf[(tyn[ - tYrO)Er(1 + I~m) + Era(1 - vf) U
Ef E m
(6) Er ] Fig. 9 shows the calculated normal stress at the interface ty, with respect to the fibre axial stress. The dashed straight line represents the solution of the perfect cylinder model as obtained from the push-out results, The slope of this line represents the important k-factor to evaluate the influence of Poisson contraction on the radial stress. -- O'zPf]
The rough surface model (solid line) predicts a nearly
~
",/
J---~c e
,
~
model
-40 Poffect cylinder
J
e-~ -8o
/~
//"
model
~ -10o '~-120 1= ~w
, --
-2,000
I
-1,000
+ 0
I
1,000
'
2,000
Axial s t r e s s in fiber
3,000
4,000
5,000
[MPa]
Fig. 9 Comparison of'perfect cylinder' and 'rough surface' models. Relationship between axial fibre stress and effective normal stress at the interface
linear correlation an = flty~)during axial compression (push-out) and at low tensile stresses. The slope in this linear region is lower than that of the perfect cylinder model and depends on the elastic and roughness parameters of the components. This indicates that, for pushout tests, the linear model with a modified factor k' can still be applied. For the material discussed here, the modified factor is k' = 0.0221, whereas the pure elastic factor is calculated to be k = 0.0284. Thus, the push-out data (Fig. 5) are newly evaluated to give tYc = - 5 5 . 8 MPa and p = 0.17. On the other hand, the clamping stress can be calculated directly from the thermoelastic and roughness material data. For this, the residual stresses due to thermal expansion mismatch are evaluated according to the exact solution of Budiansky et al.l°; the thermal residual radial stress is o-rO = - 16.1 MPa. Then, the iterative solution according to Equations (5) and (6) leads to a total clamping stress of tyc = - 51.2 MPa (see Fig. 9, the intersection of the solid line with the vertical line indicating zero axial fibre stress). The calculated clamping stress is well in the range of the result obtained from the push-out evaluation. The pull-out curves have to be evaluated numerically. With the rough surface model the predicted plateau of debonding force is much higher than that of the perfect cylinder model due to the non-linearity for low values of ty,. For the roughness data given above, the plateau is calculated to be 5000 MPa ( = 70 N), which correlates very well with the experimental results. Finally, the pull-out results have to be evaluated numerically to obtain the coefficient of friction from the pull-out data. The numerical treatment will be presented elsewhere. In this test series, a coefficient of friction of p = 0.20 was obtained.
SUMMARY
Fig. 8
Shaded AFM-imageoffibresurface
480
COMPOSITES. N U M B E R 7 . 1994
Push-out and pull-out tests have been performed with single-fibre model composites. Analysis of the results according to models based on the assumption of
perfect cylindrically s h a p e d c o m p o n e n t s revealed c o n t r a -
dictions between push-out and pull-out evaluation. Thus, a m o d e l has been p r o p o s e d to assess the effects o f interfacial r o u g h n e s s d u r i n g fibre sliding. T h e ' r o u g h surface m o d e l ' enables direct c a l c u l a t i o n o f the interfacial c l a m p i n g stress, which was s h o w n to be in g o o d
5
6 7
a g r e e m e n t with the e x p e r i m e n t a l results, 8
A d d i t i o n a l l y , the new e v a l u a t i o n o f the p u s h - o u t a n d p u l l - o u t tests revealed c o n g r u e n c e o f these different m e t h o d s in their essential results. It is c o n c l u d e d
9
clamping stress and coefficient of friction for fiber-reinforced ceramic composites' Acta Metal138 No 3 (1990) pp 403~,09 Kerans, R.J. and Parthasarathay, T.A. 'Theoretical analysis of the fiber pullout and pushout tests' J Am Ceram Soc 74 No 7 (1991) pp 1585-1596 Zhon, L.-M., Kim, J.K. and Mai, Y.-W.'Interfacialdebondingand fibre pull-out stresses' J Mater Sei 27 (1992) pp 3155-3166 Rausch, G.,Meier, B. and Grathwohl, G.'A push-out technique for the evaluation of interracial properties of fiber-reinforced materials' J Europ Ceram Soc 10 (1992) pp 229-235 Oel, H.J. and Frechette, F. 'Stress distribution in multiphase systems: II, composite disks with cylindrical interfaces' J Am Ceram Soc 69 No 4 (1986) pp 342 346 Greenwood,J.A. and Tripp, J.H. 'The contact of two nominally flat rough surfaces' Proc Inst Mech Engrs 185 No 48 (1971) pp 625 433 Budiansky, B., Hutchinson, J.W. and Evans, A.G. 'Matrix fracture in fiber-reinforced ceramics' J Mech Phys Solids 34 No 2 (1986) pp 167 189
that reliable e v a l u a t i o n o f m i c r o m e c h a n i c a l tests d e p e n d s s t r o n g l y on a d d i t i o n a l d a t a o f the interface properties,
10
REFERENCES
A UTHORS
l Evans, A.G., Zok, F.W. and Davis, J. 'The role of interfaces in fiber-reinforced brittle matrix composites' Composites Sei and Techno142 (1991)pp 3-24 2 Kim, J.K. and Mai, Y.-W. 'High strength, high fracture toughness fibre composites with interface control-- a review' Composites Sci and Techno141 (1991) pp 333-378 3 Gao, Y.-C., Mai, Y.-W. and CoUerell, B. 'Fracture of fiber reinforced materials' Z A M P 3 9 (1988) pp 550-572 4 Hsueh, C.-H. 'Evaluation of interfacial shear strength, residual
M.
Kuntz,
to
whom
correspondence
should
be
addressed, K . - H . Schlapschi a n d G. G r a t h w o h l are with
the Institut f/ir Keramik im Maschinenbau, Zentrall a b o r a t o r i u m , Universitfit K a r l s r u h e , H a i d - u n d N e u Strasse 7, 76131 K a r l s r u h e , G e r m a n y . B. M e i e r is with the Institut f/ir W e r k s t o f f k u n d e II, Universitfit K a r l s ruhe, K a i s e r s t r a s s e 12, 76128 K a r l s r u h e , G e r m a n y .
COMPOSITES. N U M B E R 7 . 1994
481
main objective is the analysis of the friction behaviour during fibre sliding in both loading cases.
friction stress.
For the pull-out test, the specimen configuration used is
Advanced models3-6 for fibre debonding and pull-out quantify the sliding resistance with two parameters: the
where cry is the interface stress relief due to Poisson's contraction of the fibre.
sketched in Fig. 1. During loading, a crack is initiated in the matrix at the circumferential notch. Thus, the classical phenomena of initial and progressive debonding, crack opening, fibre failure and pull-out are all simulated. Specimen length was varied between 8 and 60 mm, the crosshead speed was set to 2 ~m s l. The push-out tests were performed using specimens with a thickness of 200-3500 ~tm. Fig. 2 shows the principle of this technique. The diamond indenter was positioned by means of a video camera with a macro objective, axial displacement of the indenter was achieved by a piezotranslator. Experimental details were described previously7.
To correlate composite failure to the interface parameters, it is necessary first to develop highly accurate micromechanical measurement methods and second to verify the models with adequate test series. This work is dedicated to the latter case. Single-fibre model composites have been tested in pull-out and push-out tests. The
Investigation of the surface roughness was carried out using a Topometrix TMX 2000 atomic force microscope (AFM).The fibres were mounted on sample discs with thermoplastic adhesive and imaged using an Si3N4 standard cantilever. The effect of curvature of the fibres was eliminated by subtracting a fitted second-order polynomial from the data.
coefficient of friction p and an interfacial clamping stress oc, which acts perpendicular to the interface and is caused by thermal expansion mismatch or roughness interaction between fibre and matrix. The sliding resistance r is described as: r = p" (o-~ - o'v)
(1)
EXPERIMENTAL The monofilament specimens were produced from SiC fibres (SCS-6, Textron Specialty Materials, USA) and an aluminosilicate glass matrix (Supremax, Schott Glaswerke Mainz, Germany). Material data are given in Table 1. To obtain well-defined stress conditions, the specimens were treated by an annealing process above the glass transition temperature; thus only the residual stresses due to thermal expansion mismatch will remain in the composite.
0010-4361/94/07/0476-06 © 1994 Butterworth-Heinemann ktd 476 COMPOSITES.VOLUME 25. NUMBER 7. 1994
T a b l e 1. M a t e r i a l
Data
Fibre Manufacturer Trade name Chemistry Manufacturing technique Diameter, ~b (lLtm) Young's modulus, E (GPa) Poisson's ratio, v Thermal expansion coefficient, ~(10 6K 1)
Matrix
Textron Specialty Materials, USA Schott Glaswerke Mainz, Germany SCS-6 Supremax C-core, ~-SiC Si02-AI203-B20z Chemical vapour deposition Molten glass pultrusion 137 +. 1 5500 413 87 0.19 0.24 4.3
4.61
F
,::::::: ::5::::
',',',',ii',
S°,ema,,° o,, e siog,e-fib e pus,-out test
[
[
!!!!i!!i!
In Fig. 3(b), the load/displacement relation of a thick push-out specimen is shown. As expected, the critical debonding load is much higher than for the thin specimen. After the load drop, the state of pure friction is reached immediately without a second maximum.
i!!ii!iii! !!i!i!!~ i!iii:~ ~
~
develops linearly until a critical value is reached. Then a sharp load drop is observed to a lower value. Thereafter, the force increases again and reaches a second maximum. Then the force declines relatively quickly until it reaches a soft edge, which might be related to the beginning of real frictional sliding. The curve is then characterized by a low stable slope.
J
F Fig. 1 Schematicof the single-fibre pull-out test
RESULTS Load/displacement behaviour For both push-out and pull-out tests, typical phenomena can be distinguished depending on the specimen geometry. Fig. 3(a) shows the loading behaviour of a thin push-out specimen. After an initial run-in, the force
The behaviour in the pull-out test is different for short specimens, which show complete fibre pull-out, and long specimens, in which fibre failure occurs during progressive debonding. Fig. 4(a) shows the pull-out curve of a short specimen. Progressive debonding occurs only in a very short range immediately before the maximum load is reached. Then a sharp load drop occurs and the fibre is pulled out, also characterized by a low slope according to the reduction of the embedded fibre length. For long specimens (Fig. 4(b)), the first load drop indicates matrix cracking. Then, a highly extended region of progressive debonding can be observed. The maximum load indicates the strength of the embedded fibre (3500 MPa). This strength was also measured in single-fibre tensile tests with fibres that were embedded in the matrix and pulled out. As the fibre strength does not scatter very much, failure occurs within the matrix crack; thus, no final pull-out is observed.
COMPOSITES . NUMBER 7 . 1994
477
4
25
Debonding 3
20~
/I
Pure friction /
~ 15 "0
2
q lo 1
5
0
0 lO
a
20
Indenter
30
40
so
Displacement~m]
I
I
I
I
200
400
600
800
40
1,000
.13i-~nlanement [/Jm] _._~..__
a
60
Debonding
so
30
~
Z
ion
40
Z "0 30
'10 20
o,
20 10
10 0
0 ~ 0
b
I
,
10
I 20
I
30
,
I
I
i
40
50
60
,
0
70
Indenter Displacement [pro]
=
i
200
300
400
Displacement [/Jm]
b Fig. 4
Fig. 3 Typical load/displacement behaviour of push-out tests: (a) thin specimen (800 I~m); (b) thick specimen (31 70 ~,m)
I
100
Typical load/displacement behaviour of pull-out tests: (a)
short specimen (8 ram); (b) long specimen (40 ram)
where The curvature indicates a plateau of the debonding force, at which debonding progresses without further increase in load. This plateau is reached when the normal stress relief o-, eliminates the interfacial clamping stress. In the case of Fig. 4(b), the plateau cannot be reached because of premature fibre failure. From the curvature, however, the plateau can be estimated to reach a value of about 70-80 N. The debonding force plateau P* is a very important result of the pull-out test and will be discussed below, Evaluation of results It is the objective of this study to assess the influence of roughness interaction between fibre and matrix during pull-out. In order to demonstrate the necessity of such a treatment, the results are first calculated according to models based on the assumption of perfect cylinders. It will be shown that the push-out and pull-out phenomena cannot be explained by these solutions. Then it will be shown that a frictional theory based upon tribological fundamentals fits the experimental data very well. The specimen configuration in this study allows the assumption of an infinite matrix radius. Thus, the basic equations given by Kerans and Parthasarathy 5 can be applied. For push-out, the maximum frictional force Pq with respect to the specimen thickness t is given by: e q = P* (e ~' -
478
1)
COMPOSITES.NUMBER7
(2)
1994
p, _
zrr~ k o'c
t - 2pk rf Em vr k = Er(1 + Vm) -'t- Em(1 - V r ) f and m denote fibre and matrix, E and v are the Young's modulus and the coefficient of Poisson contraction, respectively. The fibre radius is indicated by rr. k is the proportionality factor of the normal stress relief due to fibre axial stress o'z; i.e., or, = k o'z. The push-out results of 30 tests and the corresponding fitting curve are illustrated in Fig. 5. The evaluation according to Equation (2) reveals a clamping stress o-c = - 75 MPa and a coefficient of friction p = 0.12. The pull-out results are evaluated by a completely different algorithm to that of the push-out tests. Concerning push-out, a sample of tests and a two-parameter regression are needed to obtain the interfacial data. Using pullout test results, the frictional parameters o'c and p are obtained independently. The clamping stress o'c is directly correlated to the debonding plateau P*, i.e., when the stress relief due to Poisson contraction o'v equals the clamping stress. From the anticipated plateau P* = 70 N, the clamping stress is immediately obtained to a value of at least 120 MPa. With this value, the coefficient of friction/~ is directly obtained from the pull-out force:
•
20
c = 75 MPa
ze 15
~p = 0.12
i . / /
E o '+"
/ °
;/
r,~2(u/s)
LI
=
.,,
(X
o ~ . .
asperity distribution and E' is defined as: 1 _I--v2+I--Vm
."_-.~' ~ ", co°
IX. 5
.".""
E'
°o~,•
S
0
Ef
2
~ m
It is now the idea of the new model that the solution of
,
,
,
2,000 3.000 Specimen thickness [pm]
1,000
0
Fig. 5
°
u/s) 5/2e-x2/2dx
t¢is the number of asperities per unit area,/3 the average asperity radius, s the Gaussian standard deviation of the
o
10
oc-
-
4,000
Frictional forces in push-out tests and two-parameter fit
curve accordingto Equation (2) PrT = e*(l - e -)a)
(3)
where l is the embedded fibre length. Evaluation of the complete pull-out data with o-~ = - 120 MPa indicates a coefficient of friction of/1 = 0.065. As can be seen, there are significant differences between the push-out and pull-out results. Additionally, the values for the coefficient of friction seem to be very low compared with literature data for frictional properties of carbon. Thus, it should be discussed whether the assumption of perfect cylinders is permissible. The following section is concerned with the effects of roughness interaction,
Kerans and Parthasarathy5, Equation (4), can still be applied with a separation u determined according to the solution of Greenwood and Tripp 9, Equation (5). Thus the roughness-induced interfacial normal stress is newly determined taking into account the actual data of the real surfaces. It is initially assumed that the asperities of fibre and matrix are intimately connected and caused by the low viscosity of the matrix during processing (Fig. 6). Then the asperity profiles of fibre and matrix have the same amplitude. A reference line to describe the separation of fibre and matrix has to be defined, which is set for convenience to the centre of each profile. Thus, the initial separation between fibre and matrix is zero. The initial state is the only one in which no roughness contribution to the interfacial clamping stress exists. During fibre sliding, there is arbitrary contact between the surfaces, which leads to the separation u. We have now two equations, (4) and (5), for the two unknowns u and an. The problem can be solved iteratively. In Fig. 7, the an/U
a)
M O D E L L I N G OF R O U G H N E S S INTERACTION
I
It has repeatedly been suggested that roughness interaction contributes to the interfacial clamping stress. Kerans and Parthasarathy5 assumed that the separation of fibre and matrix during fibre sliding equals the roughness amplitude. They applied solutions of ideal axisymmetric elastic separation of cylindrical bodies to quantify this model. The roughness-induced interfacial normal stress due to a fibre-matrix gap u was calculated usingS:
...... j
b) ~
,~' ~-i U A
......
..........
'
!
Fig. 6 Roughnessinteraction: (a) initial state, fibre and matrix rou0 nes 0ro,,,e are,n,, ate, connecte0 nose0 rat,on
an = Er(1 + Vm)-r Em(l
--
vr)
,a ,
trary state after fibre displacement, elastic contact of asperities, separation u > 0
On the other hand, it is reasonable to assume that the total separation of fibre and matrix due to roughness interaction is lower than the roughness amplitude because of the statistical distribution of the profile asperities and the larger elastic deformation of the asperities in contact. Greenwood and Tripp 9 developed a model to calculate the approach of two rough surfaces with respect to the nominal contact pressure. Their model is based on the classical theory of elastic contact (Hertzian pressure) and an appropriate statistical description of the rough surface. The approach of two rough surfaces is directly correlated to the nominal pressure o-, as:
0~
m
~" o_ -5o ~-lO0 ~ -15o ~ / ~ -20o -~ l=
separat,on
~;~hoaScU~ace
~
-250
a.(u) = 8~-(tcfls)KFh,2(u/s)
Z -3oo
0.02 0.04 0.06 0.08 0.1 0.12 0.14 separation u / approach u [pm]
where K
(5)
2X/2 (/otiS)E, ./S_
3
Fig. 7 Relationship between interfacial normal stress cro and separation u according to Equations (5) and (6)
COMPOSITES. NUMBER 7. 1994 479
relation according to Equations (4) and (5) is illustrated. It should be noted that in the following ty, means the total interface normal stress including residual clamping stresses, roughness interaction and the effect of Poisson contraction.
"-~ 0 n ~ -20
In order to apply the model, the effects of shear stresses, abrasion and plastic deformation during fibre sliding have to be neglected. The first assumption seems to be reasonable, since the calculated shear stresses are low compared with the normal stresses and do not affect the Hertzian solutions very much. The second and third assumptions are supported by a topography analysis after pull-out, which revealed no significant differences to the as-received state.
-~'~
Fig. 8 shows an AFM image of the fibre topography. The roughness parameters obtained from the AFM analysis are fl = 492 I.tm, t¢ = 14 400 per mm 2 and s = 0.040 lam. With these data, it is interesting to compare the predictions of the perfect cylinder model with this new approach, in the following named 'rough surface model'. To expand the rough surface model to fibre pull-out analysis, the elastic solution is rewritten in terms of u = f(tyn). Additionally, the radial prestrain due to residual radial stresses O'r0 and the radial strain due to Poisson contraction are also regarded. Then, the equivalent to Equation (4) with respect to these extensions is expressed as: -rf[(tyn[ - tYrO)Er(1 + I~m) + Era(1 - vf) U
Ef E m
(6) Er ] Fig. 9 shows the calculated normal stress at the interface ty, with respect to the fibre axial stress. The dashed straight line represents the solution of the perfect cylinder model as obtained from the push-out results, The slope of this line represents the important k-factor to evaluate the influence of Poisson contraction on the radial stress. -- O'zPf]
The rough surface model (solid line) predicts a nearly
~
",/
J---~c e
,
~
model
-40 Poffect cylinder
J
e-~ -8o
/~
//"
model
~ -10o '~-120 1= ~w
, --
-2,000
I
-1,000
+ 0
I
1,000
'
2,000
Axial s t r e s s in fiber
3,000
4,000
5,000
[MPa]
Fig. 9 Comparison of'perfect cylinder' and 'rough surface' models. Relationship between axial fibre stress and effective normal stress at the interface
linear correlation an = flty~)during axial compression (push-out) and at low tensile stresses. The slope in this linear region is lower than that of the perfect cylinder model and depends on the elastic and roughness parameters of the components. This indicates that, for pushout tests, the linear model with a modified factor k' can still be applied. For the material discussed here, the modified factor is k' = 0.0221, whereas the pure elastic factor is calculated to be k = 0.0284. Thus, the push-out data (Fig. 5) are newly evaluated to give tYc = - 5 5 . 8 MPa and p = 0.17. On the other hand, the clamping stress can be calculated directly from the thermoelastic and roughness material data. For this, the residual stresses due to thermal expansion mismatch are evaluated according to the exact solution of Budiansky et al.l°; the thermal residual radial stress is o-rO = - 16.1 MPa. Then, the iterative solution according to Equations (5) and (6) leads to a total clamping stress of tyc = - 51.2 MPa (see Fig. 9, the intersection of the solid line with the vertical line indicating zero axial fibre stress). The calculated clamping stress is well in the range of the result obtained from the push-out evaluation. The pull-out curves have to be evaluated numerically. With the rough surface model the predicted plateau of debonding force is much higher than that of the perfect cylinder model due to the non-linearity for low values of ty,. For the roughness data given above, the plateau is calculated to be 5000 MPa ( = 70 N), which correlates very well with the experimental results. Finally, the pull-out results have to be evaluated numerically to obtain the coefficient of friction from the pull-out data. The numerical treatment will be presented elsewhere. In this test series, a coefficient of friction of p = 0.20 was obtained.
SUMMARY
Fig. 8
Shaded AFM-imageoffibresurface
480
COMPOSITES. N U M B E R 7 . 1994
Push-out and pull-out tests have been performed with single-fibre model composites. Analysis of the results according to models based on the assumption of
perfect cylindrically s h a p e d c o m p o n e n t s revealed c o n t r a -
dictions between push-out and pull-out evaluation. Thus, a m o d e l has been p r o p o s e d to assess the effects o f interfacial r o u g h n e s s d u r i n g fibre sliding. T h e ' r o u g h surface m o d e l ' enables direct c a l c u l a t i o n o f the interfacial c l a m p i n g stress, which was s h o w n to be in g o o d
5
6 7
a g r e e m e n t with the e x p e r i m e n t a l results, 8
A d d i t i o n a l l y , the new e v a l u a t i o n o f the p u s h - o u t a n d p u l l - o u t tests revealed c o n g r u e n c e o f these different m e t h o d s in their essential results. It is c o n c l u d e d
9
clamping stress and coefficient of friction for fiber-reinforced ceramic composites' Acta Metal138 No 3 (1990) pp 403~,09 Kerans, R.J. and Parthasarathay, T.A. 'Theoretical analysis of the fiber pullout and pushout tests' J Am Ceram Soc 74 No 7 (1991) pp 1585-1596 Zhon, L.-M., Kim, J.K. and Mai, Y.-W.'Interfacialdebondingand fibre pull-out stresses' J Mater Sei 27 (1992) pp 3155-3166 Rausch, G.,Meier, B. and Grathwohl, G.'A push-out technique for the evaluation of interracial properties of fiber-reinforced materials' J Europ Ceram Soc 10 (1992) pp 229-235 Oel, H.J. and Frechette, F. 'Stress distribution in multiphase systems: II, composite disks with cylindrical interfaces' J Am Ceram Soc 69 No 4 (1986) pp 342 346 Greenwood,J.A. and Tripp, J.H. 'The contact of two nominally flat rough surfaces' Proc Inst Mech Engrs 185 No 48 (1971) pp 625 433 Budiansky, B., Hutchinson, J.W. and Evans, A.G. 'Matrix fracture in fiber-reinforced ceramics' J Mech Phys Solids 34 No 2 (1986) pp 167 189
that reliable e v a l u a t i o n o f m i c r o m e c h a n i c a l tests d e p e n d s s t r o n g l y on a d d i t i o n a l d a t a o f the interface properties,
10
REFERENCES
A UTHORS
l Evans, A.G., Zok, F.W. and Davis, J. 'The role of interfaces in fiber-reinforced brittle matrix composites' Composites Sei and Techno142 (1991)pp 3-24 2 Kim, J.K. and Mai, Y.-W. 'High strength, high fracture toughness fibre composites with interface control-- a review' Composites Sci and Techno141 (1991) pp 333-378 3 Gao, Y.-C., Mai, Y.-W. and CoUerell, B. 'Fracture of fiber reinforced materials' Z A M P 3 9 (1988) pp 550-572 4 Hsueh, C.-H. 'Evaluation of interfacial shear strength, residual
M.
Kuntz,
to
whom
correspondence
should
be
addressed, K . - H . Schlapschi a n d G. G r a t h w o h l are with
the Institut f/ir Keramik im Maschinenbau, Zentrall a b o r a t o r i u m , Universitfit K a r l s r u h e , H a i d - u n d N e u Strasse 7, 76131 K a r l s r u h e , G e r m a n y . B. M e i e r is with the Institut f/ir W e r k s t o f f k u n d e II, Universitfit K a r l s ruhe, K a i s e r s t r a s s e 12, 76128 K a r l s r u h e , G e r m a n y .
COMPOSITES. N U M B E R 7 . 1994
481